text:
Friede-Abrahamson Genealogy Site
Martha Elizabeth Rafferty
F, b. June 1849
Father* William Rafferty b. c 1810, d. 2 May 1873
Mother* Alice (?) b. c 1821, d. 29 Dec 1891
Married Name 14 November 1869 As of 14 November 1869,her married name was Denny.
Birth June 1849 Martha Elizabeth Rafferty was born in June 1849 at Ireland.
She was the daughter of William Rafferty and Alice (?).
Marriage 14 November 1869 Martha Elizabeth Rafferty married Frank Darno Denny on 14 November 1869 at Leelanau, Michigan, also reported as Oct 1869.
27 July 1870 Empire, Leelanau, Michigan Martha Elizabeth Rafferty was listed in Frank Darno Denny's household on the 1870 Census at Empire, Leelanau, Michigan.1
11 June 1880 Empire, Leelanau, Michigan Martha Elizabeth Rafferty was listed as wife in Frank Darno Denny's household on the 1880 Census at Empire, Leelanau, Michigan.2
27 June 1900 Empire, Leelanau, Michigan Martha Elizabeth Rafferty was listed as the head of a family on the 1900 Census at Empire, Leelanau, Michigan.3
Martha Elizabeth Rafferty had 5 with 5 living in 1900 children.
Children of Martha Elizabeth Rafferty and Frank Darno Denny
Frank Denny+ b. 7 Jan 1871, d. 15 Dec 1955
Edward Denny b. May 1872
Mamie Denny b. c 1875
William Denny b. Jul 1875
Charles Denny b. May 1879
[S247] U.S. Federal Census 1870 U.S. Federal Census, by Ancestry.com, Stamp 319. Dwelling 55.
[S246] U.S. Federal Census 1880 U.S. Federal Census, by Ancestry.com, NA Film Number T9-0589 Page Number 492C. ED160. Dwelling 141.
[S245] U.S. Federal Census 1900 U.S. Federal Census, by Ancestry.com, ED21. Dwelling 219.
Edward Denny
M, b. May 1872
Father* Frank Darno Denny b. c 1845, d. b 1900
Mother* Martha Elizabeth Rafferty b. Jun 1849
Birth May 1872 Edward Denny was born in May 1872 at Michigan.
He was the son of Frank Darno Denny and Martha Elizabeth Rafferty.
Marriage circa 1929 Edward Denny married Evelyn N. (?) circa 1929.
11 June 1880 Empire, Leelanau, Michigan Edward Denny was listed as a son in Frank Darno Denny's household on the 1880 Census at Empire, Leelanau, Michigan.1
27 June 1900 Empire, Leelanau, Michigan Edward Denny was listed as a son in Martha Elizabeth Rafferty's household on the 1900 Census at Empire, Leelanau, Michigan.2
4 April 1930 Empire, Leelanau, Michigan Edward Denny was listed as the head of a family on the 1930 Census at Empire, Leelanau, Michigan.3
[S250] U.S. Federal Census 1930 U.S. Federal Census, by Ancestry.com, ED5. Dwelling 38.
Charles Denny
Name Variation Charles Denny was also known as Charley.
Birth May 1879 Charles Denny was born in May 1879 at Michigan.
11 June 1880 Empire, Leelanau, Michigan Charles Denny was listed as a son in Frank Darno Denny's household on the 1880 Census at Empire, Leelanau, Michigan.1
27 June 1900 Empire, Leelanau, Michigan Charles Denny was listed as a son in Martha Elizabeth Rafferty's household on the 1900 Census at Empire, Leelanau, Michigan.2
Harry Edward Wenzel
M, b. 22 June 1899, d. 1 May 1975
Father* August Wenzel b. 13 Jun 1864, d. 15 Dec 1947
Mother* Caroline Heberlein b. 7 Apr 1866, d. 8 Nov 1934
Birth 22 June 1899 Harry Edward Wenzel was born on 22 June 1899 at Ellenboro, Grant, Wisconsin.1
He was the son of August Wenzel and Caroline Heberlein.
Marriage circa 1925 Harry Edward Wenzel married Alva M. White circa 1925.
Marriage after 1935 Harry Edward Wenzel married Lucille J. Wehnke, daughter of Edward Eide Julius Wehnke and Nellie Laura Rosina Harms, after 1935.
Death 1 May 1975 Harry Edward Wenzel died on 1 May 1975 at Cuba City, Grant, Wisconsin, at age 75.
5 June 1900 Ellenboro, Grant, Wisconsin Harry Edward Wenzel was listed as a son in August Wenzel's household on the 1900 Census at Ellenboro, Grant, Wisconsin.2
30 April 1910 Ellenboro, Grant, Wisconsin Harry Edward Wenzel was listed as a son in August Wenzel's household on the 1910 Census at Ellenboro, Grant, Wisconsin.3
18 April 1940 Platteville, Grant, Wisconsin Harry Edward Wenzel was listed as the head of a family on the 1940 Census at Platteville, Grant, Wisconsin.4
On 11 September 1918 at World War I Draft Registration, Grant, Wisconsin.
[S358] Wisconsin Pre-1907 Vital Records (published), Birth Record ID: 3280614.
[S245] U.S. Federal Census 1900 U.S. Federal Census, by Ancestry.com, ED30. Dwelling 32.
[S373] U.S. Federal Census 1940 U.S. Federal Census, by Ancestry.com, Year: 1940; Census Place: Platteville, Grant, Wisconsin; Roll: T627_4481; Page: 2B; Enumeration District: 22-44.
Alva M. White
F, b. circa 1900
Married Name circa 1925 As of circa 1925,her married name was Wenzel.
Birth circa 1900 Alva M. White was born circa 1900 at Wisconsin.
Marriage circa 1925 She married Harry Edward Wenzel, son of August Wenzel and Caroline Heberlein, circa 1925.
Dietrich Friederich Gehrke
M, b. 1 February 1850, d. 24 September 1909
Father* Dietrich Albert Gehrke b. 19 Nov 1804, d. 8 May 1850
Name Variation Dietrich Friederich Gehrke was also known as Dick F.
Name Variation Dietrich Friederich Gehrke was also known as Deid.
Name Variation Dietrich Friederich Gehrke was also known as Richard.
Name Variation Dietrich Friederich Gehrke was also known as Diet.
Name Variation Dietrich Friederich Gehrke was also known as Deidrick F.
Name Variation Dietrich Friederich Gehrke was also known as Gerke.
Birth 1 February 1850 Dietrich Friederich Gehrke was born on 1 February 1850 at Hanover, Germany.
He was the son of Dietrich Albert Gehrke.
Marriage 26 July 1878 Dietrich Friederich Gehrke married Anna Christine Bevers, daughter of Nicholas John Bevers and Johanne Helene Mehren, on 26 July 1878 at Grant, Wisconsin.1
Death 24 September 1909 Dietrich Friederich Gehrke died on 24 September 1909 at Lafayette, Wisconsin, at age 59.
Burial after 24 September 1909 He was buried after 24 September 1909 at Belmont Cemetery, Belmont, Lafayette, Wisconsin.
Dietrick F and Anna K Gehrke tombstone
5 June 1870 Platteville, Grant, Wisconsin Dietrich Friederich Gehrke was listed as farm worker for the Jonathan C. Stephens family on the 1870 Census at Platteville, Grant, Wisconsin.2
1 June 1880 Belmont, La Fayette, Wisconsin He was listed as the head of a family on the 1880 Census at Belmont, La Fayette, Wisconsin.3
20 June 1895 Belmont, Lafayette, Wisconsin He appeared on the census of 20 June 1895 at Belmont, Lafayette, Wisconsin, 5 males and 5 females in household with 1 living in Germany (Deid).4
9 June 1900 Belmont, Lafayette, Wisconsin He was listed as the head of a family on the 1900 Census at Belmont, Lafayette, Wisconsin.5
1 June 1905 Belmont, Lafayette, Wisconsin He appeared on the census of 1 June 1905 at Belmont, Lafayette, Wisconsin.6
Dietrich Friederich Gehrke immigrated in 1857.
Children of Dietrich Friederich Gehrke and Anna Christine Bevers
John N. Gehrke+ b. May 1879, d. 11 Apr 1963
Heinrich Wilhelm Gehrke b. 30 Aug 1880
Nora A. Gehrke b. Nov 1881, d. Nov 1958
Ford Gehrke b. Apr 1884, d. b 1958
Mamie Gehrke b. Mar 1886, d. 20 Jun 1958
Herman John Gehrke+ b. 27 Oct 1888, d. 28 Apr 1930
Clara H. Gehrke+ b. 26 Feb 1890, d. Mar 1975
Erma M. Gehrke b. 20 Dec 1891, d. 24 Jan 1962
Albert Gert Gehrke+ b. 30 Dec 1893, d. 12 Dec 1969
Alta Catherine Gehrke+ b. 25 May 1896, d. 26 Sep 1962
Dorothy Gehrke b. Mar 1898, d. a 1963
Ethel A. M. Gehrke b. 25 Nov 1900, d. 1962
[S5] Wisconsin, Wisconsin Marriages, 1835-1900.
[S247] U.S. Federal Census 1870 U.S. Federal Census, by Ancestry.com, Page 25. Stamp 373. Dwelling.
[S246] U.S. Federal Census 1880 U.S. Federal Census, by Ancestry.com, NA Film Number T9-1433 Page Number 23A.
[S364] Wisconsin State Census Wisconsin State Censuses, 1895 and 1905, by Ancestry.com.
[S364] Wisconsin State Census Wisconsin State Censuses, 1895 and 1905, by Ancestry.com, Dwelling 62.
Anna Christine Bevers
F, b. 27 December 1858, d. 21 March 1942
Father* Nicholas John Bevers b. c 1832, d. a Apr 1910
Mother* Johanne Helene Mehren b. c 1838, d. a Apr 1910
Name Variation Anna Christine Bevers was also known as Anna Kristine.
Name Variation Anna Christine Bevers was also known as Lizzie.
Name Variation Anna Christine Bevers was also known as Ana Katharine.
Name Variation Anna Christine Bevers was also known as Beavers.
Name Variation Anna Christine Bevers was also known as Annie.
Married Name 26 July 1878 As of 26 July 1878,her married name was Gehrke.
Birth 27 December 1858 Anna Christine Bevers was born on 27 December 1858 at Grant, Wisconsin.
She was the daughter of Nicholas John Bevers and Johanne Helene Mehren.
Baptism 6 January 1859 Anna Christine Bevers was baptized on 6 January 1859 at St Paul Evangelical Lutheran Church, Platteville, Grant, Wisconsin.
Marriage 26 July 1878 She married Dietrich Friederich Gehrke, son of Dietrich Albert Gehrke, on 26 July 1878 at Grant, Wisconsin.1
Death 21 March 1942 Anna Christine Bevers died on 21 March 1942 at age 83.
Burial after 21 March 1942 She was buried after 21 March 1942 at Belmont Cemetery, Belmont, Lafayette, Wisconsin.
12 June 1860 Platteville, Grant, Wisconsin Anna Christine Bevers was listed in Nicholas John Bevers's household on the 1860 Census at Platteville, Grant, Wisconsin.2
23 June 1870 Platteville, Grant, Wisconsin Anna Christine Bevers was listed in Nicholas John Bevers's household on the 1870 Census at Platteville, Grant, Wisconsin; living next door to Henry Gerke.3
1 June 1880 Belmont, La Fayette, Wisconsin Anna Christine Bevers was listed as wife in Dietrich Friederich Gehrke's household on the 1880 Census at Belmont, La Fayette, Wisconsin.4
9 June 1900 Belmont, Lafayette, Wisconsin Anna Christine Bevers was listed as wife in Dietrich Friederich Gehrke's household on the 1900 Census at Belmont, Lafayette, Wisconsin.5
1 June 1905 Belmont, Lafayette, Wisconsin Anna Christine Bevers is listed as wife on the census of 1 June 1905 in the household of Dietrich Friederich Gehrke at Belmont, Lafayette, Wisconsin.6
3 May 1910 Belmont, Lafayette, Wisconsin Anna Christine Bevers was listed as the head of a family on the 1910 Census at Belmont, Lafayette, Wisconsin, living next door to Nora & Hans Speth.7
January 1920 Belmont, Lafayette, Wisconsin She was listed as the head of a family on the 1920 Census at Belmont, Lafayette, Wisconsin.8
1930 Belmont, Lafayette, Wisconsin She was listed as the head of a family on the 1930 Census at Belmont, Lafayette, Wisconsin, name listed as Lizzie.9
