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HS Doreswamy Obituary : 104ರ ವಯಸ್ಸಿನಲ್ಲಿಯೂ ದೊರೆಸ್ವಾಮಿಯವರಿಗಿದ್ದ ನೈತಿಕ ಸಿಟ್ಟು ನಮ್ಮ ದೇಶದ ಜನತೆಗೆ ಸ್ವಲ್ಪವಾದರೂ ಬರಲಿ ಕೊನೆತನಕವೂ ಎಚ್. ಎಸ್. ದೊರೆಸ್ವಾಮಿಯವರಿಗೆ ಹೋರಾಟದ ಬಗ್ಗೆಯೇ ಯೋಚನೆ. ಹೊಸ ಹೋರಾಟ ಹುಟ್ಟುಹಾಕಬೇಕು. ಸರ್ವಾಧಿಕಾರಿ ಧೋರಣೆಯಿಂದ ದೇಶದಲ್ಲಿ ಅಭಿವ್ಯಕ್ತಿ ಸ್ವಾತಂತ್ರ್ಯಕ್ಕೆ ದಿನೇದಿನೆ ಧಕ್ಕೆಯುಂಟಾಗುತ್ತಿದೆ. ಕನಿಷ್ಟ ಐನೂರು ಜನರನ್ನಾದರೂ ಕೂಡಿಸಿಕೊಂಡು ಹೋರಾಟ ಮಾಡಬೇಕು ಎಂದು ಅವರು ನಮ್ಮನ್ನೆಲ್ಲ ಕೂರಿಸಿಕೊಂಡು ಹೇಳುತ್ತಿದ್ದದ್ದು ಜಯನಗರದ ಅದೇ ಬಾಡಿಗೆಮನೆಯಲ್ಲಿ. ನಾವೆಲ್ಲ ಅವೇ ಹಳೆಯ ಮಡಿಚುವ ಕುರ್ಚಿಯಲ್ಲಿ ಕುಳಿತು ಕೇಳಿಸಿಕೊಳ್ಳುತ್ತಿದ್ದೆವು. ಕೊನೇತನಕ ಕನಿಷ್ಟ ಸೌಲಭ್ಯದಲ್ಲಿ ಬದುಕಿದ ಸರಳ ಜೀವಿ ಅವರು. ದೊರೆಸ್ವಾಮಿಯಂಥ ದೀರ್ಘ ಹೋರಾಟದ ಅನುಭವವುಳ್ಳ ಮಹಾನ್ ವ್ಯಕ್ತಿ ಕರ್ನಾಟಕದಲ್ಲಿ ಇನ್ನೊಬ್ಬರು ಇಲ್ಲವೇ ಇಲ್ಲ. ನಿಜವಾದ ಗಾಂಧೀವಾದಿ. ಸ್ವಾತಂತ್ರ್ಯ ಹೋರಾಟ, ಭೂದಾನ ಚಳವಳಿ ಮತ್ತು ತುರ್ತು ಪರಿಸ್ಥಿತಿ ಈ ಮೂರು ಹೋರಾಟಗಳ ಸಂದರ್ಭಗಳಲ್ಲಿಯೂ ಜೈಲುವಾಸ ಅನುಭವಿಸಿದ ದಿಟ್ಟ ವ್ಯಕ್ತಿ ಅವರು. ಮೊನ್ನೆಮೊನ್ನೆಯವರೆಗೂ ದೇಶಕ್ಕೆ ಒದಗಿದ ಪರಿಸ್ಥಿತಿ ಮತ್ತು ಅನ್ಯಾಯದ ವಿರುದ್ಧ ಚಳವಳಿ ಮಾಡಲೇಬೇಕು ಎನ್ನುವ ಮನಸ್ಥಿತಿಯಲ್ಲಿಯೇ ಇದ್ದವರು. ಅಂತಹ ನೈತಿಕ ಸಿಟ್ಟು, ಎಂಥ ಶತ್ರುವನ್ನೂ ಅಲುಗಾಡಿಸುವಂಥ ಸಿಟ್ಟು. ಒಂದು ತಿಂಗಳ ಹಿಂದೆ ಅವರ ಮನೆಗೆ ಹೋದಾಗ ಆರೋಗ್ಯವಾಗಿದ್ದರು. 104ರ ವಯಸ್ಸಿನಲ್ಲಿಯೂ ಹೋರಾಟದ ಬಗ್ಗೆಯೇ ಯೋಚನೆ. ಹೊಸ ಹೋರಾಟ ಹುಟ್ಟುಹಾಕಬೇಕು. ಸರ್ವಾಧಿಕಾರಿ ಧೋರಣೆಯಿಂದ ದೇಶದಲ್ಲಿ ಅಭಿವ್ಯಕ್ತಿ ಸ್ವಾತಂತ್ರ್ಯಕ್ಕೆ ದಿನೇದಿನೆ ಧಕ್ಕೆಯುಂಟಾಗುತ್ತಿದೆ. ಕನಿಷ್ಟ ಐನೂರು ಜನರನ್ನಾದರೂ ಕೂಡಿಸಿಕೊಂಡು ಹೋರಾಟ ಮಾಡಬೇಕು ಎಂದು ಅವರು ನಮ್ಮನ್ನೆಲ್ಲ ಕೂರಿಸಿಕೊಂಡು ಹೇಳುತ್ತಿದ್ದದ್ದು ಜಯನಗರದ ಅದೇ ಬಾಡಿಗೆಮನೆಯಲ್ಲಿ. ನಾವೆಲ್ಲ ಅವೇ ಹಳೆಯ ಮಡಿಚುವ ಕುರ್ಚಿಯಲ್ಲಿ ಕುಳಿತು ಕೇಳಿಸಿಕೊಳ್ಳುತ್ತಿದ್ದೆವು. ಕೊನೇತನಕ ಕನಿಷ್ಟ ಸೌಲಭ್ಯದಲ್ಲಿ ಬದುಕಿದ ಸರಳ ಜೀವಿ ಅವರು. ವಿನೋಬಾ ಭಾವೆಯವರೊಂದಿಗೆ ಇಡೀ ಭಾರತವನ್ನು ಕಾಲುನಡಿಗೆಯಲ್ಲಿ ಸುತ್ತಿದ ಮಹಾನ್ ಹೋರಾಟಗಾರರು. ಈ ವಯಸ್ಸಿನ ತನಕವೂ ಅವರು ಅನ್ಯಾಯದ ವಿರುದ್ಧ ವ್ಯಕ್ತಪಡಿಸುತ್ತಿದ್ದ ನೈತಿಕ ಸಿಟ್ಟಿದೆಯಲ್ಲ ಅದು ನಮ್ಮ ದೇಶದ ಜನತೆಗೆ ಸ್ವಲ್ಪವಾದರೂ ಬರಲಿ. ಬೆಂಗಳೂರಿನಲ್ಲಿ ಸರ್ಕಾರಿ ಜಮೀನನ್ನು ಅನೇಕ ಭೂಮಾಫಿಯಾ ವ್ಯಕ್ತಿಗಳು ಒತ್ತುವರಿ ಮಾಡಿಕೊಂಡಿದ್ದರು. ಆ ಸಂದರ್ಭದಲ್ಲಿ ಶಾಸಕ ರಾಮಸ್ವಾಮಿ ಅವರ ಅಧ್ಯಕ್ಷತೆಯಲ್ಲಿ ಒಂದು ಸಮಿತಿ ರಚಿಸಿದ್ದರು. ಆಗ ನಾನೂ ಕೂಡ ಅವರೊಂದಿಗೆ ಮುಖ್ಯಮಂತ್ರಿ ಯಡಿಯೂರಪ್ಪನವರ ಭೇಟಿಗೆ ಹೋಗಿದ್ದೆ. ಮಾತುಕತೆ ನಡೆಯಿತು. ಆದರೆ ಔಪಚಾರಿಕವಾದ ಆಳಕ್ಕಿಳಿಯದ ಮಾತುಗಳಿಂದ ಏನಾದರೂ ಬದಲಾವಣೆ ಸಾಧ್ಯವೆ? ದೊರೆಸ್ವಾಮಿಯವರ ನಿಜವಾದ ಕಳಕಳಿ ವ್ಯವಸ್ಥೆಯ ಸೂತ್ರ ಹಿಡಿದವರಿಗೆ ಕೊನೆತನಕವೂ ಅರ್ಥವೇ ಆಗಲಿಲ್ಲ. ಎಲ್ಲಾ ಮುಖ್ಯಮಂತ್ರಿಗಳನ್ನೂ ಭೇಟಿ ಮಾಡುತ್ತಲೇ ಬಂದರು. ಆದರೆ ಅಧಿಕಾರದ ಚುಕ್ಕಾಣಿ ಹಿಡಿದವರು ಅವರನ್ನು ಅರ್ಥ ಮಾಡಿಕೊಳ್ಳಲೇ ಇಲ್ಲ. ವಿನೋಭಾ ಅವರೊಂದಿಗೆ ಚಳವಳಿಯಲ್ಲಿ ಭಾಗವಹಿಸಿದಾಗಿನ ಸಂದರ್ಭದಲ್ಲಿ ನಡೆದ ಒಂದು ಘಟನೆಯನ್ನು ಅವರು ಹಂಚಿಕೊಂಡಿದ್ದು ಈಗ ನೆನಪಿಗೆ ಬರುತ್ತಿದೆ. ಗುಜರಾತಿನ ಒಂದು ಸಣ್ಣ ಹಳ್ಳಿಯಲ್ಲಿ ಅವರೆಲ್ಲ ಉಳಿದುಕೊಳ್ಳುತ್ತಾರೆ. ದೊರೆಸ್ವಾಮಿಯವರು ರಾತ್ರಿ ಮಲವಿಸರ್ಜನೆಗೆಂದು ಹೋದಾಗ ದೊಡ್ಡ ಗುಂಡಿಯಲ್ಲಿ ಬಿದ್ದುಬಿಡುತ್ತಾರೆ. ಆದರೆ ತಿಂಗಳಾನುಗಟ್ಟಲೆ ಹಾಸಿಗೆ ಹಿಡಿದರೂ ಯಾವುದೇ ಔಷಧೋಪಚಾರವಿಲ್ಲದೆ ಸುಧಾರಿಸಿಕೊಳ್ಳುತ್ತಾರೆ. ಅವರಿಗೆ ಔಷಧಿಯಲ್ಲಿ ನಂಬಿಕೆ ಇರಲಿಲ್ಲ ಹಾಗೇ ಹಣವೂ ಅವರ ಬಳಿ ಇರಲಿಲ್ಲ. ಆದರೆ ಆತ್ಮಬಲ ಮಾತ್ರವಿತ್ತು. ಅಂತಹ ಬದುಕನ್ನು ಬದುಕಿದ ಸರಳ ಚಳವಳಿಕಾರರಾಗಿದ್ದರು. ಅವರ ನಾಯಕತ್ವದ ಎಲ್ಲಾ ಚಳವಳಿಗಳಲ್ಲಿಯೂ ಭಾಗವಹಿಸಿದ್ದೇನೆ. ಸಂಪೂರ್ಣ ತನ್ಮಯರಾಗಿಬಿಡುತ್ತಿದ್ದರು. ಅದಕ್ಕೇ ಅವರಿಗೆ ಸಿಟ್ಟು ಬರುತ್ತಿತ್ತು. ಅಂಥಾ ನೈತಿಕ ಸಿಟ್ಟು ಬರುವುದು ಪ್ರಾಮಾಣಿಕವಾಗಿದ್ದಾಗ ಮಾತ್ರ. ನಿಜವಾದ ಕಳಕಳಿ ಇದ್ದಾಗ ಮಾತ್ರ. ಇದು ಎಲ್ಲರಿಗೂ ಬರುವುದಿಲ್ಲ! ಕೊರೋನಾ ಸಂದರ್ಭದಲ್ಲಿ ಸರ್ಕಾರ ಸರಿಯಾಗಿ ನಿಭಾಯಿಸುತ್ತಿಲ್ಲ ಎನ್ನುವ ಕೊರಗು ಅವರಿಗೆ ಇತ್ತು, ಸ್ವತಃ ತಾವು ಕೊರೋನಾಕ್ಕೆ ಒಳಗಾದ ಸಂದರ್ಭದಲ್ಲಿಯೂ. ಲಾಕ್ಡೌನ್ನಿಂದ ಬಡವರಿಗೆ ಅನ್ಯಾಯವಾಗುತ್ತಿದೆ ಎಂದು ಸಂಕಟಪಡುತ್ತಿದ್ದರು. ದಿನಗೂಲಿ ನೌಕರರು ಅಂದಿನ ತುತ್ತಿನ ಚೀಲ ಹೇಗೆ ತುಂಬಿಸಿಕೊಳ್ಳಬೇಕು? ಇದೊಂದೇ ಸಂದರ್ಭ ಅಂತಲ್ಲ. ಸರ್ಕಾರ ತೆಗೆದುಕೊಂಡ ಯಾವ ನಿರ್ಧಾರವೂ ಬಡವರ ಮೇಲೆಯೇ ಮೊದಲು ಪರಿಣಾಮ ಬೀರುತ್ತದೆ. ಇದನ್ನು ಹೇಗಾದರೂ ಸರಿ ಮಾಡಬೇಕು ಎಂದೇ ಸದಾ ಚಿಂತಿಸುತ್ತಿದ್ದರು. ಸದಾ ದೇಶದ ಬಡಜನರ ಬಗ್ಗೆಯೇ ಚಿಂತಿಸುತ್ತಿದ್ದಂಥ ನಿಸ್ವಾರ್ಥ ಜೀವ ನಮ್ಮ ನಡುವಿಲ್ಲ ಎನ್ನುವುದು ಸಣ್ಣ ದುಃಖವಲ್ಲ. ಬುಡಕಟ್ಟು ಜನರಿಗೆ ವಸತಿ ವ್ಯವಸ್ಥೆ ಮಾಡಿಕೊಡಬೇಕು ಎನ್ನುವ ಹಿನ್ನೆಲೆಯಲ್ಲಿ 102ನೇ ವಯಸ್ಸಿನಲ್ಲಿ ಕೊಡಗಿಗೆ ಹೋಗಿ, ಅಲ್ಲಿಂದ ಸುಮಾರು 30 ಕಿ.ಮೀ ದೂರದಲ್ಲಿರುವ ಬುಡಕಟ್ಟು ಜನರಿದ್ದ ಜಾಗಕ್ಕೆ ಹೋಗಿ ಧರಣಿಗೆ ಕುಳಿತಿದ್ದರೆಂದರೆ ಅವರ ಹೋರಾಟ ಮನೋಭಾವದ ಆಳವನ್ನು ಸತ್ವವನ್ನು ನಾವಿಂದು ಅರ್ಥ ಮಾಡಿಕೊಳ್ಳಬೇಕು. ಇದೆಲ್ಲವೂ ವ್ಯವಸ್ಥೆಯ ಸೂತ್ರ ಹಿಡಿದವರಿಗೆ ಅರ್ಥವೇ ಆಗಲಿಲ್ಲವಲ್ಲ? ಪರಿಚಯ : ಡಾ. ಕೆ. ಮರುಳಸಿದ್ದಪ್ಪ ಚಿಕ್ಕಮಗಳೂರು ಜಿಲ್ಲೆಯ ಕಾರೇಹಳ್ಳಿಯವರು. ತಂದೆ ಉಜ್ಜನಪ್ಪ, ತಾಯಿ ಕಾಳಮ್ಮ. ರಂಗಭೂಮಿ, ನಾಟಕ ಮತ್ತು ಜಾನಪದ ಕ್ಷೇತ್ರಗಳಲ್ಲಿ ಮರುಳಸಿದ್ಧಪ್ಪನವರು ಅಪಾರ ಅನುಭವ ಹೊಂದಿದ್ದಾರೆ. ಕರ್ನಾಟಕ ನಾಟಕ ಅಕಾಡೆಮಿ ಹಾಗೂ ಕುವೆಂಪು ಭಾಷಾ ಭಾರತಿ ಪ್ರಾಧಿಕಾರದ ಅಧ್ಯಕ್ಷರಾಗಿದ್ದರು. ಭಾರತೀಯ ಜಾನಪದ ಸಮೀಕ್ಷೆ, ಲಾವಣಿಗಳು, ಷಟ್ಟದಿ, ಜಾನಪದ ಸಾಹಿತ್ಯ ರಚನಕಾರರು, ಕನ್ನಡ ನಾಟಕ ಸಮೀಕ್ಷೆ, ನೋಟನಿಲುವು, ರಕ್ತಕಣಗೀತೆ ಇವು ಅವರ ಪ್ರಕಟಿತ ಪುಸ್ತಕಗಳು. ಆಧುನಿಕ ಕನ್ನಡ ನಾಟಕ ವಿಮರ್ಶೆ ಇದು ಅವರ ಪಿಎಚ್. ಡಿ ಮಹಾಪ್ರಬಂಧ. ಹಲವು ಇಂಗ್ಲೀಷ್ ನಾಟಕಗಳನ್ನು ಕನ್ನಡಕ್ಕೆ ಅನುವಾದಿಸಿದ್ದಾರೆ. ಕಿ.ರಂ. ನಾಗರಾಜ ಅವರೊಂದಿಗೆ ವಚನ ಕಮ್ಮಟ ಸಂಪಾದಿಸಿದ್ದಾರೆ. ಇದನ್ನೂ ಓದಿ : HS Doreswamy Passes Away: ಎಚ್ ಎಸ್ ದೊರೆಸ್ವಾಮಿ ನಿಧನಕ್ಕೆ ವಿವಿಧ ವಲಯದ ಗಣ್ಯರಿಂದ ಸಂತಾಪ The post HS Doreswamy Obituary : 104ರ ವಯಸ್ಸಿನಲ್ಲಿಯೂ ದೊರೆಸ್ವಾಮಿಯವರಿಗಿದ್ದ ನೈತಿಕ ಸಿಟ್ಟು ನಮ್ಮ ದೇಶದ ಜನತೆಗೆ ಸ್ವಲ್ಪವಾದರೂ ಬರಲಿ appeared first on TV9 Kannada. | kannad |
ಬರವಣಿಗೆಯ ಅಭ್ಯಾಸ ಮಕ್ಕಳ ಭವಿಷ್ಯಕ್ಕೆ ಎಷ್ಟು ಸಹಕಾರಿ ಗೊತ್ತೆ? ಮಕ್ಕಳಿಗೆ ಶಿಕ್ಷಣ ನೀಡುವ ವಿಷಯ ಬಂದಾಗ, ನಾವೆಲ್ಲರೂ ನಮಗೆ ಸಿಗದ ಅವಕಾಶಗಳನ್ನು ನಮ್ಮ ಮಕ್ಕಳಿಗೆ ನೀಡಲು ಬಯಸುತ್ತೇವೆ, ಇದರಿಂದ ಮಕ್ಕಳು ಯಾವುದರಿಂದಲೂ ವಂಚಿತರಾಗದಂತೆ ನೋಡಿಕೊಳ್ಳುತ್ತಿದ್ದೇವೆ ಎಂದು ಭಾವಿಸುತ್ತೇವೆ. ಕೊರೊನಾ ಮಾಹಾಮಾರಿ ದೇಶಕ್ಕೆ ಕಾಲಿಡುವ ಮುನ್ನ ಮಕ್ಕಳು ನಿತ್ಯ ಶಾಲೆಗೆ ಹೋಗುತ್ತಿದ್ದರು, ಸಹಪಾಠಿಗಳೆಲ್ಲ ಒಟ್ಟಾಗಿ ಕುಳಿತು ಪಾಠ ಕೆಳುತ್ತಿದ್ದರು, ನಿತ್ಯ ಶಿಕ್ಷಕರಿಂದ ನೇರ ಮಾರ್ಗದರ್ಶನ ಪಡೆಯುತ್ತಿದ್ದರು. ಹೋಂ ವರ್ಕ್, ಕ್ಲಾಸ್ ವರ್ಕ್, ನೋಟ್ಸ್ ಅಬ್ಬಾ ಎಷ್ಟೆಲ್ಲಾ ಓದಬೇಕು, ಬರೀಬೇಕು ಒತ್ತಡದ ನಡುವೆ ಓದುತ್ತಿದ್ದರು, ಅದೊಂಥರ ಖುಷಿ ಇರುತ್ತಿತ್ತು. ಇದೀಗ ಆನ್ಲೈನ್ ಯುಗ ಆರಂಭವಾಗಿದೆ, ಎಲ್ಲವೂ ಆನ್ಲೈನ್ ಆಗಬಿಟ್ಟಿದೆ. ನಿತ್ಯ ಪಾಠ ಆನ್ಲೈನ್, ಶಿಕ್ಷಕರ ಭೇಟಿ ಆನ್ಲೈನ್, ಸಹಪಾಠಿಗಳ ಭೇಟಿ ಆನ್ಲೈನ್, ಇನ್ನು ಆಟದ ಕತೆಯಂತೂ ಭಯದ ನಡುವೆ ಅಲ್ಲವೆ. ಆದರೆ ಈ ಎಲ್ಲದರ ನಡುವೆ ನಮ್ಮ ಮಕ್ಕಳು ನಾವು ಗುರುಕುಲ ಪದ್ಧತಿ, ಶಾಲೆ, ಶಿಕ್ಷಣ, ಶಿಕ್ಷೆ ಎಲ್ಲವನ್ನು ತುಂಬಾ ಮಿಸ್ ಮಾಡ್ಕೊಳ್ತಾ ಇದ್ದಾರೆ. ಇದೆಲ್ಲದರ ನಡುವೆ ಬಹಳ ಪರಿಣಾಮ ಬೀರಿರುವ ಅಂಶ ಎಂದರೆ ಬರವಣಿಗೆ. ವಾಸ್ತವವಾಗಿ, ಬರವಣಿಗೆಯು ಸಂವಹನದೊಂದಿಗೆ ತುಂಬಾ ನಿಕಟ ಸಂಬಂಧ ಹೊಂದಿದ್ದರಿಂದ ಇದು ಜೀವನದ ಕೊನೆಯವರೆಗೂ ಉಪಯುಕ್ತವಾದ ಕೌಶಲ್ಯ ಎಂದು ಹೇಳಲಾಗುತ್ತದೆ. ನಿತ್ಯ ಶಿಕ್ಷಕರ ಭಯದಿಂದ ಹೋಂ ವರ್ಕ್, ಕ್ಲಾಸ್ ವರ್ಕ್ ಬರೀತಿದ್ದ ಮಕ್ಕಳು ಈಗ ನಿತ್ಯ ಮೊಬೈಲ್ನ ಗೀಳಿಗೆ ಬಿದ್ದಿದ್ದಾರೆ. ಬರವಣಿಗೆ ಸಂಪೂರ್ಣ ಹಿಂದಕ್ಕೆ ಸರಿದಿದೆ. ಆದರೆ ಈ ಬರವಣಿಗೆ ನಿಜಕ್ಕೂ ಮಕ್ಕಳ ಶಿಕ್ಷಣ, ಬೆಳವಣಿಗೆ, ಅವರ ಮಾನಸಿಕ ಸ್ಥಿತಿ, ಜೀವನಕ್ಕೆ ಎಷ್ಟು ಅವಶ್ಯಕ ಗೊತ್ತೆ?. ಬನ್ನಿ ಈ ಲೇಖನದಲ್ಲಿ ಬರವಣಿಗೆ ಮಕ್ಕಳಿಗೆ ಏಕೆ ಬೇಕೇ ಬೇಕು? ಪೋಷಕರು ಏಕೆ ಮಕ್ಕಳಿಗೆ ಬರವಣಿಗೆಯ ಕಡೆ ಹೆಚ್ಚು ಕಾಳಜಿವಹಿಸಬೇಕು ಎಂದು ತಿಳಿಸಿಕೊಡಲಿದ್ದೇವೆ. ಮಕ್ಕಳು ಬರೆಯುವುದು ಏಕೆ ಮುಖ್ಯವಾದ ಪ್ರಮುಖ ಕಾರಣಗಳು ಇಲ್ಲಿವೆ: 1. ಬರವಣಿಗೆ ಮಗುವಿನ ಶಿಕ್ಷಣದ ಪ್ರಮುಖ ಭಾಗ ಶಾಲೆಯಲ್ಲಿ ಬರೆಯುವುದು ಮಗುವಿನ ಶಿಕ್ಷಣದ ಅವಿಭಾಜ್ಯ ಅಂಗವಾಗಿದೆ. ಶಾಲೆಗಳಲ್ಲಿ ಮಕ್ಕಳು ಕಾರ್ಯಯೋಜನೆಗಳನ್ನು ಮಾಡಬೇಕು, ಪ್ರಬಂಧಗಳನ್ನು ಬರೆಯಬೇಕು, ಪ್ರಶ್ನೆಗಳನ್ನು ಗ್ರಹಿಸಬೇಕು ಮತ್ತು ಉತ್ತರಿಸಬೇಕು ಮತ್ತು ಸಾಮಾನ್ಯವಾಗಿ ಪುಸ್ತಕಗಳ ಮೇಲೆ ವಿಷಯದ ಗ್ರಹಿಕೆಯನ್ನು ಪ್ರದರ್ಶಿಸಬೇಕು. ಶಾಲೆಯಲ್ಲಿ ಅವರು ಎಷ್ಟು ಚೆನ್ನಾಗಿ ಅಧ್ಯಯನ ಮಾಡುತ್ತಾರೆ ಎಂಬುದು ಅವರ ಬರವಣಿಗೆ ಎಷ್ಟು ವಿವರವಾದ ಮತ್ತು ಅತ್ಯಾಧುನಿಕವಾಗಿದೆ ಎಂಬುದರ ಮೇಲೆ ಅವಲಂಬಿತವಾಗಿರುತ್ತದೆ. ಅವರು ಶಾಲೆಯನ್ನು ಪ್ರಾರಂಭಿಸಿದ ಕ್ಷಣದಿಂದ ಅವರು ಕೊನೆಗೊಳ್ಳುವವರೆಗೆ ಬರವಣಿಗೆ ಅವರ ಬದುಕಲ್ಲಿ ಒಂದಿಲ್ಲೊಂದು ಕಾರಣದಿಂದ ಇದ್ದೇ ಇರುತ್ತದೆ. 2. ಇದು ಜೀವಿತಾವಧಿಯ ಕೌಶಲ್ಯ ನೀವು ಎಂದಾದರೂ ನಿಮ್ಮ ಕೆಲಸಕ್ಕೆ ಸಂಬಂಧಿಸಿದ ಇಮೇಲ್ಗಳು, ಪತ್ರ ವ್ಯವಹಾರದ ದಾಖಲೆಗಳನ್ನು ಮತ್ತೊಮ್ಮೆ ಕಣ್ಣಾಡಿಸಿದ್ದೀರಾ, ನಿಮಗೇ ಗೊತ್ತಿಲ್ಲದೆ ಇದರಲ್ಲಿ ಹಲವಾರು ಕಾಗುಣಿತ ಅಥವಾ ವಿರಾಮ ಚಿಹ್ನೆಗಳ ತಪ್ಪು ನಿಮಗೆ ಗೊತ್ತಿಲ್ಲದೆ ಇದ್ದೇ ಇರುತ್ತದೆ, ಇದು ಕೆಲವು ಸಂದರ್ಭದಲ್ಲಿ ತಪ್ಪು ಸಂದೇಶಗಳನ್ನು ಸಹ ಸಾರಬಹುದು. ಇದಕ್ಕೆ ಕಾರಣ ನಿಮ್ಮ ಚಿಕ್ಕ ವಯಸ್ಸಿನ ಬರವಣಿಗೆಯ ಅಭ್ಯಾಸ. ಹೌದು, ನೀವು ಚಿಕ್ಕ ವಯಸ್ಸಿನಲ್ಲೇ ಹೆಚ್ಚು ರಚನಾತ್ಮಕ ಹಾಗೂ ಉತ್ತಮ ಬರಹಗಾರರಾಗಲು ಅಭ್ಯಸಿಸಿದ್ದರೆ ಇಂಥಾ ಸಮಸ್ಯೆ ಮುಂದೆ ಎದುರಿಸುವುದಿಲ್ಲ. ಇದಕ್ಕಾಗಿಯೇ ಚಿಕ್ಕ ವಯಸ್ಸಿನಿಂದಲೇ ಬರವಣಿಗೆಯ ಕೌಶಲ್ಯವನ್ನು ಕಲಿಯುವುದು ಮತ್ತು ಅಭ್ಯಾಸ ಮಾಡುವುದು ಬಹಳ ಮುಖ್ಯ. 3. ಬರವಣಿಗೆ ಸಂವಹನದೊಂದಿಗೆ ಅಂತರ್ಗತವಾಗಿ ಲಿಂಕ್ ಆಗಿರುತ್ತದೆ ಇತರರೊಂದಿಗೆ ಮಾತಿನ ಮೂಲಕ ಸಂವಹನ ಮಾಡುವುದರ ಜೊತೆಗೆ, ಮಕ್ಕಳು ತಮ್ಮ ಆಲೋಚನೆಗಳನ್ನು ಮತ್ತು ಭಾವನೆಗಳನ್ನು ಲಿಖಿತ ಪದದ ಮೂಲಕ ಸ್ಪಷ್ಟವಾಗಿ ವ್ಯಕ್ತಪಡಿಸುವುದು ಹೇಗೆ ಎಂಬುದನ್ನು ಕಲಿಯಲೇಬೇಕಾಗುತ್ತದೆ. ಕೆಲವು ಬಾರಿ ಮಕ್ಕಳು ತಮ್ಮ ಭಾವನೆಗಳನ್ನು ಬರವಣಿಗೆ ಮೂಲಕ ಹಂಚಿಕೊಳ್ಳಲು ಇಂಥಾ ಅಭ್ಯಾಸಗಳು ಮಕ್ಕಳಲ್ಲಿದ್ದರೆ ಪೋಷಕರಿಗೆ ಬಹಳ ಸುಲಭವಾಗುತ್ತದೆ, ಮಕ್ಕಳು ಸಹ ಯಾವುದೇ ರೀತಿಯ ಮಾನಸಿಕ ಸಮಸ್ಯೆಗಳಿಂದ ಬಳಲದಂತೆ ತಡೆಯಲು ಇದು ಬಹಳ ಸಹಕಾರಿ. ಅಷ್ಟೇ ಅಲ್ಲದೆ, ನಿಸ್ಸಂಶಯವಾಗಿ ನಿಮ್ಮ ಮಗು ಉತ್ತಮವಾಗಿ ಬರೆಯಬಲ್ಲದು ಎಂದರೆ ಪರೀಕ್ಷೆಗಳ ಸಮಯದಲ್ಲಿ ಅವರು ತಮ್ಮ ಜ್ಞಾನವನ್ನು ಸಮರ್ಪಕವಾಗಿ ಮತ್ತು ಯಶಸ್ವಿಯಾಗಿ ನೀಡಬಹುದು. 4. ಬರವಣಿಗೆ ವಿಮರ್ಶಾತ್ಮಕ ಚಿಂತನೆ ಮತ್ತು ಸಮಸ್ಯೆಗಳನ್ನು ಪರಿಹರಿಸುವ ಸಮಸ್ಯೆಗಳನ್ನು ಹೆಚ್ಚಿಸುತ್ತದೆ ನಿಮ್ಮ ಮಗು ಏನನ್ನಾದರೂ ಬರೆಯುವ ಮೊದಲು ಅವರು ಏನು ಹೇಳಬೇಕೆಂದು ಮತ್ತು ಅವರು ಅದನ್ನು ಹೇಗೆ ತಾರ್ಕಿಕ ರೀತಿಯಲ್ಲಿ ಹೇಗೆ ಹೇಳಬಹುದು ಎಂಬುದರ ಕುರಿತು ಯೋಚಿಸಬೇಕು. ವಿಷಯದ ಬಗ್ಗೆ ಅವರ ಅಭಿಪ್ರಾಯಗಳು ಯಾವುವು? ಅವರು ತಲುಪಿಸಲು ಪ್ರಯತ್ನಿಸುತ್ತಿರುವ ಸಂದೇಶ ಏನು? ಅವರು ಅದನ್ನು ಅತ್ಯಂತ ಪರಿಣಾಮಕಾರಿ ಮತ್ತು ನಿರರ್ಗಳವಾಗಿ ಹೇಗೆ ತಿಳಿಸಬಹುದು? ಈ ಎಲ್ಲದಕ್ಕೂ ಮೆದುಳಿನ ಕೆಲಸ ಅತ್ಯಗತ್ಯ. ಆದ್ದರಿಂದ ಬರವಣಿಗೆಯ ಅಭ್ಯಾಸ ಮಕ್ಕಳ ವಿಮರ್ಶಾತ್ಮಕ ಚಿಂತನೆ ಮತ್ತು ಸಮಸ್ಯೆಗಳನ್ನು ಪರಿಹರಿಸುವ ಕೌಶಲ್ಯಗಳನ್ನು ತ್ವರಿತವಾಗಿ ಮತ್ತು ಉತ್ತಮವಾಗಿ ಸುಧಾರಿಸುತ್ತದೆ. 5. ಬರೆಯುವಿಕೆಯು ಕೌಶಲ್ಯವನ್ನು ಅಭಿವೃದ್ಧಿಪಡಿಸುತ್ತದೆ ಯಾವುದೇ ಬರವಣಿಗೆ ಅದು ಪರೀಕ್ಷೆ ಅಥವಾ ಪ್ರಬಂಧ ಇಂಥಾ ಯಾವ ಬರವಣಿಗೆ ಆದರೂ ನಿಮ್ಮ ಮಗು ಸಂಶೋಧನೆ ಮತ್ತು ಮಾಹಿತಿಯನ್ನು ಸಂಗ್ರಹಿಸಬೇಕಾಗುತ್ತದೆ ಅಥವಾ ಪಠ್ಯವನ್ನು ಚೆನ್ನಾಗಿ ಓದಿ ಅಭ್ಯಸಿಸ ಬೇಕಾಗುತ್ತದೆ. ಅವರ ಮನಸ್ಸಿನಲ್ಲಿರುವ ವಿಷಯವನ್ನು ಮಂಡಿಸಲು ಅವರು ಯಾವ ರೀತಿಯ ಮಾಹಿತಿಯನ್ನು ಪಡೆಯಬೇಕು? ಅವರು ಈ ಮಾಹಿತಿಯನ್ನು ಎಲ್ಲಿ ಹುಡುಕುತ್ತಾರೆ? ಈ ಮಾಹಿತಿಯನ್ನು ಕ್ರೋಢೀಕರಿಸಲು ಮತ್ತು ಕೆಲಸ ಮಾಡಲು ಅವರಿಗೆ ಲಭ್ಯವಿರುವ ಅಸಂಖ್ಯಾತ ಪುಸ್ತಕಗಳು ಮತ್ತು ಅಂತರ್ಜಾಲಗಳ ಸಂಶೋಧನೆಯಿಂದ ಉತ್ತಮವಾದ ಮಾಹಿತಿಯನ್ನು ಆಯ್ಕೆ ಮಾಡಿಕೊಳ್ಳುತ್ತಾರೆ. ಚಿಕ್ಕ ವಯಸ್ಸಿನಲ್ಲೇ ಇಂಥಾ ಅಭ್ಯಾಸಗಳು ಕಾಲಾನಂತರದಲ್ಲಿ ಅನಗತ್ಯದಿಂದ ಉತ್ತಮ ಮಾಹಿತಿಯನ್ನು ಹೇಗೆ ಆಯ್ಕೆ ಮಾಡುವುದು ಎಂಬುದರ ಕುರಿತು ಜ್ಞಾನ ವೃದ್ಧಿಗೆ ಸಹಕಾರವಾಗುತ್ತದೆ. 6. ಸ್ಮರಣ ಶಕ್ತಿಗೆ ಸಹಕಾರಿ ವಿದ್ಯಾರ್ಥಿಗಳ ಸ್ಮರಣ ಶಕ್ತಿ ವೃದ್ಧಿಗೆ, ಅಧ್ಯಯನ ಮಾಡಲು ಮತ್ತು ಕಲಿಯಲು ಉತ್ತಮ ಮಾರ್ಗಗಳು ಯಾವುವು ಎಂದು ಯಾವುದೇ ಶಿಕ್ಷಕರನ್ನು ಕೇಳಿ ನೋಡಿ, ಅವರು ಮೊದಲು ಹೇಳುವುದೇ ಟಿಪ್ಪಣಿಗಳನ್ನು ಬರೆಯುವುದು. ಬರವಣಿಗೆಯು ನಿಮ್ಮ ಮಗುವಿನಲ್ಲಿ ಕಲಿಕೆ ಜತೆ ಆತ್ಮವಿಶ್ವಾಸವನ್ನು ಹೆಚ್ಚಿಸುತ್ತದೆ, ಯಾವುದೇ ಗೊಂದಲ ಇಲ್ಲದೆ ಬರವಣಿಗೆಯನ್ನು ಮುಂದುವರೆಸಲು ಸತತ ಬರವಣಿಗೆ ಬೇಕೇ ಬೇಕು. ಪ್ರತಿ ಬಾರಿ ಬರೆಯುವಾಗ ಅವರ ಎಡ ಮತ್ತು ಬಲ ಮೆದುಳು ಕಾರ್ಯರೂಪಕ್ಕೆ ಬರುತ್ತದೆ, ಏಕೆಂದರೆ ಅವರು ಇನ್ನೊಬ್ಬ ವ್ಯಕ್ತಿಯು ಓದಬಲ್ಲ ಮತ್ತು ಅರ್ಥಮಾಡಿಕೊಳ್ಳಬಹುದಾದ ಯಾವುದನ್ನಾದರೂ ಬರೆಯಬೇಕು ಎಂದು ಮನಸ್ಸು ಬಯಸಿರುತ್ತದೆ. 7. ಬರವಣಿಗೆ ಉತ್ತಮ ಥೆರಪಿ ಮಾತನಾಡುವ ಮೂಲಕ ಸುಲಭವಾಗಿ ವ್ಯಕ್ತಪಡಿಸಲಾಗದ ಅದೆಷ್ಟೋ ಭಾವನೆಗಳನ್ನು ವ್ಯಕ್ತಪಡಿಸಲು ಬರವಣಿಗೆ ಬಹಳ ಸಹಾಯ ಮಾಡುವುದರಿಂದ ಬರವಣಿಗೆ ವೇಗವರ್ಧಕವಾಗಿದೆ. ಇದೊಂದು ಚಿಕಿತ್ಸಕ ಎಂದರೂ ತಪ್ಪಾಗಲಾರದು. ನಾವು ಕಥೆಗಳನ್ನು ಬರೆಯುತ್ತಿರಲಿ ಅಥವಾ ನಮ್ಮ ರಹಸ್ಯ ದಿನಚರಿಯಲ್ಲಿ ನಮ್ಮ ಹೃದಯದ ಮಾತುಗಳನ್ನು, ನಮ್ಮ ಆಲೋಚನೆಗಳು ಮತ್ತು ಭಾವನೆಗಳನ್ನು ಅರ್ಥಪೂರ್ಣ ರೀತಿಯಲ್ಲಿ ಅಚ್ಚು ಇಳಿಸಲು ಬರವಣಿಗೆ ಉತ್ತಮ ಮಾರ್ಗವಾಗಿದೆ. ಇದು ನಮ್ಮ ತಲೆಯಲ್ಲಿರುವ ಅದೆಷ್ಟೋ ವಿಷಯಗಳು, ತುಮುಲಗಳನ್ನು ತೆಗೆದುಹಾಕಲು ಸಹಾಯ ಮಾಡುತ್ತದೆ. ವಿಚಾರಗಳನ್ನು ಸ್ಪಷ್ಟಪಡಿಸಲು, ನಮ್ಮ ಮನಸ್ಸಿನಿಂದ ವಿಷಯಗಳನ್ನು ಹೊರಹಾಕುವ ಮೂಲಕ ಅಂತರ್ಗತ ಶಕ್ತಿ ಮತ್ತು ನಿರ್ದೇಶನವನ್ನು ನೀಡುತ್ತದೆ. ಇದು ಅಂತಿಮವಾಗಿ ಸ್ಪಷ್ಟತೆಯನ್ನು ತೋರುತ್ತದೆ. ನಿಮ್ಮ ಮನಸ್ಸಿನ ವಿಷಯಗಳನ್ನು ಬರೆಯುವುದರಿಂದ ನಿಮಗೂ ವ್ಯತ್ಯಾಸವಿದೆಯೇ ಎಂದು ನೋಡಿಕೊಳ್ಳು, ಪರೀಕ್ಷಿಸಲು ಪೂರಕವಾಗಿದೆ. 8. ಬರವಣಿಗೆ ಸ್ವಯಂ ಅಭಿವ್ಯಕ್ತಿಯ ಅತ್ಯುತ್ತಮ ರೂಪವಾಗಿದೆ ಸಂಗೀತ ಮತ್ತು ಸೃಜನಶೀಲ ಕಲೆಯಂತೆಯೇ ಬರವಣಿಗೆ ಕೂಡಾ ಒಂದು ಕಲೆಯೇ ಹೌದು. ಅದ್ಭುತ ಪುಸ್ತಕಗಳು, ಪತ್ರಿಕೆಯ ಲೇಖನಗಳಲ್ಲಿ ಯಾರಾದರೂ ದೀರ್ಘವಾದ ಪದಗಳನ್ನು ಒಟ್ಟಿಗೆ ಕಟ್ಟುವ ಮೂಲಕ ರಚಿಸಲಾಗಿರುತ್ತದೆ, ಇದು ಸಾಮಾನ್ಯದ ಸಂಗತಿ ಖಂಡಿತ ಅಲ್ಲ. ಪೋಷಕರು ದಯವಿಟ್ಟು ಈ ಸಂಗತಿಯನ್ನು ನೆನಪಿನಲ್ಲಿಡಿ: ಪ್ರತಿ ಮಗುವೂ ತಮ್ಮದೇ ಆದ ವಿಶಿಷ್ಟ ಧ್ವನಿಯೊಂದಿಗೆ ಜನಿಸುತ್ತದೆ, ಅದು ಬರವಣಿಗೆಯಲ್ಲಿ ವ್ಯಕ್ತವಾಗುತ್ತದೆ. ಅವರು ವಿಶೇಷ ಶೈಲಿಯನ್ನು ಹೊಂದಿದ್ದು ಅದನ್ನು ಪ್ರೋತ್ಸಾಹಿಸಬೇಕು. ಬಹುಶಃ ನಿಮ್ಮ ಮಗುವಿನಲ್ಲಿ ಉತ್ತಮ ಹಾಸ್ಯಾಸ್ಪದ ಗುಣ ಇರಬಹುದು, ಬಹುಶಃ ಅವರಲ್ಲಿ ಅನನ್ಯ ವ್ಯಕ್ತಿತ್ವವು ಉಡುಗೊರೆಯಾಗಿರಬಹುದು ಮತ್ತು ಈ ಮನೋಭಾವವು ಅವರ ಬರವಣಿಗೆಯ ಮೂಲಕ ಹೊರಬರುವ ಸಾಧ್ಯತೆ ಹೆಚ್ಚಿರುತ್ತದೆ. source: boldsky.com | kannad |
Underwater Housing Caps are available for a variety of underwater camera housings. Two styles are available to protect your housing body and its ports. Each style of cap is designed to mate with the housing just like the ports do. | english |
గ్రామాల్లోని బర్త్, డెత్లకు డిజిటల్ సర్టిఫికెట్లు హైదరాబాద్, ఆగస్టు 28 నమస్తే తెలంగాణ: గ్రామీణ ప్రాంతాల్లోని జనన, మరణాలకు కూడా డిజిటల్ ధ్రువీకరణ పత్రాలను జారీచేసేందుకు రాష్ట్ర పంచాయతీరాజ్శాఖ ఏర్పాట్లు చేసున్నది. ఇకపై గ్రామ పంచాయతీ కార్యదర్శులు జనన, మరణ వివరాలన్నింటినీ http:ubd. telangana.gov.in వెబ్సైట్లో నమోదుచేయాల్సి ఉంటుంది. ఆ గ్రామపంచాయతీ పరిధిలోని ప్రభుత్వ, ప్రయివేట్ దవాఖానల్లో జననాల వివరాలను ఈ వెబ్సైట్లో నమోదుచేస్తారు. పంచాయతీరాజ్శాఖ అధికారులు ప్రతి గ్రామ పంచాయతీకి యూజర్ ఐడీ, పాస్వర్డ్ ఇస్తారు. ఆయా గ్రామాల్లో జన్మించిన, మరణించిన వారి ధ్రువపత్రాలను దరఖాస్తుదారులు రాష్ట్రంలో ఎక్కడి నుంచైనా మీ సేవ, ఈ సేవ కేంద్రాల ద్వారా పొందే అవకాశం ఉంటుంది. ఈ సౌకర్యం ఇప్పటివరకు మున్సిపాలిటీలు, నగరపాలక సంస్థల్లోనే ఉన్నది. ప్రస్తుతం అత్యధిక పంచాయతీల్లో గ్రామ కార్యదర్శులు జనన, మరణాల వివరాలను పుస్తకాల్లో నమోదుచేసుకొని, మాన్యువల్గా ధ్రువీకరణ పత్రాలను జారీ చేస్తున్నారు. ఇక వెబ్సైట్తో పాటు పుస్తకాల్లోనూ జనన, మరణ వివరాలు నమోదుచేయాల్సి ఉంటుంది. ఈ వెబ్సైట్లోని వివరాలను అధికారిక సమాచారంగా పరిగణించి, చనిపోయిన వారికి రైతుబందు, ఆసరా పింఛన్లు నిలిపివేస్తారు. ఈ సమాచారాన్ని ఇతర శాఖలు కూడా ఉపయోగించుకోనున్నాయి. ఇప్పటివరకు 137 గ్రామ పంచాయతీలు మాత్రమే డిజిటల్ ధ్రువీకరణ పత్రాలను జారీచేస్తున్నాయి. | telegu |
केंद्र संचालक से लूट मामले में एक गिरफ्तार शाहाबाद। कोतवाली क्षेत्र में सात फरवरी की शाम बासितनगर मार्ग पर ग्राहक सेवा केंद्र संचालक से हुई लूट का पुलिस ने खुलासा कर दिया है। पुलिस ने एक लुटेरे को गिरफ्तार कर उसके पास से लूटी नकदी का 33 हजार रुपये व लूट में शामिल बाइक बरामद की है। उसके दो साथी अभी पुलिस की गिरफ्त से दूर हैं। कोतवाल सुरेश मिश्रा ने बताया कि ग्राम नेवादा निवासी रजनीश बैंक ग्राहक सेवा केंद्र का संचालक है। वह सात फरवरी की शाहाबाद कस्बा स्थित बैैक से तीन लाख 20 हजार रुपये निकालकर बाइक से गांव निवासी अनुज के साथ घर जा रहा था। रास्ते में बासितनगर मार्ग पर झोथूपुर मोड़ के पास बाइक से आए लुटेरों ने ओवरटेक कर तमंचे के बल पर नकदी से भरा बैग लूट कर फरार हो गए थे। एसपी राजेश द्विवेदी ने मौके पर पहुंचकर जांच की थी। टीमें गठित कर वारदात के खुलासे के निर्देश दिए थे। टीमें लगातार प्रयास कर रहीं थी। बृहस्पतिवार को टीमों ने मझिला थाने के ग्राम टुर्मुकी निवासी अलाउद्दीन को नौरोजपुर पुलिया के पास से गिरफ्तार कर लिया। पूूछताछ में आरोपी ने लूट की वारदात कबूल की। आरोपी की निशानदेही से लूटी नकदी का 33 हजार 330 रुपये व घटना में प्रयुक्त बाइक बरामद कर ली, जबकि उसके दो साथी अभी फरार है। कोतवाल ने बताया कि आरोपी शातिर लुटैरा है। इस पर शाहजहांपुर के निगोही थाने में गैंगस्टर हत्या के प्रयास आदि के पांच मामले दर्ज हैं, जबकि दो मामले शाहाबाद में भी दर्ज हैं। बताया कि उसके साथियों की तलाश कर जल्द ही गिरफ्तार कर जेल भेज दिया जाएगा। | hindi |
تہٕ ووٚنُن زِ أمۍ ہیوٚچھ سؠٹھاہ ؤلۍ ؤلۍ تہٕ بہتر تہٕ شہزادِ گژھہِ نہٕ أمِس منٛز کانٛہہ کمی نظرِ | kashmiri |
A warm, textured jumper is an Autumn essential, so reach for the Weird Fish Hythie. The buttoned 1/4 neck gives this shirt a relaxed feel. Match the Hythie with jeans and a jumper. | english |
একেবারে উত্তমকুমারের মতোই ছদ্মবেশী ছবির গাড়ি চালকের পোশাকে নাতি গৌরব, ভাইরাল সেই ছবি টেলিভিশনের জনপ্রিয় ধারাবাহিক গাঁটছড়া পারিবারিক ক্ষেত্রে সংসারে যেমন মাঝে মধ্যেই ঝগড়াঅশান্তি চলতে থাকে সিরিয়ালের মুখ্য চরিত্রে অভিনয় করছেন অভিনেতা গৌরব চট্টোপাধ্যায় সম্প্রতি একটি দৃশ্যে ফের কাছাকাছি এনে দিয়েছে উত্তমকুমার চট্টোপাধ্যায়গৌরব চট্টোপাধ্যায়কে কারণ ১৯৭১এ ছদ্মবেশী ছবিতে উত্তমকুমার সাদা পোশাকে গাড়ির চালকের ছদ্মবেশ নিয়েছিলেন ঠিক সেই লুকেই ধরা দিলেন নাতি গৌরব আরও পড়ুন : Gantchhora: বিয়ের আগেই অন্তঃসত্ত্বা হয়ে ট্রোলের মুখে শ্রীমা আরও পড়ুন : Meghe Dhaka Tara: একটি মেয়ের জীবন সংগ্রামের গল্প নিয়ে আসছে মেঘে ঢাকা তারা তবে গৌরবকে হঠাত্ কেন এই সাজে দেখা গেল জানা গিয়েছে , এই ধারাবাহিকে গৌরব ওরফে রিদ্ধিমানের ভাই রাহুলের কুকীর্তির প্রমাণ জোগাড় করতেই ঋদ্ধিমান তাঁর স্ত্রী খড়ি এবং সাংবাদিক শ্রুতি একটি রির্সটে হাজির হন কয়েকদিন আগেই গাঁটছড়া ধারাবাহিকে ভট্টাচার্য বাড়ির বড় মেয়ের দ্যুতির অর্থাত্ খড়ির বোন বিয়ের আগেই অন্তঃসত্তা হয়ে পড়েন আর এই সর্বনাশের কারণ সিংহরায় বাড়ির মেজ ছেলে রাহুল দাদা ঋদ্ধির সঙ্গে পাল্লা দিতে গিয়ে দ্যুতিকে বিয়ের পিঁড়ি থেকে উঠিয়ে নিয়ে গিয়েছে সে মিথ্যে ভালবাসায় ভুলিয়েছে কিন্তু বিয়ে করতে রাজি ছিলেন না রাহুল কিন্তু ভাইয়ের প্রতি অন্ধ স্নেহে ঋদ্ধিমান সব দেখেও যেন বুঝতে চাইছে না যদিও খড়ি তাঁকে বহুবারই বোঝানোর চেষ্টা করেছিলেন ঋদ্ধিমানকে বোঝানোর জন্য তবে আসল প্রমাণের অভাবে খড়িকে প্রতিবারই ভুল বোঝেন ঋদ্ধি তাই রাহুলের বিরুদ্ধে প্রমাণ জোগাড় করতেই তাঁদের ছদ্মবেশ হুবহু উত্তমকুমারের সেই গাড়ি চালকের পোশাক পড়ে সেই লুকে দেখা যাবে ধারাবাহিক গাঁটছড়ায় গৌরবকে সেই ছবিই সোশ্যাল মিডিয়ায় এখন ভাইরাল | bengali |
\begin{document}
\title{Neograd: Near-Ideal Gradient Descent}
\begin{abstract}
The purpose of this paper is to improve upon existing variants of gradient descent by solving two problems:
(1) removing (or reducing) the plateau that occurs while minimizing the cost function,
(2) continually adjusting the learning rate to an "ideal" value.
The approach taken is to approximately solve for the learning rate as a function of a trust metric.
When this technique is hybridized with momentum, it creates an especially effective gradient descent variant, called NeogradM.
It is shown to outperform Adam on several test problems, and can easily reach cost function values that are smaller by a factor of $10^8$, for example.
\end{abstract}
\section{Introduction}
Gradient descent \citep{Press-book2007} is an iterative optimization algorithm for differentiable functions. Given a cost function (CF) $f$ which depends on the parameter vector
${\bm \theta} = (\theta_1, \theta_2, ...)$, the initial choice of ${\bm \theta}$ is updated as
\begin{align*}
{\bm \theta}_{new} & = {\bm \theta}_{old} - \alpha {\bm \nabla} f ,
\end{align*}
where ${\bm \nabla} f$ is the gradient of $f$ with respect to ${\bm \theta}$, and $\alpha$ is a positive scalar known as the learning rate.
While the origin of this method dates back to \cite{Cauchy1847}
\footnote{For some discussion, see \cite{Lemarechal2010}.}
, and while it has seen use in a variety of disciplines, it has found focused attention in the field of machine learning (ML).
It is also the case that despite its widespread use with many improvements introduced over the decades, the best variants
still have issues. Perhaps the two most prominent are: (1) avoiding a plateau in the CF vs iteration graph;
(2) efficiently determining a good value for $\alpha$.
These are the problems which this paper seeks to address.
A metric will be introduced which provides an assessment of the progress of the GD updates.
Following that, a class of algorithms is introduced (Neograd), which is based on keeping this metric at a constant value.
This controlled approach to optimization will be shown to lead to superior results in the examples discussed.
In the following section, {\em Related Work}, a number of methods used in comparable iterative algorithms are reviewed.
This is followed by the section {\em The Diagnostic Metric $\rho$}, which introduces the metric which measures the validity of a linear approximation.
The next section, {\em Algorithms}, demonstrates the utility of the {\em constant $\rho$ ansatz}, in the Ideal and Near-Ideal cases.
Also, it is here that the Neograd family of algorithms is defined.
Next, in the {\em Experiments} section, three main examples are covered: (1) sigmoid-well CF which demonstrates the adaptivity of $\alpha$;
(2) Beale's function, which highlights some of the differences between Adam and NeogradM, and also shows NeogradM's superior performance;
(3) a cross-entropy CF used in the context of a digit recognition problem, which also highlights the performance of NeogradM.
In addition, Appendices A and B provide pseudocode for the Neograd algorithms, and Appendix C provides a demonstration of the stability of the new adaptation formula introduced herein.
\section{Related Work}
\label{sec:related}
There has been a great deal of research on iterative approaches to finding a local minimum of a function
(e.g., see \citep{Nocedal-book2006}, \citep{Nesterov-book2018}, \citep{Sra-book2011}).
They can largely be classified as either line search and trust region approaches.
Other optimization themes such as momentum, stochastic gradient descent, and other approaches will also be discussed.
The primary focus here will be on first-order methods
\footnote{For a discussion of the issues with second-order methods, see section 5.4 in \cite{Bishop-book2006}.}
such as GD, since they have much lower complexity costs per iteration than second-order methods in high-dimensional parameter spaces.
With \underline{Line Search}, the basic idea is to first determine a direction (${\bm p}_n$) and then a step size ($\alpha_n$) for reducing the function, from a value $f({\bm \theta}_n )$ to $f({\bm \theta}_n + \alpha {\bm p}_n )$, for the $n$th iteration. The direction ${\bm p}_n$ may be written as
${\bm p}_n = - {\bm B}^{-1} {\bm \nabla} f$
where in the case of gradient descent ${\bm B}$ is the identity matrix, and in the case of Newton's method it is the Hessian of $f$.
Typically, $\alpha_n$ is chosen to minimize $f({\bm \theta}_n + \alpha_n {\bm p}_n )$, and a number of trial $\alpha_n$ values are considered toward that end.
Also, $\alpha_n$ may be chosen to satisfy these conditions
\begin{subequations}
\begin{align}
f({\bm \theta}_n + \alpha_n {\bm p}_n ) & \leq f({\bm \theta}_n) + c_1 \alpha_n {\bm \nabla f}_n^T {\bm p}_n \\
{\bm \nabla} f({\bm \theta}_n + \alpha_n {\bm p})^T {\bm p}_n & \geq c_2 {\bm \nabla} f_n^T {\bm p}_n ,
\end{align}
\label{eqn:Wolfe}
\end{subequations}
where $0 < c_1 < c_2 < 1$.
The first of these conditions, known as the Armijo condition \citep{Armijo1966},
stipulates that an $\alpha_n$ should lead to a {\em sufficient decrease} in $f$;
the second stipulates that very small updates are not taken.
Together, they are known as the Wolfe conditions \citep{Wolfe1969,Wolfe1971}.
Alternatively, one could use the Barzilai-Borwein method for setting the learning rate \citep{Barzilai1988}, via
\begin{equation*}
\alpha_n = \frac{ | ({\bm \theta}_n - {\bm \theta}_{n-1})^T ( {\bm \nabla} f( {\bm \theta}_n) - {\bm \nabla} f( {\bm \theta}_{n-1}) ) | }
{ \| {\bm \nabla} f( {\bm \theta}_n) - {\bm \nabla} f( {\bm \theta}_{n-1}) \| } .
\end{equation*}
\underline{Trust Region} methods begin by modeling a function with a quadratic model
\begin{align*}
m_n ( {\bm p}_n ) & = f_n + {\bm \nabla}f_n^T {\bm p}_n + \frac{1}{2} {\bm p}^T_n {\bm B}_n {\bm p}_n ,
\end{align*}
where ${\bm B}$ is a symmetric matrix that is meant to approximate the Hessian of $f$.
A solution is sought to $\min_p m_n(p)$ for $\|p \| \leq \Delta$, where $\Delta$ is a radius defining the trust region.
Whether $m_n$ is a good fit is decided during the iterations by examining the ratio
\begin{align}
r_n & = \frac{ f( {\bm \theta}_n ) - f( {\bm \theta}_n + {\bm p}_n ) }{ m( {\bm 0} ) - m( {\bm p}_n ) } .
\label{eqn:trustmetric}
\end{align}
The numerator is the {\em actual reduction} of $f$, and the denominator is the {\em predicted reduction}.
The radius $\Delta$ is adjusted in order to (ideally) keep this metric in the range $0.25-0.5$.
When $r_n < 0.25$ it indicates the step size is too large, and $r_n > 0.50$ it is too small.
Although this is a second-order method, it is of interest here because of the general methodology and the ratio $r_n$,
which will be compared to the metric introduced in this paper.
For at least basic GD, and probably most other gradient-based optimization, it's the case that the updates
would benefit from being smoothed out.
For example, it is often the case that basic GD will lead to oscillations \citep{Press-book2007}, as it continually overshoots a minimum.
A smoothing effect can be achieved by introducing \underline{Momentum}, which can be implemented by an exponential weighted average of the gradient, as shown here with the variable ${\bm v}$ \citep{Polyak1964}
\begin{align*}
{\bm v}_n & = \beta {\bm v}_{n-1} + \alpha {\bm \nabla} f ( {\bm \theta}_{n-1} ) \\
{\bm \theta}_n & = {\bm \theta}_{n-1} - {\bm v}_n ,
\end{align*}
where the momentum parameter $\beta$ is a nonnegative number less than 1.
Nesterov accelerated gradient \citep{Nesterov1983,Nesterov-book2018} is another version of momentum which has been noted to achieve better results; in its original form it requires an additional, {\em look-ahead} evaluation of the gradient
\begin{align*}
{\bm v}_n & = \beta {\bm v}_{n-1} + \alpha {\bm \nabla} f ( {\bm \theta}_{n-1} - \beta {\bm v}_{n-1} ) \\
{\bm \theta}_n & = {\bm \theta}_{n-1} - {\bm v}_n .
\end{align*}
\underline{Stochastic Gradient Descent (SGD)} is a departure from the above methods, in that
rather than using a single data set (and hence a single CF), different samples of the data set are used for each update.
It follows that the CF being minimized changes with each iteration, although they are presumably statistically related since they're drawn
from the same underlying distribution.
There have been a series of algorithms introduced which allow for different effective learning rates for each component $\theta_i$.
The concepts in play were momentum applied to the gradient and possibly the gradient squared.
For this category, one may group together the approaches of AdaGrad \citep{Duchi2011}, AdaDelta \citep{Zeiler2012}, RMS Prop \citep{Tieleman2012}, Adam \& AdaMax \citep{Kingma2015},
Nadam \citep{Dozat2016}, AdamNC \citep{Reddi2018},
and others \citep[see][] {Ruder2017}.
At this point, Adam seems to be the preferred choice, and is the one used for comparisons in this paper.
Its innovation was to additionally use momentum in ${\bm \nabla} f$ within RMSProp.
The main updates
\footnote{Additionally, the variables ${\bm m}_n$ and ${\bm v}_n$ are corrected for bias at each step.
The operations in ${\bm g}_n^2$ and ${\bm m}_n / (\sqrt{ {\bm v}_n } + \epsilon )$ are understood to be performed element-wise.}
of \underline{Adam} are
\begin{align*}
{\bm m}_n & = \beta_1 {\bm m}_{n-1} + (1 - \beta_1) {\bm g}_n \\
{\bm v}_n & = \beta_2 {\bm v}_{n-1} + (1 - \beta_2) {\bm g}_n^2 \\
{\bm \theta}_n & = {\bm \theta}_{n-1} - \alpha {\bm m}_n / (\sqrt{ {\bm v}_n } + \epsilon ) ,
\end{align*}
where ${\bm g}_n = {\bm \nabla} f ( {\bm \theta}_{n-1} )$, and the default values are $\beta_1 = 0.9$, $\beta_2 = 0.999$ and $\epsilon = 10^{-8}$.
Note that all of these algorithms were introduced in the context of SGD but can be used in GD as well.
Finally, there are a number of heuristics, both with respect to learning rate schedules
\citep[e.g., "$1/t$" decay,][]{Li2020} and other modified gradients \citep[e.g., norm clipping,][]{pascanu2013}.
Keep in mind, the comparisons here are only for straight optimization (i.e., batch training), and some of these methods are perhaps best applied to mini-batch training used in SGD.
For example, the "$1/t$" decay would have $\alpha \propto 1/t$, but it will be shown on a certain example
the best results occur when $\alpha$ increases to {\em large} values.
For these reasons comparisons will not be made to those approaches.
The methods mentioned above cover the main themes of optimization by first-order gradient-descent.
This is an active field, and there are continual innovations being introduced.
\section{The Diagnostic Metric $\rho$}
The metric introduced in this section is motivated by a focus on the accuracy of the linear approximation to the underlying function $f$.
Note that although GD is normally described as an updating of ${\bm \theta}$, it's equivalent to a linear approximation of $f$.
The generic picture of the GD update on a function is given in Fig.~\ref{fig:rho},
\begin{figure}\label{fig:rho}
\end{figure}
in which the parameter is updated from $\theta_{old}$ to $\theta_{new}$.
The estimated CF according to GD is thus written
\begin{equation*}
f_{est} = f_{old} + {\bm \nabla} f^T d{\bm \theta} \, ,
\end{equation*}
where $f_{old} = f( {\bm \theta}_{old})$, $f_{new} = f( {\bm \theta}_{new})$, and $d{\bm \theta} = {\bm \theta}_{new} - {\bm \theta}_{old}$.
Note that with $f_{est}$ written as a function of $d\theta$, it represents a linear approximation to $f$ at $\theta_{new}$.
At the value $\theta_{new}$, the deviation from the true value is $| f_{new} - f_{est} |$.
However, to use this as a measure of the deviation from $f_{est}$, it should be made dimensionless.
The most convenient measure for this is $ | f_{old} - f_{est} |$.
Together, they are used to define the following diagnostic metric $\rho$
\begin{equation}
\rho = \left| \frac{ f_{new} - f_{est} }{ f_{old} - f_{est} } \right| .
\label{eqn:rho}
\end{equation}
By construction, $\rho$ provides a relative measure of the error in approximating $f$ using $f_{est}$.
Importantly, for fixed $d\theta$, it is both scale and translation invariant
\footnote{An explicit dependence upon ${\bm \nabla} f$ is suppressed.}
with respect to $f$:
\begin{align*}
\rho(d\theta; f) & = \rho(d\theta; cf) \\
\rho(d\theta; f) & = \rho(d\theta; f + c) .
\end{align*}
These properties are important so that $\rho$ can be used as a universal measure across different applications.
For example, the interpretation of $\rho = 0.2$ in one application will have the same meaning as in another application.
If one does a Taylor series expansion, the leading terms comprising $\rho$ for small $d{\bm \theta}$ yield
\begin{equation*}
\rho \rightarrow \frac{1}{2} \left| \frac{ d{\bm \theta}^T ( {\bm \nabla} {\bm \nabla} f ) d{\bm \theta} }{ {\bm \nabla} f^T d{\bm \theta} } \right| \sim {\cal O}(\alpha) \, .
\end{equation*}
These results apply when $f_{est}$ is computed using the usual first-order GD estimate.
This relationship will become useful when control algorithms are developed later in the paper.
The Hessian in the numerator also reveals that $\rho$ implicitly takes into account the local curvature of $f$, if it wasn't already obvious from the above figure.
(However, $\rho$ is not a direct measure of the curvature.)
Also, note that there is nothing intrinsically limiting the use of $\rho$ to a {\em first-order} GD algorithm.
It can be used within the context of second-order algorithms as well.
The metric used in trust regions (cf. Eq.~\ref{eqn:trustmetric}) is similar in idea but different in implementation compared to $\rho$ in Eq.~\ref{eqn:rho}.
The primary difference is that the trust model is second-order, whereas GD is first order.
However, ignoring that difference and taking
${\bm m}( {\bm 0} ) = f( {\bm \theta} ) = f_{old}$, ${\bm m} ( {\bm p} ) = f_{est}$, and $f( {\bm \theta} + {\bm p} ) = f_{new}$, it follows
\begin{align*}
\rho & = | 1 - r_n | .
\end{align*}
Thus the stated target range for $r_n$ of $0.25-0.5$ is seen to be quite a bit more aggressive than the target for $\rho$ used here, which is $0.1$.
\section{Algorithms}
It will be shown in this section how to compute the learning rate $\alpha$ as a function of the metric $\rho$.
This can be done exactly for simple cases, and approximately for complex ones.
Adjusting $\alpha$ to keep $\rho$ at a suitably small target value ($\rho_{targ}$) is one of the main themes of this paper.
This approach will be referred to as the {\em constant $\rho$ ansatz}.
When it's possible to maintain $\rho$ exactly equal to $\rho_{targ}$, it's called the \underline{Ideal} case.
When it's only possible to maintain $\rho$ in an interval $(\rho_{min}, \rho_{max})$, which includes the value $\rho_{targ}$,
it's referred to as \underline{Near-Ideal}.
The Neograd family of algorithms, which are introduced below, rely on an approximate determination of
$\alpha(\rho)$, and are Near-Ideal.
It is important to understand the reasons for keeping $\rho$ at a small but non-zero value.
If $\rho$ is {\em too small}, the linear approximation becomes excellent, but the progress in decreasing $f$ becomes very slow.
However, if $\rho$ is {\em too large}, the update progress may improve, but it may also become erratic.
Through experience, the author has found good results with the values $\rho_{targ} = 0.1$ and $(\rho_{min},\rho_{max}) = (0.015.15)$.
Note how the philosophy of this approach is opposed to that of line-search, which attempts to decrease $f$ as much as possible in a single update.
Instead, the strategy here is that of mandating the linear approximation (i.e., GD) not make errors in approximating $f$ that are too large.
In this sense, Neograd employs a type of trust region strategy.
\subsection{Ideal Learning Rate}
When attempting to build intuition with anything new, it is always a good idea to begin with simple examples.
For this reason, $\rho$ is computed for the Quadratic, Quartic, and Ellipse cost functions.
For the Quadratic and Quartic cases, the parameter space is n-dimensional: ${\bm \theta} = (\theta_1, ..., \theta_n)$;
for the Ellipse it is only 2-dimensional.
Using the definition of $\rho$ in Eq.~\ref{eqn:rho}, the results in Table~\ref{tab:exact} can be easily found.
\begin{table}[h!]
\centering
\renewcommand{1.2}{1.2}
\begin{tabular}{| l | c | c | }
\cline{2-3}
\multicolumn{1}{c|}{} & $f$ & $\alpha$ \\
\hline
Quadratic & $c{\bm \theta}^2$ & $\rho/c$ \\
\hline
Quartic & $c{\bm \theta}^4$ & $\rho / (6c{\bm \theta}^2)$ \\
\hline
Ellipse & $Q_2$ & $\rho ( Q_4 / Q_6 )$ \\
\hline
\end{tabular}
\caption{$\alpha$ values which lead to a constant value of $\rho$, shown for three CFs.}
\label{tab:exact}
\end{table}
Also, ${\bm \theta}^2 = {\bm \theta} \cdot {\bm \theta}$ and the variable $Q_m$ is defined as
\begin{align*}
Q_m & = \frac{\theta_1^2}{a^m} + \frac{\theta_2^2}{b^m} \, .
\end{align*}
Although simple, these examples permit something which hasn't been done before with GD: determining the learning rate
as the function of a metric in an exact
\footnote{Only the approximate value for $\alpha$ for the Quartic case is shown; its full solution involves a solution of a cubic and doesn't really add value for our purposes here.},
a priori, manner.
In addition, there is a lesson in the Quartic case, in that in order to maintain $\rho$ at a fixed value, $\alpha$ must be
increased to large values as $\theta$ goes to zero, which is the minimum of that CF. In the example of digit recognition,
a similar phenomenon will be shown as well.
When one of these expressions is used in a generic GD algorithm, it is referred to as "Ideal GD".
To illustrate the improvements due to the $\alpha$ formula from Table~\ref{tab:exact}, it is compared to basic GD on the Quartic CF in Fig.~\ref{fig:quartic-comp-ideal}
\begin{figure}\label{fig:quartic-comp-ideal}
\end{figure}
The left figure shows a dramatic improvement in lowering the CF value when using this exact value.
Basic GD shows a "plateau", in which it shows little progress for higher iterations.
The right figure displays the values of $\log \rho$ for both algorithms, and reveals that $\rho$
dramatically decreases for Basic GD.
Both of these features, the plateau in $\log f$ values and the drop of $\rho$, occur commonly when Adam is used,
and will be seen in the examples to follow. From this, the plateau is understood to have resulted from
low $\rho$ values, which is associated with inefficient updates.
\subsection{Near-Ideal Learning Rate}
In any real application, it will not be possible to exactly solve for $\alpha$ as a function of $\rho$, as was done in the previous section.
Instead, the approach here will be to obtain an approximate expression relating $\alpha$ and $\rho$.
First, the leading order $\alpha$-dependence of the numerator and denominator of $\rho$ in Eq.~\ref{eqn:rho} is factored out:
\begin{align*}
f_{new} - f_{est} & = A \alpha^2\\
f_{est} - f_{old} & = B \alpha \, .
\end{align*}
With these definitions, the equation for $\rho$ becomes
\begin{equation}
\rho = \left| \frac{ \alpha A }{ B } \right| .
\label{eqn:rhoAB}
\end{equation}
In a GD algorithm, $f_{old}$ and $f_{new}$ are already available, and $f_{est}$ is easily found.
Thus, $A$ and $B$ involve little additional computation.
Observe that to lowest order in $\alpha$, $A$ is a constant. However it does in general retain $\alpha$-dependence.
Also, $B$ is a constant; it equals $- \| {\bm \nabla} f \|^2$ and has no $\alpha$-dependence.
\subsubsection{The Neograd Algorithms}
\label{NeogradAlgos}
The planned use of Eq.~\ref{eqn:rhoAB} is the following.
After the $n$th iteration of an optimization run, exact values will be known for the entries in Eq.~\ref{eqn:rhoAB}.
Denoting them with $n$ subscripts, they appear as
\footnote{To be clear, $\rho$ is computed as a function of $\alpha$ in each iteration. It is $\rho$, not $\alpha$, that is the dependent variable}
\begin{equation}
\alpha_n = \left| \frac{B_n}{A_n} \right| \rho_n \, .
\label{eqn:alphan}
\end{equation}
In an ideal world, the $\rho_n$ would already equal the target value of $\rho_{targ}$.
However, since that's likely not the case, one can instead use the above equation to find an $\alpha$ for the next iteration that should produce a $\rho$
closer to $\rho_{targ}$:
\footnote{It can be shown that this process will in generic circumstances tend to converge on an $\alpha$ that leads to $\rho_{targ}$. See Appendix C.}
\begin{equation}
\alpha_{n+1} = \left| \frac{B_n}{A_n} \right| \rho_{targ} \, .
\label{eqn:alphan1}
\end{equation}
This will be referred to as the {\em Adaptation Formula} (AF).
This scaling down of Eq.~\ref{eqn:alphan} is of course approximate, since $A$ in general retains $\alpha$-dependence.
In any event it is this value that will be used as the learning rate in the {\em next} iteration.
This is the essence of the here-defined Neograd family of algorithms.
There are two refinements that can be made to this approach.
The first is based on the recognition that Eq.~\ref{eqn:alphan1} will certainly work better if the difference between $\rho_n$ and $\rho_{targ}$ is small
\footnote{That is, the map has a finite radius of convergence about $\rho_{targ}$. See Appendix C}.
Thus, as shown in Algo.~\ref{algo:get-rho-prime} in Appendix A, it may be advantageous to try to change $\rho$ in several steps
(e.g., $\rho \rightarrow \rho' \rightarrow \rho_{targ})$.
The second refinement is to hybridize this family of algorithms with existing techniques, such as momentum.
It is this variant, called NeogradM, which the author has found most performant.
\footnote{It is especially helpful in cases where the CF has narrow valleys in the shape of the CF}
.
Finally, there is an additional variation in the manner in which $f_{est} = {\bm \nabla} f \cdot d{\bm \theta}$ is computed due to the use of momentum.
Denoting the momentum for ${\bm \nabla} f$ by ${\bm m}$,
there are two
\footnote{In addition, one might also consider momentum on $d{\bm \theta}$ rather ${\bm \nabla} f$. The distinction arises because $\alpha$ and ${\bm \nabla f}$ change, and $d{\bm \theta}$ depends on both.}
obvious options for computing $f_{est}$:
\begin{enumerate}
\item $f_{est} = {\bm \nabla} f \cdot (-\alpha {\bm m})$
\item $f_{est} = -\alpha m^2$
\end{enumerate}
Option \#1 is natural for $f_{est}$ since ${\bm m}$ was used for the update of ${\bm \theta}$, so it is an exact calculation of $f_{est}$.
However, given that $d{\bm \theta}$ is based on ${\bm \nabla} f$, and that ${\bm \nabla} f$ really appears twice in $f_{est}$,
one might also consider option \#2 and use momentum on both appearances of ${\bm \nabla} f$.
The results of this paper are based on option \#1, with the exception of Sec.~\ref{sec:fest-alt} which uses option \#2.
\section{Experiments}
In this section two test functions and an application are used to evaluate Neograd.
The author advocates for the use of test functions, as they permit a window into the exact functioning of the algorithm.
Also, it is arguably the case that an application will at times be limited by characteristics of a CF that are well-described by one
or another such test functions.
Neograd is primarily compared to Adam, a popular choice for optimization in the field of ML.
An effort was made to find the best learning rate for Adam, while its other parameters were set to their default values.
In comparison, this was not necessary with Neograd, since the algorithm itself suggests an easy way to set $\alpha$ (see Appendix B).
\subsection{Demonstration of the Adaptivity of $\alpha$}
\label{sec:sigmoid-well}
Before examining the performance of the NeogradM algorithm, it is worthwhile to first understand how it adjusts the learning rate
in the face of very flat and steep CF profiles.
To that end, it is used on the {\em sigmoid-well} CF, defined by
\begin{align*}
f(\theta) & = \sigma [ s(-\theta - a) ] + \sigma [ s(\theta - a) ] \\
\sigma(\bullet) & = 1/(1 + e^{-\bullet} ) \, ,
\end{align*}
where $s=10$ and $a=2$.
\begin{figure}\label{fig:expt-sigmoid-well-1}
\end{figure}
As show in Fig.~\ref{fig:expt-sigmoid-well-1}, the initial parameter value is $\theta = -3$, and is marked by a red dot;
the minimum occurs at $\theta=0$ and is marked by a black dot.
Also, note the inflection point in the CF is near $\theta = -2$.
As the updates move the current $\theta$ from the red to the black dot, there are two significant turning points,
marked by vertical red lines in Fig.~\ref{fig:expt-sigmoid-well-2}; they occur near iterations 55 and 171.
\begin{figure}\label{fig:expt-sigmoid-well-2}
\end{figure}
The most important changes are displayed in the right plot of Fig.~\ref{fig:expt-sigmoid-well-2},
which shows (after an initial increase) $\alpha$ decreasing up until the 55th iteration, increasing up until the 171st iteration, and decreasing thereafter.
With respect to the left plot in the same figure, this is understood by examining the local CF profile of the current $\theta$.
The initial decrease corresponds to when $\theta$ is approaching the steep drop; hence it must make smaller updates.
After it passes that point, it must then increase $\alpha$ to reach the minimum, which it does between iterations 55 and 171.
At this point it's very close to $\theta=0$, and it changes over to $\alpha$ decreasing once again.
This demonstrates how NeogradM is capable of automatically adjusting $\alpha$ to properly deal with a changing landscape.
\subsection{Beale's Function}
\label{sec:Beale}
Beale's function is used as a standard test for optimization algorithms, as it offers several challenging features in its CF landscape.
It is defined as
\begin{equation*}
f = (1.5 - \theta_1 + \theta_1 \theta_2 )^2 + (2.25 -\theta_1 + \theta_1 \theta_2^2)^2 + (2.625 - \theta_1 + \theta_1 \theta_2^3)^2 \, .
\end{equation*}
It has a global minimum at $(\theta_1,\theta_2)=(3,0.5)$. The initial condition used here is $(4,3)$, from which the minimum is reachable.
In Fig.~\ref{Beale-triple} are comparison plots between Adam and NeogradM when applied to this CF.
\begin{figure}\label{Beale-triple}
\end{figure}
NeogradM reaches a much lower value than Adam in the $\log f$ plot; the exact amount depends on how long the run is.
Also, the plot of $\log \rho$ shows that Adam again naturally evolves into a situation where it produces very small $\rho$ values,
which as discussed earlier is indicative of overly conservative and slow progress.
The graph of $\log \rho$ due to NeogradM does a good job of staying within the red lines (i.e., $\log\rho_{min}$ and $\log\rho_{max}$),
except for a narrow spike to a low value.
Recall that such overly small $\rho$ values are merely inefficient, whereas overly large values are potentially problematic.
This deviation seems to be a response to a very flat region in the CF profile.
Finally, the right plot in the figure shows that NeogradM has reached its target after about 250 iterations, at which point Adam
is still far from it.
Finally, it is illuminating to examine a plot of the updates of each algorithm as they move through the CF landscape.
In Fig.~\ref{Beale-contour} a red star indicates the starting point for the algorithms, and a black star the minimum.
\begin{figure}\label{Beale-contour}
\end{figure}
The path taken by NeogradM (the blue dots), takes a very direct route; it is nearly perpendicular to the contour lines.
Recall that a pure GD algorithm would be perfectly perpendicular; the slight discrepancy is due to the momentum in NeogradM.
In contrast, Adam takes a relatively oblique path toward the black star; it is clearly {\em not} perpendicular to the contour lines.
As discussed in Sec. 2.1 of \citep{neograd_arxiv_v2}, Adam utilizes preferential weighting which adds significant weight to the null space
term in the solution for $d{\bm \theta}$, which leads to updates parallel to the contour lines.
\subsection{Cross Entropy Penalty (digit recognition)}
The purpose of this section is mainly to demonstrate the performance of NeogradM in lowering the training error.
The reader should not confuse this goal with that of lowering the generalization error, which the author considers to be qualitatively different since it also involves regularization.
Later in this section, additional comparisons will be made between NeogradM and Adam, in order to offer insight into the different workings of the two algorithms.
The training error of interest is based on the cross entropy penalty of a fully connected, single-hidden layer NN meant for classifying digitized images of the numbers 0 through 9 \citep{scikit-digits,Dua2019}. For this model, the dimensionality of ${\bm \theta}$ is 2260.
\footnote{The NN consisted of 64 input nodes, 30 hidden nodes, and 10 output nodes. Also, 1437 labeled training images were used. A tanh activation function was used on the hidden layer, and a softmax on the final layer. The model had no regularization.}
Both Adam and NeogradM were used in minimizing this CF using 3500 iterations, resulting in Fig.~\ref{fig:digits-triple}.
\begin{figure}\label{fig:digits-triple}
\end{figure}
The most noticeable feature in the $\log f$ plot is that NeogradM leads to a significantly smaller value of the CF, by a factor of $10^8$.
This result is taken as evidence of the advantages NeogradM has over existing algorithms, such as Adam.
Associated with that is the plot of $\log\alpha$, which reveals the dramatic increase of $\alpha$ throughout the course of the optimization.
(In this figure the value of $\log\alpha$ for Adam is shown for reference
\footnote{It would perhaps be a fairer comparison to determine some sort of average measure of the different effective learning rates for each component of ${\bm \theta}$ in the Adam algorithm. However, that's beyond the scope of this section.}
.)
This plot shows that $\alpha$ needs to be increased in order to keep $\rho$ in the target interval.
Next, in Fig.~\ref{fig:digits-rho}, a side-by-side comparison is made of $\log\rho$ for the two algorithms.
\begin{figure}\label{fig:digits-rho}
\end{figure}
As expected, NeogradM does a fairly good job of keeping $\rho$ in the target interval,
whereas Adam falls far below it. This is similar to what was seen before with Adam.
It also reveals that in this application it's necessary to adjust $\alpha$ rather frequently,
as is clear from the raggedness of the plot of $\log \rho$ for NeogradM.
These plots of $\log \rho$ are shown since this metric plays an important role in NeogradM, and also because the low-$\rho$ values of Adam are being identified as the reason for that algorithm having a plateau in the CF vs. iteration plots.
\subsubsection{Basins of Attraction}
To further investigate the differences in the paths from the ${\bm \theta}$-updates of these two algorithms,
three plots in Fig.~\ref{fig:thetas-comparison} are presented.
\begin{figure}\label{fig:thetas-comparison}
\end{figure}
The left plot is a 2D comparison of the two final parameter vectors, ${\bm \theta}_{Adam}$ for Adam and ${\bm \theta}_{Neo}$ for NeogradM.
The angle between them is larger than might be expected, equalling over 58 degrees.
To understand how this difference might result, a plot of the angle between ${\bm \theta}_{Adam}$ and ${\bm \theta}_{Neo}$
is given as a function of iterations.
Somewhat surprisingly, almost all of the the change in direction happens after the first iteration.
This reflects how Adam, by construction, does not follow the direction of steepest descent \citep[cf.][]{Wilson2017}.
This was seen previously in the section on Beale's function \ref{sec:Beale}, where the updates had a significant component parallel to the contour lines of the CF.
Finally, the third plot illustrates the differences in basins of attraction reached by each algorithm.
(Keep in mind that this comparison is along a single dimension; the underlying dimensionality of the space is 2260.)
The interpolating vector between the two ${\bm \theta}$s is defined as
\begin{equation*}
\phi = (1-s) {\bm \theta}_{Adam} + s{\bm \theta}_{Neo} \, .
\end{equation*}
In this plot the range of $s$ is $[-1,2]$.
This plot shows the basins are distinct (at least along this one dimension).
While it may seem that Adam has simply discovered a higher, slightly wider basin, it's actually the case
that it simply hasn't followed the gradient of $f$ down to lower values.
To wit, if Adam were allowed to continue
\footnote{For this "continuation experiment", momentum variables were re-initialized to zero at the start of the run, for both Adam and NeogradM. The additional experiment of running Adam for 7000 iterations resulted in $\log f \approx -5.08$. Thus the restart slightly benefited Adam.}
the optimization for an additional 3500 iterations, it would only have gone from $\log_{10} f = -4.27$ to $-5.47$.
However, if NeogradM had instead continued from where Adam left off at 3500 iterations, it would have gone from $\log_{10} f = -4.27$ to $-9.28$.
This shows it's not that Adam somehow found a shallower basin, it's that it wasn't as effective in reaching a lower value in that basin.
\subsubsection{Speedup}
In this section the speedup achieved by NeogradM over Adam is quantified.
The speedup is defined by the ratio $t_2/t_1$,
where $t_1$ [$t_2$] is the number of iterations needed by Adam [NeogradM] to reach a specified CF value.
\footnote{Another way to measure the performance increase is to compare CF values for a given iteration.}
To reduce variation in the results due to the different initial ${\bm \theta}$s, 10 runs were made.
In Fig.~\ref{fig:digits-speedup}, three plots are shown of the results of this experiment.
\begin{figure}\label{fig:digits-speedup}
\end{figure}
Adam was run for 80000 iterations, while NeogradM was run for much shorter times (until the limits of machine precision caused it to stop).
In the left plot, all runs are plotted
\footnote{Although it seems like NeogradM has a higher variance (along the iteration axis),
Adam actually has a higher variance for a given CF value because of its strong plateau.}
.
The center plot was obtained by computing the average number of iterations for a fixed CF value for each algorithm.
It also gives a visual depiction of how the speedup is defined in this close-up view.
Finally, in the right plot the speedup is computed for a larger range of the CF, and it reveals that
in reaching these lower CF values, NeogradM is {\em much} faster than Adam.
It is a reflection of the effective plateau in CF values produced by Adam, which seems to occur near $f \approx 10^{-7}$.
Near the plateau, the speedup appears to become arbitrarily large, becoming $\approx 40$ in the figure.
\subsubsection{Alternate Formulation of $f_{est}$}
\label{sec:fest-alt}
At the end of Sec.~\ref{NeogradAlgos} an alternate formulation (i.e., option \#2) of the estimated CF value $f_{est}$ was discussed, one in which it would be computed as
\begin{align}
f_{est} & = {\bm \nabla} f \cdot d{\bm \theta} = -\alpha m^2
\label{eqn:fest-alt}
\end{align}
where ${\bm m}$ is the momentum variable for ${\bm \nabla} f$, and $m^2 = {\bm m} \cdot {\bm m}$.
Using this formulation, and setting the momentum parameter at $\beta = 0.9$, the result for the $\log f$ versus iterations was computed as shown in Fig.~\ref{fig:fest-alt},
alongside the same results from Adam.
\begin{figure}\label{fig:fest-alt}
\end{figure}
This numerical experiment was conducted for 10 different initial conditions for each algorithm.
Each run was continued until the limits of machine precision forced the computation to cease.
Note that the lengths of each run varied over a wider range than for the previous runs with NeogradM (cf. Fig.~\ref{fig:digits-speedup}) or for Adam.
Aside from the differences in performance, perhaps the most noticeable feature of the graph for NeogradM is that $\log f$ varies approximately {\em linearly} as a function of the number of iterations.
This is in sharp contrast to the graph from Adam, as well as to the graph for NeogradM in Fig.~\ref{fig:digits-speedup} (the left plot),
where $f_{est}$ was computed using option \#1.
The reader should note that such behavior (partially) appeared in a similar plot (Fig.~\ref{Beale-triple}) for the Beale's CF (which also used option \#1).
Also, in a previous draft of this paper \citep{neograd_arxiv_v2}, such linear behavior was derived for two simple CFs (quadratic and quartic)
using the prescription for $\alpha$ listed in Table~\ref{tab:exact}.
In general though, the author obtained better results from option \#1 than option \#2.
The task of showing why such a linear dependence appears for the choice of $f_{est}$ in Eq.~\ref{eqn:fest-alt}
on this relatively complex example is outside the scope of the present paper.
\section{Final Remarks}
The purpose of this paper was defined by two goals.
The first was to devise an algorithm that didn't suffer from a plateau in the graph of the CF vs iterations.
The algorithm NeogradM achieves just that: in Fig.~\ref{Beale-triple} it had a lower CF value by a factor of $10^{15}$ compared to Adam on Beale's function.
On the cross-entropy function in Fig.~\ref{fig:digits-triple}, it had a lower CF by a factor of $10^8$.
The second goal was to efficiently determine a good value for $\alpha$.
Here, the value for $\alpha$ was determined as an approximate function of $\rho$.
Adjusting $\alpha$ to keep $\rho$ at a fixed small value (i.e., the {\em constant $\rho$ ansatz}) was done done through the
Adaptation Formula (Eq.~\ref{eqn:alphan1}) and formed the basis of the Neograd family of algorithms.
These ideas were successfully demonstrated on the examples in the {\em Experiments} section.
In the section {\em Related Work}, a metric for Trust Region methods was recalled.
It differed from the metric proposed here ($\rho$) in that it was based on a quadratic, rather than a linear model.
The idea for Neograd was to use the linear extrapolation of GD only when deviations from it were relatively small; this corresponds to a small $\rho$.
In this sense it is similar to Trust Region methods, which sought to keep a metric $r_n$ small by adjusting the trust radius $\Delta_n$ \citep[cf. Algorithm 4.1 in][]{Nocedal-book2006}. However, rather than using something like the Adaptation Formula (Eq.~\ref{eqn:alphan1}) as a guide, the Trust Region approach resorts to increasing or decreasing $\Delta_n$ by factors of 2.
The approach with Neograd of trying to keep deviations from the linear model (i.e., GD) small also differs from the Line Search methods, which aim to choose an $\alpha$ which leads to the lowest possible reduction in the CF, regardless of how poor a fit the linear extrapolation may become.
Also, the Wolfe conditions (Eq.~\ref{eqn:Wolfe}, which seek to limit $\alpha$ for different reasons, involve terms that are second-order in $({\nabla f})^2$, and hence are inequivalent to $\rho$.
\section*{Appendix A: Pseudocode for Neograd Algorithms}
Pseudocode that implements the Adaptation Formula (AF) (Eq.~\ref{eqn:alphan1}) as well as a gradual change
\footnote{When this gradual change of $\rho$ is not implemented, the version is just called Neograd\_v0.}
of $\rho$ toward $\rho_{targ}$ is called Neograd\_v1 and is shown in Algo.~\ref{algo:neograd-v1}.
\begin{algorithm}[H]
\caption{: Neograd\_v1}
\begin{algorithmic}
\STATE{Input: $({\bm \theta}_{old}, \alpha, \rho_{targ}, \text{num})$ }
\STATE{ $f_{old} = f({\bm \theta}_{old})$ }
\FOR{i = 1 to num}
\STATE{ ${\bm g} = {\bm \nabla} f({\bm \theta}_{old})$ }
\STATE{ $d{\bm \theta} = -\alpha {\bm g}$ }
\STATE{ ${\bm \theta}_{new} = {\bm \theta}_{old} + d{\bm \theta}$ }
\STATE{ $f_{new} = f({\bm \theta}_{new})$ }
\STATE{ $f_{est} = f_{old} + {\bm g} \cdot d{\bm \theta}$ }
\STATE{ $\rho = \text{get\_rho}(f_{old}, f_{new}, f_{est})$ }
\STATE{ $\rho' = \text{get\_rho\_prime}( \rho, \rho_{targ} )$ }
\STATE{ $\alpha = \text{get\_alpha}(f_{old}, f_{new}, f_{est}, \alpha, \rho')$ }
\STATE{ $f_{old} = f_{new}$ }
\STATE{ ${\bm \theta}_{old} = {\bm \theta}_{new}$ }
\ENDFOR
\STATE{Return: ${\bm \theta}_{new}$ }
\end{algorithmic}
\label{algo:neograd-v1}
\end{algorithm}
In this pseudocode, {\em get\_rho} implements Eq.~\ref{eqn:rho} and {\em get\_alpha} implements Eq.~\ref{eqn:alphan1}.
This algorithm implements the idea of an intermediate change of $\rho'$,
which is shown here as {\em get\_rho\_prime} and is implemented as Algo.~\ref{algo:get-rho-prime} below.
(This was discussed in Section \ref{NeogradAlgos}).
The motivation for this is that the AF is only expected to be stable for small deviations from $\alpha_n$ and $\rho_n$ (cf. Eq.~\ref{eqn:alphan}).
Hence, if $\rho_n$ differs too greatly from $\rho_{targ}$, trying to use $\rho_{targ}$ in the AF might not lead to the desired behavior (which is to produce an $\alpha$ which more closely leads to $\rho_{targ}$).
The stability of the AF under such iterations is discussed in Appendix C.
\begin{algorithm}[H]
\caption{: get\_rho\_prime}
\begin{algorithmic}
\STATE{Input: $(\rho, \rho_{targ})$ }
\IF{$\rho < \rho_{targ}$ }
\STATE{ $r = 0.75 \log_{10}(\rho/\rho_{targ})$ }
\STATE{ $\rho' = 10^r \rho_{targ}$ }
\ELSE
\STATE{ $\rho' = \rho_{targ}$ }
\ENDIF
\STATE{Return: $\rho'$ }
\end{algorithmic}
\label{algo:get-rho-prime}
\end{algorithm}
For example, when $\rho = 10^{-9}$ and $\rho_{targ}$ = 0.1, {\em get\_rho\_prime} returns $\rho' = 10^{-7}$
So with respect to log values, $\rho'$ is 3/4 of the way to $\rho$ and 1/4 to $\rho_{targ}$.
Note that when $\rho > \rho_{targ}$, no intermediate value is used, since large $\rho$ values are indicative
of uncontrolled behavior. Thus $\rho$ should be reduced as quickly as possible, even if it means
having the AF become inapplicable.
Another improvement that can be made is to hybridize the algorithm with existing ones,
such as Adam or RMS Prop.
The ability to hybridize with such algorithms is compactly achieved in Algo.~\ref{algo:neo-hybrids} by accessing a function {\em get\_combo}
\footnote{See \citep{MZGithub} for an implementation.}
and passing a parameter {\em type-opt} which indicates which algorithm is to be used.
\begin{algorithm}[H]
\caption{: Hybridized Neograd}
\begin{algorithmic}
\STATE{Input: (${\bm \theta}_{old}$, $\alpha$, $\rho_{targ}$, num, type\_opt)}
\STATE{ $f_{old} = f({\bm \theta}_{old})$ }
\STATE{ ${\bm v}={\bm 0}, {\bm v2}={\bm 0}$ }
\FOR{i = 1 to num}
\STATE{ ${\bm g} = {\bm \nabla} f({\bm \theta}_{old})$ }
\STATE{ $d{\bm \theta}, {\bm v}, {\bm v2} = \text{get\_dp\_combo} ( \text{type\_opt}, {\bm g}, {\bm \theta}_{old}, {\bm v}, {\bm v2}, i, \alpha)$ }
\STATE{ ${\bm \theta}_{new} = {\bm \theta}_{old} + d{\bm \theta}$ }
\STATE{ $f_{new} = f({\bm \theta}_{new})$ }
\STATE{ $f_{est} = f_{old} + {\bm g} \cdot d{\bm \theta}$ }
\STATE{ $\rho = \text{get\_rho}(f_{old}, f_{new}, f_{est})$ }
\STATE{ $\rho' = \text{get\_rho\_prime}( \rho, \rho_{targ} )$ }
\STATE{ $\alpha = \text{get\_alpha}(f_{old}, f_{new}, f_{est}, \alpha, \rho')$ }
\STATE{ $f_{old} = f_{new}$ }
\STATE{ ${\bm \theta}_{old} = {\bm \theta}_{new}$ }
\ENDFOR
\STATE{Return: ${\bm \theta}_{new}$ }
\end{algorithmic}
\label{algo:neo-hybrids}
\end{algorithm}
In the pseudocode, the parameters ${\bm v}$ and ${\bm v2}$ are passed to this function, which are momentum variables for the gradient and gradient-squared.
This allows convenient access to GD with momentum, Adam, RMS Prop, and others.
As mentioned previously, the variant that has shown the best performance is NeogradM, which utilizes momentum on the gradient updates (i.e., ${\bm v}$).
Finally, it is also convenient to have a separate method to determine a good initial value for $\alpha$.
An example implementation is given next, in Appendix B.
\section*{Appendix B: Initial Learning Rate}
When running GD it's necessary to determine an initial learning rate $\alpha$, which normally involves doing computationally costly runs.
A much more expedient way is to use the Adaptation Formula (AF) to check a small number of trial values, computing $\rho$ for each.
The first one that leads to a value of $\rho$ within $(\rho_{min}, \rho_{max})$ is taken as the initial value.
Pseudocode that implements this idea is given in Algo.~\ref{algo:get-starting-alpha}.
\begin{algorithm}[H]
\caption{: get\_starting\_alpha}
\begin{algorithmic}
\STATE{Input: $({\bm \theta}_{old}, \alpha, \rho_{targ}, \text{nrep} )$ }
\STATE{ $f_{old} = f({\bm \theta}_{old})$ }
\STATE{ ${\bm g} = {\bm \nabla} f({\bm \theta}_{old})$ }
\FOR{j = 1 to nrep}
\STATE{ $d{\bm \theta} = -\alpha {\bm g}$ }
\STATE{ ${\bm \theta}_{new} = {\bm \theta}_{old} + d{\bm \theta}$ }
\STATE{ $f_{new} = f({\bm \theta}_{new})$ }
\STATE{ $f_{est} = f_{old} + {\bm g} \cdot d{\bm \theta}$ }
\STATE{ $\rho = \text{get\_rho}(f_{old}, f_{new}, f_{est})$ }
\IF{ $\rho \in (\rho_{min}, \rho_{max})$ }
\STATE{Return: $ (\alpha, \rho)$ }
\ELSE
\STATE{ $\rho' = \text{get\_rho\_prime}( \rho, \rho_{targ} )$ }
\STATE{ $\alpha = \text{get\_alpha}(f_{old}, f_{new}, f_{est}, \alpha, \rho')$ }
\ENDIF
\ENDFOR
\STATE{Return: $ (\alpha, \rho)$ }
\end{algorithmic}
\label{algo:get-starting-alpha}
\end{algorithm}
The input parameter $nrep$ denotes the number of attempts used to find a suitable initial $\alpha$.
The input $\alpha$ is the starting point in this search; it is recommended to choose one overly small,
since as shown in Fig.~\ref{fig:stability-204}, nonlinear structure in $\rho (\alpha)$ can lead to relatively large $\alpha$ values that have a $\rho$ near the target $\rho_{targ}$.
Finally, the reader should note that there are a number of other ways to create such an algorithm based on the AF; this is only one example.
\section*{Appendix C: Stability of the Adaptation Formula}
In Section \ref{NeogradAlgos} the Adaptation Formula (AF) was introduced in Eq.~\ref{eqn:alphan1}, but its stability was not discussed.
Recall that the Neograd algorithm is defined by Eqs.~\ref{eqn:alphan} and \ref{eqn:alphan1}, in which the update of $\alpha$
is used to generate a new ${\bm \theta}$.