1940 Belmont, Lafayette, Wisconsin She was listed as the head of a family on the 1940 Census at Belmont, Lafayette, Wisconsin.10
Anna Christine Bevers had 12 with 10 living in 1900 children.
She had 13 with 10 living in 1910 children.
Children of Anna Christine Bevers and Dietrich Friederich Gehrke
[S248] U.S. Federal Census 1860 U.S. Federal Census, by Ancestry.com, Page 21. Handwritten number 55. Dwelling 180.
[S247] U.S. Federal Census 1870 U.S. Federal Census, by Ancestry.com, Page 56. Stamp 405. Dwelling 407.
[S243] U.S. Federal Census 1920 U.S. Federal Census, by Ancestry.com, Year: 1920;Census Place: Belmont, Lafayette, Wisconsin; Roll: T625_1992; Page: 2A; Enumeration District: 173; Image: 1026.
[S250] U.S. Federal Census 1930 U.S. Federal Census, by Ancestry.com, Year: 1930; Census Place: Belmont, Lafayette, Wisconsin; Roll: 2579; Page: 5A; Enumeration District: 4; Image: 730.0; FHL microfilm: 2342313.
[S373] U.S. Federal Census 1940 U.S. Federal Census, by Ancestry.com, Year: 1940; Census Place: Belmont, Lafayette, Wisconsin; Roll: T627_4492; Page: 6A; Enumeration District: 33-5.
Nora A. Gehrke
F, b. November 1881, d. November 1958
Father* Dietrich Friederich Gehrke b. 1 Feb 1850, d. 24 Sep 1909
Mother* Anna Christine Bevers b. 27 Dec 1858, d. 21 Mar 1942
Name Variation Nora A. Gehrke was also known as Nora J.
Married Name 10 February 1904 As of 10 February 1904,her married name was Speth.
Married Name after 1928 As of after 1928,her married name was Nodols.
Married Name after 1928 As of after 1928,her married name was Nodolf.
Birth November 1881 Nora A. Gehrke was born in November 1881 at Wisconsin.
She was the daughter of Dietrich Friederich Gehrke and Anna Christine Bevers.
Marriage 10 February 1904 Nora A. Gehrke married Hans Paul Speth, son of Hans Adolph Speth and Sophia Marie Sievers, on 10 February 1904 at Lafayette, Wisconsin.
Marriage after 1928 Nora A. Gehrke married Ed Nodolf after 1928.
Death November 1958 Nora A. Gehrke died in November 1958.
Burial 16 November 1958 She was buried on 16 November 1958 at Hillside Cemetery, Platteville, Grant, Wisconsin.
9 June 1900 Belmont, Lafayette, Wisconsin Nora A. Gehrke was listed as a daughter in Dietrich Friederich Gehrke's household on the 1900 Census at Belmont, Lafayette, Wisconsin.1
1 June 1905 Belmont, Lafayette, Wisconsin Nora A. Gehrke appeared on the census of 1 June 1905 in the household of Hans Paul Speth at Belmont, Lafayette, Wisconsin.
3 May 1910 Belmont, Lafayette, Wisconsin Nora A. Gehrke was listed as wife in Hans Paul Speth's household on the 1910 Census at Belmont, Lafayette, Wisconsin; living next door to Ann Gehrke.2
1920 Belmont, Lafayette, Wisconsin Nora A. Gehrke was listed as wife in Hans Paul Speth's household on the 1920 Census at Belmont, Lafayette, Wisconsin.3
[S243] U.S. Federal Census 1920 U.S. Federal Census, by Ancestry.com, ED173. Dwelling 13.
Ford Gehrke
M, b. April 1884, d. before 1958
Name Variation Ford Gehrke was also known as Fred.
Birth April 1884 Ford Gehrke was born in April 1884 at Wisconsin.
He was the son of Dietrich Friederich Gehrke and Anna Christine Bevers.
Death before 1958 Ford Gehrke died before 1958.
9 June 1900 Belmont, Lafayette, Wisconsin Ford Gehrke was listed as a son in Dietrich Friederich Gehrke's household on the 1900 Census at Belmont, Lafayette, Wisconsin.1
Mamie Gehrke
F, b. March 1886, d. 20 June 1958
Married Name Her married name was Schroeder.
Name Variation Mamie Gehrke was also known as Mary.
Marriage Mamie Gehrke married Henry Schroeder.
Birth March 1886 Mamie Gehrke was born in March 1886 at Wisconsin.
Death 20 June 1958 Mamie Gehrke died on 20 June 1958 at age 72 Mrs Henry Schroeder (Mamie Gehrke) obituary.
Dubuque Telegraph Herald June 22, 1958 - Mrs Henry Schroeder (Mamie Gehrke) obituary
Burial after 20 June 1958 She was buried after 20 June 1958 at Hillside Cemetery, Platteville, Grant, Wisconsin.
9 June 1900 Belmont, Lafayette, Wisconsin Mamie Gehrke was listed as a daughter in Dietrich Friederich Gehrke's household on the 1900 Census at Belmont, Lafayette, Wisconsin.1
1 June 1905 Belmont, Lafayette, Wisconsin Mamie Gehrke is listed as daughter on the census of 1 June 1905 in the household of Dietrich Friederich Gehrke at Belmont, Lafayette, Wisconsin.2
Herman John Gehrke
M, b. 27 October 1888, d. 28 April 1930
Harry John Henry Gehrke signature 1917
Name Variation Herman John Gehrke was also known as Hermann John.
Name Variation Herman John Gehrke was also known as Harry John Henry.
Birth 27 October 1888 Herman John Gehrke was born on 27 October 1888 at Belmont, Lafayette, Wisconsin.
Marriage before 1916 Herman John Gehrke married Letha Shaw before 1916.
Marriage circa 1923 Herman John Gehrke married Amy M. Bockhop circa 1923.
Death 28 April 1930 Herman John Gehrke died on 28 April 1930 at age 41 Harry Gehrke, 40, suicide.
Harry Gehrke, 40, suicide article 29 Apr 1930 The Capital Times (Madison, WI)
Burial after 28 April 1930 He was buried after 28 April 1930 at Belmont Cemetery, Belmont, Lafayette, Wisconsin.
9 June 1900 Belmont, Lafayette, Wisconsin Herman John Gehrke was listed as a son in Dietrich Friederich Gehrke's household on the 1900 Census at Belmont, Lafayette, Wisconsin.1
1 June 1905 Belmont, Lafayette, Wisconsin Herman John Gehrke appeared on the census of 1 June 1905 in the household of Hans Paul Speth at Belmont, Lafayette, Wisconsin.
3 May 1910 Belmont, Lafayette, Wisconsin Herman John Gehrke was listed as a son in Anna Christine Bevers's household on the 1910 Census at Belmont, Lafayette, Wisconsin; living next door to Nora & Hans Speth.2
7 January 1920 Elk Grove, Lafayette, Wisconsin Herman John Gehrke was listed as the head of a family on the 1920 Census at Elk Grove, Lafayette, Wisconsin.3
1930 Grant, Wisconsin He was listed as the head of a family on the 1930 Census at Grant, Wisconsin.4
On 5 June 1917 at Lafayette, Wisconsin, WWI Draft Registration.
Gehrke, Harry John Henry - World War I Draft Registration Card
Children of Herman John Gehrke and Letha Shaw
Alice Gehrke b. 23 Mar 1915, d. 2 Sep 2002
Cleo Lorene Gehrke b. 1916, d. 13 Jul 1994
Jeannette Lenice Gehrke b. 1918, d. 1992
Willard H. Gehrke b. 21 Jan 1920, d. 1 Oct 2008
Robert R. Gehrke b. 4 Jul 1921, d. 19 Jan 2004
Children of Herman John Gehrke and Amy M. Bockhop
Mildred M. Gehrke b. 2 Jul 1924, d. 12 Nov 1990
Charles A. Gehrke+ b. 7 Dec 1927, d. 2 Dec 1989
[S250] U.S. Federal Census 1930 U.S. Federal Census, by Ancestry.com.
Clara H. Gehrke
F, b. 26 February 1890, d. March 1975
Name Variation Clara H. Gehrke was also known as Carie.
Married Name before 1913 As of before 1913,her married name was Popp.
Birth 26 February 1890 Clara H. Gehrke was born on 26 February 1890 at Wisconsin.
Marriage before 1913 Clara H. Gehrke married William Christ Popp, son of William Popp, before 1913.
Burial March 1975 Clara H. Gehrke was buried in March 1975 at Belmont Cemetery, Belmont, Lafayette, Wisconsin.
Death March 1975 She died in March 1975 at age 85.
9 June 1900 Belmont, Lafayette, Wisconsin Clara H. Gehrke was listed as a daughter in Dietrich Friederich Gehrke's household on the 1900 Census at Belmont, Lafayette, Wisconsin.1
1 June 1905 Belmont, Lafayette, Wisconsin Clara H. Gehrke is listed as daughter on the census of 1 June 1905 in the household of Dietrich Friederich Gehrke at Belmont, Lafayette, Wisconsin.2
2 March 1920 Kendall, Lafayette, Wisconsin Clara H. Gehrke was listed as wife in William Christ Popp's household on the 1920 Census at Kendall, Lafayette, Wisconsin.3
Children of Clara H. Gehrke and William Christ Popp
Esther N. Popp b. 24 Feb 1912, d. 5 Jan 2007
Helen A. Popp+ b. 24 Apr 1914, d. 7 Apr 2010
William J. Popp+ b. 20 Sep 1917, d. 22 Feb 2008
[S243] U.S. Federal Census 1920 U.S. Federal Census, by Ancestry.com, Year: 1920;Census Place: Kendall, Lafayette, Wisconsin; Roll: T625_1993; Page: 2A; Enumeration District: 183; Image: 66.
Erma M. Gehrke
F, b. 20 December 1891, d. 24 January 1962
Name Variation Erma M. Gehrke was also known as Amy.
Married Name Her married name was Popp.
Name Variation Erma M. Gehrke was also known as Irene.
Marriage Erma M. Gehrke married John Henry Popp, son of William Popp.
Birth 20 December 1891 Erma M. Gehrke was born on 20 December 1891 at Wisconsin.
Death 24 January 1962 Erma M. Gehrke died on 24 January 1962 at age 70.
Burial after 24 January 1962 She was buried after 24 January 1962 at Belmont Cemetery, Belmont, Lafayette, Wisconsin.
9 June 1900 Belmont, Lafayette, Wisconsin Erma M. Gehrke was listed as a daughter in Dietrich Friederich Gehrke's household on the 1900 Census at Belmont, Lafayette, Wisconsin.1
1 June 1905 Belmont, Lafayette, Wisconsin Erma M. Gehrke is listed as daughter on the census of 1 June 1905 in the household of Dietrich Friederich Gehrke at Belmont, Lafayette, Wisconsin.2
3 May 1910 Belmont, Lafayette, Wisconsin Erma M. Gehrke was listed as a daughter in Anna Christine Bevers's household on the 1910 Census at Belmont, Lafayette, Wisconsin; living next door to Nora & Hans Speth.3
Alta Catherine Gehrke
F, b. 25 May 1896, d. 26 September 1962
Married Name 5 December 1917 As of 5 December 1917,her married name was Richter.
Birth 25 May 1896 Alta Catherine Gehrke was born on 25 May 1896 at Wisconsin.
Marriage 5 December 1917 Alta Catherine Gehrke married Bernard Otto Richter, son of Andrew John Richter and Mary Fure, on 5 December 1917.
Death 26 September 1962 Alta Catherine Gehrke died on 26 September 1962 at Platteville, Grant, Wisconsin, at age 66 Alta (Gehrke) Richter obituary.
Dubuque Telegraph Herald September 27, 1962 - Alta (Gehrke) Richter obituary
Burial after 26 September 1962 She was buried after 26 September 1962 at Hillside Cemetery, Platteville, Grant, Wisconsin.
9 June 1900 Belmont, Lafayette, Wisconsin Alta Catherine Gehrke was listed as a daughter in Dietrich Friederich Gehrke's household on the 1900 Census at Belmont, Lafayette, Wisconsin.1
1 June 1905 Belmont, Lafayette, Wisconsin Alta Catherine Gehrke is listed as daughter on the census of 1 June 1905 in the household of Dietrich Friederich Gehrke at Belmont, Lafayette, Wisconsin.2