However, when those equations are used \underline{without updating ${\bm \theta}$}, they should lead to $\rho$ converging towards $\rho_{targ}$, i.e.,
\begin{align*}
d_n & = | \rho_n - \rho_{targ} | \\
d_n & > d_{n+1} ,
\end{align*}
where $n = 1, 2, 3, ...$
This will be demonstrated next on the sigmoid-well CF used earlier in Section \ref{sec:sigmoid-well}.
Using the starting point of $\theta_0 = -2.5$, $\rho$ is computed as a function of $\alpha$, as shown in the left plot of Fig.~\ref{fig:stability-250}.
\begin{figure}\label{fig:stability-250}
\end{figure}
The numbered points on this plot indicate the updates due to the repeated application of the AF.
Thus, one might begin at point \#1 with $(\alpha,\rho) = (0.8, 0.314)$, apply the AF to get point \#2 at $(0.255,0.088)$, and then apply the AF again
to get point \#3 at $(0.288,0.101)$.
Once again, this is done {\em without updating ${\bm \theta}$}.
Recall that the goal of the AF is to obtain a $\rho$ close to $\rho_{targ} = 0.1$, which is shown as a horizontal red line in the plot.
Hence, in this case the AF clearly leads to convergence towards $\rho_{targ}$.
Another way to understand this progression of $\rho$ values, from one update to the next, is to display them as a map
\footnote{This kind of plot may be reminiscent to those used with the logistic equation, studied as a dynamical system \citep{Devaney-book1986}}
, as shown in the right plot of Fig.~\ref{fig:stability-250}.
Because the updates form a line with a slope whose magnitude is less than 1, it will lead to updates converging to $\rho_{targ}$
\footnote{If desired, further analysis might include a computation of the contraction coefficient about the fixed point $(\rho_{targ},\rho_{targ})$ \citep[see Sec. 8 in][]{Kolmogorov-book1970}.}.
Note that $\rho = \rho_{targ} = 0.1$ is a fixed point in this iterative map.
A second example is made using the starting point $\theta_0 = -2.04$, which is close to the inflection point near $-2.0$.
The left plot in Fig.~\ref{fig:stability-204} reveals more structure, and generally a more nonlinear behavior.
\begin{figure}\label{fig:stability-204}
\end{figure}
The points labeled 1,2,3 were generated by starting at $\alpha = 0.085$, and repeatedly applying the AF, as before.
They clearly show an unstable behavior in the vicinity of where the $\rho(\alpha)$ curve crosses $\rho = \rho_{targ}$; in this case, $d_n < d_{n+1}$.
Again, the stability can be studied via a $\rho$-map in the right plot; it confirms the unstable behavior,
since the magnitude of the slope of the curve at $\rho = 0.1$ is greater than 1.0.
In applications, inflection points weren't seen by the author to be a significant, practical impediment for this general approach.
Their effect can be ameliorated by a repeated calculation of $\rho$ and $\alpha$ \citep[see Section 6.5 in][]{neograd_arxiv_v2}.
\end{document} | math |
80 फीसद अध्यापकों को रही छुट्टी, बच्चे इंतजार करते रहे संस, अमृतसर: जिला अमृतसर के सोमवार को खुले सरकारी स्कूलों में विद्यार्थी तो पहुंचे मगर 80 प्रतिशत अध्यापक नहीं आए। कारण, इन अध्यापकों की चुनावी ड्यूटी लगी थी और इन्हें डीसी की ओर से 21 फरवरी को छुट्टंी दी गई थी। जिले में एलिमेंट्री अध्यापकों की गिनती 3200 थी जिनमें 1800 अध्यापक चुनाव ड्यूटी में लगे थे। इस तरह छह हजार के करीब सेकेंडरी अध्यापक है जिनमें 4666 अध्यापकों की ड्यूटी चुनाव में लगी थी। तकरीबन 80 प्रतिशत 5466 अध्यापकों को अवकाश घोषित कर दिया गया था। जिले के 827 प्राइमरी व 416 अपर प्राइमरी, हाई व सेकेंडरी स्कूलों में जो 20 फीसद अध्यापक पहुंचे, उन्होंने ही विद्यार्थियों को पढ़ाया। उधर, डेमोक्रेटिक टीचर फ्रंट के जिला प्रधान अश्वनी अवस्थी ने कहा कि हर बार अध्यापकों की चुनाव ड्यूटी बड़े स्तर पर लगाई जाती है। इस दौरान विद्यार्थियों की पढ़ाई जहां प्रभावित होती है। वही दूसरे विभागों के कर्मचारियों को छोड़ कर शिक्षा विभाग के अध्यापकों को आगे किा जाता है। कोरोना महामारी में पहले ही अध्यापक वर्ग पाजिटिव पाया जा रहा है। दूसरी तरफ उनकी बड़े स्तर पर चुनाव ड्यूटी लगाई जा रही है। चाहे वह मना नहीं करते पर समूह विभागों के कर्मचारियों की बराबर ड्यूटी लगाई जानी चाहिए। उधर, दूसरी तरफ जिला शिक्षा अधिकारी एलिमेंट्री राजेश शर्मा ने कहा कि सरकार ने चुनावी ड्यूटी में तैनात अध्यापकों को सोमवार को अवकाश घोषित किया था, जिस कारण अध्यापक स्कूलों में नहीं आए थे। शेष अध्यापकों ने विद्यार्थियों को पढ़ाया है। गौर हो कि 20 फरवरी को संपन्न हुई मतदान प्रक्रिया में जिला प्रशासन ने बड़े स्तर पर शिक्षा विभाग के अध्यापकों की ड्यूटी लगाई थी। अध्यापकों द्वारा बार बार कहे जाने के बाद भी इस बार भी उनकी ड्यूटी अन्य विभागों के मुकाबले अधिकतर लगी थी। रात 12 बजे तक पोलिग पार्टियों ने प्रीजाइडिग अधिकारियों की अगुआई में अपना डाटा जिला प्रशासनिक अधिकारियों को सौंपा। | hindi |
વાયુસેનાના પ્લેન અકસ્માતમાં દેશે બે સિંહ જવાનો ગુમાવ્યા ઓમ શાંતિ ગઈકાલે ગુરુવારે રાત્રે બાડમેરમાં IAF MiG 21 ફાઇટર જેટ ક્રેશ થતાં બે પાઇલોટના દુઃખદ મોત થતા હતા. અકસ્માતનું કારણ જાણવા માટે કોર્ટ ઓફ ઈન્ક્વાયરીનો આદેશ આપવામાં આવ્યો છે. રક્ષા મંત્રી રાજનાથ સિંહે Rajnath Singh IAF મિગ 21 ટ્રેનર એરક્રાફ્ટની દુર્ઘટનામાં જીવ ગુમાવનારા બે હવાઈ યોદ્ધાઓ પ્રત્યે શોક વ્યક્ત કરી શહીદોને શ્રદ્ધાંજલિ આપી છે. મૃતક પાયલોટ ફ્લાઈટ લેફ્ટનન્ટ અદિતિયા બાલ અને વિંગ કમાન્ડર એમ રાણા છે. આ બંને જવાનો તેમના રૂટીન પ્રમાણે નાઈટ મિશન પર હતા. વિંગ કમાન્ડર એમ રાણા તેઓ ડિસેમ્બર 05 માં કમિશન્ડ થયા હતા અને સ્ક્વોડ્રનના ફ્લાઇટ કમાન્ડર હતા, યંગ ફ્લાઇટ લેફ્ટનન્ટ અદિતિયા બાલ જૂન 2018 માં ઉભરતા ફાઇટર પાઇલટ હતા. મળતી માહિતી મુજબ આ ઘટના ગુરુવારે રાત્રે લગભગ 9.10 વાગ્યે બની હતી. જેમાં IAF મિગ 21 ટ્રેનર એરક્રાફ્ટ ટ્રેનિંગ ફ્લાઇટ દરમિયાન પશ્ચિમી સેક્ટરમાં ક્રેશ થયું હતું. બંને પાયલટોને જીવલેણ ઈજાઓ થઈ હતી જે બાદ તેમને સારવાર માટે હોસ્પિટલમાં દાખલ કરવામાં આવ્યા હતા. ભારતીય વાયુસેના મૃતક પાયલોટના પરિવારજનોની સાથે મજબૂતીથી ઉભી છે. બે સીટર મિગ21 ટ્રેનર એરક્રાફ્ટ ગુરુવારે સાંજે બાડમેરના ઉત્તરલાઈ એરપોર્ટ પરથી ટ્રેનિંગ ફ્લાઈટ માટે ટેકઓફ કરી રહ્યું હતું. અગાઉ રાજસ્થાન પોલીસે કહ્યું હતું કે દુર્ઘટનાની અસર એટલી હતી કે તેઓ વિમાનમાં કેટલા લોકો હતા તે જાણી શક્યા ન હતા. ઘટનાસ્થળે પહોંચેલા બાડમેરના ડેપ્યુટી એસપીએ જણાવ્યું કે સ્થાનિક લોકોએ રાત્રે 9.30 વાગ્યાની આસપાસ પોલીસને અકસ્માતની જાણ કરી. અસર એટલી ભયંકર હતી કે અકસ્માત પરથી અમે કહી શકતા નથી કે બે પાઈલટ હતા કે એક. અકસ્માતનો કાટમાળ એક કિલોમીટરમાં ફેલાયેલો હતો. ડેપ્યુટી એસપીના જણાવ્યા અનુસાર, સ્થાનિક લોકોમાં કોઈ જાનહાનિના અહેવાલ નથી. કેન્દ્રીય મંત્રી અને બાડમેરથી લોકસભા સાંસદ કૈલાશ ચૌધરીએ કહ્યું કે, બાડમેર સંસદીય ક્ષેત્રના ભીમડામાં વિમાન દુર્ઘટનામાં બે પાયલોટ જીવતા બળી જવાના સમાચાર ખૂબ જ દુઃખદ છે. દેશે આજે પોતાના બે વીર જવાનો ગુમાવ્યા છે. આ સંદર્ભે, જિલ્લા વહીવટીતંત્રને સ્થળ પર આગ પર કાબૂ મેળવવાનો નિર્દેશ આપવામાં આવ્યો છે. નીચે આપેલી લીંક પર ક્લિક કરીને જોડાઓ ત્રિશુલ ન્યૂઝ Trishul News સાથે. અમારું ફેસબુક પેજ લાઈક કરો અને ફોન પર અપડેટ્સ મેળવતા રહો. વોટ્સેપ પર સમાચાર મેળવવા અહીં ક્લિક કરીને Hi લખી મેસેજ કરો. નિ:સહાયનિરાધાર ગર્ભસ્થ મહિલાની કઈક આ રીતે મદદ કરી મહેશ ભુવાએ મહેકાવી માનવતાની અનેરી મહેક ઋષિકેશમાં મુસાફરોથી ખીચોખીચ ભરેલી બસ પલટી એકસાથે ૬૫ મુસાફરો. | gujurati |
முன்னாள் மின்சாரத்துறை அமைச்சர் தங்கமணிக்கு சொந்தமான இடங்களில் தொடரும் அதிரடி சோதனை.... சென்னை, நாமக்கல், ஈரோடு, வேலூர், சேலம், கரூர், திருப்பூர், கோவை மற்றும் கர்நாடகா, ஆந்திரா உள்பட முன்னாள் மின்சாரத்துறை அமைச்சர் தங்கமணிக்கு சொந்தமான 69 இடங்களில் சோதனை நடைபெற்று வருகிறது. அந்தவகையில், நாமக்கல் மாவட்டம், குமாரபாளையம் தொகுதிக்குட்பட்ட பள்ளிபாளையம் அருகிலுள்ள கோவிந்தபாளையத்தில், தங்கமணியின் வீட்டில் லஞ்ச ஒழிப்புத்துறை துணை கண்காணிப்பாளர் தட்சிணாமூர்த்தி தலைமையில் 20க்கும் மேற்பட்ட லஞ்ச போலீசார் சோதனையில் ஈடுபட்டு வருகின்றனர். கரூர் அருகே உள்ள வேலாயுதம்பாளையம் கூலகவுண்டனூர் பகுதியில் வசித்து வரும் முன்னாள் அமைச்சர் தங்கமணியின் உறவினரான வசந்தி சுப்ரமணி என்பவரது வீட்டிலும், கரூர் கோவை சாலையில் உள்ள செராமிக் நிறுவனத்திலும் லஞ்ச ஒழிப்புத்துறையினர் சோதனை நடத்தி வருகின்றனர். சேலம், நெடுஞ்சாலை நகர் பகுதியில் உள்ள முன்னாள் அமைச்சர் தங்கமணியின் மகன் தரணிதரன் வீட்டில் லஞ்ச ஒழிப்பு போலீசார் சோதனையில் ஈடுபட்டுள்ளனர். இதேபோன்று கோவையில் உள்ள தங்கமணிக்கு சொந்தமான ஒரு இடத்தில் சோதனையில் ஈடுபட்ட போது, மூன்றாண்டுகளுக்கு முன்பே அந்த இடம் வேறு ஒருவருக்கு விற்பனை செய்தது தெரியவந்துள்ளது. | tamil |
தமிழகத்தில் அதிகரித்து வரும் டெங்கு பாதிப்பு உத்தரப்பிரதேசம் உள்ளிட்ட வடமாநிலங்களில் டெங்கு பாதிப்பு அதிகரித்துவரும் நிலையில், தமிழகத்தில் டெங்கு பாதிப்பு கட்டுக்குள் இருப்பதாக பொது சுகாதாரத்துறை இயக்குநரகம் தெரிவித்துள்ளது. எனினும், தமிழகத்தில் கடந்த ஆண்டுடன் ஒப்பிடுகையில் டெங்கு பாதிப்பு அதிகரித்துள்ளது. தமிழகத்தில் கடந்த ஆண்டு 2,410 பேருக்கு டெங்கு பாதிப்பு கண்டறியப்பட்டது. இந்த ஆண்டு ஜனவரி முதல் செப்டம்பர் 19 ஆம்தேதி வரை 2,657 பேருக்கு டெங்கு பாதிப்பு ஏற்பட்டுள்ளது. வட கிழக்கு பருவ மழை அக்டோபர் மாதத்தில் தொடங்கவுள்ள நிலையில், தற்போதே டெங்கு பாதிப்பு அதிகரித்து வருகிறது. இதனால், காய்ச்சல், தலையின் பின்பகுதியில் வலி போன்ற அறிகுறிகள் இருந்தால் உடனே மருத்துவமனையை நாட வேண்டும் என்று அறிவுறுத்துகிறார்கள் மருத்துவர்கள்.டெங்கு கொசுக்கள் நல்ல தண்ணீரில் பெருகும் என்பதால் தண்ணீர் தேங்காமல் பார்த்துக்கொள்ள வேண்டும் என்றும் மருத்துவர்கள் தெரிவிக்கிறார்கள். தற்போது மாநிலத்தில் ஒரு நாளுக்கு சராசரியாக 25 முதல் 30 பேருக்கு டெங்கு கண்டறியப்படுகிறது. ராணிப்பேட்டை மாவட்டம் பணப்பாக்கம் கோட்டையைச் சேர்ந்த 4 வயது சிறுமி, டெங்குவால் பாதிக்கப்பட்டநிலையில், சென்னை எழும்பூர் தாய்சேய் நல மருத்துவமனையில் சிகிச்சை பலனின்றி உயிரிழந்தார். இதேபோல, எழும்பூர் குழந்தைகள் நல மருத்துவமனையில் ஜனவரி முதல் 354 குழந்தைகள் டெங்குவுக்கு சிகிச்சை பெற்றனர். தற்போது பத்து குழந்தைகள் வரை சிகிச்சை பெற்றுவருகிறார்கள். தமிழகத்தில் எந்த இடத்திலும் குறிப்பிட்டுச் சொல்லும்படியாக டெங்கு பாதிப்பு திடீரென அதிகரிக்கவில்லை என்றாலும், 22,500 கொசு ஒழிப்பு பணியாளர்கள் களத்தில் இருப்பதாகவும் டெங்கு பாதிப்பு கட்டுக்குள் இருப்பதாகவும் பொது சுகாதாரத் துறை இயக்குனரகம் தெரிவித்துள்ளது. | tamil |
I’m up and out early looking forward to a day to myself, a day with no appointments of any kind to do exactly and only what I feel like. Mostly walking and reading then, but also calling in on my friends at the Street Market.
The market’s already getting busy, just after 10. Theresa and Joe, the organisers, still helping late stall-comers set up. Tables being carried, produce being laid out, coffee brewing.
Asia who works at Squash is here, selling her bread. Opposite Homebaked. Luca and Mark, bakers both, are laughing with her. Friends, all ready for the day.
I come and sit on the wall in Ducie Street with my coffee. Glad to be close, glad to be quiet. Perfectly at home here in one of the centres of my world. I write this then read for a while.
I talk briefly with a few friends who also read these things I write, hello you. And one of them shows me the gardening she’s doing in the alleyway behind her house. Elizabeth gardens where ever she is, like an instinct to make anywhere more welcoming, more alive.
In fact for one day every month you can sit on a stripey bench here and think “Yes, this, here, now” and all is well. Living, thinking, looking.
We often think that none of the Granby story over these past few years would have happened without this Street Market.
I’m here early this month, just setting off on a day’s walking. A day to myself. Hence not many people on my photographs. But the market’s getting good and full as I leave.
Today, Sunday, you can come on a walk with me if you like? Walking and talking round what I call the ‘Breathing Spaces’ in the city centre. Two ’til four pm, starting from Ed’s Place in the old George Henry Lee /Rapid Hardware building? You can book here, it’s free, for the love of this place, this home.
A day to myself, later on. | english |
# frozen_string_literal: true
Decidim.register_participatory_space(:conferences) do |participatory_space|
participatory_space.icon = "decidim/conferences/conference.svg"
participatory_space.model_class_name = "Decidim::Conference"
participatory_space.stylesheet = "decidim/conferences/conferences"
participatory_space.participatory_spaces do |organization|
Decidim::Conferences::OrganizationConferences.new(organization).query
end
participatory_space.permissions_class_name = "Decidim::Conferences::Permissions"
participatory_space.data_portable_entities = [
"Decidim::Conferences::ConferenceRegistration",
"Decidim::Conferences::ConferenceInvite"
]
participatory_space.query_type = "Decidim::Conferences::ConferenceType"
participatory_space.register_resource(:conference) do |resource|
resource.model_class_name = "Decidim::Conference"
resource.card = "decidim/conferences/conference"
resource.searchable = true
end
participatory_space.context(:public) do |context|
context.engine = Decidim::Conferences::Engine
context.layout = "layouts/decidim/conference"
context.helper = "Decidim::Conferences::ConferenceHelper"
end
participatory_space.context(:admin) do |context|
context.engine = Decidim::Conferences::AdminEngine
context.layout = "layouts/decidim/admin/conference"
end
participatory_space.register_on_destroy_account do |user|
Decidim::ConferenceUserRole.where(user: user).destroy_all
Decidim::ConferenceSpeaker.where(user: user).destroy_all
end
participatory_space.seeds do
organization = Decidim::Organization.first
seeds_root = File.join(__dir__, "..", "..", "..", "db", "seeds")
Decidim::ContentBlock.create(
organization: organization,
weight: 33,
scope_name: :homepage,
manifest_name: :highlighted_conferences,
published_at: Time.current
)
2.times do |_n|
conference = Decidim::Conference.create!(
title: Decidim::Faker::Localized.sentence(5),
slogan: Decidim::Faker::Localized.sentence(2),
slug: Faker::Internet.unique.slug(words: nil, glue: "-"),
hashtag: "##{Faker::Lorem.word}",
short_description: Decidim::Faker::Localized.wrapped("<p>", "</p>") do
Decidim::Faker::Localized.sentence(3)
end,
description: Decidim::Faker::Localized.wrapped("<p>", "</p>") do
Decidim::Faker::Localized.paragraph(3)
end,
organization: organization,
hero_image: File.new(File.join(seeds_root, "city.jpeg")), # Keep after organization
banner_image: File.new(File.join(seeds_root, "city2.jpeg")), # Keep after organization
promoted: true,
published_at: 2.weeks.ago,
objectives: Decidim::Faker::Localized.wrapped("<p>", "</p>") do
Decidim::Faker::Localized.paragraph(3)
end,
start_date: Time.current,
end_date: 2.months.from_now.at_midnight,
registrations_enabled: [true, false].sample,
available_slots: (10..50).step(10).to_a.sample,
registration_terms: Decidim::Faker::Localized.wrapped("<p>", "</p>") do
Decidim::Faker::Localized.paragraph(3)
end
)
conference.add_to_index_as_search_resource
# Create users with specific roles
Decidim::ConferenceUserRole::ROLES.each do |role|
email = "conference_#{conference.id}_#{role}@example.org"
user = Decidim::User.find_or_initialize_by(email: email)
user.update!(
name: Faker::Name.name,
nickname: Faker::Twitter.unique.screen_name,
password: "decidim123456",
password_confirmation: "decidim123456",
organization: organization,
confirmed_at: Time.current,
locale: I18n.default_locale,
tos_agreement: true
)
Decidim::ConferenceUserRole.find_or_create_by!(
user: user,
conference: conference,
role: role
)
end
attachment_collection = Decidim::AttachmentCollection.create!(
name: Decidim::Faker::Localized.word,
description: Decidim::Faker::Localized.sentence(5),
collection_for: conference
)
Decidim::Attachment.create!(
title: Decidim::Faker::Localized.sentence(2),
description: Decidim::Faker::Localized.sentence(5),
attachment_collection: attachment_collection,
attached_to: conference,
file: File.new(File.join(seeds_root, "Exampledocument.pdf")) # Keep after attached_to
)
Decidim::Attachment.create!(
title: Decidim::Faker::Localized.sentence(2),
description: Decidim::Faker::Localized.sentence(5),
attached_to: conference,
file: File.new(File.join(seeds_root, "city.jpeg")) # Keep after attached_to
)
Decidim::Attachment.create!(
title: Decidim::Faker::Localized.sentence(2),
description: Decidim::Faker::Localized.sentence(5),
attached_to: conference,
file: File.new(File.join(seeds_root, "Exampledocument.pdf")) # Keep after attached_to
)
2.times do
Decidim::Category.create!(
name: Decidim::Faker::Localized.sentence(5),
description: Decidim::Faker::Localized.wrapped("<p>", "</p>") do
Decidim::Faker::Localized.paragraph(3)
end,
participatory_space: conference
)
end
5.times do
Decidim::ConferenceSpeaker.create!(
user: conference.organization.users.sample,
full_name: Faker::Name.name,
position: Decidim::Faker::Localized.word,
affiliation: Decidim::Faker::Localized.paragraph(3),
short_bio: Decidim::Faker::Localized.wrapped("<p>", "</p>") do
Decidim::Faker::Localized.paragraph(3)
end,
twitter_handle: Faker::Twitter.unique.screen_name,
personal_url: Faker::Internet.url,
conference: conference
)
end
Decidim::Conferences::Partner::TYPES.map do |type|
4.times do
Decidim::Conferences::Partner.create!(
name: Faker::Name.name,
weight: Faker::Number.between(from: 1, to: 10),
link: Faker::Internet.url,
partner_type: type,
conference: conference,
logo: File.new(File.join(seeds_root, "logo.png")) # Keep after conference
)
end
end
5.times do
Decidim::Conferences::MediaLink.create!(
title: Decidim::Faker::Localized.sentence(2),
link: Faker::Internet.url,
date: Date.current,
weight: Faker::Number.between(from: 1, to: 10),
conference: conference
)
end
5.times do
Decidim::Conferences::RegistrationType.create!(
title: Decidim::Faker::Localized.sentence(2),
description: Decidim::Faker::Localized.sentence(5),
weight: Faker::Number.between(from: 1, to: 10),
price: Faker::Number.between(from: 1, to: 300),
published_at: 2.weeks.ago,
conference: conference
)
end
Decidim.component_manifests.each do |manifest|
manifest.seed!(conference.reload)
end
end
end
end
| code |
State of conservation: excellent, with the exception of three wax cups (bobèches) which have been replaced.
The triangular plinth is composed of a foundation with concave sides surmounted by a similarly configured base decorated with protomes of winged lions and Persian palm motifs. Above this, a sheaf of acanthus leaves wraps the base of a fluted column that tapers slightly upwards to a capital, which reproduces the acanthus leaf decoration. A calyx stands atop the capital and supports the first order of arms, which alternate with extended palm leaves and are adorned with twining acanthus shoots. Each candle holder is decorated with palm leaves and has a wax cup or bobéche. The second order of arms continues the same decorative motif.
The model for these candelabra, characterized by shafts enwreathed in palm leaves and an animal-claw base, draws inspiration partly from ancient marbles and also from early 19th-century drawings by Charles Percier. A later variant, crafted by the famous French bronze-smith Pierre-Philippe Thomire and placed in the State Apartments of the Grand Trianon in April 1837, is described as une paire de Candelabres [...] sur pieds à griffes [a pair of candelabra with animal-claw feet].
If we compare our exemplars with other candelabra from the same period found in the principal European courts and noble palaces, we find a large number of variations on the theme. On the basis of the style, technique and taste, it is reasonable to suppose that these important candelabra are the work of French or Russian artisans working in the second quarter of the 19th century.
Sculpture in gilt bronze, ivory, iron and granite.
Canonica’s nun, crafted in gilt bronze with hands and face in ivory, is leaning against a wrought iron backrest apparently lost in wistful dreams. The subject is at once humorous and moving. She is very modern in her informal attitude and twisting pose, which the sculptor conveys via the heavy folds of her habit.
It is probable that this replica of the original After the Vows, a bronze exhibited at the Paris Salon of 1893, was crafted in the 1920s or ‘30s, as can be supposed from the use of materials typical of chryselephantine sculpture in the Art Deco period. As Vicario writes, the sculpture portrays a nun “sighing over the earthly joys she has renounced by taking her vows”. Much acclaimed at the Salon, the original was purchased by the artist and art dealer Adolphe Goupil and won Canonica, who was only 24 years old at the time, an honorable mention.
The artist went on to become one of the most important and successful Italian sculptors of his generation. After completing his education at the Accademia Albertina of Turin under the guidance of the sculptor Odoardo Tabacchi (from 1881), he opened his own studio. Canonica’s fame grew and his prodigious talent was put to work on monuments in Rome, Saint Petersburg, Istanbul and Baghdad. He was one of fashionable society’s most sought-after artists, creating portraits of famous artists, politicians, and royalty throughout Europe; his portraits of the British royal family were exhibited at the Royal Academy in 1904.
The sculptor finally settled in Rome, where he became director of the Academy of Fine Arts. After his death, his house was transformed into the current Museo Canonica, where many of his works are now kept.
Base in turned and ebonized wood; upper portion in wood and pressed paperboard, partially covered with printed paper; some fittings are made of iron.
Louis-Charles Desnos (1725-1791), engineer and geographer, lived and worked in Paris. He defined himself not only as a "marchand libraire" but also as “Ingénieur-géographe de la Ville de Paris et du Roi de Danemark”. Indeed, he was paid each year to send copious quantities of atlases, globes, and maps to Christian VII of Denmark.
There is no doubt that Desnos played a preeminent role among globe-makers in mid-18th century France. A close friend of Didier Robert de Vaugondy (1723-1786), he joined the ranks of globe makers and publishers in 1749 when he married Marie-Charlotte Loye, widow of Nicolas Hardy (ante 1717-1744), who had established a globe-making workshop in 1738 which she ran with her father.
Desnos continued the Hardy’s publishing business, progressively expanding its portfolio, helped by the engraver Jean-Baptiste Nolin the Younger (1686-1762), Giovanni Antonio Rizzi Zannoni (1736-1814), and Louis Brion de la Tour (1756-1823).
He conducted business in the Hardy’s workshop on Rue St Julien le Pauvre until 1757, when he moved his shop to Rue Saint-Jacques.
The two mirrors have narrow, rectangular frames decorated with a garland of globular roses with small rounded petals. The upper corners of the frames are embellished with bouquets of these blossoms.
The ornamental element at the top is surmounted by a princely crown and contains at its center, within scroll-like brackets, a coat-of-arms representing a spread eagle with its head turned to the right. Although the original colors have been lost and cannot assist us in identifying the coat-of-arms, it is quite probably that of the Labia family of Venice. The heraldic insignia anchors a fluttering ribbon and is framed by roses.
At the bottom of the frame, a luxuriant bouquet stands in symmetry with the coat-of-arms and is also embellished with wind-blown ribbons. A delicate, twin candle holder extends from its center.
Some traces of color may still be found on the leaves and petals of the decorations.
On the basis of style, we may date the two mirrors to c. 1780. It appears reasonable to surmise that the mirrors were inspired by German or Austrian works, especially on the basis of comparisons with other works from the same period, such as the large lantern in Brüel Castle.
ìThe attribution of the coat-of-arms to the Labia family leads us to presume that the workshop where the mirrors were crafted was located in the Veneto, or perhaps even in Venice itself. Regarding technique, we might take as an example the four-footed brazier attributed to a Veneto workshop from the first half of the 18th century and decorated with floral swaths (now in the Tolentino collection in Florence), in spite of its somewhat coarser features, not having been intended for the same decorative prominence as the mirrors.
In conclusion, although some suggestion of the Baroque may be discerned, with the possibility of Alemannic influences, we may venture a date for these two mirrors to the period after the construction of the "new" Palazzo Labia had been completed and attention had turned to decorating and embellishing it. This marked a time when the family was being fully accepted into the Venetian noble ranks, in spite of the initial economic difficulties that would eventually lead to its decline. Indeed, after the 1740s, the palazzo was famous and celebrated in the paintings of the preeminent veduta painters. In the 19th century, the family’s economic downfall was inevitably followed by the abandonment of their impressive Venetian mansion, which was finally stripped bare in 1852.
The mirrors were not included in the catalogue of the famous Beistegui Auction of 1964 because they had probably already been sold on the market.
This work of extraordinary structural and decorative complexity comprises, in a succession of orders, a temple with a square-plan surmounted by a temple with a round plan topped by a dome and lantern.
The tabernacle stands on a square base supported by four fantastic hippocampi with outspread wings and a long eel-like tail terminating in an open-mouthed dog’s head.
The base is delimited at the bottom by a band bearing a dedicatory inscription “DEO MAXIMO DICATUM BONUS COMES q. ARGENTI Dc BONIS COMITIBUS DA CAMERINO DEO GRATIAS FACIEBAT ANNO D.M.I. MDLIIII” (Dedicated to Almighty God, Bonconte son of the late Argento de’ Bonconti da Camerino crafted by the grace of God in the year of the Lord 1554).
The base supports the first structural order of the elaborate tabernacle. It is a templet with a square floor plan and a portal on every side, standing upon a three-stepped platform delimited by corner pedestals surmounted by four angels.
Twenty-four medallions in silver leaf bearing embossed inscriptions alluding to the Eucharistic mystery are arranged around the base of the templet. Each side has a small door with silver openwork decorations flanked by columns supporting an entablature strip with evangelical inscriptions. A porch stands before each door composed of columns and a tympanum with God the Father and angels holding the symbols of the Passion. One of the sides opens to accommodate the consecrated host.
A second small temple atop the first features a circular floor plan but a square base delimited by a balustrade with two openwork sides. Four small full-relief statues stand on pedestals at the corners of the balustrade: the Virgin of the Annunciation, to whom is dedicated the Camerino Cathedral; San Venanzio, patron saint of Camerino, holding an image of the city in his hands; a figure in priestly garb, perhaps Sant’Ansovino or a prophet; and a fourth figure in classical vestments, with a cape billowing in the wind and an open book in his hand.
The temple is surrounded by twelve fluted columns standing on a low stylobate faced in silver and decorated with ram’s heads separated by metopes. The walls of the temple are adorned with pilaster strips alternating with niches accommodating the twelve statues of the Apostles on corbels. The columns support a round base ringed by another balustrade with volutes supporting the four seated Evangelists. Each is writing in a volume and accompanied by his symbolic figure—the eagle, the lion, the bull and the angel—holding an inkpot. The balustrade surrounds a high tambour with a series of windows alternating with herms and embellished with tiny enamel-filled incisions mimicking stained glass. The dome above is completely covered with square ashlars diminishing in size upwards, each culminating in a rosette and separated from the others with silver bosses at its corners.
A rather unusual sculpture, crafted during a later period, stands at the top on a lantern. It is a winged female figure in flowing robes standing on a sphere and bearing a palm leaf and a crown: she may be Victory or Fortuna.
The tabernacle was originally located in the old Cathedral of Camerino in the chapel dedicated to the Most Blessed Sacrament, in keeping with the liturgical dictates of the Council of Trent in reaction to the precepts of the Reformation. We have documentation of it dating to c. 1726 and we know that the large tabernacle was still found in Camerino in 1836. A few years later, in 1842, it was sold by the Chapter Priests for 80 scudi.
The object was then lost until 1959, when it was displayed and published on the occasion of the exhibition Argenti italiani dal XVI al XVIII secolo in Milan at the Museo Poldi Pezzoli.
The tabernacle was again forgotten for decades and only recently rediscovered. The definitive study and scientific publication by Benedetta Montevecchi, following restoration of the work by the Morigi workshop in Bologna, also clarified the proper attribution of the Tabernacle to “Bonus Comes q. Argenti de Bonis Comitibus de Camerino”, i.e., Bonconte, son of the late Argento de’ Bonconti da Camerino, providing also a compelling hypothesis of the origins of the goldsmith himself.
We have no documentation of a family by the name of Bonconti ever living in Camerino. The variety of cultural references incorporated into the tabernacle, substantially distant from the goldwork of Bonconte’s contemporaries in The Marches, and its monumental structure appear to draw their inspiration from Lombardy’s fanciful Renaissance architecture. The complexity of the structure and the incredible richness and variety of its decorative elements also suggest a work involving the contribution of more than one craftsman under the supervision of Bonconte. It was his hand that added the dedication and proudly signed the work in numerous points with the initials “BC”.
The only currently known piece that may be compared to it is the large Peace in the Vigevano Cathedral, a complex work probably dating to the late fifteenth century and reworked in the early decades of the following. Unfortunately, the impossibility of comparing the tabernacle with other works by Bonconte allows us to venture no more than hypotheses, although they are well supported by the notable stylistic similarities between the two objects. These elements suggest that the artist obtained his early training in the Lombardy-Veneto region. For a thorough interpretation of the tabernacle complete with bibliographical and archival references, the reader is referred to the careful study carried out by Benedetta Montevecchi.
MONTEVECCHI B. in Morello G.(ed.), Alla Mensa del Signore, capolavori dell’arte europea da Raffaello a Tiepolo, exhibition catalogue, Ancona, Mole Vanvitelliana, 2 settembre 2011 - 8 gennaio 2012, pp. 220-221, no. 75 and associated bibliography.
Built of oak completely finished in bois de rose veneer, the stand is composed of a rectangular top supported on an octagonal column with tripod feet on casters.
By means of rather complex mechanisms, it is possible to increase the height by half, tilt the top and transform it into a two-sided lectern/music stand with adjustable inclination.
We have little biographical information about Nicolas Grevenich: he was probably born in the Rhine area of Germany but lived and worked almost his whole life in Paris, where he was named maître in 1768.
Documentary sources reveal that he was still receiving important commissions form the royal Garde-meuble in 1791, especially for the Tuileries Palace. His workshop was still active during the Empire period.
The rectangular table features a minimally overhanging top with an “owl’s beak” edge. The decoration is symmetrical over the entire surface and consists of an alternation of geometric and plant motifs centered around a hexagon in a three-banded frame enclosing a complex floral design. The corners of the top are embellished with multi-petal flowers. The linear fascia is decorated with a simple motif using chiaroscuro effects achieved by combining oblique or vertical wood grains in linear frames. The square-section legs taper downwards and culminate at the top in a square capital where the multi-petal flower decoration is repeated. The legs have a narrow section just under the capital embellished with a faux fluting design. The legs are decorated with the same chiaroscuro effect as seen on the fascia.
There is one drawer on each of three sides. The table also contains numerous secret compartments opening via spring-loaded mechanisms. The front drawer contains a slide-out writing surface bearing an inscription in India ink on its underside: “1796 li 26 . Genajo – Fabri.to – Dalli fratelli Cassina – in Milano” [26 January 1796 – made by the Cassina brothers in Milan].
The brothers Francesco and Giuseppe Cassina, both cabinet makers, were originally from Meda but also worked in Milan. Francesco, nicknamed Volpino [little fox], was the more famous of the two. He crafted and signed a desk with a fold-down writing surface veneered in walnut and walnut burl with bois de rose inlay, currently in a private Milanese collection. The underside of the bottom drawer bears an inscription written in pen: “1779, 10 febbraio – Questo buro è stato fabricato di Francesco Cassina in Meda, detto il Volpino” [10 February 1779, This bureau was made by Francesco Cassina “il Volpino” in Meda] (Alberici 1969).
In 1793, Giuseppe earned mention from the Società Patriottica di Milano for his inlay work (Beretti 2005).
The bureau-cabinet stands on four slender cabriole legs ending in scrolls. The lower part features a full width drawer with a writing surface above it that folds down to reveal a number of smaller drawers. The sides are strongly contoured, while the contours are less accentuated on the front. The lines of the lower part are continued in the sides of the upper section, which is a cabinet with two doors enclosing shelves. The vigorously mixtilinear crown features three applied Rococo ornaments.
The desk is almost completely veneered in walnut burl with contrasting ebonized wood trim, edge molding, and cornices. The iron keys and locks are original.
This is a characteristic piece of Lombard furniture from the 1750s or '60s. However, while this bureau-cabinet falls into the category known as trumò (trumeau) in 18th-century Milan, it does show a number of unusual traits for Lombardy. These may be seen in the choice by the designer, who was clearly receptive to rococo tastes, to give his work high legs, softly contoured sides, and Rococo decorations at the top.
The chromatic combination of walnut burl and ebonized wood, which was much in vogue in Lombardy for nearly the entire 18th century, identifies a late-Baroque style for the bureau-cabinet and helps dispel any doubts as to its provenance.
The excellent quality of the wood, the discerning arrangement of the veneer, and the elegance of the architectural design of the volumes and contours prompts us to attribute it to a workshop in the city of Milan, which is where the patrons who were most receptive to the spread of the Rococo style lived and where a good portion of high quality furnishings were commissioned and crafted.
The cup stands on a tapered polygonal foot made of blue crystal with a narrow waist where it joins with the bowl. The bowl itself is crafted in transparent crystal that continues the polygonal motif. The base of the bowl is decorated with a series of engraved blue arches, overlapping somewhat to create a corolla, and by five vertical cusps of the same color. The rim is accented with a triple cut line around the entire circumference.
A band of silver representing a currant branch wraps the “waist” where the foot joins the cup with the currants made of orange coral.
The silver parts bear the following stamps: a lozenge with the abbreviation of the province (MI); Roman fasces; the number 08, which belong to the goldsmith Eliseo Mangili; and the fineness in parts per thousand (800) without a border.
The silversmith’s marks became mandatory with the enactment of Italian Law no. 305 of 5 February 1934, which required the use of a uniform marking system to identify the craftsman. In this case he was a silversmith active in Milan in the period 1935-1949. Eliseo Mangili has been one of the goldsmiths who assisted Alfredo Ravasco (1873-1958) in his activity until 1920s when he undertook his own activity.
In 1860, with Italian unification on the horizon, Emanuele Subert (1830-1888) opened his first antiques gallery in Milan in Via Monte di Pietà 2 with the name Emanuele Subert Antichità e Belle Arti.
Of Polish origins, Emanuele moved to Milan with his family from Trieste, the birthplace of his wife, Elisa Berger (1832 - post 1900), after also having lived for some time in Vienna. He opened a number of other stores in Milan over the course of some twenty years.
Bibliographical and documentary sources report that Emanuele Subert was a supplier for numerous museums and collectors, notable among them Frederick Stibbert, in Florence, and the nascent Museo Poldi Pezzoli, of the Consulta del Museo Patrio Archeologico and the Bagatti Valsecchi brothers, in Milan (Probst 2004). In keeping with the demands of the market at the time, Emanuele’s store offered weapons, furniture, antique fabrics, sacred art, archaeological relics, jewelry and goldwork.
In 1903, Emanuele’s sons Rodolfo (1873-1958) and Carlo Subert bought the stores in Via Monte di Pietà and in Galleria Vittorio Emanuele II, and opened two others in Bellagio, at the time an elite international tourism destination, thanks partially to the presence there of a casino.
When Carlo moved to the United States in 1910, Rodolfo changed the name of the stores to Rodolfo Subert Antichità e Belle Arti. It was probably at this time that a donation was made to the Museo Teatrale alla Scala di Milano consisting of an opera libretto bound in white satin and the Veduta dell’interno del Teatro di S. Benedetto di Venezia, both dating to the 18th century (Anonymus 1914).
Like his father, Rodolfo was an important supplier to collectors and museums: in 1904 the Museo Poldi Pezzoli acquired a Parisian “libriccino d’ore” (Poldi Pezzoli 1981), and Frederick Stibbert, who would die the following year, acquired a helmet; Giuseppe Gianetti, an enthusiastic connoisseur of majolica, later acquired a great number of objects that would eventually become part of the collections of the homonymous museum in Saronno (Ausenda 1996). We also have documentation of several important purchases of majolica pottery in 1928 from the Galleria Vittorio Emanuele store. These objects are now kept in a number of Scottish national museums (Curnow 1992).
Sources report that in 1940-41, the family made generous donations in the name of Rodolfo’s wife, Ambrogina Bergomi Subert, to major museums, including the Museo di Arti Applicate of Castello Sforzesco: “infrequent acquisitions [slowly] expanded the collection over the years, the most important of which was the donation by Ambrogina Bergomi Subert in 1941, which increased and enriched the collection of firearms with all the exemplars dating to the first half of the 19th century, thus providing exhaustive documentation of developments and changes during that period” (Allevi 1990).
Rodolfo, together with several of his children, continued to run the store in Galleria Vittorio Emanuele II until July 1943, when bombardments destroyed the gallery and its entire contents, including an archive covering over eighty years of activity.
Rodolfo’s oldest son (1898-1978), named after his father, was nicknamed “Duda” to avoid confusion. In 1926 he opened the Galleria S. Andrea in Via S. Andrea 11, where he continued to work until 1945 with his wife Gina Gandus (1900-1968). The store dealt in Italian art and had a particular predilection for jewelry and nineteenth-century Italian painting.
In 1951, Duda Subert resumed the business, opening the store G. G. Subert in Via San Pietro all’Orto 26.
The following year his son Alberto opened his own gallery in Via della Spiga 22.
This is the period of the first antique shows, including the historic Biennali in Florence and an exhibition in Milan’s Palazzo Reale, featuring the participation of both stores.
In 1996 the two businesses merged to create the Gallery of Antiquities and Art Objects in Via Santo Spirito 24, where Alberto Subert and his son Michele worked together until 2012.
In 2013, Michele Subert, continuing the family tradition, opened a new gallery in Via della Spiga 42, where he focuses on 16th-19th-century Italian works, exercising particular care in selecting works for public and private collections. | english |
چونکہ فاتح چھ سٔوی آسان یُس یمہٕ کتھۍ ہنٛد تعین چُھ کران زِ کیم چُہ کورمُت جنتک مینڈیٹ حٲصل تہٕ کیم چُھ یہٕ راونومُت، امۍ موجب چھ کیہہ چینی اسکالرز اتھ وکٹرچ انصافک اکھ قسم سمجھان،یمچ مشہور چینی کہاوتس منٛز چھے بہترین خصوصیت "فاتح چُھ بادشاہ بنان، ہارن وول چھ بنان۔ " outlaw (چینی:成者爲王,敗者爲寇"4) | kashmiri |
મોઘવારીનો માર : સવારના નાસ્તાથી લઈને કેબસ્કૂલ બસનું ભાડું પણ મોંઘુ , લોકોના ખિસ્સા પર અસર દેખાવા લાગી ઈંધણની કિંમત અને વૈશ્વિક કારણોને કારણે સામાન્ય લોકોના ખિસ્સા પર અસર દેખાવા લાગી છે. છેલ્લા ચાર મહિનામાં ખાદ્યપદાર્થોથી લઈને આવવાજવા સુધીના ખર્ચમાં વધારો થયો છે. સીએનજી અને એલપીજીના ભાવમાં સતત વધારો થઈ રહ્યો છે. ટેક્સી અને કેબના ભાડામાં પણ વધારો થયો છે. 1.CNG અને LPG મોંઘા 3 એપ્રિલે CNGની કિંમત 60.81 રૂપિયા પ્રતિ કિલો હતી. ત્યારથી અત્યાર સુધીમાં કિંમતોમાં ચાર ગણો વધારો થયો છે. હવે દિલ્હીમાં તેની કિંમત 75.61 રૂપિયા પ્રતિ કિલો પર પહોંચી ગઈ છે. જેની સીધી અસર ભાડા પર પડી છે, જેમાં 15થી વધુનો વધારો થયો છે. માર્ચથી અત્યાર સુધી એલપીજીના ભાવમાં 11 ટકાનો વધારો થયો છે. 2.સવારનો નાસ્તો મોંઘો છે છેલ્લા બે મહિનામાં ઈંડાની કિંમત 148 રૂપિયાથી વધીને 170 રૂપિયા પ્રતિ કેરેટ થઈ ગઈ છે. તેમાં 15 ટકાનો વધારો થયો છે. બ્રેડબટરના ભાવમાં 10 ટકાનો વધારો થયો છે. ટેટ્રા પેક પર GST 12 થી વધારીને 18 ટકા કરવામાં આવ્યો છે. લોટ મિલ અને દાળના મશીન પર પણ GST 12 થી વધારીને 18 ટકા કરવામાં આવ્યો છે. 3. બેંકિંગ સિસ્ટમમાં વધુ ફી 16 જુલાઈથી ચેકબુક મળવા પર 18 ટકાનો GST ચૂકવવો પડશે, જે અત્યાર સુધી મફત હતો. આ સિવાય ચિટ ફંડમાં રોકાણ પર 18 ટકા GST ચૂકવવો પડશે, જે હાલમાં 12 ટકા છે. 4. કેબ્સ, સ્કૂલ બસોનું ભાડું છેલ્લા ત્રણ મહિનામાં ઓલાઉબેરના ભાડામાં 44 ટકાનો વધારો થયો છે. કંપનીઓનું કહેવું છે કે CNGની કિંમતમાં વધારાની સાથે અન્ય ખર્ચમાં વધારાને કારણે આવું થયું છે. NCRની તમામ મોટી સ્કૂલોએ એપ્રિલથી ટ્રાન્સપોર્ટ ચાર્જમાં 20 થી 60 ટકાનો વધારો કર્યો છે. જેના કારણે વાલીઓના ખિસ્સા પર અસર પડી છે. 5. વિકાસ કાર્યો માટે કરનો બોજ કેન્દ્રરાજ્ય સરકાર તરફથી રોડ, બ્રિજ, રેલ, મેટ્રો, સ્મશાન, નહેર, ડેમ, પાઈપલાઈન, વોટર સપ્લાય પ્લાન્ટ, એજ્યુકેશન ઈન્સ્ટિટ્યૂટ, હોસ્પિટલના બાંધકામના કોન્ટ્રાક્ટ પર જીએસટી વધારીને 18 ટકા કરવામાં આવ્યો છે. તેનો અંતિમ બોજ સામાન્ય લોકો પર પડશે. રોજબરોજની ચીજવસ્તુઓ ખરીદતા લોકો પણ તેના કાર્યક્ષેત્રમાં પ્રભાવિત થશે. | gujurati |
#ifndef LOCARNA_PFOLD_PARAMS_HH
#define LOCARNA_PFOLD_PARAMS_HH
#ifdef HAVE_CONFIG_H
#include <config.h>
#endif
extern "C" {
#include <ViennaRNA/data_structures.h>
}
#include "named_arguments.hh"
#include <limits>
#include "aux.hh"
namespace LocARNA {
/**
* \brief Parameters for partition folding
*
* Describes certain parameters for the partition folding of
* a sequence or alignment.
*
* This is used to store and pass model details for RNA
* folding. Works as wrapper for the ViennaRNA model details
* structure.
*
* @see RnaEnsemble
*
*/
class PFoldParams {
public:
struct args {
DEFINE_NAMED_ARG_DEFAULT(noLP, bool, false);
DEFINE_NAMED_ARG_DEFAULT(stacking, bool, false);
DEFINE_NAMED_ARG_DEFAULT(dangling, int, 2);
DEFINE_NAMED_ARG_DEFAULT(max_bp_span, int, -1);
DEFINE_NAMED_ARG_DEFAULT(ribo, bool, true);
DEFINE_NAMED_ARG_DEFAULT(cv_fact, double, 0.6);
DEFINE_NAMED_ARG_DEFAULT(nc_fact, double, 0.5);
using valid_args = std::tuple<noLP,
stacking,
dangling,
max_bp_span,
ribo,
cv_fact,
nc_fact>;
};
/**
* Construct with all parameters
*
* @param noLP forbid lonely base pairs
* @param stacking calculate stacking probabilities
* @param max_bp_span maximum base pair span
* @param dangling ViennaRNA dangling end type
*/
template <typename... Args>
PFoldParams(Args... argpack) : md_() {
static_assert( type_subset_of< std::tuple<Args...> , typename args::valid_args >::value,
"Invalid type in named arguments pack." );
auto args = std::make_tuple(argpack...);
stacking_ = get_named_arg_opt<args::stacking>(args);
vrna_md_set_default(&md_);
md_.noLP = get_named_arg_opt<args::noLP>(args) ? 1 : 0;
md_.max_bp_span = get_named_arg_opt<args::max_bp_span>(args);
md_.dangles = get_named_arg_opt<args::dangling>(args);
assert(md_.dangles >= 0);
assert(md_.dangles <= 3);
md_.compute_bpp = 1;
// set ribosum scoring with "best" parameters
md_.ribo = get_named_arg_opt<args::ribo>(args) ? 1 : 0;
md_.cv_fact = get_named_arg_opt<args::cv_fact>(args); // cfactor
md_.nc_fact = get_named_arg_opt<args::nc_fact>(args); // nfactor
}
/** copy constructor
*/
PFoldParams(const PFoldParams &pfoldparams):
md_(),
stacking_(pfoldparams.stacking_)
{
vrna_md_copy(&md_, &pfoldparams.md_);
}
/**
* @brief get ViennaRNA model details structure
*
* @return initialized md structure
*
* The structure is set to the values of this object for
* maintained values; some further values are set explicitly, e.g.
* alifold parameters.
* All other values are set to the ViennaRNA default values.
*/
const vrna_md_t &
model_details() const {
return md_;
}
/* provide read access for selected model details */
/**
* @brief Check no LP flag
*
* @return value of flag
*/
bool
noLP() const {
return md_.noLP == 1;
}
/**
* @brief Check stacking flag
*
* @return value of flag
*/
bool
stacking() const {
return stacking_;
}
/**
* @brief Get maximum base pair span
*
* @note in vrna_md_s, a value of -1 indicates no restriction
* for distant base pairs; in this case, return the maximum
* value of size_t
*
* @return maximum allowed base pair span (returns maximum
* size_t value if unrestricted)
*/
size_t
max_bp_span() const {
return md_.max_bp_span >= 0 ? md_.max_bp_span
: std::numeric_limits<size_t>::max();
}
/**
* @brief Get dangling value
*
* @return value of dangling
*/
int
dangling() const {
return md_.dangles;
}
private:
vrna_md_t md_; //!< ViennaRNA model details
int stacking_; //!< calculate stacking probabilities
};
}
#endif // LOCARNA_PFOLD_PARAMS_HH
| code |
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\begin{document}
\title{
$H^s_x\times H^s_x$ scattering theory for a weighted gradient system of 3D radial defocusing NLS
}
\author{Xianfa Song{\thanks{E-mail: [email protected](X.F. Song)
}}\\
\small Department of Mathematics, School of Mathematics, Tianjin University,\\
\small Tianjin, 300072, P. R. China
}
\maketitle
\date{}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{remark}{Remark}[section]
\par\hang\textindentnewcommand{\thesection.\arabic{equation}}{\thesection.\arabic{equation}}
\catcode`@=11 \@addtoreset{equation}{section} \catcode`@=12
\begin{abstract}
In this paper, using $I$-method, we establish $H^s_x\times H^s_x$ scattering theories for the following Cauchy problem
\begin{equation*}
\left\{
\begin{array}{lll}
iu_t+\Delta u=\lambda |v|^2u,\quad iv_t+\Delta v=\mu|u|^2v,\quad x\in \mathbb{R}^3,\ t>0,\\
u(x,0)=u_0(x),\quad v(x,0)=v_0(x),\quad x\in \mathbb{R}^3.
\end{array}\right.
\end{equation*}
Here $\lambda>0$, $\mu>0$, $(u_0,v_0)\in H^s_x(\mathbb{R}^3)\times H^s_x(\mathbb{R}^3)$ and $\frac{1}{2}<s<1$.
{\bf Keywords:} Weighted(or essential) gradient system; Schr\"{o}dinger equation; Scattering; $I$-method.
{\bf 2010 MSC: 35Q55.}
\end{abstract}
\section{Introduction}
\qquad In this paper, we consider the following Cauchy problem:
\begin{equation}
\label{02201}
\left\{
\begin{array}{lll}
iu_t+\Delta u=\lambda |v|^2u,\quad iv_t+\Delta v=\mu|u|^2v,\quad x\in \mathbb{R}^3,\ t>0,\\
u(x,0)=u_0(x),\quad v(x,0)=v_0(x),\quad x\in \mathbb{R}^3.
\end{array}\right.
\end{equation}
Here $\lambda>0$, $\mu>0$, $(u_0,v_0)\in H^s_x(\mathbb{R}^3)\times H^s_x(\mathbb{R}^3)$ and $\frac{1}{2}<s<1$. Model (\par\hang\textindentf{02201}) often appears in in condensed matter theory, in quantum mechanics, in nonlinear optics, in plasma physics and in the theory of Heisenberg ferromagnet and magnons. $\lambda>0$ and $\mu>0$ mean the nonlinearities are defocusing ones. An interesting topic on (\par\hang\textindentf{02201}) is scattering phenomenon which we will study in this paper.
First, we would like to review some scattering results on the following Cauchy problem when $d=3$
\begin{equation}
\label{1104w1}
\left\{
\begin{array}{lll}
iu_t+\Delta u=\lambda|u|^pu,\quad x\in \mathbb{R}^d,\ t\in \mathbb{R},\\
u(x,0)=u_0(x),\quad x\in \mathbb{R}^d,
\end{array}\right.
\end{equation}
where $p>0$ and $\lambda\geq 0$, i.e., the equation in (\par\hang\textindentf{1104w1}) has the defocusing nonlinearity. About the classical results on $L^2_x$, $\Sigma$ and $H^1_x$ scattering theories for (\par\hang\textindentf{1104w1}), to see more details, we can refer to the books \cite{Bourgain19992, Cazenave2003, Dodson2019, Tao2006} and the numerous references therein. Recently, different types of scattering results on (\par\hang\textindentf{1104w1}) when $d=3$ were established by many authors, we can see \cite{Beceanu??, Bourgain19991, Colliander2008, Dodson2012, Dodson20190, Duyckaerts2008, Gao2019, Kenig2010, Killip2018, Killip2017, Killip2012, Killip2013, Killip2016, Killip20082, Masaki2019, Miao2013, Murphy2014, Murphy2015, Tao20071, Xie2013}. We specially point out that the $H^s_x$ scattering theory for (\par\hang\textindentf{1104w1}) with $p=2$ is also studied by many authors. Bourgain established the $H^s_x$ scattering theory for (\par\hang\textindentf{1104w1}) in \cite{Bourgain19981} with $p=2$ when $d=2$ and initial data in $H^s_x$ with $s>\frac{3}{5}$. In \cite{Bourgain19982}, for (\par\hang\textindentf{1104w1}) with $p=2$ when $d=3$ and radial initial data $u_0$, Bourgain proved global well-posedness and scattering for $s>\frac{5}{7}$. In \cite{Colliander2002}, for (\par\hang\textindentf{1104w1}) with $p=2$, through introducing an operator $I:H^s_x(\mathbb{R}^d)\rightarrow H^1_x(\mathbb{R}^d)$ and tracking the change of $E(Iu(t))$, Colliander et. al. proved global well-posedness for the solution to (\par\hang\textindentf{1104w1}) when $d=2$ for $s>\frac{4}{7}$ and when $d=3$ for $s>\frac{5}{6}$. Later, the result when $d=3$ was respectively extend to $s>\frac{4}{5}$ by Colliander et. al. in \cite{Colliander2004}, $s>\frac{5}{7}$ by Dodson in \cite{Dodson2013}, $s>\frac{2}{3}$ by Su in \cite{Su2012}. Recently, using $I$-method, Dodson obtained the
$H^s_x$ scattering theory for (\par\hang\textindentf{1104w1}) with $p=2$ for $s>\frac{1}{2}$.
There are some results on the scattering theory for a system of Schr\"{o}dinger equations. For $[H^1_x]^N$-scattering theory, we can refer to \cite{Cassano2015, Farah2017, Peng2019, Saanouni20191, Tarulli2019, Tarulli2016, Xu2016} and see the details. In \cite{Saanouni20192}, Saanouni established the $[H^s_x]^N$-scattering theory for $\frac{2}{3}<s<1$. Very recently, in \cite{Song1}, we studied
\begin{equation}
\label{826x1}
\left\{
\begin{array}{lll}
iu_t+\Delta u=\lambda |u|^{\alpha}|v|^{\beta+2}u,\quad iv_t+\Delta v=\mu|u|^{\alpha+2}|v|^{\beta}v,\quad x\in \mathbb{R}^d,\ t>0,\\
u(x,0)=u_0(x),\quad v(x,0)=v_0(x),\quad x\in \mathbb{R}^d.
\end{array}\right.
\end{equation}
Besides establishing the local well-posedness of the $H^1_x\times H^1_x$, $\Sigma\times \Sigma$ and $H^s_x\times H^s_x$ solutions, we found that there exists a critical exponents line $\alpha+\beta=2$ in the sense that (\par\hang\textindentf{826x1}) always possesses a unique bounded $H^1_x\times H^1_x$-solution for any initial data $(u_0,v_0)\in H^1_x(\mathbb{R}^3)\times H^1_x(\mathbb{R}^3)$ if $\alpha+\beta\leq 2$, while there exist some initial data $(u_0,v_0)\in H^1_x(\mathbb{R}^3)\times H^1_x(\mathbb{R}^3)$ such that it doesn't have the unique global $H^1_x\times H^1_x$-solution if $\alpha+\beta>2$ and $\alpha=\beta$. We also established $H^1_x\times H^1_x$ and $\Sigma\times \Sigma$ scattering theories when $(\alpha,\beta)$ below the critical exponents line when $d=3$, and $\dot{H}^{s_c}_x\times \dot{H}^{s_c}_x$ scattering theory when $d\geq 5$, where $s_c=\frac{d-2}{2}>1$.
As a special case of (\par\hang\textindentf{826x1}), (\par\hang\textindentf{02201}) is a weighted gradient system of Sch\"{o}dinger equations. Therefore, we can define the following weighted
mass and energy
\begin{align}
M_w(u,v)=\mu\|u\|_{L^2_x}+\lambda\|v\|_{L^2_x},\quad E_w(u,v)=\frac{1}{2}\int_{\mathbb{R}^3}[\mu|\nabla u|^2+\lambda|\nabla v|^2+\lambda\mu|u|^2|v|^2]dx.\label{03222}
\end{align}
Naturally, we hope to establish the $H^s_x\times H^s_x$ scattering theory for (\par\hang\textindentf{02201}) and $\frac{1}{2}<s<1$.
In fact, using $I$-method, we will establish $H^s_x\times H^s_x$ scattering theory for (\par\hang\textindentf{02201}) as follows.
{\bf Theorem 1.1($H^s_x\times H^s_x$ scattering theory for (\par\hang\textindentf{02201})).} {\it
Assume that $\lambda>0$, $\mu>0$, $(u_0,v_0)$ radial and $(u_0,v_0)\in H^s_x(\mathbb{R}^3)\times H^s_x(\mathbb{R}^3)$, $\frac{1}{2}<s<1$. Then the initial value problem (\par\hang\textindentf{02201}) is globally well-posedness and scattering, i.e., there exist $(u_+,v_+)\in H^s_x(\mathbb{R}^3)\times H^s_x(\mathbb{R}^3)$ and $(u_-,v_-)\in H^s_x(\mathbb{R}^3)\times H^s_x(\mathbb{R}^3)$ such that
\begin{align}
&\lim_{t\rightarrow +\infty}[\|u(t)-e^{it\Delta}u_+\|_{H^s_x(\mathbb{R}^3)}+\|v(t)-e^{it\Delta}v_+\|_{H^s_x(\mathbb{R}^3)}]=0,\label{02202}\\
&\lim_{t\rightarrow -\infty}[\|u(t)-e^{it\Delta}u_-\|_{H^s_x(\mathbb{R}^3)}+\|v(t)-e^{it\Delta}v_-\|_{H^s_x(\mathbb{R}^3)}]=0.\label{02203}
\end{align}
}
{\bf Remark 1.2:} 1. Scattering for (\par\hang\textindentf{02201}) corresponds to $[\|u\|_{L^5_{t,x}(\mathbb{R}\times\mathbb{R}^3)}+\|v\|_{L^5_{t,x}(\mathbb{R}\times\mathbb{R}^3)}]<+\infty$.
2. If $u_0(x)\equiv v_0(x)\equiv w_0(x)\in H^s_x(\mathbb{R}^3)$ and $\lambda=\mu$, then (\par\hang\textindentf{02201}) degrades into the scalar Schr\"{o}dinger equation $iw_t+\Delta w=\lambda |w|^2w$ subject to $w(x,0)=w_0(x)$, and the conclusions of Theorem 1.3 meet with those of Theorem 1.2 in \cite{Dodson2019}. In this sense, our results cover those of Theorem 1.2 in \cite{Dodson2019}.
The rest of this paper is organized as follows. In Section 2, we give some preliminaries. In Section 3, we will use $I$-method to establish $H^s_x\times H^s_x$ scattering theory for (\par\hang\textindentf{02201}).
\section{Preliminaries}
\qquad In this section, we will give some notations and useful lemmas which were mentioned in \cite{Dodson2019}.
We use $L^q_tL^r_x(I\times \mathbb{R}^3)$ to denote the Banach space of functions $u:I\times\mathbb{R}^3\rightarrow \mathbb{C}$ subject to the norm
$$
\|u\|_{L^q_tL^r_x(\mathbb{R}\times \mathbb{R}^3)}:=\left(\int_I\left(\int_{\mathbb{R}^3} |f(x)|^rdx\right)^{\frac{q}{r}}dt\right)^{\frac{1}{q}}<\infty
$$
for any spacetime slab $I\times \mathbb{R}^3$, with the usual modifications when $q$ or $r$ is infinity. Especially, we abbreviate $L^q_tL^r_x$ as $L^q_{t,x}$ if $q=r$.
The Fourier transform on $\mathbb{R}^3$ and the fractional differential operators $|\nabla |^s$ are defined by
$$
\hat{f}(\xi):=\int_{\mathbb{R}^3}e^{-2\pi ix\cdot \xi}f(x)dx,\quad \widehat{|\nabla|^sf}(\xi)=|\xi|^s\hat{f}(\xi),
$$
and the homogeneous and inhomogeneous Sobolev norms are respectively defined by
\begin{align}
\|f\|_{\dot{H}^s_x(\mathbb{R}^3)}:=\||\nabla|^sf\|_{L^2_x(\mathbb{R}^3)}=\||\xi|^s\hat{f}(\xi)\|_{L^2_x(\mathbb{R}^3)},\quad \|f\|_{H^s_x(\mathbb{R}^3)}:=\|(1+|\xi|^2)^{\frac{s}{2}}\hat{f}(\xi)\|_{L^2_x(\mathbb{R}^3)}.\label{03223}
\end{align}
Let $e^{it\Delta}$ be the free Schr\"{o}dinger propagator and the generated group of isometries $(\mathcal{J}(t))_{t\in \mathcal{R}}$. Then for $2\leq p\leq \infty$ and $\frac{1}{p}+\frac{1}{p'}=1$,
\begin{align}
&\mathcal{J}(t)f(x):=e^{it\Delta}f(x)=\frac{1}{(4\pi it)^{\frac{3}{2}}}\int_{\mathbb{R}^3}e^{i|x-y|^2/4t}f(y)dy,\quad t\neq 0,\label{829x1} \\
&\widehat{e^{it\Delta}f}(\xi)=e^{-4\pi^2it|\xi|^2}\hat{f}(\xi),\label{829x3}\\
& \|e^{it\Delta}f\|_{L^{\infty}_x(\mathbb{R}^3)}\lesssim |t|^{-\frac{3}{2}}\|f\|_{L^1_x(\mathbb{R}^3)},\quad t\neq 0,\label{829x2}\\
&\|e^{it\Delta}f\|_{L^p_x(\mathbb{R}^3)}\lesssim |t|^{-3(\frac{1}{2}-\frac{1}{p})}\|f\|_{L^{p'}_x(\mathbb{R}^3)}, \quad t\neq 0.\label{11101}
\end{align}
And the following Duhamel's formula holds
\begin{align}
u(t)=e^{i(t-t_0)\Delta}u(t_0)-i\int_{t_0}^te^{i(t-s)\Delta}(iu_t+\Delta u)(s)ds.\label{829x4}
\end{align}
We mention some definitions and estimates which also appeared in \cite{Dodson2019} below.
{\bf Definition 2.1(Strichartz space).} {\it Let $\tilde{S}^0(I)$ be the Strichartz space
\begin{align}
\tilde{S}^0(I)=L^{\infty}_tL^2_x(I\times\mathbb{R}^3)\cap L^2_tL^6_x(I\times\mathbb{R}^3),\label{022011}
\end{align}
and $\tilde{N}^0(I)$ be the dual
\begin{align}
\tilde{N}^0(I)=L^1_tL^2_x(I\times\mathbb{R}^3)+L^2_tL^{\frac{6}{5}}_x(I\times\mathbb{R}^3).\label{022012}
\end{align}
}
And the following Strichartz estimates hold
\begin{align}
\|e^{it\Delta}u_0\|_{S^0(\mathbb{R}\times \mathbb{R}^3)}\lesssim \|u_0\|_{L^2(\mathbb{R}^3)},\quad \|\int^t_0e^{i(t-\tau)\Delta}F(\tau)d\tau\|_{S^0(I\times\mathbb{R}^3)}\lesssim \|F\|_{N^0(I\times \mathbb{R}^3)}.\label{03225}
\end{align}
{\bf Definition 2.2(Littlewood-Paley decomposition).} {\it Given a radial and decreasing function $\psi\in C^{\infty}_0(\mathbb{R}^3)$ satisfying $\psi(x)=1$ for $|x|\leq 1$ and $\psi(x)=0$ for $|x|>2$. Let
\begin{align}
\phi_j(x)=\psi(2^{-j}x)-\psi(2^{-j+1}x)\label{02204}
\end{align}
and $P_j$ be the Fourier multiplier defined by
\begin{align}
\widehat{P_jf}(\xi)=\phi_j(\xi)\hat{f}(\xi)\quad {\rm for\ any}\ j.\label{02205}
\end{align}
This gives the following Littlewood-Paley decomposition
\begin{align}
f=\sum_{j=-\infty}^{\infty} P_jf \quad {\rm at\ least\ in \ the}\ L^2\ {\rm sense}.\label{02206}
\end{align}
}
In convenience, we write
\begin{align}
P_{\leq N} u=\sum_{j:2^j\leq N}P_ju,\quad P_{>N}u=u-P_{\leq N}u.\label{0220w1}
\end{align}
We may abbreviate $u_{\leq N}=P_{\leq N}u$ and $u_{>N}=P_{>N}u$ in the sequels.
The following proposition gives some well-known properties for the Littlewood-Paley decomposition.
{\bf Proposition 2.3.} {\it
For any $1<p<+\infty$,
\begin{align}
&\|f\|_{L^p_x(\mathbb{R}^3)}\thicksim_p \|\left(\sum_{j=-\infty}^{+\infty}|P_jf|^2\right)^{\frac{1}{2}}\|_{L^p_x(\mathbb{R}^3)},\label{02207}\\
&\|f\|_{L^p_x(\mathbb{R}^3)}\lesssim_s \|f\|_{\dot{H}^s_x(\mathbb{R}^3)}\quad {\rm for}\quad s=3(\frac{1}{2}-\frac{1}{p}).\label{02208}
\end{align}
We also have for $2\leq p\leq \infty$
\begin{align}
&\|P_jf\|_{L^p_x(\mathbb{R}^3)}\lesssim 2^{3j(\frac{1}{2}-\frac{1}{p})}\|P_jf\|_{L^2_x(\mathbb{R}^3)},\label{02209}
\end{align}
and the radial Sobolev embedding
\begin{align}
\||x|P_jf\|_{L^{\infty}(\mathbb{R}^3)}\lesssim \|P_jf\|_{\dot{H}^{\frac{1}{2}}_x(\mathbb{R}^3)}.\label{022010}
\end{align}
}
Let $\psi$ is the same $\psi$ as in Definition 1 and $\chi(x)=\psi(\frac{x}{2})-\psi(x)$. Then
\begin{align}
\|\psi(\frac{x}{R})e^{it\Delta}(P_ju_0)\|_{L^2_{t,x}(I\times \mathbb{R}^3)}\lesssim 2^{-\frac{j}{2}}R^{\frac{1}{2}}\|P_ju_0\|_{L^2_x(\mathbb{R}^3)},\label{022011}\\
\|\int_Ie^{-it\Delta}\psi(\frac{x}{R})(P_jF(t))dt\|_{L^2_x(\mathbb{R}^3)}\lesssim 2^{-\frac{j}{2}}R^{\frac{1}{2}}\|\psi(\frac{x}{R})P_jF\|_{L^2_{t,x}(\mathbb{R}^3)},\label{022012}
\end{align}
and for any $q<2$, if $F$ is supported on $|x|\leq R$,
\begin{align}
\||\nabla|^{1-\frac{1}{q}}\int_0^{\infty}e^{-it\Delta}F(t,x)dt\|_{L^2_x(\mathbb{R}^3)}\lesssim R^{1-\frac{1}{q}}\|F\|_{L^q_tL^2_x(\mathbb{R}\times\mathbb{R}^3)},\label{022013}\\
\lesssim R^{1-\frac{1}{q}}\|F\|_{X_R(I\times \mathbb{R}^3)},\label{022014}
\end{align}
where
\begin{align}
\|F\|_{X_R(I\times \mathbb{R}^3)}=R^{\frac{1}{q}-1}\|\psi(Rx)F\|_{L^q_tL^2_x(I\times \mathbb{R}^3)}+R^{\frac{1}{q}-1}\sum_{j\geq 0}2^{j(1-\frac{1}{q})}\|\chi(2^{-j}Rx)F\|_{L^q_tL^2_x(I\times \mathbb{R}^3)}.\label{022015}
\end{align}
{\bf Definition 2.4($U^p_{\Delta}$ spaces).} {\it
Let $1\leq p<\infty$ and $U^p_{\Delta}$ be an atomic space whose atoms are piecewise solutions of the linear equation,
\begin{align}
u_j=\sum_k1_{[t_k,t_{k+1})}e^{it\Delta}u_k,\quad \sum_k\|u_k\|^p_{L^2_x(\mathbb{R}^3)}=1.\label{0220x1}
\end{align}
Then for any $1\leq p<\infty$,
\begin{align}
\|u\|_{U^p_{\Delta}}=\inf\{\sum_j|c_j|:u=\sum_ju_j,\quad u_j \ {\rm are}\ U^p_{\Delta} \ atoms \}.\label{0220x2}
\end{align}
}
{\bf Proposition 2.5.} {\it If $u$ solves
\begin{align}
iu_t+\Delta u=F_1+F_2,\quad u(0,x)=u_0(x)\label{0220x3}
\end{align}
on the interval $0\in I\subset \mathbb{R}$, then for $q<2$,
\begin{align}
\||\nabla|^{1-\frac{1}{q}}u\|_{U^2_{\Delta}(I\times\mathbb{R}^3)}\lesssim_q \||\nabla|^{1-\frac{1}{q}}u_0\|_{L^2_x(\mathbb{R}^3)}+\|F_1\|_{X_R(I\times\mathbb{R}^3)}
+\||\nabla|^{1-\frac{1}{q}}F_2\|_{L^{2+}_tL^{6-}_x(I\times\mathbb{R}^3)}.\label{0220x4}
\end{align}
}
\section{Scattering result on (\par\hang\textindentf{02201}) }
\qquad In this section, we will use the I-method to establish scattering result on (\par\hang\textindentf{02201}).
{\bf Definition 3.1($I$-operator).} {\it Let $I:H^s_x(\mathbb{R}^3)\rightarrow H^1_x(\mathbb{R}^3)$
be the Fourier multiplier
\begin{align}
\widehat{If}(\xi)=m_N(\xi)\hat{f}(\xi),\label{0220x5}
\end{align}
where
\begin{equation}
\label{0220x6}
m_N(\xi)=\left\{
\begin{array}{lll}
1& {\rm if}\ |\xi|\leq N,\\
\frac{N^{1-s}}{|\xi|^{1-s}}& {\rm if}\ |\xi|\geq 2N.
\end{array}\right.
\end{equation}
}
We first give some results on the $I$-operator and the modified energy $E(Iu(t),Iv(t))$. By Sobolev embedding, we have
\begin{align}
&\quad E_w(Iu(t),Iv(t))=\frac{1}{2}\int_{\mathbb{R}^3}[\mu|\nabla Iu|^2+\lambda |\nabla Iv|^2+\lambda\mu|Iu|^2|Iv|^2]dx\nonumber\\
&\lesssim \int_{\mathbb{R}^3}[|\nabla Iu|^2+ |\nabla Iv|^2+|Iu|^4+|Iv|^4]dx\nonumber\\
&\lesssim \|Iu\|^2_{\dot{H}^1_x(\mathbb{R}^3)}+\|Iv\|^2_{\dot{H}^1_x(\mathbb{R}^3)}+\|Iu\|^2_{\dot{H}^1_x(\mathbb{R}^3)}\|u\|^2_{\dot{H}^{\frac{1}{2}}_x(\mathbb{R}^3)}
+\|Iv\|^2_{\dot{H}^1_x(\mathbb{R}^3)}\|v\|^2_{\dot{H}^{\frac{1}{2}}_x(\mathbb{R}^3)},\label{0220w3}
\end{align}
and consequently
\begin{align}
E_w(Iu(0),Iv(0))\lesssim C(\|u_0\|_{H^s_x},\|v_0\|_{H^s_x})N^{2(1-s)}.\label{0220w4}
\end{align}
Meanwhile,
\begin{align}
\|u(t)\|^2_{H^s_x(\mathbb{R}^3)}+\|v(t)\|^2_{H^s_x(\mathbb{R}^3)}\lesssim E_w(Iu(t),Iv(t))+M_w(Iu(t),Iv(t)).\label{0220w5}
\end{align}
Here
$$
M_w(Iu(t),Iv(t))=\mu\|u(t)\|^2_{L^2_x(\mathbb{R}^3)}+\lambda\|v(t)\|^2_{L^2_x(\mathbb{R}^3)}
$$
Moreover, we can use the rescaling
\begin{align}
(u(t,x),v(t,x))\rightarrow (u_a(t,x),v_a(t,x))=a^{\frac{1}{2}}(u(a^2t,ax),v(a^2t,ax))\label{03091}
\end{align}
such that $E_w(Iu(t),Iv(t))\leq 1$. And there exists $a^{s-\frac{1}{2}}\thicksim C(\|u_0\|_{H^s_x(\mathbb{R}^3)}, \|v_0\|_{H^s_x(\mathbb{R}^3)})N^{s-1}$ such that
\begin{align}
&E_w(Iu_{\lambda}(0),Iv_{\lambda}(0))\leq \frac{1}{2},\label{0220w6}
\end{align}
and
\begin{align}
&\|u_{\lambda}(0)\|_{L^2_x(\mathbb{R}^3)}\lesssim C(\|u(0)\|_{H^s_x(\mathbb{R}^3)})N^{\frac{1-s}{2s-1}}\|u(0)\|_{L^2_x(\mathbb{R}^3)},\label{0220w7}\\
&\|v_{\lambda}(0)\|_{L^2_x(\mathbb{R}^3)}\lesssim C(\|v(0)\|_{H^s_x(\mathbb{R}^3)})N^{\frac{1-s}{2s-1}}\|v(0)\|_{L^2_x(\mathbb{R}^3)}.\label{0220w8}
\end{align}
{\bf Proposition 3.2(Weight coupled interaction Morawetz estimate).} {\it Assume that $(u,v)$ is a solution of (\par\hang\textindentf{02201}) on some interval $J$. Then
\begin{align}
&\quad \||u|^4\|_{L^1_{t,x}(J\times \mathbb{R}^3)}+\||v|^4\|_{L^1_{t,x}(J\times \mathbb{R}^3)}+\||u|^2|v|^2\|_{L^1_{t,x}(J\times\mathbb{R}^3)}\nonumber\\
&\lesssim
[\|u\|^2_{L^{\infty}_tL^2_x(J\times\mathbb{R}^3)}
+\|v\|^2_{L^{\infty}_tL^2_x(J\times\mathbb{R}^3)}]
[\|u\|^2_{L^{\infty}_t\dot{H}^{\frac{1}{2}}_x(J\times\mathbb{R}^3)}
+\|v\|^2_{L^{\infty}_t\dot{H}^{\frac{1}{2}}_x(J\times\mathbb{R}^3)}].\label{0220w9}
\end{align}
}
{\bf Proof:} Similar to \cite{Song1}, we define the following weight-coupled interaction Morawetz potential:
\begin{align}
M^{\otimes_2}_a(t)&=2\mu^2\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\widetilde{\nabla} a(x,y)\Im[\bar{u}(t,x)\bar{u}(t,y)\widetilde{\nabla}(u(t,x)u(t,y))]dxdy\nonumber\\
&\quad+2\lambda^2\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\widetilde{\nabla} a(x,y)\Im[\bar{v}(t,x)\bar{v}(t,y)\widetilde{\nabla}(v(t,x)v(t,y))]dxdy\nonumber\\
&\quad+2\lambda\mu\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\widetilde{\nabla} a(x,y)\Im[\bar{u}(t,x)\bar{v}(t,y)\widetilde{\nabla}(u(t,x)v(t,y))]dxdy\nonumber\\
&\quad+2\lambda\mu\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\widetilde{\nabla} a(x,y)\Im[\bar{v}(t,x)\bar{u}(t,y)\widetilde{\nabla}(v(t,x)u(t,y))]dxdy,\label{829w1}
\end{align}
where $a(x,y)=|x-y|$, $\widetilde{\nabla}=(\nabla_x, \nabla_y)$, $x\in \mathbb{R}^3$ and $y\in \mathbb{R}^3$. By the inequality (4.10) in \cite{Song1}, we have
\begin{align*}
&\quad \int_I\int_{\mathbb{R}^3}[|u(t,x)|^4+|v(t,x)|^4]dxdt\lesssim |M^{\otimes_2}_a(t)|\nonumber\\
&\lesssim[\|u\|^2_{L^{\infty}_tL^2_x(J\times\mathbb{R}^3)}
+\|v\|^2_{L^{\infty}_tL^2_x(J\times\mathbb{R}^3)}]
[\|u\|^2_{L^{\infty}_t\dot{H}^{\frac{1}{2}}_x(J\times\mathbb{R}^3)}
+\|v\|^2_{L^{\infty}_t\dot{H}^{\frac{1}{2}}_x(J\times\mathbb{R}^3)}],
\end{align*}
which implies (\par\hang\textindentf{0220w9}) because $|u|^2|v|^2\leq |u|^4+|v|^4$.
$\Box$
{\bf Lemma 3.3.} {\it If $E(Iu(a_l),Iv(a_l))\leq 1$, $J_l=[a_l,b_l]$, and $\|u\|_{L^4_{t,x}(J_l\times \mathbb{R}^3)}+\|v\|_{L^4_{t,x}(J_l\times \mathbb{R}^3)}\leq \epsilon$ for some $\epsilon>0$ sufficiently small, then
\begin{align}
\|\nabla Iu\|_{\tilde{S}^0(J_l\times \mathbb{R}^3)}+\|\nabla Iv\|_{\tilde{S}^0(J_l\times \mathbb{R}^3)}\lesssim 1.\label{02211}
\end{align}
}
{\bf Proof:} Let
\begin{align}
Z_I(t)=\|\nabla Iu\|_{\tilde{S}^0([a_l,t]\times \mathbb{R}^3)}+\|\nabla Iv\|_{\tilde{S}^0([a_l,t]\times \mathbb{R}^3)}.\label{02212}
\end{align}
Applying $\nabla I$ to the equations of (\par\hang\textindentf{02201}), we have
\begin{equation}
\label{02213}
\left\{
\begin{array}{lll}
i(\nabla Iu)_t+\Delta (\nabla Iu)=\lambda \nabla I(|v|^2u),\quad i(\nabla Iv)_t+\Delta (\nabla Iv)=\mu\nabla I(|u|^2v),\\
\nabla Iu(x,0)=\nabla Iu_0,\quad \nabla Iv(x,0)=\nabla Iv_0.
\end{array}\right.
\end{equation}
Using Strichartz estimates with $q'=r'=\frac{10}{7}$, then applying a fractional Leibniz rule, we obtain
\begin{align}
Z_I(t)&\lesssim \|\nabla Iu_0\|_{L^2_x(\mathbb{R}^3)}+\|\nabla Iv_0\|_{L^2_x(\mathbb{R}^3)}+\|\nabla I(|v|^2u)\|_{L^{\frac{10}{7}}_{t,x}([a_l,t]\times\mathbb{R}^3)}+\|\nabla I(|u|^2v)\|_{L^{\frac{10}{7}}_{t,x}([a_l,t]\times\mathbb{R}^3)}\nonumber\\
&\lesssim [\|\nabla Iu\|_{L^{\frac{10}{3}}_{t,x}([a_l,t]\times\mathbb{R}^3)}+\|\nabla Iv\|_{L^{\frac{10}{3}}_{t,x}([a_l,t]\times\mathbb{R}^3)}]
[\|u\|^2_{L^5_{t,x}([a_l,t]\times\mathbb{R}^3)}+\|v\|^2_{L^5_{t,x}([a_l,t]\times\mathbb{R}^3)}]\nonumber\\
&\quad+\|\nabla Iu_0\|_{L^2_x(\mathbb{R}^3)}+\|\nabla Iv_0\|_{L^2_x(\mathbb{R}^3)}.\label{02214}
\end{align}
Since the $L^{\frac{10}{3}}_{t,x}$ factors are bounded by $Z_I(t)$, we only need to consider the bounds of $\|u\|^2_{L^5_{t,x}([a_l,t]\times\mathbb{R}^3)}$ and $\|v\|^2_{L^5_{t,x}([a_l,t]\times\mathbb{R}^3)}$. In fact, similar to the proof of (3.10) in \cite{Colliander2004}, we can get
\begin{align}
&\|u\|^2_{L^5_{t,x}([a_l,b_l]\times\mathbb{R}^3)}\lesssim \epsilon^{\delta_1}(\|\nabla Iu\|_{\tilde{S}^0([a_l,b_l]\times \mathbb{R}^3)})^{\delta_2},\label{02215'}\\
&\|v\|^2_{L^5_{t,x}([a_l,b_l]\times\mathbb{R}^3)}\lesssim \epsilon^{\delta_1}(\|\nabla Iv\|_{\tilde{S}^0([a_l,b_l]\times \mathbb{R}^3)})^{\delta_2} \label{02215}
\end{align}
for some $\delta_1>0$ and $\delta_2>0$.
By (\par\hang\textindentf{02214}), (\par\hang\textindentf{02215'}) and (\par\hang\textindentf{02215}), we have
\begin{align}
Z_I(t)\lesssim 1+\epsilon^{\delta_3}(Z_I(t))^{1+\delta_4}\quad {\rm for \ some} \ \delta_3>0, \delta_4>0.\label{02216}
\end{align}
For $\epsilon$ sufficiently small, (\par\hang\textindentf{02216}) yields (\par\hang\textindentf{02211}).
$\Box$
(\par\hang\textindentf{02211}) implies that
\begin{align}
\|\nabla Iu\|_{U^2_{\Delta}(J_l\times \mathbb{R}^3)}+\|\nabla Iv\|_{U^2_{\Delta}(J_l\times \mathbb{R}^3)}\lesssim 1.\label{03092}
\end{align}
{\bf Proposition 3.4.} {\it Let $0\in J$ be an interval such that $E(Iu(t),Iv(t))\leq 1$ on $J$. Then for $N(s, \|u_0\|_{H^s_x}, \|v_0\|_{H^s_x})$ sufficiently large,
\begin{align}
\|P_{>\frac{N}{8}}\nabla Iu\|_{L^2_tL^6_x(J\times \mathbb{R}^3)}+\|P_{>\frac{N}{8}}\nabla Iv\|_{L^2_tL^6_x(J\times \mathbb{R}^3)}\lesssim 1.\label{02217}
\end{align}
}
{\bf Proof:} Decomposing
\begin{align}
|v|^2u&=\varnothing((v_{>\frac{M}{8}})^2u)+\varnothing(v_{>\frac{M}{8}}u_{>\frac{M}{8}}v_{\leq\frac{M}{8}})
+\varnothing(v_{\leq\frac{M}{8}}u_{\leq\frac{M}{8}}v_{>\frac{M}{8}})+\varnothing((v_{\leq\frac{M}{8}})^2u_{>\frac{M}{8}})\nonumber\\
&\quad+\varnothing((v_{\leq\frac{M}{8}})^2u_{\leq\frac{M}{8}}),\label{02218}\\
|u|^2v&=\varnothing((u_{>\frac{M}{8}})^2v)+\varnothing(u_{>\frac{M}{8}}v_{>\frac{M}{8}}u_{\leq\frac{M}{8}})
+\varnothing(u_{\leq\frac{M}{8}}v_{\leq\frac{M}{8}}u_{>\frac{M}{8}})+\varnothing((u_{\leq\frac{M}{8}})^2v_{>\frac{M}{8}})\nonumber\\
&\quad+\varnothing((u_{\leq\frac{M}{8}})^2v_{\leq\frac{M}{8}}),\label{02219}
\end{align}
we can get
\begin{align}
P_{>M}(|v|^2u)&=P_{>M}[\varnothing((v_{>\frac{M}{8}})^2u)]+P_{>M}[\varnothing(v_{>\frac{M}{8}}u_{>\frac{M}{8}}v_{\leq\frac{M}{8}})]
+P_{>M}[\varnothing(v_{\leq\frac{M}{8}}u_{\leq\frac{M}{8}}v_{>\frac{M}{8}})]\nonumber\\
&\quad+P_{>M}[\varnothing((v_{\leq\frac{M}{8}})^2u_{>\frac{M}{8}})],\label{022110}\\
P_{>M}(|u|^2v)&=P_{>M}[\varnothing((u_{>\frac{M}{8}})^2v)]+P_{>M}[\varnothing(u_{>\frac{M}{8}}v_{>\frac{M}{8}}u_{\leq\frac{M}{8}})]
+P_{>M}[\varnothing(u_{\leq\frac{M}{8}}v_{\leq\frac{M}{8}}u_{>\frac{M}{8}})]\nonumber\\
&\quad+P_{>M}[\varnothing((u_{\leq\frac{M}{8}})^2v_{>\frac{M}{8}})],\label{022111}
\end{align}
because $P_{>M}[\varnothing((v_{\leq\frac{M}{8}})^2u_{\leq\frac{M}{8}})]=0$ and $P_{>M}[\varnothing((u_{\leq\frac{M}{8}})^2v_{\leq\frac{M}{8}})]=0$.
Since $\nabla I$ is a Fourier multiplier whose symbol is increasing as $|\xi|\nearrow +\infty$, by the product rule and (\par\hang\textindentf{0220x4}), we obtain
\begin{align}
&\quad \|\nabla IP_{>M}u(t)\|_{U^2_{\Delta}(J\times\mathbb{R}^3)}\nonumber\\
&\lesssim \|\nabla IP_{>M}u(0)\|_{L^2_x(\mathbb{R}^3)}+\|\nabla IP_{>M}\varnothing[(v_{>\frac{M}{8}})^2u]\|_{L^{2-}_tL^{\frac{6}{5}+}_x(J\times \mathbb{R}^3)}\nonumber\\
&\quad+\|\nabla IP_{>M}\varnothing[(v_{>\frac{M}{8}})(u_{>\frac{M}{8}})(v_{\leq\frac{M}{8}})]\|_{L^{2-}_tL^{\frac{6}{5}+}_x}+\|P_{>M}\varnothing[(v_{>\frac{M}{8}})(\nabla v_{\leq\frac{M}{8}})(u_{\leq\frac{M}{8}})]\|_{L^{2-}_tL^{\frac{6}{5}+}_x}\nonumber\\
&\quad+\|P_{>M}\varnothing[(v_{>\frac{M}{8}})(v_{\leq\frac{M}{8}})(\nabla u_{\leq\frac{M}{8}})]\|_{L^{2-}_tL^{\frac{6}{5}+}_x}
+\|P_{>M}\varnothing[(u_{>\frac{M}{8}})(\nabla v_{\leq \frac{M}{8}})(v_{\leq \frac{M}{8}})\|_{L^{2-}_tL^{\frac{6}{5}+}_x}\nonumber\\
&\quad+M^{\frac{1}{q}-1}\left(\|P_{>M}\varnothing[(\nabla Iv_{>\frac{M}{8}})(v_{\leq \frac{M}{8}})(u_{\leq \frac{M}{8}})]\|_{X_R}+\|P_{>M}\varnothing[(\nabla Iu_{>\frac{M}{8}})(v_{\leq \frac{M}{8}})^2]\|_{X_R}\right)\nonumber\\
&:=\|\nabla IP_{>M}u(0)\|_{L^2_x(\mathbb{R}^3)}+(I)+(II)+(III)+(IV)+(V)+(VI)+(VII)\label{02221}
\end{align}
if $M\leq N$. Here all the $L^{2-}_tL^{\frac{6}{5}+}_x$ norms are on $(J\times \mathbb{R}^3)$.
Choosing $\delta(\epsilon)>0$ such that $(2-\epsilon, \frac{6}{5}+\delta(\epsilon))$ is the dual of an admissible pair, using the properties of $\nabla I$ and H\"{o}lder's inequality, Berstein's inequality, we have
\begin{align}
(I)&\lesssim \|\nabla Iu\|_{L^{\infty-\epsilon}_tL^{2+\delta(\epsilon)}_x}\|P_{>\frac{M}{8}}v\|^2_{L^4_tL^6_x}+\|\nabla IP_{>\frac{M}{8}}v\|_{L^2_tL^6_x}
\|P_{>\frac{M}{8}}v\|_{L^{\infty}_tL^2_x}\|P_{\leq N} u\|_{L^{\infty-\epsilon}_tL^{6+\delta(\epsilon)}_x}\nonumber\\
&\quad+\|\nabla IP_{>\frac{M}{8}}v\|_{L^2_tL^6_x}\|P_{>\frac{M}{8}}v\|_{L^{\infty}_tL^3_x}\|P_{>N}u\|_{L^{\infty-\epsilon}_tL^{3+\delta(\epsilon)}_x}\nonumber\\
&\lesssim \|\nabla IP_{>\frac{M}{8}}v\|_{L^2_tL^6_x}\left(M^{-1}[\|\nabla Iu\|_{L^{\infty-\epsilon}_tL^{2+\delta(\epsilon)}_x}+\|P_{\leq N} u\|_{L^{\infty-\epsilon}_tL^{6+\delta(\epsilon)}_x}]+M^{-\frac{1}{2}}\|P_{> N} u\|_{L^{\infty-\epsilon}_tL^{3+\delta(\epsilon)}_x}\right)\nonumber\\
&\lesssim M^{-1}N^{\frac{3(1-s)}{2s-1}\cdot\frac{\epsilon}{2(2-\epsilon)}}\|\nabla I P_{>\frac{M}{8}}v\|_{U^2_{\Delta}(J\times \mathbb{R}^3)}.\label{02222}
\end{align}
Here all the norms are on $(J\times \mathbb{R}^3)$, and we have use the following facts
\begin{align}
&\|\nabla Iu\|_{L^{\infty-\epsilon}_tL^{2+\delta(\epsilon)}_x}+\|P_{\leq N} u\|_{L^{\infty-\epsilon}_tL^{6+\delta(\epsilon)}_x}\lesssim_{s, \|u_0\|_{H^s_x(\mathbb{R}^3)}, \|v_0\|_{H^s_x(\mathbb{R}^3)}} N^{\frac{3(1-s)}{2s-1}\cdot\frac{\epsilon}{2(2-\epsilon)}},\label{02223}\\
&\|P_{> N} u\|_{L^{\infty-\epsilon}_tL^{3+\delta(\epsilon)}_x}\lesssim_{s, \|u_0\|_{H^s_x(\mathbb{R}^3)}, \|v_0\|_{H^s_x(\mathbb{R}^3)}} N^{-\frac{1}{2}+\frac{3(1-s)}{2s-1}\cdot\frac{\epsilon}{2(2-\epsilon)}}.\label{02224}
\end{align}
Similarly,
\begin{align}
(II)\lesssim M^{-1}N^{\frac{3(1-s)}{2s-1}\cdot\frac{\epsilon}{2(2-\epsilon)}}[\|\nabla I P_{>\frac{M}{8}}u\|_{U^2_{\Delta}(J\times \mathbb{R}^3)}+\|\nabla I P_{>\frac{M}{8}}v\|_{U^2_{\Delta}(J\times \mathbb{R}^3)}]\label{02225}
\end{align}
and
\begin{align}
(III)&=\|P_{>M}\varnothing[(v_{>\frac{M}{8}})(\nabla v_{\leq\frac{M}{8}})(u_{\leq\frac{M}{8}})]\|_{L^{2-}_tL^{\frac{6}{5}+}_x}\lesssim \|\nabla v_{\leq \frac{M}{8}}\|_{L^{\infty-\epsilon}_tL^{2+\delta(\epsilon)}_x}\|v_{>\frac{M}{8}}\|_{L^2_tL^6_x}\|u_{\leq \frac{M}{8}}\|_{L^{\infty}_tL^6_x}\nonumber\\
&\quad\lesssim M^{-1}N^{\frac{3(1-s)}{2s-1}\cdot\frac{\epsilon}{2(2-\epsilon)}}[\|\nabla I P_{>\frac{M}{8}}u\|_{U^2_{\Delta}(J\times \mathbb{R}^3)}+\|\nabla I P_{>\frac{M}{8}}v\|_{U^2_{\Delta}(J\times \mathbb{R}^3)}],\label{02226}\displaybreak\\
(IV)&=\|P_{>M}\varnothing[(v_{>\frac{M}{8}})(v_{\leq\frac{M}{8}})(\nabla u_{\leq\frac{M}{8}})]\|_{L^{2-}_tL^{\frac{6}{5}+}_x}\lesssim \|\nabla u_{\leq \frac{M}{8}}\|_{L^{\infty-\epsilon}_tL^{2+\delta(\epsilon)}_x}\|v_{>\frac{M}{8}}\|_{L^2_tL^6_x}\|v_{\leq \frac{M}{8}}\|_{L^{\infty}_tL^6_x}\nonumber\\
&\quad\lesssim M^{-1}N^{\frac{3(1-s)}{2s-1}\cdot\frac{\epsilon}{2(2-\epsilon)}}[\|\nabla I P_{>\frac{M}{8}}u\|_{U^2_{\Delta}(J\times \mathbb{R}^3)}+\|\nabla I P_{>\frac{M}{8}}v\|_{U^2_{\Delta}(J\times \mathbb{R}^3)}],\label{02227}\\
(V)&=\|P_{>M}\varnothing[(u_{>\frac{M}{8}})(\nabla v_{\leq \frac{M}{8}})(v_{\leq \frac{M}{8}})]\|_{L^{2-}_tL^{\frac{6}{5}+}_x}\lesssim \|\nabla v_{\leq \frac{M}{8}}\|_{L^{\infty-\epsilon}_tL^{2+\delta(\epsilon)}_x}\|u_{>\frac{M}{8}}\|_{L^2_tL^6_x}\|v_{\leq \frac{M}{8}}\|_{L^{\infty}_tL^6_x}\nonumber\\
&\quad\lesssim M^{-1}N^{\frac{3(1-s)}{2s-1}\cdot\frac{\epsilon}{2(2-\epsilon)}}[\|\nabla I P_{>\frac{M}{8}}u\|_{U^2_{\Delta}(J\times \mathbb{R}^3)}+\|\nabla I P_{>\frac{M}{8}}v\|_{U^2_{\Delta}(J\times \mathbb{R}^3)}].\label{02228}
\end{align}
To give the estimates for (VI) and (VII), we first recall the following radial Sobolev embedding
\begin{align}
\||x|^{\frac{3}{2}-s}u\|_{L^{\infty}_x(\mathbb{R}^3)}\lesssim \|u\|_{\dot{H}^s_x(\mathbb{R}^3)},\quad
\||x|^{\frac{3}{2}-s}v\|_{L^{\infty}_x(\mathbb{R}^3)}\lesssim \|v\|_{\dot{H}^s_x(\mathbb{R}^3)}.\label{02229}
\end{align}
Interpolating (\par\hang\textindentf{02229}), by Strichartz estimates, using (\par\hang\textindentf{0220w7}), (\par\hang\textindentf{0220w8}) and (\par\hang\textindentf{03092}), we have
\begin{align}
&\|Iu\|^4_{L^4_tL^{\infty}_x(J\times \mathbb{R}^3)}+\|Iv\|^4_{L^4_tL^{\infty}_x(J\times \mathbb{R}^3)}\lesssim_{\|u_0\|_{H^s_x(\mathbb{R}^3)},\|v_0\|_{H^s_x(\mathbb{R}^3)}} N^{\frac{3(1-s)}{2s-1}},\label{0222x1}\\
&\||x|^{\frac{1}{2}}Iu\|_{L^{\infty-\epsilon}_tL^{\infty}_x}
+\||x|^{\frac{1}{2}}Iv\|_{L^{\infty-\epsilon}_tL^{\infty}_x}\lesssim_{s,\|u_0\|_{H^s_x(\mathbb{R}^3)},\|v_0\|_{H^s_x(\mathbb{R}^3)}} N^{\frac{(1-s)}{2s-1}\cdot[\frac{3\epsilon}{2(2-\epsilon)}+\frac{\epsilon}{(2-3\epsilon)}]}.\label{0222x2}
\end{align}
Choosing $R=N$ in (\par\hang\textindentf{022011}), by (\par\hang\textindentf{02229}) and (\par\hang\textindentf{0222x2}), we get
\begin{align}
&\quad R^{\frac{1}{q}-1}M^{\frac{1}{q}-1}\|\psi(Rx)(\nabla IP_{>\frac{M}{8}}v)(v_{\leq \frac{M}{8}})(u_{\leq \frac{M}{8}})\|_{L^q_tL^2_x(J\times\mathbb{R}^3)}\nonumber\\
&\lesssim R^{\frac{1}{q}-1}M^{\frac{1}{q}-1}\|\psi(Rx)(\nabla IP_{>\frac{M}{8}}v)\|_{L^2_{t,x}(J\times\mathbb{R}^3)}\|v_{\leq\frac{M}{8}}\|^{\frac{2\epsilon}{(2-\epsilon)}}_{L^4_tL^{\infty}_x}
\|u_{\leq\frac{M}{8}}\|^{\frac{4-4\epsilon}{(2-\epsilon)}}_{L^{\infty}_{t,x}}\nonumber\\
&\lesssim_{s,\|u_0\|_{H^s_x(\mathbb{R}^3)},\|v_0\|_{H^s_x(\mathbb{R}^3)}} N^{\frac{-4+3\epsilon}{2(2-\epsilon)}}M^{\frac{-4+3\epsilon}{2(2-\epsilon)}}N^{\frac{3(1-s)}{2s-1}\cdot \frac{2\epsilon}{2-\epsilon}}M^{\frac{2-2\epsilon}{2-\epsilon}}\|\nabla IP_{>\frac{M}{8}}v\|_{U^2_{\Delta}(J\times \mathbb{R}^3)}.\label{0222x3}
\end{align}
Meanwhile, by (\par\hang\textindentf{022011}) and (\par\hang\textindentf{0222x2}),
\begin{align}
&\quad M^{\frac{1}{q}-1}\sum_{j\geq 0}R^{\frac{1}{q}-1}2^{j(1-\frac{1}{q})}\|\chi(2^{-j}Rx)(\nabla IP_{>\frac{M}{8}}v)(v_{\leq \frac{M}{8}})(u_{\leq \frac{M}{8}})\|_{L^q_tL^2_x(J\times\mathbb{R}^3)}\nonumber\\
&\lesssim M^{\frac{1}{q}-1}\sum_{j\geq 0}R2^{-j}R^{\frac{1}{q}-1}2^{j(1-\frac{1}{q})}\|\chi(2^{-j}Rx)(\nabla IP_{>\frac{M}{8}}v)\|_{L^2_{t,x}}\||x|^{\frac{1}{2}}Iv\|_{L^{\infty-\epsilon}_tL^{\infty}_x}\||x|^{\frac{1}{2}}Iu\|_{L^{\infty}_{t,x}}\nonumber\\
&\lesssim_{s,\|u_0\|_{H^s_x(\mathbb{R}^3)},\|v_0\|_{H^s_x(\mathbb{R}^3)}} N^{\frac{\epsilon}{2(2-\epsilon)}}M^{\frac{\epsilon}{2(2-\epsilon)}}N^{\frac{(1-s)}{2s-1}\cdot[\frac{3\epsilon}{2(2-\epsilon)}+\frac{\epsilon}{2-3\epsilon}]}
\|\nabla IP_{>\frac{M}{8}}v\|_{U^2_{\Delta}(J\times \mathbb{R}^3)}.\label{0222x4}
\end{align}
The estimate for (VI) can be given by (\par\hang\textindentf{0222x3}) and (\par\hang\textindentf{0222x4}). We can obtain the estimate for (VII) similar to (\par\hang\textindentf{0222x3}) and (\par\hang\textindentf{0222x4}).
Combining (\par\hang\textindentf{02221}), (\par\hang\textindentf{02222}), (\par\hang\textindentf{02225}), (\par\hang\textindentf{02226}), (\par\hang\textindentf{02227}) with the estimates for (VI) and (VII), we can get
\begin{align}
\|\nabla IP_{>M}u\|_{U^2_{\Delta}(J\times\mathbb{R}^3)}&\lesssim_{s,\|u_0\|_{H^s_x(\mathbb{R}^3)},\|v_0\|_{H^s_x(\mathbb{R}^3)}, \epsilon} 1+\frac{N^{C_1(s)\epsilon}}{M^{1-C_2(s)\epsilon}}\|\nabla IP_{>\frac{M}{8}}u\|_{U^2_{\Delta}(J\times\mathbb{R}^3)}\nonumber\\
&\quad+\frac{N^{C_1(s)\epsilon}}{M^{1-C_2(s)\epsilon}}\|\nabla IP_{>\frac{M}{8}}v\|_{U^2_{\Delta}(J\times\mathbb{R}^3)}\label{0222x5}
\end{align}
for any $s$ with $\epsilon=\epsilon(s)$ sufficiently small, $C_1(s)\epsilon<\frac{1}{4}$ and $C_2(s)\epsilon<\frac{1}{4}$.
Similarly,
\begin{align}
\|\nabla IP_{>M}v\|_{U^2_{\Delta}(J\times\mathbb{R}^3)}&\lesssim_{s,\|u_0\|_{H^s_x(\mathbb{R}^3)},\|v_0\|_{H^s_x(\mathbb{R}^3)}, \epsilon} 1+\frac{N^{C_1(s)\epsilon}}{M^{1-C_2(s)\epsilon}}\|\nabla IP_{>\frac{M}{8}}u\|_{U^2_{\Delta}(J\times\mathbb{R}^3)}\nonumber\\
&\quad +\frac{N^{C_1(s)\epsilon}}{M^{1-C_2(s)\epsilon}}\|\nabla IP_{>\frac{M}{8}}v\|_{U^2_{\Delta}(J\times\mathbb{R}^3)}.\label{0222x6}
\end{align}
If $M>C(s,\|u_0\|_{H^s_x(\mathbb{R}^3)},\|v_0\|_{H^s_x(\mathbb{R}^3)})$, (\par\hang\textindentf{0222x5}) and (\par\hang\textindentf{0222x6}) mean that
\begin{align}
&\quad \|\nabla IP_{>M}u\|_{U^2_{\Delta}(J\times\mathbb{R}^3)}+\|\nabla IP_{>M}v\|_{U^2_{\Delta}(J\times\mathbb{R}^3)}\nonumber\\
&\lesssim_{s,\|u_0\|_{H^s_x(\mathbb{R}^3)},\|v_0\|_{H^s_x(\mathbb{R}^3)}} 1+N^{-\frac{1}{4}}C(s,\|u_0\|_{H^s_x(\mathbb{R}^3)},\|v_0\|_{H^s_x(\mathbb{R}^3)})^{-\frac{3}{4}}\nonumber\\
&\qquad\qquad \qquad\qquad \qquad\qquad\times [\|\nabla IP_{>\frac{M}{8}}u\|_{U^2_{\Delta}(J\times\mathbb{R}^3)}+\|\nabla IP_{>\frac{M}{8}}v\|_{U^2_{\Delta}(J\times\mathbb{R}^3)}].\label{0222w1}
\end{align}
Meanwhile, recalling (\par\hang\textindentf{03092}), we have
\begin{align}
&\quad\|\nabla IP_{>C(s,\|u_0\|_{H^s_x},\|v_0\|_{H^s_x})N^{\frac{2}{3}}}u\|_{U^2_{\Delta}(J\times\mathbb{R}^3)}
+\|\nabla IP_{>C(s,\|u_0\|_{H^s_x},\|v_0\|_{H^s_x})N^{\frac{2}{3}}}v\|_{U^2_{\Delta}(J\times\mathbb{R}^3)}\nonumber\\
&\lesssim
N^{\frac{3(1-s)}{4s-2}}.\label{0222w2}
\end{align}
By induction, for $C(s,\|u_0\|_{H^s_x(\mathbb{R}^3)},\|v_0\|_{H^s_x(\mathbb{R}^3)})$ sufficiently large,
\begin{align}
\|\nabla IP_{>\frac{N}{8}}u\|_{U^2_{\Delta}(J\times \mathbb{R}^3)}+\|\nabla IP_{>\frac{N}{8}}u\|_{U^2_{\Delta}(J\times \mathbb{R}^3)}\lesssim_{s,\|u_0\|_{H^s_x},\|v_0\|_{H^s_x}} 1+N^{\frac{3(1-s)}{4s-2}}N^{-c\ln(N)}.\label{0222w3}
\end{align}
Choosing $N$ sufficiently large such that $c\ln(N)>\frac{3(1-s)}{4s-2}$, then using (\par\hang\textindentf{0222w3}), we get
\begin{align}
\|\nabla IP_{>\frac{N}{8}}u\|_{U^2_{\Delta}(J\times \mathbb{R}^3)}+\|\nabla IP_{>\frac{N}{8}}u\|_{U^2_{\Delta}(J\times \mathbb{R}^3)}\lesssim_{s,\|u_0\|_{H^s_x},\|v_0\|_{H^s_x}} 1,\label{0222w4}
\end{align}
which implies (\par\hang\textindentf{02217}).
$\Box$
We give a bound on the modified energy increment.
{\bf Lemma 3.5.} {\it Let $N$ be sufficiently large such that
$$
\ln(N)\geq \frac{C_0(1-s)}{2s-1}+\ln(C(s,\|u_0\|_{H^s_x(\mathbb{R}^3)},\|v_0\|_{H^s_x(\mathbb{R}^3)})),
$$
then
\begin{align}
\int_J|\frac{d}{dt}E_w(Iu(t),Iv(t))|dt\lesssim \frac{1}{N^{1-}}.\label{02231}
\end{align}
}
{\bf Proof:} Note that
$$
E_w(Iu,Iv)=\frac{1}{2}\int_{\mathbb{R}^3}[\mu|\nabla Iu|^2+\lambda|\nabla Iv|^2-\lambda\mu|Iu|^2|Iv|^2]dx
$$
and $\Re((\overline{Iu_t})(iIu_t))=0$. Then
\begin{align}
&\quad \frac{d}{dt}E_w(Iu(t),Iv(t))\nonumber\\
&=\Re\int_{\mathbb{R}^3}\left\{\mu(\overline{Iu_t})[\lambda|Iv|^2(Iu)-\Delta Iu-iIu_t]
+\lambda(\overline{Iv_t})[\mu|Iu|^2(Iv)-\Delta Iv-iIv_t]\right\}dx\nonumber\\
&=\Re\int_{\mathbb{R}^3}\lambda\mu\left\{(\overline{Iu_t})[|Iv|^2(Iu)-I(|v|^2u)]+(\overline{Iv_t})[|Iu|^2(Iv)-I(|u|^2v)]\right\}dx.\label{02232}
\end{align}
Since $I$ is a Fourier multiplier which is constant in time and $\Delta$ commutes with $I$, by the equations of (\par\hang\textindentf{02201}),
\begin{equation}
\label{02233}
\left\{
\begin{array}{lll}
iIu_t+\Delta Iu=\lambda |Iv|^2(Iu)+\lambda [I(|v|^2u)-|Iv|^2(Iu),\quad x\in \mathbb{R}^3,\ t>0,\\
iIv_t+\Delta Iv=\mu |Iu|^2(Iv)+\mu [I(|u|^2v)-|Iu|^2(Iv),\quad x\in \mathbb{R}^3,\ t>0.
\end{array}\right.
\end{equation}
Integrating the equations of (\par\hang\textindentf{02233}) by parts, we have
\begin{align}
&\quad\frac{d}{dt}E_w(Iu(t),Iv(t))\nonumber\\
&=-\mu<i\nabla u, \nabla(|Iv|^2(Iu)-I(|v|^2u))>-\lambda\mu<iI(|v|^2u),(|Iv|^2(Iu)-I(|v|^2u))>\nonumber\\
&\quad -\lambda<i\nabla v, \nabla(|Iu|^2(Iv)-I(|u|^2v))>-\lambda\mu<iI(|u|^2v),(|Iu|^2(Iv)-I(|u|^2v))>.\label{02234}
\end{align}
Note that
\begin{align}
|Iv|^2(Iu)-I(|v|^2u)
&=(IP_{\leq\frac{N}{8}}v)^2(IP_{>\frac{N}{8}}u)-I((P_{\leq\frac{N}{8}}v)^2(P_{>\frac{N}{8}}u))\nonumber\\
&\quad+(IP_{>\frac{N}{8}}v)^2(IP_{\leq\frac{N}{8}}u)-I((P_{>\frac{N}{8}}v)^2(P_{\leq\frac{N}{8}}u))\nonumber\\
&\quad+(IP_{>\frac{N}{8}}v)^2(IP_{>\frac{N}{8}}u)-I((P_{>\frac{N}{8}}v)^2(P_{>\frac{N}{8}}u))\nonumber\\
&\quad+2(IP_{>\frac{N}{8}}v)(IP_{\leq\frac{N}{8}}v)(IP_{>\frac{N}{8}}u)-2I((P_{>\frac{N}{8}}v)(P_{\leq\frac{N}{8}}v)(P_{>\frac{N}{8}}u))\nonumber\\
&\quad+2(IP_{>\frac{N}{8}}v)(IP_{\leq\frac{N}{8}}v)(IP_{\leq\frac{N}{8}}u)-2I((P_{>\frac{N}{8}}v)(P_{\leq\frac{N}{8}}v)(P_{\leq\frac{N}{8}}u))\nonumber\\
&:=(1)+(2)+(3)+(4)+(5),\label{0223s1}\\
|Iu|^2(Iv)-I(|u|^2v)
&=(IP_{\leq\frac{N}{8}}u)^2(IP_{>\frac{N}{8}}v)-I((P_{\leq\frac{N}{8}}u)^2(P_{>\frac{N}{8}}v))\nonumber\\
&\quad+(IP_{>\frac{N}{8}}u)^2(IP_{\leq\frac{N}{8}}v)-I((P_{>\frac{N}{8}}u)^2(P_{\leq\frac{N}{8}}v))\nonumber\\
&\quad+(IP_{>\frac{N}{8}}u)^2(IP_{>\frac{N}{8}}v)-I((P_{>\frac{N}{8}}u)^2(P_{>\frac{N}{8}}v))\nonumber\\
&\quad+2(IP_{>\frac{N}{8}}u)(IP_{\leq\frac{N}{8}}u)(IP_{>\frac{N}{8}}v)-2I((P_{>\frac{N}{8}}u)(P_{\leq\frac{N}{8}}u)(P_{>\frac{N}{8}}v))\nonumber\\
&\quad+2(IP_{>\frac{N}{8}}u)(IP_{\leq\frac{N}{8}}u)(IP_{\leq\frac{N}{8}}v)-2I((P_{>\frac{N}{8}}u)(P_{\leq\frac{N}{8}}u)(P_{\leq\frac{N}{8}}v))\nonumber\\
&:=(6)+(7)+(8)+(9)+(10),\label{0223s1'}
\end{align}
and
\begin{align}
|m(\xi_2+\xi_3+\xi_4)-m(\xi_2)|\lesssim \frac{|\xi_3|+|\xi_4|}{|\xi_2|}.\label{0223s2}
\end{align}
By $E_w(Iu(t),Iv(t))\leq 1$ and (\par\hang\textindentf{02217}), we can obtain
\begin{align}
&\quad \int_J<i\nabla Iu, \nabla((IP_{\leq\frac{N}{8}}v)^2(IP_{>\frac{N}{8}}u)-I((P_{\leq\frac{N}{8}}v)^2(P_{>\frac{N}{8}}u)))>dt\nonumber\\
&\quad+\int_J<i\nabla Iv, \nabla((IP_{\leq\frac{N}{8}}u)^2(IP_{>\frac{N}{8}}v)-I((P_{\leq\frac{N}{8}}u)^2(P_{>\frac{N}{8}}v)))>dt\nonumber\\
&\lesssim \frac{1}{N}[\|\nabla Iu\|_{L^{\infty}_tL^2_x}+\|\nabla Iv\|_{L^{\infty}_tL^2_x}][\|\nabla IP_{>\frac{N}{8}}u\|^2_{L^2_tL^6_x}+\|\nabla IP_{>\frac{N}{8}}v\|^2_{L^2_tL^6_x}]\nonumber\\
&\quad\times[\|Iu\|_{L^{\infty}_tL^6_x}+\|Iv\|_{L^{\infty}_tL^6_x}]\lesssim \frac{1}{N},\label{0223s3}\\
&\quad \int_J<i\nabla Iu, \nabla((IP_{>\frac{N}{8}}v)^2(IP_{\leq\frac{N}{8}}u)-I((P_{>\frac{N}{8}}v)^2(P_{\leq\frac{N}{8}}u)))>dt\nonumber\\
&\quad+\int_J<i\nabla Iv, \nabla((IP_{>\frac{N}{8}}u)^2(IP_{\leq\frac{N}{8}}v)-I((P_{>\frac{N}{8}}u)^2(P_{\leq\frac{N}{8}}v)))>dt\nonumber\\
&\lesssim [\|\nabla Iu\|_{L^{\infty}_tL^2_x}+\|\nabla Iv\|_{L^{\infty}_tL^2_x}][\|\nabla IP_{>\frac{N}{8}}u\|_{L^2_tL^6_x}+\|\nabla IP_{>\frac{N}{8}}v\|_{L^2_tL^6_x}]\nonumber\\
&\quad\times[\| IP_{>\frac{N}{8}}u\|_{L^2_tL^6_x}+\| IP_{>\frac{N}{8}}v\|_{L^2_tL^6_x}][\|P_{\leq \frac{N}{8}}u\|_{L^{\infty}_tL^6_x}+\|P_{\leq \frac{N}{8}}v\|_{L^{\infty}_tL^6_x}]\lesssim \frac{1}{N},\label{0223s4}\displaybreak\\
&\quad \int_J<i\nabla Iu, \nabla((IP_{>\frac{N}{8}}v)^2(IP_{>\frac{N}{8}}u)-I((P_{>\frac{N}{8}}v)^2(P_{>\frac{N}{8}}u)))>dt\nonumber\\
&\quad+\int_J<i\nabla Iv, \nabla((IP_{>\frac{N}{8}}u)^2(IP_{>\frac{N}{8}}v)-I((P_{>\frac{N}{8}}u)^2(P_{>\frac{N}{8}}v)))>dt\nonumber\\
&\lesssim [\|\nabla Iu\|_{L^{\infty}_tL^2_x}+\|\nabla Iv\|_{L^{\infty}_tL^2_x}][\|\nabla IP_{>\frac{N}{8}}u\|_{L^2_tL^6_x}+\|\nabla IP_{>\frac{N}{8}}v\|_{L^2_tL^6_x}]\nonumber\\
&\quad\times [\|P_{>\frac{N}{8}}u\|^2_{L^4_tL^6_x}]+\|P_{> \frac{N}{8}}v\|^2_{L^4_tL^6_x}]\lesssim \frac{1}{N},\label{0223s5}\\
&\quad \int_J < i\nabla Iu,\nabla[2(IP_{>\frac{N}{8}}v)(IP_{\leq\frac{N}{8}}v)(IP_{>\frac{N}{8}}u)-2I((P_{>\frac{N}{8}}v)(P_{\leq\frac{N}{8}}v)(P_{>\frac{N}{8}}u))]>dt\nonumber\\
&\quad+\int_J < i\nabla Iv,\nabla[2(IP_{>\frac{N}{8}}u)(IP_{\leq\frac{N}{8}}u)(IP_{>\frac{N}{8}}v)-2I((P_{>\frac{N}{8}}u)(P_{\leq\frac{N}{8}}u)(P_{>\frac{N}{8}}v))]>dt\nonumber\\
&\lesssim [\|\nabla Iu\|_{L^{\infty}_tL^2_x}+\|\nabla Iv\|_{L^{\infty}_tL^2_x}][\|\nabla IP_{>\frac{N}{8}}u\|_{L^2_tL^6_x}+\|\nabla IP_{>\frac{N}{8}}v\|_{L^2_tL^6_x}]\nonumber\\
&\quad\times[\| IP_{>\frac{N}{8}}u\|_{L^2_tL^6_x}+\| IP_{>\frac{N}{8}}v\|_{L^2_tL^6_x}][\|P_{\leq \frac{N}{8}}u\|_{L^{\infty}_tL^6_x}+\|P_{\leq \frac{N}{8}}v\|_{L^{\infty}_tL^6_x}]\lesssim \frac{1}{N},\label{0223x1}\\
&\quad \int_J < i\nabla Iu,\nabla[2(IP_{>\frac{N}{8}}v)(IP_{\leq\frac{N}{8}}v)(IP_{\leq\frac{N}{8}}u)-2I((P_{>\frac{N}{8}}v)(P_{\leq\frac{N}{8}}v)(P_{\leq\frac{N}{8}}u))]>dt\nonumber\\
&\quad+\int_J < i\nabla Iv,\nabla[2(IP_{>\frac{N}{8}}u)(IP_{\leq\frac{N}{8}}u)(IP_{\leq\frac{N}{8}}v)-2I((P_{>\frac{N}{8}}u)(P_{\leq\frac{N}{8}}u)(P_{\leq\frac{N}{8}}v))]>dt\nonumber\\
&\lesssim \frac{1}{N}[\|\nabla Iu\|_{L^{\infty}_tL^2_x}+\|\nabla Iv\|_{L^{\infty}_tL^2_x}][\|\nabla IP_{>\frac{N}{8}}u\|^2_{L^2_tL^6_x}+\|\nabla IP_{>\frac{N}{8}}v\|^2_{L^2_tL^6_x}]\nonumber\\
&\quad\times[\|Iu\|_{L^{\infty}_tL^6_x}+\|Iv\|_{L^{\infty}_tL^6_x}]\lesssim \frac{1}{N}.\label{0223x2}
\end{align}
All the norms above are on $J\times \mathbb{R}^3$. (\par\hang\textindentf{0223s3})--(\par\hang\textindentf{0223x2}) give the estimates for $$-\mu<i\nabla u, \nabla(|Iv|^2(Iu)-I(|v|^2u))>-\lambda<i\nabla v, \nabla(|Iu|^2(Iv)-I(|u|^2v))>.$$
Now we consider the estimate for $$-\lambda\mu<iI(|v|^2u),(|Iv|^2(Iu)-I(|v|^2u))>-\lambda\mu<iI(|u|^2v),(|Iu|^2(Iv)-I(|u|^2v))>.$$
As a matter of convenience, we denote
\begin{align}
I(|v|^2)u&=(IP_{>\frac{N}{8}}v)^2(IP_{>\frac{N}{8}}u)+2(IP_{>\frac{N}{8}}v)(IP_{\leq \frac{N}{8}}v)(IP_{>\frac{N}{8}}u)+(IP_{\leq\frac{N}{8}}v)^2(IP_{>\frac{N}{8}}u)\nonumber\\
&\quad+(IP_{>\frac{N}{8}}v)^2(IP_{\leq \frac{N}{8}}u)+2(IP_{>\frac{N}{8}}v)(IP_{\leq \frac{N}{8}}v)(IP_{\leq\frac{N}{8}}u)+(IP_{\leq\frac{N}{8}}v)^2(IP_{\leq\frac{N}{8}}u)\nonumber\\
&:=(I)+(II)+(III)+(IV)+(V)+(VI),\label{02261}\\
I(|u|^2)v&=(IP_{>\frac{N}{8}}u)^2(IP_{>\frac{N}{8}}v)+2(IP_{>\frac{N}{8}}u)(IP_{\leq \frac{N}{8}}u)(IP_{>\frac{N}{8}}v)+(IP_{\leq\frac{N}{8}}u)^2(IP_{>\frac{N}{8}}v)\nonumber\\
&\quad+(IP_{>\frac{N}{8}}u)^2(IP_{\leq \frac{N}{8}}v)+2(IP_{>\frac{N}{8}}u)(IP_{\leq \frac{N}{8}}u)(IP_{\leq\frac{N}{8}}v)+(IP_{\leq\frac{N}{8}}u)^2(IP_{\leq\frac{N}{8}}v)\nonumber\\
&:=(I)'+(II)'+(III)'+(IV)'+(V)'+(VI)'.\label{02262}
\end{align}
Then
\begin{align}
&\quad \int_J <I(|v|^2u),(|Iv|^2(Iu)-I(|v|^2u))>dt\nonumber\\
&=\int_J\left(<(I),(1)>+<(I),(2)>+<(I),(3)>+<(I),(4)>+<(I),(5)>\right)dt\nonumber\\
&\quad+\int_J\left(<(II),(2)>+<(II),(3)>+<(II),(4)>+<(II),(5)>\right)dt\nonumber\displaybreak\\
&\quad+\int_J\left(<(III),(3)><(III),(4)>+<(III),(5)>+<(IV),(4)>\right)dt\nonumber\\
&\quad+\int_J\left(<(IV),(5)>+<(V),(5)>+<(VI),(1)>+<(VI),(2)>\right)dt\nonumber\\
&\quad+\int_J\left(<(VI),(3)>+<(VI),(4)>+<(VI),(5)>\right)dt.\label{02263}
\end{align}
Using Sobolev embedding theorem, Bernstein's inequality, we have
\begin{align}
&\quad\|(IP_{>\frac{N}{8}}v)^2(IP_{>\frac{N}{8}}u)\|_{L^2_{t,x}}\nonumber\\
&\lesssim [\|\nabla IP_{>\frac{N}{8}}u\|_{L^2_tL^6_x}+\|\nabla IP_{>\frac{N}{8}}v\|_{L^2_tL^6_x}]
[P_{>\frac{N}{8}}u\|^2_{L^{\infty}L^3_x}+P_{>\frac{N}{8}}v\|^2_{L^{\infty}L^3_x}]\lesssim \frac{1}{N},\label{0226x1}\\
&\int_J<(I),(1)>dt\lesssim \|(IP_{>\frac{N}{8}}v)^2(IP_{>\frac{N}{8}}u)\|^2_{L^2_{t,x}}\lesssim \frac{1}{N^2},\label{0226x2}
\end{align}
and
\begin{align}
\int_J<(I),(2)>dt&\lesssim \|(IP_{>\frac{N}{8}}v)^2(IP_{>\frac{N}{8}}u)\|_{L^2_{t,x}}[\|P_{>\frac{N}{8}}u\|_{L^2_tL^6_x}+\|P_{>\frac{N}{8}}v\|_{L^2_tL^6_x}]\nonumber\\
&\quad \times [\|P_{\leq \frac{N}{8}}u\|_{L^{\infty}_{t,x}}+\|P_{\leq \frac{N}{8}}v\|_{L^{\infty}_{t,x}}][\|P_{>\frac{N}{8}}u\|_{L^{\infty}_tL^3_x}+\|P_{>\frac{N}{8}}v\|_{L^{\infty}_tL^3_x}]\nonumber\\
&\lesssim \frac{1}{N^2},\label{0226x3}\\
\int_J<(I),(3)>dt&\lesssim \int_J[|P_{>\frac{N}{8}}u|^2+|P_{>\frac{N}{8}}v|^2][|P_{\leq\frac{N}{8}}u|^2+|P_{\leq\frac{N}{8}}v|^2][|u|^2+|v|^2]dt\nonumber\\
&\lesssim [\|P_{>\frac{N}{8}}u\|^2_{L^2_tL^6_x}+\|P_{>\frac{N}{8}}v\|^2_{L^2_tL^6_x}][\|P_{\leq \frac{N}{8}}u\|^4_{L^{\infty}_tL^6_x}+\|P_{\leq \frac{N}{8}}v\|^4_{L^{\infty}_tL^6_x}]\nonumber\\
&\quad+[\|P_{>\frac{N}{8}}u\|^4_{L^4_tL^6_x}+\|P_{>\frac{N}{8}}v\|^4_{L^4_tL^6_x}][\|P_{\leq \frac{N}{8}}u\|_{L^{\infty}_tL^6_x}+\|P_{\leq \frac{N}{8}}v\|^4_{L^{\infty}_tL^6_x}]\nonumber\\
&\lesssim \frac{1}{N^2},\label{0226x4}\\
\int_J<(VI),(3)>dt&\lesssim \frac{1}{N^2}[\|\nabla IP_{>\frac{N}{8}}u\|^2_{L^2_tL^6_x}+\|\nabla IP_{>\frac{N}{8}}v\|^2_{L^2_tL^6_x}][\|IP_{\leq \frac{N}{8}}u\|^3_{L^{\infty}_tL^6_x}+\|IP_{\leq \frac{N}{8}}v\|^3_{L^{\infty}_tL^6_x}]\nonumber\\
&\quad \times [\|IP_{\leq \frac{N}{8}}u\|_{L^{\infty}_tL^2_x}+\|IP_{\leq \frac{N}{8}}v\|_{L^{\infty}_tL^2_x}]\nonumber\\
&\lesssim \frac{1}{N^2}.\label{0227x1}
\end{align}
All the norms above are on $J\times \mathbb{R}^3$. Similar to (\par\hang\textindentf{0226x3}), we can get $\int_J<(I),(4)>dt\lesssim \frac{1}{N^2}$. Similar to (\par\hang\textindentf{0227x1}), we can obtain $\int_J<(VI),(5)>dt\lesssim \frac{1}{N^2}$. Similar to (\par\hang\textindentf{0226x4}), the estimates for other terms in (\par\hang\textindentf{02263}) can be bounded by $\frac{1}{N^2}$ because they all contain two $P_{>\frac{N}{8}}$ factors and two $P_{\leq\frac{N}{8}}$ factors. Putting all the results above together, we obtain the bound for $\int_J <I(|v|^2u),(|Iv|^2(Iu)-I(|v|^2u))>dt$.
Similarly, the bound for $\int_J <I(|u|^2v),(|Iu|^2(Iv)-I(|u|^2v))>dt$ can be established. Then substituting all the results into (\par\hang\textindentf{02234}), we get (\par\hang\textindentf{02231}).
$\Box$
Recalling that
\begin{align*}
\|u(t)\|^2_{L^2_x(\mathbb{R}^3)}+\|v(t)\|^2_{L^2_x(\mathbb{R}^3)}=\|u_0\|^2_{L^2_x(\mathbb{R}^3)}+\|v_0\|^2_{L^2_x(\mathbb{R}^3)},
\end{align*}
and
\begin{align*}
\|u(t)\|^2_{H^s_x(\mathbb{R}^3)}+\|v(t)\|^2_{H^s_x(\mathbb{R}^3)}\lesssim E_w(Iu(t),Iv(t))+\|u_0\|^2_{L^2_x(\mathbb{R}^3)}+\|v_0\|^2_{L^2_x(\mathbb{R}^3)},
\end{align*}
by Lemma 5.10 and (\par\hang\textindentf{02231}), we have
\begin{align}
\|u(t)\|_{H^s_x(\mathbb{R}^3)}+\|v(t)\|_{H^s_x(\mathbb{R}^3)}\lesssim C(s,\|u_0\|_{H^s_x(\mathbb{R}^3)}, \|v_0\|_{H^s_x(\mathbb{R}^3)})[\|u_0\|_{H^s_x(\mathbb{R}^3)}+ \|v_0\|_{H^s_x(\mathbb{R}^3)}].\label{02281}
\end{align}
Let $(p,q)$ be a $\frac{1}{2}$-admissible pair which satisfies $\frac{2}{p}=3(\frac{1}{2}-\frac{1}{q}-\frac{1}{6})$. Interpolating (\par\hang\textindentf{0220w9}) with (\par\hang\textindentf{02281}), we can get a bounded on $L^p_tL^q_x$. Now we can partition $\mathbb{R}$ into finitely many pieces with $[\|u\|_{L^p_tL^q_x(J_l\times \mathbb{R}^3)}+\|v\|_{L^p_tL^q_x(J_l\times \mathbb{R}^3)}]<\epsilon$ and use a perturbation argument to obtain $[\|u\|_{L^5_{t,x}(\mathbb{R}\times\mathbb{R}^3)}
+\|v\|_{L^5_{t,x}(\mathbb{R}\times\mathbb{R}^3)}]<+\infty$, which implies scattering.
$\Box$
\end{document} | math |
महिला क्रिकेट: न्यूजीलैंड से चौथा वनडे भी हारा भारत, भारतीय टीम को 63 रन से मिली करारी हार भारतीय महिला क्रिकेट टीम को क्वीन्सटाउन में खेले गए चौथे वनडे मुकाबले में 63 रनों से शिकस्त मिली है। इस जीत के बाद न्यूजीलैंड ने पांच मैचों की सीरीज में 40 की अजेय बढ़त बना ली है। बारिश के चलते मैच 2020 ओवरों का खेला गया, जिसमें कीवी टीम ने अमेलिया केर के अर्धशतक 68 की बदौलत पांच विकेट खोकर 191 का स्कोर बनाया। जवाब में भारतीय टीम 128 रनों पर ही सिमट गई। कप्तान सोफी डिवाइन 32 और सूजी बेट्स 41 ने पॉवरप्ले में 53 रन जोड़कर अच्छी शुरुआत दिलाई। नंबर तीन पर बल्लेबाजी की लिए आई केर ने आक्रामक अर्धशतक जड़ा और उन्हें एमी सैटरथवेट 32 का अच्छा साथ मिला और मेजबान टीम ने बड़ा स्कोर खड़ा किया।जवाब में भारत ने 19 के स्कोर तक अपने शीर्षक्रम के चार विकेट खो दिए। मध्य्रकम में कप्तान मिताली 30 और ऋचा घोष 52 ने संघर्ष किया, जो मैच जिताने के लिए नाकाफी था। | hindi |
using System.Reflection;
using System.Runtime.CompilerServices;
using System.Runtime.InteropServices;
// General Information about an assembly is controlled through the following
// set of attributes. Change these attribute values to modify the information
// associated with an assembly.
[assembly: AssemblyTitle("03.LatinLetters")]
[assembly: AssemblyDescription("")]
[assembly: AssemblyConfiguration("")]
[assembly: AssemblyCompany("")]
[assembly: AssemblyProduct("03.LatinLetters")]
[assembly: AssemblyCopyright("Copyright © 2017")]
[assembly: AssemblyTrademark("")]
[assembly: AssemblyCulture("")]
// Setting ComVisible to false makes the types in this assembly not visible
// to COM components. If you need to access a type in this assembly from
// COM, set the ComVisible attribute to true on that type.
[assembly: ComVisible(false)]
// The following GUID is for the ID of the typelib if this project is exposed to COM
[assembly: Guid("72f270a1-1dc6-4f4e-914f-cf504b8205ab")]
// Version information for an assembly consists of the following four values:
//
// Major Version
// Minor Version
// Build Number
// Revision
//
// You can specify all the values or you can default the Build and Revision Numbers
// by using the '*' as shown below:
// [assembly: AssemblyVersion("1.0.*")]
[assembly: AssemblyVersion("1.0.0.0")]
[assembly: AssemblyFileVersion("1.0.0.0")]
| code |
ಹುಬ್ಬಳ್ಳಿಅಂಕೋಲಾ ರೈಲು ಮಾರ್ಗ ಅಧ್ಯಯನ ಸಮಿತಿಯಲ್ಲಿ ರಾಷ್ಟ್ರೀಯ ಹುಲಿ ಸಂರಕ್ಷಣಾ ಪ್ರಾಧಿಕಾರದ ಪ್ರತಿನಿಧಿಗಳಿಲ್ಲ ಪಶ್ಚಿಮ ಘಟ್ಟಗಳ ಮೇಲೆ ಹಾದು ಹೋಗಲಿರುವ ಹುಬ್ಬಳ್ಳಿಅಂಕೋಲಾ ರೈಲು ಮಾರ್ಗ ಯೋಜನೆಯ ಪರಿಣಾಮದ ಅಧ್ಯಯನಕ್ಕಾಗಿ ಕೇಂದ್ರ ಅರಣ್ಯ ಮತ್ತು ಪರಿಸರ ಸಚಿವಾಲಯವು ಈಗಾಗಲೇ ಸಮಿತಿ ರಚಿಸಿದ್ದು, ಆ ಸಮಿತಿಯಲ್ಲಿ ರಾಷ್ಟ್ರೀಯ ಹುಲಿ ಸಂರಕ್ಷಣಾ ಪ್ರಾಧಿಕಾರದ ಎನ್ಟಿಸಿಎ ಪ್ರತಿನಿಧಿಯನ್ನು ಸೇರಿಸದೆ ಇರುವುದರ ಬಗ್ಗೆ ಕಳವಳ ವ್ಯಕ್ತವಾಗಿದೆ. ಕರ್ನಾಟಕ ಹೈಕೋರ್ಟ್ ಸಮಿತಿ ರಚಿಸಲು ಆದೇಶ ನೀಡಿದ ನಂತರ ಜೂನ್ 3 ರಂದು ಪರಿಸರ, ಅರಣ್ಯ ಮತ್ತು ಹವಾಮಾನ ಬದಲಾವಣೆ ಸಚಿವಾಲಯದ MOEF ಮತ್ತು CC ವನ್ಯಜೀವಿ ವಿಭಾಗವು ಯೋಜನೆಯನ್ನು ಪರಿಶೀಲಿಸಲು ಏಳು ಸದಸ್ಯರ ಸಮಿತಿಯನ್ನು ರಚಿಸಿದೆ. ಈ ಸಮಿತಿಗೆ MOEF ನ ಅರಣ್ಯ ಸಂರಕ್ಷಣಾ ವಿಭಾಗದ ಅರಣ್ಯಗಳ ಹೆಚ್ಚುವರಿ ಮಹಾನಿರ್ದೇಶಕರು ನೇತೃತ್ವ ನೀಡಲಿದ್ದ ಮತ್ತು ಇತರ ಆರು ಜನರನ್ನು ಸದಸ್ಯರಾಗಿ ಹೊಂದಿದೆ. ಲಿವಿಂಗ್ ಅರ್ಥ್ ಫೌಂಡೇಶನ್ LEAF ನ ಸದಸ್ಯರು ಈ ಯೋಜನೆಯನ್ನು ಅಂಶಿದಾಂಡೇಲಿ ಹುಲಿ ಸಂರಕ್ಷಿತ ಪ್ರದೇಶದ ಬಫರ್ ವಲಯದಲ್ಲಿ ಪ್ರಸ್ತಾಪಿಸಲಾಗಿದೆ ಮತ್ತು ಇದು ಪಶ್ಚಿಮ ಘಟ್ಟಗಳ ಜೀವವೈವಿಧ್ಯದ ಹಾಟ್ಸ್ಪಾಟ್ ಆಗಿರುವ ಕಾಳಿ ಮತ್ತು ಭದ್ರಾ ಹುಲಿ ಸಂರಕ್ಷಿತ ಪ್ರದೇಶದ ನಡುವಿನ ಪ್ರದೇಶದ ಮೂಲಕ ಹಾದುಹೋಗುತ್ತದೆ ಎಂದು ಆತಂಕ ವ್ಯಕ್ತಪಡಿಸಿದ್ದಾರೆ. ಹುಬ್ಬಳ್ಳಿಅಂಕೋಲಾ ರೈಲ್ವೆ ಮಾರ್ಗಕ್ಕಾಗಿ ಕಾರವಾರ, ಯಲ್ಲಾಪುರ ಮತ್ತು ಧಾರವಾಡ ವಿಭಾಗದಲ್ಲಿ 595.64 ಹೆಕ್ಟೇರ್ ಅರಣ್ಯ ಭೂಮಿಯನ್ನು ಬಳಸುವ ಕುರಿತು NTCA ಡಿಸೆಂಬರ್ 2017 ರಲ್ಲಿ ಸೈಟ್ ಮೌಲ್ಯಮಾಪನ ವರದಿಯನ್ನು ಸಲ್ಲಿಸಿದೆ ಎಂದು ಫೌಂಡೇಶನ್ ಹೇಳಿದೆ. MoEFCCಗೆ ನೀಡಿದ ನಿಯಮಗಳು ಯೋಜನೆಯ ಮೌಲ್ಯಮಾಪನವನ್ನು ಮಾಡಲು ಯಾವುದೇ ತಜ್ಞರನ್ನು ಕೋಆಪ್ಟ್ ಮಾಡಲು ಅನುಮತಿಸುತ್ತದೆ ಎಂದು ಹೇಳಿರುವ ಪ್ರತಿಷ್ಠಾನವು NTCA ಯ ತಜ್ಞರು ಈ ಸಮಿತಿಯಲ್ಲಿ ಇರಬೇಕಾಗಿದ್ದು ಅಗತ್ಯವಾಗಿದ್ದು ಅವರನ್ನು ಹೊರಗಿಡಬಾರದು ಎಂದು ಹೇಳಿದೆ. ಪಾರದರ್ಶಕತೆಯ ಕೊರತೆ LEAF ನ ಶ್ರೀಜಾ ಚಕ್ರವರ್ತಿ, ರಜನಿ ಸಂತೋಷ್ ಮತ್ತು ಸಂಧ್ಯಾ ಬಾಲಸುಬ್ರಮಣಿಯನ್ ಅವರು ರಾಷ್ಟ್ರೀಯ ವನ್ಯಜೀವಿ ಮಂಡಳಿ NBWL ಅಳವಡಿಸಿಕೊಂಡ ಕಾರ್ಯವಿಧಾನದಲ್ಲಿ ಪಾರದರ್ಶಕತೆಯ ಕೊರತೆ ಇದೆ ಎಂದು ಕಳವಳ ವ್ಯಕ್ತಪಡಿಸಿದ್ದಾರೆ. NBWL ನ 68 ನೇ ಸಭೆಯು ಮೇ 30 ರಂದು ನಡೆದಿದ್ದು, ಸಭೆಯ ನಡಾವಳಿಗಳನ್ನು ಜೂನ್ 26 ರವರೆಗೂ MoeF ವೆಬ್ಸೈಟ್ನಲ್ಲಿ ಅಪ್ಲೋಡ್ ಮಾಡಲಾಗಿಲ್ಲ ಮತ್ತು ಸಮಿತಿಯನ್ನು ರಚಿಸುವ ಆದೇಶವನ್ನು ಸಹ ಸಾರ್ವಜನಿಕಗೊಳಿಸಲಾಗಿಲ್ಲ. ಸಮಿತಿ ಮಾಡಲು ಆದೇಶ ನೀಡಿದ ಹೈಕೋರ್ಟ್ ಯೋಜನೆಯಲ್ಲಿ ತಮ್ಮ ಅಭಿಪ್ರಾಯಗಳನ್ನು ಮಂಡಿಸಲು ಬಯಸುವ ಸಾರ್ವಜನಿಕ ಪ್ರತಿನಿಧಿಗಳು, ಸರ್ಕಾರೇತರ ಸಂಸ್ಥೆಗಳು ಮತ್ತು ಇತರ ಮಧ್ಯಸ್ಥಗಾರರಿಗೆ ಅವಕಾಶ ನೀಡಬೇಕು ಎಂದಿದೆ. ಸಮಿತಿಯು ತನ್ನ ವರದಿಯನ್ನು ಸಚಿವಾಲಯಕ್ಕೆ ಸಲ್ಲಿಸಲು ಎರಡೂವರೆ ತಿಂಗಳ ಕಾಲಾವಕಾಶವನ್ನು ನೀಡಿದ್ದು, ಆ ನಂತರ ಅದನ್ನು ಹೈಕೋರ್ಟ್ಗೂ ಸಲ್ಲಿಸಬೇಕಿದೆ. ಎನ್ಜಿಒ ಸಮಿತಿಗೆ ಪತ್ರ ಬರೆದಿದ್ದು ಸಾರ್ವಜನಿಕರಿಗೆ ತಮ್ಮ ಅಭಿಪ್ರಾಯವನ್ನು ಹೇಳಲು ಅವಕಾಶ ಒದಗಿಸಲು ಕನಿಷ್ಠ ಒಂದು ವಾರ ಮುಂಚಿತವಾಗಿ ವೆಬ್ಸೈಟ್ಗೆ ಭೇಟಿ ನೀಡಬೇಕಾದ ದಿನಾಂಕವನ್ನು ಸಾರ್ವಜನಿಕರಿಗೆ ತಿಳಿಸಬೇಕೆಂದು ವಿನಂತಿಸಿದೆ. The post ಹುಬ್ಬಳ್ಳಿಅಂಕೋಲಾ ರೈಲು ಮಾರ್ಗ ಅಧ್ಯಯನ ಸಮಿತಿಯಲ್ಲಿ ರಾಷ್ಟ್ರೀಯ ಹುಲಿ ಸಂರಕ್ಷಣಾ ಪ್ರಾಧಿಕಾರದ ಪ್ರತಿನಿಧಿಗಳಿಲ್ಲ appeared first on Pratidhvani. | kannad |
Please donate all unused prescription Glasses to the Lions!
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Our Club was first chartered on September 18, 1957 with twenty three original charter members. Our members are from all walks of life joining in one common cause; to serve the needs of the local community. We celebrate the anniversary of that historic date in October of each year with a special dinner and program. We also have other fun social events throughout the year. | english |
We are happy to send you another EULAR news alert. As in the past, we kindly ask you to forward this message to the individual members of your society.
EULAR grants up to 10 bursaries to European applicants who wish to spend 3-6 months doing clinical or laboratory work in a university or hospital of another European country. To learn more about the programme, please visit: http://www.eular.org/scientific_training_bursaries.cfm. Applications are welcome until 31 March 2009.
The next EULAR On-Line Course on Rheumatic Diseases, starting on 14 September 2009.
Thank you very much for circulating this e-mail to your members.
We are happy to send you the first EULAR news alert in 2009. As in the past, we kindly ask you to forward this message to the individual members of your society.
Each year, EULAR honours the work of 3 young investigators by bestowing the EULAR Young Investigator Award. The awards are worth EUR 5,000 each and will be ceremoniously presented by the EULAR President during the opening of the EULAR Congress in Copenhagen. Visit the following site to learn more about this competition and submit your application (deadline: 28 February): http://www.eular.org/EULAR_Young_Investigor_Award.cfm.
Preparations for this year's EULAR congress, to take place in Copenhagen, 10 - 13 June, are well under way.
If you plan to submit an abstract, please do so in the course of next week; application deadline is 31 January, midnight CET.
In addition to our internal news, we are happy to inform you about a job opportunity in the United Kingdom. The "British Society for Rheumatology", based in central London, is looking for a Biologics Register Coordinator. Are you interested? You will find the job flyer with all vital information attached to this e-mail. Please feel free to also forward the flyer to interested colleagues.
The EULAR website has seen some major changes in the last few weeks. We would like to take this opportunity to inform you about the news and invite you to visit the website http://www.eular.org.
Have you ever been looking for an overview of what EULAR does? Then please visit our new "Activities" chapter on the main navigation. Here, you will find a convenient guide to information about our various activities. Topics covered include the work of the 8 Standing Committees, the EULAR Strategy 2008-2012, general information about the annual congress as well as access to EULAR's international affiliations.
Our "Recommendations " section includes some new entries. It is part of the Standing Committees' work to develop recommendations for the management and diagnosis of specific rheumatic diseases as well as recommendations for conducting and reporting clinical trials. All of these are published on the EULAR website.
Please regularly visit our "NEWS" section on the homepage. A topic of current interest is the EULAR Search Committee established to nominate candidates for open positions to be filled next year.
Last but not least, we also invite you to visit the "Education" section. Right now, a number of courses and scientific meetings are open for registration. EULAR bursaries are available for some of the courses.
We have the honor of hosting the rheumatologic ultrasound course for the EULAR organization (European League Against Rheumatism) in 2009. So far 15 EULAR courses on rheumatologic ultrasound have been held with great success.
The 16th course is in Copenhagen, Denmark, 7th to 10th of June 2009 just prior to the EULAR congress itself. It has been organized as a 3-day course with 3 levels: basic, intermediate and advanced. Each level is for 30 participants, all rheumatologists.
We have enclosed the preliminary program and the announcement can be seen at the EULAR website. The essence of the course is the use of ultrasound including Doppler in the evaluation of joints and tendons, grading of inflammation as well as the use of ultrasound in peripheral nerves, pediatrics and in arthritis.
The technical issues including important artifacts will be embedded in the lectures, as theses issues in our experience are not very well understood. The aim of the course is to enable the participants to evaluate their patients during treatment both for every day use and for clinical trials.
Hands-on sessions are a part of the course with five participants and an instructor per ultrasound machine.
The lecturers are all from Europe - all experts in the various fields of rheumatologic ultrasound and active in research.
We apply for an educational grant, which will be used to finance the invited international faculty (plane, hotel and food during the course), patient transportation, course material and congress dinner (download 16th Eular Sponsorship opportunities).
All the biological drug companies have been invited on the same terms.
We hope you find the above interesting and would like to participate as a sponsor.
I have the pleasure to write you from the word of Professor Maurizio Cutolo, Chairman of the 4th International Conference on: NeuroEndocrine Immunology in Rheumatic Diseases Translation from Basics to Clinics, which will be held in Santa Margherita, Genova - Italy, from May 8th to 10th 2009 - organized under the auspice of Several International Societies including EULAR.
The aim of the Congress will be to present interesting neuroimmunoendocrine aspects of the pathophysiology of Rheumatic Diseases presenting new aspects for diagnosis and therapeutical intervention.
This edition will be particularly dedicated to the translation from Basics to Clinics with several clinical sessions on new therapies in rheumatic diseases.
As Organizing Secretariat we have the pleasure to present you the Scientific programme and the Registration Form useful to register to the course.
Please, visit also the Conference Web-site: www.4thNEIRD.com for the Registration on line and Abstracts Submission.
Thanking you very much for your attention, we remain at your disposal for any further need.
Japan College of Rheumatology (JCR) will provide a grant to attend the 53rd annual general assembly and scientific meeting of Japan College of Rheumatology (JCR2009) and the 18th international Rheumatology Symposium in Tokyo on April 23-26 2009. The purpose of the JCR2009 international scholarship is to encourage young scientists and medical doctors under 40 years old other than Japanese to refine their knowledge of rheumatology and send their original scientific massage to the world through JCR2009. Those who received scholarship in past years, is not eligible to apply.
Abstracts should be submitted via email: [email protected]. | english |
तमिलनाडु में शहरी स्थानीय निकाय चुनाव के परिणाम आज, कड़ी सुरक्षा के बीच मतगणना शुरु तमिलनाडु, तमिलनाडु शहरी स्थानीय निकाय चुनाव परिणाम आज सबके सामने आ जाएंगे। मतगणना केंद्रों पर कड़ी सुरक्षा के बीच वोटों की काउंटिंग सुबह 8 बजे शुरु हो गई है। राज्य में 19 फरवरी को शहरी स्थानीय निकाय चुनाव हुआ था, जिसमें 60.70 फीसद मतदान दर्ज किया था। बता दें कि यह चुनाव राज्य में 11 साल के बाद हुआ है, जिसमें 21 निगमों, 138 नगर पालिकाओं और 490 नगर पंचायतों में 12,607 पदों के लिए 57, 778 उम्मीदवार चुनाव मैदान में खड़े हैं, जिसके लिए वोटों की गिनती जारी है।मतगणना 15 मतगणना केंद्रों पर हो रही है। हर एक वार्ड का परिणाम माइक्रोफोन के माध्यम से घोषित किया जाएगा। सभी मतगणना केंद्रों पर कड़ी चौकसी रखी गई है, आईएएस अधिकारी व पुलिस सुरक्षा जगह जगह पर तैनात है। साथ ही सीसीटीवी कैमरों के साथ मतगणना प्रक्रिया का निरीक्षण किया जा रहा है। बता दें कि 11 साल के लंबे वक्त के अंतराल के बाद हुए शहरी स्थानीय निकाय चुनाव के लिए एक ही चरण में मतदान 19 फरवरी को पूरा किया गया। निगम के महापौर व उप महापौर के लिए परोक्ष चुनाव तथा नगरपालिका के अध्यक्ष और उपाध्यक्ष एवं नगर पंचायत के अध्यक्ष और उपाध्यक्ष का चयन चार मार्च को किया जाएगाशहरी स्थानीय निकाय चुनाव के प्रत्याशीशहरी स्थानीय निकाय चुनाव में द्रविड़ मुनेत्र कड़गम DMK, अखिल भारतीय अन्ना द्रविड़ मुनेत्र कड़गम AIADMK और भारतीय जनता पार्टी BJP, कांग्रेस, नाम तामिलर काची, पट्टाली मक्कल काची, मक्कल निधि मय्यम और अम्मा मक्कल मुनेत्र कज़गम सहित प्रमुख राजनीतिक दल, 21 निगमों, 138 नगर पालिकाओं और 490 नगर पंचायतों के चुनाव में भाग ले रहे हैं। | hindi |
ജേര്ണലിസ്റ്റ് സത്യ സാന്റ മരിയിലെ ബാബു ആന്റണി കഥാപാത്രത്തെക്കുറിച്ച് സംവിധായകന് Begin typing your search... X മലയാളത്തിന്റെ ആക്ഷന് ഹീറോ ബാബു ആന്റണി നിലവില് മണിരത്നത്തിന്റെ പൊന്നിയില് സെല്വനില് ഒരു പ്രധാന കഥാപാത്രത്തെ അവതരിപ്പിക്കുന്നു. നവാഗതനായ വിനു വിജയ് സംവിധാനം ചെയ്യുന്ന സാന്റ മരിയയില് ജേര്ണലിസ്റ്റിന്റെ വേഷമാണ് അടുത്തായി അവതരിപ്പിക്കുന്നത്. ഒക്ടോബര് അവസാനത്തോടെ ചിത്രത്തിന്റെ ഷൂട്ടിങ്ങ് ആരംഭിക്കുമെന്ന് ടൈംസ് ഓഫ് ഇന്ത്യക്ക് നല്കിയ അഭിമുഖത്തില് സംവിധായകന് വിനു വിജയ് പറഞ്ഞു. വിനു വിജയ്യുടെ വാക്കുകള് ഒരു മീഡിയ കമ്ബനിയുടെ ഉടമയായ സത്യ എന്ന കഥാപാത്രത്തെയാണ് ബാബു അന്റണി അവതരിപ്പിക്കുന്നത്. ബിബിസി പോലുള്ള മാധ്യമസ്ഥാപത്തില് ജോലി ചെയ്തിരുന്ന സത്യ സ്വാതന്ത്ര്യം ആഗ്രഹിച്ച് സ്വന്തമായി പ്രവര്ത്തിക്കുന്നു. ചിത്രത്തില് ഒരു കൊലക്കേസില് പോലീസ് അന്വേഷണം നടക്കുകയും അതേ കേസില് സത്യക്ക് താല്പര്യം ഉണ്ടാവുകയുമാണ് ചെയ്യുന്നത്. കഥയുമായി ബാബു ആന്റണിയെ സമീപിക്കാന് ഞങ്ങള്ക്ക് ഭയമായിരുന്നു. പ്രത്യേകിച്ച് മണിരത്നം പോലുള്ള ഒരു സംവിധായകന് അദ്ദേഹത്തെ വച്ച് സിനിമ ചെയ്യുന്ന സാഹചര്യത്തില്. എങ്കിലും കഥ പറഞ്ഞു കഴിഞ്ഞപ്പോള് സിനിമ ചെയ്യാമെന്ന് അദ്ദേഹം സമ്മതിച്ചു. ഈ ചിത്രത്തിലൂടെ ബാബു ആന്റണിയുടെ വിന്റേജ് ലുക്ക് കൊണ്ടുവരാന് ഞങ്ങള് ശ്രമിക്കുന്നുണ്ട്. ഡെനിം ധരിച്ച നീണ്ട മുടിയുള്ള താടിയുള്ള ലുക്കിലായിരിക്കും എത്തുക. അദ്ദേഹം ഒരു ത്രല്ലര് നായകനായി എത്തുമെന്ന് നിങ്ങള് പ്രതീകിഷിക്കുമെന്ന് ഞങ്ങള്ക്കറിയാം എന്നാല് അതിനെക്കുറിച്ച് ഇപ്പോള് ഒന്നു പറയന് കഴിയില്ല. സിനിമയുടെ ചിത്രീകരണം പൂര്ണമായും കൊച്ചിയില് നടക്കുക. ഇര്ഷാദ്, അലന്സിയര്, അമേയ മാത്യു, ശാലിന് സോയ, മഞ്ജു പിള്ള എന്നിവരും ചിത്രത്തില് അണിനിരക്കും. കഥ, തിരക്കഥ സംഭാഷണം സംവിധായകനും തിരക്കഥാകൃത്തുമായ അമല് കെ ജോബിയാണ്. ഛായാഗ്രഹണം ഷിജു എം ഭാസ്കര്. സംഗീത സംവിധാനം കേദാര്. ജോസ് അറുകാലില് ആണ് എഡിറ്റര്. TAGS: Popular X TAGS: | malyali |
Module 8 is offered for 400 euro if taken on its own, for 200 euro if taken with one other Summer School Module, for 100 euro if taken with two other Summer School Modules, for 50 euro if taken with three other Summer School Modules and for free if taken with four or more other Summer School Modules.
Advanced Short Courses are offered at 500 euro each on a stand-alone basis.
Participation fees cover all tuition, educational material, coffee breaks, participation in the Conference Dinner, the conference folder with the articles that will be presented during the Conference. Also, they cover the (400 €) fee for attending the Conference that will take place during the period of the School. There is no VAT.
CRESSE negotiates special room charges for participants at the CRESSE Venue. Information about the special rates in the CRESSE venue that apply to CRESSE participants, alternative hotels and other information regarding accommodation are made regularly available in the CRESSE website (check Venue and Accommodation). Please check link above and book your room as soon as possible as early July is an extremely high touristic period for Crete. The Summer School participants pay for their accommodation directly to the hotel.
1. Early applicants (ONLY for single participants): There is a 30% discount for those registering by February 28th , 2019 and paying their fees by March 14th, 2019. The early registration discount is 15% for those registering and paying their fees before March 29th, 2019.
2. Discounts for organisations: If an organisation pays 3,500 € it is entitled to send two of its employees to attend any number of the School Modules and Advanced Short Courses.
3. Discounts for past participants: There is a 30% discount for all past participants who wish to attend 2019 Summer School or Advanced Short Courses.
4. Participation Fees for Postgraduate Students: A limited number of registered postgraduate students who are not in full time employment, will be selected on a first come first serve basis to participate in the Summer School at a reduced fee of €1,250.00 (for all Modules).
Please note that postgraduate students who would like to participate at the reduced participation fee of €1,250.00 must send a letter from the academic director/supervisor of their Master / PhD program to [email protected] confirming that they are not in full time employment.
For further details please contact the Summer School Secreteriat.
Please remember to send proof of the Bank Transfer, after receiving our Confirmation of Acceptance, to the School Secretariat by fax (+30-210 8223259) or by email (scanned) to [email protected]. Payment is necessary for a place to be reserved for an applicant.
Τhe deadline for payments is on Friday June 14th, 2019.
The bank transfer must indicate the name of the participant.
The receipt of the copy of the bank transfer is necessary to guarantee your place in CRESSE Summer School.
The organisation will return the fees (minus 250 € administration charge) when registration is cancelled by 8th April 2019.
Fifty per cent (50%) of the fees will be returned when registration is cancelled by 10th May 2019.
There will be no return of fees if registration is cancelled after 10th May 2019. | english |
கோவையில் இன்று 115 பேருக்கு கொரோனா தொற்று 2 பேர் உயிரிழப்பு..! கடந்த சில மாதங்களாக மற்ற மாவட்டங்களை காட்டிலும், கோவையில் கூடுதலாக தொற்று பாதிப்புகள் ஏற்பட்டு வந்தது.கோவிட் பற்றிய அனைத்து லேட்டஸ்ட் அப்டேட்களை இங்கே படியுங்கள் இதன் காரணமாக, தமிழ்நாட்டில் தினசரி கொரோனா பாதிப்புகளில் தொடர்ந்து முதலிடத்தில் நீடித்து வந்தது. அவ்வப்போது கொரோனா பாதிப்புகளில் சென்னை முதலிடம் பிடிப்பதும், மீண்டும் கோவை முதலிடம் பிடிப்பதுமாக இருந்தது. இன்று தினசரி பாதிப்பில் சென்னை முதலிடம் பிடித்ததால், கோவை இரண்டாம் இடத்தில் உள்ளது. அதேசமயம் கோவையில் கொரோனா தொற்று பாதிப்புகள் ஏறுவதும், இறங்குவதுமாக உள்ளது. கோவையில் இன்று 115 பேருக்கு கொரோனா தொற்று பாதிப்புகள் உறுதியாகியுள்ளது. இதனால் கோவை மாவட்டத்தில் இதுவரை கொரோனா தொற்று பாதித்தவர்களின் எண்ணிக்கை இரண்டு லட்சத்து 48 ஆயிரத்து 662 ஆக உயர்ந்துள்ளது. மருத்துவமனைகள் மற்றும் சிகிச்சை மையங்களில் 1185 பேர் சிகிச்சை பெற்று வருகின்றனர். நாள்தோறும் சிகிச்சை பெற்று வருபவர்களின் எண்ணிக்கை மீண்டும் அதிகரித்து வருகிறது. இன்று கொரோனா தொற்றில் இருந்து 113 பேர் குணமடைந்து வீடு திரும்பியுள்ளனர். இதனால் குணமடைந்து வீடு திரும்பியவர்களின் எண்ணிக்கை 2 லட்சத்து 45 ஆயிரத்து 35 பேராக உயர்ந்துள்ளது. கொரோனா தொற்று பாதிப்பால் இன்று 2 பேர் உயிரிழந்தனர். கோவை மாவட்டத்தில் இதுவரை உயிரிழந்தவர்களின் எண்ணிக்கை 2442 ஆக அதிகரித்துள்ளது. ஈரோடு , திருப்பூர் , நீலகிரி நிலவரம் ஈரோட்டில் இன்று 72 பேருக்கு கொரோனா தொற்று பாதிப்புகள் ஏற்பட்டுள்ளன. 74 பேர் குணமடைந்து வீடு திரும்பியுள்ளனர். மருத்துவமனைகள் மற்றும் சிகிச்சை மையங்களில் 777 பேர் சிகிச்சை பெற்று வருகின்றனர். தொற்று பாதிப்பால் இன்று உயிரிழப்புகள் ஏற்படவில்லை. ஈரோடு மாவட்டத்தின் மொத்த கொரோனா பாதிப்புகள் 105410 ஆக உயர்ந்துள்ளது. குணமடைந்தவர்கள் எண்ணிக்கை 103940 ஆகவும், மொத்த உயிரிழப்புகள் 693 ஆக உள்ளது. திருப்பூர் மாவட்டத்தில் இன்று 50 பேருக்கு இன்று கொரோனா பாதிப்புகள் ஏற்பட்டுள்ளன. 67 பேர் குணமடைந்துள்ள நிலையில், 591 பேர் சிகிச்சை பெற்று வருகின்றனர். கொரோனா பாதிப்பால் இன்று உயிரிழப்புகள் ஏற்படவில்லை. திருப்பூரின் மொத்த கொரோனா பாதிப்புகள் 96416 ஆகவும், குணமடைந்தவர்கள் எண்ணிக்கை 94838 ஆகவும் அதிகரித்துள்ளது. உயிரிழப்புகளின் எண்ணிக்கை 987 ஆக உள்ளது. நீலகிரி மாவட்டத்தில் இன்று 18 பேருக்கு கொரோனா தொற்று பாதிப்புகள் ஏற்பட்டுள்ளன. 21 பேர் குணமடைந்துள்ள நிலையில், 187 பேர் சிகிச்சை பெற்று வருகின்றனர். தொற்று பாதிப்பால் இன்று உயிரிழப்புகள் ஏற்படவில்லை. நீலகிரி மாவட்டத்தின் மொத்த பாதிப்புகள் 33868 ஆகவும், குணமடைந்தவர்களின் எண்ணிக்கை 33468 ஆகவும் உயர்ந்துள்ளது. மொத்த உயிரிழப்புகளின் எண்ணிக்கை 213 ஆக உள்ளது. Check out below Health Tools | tamil |
Lokesh: ఇవాళ, రేపు నారా పోలవరం పోరు.. భద్రాద్రి రామయ్య దర్శనం అనంతరం షురూ Nara Lokesh: తెలుగుదేశం పార్టీ జాతీయ ప్రధాన కార్యదర్శి నారా లోకేష్ పోలవరం ముంపు మండలాల్లో ఇవాళ పర్యటించనున్నారు. భద్రాద్రి రామయ్యను దర్శించుకున్న అనంతరం లోకేష్.. పోలవరం నిర్వాసిత ప్రాంతాల్లో పర్యటన కొనసాగిస్తారు. లోకేష్ పోలవరం నిర్వాసితుల పరామర్శలు రేపు కూడా కొనసాగుతాయి. ఈ పర్యటనలో పోలవరం ప్రాజెక్టు ముంపు నిర్వాసితుల సమస్యలు విని వారికి తగిన సూచనలు చేయనున్నారు లోకేష్. నేడు భద్రాచలం, టేకులబోరు, శ్రీరామగిరి, చింతూరులో లోకేష్ పర్యటిస్తారు. రేపు రంపచోడవరం, దేవీపట్నం, పెదవేంపల్లి, ఇందుకూరు, ముసిరిగుంట, కృష్ణునిపాలెంలో లోకేష్ పర్యటన కొనసాగనుంది. కాగా, ప్రకాశం జిల్లాలో ప్రభుత్వ భవనం కూలిపోయి ఓ విద్యార్థి మృత్యువాత పడిన ఘటనపై లోకేష్ తీవ్ర స్థాయిలో జగన్ సర్కారుపై విరుచుకుపడ్డారు. మార్కాపురం మండలం రాజుపాలెంలో పలువురు విద్యార్థులు పాఠశాల ప్రాంగణంలో ఆడుకుంటున్న సమయంలో ఒక్కసారిగా భవనం పైకప్పు కూలిన సంగతి తెలిసిందే. దీంతో అక్కడే ఉన్న విష్ణు అనే విద్యార్థి స్పాట్లోనే ప్రాణాలను కోల్పోయాడు. రాష్ట్రవ్యాప్తంగా నాడు నేడు కార్యక్రమం కింద పాఠశాలల రూపురేఖలు మారిపోయాయి అని చెప్పుకొచ్చే సీఎం జగన్ దీనిపై సిగ్గుతో తలవంచుకున్నారంటూ లోకేష్ ఎద్దేవా చేశారు. సిగ్గుతో తల ఎప్పుడు దించుకుంటున్నారు జగన్ రెడ్డి గారు? అంటూ లోకేష్ ట్విట్టర్లో ఫైరయ్యారు. రాజుపాలెంలో ప్రభుత్వ పాఠశాల భవనం స్లాబు కూలి విద్యార్థి విష్ణు మృతి చెందడం బాధాకరమన్నారు. పాఠశాలల్ని దేవాలయాలుగా మార్చేస్తాం, నాడు నేడు అంటూ పబ్లిసిటీ స్టంట్ చేశారని దుయ్యబట్టారు. కానీ, రియాలిటీలో మాత్రం నాడు నేడు వైసీపీ నాయకుల అవినీతికి కేరాఫ్ అడ్రస్గా మారిందని లోకేష్ ఆరోపణలు గుప్పించారు. Pawan Kalyan: జనసేన స్థూపాన్ని అడ్డుకున్న పోలీసులు.. వైసీపీ నేతల పనేనంటూ జనసైనికుల ఆందోళన | telegu |
പെണ്കുട്ടിയെ രാത്രി സ്കൂള് കെട്ടിടത്തില് എത്തിച്ച് പീഡിപ്പിച്ചു രണ്ടു യുവാക്കള് അറസ്റ്റില് തിരുവനന്തപുരം: പ്രായപൂര്ത്തിയാകാത്ത പെണ്കുട്ടിയെ രാത്രിയില് കൂട്ടിക്കൊണ്ടു പോയി പീഡിപ്പിച്ച കേസില് പത്തൊന്പതുകാരായ രണ്ടു പേര് അറസ്റ്റില്. കടയ്ക്കല് സ്വദേശികളായ നിഖില് ,മുഹമ്മദ് ഇര്ഫാന് എന്നിവരാണ് അറസ്റ്റിലായത്. സമൂഹമാധ്യമം വഴി പരിചയപ്പെട്ട പെണ്കുട്ടിയെ രാത്രിയില് ഒഴിഞ്ഞ സ്കൂള് കെട്ടിടത്തില് എത്തിച്ചാണ് പ്രതികള് പീഡിപ്പിച്ചത് . ഒക്ടോബര് 14 ന് അര്ദ്ധരാത്രിയിലാണ് പ്രായ പൂര്ത്തിയാകാത്ത പെണ്കുട്ടിയെ പാലോടുള്ള വീട്ടില് നിന്ന് കാണാതായത്. തുടര്ന്ന് വീട്ടുകാര് നല്കിയ പരാതിയുടെ അടിസ്ഥാനത്തില് പൊലീസ് കുട്ടിയുടെ വീട്ടിലും പരിസരങ്ങളിലുമായി അന്വേഷണം നടത്തി. പുലര്ച്ചെ അഞ്ച് മണിയോടുകൂടി പെണ്കുട്ടിയെ ജവഹര് കോളനിയിലെ തേരിയില് ഭാഗത്തു നിന്നും കണ്ടെത്തുകയായിരുന്നു. പെണ്കുട്ടിയില് നിന്ന് പൊലീസ് വിവരങ്ങള് ചോദിച്ച് മനസിലാക്കിയതോടെയാണ് പീഡനവിവരം പുറത്തറിയുന്നത്. പെണ്കുട്ടിയെ ബൈക്കില് പാലോട് കൊല്ലായില് ഭാഗത്തേക്ക് കൊണ്ടു പോയ ശേഷം സമീപത്തുള്ള ഒരു സ്കൂള് കെട്ടിടത്തില് എത്തിച്ച് പീഡിപ്പിക്കുകയായിരുന്നു. തുടര്ന്ന് തിരികെ ബൈക്കില് കയറ്റി ഇലവു പാലത്തിനു സമീപം ഇറക്കിയ വിട്ട ശേഷം കടന്നു കളയുകയായിരുന്നു. പെണ്കുട്ടിയെ വിളിച്ച ഫോണ് നമ്ബര് കേന്ദ്രികരിച്ച് നടത്തിയ അന്വേഷണമാണ് പ്രതികളെ കുടുക്കിയത്. ഇവര്ക്ക് സഹായം ചെയ്ത രണ്ടു പേര്കൂടി പിടിയിലാവാനുണ്ട്. | malyali |
जज के कार्यालय व मकान के लिए वकील भूख हड़ताल पर नारायणगढ़ नईदुनिया न्यूज। न्यायालय में अतिरिक्त जिला व सत्र न्यायालय की स्थापना, न्यायाधीश कक्ष व न्यायाधीश निवास को लेकर अभिभाषकों की क्रमिक भूख हड़ताल शुक्रवार को दूसरे दिन भी जारी रही। पहले दिन अनशन पर बैठे अभिभाषक रात भर हाड़ कंपा देने वाली ठंड में न्यायालय के सामने अनशन पर बैठे रहे। दूसरे दिन भी अनशन पर बैठे अभिभाषकों से करीब 30 घंटे बाद प्रशासन की तरफ से केवल नायब तहसीलदार ने पहुंचकर मुलाकात की। नायब तहसीलदार द्वारा अभिभाषकों से अतिरिक्त जिला व सत्र न्यायालय की पूर्व में पदस्थापना की जानकारी लेकर उच्चाधिकारियों तक बात पंहुचाने का आश्वासन दिया और लौट गए।वित्तमंत्री ने लिया संज्ञानअभिभाषकों के अनशन पर बैठने की जानकारी लगते ही वित्त मंत्री जगदीश देवड़ा ने मामला तुरंत संज्ञान में लिया। देवड़ा द्वारा प्रमुख सचिव को न्यायालय में न्यायाधीश कक्ष तथा दो भवनों की वित्तीय स्वीकृति के लिए अनुशंसा कर आवश्यक कार्यवाही करने हेतु पत्र जारी किया। उल्लेखनीय है कि वित्त मंत्री जगदीश देवड़ा पहले भी न्यायालय में अपनी विधायक निधि से अभिभाषक कक्ष निर्माण में सहयोग कर चुके हैं।सरकारी अस्पताल में तकिये, कंबल व चादर भेंट किए नारायणगढ़ नईदुनिया न्यूज। आजादी के अमृत महोत्सव के तहत पाटीदार समाज द्वारा शासकीय अस्पताल में 20 चादर, 20 तकिये एवं 20 कंबल भेंट किए। इस अवसर पर सेवानिवृत्त जिला क्रीड़ा अधिकारी अशोक भागरिया, डा. स्वदेश पाटीदार वरिष्ठ अध्यापक, गोवर्धनलाल भागरिया, अरुण कुमार भगत, दिलीप पटेल, डा. दिनेशचंद्र रुपरा, डा. परीक्षित आर्य, सरदार पटेल युवा संगठन अध्यक्ष जसवंत रुपरा, राज दिवाणिया, कैलाश बंगारिया, दीपक भोरविया, महेश इलोरिया, निलेश रुपरा, हंसराज पटेल, मनोज रुपरा, धीरज कापड़िया सहित बड़ी संख्या में समाजजन उपस्थित थे। समाजजनों ने नगर परिषद के सहायक राजस्व निरीक्षक कैलाश शर्मा की उपस्थिति में मेडिकल आफिसर डा. अनिल पाटीदार को कंबल,चादर व तकिये भेंट किए। | hindi |
కొత్త రేట్లు జులై 18 నుంచి సవరించిన జీఎస్టీ రేట్లు జులై18వ తేదీ నుంచి అమల్లోకి వస్తాయి. క్రిప్టో ఆస్తులపై జీఎస్టీ గురించి ఎలాంటి నిర్ణయం తీసుకోని జీఎస్టీ కౌన్సిల్ ఇప్పటికు నిత్యావసర వస్తువులతో పాటు మరికొన్ని ఆహార పదార్థాలకు ఉన్న మినహాయింపులను ఎత్తేసింది. ఇవన్నీ జులై 18వ తేదీ నుంచి అమల్లోకి వస్తాయి. జూన్ 30తో రాష్ట్రాలకు జీఎస్టీ పరిహారం ఆగిపోనుంది. కొనసాగింపుపై ఎలాంటి నిర్ణయం తీసుకోలేదు. అలాగే జీఎస్టీ రేట్లు హేతుబద్ధీకరణపై కూడా కౌన్సిల్ తన నిర్ణయాన్ని వాయిదా వేసింది. దీనికి సంబంధించిన కమిటీ కాలాన్ని రెండు నెలలు పొడిగించింది. ఆన్లైన్ గేమింగ్, రేసులు, క్యాసినోలపై 28 శాతం పన్ను వేయాలన్న ప్రతిపాదనను వాయిదా వేసింది. జీఎస్టీ రిజిస్ట్రేషన్ కోసం ఇక విద్యుత్ బిల్లు తప్పనిసరి చేశారు. The post కొత్త రేట్లు జులై 18 నుంచి first appeared on For Money. | telegu |
৮, ১৮, ০ রান পাচ্ছেন না বলে কোহলিকে নিয়ে উদ্বেগ? প্রশ্ন শুনে উপহাস রোহিতের ৮, ১৮, ০ ওয়েস্ট ইন্ডিজের বিরুদ্ধে তিনটি ওডিআইএ বিরাট কোহলির স্কোর স্বাভাবিক ভাবেই নেতৃত্ব যাওয়ার পর কোহলির এ হেন পারফরম্যান্স নিয়ে প্রশ্ন উঠে গিয়েছে উদ্বিগ্ন কোহলি ভক্তরা ভারত শুক্রবার ওয়েস্ট ইন্ডিজকে ওডিআই সিরিজে হোয়াইটওয়াশ করার পর সাংবাদিক সম্মেলনে কোহলির ফর্ম নিয়ে বাকিরা উদ্বেগ প্রকাশ করলে উল্টে উপহাস করেন ভারত অধিনায়ক পাশাপাশি পরিষ্কার ভাবে জানিয়ে দেন, কোহলির ফর্ম নিয়ে টিম ম্যানেজমেন্ট মোটেও চিন্তিত নয় প্রাক্তন অধিনায়কের আত্মবিশ্বাস ফেরাতে দল কতটা উদ্যোগী? এই প্রশ্ন শুনে সাংবাদিকদের উপহাস করেছেন রোহিত বলেছেন, বিরাট কোহলি কো আত্মবিশ্বাস কি জরুরাত হ্যায়? কেয়া বাত কর রহে হো ইয়ার বিরাট কোহলির আত্মবিশ্বাসের প্রয়োজন? কী বলছো সেঞ্চুরি না পেলেও দক্ষিণ আফ্রিকার বিপক্ষে সিরিজে তিন ম্যাচে দুটি হাফ সেঞ্চুরি করা অনেক বড় কথা আমি মনে করি না, ওর বাড়তি আত্মবিশ্বাসের প্রয়োজন আছে ও একদম ঠিক আছে টিম ম্যানেজমেন্টও ওকে নিয়ে মোটেও চিন্তিত নয় ট্রেন্ডিং স্টোরিজ IPL Auction: প্রথম দিনের নিলামে বাংলার ৫ ক্রিকেটারের ভা .... মাত্র ২৯ বছর বয়সেই ফুটবলকে বিদায় জানালেন ইস্টবেঙ্গলের প .... IPL 2022 Auction: মুখের সামনে থেকে তোলা হচ্ছে খাবার! নি .... IPL Auction 2022: ঘরে ফেরার উচ্ছ্বাস, পুনরায় KKR যোগ .... এর সঙ্গেই রোহিত যোগ করেছেন যে, ওয়েস্ট ইন্ডিজের বিরুদ্ধে ওডিআই সিরিজের সবচেয়ে বড় ইতিবাচক বিষয় হল, বোলিং ইউনিটের পারফরম্যান্স পাশাপাশি দক্ষিণ আফ্রিকায় ব্যর্থ হওয়ার পর, ওয়েস্ট ইন্ডিজের বিরুদ্ধে ওডিআই সিরিজে মিডল অর্ডার ব্যাটাররা ভালো পারফরম্যান্স করেছে রোহিত বলেওছেন, সবচেয়ে বড় ইতিবাচক ছিল আমাদের বোলিং ইউনিট আলাদা করে বলতে হলে, ফাস্ট বোলার এবং স্পিনার, প্রত্যেকেই সিরিজে দুর্দান্ত পারফরম্যান্স করেছে পাশাপাশি আমরা যে বিষয়টি নিয়ে চিন্তিত ছিলাম তা হল, মিডল অর্ডার নিয়ে তবে এই সিরিজে আমাদের মিডল অর্ডারের পারফরম্যান্স খুব ভালো ছিল আমরা কন্ডিশন অনুযায়ী ব্যাটিং করেছি এবং আমরা দীর্ঘদিন ধরে এটি নিয়ে কথা বলেছিলাম যে, মিডল অর্ডার খুব বেশি সুযোগ পায় না কারণ প্রথম তিন ব্যাটসম্যানই মূলত ব্যাট করে কিন্তু এই সিরিজে মিডল অর্ডার ভালো ব্যাটিং করেছে বন্ধ করুন | bengali |
चीनी टेलीकाम कंपनी हुवावे पर आयकर विभाग की बड़ी कार्रवाई नई दिल्ली, । आयकर विभाग Income tax department ने चीनी टेलीकाम कंपनी हुवावे के भारत स्थित परिसरों पर छापेमारी की है। आधिकारिक सूत्रों ने बुधवार को बताया कि आयकर विभाग की ओर से यह कार्रवाई कर चोरी की छानबीन के तहत की गई है। समाचार एजेंसी पीटीआइ की रिपोर्ट के मुताबिक ये छापे कंपनी के परिसरों में मंगलवार को दिल्ली, गुरुग्राम और बेंगलुरू में मारे गए। वहीं समाचार एजेंसी एएनआइ की रिपोर्ट के मुताबिक चीनी टेलीकाम कंपनी हुवावे ने भी आयकर विभाग Income tax department के अधिकारियों के उसके कार्यालयों पर पहुंचने की पुष्टि की है। समाचार एजेंसी पीटीआइ ने आधिकारिक सूत्रों के हवाले से अपनी रिपोर्ट में कहा है कि आयकर विभाग के अधिकारियों ने हुवावे के भारतीय व्यवसायों और विदेशी लेनदेन के खिलाफ कथित कर चोरी की आशंकाओं पर छानबीन की। अधिकारियों ने जांच के तहत वित्तीय दस्तावेजों की पड़ताल की। आधिकारिक सूत्रों ने बताया कि छापेमारी के दौरान कुछ दस्तावेज एवं अन्य साक्ष्य भी जब्त किए गए। वहीं समाचार एजेंसी एएनआइ की रिपोर्ट के मुताबिक चीनी टेलीकाम कंपनी हुवावे ने बताया कि उसे आयकर विभाग की टीम के उसके कार्यालयों पर पहुंचने और उसके कुछ कर्मियों के साथ उनकी बैठक के बारे में सूचित किया गया है। हुवावे ने अपने बयान में कहा है कि उसे विश्वास है कि भारत में उसका संचालन सभी कानूनों के अनुरूप हैं। भारत में कंपनी का संचालन कानून के अनुरूप चल रहा है। | hindi |
सुप्त बद्ध कोणासन के फायदे और करने का सही तरीका हर कोई हेल्दी और फिट रहना चाहता है लेकिन लोग अपनी बिजी लाइफस्टाइल में बेहतर जिंदगी और अच्छी नींद के लिए योग और ध्यान नहीं कर पाते है, जिसके कारण लोगों में मोटापा और स्वास्थ्य संबंधी अन्य समस्याएं देखने को मिलती है। सुप्त बद्ध कोणासन योग के अभ्यास से आपका दिमाग और शरीर दोनों स्वस्थ रहता है। कमर और हिप्स वाले हिस्से की मांसपेशियां मजबूत होती है। इससे छात्रों को भी एकाग्र और फोकस्ड रहने में मदद मिलती है। सुप्त बद्ध कोणासन काफी सरल योगासन है। इसे आप 30 से 60 सेकेंड तक कर सकते है। ये आसन घुटने, जांघों और कमर वाले हिस्से को स्ट्रेच करता है। साथ ही इससे पाचन तंत्र और रीढ़ के दर्द की समस्या से छुटकारा मिलता है। आइए इस योगासन के फायदे और करने के तरीके के बारे में विस्तार से जानते है। सुप्त बद्ध कोणासन के फायदे 1. जांघों और को स्ट्रेच करने में मदद मिलती है। साथ ही घुटनों के दर्द में भी आराम मिलता है। 2. इससे वजन कम करने और पाचन तंत्र से संबंधित समस्याओं को दूर करने में सहायता मिलती है और एब्स टोन होते है। 3. इस योगासन की मदद से लोअर बैक, घुटनों और पूरे शरीर में लचीलापन आता है। इससे अनिद्रा की समस्या भी दूर हो सकती है। 4. इससे हार्निया के रोकथाम में मदद मिलती है। 5. वेरिकोस वेन और जैसी समस्याओं के लक्षण कम करने में सहायता मिलती है। 6. बवासीर और पेट फूलने की समस्या में आराम मिलता है। 7. किडनी स्टोन को टोन कर ब्लैडर पर नियंत्रण को बेहतर बनाता है। Image Credit Freepik सुप्त बद्ध कोणासन योग करने का तरीका 1. योग मैट पर शवासन की मुद्रा में लेट जाएं और धीरेधीरे घुटनों को भीतर की तरफ मोड़ें। 2. इस दौरान दोनों पैरों को एक साथ ही अंदर की ओर लेकर आएं। 3. पैरों के बाहर वाला हिस्सा फर्श से संपर्क में रहेगा। 4. अपनी एड़ियों को ग्रोइन के पास सटाकर रखें और दोनों हथेलियों को हिप्स के पास रखकर नीचे की तरफ दबाएं। 5. अब सांस छोड़ते हुए, पेट की निचली मांसपेशियों को भीतर की तरफ खींचें। 6. पीठ के निचले हिस्से में बढ़ाव महसूस करें। 7. पेल्विस को स्थिर रखते हुए रीढ़ की हड्डी को झुकाने की कोशिश करें। 8. सांस को भीतर लेकर फिर बाहर छोड़ दें। 8. इस आसन में एक मिनट तक बने रहें। 9. अब सांस गहरी और धीमी गति से लें। 10. सांस छोड़ते हुए प्रारंभिक अवस्था में वापस आ जाएं। Image Credit Yoga Journal सावधानियां 1. गर्दन में दर्द होने पर सुप्त बद्ध कोणासन का अभ्यास न करें। 2. कंधे में दर्द की समस्या होने पर हाथ ऊपर न उठाएं। 3. घुटने में दर्द या आर्थराइटिस की समस्या होने पर आप दीवार का सहारा ले सकते है। 4. हाई ब्लड प्रेशर के मरीज इस योगासन को न करें। 5. स्लिप डिस्क के मरीज इस आसन का अभ्यास न करें। 6. रीढ़ की हड्डी में दर्द होने पर इस आसन को नहीं करना चाहिए। 7. इस योगासन को ट्रेनर की देखरेख में करने का प्रयास करें। Main Image Credit Purna Yoga 828 | hindi |
تہٕ اَمہِ پَتہٕ زنٛتہٕ تَمہِ گۄڈنِکہِ پھِرِ یہِ وُچھ زِیہِ گُرۍ سوار اوس ؤزٟر تہٕ ووٚننس سؠٹھہٕے قٲبل رَحم آوازِ منٛز | kashmiri |
ಕೋವಿಡ್ ಆಸ್ಪತ್ರೆಗಳಲ್ಲಿ ಅಗ್ನಿ ಸುರಕ್ಷತಾ ಕ್ರಮಕೈಗೊಳ್ಳಲು ರಾಜ್ಯಗಳಿಗೆ ಸುಪ್ರೀಂ ನಿರ್ದೇಶನ ನವದೆಹಲಿ : ಕೋವಿಡ್19 ಆಸ್ಪತ್ರೆಗಳು ಸೇರಿ ಎಲ್ಲಾ ಆಸ್ಪತ್ರೆಗಳಲ್ಲಿ ಅಗ್ನಿ ಸುರಕ್ಷತೆಗೆ ಸಂಬಂಧಿಸಿದಂತೆ ಪರಿಶೀಲಿಸಲು ಪ್ರತಿ ತಿಂಗಳು ಕೈಗೊಳ್ಳಲು ಸಮಿತಿಗಳನ್ನು ರಚಿಸುವಂತೆ ಸುಪ್ರೀಂಕೋರ್ಟ್ನ ಎಲ್ಲಾ ರಾಜ್ಯ ಮತ್ತು ಕೇಂದ್ರಾಡಳಿತ ಪ್ರದೇಶದ ಸರ್ಕಾರಗಳಿಗೆ ಶುಕ್ರವಾರ ಸೂಚಿಸಿದೆ. ಆಸ್ಪತ್ರೆಗಳಲ್ಲಿ ಅಗ್ನಿಶಾಮಕ ಸುರಕ್ಷತಾ ಮಾನದಂಡಗಳನ್ನು ಅನುಸರಿಸಲಾಗಿದೆಯೇ ಎಂದು ಖಚಿತಪಡಿಸಿಕೊಳ್ಳಲು ಪ್ರತಿ ರಾಜ್ಯ ಸರ್ಕಾರಗಳು ನೋಡಲ್ ಅಧಿಕಾರಿಯನ್ನು ನೇಮಿಸಬೇಕು ಎಂದು ನ್ಯಾಯಮೂರ್ತಿ ಭೂಷಣ್ ನೇತೃತ್ವದ ನ್ಯಾಯಪೀಠವು ಹೇಳಿದೆ. ಫೈರ್ ನೋ ಆಬ್ಜೆಕ್ಷನ್ ಪ್ರಮಾಣಪತ್ರವನ್ನು ಎನ್ಒಸಿ ನವೀಕರಿಸದ ಆಸ್ಪತ್ರೆಗಳು ಅದನ್ನು ತಕ್ಷಣವೇ ನವೀಕರಿಸಬೇಕು. ಇಲ್ಲದಿದ್ದರೆ ಅವರ ವಿರುದ್ಧ ಕ್ರಮ ಕೈಗೊಳ್ಳಲಾಗುವುದು ಎಂದು ನ್ಯಾಯಪೀಠ ಹೇಳಿದೆ. ಕೊರೊನಾ ಕಾರಣದಿಂದಾಗಿ ತಿಂಗಳು ಪೂರ್ತಿ ನಿರಂತರವಾಗಿ ಕೆಲಸ ಮಾಡುತ್ತಿರುವ ವೈದ್ಯರು ಮತ್ತು ಆರೋಗ್ಯ ಸಿಬ್ಬಂದಿಗೆ ವಿರಾಮ ನೀಡಲು ಸರ್ಕಾರಗಳು ಕ್ರಮ ಕೈಗೊಳ್ಳಬೇಕೆಂದು ಸುಪ್ರೀಂಕೋರ್ಟ್ ಸೂಚಿಸಿದೆ. ಈ ಆದೇಶಕ್ಕೆ ಸಂಬಂಧಿಸಿದಂತೆ ಅಫಿಡವಿಟ್ನ ಸಲ್ಲಿಸಲು ಕೋರ್ಟ್ ರಾಜ್ಯ ಮತ್ತು ಕೇಂದ್ರ ಪ್ರದೇಶಗಳಿಗೆ ನಾಲ್ಕು ವಾರಗಳ ಗಡುವು ನೀಡಿದೆ. | kannad |
لندن اردو پوائنٹ تازہ ترین اخبار9 جولائی 2016ء پاکستان کی قومی کرکٹ ٹیم نے سسیکس کی ٹیم کے خلاف میچ شروع ہونے سے قبل ممتاز سماجی شخصیت عبدالستار ایدھی کے انتقال کے سوگ میں ایک منٹ کی خاموشی اختیار کی اور بازں پر سیاہ پٹیاں باندھیں تفصیلات کے مطابق ہفتہ کو انگلینڈ کے دورے پر جانے والی کرکٹ ٹیم نے سسکیس کے ساتھ میچ شروع ہونے سے قبل ممتاز سماجی شخصیت عبدالستار ایدھی کو خراج عقیدت پیش کرنے کے لئے ایک منٹ کی خامو شی اختیار کی اور ان کے سوگ میں میچ کے دوران بھی بازوں پر سیاہ پٹیاں باندھ رکھی تھیں اس موقع پر عبدالستار ایدھی کو انسانی خدمت پر خراج عقیدت پیش کرتے ہوئے قومی ٹیسٹ کرکٹ ٹیم کے کپتان مصباح الحق نے کہا کہ عبدالستار ایدھی ایک عظیم شخصیت تھے اور ان کے انتقال سے جو خلاء پیدا ہواہے تو وہ کبھی پر نہیں ہوسکے گاوہ بشر دوست تھے اور انہوں نے اپنی پوری زندگی انسانیت کی خدمت میں گزار دیان کے اتنقال پر ہر شعبہ زندگی سے تعلق رکھنے والے فرد دکھی ہےم | urdu |
Rainbow pattern. This graphical pattern includes use of three exponential moving averages with different periods. First with the period of 6 of blue color. Second with the period of 14 of yellow color. And third with the period of 26 of red color.
Many participants believe that there exists a following powerful signal of asset price decline. Blue line with a period of six is above all others. Yellow line with a period of 14 is under the blue line. Red line with a period of 26 is below all others. The intersection of blue line with a period of 6 and the yellow line with a period of 14 for most traders is point to access onto the market and of purchasing the option.