3 May 1910 Belmont, Lafayette, Wisconsin Alta Catherine Gehrke was listed as a daughter in Anna Christine Bevers's household on the 1910 Census at Belmont, Lafayette, Wisconsin; living next door to Nora & Hans Speth.3
22 April 1930 Platteville, Grant, Wisconsin Alta Catherine Gehrke was listed as wife in Bernard Otto Richter's household on the 1930 Census at Platteville, Grant, Wisconsin.4
Children of Alta Catherine Gehrke and Bernard Otto Richter
Anna Mae Richter+ b. c 1919, d. Oct 1949
Margaret M. Richter+ b. c 1921, d. 19 Feb 2002
[S250] U.S. Federal Census 1930 U.S. Federal Census, by Ancestry.com, Year: 1930; Census Place: Platteville, Grant, Wisconsin; Roll: 2574; Page: 3B; Enumeration District: 40; Image: 608.0; FHL microfilm: 2342308.
Dorothy Gehrke
F, b. March 1898, d. after 1963
Name Variation Dorothy Gehrke was also known as Dora.
Married Name Her married name was Martens.
Marriage Dorothy Gehrke married Clarence Martens.
Birth March 1898 Dorothy Gehrke was born in March 1898 at Wisconsin.
Death after 1963 Dorothy Gehrke died after 1963.
9 June 1900 Belmont, Lafayette, Wisconsin Dorothy Gehrke was listed as a daughter in Dietrich Friederich Gehrke's household on the 1900 Census at Belmont, Lafayette, Wisconsin.1
1 June 1905 Belmont, Lafayette, Wisconsin Dorothy Gehrke is listed as daughter on the census of 1 June 1905 in the household of Dietrich Friederich Gehrke at Belmont, Lafayette, Wisconsin.2
3 May 1910 Belmont, Lafayette, Wisconsin Dorothy Gehrke was listed as a daughter in Anna Christine Bevers's household on the 1910 Census at Belmont, Lafayette, Wisconsin; living next door to Nora & Hans Speth.3
January 1920 Belmont, Lafayette, Wisconsin Dorothy Gehrke was listed as a daughter in Anna Christine Bevers's household on the 1920 Census at Belmont, Lafayette, Wisconsin.4
Ethel A. M. Gehrke
F, b. 25 November 1900, d. 1962
Married Name 28 April 1919 As of 28 April 1919,her married name was Havry.
Married Name 28 April 1919 As of 28 April 1919,her married name was Houy.1
Married Name after 1920 As of after 1920,her married name was Smith.
Birth 25 November 1900 Ethel A. M. Gehrke was born on 25 November 1900 at Lafayette, Wisconsin.2
Marriage 28 April 1919 Ethel A. M. Gehrke married George Robert Houy, son of George D. Houy and unknown (?), on 28 April 1919 at Minnehaha, South Dakota, her residence: Belmont, Lafayette, Wisconsin; his residence: Galena, Jo Daviess, Illinois.1
Marriage after 1920 Ethel A. M. Gehrke married Van Byron Smith after 1920.
Burial 1962 Ethel A. M. Gehrke was buried in 1962 at Belmont Cemetery, Belmont, Lafayette, Wisconsin.
Death 1962 She died in 1962.
1 June 1905 Belmont, Lafayette, Wisconsin Ethel A. M. Gehrke is listed as daughter on the census of 1 June 1905 in the household of Dietrich Friederich Gehrke at Belmont, Lafayette, Wisconsin.3
3 May 1910 Belmont, Lafayette, Wisconsin Ethel A. M. Gehrke was listed as a daughter in Anna Christine Bevers's household on the 1910 Census at Belmont, Lafayette, Wisconsin; living next door to Nora & Hans Speth.4
January 1920 Belmont, Lafayette, Wisconsin Ethel A. M. Gehrke was listed as a daughter in Anna Christine Bevers's household on the 1920 Census at Belmont, Lafayette, Wisconsin.5
[S342] South Dakota, South Dakota Marriage Certificates, 1905-1949.
John N. Gehrke
M, b. May 1879, d. 11 April 1963
Birth May 1879 John N. Gehrke was born in May 1879 at Wisconsin.
Marriage 15 August 1906 John N. Gehrke married Wilhelmina Minnie Bose, daughter of Dietrich Bose and Sophie (?), on 15 August 1906.
Death 11 April 1963 John N. Gehrke died on 11 April 1963 at Weld, Colorado, at age 83 John N Gehrke obituary.
Greeley Daily Tribune April 11, 1963 - John N Gehrke obituary
Burial after 12 April 1963 He was buried after 12 April 1963 at Linn Grove Cemetery, Greeley, Weld, Colorado.
1 June 1880 Belmont, La Fayette, Wisconsin John N. Gehrke was listed as a son in Dietrich Friederich Gehrke's household on the 1880 Census at Belmont, La Fayette, Wisconsin.1
23 June 1900 Syracuse, Otoe, Nebraska John N. Gehrke was listed as a nephew in John Gehrke's household on the 1900 Census at Syracuse, Otoe, Nebraska.2
20 April 1910 Russell, Otoe, Nebraska John N. Gehrke was listed as the head of a family on the 1910 Census at Russell, Otoe, Nebraska.3
6 February 1920 Stratton, Kit Carson, Colorado He was listed as the head of a family on the 1920 Census at Stratton, Kit Carson, Colorado.4
17 April 1930 Precinct 9, Kit Carson, Colorado He was listed as the head of a family on the 1930 Census at Precinct 9, Kit Carson, Colorado.5
Children of John N. Gehrke and Wilhelmina Minnie Bose
Esther A. Gehrke b. 16 Mar 1908, d. 2 Mar 2003
Ruth M. Gehrke b. 8 Jan 1910, d. 1 Dec 2003
Paul Gehrke b. a 1910, d. b 2006
Erma L. Gehrke b. 17 Jul 1911, d. 4 Dec 1991
Lucia Gehrke+ b. 9 Jan 1917, d. 4 Jan 2006
Ruby L. Gehrke b. 22 Sep 1920, d. 1 Jul 1994
[S245] U.S. Federal Census 1900 U.S. Federal Census, by Ancestry.com, ED123. Dwelling 329.
Wilhelmina Minnie Bose
F, b. July 1884, d. 17 October 1967
Father* Dietrich Bose b. Aug 1848, d. b 1930
Mother* Sophie (?) b. Aug 1847, d. a Apr 1930
Name Variation Wilhelmina Minnie Bose was also known as Minnie.
Married Name 15 August 1906 As of 15 August 1906,her married name was Gehrke.
Birth July 1884 Wilhelmina Minnie Bose was born in July 1884 at Nebraska.
She was the daughter of Dietrich Bose and Sophie (?).
Marriage 15 August 1906 Wilhelmina Minnie Bose married John N. Gehrke, son of Dietrich Friederich Gehrke and Anna Christine Bevers, on 15 August 1906.
Death 17 October 1967 Wilhelmina Minnie Bose died on 17 October 1967 at age 83.
Burial after 17 October 1967 She was buried after 17 October 1967 at Linn Grove Cemetery, Greeley, Weld, Colorado.
11 June 1900 Syracuse, Otoe, Nebraska Wilhelmina Minnie Bose was listed as a daughter in Dietrich Bose's household on the 1900 Census at Syracuse, Otoe, Nebraska.1
20 April 1910 Russell, Otoe, Nebraska Wilhelmina Minnie Bose was listed as wife in John N. Gehrke's household on the 1910 Census at Russell, Otoe, Nebraska.2
6 February 1920 Stratton, Kit Carson, Colorado Wilhelmina Minnie Bose was listed as wife in John N. Gehrke's household on the 1920 Census at Stratton, Kit Carson, Colorado.3
17 April 1930 Precinct 9, Kit Carson, Colorado Wilhelmina Minnie Bose was listed as wife in John N. Gehrke's household on the 1930 Census at Precinct 9, Kit Carson, Colorado.4
Wilhelmina Minnie Bose had 2 with 2 living in 1910 children.
Children of Wilhelmina Minnie Bose and John N. Gehrke
Esther A. Gehrke
F, b. 16 March 1908, d. 2 March 2003
Father* John N. Gehrke b. May 1879, d. 11 Apr 1963
Mother* Wilhelmina Minnie Bose b. Jul 1884, d. 17 Oct 1967
Married Name circa 1929 As of circa 1929,her married name was Scheierman.
Birth 16 March 1908 Esther A. Gehrke was born on 16 March 1908 at Nebraska.
She was the daughter of John N. Gehrke and Wilhelmina Minnie Bose.
Marriage circa 1929 Esther A. Gehrke married Soloman Scheierman circa 1929.
Death 2 March 2003 Esther A. Gehrke died on 2 March 2003 at age 94.
20 April 1910 Russell, Otoe, Nebraska Esther A. Gehrke was listed as a daughter in John N. Gehrke's household on the 1910 Census at Russell, Otoe, Nebraska.1
6 February 1920 Stratton, Kit Carson, Colorado Esther A. Gehrke was listed as a daughter in John N. Gehrke's household on the 1920 Census at Stratton, Kit Carson, Colorado.2
3 April 1930 Greeley, Weld, Colorado Esther A. Gehrke was listed as wife in Soloman Scheierman's household on the 1930 Census at Greeley, Weld, Colorado.3
Esther A. Gehrke obtained US Social Security Card at Nebraska; last residence Sutton, Clay, Nebraska.
She lived in 2000 at 200 E. Hickory St. Apt 13, Sutton, Nebraska.
[S250] U.S. Federal Census 1930 U.S. Federal Census, by Ancestry.com, ED62-102. Dwelling 66-70.
Ruth M. Gehrke
F, b. 8 January 1910, d. 1 December 2003
Birth 8 January 1910 Ruth M. Gehrke was born on 8 January 1910 at Syracuse, Otoe, Nebraska.
Death 1 December 2003 Ruth M. Gehrke died on 1 December 2003 at Hospice and Palliative Care of Northern Colorado in-patient unit, Greeley, Colorado, at age 93 Ruth Gehrke, 93, of Greeley died Monday, Dec. 1, at the Hospice and Palliative Care of Northern Colorado inpatient unit, Greeley. She was born Jan. 8, 1910, in Syracuse, Neb., to John N. and Minnie (Bose) Gehrke. Miss Gehrke was baptized in Zion Lutheran Church in Syracuse in February 1910. She moved with her family in 1919 to Stratton. She was confirmed in St. Paul's Lutheran Church in Stratton in April 1924. She attended schools in Stratton and graduated from Barnes Business College in Denver. She was a legal secretary for Attorney John W. Henderson for 49 years until his death March 6, 1986. Miss Gehrke never actively participated in sports, but was always a fan. A member Trinity Lutheran Church for more than 65 years, she taught Sunday school for 18 years and was Sunday school superintendent for two years. She was a member of Beta Sigma Phi International Business and Cultural Sorority with 53 years of perfect attendance. Survivors are a sister, Lucia "Sally" Cowles of Greeley; six nieces and nephews; and 12 grandnieces and grandnephews. Her parents and three sisters, Esther A. Scheierman, Erma Beisner and Ruby L. Bates, are deceased. Services will be at 2 p.m. Friday at Trinity Lutheran Church. Inurnment will be in Linn Grove Cemetery, Greeley, at a later date. Friends may visit from 4-6 p.m. today at the Adamson Memorial West Chapel. In lieu of flowers, memorial gifts may be made to Trinity Lutheran in care of the West Chapel, 2000 47th Ave., Greeley, CO 80634.
20 April 1910 Russell, Otoe, Nebraska Ruth M. Gehrke was listed as a daughter in John N. Gehrke's household on the 1910 Census at Russell, Otoe, Nebraska.1
6 February 1920 Stratton, Kit Carson, Colorado Ruth M. Gehrke was listed as a daughter in John N. Gehrke's household on the 1920 Census at Stratton, Kit Carson, Colorado.2
17 April 1930 Precinct 9, Kit Carson, Colorado Ruth M. Gehrke was listed as a daughter in John N. Gehrke's household on the 1930 Census at Precinct 9, Kit Carson, Colorado.3
Ruth M. Gehrke obtained US Social Security Card at Colorado; last residence Greeley, Weld, Colorado.
Erma L. Gehrke
F, b. 17 July 1911, d. 4 December 1991
Married Name Her married name was Beisner.
Birth 17 July 1911 Erma L. Gehrke was born on 17 July 1911 at Nebraska.