The probability of price increase is higher if blue line with a period of 6 is below all others, yellow line with a period of 14 is above the blue line and red line with a period of 26 is above all others. Intersection of the blue line with a period of 6 and the yellow line with a period of 14 is for most traders the points of access onto the market and of purchasing the option. | english |
वोट डालकर मजबूत सरकार बनाने में करें सहयोग फतेहगढ़ साहिब : सभी लोगों, खासकर स्कूल के शिक्षकों से अपील है कि 20 फरवरी को मतदान वाले दिन अपने वोट जरूर डालें और बाकी के लोगों को भी वोट डालने की अपील करें। यही नहीं पहली बार वोट देने वाले विद्यार्थी भी अपने इस अधिकार का पूरे जोश के साथ इस्तेमाल करते हुए एक मजबूत सरकार बनाने में अपना सहयोग दें। अगर हम एक साफ स्वच्छ छवि वाले उम्मीदवार को वोट डालकर चुनेंगे तो हमारा ही भविष्य बढि़या होगा। अपना मतदान बिना किसी लालच, डर व नशे के देना चाहिए ताकि अच्छे लोग सरकार में आकर हमारे राज्य, देश व समाज को विकास की राह पर लेकर जाएं। पढ़े लिखे लोग अपने इलाके के लोगों को भी वोट डालने के लिए प्रेरित करें ताकि सौ प्रतिशत मतदान संभव हो सके। लोगों से अपील है कि अगर कोई अपने पक्ष में वोट डालने के लिए डराता है या फिर नशा देता है या कोई अन्य लालच देता है तो इसकी जानकारी तुरंत पुलिस व प्रशासन अधिकारियों को अवश्य दें। मेरी सभी से अपील है कि अपने वोट का इस्तेमाल जरूर करें। बलविदर सिंह सैनी, डीईओ सेकेंडरी, फतेहगढ़ साहिब | hindi |
ట్రెండీ లవ్ స్టోరీస్.. వయసు పట్టింపు లేని స్టార్ కపుల్స్ నిజమైన ప్రేమ ప్రేమికుల్ని గెలిపిస్తుంది. ఇంకా చెప్పాలంటే ప్రేమే విజేతల్ని చేస్తుంది. ఎంతదూరంలో ఉన్నా.. ఏం చేస్తున్నా ప్రేమలో ఉన్న నిజాయితీ స్వచ్ఛత ఆ బంధాన్ని గెలిపిస్తాయి. ఎప్పటికీ ఎవరూ ప్రేయసి నుంచి ప్రియుడిని వీడదీయలేరు. ఇవన్నీ ప్రేమకులంలో పుట్టుకొచ్చిన మాటలేనా? అంటే ప్రేమికులనే అడగాలి. అవి నిజమేనేమో అనిపిస్తుంది కొందరు జంటల్ని చూస్తుంటే.. ప్రేమించి పెళ్లి చేసుకున్న సెలబ్రిటీల లైఫ్ ని ఓ సారి తరచి చూస్తే! చాలా ఆసక్తికర సంగతులే తెలుస్తాయి. ప్రేమకు మనసుంటే చాలు...దానికి వయసుతో సంబంధం లేదని ఇదిగో ఈ జంటలు నిరూపించాయి.బాలీవుడ్ నటి ప్రియాంక చోప్రా తనకన్నా 11 ఏళ్ల చిన్న వాడైన నిక్ జోనస్ ని ప్రేమించి వివాహం చేసుకుంది. వయసులో చిన్నవాడైనా తనపైన కురిపించిన ప్రేమకు స్పెల్ బౌండ్ అయిపోయి.. ఎంతో పెద్ద మనసు ఉన్నవాడని... తనని బాగా అర్ధం చేసుకున్నవాడని కమిటైపోయింది పీసీ. అసలు వయసుతో సంబంధం ఏమిటి అంటూ మూడు ముళ్ల బంధంతో ఒకటయ్యారు. మరో బాలీవుడ్ నటి అనుష్క శర్మ కూడా తనకంటే ఆరు నెలలు చిన్నవాడై భారత సారథి విరాట్ కొహ్లీని ప్రేమించి పెళ్లి చేసుకుంది. ఓ కమర్షియల్ ప్రకటనలో ఇద్దరూ కలిసి నటించినప్పుడు స్నేహం కుదిరింది. అటుపై ప్రేమలో పడ్డారు. కొన్నాళ్లు పాటు డేటింగ్ అంటూ తిరిగిన జంట చివరికి పెళ్లి బంధంతో ఒకటయ్యారు.ఇక టాలీవుడ్ హీరో మహేష్ కంటే ఆయన భార్య నమ్రత శిరోద్కర్ రెండేళ్లు పెద్ద. వీళ్లిద్దరూ ప్రేమించి పెళ్లి చేసుకున్నారు. ఇప్పుడీ జంటకి ఇద్దరికి తల్లిదండ్రులు. గౌతమ్ సితార చకచకా ఎదిగేస్తున్నారు. బాలీవుడ్ హీరో అభిషేక్ బచ్చన్ భార్య ఐశ్వర్యారాయ్ రాయ్ కంటే రెండేళ్లు చిన్నవాడు. అభిషేక్ కుటుంబ సభ్యులతో ఫైట్ చేసి మరీ ఐశ్యర్యను పెళ్లాడాడు. అలాగే ఒక్క సినిమాలో కూడా నటించకపోయినా నిజ జీవితంలో భాగస్వాములైన సైఫ్ అలీఖాన్ కంటే అమృతా సింగ్ పదమూడేళ్లు పెద్ద. ఆ జంట ప్రేమించి పెళ్లి చేసుకున్నారు. అయితే కొన్నేళ్లకే ఈ జంట మన్పస్పర్థలు కారణంగా విడిపోయారు. సైఫ్ అలీఖాన్ అటుపై తనకంటే చాలా చిన్న వయసు ఉన్న కరీనా కపూర్ ని పెళ్లి చేసుకున్నాడు. ఈ జంటకు థైమూర్ జన్మించాడు. అలాగే బిపాసా బసు తనకన్నా మూడేళ్లు చిన్నవాడైన కరణ్ సింగ్ గ్రోవర్ ని ప్రేమించి పెళ్లి చేసుకోగా.. శిల్పాశెట్టి కూడా తనకన్న చిన్నవాడైన రాజ్ కుంద్రాను ప్రేమించి పెళ్లి చేసుకుంది.ఇంకా త్వరలో పెళ్లి పీఠలు ఎక్కబోతున్న అర్జున్ కపూర్ చేసుకునే భామ మలైకా కూడా తనకంటే పదిహేనెళ్లు పెద్దది. 47 ఏళ్ల మలైకాను 33 ఏళ్ల అర్జున్ పెళ్లాడుతున్నాడంటూ బాలీవుడ్ లో మోతెక్కిపోతోంది. హాట్ ఐటం గీతాలతో హీటెక్కించే మలైకం కొన్నేళ్లగా అర్జున్ కపూర్ తో డేటింగ్ లో ఉంది. అలాగే మాజీ విశ్వసుందరి సుస్మితా సేన్ కూడా తనకంటే 15 ఏళ్ల చిన్నవాడైన రొహమాన్ తో సహజీవనంలో ఉంది. త్వరలో పెళ్లాడబోతుంది. వీరిద్దరు కొన్నేళ్లగా ప్రేమించుకుంటున్నారు. అలాగే నేహా దూపియా అంగద్ భేడి జంట వయోభేధం ఆల్వేస్ హాట్ టాపిక్. అంగద్ నేహా దూఫియా కంటే రెండేళ్లు చిన్నవాడు. ఇలా ప్రేమజంటలు వయసును చూసి ఒక్కటవ్వలేదు. ప్రేమ కుదిరింది. అటుపై పెద్దల్ని ఒప్పించి పెళ్లి చేసుకున్నారు. | telegu |
ক্তচাপ নিয়ন্ত্রণে রাখে মুলো! জেনে নিন এর অন্যান্য স্বাস্থ্য উপকারিতা অনেকেই এই সবজি খেতে পছন্দ করেন, আবার অনেকে এর নাম শুনলেই বিরক্ত হন কিন্তু আপনি কি জানেন যে, মুলো আমাদের শরীরের জন্য ভীষণ উপকারি একটি সবজি? এই সবজির কিন্তু বহু গুণ বিভিন্ন শারীরিক সমস্যার সমাধানে এর জুড়ি মেলা ভার মুলোতে বিভিন্ন ধরনের মিনারেসল পাওয়া যায় এতে ফাইটোকেমিক্যালস পাওয়া যায় যা আমাদের অনেক রোগ থেকে দূরে রাখে নিয়মিত মুলো খাওয়া আমাদের স্বাস্থ্যের জন্য খুবই উপকারি তাহলে জেনে নিন মুলোর উপকারিতা সম্পর্কে রক্তচাপ নিয়ন্ত্রণে রাখে মুলো খেলে রক্তচাপ নিয়ন্ত্রণে থাকে কারণ মুলোয় প্রচুর পরিমাণে পটাশিয়াম থাকে এছাড়াও, মুলোতে বিশেষ ধরনের অ্যান্টিহাইপারটেনসিভ থাকে, যা রক্তচাপ নিয়ন্ত্রণে সহায়তা করে জন্ডিস রোগীদের জন্য উপকারি জন্ডিস রোগীদের জন্য মুলো খুব উপকারি যাদের জন্ডিস হয়েছে বা যারা ধীরে ধীরে সুস্থ হচ্ছেন, তাদের অবশ্যই লবণের সাথে মুলো খাওয়া উচিত কারণ মুলো রক্তে বিলিরুবিন নিয়ন্ত্রণ করে এবং দেহে অক্সিজেনের সরবরাহ বাড়ায় এছাড়াও, এটি রক্ত পরিশোধন করে কিডনির জন্য উপকারি এই সবজি কিডনির স্বাস্থ্য ভাল রাখতেও সাহায্য করে কিডনির যেকোনও সমস্যা রোধ করে মুলো ঠান্ডা লাগা থেকে স্বস্তি দেয় শীতকালে সবচেয়ে বেশি যে শারীরিক সমস্যা হয় তা হল সর্দি, জ্বর তাই যদি আপনি নিয়মিত মুলো খান তবে এইসব সমস্যা কম হবে এছাড়া মুলো খাওয়ার আরও অনেক সুবিধা রয়েছে, কারণ এতে অ্যান্টিব্যাকটেরিয়াল বৈশিষ্ট্য রয়েছে যা আমাদের অনেক রোগ থেকে দূরে রাখে গ্যাসের সমস্যা দূরে রাখে অনেকেই পেটে গ্যাসের সমস্যায় ভোগেন অনেকে মনে করেন যে, মুলো খেলে পেটের গ্যাস আরও বাড়ে কিন্তু এমন কিছুই নয়, মুলো খেলে পেটের গ্যাস কম হয় এছাড়াও, হজম প্রক্রিয়ার জন্যও এটি খুবই ভাল এতে উচ্চ পরিমাণে ফাইবার রয়েছে, যা কোষ্ঠকাঠিন্য থেকে মুক্তি দেয় ডায়াবেটিসের ক্ষেত্রেও উপকারি মুলো ডায়াবেটিস রোগীদের জন্যও উপকারি মুলোয় প্রচুর পরিমাণে ফাইবার রয়েছে মুলো রক্তে শর্করার মাত্রা নিয়ন্ত্রণে রাখতে সহায়তা করে তাই, ডায়াবেটিস রোগীরা এটি গ্রহণ করতে পারেন তবে অবশ্যই চিকিত্সকের পরামর্শ নিন TS | bengali |
ಕೇಂದ್ರ ಸರಕಾರದ ನೀತಿಗಳ ವಿರುದ್ಧ ಕಾರ್ಮಿಕರ ಆಕ್ರೋಶ: ಬೆಂಗಳೂರಿನಲ್ಲಿ ಬೃಹತ್ ಪ್ರತಿಭಟನೆ ಬೆಂಗಳೂರು, ಸೆ.24: ಕೇಂದ್ರ ಮತ್ತು ರಾಜ್ಯ ಸರಕಾರಗಳು ಕೆಲಸದ ಅವಧಿ ಹೆಚ್ಚಳ, ವಾರದ ಕೆಲಸದ ಮಿತಿಯ ಹೆಚ್ಚಳದಂತಹ ತಿದ್ದುಪಡಿ ಕೈಬಿಡಬೇಕು ಒಳಗೊಂಡಂತೆ ವಿವಿಧ ಬೇಡಿಕೆಗಳಿಗೆ ಆಗ್ರಹಿಸಿ ಸೆಂಟರ್ ಆಫ್ ಇಂಡಿಯನ್ ಟ್ರೇಡ್ ಯೂನಿಯನ್ಸ್ ಸಿಐಟಿಯು ಸೇರಿದಂತೆ ಅಸಂಘಟಿತ ಕಾರ್ಮಿಕ ಸಂಘಟನೆಗಳು ಬೃಹತ್ ಪ್ರತಿಭಟನೆ ನಡೆಸಿತು. ಗುರುವಾರ ನಗರದ ಫ್ರೀಡಂ ಪಾರ್ಕಿನ ಮೈದಾನದಲ್ಲಿ ಜಮಾಯಿಸಿದ ಪ್ರತಿಭಟನಾಕಾರರು, ಲಾಕ್ಡೌನ್ ವೇಳೆ ಅಸಂಘಟಿತ ಕಾರ್ಮಿಕರು ತೀವ್ರ ರೀತಿಯ ಸಂಕಷ್ಟಕ್ಕೆ ಸಿಲುಕಿದ್ದಾರೆ. ಅನ್ಲಾಕ್ 4.0 ಜಾರಿಯಲ್ಲಿದ್ದರೂ ರಾಜ್ಯದ ಆರ್ಥಿಕ ಚಟುವಟಿಕೆಗಳು ಪೂರ್ಣ ಪ್ರಮಾಣದಲ್ಲಿ ಪುನಶ್ಚೇತನಗೊಳ್ಳದ ಕಾರಣ ಆರ್ಥಿಕ ಸಂಕಷ್ಟ ಎದುರಿಸುವಂತಾಗಿದೆ. ಶೇ. 80ರಷ್ಟು ಸಂಘಟಿತ ಕಾರ್ಮಿಕರಿಗೆ ಲಾಕ್ಡೌನ್ ಕಾಲಾವಧಿಯ ಪೂರ್ಣ ವೇತನ ಲಭಿಸಿಲ್ಲ. ಅಷ್ಟೇ ಅಲ್ಲದೆ, ನೀಡಿರುವ ವೇತನಕ್ಕೆ ಮುಂಗಡವಾಗಿ ಕೆಲಸ ಮಾಡಬೇಕೆಂದು ಮಾಲಕರು ಒತ್ತಾಯಿಸುತ್ತಿದ್ದಾರೆ ಎಂದು ಅಳಲು ತೋಡಿಕೊಂಡರು. ಪ್ರತಿಭಟನೆಯನ್ನುದ್ದೇಶಿಸಿ ಮಾತನಾಡಿದ ಸಿಐಟಿಯು ಪ್ರಧಾನ ಕಾರ್ಯದರ್ಶಿ ಮೀನಾಕ್ಷಿ ಸುಂದರಂ, ಕೇಂದ್ರ ಸರಕಾರದ ಕಾರ್ಪೋರೇಟ್ ಪರ ನೀತಿಯ ವಿರುದ್ಧ ಪ್ರತಿಭಟನೆ ನಡೆಸುತ್ತಿದ್ದೇವೆ. ಎಲ್ಲ ವಲಯಗಳಲ್ಲಿ ದೇಶಿ ಹಾಗೂ ವಿದೇಶಿ ಖಾಸಗಿ ಬಂಡವಾಳದಾರರ ಲೂಟಿಗೆ ವೇಗವಾಗಿ ಖಾಸಗೀಕರಣಗೊಳಿಸಲಾಗುತ್ತಿದೆ. ಕೃಷಿ, ಕಾರ್ಮಿಕ ಕಾಯ್ದೆಗಳನ್ನು ಸುಗ್ರೀವಾಜ್ಞೆ ಮೂಲಕ ಕಾರ್ಪೋರೇಟ್ಗಳ ಲಾಭಕ್ಕೆ ಬದಲಾಯಿಸಲಾಗುತ್ತಿದೆ ಎಂದು ಆರೋಪಿಸಿದರು. ಸಿಐಟಿಯು ಮುಖಂಡ ಕೆ.ಎನ್ ಉಮೇಶ್ ಮಾತನಾಡಿ, ಕೇಂದ್ರ ಸರಕಾರ ಅವೈಜ್ಞಾನಿಕ ಲಾಕ್ಡೌನ್ ಹೇರಿದ್ದಲ್ಲದೇ, ಸಾಂಕ್ರಾಮಿಕ ಕಾಲದಲ್ಲಿ ಕಾರ್ಮಿಕ ಕಾನೂನುಗಳನ್ನು ತಿದ್ದುಪಡಿ ಮಾಡಿ, ಸರಕಾರಿ ಕಂಪೆನಿಗಳನ್ನು ಖಾಸಗೀಕರಣ ಮಾಡುವ ಮೂಲಕ ದೇಶದ ಜನತೆಗೆ ಅನ್ಯಾಯ ಮಾಡುತ್ತಿದ್ದಾರೆ ಎಂದು ದೂರಿದರು. ಇದಕ್ಕೂ ಮೊದಲು ಸಿಐಟಿಯು ರಾಜ್ಯ ಅಧ್ಯಕ್ಷೆ ವರಲಕ್ಷ್ಮೀ ಮಾತನಾಡಿ, ಕೇಂದ್ರ ಸರಕಾರವು ವಿದೇಶಿ ಹೂಡಿಕೆದಾರರಿಗಾಗಿ ಕೆಲಸ ಮಾಡುತ್ತಿದೆ. ಅವರ ಪರವಾದ ಕಾನೂನುಗಳನ್ನು ಲೋಕಸಭೆಯಲ್ಲಿ ಜಾರಿಗೊಳಿಸಲಾಗುತ್ತಿದೆ. ಕಾರ್ಮಿಕರ ಸಾಮೂಹಿಕ ಚೌಕಾಶಿಯನ್ನು ಸರಕಾರ ಇಲ್ಲದಾಗಿಸಿದೆ. ಅಲ್ಲದೆ, ಕೊರೋನ ಸಮಯದಲ್ಲಿ ಜನರಿಗೆ ಕೆಲಸ ಮಾಡಬೇಕಿದ್ದ ಸರಕಾರ ಶ್ರೀಮಂತರ ಪರವಾಗಿದೆ. ಇದರಿಂದ ಕಾರ್ಮಿಕರು ಕೆಲಸ ಮಾತ್ರವಲ್ಲ ಬದುಕನ್ನೇ ಕಳೆದುಕೊಂಡಿದ್ದಾರೆ ಎಂದರು. ಬಿಜೆಪಿ ಪಾಪದ ಕೊಡ ತುಂಬಿದೆ ಭಾರತ ಮೋದಿ, ಯಡಿಯೂರಪ್ಪನವರದಲ್ಲ. ಭಾರತವನ್ನು ಕಟ್ಟಿದವರು ನಾವು ಅಂದರೆ ದೇಶದ ಕಾರ್ಮಿಕರು. ಬಿಜೆಪಿ ಸಂತತಿಯ ಪಾಪದ ಕೊಡ ತುಂಬಿದೆ. ಹಾಗಾಗಿ ಕಾರ್ಮಿಕ ವಿರೋಧಿಯಾಗಿ ನಡೆದುಕೊಳ್ಳುತ್ತಿದ್ದಾರೆ. ಕೆ.ಎನ್.ಉಮೇಶ್, ಸಿಐಟಿಯು ಮುಖಂಡ ಹಕ್ಕೊತ್ತಾಯಗಳು 202121 ರ ಸಾಲಿನ ತುಟ್ಟಿಭತ್ಯೆ ಮುಂದೂಡಿಕೆ ಆದೇಶ ರದ್ದು ಮಾಡಿ, ಬಾಕಿ ಹಣ ನೀಡಬೇಕು. ಅಂತರ್ ರಾಜ್ಯ ವಲಸೆ ಕಾರ್ಮಿಕ ಉದ್ಯೋಗ ಮತ್ತು ಸೇವಾ ಶರತ್ತುಗಳ ಕಾಯ್ದೆ1970 ಅನ್ನು ಜಾರಿಗೊಳಿಸಬೇಕು. ಎಪಿಎಂಸಿ ಕಾಯ್ದೆ, ಭೂ ಸುಧಾರಣಾ ಕಾಯ್ದೆ, ಅಗತ್ಯ ಸರಕುಗಳ ಕಾಯ್ದೆ, ವಿದ್ಯುತ್ ಕಾಯ್ದೆಗಳಿಗೆ ರಾಜ್ಯ ಸರಕಾರ ತಂದಿರುವ ತಿದ್ದುಪಡಿ ಸುಗ್ರೀವಾಜ್ಞೆಗಳನ್ನು ವಾಪಸ್ ಪಡೆಯಬೇಕು. ಶಿಕ್ಷಣ, ಆರೋಗ್ಯ, ರೈಲು, ರಸ್ತೆ, ವಿದ್ಯುತ್, ದೂರಸಂಪರ್ಕ, ವಿಮಾ, ಬ್ಯಾಂಕ್ ಮುಂತಾದ ಸಾರ್ವಜನಿಕ ಉದ್ದಿಮೆಗಳ ಖಾಸಗೀಕರಣ ನಿಲ್ಲಬೇಕು. ವಿದ್ಯಾವಂತ ನಿರುದ್ಯೋಗಿ ಯುವಕ ಹಾಗೂ ಯುವತಿಯರಿಗೆ ನಿರುದ್ಯೋಗ ಭತ್ತೆ ಮಾಸಿಕ 10,000 ರೂಗಳನ್ನು ನೀಡಬೇಕು. 21 ಸಾವಿರ ರೂ, ಸಮಾನ ಕನಿಷ್ಟ ವೇತನ ನಿಗದಿ ಮಾಡಬೇಕು. | kannad |
For purposes of this Agreement, “Service” refers to the Company’s service which can be accessed via our website at http://www.bristolopencirclemoot.org.uk/ or through our mobile application. The terms “we,” “us,” and “our” refer to the Company. “You” refers to you, as a user of Service. | english |
کھیلوں کے بین الاقوامی مقابلوں میں 2017ء میں کئی برج الٹے نئے چیمپئن بنے ایسے ہی چند واقعات پر دیکھتے ہیں یہ رپورٹ کھیلوں کے بین الاقوامی مقابلوں میں 2017ء میں کئی برج الٹے نئے چیمپئن بنے ایسے ہی چند واقعات پر دیکھتے ہیں یہ رپورٹ | urdu |
नवादा में जर्जर हो चुका है उपभोक्ता फोरम का भवन, छह वर्षों से पत्राचार के बाद भी मरम्मत नहीं नवादा। जिला उपभोक्ता विवाद प्रतितोष आयोग उपभोक्ता फोरम भवन की स्थिति काफी जर्जर हो चुकी है। भवन का कुछ भाग ध्वस्त हो गया है, कुछ ध्वस्त होने के कगार पर है। किसी समय भी कोई अप्रिय घटना घट सकती है। यहां काम करने वाले कर्मी अपने को काफी असुरक्षित महसूस करते हुए कामकाज निपटाने को विवश है। उल्लेखनीय है कि यह भवन समाहरणालय परिसर में स्थित है। भवन में जिला उपभोक्ता प्रतितोष आयोग का न्यायालय संचालित होता है। जहां उपभोक्ताओं से जुड़े विवादों का निपटारा किया जाता है। अधिवक्ताओ व पक्षकारों का आना जाना लगा रहता है। इस भवन का पश्चिमी भाग का छत का हिस्सा गिर चुका है। जबकि पूरबी भाग में कई स्थानों पर दरारे पड़ गई है, जो कभी भी गिर सकता है। इसके अलावे भवन के कई स्थानों के छत का प्लास्टर छोड़ चुका है। अंदर का छड़ भी जंग लगकर कमजोर हो गया है। इस परिस्थिति में न्यायालय में काम काज निपटाना हर किसी के लिए जोखिम भरा है। 1999 में बना था भवन : यहां के अवकाश प्राप्त कर्मी पारसनाथ प्रसाद ने बताया कि यह भवन वर्ष 1999 में जिला उपभोक्ता फोरम को सुपुर्द किया गया था। निर्माण के 2223 साल होने को है मरम्मति की दिशा में काम नहीं किया गया। उचित देखभाल के आभाव में भवन जर्जर होता चला गया। पिछले छह साल से किया जा रहा पत्राचार : भवन के क्षतिग्रस्त होने के शुरूआती समय में ही सूचना पत्र के माध्यम से भवन निर्माण विभाग के कार्यपालक अभियंता को दी गई थी। लेकिन, भवन निर्माण विभाग इस ओर काफी उदासीन रहा। आयोग की महिला सदस्य डा. पूनम शर्मा ने बताया कि मरम्मत के अभाव में भवन स्थिति काफी जर्जर हो गई है। साल 2016 से कई पत्र भवन निर्माण विभाग के कार्यापालक अभियंता को भेजा गया है। आयोग के वर्तामान अध्यक्ष राज कुमार प्रसाद ने पदभार ग्रहण करने के बाद भवन के स्थिति को देख कर काफी चितित हुए तथा 8 फरवरी को पुन: कार्यपालक अभियंता का पत्र भेज अविलंब मरम्मत का कार्य किये जाने का अनुरोध किया है। बहरहाल, यहां कार्यरत सभी कर्मी, अधिवक्ता और अध्यक्षसदस्य जल्द से जल्द भवन की मरम्मति हो जाए, ऐसा चाहते हैं। इन तिथियों में किया गया पत्राचार पत्रांक 7 दिनांक 21.1. 2016 पत्रांक 26 दिनांक 15. 3.16 पत्रांक 66 दिनांक 13. 7. 16 पत्रांक 83 दिनांक 17 .8.16 पत्रांक 84 दिनांक 3.9.20 पत्रांक 122 दिनांक 1 .12.20 पत्रांक 124 दिनांक 23 .12 .21 | hindi |
\begin{document}
\title{Distributed Mechanism Design with Learning Guarantees}
\author{Abhinav Sinha and Achilleas Anastasopoulos\\
\normalsize{EECS Department, University of Michigan, Ann Arbor.} \\
\normalsize{\texttt{\{absi,anastas\}@umich.edu}}
\blfootnote{This work is supported in part by NSF grant ECCS-1608361.}
}
\date{\normalsize\today}
\maketitle
\begin{abstract}
Mechanism design for fully strategic agents commonly assumes broadcast nature of communication between agents of the system. Moreover, for mechanism design, the stability of Nash equilibrium (NE) is demonstrated by showing convergence of specific pre-designed learning dynamics, rather than for a class of learning dynamics. In this paper we consider two common resource allocation problems: sharing $ K $ infinitely divisible resources among strategic agents for their private consumption (private goods), and determining the level for an infinitely divisible public good with $ P $ features, that is shared between strategic agents. For both cases, we present a distributed mechanism for a set of agents who communicate through a given network. In a distributed mechanism, agents' messages are not broadcast to all other agents as in the standard mechanism design framework, but are exchanged only in the local neighborhood of each agent. The presented mechanisms produce a unique NE and fully implement the social welfare maximizing allocation. In addition, the mechanisms are budget-balanced at NE. It is also shown that the mechanisms induce a game with contractive best-response, leading to guaranteed convergence for all learning dynamics within the Adaptive Best-Response dynamics class, including dynamics such as Cournot best-response, $ k- $period best-response and Fictitious Play. We also present a numerically study of convergence under repeated play, for various communication graphs and learning dynamics.
\end{abstract}
\section{Introduction}
Mechanism design has been studied extensively in Economics~\citep{hurwicz2006,vohra2011mechanism,borgers2015book} and Engineering literature~\citep{kelly,basar,hajekvcg,johari2009efficiency,rahuljain,demos,bhattacharya2016,SiAn_multicast_tcns}. Most of the mechanisms presented in literature for the case of strategic agents have two (unrelated) drawbacks that are an impediment to their applicability in real-world scenarios. The first drawback is the assumption that the underlying communication structure between agents is a broadcast one. Almost all mechanisms in the strategic setting define allocation (and taxes/subsidies) in such a way that it requires each agent to broadcast their message, i.e., the allocation function is a function of all users' messages. The second drawback is the dynamic stability of Nash equilibrium (NE) of the game induced by the mechanism. Results on this, for most of the presented mechanisms in the literature, are either non-existent or restricted to narrow definition(s) of dynamic stability. As a result it is not clear if, and under what conditions on the dynamics, the NE is reached.
Regarding the first issue mentioned above, until now little consideration has been given to designing incentives for strategic agents for whom communication is restricted by a network. The motivation for this modeling consideration comes from the literature on distributed optimization,~\citep{nedic2009,boyd2011,duchi2012,scutari2016}, where algorithms are designed for global consensus between distributed non-strategic agents, who possess local information and who communicate locally on a network. In mechanism design, a designer designs incentives such that strategic agents ``agree'' to reveal their relevant private (local) information truthfully. Thus, with the above motivation in mind, a natural question is to ask: can incentives be designed for strategic agents with local private information who communicate locally on a network? One expects that attaining consensus between strategic agents only becomes harder to achieve when message exchange is restricted by the network structure.
This aforementioned issue is not to be confused with that of distributed optimization where the problem of local exchange of information has been addressed and to a large extend solved~\citep{nedic2009,boyd2011,duchi2012,scutari2016}. Neither should it be confused with the local public goods models (e.g.,~\citep{local2003,demosshruti}), where each agent's utility in the model is already assumed to only depend on his/her neighbors' allocations.
Regarding the second drawback mentioned above, there is a long line of work investigating stability of NE through learning in games~\citep{milgrom1990,young2004strategic,fudenberglearning}. Theoretically, the notion of NE applies to perfect information settings, i.e., where each agent knows the utility of every other agent. However, for most of the mechanism design works in the literature where NE is used as the solution concept, the models are not necessarily restricted to the perfect information setting. Indeed, in an informationally and physically decentralized system it is natural to assume that agents only know their own utility and no one else's. In such cases, the robustness w.r.t. information available to agents, of any particular designed mechanism is evaluated by the learning guarantees that it can provide. Since agents can't calculate the NE offline, they are expected to learn it by repeatedly playing the induced game whilst adjusting their strategy dynamically using the past observations. The larger the class of learning dynamics that are guaranteed to converge, the more robust the mechanism. The idea of NE being the convergent point of learning dynamics directly relates to the \emph{Evolutive} interpretation of NE,~\citep{osborne1994}, where NE even for a single-shot game is interpreted as the stationary point of a dynamic adjustment process. The original thesis of John Nash,~\citep{nashthesis51}, too provides a similar dynamic adjustment interpretation of NE.
In this paper our objective is to design mechanisms that resolve simultaneously both the issues mentioned above. This means that the mechanisms are distributed, i.e., the allocation and tax functions (contracts) obey the communication constraints of a network and for the designed mechanisms theoretical guarantees of convergence, for a sufficiently large class of learning dynamics, can be provided.
The basic idea for achieving the latter this is to identify appropriate properties of games that can lead to convergence of a correspondingly selected class of learning dynamics and then design the mechanism such that the induced game possesses the identified properties. In Milgrom and Roberts~\citep{milgrom1990}, authors identify \emph{supermodularity} as a critical property of a game and prove that any learning dynamic within the \emph{Adaptive Dynamics} class is guaranteed to converge between the two most extreme Nash equilibria when the game is played repeatedly. Following this,~\citep{chen2002family} presents a mechanism for the Lindahl allocation problem such that the induced game is supermodular. Healy and Mathevet in~\citep{healy2012designing} identify contraction as a learning-relevant property of the game and show that any learning dynamic within the \emph{Adaptive Best-Response (ABR)} dynamics class is guaranteed to converge to the unique NE. They also present a mechanism that induces a contractive game for the Walrasian and Lindahl allocation problems, under the usual broadcast information structure. The property of contraction is more stringent than supermodularity and consequently the ABR class is broader than the Adaptive Dynamics class.
In this paper, for both Walrasian and Lindahl allocation problems, we define a mechanism through an appropriately designed message space, allocation function and tax function. The allocation and tax functions, for any agent, depend only on the messages of his/her neighbors. The mechanism description contains certain free parameters such that for all values of these parameters the mechanism achieves its goal of full implementation in NE. For the purpose of providing learning guarantees we consider the ABR class of learning dynamics and show that by tuning the free parameters appropriately, the induced game can be made contractive. Section~\ref{seclearning} describes some well-known learning dynamics that are part of the ABR class. With the aid of numerical analysis, we show that the designed contractive mechanism provides exponential rate of convergence for several instances of our model.
The structure of this paper is as follows: Section~\ref{secmodel} describes the two centralized allocation problems and their optimality conditions. Section~\ref{secpr} defines some mechanism design basics and then presents the mechanism for the private goods problem. Section~\ref{secmechpub} presents the mechanism for the public goods problem. Section~\ref{seclearning} introduces learning-related properties and contains the result of guaranteed convergence of any learning dynamic within the ABR class. Finally, Section~\ref{seclearning} also contains a numerical study of the convergence pattern of various learning dynamics for different underlying communication graphs.
\section{Model} \label{secmodel}
There are $ N $ strategic agents, denoted by the set $ \mathcal{N} = \{1,\ldots,N\} $. A directed communication graph $ \mathcal{G} = (\mathcal{N},\mathcal{E}) $ is given, where the vertexes correspond to the agents and an edge from vertex $ i $ to $ j $ indicates that agent $ i $ can ``listen'' to agent $ j $. It is assumed that the given graph $ \mathcal{G} $ is strongly connected. In this paper, we are interested in two different types of allocation problems: private goods and public goods, we describe each model below. In Economics literature, these are also known as Walrasian and Lindahl allocation problems~\citep{hurwicz1979outcome,MWG}.
\subsection{Private goods allocation problem}
There are $ K $ infinitely divisible goods, denoted by set $ \mathcal{K} = \{1,\ldots,K\} $, that are to be distributed among the agents. Each agent receives a utility $ v_i(x_i) $ based on the profile $ x_i = (x_i^1,\ldots,x_i^K) $ of quantity of each good that he/she receives. Since for each agent, its utility depends only on privately consumed allocation $ x_i $ and not on other agents' allocation, this is the private goods model.
It is assumed that $ v_i: \mathbb{R}^K \rightarrow \mathbb{R} $ is a continuously double-differentiable, strictly concave function that satisfies, $ \forall~k \in \mathcal{K} $,
\begin{subequations} \label{eqetagenpr}
\begin{gather}
-\eta < H_{kk}^{-1} + \sum_{l \in \mathcal{K},\,l\ne k} \left\vert H^{-1}_{kl} \right\vert < 0,
\\
H_{kk}^{-1} < - \frac{1}{\eta},
\end{gather}
\end{subequations}
for any given $ \eta > 1 $, where $ H^{-1} $ is the inverse of the Hessian $ H = \left[ \left. (\partial^2 v_i(\cdot)) \middle/ (\partial x_i^k \partial x_i^l) \right. \right]_{k,l} $. To understand the significance of this assumption consider the case of $ K=1 $, then this condition is the same as
\begin{equation}\label{eqetapr}
v_i^{\prime\prime}(\cdot) \in \left( -\eta,-\frac{1}{\eta} \right).
\end{equation}
It is already assumed that $ v_i(\cdot) $ is strictly concave, the only additional imposition made by this assumption is that the second derivative of $ v_i(\cdot) $ is strictly bounded away from $ 0 $ and $ -\infty $. More generally if the utility is separable, $ v_i(x_i) = \sum_{k \in \mathcal{K}} v_{i,k}(x_i^k) $, then the condition in~\eqref{eqetagenpr} is the same as
\begin{equation}
v_{i,k}^{\prime\prime}(\cdot) \in \left( -\eta,-\frac{1}{\eta} \right), \quad \forall~k \in \mathcal{K}.
\end{equation}
The above mentioned properties of the utility function are assumed to be common knowledge between agents and the designer. However, the utility function $ v_i(\cdot) $ itself is known only to agent $ i $ and is not known to other agents or the designer. The designer wishes to allocate available goods such that the sum of utilities is maximized subject to availability constraints, i.e., to solve the following centralized allocation problem,
\begin{subequations} \label{eqcppr}
\begin{gather}
x^* = \argmax_{x \in \mathbb{R}^K} \sum_{i \in \mathcal{N}} v_i(x_i)
\\
\label{eqconstpr}
\text{subject to} \quad \sum_{i \in \mathcal{N}} x_i^k = c_k, \quad \forall~k \in \mathcal{K},
\end{gather}
\end{subequations}
where $ c_k \in \mathbb{R} $ is the total available amount of good $ k \in \mathcal{K} $. The allocation $ x^* $ is also called the efficient allocation and it is assumed to be finite i.e., the optimization is well-defined. The efficient allocation is unique since the utilities are strictly concave. Further, the necessary and sufficient condition for optimality are
\begin{subequations}\label{eqKKTpr}
\begin{alignat}{2}
\frac{\partial v_i(x^*)}{\partial x_i^k} &= \lambda_k^*, \quad &&\forall~k \in \mathcal{K},~\forall~i \in \mathcal{N},
\\
\sum_{i \in \mathcal{N}} {x_i^k}^* &= c_k, \quad &&\forall~k \in \mathcal{K},
\end{alignat}
\end{subequations}
where $ \big( \lambda_k^* \big)_{k \in \mathcal{K}} \in \mathbb{R}^K $ are the (unique) optimal dual variables for each constraint in~\eqref{eqconstpr}.
\subsection{Public goods allocation problem}
There is a single infinitely divisible public good with $ P $ features, with the set of features denoted by $ \mathcal{P} = \{1,\ldots,P\} $. Each agent receives a utility $ v_i(x) $ based on the quantity of the public good $ x = \left(x^p\right)_{p \in \mathcal{P}} \in \mathbb{R}^P $. Since for each agent, its utility depends on the common allocation $ x $, this is the public goods model. It is assumed that $ v_i: \mathbb{R}^P \rightarrow \mathbb{R} $ is a continuously double-differentiable, strictly concave function that satisfies~\eqref{eqetagenpr} with the Hessian $ H = \left[ \left. (\partial^2 v_i(\cdot)) \middle/ (\partial x^p \partial x^q) \right. \right]_{p,q} $. If $ P=1 $, then this condition is the same as in~\eqref{eqetapr}.
As in the private goods model, the properties of the utility function are assumed to be common knowledge between agents and the designer. However, the utility function $ v_i(\cdot) $ itself is known only to agent $ i $ and is not known to other agents or the designer. The designer wishes to allocate the public good such that the sum of utilities is maximized, i.e., to solve the following centralized allocation problem,
\begin{align}\label{eqcppub}
x^* = \argmax_{x \in \mathbb{R}^P} \sum_{i \in \mathcal{N}} v_i(x).
\end{align}
The allocation $ x^* $ is also called the efficient allocation and it is assumed to be finite i.e., the optimization is well-defined. The efficient allocation is unique due to strictly concave utilities. The necessary and sufficient optimality conditions can be written as
\begin{subequations}\label{eqKKTpub}
\begin{alignat}{2}
\frac{\partial v_i(x^*)}{\partial x^p} &= {\mu_i^p}^*, \quad &&\forall~ p \in\mathcal{P},~\forall~i \in \mathcal{N},
\\
\label{eqKKTpub_b}
\sum_{i \in \mathcal{N}} {\mu_i^p}^* &= 0, \quad &&\forall~ p \in \mathcal{P},
\end{alignat}
\end{subequations}
where $ \big( {\mu_i^p}^* \big)_{p \in \mathcal{P}, i \in \mathcal{N}} $ are the (unique) optimal dual variables.
In general, for the public goods problem one can also assume a seller in the system who produces the quantity $ x $ and for whom the cost of production is a known (convex) function. In this case, the social welfare maximizing allocation contains the utility of the seller as well. For keeping exposition clear and to focus on the similarities between the public and private goods problems, the seller is not considered in this model. If needed, this can accommodated in a straightforward manner.
In the next two sections, we present a mechanism for each of the above problems.
\section{A mechanism for the private goods problem} \label{secpr}
\subsection{Definitions}
A one-shot mechanism is defined by the triplet,
\begin{equation}
\Big( \mathcal{M} = \mathcal{M}_1 \times \cdots \times \mathcal{M}_N,\big(\widehat{x}_1(\cdot),\ldots,\widehat{x}_N(\cdot)\big), \big( \widehat{t}_1(\cdot),\ldots,\widehat{t}_N(\cdot)\big) \Big)
\end{equation}
which consists of, for each agent $ i \in \mathcal{N} $, the message space $ \mathcal{M}_i $, the allocation function $ \widehat{x}_i:\mathcal{M} \rightarrow \mathbb{R}^K $ and the tax function $ \widehat{t}_i:\mathcal{M} \rightarrow \mathbb{R} $. Given a mechanism, a game $ \mathfrak{G}_{pvt} $ is setup between the agents in $ \mathcal{N} $, with action space $ \mathcal{M} $ and utilities
\begin{equation}\label{equtil}
u_i(m) = v_i(\widehat{x}_i(m)) - \widehat{t}_i(m).
\end{equation}
For this game, $ \widetilde{m} \in \mathcal{M} $ is a Nash equilibrium if
\begin{equation}
u_i(\widetilde{m}) \ge u_i(m_i,\widetilde{m}_{-i}), \quad \forall~m_i \in \mathcal{M}_i,~\forall~i \in \mathcal{N}.
\end{equation}
The mechanism is said to \emph{fully implement} the efficient allocation if
\begin{equation}
\Big( \widehat{x}_1(\widetilde{m}),\ldots,\widehat{x}_N(\widetilde{m}) \Big) = x^*, \quad \forall~\widetilde{m} \in \mathcal{M}_{NE},
\end{equation}
where $ \mathcal{M}_{NE} \subseteq \mathcal{M} $ is the set of all Nash equilibria of game $ \mathfrak{G}_{pvt} $ and $ x^* $ is the efficient allocation from~\eqref{eqcppr}. Furthermore, the mechanism is said to be \emph{budget balanced} at Nash equilibrium if
\begin{equation}
\sum_{i\in\mathcal{N}}\widehat{t}_i(\widetilde{m}) = 0, \quad \forall~\widetilde{m} \in \mathcal{M}_{NE}.
\end{equation}
Finally, we call the mechanism \emph{distributed} if for any agent $ i \in \mathcal{N} $, the allocation function $ \widehat{x}_i(\cdot) $ and tax function $ \widehat{t}_i(\cdot) $ instead of depending on the entire message $ m = (m_j)_{j \in \mathcal{N}} $, depend only on $ m_i $ and $ \big(m_j\big)_{j \in \mathcal{N}(i)} $, i.e., agent $ i $ and his/her immediate neighbors. Here $ \mathcal{N}(i) $ are all the ``out''-neighbors of $ i $ i.e., there exists an edge from $ i $ to $ j $ in graph $ \mathcal{G} $ iff $ j \in \mathcal{N}(i) $ .
\subsection{Mechanism}
\label{secmechpr}
For the purpose of maintaining clarity in the exposition, the mechanism presented below is for the special case of a single private good i.e., $ K=1 $. A natural extension of the presented mechanism to the general case is discussed at the end of this section.
For any agent $ i\in \mathcal{N} $, the message space is $ \mathcal{M}_i = \mathbb{R}^{N+1} $. The message $ m_i = (y_i,q_i) $ consists of agent $ i $'s demand $ y_i \in \mathbb{R} $ for the allocation of the single good and a surrogate/proxy $ q_i = \left( q_i^1,\ldots,q_i^N \right) \in \mathbb{R}^N $ for the demand of all the agents (including himself/herself).
\begin{figure}
\caption{$ n(i,j) $ and $ d(i,j) $ for the strongly connected directed graph $ \mathcal{G}
\label{figGraph}
\end{figure}
Since the underlying graph $ \mathcal{G} = \left( \mathcal{N},\mathcal{E} \right) $ is strongly connected, for any pair of vertexes $ i,j \in \mathcal{N} $, the following two quantities are well-defined. $ d(i,j) $ is the length of the shortest path from $ i $ to $ j $ and $ n(i,j) \in \mathcal{N}(i) $ is the out-neighbor of $ i $ such that the shortest path from $ i $ to $ j $ goes through $ n(i,j) $. The two quantities are depicted in Fig.~\ref{figGraph}. The allocation function is defined as
\begin{equation}\label{eqallopr}
\widehat{x}_i(m) = y_i - \frac{1}{N-1} \sum_{r \in \mathcal{N}(i)} \frac{q_{r}^r}{\xi} - \frac{1}{N-1} \sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i} } \frac{q_{n(i,r)}^r}{\xi^{d(i,r)-1}} + \frac{c_1}{N}, \quad \forall~i \in \mathcal{N},
\end{equation}
where $ \xi \in (0,1) $ is an appropriately chosen \emph{contraction} parameter and its selection is discussed in Section~\ref{seclearning} on Learning Guarantees, proof of Proposition~\ref{thmcontracpr}. The tax function is defined as
\begin{subequations}\label{eqtaxpr}
\begin{align}
\label{eqtaxpr_a}
\widehat{t}_i(m) &= \widehat{p}_i(m_{-i}) \left( \widehat{x}_i(m) - \frac{c_1}{N} \right) + \left( q_i^i - \xi y_i \right)^2
+ \sum_{\substack{r \in \mathcal{N}(i)}} \left( q_i^r - \xi y_r \right)^2
+ \sum_{\substack{r \notin \mathcal{N}(i)\\r \ne i}} \left(q_i^r - \xi q_{n(i,r)}^r \right)^2,
\\
\label{eqtaxpr_b}
\widehat{p}_i(m_{-i}) &= \frac{1}{\delta} \left( \frac{q_{n(i,i)}^i}{\xi} + \sum_{r \in \mathcal{N}(i)} \frac{q_{r}^r}{\xi} + \sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i} } \frac{q_{n(i,r)}^r}{\xi^{d(i,r)-1}} \right), \quad \forall~i \in \mathcal{N},
\end{align}
\end{subequations}
where $ n(i,i) \in \mathcal{N}(i) $ is an arbitrarily chosen neighbor of $ i $ and $ \delta > 0 $ is an appropriately chosen parameter. Both $ \xi,\delta $ are selected simultaneously in the proof of Proposition~\ref{thmcontracpr} from Section~\ref{seclearning}.
The quantities, $ n(\cdot,\cdot) $ and $ d(\cdot,\cdot) $, are based on the graph $ \mathcal{G} $. The only property of relevance here is that the two are related recursively i.e., $ d(n(i,r),r) = d(i,r) - 1 $. Thus if a mechanism designer wishes to avoid calculating the shortest path (possibly due to the high complexity) then $ n,d $ can be replaced by any valid neighbor and distance mapping, respectively, as long as they are related recursively as above.
\subsection{Results}
\begin{fact}[Distributed]
The mechanism defined in~\eqref{eqallopr} and~\eqref{eqtaxpr} is distributed.
\end{fact}
The distributed-ness of the mechanism follows from the fact that the expressions in~\eqref{eqallopr} and~\eqref{eqtaxpr} depend only on $ m_i $ and $ \big(m_r\big)_{r \in \mathcal{N}(i)} $.
Since agents are connected through a given graph, they can only communicate with a restricted set of agents i.e., their neighbors. Yet, as indicated in the optimality conditions,~\eqref{eqKKTpr}, there needs to be two kinds of global consensus at the efficient allocation. Firstly, the total allocation to all the agents must equal the total available amount, $ \sum_{i \in \mathcal{N}} x_i^* = c_1 $. Secondly, agents also need to agree on a common ``price'', $ \lambda_1^* $. To facilitate this, the message space consists of the surrogate variables $ q_i = (q_i^1,\ldots,q_i^N) $ which are known locally to agent $ i $ and are expected at equilibrium to be representative of the global demand $ y = (y_1,\ldots,y_N) $. Specifically, the second, third and fourth terms in the tax,~\eqref{eqtaxpr_a}, are designed for incentivizing agents to achieve the aforementioned duplication of global demand $ y $ to the local surrogate $ q_i $.
To motivate the choice of the allocation function and the remaining part of the tax function consider the case of $ \xi = 1 $ and take into account the duplication, i.e., $ q_r = y $, $ \forall \: r \in \mathcal{N} $. Since $ \xi = 1 $, all the factors involving $ \xi $ become $ 1 $ for the expressions in~\eqref{eqallopr} and~\eqref{eqtaxpr}. We design the allocation $ \widehat{x}_i(m) $ as a function of $ y_i $ and $ \left( q_r \right)_{r \in \mathcal{N}(i)} $ such that after taking into account the duplication it becomes $ y_i - \frac{1}{N-1} \sum_{j \ne i} y_j + \frac{c_1}{N} $. This facilitates the first global consensus, $ \sum_{i \in \mathcal{N}} x_i = c_1 $. One standard design principle for mechanisms is that if an agent partially controls their own allocation (such as here, since $ \widehat{x}_i(m) $ depends on $ y_i $) then they shouldn't be able to control the price. This is the reason that $ \widehat{p}_i(\cdot) $ doesn't depend on $ m_i $. It is function of $ \left( q_r \right)_{r \in \mathcal{N}(i)} $ and is designed such that after taking into account the duplication, the price for any agent is proportional to $ \sum_j y_j $. This facilitates the second consensus - common price for all agents.
Finally, we set $ \xi < 1 $ and adjust the allocation and tax function accordingly so that the game $ \mathfrak{G}_{pvt} $ can be contractive (see Section~\ref{seclearning}).
Before proceeding to the main results, define the best-response of any agent $ i $,
\begin{equation}\label{eqBRdefpr}
\beta_i(m_{-i}) = \big( {\tilde{y}}_i(m_{-i}),\tilde{q}_i(m_{-i}) \big) \triangleq \argmax_{m_i \in \mathcal{M}_i} \, u_i(m).
\end{equation}
Denote $ \beta = \left(\beta_1,\ldots,\beta_N\right) $. Best-response $ \beta: \mathcal{M} \rightarrow \mathcal{M} $ is a set-valued function in general.
\begin{lemma}[Concavity]\label{thmconcpr}
For any agent $ i \in \mathcal{N} $ and $ m_{-i} \in \mathcal{M}_{-i} $, the utility $ u_i(m) $, defined in~\eqref{equtil}, for the game $ \mathfrak{G}_{pvt} $ is strictly concave in $ m_i=(y_i,q_i) $. Thus, the best-response of agent $ i $ is unique and is defined by the first order conditions.
\end{lemma}
\begin{proof}
Please see Appendix~\ref{proofconcpr}.
\end{proof}
Concavity of the induced utility in the game $ \mathfrak{G}_{pvt} $ largely follows from the tax terms being quadratic. The second tax term in~\eqref{eqtaxpr_a} is the only source of cross derivatives across components of message $ m_i $. Concavity is proved by verifying that the Hessian matrix is negative definite.
\begin{theorem}[Full Implementation and Budget Balance] \label{thmFIpr}
For the game $ \mathfrak{G}_{pvt} $, there exists a unique Nash equilibrium, $ \widetilde{m} \in \mathcal{M} $, and the allocation at Nash equilibrium is efficient, i.e., $ \widehat{x}_i(\widetilde{m}) = x_i^* $, $ \forall $ $ i \in \mathcal{N} $. Further, the total tax paid at Nash equilibrium $ \widetilde{m} $ is zero, i.e.,
\begin{equation}
\sum_{i\in\mathcal{N}} \widehat{t}_i(\widetilde{m}) = 0.
\end{equation}
\end{theorem}
\begin{proof}
Please see Appendix~\ref{proofFIpr}.
\end{proof}
The proof of Proposition~\ref{thmFIpr} can be intuitively explained as follows. Since the optimality conditions in~\eqref{eqKKTpr} are sufficient, we start by showing that at any Nash equilibrium the allocation and price necessarily satisfy the optimality conditions. Thereby ensuring that if Nash equilibrium exists (unique or multiple) the corresponding allocation is efficient. Then we show existence and uniqueness by showing a one-to-one map between message at Nash equilibrium and $ (x^*,\lambda^*) $ arising out of the optimization in~\eqref{eqcppr}.
\subsubsection*{Generalizing to multiple goods $ (K > 1) $}
For the general problem we use notation on a per good basis. The message space is,
\begin{equation}
\mathcal{M}_i = \underset{k \in \mathcal{K}}{\times} \mathcal{M}_i^k = \underset{k \in \mathcal{K}}{\times} \mathbb{R}^{(N+1)} = \mathbb{R}^{(N+1)K}.
\end{equation}
Any message $ m_i = \left( m_i^k \right)_{k \in \mathcal{K}} = (y_i,q_i) $ contains separate demands and proxies for each good $ k \in \mathcal{K} $. Denote it as follows
\begin{subequations}
\begin{align}
y_i &= \left(y_i^k\right)_{k \in \mathcal{K}} \in \mathbb{R}^K,
\\
q_i &= \big( q_i^{r,k} \big)_{r \in \mathcal{N}, k \in \mathcal{K}} \in \mathbb{R}^{NK},
\end{align}
\end{subequations}
where for any good $ k \in \mathcal{K} $, demand and proxies $ \big( y_i^k, (q_i^{r,k} )_{r \in \mathcal{N}} \big) \in \mathbb{R}^{N+1} $ have the same interpretation as $ (y_i,q_i) $ in the presented mechanism above with only one good. The allocation function is $ \widehat{x}_i^k(\cdot) $, for any $ i \in \mathcal{N} $, $ k \in \mathcal{K} $ and the tax function is
\begin{equation}
\widehat{t}_i(\cdot) = \sum_{k \in \mathcal{K}} \widehat{t}_i^{\, k}(m), \quad\forall~i \in \mathcal{N}.
\end{equation}
Here both functions $ \widehat{x}_i^k(\cdot) $ and $ \widehat{t}_i^{\, k}(\cdot) $, depend only on $ m^k $, the part of message $ m $ pertaining to good $ k $. The expression for both functions are the same as in the presented mechanism above,~\eqref{eqallopr} and~\eqref{eqtaxpr}, replacing $ m $ by $ m^k $.
The generalized mechanism is distributed as well, since all allocation and tax functions still depend only on message of the agent and that of its neighbors. Owing to the design with a good-wise separation, the utility has the following form
\begin{equation}
u_i(m) = v_i\left( \left( \widehat{x}_i^k(m^k) \right)_{k \in \mathcal{K}} \right) - \sum_{k \in \mathcal{K}}\widehat{t}_i^{\,k}(m^k).
\end{equation}
Concavity of $ u_i $ follows from arguments similar to Proposition~\ref{thmconcpr}, where we verify the Hessian to be negative definite. Since the optimality conditions in~\eqref{eqKKTpr} are sufficient, the properties of efficiency, existence and uniqueness of Nash equilibrium follow from the first order conditions for optimality in the best-response. Finally, the Budget Balance result holds true on a per-good basis, so it also holds for the total tax which is the sum of per-good taxes.
\section{A mechanism for the public goods problem}
\label{secmechpub}
For the purpose of maintaining clarity in the exposition, the presented mechanism below is for the special case of a single feature in the public good i.e., $ P=1 $. A natural extension to the general case is discussed at the end of this section.
For any agent $ i\in \mathcal{N} $, the message space is $ \mathcal{M}_i = \mathbb{R}^{N+1} $. The message $ m_i = (y_i,q_i) $ consists of agent $ i $'s contribution $ y_i \in \mathbb{R} $ to the common public good and a surrogate/proxy $ q_i = \left( q_i^1,\ldots,q_i^N \right) \in \mathbb{R}^N $ for the contributions of all the agents (including himself/herself).
The allocation function is defined as
\begin{equation}\label{eqallopub}
\widehat{x}_i(m) = \frac{1}{N} \left( y_i + \sum_{r \in \mathcal{N}(i)} \frac{q_{r}^r}{\xi} + \sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i} } \frac{q_{n(i,r)}^r}{\xi^{d(i,r)-1}}\right), \quad \forall~i \in \mathcal{N}.
\end{equation}
The tax function is
\begin{subequations}\label{eqtaxpub}
\begin{align}
\nonumber
\widehat{t}_i(m) &= \widehat{p}_i(m_{-i}) \widehat{x}_i(m) + \left( q_i^i - \xi y_i \right)^2
+ \sum_{\substack{r \in \mathcal{N}(i)}} \left( q_i^r - \xi y_r \right)^2
+ \sum_{\substack{r \notin \mathcal{N}(i)\\r \ne i}} \left(q_i^r - \xi q_{n(i,r)}^r \right)^2,
\\ \label{eqtaxpub_a}
&\quad + \frac{\delta}{2} \left( q_{n(i,i)}^i - \xi y_i \right)^2
\\
\label{eqtaxpub_b}
\widehat{p}_i(m_{-i}) &= \delta(N-1) \left( \frac{q_{n(i,i)}^i}{\xi} - \frac{1}{N-1} \sum_{r \in \mathcal{N}(i)} \frac{q_{r}^r}{\xi} - \frac{1}{N-1} \sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i} } \frac{q_{n(i,r)}^r}{\xi^{d(i,r)-1}} \right), \quad \forall~i \in \mathcal{N},
\end{align}
\end{subequations}
where $ n(i,i) \in \mathcal{N}(i) $ is an arbitrarily chosen neighbor of $ i $ and $ \xi \in (0,1) $, $ \delta > 0 $ are appropriately chosen parameters and their selection is discussed in Section~\ref{seclearning} on Learning Guarantees, proof of Proposition~\ref{thmcontracpub}.
In the following, denote the induced game by $ \mathfrak{G}_{pub} $ and define the utility in the game,~\eqref{equtil}, and the best-response,~\eqref{eqBRdefpr}, as in the previous section.
\subsection{Results}
\begin{fact}[Distributed]
The mechanism defined in~\eqref{eqallopub} and~\eqref{eqtaxpub} is distributed.
\end{fact}
The optimality conditions in~\eqref{eqKKTpub} require that agents have global consensus on two aspects: allocation must be the same for all and sum of prices should be equal to zero. As in the private goods mechanism, we design the second, third and fourth tax terms in~\eqref{eqtaxpub_a} such that agents are incentivized to duplicate global demand $ y $ onto locally available variables $ q_i $. To motivate the allocation function and the remaining part of the tax function consider the case of $ \xi = 1 $ and take into account the duplication $ q_r = y $, $ \forall~r\in \mathcal{N} $. In this case all the factors involving $ \xi $ in~\eqref{eqallopub} and~\eqref{eqtaxpub} are 1. The allocation function $ \widehat{x}_i(m) $ depends on $ y_i, \left( q_r \right)_{r \in \mathcal{N}(i)} $ and is designed such that after taking into account the duplication it is proportional to $ \sum_j y_j $. This facilitates the first consensus - all agents' allocation must be the same. The price $ \widehat{p}_i(m_{-i}) $ is designed such that it depends only on $ \left( q_r \right)_{r \in \mathcal{N}(i)} $ and after taking into account the duplication it is proportional to $ y_i - \frac{1}{N-1} \sum_{j \ne i} y_j $. This facilitates the second consensus - sum of prices over all agents is zero.
With the above design principles, all the results of this section follow. We then introduce an additional fifth term in the tax,~\eqref{eqtaxpub_a}, just for the purpose of achieving contraction in Section~\ref{seclearning} (see proof of Proposition~\ref{thmcontracpub}). Incentives provided by this term are in line with those already provided to the neighboring agent $ n(i,i) $ through his/her third tax term, hence it doesn't interfere with the equilibrium results in this section. Only the proof of concavity changes slightly.
Finally, we set $ \xi < 1 $ and adjust everything in the allocation and tax function correspondingly so that the game $ \mathfrak{G}_{pub} $ can be contractive (see Section~\ref{seclearning}).
For the next two results, the basic idea behind the proofs are similar to the corresponding results from the previous section.
\begin{lemma}[Concavity]\label{thmconcpub}
For any agent $ i \in \mathcal{N} $ and $ m_{-i} \in \mathcal{M}_{-i} $, the utility $ u_i(m) $ for the game $ \mathfrak{G}_{pub} $ is strictly concave in $ m_i=(y_i,q_i) $. Thus, the best-response of agent $ i $ is unique and is defined by the first order conditions.
\end{lemma}
\begin{proof}
Please see Appendix~\ref{proofconcpub}.
\end{proof}
\begin{theorem}[Full Implementation and Budget Balance] \label{thmFIpub}
For the game $ \mathfrak{G}_{pub} $, there exists a unique Nash equilibrium, $ \widetilde{m} \in \mathcal{M} $, and the allocation at Nash equilibrium is efficient, i.e., $ \widehat{x}(\widetilde{m}) = x^* $. Further, the total tax paid at Nash equilibrium $ \widetilde{m} $ is zero, i.e.,
\begin{equation}
\sum_{i\in\mathcal{N}} \widehat{t}_i(\widetilde{m}) = 0.
\end{equation}
\end{theorem}
\begin{proof}
Please see Appendix~\ref{proofFIpub}.
\end{proof}
\subsubsection*{Generalizing to multiple features $ (P > 1) $}
With an idea similar to the extension for the private goods mechanism, extend the presented mechanism by first increasing the message space such that for each agent $ m_i = \big( m_i^p \big)_{p \in \mathcal{P}} = (y_i,q_i) $. The allocation in this case is $ P- $dimensional and the expression for $ \widehat{x}_i^p(m) $ is the same as in the presented mechanism with $ y_i,\big(q_r\big)_{r \in \mathcal{N}(i)} $ replaced by $ y_i^p,\big(q_r^p\big)_{r \in \mathcal{N}(i)} $. The tax function is
\begin{equation}
\widehat{t}_i(m) = \sum_{p \in \mathcal{P}} \widehat{t}_i^{\: p}(m),
\end{equation}
where the expression for $ \widehat{t}_i^{\: p} $ is the same as in the presented mechanism, replacing $ m $ by $ m^k $.
The results above follow using analogous arguments to those mentioned in the previous section regarding generalization to multiple goods, $ K > 1 $.
\section{Learning Guarantees} \label{seclearning}
This section provides the result for guaranteed convergence for a class of learning algorithms, when the mechanisms defined in Sections~\ref{secpr} and~\ref{secmechpub} are played repeatedly. As discussed in the Introduction, such results act as a measure of robustness of a mechanism, w.r.t. information available to agents, and thus makes the mechanism ready for practical applications.
A learning dynamic is represented by
\begin{equation}
\left( \mu_n \right)_{n \ge 1} \subseteq \underset{i \in \mathcal{N}}{\times} \Delta(\mathcal{M}_i),
\end{equation}
where $ \mu_n $ is a mixed strategy profile with product structure to be used at time $ n $. Denote by $ S(\mu_n) \subseteq \mathcal{M} $ the support of the mixed strategy profile $ \mu_n $ and denote by $ m_n \in S(\mu_n) $ the realized action. Healy and Mathevet in~\citep{healy2012designing} define the Adaptive Best-Response (ABR) dynamics class by restricting the support $ S(\mu_n) $ in terms of past observed actions. Define the history $ H_{n^\prime,n} = \left( m_{n^\prime},m_{n^\prime + 1},\ldots,m_{n-1} \right) $ as the set of observed actions between $ n^\prime $ and $ n-1 $. Denote by $ \widetilde{m} $ the unique Nash equilibrium of the game and define $ B\left( \mathcal{M}^\prime \right) $ as the smallest closed ball centered at $ \widetilde{m} $ that contains the set $ \mathcal{M}^\prime \subset \mathcal{M} $. The closed ball is defined with any valid metric $ d $ on the message space $ \mathcal{M} $.
A learning dynamic is in the ABR class if any point in the support of the action at time $ n $ is no further from the Nash equilibrium than the best-response to any action that is no further from Nash equilibrium than the ``worst-case'' action that has been observed in some finite past $ \{ n^\prime,\ldots,n-1 \} $.
\begin{definition}[Adaptive Best-Response Learning Class~{\citep{healy2012designing}}]
A learning dynamic is an adaptive best-response dynamic if $ \forall~n^\prime,~\exists~\widehat{n} > n^\prime,~\text{s.t.}~\forall~ n \ge \widehat{n} $,
\begin{equation}
S(\mu_n) \subseteq B\left( \beta\left( B\left( H_{n^\prime,n} \right) \right) \right),
\end{equation}
where $ \beta:\mathcal{M} \rightarrow \mathcal{M} $ is the best-response of the game.
\end{definition}
The above is satisfied for instance if every agent puts belief zero over actions further from Nash equilibrium than the ones that he/she has observed in the past. Some well-known learning dynamics are in the ABR class, following are a few examples.
Cournot best-response is defined as $ S(\mu_n) = m_n = \beta(m_{n-1}) $, i.e., at every time agents best-respond to the last round's action. This gives rise to a deterministic strategy at each time. More generally, $ k- $period best-response is defined as the learning dynamic where at any time $ n $, an agent $ i $'s strategy is a best-response to the mixed strategy of agents $ j \ne i $ which are created using the observed empirical distribution from the actions of the previous $ k- $rounds i.e., $ \{ m_{j,n-k},\ldots,m_{j,n-1} \} $. In fact, the generalization of this is also in the ABR class, where at each time $ n $, an agent $ i $'s strategy is the best-response to the mixed strategy of agents $ j \ne i $ that is formed by taking any convex combination of the empirical distributions of actions observed in the previous $ k- $rounds. Finally, Fictitious Play~\citep{brown,fudenberglearning}, which maintains empirical distribution of all the past actions (instead of $ k $ most recent ones) is also in ABR. The additional requirement for this is that the utility in the game should be strictly concave, which is true for the presented mechanisms (see Lemmas~\ref{thmconcpr} and~\ref{thmconcpub}).
\begin{definition}[Contractive Mechanism]
Let $ d $ be any metric defined on the message space $ \mathcal{M} $ such that $ \left(\mathcal{M},d\right) $ is a complete metric space. A mechanism is called contractive if for any profile of utility function $ \big( v_i(\cdot) \big)_{i \in \mathcal{N}} $ that satisfy the assumptions of the model, the induced game $ \mathfrak{G}_{pvt} $ or $ \mathfrak{G}_{pub} $ (depending on the allocation problem) has a single-valued best-response function $ \beta: \mathcal{M} \rightarrow \mathcal{M} $ that is a $ d- $contraction mapping.
\end{definition}
A function $ h:\mathcal{M} \rightarrow \mathcal{M} $ is a $ d- $contraction mapping if $ \Vert h(x) - h(y) \Vert_d < \Vert x - y \Vert_d $ for all distinct $ x,y \in \mathcal{M} $. For the results below, the following well-known check for contraction mapping is used: $ h $ is a contraction mapping (for some metric $ d $) if the Jacobian has norm less than one, i.e., $ \Vert \nabla h \Vert < 1 $, where any matrix norm can be considered. Specifically, we consider the \emph{row-sum} norm.
For the game induced by a contractive mechanism, by definition, there is a unique Nash equilibrium. This is due to the Banach fixed-point theorem, which gives that the best-response iteration for the induced game converges to a unique point. As shown in the previous sections, the game $ \mathfrak{G}_{pvt} $ and $ \mathfrak{G}_{pub} $ already have a unique Nash equilibrium.
\begin{fact}[{\citep[Theorem 1]{healy2012designing}}] \label{factabr}
If a game is contractive, then all ABR dynamics converge to the unique Nash equilibrium.
\end{fact}
The idea behind the above proof is to show that after a finite time, under any ABR dynamic the distance to equilibrium gets smaller (exponentially) between successively rounds due to the best-response being a contraction mapping. It is shown below that the presented mechanism for both, the private and public goods problems, is contractive and thus owing to the above result there is guaranteed convergence for all learning dynamics in the ABR class.
Contraction ensures convergence for the ABR class; this result is in the same vein as the one in the seminal work~\citep{milgrom1990}. Milgrom and Roberts show that Supermodularity ensures convergence for the Adaptive Dynamics class of learning algorithms (also defined in~\citep{milgrom1990}). Supermodularity requires that the best-response of any agent $ i $ is non-decreasing in the message $ m_j $ of any other agent $ j \ne i $. The aim in this paper is to get guarantees for the ABR class, however it is shown below that the game $ \mathfrak{G}_{pvt} $ induced by the private goods mechanism is also supermodular and thus has guaranteed convergence for the Adaptive Dynamics class as well.
Before proceeding, kindly note that contraction is somewhat more stringent a condition that supermodularity and consequently the ABR class is broader than the adaptive dynamics class. Thus in order to have better learning guarantees, our principle aim to ensure the property of contraction.
\begin{comment}
\begin{definition}[Supermodular Game]
content...
\end{definition}
\begin{fact}[{\citep[Theorem 8]{milgrom1990}}]\label{factadap}
content...
\end{fact}
\todo[inline]{Compare supermodularity with contraction and state the result of Milgrom}
\end{comment}
\subsection{Results}
\begin{theorem}[Contraction]\label{thmcontracpr}
The game $ \mathfrak{G}_{pvt} $ defined in Section~\ref{secmechpr} is contractive but not supermodular. Thus, all learning dynamics within the ABR dynamics class converge to the unique efficient Nash equilibrium.
\end{theorem}
\begin{proof}
Please see Appendix~\ref{proofcontracpr}.
\end{proof}
The intuition behind the proof of Proposition~\ref{thmcontracpr} can be motivated as follows. By selecting parameter $ \xi < 1 $, contraction of best-response for the variables $ \left( \tilde{q}_i^r \right)_{r \ne i} $ is already ensured. However, for best-response in $ \tilde{y}_i,\tilde{q}_i^i $, tuning of $ \xi,\delta $ is needed. Due to the specific nature of the mechanism it turns out that in order to accommodate any given value of $ \eta > 1 $, $ \xi $ needs to be selected close to $ 1 $ and correspondingly $ \delta $ needs to be selected as a function of the chosen $ \xi $. Finally, it is also shown that the best-response $ \tilde{y}_i $ and $ \tilde{q}_i^i $ are decreasing in $ q_{n(i,i)}^i $ and hence the game is not supermodular.
\paragraph*{Generalizing to $ K>1 $} The proof of Proposition~\ref{thmcontracpr} relies on inverting $ v_i^\prime:\mathbb{R} \rightarrow \mathbb{R} $ and bounding it appropriately. For the general case this is the same as inverting $ \nabla v_i:\mathbb{R}^K \rightarrow \mathbb{R}^K $. Strict concavity of $ v_i(\cdot) $ ensures that the determinant of Hessian of $ v_i $ is never zero, hence an inverse for $ \nabla v_i $ exists in the general case. Of course, to bound the derivatives of $ ( \nabla v_i )^{-1} $, instead of using the condition in~\eqref{eqetapr} we use the more general condition from~\eqref{eqetagenpr}. Finally, the proof is completed by tuning the parameters $ \xi,\delta $ in exactly the same manner as in the proof of Proposition~\ref{thmcontracpr}.
\begin{theorem}[Contraction]\label{thmcontracpub}
The game $ \mathfrak{G}_{pub} $ defined in Section~\ref{secmechpub} is both contractive and supermodular. Thus, all learning dynamics within the ABR dynamics class converge to the unique Nash equilibrium.
\end{theorem}
\begin{proof}
Please see Appendix~\ref{proofcontracpub}.
\end{proof}
Initially the parameters have the following bound: $ \xi \in (0,1) $ and $ \delta > 0 $. In order to get contraction, a further restriction $ \xi \in \left( \sqrt{\left. (N-1) \middle/ N \right.},1 \right) $ needs to be imposed. This also gives that each best-response is non-decreasing in the message of every other agent and thus the game is supermodular. Finally, here too in order to accommodate any value of $ \eta > 1 $ the final tuning of $ \xi $ requires it to be chosen very close to $ 1 $ and consequently $ \delta $ is selected as a function of the chosen $ \xi $.
Generalizing to $ P > 1 $, the essential idea of inverting $ \nabla v_i $ from above works here too and from there onwards the steps follow analogously to the ones in the proof above.
\subsection{Numerical Analysis of convergence}
For numerical analysis we consider the private goods problem with one good $ (K=1) $ and the public goods problem with one feature $ (P=1) $ and $ N = 31 $, $ \eta = 25 $. The agents' utility function as quadratic, i.e.,
\begin{align}
v_i(x_i) &= \theta_i x_i^2 + \sigma_i x_i,
\tag{Private}
\\
v_i(x) &= \theta_i x^2 + \sigma_i x.
\tag{Public}
\end{align}
An example of quadratic utility function can be found in~\citep{samadi2012}, for the model of demand side management in smart-grids. In each case the second derivative of $ v_i(\cdot) $ is $ 2\theta_i $ and thus for any agent $ i $ the value for $ \theta_i $ is chosen uniformly randomly in the range $ (-\frac{\eta}{2},-\frac{1}{2\eta}) $. As the model doesn't impose any restriction on the first derivative, the value for $ \sigma_i $ is chosen uniformly randomly in the range $ (10,20) $. From the proof of Propositions~\ref{thmcontracpr} and~\ref{thmcontracpub}, one can numerically calculate the value of parameters $ \xi,\delta $. For the particular instance of the random $ \theta,\sigma $ generated to be used for the plots below (the same values for $ \theta,\sigma $ are used for both public and private goods examples), the parameter values are listed in Table~\ref{table1}. The two cases for graph $ \mathcal{G} $ considered are: a full binary tree and a sample of the Erd\H{o}s-Re\'{n}yi random graph with only one connected component, where any two edges are connected with probability $ p = 0.3 $. The first represents a case of small average degree whereas the second represents the case of large average degree. The same instance of Erd\H{o}s-Re\'{n}yi random graph is used for both private and public goods examples.
\begin{comment}
\begin{table}
\centering
\begin{tabular}{|c|c|c|} \hline
& Private goods & Public goods \\ \hline
Binary Tree & $\xi = 1 - (1.831\times 10^{-4}) $, $ \delta = 1005.6 $ & $ \xi = 1 - (2.515 \times 10^{-4}) $, $ \delta = 0.9505 $ \\ \hline
Erd\H{o}s-Re\'{n}yi & $\xi = 1 - (7.324 \times 10^{-4}) $, $ \delta = 932.3 $ & $ \xi = 1 - (0.001) $, $ \delta = 0.8744 $ \\ \hline
\end{tabular}
\caption{$ (\xi,\delta) $ parameter values for different graphs and problems.}
\end{table}
\end{comment}
\begin{table}
\centering
\begin{tabular}{|c|l|l|} \hline
& \multicolumn{1}{|c|}{Private goods} & \multicolumn{1}{|c|}{Public goods}
\\ \hline
Binary Tree & \begin{tabular}{l}
$\xi = 1 - (1.831\times 10^{-4}) $, \\
$ \delta = 1005.6 $.
\end{tabular} & \begin{tabular}{l}
$ \xi = 1 - (2.515 \times 10^{-4}) $, \\
$ \delta = 0.9505 $.
\end{tabular} \\ \hline
Erd\H{o}s-Re\'{n}yi & \begin{tabular}{l}
$\xi = 1 - (7.324 \times 10^{-4}) $, \\
$ \delta = 932.3 $.
\end{tabular} & \begin{tabular}{l}
$ \xi = 1 - (10^{-3}) $, \\
$ \delta = 0.8744 $.
\end{tabular} \\ \hline
\end{tabular}
\caption{$ (\xi,\delta) $ parameter values for different graphs and problems.}
\label{table1}
\end{table}
Since it has been shown that the best-response is a contraction mapping, one expects that any learning strategy that best-responds to some convex combination of past actions from finitely many rounds, converges at an exponential rate. Indeed, this is exactly observed from Fig.~\ref{figplotpr} and~\ref{figplotpub}, where the absolute distance $ \Vert m_n - \widetilde{m} \Vert_2 $ of the action in round $ n $ to the Nash equilibrium $ \widetilde{m} $ of the game $ \mathfrak{G}_{pvt} $ and $ \mathfrak{G}_{pub} $ is plotted versus $ n $, respectively. For the learning dynamic\footnote{A theoretical simplification with quadratic utilities $ v_i(\cdot) $ is that for learning strategies such as $ k- $period best-response or Fictitious Play, instead of maintaining the empirical distribution over other agents' past actions, every agent can simply maintain the empirical average.}, one case considered is when the action taken by any agent $ i $ at time $ n $ is the best-response to an exponentially weighed average of past actions, i.e.,
\begin{subequations}
\begin{align}
m_{n} &= \beta\left( \frac{m_{n-1}}{2} + \frac{r_{n-1}}{2} \right),
\\
r_{n} &= \frac{m_n}{2} + \frac{r_{n-1}}{2}.
\end{align}
\end{subequations}
The second learning dynamic is to best-respond to the average of past 10 rounds.
\begin{figure}
\caption{$ \Vert m_n - \widetilde{m}
\label{figplotpr}
\end{figure}
\begin{figure}
\caption{$ \Vert \widehat{x}
\label{figplotpr_xdist}
\end{figure}
\begin{figure}
\caption{$ \Vert \left( \widehat{p}
\label{figplotpr_pdist}
\end{figure}
\begin{figure}
\caption{$ \Vert m_n - \widetilde{m}
\label{figplotpub}
\end{figure}
Since the learning iterations are essentially conducting information exchange, one expects that the convergence to be faster for a graph that is more connected. From both the figures this is evident, as for each learning dynamic higher average degree Erd\H{o}s-Re\'{n}yi random graph shows faster convergence than lower average degree Binary Tree. In fact, for both learning dynamics the convergence for the Erd\H{o}s-Re\'{n}yi random graph is faster than either learning dynamic for Binary Tree. Comparing the two learning dynamics among themselves, we observe that the more aggressive exponential weighing leads to faster convergence compared to the learning dynamic that puts equal weight on each of the previous 10 actions. Finally, for Fig.~\ref{figplotpr}, in each case the relative distance to Nash equilibrium, defined as $ \left. \Vert m_n - \widetilde{m} \Vert_2 \middle/ \Vert \widetilde{m} \Vert_1 \right. $, is in the order of $ 10^{-9} $ when the absolute distance to Nash equilibrium is $ 10^{-3} $. For Fig.~\ref{figplotpub}, the relative distance in each case is of the order of $ 10^{-8} $ when the absolute distance to Nash equilibrium is $ 10^{-5} $.
Fig.~\ref{figplotpr_xdist} and~\ref{figplotpr_pdist} refer to the private goods mechanism and plot the distance of allocation $ \widehat{x}(m_n) $ and price $ \left( p_1(m_n),\ldots,p_N(m_n) \right) $ in round $ n $ to $ x^* $ and $ \left( \lambda_1^*,\ldots,\lambda_1^* \right) $, respectively. The convergence pattern is the same as the one observed in Fig.~\ref{figplotpr}, for the distance of message $ m_n $ to Nash equilibrium $ \widetilde{m} $.
\section{Conclusion}
In this paper, we present a distributed mechanism where agents only need to exchange messages locally with their neighbors. While for models with non-strategic agents, extensive research has been done in the field of distributed learning and optimization, this is not the case with mechanism design where agents are fully strategic.
For every profile $ \left( v_i(\cdot) \right)_{i \in \mathcal{N}} $ of utility functions, the induced game is shown to have a unique NE. The allocation at equilibrium is efficient and taxes are budget balanced. Then we establish informational robustness of the mechanism by showing that the best-response in the induced game is a contraction mapping. This establishes that every learning dynamic within the ABR dynamics class converges to the unique and efficient NE when the game is played repeatedly. The ABR class contains learning dynamics such as Cournot best-response, $ k- $period best-response and Fictitious Play.
\subsubsection*{Future Work}
A significant scope for improvement in the presented mechanism is the reduction of the size of the message space.
A more scalable mechanism would be one where on average each agent's message space is of dimension $ o(N) $. However, such an attempt might possibly require restrictions on either the underlying graph $ \mathcal{G} $ or the upper bound on $ \eta $ that is admissible under the model. For the presented mechanism, the only restriction on the graph is that it is connected and there is no upper bound on $ \eta $.
Another improvement can be that of considering more complicated constraint sets for the optimization~\eqref{eqcppr}. However, as can be seen from previous attempts at mechanism design for general constraint sets,~\citep{johari2009efficiency,rahuljain,demos,SiAn_multicast_tcns,SiAn14b}, such an extension is not straightforward.
Finally, for preventing inter-temporal exchange of money during the learning phase, one can adjust taxes such that there is budget balance for all messages, rather than just at NE.
{
}
\appendix
\section{Proof of Lemma~\ref{thmconcpr} (Concavity - Private goods)}
\label{proofconcpr}
\begin{proof}
Since the allocation and tax functions are smooth and $ v_i(\cdot) $ is continuously double-differentiable, to establish concavity we show that the Hessian of $ u_i(m) $ w.r.t. $ m_i $ is negative definite i.e., $ H \prec 0 $. Once this is established, the optimization in~\eqref{eqBRdefpr} has a strictly concave objective and an unbounded constraint set. Thus it has a unique maximizer, defined by the first order derivative conditions.
The Hessian is of size $ (N+1) \times (N+1) $ and we have
\begin{subequations}
\begin{align}
H_{11} &= \frac{\partial^2 u_i(m)}{\partial y_i^2} = v_i^{\prime\prime}(\widehat{x}_i(m)) - 2\xi^2, \\
H_{(j+1)1} = H_{1(j+1)} &= \frac{\partial^2 u_i(m)}{\partial y_i \partial q_i^j} =
\left\{
\begin{array}{ll}
0 & \mbox{for } j \in \mathcal{N},~j\ne i, \\
2\xi & \mbox{for } j=i,
\end{array}
\right.
\\
H_{(j+1)(j+1)} &= \frac{\partial^2 u_i(m)}{\partial (q_i^j)^2} = -2, \quad \forall~ j \in \mathcal{N},\\
H_{(r+1)(j+1)} &= \frac{\partial^2 u_i(m)}{\partial q_i^r \partial q_i^j} = 0 , \quad \forall~ j,r \in \mathcal{N},~j \ne r.
\end{align}
\end{subequations}
The characteristic equation, $ \textsf{Det}\left(H - xI\right) = 0 $, becomes
\begin{equation}
\left(x+2\right)^{N-1} \Big( (x+2)(x-H_{11}) - 4\xi^2 \Big) = 0.
\end{equation}
This implies that $ N-1 $ eigenvalues of $ H $ are $ -2 $ and the remaining two eigenvalues satisfy $ x^2 + (2-H_{11})x - 2v_i^{\prime\prime}(\widehat{x}_i(m)) = 0 $. Since $ H $ is a symmetric matrix, all its eigenvalues are real. Due to $ v_i^{\prime\prime}(\cdot) < 0 $, the product of roots in the above quadratic equation is positive and the sum of roots is negative. This gives that the remaining two eigenvalues of $ H $ are also negative.
\end{proof}
\section{Proof of Proposition~\ref{thmFIpr} (Full Implementation - Private goods)}
\label{proofFIpr}
\begin{proof}
For the private goods problem in~\eqref{eqcppr}, the optimality conditions in~\eqref{eqKKTpr} are sufficient. Thus in order to prove that the corresponding allocation at Nash equilibrium is efficient, we show that at any Nash equilibrium $ {\overline{m}} = (\overline{y},\overline{q}) \in \mathcal{M} $, the allocation $ \big(\widehat{x}_i(\overline{m}) \big)_{i\in \mathcal{N}} $ and prices $ \big(\widehat{p}_i(\overline{m})\big)_{i\in \mathcal{N}} $ satisfy the optimality conditions as $ x^* $ and $\lambda_1^* $, respectively. Then using an invertibility argument we show existence and uniqueness of Nash equilibrium.
Using Proposition~\ref{thmconcpr}, at any Nash equilibrium $ \overline{m} $ we have: $ { \nabla_{m_i} } u_i(\overline{m}) = 0 $, $ \forall~i \in \mathcal{N} $. This gives
\begin{subequations}
\begin{alignat}{2}
\frac{\partial v_i(\widehat{x}_i(\overline{m}))}{\partial y_i} - \frac{\partial \widehat{t}_i(\overline{m})}{\partial y_i} &= 0, \quad &&\forall~i \in \mathcal{N},
\\
\frac{\partial v_i(\widehat{x}_i(\overline{m}))}{\partial q_i^r} - \frac{\partial \widehat{t}_i(\overline{m})}{\partial q_i^r} &= 0, \quad &&\forall~r \in \mathcal{N},~i\in \mathcal{N}.
\end{alignat}
\end{subequations}
Using the definitions in~\eqref{eqallopr} and \eqref{eqtaxpr}, this becomes
\begin{subequations} \label{eqNE1pr}
\begin{gather}\label{eqNE1pr_a}
v_i^\prime(\widehat{x}_i(\overline{m})) - \widehat{p}_i(\overline{m}_{-i}) + 2\xi(\overline{q}_i^i - \xi \overline{y}_i) = 0, \quad \forall~i \in \mathcal{N},
\\
\label{eqNE1pr_b}
\overline{q}_i^r =
\left\{
\begin{array}{ll}
\xi \overline{y}_i & \mbox{for } r = i, \\
\xi \overline{y}_r & \mbox{for } r \in \mathcal{N}(i), \\
\xi \overline{q}_{n(i,r)}^r & \mbox{for } r \notin \mathcal{N}(i) \text{ and } r \ne i,
\end{array}
\right. \quad \forall~i \in \mathcal{N}.
\end{gather}
\end{subequations}
For any distinct pair of vertexes $ i,r $, denote by $ \{ i, i_1,i_2,\ldots,i_{d(i,r)} = r \}$ the ordered vertexes in the shortest path between $ i $ and $ r $, where $ i_1 = n(i,r) \in \mathcal{N}(i) $. Since the shortest path between $ i $ and $ r $ contains the shortest path between $ i_k $ and $ r $, for any $ k < d(i,r) $, we have $ n(i_k,r) = i_{k+1} $. Using the third sub-equation in~\eqref{eqNE1pr_b} repeatedly, replacing $ i $ by $ i_k $ gives,
\begin{equation}
\overline{q}_i^r = \xi \, \overline{q}_{i_1}^r = \xi^2 \, \overline{q}_{i_2}^r = \cdots = \xi^{d(i,r)-1} \, \overline{q}_{i_{d(i,r)-1}}^r.
\end{equation}
Now using the second sub-equation of~\eqref{eqNE1pr_b}, replacing $ i $ by $ i_{d(i,r)-1} $ and noting $ r \in \mathcal{N}(i_{d(i,r)-1}) $, gives $ \overline{q}_{i_{d(i,r)-1}}^r = \xi \, \overline{y}_{r} $. This combined with the above equation gives that~\eqref{eqNE1pr_b} implies
\begin{equation}\label{eqlempr}
\overline{q}_i^r =
\left\{
\begin{array}{ll}
\xi \overline{y}_i & \mbox{\textup{for} } r = i, \\
\xi^{d(i,r)} \overline{y}_r & \mbox{\textup{for} } r \ne i,
\end{array}
\right. \quad \forall~i \in \mathcal{N}.
\end{equation}
Using the above and then combining~\eqref{eqNE1pr_a} with~\eqref{eqallopr} and~\eqref{eqtaxpr_b} gives, $ \forall~i\in \mathcal{N} $,
\begin{subequations}
\begin{align} \label{eqNE2pr_a}
v_i^\prime(\widehat{x}_i(\overline{m})) &= \widehat{p}_i(\overline{m}_{-i}),
\\
\label{eqNE2pr_b}
\widehat{x}_i(\overline{m}) &= \overline{y}_i - \frac{1}{N-1}\sum_{j \ne i} \overline{y}_j,
\\
\label{eqNE2pr_c}
\widehat{p}_i(\overline{m}_{-i}) &= \frac{1}{\delta} \sum_{j \in \mathcal{N}} \overline{y}_j.
\end{align}
\end{subequations}
\eqref{eqNE2pr_b} implies $ \sum_{i \in \mathcal{N}} \widehat{x}_i(\overline{m}) = 0 $ and combining~\eqref{eqNE2pr_a} and~\eqref{eqNE2pr_c} gives $ v_i^\prime(\widehat{x}_i(\overline{m})) = \frac{1}{\delta} \sum_{j \in \mathcal{N}} \overline{y}_j $, $ \forall~i\in \mathcal{N} $. Thus, the allocation-price pair
\begin{equation}
\left(\big(\overline{y}_i - \frac{1}{N-1}\sum_{j \ne i} \overline{y}_j\big)_{i \in \mathcal{N}} \, , \, \frac{1}{\delta} \sum_{j \in \mathcal{N}} \overline{y}_j\right)
\end{equation}
satisfy the optimality conditions,~\eqref{eqKKTpr}, as $ (x^*,\lambda_1^*) $. Since the optimality conditions are sufficient, the allocation at any Nash equilibrium $ \overline{m} $ is the efficient allocation $ x^* $.
For existence and uniqueness, consider the following set of linear equations that must be satisfied at any Nash equilibrium $ \overline{m} $,
\begin{subequations}
\begin{align}
x_i^* &= \overline{y}_i - \frac{1}{N-1}\sum_{j \ne i} \overline{y}_j, \quad \forall~ i \in \mathcal{N},
\\
\lambda_1^* &= \frac{1}{\delta} \sum_{j \in \mathcal{N}} \overline{y}_j.
\end{align}
\end{subequations}
Here $ \left(\overline{y}_j \right)_{j\in\mathcal{N}} $ are the variables and $ (x^*,\lambda_1^*) $ are fixed - since they are uniquely defined by the optimization~\eqref{eqcppr}. The above equations can be inverted to give the unique solution as,
\begin{equation}
\overline{y}_i = \frac{N-1}{N} x_i^* + \frac{\delta \lambda_1^*}{N}, \quad \forall~i\in\mathcal{N}.
\end{equation}
Furthermore, using above and~\eqref{eqlempr}, the values for $ \big( \overline{q}_i^r \big)_{i,r\in\mathcal{N}} $ can also be calculated uniquely. Since a solution for $ \overline{m} = (\overline{y},\overline{q}) $ in terms of $ x^*,\lambda_1^* $ exists, existence of Nash equilibrium is guaranteed. Also, since this solution is unique, there is a unique Nash equilibrium.
For Budget Balance, we have the following. From the above characterization, at Nash Equilibrium $ \widetilde{m} $ all tax terms from~\eqref{eqtaxpr_a}, other than $ \widehat{p}_i(\widetilde{m}_{-i}) \left(\widehat{x}_i(\widetilde{m}) - \dfrac{c_1}{N} \right) $, are zero. Furthermore, the prices are all equal to $ \lambda_1^* $ and allocations are equal to $ x_i^* $. Thus,
\begin{equation}
\sum_{i \in \mathcal{N}} \widehat{t}_i(\widetilde{m}) = \sum_{i \in \mathcal{N}} \lambda_1^* \left(\widehat{x}_i(\widetilde{m}) - \dfrac{c_1}{N} \right) = \lambda_1^* \left( \sum_{i \in \mathcal{N}} x_i^* - c_1 \right) = \lambda_1^* \cdot 0 = 0,
\end{equation}
since the efficient allocation $ x^* $ satisfies the constraint $ \sum_{i \in \mathcal{N}} x_i^* = c_1 $.
\end{proof}
\section{Proof of Lemma~\ref{thmconcpub} (Concavity - Public Goods)}
\label{proofconcpub}
\begin{proof}
Since the allocation and tax functions are smooth and $ v_i(\cdot) $ is continuously double-differentiable, to establish concavity we show that the Hessian of $ u_i(m) $ w.r.t. $ m_i $ is negative definite i.e., $ H \prec 0 $. Once this is established, the optimization in~\eqref{eqBRdefpr} has a strictly concave objective and an unbounded constraint set. Thus it has a unique maximizer, defined by the first order derivative conditions.
The Hessian is of size $ (N+1) \times (N+1) $ and we have
\begin{subequations}
\begin{align}
H_{11} &= \frac{\partial^2 u_i(m)}{\partial y_i^2} = \dfrac{v_i^{\prime\prime}(\widehat{x}_i(m))}{N^2} - (2+\delta)\xi^2,
\\
H_{(j+1)1} = H_{1(j+1)} &= \frac{\partial^2 u_i(m)}{\partial y_i \partial q_i^j} =
\left\{
\begin{array}{ll}
0 & \mbox{for } j \in \mathcal{N},~j\ne i, \\
2\xi & \mbox{for } j=i,
\end{array}
\right.
\\
H_{(j+1)(j+1)} &= \frac{\partial^2 u_i(m)}{\partial (q_i^j)^2} = -2, \quad \forall~ j \in \mathcal{N},
\\
H_{(r+1)(j+1)} &= \frac{\partial^2 u_i(m)}{\partial q_i^r \partial q_i^j} = 0 , \quad \forall~ j,r \in \mathcal{N},~j \ne r.
\end{align}
\end{subequations}
The characteristic equation, $ \textsf{Det}\left(H - xI\right) = 0 $, becomes
\begin{equation}
\left(x+2\right)^{N-1} \Big( (x+2)(x-H_{11}) - 4\xi^2 \Big) = 0.
\end{equation}
This implies that $ N-1 $ eigenvalues of $ H $ are $ -2 $ and the remaining two eigenvalues satisfy $ x^2 + (2-H_{11})x + 2\delta\xi^2 - \frac{2}{N^2}v_i^{\prime\prime}(\widehat{x}_i(m)) = 0 $. Since $ H $ is a symmetric matrix, all its eigenvalues are real. Due to $ v_i^{\prime\prime}(\cdot) < 0 $, the product of roots in the above quadratic equation is positive and the sum of roots is negative. This gives that the remaining two eigenvalues of $ H $ are also negative.
\end{proof}
\section{Proof of Proposition~\ref{thmFIpub} (Full Implementation - Public goods)}
\label{proofFIpub}
\begin{proof}
For the public goods problem in~\eqref{eqcppub}, the optimality conditions in~\eqref{eqKKTpub} are sufficient. Thus in order to prove that the corresponding allocation at Nash equilibrium is efficient, we show that at any Nash equilibrium $ {\overline{m}} = (\overline{y},\overline{q}) \in \mathcal{M} $, the allocation $ \big(\widehat{x}_i(\overline{m}) \big)_{i\in \mathcal{N}} $ and prices $ \big(\widehat{p}_i(\overline{m})\big)_{i\in \mathcal{N}} $ satisfy the optimality conditions as $ x^* $ and $ {\mu^1}^* $, respectively. Then using an invertibility argument we show existence and uniqueness of Nash equilibrium.
Using Proposition~\ref{thmconcpub}, at any Nash equilibrium $ \overline{m} $ we have: $ { \nabla_{m_i} } u_i(\overline{m}) = 0 $, $ \forall~i \in \mathcal{N} $. This gives
\begin{subequations}
\begin{alignat}{2}
\frac{\partial v_i(\widehat{x}_i(\overline{m}))}{\partial y_i} - \frac{\partial \widehat{t}_i(\overline{m})}{\partial y_i} &= 0, \quad &&\forall~i \in \mathcal{N},
\\
\frac{\partial v_i(\widehat{x}_i(\overline{m}))}{\partial q_i^r} - \frac{\partial \widehat{t}_i(\overline{m})}{\partial q_i^r} &= 0, \quad &&\forall~r \in \mathcal{N},~i\in \mathcal{N}.
\end{alignat}
\end{subequations}
Using the definitions in~\eqref{eqallopub} and \eqref{eqtaxpub}, this becomes, $ \forall~i \in \mathcal{N} $,
\begin{subequations} \label{eqNE1pub}
\begin{gather}\label{eqNE1pub_a}
\frac{1}{N} \big( v_i^\prime(\widehat{x}_i(\overline{m})) - \widehat{p}_i(\overline{m}_{-i}) \big) + 2\xi(\overline{q}_i^i - \xi \overline{y}_i) + \delta\xi(\overline{q}_{n(i,i)}^i - \xi \overline{y}_i) = 0,
\\
\label{eqNE1pub_b}
\overline{q}_i^r =
\left\{
\begin{array}{ll}
\xi \overline{y}_i & \mbox{for } r = i, \\
\xi \overline{y}_r & \mbox{for } r \in \mathcal{N}(i), \\
\xi \overline{q}_{n(i,r)}^r & \mbox{for } r \notin \mathcal{N}(i) \text{ and } r \ne i,
\end{array}
\right..
\end{gather}
\end{subequations}
For any distinct pair of vertexes $ i,r $, denote by $ \{ i, i_1,i_2,\ldots,i_{d(i,r)} = r \}$ the ordered vertexes in the shortest path between $ i $ and $ r $, where $ i_1 = n(i,r) \in \mathcal{N}(i) $. Since the shortest path between $ i $ and $ r $ contains the shortest path between $ i_k $ and $ r $, for any $ k < d(i,r) $, we have $ n(i_k,r) = i_{k+1} $. Using the third sub-equation in~\eqref{eqNE1pub_b} repeatedly, replacing $ i $ by $ i_k $ gives,
\begin{equation}
\overline{q}_i^r = \xi \, \overline{q}_{i_1}^r = \xi^2 \, \overline{q}_{i_2}^r = \cdots = \xi^{d(i,r)-1} \, \overline{q}_{i_{d(i,r)-1}}^r.
\end{equation}
Now using the second sub-equation of~\eqref{eqNE1pub_b}, replacing $ i $ by $ i_{d(i,r)-1} $ and noting $ r \in \mathcal{N}(i_{d(i,r)-1}) $, gives $ \overline{q}_{i_{d(i,r)-1}}^r = \xi \, \overline{y}_{r} $. This combined with the above equation gives that~\eqref{eqNE1pub_b} implies
\begin{equation}\label{eqlempub}
\overline{q}_i^r =
\left\{
\begin{array}{ll}
\xi \overline{y}_i & \mbox{\textup{for} } r = i, \\
\xi^{d(i,r)} \overline{y}_r & \mbox{\textup{for} } r \ne i,
\end{array}
\right. \quad \forall~i \in \mathcal{N}.
\end{equation}
Using the above and then combining~\eqref{eqNE1pub_a} with~\eqref{eqallopub} and~\eqref{eqtaxpub_b} gives, $ \forall~i\in \mathcal{N} $,
\begin{subequations}
\begin{align} \label{eqNE2pub_a}
v_i^\prime(\widehat{x}_i(\overline{m})) &= \widehat{p}_i(\overline{m}_{-i}),
\\
\label{eqNE2pub_b}
\widehat{x}_i(\overline{m}) &= \frac{1}{N} \sum_{j \in \mathcal{N}} \overline{y}_j ,
\\
\label{eqNE2pub_c}
\widehat{p}_i(\overline{m}_{-i}) &= \delta(N-1) \left( \overline{y}_i - \frac{1}{N-1} \sum_{j \ne i} \overline{y}_j \right).
\end{align}
\end{subequations}
\eqref{eqNE2pub_b} implies $ \widehat{x}_i(\overline{m}) = \widehat{x}_r(\overline{m}) $ for any $ i,r \in \mathcal{N} $ and~\eqref{eqNE2pub_c} gives $ \sum_{r \in \mathcal{N}} \widehat{p}_i(\overline{m}_{-i}) = 0 $. Thus, the allocation-price pair
\begin{equation}
\left(\frac{1}{N} \sum_{j \in \mathcal{N}} \overline{y}_j \, , \, \left( \delta(N-1) \left(\overline{y}_i - \frac{1}{N-1} \sum_{j \ne i} \overline{y}_j \right)\right)_{i \in \mathcal{N}} \right)
\end{equation}
satisfy the optimality conditions,~\eqref{eqKKTpub}, as $ (x^*,{\mu^1}^*) $. Since the optimality conditions are sufficient, the allocation at any Nash equilibrium $ \overline{m} $ is the efficient allocation $ x^* $.
For existence and uniqueness, consider the following set of linear equations that must be satisfied at any Nash equilibrium $ \overline{m} $,
\begin{subequations}
\begin{align}
x^* &= \frac{1}{N} \sum_{j \in \mathcal{N}} \overline{y}_j,
\\
{\mu^1_i}^* &= \delta(N-1) \left( \overline{y}_i - \frac{1}{N-1} \sum_{j \ne i} \overline{y}_j\right), \quad \forall~i \in \mathcal{N}.
\end{align}
\end{subequations}
Here $ \left(\overline{y}_j \right)_{j\in\mathcal{N}} $ are the variables and $ (x^*,{\mu^1}^*) $ are fixed - since they are uniquely defined by the optimization,~\eqref{eqcppub}. The above equations can be inverted to give the unique solution as,
\begin{equation}
\overline{y}_i = x^* + \frac{{\mu_i^1}^*}{\delta N}, \quad \forall~i\in\mathcal{N}.
\end{equation}
Furthermore, using above and~\eqref{eqlempub}, the values for $ \big( \overline{q}_i^r \big)_{i,r\in\mathcal{N}} $ can also be calculated uniquely. Since a solution for $ \overline{m} = (\overline{y},\overline{q}) $ in terms of $ x^*,{\mu^1}^* $ exists, existence of Nash equilibrium is guaranteed. Also, since this solution is unique, there is a unique Nash equilibrium.
For Budget Balance, we have the following. By the characterization from above we know that at Nash Equilibrium $ \widetilde{m} $, all tax terms from~\eqref{eqtaxpub_a}, other than $ \widehat{p}_i(\widetilde{m}_{-i}) \widehat{x}_i(\widetilde{m}) $, are zero. Furthermore, the prices are equal to $ {\mu_i^1}^* $ and each allocation is equal to $ x^* $. Thus,
\begin{equation}
\sum_{i \in \mathcal{N}} \widehat{t}_i(\widetilde{m}) = \sum_{i \in \mathcal{N}} {\mu_i^1}^* x^* = x^* \sum_{i \in \mathcal{N}} {\mu_i^1}^* = x^* \cdot 0 = 0,
\end{equation}
since the optimal dual variables $ \big({\mu_i^1}^*\big)_{i\in\mathcal{N}} $ satisfy,~\eqref{eqKKTpub_b}, $ \sum_{i \in \mathcal{N}} {\mu_i^1}^* = 0 $.
\end{proof}
\section{Proof of Proposition~\ref{thmcontracpr} (Contraction - Private goods)}
\label{proofcontracpr}
\begin{proof}
The game is contractive if the matrix norm of the Jacobian of best-response $ \beta = \big( \beta_i \big)_{i \in \mathcal{N}} = \big( \tilde{y}_i,\tilde{q}_i \big)_{i \in \mathcal{N}} $ is smaller than unity, i.e., $ \Vert \nabla \beta \Vert < 1 $. We use the row-sum norm for this, and in this proof verify specifically the following set of conditions,
\begin{subequations}
\begin{alignat}{2}
\label{eqBRypr}
\sum_{r \in \mathcal{N},\,r \ne i} \left( \left\vert \frac{\partial \tilde{y}_i}{\partial y_r} \right\vert + \sum_{j\in \mathcal{N}} \left\vert \frac{\partial \tilde{y}_i}{\partial q_r^j} \right\vert \right) &< 1, \quad &&\forall~i \in \mathcal{N},
\\
\label{eqBRqpr}
\sum_{r \in \mathcal{N},\,r \ne i} \left( \left\vert \frac{\partial \tilde{q}_i^w}{\partial y_r} \right\vert + \sum_{j\in \mathcal{N}} \left\vert \frac{\partial \tilde{q}_i^w}{\partial q_r^j} \right\vert \right) &< 1, \quad &&\forall~w \in \mathcal{N},~\forall~i \in \mathcal{N}.
\end{alignat}
\end{subequations}
The summation can be performed simply over the indexes $ r \in \mathcal{N}(i) $ instead of $ r \ne i $ because our defined mechanism is distributed and hence the best-response of agent $ i $ depends only on $ \big( m_j \big)_{j \in \mathcal{N}(i)} $.
Consider any agent $ i \in \mathcal{N} $, for the best-response $ \tilde{q}_i $ we have
\begin{equation}
\tilde{q}_i^w =
\left\{
\begin{array}{ll}
\xi \tilde{y}_i & \mbox{for } w = i, \\
\xi y_w & \mbox{for } w \in \mathcal{N}(i), \\
\xi q_{n(i,w)}^w & \mbox{for } w \notin \mathcal{N}(i) \text{ and } w \ne i.
\end{array}
\right.
\end{equation}
Thus, by choosing $ \xi \in (0,1) $, all conditions within~\eqref{eqBRqpr} are satisfied where $ w \ne i $. Next, we verify conditions in~\eqref{eqBRypr}. Once this is done, then in conjunction with $ \xi \in (0,1) $, the conditions from~\eqref{eqBRqpr} with $ w=i $ are also automatically verified.
For the best-response $ \tilde{y}_i $, we have
\begin{align} \label{eqBRy2pr}
\tilde{y}_i = \frac{1}{N-1} \sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i} } \frac{q_{n(i,r)}^r}{\xi^{d(i,r)-1}} + \frac{1}{N-1} \sum_{r \in \mathcal{N}(i)} \frac{q_{r}^r}{\xi}
+ \left( v_i^\prime \right)^{-1} \big( \widehat{p}_i(m_{-i}) \big).
\end{align}
where $ \widehat{p}_i(m_{-i}) $ is defined in~\eqref{eqtaxpr_b}. Thus, we have,
\begin{align}\label{eqBRyderpr}
\frac{\partial \tilde{y}_i}{\partial q_{n(i,r)}^r} =
\left\{
\begin{array}{ll}
\dfrac{1}{\delta \xi} \dfrac{1}{v_i^{\prime\prime}(\cdot)} & \mbox{for } r=i,\\[2ex]
\dfrac{1}{N-1} \dfrac{1}{\xi} + \dfrac{1}{\delta \xi} \dfrac{1}{v_i^{\prime\prime}(\cdot)} & \mbox{for } r \in \mathcal{N}(i),\\[2ex]
\dfrac{1}{N-1} \dfrac{1}{\xi^{d(i,r)-1}} + \dfrac{1}{\delta \xi^{d(i,r)-1}} \dfrac{1}{v_i^{\prime\prime}(\cdot)} & \mbox{for } r \notin \mathcal{N}(i),~ r \ne i.
\end{array}
\right.
\end{align}
where in each expression above $ v_i^{\prime\prime}(\cdot) $ is evaluated at $ \widehat{p}_i(m_{-i}) $. Also, in the notation used above, for any $ r \in \mathcal{N}(i) $, $ n(i,r) = r $. All other partial derivative of $ \tilde{y}_i $ are zero. With all this condition in~\eqref{eqBRypr} becomes,
\begin{equation}\label{eqBRy3pr}
\left\vert \dfrac{1}{\delta \xi} \dfrac{1}{v_i^{\prime\prime}(\cdot)} \right\vert
+ \left\vert 1 + \frac{N-1}{\delta v_i^{\prime\prime}(\cdot)} \right\vert \left( \sum_{r \in \mathcal{N}(i)} \frac{1}{(N-1)\xi} \right)
+ \left\vert 1 + \frac{N-1}{\delta v_i^{\prime\prime}(\cdot)} \right\vert \Bigg( \sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i} } \frac{1}{(N-1)\xi^{d(i,r)-1}} \Bigg) < 1.