Death 4 December 1991 Erma L. Gehrke died on 4 December 1991 at age 80.
6 February 1920 Stratton, Kit Carson, Colorado Erma L. Gehrke was listed as a daughter in John N. Gehrke's household on the 1920 Census at Stratton, Kit Carson, Colorado.1
17 April 1930 Precinct 9, Kit Carson, Colorado Erma L. Gehrke was listed as a daughter in John N. Gehrke's household on the 1930 Census at Precinct 9, Kit Carson, Colorado.2
Erma L. Gehrke obtained US Social Security Card at Colorado; last residence Rapid City, Pennington, South Dakota.
Lucia Gehrke
F, b. 9 January 1917, d. 4 January 2006
Name Variation Lucia Gehrke was also known as Sally.
Married Name 25 December 1936 As of 25 December 1936,her married name was Cowles.
Birth 9 January 1917 Lucia Gehrke was born on 9 January 1917 at Syracuse, Otoe, Nebraska.
Marriage 25 December 1936 Lucia Gehrke married Melbourne E. Cowles, son of Eddie M. Cowles and Lillian E. Reynolds, on 25 December 1936 at Stratton.
Death 4 January 2006 Lucia Gehrke died on 4 January 2006 at age 88 Lucia "Sally" Cowles, 88, of Monte Vista, formerly of Greeley, died Wednesday, Jan. 4, in Alamosa.
She was born Jan. 9, 1917, in Syracuse, Neb., to John and Wilhelmina (Bose) Gehrke.
On Dec. 25, 1936, she married Melbourne Cowles in Stratton. He is deceased.
Mrs. Cowles was a secretary for many years. She knitted caps to donate to the hospital for newborn babies. She was a member of Gloria Christi Church Lutheran Church.
Survivors are a son, Paul Cowles and wife Garienan of Creede; three grandchildren, Anthony, Cristin and Dawn; and three great-grandchildren, Kellie, Dylan and Alexander.
In addition to her husband, she was preceded in death by her parents, four sisters, Esther, Ruth, Erma and Ruby, and a brother, Paul.
Memorial services will be at 11 a.m. Monday at Gloria Christi in Greeley. Inurnment will be in Sunset Memorial Gardens, Greeley.
Contributions may be made to the Gloria Christi Building Fund in care of Stoddard Funeral Home, 3205 28th St., Greeley, CO 80634.
6 February 1920 Stratton, Kit Carson, Colorado Lucia Gehrke was listed as a daughter in John N. Gehrke's household on the 1920 Census at Stratton, Kit Carson, Colorado.1
17 April 1930 Precinct 9, Kit Carson, Colorado Lucia Gehrke was listed as a daughter in John N. Gehrke's household on the 1930 Census at Precinct 9, Kit Carson, Colorado.2
Lucia Gehrke lived in December 2003 at Greeley, Weld, Colorado.
Ruby L. Gehrke
F, b. 22 September 1920, d. 1 July 1994
Married Name Her married name was Bates.
Birth 22 September 1920 Ruby L. Gehrke was born on 22 September 1920 at Colorado.
Death 1 July 1994 Ruby L. Gehrke died on 1 July 1994 at age 73.
17 April 1930 Precinct 9, Kit Carson, Colorado Ruby L. Gehrke was listed as a daughter in John N. Gehrke's household on the 1930 Census at Precinct 9, Kit Carson, Colorado.1
Ruby L. Gehrke obtained US Social Security Card at Colorado; last residence Greeley, Weld, Colorado.
Sophie (?)
F, b. August 1847, d. after April 1930
Married Name circa 1882 As of circa 1882,her married name was Bose.
Birth August 1847 Sophie (?) was born in August 1847 at Germany.
Marriage circa 1882 She married Dietrich Bose circa 1882.
Death after April 1930 Sophie (?) died after April 1930.
11 June 1900 Syracuse, Otoe, Nebraska Sophie (?) was listed as wife in Dietrich Bose's household on the 1900 Census at Syracuse, Otoe, Nebraska.1
17 April 1930 Precinct 9, Kit Carson, Colorado Sophie (?) was listed as mother-in-law in John N. Gehrke's household on the 1930 Census at Precinct 9, Kit Carson, Colorado.2
Sophie (?) immigrated in 1880.
She had 2 with 1 living in 1900 children.
Child of Sophie (?) and Dietrich Bose
Wilhelmina Minnie Bose+ b. Jul 1884, d. 17 Oct 1967
Dietrich Bose
M, b. August 1848, d. before 1930
Birth August 1848 Dietrich Bose was born in August 1848 at Germany.
Marriage circa 1882 He married Sophie (?) circa 1882.
Death before 1930 Dietrich Bose died before 1930.
11 June 1900 Syracuse, Otoe, Nebraska Dietrich Bose was listed as the head of a family on the 1900 Census at Syracuse, Otoe, Nebraska.1
Child of Dietrich Bose and Sophie (?)
Melbourne E. Cowles
M, b. 13 April 1914, d. 4 December 2005
Father* Eddie M. Cowles d. b Dec 2005
Mother* Lillian E. Reynolds d. b Dec 2005
Name Variation Melbourne E. Cowles was also known as Mel.
Birth 13 April 1914 Melbourne E. Cowles was born on 13 April 1914 at Stockton, Illinois.
He was the son of Eddie M. Cowles and Lillian E. Reynolds.
Marriage 25 December 1936 Melbourne E. Cowles married Lucia Gehrke, daughter of John N. Gehrke and Wilhelmina Minnie Bose, on 25 December 1936 at Stratton.
Death 4 December 2005 Melbourne E. Cowles died on 4 December 2005 at Mountain Meadows Nursing Center at age 91 Melbourne E. "Mel" Cowles, 91, of Monte Vista, formerly of Weld County, died Sunday, Dec. 4, at Mountain Meadows Nursing Center.
He was born April 13, 1914, in Stockton, Ill. to Eddie M. and Lillian E. (Reynolds) Cowles.
The Cowles family moved from Stockton to Maquoketa, Iowa, and subsequently to Platteville, Wis., where they bought and operated a photographic studio.
He graduated from high school in Platteville, Wis., in 1932, ranked third in a class of 200. He went to Platteville State Teachers College, now the University of Wisconsin-Platteville, and earned a bachelor's degree in 1936.
On Dec. 25, 1936, he married Lucia (Sally) Gehrke in Stratton.
After returning from the Navy in 1946, he earned a master's degree from Colorado State College of Education, now the University of Northern Colorado, in Greeley. He was a member of Lambda Sigma Tau and Phi Delta Kappa.
Mr. Cowles taught in the public schools in Seibert, Willard, Greeley, Estes Park and Fort Lupton and in Dodgeville, Wis.
After leaving teaching, Mr. Cowles worked at Greeley Gas Co. He retired from Greeley Gas in 1979 as assistant manager, followed by 20 years as a volunteer at North Colorado Medical Center. He was active in civic affairs, Jaycees and Gloria Christi Lutheran Church.
Survivors include his wife, Sally Cowles of Monte Vista; a son, Paul Cowles and wife Garienan of Creede; three grandchildren; and three great-grandchildren.
He was preceded in death by his parents; and a sister, Mary Cowles.
Memorial services will be at 10 a.m. Friday at Gloria Christi. A private inurnment will be in Sunset Memorial Gardens, Greeley.
Contributions may be made to the Gloria Christi Lutheran Church Building Fund in care of Stoddard Funeral Home.
Soloman Scheierman
M, b. 9 August 1906, d. May 1979
Name Variation Soloman Scheierman was also known as Sam.
Birth 9 August 1906 Soloman Scheierman was born on 9 August 1906 at Nebraska.
Marriage circa 1929 He married Esther A. Gehrke, daughter of John N. Gehrke and Wilhelmina Minnie Bose, circa 1929.
Death May 1979 Soloman Scheierman died in May 1979 at age 72.
3 April 1930 Greeley, Weld, Colorado Soloman Scheierman was listed as the head of a family on the 1930 Census at Greeley, Weld, Colorado.1
Soloman Scheierman obtained US Social Security Card at Nebraska; last residence Sutton, Clay, Nebraska.
Letha Shaw
F, b. 30 March 1890, d. 5 January 1922
Name Variation Letha Shaw was also known as Lina.
Married Name before 1916 As of before 1916,her married name was Gehrke.
Birth 30 March 1890 Letha Shaw was born on 30 March 1890 at Wisconsin.
Marriage before 1916 She married Herman John Gehrke, son of Dietrich Friederich Gehrke and Anna Christine Bevers, before 1916.
Death 5 January 1922 Letha Shaw died on 5 January 1922 at age 31.
Burial after 5 January 1922 She was buried after 5 January 1922 at Belmont Cemetery, Belmont, Lafayette, Wisconsin.
7 January 1920 Elk Grove, Lafayette, Wisconsin Letha Shaw was listed as wife in Herman John Gehrke's household on the 1920 Census at Elk Grove, Lafayette, Wisconsin.1
Children of Letha Shaw and Herman John Gehrke
Alice Gehrke
F, b. 23 March 1915, d. 2 September 2002
Father* Herman John Gehrke b. 27 Oct 1888, d. 28 Apr 1930
Mother* Letha Shaw b. 30 Mar 1890, d. 5 Jan 1922
Married Name 3 January 1949 As of 3 January 1949,her married name was McQuillan.
Birth 23 March 1915 Alice Gehrke was born on 23 March 1915 at Wisconsin.
She was the daughter of Herman John Gehrke and Letha Shaw.
Marriage 3 January 1949 Alice Gehrke married Paul F. McQuillan on 3 January 1949.
Death 2 September 2002 Alice Gehrke died on 2 September 2002 at age 87.
7 January 1920 Elk Grove, Lafayette, Wisconsin Alice Gehrke was listed as a daughter in Herman John Gehrke's household on the 1920 Census at Elk Grove, Lafayette, Wisconsin.1
1930 Grant, Wisconsin Alice Gehrke was listed as a daughter in Herman John Gehrke's household on the 1930 Census at Grant, Wisconsin.2
Cleo Lorene Gehrke
F, b. 1916, d. 13 July 1994
Cleo L Gehrke 1937 senior portrait State Teachers College, Platteville, Grant, WI
Cleo Lorene (Gehrke) DuBois
Married Name 28 January 1943 As of 28 January 1943,her married name was DuBois.
Birth 1916 Cleo Lorene Gehrke was born in 1916 at Belmont, Lafayette, Wisconsin.
Marriage 28 January 1943 Cleo Lorene Gehrke married Charles Frederick DuBois on 28 January 1943.
Death 13 July 1994 Cleo Lorene Gehrke died on 13 July 1994 Cleo Lorene (Gehrke) DuBois obituary.
Madison Wisconsin State Journal July 17, 1994 - Cleo Lorene (Gehrke) DuBois obituary
Burial after 13 July 1994 She was buried after 13 July 1994 at Belmont Cemetery, Belmont, Lafayette, Wisconsin.
7 January 1920 Elk Grove, Lafayette, Wisconsin Cleo Lorene Gehrke was listed as a daughter in Herman John Gehrke's household on the 1920 Census at Elk Grove, Lafayette, Wisconsin.1
1930 Grant, Wisconsin Cleo Lorene Gehrke was listed as a daughter in Herman John Gehrke's household on the 1930 Census at Grant, Wisconsin.2
Cleo Lorene Gehrke and Jeannette Lenice Gehrke lived in 1936 at 318 N. Bonson St., Platteville, Grant, Wisconsin, property owner Mrs. Charles Shaw.
Cleo Lorene Gehrke graduated in 1937 at State Teachers College, Platteville, Grant, Wisconsin.
http://www.friede-abrahamson-genealogy.com/
Researcher: Heather Abrahamson, 5818 Manor Lane, Parkville, MO 64152
Site updated on 20 Dec 2020 at 9:57:46 PM from friede-abrahamson; 73,733 people
Page created by John Cardinal's Second Site v5.3.7. | Web site design by John Cardinal based on an idea by Jeremy D


text:
"""
Root object

Format:

"""

import Lexer
from Command import Command
from Root import Root

class SquareRoot(Command):
	def __init__(self):
		super(self.__class__, self).__init__(1)

	def apply_args(self,args):
		return Root().apply_args([" "]+args)

redpajama_set_name: RedPajamaGithub


text:
Shoreview is een plaats (city) in de Amerikaanse staat Minnesota, en valt bestuurlijk gezien onder Ramsey County.

Demografie
Bij de volkstelling in 2000 werd het aantal inwoners vastgesteld op 25.924.
In 2006 is het aantal inwoners door het United States Census Bureau geschat op 26.726, een stijging van 802 (3.1%).

Geografie
Volgens het United States Census Bureau beslaat de plaats een oppervlakte van
33,0 km², waarvan 29,0 km² land en 4,0 km² water.

Plaatsen in de nabije omgeving
De onderstaande figuur toont nabijgelegen plaatsen in een straal van 8 km rond Shoreview.

Externe link

Plaats in Minnesota

redpajama_set_name: RedPajamaWikipedia


text:
<!doctype html>

  <meta charset="utf-8">
  <title>commonmark.js demo</title>

  <link href="//maxcdn.bootstrapcdn.com/bootstrap/3.1.1/css/bootstrap.min.css" rel="stylesheet">

commonmark.js dingus

clear permalink

      <textarea id="text"></textarea>

Parsed in
      ms.  Rendered in  ms.

Preview

HTML

AST


The weather can't keep Voltage Vapin' down! We're open today until 5pm!
Voltage Vapin' is open for business today until 5PM! Stop in to stock up on liquids, coils, and accessories. Also, ask an associate how you can save 15% off all of your liquids purchases through the end of the year by becoming a consumer member of the Virginia Smoke Free Association.
Stay safe on the roads and we'll see you soon!
Voltage Vapin's Winter Weather Update!
Voltage Vapin' will be closing tonight (January 7th) at 5pm but will reopen tomorrow (January 8th) for full hours from 12-5!
As a precaution to all of this winter weather we will close tonight at 5pm but reopen at 12 tomorrow for full hours. We want everyone to stay safe and stay dry so if you don't have a car capable of handling snow very well or don't feel safe driving please make the right choice and don't put yourself in any danger. It's going to freeze over pretty hard tonight so if you do go somewhere make sure you plan ahead and drive safe!
Remember, without you guys all of this wouldn't be possible so stay safe and stay warm!
Stay safe and stay warm with a nice warm flavor like Banana nut bread!
Hohm Baked has produced one of the most accurate Banana Nut Bread blends ever tasted! This sweet nutty mixture is a max vg e-liquid that will make your jaw drop with how accurate this comes to a freshly baked loaf of the flavor you crave! Available in 0, 3, and 6 mg in a 60 ml bottle.
Enjoy a comfort Vape during this snow storm with Banana Nut Bread… at Voltage Vapin'!
UPDATE: Due to the winter weather advisory, while you are out stocking up on bread & milk, etc. Don't forget to stock up on your liquid, coils & other vaping supplies at Voltage Vapin'!! We will keep you updated as the weather comes in tomorrow, Saturday January 7th, with any changes to our normal business hours of 10am – 7pm.
Stay tuned for updated info! You can also check nbc12.com or wtvr.com for updates!
We want everyone including our staff to be safe under these conditions!!Thank you for understanding!
Make Sure your batteries are ready for the winter storm. If they have tears in their wrapping come in and get them re-wrapped or if you just simply need another pair, we have you covered with the LG Chocolate batteries to get you through.
This High-drain Dynamo is another Flat top battery that puts out a constant 20 amp with a 35 amp peak. Fully charged with 3000 mah, it can compete with other 18650 batteries of it's kind. Great for cloud chasers and flavor tasters alike, it's compatible with any device that uses 18650 batteries, but better with devices that are regulated.
Match a pair of Chocolate LGs with your favorite mod today at Voltage Vapin' and feel the power!
The LUC Blu charger is perfect for the family especially during the storm. The 6 bays give you plenty of room to keep extra batteries charging to ensure that you never have a dead mod.
Come on by Voltage Vapin' and check out the Efest LUC BLU6 OLED Intelligent Charger! We know a lot of people that try to share a simple 2 or 4 bay battery charger among friends or family and sometimes that doesn't always work. Now you can put all your batteries on one charger and make everyone happy! Come check out the best charger we have in stock!


text:
The aim of this study was the purification procedure for polyphenols
The aim of this study was the purification procedure for polyphenols from (chokeberry), as well as the purification parameters were optimised by adsorption and desorption tests. of polyphenols in fruit, polyphenols from chokeberry represent a very valuable natural antioxidant source with antiproliferative products. (Rosaceae), also called chokeberry, is the small berry native to North America that has a high economic value. The flesh is deep red with a bitter and astringent taste. Chokeberry fruit is a dark blue and round berry. Generally, chokeberries is used for processing into fruit jams, fruit wines, fruit juices, dried fruit, canned-foods and other products. Plant polyphenols are known for their unique antioxidant and Gadodiamide supplier antiproliferative activities. But there is a great difference between the components and the content of polyphenols in different kinds of plants. Due to its rich content of polyphenols, berries can be a good source of natural antioxidants [1,2,3]. In previous reports, it was confirmed that berry polyphenol substances can remove free radicals from the body effectively, reduce bloodstream lipid amounts and regulate the bodys disease fighting capability efficiently, to maintain the standard function of the body [4,5,6,7]. Early outcomes showed how the fruits of (chokeberry) was abundant with polyphenols. Reaching a complete polyphenols content material of Gadodiamide supplier 7.85 g/100 g (DW) [8]. The new reddish colored flesh displays a big percentage of polyphenols in fruits pomace also, however the astringent and bitter flavor, it creates it hard to get a full great deal for folks to accept, but it could be utilized as an excellent source of organic antioxidants, to efficiently utilize the wealthy polyphenol content material of (chokeberry) [9,10]. The analysis demonstrates the free of charge radicals produced through the oxidation of the body are strongly oxidizing and can damage tissues and cells in the body, which can lead to chronic diseases and aging effects [11]. Polyphenols, due to their redox properties, can be used as DLL3 hydrogen donors, reductants, and singlet oxygen quenchers, thus playing a key anti-oxidation role [12]. Hwang found that, because it contains a lot of polyphenols, flavonoids and procyanidins, chokeberry extract showed stronger antioxidant activity than blueberry Gadodiamide supplier extract, allowing the body to reduce the damage from active oxygen free radicals [13]. In in vivo experiments, chokeberry extract also exhibited significant antioxidant activity in plasma and liver. Every 100 g of chokeberry fruit (dry weight) showed a DPPH free radical scavenging capacity of 279.38 M Trolox equivalent. For ABTS free radical the scavenging capacity was 439.49 M Trolox equivalent [8]. A thorough comparison of the anthocyanin content of nine kinds of blackberry, three kinds of raspberry, nine kinds of black currant, four kinds of red currant and chokeberry show that anthocyanin content of chokeberry was significantly higher than that Gadodiamide supplier of the other berries. The reasons for the high antioxidant activity of the extract were confirmed by comparing the antioxidant capacity of chokeberry fruit in different forms. It was found that the antioxidant capacity of fruit residue was significantly higher than that of fruit and fruit juice, which had a great relationship with the content and distribution of polyphenol in various elements of the fruit [8]. Gadodiamide supplier The hypolipidemic function of chokeberry continues to be praised in medical study broadly, and a scholarly research discovered that in rats given a diet plan including high degrees of cholesterol, chokeberry juice can hinder the boost of total cholesterol considerably, low denseness lipoprotein and triglycerides in vivo, reducing the incidence of coronary disease [6] thus. Furthermore, Kim et al.s study demonstrated how the mechanism from the chokeberry draw out reduced amount of cholesterol content material is in addition to the manifestation of genes linked to cholesterol rate of metabolism in the liver organ [14]. Because of the wealthy flavonoid content material of the fruit, it can effectively inhibit pancreatic lipase, -amylase and -glucosidase so as to regulate the digestion and absorption of lipids and carbohydrates in the gastrointestinal tract, therefore chokeberry can be used as a natural material for functional foods, and the prevention of hyperglycemia and obesity, and the extract can be used for hyperglycemia and.
← Angiogenesis, the forming of new arteries, is crucial for success and → Information over the defense response against H5N1 inside the lung is

redpajama_set_name: RedPajamaCommonCrawl


text:
'Consider the multiplier effect' — Sirika appeals to airlines over planned shutdown
Wasilat Azeez
Hadi Sirika, minister of aviation, has appealed to domestic airlines to suspend the planned shutdown of operations from Monday.
Sirika made the appeal in a statement issued by James Odaudu, his special assistant on public affairs, on Saturday, in Lagos.
On Friday, airlines under the aegis of the Airline Operators of Nigeria (AON), announced plans to shut down flight operations over the high cost of aviation fuel, also known as Jet A1.
The minister urged the airlines to consider the multiplier effect that shutting down operations would have on travellers.
"The attention of the Minister of Aviation, Sen. Hadi Sirika, has been drawn to reports that the nation's air transportation system will be disrupted as from Monday," the statement reads.
"This follows the threats by airline operators to shut down operations as a result of rising cost of Jet A1, otherwise known as aviation fuel.
"As the ministry charged with the industry, we are greatly concerned about the difficulties being faced by the airlines in procuring aviation fuel which has resulted in spiraling costs in air transportation in the immediate past.
"We also acknowledge that the airlines are in the business to make profits, while servicing the very critical sector that is not only the preferred mode of transport for most Nigerians, but the main international gateway to the nation.
"Unfortunately, the issue of fuel supply is not within the purview of the ministry, so the much it can do in the present situation is to engage with agencies, institutions and individuals in positions to provide succour to the airlines. This is already being done by the relevant team led by the minister.
"While the efforts to assuage the situation are on, we wish to appeal to the airline operators, even in the difficult situation, to consider the multiplier effect of shutting down operations, on Nigerians and global travellers, in taking their business-informed decisions and actions."
The aviation minister assured Nigerians that the Buhari administration remains commitment to the creation and sustenance of an environment that promotes the growth of the aviation industry.
He also said the president was committed to the development of the industry where major players such as the airlines could operate in a profitable and competitive market.
Next Temi Otedola: How I knew Mr Eazi is right man for me »
Previous « Ibom Air backs out of airlines' strike
Tags: Airline Operators of Nigeria (AON)Aviation fuelHadi Sirika
FG on NAHCO strike: We'll no longer tolerate it — aviation is an essential service
Airline operators warn passengers of imminent flight delays over aviation fuel scarcity
AON to FAAN: Increase check-in counters in airports to reduce flight delays
Nigeria Air: Court adjourns case till Feb 13, orders parties to maintain status quo
'It is at Nigeria's detriment' — airline operators explain position on Ethiopian Air's stake in national carrier
Sirika: No going back on planned demolition of structures at Lagos airport
Court stops FG from selling Nigeria Air's shares to Ethiopian Airlines
FG: We're not threatened by foreign airlines' withdrawal of operations
'We must learn from past' — airline operators oppose Ethiopian Air's stake in Nigeria Air
Trapped funds: Gbaja meets foreign airlines, aviation stakeholders to resolve dispute


FAQ — Fireplaces of Lake Hartwell. Your resource for all things fireplaces, stoves and accessories.
Fireplaces of Lake Hartwell. Your resource for all things fireplaces, stoves and accessories.
Please refer to our manufacturers for specific product inquiries & user guides. If you can't find what you are looking for let us know.
Can you deliver and install in one day?
No. But, we can within one week of the order, depending on the type of product you choose.
The types of fuel range from raw wood to natural gas. All of our heating methods are energy-efficient.
Will you train us how to use our new fireplace?
Of course! You will be instructed by our technicians on the proper operation of your fireplace, stove or gas logs. And, we stand behind all the products we recommend. Additionally, all our manufacturers have toll free numbers for a Help Line.
I want to get a fireplace but don't want to worry about maintaining it. Do you offer maintenance?
We have service and maintenance plans to fit your needs.
I heard people can cook on top of some stove models. Is this true?
Heat a house, boil water, cook breakfast, lunch and dinner …. Our stoves are practical in more ways than one.
Before it gets too cold!
1 ) Supervise children, the elderly, disabled and pets near a gas fireplace, stove or insert that is in use or was recently turned off.
5) Wait for the appliance and glass panel to cool down before allowing anyone near it. Cool down can take an hour or more.
Some appliances turn on and off automatically with a thermostat, so it may not be clear when a fire is turned off.
Contact Us and we'll get right on it.