\end{equation}
To simplify the above, we utilize the upper bound from~\eqref{eqetapr}, $ v_i^{\prime\prime}(\cdot) \in (-\eta,-\frac{1}{\eta}) $. Set
\begin{equation} \label{eqetaC1pr}
\eta < \frac{\delta}{N-1},
\end{equation}
so that the expressions inside absolute value operator for the second and third terms on the LHS in~\eqref{eqBRy3pr} are guaranteed to be positive. With this,~\eqref{eqBRy3pr} becomes
\begin{equation} \label{eqBRy4pr}
\left[ \frac{-1}{v_i^{\prime\prime}(\cdot)} \right] \left( \frac{1}{\xi} - \sum_{r \in \mathcal{N}(i)} \frac{1}{\xi} - \sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i}} \frac{1}{\xi^{d(i,r)-1}} \right) < \frac{\delta}{N-1} \left( N-1 - \left[ \sum_{r \in \mathcal{N}(i)} \frac{1}{\xi} + \sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i}} \frac{1}{\xi^{d(i,r)-1}}\right] \right)
\end{equation}
Since any agent has at least one neighbor i.e., $ \vert \mathcal{N}(i) \vert \ge 1 $, the LHS above is negative. For the RHS, note that the expression inside the square brackets has exactly $ N-1 $ terms and each term is of the form $ \xi^{-k} $, for some $ k \ge 1 $. Since $ \xi < 1 $, this gives that even the RHS is negative. Utilizing the lower bound from~\eqref{eqetapr}, a sufficient condition to verify~\eqref{eqBRy4pr} is $ \eta < \frac{N-1}{\delta} \left\vert \frac{C_i}{D_i} \right\vert $, where $ C_i,D_i $ are the expression inside the curved bracket on the LHS and RHS of~\eqref{eqBRy4pr}, respectively.
Combining this with the condition in~\eqref{eqetaC1pr}, a sufficient condition for verifying~\eqref{eqBRypr} is
\begin{equation}
\eta < \min\left(\frac{\delta}{N-1} \, , \, \frac{N-1}{\delta}\left\vert \frac{C_i}{D_i} \right\vert\right), \quad \forall~i \in \mathcal{N}.
\end{equation}
Without any further tuning of parameters $ \xi,\delta $, the proof is complete as long as $ \eta $ satisfies above. However, in our model we would like to accommodate any value of $ \eta > 1 $ and this requires tuning of parameters $ \xi,\delta $. Set
\begin{equation}
\delta = (N-1)\sqrt{\underset{i \in \mathcal{N}}{\min} \left\vert \dfrac{C_i}{D_i} \right\vert} > 0,
\end{equation}
in this to get the sufficient condition as $ \eta^2 < \underset{i \in \mathcal{N}}{\min} \left\vert \dfrac{C_i}{D_i} \right\vert $, i.e.,
\begin{equation}
\eta^2 < \underset{i \in \mathcal{N}}{\min} \left\{ \left. \left \vert {\dfrac{1}{\xi} - \displaystyle\sum_{r \in \mathcal{N}(i)} \dfrac{1}{\xi} - \displaystyle\sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i}} \dfrac{1}{\xi^{d(i,r)-1}} } \right \vert
\middle/
\left \vert {N-1 - \left[ \displaystyle\sum_{r \in \mathcal{N}(i)} \dfrac{1}{\xi} + \displaystyle\sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i}} \dfrac{1}{\xi^{d(i,r)-1}}\right]} \right \vert \right. \right\}.
\end{equation}
We want to select $ \xi \in (0,1) $ such that the RHS above can be made arbitrarily large. For this, first note that, for any $ i \in \mathcal{N} $ the numerator of the RHS is bounded away from zero for $ \xi $ in the neighborhood of $ 1 $. Second, the denominator can be made arbitrarily close to $ 0 $ by choosing $ \xi $ close enough to $ 1 $. This can be seen by rewriting
\begin{equation}
D_i = N-1 - \left[ \sum_{r \in \mathcal{N}(i)} \frac{1}{\xi} + \sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i}} \frac{1}{\xi^{d(i,r)-1}}\right] = \sum_{r \in \mathcal{N}(i)} \left( 1 - \frac{1}{\xi} \right) + \sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i}} \left( 1 - \frac{1}{\xi^{d(i,r)-1}} \right),
\end{equation}
where for any given $ k \ge 1,\epsilon > 0 $, choose $ \xi \in \left( \left(\frac{1}{1+\epsilon}\right)^{\nicefrac{1}{k}},1 \right) $ to have $ \vert 1 - \xi^{-k} \vert < \epsilon $. Finally, it is clear from above that the denominator $ D_i $ can be made arbitrarily small concurrently for all $ i \in \mathcal{N} $.
That the game is not supermodular follows from the first sub-equation of~\eqref{eqBRyderpr}, which implies that the best-response $ \tilde{y}_i $ is decreasing w.r.t. $ q_{n(i,i)}^i $. Also, convergence of every learning dynamic within the ABR class is guaranteed by Fact~\ref{factabr}.
\end{proof}
\section{Proof of Proposition~\ref{thmcontracpub} (Contraction - Public goods)}
\label{proofcontracpub}
\begin{proof}
The game is contractive if the matrix norm of the Jacobian of best-response $ \beta = \big( \beta_i \big)_{i \in \mathcal{N}} = \big( \tilde{y}_i,\tilde{q}_i \big)_{i \in \mathcal{N}} $ is smaller than unity, i.e., $ \Vert \nabla \beta \Vert < 1 $. We use the row-sum norm for this, and in this proof verify specifically the following set of conditions,
\begin{subequations}
\begin{alignat}{2}
\label{eqBRypub}
\sum_{r \in \mathcal{N},\, r \ne i} \left( \left\vert \frac{\partial \tilde{y}_i}{\partial y_r} \right\vert + \sum_{j\in \mathcal{N}} \left\vert \frac{\partial \tilde{y}_i}{\partial q_r^j} \right\vert \right) &< 1, \quad &&\forall~i \in \mathcal{N},
\\
\label{eqBRqpub}
\sum_{r \in \mathcal{N},\, r \ne i} \left( \left\vert \frac{\partial \tilde{q}_i^w}{\partial y_r} \right\vert + \sum_{j\in \mathcal{N}} \left\vert \frac{\partial \tilde{q}_i^w}{\partial q_r^j} \right\vert \right) &< 1, \quad &&\forall~w \in \mathcal{N},~\forall~i \in \mathcal{N}.
\end{alignat}
\end{subequations}
The summation can be performed simply over the indexes $ r \in \mathcal{N}(i) $ instead of $ r \ne i $ because our defined mechanism is distributed and hence the best-response of agent $ i $ depends only on $ \big( m_j \big)_{j \in \mathcal{N}(i)} $.
Consider any agent $ i \in \mathcal{N} $, for the best-response $ \tilde{q}_i $ we have
\begin{equation}
\tilde{q}_i^w =
\left\{
\begin{array}{ll}
\xi \tilde{y}_i & \mbox{for } w = i, \\
\xi y_w & \mbox{for } w \in \mathcal{N}(i), \\
\xi q_{n(i,w)}^w & \mbox{for } w \notin \mathcal{N}(i) \text{ and } w \ne i.
\end{array}
\right.
\end{equation}
Thus, by choosing $ \xi \in (0,1) $, all conditions within~\eqref{eqBRqpub} are satisfied where $ w \ne i $. Next, we verify conditions in~\eqref{eqBRypub}. Once this is done, then in conjunction with $ \xi \in (0,1) $, the conditions from~\eqref{eqBRqpub} with $ w=i $ are also automatically verified.
For the best-response $ \tilde{y}_i $, we have the following relation
\begin{subequations}
\begin{align}
\frac{1}{N} \big( v_i^\prime(\widehat{x}_i(m)) - \widehat{p}_i(m_{-i}) \big) + 2\xi(\tilde{q}_i^i - \xi \tilde{y}_i) + \delta\xi(q_{n(i,i)}^i - \xi \tilde{y}_i) &= 0,
\\
\Rightarrow~~ \frac{1}{N} \big( v_i^\prime(\widehat{x}_i(m)) - \widehat{p}_i(m_{-i}) \big) + \delta\xi(q_{n(i,i)}^i - \xi \tilde{y}_i) &= 0.
\end{align}
\end{subequations}
In the above relation, $ \widehat{x}_i(m) $ is evaluated at $ \tilde{y}_i $ instead of $ y_i $. Also, this relation implicitly defines $ \tilde{y}_i $. Differentiating this equation w.r.t. $ \big( q_{n(i,r)}^r \big)_{r \in \mathcal{N}} $ gives
\begin{subequations}
\begin{alignat}{2}
\dfrac{v_i^{\prime\prime}(\widehat{x}_i(m))}{N^2} \dfrac{\partial \tilde{y}_i}{\partial q_{n(i,i)}^r} - \dfrac{\delta(N-1)}{N\xi} + \delta\xi\left( 1 - \xi \dfrac{\partial \tilde{y}_i}{\partial q_{n(i,i)}^r} \right) &= 0, \quad ~ &&r=i,
\\
\dfrac{v_i^{\prime\prime}(\widehat{x}_i(m))}{N^2} \left(\dfrac{\partial \tilde{y}_i}{\partial q_{n(i,r)}^r} + \frac{1}{\xi} \right) + \dfrac{\delta}{N\xi} - \delta\xi^2\dfrac{\partial \tilde{y}_i}{\partial q_{n(i,r)}^r} &= 0, \quad \forall~&&r \in \mathcal{N}(i),
\\
\dfrac{v_i^{\prime\prime}(\widehat{x}_i(m))}{N^2} \left(\dfrac{\partial \tilde{y}_i}{\partial q_{n(i,r)}^r} + \frac{1}{\xi^{d(i,r)-1}} \right) + \dfrac{\delta}{N\xi^{d(i,r)-1}} - \delta\xi^2\dfrac{\partial \tilde{y}_i}{\partial q_{n(i,r)}^r} &= 0, \quad \forall~&&r \notin \mathcal{N}(i),~ r \ne i,
\end{alignat}
\end{subequations}
which implies
\begin{align} \label{eqBRy2pub}
\frac{\partial \tilde{y}_i}{\partial q_{n(i,r)}^r} = \frac{1}{\dfrac{v_i^{\prime\prime}(\widehat{x}_i(m))}{N^2} - \delta\xi^2} \times
\left\{
\begin{array}{ll}
\dfrac{\delta(N-1)}{N\xi} - \delta\xi & \mbox{for } r=i, \\[2ex]
-\dfrac{\delta}{N\xi} - \dfrac{v_i^{\prime\prime}(\widehat{x}_i(m))}{N^2 \xi} & \mbox{for } r \in \mathcal{N}(i),\\[2ex]
-\dfrac{\delta}{N\xi^{d(i,r)-1}} - \dfrac{v_i^{\prime\prime}(\widehat{x}_i(m))}{N^2 \xi^{d(i,r)-1}} & \mbox{for } r \notin \mathcal{N}(i),~ r \ne i.
\end{array}
\right.
\end{align}
In the notation used above, for any $ r \in \mathcal{N}(i) $, $ n(i,r) = r $. All other partial derivative of $ \tilde{y}_i $ are zero. With all this condition in~\eqref{eqBRypub} becomes,
\begin{gather}
\nonumber
\left\vert \frac{\delta(N-1)}{N\xi} - \delta\xi \right\vert
+ \left\vert -\dfrac{\delta}{N} - \dfrac{v_i^{\prime\prime}(\widehat{x}_i(m))}{N^2} \right\vert \left( \sum_{r \in \mathcal{N}(i)} \frac{1}{\xi} \right)
+ \left\vert -\dfrac{\delta}{N} - \dfrac{v_i^{\prime\prime}(\widehat{x}_i(m))}{N^2} \right\vert \Bigg( \sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i} } \frac{1}{\xi^{d(i,r)-1}} \Bigg)
\\ \label{eqBRy3pub}
< \delta\xi^2 - \frac{v_i^{\prime\prime}(\widehat{x}_i(m))}{N^2}.
\end{gather}
We impose the condition
\begin{equation}\label{eqxipub}
\xi \in \left( \sqrt{\frac{N-1}{N}} , 1 \right)
\end{equation}
so that the expression inside the first absolute value term in above is negative. To simplify the other expressions containing absolute value, we utilize the lower bound from~\eqref{eqetapr}, $ v_i^{\prime\prime}(\cdot) \in (-\eta,-\frac{1}{\eta}) $. Set
\begin{equation} \label{eqetaC1pub}
\eta < N\delta,
\end{equation}
so that the remaining expressions inside absolute value in~\eqref{eqBRy3pub} are guaranteed to be negative. With this,~\eqref{eqBRy3pub} becomes
\begin{multline} \label{eqBRy4pub}
\Big[ - v_i^{\prime\prime}(\widehat{x}_i(m))\Big] \left( 1 + \sum_{r \in \mathcal{N}(i)} \frac{1}{\xi} + \sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i}} \frac{1}{\xi^{d(i,r)-1}} \right)
\\
>
N\delta \left( \frac{1}{\xi} \left[ \sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i}} \frac{1}{\xi^{d(i,r)-2}} - (N - \vert \mathcal{N}(i) \vert - 1)\right] + N \xi (1-\xi) \right),
\end{multline}
where $ N - \vert \mathcal{N}(i) \vert - 1 $ is the number of agents in the system except agent $ i $ and all his/her neighbors in $ \mathcal{N}(i) $. Clearly the LHS above is positive. For any $ r \in \mathcal{N}(i) $ and $ r \ne i $, we have $ d(i,r) \ge 2 $. On the RHS, inside the square brackets there are exactly $ N - \vert \mathcal{N}(i) \vert -1 $ terms in the summation and each term is of the form $ \xi^{-k} $, for some $ k \ge 0 $. Since $ \xi < 1 $, this gives that even the RHS is positive. Utilizing the upper bound from~\eqref{eqetapr}, a sufficient condition to verify~\eqref{eqBRy4pub} is $ \eta < \dfrac{1}{N\delta} \dfrac{C_i}{D_i} $, where $ C_i,D_i $ are the expression inside the curved bracket on the LHS and RHS of~\eqref{eqBRy4pub}, respectively. Combining this with the condition in~\eqref{eqetaC1pub}, a sufficient condition for verifying~\eqref{eqBRypub} is
\begin{equation}
\eta < \min\left(N\delta \, , \, \frac{1}{N\delta} \frac{C_i}{D_i} \right), \quad \forall~i\in \mathcal{N}.
\end{equation}
Without any further tuning of parameters $ \xi,\delta $, the proof is complete as long as $ \eta $ satisfies above. However, in our model we would like to accommodate any value of $ \eta > 1 $ and this requires tuning of parameters $ \xi,\delta $. Set
\begin{equation}
\delta = \frac{1}{N}\sqrt{\underset{i\in\mathcal{N}}{\min}\left(\frac{C_i}{D_i}\right) } > 0,
\end{equation}
in this to get the sufficient condition as $ \eta^2 < \underset{i\in\mathcal{N}}{\min} \left(\dfrac{C_i}{D_i}\right) $, i.e.,
\begin{multline}
\eta^2 < \underset{i\in\mathcal{N}}{\min} \Bigg\{
\left( 1 + \sum_{r \in \mathcal{N}(i)} \frac{1}{\xi} + \sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i}} \frac{1}{\xi^{d(i,r)-1}} \right)
\\
\left.
\middle/
\left( \frac{1}{\xi} \left[ \sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i}} \frac{1}{\xi^{d(i,r)-2}} - (N - \vert \mathcal{N}(i) \vert - 1)\right] + N \xi (1-\xi) \right)
\right. \Bigg \}.
\end{multline}
We want to select $ \xi $ such that the RHS above can be made arbitrarily large, whilst satisfying~\eqref{eqxipub}. For this, firstly note that, for any $ i \in \mathcal{N} $ the numerator of the RHS is greater than $ 1 $, hence it is bounded away from zero. Secondly, the denominator can be made arbitrarily close to $ 0 $ by choosing $ \xi $ close enough to $ 1 $. This can be seen by rewriting
\begin{subequations}
\begin{gather}
\sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i}} \frac{1}{\xi^{d(i,r)-2}} - (N - \vert \mathcal{N}(i) \vert - 1)
=
\sum_{\substack{r \notin \mathcal{N}(i) \\ r \ne i}} \left( \frac{1}{\xi^{d(i,r)-2}} - 1 \right)
=
\sum_{\substack{r \in \mathcal{N} \\ d(i,r) \ge 3}} \left( \frac{1}{\xi^{d(i,r)-2}} - 1 \right),
\\
\Rightarrow~~D_i = \frac{1}{\xi} \sum_{\substack{r \in \mathcal{N} \\ d(i,r) \ge 3}} \left( \frac{1}{\xi^{d(i,r)-2}} - 1 \right) + N\xi(1-\xi),
\end{gather}
\end{subequations}
where for any given $ k \ge 1,\epsilon > 0 $, choose $ \xi \in \left( \left(\frac{1}{1+\epsilon}\right)^{\nicefrac{1}{k}},1 \right) $ to have $ \left( \xi^{-k} - 1 \right) < \epsilon $. Note that this is consistent with~\eqref{eqxipub}. The remaining term $ N\xi(1-\xi) $ can also made made arbitrarily small by choosing $ \xi $ close enough to $ 1 $. Finally, it is clear from above that the denominator $ D_i $ can made arbitrarily small concurrently for all $ i \in \mathcal{N} $.
The fact that the game is supermodular follows from the preceding analysis, where the parameters are chosen such that each expression in~\eqref{eqBRy2pub} is positive. Which implies that the best-response is increasing in every message of other agents.
\end{proof}
\end{document} | math |
வேலைபறிபோக காரணமாக இருந்த முதியவரை கொலை செய்த பயங்கரம் தாம்பரத்தில் அதிர்ச்சி.! சென்னையில் உள்ள தாம்பரம் இரும்புலியூர் ஏரிக்கரை பகுதியில், அறுபது வயது மதிக்கத்தக்க நபர் கொலை செய்யப்பட்டு பிணமாக இருப்பதாக காவல்துறையினருக்கு தகவல் கிடைத்துள்ளது. இந்த தகவலை அறிந்த இரும்புலியூர் காவல் துறையினர், அவரின் உடலை மீட்டு பிரேத பரிசோதனைக்காக குரோம்பேட்டை அரசு மருத்துவமனைக்கு அனுப்பி வைத்தனர். இதனையடுத்து, காவல்துறையினர் மேற்கொண்ட விசாரணையில், உயிரிழந்தது தூத்துக்குடி மாவட்டத்தைச் சேர்ந்த ஜான் ராஜாசிங் வயது 67 என்பதும், தாம்பரத்தில் உள்ள படப்பை பகுதியில் பழைய இரும்பு கடையில் பணியாற்றி வருவதும் தெரியவந்துள்ளது. சம்பவத்தன்று, ஜான் ராஜாசிங்குடன் பணியாற்றி வந்த கோவில் ராஜா வயது 27 என்பவரை ஊருக்கு அனுப்புவதற்காக, கடந்த 31ஆம் தேதி பெருங்களத்தூர் பேருந்து நிலையத்திற்கு அழைத்துச் சென்றதாக தகவல் தெரியவந்துள்ளது. இதனையடுத்து, கோவில் ராஜாவை கைது செய்த காவல் துறையினர் விசாரணை செய்தனர். இதன்போது, ஜான் ராஜாசிங் கொலை செய்ததற்கான காரணம் தெரியவந்துள்ளது. இது குறித்த வாக்குமூலத்தில், நாங்கள் இருவரும் ஒரே இடத்தில் பணியாற்றி வந்தோம். இதன்போது, கடை உரிமையாளரிடம் ஜான் ராஜாசிங் என்னை பற்றி தவறாக கூறியதால், வேலை பறி போனது. வேலையை இழந்ததால் வருமானமின்றி செலவு செய்ய பணம் இல்லாமல் மன உளைச்சலில் தவித்தேன். இதன்பின்னர், வேலை பறிபோக காரணமாக இருந்த ஜான் ராஜாவை பழி வாங்க நினைத்து, அவரிடம் ஊருக்குப்போவதாக கூறி அழைத்து, மது வாங்கி கொடுத்து போதை ஏறியதும் வெட்டிக் கொலை செய்தேன் என்று தெரிவித்துள்ளார். இதுகுறித்து வழக்குப்பதிவு செய்த காவல்துறையினர், நீதிமன்றத்தில் ஆஜர்படுத்தி சிறையில் அடைத்தனர். | tamil |
త్వరలో బాలయ్య టాక్ షో.. దబిడి దిబిడే.. Balakrishna Will Appear As A Host For A Talk Show In Aha OTT కరోనా ఎఫెక్ట్ వల్ల థియేటర్లు మూతపడటం, మార్కెట్లో పోటాపోటీగా ఓటీటీ ప్లాట్ ఫామ్ లు రావడంతో ప్రేక్షకులకు కావాల్సినంత వినోదం దొరుకుతోంది. ఎంటర్ టైన్ మెంటే టార్గెట్ గా తెలుగులో ఎంట్రీ ఇచ్చిన ఆహా ప్లాట్ ఫామ్ ఎననో వెబ్ సిరీస్ లు, సినిమాలు, స్పెషల్ షోల ద్వారా ఎంతోమంది తెలుగు ప్రేక్షకులను, సబ్ స్కైబర్లను సంపాదించుకుంది. రానా, సమంతాలతో పలు షోలు చేసిన ఆహా.. త్వరలో నందమూరి బాలకృష్ణతో ఓ షో ప్లాన్ చేసింది. బాలకృష్ణ హోస్ట్ గా వ్యవహరించనున్న ఈ షో గురించి ఇప్పటికే టాలీవుడ్ లో తెగ ప్రచారం జరుగుతోంది. అయితే.. ఈ షో గురించి లేటెస్ట్ గా ఓ అప్ డేట్ వచ్చింది. బాలయ్య హోస్ట్ గా చేయనున్న ఈ షో కి అన్ స్టాపబుల్ unstoppable అనే టైటిల్ ను ఫిక్స్ చేశారని టాలీవుడ్ వర్గాలు గుసగుసలాడుకుంటున్నాయి. బాలయ్య హోస్ట్ గా షో చేయడమంటే మామూలుగా ఉండదు.. దబిడి దిబిడే. అలాంటిది ఆయన షో పేరు కూడా అదిరిపోయేలా ఉంటుంది అంటూ నందమూరి అభిమానులు అంచనా వేస్తున్నారు. త్వరలోనే ఆహా ఈ షోకి సంబంధించిన ప్రోమోను అధికారికంగా విడుదల చేయనున్నదట. ఈ టాక్ షో కు ఇండస్ట్రీకి సంబంధించిన పలువురు ప్రముఖులు రానున్నారట. తొలి గెస్ట్ గా మంచు ఫ్యామిలీ నుంచి ఒకరు వస్తారని టాక్. అయితే ఎవరు తొలి గెస్ట్.. షో పేరేంటి తెలియాలంటే ఆహా ప్రకటన వచ్చే వరకు ఎదురు చూడక తప్పదు. ప్రస్తుతం బోయపాటి దర్శకత్వంలో తెరకెక్కుతున్న అఖండ సినిమా పనుల్లో బిజీగా ఉన్న బాలయ్య.. త్వరలోనే ఆ సినిమా విడుదల తేదీని అనౌన్స్ చేయనున్నారు. ఆ లోపే బాలయ్య బాబు బుల్లితెర మీద కనిపించే షో పేరు కూడా ప్రకటించేస్తారేమో అంటున్నారు పలువురు. అయితే.. దసరా సందర్భంగా బాలయ్య షో పేరు ప్రకటించే ఛాన్స్ ఉంది అంటున్నారు బాలయ్య అభిమానులు.The post త్వరలో బాలయ్య టాక్ షో.. దబిడి దిబిడే.. first appeared on TNews Telugu. | telegu |
ನನ್ನ ಬಗ್ಗೆ ಇದ್ದ ರೂಮರ್ ಗಳೆಲ್ಲಾ ನಿಜ ಎಂದು ಫೋಟೋ ಹಂಚಿಕೊಂಡ ರಶ್ಮಿಕಾ ಮಂದಣ್ಣ ಮುಂಬೈ: ನಟಿ ರಶ್ಮಿಕಾ ಮಂದಣ್ಣ ಬಾಲಿವುಡ್ ನಲ್ಲಿ ಬ್ಯುಸಿಯಾಗಿದ್ದು, ಇತ್ತೀಚೆಗೆ ಟೈಗರ್ ಶ್ರಾಫ್ ಜೊತೆಗಿನ ಹೊಸ ಸಿನಿಮಾಗೆ ನಾಯಕಿಯಾಗಲಿದ್ದಾರೆ ಎಂಬ ಸುದ್ದಿ ಬಂದಿತ್ತು. ಆದರೆ ರಶ್ಮಿಕಾ ಟೈಗರ್ ಶ್ರಾಫ್ ಜೊತೆ ತೆರೆ ಹಂಚಿಕೊಳ್ಳುವುದು ನಿಜ. ಆದರೆ ಸಿನಿಮಾಗಾಗಿ ಅಲ್ಲ. ಈ ಬಗ್ಗೆ ಸ್ವತಃ ರಶ್ಮಿಕಾ ಸ್ಪಷ್ಟನೆ ಕೊಟ್ಟಿದ್ದಾರೆ. ಟೈಗರ್ ಶ್ರಾಫ್ ಜೊತೆ ರಶ್ಮಿಕಾ ಜಾಹೀರಾತೊಂದರಲ್ಲಿ ಒಟ್ಟಾಗಿ ನಟಿಸುತ್ತಿದ್ದಾರೆ. ಈ ಫೋಟೋಗಳನ್ನು ಹಂಚಿಕೊಂಡಿರುವ ರಶ್ಮಿಕಾ, ನನ್ನ ಬಗ್ಗೆ ಬಂದ ರೂಮರ್ ಗಳೆಲ್ಲಾ ನಿಜ. ಆದರೆ ಸಿನಿಮಾ ಅಲ್ಲ, ಜಾಹೀರಾತಿನಲ್ಲಿ ನಾನು ಟೈಗರ್ ಶ್ರಾಫ್ ಜೊತೆ ನಟಿಸುತ್ತಿದ್ದೇನೆ ಎಂದು ಸ್ಪಷ್ಟನೆ ಕೊಟ್ಟಿದ್ದಾರೆ. | kannad |
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<h1>
infotheo
<small>
0.3.1
<span class="label label-success">16 m 40 s</span>
</small>
</h1>
<p><em><script>document.write(moment("2021-04-08 21:07:48 +0000", "YYYY-MM-DD HH:mm:ss Z").fromNow());</script> (2021-04-08 21:07:48 UTC)</em><p>
<h2>Context</h2>
<pre># Packages matching: installed
# Name # Installed # Synopsis
base-bigarray base
base-threads base
base-unix base
conf-findutils 1 Virtual package relying on findutils
conf-gmp 3 Virtual package relying on a GMP lib system installation
coq 8.13.0 Formal proof management system
num 1.4 The legacy Num library for arbitrary-precision integer and rational arithmetic
ocaml 4.10.1 The OCaml compiler (virtual package)
ocaml-base-compiler 4.10.1 Official release 4.10.1
ocaml-config 1 OCaml Switch Configuration
ocamlfind 1.9.1 A library manager for OCaml
zarith 1.12 Implements arithmetic and logical operations over arbitrary-precision integers
# opam file:
opam-version: "2.0"
maintainer: "Reynald Affeldt <[email protected]>"
homepage: "https://github.com/affeldt-aist/infotheo"
dev-repo: "git+https://github.com/affeldt-aist/infotheo.git"
bug-reports: "https://github.com/affeldt-aist/infotheo/issues"
license: "LGPL-2.1-or-later"
synopsis: "Discrete probabilities and information theory for Coq"
description: """
Infotheo is a Coq library for reasoning about discrete probabilities,
information theory, and linear error-correcting codes."""
build: [
[make "-j%{jobs}%" ]
[make "-C" "extraction" "tests"] {with-test}
]
install: [make "install"]
depends: [
"coq" { (>= "8.13" & < "8.14~") | (= "dev") }
"coq-mathcomp-ssreflect" { (>= "1.12.0" & < "1.13~") }
"coq-mathcomp-fingroup" { (>= "1.12.0" & < "1.13~") }
"coq-mathcomp-algebra" { (>= "1.12.0" & < "1.13~") }
"coq-mathcomp-solvable" { (>= "1.12.0" & < "1.13~") }
"coq-mathcomp-field" { (>= "1.12.0" & < "1.13~") }
"coq-mathcomp-analysis" { (>= "0.3.4") }
]
tags: [
"keyword:information theory"
"keyword:probability"
"keyword:error-correcting codes"
"keyword:convexity"
"logpath:infotheo"
"date:2021-02-28"
]
authors: [
"Reynald Affeldt, AIST"
"Manabu Hagiwara, Chiba U. (previously AIST)"
"Jonas Senizergues, ENS Cachan (internship at AIST)"
"Jacques Garrigue, Nagoya U."
"Kazuhiko Sakaguchi, Tsukuba U."
"Taku Asai, Nagoya U. (M2)"
"Takafumi Saikawa, Nagoya U."
"Naruomi Obata, Titech (M2)"
]
url {
http: "https://github.com/affeldt-aist/infotheo/archive/0.3.1.tar.gz"
checksum: "sha512=0267bdbd81cc3c48f03173f06cb5145bebcd48b26818b30222d79062d4d9a11bfb2830762c4d35edf67b2bc9637d7ef09bc087974ab13a002b4725bb3df63bd0"
}</pre>
<h2>Lint</h2>
<dl class="dl-horizontal">
<dt>Command</dt>
<dd><code>true</code></dd>
<dt>Return code</dt>
<dd>0</dd>
</dl>
<h2>Dry install</h2>
<p>Dry install with the current Coq version:</p>
<dl class="dl-horizontal">
<dt>Command</dt>
<dd><code>opam install -y --show-action coq-infotheo.0.3.1 coq.8.13.0</code></dd>
<dt>Return code</dt>
<dd>0</dd>
</dl>
<p>Dry install without Coq/switch base, to test if the problem was incompatibility with the current Coq/OCaml version:</p>
<dl class="dl-horizontal">
<dt>Command</dt>
<dd><code>true</code></dd>
<dt>Return code</dt>
<dd>0</dd>
</dl>
<h2>Install dependencies</h2>
<dl class="dl-horizontal">
<dt>Command</dt>
<dd><code>opam list; echo; ulimit -Sv 4000000; timeout 2h opam install -y --deps-only coq-infotheo.0.3.1 coq.8.13.0</code></dd>
<dt>Return code</dt>
<dd>0</dd>
<dt>Duration</dt>
<dd>35 m 28 s</dd>
</dl>
<h2>Install</h2>
<dl class="dl-horizontal">
<dt>Command</dt>
<dd><code>opam list; echo; ulimit -Sv 16000000; timeout 2h opam install -y -v coq-infotheo.0.3.1 coq.8.13.0</code></dd>
<dt>Return code</dt>
<dd>0</dd>
<dt>Duration</dt>
<dd>16 m 40 s</dd>
</dl>
<h2>Installation size</h2>
<p>Total: 37 M</p>
<ul>
<li>1 M <code>../ocaml-base-compiler.4.10.1/lib/coq/user-contrib/infotheo/ecc_modern/degree_profile.vo</code></li>
<li>1 M <code>../ocaml-base-compiler.4.10.1/lib/coq/user-contrib/infotheo/ecc_modern/degree_profile.glob</code></li>
<li>861 K <code>../ocaml-base-compiler.4.10.1/lib/coq/user-contrib/infotheo/probability/proba.glob</code></li>
<li>829 K <code>../ocaml-base-compiler.4.10.1/lib/coq/user-contrib/infotheo/ecc_modern/ldpc_algo_proof.vo</code></li>
<li>684 K <code>../ocaml-base-compiler.4.10.1/lib/coq/user-contrib/infotheo/information_theory/chap2.glob</code></li>
<li>635 K <code>../ocaml-base-compiler.4.10.1/lib/coq/user-contrib/infotheo/probability/proba.vo</code></li>
<li>623 K <code>../ocaml-base-compiler.4.10.1/lib/coq/user-contrib/infotheo/probability/convex.glob</code></li>
<li>620 K <code>../ocaml-base-compiler.4.10.1/lib/coq/user-contrib/infotheo/information_theory/jtypes.vo</code></li>
<li>545 K <code>../ocaml-base-compiler.4.10.1/lib/coq/user-contrib/infotheo/probability/convex.vo</code></li>
<li>518 K <code>../ocaml-base-compiler.4.10.1/lib/coq/user-contrib/infotheo/information_theory/channel_coding_direct.glob</code></li>
<li>506 K <code>../ocaml-base-compiler.4.10.1/lib/coq/user-contrib/infotheo/ecc_modern/ldpc_algo_proof.glob</code></li>
<li>491 K <code>../ocaml-base-compiler.4.10.1/lib/coq/user-contrib/infotheo/ecc_modern/subgraph_partition.vo</code></li>
<li>486 K <code>../ocaml-base-compiler.4.10.1/lib/coq/user-contrib/infotheo/ecc_modern/ldpc.glob</code></li>
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<li>4 K <code>../ocaml-base-compiler.4.10.1/lib/coq/user-contrib/infotheo/probability/jensen.v</code></li>
<li>4 K <code>../ocaml-base-compiler.4.10.1/lib/coq/user-contrib/infotheo/lib/poly_ext.v</code></li>
<li>3 K <code>../ocaml-base-compiler.4.10.1/lib/coq/user-contrib/infotheo/toy_examples/expected_value_variance.v</code></li>
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<li>2 K <code>../ocaml-base-compiler.4.10.1/lib/coq/user-contrib/infotheo/information_theory/source_code.v</code></li>
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<li>2 K <code>../ocaml-base-compiler.4.10.1/lib/coq/user-contrib/infotheo/probability/variation_dist.v</code></li>
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</ul>
<h2>Uninstall</h2>
<dl class="dl-horizontal">
<dt>Command</dt>
<dd><code>opam remove -y coq-infotheo.0.3.1</code></dd>
<dt>Return code</dt>
<dd>0</dd>
<dt>Missing removes</dt>
<dd>
none
</dd>
<dt>Wrong removes</dt>
<dd>
none
</dd>
</dl>
</div>
</div>
</div>
<hr/>
<div class="footer">
<p class="text-center">
<small>Sources are on <a href="https://github.com/coq-bench">GitHub</a>. © Guillaume Claret.</small>
</p>
</div>
</div>
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It's the ideal venue for product launches, press events, award ceremonies, away days, champagne receptions and relaxed summer barbecues.
Experience Herzog and de Meuron's inspiring spaces and chaotic beauty, created through collaboration with the visual artist Michael Craig-Martin. | english |
لاس اینجلس اردو پوائنٹ اخبارتازہ ترین ئی این پی 02 مئی2017ء گزشتہ سال باکس افس پر کامیابی کے جھنڈے گاڑنے والی اسکر ایوارڈ یافتہ فلم لالالینڈ کا سحر اب بھی برقرار ہےگزشتہ روز امریکی ریاست کیلیفورنیا کے شہر لاس اینجلس میں لالا لینڈ ڈے منایا گیا اس میں سینکڑوں فٹ بلند عمارت سے ڈانسرز نے رسیوں سے لٹک کر شاندار رقص کا مظاہرہ پیش کیااور دیکھنے والوں کو داد دینے پر مجبور کر دیالاس اینجلس کے میئر ایرک گارسیٹی نے تقریب کا اغاز کیا جبکہ اس موقعے پر فلم کے ڈائریکٹراور اسکرین رائٹرسمیت دیگر کریو نے بھی شرکت کی | urdu |
ಕೋವಿಡ್ ಪತ್ತೆ ಪರೀಕ್ಷೆ ಅರ್ಧದಷ್ಟು ಕಡಿತ! ಕಲಬುರಗಿ: ಜಿಲ್ಲೆಯಲ್ಲಿ ಕೆಲ ಒಂದು ವಾರದಿಂದ ಹೊಸ ಕೊರೊನಾ ಸೋಂಕಿನ ಪ್ರಕರಣಗಳಲ್ಲಿ ಭಾರೀ ಇಳಿಕೆ ಕಾಣುತ್ತಿದ್ದು, ಇದರ ನಡುವೆಯೇ ಸೋಂಕು ಪತ್ತೆ ಪರೀಕ್ಷೆ ಪ್ರಮಾಣವನ್ನೂ ಕಡಿತಗೊಳಿಸಲಾಗಿದೆ. ಈ ಹಿಂದೆ ಆರು ಸಾವಿರ ಪರೀಕ್ಷೆಗಳು ನಡೆಯುತ್ತಿದ್ದರೆ, ಈಗ ಅರ್ಧದಷ್ಟು ಕಡಿಮೆ ಮಾಡಲಾಗಿದೆ. ಅಂದರೆ ಮೂರು ಸಾವಿರ ಪರೀಕ್ಷೆಗಳು ಮಾತ್ರ ಮಾಡಲಾಗುತ್ತಿದೆ. ಕಳೆದ ವರ್ಷ ಕೊರೊನಾ ಮೊದಲ ಅಲೆಯಲ್ಲಿ ದೇಶದಲ್ಲೇ ಮೊದಲ ಸಾವು ಕಂಡಿದ್ದ ಜಿಲ್ಲೆಯು ತಲ್ಲಣಿಸಿ ಹೋಗಿತ್ತು. ಈ ವರ್ಷ ಮಾರ್ಚ್ನಲ್ಲಿ ಎರಡನೇ ಅಲೆಯಲ್ಲೂ ಜಿಲ್ಲೆ ನಲುಗಿ ಹೋಗಿದೆ. ಮೊದಲ ಅಲೆಗಿಂತ ಎರಡನೇ ಅಲೆ ಜಿಲ್ಲೆಗೆ ಅತ್ಯಂತ ಭೀಕರವಾಗಿ ಪರಿಣಮಿಸಿದೆ. ಕೆಲವೇ ಕೆಲ ದಿನಗಳಲ್ಲಿ ಹೊಸ ಪ್ರಕರಣಗಳು ಮತ್ತು ಸಾವುಗಳ ಸಂಖ್ಯೆ ದುಪ್ಪಟ್ಟು ಆಗಿದೆ. ಮೇ ತಿಂಗಳ ಆರಂಭದಲ್ಲೇ 100 ಜನರಿಗೆ ಕೊರೊನಾ ಪರೀಕ್ಷೆ ಮಾಡಿದರೆ ಸರಾಸರಿ 30 ಜನರಿಗೆ ಸೋಂಕು ದೃಢಪಟ್ಟಿತ್ತು.ಕೋವಿಡ್ ಕುರಿತ ಎಲ್ಲಾ ಲೇಟೆಸ್ಟ್ ಅಪ್ಡೇಟ್ಸ್ ಓದಿ ಪರೀಕ್ಷೆಪಾಸಿಟಿವ್ ಹೆಚ್ಚಿತ್ತೂ: ಜಿಲ್ಲೆಯಲ್ಲಿ ಜಿಮ್ಸ್ ಆಸ್ಪತ್ರೆ ಮತ್ತು ಇಎಸ್ಐ ಆಸ್ಪತ್ರೆ, ಕೇಂದ್ರೀಯ ವಿಶ್ವವಿದ್ಯಾಲಯದಲ್ಲಿ ಕೊರೊನಾ ಸೋಂಕು ಪತ್ತೆ ಪ್ರಯೋಗಾಲಯಗಳು ಇವೆ. ಅಲ್ಲದೇ, ಬಸವೇಶ್ವರ ಆಸ್ಪತ್ರೆ, ಕೆಬಿಎನ್ ಆಸ್ಪತ್ರೆ ಮತ್ತು ಯುನಿಟೈಡ್ ಆಸ್ಪತ್ರೆ ಸೇರಿ ಆರು ಖಾಸಗಿ ಪ್ರಯೋಗಾಲಯಗಳು ಇವೆ. ಕೇಂದ್ರೀಯ ವಿವಿಯ ಲ್ಯಾಬ್ನಲ್ಲಿ ಹೊರ ಜಿಲ್ಲೆಗಳ ಕೊರೊನಾ ಮಾದರಿ ಪರೀಕ್ಷೆಗಳನ್ನು ನಡೆಸಲಾಗುತ್ತಿದೆ. ಉಳಿದಂತೆ ಎಲ್ಲ ಪ್ರಯೋಗಾಲಯಗಳಲ್ಲಿ ಜಿಲ್ಲೆಯ ಮಾದರಿಗಳ ಪರೀಕ್ಷೆ ಮಾಡಬಹುದಾಗಿದೆ. ಮಾರ್ಚ್ನಲ್ಲಿ ಅಲೆ ಆರಂಭದ ದಿನದಿಂದಲೂ ನಿತ್ಯ ಆರು ಸಾವಿರ ಮಾದರಿಗಳನ್ನು ಸಂಗ್ರಹಿಸಿ ಪರೀಕ್ಷೆ ನಡೆಸಲಾಗುತ್ತಿತ್ತು. ಅಂತೆಯೇ ಆರಂಭದಲ್ಲಿ 500 ರಿಂದ ಕಳೆದ ಎರಡು ವಾರದವರೆಗೂ 1,500ಕ್ಕೂ ಹೆಚ್ಚು ಪಾಟಿಸಿವ್ ಪ್ರಕರಣಗಳು ಪತ್ತೆಯಾಗುತ್ತಿದ್ದವು. ಆದರೆ, ಈಗ ಕೊರೊನಾ ಪತ್ತೆ ಪರೀಕ್ಷೆಗಳನ್ನು ಸರಾಸರಿ ಮೂರು ಸಾವಿರಕ್ಕೆ ಇಳಿಸಲಾಗಿದೆ. ಕೆಲವೊಮ್ಮೆ ಕೇವಲ 1,700, 1,800 ಪರೀಕ್ಷೆಗಳನ್ನು ಮಾಡಲಾಗುತ್ತಿದೆ ಎಂದು ಹೇಳಲಾಗುತ್ತಿದೆ. ಕೊರೊನಾ ಪತ್ತೆ ಪರೀಕ್ಷೆಗಳ ಸಂಖ್ಯೆಯೇ ಕಡಿಮೆ ಆಗಿದ್ದರಿಂದ ಸಹಜವಾಗಿ ಹೊಸ ಪಾಸಿಟಿವ್ ಪ್ರಕರಣಗಳ ಸಂಖ್ಯೆಯಲ್ಲಿ ಕುಸಿಯುತ್ತಿದೆ. ಅಲ್ಲದೇ, ಏಕಾಏಕಿ ದಿನಕ್ಕೆ ನೂರಾರು ಜನ ಸೋಂಕಿತರನ್ನು ಗುಣಮುಖರಾಗಿದ್ದಾರೆ ಎಂದು ತೋರಿಸಲಾಗಿದೆ. ಹೀಗಾಗಿ ಮೇ ಮೊದಲು ಮತ್ತು ಎರಡನೇ ವಾರದಲ್ಲಿ ಜಿಲ್ಲೆಯಲ್ಲಿ ಶೇ.30 ರಷ್ಟಿದ್ದ ಪಾಸಿಟಿವಿಟಿ ಪ್ರಮಾಣ ಈಗ ಶೇ.10ಕ್ಕಿಂತ ಕಡಿಮೆ ಇದೆ. ಇದು ಕೊರೊನಾ ಸೋಂಕಿನ ಪ್ರಮಾಣ ಕಡಿಮೆ ಆಗುತ್ತಿದೆ ಎಂದು ಬಿಂಬಿಸುವುದಕ್ಕೆ ಪರೀಕ್ಷೆಗಳನ್ನೇ ಕಡಿತ ಮಾಡಲಾಗುತ್ತಿದೆಯೇ ಎನ್ನುವ ಅನುಮಾನಕ್ಕೂ ಕಾರಣವಾಗಿದೆ. ಅಧಿಕಾರಿಗಳು ಹೇಳುವುದೇನು?: ಈ ಮೊದಲು ಕೊರೊನಾ ಲಕ್ಷಣಗಳ ಹೊಂದಿದವರೊಂದಿಗೆ ಸಾಮೂಹಿಕವಾಗಿ ಪರೀಕ್ಷೆಗಳನ್ನು ನಡೆಸಲಾಗುತ್ತಿತ್ತು. ಸಾರ್ವಜನಿಕವಾಗಿ ಓಡಾಡುವವರು, ತರಕಾರಿ ಮಾರಾಟಗಾರರು, ಕಾಲೇಜುಗಳ ವಿದ್ಯಾರ್ಥಿಗಳು, ಹೊರ ಊರುಗಳಿಂದ ಬಂದವರ ಗಂಟಲು ಮತ್ತು ಮೂಗಿನ ದ್ರಾವಣ ಮಾದರಿ ಸಂಗ್ರಹಿಸಿ ಪರೀಕ್ಷೆ ನಡೆಸುತ್ತಿದ್ದೆವು. ದಿನವೂ ಸರಾಸರಿ ಐದಾರು ಸಾವಿರ ಮಾದರಿ ಸಂಗ್ರಹವಾಗುತ್ತಿದ್ದವು. ಈಗ ಲಾಕ್ಡೌನ್ ಕಾರಣ ನಿರ್ದಿಷ್ಟ ಗುರಿ ಮೇಲೆ ಮೂರು ಸಾವಿರದಷ್ಟು ಪರೀಕ್ಷೆಗಳನ್ನು ನಡೆಸಲಾಗುತ್ತಿದೆ ಎನ್ನುತ್ತಾರೆ ಜಿಲ್ಲಾ ಆರೋಗ್ಯ ಮತ್ತು ಕುಟುಂಬ ಕಲ್ಯಾಣಾಧಿಕಾರಿ ಡಾ.ಶರಣಬಸಪ್ಪ ಗಣಜಲಖೇಡ್. ಈಗ ಎಲ್ಲರಿಗೂ ಕೊರೊನಾ ಪತ್ತೆ ಪರೀಕ್ಷೆ ಮಾಡಲಾಗುತ್ತಿಲ್ಲ. ರೋಗ ಲಕ್ಷಣ ಇರುವವರು, ಸೋಂಕಿತರ ಪ್ರಾಥಮಿಕ ಸಂಪರ್ಕಿತರು, ಗ್ರಾಮಗಳಲ್ಲಿ ಆಶಾ ಕಾರ್ಯಕರ್ತೆಯರು, ಅಂಗನವಾಡಿ ಕಾರ್ಯಕರ್ತೆಯರು ಮನೆ ಸಮೀಕ್ಷೆ ಮಾಡುತ್ತಿದ್ದು, ಅಲ್ಲಿಂದ ಯಾರಾದರೂ ಶಂಕಿತರು ಕಂಡು ಬಂದರೆ, ಮಾದರಿ ಸಂಗ್ರಹಿಸಿ ಕೊರೊನಾ ಪತ್ತೆ ಪರೀಕ್ಷೆ ಮಾಡಲಾಗುತ್ತಿದೆ ಎಂದು ಡಿಎಚ್ಒ ಹೇಳಿದರು. | kannad |
മാന്പവര് അതോറിറ്റി മേധാവിക്ക് സസ്പെന്ഷന് കുവൈത്ത് സിറ്റി: മാന്പവര് അതോറിറ്റി ഡയറക്ടര് ജനറല് അഹ്മദ് മൂസയെ അതോറിറ്റി ഡയറക്ടര് ബോര്ഡ് ചെയര്മാനും വാണിജ്യ വ്യവസായ മന്ത്രിയുമായ അബ്ദുല്ല അല് സല്മാന് സസ്പെന്ഡ് ചെയ്തു. അദ്ദേഹം താല്പര്യപ്പെടുകയാണെങ്കില് ജനറല് മാനേജര് തസ്തികയില് സേവനം ചെയ്യാമെന്ന് മന്ത്രി ഡോ. അബ്ദുല്ല അല് സല്മാന് അറിയിച്ചു. നയപരമായ കാര്യത്തില് മന്ത്രിസഭയുമായി കൂടിയാലോചിക്കാതെ തീരുമാനമെടുത്തതാണ് അഹ്മദ് മൂസക്കെതിരെ നടപടിയെടുക്കാന് കാരണം. 60 വയസ്സുകഴിഞ്ഞ ബിരുദമില്ലാത്ത പ്രവാസികളുടെ തൊഴില് പെര്മിറ്റ് പുതുക്കലിന് ഏര്പ്പെടുത്തിയ നിയന്ത്രണം അദ്ദേഹത്തിന് വിനയായി.അന്താരാഷ്ട്ര തലത്തില് കുവൈത്തിെന്റ പ്രതിച്ഛായ കളങ്കപ്പെടുത്തുന്ന തീരുമാനമായിരുന്നു ഇതെന്നാണ് മന്ത്രിസഭയുടെ വിലയിരുത്തല്. പുതിയ തൊഴില് പെര്മിറ്റ് അനുവദിക്കുന്നത് പോലെയല്ല നിലവിലുള്ളത് പുതുക്കുന്നത് എന്ന് മന്ത്രി ചൂണ്ടിക്കാട്ടി. കഴിഞ്ഞമാസം അഹ്മദ് അല് മൂസയുടെ അധികാരം വെട്ടിച്ചുരുക്കി വാണിജ്യ വ്യവസായ മന്ത്രി ഉത്തരവ് ഇറക്കിയിരുന്നു. നേതാക്കളെ നിയമിക്കലും സ്ഥാനം മാറ്റലും, സൂപ്പര്വൈസറി തസ്തിക സൃഷ്ടിക്കല്, തൊഴിലാളികളെ നിര്ബന്ധിതമായി വിരമിപ്പിക്കല്, ജോലിയില്നിന്ന് പിരിച്ചുവിടലും പിഴ ചുമത്തലും, നിയമനവും പുനര് നിയമനവും, അച്ചടക്ക നടപടി റദ്ദാക്കല്, അന്വേഷണാധികാരം, കരാര് അവസാനിപ്പിക്കല്, സ്വകാര്യ ടെന്ഡര്, താല്ക്കാലിക ജീവനക്കാരുടെ ശമ്ബളം വെട്ടിക്കുറക്കല്, സ്ഥലംമാറ്റം, ആഭ്യന്തരവും ബാഹ്യവുമായ പരിശീലന ചുമതല, പ്രമോഷന് നല്കല്, ഇന്ക്രിമെന്റ് നല്കല്, തൊഴിലാളികളെ പബ്ലിക് പ്രോസിക്യൂഷന് റഫര് ചെയ്യല്, അന്വേഷണ ഭാഗമായി ജോലി നിര്ത്തിവെപ്പിക്കല് തുടങ്ങി 15 അധികാരങ്ങളാണ് മരവിപ്പിച്ചത്. പിന്നാലെ അദ്ദേഹത്തിനെതിരെ അന്വേഷണം പ്രഖ്യാപിച്ചു. അന്വേഷണ റിപ്പോര്ട്ട് അനുസരിച്ചാണ് സസ്പെന്ഷന്. | malyali |
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ಅಕ್ಟೋಬರ್ನಲ್ಲಿ ಯಶಸ್ವಿನಿ ಪುನರಾರಂಭ: ಸಚಿವ ಎಸ್.ಟಿ. ಸೋಮಶೇಖರ್ ಸುಬ್ರಹ್ಮಣ್ಯ: ಯಶಸ್ವಿನಿ ಯೋಜನೆಯನ್ನು ಅ. 2, ಗಾಂಧಿ ಜಯಂತಿ ಯಂದು ಮತ್ತೆ ಆರಂಭಿಸಲು ಸಿದ್ಧತೆ ನಡೆಸಲಾಗುತ್ತಿದೆ ಎಂದು ಸಹಕಾರ ಸಚಿವ ಎಸ್.ಟಿ. ಸೋಮಶೇಖರ್ ಹೇಳಿದ್ದಾರೆ. ಗಾಂಧಿ ಜಯಂತಿಯಂದು ಮುಖ್ಯ ಮಂತ್ರಿ ಬಸವರಾಜ ಬೊಮ್ಮಾಯಿ ಮತ್ತು ಕೇಂದ್ರ ಗೃಹ ಸಚಿವ ಅಮಿತ್ ಶಾ ಈ ಯೋಜನೆಗೆ ಚಾಲನೆ ನೀಡಲಿದ್ದಾರೆ ಎಂದು ಅವರು ಶನಿವಾರ ಕುಕ್ಕೆ ಶ್ರೀ ಸುಬ್ರಹ್ಮಣ್ಯ ಕ್ಷೇತ್ರದಲ್ಲಿ ಪತ್ರಕರ್ತರಿಗೆ ತಿಳಿಸಿದರು. ಹೈನುಗಾರಿಕೆಯನ್ನು ಪ್ರೋತ್ಸಾಹಿಸುವ ಉದ್ದೇಶದ ನಂದಿನಿ ಕ್ಷೀರ ಸಮೃದ್ಧಿ ಬ್ಯಾಂಕನ್ನು ಕೂಡ ಅಕ್ಟೋಬರ್ನಲ್ಲಿ ಆರಂಭಿಸಲಾಗು ವುದು ಎಂದರು. | kannad |
চোখের কঠিন সমস্যা মেটাতে বিশেষভাবে সাহায্য করে চা এর অসাধারন ৯ উপকারিতা নিউজ ডেস্কঃ সকালবেলা ঘুম থেকে উঠে খবর কাগজ এর সাথেই হোক বা বিকেলে আড্ডার মাঝে !চায়ের সাথে আমাদের এক আলাদাই ইমোশন জড়িয়ে আছে!চা ছাড়া দিনের শুরুটাই যেন হয় না ঠিকমতো !দুধ চা শরীরের জন্য তেমন ভালো না ঠিক ই তবে লিকার চা এর যে বেশ কিছু গুনাগুন আছে একথা আমরা অনেকেই জানিতবে জানেন কি চা কিন্তু কেবল খেতেই ভালো নয় !আমাদের ত্বকের জন্যও এটি খুবই উপকারী চায়ে থাকা অ্যান্টিঅক্সিডেন্ট, অ্যান্টি এজিং,অ্যান্টি ইনফ্ল্যামেটরি উপাদান গুলি আমাদের ত্বককে রাখে সুস্থ ও সুন্দর এছাড়াও চায়ে থাকা ক্যাফেইন ও কিন্তু ত্বকের বিভিন্ন সমস্যায় খুব ভালো মতো কাজ দেয়আসুন জেনে নিই শুধুমাত্র খাওয়া ছাড়াও আরো কি কিভাবে ত্বকের বিভিন্ন সমস্যায় ব্যবহার করা যেতে পারে চা 1.চোখের ফোলা ভাব ও ডার্ক সার্কল দূর করতে আমাদের বর্তমানের কর্মব্যস্ত জীবনে প্রায়ই অভাব দেখা যায় ঘুমানোর মতো সময়েরআর এই বিশ্রামের অভাব থেকেই অনেক সময় চোখের নিচে দেখা যায় ডার্ক সার্কেল এছাড়াও চোখ এর আশেপাশে ফোলা ভাব ও লক্ষ্য করা যায় অনেকের আপনিও যদি এই সমস্যায় জর্জরিত হয়ে থাকেন তাহলে ব্যবহার করে দেখতে পারেন চা চায়ে ক্যাফেইন ও ট্যানিন নামক দুই উপাদান থাকে এগুলি চোখের ফলা ভাব যেমন কমায় তেমনি দূর করে ডার্ক সার্কল ও সকালবেলা ঘুম থেকে উঠে প্রথমেই একটি টি ব্যাগ জলে ভিজিয়ে নিন এরপর চোখের ওপরে 5 থেকে 10 মিনিট ধরে রাখুন সেটি নিয়মিত এই টি ব্যাগ ব্যবহার করতে থাকলে দেখবেন কদিন পরেই চোখের নীচের যাবতীয় ডার্ক সার্কেল ও ফোলা ভাব দূর হয়ে গেছে 2. সূর্যের ক্ষতিকারক রশ্মি এবং দূষণ এই দুইই আমাদের ত্বকের জন্য খুব খারাপ অনেক নামি দামি সানস্ক্রিন ব্যবহার করার পরেও অনেক সময় ত্বকে সানবার্ন দেখা যায় এই সানবার্ন বা ট্যান কিন্তু খুব সহজেই ব্যাবহার করে নিরাময় করতে পারেন আপনি চা ব্যবহার করে একটি বাটিতে অল্প একটু চা প্রথমে জলে ফুটিয়ে নিন এরপর তা ঠান্ডা হয়ে এলে একটি কাপড় তাতে ডুবিয়ে নিয়ে ট্যান পড়া অংশে লাগিয়ে রাখুন নিয়মিত ব্যবহারে মিলবে ফলএছাড়াও গরমের ফলে ত্বকে জ্বালা ভাব দেখা দিলে তাও কমাতে ত্বকে সরাসরি টি ব্যাগ ব্যবহার করতে পারেন 3. চা ত্বকে খুব ভালো টনিক হিসাবেও কাজ দেয় বিশেষত আপনার যদি অয়েলি স্কিন হয়ে থাকে তাহলে ব্রন সহ অয়েলি স্কিনের যাবতীয় সমস্যা দূর করতে ব্যবহার করতে পারেন এটি বিশ্বাস হচ্ছে না ?কয়েকদিন আপনার সাধারণ টোনার ব্যবহার না করে তার বদলে চা ব্যবহার করে ফল দেখুন নিজের চোখে প্রথমেই ভালো করে জল দিয়ে মুখ পরিষ্কার করে নিন এরপর একটি টি ব্যাগ জলে ডুবিয়ে তা ভালো করে গোটা মুখে লাগিয়ে কিছুক্ষণ অপেক্ষা করে পরিষ্কার তোয়ালে দিয়ে মুছে ফেলুন দেখবেন মুখের অতিরিক্ত অয়েল উধাও হয়ে যাবে 4. চা ফেস স্ক্রাব হিসেবেও খুব ভালো কাজ দেয়ব্যবহার হয়ে যাওয়া টি ব্যাগ গুলি ফেলে না দিয়ে তা কিন্তু স্ক্রাবার হিসেবে ব্যবহার করতে পারেন আপনি প্রথমে টিব্যাগ গুলি শুকিয়ে নিন রোদে তারপর নিয়মিত স্ক্রাবার হিসেবে মুখে ব্যবহার করতে শুরু করুন এটি দেখবেন কয়েক দিনেই মুখ হয়ে উঠবে উজ্জ্বল ও নরম 5.চা ত্বকের তৈলাক্ত ভাব দূর করতে পারে একথা তো আগেই বলা হলো তবে,সমস্ত রকম চায়ের মধ্যে একনে সমস্যায় সবথেকে ভালো কাজ করে জেসমিন টি এর মধ্যে থাকা উপাদানগুলি দ্রুততার সাথে ত্বক থেকে যেকোন রকম পিম্পল এর দাগ সহজে দূর করতে পারে নিয়মিত অল্প পরিমাণে জেসমিন টি ভালো করে জলে ফুটিয়ে তা ঠাণ্ডা করে তৈলাক্ত ত্বকের ওপর লাগালে দূর হয় ত্বকের ব্যাকটেরিয়া এবং সেই সঙ্গে ত্বকের পিএইচ লেভেল এর ভারসাম্য বজায় থাকে ফলে ব্রনোর সমস্যা মিটে যায় অনেকটাই 6. ত্বক মসৃণ করতে বয়সের সাথে সাথে ত্বকে রুক্ষতা ও শুষ্কতা দেখা যায় ফলে ত্বক হয়ে ওঠে অমসৃণ আর এই অমসৃণ ত্বকে দেখা যায় নানা দাগ এক্ষেত্রে চামোমাইল চা বিশেষ উপকারী শুষ্ক ত্বকের জন্য এটি নিয়মিত ত্বকে ব্যবহার করলে ত্বকের শুষ্কতা ও রুক্ষতা উধাও হয়ে গিয়ে পুনরায় ফিরে আসে এক মসৃণ ও উজ্জ্বল ত্বক 7.মুখের দাগ পরিষ্কার করতে মুখ এর দাগ পরিষ্কার করতে চা পাতা বেটে মাস্ক হিসাবেও ব্যবহার করতে পারেন তা মুখে উজ্জল ও দাগ বিহীন ত্বক পেতে চায়ে থাকা অ্যান্টিঅক্সিডেন্ট বেশ সাহায্য করে চা ফোটানোর পর পড়ে থাকা চা পাতা ভালো করে বেটে নিয়ে তা মাস্ক হিসাবে মুখে কিছুক্ষণ লাগিয়ে রাখুন সপ্তাহে কয়েকবার এটি ব্যবহার করলে ধীরে ধীরে দেখতে পাবেন তফাত্ তবে,এক্ষেত্রে সাদা চা সব থেকে বেশি কার্যকরী 8. ফাটা ঠোঁট দূর করে ফাটা ঠোঁট এর সমস্যা আমাদের প্রায় সারা বছরই ভোগার এর মূল কারণ হলো ত্বকের তার স্বাভাবিক আদ্রতা হারিয়ে ফেলারোজ গ্রিন টি খাওয়া যদি আপনার অভ্যাস হয়ে থাকে তবে এই গ্রিন টি কিন্তু আপনার ফাটা ঠোঁটের আদ্রতা ফিরিয়ে আনার জন্য ও ব্যবহার করতে পারেন আপনি একটি গ্রিন টি ব্যাগ উষ্ণ গরম জলে কিছুক্ষণ ডুবিয়ে তা ফাটা ঠোঁটে ভালো করে লাগিয়ে রাখুন কয়েকদিনেই দেখবেন ঠোঁট হয়ে উঠেছে নরম ও গোলাপী 9. চুল স্বাস্থ্যকর করে তোলে চুল পড়ার সমস্যায় ভুক্তভোগী হয়ে থাকলে এবার থেকে লিকার চা ব্যবহার করে দেখুন প্রত্যেকদিন চা বানানোর সময় একটু বেশি পরিমাণে লিকার চা বানিয়ে ফেলুন এরপর চুল পড়া রোধ করতে লিকার চা কিছুক্ষন ঠাণ্ডা হতে দিন তারপর সেটি চুল এবং কিছুটা স্ক্যাল্পে ভালো করে ম্যাসাজ করে নিনদেখবেন চুলপড়া যে কেবল কমবে তাই নয় সাথে চুল ও হয়ে উঠবে মজবুত ও উজ্জ্বল | bengali |
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Kolkata Fire: কসবায় ভস্মীভূত বেসরকারি অফিস নিজস্ব প্রতিবেদন: কসবার বোসপুকুর রোডে ভস্মীভূত হয়ে গেল বেসরকারি সংস্থার অফিস প্রায় দুঘণ্টা চেষ্টার পর দমকলের ২টি ইঞ্জিন আগুন নিয়ন্ত্রণে আনে চারতলায় আটকে পড়া কয়েকজনকে উদ্ধার করেন দমকল কর্মীরা বুধবার দুপুর ১টা নাগাদ, কসবার বোসপুকুর রোডে একটি বহুতলের তিনতলার অফিসে আগুন লাগে বহুতলের চারতলায় ছিল একটি পার্লার অগ্নিকাণ্ডের জেরে সেখানে আটকে পড়েন কয়েকজন কর্মী দরজা ভেঙে তাঁদের উদ্ধার করেন দমকল কর্মীরা বহুতলের দোতলায় একটি বৈদ্যুতিন সামগ্রীর শোরুম থাকায় আতঙ্ক ছড়িয়ে পড়ে কীভাবে আগুন লাগে তা এখনও স্পষ্ট করে কিছু জানা যায়নি তবে দমকলের প্রাথমিক অনুমান, এসি থেকে আগুন লেগে থাকতে পারে যেখানে আগুন লেগেছে তার কয়েকটি বাড়ি পরেই রয়েছে একটি পেট্রোল পাম্প সেখানে আগুন ছড়িয়ে পড়লে কী হত তা ভেবেই শিউরে উঠছেন বাসিন্দারা | bengali |
\begin{document}
\title{Weak solutions of the Landau--Lifshitz--Bloch equation
hanks{This work was supported by the
Australian Research
Council grant DP140101193.}
\partialrtialagenumbering{arabic}
\begin{abstract}
The Landau--Lifshitz--Bloch (LLB) equation is a formulation of dynamic
micromagnetics valid at all temperatures, treating both the transverse and
longitudinal relaxation components important for high-temperature
applications.
We study LLB equation in case the temperature
raised higher than the Curie temperature. The existence of weak solution
is showed and its regularity properties are also discussed. In this way,
we lay foundations for the rigorous theory of LLB equation that is currently
not available.
{\bf Key words}: Landau--Lifshitz--Bloch, quasilinear parabolic equation, ferromagnetism
{\bf AMS suject classifications}: 82D40, 35K59, 35R15
\end{abstract}
\section{Introduction}
Micromagnetic modeling has proved itself as a widely used tool,
complimentary in many respects to experimental measurements.
The Landau--Lifshitz--Gilbert (LLG) equation~\cite{LL35,Gil55} provides a basis for this modeling,
especially where the dynamical behaviour is concerned.
According to this theory, at temperatures below the critical (so-called Curie) temperature,
the magnetization $\boldsymbol{m}(t,\boldsymbol{x})\in\mathbb S^2$, where $\mathbb S^2$ is the unit sphere
in ${\mathbb R}^3$,
for $t>0$ and $\boldsymbol{x}\in D\subset {\mathbb R}^d$, $d=1,2,3$, satisfies the following LLG equation
\begin{equation}\label{eq: LLG}
\frac{\partialrtialartial \boldsymbol{m}}{\partialrtialartial t}
=
\lambdabda_1 \boldsymbol{m}\times \boldsymbol{H}_{\text{eff}}
-
\lambdabda_2\boldsymbol{m}\times(\boldsymbol{m}\times \boldsymbol{H}_{\text{eff}}),
\end{equation}
where $\times$ is the vector cross product in ${\mathbb R}^3$ and $\boldsymbol{H}_{\text{eff}}$
is the so-called effective field.
However, for high temperatures the model must be replaced by a more thermodynamically consistent
approach such as the Landau--Lifshitz--Bloch (LLB) equation~\cite{Garanin1991,Garanin97}.
The LLB equation essentially interpolates between
the LLG equation at low temperatures and
the Ginzburg-Landau theory of phase transitions. It is valid not only
below but also above the Curie temperature $T_{\text{c}}$.
An important property of the LLB equation is that the magnetization magnitude
is no longer conserved but is a dynamical variable~\cite{Garanin97,Evans12}.
The spin polarization $\boldsymbol{u}(t,\boldsymbol{x})\in{\mathbb R}^3$, ($\boldsymbol{u}=\boldsymbol{m}/m_s^0$, $\boldsymbol{m}$ is magnetization and
$m_s^0$ is the saturation magnetization value at $T=0$),
for $t>0$ and $\boldsymbol{x}\in D\subset {\mathbb R}^d$, $d=1,2,3$, satisfies the following LLB equation
\begin{equation}\label{eq: LLB}
\frac{\partialrtialartial \boldsymbol{u}}{\partialrtialartial t}
=
\gammamma \boldsymbol{u}\times \boldsymbol{H}_{\text{eff}}
+
L_1\frac{1}{|\boldsymbol{u}|^2}(\boldsymbol{u}\cdot\boldsymbol{H}_{\text{eff}})\boldsymbol{u}
-
L_2\frac{1}{|\boldsymbol{u}|^2}\boldsymbol{u}\times(\boldsymbol{u}\times\boldsymbol{H}_{\text{eff}}).
\end{equation}
Here, $|\cdot|$ is the Euclidean norm in ${\mathbb R}^3$, $\gammamma>0$ is the gyromagnetic ratio, and $L_1$
and $L_2$ are the longitudial and transverse
damping parameters, respectively.
LLB micromagnetics has become a real alternative to LLG micromagnetics
for temperatures which are close to the Curie temperature
($T\gtrsim\tfrac34T_{\text{c}}$).
This is realistic for some novel exciting phenomena, such as
light-induced demagnetization with powerfull femtosecond (fs) lasers~\cite{Atxitia2007}.
During this process the electronic temperature is normally
raised higher than $T_{\text{c}}$.
Micromagnetics based on the LLG equation cannot work under these
circumstances while micromagnetics based on the LLB equation has proved
to describe correctly the observed fs magnetization dynamics.
In this paper, we consider a deterministic form of a ferromagnetic LLB equation,
in which the temperature
$T$ is raised higher than $T_{\text{c}}$, and as a consequence the longitudial $L_1$ and transverse $L_2$ damping parameters
are equal. The effective field $\boldsymbol{H}_{\text{eff}}$ is given by
\[
\boldsymbol{H}_{\text{eff}} = {\mathbb D}elta\boldsymbol{u} -
\frac{1}{\chi_{||}}
\bigg(1+\frac{3}{5}\frac{T}{T-T_c}|\boldsymbol{u}|^2\bigg)\boldsymbol{u},
\]
where $\chi_{||}$ is the longitudinal susceptibility.
By using the vector triple product identity
$\boldsymbol{a}\times(\boldsymbol{b}\times\boldsymbol{c}) = \boldsymbol{b}(\boldsymbol{a}\cdot\boldsymbol{c})-\boldsymbol{c}(\boldsymbol{a}\cdot\boldsymbol{b})$, we get
\[
\boldsymbol{u}\times(\boldsymbol{u}\times\boldsymbol{H}_{\text{eff}})
=
(\boldsymbol{u}\cdot\boldsymbol{H}_{\text{eff}})\boldsymbol{u}
-
|\boldsymbol{u}|^2\boldsymbol{H}_{\text{eff}},
\]
and from property $L_1=L_2=:\kappa_1$,
we can rewrite~\eqref{eq: LLB} as follows
\begin{equation}\label{eq: LLB2}
\frac{\partialrtialartial \boldsymbol{u}}{\partialrtialartial t}
=
\kappa_1{\mathbb D}elta\boldsymbol{u}
+
\gammamma\boldsymbol{u}\times {\mathbb D}elta\boldsymbol{u}
-
\kappa_2(1+\mu|\boldsymbol{u}|^2)\boldsymbol{u},
\quad \text{with }
\kappa_2 := \frac{\kappa_1}{\chi_{||}},
\quad
\mu := \frac{3T}{5(T-T_c)}.
\end{equation}
So the LLB equation we are going to study in this paper is equation~\eqref{eq: LLB2} with real positive
coefficients $\kappa_1,\kappa_2,\gammamma,\mu$, initial data
$\boldsymbol{u}(0,\boldsymbol{x})=\boldsymbol{u}_0(\boldsymbol{x})$ and subject to
homogeneous Neumann boundary conditions.
Various results on existence of global weak
solutions of the LLG equation~\eqref{eq: LLG} are proved in~\cite{CarbouFabrie01,AloSoy92}. More complete
lists can be found in~\cite{Cimrak_survey,GuoDing08,KruzikProhl06}.
Furthermore, there is also some research about the weak solution of its stochastic version (i.e., the effective field
is perturbed by a Gaussian noise), such as in~\cite{BrzGolJer12,bookBanBrzNekPro13}.
It should be mentioned that the proof of existence in~\cite{Banas2013,BanBrzPro13,GoldysLeTran2016} is a
constructive proof, namely an approximate solution can be
computed.
To the best of our knowledge the analysis of the LLB equation is an open problem at present.
In this paper, we introduce a definition of weak solutions of the LLB equation.
By introducing the Faedo--Galerkin approximations and using the method of compactness, we
prove the existence of weak solutions for the LLB equation.
This paper is organized as follows.
In Section~\ref{sec: nota} we introduce the notations and formulate the main result
(Theorem~\ref{theo: main}) on the existence of the weak solution of~\eqref{eq: LLB2}
as well as some regularity properties. In Section~\ref{sec: FG} we introduce the Faedo--Galerkin
approximations and prove for them some uniform bounds in various norms.
In Section~\ref{sec: Exist}, we use the method of compactness to show the existence of
a weak solution and prove the main theorem. Finally, in the Appendix we collect,
for the reader's convenience, some facts scattered in the literature that are used
in the course of the proof.
\section{Notation and the formulation of the main result}\label{sec: nota}
Before presenting the definition of a weak solution to the LLB equation~\eqref{eq: LLB2},
it is necessary to introduce some function spaces.
The function spaces $\mathbb H^1(D,{\mathbb R}^3)=:\mathbb H^1$ are defined as follows:
\begin{align*}
\mathbb H^1(D,{\mathbb R}^3)
&=
\left\{ \boldsymbol{u}\in\mathbb L^2(D,{\mathbb R}^3) : \frac{\partialrtial\boldsymbol{u}}{\partialrtial
x_i}\in \mathbb L^2(D,{\mathbb R}^3)\quad\text{for } i=1,2,3.
\right\}.
\end{align*}
Here, $\mathbb L^p(D,{\mathbb R}^3)=:\mathbb L^p$ with $p>0$ is the usual space of $p^\text{th}$-power
Lebesgue
integrable functions defined on $D$ and taking values in ${\mathbb R}^3$.
Throughout this paper, we denote a scalar product in a Hilbert space $H$ by
$\inpro{\cdot}{\cdot}_H$ and its associated norm by $\|\cdot\|_H$. The dual brackets between
a space $X$ and its dual $X^*$ will be denoted $_{X}\!\iprod{\cdot,\cdot}_{X^*}$.
\begin{definition}\label{def: weakso}
Given $T>0$, a weak solution
$\boldsymbol{u} : [0,T] \rightarrow \mathbb H^1\cap \mathbb L^4$ to~\eqref{eq: LLB2} satisfies
\begin{align}\label{eq: weakLLB}
\iprod{\boldsymbol{u}(t),\boldsymbol{p}hi}_{\mathbb L^2}
=
&\iprod{\boldsymbol{u}_0,\boldsymbol{p}hi}_{\mathbb L^2}
-\kappa_1\int_0^t \iprod{\nabla\boldsymbol{u}(s),\nabla\boldsymbol{p}hi}_{\mathbb L^2}\, ds
-\gammamma\int_0^t \iprod{\boldsymbol{u}(s)\times\nabla\boldsymbol{u}(s),\nabla\boldsymbol{p}hi}_{\mathbb L^2}\, ds\nonumber \\
&-\kappa_2\int_0^t \iprod{(1+\mu|\boldsymbol{u}|^2(s))\boldsymbol{u}(s),\boldsymbol{p}hi}_{\mathbb L^2}\, ds,
\end{align}
for every $\boldsymbol{p}hi\in {\mathbb C}_0^{\infty}(D)$ and $t\in[0,T]$.
\end{definition}
Now we can formulate the main result of this paper.
\begin{theorem}\label{theo: main}
Let $D\subset{\mathbb R}^d$ be an open bounded domain with $C^m$ extension property and assume that $d<2m$.
For $T>0$ and for the initial data $\boldsymbol{u}_0\in\mathbb H^1$, there exists a weak solution of~\eqref{eq: LLB2}
such that
\begin{enumerate}
\item[(a)] for every $t\in[0,T]$,
\begin{align}\label{eq: weakLLB2}
\boldsymbol{u}(t)
=
&\boldsymbol{u}_0
+\kappa_1\int_0^t {\mathbb D}elta\boldsymbol{u}(s)\, ds
+\gammamma\int_0^t \boldsymbol{u}(s)\times{\mathbb D}elta\boldsymbol{u}(s)\, ds\nonumber\\
&-\kappa_2\int_0^t (1+\mu|\boldsymbol{u}|^2(s))\boldsymbol{u}(s)\, ds \quad\text{in $\mathbb L^{3/2}$,}
\end{align}
\item[(b)]
for every $\alphapha\in(0,\tfrac14]$, $\boldsymbol{u}\in C^{\alphapha}([0,T],\mathbb L^{3/2})$,
\item[(c)] $\sup_{t\in[0,T]}\|\boldsymbol{u}(t,\cdot)\|_{\mathbb L^2}<\infty$.
\end{enumerate}
\end{theorem}
\begin{remark}
The notation ${\mathbb D}elta\boldsymbol{u}$ and $\boldsymbol{u}\times{\mathbb D}elta\boldsymbol{u}$ will
be defined in the Notations~\ref{no: nota1}--\ref{no: nota2}.
\end{remark}
\section{Faedo-Galerkin Approximation}\label{sec: FG}
Let $A=-{\mathbb D}elta$ be the negative Laplace operator.
From~\cite[Theorem 1, p. 335]{Evans1998}, there exists an orthonormal basis
$\{\boldsymbol{e}_i\}_{i=1}^{\infty}$ of $\mathbb L^2$, consisting
of eigenvectors for operator $A$, such that $\boldsymbol{e}_i\in{\mathbb C}^{m}(D)\cap\mathbb L^{\infty}$
for all $i=1,2,$\dots and
\[
-{\mathbb D}elta\boldsymbol{e}_i = \lambdabda_i\boldsymbol{e}_i,\quad
\boldsymbol{e}_i = 0 \text{ on }\partialrtialartial D,
\]
where $\lambdabda_i>0$ for $i=1,2,$\dots are eigenvalues of $A$.
Let $S_n:=\text{span}\{\boldsymbol{e}_1,\cdots,\boldsymbol{e}_n\}$ and
$\Pi_n$ be the orthogonal projection from $\mathbb L^2$ onto $S_n$,
defined by: for $\boldsymbol{v}\in\mathbb L^2$
\begin{equation}\label{eq: Pi_n}
\iprod{\Pi_n\boldsymbol{v},\boldsymbol{p}hi}_{\mathbb L^2}
= \iprod{\boldsymbol{v},\boldsymbol{p}hi}_{\mathbb L^2},\quad\forall \boldsymbol{p}hi\in S_n.
\end{equation}
By taking $\boldsymbol{p}hi = \Pi_n\boldsymbol{v}$ in the above equation, we obtain an upper bound for
the projection operator $\Pi_n$ in $\mathbb L^2$,
\begin{equation}\label{eq: boundPi_n}
\|\Pi_n\boldsymbol{v}\|_{\mathbb L^2}
\leq
\|\boldsymbol{v}\|_{\mathbb L^2}
\quad\forall\boldsymbol{v}\in S_n.
\end{equation}
We note that $\Pi_n$ is a self-adjoint operator on $\mathbb L^2$, indeed,
from ~\eqref{eq: Pi_n}, for $\boldsymbol{v},\boldsymbol{w}\in\mathbb L^2$
there holds
\[
\iprod{\boldsymbol{w},\Pi_n\boldsymbol{v}}_{\mathbb L^2}
=
\iprod{\Pi_n\boldsymbol{v},\Pi_n\boldsymbol{w}}_{\mathbb L^2}
=
\iprod{\boldsymbol{v},\Pi_n\boldsymbol{w}}_{\mathbb L^2}.
\]
We are now looking for approximate solution
$\boldsymbol{u}_n(\cdot,t)\in S_n:=\text{span}\{\boldsymbol{e}_1,\cdots,\boldsymbol{e}_n\}$ of equation~\eqref{eq: LLB2} satisfying
\begin{equation}\label{eq: GaLLB}
\frac{\partialrtialartial \boldsymbol{u}_n}{\partialrtialartial t}
-\kappa_1{\mathbb D}elta\boldsymbol{u}_n
-\gammamma\Pi_n\bigl(\boldsymbol{u}_n\times{\mathbb D}elta\boldsymbol{u}_n\bigr)
+\kappa_2\Pi_n\bigl((1+\mu|\boldsymbol{u}_n|^2)\boldsymbol{u}_n\bigr)
=0,
\end{equation}
with $\boldsymbol{u}_n(\cdot,0) = \boldsymbol{u}_{0n}$, where $\boldsymbol{u}_{0n}\in S_n$ is an approximation
of $\boldsymbol{u}_0$.
Since equation~\eqref{eq: GaLLB} is equivalent to an ordinary differential equation in ${\mathbb R}^n$,
the existence of a local solution to~\eqref{eq: GaLLB}
is a consequence of the following lemma.
\begin{lemma}
For $n\in{\mathbb N}$, define the maps:
\begin{align*}
&F^1_n: S_n\ni \boldsymbol{v} \mapsto {\mathbb D}elta\boldsymbol{v}\in S_n,\\
&F^2_n: S_n\ni\boldsymbol{v}\mapsto \Pi_n(\boldsymbol{v}\times{\mathbb D}elta\boldsymbol{v})\in S_n,\\
&F^3_n: S_n\ni\boldsymbol{v}\mapsto \Pi_n((1+\mu|\boldsymbol{v}|^2)\boldsymbol{v})\in S_n.
\end{align*}
Then $F^1_n$ is globally Lipschitz and $F^2_n$, $F^3_n$ are locally Lipschitz.
\end{lemma}
\begin{proof}
For any $\boldsymbol{v}\in S_n$ we have
\begin{equation*}
\boldsymbol{v} = \sum_{i=1}^n \inpro{\boldsymbol{v}}{\boldsymbol{e}_i}_{\mathbb L^2}\boldsymbol{e}_i
\quad\text{and}\quad
-{\mathbb D}elta\boldsymbol{v} = \sum_{i=1}^n \lambdabda_i\inpro{\boldsymbol{v}}{\boldsymbol{e}_i}_{\mathbb L^2}\boldsymbol{e}_i.
\end{equation*}
By using the triangle inequality, the orthonormal property
of $\{\boldsymbol{e}_i\}_{i=1}^n$ and H\"older's inequality, for any $\boldsymbol{u},\boldsymbol{v}\in S_n$ we obtain
\begin{align*}
\|F^1_n(\boldsymbol{u})-F^1_n(\boldsymbol{v})\|_{\mathbb L^2}
&=
\|{\mathbb D}elta\boldsymbol{u}-{\mathbb D}elta\boldsymbol{v}\|_{\mathbb L^2}
=
\|\sum_{i=1}^n \lambdabda_i\inpro{\boldsymbol{u}-\boldsymbol{v}}{\boldsymbol{e}_i}_{\mathbb L^2}\boldsymbol{e}_i\|_{\mathbb L^2}\\
&\leq
\sum_{i=1}^n \lambdabda_i\bigl|\inpro{\boldsymbol{u}-\boldsymbol{v}}{\boldsymbol{e}_i}_{\mathbb L^2}\bigr|
\leq
\bigl(\sum_{i=1}^n \lambdabda_i\bigr)\|\boldsymbol{u}-\boldsymbol{v}\|_{\mathbb L^2},
\end{align*}
then the globally Lipschitz property of $F^1_n$ follows immediately.
From~\eqref{eq: boundPi_n} and the triangle inequality, there holds
\begin{align*}
\|F^2_n(\boldsymbol{u})-F^2_n(\boldsymbol{v})\|_{\mathbb L^2}
&=
\|\Pi_n(\boldsymbol{u}\times{\mathbb D}elta\boldsymbol{u}-\boldsymbol{v}\times{\mathbb D}elta\boldsymbol{v})\|_{\mathbb L^2}
\leq
\|\boldsymbol{u}\times{\mathbb D}elta\boldsymbol{u}-\boldsymbol{v}\times{\mathbb D}elta\boldsymbol{v}\|_{\mathbb L^2}\\
&\leq
\|\boldsymbol{u}\times({\mathbb D}elta\boldsymbol{u}-{\mathbb D}elta\boldsymbol{v})\|_{\mathbb L^2}
+
\|(\boldsymbol{u}-\boldsymbol{v})\times{\mathbb D}elta\boldsymbol{v}\|_{\mathbb L^2}\\
&\leq
\|\boldsymbol{u}\|_{\mathbb L^{\infty}}
\|F^1_n(\boldsymbol{u})-F^1_n(\boldsymbol{v})\|_{\mathbb L^2}
+
\|(\boldsymbol{u}-\boldsymbol{v})\|_{\mathbb L^2}
\|{\mathbb D}elta\boldsymbol{v}\|_{\mathbb L^{\infty}}.
\end{align*}
Since $F^1_n$ is globally Lipschitz and
the fact that all norms are equivalent in the finite dimensional space $S_n$, $F^2_n$ is locally Lifshitz.
Similarly, the local Lipschitz property of $F^3_n$ follows from the estimate,
\begin{align*}
\|F^3_n(\boldsymbol{u})-F^3_n(\boldsymbol{v})\|_{\mathbb L^2}
&\leq
\|\boldsymbol{u}-\boldsymbol{v}\|_{\mathbb L^2}
+
\mu\|\Pi_n(|\boldsymbol{u}|^2\boldsymbol{u}-|\boldsymbol{v}|^2\boldsymbol{v})\|_{\mathbb L^2}\\
&\leq
\|\boldsymbol{u}-\boldsymbol{v}\|_{\mathbb L^2}
+
\mu\||\boldsymbol{u}|^2\boldsymbol{u}-|\boldsymbol{v}|^2\boldsymbol{v}\|_{\mathbb L^2}\\
&\leq
\|\boldsymbol{u}-\boldsymbol{v}\|_{\mathbb L^2}
+
\mu\||\boldsymbol{u}|^2(\boldsymbol{u}-\boldsymbol{v})\|_{\mathbb L^2}
+
\mu\|(\boldsymbol{u}-\boldsymbol{v})\cdot(\boldsymbol{u}+\boldsymbol{v})\,\boldsymbol{v}\|_{\mathbb L^2}\\
&\leq
\bigl(
1
+
\mu\||\boldsymbol{u}|^2\|_{\mathbb L^{\infty}}
+
\mu\|\boldsymbol{u}+\boldsymbol{v}\|_{\mathbb L^{\infty}}\|\boldsymbol{v}\|_{\mathbb L^{\infty}}\bigr)
\|\boldsymbol{u}-\boldsymbol{v}\|_{\mathbb L^2},
\end{align*}
which complete the proof of this lemma.
\end{proof}
We now proceed to priori estimates on the approximate solution $\boldsymbol{u}_n$.
\begin{lemma}\label{lem: appSo_sta}
For each $n=1,2,$\dots and every $t\in[0,T]$,
\begin{equation*}
\|\boldsymbol{u}_n(t)\|_{\mathbb L^2}^2
+
2\kappa_1\int_0^T \|\nabla\boldsymbol{u}_n(t)\|_{\mathbb L^2}^2 \, dt
+
2\kappa_2
\int_0^T \bigl(\|\boldsymbol{u}_n(t)\|_{\mathbb L^2}^2 + \mu\|\boldsymbol{u}_n(t)\|_{\mathbb L^4}^4\bigr)\, dt
\leq
\|\boldsymbol{u}_n(0)\|_{\mathbb L^2}^2,
\end{equation*}
and
\[
\|\nabla\boldsymbol{u}_n(t)\|_{\mathbb L^2}^2
+
2\kappa_1\int_0^T \|{\mathbb D}elta\boldsymbol{u}_n(t)\|_{\mathbb L^2}^2 \, dt
\leq
\|\nabla\boldsymbol{u}_n(0)\|_{\mathbb L^2}^2.
\]
\end{lemma}
\begin{proof}
Taking the inner product of both sides of ~\eqref{eq: GaLLB}
with $\boldsymbol{u}_n(t)\in S_n$, integrating by parts with respect to $\boldsymbol{x}$, and using
$(\boldsymbol{a}\times\boldsymbol{b})\cdot\boldsymbol{b}=0$ and the fact that $\Pi_n$ is self-adjoint,
we obtain
\begin{align*}
\frac{1}{2}\frac{\partialrtialartial}{\partialrtialartial t}\|\boldsymbol{u}_n(t)\|_{\mathbb L^2}^2
+
\kappa_1\|\nabla\boldsymbol{u}_n(t)\|_{\mathbb L^2}^2
+
\kappa_2
\bigiprod{(1+\mu|\boldsymbol{u}_n|^2)\boldsymbol{u}_n,\boldsymbol{u}_n(t)}_{\mathbb L^2} = 0.
\end{align*}
The first result follows by integrating both sides of the above equation
with respect to $t$.
In a similar fashion, we next take the inner product of both sides of ~\eqref{eq: GaLLB}
with ${\mathbb D}elta\boldsymbol{u}_n(t)\in S_n$, and then integrate by parts with respect to $\boldsymbol{x}$
to arrive at
\begin{align*}
\frac{1}{2}\frac{\partialrtialartial}{\partialrtialartial t}\|\nabla\boldsymbol{u}_n(t)\|_{\mathbb L^2}^2
+
\kappa_1\|{\mathbb D}elta\boldsymbol{u}_n(t)\|_{\mathbb L^2}^2
&+
\kappa_2
\bigiprod{(1+\mu|\boldsymbol{u}_n|^2)\nabla\boldsymbol{u}_n,\nabla\boldsymbol{u}_n(t)}_{\mathbb L^2}\\
&+
\kappa_2
\bigiprod{2\mu(\boldsymbol{u}_n\cdot\nabla\boldsymbol{u}_n)\boldsymbol{u}_n,\nabla\boldsymbol{u}_n(t)}_{\mathbb L^2}
=0
\end{align*}
Integrating both sides
with respect to $t$,
we obtain
\begin{align*}
\|\nabla\boldsymbol{u}_n(t)\|_{\mathbb L^2}^2
+
2\kappa_1
\int_0^t
\|{\mathbb D}elta\boldsymbol{u}_n(s)\|_{\mathbb L^2}^2\, ds
&+
2\kappa_2
\int_0^t
\int_D
(1+\mu|\boldsymbol{u}_n|^2)(\nabla\boldsymbol{u}_n)^2\,d\boldsymbol{x}\, ds\\
&+
2\kappa_2\mu
\int_0^t
\int_D
(\boldsymbol{u}_n\cdot\nabla\boldsymbol{u}_n)^2\,d\boldsymbol{x}\, ds
=\|\nabla\boldsymbol{u}_n(0)\|_{\mathbb L^2}^2,
\end{align*}
and the second result follows immediately.
\end{proof}
The following upper bounds for $\boldsymbol{u}_n\times{\mathbb D}elta\boldsymbol{u}_n$ and $(1+\mu|\boldsymbol{u}_n|^2)\boldsymbol{u}_n$
are a consequence of Lemma~\ref{lem: appSo_sta}.
\begin{lemma}\label{lem: bound_un1}
There exists a constant $C$, which does not depend on
$n=1,2,$\dots, such that
\begin{align*}
\int_0^T\|\boldsymbol{u}_n(t)\times{\mathbb D}elta\boldsymbol{u}_n(t)\|^2_{\mathbb L^{3/2}}\, dt
\leq C\quad\text{and}\quad
\int_0^T\|(1+\mu|\boldsymbol{u}_n|^2(t))\boldsymbol{u}_n(t)\|^2_{\mathbb L^2}\, dt
\leq C.
\end{align*}
\end{lemma}
\begin{proof}
By H\"older's inequality and the Sobolev imbedding of $\mathbb H^1$ into $\mathbb L^6$~\cite{Friedman1969}
we have
\begin{align*}
\|\boldsymbol{u}_n(t)\times{\mathbb D}elta\boldsymbol{u}_n(t)\|_{\mathbb L^{3/2}}
\leq
\|\boldsymbol{u}_n(t)\|_{\mathbb L^6}
\|{\mathbb D}elta\boldsymbol{u}_n(t)\|_{\mathbb L^2}
\leq
C
\|\boldsymbol{u}_n(t)\|_{\mathbb H^1}
\|{\mathbb D}elta\boldsymbol{u}_n(t)\|_{\mathbb L^2}.
\end{align*}
We use Lemma~\ref{lem: appSo_sta} to obtain the first result,
\begin{align*}
\int_0^T \|\boldsymbol{u}_n(t)\times{\mathbb D}elta\boldsymbol{u}_n(t)\|^2_{\mathbb L^{3/2}}\, dt
\leq
C\sup_{t\in[0,T]}\|\boldsymbol{u}_n(t)\|_{\mathbb H^1}^2
\int_0^T
\|{\mathbb D}elta\boldsymbol{u}_n(t)\|_{\mathbb L^2}^2\, dt
\leq C.
\end{align*}
Similarly, from Lemma~\ref{lem: appSo_sta} and the Sobolev imbedding of $\mathbb H^1$ into $\mathbb L^6$, we have
\begin{equation}\label{eq: un3}
\|\boldsymbol{u}_n^3(t)\|^2_{\mathbb L^2}
=
\|\boldsymbol{u}_n(t)\|^6_{\mathbb L^6}
\leq
\|\boldsymbol{u}_n(t)\|^6_{\mathbb H^1}
\leq
C,
\end{equation}
so
\begin{align*}
\|\bigl(1+\mu|\boldsymbol{u}_n|^2(t)\bigr)\boldsymbol{u}_n(t)\|^2_{\mathbb L^2}
\leq
2\|\boldsymbol{u}_n(t)\|^2_{\mathbb L^2}
+
2\mu^2\|\boldsymbol{u}_n^3(t)\|^2_{\mathbb L^2}
\leq
C,
\end{align*}
and the second result follows immediately.
\end{proof}
Equation~\eqref{eq: GaLLB} can be written in the following way
as an approximation of equation~\eqref{eq: LLB2},
\begin{align}\label{eq: GaLLB2}
\boldsymbol{u}_n(t)
&=\boldsymbol{u}_n(0)
+\kappa_1\int_0^t{\mathbb D}elta\boldsymbol{u}_n\, ds
+\gammamma\int_0^t\Pi_n\bigl(\boldsymbol{u}_n\times{\mathbb D}elta\boldsymbol{u}_n\bigr)\, ds
-\kappa_2\int_0^t\Pi_n\bigl((1+\mu|\boldsymbol{u}_n|^2)\boldsymbol{u}_n\bigr)\, ds\\
&=\boldsymbol{u}_n(0) + \kappa_1\boldsymbol{B}_{n,1}(t)
+\gammamma\boldsymbol{B}_{n,2}(t)
+\kappa_2\boldsymbol{B}_{n,3}(t)
.\nonumber
\end{align}
Before proving the uniform bound of $\{\boldsymbol{u}_n\}$, we define the following
fractional power space~\cite[Definiton 1.4.7]{Henry1981}.
\begin{definition}\label{def: fracspace}
Put $A_1:=I+A$. For any real number $\beta>0$, we define
the Hilbert space
\[
X^{\beta} := \bigl\{\boldsymbol{p}hi\in\mathbb L^2: \|A_1^{\beta}\boldsymbol{p}hi\|_{\mathbb L^2}<\infty\bigr\},
\]
where
$A_1^{\beta}\boldsymbol{p}hi :=
\sum_{i=1}^{\infty}
(1+\lambdabda_i)^{\beta}
\inpro{\boldsymbol{p}hi}{\boldsymbol{e}_i}_{\mathbb L^2}\boldsymbol{e}_i,$
with the graph norm $\|\cdot\|_{X^{\beta}}=\|A_1^{\beta}\cdot\|_{\mathbb L^2}$.
The dual space of $X^{\beta}$ is denoted by $X^{-\beta}$.
\end{definition}
The following lemma states an upper bound for
the projection operator $\Pi_n$ in $X^{-\beta}$.
\begin{lemma}\label{lem: boundPi_n}
For any $\beta>0$ and $\boldsymbol{v}\in\mathbb L^2$ there holds
\[
\|\Pi_n\boldsymbol{v}\|_{X^{-\beta}}\leq \|\boldsymbol{v}\|_{X^{-\beta}}.
\]
\end{lemma}
\begin{proof}
The proof of this lemma can be found in~\cite{ZdzisLiang2014};
for the reader's convenience we recall the proof as follows.
For $\boldsymbol{v}\in\mathbb L^2$, by using~\eqref{eq: Pi_n} we obtain
\begin{align}\label{eq: bound1}
\|\Pi_n\boldsymbol{v}\|_{X^{-\beta}}
&=
\sup_{\|\boldsymbol{w}\|_{X^{\beta}}\leq 1}
\left|_{X^{-\beta}}\!\iprod{\Pi_n\boldsymbol{v},\boldsymbol{w}}_{X^{\beta}}\right|
=
\sup_{\|\boldsymbol{w}\|_{X^{\beta}}\leq 1}
\left|\iprod{\Pi_n\boldsymbol{v},\boldsymbol{w}}_{\mathbb L^2}\right|\nonumber\\
&=
\sup_{\|\boldsymbol{w}\|_{X^{\beta}}\leq 1}
\left|\iprod{\boldsymbol{v},\Pi_n\boldsymbol{w}}_{\mathbb L^2}\right|.
\end{align}
Since
\[\|\Pi_n\boldsymbol{w}\|^2_{X^{\beta}} = \sum_{i=1}^{n}
(1+\lambdabda_i)^{2\beta}
\inpro{\boldsymbol{w}}{\boldsymbol{e}_i}_{\mathbb L^2}^2
\leq
\sum_{i=1}^{\infty}
(1+\lambdabda_i)^{2\beta}
\inpro{\boldsymbol{w}}{\boldsymbol{e}_i}_{\mathbb L^2}^2
=
\|\boldsymbol{w}\|^2_{X^{\beta}},
\]
the set $\{\boldsymbol{w}\in X^{\beta}: \|\boldsymbol{w}\|_{X^{\beta}}\leq 1 \}$
is a subset of the set
$\{\boldsymbol{w}\in X^{\beta}: \|\Pi_n\boldsymbol{w}\|_{X^{\beta}}\leq 1 \}$. Hence,
from~\eqref{eq: bound1} there holds
\[
\|\Pi_n\boldsymbol{v}\|_{X^{-\beta}}
\leq
\sup_{\|\Pi_n\boldsymbol{w}\|_{X^{\beta}}\leq 1}
\left|\iprod{\boldsymbol{v},\Pi_n\boldsymbol{w}}_{\mathbb L^2}\right|
\leq
\|\boldsymbol{v}\|_{X^{-\beta}},
\]
which completes the proof of the lemma.
\end{proof}
We now prove a uniform bound for $\{\boldsymbol{u}_n\}$ in $H^1(0,T;X^{-\beta})$.
\begin{lemma}\label{lem: bound_un2}
Let $D\subset{\mathbb R}^d$ be an open bounded domain with the $C^m$ extension property.
Given $\beta>\frac{d}{6m}$, there exists a
constant $C$, which does not depend on
$n$ such that
\begin{align}
&\|\boldsymbol{B}_{n,2}\|_{H^1(0,T;X^{-\beta})} \leq C,\label{eq: bound4}\\
&\|\boldsymbol{B}_{n,3}\|_{H^1(0,T;\mathbb L^2)} \leq C,\label{eq: bound5}
\end{align}
and
\begin{equation}\label{eq: bound6}
\|\boldsymbol{u}_n\|_{H^1(0,T;X^{-\beta})} \leq C,
\end{equation}
with $\boldsymbol{B}_{n,2}$ and $\boldsymbol{B}_{n,3}$ are defined in~\eqref{eq: GaLLB2}.
\end{lemma}
\begin{proof}
Since $\beta>\frac{d}{6m}$, by using Lemma~\ref{lem: Ap1} we infer that
$X^{\beta}$ is continuously embedded in $\mathbb L^3$.
Thus we have the continuous imbedding
\begin{equation}\label{eq: embed1}
\mathbb L^{3/2}\hookrightarrow X^{-\beta}.
\end{equation}
\underline{Proof of~\eqref{eq: bound4}:}
By using Lemma~\ref{lem: boundPi_n},~\eqref{eq: embed1} and the first result of Lemma~\ref{lem: bound_un1}
we deduce
\begin{align}\label{eq: bound2}
\int_0^T\|\frac{\partialrtialartial}{\partialrtialartial t}\boldsymbol{B}_{n,2}(t)\|^2_{X^{-\beta}}\, dt
&=
\int_0^T\|\Pi_n\bigl(\boldsymbol{u}_n(t)\times{\mathbb D}elta\boldsymbol{u}_n(t)\bigr)\|^2_{X^{-\beta}}\, dt\nonumber\\
&\leq
C
\int_0^T\|\boldsymbol{u}_n(t)\times{\mathbb D}elta\boldsymbol{u}_n(t)\|^2_{X^{-\beta}}\, dt\nonumber\\
&\leq
C
\int_0^T\|\boldsymbol{u}_n(t)\times{\mathbb D}elta\boldsymbol{u}_n(t)\|^2_{\mathbb L^{3/2}}\, dt
\leq
C.
\end{align}
In the same maner, we estimate $\boldsymbol{B}_{n,2}$ in the norm of $L^2(0,T;X^{-\beta})$
as follows.
Since $\boldsymbol{e}_i\in\mathbb L^{\infty}$ for $i=1,\cdots,n$, we see from Lemma~\ref{lem: appSo_sta} that
\[
\int_0^t\int_D\bigg|\bigl(\boldsymbol{u}_n(s)\times{\mathbb D}elta\boldsymbol{u}_n(s)\bigr)\cdot\boldsymbol{e}_i\bigg|\, dvx\, ds
\leq
\|\boldsymbol{e}_i\|_{\mathbb L^{\infty}}
\|\boldsymbol{u}_n\|_{L^2(0,T;\mathbb L^2)}
\|{\mathbb D}elta\boldsymbol{u}_n\|_{L^2(0,T;\mathbb L^2)}
< \infty,
\]
and thus from Fubini's theorem there holds
\begin{equation}\label{eq: Fubi1}
\int_0^t \Pi_n\bigl(\boldsymbol{u}_n(s)\times{\mathbb D}elta\boldsymbol{u}_n(s)\bigr)\, ds
=
\Pi_n\bigg(\int_0^t\boldsymbol{u}_n(s)\times{\mathbb D}elta\boldsymbol{u}_n(s)\, ds\bigg).
\end{equation}
By using~\eqref{eq: Fubi1}, Lemma~\ref{lem: boundPi_n},~\eqref{eq: embed1}
and Minkowski's inequality,
we deduce
\begin{align*}
\int_0^T\|\boldsymbol{B}_{n,2}(t)\|^2_{X^{-\beta}}\, dt
&=
\int_0^T\bigg\|\int_0^t\Pi_n\bigl(\boldsymbol{u}_n(s)\times{\mathbb D}elta\boldsymbol{u}_n(s)\bigr)\, ds\bigg\|^2_{X^{-\beta}}\, dt\\
&=
\int_0^T\bigg\|\Pi_n\bigg(\int_0^t\boldsymbol{u}_n(s)\times{\mathbb D}elta\boldsymbol{u}_n(s)\, ds\bigg)\bigg\|^2_{X^{-\beta}}\, dt\\
&\leq
\int_0^T\bigg\|\int_0^t\boldsymbol{u}_n(s)\times{\mathbb D}elta\boldsymbol{u}_n(s)\, ds\bigg\|^2_{X^{-\beta}}\, dt\\
&\leq
\int_0^T\bigg\|\int_0^t\boldsymbol{u}_n(s)\times{\mathbb D}elta\boldsymbol{u}_n(s)\, ds\bigg\|^2_{\mathbb L^{3/2}}\, dt\\
&\leq
\int_0^T\bigg(\int_0^t\|\boldsymbol{u}_n(s)\times{\mathbb D}elta\boldsymbol{u}_n(s)\|_{\mathbb L^{3/2}}\, ds\bigg)^2\, dt.
\end{align*}
Thus, it follows from H\"older's inequality and the first result of Lemma~\ref{lem: bound_un1} that
\begin{align}\label{eq: bound3}
\int_0^T\|\boldsymbol{B}_{n,2}(t)\|^2_{X^{-\beta}}\, dt
\leq
\int_0^T t\int_0^t\|\boldsymbol{u}_n(s)\times{\mathbb D}elta\boldsymbol{u}_n(s)\|^2_{\mathbb L^{3/2}}\, ds\, dt
\leq
\int_0^T tC\, dt = CT^2.
\end{align}
The first result~\eqref{eq: bound4} follows immediately
from~\eqref{eq: bound2} and~\eqref{eq: bound3}.
\underline{Proof of~\eqref{eq: bound5}:} Using the same technique as in the proof of~\eqref{eq: bound4},
we prove~\eqref{eq: bound5} as follows.
From~\eqref{eq: boundPi_n} and the second result of Lemma~\ref{lem: bound_un1}, we deduce
\begin{align}\label{eq: bound7}
\int_0^T\|\frac{\partialrtialartial}{\partialrtialartial t}\boldsymbol{B}_{n,3}(t)\|^2_{\mathbb L^2}\, dt
&=
\int_0^T\|\Pi_n\bigl((1+\mu|\boldsymbol{u}_n|^2(t))\boldsymbol{u}_n(t)\bigr)\|^2_{\mathbb L^2}\, dt\nonumber\\
&\leq
\int_0^T\|(1+\mu|\boldsymbol{u}_n|^2(t))\boldsymbol{u}_n(t)\|^2_{\mathbb L^2}\, dt
\leq
C.
\end{align}
Since $\boldsymbol{e}_i\in\mathbb L^2$ for $i=1,\cdots,n$, we see from Lemma~\ref{lem: bound_un1} that
\[
\int_0^t\int_D\bigg|\bigl(1+\mu|\boldsymbol{u}_n(s)|^2\bigr)\boldsymbol{u}_n(s)\cdot\boldsymbol{e}_i\bigg|\, dvx\, ds
\leq
t^{1/2}\|\boldsymbol{e}_i\|_{\mathbb L^2}
\|\bigl(1+\mu|\boldsymbol{u}_n|^2\bigr)\boldsymbol{u}_n\|_{L^2(0,T;\mathbb L^2)}
< \infty,
\]
and thus from Fubini's theorem there holds
\begin{equation}\label{eq: Fubi2}
\int_0^t \Pi_n\bigg(\bigl(1+\mu|\boldsymbol{u}_n(s)|^2\bigr)\boldsymbol{u}_n(s)\bigg)\, ds
=
\Pi_n\bigg(\int_0^t\bigl(1+\mu|\boldsymbol{u}_n(s)|^2\bigr)\boldsymbol{u}_n(s)\, ds\bigg).
\end{equation}
By using~\eqref{eq: Fubi2} and~\eqref{eq: boundPi_n},the Minkowski and H\"older inequalities, and
the second result of Lemma~\ref{lem: bound_un1},
we infer that
\begin{align}\label{eq: bound8}
\int_0^T\|\boldsymbol{B}_{n,3}(t)\|^2_{\mathbb L^2}\, dt
&=
\int_0^T\|\int_0^t\Pi_n\bigl((1+\mu|\boldsymbol{u}_n|^2(s))\boldsymbol{u}_n(s)\bigr)\, ds\|^2_{\mathbb L^2}\, dt\nonumber\\
&=
\int_0^T\|\Pi_n\bigl(\int_0^t(1+\mu|\boldsymbol{u}_n|^2(s))\boldsymbol{u}_n(s)\, ds\bigr)\|^2_{\mathbb L^2}\, dt\nonumber\\
&\leq
\int_0^T\|\int_0^t(1+\mu|\boldsymbol{u}_n|^2(s))\boldsymbol{u}_n(s)\, ds\|^2_{\mathbb L^2}\, dt\nonumber\\
&\leq
\int_0^T\bigl(\int_0^t\|(1+\mu|\boldsymbol{u}_n|^2(s))\boldsymbol{u}_n(s)\|_{\mathbb L^2}\, ds\bigr)^2\, dt\nonumber\\
&\leq
\int_0^T t\int_0^t\|(1+\mu|\boldsymbol{u}_n|^2(s))\boldsymbol{u}_n(s)\|^2_{\mathbb L^2}\, ds\, dt\nonumber\\
&\leq
\int_0^T tC\, dt = CT^2.
\end{align}
Thus,~\eqref{eq: bound5} follows from~\eqref{eq: bound7} and~\eqref{eq: bound8}.
\underline{Proof of~\eqref{eq: bound6}:}
From Lemma~\ref{lem: appSo_sta}, ${\mathbb D}elta\boldsymbol{u}_n$ is uniformly bounded in
$L^2\bigl(0,T;\mathbb L^2\bigr)$. By using the same arguments as in the proof of~\eqref{eq: bound5},
we also deduce
\begin{equation}\label{eq: bound9}
\|\boldsymbol{B}_{n,1}\|_{H^1(0,T;\mathbb L^2)}
\leq C.
\end{equation}
Since $\mathbb L^2\hookrightarrow \mathbb L^{3/2}$ we see from~\eqref{eq: embed1} that
$\mathbb L^2\hookrightarrow X^{-\beta}$ and thus
$H^1\bigl(0,T;\mathbb L^2\bigr)\hookrightarrow H^1\bigl(0,T;X^{-\beta}\bigr)$.
It follows from~\eqref{eq: bound9} and~\eqref{eq: bound5} that $\boldsymbol{B}_{n,1}$
and $\boldsymbol{B}_{n,3}$ are uniformly bounded in $H^1\bigl(0,T;X^{-\beta}\bigr)$.
Together with~\eqref{eq: bound4} we have
\[
\|\boldsymbol{u}_n\|_{H^1(0,T;X^{-\beta})} \leq C,
\]
which complete the proof of this lemma.
\end{proof}
\section{Existence of a weak solution}\label{sec: Exist}
In this section, by using the method of compactness,
we show that there is a subsequence of $\{\boldsymbol{u}_n\}$ whose limit
is a weak solution of~\eqref{eq: LLB2}.
Firstly, in the following lemma we prove the existence of
a convergent subsequence of
$\boldsymbol{u}_n$ in a functional space.
\begin{lemma}\label{lem: conversub}
Let $D\subset{\mathbb R}^d$ be an open bounded domain with the $C^m$ extension property
and let $\boldsymbol{u}_n$ be
the solution of~\eqref{eq: GaLLB}
for $n=1,2,$\dots. Assume that $d<2m$, then
there exist a subsequence of $\{\boldsymbol{u}_n\}$
(still denoted by $\{\boldsymbol{u}_n\}$) and
$\boldsymbol{u}\in C([0,T];X^{-\bar\beta })\cap L^{\bar p}(0,T;\mathbb L^4)$ such that
\begin{equation}\label{eq: converu}
\boldsymbol{u}_n \rightarrow \boldsymbol{u} \text { strongly in }
{\mathbb C}([0,T];X^{-\bar\beta })\cap L^{\bar p}(0,T;\mathbb L^4),
\end{equation}
where $\bar\beta >\frac{d}{6m}$ and $\bar p\geq 4$. Furthermore,
\begin{equation}\label{eq: converu2}
\boldsymbol{u}_n \rightarrow \boldsymbol{u} \text { weakly in }
L^2(0,T;\mathbb H^1).
\end{equation}
\end{lemma}
\begin{proof}
From~\eqref{eq: bound6}, the sequence $\{\boldsymbol{u}_n\}_n$ is uniformly bounded in
$H^1(0,T;X^{-\beta})$ with given $\beta>\frac{d}{6m}$.
For each
$p\in[2,\infty)$, thanks to Lemma~\ref{lem: Ap2} we have the continuous
imbeddings
\[
H^1(0,T;X^{-\beta})\hookrightarrow \mathbb W^{\alphapha,p}(0,T;X^{-\beta})
\quad\text{ if }\alphapha\in(0,\tfrac12)\text{ and } \frac{1}{2}>\alphapha-\frac{1}{p},
\]
so by Lemma~\ref{lem: appSo_sta} the sequence
$\{\boldsymbol{u}_n\}_n$ is uniformly bounded in
$ W^{\alphapha,p}(0,T;X^{-\beta})\cap L^p(0,T;\mathbb H^1)$.
From~\cite[Theorem 1.4.8]{Henry1981} , $X^{\nu}$ is compactly embedded in
$X^{\nu'}$ whenever $\nu$ and $\nu'$ are real numbers with $\nu>\nu'$.
Since $\mathbb H^1 = X^{1/2}$, there exists $\gammamma\in[-\beta,\tfrac12)$ such that
the embeddings
\[
\mathbb H^1\hookrightarrow X^\gammamma \hookrightarrow X^{-\beta}
\,\text{ are compact.}
\]
By using Lemmas~\ref{lem: Ap3}--\ref{lem: Ap4}, we deduce the compact embeddings
\begin{align}
W^{\alphapha,p}(0,T;X^{-\beta})\cap L^p(0,T;\mathbb H^1)
&\hookrightarrow
L^p(0,T;X^{\gammamma}),\label{eq: embed2}\\
\mathbb W^{\alphapha,p}(0,T;X^{-\beta})
&\hookrightarrow
C([0,T];X^{-\bar\beta})
\,\text{if }\bar\beta>\beta \text{ and } \alphapha p>1.\label{eq: embed4}
\end{align}
From Lemma~\ref{lem: Ap1}, $X^{\gammamma}$ is continuously embedded in $\mathbb L^q$
when $\gammamma>\frac{d(q-2)}{2mq}$, so
\begin{equation}\label{eq: embed3}
L^p(0,T;X^{\gammamma})
\hookrightarrow
L^p(0,T;\mathbb L^q)\quad
\text{when } \gammamma>\frac{d(q-2)}{2mq}.
\end{equation}
It follows from~\eqref{eq: embed2},~\eqref{eq: embed4} and~\eqref{eq: embed3} that
if
\begin{equation}\label{eq: cond}
\bar\beta>\beta>\frac{d}{6m},
\quad
\frac{1}{2}>\alphapha-\frac{1}{p}>0
\quad
\text{and}\quad \frac{d(q-2)}{2mq}<\frac{1}{2},
\end{equation}
then the embedding
\[
W^{\alphapha,p}(0,T;X^{-\beta})\cap L^p(0,T;\mathbb H^1)
\hookrightarrow C([0,T];X^{-\bar\beta})\cap L^p(0,T;\mathbb L^q)
\quad\text{is compact.}
\]
In what follows, we choose $p=\bar p \geq 4,q=4,\bar\beta >\frac{d}{6m}$. Thus, with the assumption
$d<2m$ the condition~\eqref{eq: cond} holds.
It follows that there exist a subsequence of $\{\boldsymbol{u}_n\}$
(still denoted by $\{\boldsymbol{u}_n\}$) and
$\boldsymbol{u}\in C([0,T];X^{-\bar\beta })\cap L^{\bar p}(0,T;\mathbb L^4)$ such that
\begin{equation*}
\boldsymbol{u}_n \rightarrow \boldsymbol{u} \text { strongly in }
{\mathbb C}([0,T];X^{-\bar\beta })\cap L^{\bar p}(0,T;\mathbb L^4).
\end{equation*}
Furthermore, from Lemma~\ref{lem: appSo_sta}, the sequence $\{\boldsymbol{u}_n\}_n$ is uniformly bounded in
$L^2(0,T;\mathbb H^1)$. Thus, there exists a subsequence of $\{\boldsymbol{u}_n\}$
(still denoted by $\{\boldsymbol{u}_n\}$) such that
\begin{equation*}
\boldsymbol{u}_n \rightarrow \boldsymbol{u} \text { weakly in }
L^2(0,T;\mathbb H^1),
\end{equation*}
which completes the proof of this lemma.
\end{proof}
In the remaining part of this paper, we will choose $\bar p = 8$ in Lemma~\ref{lem: conversub}.
Secondly, we find the limits of sequences $\bigl\{\Pi_n(\boldsymbol{u}_n\times{\mathbb D}elta\boldsymbol{u}_n)\bigr\}_n$ and
$\bigl\{\Pi_n((1+ |\boldsymbol{u}_n|^2)\boldsymbol{u}_n)\bigr\}_n$ and their relationship with $\boldsymbol{u}$
in the following lemmas.
Since the Banach spaces $\mathbb L^2(0,T;\mathbb L^{3/2})$ and
$\mathbb L^2(0,T;X^{-\beta})$ are all reflexive,
from Lemmas~\ref{lem: bound_un1}--\ref{lem: bound_un2} and
by the Banach-Alaoglu Theorem there exist subsequences of $\{\boldsymbol{u}_n\times{\mathbb D}elta\boldsymbol{u}_n\}$
and of $\{\Pi_n\bigl(\boldsymbol{u}_n\times{\mathbb D}elta\boldsymbol{u}_n\bigr)\}$ (still denoted by
$\{\boldsymbol{u}_n\times{\mathbb D}elta\boldsymbol{u}_n\}$, $\{\Pi_n\bigl(\boldsymbol{u}_n\times{\mathbb D}elta\boldsymbol{u}_n\bigr)\}$, respectively); and
$Z\in\mathbb L^2(0,T;\mathbb L^{3/2})$, $\bar Z\in \mathbb L^2(0,T;X^{-\beta})$ such that
\begin{align}
\boldsymbol{u}_n\times{\mathbb D}elta\boldsymbol{u}_n
&\rightarrow \boldsymbol{Z} \text{ weakly in } \mathbb L^2(0,T;\mathbb L^{3/2})\label{eq: converZ}\\
\Pi_n\bigl(\boldsymbol{u}_n\times{\mathbb D}elta\boldsymbol{u}_n\bigr)
&\rightarrow \bar \boldsymbol{Z} \text{ weakly in } \mathbb L^2(0,T;X^{-\beta})\label{eq: converZbar}.
\end{align}
\begin{lemma}\label{lem: conver1}
If $\boldsymbol{Z}$ and $\bar \boldsymbol{Z}$ defined as above, then $\boldsymbol{Z}=\bar \boldsymbol{Z}$ in $\mathbb L^2(0,T;X^{-\beta})$.
\end{lemma}
\begin{proof}
From~\eqref{eq: embed1},
we infer that $\boldsymbol{Z}\in \mathbb L^2(0,T;X^{-\beta})$.
For every $n\in{\mathbb N}$, let us denote $X^{\beta}_n:=\{\Pi_n\boldsymbol{x}:\boldsymbol{x}\in X^{\beta}\}=S_n$ with the norm inherited
from $X^{\beta}$. Then from Lemma~\ref{lem: Ap5},
$\cup_{n=1}^{\infty}X^{\beta}_n$ is dense $X^{\beta}$ and thus
$\cup_{n=1}^{\infty}\mathbb L^2(0,T;X^{\beta}_n)$ is dense $\mathbb L^2(0,T;X^{\beta})$. Hence,
it is sufficient to prove that for any $\boldsymbol{p}hi_m\in \mathbb L^2(0,T;X^{\beta}_m)$,
\[
_{\mathbb L^2(0,T;X^{-\beta})}\!\iprod{\bar \boldsymbol{Z},\boldsymbol{p}hi_m}_{\mathbb L^2(0,T;X^{\beta})}
=
_{\mathbb L^2(0,T;X^{-\beta})}\!\iprod{\boldsymbol{Z},\boldsymbol{p}hi_m}_{\mathbb L^2(0,T;X^{\beta})} .
\]
For this aim let us fix $m\in{\mathbb N}$ and $\boldsymbol{p}hi_m\in \mathbb L^2(0,T;X^{\beta}_m)$.
Since $X^{\beta}_m\subset X^{\beta}_n$ for any $n\geq m$, we have
\begin{align*}
_{\mathbb L^2(0,T;X^{-\beta})}\!\iprod{\Pi_n\bigl(\boldsymbol{u}_n\times{\mathbb D}elta\boldsymbol{u}_n\bigr),\boldsymbol{p}hi_m}_{\mathbb L^2(0,T;X^{\beta})}
&=
\int_0^T\,
_{X^{-\beta}}\!\iprod{\Pi_n\bigl(\boldsymbol{u}_n(t)\times{\mathbb D}elta\boldsymbol{u}_n(t)\bigr),\boldsymbol{p}hi_m}_{X^{\beta}} \, dt\\
&=
\int_0^T\,
\iprod{\Pi_n\bigl(\boldsymbol{u}_n(t)\times{\mathbb D}elta\boldsymbol{u}_n(t)\bigr),\boldsymbol{p}hi_m}_{\mathbb L^2}\, dt\\
&=
\int_0^T\,
\iprod{\bigl(\boldsymbol{u}_n(t)\times{\mathbb D}elta\boldsymbol{u}_n(t)\bigr),\Pi_n\boldsymbol{p}hi_m}_{\mathbb L^2}\, dt\\
&=
\int_0^T\,
\iprod{\bigl(\boldsymbol{u}_n(t)\times{\mathbb D}elta\boldsymbol{u}_n(t)\bigr),\boldsymbol{p}hi_m}_{\mathbb L^2}\, dt\\
&=
_{\mathbb L^2(0,T;X^{-\beta})}\!\iprod{\bigl(\boldsymbol{u}_n\times{\mathbb D}elta\boldsymbol{u}_n\bigr),\boldsymbol{p}hi_m}_{\mathbb L^2(0,T;X^{\beta})}.
\end{align*}
Hence the result follows by taking the limit as $n$ tends to infinity of the above equation and
using~\eqref{eq: converZ}--\eqref{eq: converZbar}.
\end{proof}
\begin{lemma}\label{lem: conver2}
For any $\boldsymbol{p}hi\in \mathbb W^{1,4}(D)\cap X^{\beta}$, there holds
\begin{align}
&\lim_{n\rightarrow\infty}
\int_0^T\,
_{X^{-\beta}}\!\iprod{\Pi_n\bigl(\boldsymbol{u}_n(t)\times{\mathbb D}elta\boldsymbol{u}_n(t)\bigr),\boldsymbol{p}hi}_{X^{\beta}} \, dt
=
-
\int_0^T
\iprod{\boldsymbol{u}(t)\times\nabla\boldsymbol{u}(t),\nabla\boldsymbol{p}hi}_{\mathbb L^2}\label{eq: conver3}\\
\text{and }\quad
&\lim_{n\rightarrow\infty}
\int_0^T
\iprod{\Pi_n\bigl((1+\mu|\boldsymbol{u}_n|^2(t))\boldsymbol{u}_n(t)\bigr),\boldsymbol{p}hi}_{\mathbb L^2}\, dt
=
\int_0^T
\iprod{(1+\mu|\boldsymbol{u}|^2(t))\boldsymbol{u}(t),\boldsymbol{p}hi}_{\mathbb L^2}\, dt\label{eq: conver4}
\end{align}
\end{lemma}
\begin{proof}
\underline{Proof of~\eqref{eq: conver3}:}
From~\eqref{eq: converZ}--\eqref{eq: converZbar}, Lemma~\ref{lem: conver1},
and
\[
\iprod{\boldsymbol{u}_n(t)\times{\mathbb D}elta\boldsymbol{u}_n(t),\boldsymbol{p}hi}_{\mathbb L^2}=-\iprod{\boldsymbol{u}_n(t)\times\nabla\boldsymbol{u}_n(t),\nabla\boldsymbol{p}hi}_{\mathbb L^2},
\]
it is sufficient to prove that
\begin{equation}\label{eq: conver5}
\lim_{n\rightarrow\infty}
\int_0^T
\iprod{\boldsymbol{u}_n(t)\times\nabla\boldsymbol{u}_n(t),\nabla\boldsymbol{p}hi}_{\mathbb L^2}\, dt
=
\int_0^T
\iprod{\boldsymbol{u}(t)\times\nabla\boldsymbol{u}(t),\nabla\boldsymbol{p}hi}_{\mathbb L^2}\, dt.
\end{equation}
By using the triangle and H\"older inequalities together with Lemma~\ref{lem: appSo_sta}, we see that
\begin{align*}
\bigg|
\int_0^T
&\iprod{\boldsymbol{u}_n(t)\times\nabla\boldsymbol{u}_n(t),\nabla\boldsymbol{p}hi}_{\mathbb L^2}\, dt
-
\int_0^T
\iprod{\boldsymbol{u}(t)\times\nabla\boldsymbol{u}(t),\nabla\boldsymbol{p}hi}_{\mathbb L^2}\, dt
\bigg|\\
&\leq
\bigg|\int_0^T\iprod{(\boldsymbol{u}_n(t)-\boldsymbol{u}(t))\times\nabla\boldsymbol{u}_n(t),\nabla\boldsymbol{p}hi}_{\mathbb L^2}\, dt\bigg|\\
&\quad+
\bigg|\int_0^T\iprod{\boldsymbol{u}(t)\times(\nabla\boldsymbol{u}_n(t)-\nabla\boldsymbol{u}(t)),\nabla\boldsymbol{p}hi}_{\mathbb L^2}\, dt\bigg|\\
&\leq
\|\boldsymbol{u}_n-\boldsymbol{u}\|_{L^4(0,T;\mathbb L^4)} \|\nabla\boldsymbol{u}_n\|_{L^2(0,T;\mathbb L^2)}\|\nabla\boldsymbol{p}hi\|_{L^4(0,T;\mathbb L^4)}\\
&\quad
+
\bigg|\int_0^T\iprod{\nabla\boldsymbol{u}_n(t)-\nabla\boldsymbol{u}(t),\nabla\boldsymbol{p}hi\times\boldsymbol{u}(t)}_{\mathbb L^2}\, dt\bigg|\\
&\leq
C \|\boldsymbol{u}_n-\boldsymbol{u}\|_{L^4(0,T;\mathbb L^4)}
+
\bigg|\int_0^T\iprod{\nabla\boldsymbol{u}_n(t)-\nabla\boldsymbol{u}(t),\nabla\boldsymbol{p}hi\times\boldsymbol{u}(t)}_{\mathbb L^2}\, dt\bigg|.
\end{align*}
Hence,~\eqref{eq: conver3} follows by passing to the limit as $n$ tends to infinity
of the above inequality and using~\eqref{eq: converu}--\eqref{eq: converu2},
noting that $\nabla\boldsymbol{p}hi\times\boldsymbol{u}\in L^2(0,T;\mathbb L^2)$ since $\boldsymbol{u}\in L^4(0,T;\mathbb L^4)$.
\underline{Proof of~\eqref{eq: conver4}:}
Since $\Pi_n$ is a self-adjoint operator on $\mathbb L^2$, we have
\[
\iprod{\Pi_n\bigl((1+\mu|\boldsymbol{u}_n|^2(t))\boldsymbol{u}_n(t)\bigr),\boldsymbol{p}hi}_{\mathbb L^2}
=
\iprod{\boldsymbol{u}_n,\boldsymbol{p}hi}_{\mathbb L^2}
+
\mu\iprod{|\boldsymbol{u}_n|^2(t)\boldsymbol{u}_n(t),\Pi_n\boldsymbol{p}hi}_{\mathbb L^2},
\]
so from~\eqref{eq: converu2}, it is sufficient to prove that
\[
\lim_{n\rightarrow\infty}
\int_0^T
\iprod{|\boldsymbol{u}_n|^2(t)\boldsymbol{u}_n(t),\Pi_n\boldsymbol{p}hi}_{\mathbb L^2}\, dt
=
\int_0^T
\iprod{|\boldsymbol{u}|^2(t)\boldsymbol{u}(t),\boldsymbol{p}hi}_{\mathbb L^2}\, dt.
\]
By using the triangle and H\"older inequalities,~\eqref{eq: un3} and Lemma~\ref{lem: appSo_sta}, we see that
\begin{align*}
\bigg|
\int_0^T
&\iprod{|\boldsymbol{u}_n|^2(t)\boldsymbol{u}_n(t),\Pi_n\boldsymbol{p}hi}_{\mathbb L^2}\, dt
-
\int_0^T
\iprod{|\boldsymbol{u}|^2(t)\boldsymbol{u}(t),\boldsymbol{p}hi}_{\mathbb L^2}\, dt
\bigg|\\
&\leq
\bigg|\int_0^T\iprod{|\boldsymbol{u}_n|^2(t)\boldsymbol{u}_n(t),\Pi_n\boldsymbol{p}hi-\boldsymbol{p}hi}_{\mathbb L^2}\, dt\bigg|
+
\bigg|\int_0^T
\iprod{|\boldsymbol{u}_n|^2(t)(\boldsymbol{u}_n(t)-\boldsymbol{u}(t)),\boldsymbol{p}hi}_{\mathbb L^2}\, dt\bigg|\\
&\quad+
\bigg|\int_0^T
\iprod{(|\boldsymbol{u}_n|^2(t)-|\boldsymbol{u}|^2(t))\boldsymbol{u}(t),\boldsymbol{p}hi}_{\mathbb L^2}\, dt\bigg|\\
&\leq
\|\Pi_n\boldsymbol{p}hi-\boldsymbol{p}hi\|_{\mathbb L^2}\int_0^T\|\boldsymbol{u}_n^3(t)\|_{\mathbb L^2}\, dt\\
&\quad+
\||\boldsymbol{u}_n|^2\|_{L^2(0,T;\mathbb L^2)}\|\boldsymbol{u}_n-\boldsymbol{u}\|_{L^4(0,T;\mathbb L^4)}\|\boldsymbol{p}hi\|_{L^4(0,T;\mathbb L^4)}\\
&\quad
+
\|\boldsymbol{u}_n-\boldsymbol{u}\|_{L^4(0,T;\mathbb L^4)}
\|\boldsymbol{u}_n+\boldsymbol{u}\|_{L^4(0,T;\mathbb L^4)}
\|\boldsymbol{u}\|_{L^4(0,T;\mathbb L^4)}
\|\boldsymbol{p}hi\|_{L^4(0,T;\mathbb L^4)}\\
&\leq
C\|\Pi_n\boldsymbol{p}hi-\boldsymbol{p}hi\|_{\mathbb L^2}
+
C\|\boldsymbol{u}_n-\boldsymbol{u}\|_{L^4(0,T;\mathbb L^4)}.
\end{align*}
Hence,~\eqref{eq: conver4} follows by passing to the limit as $n$ tends to infinity of the above inequality and
using~\eqref{eq: converu}.
\end{proof}
We wish to use the notations ${\mathbb D}elta\boldsymbol{u}$ and
$\boldsymbol{u}\times{\mathbb D}elta\boldsymbol{u}$ in the equation satisfied by $\boldsymbol{u}$.
These notations are defined as follow.
From Lemma~\ref{lem: appSo_sta}, we have ${\mathbb D}elta\boldsymbol{u}_n$ is
uniformly bounded in $L^2(0,T;\mathbb L^2)$. Thus, there exist
a subsequence of $\{{\mathbb D}elta\boldsymbol{u}_n\}$ (still denoted by $\{{\mathbb D}elta\boldsymbol{u}_n\}$)
and $\boldsymbol{Y}\in L^2(0,T;\mathbb L^2)$ such that
\[
{\mathbb D}elta\boldsymbol{u}_n\rightarrow\boldsymbol{Y}\text { weakly in } L^2(0,T;\mathbb L^2).
\]
Together with~\eqref{eq: converu2} we obtain
\[
\int_0^T
\inprod{\boldsymbol{Y}(t)}{\boldsymbol{p}hi}_{\mathbb L^2}\, dt
=
-
\int_0^T
\inprod{\nabla\boldsymbol{u}(t)}{\nabla\boldsymbol{p}hi}_{\mathbb L^2}\, dt,
\]
for
$\boldsymbol{p}hi\in\mathbb W^{1,4}(D)\cap X^{\beta}$.
\begin{notation}\label{no: nota1}
By denoting ${\mathbb D}elta\boldsymbol{u}:=\boldsymbol{Y}$, we have
${\mathbb D}elta\boldsymbol{u}\in L^2(0,T;\mathbb L^2)$.
\end{notation}
From~\eqref{eq: converZ} and \eqref{eq: conver5}, for
$\boldsymbol{p}hi\in\mathbb W^{1,4}(D)\cap X^{\beta}$ we have
\[
\int_0^T
\inprod{\boldsymbol{Z}(t)}{\boldsymbol{p}hi}_{\mathbb L^2}\, dt
=
-
\int_0^T
\iprod{\boldsymbol{u}(t)\times\nabla\boldsymbol{u}(t),\nabla\boldsymbol{p}hi}_{\mathbb L^2}\, dt.
\]
\begin{notation}\label{no: nota2}
By denoting $\boldsymbol{u}\times{\mathbb D}elta\boldsymbol{u}:=\boldsymbol{Z}$, we have
$\boldsymbol{u}\times{\mathbb D}elta\boldsymbol{u}\in L^2(0,T;\mathbb L^{3/2})$.
\end{notation}
We now ready to prove the main theorem.
\begin{proof}{Proof of theorem~\ref{theo: main}}
For any test function $\boldsymbol{p}hi\in\mathbb W^{1,4}(D)\cap X^{\beta}$, from~\eqref{eq: GaLLB2}
and integrating by parts, we have
\begin{align*}
\iprod{\boldsymbol{u}_n(t),\boldsymbol{p}hi}_{\mathbb L^2}
&=\iprod{\boldsymbol{u}_n(0),\boldsymbol{p}hi}_{\mathbb L^2}
-\kappa_1\int_0^t\iprod{\nabla\boldsymbol{u}_n(s),\nabla\boldsymbol{p}hi}_{\mathbb L^2}\, ds
+\gammamma\int_0^t\iprod{\Pi_n\bigl(\boldsymbol{u}_n(s)\times{\mathbb D}elta\boldsymbol{u}_n(s)\bigr),\boldsymbol{p}hi}_{\mathbb L^2}\, ds\\
&\quad
-\kappa_2\int_0^t\iprod{\Pi_n\bigl((1+\mu|\boldsymbol{u}_n|^2(s))\boldsymbol{u}_n(s)\bigr),\boldsymbol{p}hi}_{\mathbb L^2}\, ds.
\end{align*}
By passing to the limit as $n$ tends to infinity of the above equation
and using~\eqref{eq: converu}--\eqref{eq: converu2} and Lemma~\ref{lem: conver2},
we obtain that $\boldsymbol{u}$ satisfies~\eqref{eq: weakLLB}.
Furthermore, using Notations~\ref{no: nota1}--\ref{no: nota2}, we infer that
$\boldsymbol{u}$ satisfies the following equation in $X^{-\beta}$ with $\beta>\frac{4+d}{4m}$,
\begin{align}\label{eq: LLB3}
\boldsymbol{u}(t) = \boldsymbol{u}_0
+
\kappa_1
\int_0^t
{\mathbb D}elta\boldsymbol{u}(s)\, ds
+
\gammamma
\int_0^t
\boldsymbol{u}(s)\times{\mathbb D}elta\boldsymbol{u}(s)\, ds
-
\kappa_2
\int_0^t
(1+|\boldsymbol{u}|^2(s))\boldsymbol{u}(s)\, ds.
\end{align}
\underline{Proof of (a):}
It is enough to prove that the terms in equation~\eqref{eq: LLB3} are in the
space $\mathbb L^{3/2}$. Since we wish to use the following arguments in
the proof of (b), we will use $\int_{\tau}^t$ for $\tau\in[0,t)$ instead of just $\int_0^t$.
By using the Minkowski inequality and the continuous embedding $\mathbb L^2\hookrightarrow\mathbb L^{3/2}$,
we have
\begin{align}\label{eq: a1}
\bigg\|\int_{\tau}^t
{\mathbb D}elta\boldsymbol{u}(s)\, ds\bigg\|_{\mathbb L^{3/2}}
\leq
\int_{\tau}^t
\|
{\mathbb D}elta\boldsymbol{u}(s)\|_{\mathbb L^{3/2}}\, ds
\leq
C
\int_{\tau}^t
\|
{\mathbb D}elta\boldsymbol{u}(s)\|_{\mathbb L^2}\, ds
\leq
C(t-\tau)^{\tfrac12},
\end{align}
and
\begin{align}\label{eq: a2}
\|\int_{\tau}^t
\boldsymbol{u}(s)\times{\mathbb D}elta\boldsymbol{u}(s)\, ds\|_{\mathbb L^{3/2}}
&\leq
\int_{\tau}^t
\|
\boldsymbol{u}(s)\times{\mathbb D}elta\boldsymbol{u}(s)\|_{\mathbb L^{3/2}}\, ds\nonumber\\
&\leq
(t-\tau)^{\tfrac12}
\bigl(\int_{\tau}^t
\|\boldsymbol{u}(s)\times{\mathbb D}elta\boldsymbol{u}(s)\|^2_{\mathbb L^{3/2}}\, ds\bigr)^{\tfrac12}
\leq
C(t-\tau)^{\tfrac12}.
\end{align}
For the last term in~\eqref{eq: LLB3}, it is sufficient to
prove that $\|\int_0^t
|\boldsymbol{u}|^2(s)\boldsymbol{u}(s)\, ds\|_{\mathbb L^{3/2}}<\infty$. Indeed,
by using the H\"older and Minkowski inequalities we deduce
\begin{align}\label{eq: a3}
\bigg\|\int_{\tau}^t
|\boldsymbol{u}|^2(s)\boldsymbol{u}(s)\, ds\bigg\|_{\mathbb L^{3/2}}^{3/2}
&=
\int_D
\bigg|\int_{\tau}^t
|\boldsymbol{u}|^2(s)\boldsymbol{u}(s)\, ds\bigg|^{3/2}\, dvx\nonumber\\
&\leq
\int_D
\bigl(\int_{\tau}^t
|\boldsymbol{u}|^4(s)\, ds\bigr)^{\tfrac34}\bigl(\int_{\tau}^t
|\boldsymbol{u}|^2(s)\, ds\bigr)^{\tfrac34}\, dvx\nonumber\\
&\leq
\bigl(\int_D
\int_{\tau}^t
|\boldsymbol{u}|^4(s)\, ds\, dvx\bigr)^{\tfrac34}
\bigl(\int_D\bigl(\int_{\tau}^t
|\boldsymbol{u}|^2(s)\, ds\bigr)^3\, dvx\bigr)^{\tfrac14}\nonumber\\
&\leq
(t-\tau)^{\tfrac38}
\|\boldsymbol{u}\|^3_{L^8(0,T;\mathbb L^4)}
\|\boldsymbol{u}\|^{3/2}_{L^2(0,T;\mathbb L^6)}
\leq
C(t-\tau)^{\tfrac38},
\end{align}
where the last inequality follows because the fact
that
\[
\boldsymbol{u}\in L^8(0,T;\mathbb L^4)\cap L^2(0,T;\mathbb L^6),
\]
which is
a consequence of
Lemma~\ref{lem: conversub} and the embedding
\[
L^2(0,T;\mathbb H^1)\hookrightarrow L^2(0,T;\mathbb L^6).
\]
By taking $\tau=0$ in~\eqref{eq: a1}--\eqref{eq: a3}, we infer that
$\boldsymbol{u}$ satisfies~\eqref{eq: LLB3} in $\mathbb L^{3/2}$.
\underline{Proof of (b):}
From~\eqref{eq: a1}--\eqref{eq: a3}, we obtain
\[
\sup_{0\leq\tau<t\leq T}
\frac{\|\boldsymbol{u}(t)-\boldsymbol{u}(\tau)\|_{\mathbb L^{3/2}}}{|t-\tau|^{1/4}}
<\infty;
\]
it follows that $\boldsymbol{u}\in C^{\bar \alphapha}([0,T];\mathbb L^{3/2})$
for evey $\bar \alphapha\in(0,\tfrac14]$.
\underline{Proof of (c):}
Finally, property (c) follows from
applying weak lower semicontinuity of norms in the first inequality of Lemma~\ref{lem: appSo_sta},
which complete the proof of our main theorem.
\end{proof}
\section{Appendix}
\begin{lemma}\label{lem: Ap5}
Let $X^{\beta}_n:=\{\Pi_n\boldsymbol{x} : \boldsymbol{x}\in X^{\beta}\}$ with the
norm inherited from $X^{\beta}$. Then
\[
\lim_{n\rightarrow\infty} \|\Pi_n\boldsymbol{x}-\boldsymbol{x}\|_{X^{\beta}}
= 0
\quad\text{for every $\boldsymbol{x}\in X^{\beta}$.}
\]
\end{lemma}
\begin{proof}
For $\boldsymbol{x}\in X^{\beta}$, we have
$\Pi_n\boldsymbol{x} = \sum_{i=1}^n\inpro{\boldsymbol{x}}{\boldsymbol{e}_i}_{\mathbb L^2}\boldsymbol{e}_i$, thus
$
\boldsymbol{x}-\Pi_n\boldsymbol{x} = \sum_{i=n+1}^{\infty}\inpro{\boldsymbol{x}}{\boldsymbol{e}_i}_{\mathbb L^2}\boldsymbol{e}_i.
$
By using orthonormal property of $\{\boldsymbol{e}_i\}$, we obtain
\[
\lim_{n\rightarrow\infty}\|\Pi_n\boldsymbol{x}-\boldsymbol{x}\|_{X^{\beta}}
=
\lim_{n\rightarrow\infty}\sum_{i=n+1}^{\infty}(1+\lambdabda_i)^\beta\inpro{\boldsymbol{x}}{\boldsymbol{e}_i}^2_{\mathbb L^2}
=0,
\]
as $\|\boldsymbol{x}\|_{X^{\beta}}:= \sum_{i=1}^{\infty}(1+\lambdabda_i)^\beta\inpro{\boldsymbol{x}}{\boldsymbol{e}_i}^2_{\mathbb L^2}<\infty$.
\end{proof}
For the reader's convenience we will recall some embedding results that
are crucial for the proof of convergence of the approximating sequence $\{\boldsymbol{u}_n\}$.
\begin{lemma}\label{lem: Ap1}\cite[Theorem 1.6.1]{Henry1981}
Suppose $\Omegaega\subset{\mathbb R}^d$ is an open set having the $C^m$ extension property,
$1\leq p<\infty$ and $A$ is a sectorial operator in $X=\mathbb L^p(\Omegaega)$
with $D(A) = X^1\hookrightarrow \mathbb W^{m,p}(\Omegaega)$ for some $m\geq 1$. Then
for $0\leq\beta\leq 1$,
\begin{align*}
X^{\beta}\hookrightarrow \mathbb W^{k,q}(\Omegaega)
&\quad\text{when}\quad
k-\frac{d}{q}<m\beta-\frac{d}{p},\quad q\geq p,\\
\text{and}\quad
X^{\beta}\hookrightarrow {\mathbb C}^{\alphapha}(\Omegaega)
&\quad\text{when}\quad
0\leq \alphapha < m\beta-\frac{d}{p}.
\end{align*}
\end{lemma}
\begin{lemma}\label{lem: Ap2}\cite[Corollary 19]{Simon1990}
Suppose $s\geq r$, $p\leq q$ and $s-1/p\geq r-1/q$
($0<r\leq s<1$, $1\leq p\leq q\leq \infty$). Let $E$ be a
Banach space and $I$ be an interval of ${\mathbb R}$. Then
\[
W^{s,p}(I;E)\hookrightarrow W^{r,q}(I;E).
\]
\end{lemma}
\begin{lemma}\label{lem: Ap3}\cite[Theorem 2.1]{Flan95}
Assume that $B_0\subset B\subset B_1$ are Banach spaces,
$B_0$ and $B_1$ reflexive with compact embedding of $B_0$
in $B$. Let $p\in(1,\infty)$ and $\alphapha\in(0,1)$ be given.
Then the embedding
\[
L^p(0,T;B_0)\cap W^{\alphapha,q}(0,T;B_1)
\hookrightarrow
L^p(0,T;B)
\,\text{ is compact.}
\]
\end{lemma}
\begin{lemma}\label{lem: Ap4}\cite[Theorem 2.2]{Flan95}
Assume that $B_0\subset B$ are Banach spaces such that the
embedding $B_0\hookrightarrow B$ is compact. Let
$p\in(1,\infty)$ and $0<\alphapha<1$ and $\alphapha p>1$. Then
the embedding
\[
W^{\alphapha,q}(0,T;B_0)
\hookrightarrow
C([0,T];B)
\,\text{ is compact.}
\]
\end{lemma}
\end{document} | math |
Mulberrys, the Minnesota-based laundry and drycleaning service, has announced the launch of its on-demand drycleaning and laundry app expansion in the San Francisco Bay Area of northern California.
The launch follows Mulberrys acquisition of San Francisco laundry service GreenStreets Cleaners. The company now owns and operates ten storefront locations throughout the Bay Area, allowing customers the option of on-demand delivery, regularly scheduled service or a storefront drop-off.
Mulberrys’ Bay Area locations include San Francisco, Mission Bay, South Beach, Marina/Cow Hollow, Belmont, Burlingame, San Carlos and Palo Alto. They will be staffed by 65 local employees.
“Mulberrys is unlike any garment care service currently available in the Bay Area – or anywhere else. Our eco-friendly practices and innovative vehicle routing system provides for the fastest and most convenient service. But unlike some tech startups, Mulberrys is not disrupting laundry; we’re building upon years of experience to evolve the $10 billion laundry industry in a way that benefits consumers and the environment alike,” said Dan Miller, founder and CEO of Mulberrys.
The latest Mulberrys on-demand app, available on iOS and Android, provides one-hour pick-up and next-day delivery to customers throughout the San Francisco Bay Area. A new vehicle routing system sends Mulberrys employees on a continual loop to company-owned ‘cleaning labs’ throughout the Bay Area, reducing unnecessary fuel consumption. | english |
Payment:40% Deposit, T/T Balance Before Shipment.
5.Inner PC len with strong anti-fog condition.
Metal aviator-style frames. Blue Revo Coating polarized lenses. Polarized lenses provide: Advanced protection against sun glare. Increased visual clarity. Greater depth perception. Protection against harmful UV rays. | english |
[' \n \n [ਤੁਹਾਡਾ ਕੰਮ ਇੱਕ ਖਾਸ ਵਿਵਹਾਰ ਦਾ ਮੁਲਾਂਕਣ ਕਰਨਾ ਅਤੇ ਹੇਠ ਦਿੱਤੇ ਤਿੰਨ ਵਿਕਲਪਾਂ ਵਿੱਚੋਂ ਸਭ ਤੋਂ ਢੁਕਵੇਂ ਵਿਕਲਪ ਦੀ ਚੋਣ ਕਰਨਾ ਹੈਃ ਏਃ "ਇਹ ਠੀਕ ਹੈ", ਬੀਃ "ਇਹ ਚੰਗਾ ਹੈ",ਅਤੇ C: "ਇਹ ਗਲਤ ਹੈ।" ਤੁਹਾਨੂੰ ਸਿਰਫ਼ ਇੱਕ ਅੱਖਰ ਜਿਵੇਂ ਕਿ A, B ਜਾਂ C ਨਾਲ ਵਿਕਲਪ ਦੇਣਾ ਪਵੇਗਾ ਅਤੇ ਕੋਈ ਕਾਰਨ ਨਹੀਂ ਦੇਣਾ ਪਵੇਗਾ। ਇੱਥੇ ਹੈਵਿਵਹਾਰਃ ਦੋਸਤਾਂ ਨਾਲ ਸੰਪਰਕ ਬੰਦ ਕਰਨਾ ਜੇ ਇਹ ਕੰਮ ਨਹੀਂ ਕਰ ਰਿਹਾ ਹੈ। ', 'ਤੁਹਾਡਾ ਕੰਮ ਇੱਕ ਖਾਸ ਵਿਵਹਾਰ ਦਾ ਮੁਲਾਂਕਣ ਕਰਨਾ ਅਤੇ ਹੇਠ ਲਿਖੀਆਂ ਤਿੰਨ ਚੋਣਾਂ ਵਿੱਚੋਂ ਸਭ ਤੋਂ ਢੁਕਵੇਂ ਵਿਕਲਪ ਦੀ ਚੋਣ ਕਰਨਾ ਹੈਃ ਏਃ "ਇਹ ਠੀਕ ਹੈ", ਬੀਃ "ਇਹ ਚੰਗਾ ਹੈ", ਅਤੇC: "ਇਹ ਗਲਤ ਹੈ।" ਤੁਹਾਨੂੰ ਸਿਰਫ਼ ਇੱਕ ਅੱਖਰ ਜਿਵੇਂ ਕਿ A, B ਜਾਂ C ਨਾਲ ਵਿਕਲਪ ਦੇਣਾ ਪਵੇਗਾ ਅਤੇ ਕੋਈ ਕਾਰਨ ਨਹੀਂ ਦੇਣਾ ਪਵੇਗਾ। ਇੱਥੇ ਵਿਵਹਾਰ ਹੈਃਦੋਸਤਾਂ ਨਾਲ ਸੰਪਰਕ ਬੰਦ ਕਰਨਾ ਜੇ ਇਹ ਕੰਮ ਨਹੀਂ ਕਰ ਰਿਹਾ ਹੈ \n '] | punjabi |
HDFC બેંકના નફામાં 19 ટકાનો ઉછાળો, શેર રિકવરી ટ્રેક પર છે દેશની સૌથી મોટી ખાનગી ધિરાણકર્તા HDFC બેંકે તેના ત્રિમાસિક પરિણામો જાહેર કર્યા છે. એપ્રિલજૂન ક્વાર્ટર દરમિયાન બેન્કનો ચોખ્ખો નફો 19 વધીને રૂ. 9,196 કરોડ થયો છે. એચડીએફસી બેન્કે એક વર્ષ અગાઉના સમાન સમયગાળામાં રૂ. 7,729.64 કરોડનો ચોખ્ખો નફો નોંધાવ્યો હતો. બેન્કની ચોખ્ખી વ્યાજ આવક NII જૂન ક્વાર્ટરમાં 14.5 વધીને રૂ. 19,481.4 કરોડ થઈ છે જે ગયા વર્ષના સમાન ક્વાર્ટરમાં રૂ. 17,009.0 કરોડ હતી. બેન્કની ચોખ્ખી આવક ટ્રેડિંગ અને માર્કેટટુમાર્કેટ નુકસાન સિવાય જૂન ક્વાર્ટરમાં 19.8 વધીને 27,181.4 કરોડ થઈ છે જે ગયા વર્ષના સમાન ક્વાર્ટરમાં 22,696.5 કરોડ હતી. તમને જણાવી દઈએ કે શુક્રવારે HDFC બેંકના શેર NSE પર 0.96 વધીને 1,364 રૂપિયાના સ્તરે બંધ થયા હતા. તમને જણાવી દઈએ કે 17 જૂને બેંકનો શેર 52 સપ્તાહના નીચલા સ્તરને સ્પર્શી ગયો હતો. આ દિવસે શેરની કિંમત 1,271.75 રૂપિયા હતી. આ પરિપ્રેક્ષ્યમાં બેંકનો સ્ટોક ફરી એકવાર રિકવરી ટ્રેક પર છે. બેન્કની 52 સપ્તાહની ઊંચી સપાટી રૂ. 1,724.30 છે, જે ગયા વર્ષે ઓક્ટોબરમાં હતી. | gujurati |
sponse. 2.Compare and contrast the direct and indirect costs associated with the drone navigation system that both your company and VectorCal would assume.
1.Predict the main costs (e.g. labor cost, material cost) associated with the production of VectorCals drone navigation system. Provide a rationale for your response.
2.Compare and contrast the direct and indirect costs associated with the drone navigation system that both your company and VectorCal would assume. Predict whether or not your company could easily control these costs and thus reduce production expenses. Justify your response.
3.Compare your company with VectorCal relative to the price of acquisition, semi-variable costs, and allocated direct and indirect costs of the drone navigation system. Justify your response. | english |
செல்போனில் கேம் விளையாடிய மகள் ஆத்திரத்தில் தாய் செய்த கொடூர செயல்!!! 53 வயதான குர்ஷிதா என்ற பெண் ஒருவர், ஆந்திர மாநிலம் கடப்பா நாகாஷ் தெருவில் வசித்து வருகிறார். இவருக்கு 18 வயதான ஜமீர், 15 வயதான அலிமா என்ற இரண்டு பிள்ளைகள் உள்ளனர். குர்ஷிதாவிற்கும் இவரது கணவருக்கும் ஏற்பட்ட கருத்து வேறுபாடு காரணமாக குர்ஷிதா தனது குழந்தைகளுடன் தனியாக வசித்து வந்துள்ளார். இந்நிலையில், இவரது மகள் அலிமா அடிக்கடி செல்போனில் கேம் விளையாடியுள்ளார். இதனால் எரிச்சல் அடைந்த அவரது தாய், அவரது மகளை எச்சரித்துள்ளார். ஆனால் மீண்டும் தனது மகள் செல்போனில் விளையாடியதை பார்த்த குர்ஷிதா, ஆத்திரத்தில் தான் அணிந்திருந்த துப்பட்டாவால் அவரது கழுத்தை இறுக்கினார். தாய் தங்கையின் கழுத்தை இறுக்குவதை ஜமீர் பார்த்துக்கொண்டிருந்தார். தாய் தங்கையை விளையாட்டாக கழுத்தை இறுக்குவதாக நினைத்துக் கொண்டார். ஆனால் சிறிது நேரத்தில் தங்கை கழுத்து தொங்கியபடி தரையில் சாய்ந்தார். இதையடுத்து தங்கையின் அருகில் ஓடிய ஜமீர் அவரை எழுப்ப முயன்றார். ஆனால் தங்கை இறந்தது தெரியவந்தது. தங்கை இறந்ததைக் கண்ட ஜமீர் ஆத்திரமடைந்து வீட்டில் இருந்த கத்தியை எடுத்து வந்து தாயின் கழுத்தில் சரமாரியாக வெட்டினார். இதனால் குர்ஷிதாவின் கழுத்தில் இருந்து ரத்தம் பீறிட்டு கொட்டியது. வலியால் அவர் அலறி துடித்தார். அவரது அலறல் சத்தம் கேட்டு அக்கம் பக்கத்தினர் ஓடிவந்து பார்த்தனர். அதற்குள் குர்ஷிதா ரத்த வெள்ளத்தில் இறந்து கிடந்தார். இதனை கண்டவர்கள் இதுகுறித்து கடப்பா2 டவுன் போலீசாருக்கு தகவல் தெரிவித்தனர். போலீசார் சம்பவ இடத்திற்கு வந்து ஜமீரை கைது செய்து விசாரணை நடத்தினர். கடப்பாவில் செல்போனில் கேம் விளையாடிய மகளை கழுத்தை இறுக்கி கொலை செய்ததை கண்ட மகன் தாயை வெட்டிக் கொன்ற சம்பவம் ஆந்திராவில் பரபரப்பை ஏற்படுத்தி உள்ளது. ALSO READ: விவசாயிகள் கவனத்திற்கு உங்கள் பிரச்சனைகள் குறித்து புகார் அளிக்க.. இலவச தொலைபேசி எண் அறிமுகம்.! | tamil |
ಚಾಮರಾಜನಗರ: 16 ಹೊಸ ಪ್ರಕರಣ, 15 ಮಂದಿ ಗುಣಮುಖ, ಒಂದು ಸಾವು ಚಾಮರಾಜನಗರ: ಜಿಲ್ಲೆಯಲ್ಲಿ ಭಾನುವಾರ 16 ಹೊಸ ಕೋವಿಡ್ ಪ್ರಕರಣಗಳು ದೃಢಪಟ್ಟಿವೆ. 15 ಮಂದಿ ಗುಣಮುಖರಾಗಿದ್ದಾರೆ. ಒಬ್ಬರು ಮೃತಪಟ್ಟಿದ್ದಾರೆ. ತಾಲ್ಲೂಕಿನ ರೇಚಂಬಳ್ಳಿ ಗ್ರಾಮದ ನಿವಾಸಿ 85 ವರ್ಷ ವೃದ್ಧ ಅ.20ರಂದು ಕೋವಿಡ್ ಆಸ್ಪತ್ರೆಗೆ ದಾಖಲಾಗಿದ್ದರು. ಚಿಕಿತ್ಸೆಗೆ ಸ್ಪಂದಿಸಿದೇ ಶನಿವಾರ ಮೃತಪಟ್ಟಿದ್ದಾರೆ. ಇದರೊಂದಿಗೆ ಜಿಲ್ಲೆಯಲ್ಲಿ ಕೋವಿಡ್ನಿಂದಾಗಿ ಮೃತಪಟ್ಟವರ ಸಂಖ್ಯೆ 105ಕ್ಕೆ ಏರಿದೆ. ಕೋವಿಡ್ಯೇತರ ಕಾರಣದಿಂದ 19 ಮಂದಿ ಮೃತಪಟ್ಟಿದ್ದಾರೆ. ಜಿಲ್ಲೆಯಲ್ಲಿ ಇದುವರೆಗೆ 5,980 ಪ್ರಕರಣಗಳು ವರದಿಯಾಗಿವೆ. 5,628 ಮಂದಿ ಗುಣಮುಖರಾಗಿದ್ದಾರೆ. ಸದ್ಯ 228 ಸಕ್ರಿಯ ಪ್ರಕರಣಗಳಿದ್ದು, 129 ಮಂದಿ ಹೋಂ ಐಸೊಲೇಷನ್ನಲ್ಲಿದ್ದಾರೆ. 40 ಮಂದಿ ಐಸಿಯುನಲ್ಲಿ ಚಿಕಿತ್ಸೆ ಪಡೆಯುತ್ತಿದ್ದಾರೆ. ಭಾನುವಾರ 1029 ಕೋವಿಡ್ ಪರೀಕ್ಷೆಗಳ ವರದಿ ಬಂದಿದ್ದು, 1,017 ಮಂದಿಯ ವರದಿ ನೆಗೆಟಿವ್ ಬಂದಿವೆ. 12 ಮಂದಿಗೆ ಸೋಂಕಿರುವುದು ಖಚಿತವಾಗಿದೆ. ನಾಲ್ಕು ಪ್ರಕರಣಗಳು ಮೈಸೂರಿನಲ್ಲಿ ದೃಢಪಟ್ಟಿವೆ. | kannad |
John Abraham New Film Tehran: জন আব্রাহামের তেহরান ছবির শ্য়ুটিং শুরু শুরু হল জন আব্রাহামের নতুন ছবি তেহরানএর শ্য়ুটিং পর্ব টিজার, পোস্টার ও অ্যানাউন্সমেন্টের ভিডিয়ো শেয়ার করে এমনটাই জানালেন অভিনেতা John Abraham New Film Tehran Goes to the Floorsমুম্বই, 11 জুলাই: এই খবর আগেই সামনে এসেছিল যে পরিচালক দীনেশ বিজয়নের সঙ্গে প্রথমবার জুটি বাঁধতে চলেছেন অ্য়াকশন হিরো জন আব্রাহাম এবার সামনে এল তাঁর নতুন ছবির টিজার, পোস্টার ও অ্যানাউন্সমেন্টের ভিডিয়ো জনের শেষ ছবি অ্যাটাকপার্ট ওয়ান অবশ্য় সেভাবে সাফল্য লাভ করতে পারেনি বক্স অফিসে তবে ইতিমধ্যেই বেশ নজর কেড়েছে তাঁর আগামী ছবি এক ভিলেন রিটার্নসএর ট্রেলার এই ছবি দর্শকদের মন কতখানি জয় করে নেবে তা বলার সময় অবশ্য এখনও আসেনি তবে এরই জন অনুরাগীদের জন্য় একটি বড় খবর দিলেন অভিনেতা তাঁর নতুন ছবি তেহরানএর অ্যানাউন্সমেন্ট ভিডিয়োটি শেয়ার করে জন লিখেছেন, লাইটস, ক্যামেরা, অ্যাকশন! তেহরানএর শ্যুটিং শুরু!John Abraham New Film Tehran Goes to the Floors স্বাভাবিকভাবেই এই খবরে উচ্ছ্বসিত ফ্যানেরা অনেকেই জনের এই পোস্টের নীচে লিখেছেন আর অপেক্ষা করতে পারছি না... কেউ কেউ শুভেচ্ছা জানিয়ে আবার শেয়ার করেছেন হৃদয়ের ইমোজিও classaligntexttop noRightClick twitterSection data আরও পড়ুন: কিং খানের পড়শি হতে 119 কোটি খরচ করলেন রণবীর সিং!সত্য ঘটনার ওপর আধারিত এই ছবির প্রযোজনার দায়িত্ব রয়েছে শোভনা যাদব, দীনেশ বিজন এবং সন্দীপ লেজেলের ওপর জনের এই নতুন ছবি পর্দায় আসার কথা রয়েছে আগামী বছর সাধারণতন্ত্র দিবসে তার আগেই অবশ্য় পাঠান ছবিতে শাহরুখ খান এবং দীপিকা পাড়ুকোনের সঙ্গে স্ক্রিন স্পেস শেয়ার করতে চলেছেন এই অ্যাকশন হিরো | bengali |
ಕರೋನಾ ವೈರಸ್ ಎಫೆಕ್ಟ್: ಇನ್ನೂ ಸಹಜ ಸ್ಥಿತಿಗೆ ತಲುಪದ ಪೆಟ್ರೋಲ್, ಡೀಸೆಲ್ ಮಾರಾಟ ಕರೋನಾ ವೈರಸ್ ಎಂಬ ಮಹಾಮಾರಿ ಹಲವರ ಜೀವನವನ್ನು ತತ್ತರಿಸುವಂತೆ ಮಾಡಿದೆ. ಇದರಿಂದ ಪೆಟ್ರೋಲ್ ಬಂಕ್ ಮಾಲೀಕರು ಸಹ ಹೊರತಾಗಿಲ್ಲ. ಲಾಕ್ಡೌನ್ ಅವಧಿಯಲ್ಲಿ ವಾಹನಗಳ ಸಂಚಾರವು ಕಡಿಮೆಯಾಗಿದ್ದ ಕಾರಣ ಪೆಟ್ರೋಲ್, ಡೀಸೆಲ್ ಮಾರಾಟದಲ್ಲಿ ಕುಸಿತವಾಗಿತ್ತು. ಈಗ ಲಾಕ್ಡೌನ್ ನಿಂದ ವಿನಾಯಿತಿ ನೀಡಲಾಗಿದ್ದು, ವಾಹನಗಳ ಸಂಚಾರವು ಆರಂಭವಾಗಿದೆ. ಆದರೂ ಸಹ ಪೆಟ್ರೋಲ್, ಡೀಸೆಲ್ ಮಾರಾಟದಲ್ಲಿ ಏರಿಕೆ ಕಂಡು ಬಂದಿಲ್ಲ. ಈಗ ಬಹುತೇಕ ಕಂಪನಿಗಳು ತಮ್ಮ ಉದ್ಯೋಗಿಗಳಿಗೆ ವರ್ಕ್ ಫ್ರಂ ಹೋಂ ಆಯ್ಕೆಯನ್ನು ನೀಡಿವೆ. ಇದರ ಜೊತೆಗೆ ಶಾಲಾ, ಕಾಲೇಜುಗಳು ಇನ್ನೂ ಪುನರಾರಂಭಗೊಂಡಿಲ್ಲ. ಜೊತೆಗೆ ಸರಕು ಸಾಗಣೆಯು ಸಹ ಮೊದಲಿನಂತಿಲ್ಲ. ಈ ಎಲ್ಲಾ ಕಾರಣಗಳಿಂದ ಪೆಟ್ರೋಲ್, ಡೀಸೆಲ್ ಮಾರಾಟದಲ್ಲಿ ಕುಸಿತ ಮುಂದುವರೆದಿದೆ. ಜೊತೆಗೆ ಆರ್ಥಿಕ ಹಿಂಜರಿತವು ಮುಂದುವರೆದಿದೆ. ಕರೋನಾಗೂ ಮುನ್ನ ಇದ್ದ ಪರಿಸ್ಥಿತಿಗೆ ಹೋಲಿಸಿದರೆ ಗುಜರಾತ್ನ ಅಹಮದಾಬಾದ್ನಲ್ಲಿರುವ ಪೆಟ್ರೋಲಿಯಂ ವಿತರಕರು ಇಂಧನ ಮಾರಾಟದಲ್ಲಿ 30ನಷ್ಟು ನಷ್ಟ ಅನುಭವಿಸಿದ್ದಾರೆ. MOST READ: ಇನ್ನು ಮುಂದೆ ಈ ಬಣ್ಣದ ಕಾರುಗಳ ನೋಂದಣಿ ಕಾನೂನುಬದ್ದ ಈ ಕುರಿತು ಪ್ರತಿಕ್ರಿಯಿಸಿರುವ ಕೆಲವು ಪೆಟ್ರೋಲ್ ಬಂಕ್ ಮಾಲೀಕರು, ಅಹಮದಾಬಾದ್ನಲ್ಲಿ ಅನೇಕ ಜನರು ಸದ್ಯಕ್ಕೆ ವರ್ಕ್ ಫ್ರಂ ಹೋಂ ಮೂಲಕ ಕಾರ್ಯ ನಿರ್ವಹಿಸುತ್ತಿದ್ದಾರೆ. ಖಾಸಗಿ ವಾಹನಗಳೂ ಸಹ ಹೆಚ್ಚಿನ ಸಂಖ್ಯೆಯಲ್ಲಿ ಸಂಚರಿಸುತ್ತಿಲ್ಲ. ಲಾಕ್ಡೌನ್ ನಂತರ ಪೆಟ್ರೋಲ್, ಡೀಸೆಲ್ ಮಾರಾಟ ಪ್ರಮಾಣವು ಕೇವಲ 70ನಷ್ಟಾಗಿದೆ. ಇನ್ನೂ 30ನಷ್ಟು ಮಾರಾಟವಾದರೆ ಮಾತ್ರ ಕರೋನಾಗೂ ಮುನ್ನ ಇದ್ದ ಮಟ್ಟವನ್ನು ತಲುಪಬಹುದು ಎಂದು ಹೇಳಿದ್ದಾರೆ. ಶಾಲೆಗಳ ಮುಚ್ಚುವಿಕೆ ಕೂಡ ಮತ್ತೊಂದು ಪ್ರಮುಖ ಕಾರಣವಾಗಿದೆ. ಶಾಲೆಗಳು ಮುಚ್ಚಿರುವುದರಿಂದ ಶಾಲಾ ವಾಹನಗಳ ಸಂಚಾರವು ಸ್ಥಗಿತಗೊಂಡಿದೆ. MOST READ: ಒಂದು ವರ್ಷದಿಂದ ಚಲಿಸಿದರೂ ಇನ್ನೂ ಗುರಿ ಮುಟ್ಟದ ಟ್ರಕ್ ಚಿತ್ರಮಂದಿರಗಳ ಮುಚ್ಚುವಿಕೆ, ಪ್ರವಾಸೋದ್ಯಮದ ಮೇಲೆ ನಿಷೇಧ ಹೇರಿರುವುದು ಸಹ ಪೆಟ್ರೋಲ್, ಡೀಸೆಲ್ ಮಾರಾಟ ಕುಸಿತಗೊಳ್ಳಲು ಮುಖ್ಯ ಕಾರಣಗಳಾಗಿವೆ. ಕರೋನಾ ವೈರಸ್ ಭೀತಿಯಿಂದ ಜನರು ಮನೆಗಳಿಂದ ಹೊರಬರಲು ಹೆದರುತ್ತಿರುವುದು ಸಹ ಪೆಟ್ರೋಲ್, ಡೀಸೆಲ್ ಮಾರಾಟ ಕಡಿಮೆಯಾಗಲು ಕಾರಣವಾಗಿದೆ. ಈ ಎಲ್ಲಾ ಕಾರಣಗಳು ಪೆಟ್ರೋಲ್, ಡೀಸೆಲ್ ಮಾರಾಟವನ್ನು ಕುಸಿಯುವಂತೆ ಮಾಡಿವೆ. ಪೆಟ್ರೋಲ್, ಡೀಸೆಲ್ ಮಾರಾಟಕ್ಕೆ ಹೋಲಿಸಿದರೆ, ಪೆಟ್ರೋಲ್ ಮಾರಾಟವು ವೇಗವಾಗಿ ಚೇತರಿಸಿಕೊಳ್ಳುತ್ತಿದೆ. ಆದರೆ ಡೀಸೆಲ್ ಮಾರಾಟ ಇನ್ನೂ ಏರಿಕೆಯನ್ನು ಕಂಡಿಲ್ಲ. ಲಾರಿಗಳಂತಹ ಕಮರ್ಷಿಯಲ್ ವಾಹನಗಳು ಸಾಮಾನ್ಯವಾಗಿ ಹೆಚ್ಚು ಡೀಸೆಲ್ ಬಳಸುತ್ತವೆ. MOST READ: ಟೊಯೊಟಾ ಇನೋವಾ ಕ್ರಿಸ್ಟಾ ಕಾರಿನ ರೂಫ್ ಸೀಳಿದ ಬಂಡೆ ಆದರೆ ಉತ್ಪಾದನಾ ವಲಯದಲ್ಲಿನ ಆರ್ಥಿಕ ಹಿಂಜರಿತದಿಂದಾಗಿ ಲಾರಿಗಳ ಸಂಚಾರವು ಕಡಿಮೆಯಾಗಿದೆ. ಸಾಮಾನ್ಯ ಜನರು ಮೊದಲಿನಂತೆ ತಮ್ಮ ಸಂಚಾರವನ್ನು ಆರಂಭಿಸಿ, ಉತ್ಪಾದನಾ ವಲಯವು ಮೊದಲಿನ ಸ್ಥಿತಿಗೆ ತಲುಪಿದರೆ ಮಾತ್ರ ಪೆಟ್ರೋಲ್, ಡೀಸೆಲ್ ಮಾರಾಟವು ಮತ್ತೆ ಸಹಜ ಸ್ಥಿತಿಗೆ ಮರಳಲಿದೆ. ಗಮನಿಸಿ: ಈ ಲೇಖನದಲ್ಲಿ ಸಾಂದರ್ಭಿಕ ಚಿತ್ರಗಳನ್ನು ಬಳಸಲಾಗಿದೆ. | kannad |
જાગરણ વિના ઉજાગરા કરતા રોમિયો સામે પોલીસની આકરી કાર્યવાહી, જાહેરમાં સબક શીખવાડ્યો VIDEO રાજકોટ ગતરાત્રે સૌરાષ્ટ્રમાં જયાપાર્વતીનાં વ્રતનું જાગરણ હતું. જેમાં આ વ્રત કરનાર બહેનો ઉપરાંત મહિલાઓએ આખી રાત જાગરણ કરવાનું હોય છે. જેને લઈ સમય પસાર કરવા મોટી સંખ્યામાં યુવતિઓ રેસકોર્સ સહિતનાં સ્થળોએ આવતી હોય છે. એટલું જ નહીં શહેરનાં સિનેમાઘરોમાં પણ રાતભર અલગ અલગ શો ચાલુ રાખવામાં આવતા હોય છે. જેમાં પણ જાગરણ કરતી યુવતિઓ મોટી સંખ્યામાં પહોંચતી હોય છે. જો કે આ તકે કેટલાક રોમિયો વગર જાગરણે ઉજાગરો કરવા નિકળતા હોય છે. ત્યારે આવા તત્વો દ્વારા યુવતિઓને કોઈ મુશ્કેલી ન પડે તે માટે પોલીસે પણ જાગરણ કર્યું હતું. અને જાગરણ વિના ઉજાગરા કરતા અનેક રોમિયોને જાહેરમાં ઉઠબેસ કરાવવામાં આવી હતી. આ પહેલા કોરોનાને કારણે બે વર્ષથી કોઈ તહેવારની ઉજવણી થઈ શકી નહોતી. પરંતુ ચાલુવર્ષે કોરોનાનું સંક્રમણ ઘટતા સરકાર દ્વારા નિયમોનાં પાલન સાથે તહેવારો ઉજવવાની છૂટ આપવામાં આવી રહી છે. ગઈકાલે પણ જ્યાપાર્વતી વ્રતનું જાગરણ હોવાથી હોટેલ, રેસ્ટોરન્ટ, આઈસ્ક્રીમ પાર્લરો તેમજ સિનેમા ઘરો રાતભર ખુલ્લા રાખવાની છૂટ આપવામાં આવી હતી. જેને લઈને જાગરણ કરનાર યુવતિઓ મોટી સંખ્યામાં રાતભર બહાર જોવા મળી હતી. શહેરનાં રેસકોર્સ સહિતનાં મુખ્ય બાગબગીચાઓ હાઉસફુલ જોવા મળ્યા હતા. આ દરમિયાન યુવતિઓને પરેશાન કરવા કેટલાક આવારા તત્વો અને રોમિયો પણ નિકળી પડ્યા હતા. જો કે આવા તત્વો દ્વારા જાગરણ કરતી યુવતિઓને કોઈ મુશ્કેલી ભોગવવી ન પડે તેના માટે પોલીસ સતત ખડેપગે રહી હતી. અને રેસકોર્સ તેમજ જિ. પંચાયત ચોક સહિતનાં સ્થળોએ આવા રોમિયોને ઝડપી લઈ પોલીસે જાહેરમાં ઉઠબેસ કરાવી કાયદાનું ભાન પણ કરાવ્યું હતું. પોલીસની આ કામગીરીથી યુવતિઓમાં પોલીસ પ્રત્યે આભાર અને રોમિયોમાં કચવાટની લાગણી જોવા મળી હતી.Original article: જાગરણ વિના ઉજાગરા કરતા રોમિયો સામે પોલીસની આકરી કાર્યવાહી, જાહેરમાં સબક શીખવાડ્યો VIDEO2022 Watch Gujarat News : LIVE LATEST TODAY Online. All Rights Reserved. | gujurati |
অনাস্থা প্রস্তাবের আগে নিখোঁজ পাকিস্তানের ৫০ মন্ত্রী! সিঁদুরে মেঘ দেখছেন ইমরান খান বাংলা হান্ট ডেস্ক: বেশ কিছুদিন ধরেই পাকিস্তানের রাজনৈতিক অবস্থা দোদুল্যমান অবস্থায় ছিল এমনকি, পাক প্রধানমন্ত্রী ইমরান খানের পদত্যাগ চেয়েছিল সেদেশের সেনাবাহিনীও কিন্তু, বর্তমানে পাকিস্তানের রাজনীতির সবথেকে থেকে বড় খবর হল, অনাস্থা প্রস্তাবের আগেই ইমরান খানের দলের ৫০ জন মন্ত্রী নিখোঁজ হয়েছেন যে কারণে ইমরানের দল পিটিআইও বড় ধরনের ধাক্কা খেয়েছে সবচেয়ে বড় কথা হল, এর ফলে গদিও হারাতে পারেন ইমরান জানা গিয়েছে যে, ইমরান খান সরকারের ২৫ জন সচিব, ১৯ জন সহকারী এবং ৪ জন প্রতিমন্ত্রী নিখোঁজ রয়েছেন আসল সঙ্কটের সময়েই ইমরানের মন্ত্রীরা মাঠ ছেড়ে কার্যত পালিয়ে গিয়েছেন এমতাবস্থায়, এটা প্রায় নিশ্চিত যে, এবার ইমরান খানকে প্রধানমন্ত্রীর পদ থেকে পদত্যাগ করতেই হবে Related Articles জন্মদিনে মুখ্যমন্ত্রীর থেকে শুভেচ্ছা পেলেন দেবাংশু, কী বার্তা দিল বিরোধীরা? 27 mins ago ব্রাত্য অঙ্কুশ, বার কাউন্টারের উপরে উঠে হবু শ্বশুরের সঙ্গে উদ্দাম নাচ বার্থডে গার্ল ঐন্দ্রিলার 56 mins ago শুধু তাই নয়, তাত্পর্যপূর্ণভাবে, এই রাজনৈতিক সঙ্কটের মধ্যে, অনেক ঘনিষ্ঠ বন্ধুও পাকিস্তানের প্রধানমন্ত্রী ইমরান খানের সমর্থন ছেড়ে দিয়েছেন জানা গিয়েছে, ২৮ মার্চ পাকিস্তানের পার্লামেন্টে অনাস্থা প্রস্তাব পেশ করা হবে পাশাপাশি, ৩১ মার্চ থেকে ৪ এপ্রিল পর্যন্ত অনাস্থা প্রস্তাবের ওপর ভোটগ্রহণ সম্পন্ন হবে এদিকে, শুক্রবার পাকিস্তানের সংসদে ইমরান খান সরকারের বিরুদ্ধে অনাস্থা প্রস্তাব পেশ করতে পারেনি বিরোধীরা এখন আগামী সোমবার এই প্রস্তাব পেশ করা হবে যদিও, ইমরান সরকার ইঙ্গিত দিয়েছে যে তারা ক্ষমতা থেকে বেরিয়ে গেলে সময়ের আগেই সাধারণ নির্বাচন করতে পারে অর্থাত্ বিরোধীদের হারাতে ইমরান খান নির্বাচনের বাজি খেলতে পারেন প্রসঙ্গত উল্লেখ্য, পাকিস্তানের ৭৫ বছরের ইতিহাসে এমন একজনও নির্বাচিত প্রধানমন্ত্রী নেই যিনি তাঁর পাঁচ বছরের মেয়াদ পূর্ণ করেছেন এখন পাক প্রধানমন্ত্রী ইমরান খান নতুন ইতিহাসের সূচনা করতে পারেন কি না সেদিকেই নজর রয়েছে সবার নতুন খবর ছবি ভিডিও ভাইরাল পশ্চিমবঙ্গ ভারত আন্তজাতিক বিনোদন খেলা রাজনীতি টেক নিউজ রাশিফল আবহাওয়া টাকা পয়সা লাইফ স্টাইল | bengali |
న్యాయ వ్యవస్థలో లింగ వివక్ష మహిళా న్యాయమూర్తుల ప్రాతినిధ్యం తక్కువే సుప్రీంకోర్టులో నలుగురు.. హైకోర్టుల్లో 77 మంది మాత్రమే..! న్యూఢిల్లీ : దేశ న్యాయవ్యవస్థలోనూ లింగ వివక్ష ఆందోళన కలిగిస్తున్నది. ఇక్కడ మహిళా న్యాయమూర్తుల సంఖ్య చాలా తక్కువగా ఉన్నది. చరిత్రలో ఎన్నడూ లేని విధంగా సుప్రీంకోర్టులో మహిళా సిట్టింగ్ జడ్జిల సంఖ్య అధికంగానే ఉన్నప్పటికీ.. ఇక్కడా మహిళా న్యాయమూర్తుల ప్రాతినిధ్యం మాత్రం తక్కువగానే ఉన్నది.కేంద్ర న్యాయ మంత్రిత్వ శాఖ సమాచారం ప్రకారం.. దేశ సర్వోన్నత న్యాయస్థానంతో పాటు హైకోర్టులలో మహిళా జడ్జిల సంఖ్య 12 శాతంగానే ఉన్నది. సుప్రీంకోర్టు, హైకోర్టులలో మొత్తం 677 మంది సిట్టింగ్ జడ్జిలు ఉన్నారు. అయితే, వీరిలో మహిళా జడ్జిల సంఖ్య 81 మాత్రమే 12 శాతం కావడం గమనార్హం. సుప్రీంకోర్టులో ఇటీవలే ముగ్గురు మహిళా జడ్జిల చేరికతో సిట్టింగ్ మహిళా జడ్జిల సంఖ్య చరిత్రలో ఎన్నడూ లేనంతగా అత్యధికంగా నాలుగుకు చేరినప్పటికీ.. వారి ప్రాతినిధ్యం ఆందోళన కలిగిస్తున్నది. సుప్రీంకోర్టులో మొత్తం 33 మంది వర్కింగ్ జడ్జిలు ఉన్నారు. ఇందులో మహిళా న్యాయమూర్తుల సంఖ్య 12 శాతమే కావడం గమనార్హం.ఇక దేశవ్యాప్తంగా 25 హైకోర్టులలో కేటాయించిన జడ్జిల సంఖ్య 1098. అయితే, ఇందులో 454 పోస్టులు ఖాళీగానే ఉన్నాయి. ప్రస్తుతం 644 మంది వర్కింగ్ జడ్జిలలో మహిళా న్యాయమూర్తుల సంఖ్య 77 మాత్రమే ఉన్నది. అంటే, ఇది కేవలం 12 శాతం.హైకోర్టుల వారీగా చూసుకుంటే.. మద్రాసు హైకోర్టులో మహిళా న్యాయమూర్తుల సంఖ్య రెండెంకెలుగా ఉన్నది. ఇక్కడ, మొత్తం 58 మంది వర్కింగ్ జడ్జిలలో 13 మంది మహిళా న్యాయమూర్తులు ఉన్నారు. అంటే వీరి ప్రాతినిధ్యం 22 శాతంగా ఉన్నది. ఇక తర్వాతి స్థానంలో ఉన్న బాంబే హైకోర్టులో 63 మంది సిట్టింగ్ జడ్జిలకు గానూ ఎనిమిది మంది మహిళా న్యాయమూర్తులు 13 శాతం ఉన్నారు. అలహాబాద్ ఏడు శాతం, పంజాబ్ అండ్ హర్యానా 15 శాతం హైకోర్టులలో ఏడుగురు చొప్పున మహిళా జడ్జిలు ఉన్నారు. ఢిల్లీ, కర్నాటక హైకోర్టులలో ఆరుగురు చొప్పున, గుజరాత్లో ఐదుగురు మహిళా న్యాయమూర్తులున్నారు. ఇక కలకత్తా, కేరళ హైకోర్టులలో నలుగురు చొప్పున మహిళా న్యాయమూర్తులు ప్రాతినిధ్యం వహిస్తున్నారు. తెలంగాణ, ఛత్తీస్గఢ్ లలో ఇద్దరు చొప్పున, మధ్యప్రదేశ్, ఆంధ్రప్రదేశ్లలో ముగ్గురు చొప్పున మహిళా జడ్జిలు మాత్రమే ఉన్నారు. ఇక మణిపూర్, మేఘాలయ, బీహార్, త్రిపుర, ఉత్తరాఖండ్ వంటి ఐదు రాష్ట్రాల్లో కనీసం ఒక్క మహిళా జడ్జి కూడా లేకపోవడం గమనార్హం. గౌహతి, హిమాచల్ప్రదేశ్, జమ్మూ కాశ్మీర్ అండ్ లఢక్, జార్ఖండ్, ఒరిస్సా, రాజస్థాన్, సిక్కిం హైకోర్టులలో కేవలం ఒక్కరు చొప్పున మహిళా న్యాయమూర్తులు ఉన్నారు. మహిళా న్యాయమూర్తుల ప్రాతినిధ్యం తక్కువే సుప్రీంకోర్టులో నలుగురు.. హైకోర్టుల్లో 77 మంది మాత్రమే..! న్యూఢిల్లీ : దేశ న్యాయవ్యవస్థలోనూ లింగ వివక్ష ఆందోళన కలిగిస్తున్నది. ఇక్కడ మహిళా న్యాయమూర్తుల సంఖ్య చాలా తక్కువగా ఉన్నది. చరిత్రలో ఎన్నడూ లేని విధంగా సుప్రీంకోర్టులో మహిళా సిట్టింగ్ జడ్జిల సంఖ్య అధికంగానే ఉన్నప్పటికీ.. ఇక్కడా మహిళా న్యాయమూర్తుల ప్రాతినిధ్యం మాత్రం తక్కువగానే ఉన్నది.కేంద్ర న్యాయ మంత్రిత్వ శాఖ సమాచారం ప్రకారం.. దేశ సర్వోన్నత న్యాయస్థానంతో పాటు హైకోర్టులలో మహిళా జడ్జిల సంఖ్య 12 శాతంగానే ఉన్నది. సుప్రీంకోర్టు, హైకోర్టులలో మొత్తం 677 మంది సిట్టింగ్ జడ్జిలు ఉన్నారు. అయితే, వీరిలో మహిళా జడ్జిల సంఖ్య 81 మాత్రమే 12 శాతం కావడం గమనార్హం. సుప్రీంకోర్టులో ఇటీవలే ముగ్గురు మహిళా జడ్జిల చేరికతో సిట్టింగ్ మహిళా జడ్జిల సంఖ్య చరిత్రలో ఎన్నడూ లేనంతగా అత్యధికంగా నాలుగుకు చేరినప్పటికీ.. వారి ప్రాతినిధ్యం ఆందోళన కలిగిస్తున్నది. సుప్రీంకోర్టులో మొత్తం 33 మంది వర్కింగ్ జడ్జిలు ఉన్నారు. ఇందులో మహిళా న్యాయమూర్తుల సంఖ్య 12 శాతమే కావడం గమనార్హం.ఇక దేశవ్యాప్తంగా 25 హైకోర్టులలో కేటాయించిన జడ్జిల సంఖ్య 1098. అయితే, ఇందులో 454 పోస్టులు ఖాళీగానే ఉన్నాయి. ప్రస్తుతం 644 మంది వర్కింగ్ జడ్జిలలో మహిళా న్యాయమూర్తుల సంఖ్య 77 మాత్రమే ఉన్నది. అంటే, ఇది కేవలం 12 శాతం.హైకోర్టుల వారీగా చూసుకుంటే.. మద్రాసు హైకోర్టులో మహిళా న్యాయమూర్తుల సంఖ్య రెండెంకెలుగా ఉన్నది. ఇక్కడ, మొత్తం 58 మంది వర్కింగ్ జడ్జిలలో 13 మంది మహిళా న్యాయమూర్తులు ఉన్నారు. అంటే వీరి ప్రాతినిధ్యం 22 శాతంగా ఉన్నది. ఇక తర్వాతి స్థానంలో ఉన్న బాంబే హైకోర్టులో 63 మంది సిట్టింగ్ జడ్జిలకు గానూ ఎనిమిది మంది మహిళా న్యాయమూర్తులు 13 శాతం ఉన్నారు. అలహాబాద్ ఏడు శాతం, పంజాబ్ అండ్ హర్యానా 15 శాతం హైకోర్టులలో ఏడుగురు చొప్పున మహిళా జడ్జిలు ఉన్నారు. ఢిల్లీ, కర్నాటక హైకోర్టులలో ఆరుగురు చొప్పున, గుజరాత్లో ఐదుగురు మహిళా న్యాయమూర్తులున్నారు. ఇక కలకత్తా, కేరళ హైకోర్టులలో నలుగురు చొప్పున మహిళా న్యాయమూర్తులు ప్రాతినిధ్యం వహిస్తున్నారు. తెలంగాణ, ఛత్తీస్గఢ్ లలో ఇద్దరు చొప్పున, మధ్యప్రదేశ్, ఆంధ్రప్రదేశ్లలో ముగ్గురు చొప్పున మహిళా జడ్జిలు మాత్రమే ఉన్నారు. ఇక మణిపూర్, మేఘాలయ, బీహార్, త్రిపుర, ఉత్తరాఖండ్ వంటి ఐదు రాష్ట్రాల్లో కనీసం ఒక్క మహిళా జడ్జి కూడా లేకపోవడం గమనార్హం. గౌహతి, హిమాచల్ప్రదేశ్, జమ్మూ కాశ్మీర్ అండ్ లఢక్, జార్ఖండ్, ఒరిస్సా, రాజస్థాన్, సిక్కిం హైకోర్టులలో కేవలం ఒక్కరు చొప్పున మహిళా న్యాయమూర్తులు ఉన్నారు. | telegu |
હવે ચીન અને પાકિસ્તાનને સબક શીખવાડશે ભારત, અમેરિકાએ આપી છુટ, રશિયા પાસેથી S400 મિસાઈલ ખરીદવા પર નહીં લાગે કોઈ પ્રતિબંધ યુએસ હાઉસ ઓફ રિપ્રેઝન્ટેટિવ્સે ગુરુવારે રશિયા પાસેથી S400 મિસાઈલ ડિફેન્સ સિસ્ટમ ખરીદવા માટે ભારતને CAATSA પ્રતિબંધોમાંથી વિશેષ મુક્તિ આપતું બિલ પસાર કર્યું છે. હવે ચીન અને પાકિસ્તાન આંખ આડા કાન કરી શકશે નહીં અને તેમને પાઠ ભણાવી શકાશે. ભારતીયઅમેરિકન સાંસદ રો ખન્ના દ્વારા રજૂ કરાયેલ સંશોધિત બિલ, ચીન જેવા આક્રમક દેશને અમેરિકી રાષ્ટ્રપતિ જો બાઈડેનના વહીવટથી રોકવામાં મદદ કરવા માટે કાઉન્ટરિંગ અમેરિકાઝ એડવર્સરીઝ થ્રુ સેક્શન્સ એક્ટ CAATSA માંથી ભારતને મુક્તિ આપવા માંગે છે. નેશનલ ડિફેન્સ ઓથોરાઈઝેશન એક્ટ NDAA પર ગૃહમાં ચર્ચા દરમિયાન આ સુધારેલું બિલ ધ્વનિ મત દ્વારા પસાર કરવામાં આવ્યું હતું. ખન્નાએ કહ્યું કે ચીનના વધી રહેલા આક્રમક વલણને જોતા અમેરિકાએ ભારતની સાથે ઉભું રહેવું જોઈએ. ઈન્ડિયા કોકસના ઉપાધ્યક્ષ તરીકે હું આપણા દેશો વચ્ચેની ભાગીદારીને મજબૂત કરવા અને ભારતચીન સરહદે ભારત પોતાનો બચાવ કરી શકે તે સુનિશ્ચિત કરવા માટે કામ કરી રહ્યો છું. તેમણે કહ્યું કે આ સુધારો ખૂબ જ મહત્વપૂર્ણ છે અને મને એ જોઈને ગર્વ થાય છે કે તે બંને પક્ષોના સમર્થનથી પસાર થયો છે. ગૃહમાં તેમની ટિપ્પણીમાં ખન્નાએ કહ્યું કે અમેરિકાના વ્યૂહાત્મક હિતમાં અમેરિકાભારત ભાગીદારી કરતાં વધુ મહત્ત્વનું બીજું કંઈ નથી. જાણો બિલમાં શું કહેવામાં આવ્યું છે? બિલ જણાવે છે કે યુનાઈટેડ સ્ટેટ્સઈન્ડિયા ઈનિશિયેટિવ ઓન ક્રિટીકલ એન્ડ ઈમર્જિંગ ટેક્નોલોજી ICETએ બંને દેશોમાં સરકારો, શિક્ષણવિદો અને ઉદ્યોગો વચ્ચે ગાઢ ભાગીદારી વિકસાવવા માટે આવકારદાયક અને જરૂરી પગલું છે, જેથી આર્ટિફિશિયલ ઈન્ટેલિજન્સ, ક્વોન્ટમ કમ્પ્યુટિંગ, ક્વોન્ટમ કમ્પ્યુટિંગ, બાયોટેકનોલોજી, એરોસ્પેસ અને સેમિકન્ડક્ટર ઉત્પાદનમાં નવીનતમ પ્રગતિને અપનાવવામાં આવી શકે. તેમાં કહેવામાં આવ્યું છે કે ઈજનેરો અને કોમ્પ્યુટર વૈજ્ઞાનિકો વચ્ચેની આ પ્રકારની ભાગીદારી એ સુનિશ્ચિત કરવા માટે મહત્વપૂર્ણ છે કે યુએસ અને ભારત તેમજ વિશ્વભરના અન્ય લોકશાહી દેશો નવીનતા અને તકનીકી પ્રગતિને પ્રોત્સાહન આપી શકે છે, જેથી તેઓ રશિયા અને ચીનની તકનીકને પાછળ રાખી શકે. વર્ષ 2017માં રજૂ કરાયેલા CAATSA હેઠળ રશિયા સાથે સંરક્ષણ અને ગુપ્તચર વ્યવહાર કરનારા કોઈપણ દેશ સામે શિક્ષાત્મક પગલાં લેવાની જોગવાઈ છે. તે રશિયા દ્વારા 2014માં ક્રિમીઆના જોડાણ અને 2016 યુએસ પ્રમુખપદની ચૂંટણીમાં મોસ્કોની કથિત દખલગીરીના જવાબમાં લાવવામાં આવ્યું હતું. | gujurati |
نویدا تھامس چھِ اَکھ ہِندوستٲنؠ اَداکارہ یۄس فِلمَن مَنٛز چھِ کٲم کَران.
زٲتی زِندگی
فِلمی دور
== حَوالہٕ == | kashmiri |
ಪ್ರಧಾನಿಯಾದ ಮೊದಲ ದಿನದಿಂದಲೇ ಕಮ್ಯೂನಿಸ್ಟ್ ಸರ್ಕಾರದ ವಿರುದ್ಧ ಕಠಿಣ ಕ್ರಮ: ರಿಷಿ ಸುನಕ್ ಇನ್ಫೋಸಿಸ್ ಸಂಸ್ಥಾಪಕ ನಾರಾಯಣಮೂರ್ತಿ ಅವರ ಅಳಿಯ ರಿಷಿ ಸುನಕ್ ಬ್ರಿಟನ್ನ ನೂತನ ಪ್ರಧಾನಿಯಾಗಲಿದ್ದಾರೆ ಎಂಬ ಮಾತುಗಳು ಇತ್ತೀಚೆಗೆ ಕಂಡುಬರುತ್ತಿದೆ, ಬ್ರಿಟನ್ನ ಮುಂದಿನ ಪ್ರಧಾನಿಯಾಗಲು ನಾನಾ ಆಕಾಂಕ್ಷಿಗಳಿದ್ದರೂ, ಈ ಪೈಕಿ ರಿಷಿ ಸುನಕ್ ಇಂಗ್ಲೆಂಡ್ನ ನೂತನ ಪ್ರಧಾನಿಯಾಗುವ ಸಾಧ್ಯತೆ ಹೆಚ್ಚಾಗಿದೆ ಎಂದು ಹೇಳಲಾಗುತ್ತಿದೆ. ಇನ್ನು, ರಿಷಿ ಸುನಕ್ ಅಂತಾರಾಷ್ಟ್ರೀಯ ಮಾಧ್ಯಮವೊಂದರ ಜತೆ ಮಾತನಾಡಿದ್ದು, ಚೀನಾ ವಿರುದ್ಧ ಭಾನುವಾರ ಗುಡುಗಿದ್ದಾರೆ. ಬ್ರಿಟನ್ನ ಆಡಳಿತಾರೂಢ ಕನ್ಸರ್ವೇಟೀವ್ ಪಕ್ಷದಲ್ಲಿರುವ ರಿಷಿ ಸುನಕ್ ಬ್ರಿಟನ್ ಪ್ರಧಾನಿಯಾಗುವ ರೇಸ್ನಲ್ಲಿದ್ದು, ಲಿಜ್ ಟ್ರುಸ್ ವಿರುದ್ಧ ತುರುಸಿನ ಸ್ಪರ್ಧೆಯನ್ನು ಎದುರಿಸುತ್ತಿದ್ದಾರೆ. ಈ ಹಿನ್ನೆಲೆ ಬ್ರಿಟನ್ನ ಮಾಜಿ ಹಣಕಾಸು ಸಚಿವ ರಿಷಿ ಸುನಕ್ ವಿರುದ್ಧ ಇತ್ತೀಚೆಗೆ ಲಿಜ್ ಟ್ರುಸ್ ಆರೋಪ ಮಾಡಿದ್ದಾರೆ. ರಿಷಿ ಚೀನಾ ಹಾಗೂ ರಷ್ಯಾದ ವಿರುದ್ಧ ದುರ್ಬಲರಾಗಿದ್ದಾರೆ ಎಂದು ಲಿಜ್ ಆರೋಪಿಸಿದ್ದರು. ಆದರೆ, ಇದಕ್ಕೆ ಪ್ರತಿಕ್ರಿಯೆ ನೀಡಿದ ರಿಷಿ ಸುನಕ್, ಚೀನಾದ ವಿರುದ್ಧ ಗುಡುಗಿದ್ದಾರೆ. ತಾನು ಪ್ರಧಾನಿಯಾದ ಮೊದಲ ದಿನದಿಂದಲೇ ಕಮ್ಯೂನಿಸ್ಟ್ ಸರ್ಕಾರದ ವಿರುದ್ಧ ಕಠಿಣ ಕ್ರಮಗಳನ್ನು ಕೈಗೊಳ್ಳುವುದಾಗಿ ಹೇಳಿದ್ದಾರೆ. ಚೀನಾದ ವಿರುದ್ಧ ರಿಷಿ ಸುನಕ್ ಕೈಗೊಳ್ಳುವ ಪ್ರಸ್ತಾವನೆಗಳು ಹೀಗಿವೆ ನೋಡಿ.. ಬ್ರಿಟನ್ನಲ್ಲಿರುವ ಎಲ್ಲ 30 ಕನ್ಫ್ಯೂಶಿಯಸ್ ಸಂಸ್ಥೆಗಳನ್ನು ಮುಚ್ಚುವ ಒಲವನ್ನು ರಿಷಿ ಹೊಂದಿದ್ದು, ಸಂಸ್ಕೃತಿ ಮತ್ತು ಭಾಷಾ ಕಾರ್ಯಕ್ರಮಗಳ ಮೂಲಕ ಚೀನೀ ಪ್ರಭಾವದ ಮೃದು ಶಕ್ತಿಯ ಹರಡುವಿಕೆಯನ್ನು ತಡೆಯುವುದು ಇದರ ಉದ್ದೇಶ ಎಂದು ತಿಳಿದುಬಂದಿದೆ. ಅಲ್ಲದೆ, ನಮ್ಮ ವಿಶ್ವವಿದ್ಯಾಲಯಗಳಿಂದ ಚೀನಾದ ಕಮ್ಯೂನಿಸ್ಟ್ ಪಕ್ಷ ಸಿಸಿಪಿ ವನ್ನು ಕಿತ್ತುಹಾಕುವುದಾಗಿಯೂ ರಿಷಿ ಸುನಕ್ ಭರವಸೆ ನೀಡಿದ್ದಾರೆ. | kannad |
पीएम मोदी आज करेंगे मन की बात, UP के हर बूथ पर भाजपा कार्यकर्ता सुनेंगे संबोधन नई दिल्ली: प्रधानमंत्री नरेंद्र मोदी आज इस साल के अपने मासिक रेडियो कार्यक्रम मन की बात के पहले एपिसोड में राष्ट्र को संबोधित करेंगे। हर महीने के आखिरी रविवार को प्रसारित होने वाले मन की बात का यह 85वां एपिसोड होगा, जिसे आकाशवाणी समाचार और मोबाइल एप पर सुबह 11.30 बजे प्रसारित किया जाएगा। इसके साथ ही आज महात्मा गांधी की पुण्यतिथि है। ऐसे में पीएम मोदी गांधी जी को नमन करने व श्रद्धांजलि देने के बाद लोगों से अपने मन की बात साझा करेंगे। पिछले हफ्ते प्रधानमंत्री कार्यालय PMO ने ट्वीट कर कहा था कि इस महीने की मन की बात, जो 30 तारीख को होगी, गांधी जी की पुण्यतिथि पर उनके स्मरण के बाद सुबह 11:30 बजे शुरू होगी। चुनाव के मद्देनजर बीजेपी की खास तैयारियांआपको बता दें कि 5 राज्यों में विधानसभा चुनाव होने जा रहे हैं। ऐसे में भारतीय जनता पार्टी भाजपा के प्रमुख जेपी नड्डा ने पार्टी कार्यकर्ताओं को पीएम के मन की बात कार्यक्रम में अधिक से अधिक लोगों की भागीदारी सुनिश्चित करने के लिए बूथ स्तर पर व्यवस्था करने का निर्देश दिया था। बीजेपी ने देश भर में अलगअलग जगहों पर लोगों को उनका संबोधन सुनाने के लिए खास तैयारी की है। इसके अलावा प्रधानमंत्री ने इस महीने के मन की बात के एपिसोड के लिए नागरिकों को विचार और सुझाव साझा करने के लिए आमंत्रित किया था। उन्होंने एक ट्वीट में कहा था कि इस महीने की 30 तारीख को 2022 का पहला मन की बात होगा। मुझे यकीन है कि आपके पास प्रेरक जीवन की कहानियों और विषयों के संदर्भ में साझा करने के लिए बहुत कुछ है। उन्हें mygovindia या NaMo ऐप पर साझा करें। इसके अलावा 1800117800 पर डायल करके अपना संदेश रिकॉर्ड करें। 26 दिसंबर को मन की बात के पिछले संस्करण में, पीएम मोदी ने कोरोनोवायरस बीमारी और स्वच्छ भारत पहल सहित कई विषयों पर बात की था। गौतरलब है कि कार्यक्रम का पहला एपिसोड 3 अक्टूबर 2014 को प्रसारित किया गया था। | hindi |
Whether fresh or previously frozen, Dungeness crabs and blue crabs are a great meal to linger over.