THE JACKSON COUNTY SCHOOL SYSTEM IS COMMITTED TO PROVIDING REMARKABLE EXPERIENCES FOR ITS STUDENTS & STAFF.
To promote student success through leadership, character, and performance.
To offer a challenging and rigorous educational experience for all students in a safe and caring learning environment.


Located 2 km from Fresse-sur-Moselle, Chambres d'Hôtes le Clos des Lesses offers B&B accommodations. You can enjoy the garden, relax on the furnished terrace and organise BBQs. Rouge Gazon Ski Resort is 14 km away. Decorated with antique furniture and a parquet floor, all of the rooms at Chambres d'Hôtes le Clos des Lesses are soundproofed. Each room features a desk, a wardrobe and a private bathroom with a hairdryer and a shower. A continental breakfast is included and served daily in the dining room. The hosts can also prepare traditional dishes from the region for dinner, upon prior reservation. Chambres d'Hôtes le Clos des Lesses is equipped with free Wi-Fi. Free private parking is also possible on site and Remiremont is 25 km from the B&B. There is a hiking path, which starts from the property.
Decorated with antique furniture and a parquet floor, this soundproofed room offers a view of the mountain. It features a desk, a wardrobe and a private bathroom with a hairdryer and a shower.
Offering a garden view, this room is decorated with natural and red tones. The private bathroom features a walk-in shower.
This room has a mountain-style décor with wooden panelling and antique furniture. The children's beds are set in a wooden cabin while the bathroom is fitted with a walk-in shower.


An extremely easy to extend dining table with a chunky profile in beautiful walnut veneer. The extension leaves are stored underneath the table top which separates at the centre to comfortably accommodate 6-8 people or more. The profile of the table top is matched by the angular legs.


The reason being is the air filter helps to keep debris from getting into your vehicle's engine, thus optimizing the output of the engine in your particular model Honda, such as the Accord, the Civic, or the Odyssey. If you notice lower gas mileage or an air filter that is visibly dirty, it may already be time for new air filter.
We know that sufficient airflow and adequate dirt protection are equally important for optimal engine performance, that's why we carry only high-quality Honda performance air filters that maintain unrestricted flow of air without compromising filtration efficiency.
It is widely known who makes the OEM oil filters (both domestic and import versions) for Honda and Toyota. Anyone know who makes the OEM air filters for Honda and Toyota?
No results found for " air filter ". Here are some helpful search tips: Search by a part name. Example: water pump. Be less specific. Example: brake instead of ceramic brake. Remove information such as make, year or model. Can't find the part you are looking for? Contact us and we'll find it for you!
Honda Air Filters: Jacks is your Place! We have the Air Filters you need, with fast shipping and great prices! We have the Air Filters you need, with fast shipping and great prices! For lawn mower parts and accessories, think Jacks!
OEM Honda Air Filters vs. Aftermarket Air Filters. No matter the model or year of your Honda, we carry an air filter for it here at All Discount Parts. We know that you may be tempted by a cheaper aftermarket air filter for your Honda, but we recommend you only use OEM products, especially when it comes to your air filter.
Keeping your cabin and engine clean is very important. One of the best ways to do this is by replacing your air filters when necessary. For those that are driving a Honda, getting a car cabin air filter or engine air filter designed for Honda vehicles could be a very good option.
This type of Honda air filter replacement is reusable, making changing a dirty filter as simple and affordable as washing it out, letting it dry and reinstalling it. It is a terrific bargain, as you enjoy the features of a higher performing Honda air filter, while saving money from no longer buying the throw-aways.


text:
\section*{Introduction}\label{S1}

The problem of decomposing a tensor into a minimal sum of rank-1 terms, is raising interest and attention from many applied areas as signal processing for telecommunications \cite{deLC}, independent component analysis \cite{Comon}, complexity of matrix multiplication \cite{Stra83:laa},  complexity problem of  P versus NP \cite{Vali},  quantum physics \cite{entanglement}, \cite{bi} and phylogenetic  \cite{ar}.
The particular instance in which the tensor is symmetric and hence representable by a homogeneous polynomial, is one of the most studied and developed ones (cfr. \cite{l} and references therein). In this last case we say that the rank of a homogeneous polynomial $P$ of degree $d$ is the minimum integer $r$ needed to write it as a linear combination of pure powers of linear forms $L_1, \ldots , L_r$:
\begin{equation}\label{F}
P=c_1L_1^d + \cdots +  c_dL_r^d.
\end{equation}

with $c_i\ne 0$. Most of the papers concerning the abstract theory of the symmetric tensor rank, require that the base field is algebraically closed. In this case
we may take $c_i=1$ for all $i$ without loss of generality. However, for the applications, it is very important to consider the case of real polynomials and look at their real decomposition. Namely, one can study separately the case in which the linear forms appearing in (\ref{F}) are complex or reals. In the
real case we may take $c_i=1$ for all $i$ if $d$ is odd, while we take $c_i\in \{-1,1\}$ if $d$ is even. When we look for a minimal complex (resp. real) decomposition as in (\ref{F}) we  say that we are computing the \emph{complex symmetric rank} (resp. \emph{real symmetric rank}) of $P$ and we will indicate it $r_{\mathbb{C}}(P)$ (resp. $r_{\mathbb{R}}(P)$). Obviously
$$r_{\mathbb{C}}(P)\leq r_{\mathbb{R}}(P),$$
and in many cases such an equality is strict.

In \cite{co} P. Comon and G. Ottaviani studied the real case for bivariate symmetric tensors. Even in this case there are many open conjectures and, up to now, few cases are completely settled (\cite{co}, \cite{cr}, \cite{b}, \cite{bl}).

In this paper we want to study the relation among $r_{\mathbb{C}}(P)$ and $r_{\mathbb{R}}(P)$ in the special circumstance in which $r_{\mathbb{C}}(P) < r_{\mathbb{R}}(P)$. In particular we will show that, in a certain range (say, $r_{\mathbb{C}} (P) +r_{\mathbb{R}} (P) \le 3\deg (P)-1$),
all homogeneous polynomials $P$ of that degree with $r_{\mathbb{R}} (P) \ne r_{\mathbb{C}} (P)$ are characterized by the existence of a curve with the property that the sets evincing the real and the complex ranks coincide out of it (see Theorem \ref{i1} for the precise statement). More precisely: let $P\in S^d\mathbb{R}^{m+1}$ be a real homogeneous polynomial of degree $d$ in $m+1$ variables such that $r_{\mathbb{C}}(P) < r_{\mathbb{R}}(P)$ and $r_{\mathbb{C}} (P) +r_{\mathbb{R}} (P) \le 3\deg (P)-1$; therefore its real and complex decomposition are
$$P=a_1L_1^d+ \cdots + a_kL_k^d+c_1M_1^d+ \cdots +c_sM_s^d,$$
$$P=N_1^d+ \cdots + N_h^d+c_1M_1^d+ \cdots +c_sM_s^d$$
respectively, with $k>h$, $a_i\in \{-1,1\}$, $c_j\in \{-1,1\}$, $M_1, \ldots , M_s\in S^1\mathbb{R}^{m+1}$, $h+k>d+2$. Moreover there exists a curve $C' \subset \mathbb {P}^m$ such that $[L_1], \ldots , [L_k], [N_1], \ldots , [N_h]$
``~depends only from the variables of $C'$~'' and $C'$ is either a line or a reduced conic or a disjoint union of two lines.
If $C'$ is a line (item (\ref{a}) in Theorem \ref{i1}) then both the $L_i$'s and the $N_i$'s are linear forms in the same two ``variables". If $C'$ is a conic, then $L_i$'s and $N_i$'s depend on 3 ``variables" and their projectivizations lie on $C'$. See item (c) of Theorem \ref{i1} for the geometric interpretation
of the reduction of $L_1,\dots ,L_k$ and $N_1,\dots ,N_h$ to bivariate forms involved with
$C'$ when $C' =l\sqcup r$ is a disjoint union of two lines $l$ and $r$ (we have two sets of bivariant forms, one for the variables of $l$ and one for the variables of $r$).

\section{Notation and statements}

Before giving the precise statement of Theorem \ref{i1} we need to introduce the main algebraic geometric tools that we will use all along the paper.

Let $\nu _d: \mathbb {P}^m \to \mathbb {P}^N$, $N:= {m+d \choose d}-1$,
denote the degree $d$ Veronese embedding of $\mathbb {P}^m$ (say, defined over $\mathbb {C})$.
Set $X_{m,d}:= \nu _d(\mathbb {P}^m)$.
For any $P\in \mathbb {P}^N$, the {\emph {symmetric rank}} or {\emph
{symmetric tensor rank}} or, just, the {\emph {rank}}
$r_{\mathbb{C}} (P)$ of $P$ is the minimal cardinality of a finite set
$S\subset \mathbb {P}^m(\mathbb {C})$ such that $P\in \langle \nu _d(S)\rangle$, where
$\langle \ \ \rangle$ denote the linear span (here the linear span is with respect to complex coefficients), and we will say that $S$ \emph{evinces} $r_{\mathbb{C}}(P)$.
Notice that the Veronese embedding $\nu _d$ is defined over $\mathbb {R}$, i.e.
$\nu _d(\mathbb {P}^m(\mathbb {R})) \subset \mathbb {P}^N(\mathbb {R})$. For each $P\in \mathbb {P}^N(\mathbb {R})$ the \emph{real symmetric rank} $r_{\mathbb{R}} (P)$ of $P$ is the minimal cardinality
of a finite set $S\subset \mathbb {P}^m(\mathbb {R})$ such that $P\in \langle \nu _d(S)\rangle _{\mathbb {R}}$, where $\langle \ \ \rangle _{\mathbb {R}}$ means the linear span with real coefficients, and we will say that $S$ evinces $r_{\mathbb{R}}(P)$. The integer $r_{\mathbb{R}} (P)$ is well-defined, because $\nu _d(\mathbb {P}^m(\mathbb {R}))$ spans $\mathbb {P}^N(\mathbb {R})$.