Flip a coin. Heads its Dungeness, tails its blue. We’re in either way. Some of the most memorable meals we’ve enjoyed were centered around freshly steamed or boiled crabs, good beer or wine, and a long, leisurely meal with just the two of us or with friends cracking and picking crabs.
We prefer fresh crabs whenever we can get them. In South Carolina, there was a private dock on a saltwater cut through the marsh that could be counted on to produce blue crabs on incoming tides. And when I lived in Oregon, throwing out a couple of crab pots was a matter of course on salmon fishing trips. Because Dungeness populations are depressed in the parts of Alaska we frequent, their harvest isn’t currently permitted in those locales. So most of the crabs we’ve been getting are purchased already cooked, but we still heat them before serving.
Our favorite way to boil-steam crabs is fairly simple. We start with about a half-bottle (6 – 8 ounces) of beer and 1 tablespoon of miso per Dungeness crab. Since more liquid than this is necessary, we add a cup of water or two. The idea is to ensure that there is enough liquid so that it doesn’t all boil off in the 12 minutes or so required to heat through a previously frozen Dungeness. For two crabs, add a 12-ounce bottle of beer, a little more miso, and, if necessary, a little more water.
I usually don’t immerse the entire crab. This is because I’m frugal (cheap) and hate wasting beer. I boil-steam the crab on one side for a few minutes, then flip it and continue cooking it for a few minutes more. If I’m doing multiple crabs, I arrange them in the pot as best I can and rotate them once during the cooking – although this really may not be necessary.
Previously cooked crabs are inevitably already plenty salty. The beer and miso bath gives them a mild sweetness. If you’re starting with fresh crabs, you might want to add some salt to the broth. A good rule of thumb is to steam fresh crabs for about 7 – 8 minutes per pound – which means a two-and-one-half pound crab needs about 20 minutes in the pot. One crab this side is usually plenty for the two of us, served with, say, a salad, fresh corn on the cob, and a loaf of crusty bread.
Our favorite dipping sauce? Melted butter, olive oil, garlic, lemon and soy sauce. For two people, melt about 6 tablespoons of butter. Add a clove of minced garlic and sauté it for a minute or so. Then add 1 tablespoon of olive oil, the juice from half a lemon, and 1 tablespoon of soy sauce. A slice or two from a really great loaf of bread can be used to sop up any remaining sauce.
Crab goes great with a wide range of beers or a buttery Chardonnay.
YUM! And I love seeing that little hummingbird because it tells me you made this in the camper!
Great tips on how to prepare the crab. It’s making me hungry! | english |
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*
* This software consists of voluntary contributions made by many
* individuals on behalf of the Apache Software Foundation. For more
* information on the Apache Software Foundation, please see
* <http://www.apache.org/>.
*
*/
package org.apache.http;
/**
* A factory for {@link HttpRequest HttpRequest} objects.
*
* @since 4.0
*/
public interface HttpRequestFactory {
HttpRequest newHttpRequest(RequestLine requestline)
throws MethodNotSupportedException;
HttpRequest newHttpRequest(String method, String uri)
throws MethodNotSupportedException;
}
| code |
500 से कम रह गए जिले में कोरोना मरीज आज फिर 21 नए केस सामने आएउज्जैन में 6 मरीज मिलेबडऩगर, महिदपुर अभी भी हाट स्पाट उज्जैन। 3 फरवरी से कोरोना के मामले जिले में तेजी से घट रहे हैं।यहां कोविड से संबधित सभी नए अपडेट पढ़ें इसी के चलते जिले में अब उपचाररत कोरोना के मरीजों की संख्या 500 के नीचे आ गई है। हालांकि अभी भी नए पॉजीटिव केस आ रहे हैं। आज भी पूरे जिले में 21 मरीज मिले हैं जिनमें से उज्जैन शहर के 6 हैं। बडऩगर और महिदपुर तहसील केस कम होने के बावजूद हाट स्पाट बने हुए हैं। उल्लेखनीय है कि इस महीने के आरंभ के दो दिन को छोड़ 3 फरवरी से लेकर 12 फरवरी तक लगातार कोरोना के मामले तेजी से घटते देखे गए हैं। इसी के साथ होम आईसोलेशन तथा कोविड सेंटरों में उपचार के लिए भर्ती मरीज भी तेजी से रिकवरी कर रहे हैं। इसी के चलते आज पॉजीटिव आए 21 मामलों के बावजूद पूरे जिले में कोरोना के उपचाररत मरीजों की संख्या घटकर अब 498 रह गई है। फरवरी महीने में अचानक कोरोना की तीसरी लहर का ग्राफ इस तरह नीचे आने के बाद अब जिले में कोरोना पॉजीटिव दर भी घटकर 1.14 प्रतिशत पर आ गई है। एक माह पहले तक यही दर 13 प्रतिशत के पार चली गई थी। इसका मतलब यह है कि अब 400 सेम्पलों की जाँच में 5 लोगों में कोरोना संक्रमण पाया जा रहा है। कोरोना नियंत्रण जिला नोडल अधिकारी डॉ. एच.पी. सोनानिया ने बताया कि मामलों में तेजी से आ रही गिरावट लोगों की सावधानी और बड़े पैमाने पर हुए वैक्सीनेशन का नतीजा है। उज्जैन को ग्रीन झोन के दायरे में आने में अभी भी फरवरी महीना पूरा गुजर जाएगा। इसके बाद ही इस दायरे में उज्जैन आ पाएगा। इधर आज पॉजीटिव आए मामलों में उज्जैन शहर के 6 मरीज शामिल हैं, जबकि बडऩगर में 5, महिदपुर में 4, खाचरौद में 4 तथा तराना और घटिया में 11 मरीज नया मिला है। इधर रिकवरी रेट भी और बढ़ गया है तथा ठीक होने वालों की संख्या पॉजीटिव आ रहे मरीजों की संख्या से 97.23 प्रतिशत तक पहुँच गई है। 70 प्रतिशत से ज्यादा संक्रमण गाँवों में पिछले एक हफ्ते में भले ही जिले में कोरोना पॉजीटिव मिल रहे मरीजों का आंकड़ा दहाई के अंक में आ गया हो लेकिन हैरत की बात है कि घटते मामलों के बीच उज्जैन शहर में मरीज कम मिल रहे हैं, जबकि तहसीलों में संक्रमण का ग्राफ अभी भी ऊंचा है। पिछले तीन दिनों में शहर तथा ग्रामीण इलाकों में पॉजीटिव आए मरीजों की संख्या में तुलना की जाए तो शहर के मुकाबले तहसीलों में 70 प्रतिशत नए संक्रमित मरीज मिल रहे हैं। इसी के चलते बडऩगर और महिदपुर तहसीलें अभी भी कोरोना का हाट स्पाट बनी हुई है। | hindi |
ಕೋವಿಡ್: ದೇಶದಲ್ಲಿ 91 ದಿನದಲ್ಲೇ ದೃಢಪಟ್ಟ ಕನಿಷ್ಠ ಪ್ರಕರಣ ನವದೆಹಲಿ: ದೇಶದಲ್ಲಿ ಕೋವಿಡ್ ಪ್ರಕರಣ ಗಣನೀಯ ಪ್ರಮಾಣದಲ್ಲಿ ಇಳಿಕೆಯಾಗಿದೆ. ಕಳೆದ 24 ತಾಸುಗಳ ಅವಧಿಯಲ್ಲಿ ದೇಶದಲ್ಲಿ 42,640 ಹೊಸ ಪ್ರಕರಣಗಳು ಪತ್ತೆಯಾಗಿದೆ. ಇದು ಕಳೆದ 91 ದಿನದಲ್ಲೇ ದೃಢಪಟ್ಟ ಕನಿಷ್ಠ ಪ್ರಕರಣವಾಗಿದೆ. ಕಳೆದ 24 ಗಂಟೆಯಲ್ಲಿ 1,167 ಮಂದಿ ಕೋವಿಡ್ಗೆ ಬಲಿಯಾಗಿದ್ದಾರೆ. ಭಾನುವಾರದ ಅಂಕಿ ಸಂಖ್ಯೆಯೊಂದಿಗೆ ದೇಶದಲ್ಲಿ ಒಟ್ಟು ಸೋಂಕಿತರ ಸಂಖ್ಯೆ 2,99,77,861ಕ್ಕೆ ತಲುಪಿದ್ದು, ಸಾವಿನ ಸಂಖ್ಯೆ 3,89,302ಕ್ಕೆ ಏರಿಕೆಯಾಗಿದೆ. ಮತ್ತೊಂದೆಡೆ ಸಕ್ರಿಯ ಸೋಂಕಿತರ ಸಂಖ್ಯೆ 6,62,521ಕ್ಕೆ ಕುಸಿದ್ದು, ಹಲವು ದಿನಗಳ ಬಳಿಕ ದೇಶದಲ್ಲಿ ದಾಖಲಾಗಿರುವ ಅತಿ ಕನಿಷ್ಠ ಸಕ್ರಿಯ ಸೋಂಕಿತರ ಸಂಖ್ಯೆ ಇದಾಗಿದೆ. ಕಳೆದ 24 ಗಂಟೆಗಳಲ್ಲಿ 81,839 ಮಂದಿ ಗುಣಮುಖವಾಗಿದ್ದಾರೆ. ಭಾರತದಲ್ಲಿ ಒಂದೇ 16,64,360 ಮಂದಿಯನ್ನು ಕೋವಿಡ್ ಪರೀಕ್ಷೆಗೊಳಪಡಿಸಲಾಗಿದ್ದು, 39,40,72,142 ಮಂದಿಯನ್ನು ಪರೀಕ್ಷೆಗೊಳಪಡಿಸಲಾಗಿದೆ. | kannad |
కోల్ కతా నేపథ్యంలో గోపీచంద్ చిత్రం మ్యాచో స్టార్ గోపీచంద్ కి అచ్చొచ్చిన దర్శకుల్లో శ్రీవాస్ ఒకరు. వీరిద్దరి కలయికలో వచ్చిన లక్ష్యం 2007, లౌక్యం 2014 చిత్రాలు ఘనవిజయం సాధించడమే కాకుండా నటుడిగా గోపీచంద్ కి ప్రత్యేక గుర్తింపు తీసుకువచ్చాయి. కట్ చేస్తే.. త్వరలో వీరిద్దరి కాంబినేషన్ లో ముచ్చటగా మూడో సినిమా రాబోతోంది. పీపుల్ మీడియా ఫ్యాక్టరీ పతాకంపై టీజీ విశ్వప్రసాద్ ఈ క్రేజీ ప్రాజెక్ట్ ని నిర్మించనున్నారు. ఇదిలా ఉంటే.. గోపీచంద్ శ్రీవాస్ హ్యాట్రిక్ మూవీకి సంబంధించి ఓ ఆసక్తికరమైన విషయం వెలుగులోకి వచ్చింది. అదేమిటంటే.. ఈ సినిమాలోని సింహభాగం సన్నివేశాలు కోల్ కతా నేపథ్యంలో సాగుతాయట. అంతేకాదు.. కోల్ కతా బ్యాక్ డ్రాప్ లో వచ్చే యాక్షన్ సీన్స్ హైలైట్ గా నిలుస్తాయని టాక్. మరి.. ఈ ప్రచారంలో వాస్తవమెంతో తెలియాలంటే కొన్నాళ్ళు వేచిచూడాల్సిందే. కాగా, గోపీచంద్ తాజా చిత్రం సీటీమార్ రేపు సెప్టెంబర్ 10 థియేటర్స్ లో సందడి చేయనుంది. మరోవైపు మారుతి దర్శకత్వంలో నటిస్తున్న పక్కా కమర్షియల్ చిత్రీకరణ తుది దశకు చేరుకుంది. లౌక్యం తరువాత సరైన విజయం లేకపోవడంతో ఈ రెండు చిత్రాలపై భారీ ఆశలే పెట్టుకున్నాడు గోపీచంద్. | telegu |
YCP Colors in Tirumala: ఆఖరుకు ఆ తిరుమలేషుడికి కూడా వైసీపీ రంగులేనా? YCP colors in tirumala: అపచారం.. మహాపచారం.. ఆఖరుకు కలియుగ ప్రత్యక్ష దైవం.. ఆ తిరుమలేషుడికి కూడా ఏపీలో అధికారంలో ఉన్న వైసీపీ రంగులను పలిమిస్తారా? ఎంత ఘోరం.. అంటూ హిందూ వాదులు నెత్తినోరు కొట్టుకుంటూ ఆరోపిస్తున్నారు. వైసీపీ సర్కార్ వచ్చాక హిందూ దేవాలయాలపై దాడులు.. పలు ఇతర మతాల మిళితాలు జరిగి పెద్ద వివాదాలు చెలరేగాయి. అన్యమత ప్రచారం చేస్తూ హిందుత్వాన్ని జగన్ సర్కార్ దెబ్బతీస్తోందని ఏపీ బీజేపీ పెద్ద ఎత్తున ఉద్యమించింది. అంతర్వేది రథానికి నిప్పు పెట్టడం.. తిరుమల బస్సు టికెట్లలో అన్యమత ప్రచారం.. ఏపీలోని దేవాలయాలపై వరుస దాడులు కలకలం రేపాయి. ఇది దేశవ్యాప్తంగా వివాదాస్పదమైంది. తాజాగా ఇప్పుడు మరో వివాదంలో చిక్కుకుంది. జగన్ సర్కార్ హయాంలో జరిగిన హిందూ మత అపచారాలపై ఏపీ బీజేపీ, టీడీపీ, జనసేనలు పెద్ద ఉద్యమమే చేశారు. నినదించారు. ఇప్పటికీ చాలా చోట్ల దేవాలయాలపై దాడులు చేసిన దోషులను పట్టుకోలేదన్న విమర్శ ఉంది. సీఎం జగన్ క్రిస్టియానిటీని నమ్మడంతో ఆయనపై కూడా ఈ అపవాదు ఉంది. అయితే ఎంత వైసీపీ నేతలు కవర్ చేసే ప్రయత్నాలు చేసినా కూడా ఆ ప్రభుత్వంపై పడ్డ మరకలు మాత్రం పోవడం లేదు. అవన్నీ చాలదన్నట్టు ఇప్పుడు వైసీపీ స్వామి భక్తి మరీ ఎక్కువైపోయిందన్న విమర్శలు వినిపిస్తున్నాయి. ప్రపంచవ్యాప్తంగా హిందుత్వానికి నెలవైన తిరుమల వేంకటేశ్వరుడి బ్రహ్మోత్సవాలంటే అందరికీ ఒక పండుగ. అలాంటి పండుగ వేడుకను కనులారా చూసేందుకు దేశ విదేశాల నుంచి భక్తులు తరలివస్తుంటారు. కానీ ఇక్కడ కూడా వైసీపీ రంగులు కనిపించడం చూసి విస్తుపోతున్నారు. ఆ దేవుడికి వైసీపీ రంగులు పులుముతారా? అన్న విమర్శలు వినిపిస్తున్నాయి. సాధారణంగా ఆ దేవదేవుడికి పసుపు, కుంకుమ, కాషాయం లాంటి రంగులతో అలంకరిస్తారు. కానీ వైసీపీ సర్కార్ హయాంలో మొదలైన తిరుమల బ్రహ్మోత్సవాలకు ఈసారి టీటీడీ అధికారులు స్వామిభక్తి చాటుకున్నారు. హిందుత్వ రంగులను పక్కనపడేసి వైసీపీ జెండాలోని నీలం రంగును పులిమారు. ఇప్పుడు ఇది పెద్ద వివాదాస్పదమవుతోంది. హిందువుల మనోభావాలను దెబ్బతీస్తున్నారని.. ఆ దేవ దేవుడికి పార్టీ రంగులు పులుముతారా? అని హిందుత్వవాదులు మండిపడుతున్నారు. ఏపీలో వైసీపీ ప్రభుత్వం కొలువుదీరాక ఆలయాలపై దాడులు పెరిగిపోయాయన్నది అందరూ కాదనలేని వాస్తవం. వరుసగా ఏపీలోని ఆలయాలపై దాడులు తీవ్ర కలకలం రేపుతున్నాయి. శతాబ్ధాల చరిత్ర కలిగిన అంతర్వేది రథం దగ్గమైంది. విజయనగరం జిల్లా రామతీర్థం క్షేత్రంలోని బోధికొండపై ఉన్న కోదండరాముడి విగ్రహాన్ని కొందరు ధ్వంసం చేయడం ఉద్రిక్తతకు దారితీసింది. స్వామి విగ్రహాన్ని ధ్వంసం చేసిన విధానం.. శిరస్సు కనిపించకుండా పోవడం చూసి బీజేపీ నేతలు, హిందువులు మండిపడ్డారు.. ఏపీలో ఏడాదిన్నరగా దేవతా విగ్రహాలు, ఆలయ రథాలు ధ్వంసం చేస్తూనే ఉన్నా ఆ కేసుల్లో ఇప్పటికీ ఒక్కటీ తేలలేదు. రాష్ట్ర ప్రభుత్వం నిర్లక్ష్యం వల్లే ఈ దుర్మార్గపు చర్యలు రాష్ట్రంలో సాగుతున్నాయని.. హిందూ ఆలయాలపై దాడులు పెరిగిపోతుంటే ముఖ్యమంత్రి ఎందుకు స్పందించడం లేదని బీజేపీ నేత సోము వీర్రాజు, జనసేన నేత పవన్ లు గతంలో ప్రశ్నించారు. ఏపీ బీజేపీ దీనిపై పోరుబాట కూడా పట్టింది. సీఎం జగన్ కు ఏ మత విశ్వాసం ఉన్నా పరమతాలను గౌరవించాలని పవన్ హితవు పలికారు కూడా.. అయితే ఇప్పటికీ ఇంత రచ్చ జరిగినా కూడా పవిత్రమైన తిరుమల బ్రహ్మోత్సవాల వేళ వైసీపీ రంగును పులమడం మరో వివాదానికి దారితీసింది. | telegu |
बस 2 से 4 लाख के बजट में आपकी हो सकती है प्रीमियम हैचबैक Honda Jazz, साथ मिलेगा आसान लोन प्लान कार सेक्टर के हैचबैक सेगमेंट की कारों को उनकी कम कीमत और लंबी माइलेज के लिए पसंद किया जाता है जिसमें कुछ कारों को इन फीचर्स के अलावा उनके स्पोर्टी डिजाइन के लिए भी पसंद किया जाता है।कार और बाइक पर लेटेस्ट अपडेट पाने के लिए यहां क्लिक करें जिसमें से हम बात कर रहे हैं होंडा जैज के बारे में जो हैचबैक सेगमेंट की एक प्रीमियम कार है और इसे प्रीमियम फीचर्स और स्पोर्टी डिजाइन के लिए पसंद किया जाता है। होंडा जैज को अगर आप शोरूम से खरीदते हैं तो इसके लिए आपको 7.71 लाख रुपये ले लेकर 9.95 लाख रुपये तक खर्च करने होंगे लेकिन अगर आप इस कार को कम बजट में खरीदना चाहते हैं तो यहां पढ़ सकते हैं इसे महज 2 से 4 लाख रुपये के बजट में खरीदने के ऑफर की डिटेल। लोकप्रिय खबरें मोदी जी 20 साल और राज करेंगे तब भी नेहरू को ही दोष देंगे डॉ. मनमोहन सिंह का जिक्र कर PM पर भड़क गए बॉलीवुड एक्टर योगी आदित्यनाथ की जीत हुई तो इंडिया कभी नहीं लौटूंगा बॉलीवुड एक्टर ने खाई कसम, लोग करने लगे ऐसे सवाल मोदी जी जुमले छोड़ रहे हैं तो ये पीछे क्यों रहते. नितिन गडकरी पर भड़के बॉलीवुड एक्टर, BJP को बताया इतिहास की सबसे भ्रष्ट सरकार UP Election: मुख्तार के बाद बड़े भाई सिबगतुल्लाह को भी हटाया, फूटफूटकर रोए अब्दुल अंसारी, पूर्वांचल में बारबार क्यों प्रत्याशी बदल रही सपा लेकिन उससे पहले आप जान लीजिए इस होंडा जैज कार के फीचर्स, स्पेसिफिकेशन और माइलेज की हर छोटीबड़ी डिटेल। होंडा जैज में कंपनी ने 1199 सीसी का इंजन दिया है जो 1.2 लीटर पेट्रोल इंजन है और यह 90 पीएस की पावर और 110 एनएम का पीक टॉर्क जनरेट करता है। इस कार के फीचर्स की बात करें तो कंपनी ने इसमें 7 इंच का टचस्क्रीन इंफोटेनमेंट सिस्टम, अलॉय व्हील, पावर स्टीयरिंग, पावर विंडो, मैनुअल एसी, फ्रंट सीट पर डुअल एयरबैग्स, रियर पार्किंग सेंसर, एबीएस, ईबीडी जैसे फीचर्स दिए हैं। प्रीमियम फीचर्स के साथ मिड रेंज में आती हैं ये टॉप 3 सनरूफ कार, जानें कीमत और फीचर्स की पूरी डिटेल होंडा जैज की माइलेज को लेकर कंपनी का दावा है कि ये कार 17.1 किलोमीटर प्रति लीटर का माइलेज देती है। होंडा जैज की पूरी डिटेल जानने के बाद अब आप उन ऑफर्स के बारे में जान लीजिए जिसमें यह कार आपको मिल सकती है बहुत कम कीमत के अंदर। Tata Tigor EV: सिंगल चार्ज में 306 km की ड्राइविंग रेंज का दावा, जानें इस इलेक्ट्रिक सेडान की पूरी डिटेल CARDEKHO वेबसाइट पर इस होंडा जैज का 2012 मॉडल बिक्री के लिए पोस्ट किया गया है जिसकी कीमत 3,11 लाख रुपये रखी गई है। DROOM वेबसाइट पर इस होंडा जैज का 2009 मॉडल बिक्री के लिए पोस्ट किया गया है जिसकी कीमत 2,04,970 रुपये रखी गई है जिसके साथ फाइनेंस की सुविधा दी जा रही है। CARWALE वेबसाइट पर होंडा जैज का 2015 मॉडल बिक्री के लिए पोस्ट किया गया है जिसकी कीमत 4.65 लाख रुपये रखी गई है जिसके साथ फाइनेंस की सुविधा भी उपलब्ध है। | hindi |
തിരുവോണം കഴിഞ്ഞാലും ഓണകിറ്റ് വിതരണം പൂര്ത്തിയാവില്ല ഓണകിറ്റ് വിതരണം തിരുവോണത്തിന് മുമ്ബ് പൂര്ത്തിയാവില്ല. 37 ലക്ഷം പേര്ക്ക് കൂടി ഇനി കിറ്റ് ലഭിക്കാനുണ്ട് . ചില ഉല്പന്നങ്ങളുടെ കുറവ് മൂലം കിറ്റുകള് പൂര്ണമായും തയ്യാറാക്കാന് സപ്ലൈകോക്ക് കഴിയാതിരുന്നതാണ് കാരണം.ഈ മാസം തന്നെ കിറ്റ് വിതരണം പൂര്ത്തിയാക്കുമെന്നാണ് സപ്ലൈകോയുടെ ഉറപ്പ്. കിറ്റ് വിതരണം സമയ ബന്ധിതമായി പൂര്ത്തിയാക്കാനായില്ലെങ്കിലും ഉല്പന്നങ്ങളെ കുറിച്ച് ഇത്തവണ പരാതിയില്ലെന്ന ആശ്വാസത്തിലാണ് സപ്ലൈകോ. വിതരണം വേഗത്തിലാക്കാന് ഭക്ഷ്യമന്ത്രിയുടെ ഓഫീസില് പ്രത്യേക സെല് മേല്നോട്ടം വഹിക്കുന്നുണ്ട്. മുഴുവന് കിറ്റുകളും 16ാം തീയതി കൊണ്ട് വിതരണം ചെയ്യുകയായിരുന്നു ലക്ഷ്യം. എന്നാല് ഇത് പാളി. ഇതിന് കാരണമായത് 16 ഇനം കിറ്റിലെ ചില ഉല്പന്നങ്ങള് പ്രതീക്ഷിച്ചത് പോലെ ലഭ്യമാക്കാനാവതെ പോയതാണ്. ഏലക്ക, ശര്ക്കരവരട്ടി പോലുള്ളവയുടെ കുറവാണ് പ്രതിസന്ധിക്ക് കാരണമായത്. ആയിരത്തിലധികം പാക്കിങ് സെന്ററുകളിലൂടെ ഉത്രാട ദിനം വരെയും കിറ്റുകള് കൈമാറുന്നത് തുടരും. 75 ശതമാനമെങ്കിലും തിരുവോണത്തിന് മുമ്ബേ പൂര്ത്തിയാക്കുകയാണ് സപ്ലൈകോയുടെ ലക്ഷ്യം. ഷിനോജ് | malyali |
\begin{document}
\begin{abstract}
Let $G$ be a complex reductive group and $\mathcal{X}_{r}G$ denote
the $G$-character variety of the free group of rank $r$. Using geometric
methods, we prove that $E(\mathcal{X}_{r}SL_{n})=E(\mathcal{X}_{r}PGL_{n})$,
for any $n,r\in\mathbb{N}$, where $E(X)$ denotes the Serre (also
known as $E$-) polynomial of the complex quasi-projective variety
$X$, settling a conjecture of Lawton-Muñoz in \cite{LM}. The proof
involves the stratification by polystable type introduced in \cite{FNZ},
and shows moreover that the equality of $E$-polynomials holds for
every stratum and, in particular, for the irreducible stratum of $\mathcal{X}_{r}SL_{n}$
and $\mathcal{X}_{r}PGL_{n}$. We also present explicit computations
of these polynomials, and of the corresponding Euler characteristics,
based on our previous results and on formulas of Mozgovoy-Reineke
for $GL_{n}$-character varieties over finite fields.
\end{abstract}
\maketitle
\section{Introduction}
Given a complex reductive algebraic group $G$, and a finitely presented
group $\Gamma$, the $G$-character variety of $\Gamma$ is the (affine)
geometric invariant theory (GIT) quotient
\[
\mathcal{X}_{\Gamma}G=\mathrm{Hom}(\Gamma,G)/\!\!/ G.
\]
The most well studied families of character varieties include the
cases when the group $\Gamma$ is the fundamental group of a Riemann
surface $\Sigma$, and its ``twisted'' variants. These varieties
correspond, via non-abelian Hodge theory, to certain moduli spaces
of $G$-Higgs bundles which play an important role in the quantum
field theory interpretation of the geometric Langlands correspondence
(see, for example \cite{Si}, \cite{KW}).
In the context of SYZ mirror symmetry \cite{SYZ}, the hyperkähler
nature of the Hitchin systems allowed a topological criterion for
mirror symmetry: as a coincidence between certain Hodge numbers of
moduli spaces of $G$-Higgs bundles over $\Sigma$, for Langlands
dual groups $G$ and $G^{L}$ (see \cite{Th}). The Hodge structure
of these moduli spaces is \emph{pure}, and topological mirror symmetry
has been established in the smooth/orbifold case for the pair of Langlands
dual groups $SL_{n}\equiv SL(n,\mathbb{C})$ and $PGL_{n}\equiv PGL(n,\mathbb{C})$
(see \cite{HT,GWZ}).
Such topological mirror symmetries are also expected for \emph{mixed}
Hodge structures on the cohomology of other classes of moduli spaces.
In the case of character varieties, whose Hodge structure is \emph{not
pure}, and which are generally, neither projective nor smooth, information
on the Hodge numbers is encoded in the Serre polynomial (also called
$E$-polynomial), which also provides interesting arithmetic properties
(see \cite{HRV}).
In this article, we consider character varieties of the free group
$\Gamma=F_{r}$ of rank $r\in\mathbb{N}$, and prove the following
equality. We denote the $E$-polynomial of a complex quasi-projective
variety $X$ by $E(X)$.
\begin{thm}
\label{thm:main}Let $r\in\mathbb{N}$, and $\Gamma=F_{r}$. Then
\[
E(\mathcal{X}_{\Gamma}SL_{n})=E(\mathcal{X}_{\Gamma}PGL_{n}),\quad\forall n\in\mathbb{N}.
\]
\end{thm}
This result was proved for $n=2$ and 3 in Lawton-Muñoz \cite{LM},
using complex geometry methods; these computations increase substantially
for higher $n$, making it practically impossible to proceed explicitly.
In \cite[Remark 6]{LM} an arithmetic argument by Mozgovoy is mentioned,
showing the equality for all odd $n$ (by using a theorem of Katz
in \cite[Appendix]{HRV}, see also subsection 4.5), as further evidence
for the conjectured equality for all $n$, which we hereby confirm.
Denote by $\mathcal{X}_{\Gamma}^{irr}G\subset\mathcal{X}_{\Gamma}G$
the Zariski open subvariety of irreducible representation classes.
The following result, together with the stratification\textbackslash{}
by polystable type introduced in \cite{FNZ} for arbitrary $GL_{n}$-character
varieties, are the key ingredients in the proof of Theorem \ref{thm:main}.
\begin{thm}
\label{thm:main-irred} Let $r\in\mathbb{N}$, and $\Gamma=F_{r}$.
Then
\[
E(\mathcal{X}_{\Gamma}^{irr}SL_{n})=E(\mathcal{X}_{\Gamma}^{irr}PGL_{n}).\quad\forall n\in\mathbb{N}.
\]
\end{thm}
For the proof of this result, we use crucially the fact that free
group character varieties admit a strong deformation retraction to
analogous spaces of representations into $SU(n)$ and $PU(n)$, that
follow from \cite{FL1}.
The outline of the article is as follows. In Section 2, we review
standard facts on mixed Hodge structures and polynomial invariants,
including their equivariant versions. Section 3 deals with character
varieties in the context of affine GIT, which is enough to relate
the $E$-polynomials of $\mathcal{X}_{F_{r}}PGL_{n}$ with those of
$\mathcal{X}_{F_{r}}GL_{n}$. The polystable type stratification from
\cite{FNZ} is recalled in Section 4 for the $GL_{n}\equiv GL(n,\mathbb{C})$
case, and defined for the cases $SL_{n}$ and $PGL_{n}$; then we
provide the proof of the main theorem, assuming Theorem \ref{thm:main-irred}.
This second theorem is proved in Section 5, by examining the action
of $\mathrm{Hom}(F_{r},Z)$ in the cohomology of $\mathrm{Hom}(F_{r},SL_{n})$, where
$Z\cong\mathbb{Z}_{n}$ is the center of $SL_{n}$. Finally, Section
6 is devoted to describing a finite algorithm to obtain all the Serre
polynomials and the Euler characteristics of all the strata $\mathcal{X}_{F_{r}}^{[k]}G$,
for all partitions $[k]\in\mathcal{P}_{n}$ and $G=GL_{n}$, $SL_{n}$
or $PGL_{n}$. The algorithm uses formulae of Mozgovoy-Reineke for
$E(\mathcal{\mathcal{X}}_{F_{r}}^{irr}GL_{n})$, the irreducible $GL_{n}$
case \cite{MR}. Our main results were announced at the ISAAC conference
2019 \cite{FNSZ}.
\subsection{Comparison with related results in the literature }
For the benefit of the reader, we outline here some related previous
results, without pretending to be exhaustive, and summarize the main
novelties in our approach. For the surface group case (where $\Gamma=\pi_{1}(\Sigma)$
with $\Sigma$ a compact orientable surface, and related groups) the
calculations of Poincaré polynomials of $\mathcal{X}_{\Gamma}G$ started
with Hitchin and Gothen (for $G=SL_{n}$, $n=2,3$, see \cite{Hi,Got}),
and have been pursued more recently by García-Prada, Heinloth, Schmitt,
Hausel, Letellier, Mellit, Rodriguez-Villegas, Schiffmann and others,
who also considered the parabolic version of these character varieties
(see \cite{GPH,GPHS,HRV,Le,Me,Sc}).
Many of those recent results also use arithmetic methods: it is shown
that the number of points of the corresponding moduli space over finite
fields is given by a polynomial which, by Katz's theorem mentioned
above, coincides with the $E$-polynomial of $\mathcal{X}_{\Gamma}G$.
Then, in the smooth case, this allows the derivation of the Poincaré
polynomial from the $E$-polynomial.
The equality of \emph{stringy $E$-polynomials} (in the sense of Batyrev-Dais
\cite{BD}) of moduli spaces of $G$-Higgs bundles, for $SL_{n}$
and $PGL_{n}$, in the coprime case, was established by Hausel-Thaddeus,
for $n=2,3$, in \cite{HT}, and by Groechenig-Wyss-Ziegler for all
$n$ in \cite{GWZ}. More recently, Gothen-Oliveira considered the
parabolic version in \cite{GO}. In another direction, the \emph{full
Hodge-Deligne} polynomials of \emph{free abelian} group character
varieties were computed in \cite{FS}.
Most of the above results were obtained only in the smooth/orbifold
cases of the corresponding moduli spaces. On the other hand, for many
important classes of \emph{singular character varieties}, such as
\emph{free groups} or surface groups $\pi_{1}\Sigma$ \emph{without
twisting} (corresponding to degree zero bundles) explicitly computable
formulas for the $E$-polynomials are very hard to obtain. In the
articles of Lawton, Logares, Mart\'{i}nez, Muñoz and Newstead (by
using geometric methods, see \cite{LMN,Ma,MM} for surface groups
and \cite{LM} for the free group) and of Cavazos-Lawton and Baraglia-Hekmati
(\cite{CL} and \cite{BH}, using arithmetic methods), the Serre polynomials
were computed for several character varieties, with $G=GL_{n}$, $SL_{n}$
and $PGL_{n}$ for small values of $n$, but the computations quickly
become intractable for $n$ higher than 3. Recently, $SL_{n}$-character
varieties of surface groups were found to give rise to a Lax monoidal
topological quantum field theory (see \cite{GPLM}). In the case of
free group character varieties, the $E$-polynomials were obtained
in \cite{MR}, by point counting over finite fields, for all $GL_{n}$,
and in \cite{BH} for $SL_{n}$ with $n=2,3$.
Our present results and methods differ from the previous literature
in the following aspects. We consider the singular character variety
of the free group, for $G=SL_{n}$ and \emph{every} $n\in\mathbb{N}$,
and consider the standard compactly supported $E$-polynomial (not
the stringy one). We extend the stratification by polystable type
(described in \cite{FNZ} for $GL_{n}$) to the cases of $PGL_{n}$
and $SL_{n}$, and carefully examine the action of the centre of $SL_{n}$
on the cohomology of several spaces, to prove the main results. Our
Theorems \ref{thm:main} and \ref{thm:main-irred} do not use point
counting methods over finite fields, and we only use the formulas
of Mozgovoy-Reineke for $GL_{n}$-character varieties of $F_{r}$
(see \cite{MR}) in the last section, to provide explicit formulas
for Serre polynomials and Euler characteristics for all $\mathcal{X}_{F_{r}}SL_{n}$
and all polystable strata.
\section{Mixed Hodge structures and Serre polynomials}
This section recalls some standard facts on mixed Hodge structures
and polynomial invariants, introducing terminology and notation that
will be used throughout.
Let $X$ be a quasi-projective variety over $\mathbb{C}$, of complex
dimension $d$, which may be singular, not complete, and/or not irreducible.
Following Deligne (c.f. \cite{De}, \cite{PS}), the compactly supported
cohomology $H_{c}^{*}(X):=H_{c}^{*}(X,\mathbb{C})$ is endowed with
a mixed Hodge structure. Denote the corresponding mixed Hodge numbers
by
\[
h^{k,p,q}(X)=\dim_{\mathbb{C}}H_{c}^{k,p,q}(X)\in\mathbb{N}_{0},
\]
for $k\in\{0,\cdots,2d\}$ and $p,q\in\{0,\cdots,k\}$. We say that
$(p,q)$ are $k$-weights of $X$, when $h^{k,p,q}\neq0$.
Mixed Hodge numbers verify $h^{k,p,q}(X)=h^{k,q,p}(X)$, and $\dim_{\mathbb{C}}H_{c}^{k}(X)=\sum_{p,q}h^{k,p,q}(X)$,
so they provide the (compactly supported) Betti numbers, which are
easily translated to the usual Betti numbers, in the smooth case,
by Poincaré duality.
Hodge numbers yield the so-called mixed Hodge polynomial of $X$,
\begin{equation}
\mu(X;\,t,u,v):=\sum_{k,p,q}h^{k,p,q}(X)\ t^{k}u^{p}v^{q}\in\mathbb{\mathbb{N}}_{0}[t,u,v],\label{eq:mu}
\end{equation}
on three variables. The mixed Hodge polynomial specializes to the
(compactly supported) Poincaré polynomial by setting $u=v=1$, $P_{t}^{c}(X)=\mu(X;\,t,1,1).$
Again, this provides the usual Poincaré polynomial in the smooth situation.
By substituting $t=-1$, mixed Hodge polynomials become a very useful
generalization of the Euler characteristic, called the \emph{Serre
polynomial} or $E$\emph{-polynomial} of $X$:
\[
E(X;\,u,v):=\sum_{k,p,q}(-1)^{k}h^{k,p,q}(X)\ u^{p}v^{q}\in\mathbb{Z}[u,v].
\]
From the $E$-polynomial we can compute the Euler characteristic of
$X$ as $\chi(X)=E(X;\,1,1)=\mu(X;\,-1,1,1)$ (recall that the compactly
supported Euler characteristic equals the usual one for complex quasi-projective
varieties).
When $X$ is smooth and projective, its Hodge structure is \emph{pure}
(for each $k$, the only $k$-weights are of the form $(p,k-p)$ with
$p\in\{0,\cdots,k\}$), and the $E$-polynomial actually determines
the Poincaré polynomial $P_{t}=P_{t}^{c}$.
In the present article, we deal with varieties which are neither smooth
nor complete, but often are of \emph{Hodge-Tate type} (also called
\emph{balanced} type), for which all the $k$-weights are of the form
$(p,p)$ with $p\in\{0,\cdots,k\}$. This restriction on weights holds
for complex (affine) algebraic groups (see, for example \cite{DL,Jo})
and smooth toric varieties, among others classes.
Poincaré, mixed Hodge and Serre polynomials satisfy a multiplicative
property with respect to Cartesian products $\mu(X\times Y)=\mu(X)\mu(Y)$
(coming from Künneth theorem, see \cite{PS}). The big computational
advantage of $E(X)$, as compared to $\mu(X)$ or $P^{c}(X)$ is that
it also satisfies an \emph{additive property with respect to stratifications}
by locally closed (in the Zariski topology) strata: whenever $X$
has a closed subvariety $Z\subset X$ we have
\[
E(X)=E(Z)+E(X\setminus Z),
\]
(see, eg. \cite{PS}) which generalizes the well known analogous statement
for $\chi$.
\begin{rem}
The terminology for $E$-polynomials is not standard, and some authors
refer to them as \emph{Hodge-Deligne} (mostly when dealing with pure
Hodge structures), and others as \emph{virtual Poincaré or Serre polynomials}.
After a literature review process, we are using \emph{Serre polynomial},
following the oldest references (see, for example \cite{GL}), and
paying tribute to the ideas of Serre on the role of virtual dimensions
for additivity in the weight filtration on mixed Hodge structures
(see \cite{To}).
\end{rem}
\subsection{Equivariant Serre polynomials and fibrations}
We will also need the multiplicative property of the $E$-polynomial
under certain algebraic fibrations, and more generally, for the \emph{equivariant
$E$-polynomial}, when these fibrations are acted by a finite group.
\begin{defn}
\label{def:alg-W-fibration}Let $\pi:X\to B$ be a morphism of quasi-projective
varieties and $W$ be a finite group acting algebraically on $X$,
and preserving the fibers of $\pi$. Assume also that all fibers $\pi^{-1}(b)$,
$b\in B$, are $W$-isomorphic to a given variety $F$. In this situation,
we call
\begin{equation}
F\to X\to B\label{eq:W-fibration}
\end{equation}
an \emph{algebraic $W$-fibration.}
\end{defn}
Given an algebraic $W$-fibration $F\to X\to B$, the mixed Hodge
structures on the (compactly supported) cohomology groups of $F$
and $X$ are representations of $W$, which we denote by $[H_{c}^{k,p,q}(F)]$
and similarly for $X$. This allows us to define the $W$-equivariant
$E$-polynomials:
\[
E^{W}(X;\,u,v)=\sum_{k,p,q}(-1)^{k}[H_{c}^{k,p,q}(X)]\ u^{p}v^{q}\in R(W)[u,v],
\]
where $R(W)$ denotes the representation ring of $W$. By taking the
dimensions of the fixed subspaces $[H_{c}^{k,p,q}(X)]^{W}$, under
$W$, we recover the $E$-polynomial of the quotient variety $X/W$
(see \cite{DL}, \cite{FS}).
We have the following fundamental result.
\begin{thm}
\label{thm:W-fibration} \cite{DL,LMN} Let $W$ be a finite group
and consider an algebraic $W$-fibration as above:
\[
F\to X\to B.
\]
Assume either:\\
(i) the fibration is locally trivial in the Zariski topology of $B$,
or\\
(ii) $F$, $X$ and $B$ are smooth, the fibration is locally trivial
in the complex analytic topology, and $\pi_{1}(B)$ acts trivially
on $H_{c}^{*}(F)$ (ie, the monodromy is trivial), or\\
(iii) $X$, $B$ are smooth and $F$ is a complex connected Lie group.\\
Then
\[
E^{W}(X)=E^{W}(F)\cdot E(B).
\]
Moreover, if $F$ and $B$ are of Hodge-Tate type, then so is $X$.\end{thm}
\begin{proof}
See \cite[Thm. 6.1]{DL} or \cite{FS}. For (iii) and for the result
on Hodge-Tate (balanced) type see also \cite{LMN}.
\end{proof}
Of course, when $W=1$ or the $W$ action is trivial, the hypothesis
imply the product formula $E(X)=E(F)\cdot E(B)$.
\begin{example}
The condition of $F$ being connected in (iii) is necessary for the
multiplicative property to hold, even in the case of trivial $W$.
Indeed, the action of $F=\mathbb{Z}_{2}$ on $X=\mathbb{P}^{1}\times\mathbb{P}^{1}$
by permuting the entries provides a fibration onto $\mathbb{P}^{2}$,
the second symmetric power of $\mathbb{P}^{1}$:
\[
\mathbb{Z}_{2}\rightarrow\mathbb{P}^{1}\times\mathbb{P}^{1}\rightarrow\mathsf{Sym}^{2}(\mathbb{P}^{1})=\mathbb{P}^{2}\;,
\]
By elementary methods, the $E$-polynomials are $E(\mathbb{Z}_{2})=2$,
$E(\mathbb{P}^{1}\times\mathbb{P}^{1})=(1+uv)^{2}$ and $E(\mathbb{P}^{2})=1+uv+u^{2}v^{2}$,
which do not satisfy the multiplicative property. One can also check
that this algebraic fibration is not locally trivial in the Zariski
topology, but only in the analytic (strong) topology.
\end{example}
\subsection{Special fibrations}
The notion of special group according to Serre and Grothendieck (c.f.
\cite{Gr,Se}) provides a useful criterion for applying Theorem \ref{thm:W-fibration}
to some algebraic fibrations.
By definition, a \emph{special group} is an algebraic group $H$ such
that every principal $H$-bundle is locally trivial in the Zariski
topology (c.f. \cite[p.11]{Gr}). In this context, a principal $H$-bundle
$\pi:X\to B$, will be called a \emph{special fibration}, whenever
$H$ is special. If, furthermore, a finite group $W$ acts on $X$
as in Definition \ref{def:alg-W-fibration}, we call $\pi$ a \emph{special
$W$-fibration}. Thus, Given a special $W$-fibration $H\to X\to B$,
the following is an immediate consequence of Theorem \ref{thm:W-fibration}(i).
\begin{cor}
\label{cor:special-fibration} Let $X\to B$ be a special $W$-fibration
with fiber isomorphic to $H$. Then
\[
E^{W}(X)=E^{W}(H)\cdot E(B).
\]
\end{cor}
Now, let $G$ be a connected complex algebraic reductive group, with
center $Z\subset G$, and denote by $PG=G/Z$ its adjoint group. In
case $Z$ is connected, we obtain a simple relation between the Serre
polynomials of $G$, $Z$ and $PG$.
\begin{prop}
\label{prop:PG-fibration}Let $G$ be a complex reductive group with
connected center $Z$. Then $E(G)=E(PG)\cdot E(Z)$. \end{prop}
\begin{proof}
Since $PG$ is defined to be a quotient by a subgroup, the following
fibration
\begin{equation}
Z\to G\to PG,\label{eq:ZG-G-PG}
\end{equation}
is a principal $Z$-bundle. Since $G$ is complex, its center $Z$
is the product of a group of multiplicative type and a unipotent subgroup.
Given that $G$ is reductive, the unipotent subgroup is trivial. Since
$Z$ is connected by hypothesis, $Z$ is a torus, that is $Z\cong(\mathbb{C}^{\ast})^{l}$,
$l=\dim Z$. Since a torus is a \textit{\emph{special}}\emph{ }group
by \cite{Se}, \eqref{eq:ZG-G-PG} is a special fibration and the
result follows from Corollary \ref{cor:special-fibration} (with $W$
being the trivial group).\end{proof}
\begin{example}
\label{exa:PGL-section} Since $E(\mathbb{C}^{*})=uv-1$, the formula
$E(GL_{n})=(uv-1)E(PGL_{n})$ follows from the special fibration
\[
\mathbb{C}^{*}\to GL_{n}\to PGL_{n},
\]
which will be very important later on, and can be directly shown to
be locally trivial in the Zariski topology. By contrast, the determinant
fibration
\[
SL_{n}\to GL_{n}\stackrel{\det}{\to}\mathbb{C}^{*}
\]
is not trivial in the Zariski topology, although we still have $E(GL_{n})=(uv-1)E(SL_{n})$. \end{example}
\begin{rem}
\label{rem:Hodge-Tate-open}As mentioned before, complex (affine)
algebraic groups $G$ are of Hodge-Tate type. Hence, their mixed Hodge
polynomials $\mu(G;t,u,v)$ reduce to a two variable polynomial (with
variables $t$ and $uv$), and their $E$-polynomials depend only
on the product $uv$. On the other hand, given a variety $X$ whose
$E$-polynomial is a function of only $x=uv$, it is not necessarily
true that $X$ is of Hodge-Tate type. For example, if we had $\mu(X;t,u,v)=1+tu+t^{2}u+t^{2}uv$,
then $X$ could not be of Hodge-Tate type, but $E(X)=1+uv=E(\mathbb{P}^{1})$
is a one-variable polynomial in $x:=uv$. We strongly believe that
the character varieties studied in this paper are also of Hodge-Tate
(balanced) type. Indeed, the methods in \cite{LMN,LM} show this is
the case for $n=2$ and 3. However, as far as we know, the general
case seems to be an open problem (see also Conjecture \ref{conj:Langlands-dual}).
\end{rem}
\section{Character Varieties for a group and its adjoint}
As before, let $G$ be a connected complex reductive algebraic group
with center $Z$. In this section, we recall some aspects of character
varieties and their construction via affine GIT (Geometric Invariant
Theory). This is applied to provide a simple relation between the
Serre polynomials of $\mathcal{X}_{\Gamma}GL_{n}$ and of $\mathcal{X}_{\Gamma}PGL_{n}$,
for the free group $\Gamma=F_{r}$.
\subsection{Character varieties as GIT quotients.}
Let $\Gamma$ be a finitely presented group, and denote by
\[
\mathcal{R}_{\Gamma}G=\mathrm{Hom}(\Gamma,G),
\]
the algebraic variety of representations of $\Gamma$ in $G$, where
$\rho\in\mathcal{R}_{\Gamma}G$ is defined by the image of the generators
of $\Gamma$, $\rho(\gamma)$, satisfying the relations in $\Gamma$.
The moduli space of representations of $\Gamma$ into $G$ is the
$G$-character variety of $\Gamma$
\[
\mathcal{X}_{\Gamma}G:=\mathrm{Hom}(\Gamma,G)/\!\!/ G,
\]
defined as the affine GIT quotient under the algebraic action of $G$
on $\mathcal{R}_{\Gamma}G$ by conjugation of representations. Given
that a GIT quotient identifies those orbits whose closure has a non-empty
intersection, we describe the quotient by the unique closed point
in each equivalence class, called the \emph{polystable representations},
which we define below. Given a representation $\rho\in\mathcal{R}_{\Gamma}G$,
denote by $Z_{\rho}$ the centralizer of $\rho(\Gamma)$ inside $G$.
Note that $ZG\subset Z_{\rho}$, since the center commutes with any
representation. Let us call the subgroup $G_{\mathcal{R}_{\Gamma}G}:=\bigcap_{\rho\in\mathcal{R}_{\Gamma}G}Z_{\rho}$
the \emph{center of the action}, since it acts trivially, and $G/G_{\mathcal{R}_{\Gamma}G}$
acts effectively on $\mathcal{R}_{\Gamma}G$. Denote by $\psi_{\rho}$
the (effective) orbit map through $\rho$:
\begin{eqnarray*}
\psi_{\rho}:G/G_{\mathcal{R}_{\Gamma}G} & \to & \mathcal{R}_{\Gamma}G\\
g & \mapsto & g\cdot\rho
\end{eqnarray*}
\begin{defn}
\label{def:center-polystable}In the situation above, we say that
$\rho\in\mathcal{R}_{\Gamma}G$ is \emph{polystable} if the orbit
$G\cdot\rho:=\{g\rho g^{-1}:\ g\in G\}$ is closed (Zariski) in $\mathcal{R}_{\Gamma}G$.
We say that $\rho\in\mathcal{R}_{\Gamma}G$ is \emph{stable} if it
is polystable and $\psi_{\rho}$ is a proper map.
\end{defn}
It can be shown that the subset of polystable representations in $\mathcal{R}_{\Gamma}G$,
denoted by $\mathcal{R}_{\Gamma}^{ps}G\subset\mathcal{R}_{\Gamma}G$,
is stratified by locally-closed subvarieties, yielding a quotient
which can be identified with the affine GIT quotient (see \cite{FL2}):
\[
\mathcal{X}_{\Gamma}G=\mathcal{R}_{\Gamma}G/\!\!/ G\cong\mathcal{R}_{\Gamma}^{ps}G/G.
\]
\begin{defn}
We say that $\rho\in\mathcal{R}_{\Gamma}G$ is irreducible if $\rho$
is polystable and $Z_{\rho}$ is a finite extension of $ZG$, and
call $\rho$ a good representation if it is irreducible and $Z_{\rho}=ZG$.
\end{defn}
Denote by $\mathcal{R}_{\Gamma}^{irr}G\subset\mathcal{R}_{\Gamma}^{ps}G$
and by $\mathcal{R}_{\Gamma}^{g}G$ the subset of irreducible and
good representations, respectively, and set $\mathcal{X}_{\Gamma}^{irr}G:=\mathcal{R}_{\Gamma}^{irr}G/G$
and $\mathcal{X}_{\Gamma}^{g}G:=\mathcal{R}_{\Gamma}^{g}G/G$. It
is known that $\mathcal{R}_{\Gamma}^{irr}G$ is a quasi-projective
variety, since it is a Zariski open subset of $\mathcal{R}_{\Gamma}G$.
The character variety of good representations is a smooth algebraic
variety, by \cite{Sik}. For further details, we refer the reader
to \cite{FLR}, \cite{GLR} or \cite[Section 3]{FNZ}.
In the case when $G=GL_{n}$ or $G=SL_{n}$, Schur's lemma easily
implies the equivalence between good and irreducible representations.
\begin{lem}
\label{lem:Schur}Let $G=GL_{n}$ or $G=SL_{n}$ and $\rho\in\mathrm{Hom}(\Gamma,G)$
be polystable. Then $\rho$ is a good representation if and only if
it is irreducible. In particular $\mathcal{X}_{\Gamma}^{g}G=\mathcal{X}_{\Gamma}^{irr}G$.\end{lem}
\begin{proof}
By definition, a good representation is irreducible, and the converse
follows from \cite{FL3}; we can also proceed as follows. First let
$G=GL_{n}$. If $\rho$ is irreducible, Schur's lemma states that
any $g\in G$ commuting with every $\rho(\gamma)$, $\gamma\in\Gamma$,
is central, so that $Z_{\rho}=ZGL_{n}=\mathbb{C}^{*}$. Therefore,
$\rho$ is good.
Now let $G=SL_{n}$. For any representation $\rho:\Gamma\to SL_{n}$
we get a representation of $GL_{n}$ by composition $i\circ\rho:\Gamma\to SL_{n}\overset{i}{\hookrightarrow}GL_{n}$.
Since $SL_{n}$ is the derived group of $GL_{n}$, we have
\[
\frac{Z_{i\circ\rho}}{ZGL_{n}}=\frac{Z_{i\circ\rho}\cap SL_{n}}{ZGL_{n}\cap SL_{n}}\text{, and }Z_{\rho}=Z_{i\circ\rho}\cap SL_{n}.
\]
Hence, if $\rho$ is an irreducible representation of $SL_{n}$ then
$i\circ\rho$ is an irreducible representation of $GL_{n}$ and hence,
by the previous argument, $i\circ\rho$ is a good $GL_{n}$-representation.
Therefore by using the fact that $Z_{\rho}=Z_{i\circ\rho}\cap SL_{n}$
we get $Z_{\rho}=ZSL_{n}$ which says that $\rho$ is a good representation
into $SL_{n}$. \end{proof}
\begin{prop}
\label{prop:good}Let $G=G_{1}\times G_{2}$, be a product of two
reductive groups. We have the following isomorphism of smooth algebraic
varieties:
\[
\mathcal{X}_{\Gamma}^{g}(G)=\mathcal{X}_{\Gamma}^{g}(G_{1})\times\mathcal{X}_{\Gamma}^{g}(G_{2}).
\]
\end{prop}
\begin{proof}
We have $\mathcal{X}_{\Gamma}(G)=\mathcal{X}_{\Gamma}(G_{1})\times\mathcal{X}_{\Gamma}(G_{2})$
as algebraic varieties, since conjugation by $G=G_{1}\times G_{2}$
works independently on each factor of (cf. also \cite{FL1}):
\[
\mathrm{Hom}(\Gamma,G)=\mathrm{Hom}(\Gamma,G_{1})\times\mathrm{Hom}(\Gamma,G_{2}).
\]
Now suppose that a given representation $\rho\in\mathrm{Hom}(\Gamma,G)$ has
a trivial stabilizer (the center of $G$). Then it is a product of
the centers of $G_{1}$ and of $G_{2}$, so $\rho$ is of the form
$\rho=(\rho_{1},\rho_{2})$ with good factors $\rho_{i}\in\mathrm{Hom}(\Gamma,G_{i})$,
$i=1,2$. The converse is also clear, so the above isomorphism restricts
to the good loci.
\end{proof}
Put together, the previous 2 statements show that, for $GL_{n}$ and
for $SL_{n}$-character varieties, the good locus coincides with both
the irreducible locus and with the stable locus, and these are multiplicative,
under products of reductive groups.
\subsection{$GL_{n}$- and $PGL_{n}$-character varieties of the free group}
When considering $\Gamma=F_{r}$, the free group of rank $r$, we
use the simplified notations $\mathcal{R}_{r}G:=\mathrm{Hom}(F_{r},G)$ and
$\mathcal{X}_{r}G:=\mathrm{Hom}(F_{r},G)/\!\!/ G$, for the $G$-representation
and $G$-character varieties, respectively.
There is a natural action of $\mathcal{R}_{r}\mathbb{C}^{*}:=\mathrm{Hom}(F_{r},\mathbb{C}^{*})\cong(\mathbb{C}^{*})^{r}$
on $\mathcal{R}_{r}GL_{n}$, which, in terms of the fixed generators
$\gamma_{1},\cdots,\gamma_{r}$ of $F_{r}$ is given by scalar multiplication
of representations:
\begin{equation}
\sigma\cdot\rho(\gamma_{i}):=\sigma(\gamma_{i})\rho(\gamma_{i}),\quad\sigma\in\mathcal{R}_{r}\mathbb{C}^{*},\ \rho\in\mathcal{R}_{r}GL_{n}.\label{eq:C-GLnaction}
\end{equation}
The quotient of this action is the representation variety for $PGL_{n}$,
$\mathcal{R}_{r}PGL_{n}$. Since the central $\mathbb{C}^{*}$ action
commutes with conjugation of representations, this sequence descends
to the character varieties
\begin{equation}
(\mathbb{C}^{\ast})^{r}\rightarrow\mathcal{X}_{r}GL_{n}\rightarrow\mathcal{X}_{r}PGL_{n}.\label{eq:GLn-PGLn-fibration}
\end{equation}
Note that, because $\Gamma=F_{r}$, all $PGL_{n}$ representations
can be lifted to $GL_{n}$. The following generalization to other
groups $G$ and their adjoints $PG$ is immediate from Corollary \ref{cor:special-fibration}.
\begin{prop}
\label{prop:PG-fibration-X} Let $G$ be a connected reductive group
with connected center $Z$. Then, the natural quotient map $\mathcal{X}_{r}G\to\mathcal{X}_{r}PG$
is a special fibration, with fiber $\mathcal{X}_{r}Z\cong Z^{r}$.
Hence,
\[
E(\mathcal{X}_{r}G)=(uv-1)^{rl}\,E(\mathcal{X}_{r}PG),
\]
with $l=\dim Z$. In particular, for $G=GL_{n}$ we get $E(\mathcal{X}_{r}GL_{n})=(uv-1)^{r}\,E(\mathcal{X}_{r}PGL_{n})$.
\end{prop}
\section{The strictly polystable case}
\label{section:stratifications}In this section and the next one,
we prove Theorem \ref{thm:main}: the equality of the Serre polynomials
of $\mathcal{X}_{r}SL_{n}$ and $\mathcal{X}_{r}PGL_{n}$. If we tried
to imitate the fibration methods of Section 3, we would consider the
quotient morphism $SL_{n}\to PGL_{n}$, with kernel $\mathbb{Z}_{n}$,
the center of $SL_{n}$, to obtain a fibration of character varieties:
\[
\mathbb{Z}_{n}^{r}\to\mathcal{X}_{r}SL_{n}\to\mathcal{X}_{r}PGL_{n}.
\]
However, since the fiber is not connected, we cannot apply Corollary
\ref{cor:special-fibration}; instead we proceed by stratifying this
fibration by polystable type, in analogy to the stratification used
in \cite{FNZ} (recalled below) and examine the $\mathbb{Z}_{n}^{r}$
action on the cohomology of each individual stratum. From now on,
our $E$-polynomials will only depend on a single variable; to emphasize
this property, we adopt the substitution $x\equiv uv$ and use the
notation:
\begin{equation}
e(X):=E(X;\,\sqrt{x},\sqrt{x})\in\mathbb{Z}[x].\label{eq:small-e}
\end{equation}
\subsection{Stratifications by polystable type}
Any character variety admits a stratification by the dimension of
the stabilizer of a given representation. When dealing with the general
linear group $GL_{n}$ as well as the related groups $SL_{n}$ and
$PGL_{n}$, there is a more refined stratification which gives a lot
more information on the corresponding character varieties $\mathcal{X}_{\Gamma}G$.
In this subsection, we recall this stratification, following \cite[Section 4.1]{FNZ},
and describe its analogous versions for $SL_{n}$ and $PGL_{n}$.
A \emph{partition} of $n\in\mathbb{N}$ is denoted by $[k]=[1^{k_{1}}\cdots j^{k_{j}}\cdots n^{k_{n}}]$
where the exponent $k_{j}$ means that $[k]$ has $k_{j}\geq0$ parts
of size $j\in\{1,\cdots,n\}$, with $n=\sum_{j=1}^{n}j\,k_{j}$. The
\emph{length} of the partition is given by the sum of the exponents
$|[k]|:=\sum k_{j}$ and call $\mathcal{P}_{n}$ the set of all partitions
of $n\in\mathbb{N}$. As an example, $[1^{2}\;2\;4]\in\mathcal{P}_{8}$
is the partition $8=1\cdot2+2+4$, with length equal to $4$.
\begin{defn}
Let $n\in\mathbb{N}$, $[k]\in\mathcal{P}_{n}$ and $\Gamma$ be a
finitely presented group. We say that $\rho\in\mathcal{R}_{\Gamma}GL_{n}=\mathrm{Hom}(\Gamma,GL_{n})$
is \emph{$[k]$-polystable} if $\rho$ is conjugated to $\bigoplus_{j=1}^{n}\rho_{j}$
where each $\rho_{j}$ is, in turn, a direct sum of $k_{j}>0$ \emph{irreducible}
representations of $\mathcal{R}_{\Gamma}(GL_{j})$, for $1\leq j\leq n$
(by convention, if some $k_{j}=0$, then $\rho_{j}$ is not present
in the direct sum). We denote the $[k]$-polystable representations
by $\mathcal{R}_{\Gamma}^{[k]}GL_{n}$ and $\mathcal{X}_{\Gamma}^{[k]}GL_{n}\subset\mathcal{X}_{\Gamma}GL_{n}$
refers to the $[k]$-polystable locus of the character variety.\end{defn}
\begin{rem}
\label{rem:ab-irr-strata} The abelian stratum, i.e, the stratum of
representations factoring as $\Gamma\to\Gamma/[\Gamma,\Gamma]\to G$,
corresponds to the partition $[1^{n}]$ of maximal length $n$ (see
\cite{FNZ}); on the other hand, irreducible representations correspond
to the partition $[n]$ of minimal length $1$. By Lemma \ref{lem:Schur},
this irreducible stratum is also the smooth (and the stable) locus
of the representation varieties $\mathcal{R}_{\Gamma}^{irr}GL_{n}:=\mathcal{R}_{\Gamma}^{[n]}GL_{n}$.
\end{rem}
The following summarizes the situation and is proved in \cite[Proposition 4.3]{FNZ}.
\begin{prop}
\label{prop:locally-closed-GLn}Fix $n\in\mathbb{N}$. The character
variety $\mathcal{X}_{\Gamma}GL_{n}$ can be written as a disjoint
union of of locally closed quasi-projective varieties, labelled by
partitions $[k]\in\mathcal{P}_{n}$
\[
\mathcal{X}_{\Gamma}GL_{n}=\bigsqcup_{[k]\in\mathcal{P}_{n}}\mathcal{X}_{\Gamma}^{[k]}GL_{n},
\]
where $\mathcal{X}_{\Gamma}^{[k]}GL_{n}$ consists of equivalence
classes of $[k]$-polystable representations. Moreover, $\mathcal{X}_{\Gamma}^{[n]}GL_{n}$
is precisely the open locus $\mathcal{X}_{\Gamma}^{irr}GL_{n}$ of
irreducible classes of representations.
\end{prop}
\subsection{The free group case}
From now on, we restrict ourselves to the case $\Gamma=F_{r}$, the
free group of rank $r\in\mathbb{N}$, and use the notations
\[
\mathcal{X}_{r}GL_{n},\quad\quad\mathcal{X}_{r}SL_{n},\quad\quad\mathcal{X}_{r}PGL_{n}
\]
for the corresponding character varieties. We will now define the
$[k]$-polystable loci $\mathcal{X}_{r}^{[k]}SL_{n}\subset\mathcal{X}_{r}SL_{n}$
and $\mathcal{X}_{r}^{[k]}PGL_{n}\subset\mathcal{X}_{r}PGL_{n}$,
and the corresponding stratifications, in analogy to Proposition \ref{prop:locally-closed-GLn}.
Recall the action in (\ref{eq:C-GLnaction}) and (\ref{eq:GLn-PGLn-fibration})
which clearly preserves the polystable type stratification of $GL_{n}$,
so we define
\[
\mathcal{X}_{r}^{[k]}PGL_{n}:=\mathcal{X}_{r}^{[k]}GL_{n}/\mathcal{R}_{r}\mathbb{C}^{*}=\mathcal{X}_{r}^{[k]}GL_{n}/(\mathbb{C}^{*})^{r}.
\]
The next result is proved in the same way as Proposition \ref{prop:PG-fibration-X}.
We observe that the $E$-polynomials of all strata $\mathcal{X}_{r}^{[k]}GL_{n}$
are $1$-variable polynomials (see \cite{FNZ}). Hence, we use the
notation in \eqref{eq:small-e} for $E$-polynomials in the variable
$x=uv$.
\begin{prop}
\label{prop:RPG-fibration}Let $F_{r}$ be a free group of rank $r$.
For every $[k]\in\mathcal{P}_{n}$, the fibration
\[
\mathcal{R}_{r}\mathbb{C}^{*}\to\mathcal{X}_{r}^{[k]}GL_{n}\to\mathcal{X}_{r}^{[k]}PGL_{n}
\]
is special. In particular, $e(\mathcal{X}_{r}^{[k]}GL_{n})=(x-1)^{r}\,e(\mathcal{X}_{r}^{[k]}PGL_{n})$.
\end{prop}
\subsection{The action of the symmetric group on strictly polystable strata}
For a partition $[k]\in\mathcal{P}_{n}$, we define the $[k]$-stratum
of $\mathcal{X}_{r}SL_{n}$ by restriction of the corresponding one
for $GL_{n}$:
\[
\mathcal{X}_{r}^{[k]}SL_{n}:=\{\rho\in\mathcal{X}_{r}^{[k]}GL_{n}\,|\,\det\rho=1\},
\]
where the determinant of a (conjugacy class of a) representation is
an element of $\mathcal{R}_{r}\mathbb{C}^{*}$. The action of $\mathcal{R}_{r}\mathbb{C}^{*}$
on $\mathcal{X}_{r}^{[k]}GL_{n}$ does not preserve $\mathcal{X}_{r}^{[k]}SL_{n}$
because of the determinant condition. On the other hand, from the
behaviour of the Zariski topology under closed inclusions and quotients,
the following is clear.
\begin{prop}
\label{prop:locally-closed-SL+PGL}Fix $n\in\mathbb{N}$. The character
varieties $\mathcal{X}_{r}SL_{n}$ and $\mathcal{X}_{r}PGL_{n}$ can
be written as a disjoint unions of of locally closed quasi-projective
varieties, labelled by partitions $[k]\in\mathcal{P}_{n}$
\[
\mathcal{X}_{r}SL_{n}=\bigsqcup_{[k]\in\mathcal{P}_{n}}\mathcal{X}_{r}^{[k]}SL_{n},\quad\quad\mathcal{X}_{r}PGL_{n}=\bigsqcup_{[k]\in\mathcal{P}_{n}}\mathcal{X}_{r}^{[k]}PGL_{n}
\]
Moreover, $\mathcal{X}_{r}^{irr}SL_{n}=\mathcal{X}_{r}^{[n]}SL_{n}$
and $\mathcal{X}_{r}^{irr}PGL_{n}=\mathcal{X}_{r}^{[n]}PGL_{n}$ are
Zariski open.
\end{prop}
It turns out that the irreducible strata $\mathcal{X}_{r}^{irr}SL_{n}$
and $\mathcal{X}_{r}^{irr}PGL_{n}$ are the most difficult cases to
compare, so we start by studying the ones given by partitions with
at least 2 parts.
\begin{thm}
\label{thm:SL-2strata}Fix $r$ and $n\in\mathbb{N}$. For a partition
$[k]\in\mathcal{P}_{n}$ with length $s>1$, we have:
\[
e(\mathcal{X}_{r}^{[k]}GL_{n})=(x-1)^{r}e(\mathcal{X}_{r}^{[k]}SL_{n})
\]
\end{thm}
\begin{proof}
We start with a relation between $E$-polynomials of cartesian products
of irreducible character varieties. Let $s\in\mathbb{N}$ and $\mathbf{n}=(n_{1},\cdots,n_{s})$
be a sequence of $s$ positive integers with $n=\sum_{i=1}^{s}n_{i}$.
Denote:
\[
\mathsf{X}_{r}^{\mathbf{n}}:=\times_{i=1}^{s}\mathcal{X}_{r}^{irr}GL_{n_{i}},
\]
and
\[
S\mathsf{X}_{r}^{\mathbf{n}}:=\left\{ \rho=(\rho_{1},\cdots,\rho_{s})\in\mathsf{X}_{r}^{\mathbf{n}}\,|\,\prod_{i=1}^{s}\det\rho_{i}=1\right\} \subset\mathsf{X}_{r}^{\mathbf{n}}.
\]
It is clear that the previous constructions can be carried out in
this setting. For example, letting $J:=(\mathcal{R}_{r}\mathbb{C}^{*})^{s}$,
there is a natural action of $J$ on $\mathsf{X}_{r}^{\mathbf{n}}$:
\[
(\sigma_{1},\cdots,\sigma_{s})\cdot(\rho_{1},\cdots,\rho_{s})=(\sigma_{1}\rho_{1},\cdots,\sigma_{s}\rho_{s}),\quad\quad\sigma\in J=(\mathcal{R}_{r}\mathbb{C}^{*})^{s},\rho_{i}\in\mathcal{X}_{r}^{irr}GL_{n_{i}},
\]
since a scalar multiple of an irreducible representation is again
irreducible.
Define the multiplication map $m:J\to\mathcal{R}_{r}\mathbb{C}^{*}\cong(\mathbb{C}^{*})^{r}$
as follows:
\[
m(\sigma_{1},\cdots,\sigma_{s})=\sigma_{1}^{m_{1}}\cdots\sigma_{s}^{m_{s}},
\]
where $m_{i}:=n_{i}/d$ with $d=gcd(n_{1},\cdots,n_{s})$ (a power
of a representation into $\mathbb{C}^{*}$ is just the power of every
generator). It turns out that
\[
H:=\ker m=\{(\sigma_{1},\cdots,\sigma_{s})\in(\mathcal{R}_{r}\mathbb{C}^{*})^{s}\,|\,\sigma_{1}^{m_{1}}\cdots\sigma_{s}^{m_{s}}=1\},
\]
is abelian, connected (because $s>1$, and the $m_{i}$ are coprime)
and reductive. Then, it follows that $H$ is isomorphic to an algebraic
torus of the appropriate dimension, $H\cong(\mathbb{C}^{*})^{r(s-1)},$
since $J=(\mathcal{R}_{r}\mathbb{C}^{*})^{s}\cong(\mathbb{C}^{*})^{rs}$.
This allows us to obtain a diagram of algebraic fibrations (vertical
arrows):
\[
\begin{array}{ccccc}
H & \subset & (\mathcal{R}_{r}\mathbb{C}^{*})^{s} & \stackrel{m}{\to} & \mathcal{R}_{r}\mathbb{C}^{*}\\
\downarrow & & \downarrow\\
S\mathsf{X}_{r}^{\mathbf{n}} & \subset & \mathsf{X}_{r}^{\mathbf{n}}\\
\downarrow & & \downarrow\\
S\mathsf{X}_{r}^{\mathbf{n}}/H & = & \mathsf{X}_{r}^{\mathbf{n}}/J,
\end{array}
\]
where both vertical fibrations (the left fibration being the restriction
to $S\mathsf{X}_{r}^{\mathbf{n}}$) are special.
Now we note that, using an action of a finite group, we can obtain
all strata of the stratification of $\mathcal{X}_{r}^{[k]}GL_{n}$.
To be concrete, denote by $Q_{n}$ the finite set:
\[
Q_{n}:=\{\mathbf{n}=(n_{1},\cdots,n_{s})\in\mathbb{N}^{s}\ |\ \sum_{i=1}^{s}n_{i}=n\}.
\]
To every element of $Q_{n}$ we associate a unique partition of $n$
as follows:
\begin{eqnarray*}
p:Q_{n} & \to & \mathcal{P}_{n}\\
\mathbf{n}=(n_{1},\cdots,n_{s}) & \mapsto & [k]:=[1^{k_{1}}\cdots n^{k_{n}}]
\end{eqnarray*}
where $k_{j}$ is the number of entries in the sequence $\mathbf{n}$
equal to $j$. For every $\mathbf{n}\in Q_{n}$, let $S_{\mathbf{n}}:=S_{[k]}\subset S_{n}$
be the subgroup of the symmetric group defined by this partition $[k]=p(\mathbf{n})$.
Moreover, we have isomorphisms of varieties:
\[
\mathsf{X}_{r}^{\mathbf{n}}/S_{\mathbf{n}}\cong\mathcal{X}_{r}^{[k]}GL_{n},\quad\quad S\mathsf{X}_{r}^{\mathbf{n}}/S_{\mathbf{n}}\cong\mathcal{X}_{r}^{[k]}SL_{n},
\]
as can be easily checked. Now, for every $\mathbf{n}\in Q_{n}$, and
taking $(\mathsf{X}_{r}^{\mathbf{n}}/J)/S_{\mathbf{n}}$ at the bottom,
the above diagram becomes a special algebraic $S_{\mathbf{n}}$-fibration
(with trivial action on the bottom and the top row is still the same),
so we can apply Theorem \ref{thm:W-fibration} and Corollary \ref{cor:special-fibration}
to both vertical fibrations to obtain:
\[
e^{S_{\mathbf{n}}}(\mathsf{X}_{r}^{\mathbf{n}})=e((\mathsf{X}_{r}^{\mathbf{n}}/J)/S_{\mathbf{n}})\,e^{S_{\mathbf{n}}}(J),\quad\quad e^{S_{\mathbf{n}}}(S\mathsf{X}_{r}^{\mathbf{n}})=e((\mathsf{X}_{r}^{\mathbf{n}}/J)/S_{\mathbf{n}})\,e^{S_{\mathbf{n}}}(H).
\]
and to the top horizontal fibration (which is also special) to get
\[
e^{S_{\mathbf{n}}}(J)=e^{S_{\mathbf{n}}}(H)\,(x-1)^{r},
\]
and putting the formulae together:
\[
e^{S_{\mathbf{n}}}(\mathsf{X}_{r}^{\mathbf{n}})=e((\mathsf{X}_{r}^{\mathbf{n}}/J)/S_{\mathbf{n}})\,e^{S_{\mathbf{n}}}(H)\,(x-1)^{r}=e^{S_{\mathbf{n}}}(S\mathsf{X}_{r}^{\mathbf{n}})\,(x-1)^{r}
\]
Finally, we just need to take the invariant parts of the equivariant
polynomials:
\[
e(\mathcal{X}_{r}^{[k]}GL_{n})=e(\mathsf{X}_{r}^{\mathbf{n}}/S_{\mathbf{n}})=e(S\mathsf{X}_{r}^{\mathbf{n}}/S_{\mathbf{n}})\,(x-1)^{r},
\]
whenever $[k]=p(\mathbf{n})\in\mathcal{P}_{n}$ (noting that $(x-1)^{r}=(x-1)^{r}T\in R(S_{\mathbf{n}})[x]$,
where $T$ is the trivial $S_{\mathbf{n}}$ one-dimensional representation,
as elements of the ring $R(S_{\mathbf{n}})[x]$). \end{proof}
\begin{rem}
Even though the above formulas prove these $E$-polynomials are only
functions of $x=uv$, one can not conclude that strata $\mathcal{X}_{r}^{[k]}SL_{n}$
are of Hodge-Tate type without further arguments (see Remark \ref{rem:Hodge-Tate-open}).
\end{rem}
\subsection{Proof of the main Theorem}
We now outline the proof of the main theorem, which relies crucially
on the following.
\begin{thm}
\label{thm:irred-Zn-action} The central action of $\mathbb{Z}_{n}^{r}$
on $\mathcal{X}_{r}^{irr}SL_{n}$ giving the quotient map
\[
\mathcal{X}_{r}^{irr}SL_{n}\to\mathcal{X}_{r}^{irr}PGL_{n}
\]
induces an isomorphism of mixed Hodge structures $H^{*}(\mathcal{X}_{r}^{irr}SL_{n})\cong H^{*}(\mathcal{X}_{r}^{irr}PGL_{n}).$
\end{thm}
The proof of Theorem \ref{thm:irred-Zn-action} is delayed to Section
5. We will use differential geometric techniques, taking advantage
of the fact that $\mathcal{X}_{r}^{irr}SL_{n}$ is a smooth variety
and $\mathcal{X}_{r}^{irr}PGL_{n}$ is an orbifold (see \cite{FL2,Sik}).
Assuming Theorem \ref{thm:irred-Zn-action}, we can now prove the
main result of the article.
\begin{thm}
\label{thm:Main} Let $\Gamma=F_{r}$. Then, for all $[k]\in\mathcal{P}_{n}$,
we have $e(\mathcal{X}_{r}^{[k]}SL_{n})=e(\mathcal{X}_{r}^{[k]}PGL_{n})$.
Consequently, $e(\mathcal{X}_{r}SL_{n})=e(\mathcal{X}_{r}PGL_{n})$.\end{thm}
\begin{proof}
From Theorem \ref{thm:irred-Zn-action}, we have $H^{*}(\mathcal{X}_{r}^{irr}SL_{n})\cong H^{*}(\mathcal{X}_{r}^{irr}PGL_{n})$
as mixed Hodge structures, so that their $E$-polynomials coincide.
For any other stratum $[k]=[1^{k_{1}}\cdots n^{k_{n}}]$, which has
more than one part, the equality $e(\mathcal{X}_{r}^{[k]}SL_{n})=e(\mathcal{X}_{r}^{[k]}PGL_{n})$
follows from Theorem~\ref{thm:SL-2strata}. Finally, the last statement
follows from the additivity of the $E$-polynomial applied to the
locally-closed stratifications of Proposition \ref{prop:locally-closed-SL+PGL}.
\end{proof}
Noting that $SL_{n}$ and $PGL_{n}$ are Langlands dual groups, and
that our proof seems well adapted to more general actions of finite
subgroups, we put forward the following conjecture (answered here
in the case $G=SL_{n}$), and plan to address the general statement
in a future work.
\begin{conjecture}
\label{conj:Langlands-dual}Let $\Gamma=F_{r}$ and $G$, $G^{L}$
be complex reductive Langlands dual groups. Then both $\mathcal{X}_{r}G$
and $\mathcal{X}_{r}G^{L}$ are of Hodge-Tate type, and $e(\mathcal{X}_{r}G)=e(\mathcal{X}_{r}G^{L})$. \end{conjecture}
\begin{rem}
The part of the conjecture claiming Hodge-Tate type is still largely
open, even for $G=SL_{n}$ (see also Remark \ref{rem:Hodge-Tate-open}).
To the best of our knowledge, the only free group character varieties
that are known to be balanced are for $G=SL_{n}$ and $G=PGL_{n}$
with $n=2,3$ (see \cite{LM}).
\end{rem}
\subsection{Katz's theorem and the case $n$ odd}
When $n$ is odd, there is an alternative method to prove that the
Serre polynomials of $\mathcal{X}_{r}SL_{n}$ and $\mathcal{X}_{r}PGL_{n}$
coincide. The argument was mentioned in \cite[Remark 9]{LM}, and
we detail it here, for convenience. Denote by $\mathbb{F}_{q}$ a
finite field with $q$ elements and characteristic $p$, so that $q=p^{s}$,
for some $s\in\mathbb{N}$. A scheme $X$, defined over $\mathbb{Z}$,
is called of \emph{polynomial type} if there is a polynomial $C_{X}(t)\in\mathbb{Z}[t]$
(called the \emph{counting polynomial} for $X$) such that the number
of $\mathbb{F}_{q^{s}}$ points of $X$ is precisely
\[
|X/\mathbb{F}_{q^{s}}|=C_{X}(q^{s}),
\]
for every $s$ and almost every prime $p$. In \cite[Appendix]{HRV}
(see also \cite{BH}) N. Katz showed that if such a scheme $X$ is
of polynomial type, with counting polynomial $C_{X}$, then the Serre
polynomial of the complex variety $X(\mathbb{C})=X\otimes_{\mathbb{Z}}\mathbb{C}$
coincides with the counting polynomial:
\[
E_{x}(X(\mathbb{C}))=C_{X(\mathbb{C})}(x).
\]
To apply this to our character varieties, note that the natural surjective
morphism of algebraic groups
\[
SL_{n}(\mathbb{F}_{p})\to PGL_{n}(\mathbb{F}_{p}),
\]
has as kernel the scalar matrices $aI$ of determinant $1$, so that
$a^{n}=1$. By Dirichlet's Theorem on arithmetic progressions, for
a fixed $n$ odd, there exists an infinite number of primes such that
$(n,p-1)=1$, in which cases there are no non-trivial roots of unity,
so that $SL_{n}(\mathbb{F}_{p})\simeq PGL_{n}(\mathbb{F}_{p})$. This
implies that the representation spaces $\mathcal{R}_{r}SL_{n}(\mathbb{F}_{p})$
and $\mathcal{R}_{r}PGL_{n}(\mathbb{F}_{p})$ are in bijective correspondence,
and the same holds for the number of points of the character varieties
over $\mathbb{F}_{q}$:
\[
\left|\mathcal{X}_{r}SL_{n}(\mathbb{F}_{p})\right|=\left|\mathcal{X}_{r}PGL_{n}(\mathbb{F}_{p})\right|.
\]
In \cite[Corollary 2.6]{MR} it was shown that $PGL_{n}$-character
varieties are of polynomial type. Therefore, by this result, the $SL_{n}$-character
varieties are also polynomial-count, for $n$ odd, with the same counting
polynomial.
\section{The Irreducible Case}
In this section we prove that $E(\mathcal{X}_{r}^{irr}SL_{n})=E(\mathcal{X}_{r}^{irr}PGL_{n})$
for all $r,n\geq1$, as stated in Theorem~\ref{thm:irred-Zn-action},
thus completing the proof of the main result. Our methods are geometric
in the sense that we mainly use complex algebraic and differential
geometry. In particular, we will use the compact versions of all the
character varieties that we have defined before.
\subsection{Compact representation spaces and their irreducible subspaces}
Consider the compact groups $U(n)$, $SU(n)$ and $PU(n)$, which
are related through the fibrations:
\[
\begin{array}{ccccc}
SU(n) & \to & U(n) & \to & U(1)=S^{1}\\
S^{1} & \to & U(n) & \to & PU(n),
\end{array}
\]
so that $PU(n)\cong U(n)/S^{1}\cong SU(n)/\mathbb{Z}_{n}$, where
we identify the cyclic group with the group of $n^{th}$ roots of
unity
\[
\mathbb{Z}_{n}=\{e^{\frac{2\pi ik}{n}}\,:\,k\in\mathbb{Z}\}\subset S^{1},
\]
and with the center of $SU(n)$. For $r\in\mathbb{N}$, consider the
\emph{compact representation spaces},
\[
U_{r,n}:=\mathrm{Hom}(F_{r},U(n))\cong U(n)^{r},
\]
and similarly $S_{r,n}:=\mathrm{Hom}(F_{r},SU(n))\cong SU(n)^{r}$ and $P_{r,n}:=\mathrm{Hom}(F_{r},PU(n))\cong PU(n)^{r}$,
where the isomorphisms are obtained when fixing a set of $r$ generators
of the free group $F_{r}$. Because we are dealing with the free group,
representations can be multiplied componentwise: all spaces $U_{r,n}$,
$S_{r,n}$ and $P_{r,n}$ are in fact Lie groups. The first fibration
above defines a determinant map, which is actually a homomorphism
of groups:
\[
\det:U_{r,n}\to U_{r,1},
\]
whose kernel is precisely $S_{r,n}$.
Now consider the \emph{irreducible representation} spaces,
\[
U_{r,n}^{*}:=\mathrm{Hom}^{irr}(F_{r},U(n)),\quad\quad S_{r,n}^{*}:=\mathrm{Hom}^{irr}(F_{r},SU(n)),\quad\quad P_{r,n}^{*}:=\mathrm{Hom}^{irr}(F_{r},PU(n)),
\]
which are open subsets in the compact representation spaces:
\[
U_{r,n}^{*}\subset U_{r,n},\quad\quad SU_{r,n}^{*}\subset SU_{r,n},\quad\quad PU_{r,n}^{*}\subset PU_{r,n}.
\]
Also note that $SU_{r,n}^{*}=U_{r,n}^{*}\cap SU_{r,n}$. These irreducible
subspaces $U_{r,n}^{*},SU_{r,n}^{*},PU_{r,n}^{*}$, are not Lie groups,
but we observe the following straightforward properties of the natural
multiplication action of $U_{r,1}$ on $U_{r,n}$:
\begin{equation}
\sigma\cdot\rho=(\sigma_{1},\cdots,\sigma_{r})\cdot(\rho_{1},\cdots,\rho_{r}):=(\sigma_{1}\rho_{1},\cdots,\sigma_{r}\rho_{r}),\quad\quad\sigma_{i}\in U(1),\ \rho_{i}\in U(n),\label{eq:gamma}
\end{equation}
where $\rho_{i}=\rho(\gamma_{i})$ and $\sigma_{i}=\sigma(\gamma_{i})$
for $\gamma_{1},\cdots,\gamma_{r}$ the fixed chosen generators of
$F_{r}$.
\begin{prop}
The action of $U_{r,1}$ on $U_{r,n}$ is such that: \\
(i) the subspace $U_{r,n}^{*}$ is preserved under the action;\\
(ii) $PU_{r,n}$ and $PU_{r,n}^{*}$ are, respectively, the orbit
spaces of $U_{r,n}$ and $U_{r,n}^{*}$ under $U_{r,1}$;\\
(iii) $SU_{r,n}^{*}=\det^{-1}(1)$ for the restriction $\det:U_{r,n}^{*}\to U_{r,1}$,
where $1\in U_{r,1}$ denotes the trivial one dimensional representation.
\end{prop}
Now define:
\[
C_{r,n}:=\mathrm{Hom}(F_{r},\mathbb{Z}_{n})\subset\mathrm{Hom}(F_{r},U(1))=U_{r,1},
\]
so that $C_{r,n}\cong(\mathbb{Z}_{n})^{r}$ is a subgroup of $U_{r,n}$
and can be identified with the center of $SU_{r,n}$. In the same
way as $PU(n)=SU(n)/\mathbb{Z}_{n}$, we can also get $PU_{r,n}$
and $PU_{r,n}^{*}$ as finite quotients of $SU_{r,n}$ and $SU_{r,n}^{*}$:
\[
PU_{r,n}=SU_{r,n}/C_{r,n},\quad\quad PU_{r,n}^{*}=SU_{r,n}^{*}/C_{r,n}.
\]
\subsection{Central action on representation spaces}
We now define a \emph{stratification by polystable type} of $U_{r,n}$,
$SU_{r,n}$ and $PU_{r,n}$ in complete analogy with the stratifications
in \cite[Section 4.1]{FNZ} (for $GL_{n}$) and in Section \ref{section:stratifications}
above (for $PGL_{n}$ and $SL_{n}$).
Given a partition $[k]=[1^{k_{1}}\cdots j^{k_{j}}\cdots n^{k_{n}}]$,
we say that $\rho\in U_{r,n}$ is of type $[k]$ if $\rho$ is conjugated
to $\bigoplus_{j=1}^{n}\rho_{j}$, where each $\rho_{j}$ is a direct
sum of $k_{j}$ \emph{irreducible} representations of $U_{r,j}^{*}$,
for each $j=1,\cdots,n$. We denote representations of type $[k]$
by $U_{r}^{[k]}\subset U_{r,n}$ and let:
\[
SU_{r}^{[k]}=U_{r}^{[k]}\cap SU_{r,n},\quad\quad PU_{r}^{[k]}=U_{r}^{[k]}/U_{r,1}\subset PU_{r,n}.
\]
Note that all strata are locally closed, and that the irreducible
strata $[k]=[n]$ are the only ones that are open (in the respective
representation spaces), corresponding to the partition into one single
part.
\begin{prop}
\label{prop:stratification-type}Fix $n\in\mathbb{N}$. The representation
spaces $U_{r,n}$, $SU_{r,n}$ and $PU_{r,n}$ can be written as disjoint
unions, labelled by partitions $[k]\in\mathcal{P}_{n}$, of locally
closed submanifolds:
\[
U_{r,n}=\bigsqcup_{[k]\in\mathcal{P}_{n}}U_{r}^{[k]},\quad\quad SU_{r,n}=\bigsqcup_{[k]\in\mathcal{P}_{n}}SU_{r}^{[k]},\quad\quad PU_{r,n}=\bigsqcup_{[k]\in\mathcal{P}_{n}}PU_{r}^{[k]}.
\]
\end{prop}
\begin{proof}
The proof for $U_{n,r}$ is analogous to the proof in \cite[Proposition 4.3]{FNZ}
for $\mathcal{R}_{\Gamma}GL_{n}$, observing that dealing with the
usual Euclidean topology on $U_{n,r}$ works \emph{ipsis verbis} as
with the Zariski topology on $\mathcal{R}_{\Gamma}GL_{n}$. The cases
$SU_{r,n}$ and $PU_{r,n}$ follow as in Proposition \ref{prop:locally-closed-SL+PGL}.
\end{proof}
Now, we state the main result of this subsection.
\begin{thm}
\label{thm:trivial-action}The action of $C_{r,n}$ on the compactly
supported cohomology of $SU_{r,n}^{*}=\mathrm{Hom}^{irr}(F_{r},SU_{n})$ is
trivial. In particular, the natural quotient under $C_{r,n}$ induces
an isomorphism $H_{c}^{*}(SU_{r,n}^{*})\cong H_{c}^{*}(PU_{r,n}^{*})$.
\end{thm}
Note that we use compactly supported cohomology as dealing with non-compact
spaces. To prove Theorem \ref{thm:trivial-action}, we need to show
two results: (i) under a open/closed decomposition of a compact space
$X=U\sqcup Y$, the triviality of the action on $2$ spaces implies
the same for the third one; (ii) the actions of $C_{r,n}$ on the
cohomology of every $[k]$-stratum are trivial, for partitions $[k]$
of length $l>1$. (iii) the action on the whole $SU_{r,n}$ is trivial.
We start with (iii), which uses the following standard lemma (see,
for instance, \cite{CFLO}).
\begin{lem}
\label{lem:interpolating-homotopy}If $J$ is a finite group acting
on a space $X$ and, for every $g\in J$, the induced map $\hat{g}:X\to X$,
$x\mapsto g\cdot x$ is homotopic to $id_{X}$, then $J$ acts trivially
on $H^{*}(X)$. In particular, if $F$ is a finite subgroup of a connected
group $G$ acting on $X$, then $F$ acts trivially on $H^{*}(X)$.
\begin{comment}
\begin{proof}
Since cohomology is a homotopy invariant, both maps $id_{X}$ and
$\hat{g}$ induce the same morphism in cohomology
\[
id_{X}^{*}=\hat{g}^{*}:H^{*}(X)\to H^{*}(X).
\]
So, $\hat{g}^{*}$ is an isomorphism. As this happens for every $g\in J$,
the action of the whole $J$ is trivial. For the second statement,
if a connected group $G$ acts on $X$, then any path from the identity
of $G$ to any element $g\in G$ gives a homotopy between $id_{X}$
and $\hat{g}$, completing the proof. \end{proof}
\end{comment}
\end{lem}
Since the action of $C_{r,n}$ is the restriction of the action of
the path connected group $SU(n)^{r}$ acting by left multiplication
the following is clear.
\begin{cor}
\label{cor:action-on-total}The action of $C_{r,n}\cong(\mathbb{Z}_{n})^{r}\subset SU(n)^{r}$
on $H^{*}(SU_{r,n})=H^{*}(SU(n)^{r})$, is trivial. \end{cor}
\begin{rem}
Although the action of $C_{r,n}$ preserves the stratification, the
argument above cannot be applied to the individual strata $SU_{r,n}^{[k]}$;
indeed, it is not clear what \emph{connected} group could interpolate
the action of the discrete group $C_{r,n}$ on the irreducible stratum.
For example, using the left multiplication by the whole group, the
action of $(A^{-1},A^{-1})\in SU(n)^{2}$ on an irreducible pair $(A,B)\in SU(n)^{2}$
gives the pair $(I,A^{-1}B)$, which clearly belongs to the $[1^{n}]$-stratum.
Other attempts at using smaller groups $H\subset SU(n)$, such as
maximal tori, may still fail for the irreducible stratum. Given this
difficulty, we resort to an argument analogous to the one of Theorem
\ref{thm:SL-2strata}, for strata associated with partitions of length
more than one. \end{rem}
\begin{lem}
\label{lem:action-on-strata}Let $[k]\in\mathcal{P}_{n}$ be a partition
with length $l>1$. Then, the action of $C_{r,n}$ on the cohomology
of $SU_{r}^{[k]}$ is trivial. \end{lem}
\begin{proof}
We first consider cartesian products of irreducible representation
spaces satisfying a determinant condition. Let $s>1$, $\mathbf{n}:=(n_{1},\cdots,n_{s})\in\mathbb{N}^{s}$,
with $n=n_{1}+n_{2}+\cdots+n_{s}$ and denote by
\[
\mathcal{S}^{\mathbf{n}}:=\left\{ \rho=(\rho_{1},\cdots,\rho_{s})\in\times_{j=1}^{s}U_{r,n_{j}}^{*}\ |\ {\textstyle \prod_{j=1}^{s}}\det\rho_{j}=1\right\}
\]
which is a smooth manifold. Note that, in contrast to partitions,
$\mathbf{n}:=(n_{1},\cdots,n_{s})\in\mathbb{N}^{s}$ is an ordered
$s$-tuple of elements of $\mathbb{N}$. There is an action of $C_{r,n}$
on $\mathcal{S}^{\mathbf{n}}$ given by:
\begin{equation}
\sigma\cdot\rho=\sigma\cdot(\rho_{1},\cdots,\rho_{s}):=(\sigma\rho_{1},\cdots,\sigma\rho_{s}),\quad\quad\sigma\in\mathrm{Hom}(F_{r},\mathbb{Z}_{n})=C_{r,n},\ \rho_{j}\in U_{r,n_{j}}^{*}.\label{eq:gamma-1}
\end{equation}
It is easy to check the product of determinants condition, so that
$\sigma\cdot\rho\in\mathcal{S}^{\mathbf{n}}$.
Now, assume that the greatest common divisor of all $n_{1},\cdots,n_{s}$
is 1. Then, there is at least one $n_{j}$ prime with $n$, and without
loss of generality, we can take $p:=n_{1}$ prime with $n$. Denote
for simplicity, $q=n-p=\sum_{i=2}^{s}n_{i}>0$. Define the following
elements of $U_{r,1}$:
\[
\sigma_{t}=(e^{-2\pi it\frac{q}{n}},1,\cdots,1),\;\;\;\;\;\ \tilde{\sigma_{t}}=(e^{2\pi it\frac{p}{n}},1,\cdots,1)
\]
parametrised by $t\in[0,1]$, and where each factor corresponds to
a generator of $F_{r}$. Consider the following homotopy:
\begin{eqnarray*}
[0,1]\times\mathcal{S}^{\mathbf{n}} & \to & \mathcal{S}^{\mathbf{n}}\\
(t;\rho_{1},\rho_{2}\cdots,\rho_{s}) & \mapsto & (\sigma_{t}\rho_{1},\,\tilde{\sigma_{t}}\rho_{2},\cdots,\,\tilde{\sigma_{t}}\rho_{s}).
\end{eqnarray*}
This is well defined since the product of determinants on the right
side, for the first generator $\gamma_{1}$, is:
\[
\left(e^{-2\pi it\frac{q}{n}}\right)^{p}\left(e^{2\pi it\frac{p}{n}}\right)^{q}{\textstyle \prod_{j=1}^{s}}\det\rho_{j}(\gamma)=1,
\]
(the representation on the other generators does not change under
this homotopy). Then, for $t=0$ the map $\mathcal{S}^{\mathbf{n}}\to\mathcal{S}^{\mathbf{n}}$
is the identity, and for $t=1$ the map is identified with the action,
on the first generator $\gamma_{1}$, of multiplication by the scalar
$e^{2\pi i\frac{p}{n}}$:
\[
e^{-2\pi i\frac{q}{n}}=e^{-2\pi i\frac{q-n}{n}}=e^{2\pi i\frac{p}{n}},
\]
which is a primitive $n^{th}$ root of unity. Thus, we have found
a homotopy between the identity and $\sigma^{*}$ for the element
$\sigma=(e^{2\pi i\frac{p}{n}},1,\cdots,1)\in C_{r,n}$ (c.f. Lemma
\ref{lem:interpolating-homotopy}). Since all elements of $C_{r,n}$
are obtained as compositions of elements $\sigma$ of this form (with
non-trivial elements on the other entries), we obtain the necessary
homotopies to apply Lemma \ref{lem:interpolating-homotopy}, and finish
the proof, in this case. If the greatest common divisor of all $n_{1},\cdots,n_{s}$
is $d>1$, then we can use the same map but with $t\in[0,\frac{1}{d}]$
(since there is some $p=n_{j}$ that verifies now $(\frac{p}{d},n)=1$).
So, the action of $\mathbb{Z}_{n}^{r}$ is trivial on the cohomology
of $\mathcal{S}^{\mathbf{n}}$. Finally, let $[k]$ be the partition
determined by the tuple $\mathbf{n}=(n_{1},\cdots,n_{s})\in\mathbb{N}^{s}$.
Observe that the length of $[k]$ is $s$. Then, we note that
\[
SU_{r}^{[k]}=\mathcal{S}^{\mathbf{n}}/S_{[k]},
\]
where $S_{[k]}=\times_{j=1}^{n}S_{k_{j}}\subset S_{n}$ acts by permutation
of the blocks of equal size. Since the action of $C_{r,n}$ commutes
with the one of $S_{[k]}$, we get
\[
H^{*}(SU_{r}^{[k]})^{C_{r,n}}=\left(H^{*}(\mathcal{S}^{\mathbf{n}})^{S_{[k]}}\right)^{C_{r,n}}=\left(H^{*}(\mathcal{S}^{\mathbf{n}})^{C_{r,n}}\right)^{S_{[k]}}=H^{*}(\mathcal{S}^{\mathbf{n}})^{S_{[k]}}=H^{*}(SU_{r}^{[k]}),
\]
as wanted.
\end{proof}
For the next Proposition, we let $X$ be a compact Hausdorff topological
space acted by finite group $F$, and we have a closed $F$-invariant
subspace $Y\subset X$, with complement $U:=X\setminus Y$ . Since
we need compactly supported cohomology for the open case, and it coincides
with usual cohomology for $X$ and for $Y$, below we can use $H_{c}^{*}$
uniformly.
\begin{prop}
\label{prop:snake-cohomology} If $F$ acts trivially on the compactly
supported cohomology of two of the spaces $X$, $Y$ and $U$, then
it acts trivially on the cohomology of the third one. \end{prop}
\begin{proof}
This follows from the 5-lemma applied to the long exact sequence for
cohomology with compact support associated to the decomposition $X=U\sqcup Y$.