Let us fix some notation: If $C\subset \mathbb{P}^m$ is either a curve or a subspace and $S\subset \mathbb{P}^m$ is a finite set, we will use the following abbreviations:
$$S_C:=S\cap C,$$
$$S_{\hat{C}}:=S\setminus (S\cap C).$$

\begin{theorem}\label{i1}
Let $P\in \mathbb {P}^N(\mathbb {R})$ be such that $r_{\mathbb{C}} (P)+r_{\mathbb{R}} (P) \le 3d-1$
and $r_{\mathbb{C}} (P) \ne r_{\mathbb{R}} (P)$. Fix any set $S_{\mathbb{C}}\subset \mathbb {P}^m(\mathbb {C})$ and ${S_{\mathbb{R}}}\subset \mathbb {P}^m(\mathbb {R})$  evincing $r_{\mathbb{C}} (P)$ and  $r_{\mathbb{R}} (P)$ respectively. Then
one of the following cases (a), (b), (c) occurs:

\begin{enumerate}[(a)]

\item\label{a} There is a line $l\subset \mathbb {P}^m$ defined
over $\mathbb {R}$ and with the following properties:

\begin{enumerate}[(i)]

	\item\label{ai} $S_{\mathbb{C}}$ and $S_{\mathbb{R}}$ coincide out of the line $l$ in a set $S_{\hat{l}}$: $$S_{\mathbb{C}}\setminus S_{\mathbb{C}}\cap l = {S_{\mathbb{R}}}\setminus {S_{\mathbb{R}}}\cap l=:S_{\hat{l}};$$

	\item\label{aii} there is a point $P_l\in \langle \nu _d(S_{\mathbb{C},l})\rangle \cap \langle \nu _d(S_{\mathbb{R},l})\rangle$ such that $S_{\mathbb{C},l}$ evinces $r_{\mathbb{C}} (P_l)$ and $S_{\mathbb{R},l}$ evinces $r_{\mathbb{R}} (P_l)$;

	\item\label{aiii} $\sharp (S_{\mathbb{C},l}\cup {S_{\mathbb{R},l}}) \ge d+2$ and $\sharp (S_{\mathbb{C},l})< \sharp (S_{\mathbb{R},l})$.

\end{enumerate}

\item\label{b} There is a conic $C\subset \mathbb {P}^m$ defined
over $\mathbb {R}$ and with the following properties:

\begin{enumerate}[(i)]

	\item\label{bi} $S_{\mathbb{C}}$ and $S_{\mathbb{R}}$ coincide out of the conic $C$ in a set  $S_{\hat{C}}$: $$S_{\mathbb{C}}\setminus S_{\mathbb{C},C}= {S_{\mathbb{R}}}\setminus {S_{\mathbb{R}}}\cap C=:S_{\hat{C}};$$

	\item\label{bii}  there is a point $P_C\in \langle \nu _d(S_{\mathbb{C},C})\rangle \cap \langle \nu _d(S_{\mathbb{R},C})\rangle$ such that $S_{\mathbb{C},C}$ evinces $r_{\mathbb{C}} (P_C)$ and $S_{\mathbb{R},C}$ evinces $r_{\mathbb{R}} (P_C)$;

	\item\label{biii} $\sharp (S_{\mathbb{C},C}\cup {S_{\mathbb{R},C}}) \ge 2d+2$ and $\sharp (S_{\mathbb{C},C})< \sharp (S_{\mathbb{R},C})$;

	\item\label{biv} if $C$ is reducible, say $C = l_1\cup l_2$ with $Q=l_1\cap l_2$, then $\sharp ((S_{\mathbb{C}}\cup {S_{\mathbb{R}}})\cap (l_i\setminus Q)) \ge d+1$ for $i\in \{1,2\}$.

\end{enumerate}

\item\label{c} $m\ge 3$ and there are $2$ disjoint lines $l, r \subset \mathbb {P}^m$ defined over $\mathbb {R}$  with the following properties:

\begin{enumerate}[(i)]

	\item\label{ci} $S_{\mathbb{C}}$ and $S_{\mathbb{R}}$ coincide out of the union $\Gamma:=l\cup r$ in a set $S_{\hat{\Gamma}}$:
	$$S_{\mathbb{C}}\setminus S_{\mathbb{C}}\cap (l\cup r) =
	S_{\mathbb{R}}\setminus S_{\mathbb{R}}\cap (l\cup r):=S_{\hat{\Gamma}};$$

	\item\label{cii} $\sharp (S_{\mathbb{C},l}\cup {S_{\mathbb{R},l}}) \ge d+2$ and $\sharp (S_{\mathbb{C},r}\cup {S_{\mathbb{R},r}}) \ge d+2$;

		\item\label{ciii} the set  $\langle  \nu_d(S_{\hat{\Gamma}})\rangle\cap \langle\nu _d(\Gamma)\rangle$
is a single point, $O_{\Gamma}\in \mathbb {P}^N(\mathbb {R})$; \\$S_{\mathbb{C},\Gamma}$ evinces $r_{\mathbb{C}} (O_{\Gamma})$ and ${S_{\mathbb{R},\Gamma}}$ evinces $O_{\Gamma}$;
		\item\label{civ} the set $\langle \{O_\Gamma\}\cup \nu _d(l)\rangle \cap \langle \nu
_d(l)\rangle$ (resp. $\langle \{O_{\Gamma}\}\cup \nu _d(r)\rangle \cap \langle \nu
_d(r)\rangle$) is formed by a unique point $O_l\in \mathbb {P}^N(\mathbb {R})$ (resp. $O_r\in \mathbb {P}^N(\mathbb {R})$);\\
$ S_{\mathbb{C},l}$ (resp. $S_{\mathbb{C},r}$) evinces $r_{\mathbb{C}} (O_l)$ (resp.  $r_{\mathbb{C}} (O_r)$);\\
${S_{\mathbb{R},l}}$ (resp. $ {S_{\mathbb{R},r}}$) evinces $r_{\mathbb{R}} (O_l)$ (resp $r_{\mathbb{R}} (O_r)$).
 \end{enumerate} \end{enumerate} \end{theorem}

\section{The proof}

\begin{remark}\label{d0} Let $S\subset \mathbb{P}^N(\mathbb{R})$. It will be noteworthy in the sequel  that  $S$ can be used to span both a real  space $\langle S \rangle_{\mathbb{R}}\subset \mathbb{P}^N(\mathbb{R})$ and a complex space $\langle S \rangle_{\mathbb{C}}\subset \mathbb{P}^N(\mathbb{C})$ of the same dimension and $\langle S \rangle_{\mathbb{C}}\cap \mathbb{P}^N(\mathbb{R})=
\langle S \rangle_{\mathbb{R}}$.
In the following we will always use $ \langle \ \ \rangle$
to denote $\langle \ \  \rangle_{\mathbb{C}}$.
\end{remark}

\begin{remark}\label{d1}
Fix $P\in \mathbb {P}^N$ and a finite set $S\subset \mathbb {P}^N$ such that
$S$ evinces $r_{\mathbb{C}} (P)$. Fix
any $E\subsetneq S$. Then the set $\langle \{P\} \cup E \rangle \cap \langle S\setminus E\rangle$ is a single point (call it $P_1$) and $S\setminus
E$ evinces $r_{\mathbb{C}} (P_1)$. Now assume $P\in \mathbb {P}^N(\mathbb {R})$ and $S\subset  \mathbb {P}^N(\mathbb {R})$. Then $P_1\in \mathbb {P}^N(\mathbb {R})$. If  $S$ evinces $r_{\mathbb{R}} (P)$, then $S\setminus E$ evinces $r_{\mathbb{R}} (P_1)$.
\end{remark}

\begin{lemma}\label{c2} Let $C\subset \mathbb {P}^m$ be a reduced curve of degree $t$ with $t=1,2$. Fix finite sets $A,B \subset \mathbb {P}^m$. Fix an integer $d>t$ such
that:
$$h^1(\mathcal {I}_{(A\cup B)_{\hat{C}}}(d-t))=0.$$
Assume the existence of $P\in \langle \nu _d(A)\rangle \cap \langle \nu _d(B)\rangle$ and $P\notin \langle \nu _d(S')\rangle$ for any $ S' \subsetneq A$ and any $S' \subsetneq {B}$. Then
$$A_{\hat{C}}=B_{\hat{C}}.$$
\end{lemma}

\begin{proof} The case $t=1$ is \cite[Lemma 8]{bb1}.
If $t=2$, then either $C$ is a conic or $m\ge 3$ and $C$ is a disjoint union of $2$ lines.
In both cases we have $h^0(\mathcal {I}_C(t)) >0$
and the linear system $\vert \mathcal {I}_C(t)\vert$ has no base points outside $C$. Since
$A\cup B$ is a finite set,
there is $M\in \vert \mathcal {I}_C(t)\vert$ such that $M\cap (A\cup  B) = C\cap (A\cup B)$. Look
at the residual exact sequence (also called the Castelnuovo's exact sequence):
\begin{equation}\label{eqa1}
0 \to \mathcal {I}_{(A\cup B)_{\hat{C}}}(d-t) \to \mathcal {I}_{A\cup B}(d)
\to \mathcal {I}_{(A\cup B)\cap M,M}(d) \to 0
\end{equation}
We can now repeat the same proof of \cite[Lemma 8]{bb1} but starting with (\ref{eqa1}) instead of the exact sequence used there (cfr. first displayed formula in the proof of \cite[Lemma 8]{bb1}).

We will therefore get
$A_{\hat{M}} = B_{\hat{M}}$. Now, since $M\cap (A\cup  B) = C\cap (A\cup B)$, we
are done.
\end{proof}

We are now going to prove Theorem \ref{i1}.

\vspace{0.3cm}

\quad {\emph {Proof of Theorem \ref{i1}}}

Fix $P\in \mathbb {P}^N(\mathbb {R})$ such that $r_{\mathbb{C}} (P)+r_{\mathbb{R}} (P) \le 3d-1$
and $r_{\mathbb{C}} (P) \ne r_{\mathbb{R}} (P)$.
\\ Fix any set $S_{\mathbb{C}}\subset \mathbb {P}^m(\mathbb {C})$
evincing $r_{\mathbb{C}} (P)$ and any ${S_{\mathbb{R}}}\subset \mathbb {P}^m(\mathbb {R})$ evincing $r_{\mathbb{R}} (P)$.
\\
By applying \cite{bb}, Lemma 1, we immediately get that
$$h^1(\mathcal {I}_{S_{\mathbb{C}}\cup {S_{\mathbb{R}}}}(d)) >0.$$
Since $\sharp (S_{\mathbb{C}})+\sharp ({S_{\mathbb{R}}}) \le 3d-1$, either there is a line $l\subset \mathbb {P}^m$ such that $\sharp (S_{\mathbb{C},l}\cup {S_{\mathbb{R},l}}) \ge d+2$ or there is a conic $C$ such that $\sharp  (S_{\mathbb{C},C}\cup {S_{\mathbb{R},C}}) \ge 2d+2$ (\cite{c2}, Theorem 3.8). We are going to study separately these two cases in items (\ref{1}) and (\ref{2}) below.

\begin{enumerate}

\item\label{1} In this step we assume the existence of a line $l \subset \mathbb {P}^m$ such that
$$\sharp ( S_{\mathbb{C},l}\cup {S_{\mathbb{R},l}})\ge d+2.$$

\newcounter{mycounter}
\setcounter{mycounter}{\theenumi}
\end{enumerate}

This hypothesis, together with $r_{\mathbb{R}}(P)\neq r_{\mathbb{C}}(P)$,  immediately implies property (\ref{aiii}) of the statement of the theorem.

We are now going to distinguish the case $h^1(\mathcal {I}_{S_{\mathbb{C}}\cup {S_{\mathbb{R}}}}(d-1))=0$ (item (1.1) below) from the case $h^1(\mathcal {I}_{S_{\mathbb{C}}\cup {S_{\mathbb{R}}}}(d-1))>0$ (item (1.2) below).

\vspace{0.1cm}
\quad (1.1)  Assume $h^1(\mathcal {I}_{S_{\mathbb{C}}\cup {S_{\mathbb{R}}}}(d-1))=0$.

First of all, observe that the line $l\subset \mathbb{P}^m$ is well defined over $\mathbb{R}$ since it contains at least 2 points of ${S_{\mathbb{R}}}$ (Remark \ref{d0}). Then, by Lemma \ref{c2}, we have that ${S_{\mathbb{C}}}$ and ${S_{\mathbb{R}}}$ have to coincide out of the line $l$:
$${S_{\mathbb{C}}}\setminus S_{\mathbb{C},l} = {S_{\mathbb{R}}}\setminus S_{\mathbb{R},l}:=S_{\hat{l}},$$
and this proves  (\ref{ai}) of the statement of the theorem in this case (1.1).