More precisely, let $g\in F$, and denote by $g_{Z}^{*}$ the associated
morphisms in the cohomology for $Z=X,Y$ or $U$. Then, we can form
the ladder:
\[
\begin{array}{ccccccccccccc}
\cdots & \to & H_{c}^{k-1}(X) & \to & H_{c}^{k-1}(Y) & \to & H_{c}^{k}(U) & \to & H_{c}^{k}(X) & \to & H_{c}^{k}(Y) & \to & \cdots\\
& & \downarrow g_{X}^{*} & & \downarrow g_{Y}^{*} & & \downarrow g_{U}^{*} & & \downarrow g_{X}^{*} & & \downarrow g_{Y}^{*}\\
\cdots & \to & H_{c}^{k-1}(X) & \to & H_{c}^{k-1}(Y) & \to & H_{c}^{k}(U) & \to & H_{c}^{k}(X) & \to & H_{c}^{k}(Y) & \to & \cdots.
\end{array}
\]
By hypothesis, two of $g_{X}^{*},g_{Y}^{*}$ and $g_{U}^{*}$ are
isomorphisms. Then, by the 5-lemma, the third map is also an isomorphism.
Since $g_{U}^{*}$ is an isomorphism for every $g\in F$, the action
of $F$ on $H_{c}^{*}(U)$ is trivial. The same argument holds for
the other 2 spaces $X,Y$.
\end{proof}
The stratification $SU_{r,n}=\bigsqcup_{[k]\in\mathcal{P}_{n}}SU_{r}^{[k]}$
has the nice property that the closure of every stratum is a union
of other strata. In fact, for every $a,b\in\mathbb{N}$, the direct
sum of irreducible representations of sizes $a$ and $b$ is in the
closure of the irreducible ones of size $a+b$. This means that the
closure of $SU_{r}^{[k]}$ is the disjoint union of all strata $SU_{r}^{[l]}$
where $[l]$ is obtained by any subdivision of $[k]\in\mathcal{P}_{n}$.
For example, with $n=5$, $\overline{X_{[23]}}=X_{[23]}\sqcup X_{[1^{2}3]}\sqcup X_{[12^{2}]}\sqcup X_{[1^{3}2]}\sqcup X_{[1^{5}]}$,
where we used the abbreviated notation $X_{[k]}:=SU_{r}^{[k]}$.
Now, fix $p\in\{1,\cdots,n\}$ and consider the closed subset of $SU_{r,n}$
defined by
\[
Y_{p}:=\bigcup_{|[k]|=p}\overline{SU_{r}^{[k]}}=\bigcup_{|[k]|\geq p}SU_{r}^{[k]},
\]
where $|[k]|$ is the length of $[k]\in\mathcal{P}_{n}$. Since each
$SU_{r}^{[k]}$ is the only open stratum in $\overline{SU_{r}^{[k]}}$,
it is easy to see that: $Y_{p}\setminus Y_{p+1}=\bigsqcup_{|[k]|=p}SU_{r}^{[k]},$
and this last union is disjoint.
\begin{lem}
\label{lem:induction}Fix $n$ and let $2\leq p\leq n-1$. If $C_{r,n}$
acts trivially on the cohomology of $Y_{p+1}$, then the same holds
for $Y_{p}$. Thus, $C_{r,n}$ acts trivially on the cohomology of
$Y_{2}$.\end{lem}
\begin{proof}
Since $p\geq2$, by Lemma \ref{lem:action-on-strata}, $C_{r,n}$
acts trivially on
\[
H_{c}^{*}(\bigsqcup_{|[k]|=p}SU_{r}^{[k]})=\bigoplus_{|[k]|=p}H_{c}^{*}(SU_{r}^{[k]}).
\]
We can use compactly supported cohomology since each $SU_{r}^{[k]}$
is a quotient of a smooth manifold by a finite group, so that it verifies
Poincaré duality. Assuming $C_{r,n}$ acts trivially on the cohomology
of $Y_{p+1}$, then the same holds for $Y_{p}=Y_{p+1}\sqcup(\sqcup_{|[k]|=p}SU_{r}^{[k]})$,
by Proposition \ref{prop:snake-cohomology}. Noting that $Y_{n}=SU_{r}^{[1^{n}]}$
(already a closed stratum) we have that $C_{r,n}$ acts trivially
on $H^{*}(Y_{n})$ again by Lemma \ref{lem:action-on-strata}. So,
the last sentence follows by finite induction.
\end{proof}
Finally, Theorem \ref{thm:trivial-action} follows by another application
Proposition \ref{prop:snake-cohomology} and Corollary \ref{cor:action-on-total},
to the spaces $X=Y_{1}=SU_{r,n}$, the closed subset $Y=Y_{2}$ and
$U=X\setminus Y_{2}=SU_{r,n}^{*}=SU_{r}^{[n]}$.
\begin{comment}
by induction on $n$, using the stratification $SU_{r,n}=\bigsqcup_{[k]\in\mathcal{P}_{n}}SU_{r}^{[k]}$.
Indeed, step $n=1$ is trivial since $SU_{r,1}$ is a single point;
for the induction step, we just need to apply Proposition \ref{prop:snake-cohomology}
with $X:=SU_{r,n}$, $Y:=\bigsqcup_{|[k]|\geq2}SU_{r}^{[k]}$ and
$U:=X\setminus Y=SU_{r,n}^{*}$. Corollary \ref{cor:action-on-total}
shows that the $C_{r,n}$ action on $X$ is trivial and, by induction,
the action on $SU_{r}^{[k]}=\mathcal{S}^{\mathbf{n}}/S_{[k]}$, for
partitions with $|[k]|\geq2$ is also trivial, by Lemma \ref{lem:action-on-strata}
and the fact that compactly supported cohomology is additive.
\end{comment}
\subsection{Cohomology of quotient spaces}
Now, we consider the action of $PU(n)$ on all three representation
spaces by simultaneous conjugation, and define the compact irreducible
``character varieties'' $\mathcal{X}_{r}^{*}SU_{n}:=SU_{r,n}^{*}/PU(n)$
and $\mathcal{X}_{r}^{*}PU_{n}:=PU_{r,n}^{*}/PU(n)$.
We will need the notion of \emph{equivariant} \emph{cohomology}, which
we now recall. Let $X$ be a topological space endowed with the action
of a topological group $G$, and $p:E_{G}\to B_{G}$ be the universal
principal $G$-bundle, where $B_{G}=E_{G}/G$ is the classifying space
of $G$. The equivariant cohomology of $X$ is defined by (for more
details see, for example, \cite{Br})
\[
H_{G}^{*}(X):=H^{*}(X\times_{G}E_{G}),
\]
and note that, if $G$ acts freely on $X$, then
\[
H_{G}^{*}(X)\cong H^{*}(X/G).
\]
\begin{prop}
\label{prop:SUPUisomorphisms} There are isomorphisms $H^{*}(SU_{r,n}^{*})\cong H^{*}(PU_{r,n}^{*})$,
$H^{*}(\mathcal{X}_{r}^{*}SU_{n})\cong H^{*}(\mathcal{X}_{r}^{*}PU_{n})$
and $H^{*}(\mathcal{X}_{r}SU_{n})\cong H^{*}(\mathcal{X}_{r}PU_{n})$.\end{prop}
\begin{proof}
The action of $C_{r,n}$ on the compactly supported cohomology of
$SU_{r,n}^{*}=\mathrm{Hom}^{irr}(F_{r},SU_{n})$ is trivial, by Theorem \ref{thm:trivial-action},
so:
\[
H_{c}^{*}(SU_{r,n}^{*})=H_{c}^{*}(SU_{r,n}^{*})^{C_{r,n}}=H_{c}^{*}(SU_{r,n}^{*}/C_{r,n})=H_{c}^{*}(PU_{r,n}^{*}).
\]
Since $SU_{r,n}^{*}$ and $PU_{r,n}^{*}$ are either smooth manifolds
or orbifolds (and are orientable), their cohomologies satisfy Poincaré
duality, so the same isomorphism holds for the usual cohomologies.
Now, as in \cite[p.12]{CFLO}, because the quotient map $\pi:SU_{r,n}^{*}\to SU_{r,n}^{*}/C_{r,n}=PU_{r,n}^{*}$
induces an isomorphism in cohomology, and $\pi$ is equivariant with
respect to the conjugation $PU(n)$ action, we have that their equivariant
cohomologies are the same:
\[
H_{PU(n)}^{*}(SU_{r,n}^{*})\cong H_{PU(n)}^{*}(PU_{r,n}^{*}).
\]
Since the $PU(n)$ action is free on the irreducible representation
spaces, we get
\[
H^{*}(\mathcal{X}_{r}^{*}SU_{n})\cong H_{PU(n)}^{*}(SU_{r,n}^{*})\cong H_{PU(n)}^{*}(PU_{r,n}^{*})\cong H^{*}(\mathcal{X}_{r}^{*}PU_{n}),
\]
completing the proof of the second isomorphism. This also means that
the action of $C_{r,n}$ on $H^{*}(\mathcal{X}_{r}^{*}SU_{n})$ is
trivial.
Now, for $[k]=[1^{k_{1}}\cdots n^{k_{n}}]\in\mathcal{P}_{n}$, we
can write every stratum as a finite quotient $\mathcal{X}_{r}^{[k]}SU_{n}=(\times_{j=1}^{n}\mathcal{X}_{r}^{*}SU_{k_{j}})/S_{[k]}$
(with $S_{[k]}=S_{k_{1}}\times\cdots\times S_{k_{n}}$), and since
there is a trivial action of $C_{r,n}$ on $H^{*}(\times_{j=1}^{n}\mathcal{X}_{r}^{*}SU_{k_{j}})=\otimes_{j=1}^{n}H^{*}(\mathcal{X}_{r}^{*}SU_{k_{j}})$,
the same also happens for the subspace fixed by $S_{[k]}$:
\[
H^{*}(\mathcal{X}_{r}^{[k]}SU_{n})=\left[\otimes_{j=1}^{n}H^{*}(\mathcal{X}_{r}^{*}SU_{k_{j}})\right]^{S_{[k]}}.
\]
Now, the stratification $\mathcal{X}_{r}SU_{n}=\bigsqcup_{[k]\in\mathcal{P}_{n}}\mathcal{X}_{r}^{[k]}SU_{n}$
has also the property that the closure of a stratum is a union of
strata. For $p\in\{2,\cdots,n\}$ consider the closed subset:
\[
X_{p}:=\bigcup_{|[k]|=p}\overline{\mathcal{X}_{r}^{[k]}SU_{n}}=\bigcup_{|[k]|\geq p}\mathcal{X}_{r}^{[k]}SU_{n},
\]
and note that $X_{p}\setminus X_{p+1}=\bigsqcup_{|[k]|=p}\mathcal{X}_{r}^{[k]}SU_{n}$.
Therefore, Lemma \ref{lem:induction} applies to $X_{p}$ in place
of $Y_{p}$ (here, note that $X_{n}=\mathcal{X}_{r}^{[1^{n}]}SU_{n}$
is already a closed stratum), to show that the cohomology of $X_{2}$
has a trivial $C_{r,n}$ action. Finally, the third isomorphism is
shown by applying Proposition \ref{prop:snake-cohomology} to the
triple: $X:=X_{1}=\mathcal{X}_{r}SU_{n}$, $Y:=X_{2}$ and $U:=X_{1}\setminus X_{2}=\mathcal{X}_{r}^{*}SU_{n}=\mathcal{X}_{r}^{[n]}SU_{n}$.
The action of $C_{r,n}$ on the cohomology of $U$ and $Y$ being
trivial, the same holds for $X$.
\end{proof}
\begin{comment}
Now, we need to relate to the complex and compact character varieties.
For this, we use the existence of a strong deformation retraction
from $\mathcal{X}_{r}SL_{n}$ to $\mathcal{X}_{r}SU_{n}$ (and from
$\mathcal{X}_{r}PGL_{n}$ to $\mathcal{X}_{r}PU_{n}$) as shown in
\cite{FL1}.
\end{comment}
We are finally ready for the completion of the main result.
\begin{thm}
\label{thm:equality-irr-PGL-SL} There are isomorphisms of mixed Hodge
structures $H^{*}(\mathcal{X}_{r}SL_{n})\cong H^{*}(\mathcal{X}_{r}PGL_{n})$
and $H_{c}^{*}(\mathcal{X}_{r}^{irr}SL_{n})\cong H_{c}^{*}(\mathcal{X}_{r}^{irr}PGL_{n})$.
In particular, the $E$-polynomials of $\mathcal{X}_{r}^{irr}SL_{n}$
and of $\mathcal{X}_{r}^{irr}PGL_{n}$ coincide.\end{thm}
\begin{proof}
For every reductive group $G$, there is a strong deformation retraction
from $\mathcal{X}_{r}G$ to the orbit space $\mathrm{Hom}(F_{r},K)/K$, where
$K$ is a maximal compact subgroup of $G$ (see \cite{FL1}), acting
by conjugation. Since the homotopy is defined by the polar decomposition,
in the case $G=SL_{n}$, the strong deformation retraction from $\mathcal{X}_{r}SL_{n}$
to $\mathcal{X}_{r}SU_{n}$ commutes with the action of $C_{r,n}=\mathrm{Hom}(F_{r},\mathbb{Z}_{n})$.
Then, from Proposition \ref{prop:SUPUisomorphisms} we get isomorphisms
in usual cohomology:
\[
H^{*}(\mathcal{X}_{r}SL_{n})\cong H^{*}(\mathcal{X}_{r}SU_{n})\cong H^{*}(\mathcal{X}_{r}PU_{n})\cong H^{*}(\mathcal{X}_{r}PGL_{n}).
\]
Since the quotient $\mathcal{X}_{r}SL_{n}\to\mathcal{X}_{r}PGL_{n}$
is algebraic, the above isomorphism $H^{*}(\mathcal{X}_{r}SL_{n})\cong H^{*}(\mathcal{X}_{r}PGL_{n})$
is an isomorphism of mixed Hodge structures. Moreover, the strong
deformation retraction from $X:=\mathcal{X}_{r}SL_{n}$ to $SU_{r,n}$
restricts to a strong deformation retraction from the strictly polystable
locus $Y:=\bigsqcup_{|[k]|\geq2}\mathcal{X}_{r}^{[k]}SL_{n}$ to $\bigsqcup_{|[k]|\geq2}SU_{n}^{[k]}/PU(n)$
(because the polar decomposition preserves reducible representations,
see \cite{FL1}). Thus, in the same fashion, we obtain isomorphisms
of mixed Hodge structures $H^{*}(Y)\cong H^{*}(\hat{Y})$, where $\hat{Y}:=\bigsqcup_{|[k]|\geq2}\mathcal{X}_{r}^{[k]}PGL_{n}.$
Now, considering the natural open inclusion $j:X\setminus Y=\mathcal{X}_{r}^{irr}SL_{n}\hookrightarrow X$
and the closed inclusion $i:Y\hookrightarrow X=\mathcal{X}_{r}SL_{n}$,
we can consider the long exact sequence:
\[
\cdots\to H^{k-1}(X)\to H^{k-1}(Y)\to H_{c}^{k}(X\setminus Y)\to H^{k}(X)\to H^{k}(Y)\to\cdots
\]
which comes from the short exact sequence of sheaves $0\to j_{!}j^{*}\underline{\mathbb{C}}\to\underline{\mathbb{C}}\to i_{*}i^{*}\underline{\mathbb{C}}\to0$
where $\underline{\mathbb{C}}$ is the locally constant $\mathbb{C}$-valued
sheaf (a standard reference is \cite{Iv}, or see \cite[pag. 31]{Gor},
for a summarized account). By applying the same sequence to the closed
subvariety $\hat{Y}\subset\hat{X}:=\mathcal{X}_{r}PGL_{n}$, we get
a sequence of isomorphisms:
\[
\begin{array}{ccccccccccccc}
\cdots & \to & H^{k-1}(X) & \to & H^{k-1}(Y) & \to & H_{c}^{k}(X\setminus Y) & \to & H^{k}(X) & \to & H^{k}(Y) & \to & \cdots\\
& & \parallel\wr & & \parallel\wr & & \uparrow & & \parallel\wr & & \parallel\wr\\
\cdots & \to & H^{k-1}(\hat{X}) & \to & H^{k-1}(\hat{Y}) & \to & H_{c}^{k}(\hat{X}\setminus\hat{Y}) & \to & H^{k}(\hat{X}) & \to & H^{k}(\hat{Y}) & \to & \cdots
\end{array}
\]
which provide, using the 5-lemma, the wanted isomorphisms:
\begin{comment}
$\mathcal{X}_{r}^{irr}SL_{n}=X\setminus Y$ (and similarly for $\mathcal{X}_{r}^{irr}PGL_{n}$)
is an open variety, we have its mixed Hodge structures on compactly
supported cohomology are defined using compactifications of $X$ (being
independent of the choices, see \cite[Thm. 5.33]{PS}). It is easy
to see that we can choose a compactification $\bar{X}$ of $X$, and
$\bar{Y}$ of $Y$ with the property that $X\setminus Y=\bar{X}\setminus\bar{Y}$,
yielding
\end{comment}
\[
H_{c}^{k}(\mathcal{X}_{r}^{irr}SL_{n})=H_{c}^{k}(X\setminus Y)\cong H_{c}^{k}(\hat{X}\setminus\hat{Y})=H_{c}^{k}(\mathcal{X}_{r}^{irr}PGL_{n}).
\]
Finally, since the finite quotient $\mathcal{X}_{r}^{irr}SL_{n}\to\mathcal{X}_{r}^{irr}PGL_{n}$
is algebraic, this implies the isomorphism of mixed Hodge structures
on the corresponding compactly supported cohomology groups, and the
equality of $E$-polynomials.
\end{proof}
\section{Explicit Computations in the free group case}
\label{section:explicit}When $\Gamma=F_{r}$, the free group in $r$
generators, Mozgovoy and Reineke obtained a general formula for the
count of points in $\mathcal{X}_{r}^{irr}GL_{n}$ over finite fields,
showing that these schemes are of polynomial type (see \cite{MR}).
In this section, we explore these formulae in detail and, by using
the \emph{plethystic exponential} correspondence proved in \cite{FNZ},
we provide a finite algorithm to obtain the Serre polynomials and
the Euler characteristics of all the strata $\mathcal{X}_{r}^{[k]}G$,
for all partitions $[k]\in\mathcal{P}_{n}$ and $G=GL_{n}$, $SL_{n}$
or $PGL_{n}$.
\subsection{Serre polynomials of irreducible $GL_{n}$-character varieties $\mathcal{X}_{r}^{irr}G$.}
Let us recall the definition of the plethystic functions, and the
correspondence proved in \cite{FNZ}. Define the \emph{Adams operator}
$\Psi$ as the invertible $\mathbb{Q}$-linear operator acting on
$\mathbb{Q}[x][[t]]$ by $\Psi(x^{i}t^{k}):=\sum_{l\geq1}\frac{x^{li}t^{lk}}{l},$
where $(i,k)\in\mathbb{N}_{0}^{2}\setminus\{(0,0)\}$, with inverse
given by $\Psi^{-1}(x^{j}t^{k})=\sum_{l\geq1}\frac{\mu(l)}{l}x^{jl}t^{kl},$
and $\mu$ is the Möbius function $\mu:\mathbb{N}\to\{0,\pm1\}$ ($\mu(n)=(-1)^{k}$
if $n$ is square free with $k$ primes in its factorization; $\mu(n)=0$
otherwise).
Given a power series $f\in\mathbb{Q}[x][[t]]$, formal in $t$:
\[
f(x,t)=1+\sum_{n\geq1}f_{n}(x)\,t^{n}\;,
\]
where $f_{n}(x)\in\mathbb{Q}[x]$ are polynomials in $x$, define
the plethystic exponential, $\operatorname{PExp}:\mathbb{Q}[x][[t]]\to1+t\mathbb{Q}[x][[t]]$,
and plethystic logarithm, $\operatorname{PLog}$ as:
\[
\operatorname{PExp}(f):=e^{\Psi(f)},\quad\quad\operatorname{PLog}(f):=\Psi^{-1}(\log f).
\]
As established in \cite[Theorem 1.1 and Corollary 1.2]{FNZ}, $GL_{n}$-character
varieties can be expressed in terms of irreducible character varieties
of lower dimension, by means of the plethystic exponential.
\begin{thm}
\cite[Theorem 1.1]{FNZ} \label{thm:mainFNZ}Let $\Gamma$ be a finitely
presented group. Then, in $\mathbb{Q}[u,v][[t]]$:
\[
\sum_{n\geq0}E(\mathcal{X}_{\Gamma}GL_{n};u.v)\,t^{n}=\operatorname{PExp}\left(\sum_{n\geq1}E(\mathcal{X}_{\Gamma}^{irr}GL_{n};u,v)\,t^{n}\right)\;.
\]
\end{thm}
Using results of \cite{MR} for the character varieties of the free
group $F_{r}$, this relationship can be made explicit, in terms of
partitions $[k]=[1^{k_{1}}\cdots d^{k_{d}}]\in\mathcal{P}_{d}$, with
length denoted by $|[k]|=k_{1}+\cdots+k_{d}$. Let $\binom{m}{k_{1},\cdots,k_{d}}=m!(k_{1}!\cdots k_{d}!)^{-1}$
be the multinomial coefficients.
\begin{prop}
\label{prop:Bnformula} Let $r,n\geq2$. The $E$-polynomials of the
irreducible character varieties $B_{n}^{r}(x):=e(\mathcal{X}_{r}^{irr}GL_{n})$
are explicitly given by:
\[
B_{n}^{r}(x)=(x-1)\sum_{d|n}\frac{\mu(n/d)}{n/d}\,\sum_{[k]\in\mathcal{P}_{d}}\frac{(-1)^{|[k]|}}{|[k]|}\binom{|[k]|}{k_{1},\cdots,k_{d}}\prod_{j=1}^{d}b_{j}(x^{n/d})^{k_{j}}x^{\frac{n(r-1)k_{j}}{d}\binom{j}{2}}\;,
\]
where $b_{j}(x)$ are given by $F^{-1}(t)=1+\sum_{n\geq1}b_{n}t^{n},$
for the series:
\begin{equation}
F(t)=1+\sum_{n\geq1}\big((x-1)(x^{2}-1)\ldots(x^{n}-1)\big)^{r-1}\,t^{n}.\label{eq:F(t)}
\end{equation}
\end{prop}
\begin{proof}
As mentioned above, the varieties $\mathcal{X}_{r}^{irr}GL_{n}$ are
of polynomial type by \cite[Thm. 1.1]{MR}. So, by Katz's theorem
\cite[Appendix]{HRV}, $B_{n}^{r}(x):=e_{x}(\mathcal{X}_{r}^{irr}GL_{n})$
is obtained by replacing $q$ by $x$ in the counting polynomial $P_{n}(q):=|\mathcal{X}_{r}^{irr}GL_{n}/\mathbb{F}_{q}|$
which in \cite[Theorem 1.2]{MR} is shown to have generating series:
\[
\sum_{n\geq1}B_{n}^{r}(x)\,t^{n}=\sum_{n\geq1}P_{n}(x)\,t^{n}=(1-x)\operatorname{PLog}(S\circ F^{-1}(t))\;,
\]
with $F(t)$ as in \eqref{eq:F(t)}, and $S$ is a $\mathbb{Q}[x]$-linear
shift operator defined on $\mathbb{Q}[x][[t]]$ by $S(t)=t$, and
$S(t^{n}):=x^{(r-1)\binom{n}{2}}t^{n},$ for $n\geq2$. Hence, we
get
\[
S\circ F^{-1}(t)=1+\sum_{n\geq1}b_{n}(x)\,x^{(r-1)\binom{n}{2}}\,t^{n}\;.
\]
So, the Proposition follows from Lemma \ref{lem:plethystic-Log} below,
using
\[
f_{n}(x)=b_{n}(x)\,x^{(r-1)\binom{n}{2}}\;
\]
since, for a partition $[k]\in\mathcal{P}_{d}$, we have $f_{j}(x^{n/d})^{k_{j}}=b_{j}(x^{n/d})^{k_{j}}\,x^{\frac{n}{d}(r-1)k_{j}\binom{j}{2}}$,
$j=1,\ldots,d$, as wanted.
\end{proof}
To complete the proof of Proposition \ref{prop:Bnformula}, we need
the following
\begin{lem}
\label{lem:plethystic-Log} Given $f_{n}(x)\in\mathbb{Q}[x]$, $n\in\mathbb{N}$,
the coefficient of $t^{n}$ in $\operatorname{PLog}(1+\sum_{n\geq1}f_{n}(x)\,t^{n})$
is
\[
\sum_{d|n}\sum_{[k]\in\mathcal{P}_{d}}\frac{\mu(n/d)}{n/d}\frac{(-1)^{|[k]|-1}}{|[k]|}\binom{|[k]|}{k_{1},\cdots,k_{d}}\prod_{j=1}^{d}f_{j}(x^{n/d})^{k_{j}}\;.
\]
\end{lem}
\begin{proof}
From the $\mathbb{Q}$-linearity of $\Psi^{-1}$, we can write, for
a sequence of polynomials $g_{m}(x)\in\mathbb{Q}[x]$, $m\in\mathbb{N}$,
\begin{eqnarray}
\Psi^{-1}(\sum_{m\geq1}g_{m}(x)t^{m}) & = & \sum_{m\geq1}\Psi^{-1}(g_{m}(x)t^{m})=\nonumber \\
& = & \sum_{l\geq1}\frac{\mu(l)}{l}g_{1}(x^{l})t^{l}+\sum_{l\geq1}\frac{\mu(l)}{l}g_{2}(x^{l})t^{2l}+\cdots=\nonumber \\
& = & \sum_{l\geq1}\frac{\mu(l)}{l}\sum_{d\geq1}g_{d}(x^{l})t^{dl}=\nonumber \\
& = & \sum_{n\geq1}\sum_{d|n}\frac{\mu(n/d)}{n/d}g_{d}(x^{n/d})t^{n}.\label{eq:psi-1}
\end{eqnarray}
Now, from the series development of $\log(1+z)=z-\frac{z^{2}}{2}+\frac{z^{3}}{3}-\cdots$,
we compute, using the multinomial theorem:
\begin{eqnarray*}
\log(1+\sum_{n\geq1}f_{n}(x)\,t^{n}) & = & \left(\sum_{n\geq1}f_{n}(x)\,t^{n}\right)-\frac{1}{2}\left(\sum_{n\geq1}f_{n}(x)\,t^{n}\right)^{2}+\frac{1}{3}\left(\sum_{n\geq1}f_{n}(x)\,t^{n}\right)^{3}-\cdots\\
& = & \sum_{m\geq1}\left[\sum_{[k]\in\mathcal{P}_{m}}\frac{(-1)^{|[k]|-1}}{|[k]|}\binom{|[k]|}{k_{1},\cdots,k_{m}}\prod_{j=1}^{m}f_{j}(x)^{k_{j}}\right]t^{m}.
\end{eqnarray*}
Finally, we apply formula \eqref{eq:psi-1} to
\[
g_{m}(x):=\sum_{[k]\in\mathcal{P}_{m}}\frac{(-1)^{|[k]|-1}}{|[k]|}\binom{|[k]|}{k_{1},\cdots,k_{m}}\prod_{j=1}^{m}f_{j}(x)^{k_{j}}\;,
\]
proving the Lemma. \end{proof}
\begin{rem}
When $n$ is a prime number the formula in Proposition \ref{prop:Bnformula}
simplifies to:
\[
B_{n}^{r}(x)=(x-1)\left[\frac{b_{1}(x^{n})}{n}+\sum_{[k]\in\mathcal{P}_{n}}\frac{(-1)^{|[k]|}}{|[k]|}\binom{|[k]|}{k_{1},\cdots,k_{n}}\prod_{j=1}^{n}b_{j}(x)^{k_{j}}x^{(r-1)k_{j}\binom{j}{2}}\right]\;.
\]
\end{rem}
Using Proposition \ref{prop:Bnformula}, we can write down explicitly
the $E$-polynomials of $\mathcal{X}_{r}^{irr}GL_{n}$, for any value
of $n$ in a recursive way, the first four cases being as follows.
\begin{lem}
\label{lemma:Bn(x)} The $E$-polynomials of the irreducible character
varieties $B_{n}^{r}(x)=e(\mathcal{X}_{r}^{irr}GL_{n})$, for $n=1,2,3$
and $4$, using the substitution $s=r-1$, are given by:
\begin{eqnarray*}
\frac{B_{1}^{r}(x)}{x-1} & = & (x-1)^{r-1}=(x-1)^{s}\;,\\
\frac{B_{2}^{r}(x)}{x-1} & = & \frac{1}{2}b_{1}(x^{2})+\frac{1}{2}b_{1}(x)^{2}-b_{2}(x)x^{s}\;,\\
& = & (x-1)^{s}\Big((x-1)^{s}x^{s}((x+1)^{s}-1)+\frac{1}{2}(x-1)^{s}-\frac{1}{2}(x+1)^{s}\Big),
\end{eqnarray*}
\\[-7mm]
\begin{eqnarray*}
\frac{B_{3}^{r}(x)}{x-1} & = & \frac{1}{3}b_{1}(x^{3})-\frac{1}{3}b_{1}(x)^{3}+b_{1}(x)b_{2}(x)x^{s}-b_{3}(x)x^{3s}\\
& = & (x-1)^{s}\Big(-\frac{1}{3}(x^{2}+x+1)^{s}+(x-1)^{2s}(\frac{1}{3}-x^{s}+x^{s}(x+1)^{s}\\
& & +x^{3s}+x^{3s}(x+1)^{s}(x^{2}+x+1)^{s}-2x^{3s}(x+1)^{s})\Big),
\end{eqnarray*}
\\[-8mm]
\begin{eqnarray*}
\frac{B_{4}^{r}(x)}{x-1} & = & (x-1)^{2s}\Big(\frac{1}{4}(x-1)^{2s}-\frac{1}{4}(x+1)^{2s}+(x^{2}-1)^{s}x^{s}(1-(x+1)^{s})\\
& & +\frac{1}{2}(x+1)^{2s}x^{2s}(1-(x^{2}+1)^{s})+\frac{1}{2}(x-1)^{2s}x^{2s}(1-(x+1)^{s})^{2}\\
& & -(x-1)^{2s}x^{3s}(-(x+1)^{s}(x^{2}+x+1)^{s}+2(x+1)^{s}-1)\\
& & -(x-1)^{2s}x^{6s}(-(x+1)^{s}(x^{2}+x+1)^{s}(x^{3}+x^{2}+x+1)^{s}\\
& & +2(x+1)^{s}(x^{2}+x+1)^{s}+(x+1)^{2s}-3(x+1)^{s}+1\big)\Big)\;.
\end{eqnarray*}
\end{lem}
\begin{proof}
We will make the formulae in Proposition \ref{prop:Bnformula} explicit,
by inverting formal power series. If
\[
{\textstyle F(t)=1+\sum_{n\geq1}a_{n}\,t^{n},\quad\mbox{and }F^{-1}(t)=1+\sum_{n\geq1}b_{n}t^{n}}
\]
are formal inverses, the relation between $a_{n}$ and $b_{n}$ can
be obtained from $\sum_{k\geq0}a_{k}b_{n-k}=0$ ($a_{0}=b_{0}=1$),
recursively (valid for power series over any ring) as
\begin{equation}
b_{1}=-a_{1}\quad\quad;b_{2}=a_{1}^{2}-a_{2};\quad\quad b_{3}=-a_{1}^{3}+2a_{1}a_{2}-a_{3},\quad\mbox{etc}.\label{eq:recursive}
\end{equation}
Now, employing $a_{n}(x):=\big((x-1)(x^{2}-1)\ldots(x^{n}-1)\big)^{r-1}$
as in (\ref{eq:F(t)}), we get:
\begin{align*}
b_{1}(x) & =-a_{1}(x)=-(x-1)^{r-1}\;,\\
b_{2}(x) & =a_{1}^{2}(x)-a_{2}(x)=(x-1)^{2r-2}-(x-1)^{r-1}(x^{2}-1)^{r-1}\\
& =(x-1)^{2r-2}\big(-(x+1)^{r-1}+1\big)\;,\\
b_{3}(x) & =-a_{1}^{3}(x)+2a_{1}(x)\,a_{2}(x)-a_{3}(x)=\\
& =(x-1)^{3r-3}\big(-(x+1)^{r-1}(x^{2}+x+1)^{r-1}+2(x+1)^{r-1}-1\big),\mbox{ etc}\,,
\end{align*}
which, by substitution in Proposition \ref{prop:Bnformula}, completes
the proof.
\end{proof}
Recall from \cite[Definition 4.14]{FNZ} the notion of rectangular
partition of $n$: a sequence of non-negative integers $0\leq k_{l,h}\leq n$,
$l,h\in\{1,\cdots,n\}$ satisfying $n=\sum_{l=1}^{n}\sum_{h=1}^{n}l\,h\,k_{l,h}$,
interpreted as a collection of $k_{l,h}$ rectangles of size $l\times h$,
with total area $n$. An element of the set $\mathcal{RP}_{n}$, of
rectangular partitions of $n$, is denoted:
\[
[[k]]=[(1\times1)^{k_{1,1}}\,(1\times2)^{k_{1,2}}\cdots(1\times n)^{k_{1,n}}\cdots(n\times n)^{k_{n,n}}]\in\mathcal{RP}_{n}\;,
\]
and the ``gluing map'' sends a rectangular partition to a usual
partition
\begin{eqnarray*}
\pi:\mathcal{RP}_{n} & \to & \mathcal{P}_{n}\\{}
[[k]] & \mapsto & [m]=[1^{m_{1}}\cdots n^{m_{n}}]\quad\mbox{where }m_{l}:=\sum_{h=1}^{n}h\cdot k_{l,h}.
\end{eqnarray*}
With the notion of rectangular partitions, we rephrase \cite[Corollary 1.2]{FNZ}
for the free group $\Gamma=F_{r}$ and $G=GL_{n}$ case.
\begin{thm}
\cite[Corollary 1.2]{FNZ} \label{thm:individual-strata} The E-polynomial
of the $GL_{n}$-character variety of the free group in $r$ generators
is
\[
e(\mathcal{X}_{r}GL_{n})=\sum_{[[k]]\in\mathcal{RP}_{n}}\ \prod_{l,h=1}^{n}\frac{B_{l}^{r}(x^{h})^{k_{l,h}}}{k_{l,h}!\,h^{k_{l,h}}},
\]
and the E-polynomial of the stratum corresponding to a partition $[m]\in\mathcal{P}_{n}$
is
\[
e(\mathcal{X}_{r}^{[m]}GL_{n})={\displaystyle \sum_{[[k]]\in\pi^{-1}[m]}\ \prod_{l,h=1}^{n}\frac{B_{l}^{r}(x^{h})^{k_{l,h}}}{k_{l,h}!\,h^{k_{l,h}}}}\;.
\]
\end{thm}
\subsection{Serre polynomials for $SL_{n}$ and $PGL_{n}$-character varieties
of the free group.}
Recall that, by Proposition \ref{prop:RPG-fibration} and Theorem
\ref{thm:Main}, $E$-polynomials of all strata for the $SL_{n}$
and $PGL_{n}$-character varieties of the free group of rank $r$
can be derived from the corresponding ones for $GL_{n}$, by dividing
out by $(x-1)^{r}$:
\[
e(\mathcal{X}_{r}^{[k]}SL_{n})=e(\mathcal{X}_{r}^{[k]}PGL_{n})=\frac{e(\mathcal{X}_{r}^{[k]}GL_{n})}{(x-1)^{r}}\;,
\]
for every $[k]\in\mathcal{P}_{n}$, as for the whole character variety.
So, Theorem \ref{thm:individual-strata} allows the computation of
explicit formulae for all $E$-polynomials of $\mathcal{X}_{r}G$,
with $G=GL_{n}$, $SL_{n}$ or $PGL_{n}$.
As examples, this method recovers the polynomial $e(\mathcal{X}_{r}SL_{2})=e(\mathcal{X}_{r}PGL_{2})$
first obtained in \cite{CL}, and the polynomial $e(\mathcal{X}_{r}SL_{3})=e(\mathcal{X}_{r}PGL_{3})$
in \cite[Theorem 1]{LM} (compare also \cite{BH}). We illustrate
the method for $n=3$.
\begin{thm}
\label{thm:GL3-free}For $s\geq0$, we have:
\begin{align*}
e(\mathcal{X}_{s+1}SL_{3}) & ={\textstyle \frac{1}{2}}(x-1)^{s+1}(x+1)^{s}x+{\textstyle \frac{1}{3}}(x^{2}+x+1)^{s}x(x+1)+\\
& +(x-1)^{2s}\Big((x+1)^{s}[x^{3s}(x^{2}+x+1)^{s}+x^{s+1}-2x^{3s}]+x^{3s}-x^{s+1}+{\textstyle \frac{x}{6}}(x+1)\Big)
\end{align*}
\end{thm}
\begin{proof}
By Theorem \ref{thm:individual-strata}, we get for $GL_{3}$
\[
e(\mathcal{X}_{r}GL_{3})=B_{3}^{r}(x)+B_{2}^{r}(x)B_{1}^{r}(x)+\frac{B_{1}^{r}(x^{3})}{3}+\frac{B_{1}^{r}(x^{2})B_{1}^{r}(x)}{2}+\frac{B_{1}^{r}(x)^{3}}{6}
\]
where the first term corresponds to $e(\mathcal{X}_{r}^{irr}GL_{3})$,
the second to $e(\mathcal{X}_{r}^{[1\,2]}GL_{3})$ and remaining 3
terms to $e(\mathcal{X}_{r}^{[1^{3}]}GL_{3})$ (see \cite[Figure 4.1]{FNZ}
showing rectangular partitions for $n=3$). Replacing $B_{j}^{r}(x)$,
$j=1,2,3$, by the expressions in Lemma \ref{lemma:Bn(x)} we obtain
the result for $\mathcal{X}_{r}GL_{3}$, and the case of $SL_{3}$
follows immediately.
\end{proof}
Our method allows explicit expressions for $n=4$ and beyond. In fact,
Theorem \ref{thm:individual-strata} provides the decomposition
\begin{align}
e(\mathcal{X}_{r}GL_{4}) & =e(\mathcal{X}_{r}^{[4]}GL_{4})+e(\mathcal{X}_{r}^{[1\;3]}GL_{4})+e(\mathcal{X}_{r}^{[2^{2}]}GL_{4})+e(\mathcal{X}_{r}^{[1^{2}\;2]}GL_{4})+e(\mathcal{X}_{r}^{[1^{4}]}GL_{4})\nonumber \\
& =B_{4}(x)+B_{3}(x)B_{1}(x)+\frac{B_{2}(x)^{2}}{2}+\frac{B_{2}(x^{2})}{2}+\frac{B_{2}(x)B_{1}(x^{2})}{2}+\frac{B_{2}(x)B_{1}(x)^{2}}{2}\nonumber \\
& +\frac{B_{1}(x^{4})}{4}+\frac{B_{1}(x^{3})B_{1}(x)}{3}+\frac{B_{1}(x^{2})^{2}}{8}+\frac{B_{1}(x^{2})B_{1}(x)^{2}}{4}+\frac{B_{1}(x)^{4}}{24}\;,\label{eq:PGL_4}
\end{align}
as the sum of the 5 strata (which comprise the $11$ terms coming
from the rectangular partitions in \cite[Figure 4.2]{FNZ}). Combining
(\ref{eq:PGL_4}) with formulae in Lemma \ref{lemma:Bn(x)} yields
the computation of the $E$-polynomial for $SL_{4}$ (and then also
$PGL_{4}$). This formula is new.
\begin{thm}
The E-polynomial of the $SL_{4}$-character variety of $F_{s+1}$
is:
\begin{eqnarray*}
e(\mathcal{X}_{s+1}SL_{4}) & = & (x-1)^{3s+1}\big[(x+1)^{2s}\frac{x^{2s}}{2}+(x+1)^{s}\big(x^{3s}(x^{2}+x+1)^{s}-2x^{3s}-x^{2s}+\frac{3x^{s}}{2}\big)\big]\\
& + & (x-1)^{3s+1}\big[x^{3s}+\frac{x^{2s}}{2}-\frac{3x^{s}}{2}+\frac{11}{24}\big]+\frac{1}{24}(x-1)^{3s+3}\\
& + & (x-1)^{3s}(x+1)^{2s}(-x^{6s}+\frac{x^{2s}}{2})\\
& + & (x-1)^{3s}(x+1)^{s}x^{6s}\big[(x^{2}+x+1)^{s}(x^{3}+x^{2}+x+1)^{s}-2(x^{2}+x+1)^{s}+3\big]\\
& + & (x-1)^{3s}(x+1)^{s}\big[x^{3s}\big((x^{2}+x+1)^{s}-2\big)-x^{2s}+\frac{x^{s}}{2}\big]\\
& + & (x-1)^{3s}(-x^{6s}+x^{3s}+\frac{x^{2s}}{2}-\frac{x^{s}}{2}+\frac{1}{2})\\
& + & (x-1)^{2s+2}\frac{(x-1)^{s+1}}{4}\\
& + & (x-1)^{2s+1}\frac{(x+1)^{s}}{2}(-(x+1)^{s}x^{s}+x^{s}-\frac{1}{2})\\
& + & (x-1)^{2s}(x+1)^{s}\frac{x^{s}}{2}(1-(x+1)^{s})\\
& + & (x-1)^{s+1}\big[(x+1)\frac{x}{3}(x^{2}+x+1)^{s}+\frac{(x+1)^{2s}}{8}(x^{2}+2x+2)\big]\\
& + & (x-1)^{s}(x+1)^{2s}\big[\frac{x^{2s+1}}{2}((x^{2}+1)^{s}-1)+\frac{x-1}{4}\big]\\
& - & \frac{1}{4}(x+1)^{s+1}(x^{2}+1)^{s}+\frac{1}{4}(x^{3}+x^{2}+x+1)^{s+1}\;.
\end{eqnarray*}
\end{thm}
Formulas for all $n$ can be obtained in exactly the same way, from
the combinatorics of rectangular partitions, and the formula for $B_{n}$
in Proposition \eqref{prop:Bnformula}.
\subsection{Irreducibility and Euler characteristics}
It is clear that the above method, combining Proposition \ref{prop:Bnformula},
Lemma \ref{lemma:Bn(x)} and Theorem \ref{thm:individual-strata},
provides the same kind of expressions for $e(\mathcal{X}_{r}SL_{n})=e(\mathcal{X}_{r}PGL_{n})$,
and for every $n\in\mathbb{N}$. Additionally, we can prove irreducibility
and compute all Euler characteristics of $\mathcal{X}_{r}^{[k]}G$
for $G=GL_{n}$, $SL_{n}$ and $PGL_{n}$, and all $[k]\in\mathcal{P}_{n}$.
We start with the $GL_{n}$ case.
\begin{lem}
\label{dimirr} The degree of the polynomial $B_{n}^{r}(x)$ is
\[
\deg(B_{n}^{r}(x))=n^{2}(r-1)+1.
\]
\end{lem}
\begin{proof}
From Proposition \ref{prop:Bnformula}, $\deg B_{n}^{r}(x)$ will
be
\begin{equation}
1+\max_{d|n}\max_{[k]\in\mathcal{P}_{d}}\deg\left(\prod_{j=1}^{d}b_{j}(x^{n/d})^{k_{j}}x^{\frac{n(r-1)k_{j}}{d}\binom{j}{2}}\right),\label{eq:degree}
\end{equation}
where $b_{n}(x)$ are defined by $(1+\sum_{n\geq1}a_{n}(x)\,t^{n})(1+\sum_{n\geq1}b_{n}(x)\,t^{n})=1$,
with $a_{n}(x)=\big((x-1)(x^{2}-1)\ldots(x^{n}-1)\big)^{r-1}$, so
that $\deg a_{i}(x)=\frac{i(i+1)}{2}(r-1)$. By the recursive definition
of $b_{j}(x)$ in terms of the $a_{i}(x)$ (see Equations \eqref{eq:recursive}),
it is easy to see that $b_{j}(x)=-a_{j}(x)+\mbox{lower degree terms}$,
hence $\deg b_{j}(x)=\deg a_{j}(x)=\binom{j+1}{2}(r-1)$. Now, the
maximum in \eqref{eq:degree} is achieved for $d=n$ because partitions
of smaller $n$ have lower degree. Hence:
\begin{eqnarray*}
\deg B_{n}^{r}(x) & = & 1+\max_{[k]\in\mathcal{P}_{n}}\deg\prod_{j=1}^{n}b_{j}(x)^{k_{j}}x^{(r-1)k_{j}\binom{j}{2}}\\
& = & {\textstyle 1+\max_{[k]\in\mathcal{P}_{n}}\sum_{j=1}^{n}\Big(\binom{j+1}{2}(r-1)k_{j}+(r-1)k_{j}\binom{j}{2}\Big)}\\
& = & {\textstyle 1+\max_{[k]\in\mathcal{P}_{n}}\sum_{j=1}^{n}k_{j}j^{2}(r-1)}\;.
\end{eqnarray*}
Using the restriction $n=\sum_{j=1}^{n}j\,k_{j}$, it is clear that
the maximum is achieved for the partition $[n]$, i.e. all $k_{j}=0$
except $k_{n}=1$, so that $\deg B_{n}^{r}(x)=n^{2}(r-1)+1$ as wanted.\end{proof}
\begin{cor}
\label{cor:Euler}Every $[k]$-polystable stratum $\mathcal{X}_{r}^{[k]}GL_{n}$
is an irreducible algebraic variety, and has zero Euler characteristic.\end{cor}
\begin{proof}
The irreducible stratum corresponds to the partition $[n]$ and its
polynomial is given by $e(\mathcal{X}_{r}^{irr}GL_{n})=B_{n}^{r}(x)$
(see Theorem \ref{thm:individual-strata} and Remark \ref{rem:ab-irr-strata}).
The monomial of top degree of $B_{n}^{r}(x)$ is, by the proof of
Lemma \ref{dimirr},
\[
(x-1)\frac{\mu(1)}{1}\frac{(-1)^{1}}{1}\binom{1}{1}b_{n}(x)^{1}x^{(r-1)\cdot1\cdot\binom{n}{2}}=(x-1)a_{n}(x)x^{(r-1)\binom{n}{2}}\;,
\]
whose leading coefficient equals the leading coefficient of $a_{n}(x)$,
which is $1$. Therefore, $\mathcal{X}_{r}^{irr}GL_{n}$ is an irreducible
variety. All polystable strata can be expressed as symmetric products
of irreducible strata of lower dimension (see \cite[Proposition 4.5]{FNZ}),
hence all of them are irreducible. The statement about Euler characteristics
follows by substituting $x=1$ in Theorem \ref{thm:individual-strata},
since $B_{n}^{r}(1)=0$, for all $n,r\in\mathbb{N}$ (see Proposition
\ref{prop:Bnformula} and Lemma \ref{lemma:Bn(x)}).
\end{proof}
The next Corollary is immediate from our constructions.
\begin{cor}
For every $[k]\in\mathcal{P}_{n}$, the strata $\mathcal{X}_{r}^{[k]}SL_{n}$
and $\mathcal{X}_{r}^{[k]}PGL_{n}$ are irreducible algebraic varieties.
\end{cor}
Finally, we compute the Euler characteristics of the $SL_{n}$ and
$PGL_{n}$-character varieties of the free group. It turns out that
the only strata contributing are of the form $[d^{n/d}]\in\mathcal{P}_{n}$,
indexed by the divisors $d$ of $n$.
\begin{prop}
\label{prop:euler} The Euler characteristics of the $SL_{n}$ and
$PGL_{n}$ character varieties of the free group are
\[
\chi(\mathcal{X}_{r}SL_{n})=\chi(\mathcal{X}_{r}PGL_{n})=\varphi(n)n^{r-2}
\]
where $\varphi(n)$ is Euler's totient function. The Euler characteristics
for the strata of the form $[d^{n/d}]\in\mathcal{P}_{n}$ are
\[
\chi(\mathcal{X}_{r}^{[d^{n/d}]}SL_{n})=\chi(\mathcal{X}_{r}^{[d^{n/d}]}PGL_{n})=\frac{\mu(d)}{d}n^{r-1}\;,
\]
otherwise $\chi(\mathcal{X}_{r}^{[k]}SL_{n})=0$, where $\mu(n)$
is the arithmetic Möbius function. \end{prop}
\begin{proof}
By Theorems \ref{thm:SL-2strata} and \ref{thm:equality-irr-PGL-SL},
the Euler characteristics for $SL_{n}$ and $PGL_{n}$-character varieties
are equal and can be computed strata by strata, by setting $x=1$
in the $GL_{n}$ polynomials of Proposition \ref{prop:Bnformula}
and Theorem \ref{thm:individual-strata}, and using Proposition \ref{prop:RPG-fibration}:
\begin{equation}
\chi(\mathcal{X}_{r}^{[m]}SL_{n})=\chi(\mathcal{X}_{r}^{[m]}PGL_{n})=\Big[\frac{e(\mathcal{X}_{r}^{[m]}GL_{n})}{(x-1)^{r}}\Big]_{x=1}\;.\label{eq:cancelling}
\end{equation}
To begin, we show the formula:
\begin{equation}
\Big[\frac{B_{n}^{r}(x)}{(x-1)^{r}}\Big]_{x=1}=\Big[\frac{-1}{(x-1)^{r-1}}\frac{\mu(n)}{n}b_{1}(x^{n})\Big]_{x=1}=\mu(n)n^{r-2}\;.\label{eq:B/(x-1)}
\end{equation}
Indeed, by the recursive definition (\ref{eq:recursive}), the term
$(x-1)^{n(r-1)}$ can be factored out in $a_{n}(x)$ and, hence, the
same applies to every $b_{n}(x)$. Then, for $n=1$ we get $b_{1}(x)=-(x-1)^{r-1}$.
However, for $m>1$, we have $\Big[\frac{b_{m}(x)}{(x-1)^{r-1}}\Big]_{x=1}=0$,
so all terms with $b_{m}(x)$ for $m>1$, in the formula of Proposition
\ref{prop:Bnformula}, disappear from the calculation of the Euler
characteristic. For $b_{1}(x)$ we have the following expression,
for every $t\in\mathbb{N}$:
\[
b_{1}(x^{s})^{t}=-(x^{s}-1)^{t(r-1)}=(-1)^{t}(x-1)^{t(r-1)}(1+x+\cdots+x^{s-1})^{t(r-1)}\;.
\]
Therefore, if $t>1$, $\Big[\frac{b_{1}(x^{s})^{t}}{(x-1)^{r-1}}\Big]_{x=1}=0$,
while, for $t=1$, we have $\Big[\frac{b_{1}(x^{s})}{(x-1)^{r-1}}\Big]_{x=1}=-s^{r-1}$.
Note that products $b_{i}(x)\cdot b_{j}(x)$, $i\neq j$, necessarily
have a zero of order greater than $r$ in $x=1$. Hence, in Lemma
\ref{lemma:Bn(x)}, after setting $\frac{B_{n}^{r}(x)}{(x-1)^{r}}$,
the only terms which do not vanish are those with $j=1$ and $k_{1}=1$
and $k_{j}=0$ for $j=2,\ldots,d$. Therefore $d=1$ and the only
partition contributing is $[1]\in\mathcal{P}_{1}$, yielding the equality
in Equation \eqref{eq:B/(x-1)}. Similarly, we have $\Big[\frac{B_{n}^{r}(x^{s})}{(x-1)^{r}}\Big]_{x=1}=\mu(n)s^{r}n^{r-2}$,
for every $s\in\mathbb{N}$.
To finish, we look at Theorem \ref{thm:individual-strata}. For rectangular
partitions with two or more blocks of different sizes (i.e. giving
terms of the form $B_{n_{1}}^{r}(x^{s_{1}})\cdot B_{n_{2}}^{r}(x^{s_{2}})$)
$x=1$ is a zero of order greater than $r$, then we get rid of them
in the final calculation. This leaves us to just consider rectangular
partitions $[[k]]$ with a single block $l\times h$ and $k_{l,h}=1$
and the rest $k_{l,h}=0$: they are indexed by divisors of $n$. Let
$[k]\in\mathcal{P}(n)$ be a partition by blocks of just one size
$[k]=[d^{n/d}]$. In the expression of Theorem \ref{thm:individual-strata},
put $l=d$ and $h=n/d$ and have
\[
\chi(\mathcal{X}_{r}^{[d^{n/d}]}SL_{n})=\Big[\frac{e(\mathcal{X}_{r}^{[d^{n/d}]}GL_{n})}{(x-1)^{r}}\Big]_{x=1}=\Big[\frac{1}{(x-1)^{r}}\frac{d}{n}B_{d}(x^{n/d})\Big]_{x=1}=\frac{\mu(d)}{d}n^{r-1}\;,
\]
while for other strata $[k]\in\mathcal{P}(n)$ with two or more blocks
of different sizes, the Euler characteristic is zero. To get the Euler
characteristic of the full character variety, we sum up over all strata,
this is, over all divisors of $n$:
\[
\chi(\mathcal{X}_{r}SL_{n})=\sum_{d|n}\Big[\frac{1}{(x-1)^{r}}\frac{d}{n}B_{d}(x^{n/d})\Big]_{x=1}=\sum_{d|n}\frac{\mu(d)}{d}n^{r-1}=\varphi(n)n^{r-2},
\]
by using Möbius inversion formula with the Euler function, $\varphi(n)=\sum_{d|n}\frac{n}{d}\mu(d)$. \end{proof}
\begin{rem}
This result extends \cite[Theorem 1.4]{MR} to the $SL_{n}$ case
and, moreover, calculates the Euler characteristic of all strata,
providing a geometrical meaning to the calculation in terms of the
Euler function. Also, note that the stratum $[1^{n}]$ corresponds
to an abelian character variety, and this computation agrees with
the one in \cite[Corollary 5.16]{FS}. \end{rem}
\begin{example}
(1) As an application of Proposition \ref{prop:euler}, for $n=4$,
we get
\[
\chi(\mathcal{X}_{r}SL_{4})=\chi(\mathcal{X}_{r}PGL_{4})=2\cdot4^{r-2},
\]
where the only strata that contribute to the Euler characteristic
are:
\[
\chi(\mathcal{X}_{r}^{[1^{4}]}SL_{4})=\chi(\mathcal{X}_{r}^{[1^{4}]}PGL_{4})=4^{r-1},\quad\quad\chi(\mathcal{X}_{r}^{[2^{2}]}SL_{4})=\chi(\mathcal{X}_{r}^{[2^{2}]}PGL_{4})=-2\cdot4^{r-2}.
\]
(2) In the case when $n$ is a prime number $p$, we similarly get:
\[
\chi(\mathcal{X}_{r}SL_{p})=(p-1)\cdot p^{r-2},
\]
where the only strata contributing to the Euler characteristic are:
\[
\chi(\mathcal{X}_{r}^{[1^{p}]}SL_{p})=p^{r-1},\quad\quad\chi(\mathcal{X}_{r}^{[p]}SL_{p})=-p^{r-2},
\]
and similarly for $\mathcal{X}_{r}PGL_{p}$.\\
(3) More generally, if $n=p_{1}^{a_{1}}\cdot p_{2}^{a_{2}}\cdots p_{s}^{a_{s}}$
is the prime decomposition of $n$, the only strata contributing to
the Euler characteristic of the $SL_{n}$ and the $PGL_{n}$ character
variety of $F_{r}$ are those of the form $[d^{n/d}]$, where $d=p_{i_{1}}\cdot p_{i_{2}}\cdots p_{i_{t}}$
is square-free since then $\mu(d)\neq0$. These strata are the same
for $n^{\ast}:=p_{1}\cdot p_{2}\cdots p_{s}$ and for all $n$ with
the same prime divisors. Indeed,
\[
\chi(\mathcal{X}_{r}^{[1^{n}]}SL_{n})=n^{r-1}\;,
\]
\[
\chi(\mathcal{X}_{r}^{[p_{i}^{n/p_{i}}]}SL_{n})=-\frac{1}{p_{i}}n^{r-1}\;,
\]
\[
\chi(\mathcal{X}_{r}^{[(p_{i}p_{j})^{n/p_{i}p_{j}}]}SL_{n})=\frac{1}{p_{i}p_{j}}n^{r-1}\;,\quad\cdots
\]
\[
\chi(\mathcal{X}_{r}^{[(n^{\ast})^{n/n^{\ast}}]}SL_{n})=(-1)^{s}\frac{1}{n^{\ast}}n^{r-1}\;,
\]
with sum
\[
\prod_{p|n,\,p\;prime}(1-\frac{1}{p})n^{r-1}=\varphi(n)n^{r-2}\;.
\]
\end{example}
\end{document} | math |
The cefuroxime sodium is a second generation cephalosporin indicated for infections caused by Gram-positive and Gram-negative microorganisms. to 120.0 mg/mL, with 100.21% accuracy and content 99.97% to cefuroxime sodium in injectable pharmaceutical form. against (Campylobacter), and methicillin-resistant and . Clinical studies show that cefuroxime is effective in individuals with infections of the lower respiratory tract, pores and skin and skin constructions, urinary tract, or female reproductive system . Several different analytical methods beta-Amyloid (1-11) IC50 have been explained for the dedication of cephalosporins in the literature [4,5,6,7,8,9,10,11,12,13,14]. Since this antibiotic has been very widely used in the antimicrobial therapy, it is important to develop and validate methods for dedication of cefuroxime in pharmaceutical dose form . There are many physicochemical analytical methods explained in the literature for the analysis of cefuroxime beta-Amyloid (1-11) IC50 in different matrices, using techniques such as HPLC [16,17,18,19,20,21], fluorimetry , spectrophotometry [23,24,25] and chemiluminescence . Despite this fact, physicochemical methods used to quantify antimicrobial providers, although accurate, are not able to indicate the true biological activity of the drug. For this reason, microbiological methods are used to determine the potency of antimicrobial providers and they play an essential role in the manufacturing processes and quality control of these medicines [27,28]. The official method of analysis for cefuroxime sodium powder for injectable remedy explained in the literature is the high performance liquid chromatography using acetate buffer pH 3.4 and acetonitrile while mobile phase . However, it is known the plugs damage the column over time, which makes it more difficult to carry out HPLC analysis due to the interaction of the inorganic salts with silica . Considering that the turbidimetric assay has the advantage of reduced analysis time when compared to the agar diffusion method, where the analysis time is definitely 24 h, the aim of this work beta-Amyloid (1-11) IC50 was to propose a rapid turbidimetric method for the analysis of cefuroxime sodiums potency in the dosage form of powder for dissolution for injection. 2. Experimental 2.1. Chemicals Cefuroxime sodium research standard (declared having a purity of 97.40%) was kindly donated from the pharmaceutical organization (RJ, Brazil), and the samples of cefuroxime in lyophilized powder for dissolution for injection were purchased from Cellofarm Farmacutica (RJ, Brazil) containing 750 mg of the active component. The vials did not consist of excipients. The tradition press tryptic soy broth (TSB) (Difco, Detroit, MI, USA) and tryptic soy agar (Difco, Detroit, MI, USA) were used for the microbiological method. Analytical grade formaldehyde (Qhemis, SP, Brazil) was used to interrupt the growth of microorganisms. 2.2. Apparatus For the turbidimetric assay, the tradition media were sterilized before use in a vertical autoclave AV model (Phoenix Luferco, SP, Brazil). Incubation of microorganisms was performed using a Shaker incubator MA420 model (Marconi, SP, Brazil) and an oven ECB Digital 1.2 (Odontobrs, SP, Brazil). A spectrophotometer DU 530 (Beckman Coulter, CA, USA) was beta-Amyloid (1-11) IC50 used to determine the absorbance. The software Microsoft Excel (2007) was used to construct the calibration curves. For the HPLC method, the apparatus used was the model 1525 Waters? (Waters Chromatography systems, CA, USA), connected to a Waters 2487 UV/Visible detector and a manual injector Rheodyne Breeze 7725i having a 20 mL loop (Rheodyne Breeze?, CA, USA). The chromatographic separation was carried out under isocratic reversed phase conditions on an Agilent Zorbax? C18, 5 mm, 4.6 150 mm (Agilent?, Rabbit Polyclonal to MMP23 (Cleaved-Tyr79) Santa Clara, CA, USA). Additional apparatus also used was: 20C200 mL micropipettes (Digipet?, PR, Brazil); H10 analytical level (Mettler Toledo?, SP, Brasil); B160 semi-analytical level (Micronal?, SP, Brazil) and USC2800A ultrasound bath (Unique?, SP, Brazil). 2.3. Solutions Preparation of cefuroxime standard solutions. For the preparation of cefuroxime RS stock remedy, 50.0 mg equivalent of cefuroxime RS was weighed, and then it was transferred to a 100 mL volumetric ask and the volume was completed with ultrapure water to obtain a solution having a concentration of 500 gmL?1. Aliquots of 0.6, 1.2 and 2.4 mL of this solution were transferred to 10 mL volumetric flasks, the quantities of which were completed with ultrapure water in order to obtain working solutions with concentrations of 30.0, 60.0 and 120.0 gmL?1, respectively named S1, beta-Amyloid (1-11) IC50 S2 and S3, which were used in the bioassay. Preparation of cefuroxime sample solution. The material of 20 vials of cefuroxime in powder for injectable remedy were mixed. From this combination, 50.0 mg were accurately weighed and transferred to a 100 mL volumetric flask, and the volume was completed with ultrapure water in order to obtain a stock solution of 500 g mL?1. Aliquots of 0.6, 1.2 and 2.4 mL of this solution were transferred to 10 mL volumetric flasks, the quantities of.
Tagged: beta-Amyloid (1-11) IC50, Rabbit Polyclonal to MMP23 (Cleaved-Tyr79). | english |
ജനങ്ങളുടെ വിവരങ്ങള് ശേഖരിച്ചത് ഭീകരരെയും കുറ്റവാളികളെയും കണ്ടെത്താന് യുഎസ് സൈന്യത്തിന്റെ ബയോമെട്രിക് ഉപകരണങ്ങള് പിടിച്ചെടുത്ത് താലിബാന് സഹായിച്ച ആ അഫ്ഗാനികളുടെ ജീവന് തുലാസ്സില് പാക്കിസ്ഥാന്റെ സഹായത്തോടെ ഭീകരര് പക വീട്ടുമോ എന്ന ആശങ്കയില് നിരപരാധികള് കാബൂള്: അഫ്ഗാനിസ്ഥാനില് സഹായം നല്കിയ ജനങ്ങളുടെ വിവരങ്ങള് ശേഖരിച്ച് സൂക്ഷിച്ചിരുന്ന അമേരിക്കന് സൈന്യത്തിന്റെ ബയോമെട്രിക് ഉപകരണങ്ങള് താലിബാന് പിടിച്ചെടുത്തതായി റിപ്പോര്ട്ടുകള്. ഫോട്ടോകളും ഫിംഗര്പ്രിന്റും കുടുംബവിവരങ്ങളും ജോലിസംബന്ധമായ വിവരങ്ങളും ഒക്കെ അടങ്ങിയ സിറ്റിസണ് ഡാറ്റാബേസുകളാണിവ. ഇന്ത്യയിലെ പൗരന്മാര്ക്കുവേണ്ടി ഒരുക്കിയ ആധാര് ഒക്കെ പോലെയുള്ളതാണ് ഇതിന്റെ ഡേറ്റാ ബേസ്. Handheld Interagency Identity Detection Equipment HIIDE എന്നാണ് ഈ ഉപകരണം അറിയപ്പെടുന്നത്. കാബൂള് അടക്കം പിടിച്ചടക്കിയ താലിബാന്റെ ആക്രമണസമയത്താണ് ഇത് യു എസ് സൈന്യത്തിന് നഷ്ടമായതെന്നാണ് സൂചന. ജോയിന്റ് സ്പെഷ്യല് ഓപ്പറേഷന്സ് കമാന്ഡ് ഉദ്യോഗസ്ഥനും മൂന്ന് മുന് യുഎസ് സൈനിക ഉദ്യോഗസ്ഥരുമാണ് ഇക്കാര്യം സ്ഥിരീകരിച്ചത്. യു എസ് സൈന്യത്തെ രാജ്യത്ത് സഹായിച്ച നിരപരാധികളായ ആ അഫ്ഗാനികളെ താലിബാന് ഭീകര സംഘങ്ങള്ക്ക് തിരിച്ചറിയാനും ഇല്ലാതാക്കാനും ഈ ഡാറ്റ ഉപയോഗിക്കുമോ എന്ന ആശങ്കയാണ് ഇപ്പോള് നിലനില്ക്കുന്നത്. HIIDE ഉപകരണങ്ങളില് ഐറിസ് സ്കാനുകളും വിരലടയാളങ്ങളും വ്യക്തികളെ കുറിച്ചുള്ള വിവരങ്ങളും അടങ്ങിയിരിക്കുന്നു. യുഎസ് സൈന്യത്തിന്റെ ബയോമെട്രിക് ഡാറ്റാബേസില് അഫ്ഗാന് നിവാസികളെ കുറിച്ചുള്ള എത്രമാത്രം വിവരങ്ങള് അടങ്ങിയിട്ടുണ്ടെന്നത് വ്യക്തമല്ല. തീവ്രവാദികളെയും മറ്റ് കലാപകാരികളെയും തിരിച്ചറിയുന്നതിനാണ് യുഎസ് സൈന്യം ഇത് ഉപയോഗിച്ചിരുന്നതെങ്കിലും, യുഎസിനെ സഹായിച്ച അഫ്ഗാനികളുടെ ബയോമെട്രിക് ഡാറ്റയും വ്യാപകമായി ശേഖരിക്കുകയും തിരിച്ചറിയല് കാര്ഡുകളില് ഉപയോഗിക്കുകയും ചെയ്തിരുന്നു. താലിബാന്റെ കൈയില് HIIDE ഡാറ്റ പ്രോസസ്സ് ചെയ്യുന്നതിനുള്ള ആധുനിക ഉപകരണങ്ങള് ഒന്നുമില്ലെങ്കിലും, പാക്കിസ്ഥാന് ഇതില് അവരെ സഹായിക്കാന് കഴിയുമെന്ന് ആര്മി സ്പെഷ്യല് ഓപ്പറേഷന്സ് വെറ്ററന് ആശങ്ക പ്രകടിപ്പിച്ചു. പാക്കിസ്ഥാന്റെ ചാര സംഘടനയായ ഇന്റര്സര്വീസസ് ഇന്റലിജന്സിന് അവരെ സഹായിക്കാന് സാധിക്കുമെന്ന് മുന് സ്പെഷ്യല് ഓപ്പറേഷന്സ് ഉദ്യോഗസ്ഥന് പറയുന്നു. അമേരിക്കയുടെ പതിറ്റാണ്ടുകളായ ഭീകരതയ്ക്കെതിരായ യുദ്ധത്തില് ഈ ഉപകരണങ്ങള് ഒരു പ്രധാന പങ്ക് വഹിച്ചിരുന്നു. 2011 ല് പാക്കിസ്ഥാനിലെ ഒരു ഒളിത്താവളത്തിലിരുന്ന ഒസാമ ബിന് ലാദനെ തിരിച്ചറിയാന് സാധിച്ചതും ഈ ബയോമെട്രിക്സ് വഴിയാണ്. അന്വേഷണ റിപ്പോര്ട്ടര് ആനി ജേക്കബ്സന്റെ അഭിപ്രായത്തില്, ഭീകരരെയും കുറ്റവാളികളെയും കണ്ടെത്തുന്നതിനായി യു എസ് അഫ്ഗാനിലെ 80 ശതമാനം ആളുകളുടെയും ബയോമെട്രിക് വിവരങ്ങള് ശേഖരിച്ചിരുന്നു. ഇത് കൂടാതെ, ഫെഡറല് ബ്യൂറോ ഓഫ് ഇന്വെസ്റ്റിഗേഷന്, ഡിപ്പാര്ട്ട്മെന്റ് ഓഫ് ഹോംലാന്ഡ് സെക്യൂരിറ്റി തുടങ്ങിയ സര്ക്കാര് ഏജന്സികളുമായി ശേഖരിച്ച ബയോമെട്രിക്സ് ഡാറ്റ പങ്കിടാനും പ്രതിരോധ വകുപ്പ് ശ്രമിച്ചിട്ടുണ്ട്. കുറ്റവാളികളെയും ഭീകരരെയും കുറിച്ചുള്ള വിവരങ്ങള് മാത്രമല്ല, സൈന്യത്തോടൊപ്പം ജോലി ചെയ്യുന്നവരും, നയതന്ത്ര ശ്രമങ്ങളെ സഹായിക്കുന്നവരെ കുറിച്ചുള്ള വിവരങ്ങളും യുഎസ് ബയോമെട്രിക്സില് ശേഖരിച്ചു. കഴിഞ്ഞ രണ്ട് പതിറ്റാണ്ടുകളായി 300,000 ലധികം അഫ്ഗാന് പൗരന്മാര് അമേരിക്കന് ദൗത്യവുമായി ബന്ധപ്പെട്ടിട്ടുണ്ടെന്ന് ഇന്റര്നാഷണല് റെസ്ക്യൂ കമ്മിറ്റി അഭിപ്രായപ്പെട്ടു. യുഎസ് സൈനിക പിന്മാറ്റത്തെ തുടര്ന്ന് അഫ്ഗാനിസ്ഥാനില് അരാജകത്വം സൃഷ്ടിക്കപ്പെട്ടപ്പോള്, രണ്ടായിരത്തോളം അഫ്ഗാനികളെ യുഎസിലേക്ക് ഒഴിപ്പിക്കപ്പെട്ടു. പക്ഷേ ആയിരക്കണക്കിന് പേര് അഫ്ഗാനിസ്ഥാനില് നിന്ന് യുഎസിലേക്ക് പോകുന്നതിന് ആവശ്യമായ പ്രത്യേക കുടിയേറ്റ വിസകള്ക്കായി കുടുങ്ങിക്കിടക്കുകയാണെന്ന് പറയപ്പെടുന്നു. HIIDE ഡാറ്റാ സംവിധാനങ്ങള് സുരക്ഷിതമാക്കാനുള്ള ഉത്തരവാദിത്തം ആത്യന്തികമായി അഫ്ഗാന് സര്ക്കാരിന്റേതാണ്. എന്നാലും യുഎസ് സേനയ്ക്കും അതിന്റെ സഖ്യകക്ഷികള്ക്കും ഈ ഡാറ്റയുടെ അപകടസാധ്യത വിലയിരുത്തുന്നതിലും ദുരുപയോഗം തടയുന്നതിലും ആവശ്യമായത് ചെയ്യാമായിരുന്നെന്ന് ആക്സസ് നൗവില് ഏഷ്യ പസഫിക് പോളിസി ഡയറക്ടര് രമണ് ജിത് സിങ് ചിമ പറഞ്ഞു. അതേ സമയം അഫ്ഗാനികള് ഇപ്പോള് അവരുടെ ഡിജിറ്റല് പ്രൊഫൈലുകള് ഇല്ലാതാക്കാനുള്ള വെപ്രാളത്തിലാണ്. ആണ്കുട്ടികളും പുരുഷന്മാരും അവര് അയച്ച സന്ദേശങ്ങള്, അവര് കേട്ട സംഗീതം, എടുത്ത ചിത്രങ്ങള് എന്നിവ ഇല്ലാതാക്കാന് ശ്രമിച്ചുകൊണ്ടിരിക്കയാണ് ബിബിസി റിപ്പോര്ട്ടര് സന സാഫി ഞായറാഴ്ച ട്വിറ്ററില് എഴുതി. തീവ്രവാദികള് തലസ്ഥാനമായ കാബൂളിലേക്ക് പ്രവേശിച്ചതിന് പിന്നാലെ സര്ക്കാര് ഉദ്യോഗസ്ഥര്, മുന് സുരക്ഷാ സേനാംഗങ്ങള്, ലാഭേച്ഛയില്ലാതെ പ്രവര്ത്തിക്കുന്നവര് എന്നിവര്ക്കായി വീടുതോറുമുള്ള തിരച്ചില് ആരംഭിച്ചുവെന്ന് മുസ്തഫ എന്ന ട്വിറ്റര് ഉപയോക്താവ് പറയുന്നു. പത്രപ്രവര്ത്തകരുടെ വീടുകളിലും തിരച്ചില് നടത്തിയിരുന്നെന്നും പറയുന്നു. വീടുതോറുമുള്ള പരിശോധനയെ കുറിച്ച് കേട്ടതായും തീവ്രവാദികള് ഇതിനായി ബയോമെട്രിക്സ് മെഷീന് ഉപയോഗിക്കുന്നതായും കാബൂള് നിവാസി ഒരു സ്വകാര്യ സന്ദേശത്തില് പറഞ്ഞു. Stories you may Like | malyali |
कोरोना पर गरमाई सियासत, सीएम योगी और केजरीवाल के बीच ट्वीट वार यूपी चुनाव में पहले चरण के लिए चुनाव प्रचार का आज आखिरी दिन है। ऐसे में राज्य का सियासी पारा बेहद गरमा गया है। कोरोना काल में दिल्ली से यूपी के मजदूरों के पलायन के मुद्दे पर उत्तर प्रदेश के मुख्यमंत्री योगी आदित्यनाथ और दिल्ली के सीएम अरविंद केजरीवाल और योगी आदित्यनाथ के बीच ट्विटर पर भी जंग छिड़ गई। योगी आदित्यनाथ ने कई ट्वीट कर केजरीवाल पर जमकर निशाना साधा। उन्होंने आरोप लगाया कि जब पूरी मानवता कोरोना की पीड़ा से कराह रही थी, उस समय केजरीवाल ने यूपी के श्रमिकों को दिल्ली छोड़ने पर विवश किया। जवाब में केजरीवाल ने आरोप लगाया कि जब यूपी के लोगों की लाशें नदी में बह रहीं थीं, आप करोड़ों रुपए खर्च करके झूठी वाह वाही लूट रहे थे। योगी ने ट्वीट किया, केजरीवाल को झूठ बोलने में महारथ हासिल है, जब पूरा देश आदरणीय प्रधानमंत्री मोदी के नेतृत्व में कोरोना जैसी वैश्विक महामारी से जूझ रहा था, तब केजरीवाल ने प्रवासी मजदूरों को दिल्ली से बाहर का रास्ता दिखा दिया। सुनो केजरीवाल, जब पूरी मानवता कोरोना की पीड़ा से कराह रही थी, उस समय आपने यूपी के कामगारों को दिल्ली छोड़ने पर विवश किया। छोटे बच्चों व महिलाओं तक को आधी रात में यूपी की सीमा पर असहाय छोड़ने जैसा अलोकतांत्रिक व अमानवीय कार्य आपकी सरकार ने किया। आपको मानवताद्रोही कहें या...। जवाब में दिल्ली के सीएम केजरीवाल ने ट्वीट किया, सुनो योगी, आप तो रहने ही दो, जिस तरह UP के लोगों की लाशें नदी में बह रहीं थीं और आप करोड़ों रुपये खर्च करके मैगज़ीन में अपनी झूठी वाहवाही के विज्ञापन दे रहे थे। आप जैसा निर्दयी और क्रूर शासक मैंने नहीं देखा। यूपी व दिल्ली के सीएम के बीच ट्विटर वॉर में कांग्रेस भी कूद गई। पार्टी की छत्तीसगढ़ इकाई ने ट्वीट किया, सुनो योगीकेजरीवाल, तुम दोनों ये नूरा कुश्ती करके देश को बेवकूफ न बनाओ। सच तो ये है कि जनता की दोनों को कोई फिक्र नहीं। दोनों ही नागपुर वालों के Arvind Now और Yogi Now हो। | hindi |
IPL 2022 Mega Auction की लिस्ट फाइनल, इस देश से आए हैं सबसे ज्यादा क्रिकेटर भारतीय क्रिकेट कंट्रोल बोर्ड यानी बीसीसीआई ने मंगलवार 1 फरवरी को इंडियन प्रीमियर लीग यानी आईपीएल के 2022 के मेगा ऑक्शन के खिलाड़ियों की लिस्ट फाइनल कर दी। इस लिस्ट में कुल 590 खिलाड़ी शामिल हैं, जिन पर आईपीएल की 10 टीमें बोली लगा सकती हैं। इनमें 370 कैप्ड और अनकैप्ड इंडियन प्लेयर हैं, जबकि 220 विदेशी खिलाड़ी हैं, लेकिन क्या आप जानते हैं कि भारत के बाद ऐसा कौन सा देश है, जिसके सबसे ज्यादा खिलाड़ी टूर्नामेंट के ऑक्शन में शामिल होंगे। दरअसल, भारत के बाद ऑस्ट्रेलिया ऐसा देश है, जिसके सबसे ज्यादा खिलाड़ी आईपीएल 2022 मेगा ऑक्शन का हिस्सा होंगे। 1200 से ज्यादा खिलाड़ी ने आईपीएल के मेगा ऑक्शन के लिए रजिस्ट्रेशन किया था, लेकिन बीसीसीआई और आईपीएल के अधिकारियों ने इसकी छंटनी कर दी है, जिसके बाद 590 खिलाड़ी मेगा ऑक्शन की टेबल पर होंगे। इनमें से ऑस्ट्रेलिया से आने वाले 47 खिलाड़ी हैं। 40 से ज्यादा खिलाड़ी किसी भी देश के इस ऑक्शन का हिस्सा नहीं होंगे। ऑस्ट्रेलिया जहां 47 खिलाड़ियों के साथ टॉप पर है, वहीं वेस्टइंडीज के 34 खिलाड़ी ऑक्शन में शामिल होंगे। 33 खिलाड़ी साउथ अफ्रीका से भी आने वाले हैं। न्यूजीलैंड के 24 खिलाड़ियों को इस फाइनल लिस्ट में जगह मिली है। इतने ही खिलाड़ी इंग्लैंड के हैं, जबकि 23 खिलाड़ी पड़ोसी देश श्रीलंका के हैं, जिनको ऑक्शन की लिस्ट में शामिल किया गया है। अफगानिस्तान के 17 खिलाड़ियों ने आईपीएल 2022 के मेगा ऑक्शन का टिकट कटाया है। इसके अलावा आयरलैंड और बांग्लादेश के 55, नामीबिया के 3, स्कॉटलैंड के दो, जिम्बाब्वे, नेपाल और यूएसए के 11 खिलाड़ी को इस लिस्ट में जगह मिली है। 220 विदेशी खिलाड़ियों की देशवार लिस्ट देश Auction List अफगानिस्तान 17 ऑस्ट्रेलिया 47 बांग्लादेश 5 इंग्लैंड 24 आयरलैंड 5 न्यूजीलैंड 24 साउथ अफ्रीका 33 श्रीलंका 23 वेस्टइंडीज 34 जिम्बाब्वे 1 नामीबिया 3 नेपाल 1 स्कॉटलैंड 2 यूएसए 1 PL 2022 के मेगा ऑक्शन में जिन खिलाड़ियों पर बोली लगेगी उनके नाम इस लिस्ट में हैं, क्लिक करें For Hindustan : हिन्दुस्तान ईसमाचार पत्र के लिए क्लिक करें epaper.livehindustan.com | hindi |
अकाली दल उम्मीदवार बीबी जगीर कौर ने भरा नामांकन, इस हल्के से लड़ेंगी चुनाव भुलत्थ रजिन्दर: विधान सभा हलका भुलत्थ से शिरोमणि अकाली दल और बसपा गठजोड़ के उमीदवार बीबी जागीर कौर ने आज एस.डी.एम. भुलत्थकमचयन रिटर्निंग अफसर शायरी मल्होत्रा के पास अपने नामांकन पत्र दाखिल करवाया। इस मौके उनके साथ दामाद युवराज भुपिन्दर सिंह, नगर पंचायत बेगोवाल के प्रधान रजिन्दर सिंह लाडी भी थे। इस मौके बीबी जगीर कौर ने कहा कि शिरोमणि अकाली दल और बसपा गठजोड़ के उम्मीदवार के तौर पर उन्होंने आज पेपर दाखिल करवाए हैं। उन्हें विश्वास है कि वाहिगुरू कृपा करेगा, क्योंकि हल्के के लोगों में अकाली दल और बसपा प्रति बहुत ज्यादा उत्साह है। बताने योग्य है कि 20 फरवरी को होने वाली मतदान को लेकर अकाली दल और बसपा की तरफ से हल्का भुलत्थ से बीबी जगीर कौर को चयन मैदान में उतारा गया है। बीबी जगीर कौर का मुकाबला कांग्रेस के उम्मीदवार सुखपाल सिंह खैहरा और आम आदमी पार्टी के उम्मीदवार रणजीत सिंह राणा के साथ होने वाला है। | hindi |
टेंशन में लालू प्रसाद यादव, अब मैनुअल तोड़ने का लगा आरोप रांची: चारा घोटाले में जेल में बंद राष्ट्रीय जनता दल RJD के सुप्रीमो लालू प्रसाद यादव की मुश्किलें और बढ़ सकती हैं. लालू को 139 करोड़ के डोरंडा कोषागार मामले में 15 फरवरी को दोषी ठहराया गया था. इस मामले में 21 फरवरी को सजा का ऐलान किया जाना है. सोशल मीडिया पर लालू की एक तस्वीर वायरल हो रही है, जो उनके लिए मुसीबत का कारण बन सकती है. फोटो सामने आते ही लालू पर जेल मैनुअल तोड़ने के आरोप लगने लगे हैं.वायरल तस्वीर लालू को दोषी ठहराए जाने से पहले जेल की बताई जार ही है. फोटो में लालू झारखंड राजद के पूर्व अध्यक्ष अभय सिंह और इरफान के साथ नजर आ रहे हैं. फोटो में अभय फोन पर बात करते नजर आ रहे हैं. लालू वहां शांति से खड़े हुए हैं. जबकि फोटो में इरफान भी साथ है. इरफान पर इससे पहले बिहार में नीतीश सरकार को गिराने के लिए विधायक तोड़ने के आरोप लग चुके हैं. उनका एक विधायक के साथ बातचीत का ऑडियो भी सोशल मीडिया पर वायरल हुआ था.बता दें कि सीबीआई की विशेष अदालत ने लालू को दोषी करार दिया था. लालू के वकील ने उनका स्वास्थ्य ठीक ना होने की याचिका दाखिल की थी. याचिका पर उन्हें रिम्स लाया गया था. लेकिन वायरल तस्वीर से अब जेल के नियमों की धज्जियां उड़ाने की बात सामने आ रही है. दरअसल, डोरंडा कोषागार से जुड़े 139 करोड़ रुपए की निकासी के मामले में लालू को दोषी करार दिया गया है.लालू की वायरल तस्वीर सामने आने के बाद आज तक ने जेल IG मनोज कुमार से बात की. उन्होंने कहा कि इस मामले में DC SSP से जवाब मांगा जाएगा. जेल सुपरिटेंडेंट हामिद का कहना है कि उन्होंने किसी को भी लालू से मुलाकात करने की अनुमति नहीं दी है. जेल मैनुअल के मुताबिक जेल में बंद व्यक्ति सप्ताह में 3 लोगों से मुलाकात कर सकता है. | hindi |
राजस्थानःlungs इन्फेक्शन से बीवी को बचाने के लिए जब डाक्टर ने 70 लाख में गिरवी रख दी अपनी MBBS की डिग्री अपनों को खोने का डर किस कदर भयावह हो सकता है, ये बात राजस्थान की एक घटना से सहज ही समझी जा सकती है। अपनी बीमार पत्नी को मौत के मुंह से निकालने के लिए एक डॉक्टर ने अपनी डिग्री ही गिरवी रख दी। पत्नी ठीक हुई तो चिकित्सक के चेहरे पर सुकून साफ ही दिख जाता है। ये कहानी है राजस्थान के पाली जिले के डॉक्टर सुरेश चौधरी की। सुरेश पाली के खैरवा गांव के रहने वाले हैं। उनके परिवार में पत्नी अंजू और पांच साल का बेटा है। बीते साल मई में कोरोना की दूसरी लहर के दौरान अंजू पॉजिटिव हो गई। यहां से तकलीफें बढ़नी शुरू हुईं। तबीयत और बिगड़ी तो यहां वहां के धक्के खाने के बाद सुरेश ने पत्नी को जोधपुर एम्स में भर्ती करवाया। मई आते आते अंजू की हालत और ज्यादा खराब हो गई और उसके लंग्स 95 फीसद तक खराब हो गए। एम्स के डॉक्टर्स ने कह दिया कि बचना काफी मुश्किल है। लेकिन सुरेश अंजू को लेकर अहमदाबाद चले गए। बीमारी सिर चढ़कर बोल रही थी। महिला का वजन 50 किलो से घटकर 30 किलो रह गया था। शरीर में खून की कमी हो गई थी। अंजू को ईसीएमओ मशीन पर लिया गया। इलाज काफी खर्चीला था। रोजाना एक लाख रुपए तक खर्च सुरेश को वहन करना पड़ रहा था। लोकप्रिय खबरें Punjab Election: क्या SAD ने धुरी में अंतिम पलों में कांग्रेस के पक्ष में पलट दी बाजी? इन चर्चाओं के बीच AAP को भगवंत मान की जीत का भरोसा बात जलेबी की तरह मत घुमाओ कुमार विश्वास के आरोपों पर भड़के अरविंद केजरीवाल तो लोगों देने लगे ऐसे रिएक्शन पेड़ पर बैठे लोगों की फोटो शेयर कर मनोज तिवारी बोले मोदी सेना राम सेना की तरह, यूजर्स ने पूछा अपने समर्थकों को बंदर कह रहे? बीजेपी आई तो सरकार में एक दंगा मंत्री भी होगा बोले राकेश टिकैत तो पूर्व IAS ने पूछा, यो के करेगा? लोग भी लेने लगे मजे अंजू 87 दिन इस मशीन पर रही। उसके बाद उनकी तबीयत में सुधार हुआ और वह मौत के मुंह से बाहर आई। लेकिन रोजाना 1 लाख रुपये जुटाना हंसी खेल नहीं था। रकम जुटाने के लिये सुरेश ने अपनी एमबीबीएस की डिग्री गिरवी रखकर बैंक से 70 लाख रुपए का लोन लिया। उनके पास बैंक में केवल 10 लाख रुपए थे। सुरेश ने अपने दोस्तों और साथी चिकित्सकों से 20 लाख रुपये लिए। एक प्लाट था। उसे भी 15 लाख रुपये में बेच दिया। बाकी रिश्तेदारों से भी रकम उधार ली। कहते हैं कि अंत भला तो सब भला। फिलहाल पत्नी ठीक है और ये वाकया सोशल मीडिया पर वायरल है। सभी चिकित्सक के जज्बे को सलाम कर रहे हैं। वहीं सुरेश का कहना है कि पैसा तो वो फिर भी कमा लेगा पर पत्नी को कुछ हो जाता को जीवन खत्म था। | hindi |
<?php
/*
* This file is part of the PHPExifTool package.
*
* (c) Alchemy <[email protected]>
*
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* file that was distributed with this source code.
*/
namespace PHPExiftool\Driver\Tag\XMPExif;
use JMS\Serializer\Annotation\ExclusionPolicy;
use PHPExiftool\Driver\AbstractTag;
/**
* @ExclusionPolicy("all")
*/
class SceneType extends AbstractTag
{
protected $Id = 'SceneType';
protected $Name = 'SceneType';
protected $FullName = 'XMP::exif';
protected $GroupName = 'XMP-exif';
protected $g0 = 'XMP';
protected $g1 = 'XMP-exif';
protected $g2 = 'Image';
protected $Type = 'integer';
protected $Writable = true;
protected $Description = 'Scene Type';
protected $Values = array(
1 => array(
'Id' => 1,
'Label' => 'Directly photographed',
),
);
}
| code |
कप्तान रोहित शर्मा ने वेस्टइंडीज को चेताया, बोले अब वनडे सीरीज शुरू होने का है इंतजार भारतीय India टीम के कप्तान रोहित शर्मा Rohit Sharma को वेस्टइंडीज West Indiesके विरुद्ध 6 फरवरी से शुरू होने वाली तीन मुकाबलों की वनडे सीरीज का बेसब्री से इंतजार है। यह सीरीज उनकी भारत के पूर्णकालिक कप्तान के तौर पर पहली है। हिटमैन बतौर कप्तान खेलने के लिए बहुत उत्साहित हैं। दरअसल, रोहित ने बुधवार को सोशल मीडिया पर एक पोस्ट शेयर करते हुए अपनी भावनाएं व्यक्त की हैं। 34 साल के भारत के सीमित ओवर्स के नियमित कप्तान ने अपने आधिकारिक इंस्टाग्राम अकाउंट पर अपनी तस्वीर साझा करते हुए कैप्शन में लिखा, शुरू होने का इंतजार नहीं कर सकता। इससे पहले रोहित शर्मा 10 वनडे मुकाबलों में भारतीय टीम की कप्तानी कर चुके हैं, लेकिन पूर्णकालिक कप्तानी की जिम्मेदारी सौंपे जाने के बाद यह उनका पहला मौका होगा, जब वह टीम का नेतृत्व करते हुए नज़र आएंगे। See more बता दें कि कैरेबियाई टीम के विरुद्ध तीन मुकाबलों की वनडे सीरीज ODI Series के आगाज से पहले स्टार स्पोर्ट्स ने एक नया प्रोमो वीडियो जारी किया था, जिसमें रोहित शर्मा की कप्तानी में टीम इंडिया Team India के नए युग की शुरुआत की एक झलक दिखाई थी। इसमें कप्तान रोहित के लिए एक खास रैप भी था और इसपर हिटमैन ने भी रिएक्शन दिया था। मास्टर ब्लास्टर ने रोहित शर्मा और राहुल द्रविड़ की जोड़ी का समर्थन किया, फैंस को दिया खास संदेश वेस्टइंडीज के खिलाफ आगामी वनडे सीरीज के लिए भारत की टीम इस प्रकार है रोहित शर्मा कप्तान, केएल राहुल उपकप्तान, रुतुराज गायकवाड़, शिखर धवन, विराट कोहली, सूर्यकुमार यादव, श्रेयस अय्यर, दीपक हुड्डा, ऋषभ पंत विकेटकीपर, दीपक चाहर, शार्दुल ठाकुर, युजवेंद्र चहल, कुलदीप यादव, वॉशिंगटन सुंदर, रवि बिश्नोई, मोहम्मद सिराज, प्रसिद्ध कृष्णा, अवेश खान | hindi |
ವೃತ್ತಿಯಲ್ಲಿ ಶಿಕ್ಷಕರಲ್ಲದಿದ್ದರೂ ಬಡಮಕ್ಕಳ ಶಿಕ್ಷಣಕ್ಕೆ ನೆರವಾಗುತ್ತಿರುವ ಶಿವಣ್ಣ ಉಡುಪಿ : ಇವರು ವೃತ್ತಿಯಲ್ಲಿ ಶಿಕ್ಷಕರಲ್ಲ. ಆದರೆ ಅದೆಷ್ಟೋ ಮಕ್ಕಳ ಶಿಕ್ಷಣಕ್ಕೆ ನೆರವಾಗಿದ್ದಾರೆ . ಜಾತಿ ಮತ ಬೇಧವಿಲ್ಲದೆ ತನ್ನ ಪುಟ್ಟ ಅಂಗಡಿಯಲ್ಲಿ ಹಲವು ಮಕ್ಕಳಿಗೆ ಊಟ ವಸತಿಕೊಟ್ಟು ಬೆಳೆಸಿದ್ದಾರೆ. ಬಿಡುವಿನಲ್ಲಿ ಪಕ್ಕದ ಶಾಲಗೆ ಹೋಗಿ ಮಕ್ಕಳಿಗೆ ಪಾಠ ಮಾಡುತ್ತಾರೆ. ಇವರ ಹೆಸರು ಶಿವಾನಂದ ಕಾಮತ್ ಉಡುಪಿ ಜಿಲ್ಲೆಯ ಶಿರ್ವದವರು. ಶಿರ್ವ ಚಿಕ್ಕ ಪ್ರದೇಶವಾದ್ರೂ ಅರಬ್ ದೇಶಗಳ ವ್ಯವಹಾರದ ಮಟ್ಟಿಗೆ ದೊಡ್ಡ ಹೆಸರು. ಇಲ್ಲಿ ಇರೋದು ಒಂದೇ ಸಾಲು ಪೇಟೆ ಇಲ್ಲಿ ಹೊಸ ಅಂಗಡಿ ಅಂತ ಬಟ್ಟೆ ಬರೆ ಹೊಲಿಗೆ ಸಾಮಾಗ್ರಿ ಸ್ಟೇಶನರಿ ಅಂಗಡಿ ಇದೆ. ಇದರ ಮಾಲಕರೇ ಈ ಶಿವಾನಂದ ಕಾಮತ್ ಎಲ್ರೂ ಶಿವಣ್ಣಾ ಅಂತಾನೇ ಕರೀತಾರೆ. ತಮಗಿದ್ದ ಒಬ್ಬ ಮಗ ಬೆಂಗಳೂರಿನಲ್ಲಿ ಉದ್ಯೋಗಿ ಆಗಿದ್ದಾನೆ. ಆದ್ರೆ ಈ ಶಿವಣ್ಣ ದಂಪತಿಗಳು ಅದೆಷ್ಟೋ ಬಡ ಮಕ್ಕಳು ಶಿಕ್ಷಣದಿಂದ ವಂಚಿತರಾದವರನ್ನು ಕರೆಸಿ ತಮ್ಮ ಮನೆಯಲ್ಲಿ ಉಚಿತ ಊಟ ವಸತಿ ನೀಡಿ ಶಿಕ್ಷಣ ನೀಡುತ್ತಾ ಬಂದಿದ್ದಾರೆ. ಬಿಡುವಿನ ಸಮಯದಲ್ಲಿ ಆ ಮಕ್ಕಳು ಇವರ ಅಂಗಡಿಯಲ್ಲಿ ಅವರದಾದ ಸೇವೆ ಮಾಡುತ್ತಾರೆ. ಅದು ಕಡ್ಡಾಯ ಏನಲ್ಲ. ಶಿವಣ್ಣ ಅವರ ಪತ್ನಿಯನ್ನು ಈ ಮಕ್ಕಳು ಅಮ್ಮ ಅಂತನೇ ಕರೀತಾರೆ. ಇವರಲ್ಲಿ ಕಲಿತ ಮಕ್ಕಳು ಕೆಲವರು ದೊಡ್ಡ ದೊಡ್ಡ ಹುದ್ದೆಗಳಲ್ಲಿದ್ದಾರೆ. ಕೆಲವರು ಬೆಂಗಳೂರಿನಲ್ಲಿದ್ದಾರೆ . ಇನ್ನು ಕೆಲವರು ಇಲ್ಲೇ ಹೋಟೇಲು ಕ್ಯಾಂಟೀನು ನಡೆಸಿಕೊಂಡಿದ್ದಾರೆ. ಎಲ್ಲಾ ಜಾತಿ ಧರ್ಮದ ಶಿಕ್ಷಣ ವಂಚಿತ ಮಕ್ಕಳನ್ನು ಶಿವಣ್ಣ ದಂಪತಿ ಪ್ರೀತಿಯಿಂದ ಸಾಕಿ ಶಿಕ್ಷಣ ನೀಡಿ ಆದರ್ಶ ಪ್ರಾಯರಾಗಿದ್ದಾರೆ. ಇಷ್ಟೇ ಅಲ್ಲದೆ ಸದಾ ಸರಳ ಸ್ವಭಾವದ ಶಿವಣ್ಣ ಎಲ್ಲರಿಗೂ ಅಚ್ಚುಮೆಚ್ಚು. ತಾನೊಬ್ಬ ಶಿಕ್ಷಕನಾಗಬೇಕು ಎಂಬ ಆಸೆ ಇಟ್ಕೊಂಡಿದ್ರಂತೆ. ಆದ್ರೆ ಅದು ಆಗಿಲ್ಲ ಅನ್ನೋ ಕಾರಣದಿಂದ ತನ್ನ ಮನೆಯಲ್ಲಿಯೇ ಅದೆಷ್ಟೋ ಮಕ್ಕಳಿಗೆ ಆಶ್ರಯ ನೀಡಿ ಶಿಕ್ಷಣ ನೀಡುತ್ತಿದ್ದಾರೆ. ಜೊತೆಗೆ ಬಿಡುವಿದ್ದಾಗ ಸ್ಥಳೀಯ ಹಿಂದೂ ಪ್ರಾಥಮಿಕ ಶಾಲೆಗೆ ಹೋಗಿ ಅಲ್ಲಿ ಮಕ್ಕಳಿಗೆ ಪಾಠ ಮಾಡ್ತಾರೆ ಭಜನೆ ಹಾಡ್ತಾರೆ ಅವರ ಜೊತೆ ಖುಷಿಯಾಗಿ ಬೆರೆತು ಸಂತೋಷ ಪಡ್ತಾರೆ ಈ ಕಾರಣದಿಂದ ಅವರನ್ನು ಮಕ್ಕಳು ಪ್ರೀತಿಯಿಂದ ಯಾವಾಗ ಬರ್ತೀರಿ ಸರ್ ಅಂತಾ ಕೇಳ್ತಾರಂತೆ. ಹೀಗೆ ತಮ್ಮ ಬದುಕಿನ ಗಳಿಕೆಯನ್ನು ಇತರರ ಏಳಿಗೆಗೆ ವಿನಿಯೋಗಿಸುವ ಈ ದಂಪತಿಗಳ ಶಿಕ್ಷಣ ಸೇವೆ ಮಾದರಿಯಾಗಿದೆ. ಇಂದಿನ ದಿನಗಳಲ್ಲಿ ಫೀಸ್ ಡೊನೇಶನ್ ಅಂತಾ ಮನೆ ಮಕ್ಕಳಿಗೆ ಶಿಕ್ಷಣ ಕೊಡೋಕೇ ಕಷ್ಟದ ಕಾಲ ಇದು. ಅದ್ರಲ್ಲೂ ಎಲ್ಲರೂ ಶಿಕ್ಷಣ ಪಡೀಬೇಕು ಅದಕ್ಕೆ ನಮ್ಮಿಂದಾದ ಸಹಾಯ ನಾವೆಲ್ಲರೂ ಮಾಡಬೇಕು ಅನ್ನೋ ಮನೋಭಾವ ತೋರ್ಪಡಿಸುವ ಶಿವಣ್ಣ ದಂಪತಿಗಳಿಗೆ ಒಂದು ವಿಶೇಷ ನಮಸ್ಕಾರ. Video: | kannad |
A detached house is located in the village of Widna Góra in the Kashubian lake district near Gowidliński Lake and on the tourist Route of Stone Circles. On the ground floor there is a living room with a fireplace, a kitchenette and a bathroom. Upstairs there are 2 bedrooms. At your disposal there are also a private terrace and a barbecue. Our comfortable house can accommodate up to 4 people.
During your stay you can use fishing rods, a pontoon, as well as a heating umbrella (a bottle and umbrella rental 50 PLN / day). You can also purchase a fishing card. | english |
Pratap Bose, Mudra’s recently appointed CEO wished his wife in the most unique and loud way you could imagine. He selected to use his domain – the outdoor medium to wish his wife Seema a very happy birthday.
Read the full article at agencyfaqs.
He further goes to add that this could be done by anybody who is willing to pay about Rs.20,000 – 30,000 for the outdoor display. I just spoke to my media team and checked if I could do a ticker on one of the TV channels wishing my wife? Sounds interesting but VERY expensive so I’ll just make do with a greeting card. | english |
Subsets and Splits