The fact that $\sharp (S_{\mathbb{C}}) < \sharp ({S_{\mathbb{R}}})$  implies that $\sharp (S_{\mathbb{R},l}) > (d+2)/2$ and $\sharp (S_{\hat{l}}) \le d$,
hence
$h^1(\mathcal {I}_{S_{\hat{l}}\cup l}(d))=0.$

Therefore we have that $\dim (\langle \nu _d(S_{\hat{l}} \cup l)\rangle )= \sharp (S_{\hat{l}})+d+1$, $\dim (\langle \nu
_d(S_{\hat{l}})\rangle ) =\sharp (S_{\hat{l}})-1$ and $\dim (\langle \nu _d(l)\rangle )=d+1$, and Grassmann's formula
gives $\langle \nu _d(S_{\hat{l}})\rangle \cap \langle \nu _d(l)\rangle = \emptyset$. \\

Since $P\in \langle \nu _d(S_{\hat{l}}\cup S_{\mathbb{C}})\rangle$ and $S_{\mathbb{C},l}\subset l$, the
set $\langle \nu _d(S_{\hat{l}})\cup \{P\}\rangle \cap \langle \nu _d(l)\rangle$ is a single point, $P_l\in \mathbb {P}^N(\mathbb {R})$. \\

Since $P\in \langle \nu _d(S_{\mathbb{C}})\rangle$ and $P\notin \langle \nu _d(S_{\mathbb{C}}')\rangle$ for any $S'\subsetneq S_{\mathbb{C}}$,
the set $\langle \nu _d(S_{\hat{l}})\cup \{P\}\rangle \cap \langle \nu _d(S_{\mathbb{C},l})\rangle$ is a single point, $P_C$
(Remark \ref{d1}).

Then obviously:
$$P_C=P_l\in \mathbb{P}^N(\mathbb{R}).$$
Since $S_{\mathbb{C}}$ evinces $r_{\mathbb{C}} (P)$, then $S_{\mathbb{C},l}$ evinces $P_l$ (Remark \ref{d1}). In the same way we see that  $\langle \nu _d(S_{\hat{l}})\cup \{P\}\rangle \cap \langle \nu _d(S_{\mathbb{R},l})\rangle =\{P_l\}$ and that $S_{\mathbb{R},l}$ evinces $r_{\mathbb{R}} (P_l)$.
This proves (\ref{aii}) of Theorem \ref{i1} in this case (1.1).

\vspace{0.1cm}
\quad (1.2) Assume $h^1(\mathcal {I}_{S_{\mathbb{C}}\cup {S_{\mathbb{R}}}}(d-1))>0$.

First of all, observe that there exists a line $r\subset \mathbb {P}^m$
such that $\sharp (r\cap (S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{\hat{l}})\ge d+1$, because $\sharp (S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{\hat{l}} \le 3d-1-d-2 \le 2(d-1)+1$.
\\
By the same reason, if we write: $C:=l\cup r$, we get that $\sharp (S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{\hat{C}}\le 3d-1 -d-2-d-1 \le d-2$ and hence   $h^1(\mathcal {I}_{(S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{\hat{C}}}(d-2))=0$
(e.g. by
\cite{bgi}, Lemma 34, or by \cite{c2}, Theorem 3.8). Lemma \ref{c2}
gives:
\begin{equation}\label{SChat}
S_{\mathbb{C},{\hat{C}}}= {S_{\mathbb{R},{\hat{C}}}}.
\end{equation}

-- Assume for the moment $l\cap r \ne \emptyset$.
In this case, Remark \ref{d1} indicates that we can consider case (\ref{b}) of the statement of the theorem. Therefore (\ref{SChat}) proves (\ref{bi}) in the case that the conic $C$ in (\ref{b}) in the statement of the theorem is reduced.
Moreover, condition (\ref{biv}) is satisfied, because $\sharp (r\cap (S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{\hat{l}}) \ge d+1$.

-- Now assume $l\cap r = \emptyset$.  We will check that we are in case (\ref{a}) with respect to the line $l$ if $\sharp (S_{\mathbb{C},r}\cup {S_{\mathbb{R},r}}) =d+1$, while we are in case (\ref{c}) with respect to the lines
$l$ and $r$ if $\sharp  (S_{\mathbb{C},r}\cup {S_{\mathbb{R},r}}) \ge d+2$, and the case $\sharp (S_{\mathbb{C},l}\cup {S_{\mathbb{R},l}}) = \sharp (S_{\mathbb{C},r}\cup {S_{\mathbb{R},r}})
= d+1$ cannot occur.

Set $\Gamma:=  l\cup r$. Since $r\cap l =\emptyset$, we have $\dim \langle \Gamma \rangle =3$ and hence
$m \ge 3$.

--- Assume for the moment $m\ge 4$.
Hence $\sharp (S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{\widehat{\langle \Gamma \rangle}} \le d$ and $h^1(\mathcal {I}_{ (S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{\widehat{\langle \Gamma \rangle}}}(d-1))=0$. Therefore:

\begin{equation}\label{CRGamma}
S_{\mathbb{C}, \widehat{\langle\Gamma\rangle}}= {S_{\mathbb{R}, \widehat{\langle\Gamma\rangle}}}
\end{equation}
and the set $\langle \{P\}\cup \nu _d(S_{\mathbb{C}, \widehat{\langle\Gamma\rangle}})\rangle \cap
\langle \nu _d(\langle \Gamma \rangle)\rangle$ is a single real point:

\begin{equation}\label{Oset}
O:=\langle \{P\}\cup \nu _d(S_{\mathbb{C}, \widehat{\langle\Gamma\rangle}})\rangle \cap
\langle \nu _d(\langle \Gamma \rangle)\rangle \in \mathbb{P}^N(\mathbb{R}),
\end{equation}
 $S_{\mathbb{C},\Gamma}$ evinces $r_{\mathbb{C}} (O)$
and ${S_{\mathbb{R}, \Gamma}}$ evinces $r_{\mathbb{R}} (O)$. Now, (\ref{CRGamma}) implies that if we are either in case (\ref{a}) or in case (\ref{c}) of the theorem, we can simply study what happens at $S_{\mathbb{C}, {\langle\Gamma\rangle}}$ and at $ {S_{\mathbb{R},{\langle\Gamma\rangle}}}$, which means that we can reduce our study to  the case  $m=3$,  since $\langle \Gamma \rangle= \mathbb{P}^3$.

--- Until step (\ref{2}) below, we will assume $m=3$.

The linear system $\vert \mathcal {I}_{\Gamma}(2)\vert$ on $\langle \Gamma \rangle$ has no base points outside $\Gamma$ itself. Since
$S_{\mathbb{C}}\cup {S_{\mathbb{R}}}$ is finite, there is a smooth quadric surface $W$ containing $\Gamma$ such that
$$S_{\mathbb{C},W}\cup {S_{\mathbb{R},W}} = S_{\mathbb{C},\Gamma}\cup {S_{\mathbb{R},\Gamma}}.$$
Moreover, such a $W$ can be found among the real smooth quadrics,  since $l$ and $r$ are real lines.\\

Since $\sharp (S_{\mathbb{C},\langle \Gamma \rangle}\cup {S_{\mathbb{R}, \langle \Gamma \rangle}})_{\hat{W}}
\le d-1$, we have $h^1(\mathcal {I}_{(S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{\hat{W}}}(d-2))=0$. Hence
Lemma \ref{c2} applied to the point $O$ defined in (\ref{Oset}),
gives:
$$(S_{\mathbb{C}, \langle \Gamma \rangle})_{\hat{W}}= (S_{\mathbb{R},\langle \Gamma \rangle})_{\hat{W}},$$
$\langle \{O\}\cup \nu _d(S_{\mathbb{C},\langle \Gamma , \rangle})_{\hat{W}}\rangle \cap \langle \nu _d(W)\rangle$ is a single real point
\begin{equation}\label{Opset}
O'=\langle \{O\}\cup \nu _d(S_{\mathbb{C},\langle \Gamma , \rangle})_{\hat{W}}\rangle \cap \langle \nu _d(W)\rangle\in \mathbb {P}^N(\mathbb {R}),
\end{equation}
and $S_{\mathbb{C},W}$ evinces
$r_{\mathbb{C}} (O')$.
If $(O',S_{\mathbb{C},W},{S_{\mathbb{R},W}})$ is either as in case (\ref{a}) or in case (\ref{c}) of the statement of the theorem, then $(O,S_{\mathbb{C}, \langle \Gamma \rangle},{S_{\mathbb{R}, \langle \Gamma \rangle}})$ is in the same case.
Consider the system $\vert (1,0)\vert $ of lines on the smooth quadric surface $W$ containing $\Gamma$. We have that $h^1(W,\mathcal {O}_W(d-2,d))=0 $, and hence the restriction map $H^0(W,\mathcal {O}_W(d))\to
H^0(\Gamma,\mathcal {O}_{\Gamma}(d))$ is surjective. Therefore
$$h^1(W,\mathcal {I}_{W\cap (S_{\mathbb{C}}\cup S_{\mathbb{R}})}(d)) = h^1(l,\mathcal {I}_{l\cap (S_{\mathbb{C}}\cup {S_{\mathbb{R}}}),l}(d))+h^1(r,\mathcal {I}_{r\cap (S_{\mathbb{C}}\cup {S_{\mathbb{R}}}),r}(d)).$$
Now, this last equality, together with the facts that $\nu _d(S_{\mathbb{C},W})$ and $\nu _d({S_{\mathbb{R},W}})$ are linearly independent and $(S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{W}
\subset \Gamma$, gives:
\begin{eqnarray}\label{eqa}
&\dim (\langle \nu _d(S_{\mathbb{C},W})\rangle \cap \langle \nu _d({S_{\mathbb{R},W}})\rangle) =
\sharp (S_{\mathbb{C}}\cap S_{\mathbb{R}})_l
+ \sharp (S_{\mathbb{C}}\cap S_{\mathbb{R}})_r+&\nonumber  \\
&h^1(l,\mathcal {I}_{l\cap (S_{\mathbb{C}}\cup {S_{\mathbb{R}}}),l}(d))+h^1(r,\mathcal {I}_{r\cap (S_{\mathbb{C}}\cup {S_{\mathbb{R}}}),r}(d)).
\end{eqnarray}

\vspace{0.1cm}
\quad (1.2.1) Observe that (\ref{eqa}) implies that the case $\sharp (S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{r}= \sharp (S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{l} =d+1$ cannot happen because there is no contribution from $h^1(l,\mathcal {I}_{l\cap (S_{\mathbb{C}}\cup {S_{\mathbb{R}}}),l}(d))+h^1(r,\mathcal {I}_{r\cap (S_{\mathbb{C}}\cup {S_{\mathbb{R}}}),r}(d))$ since both terms, in this case, are equal to 0. So, we can assume that at least $\sharp (S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{l} >d+1$.

\vspace{0.1cm}
\quad (1.2.2) Assume $\sharp (S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{r} =d+1$ and $\sharp (S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{l} >d+1$.
\\
To prove that we are in case (\ref{a}) with respect to $l$ it is sufficient to prove $S_{\mathbb{C},r} = {S_{\mathbb{R},r}}$.
\\
We have
$\dim \langle \nu _d(S_{\mathbb{C}}\cup {S_{\mathbb{R}}})\rangle = \sharp (S_{\mathbb{C}}\cup {S_{\mathbb{R}}})-1 -h^1(\mathcal {I}_{S_{\mathbb{C}}\cup {S_{\mathbb{R}}}}(d))$.
Since
$\nu _d(S_{\mathbb{C}})$ and $\nu _d({S_{\mathbb{R}}})$ are linearly independent, $\sharp (S_{\mathbb{C}}\cup {S_{\mathbb{R}}})=\sharp (S_{\mathbb{C}}) +\sharp ({S_{\mathbb{R}}})
-\sharp (S_{\mathbb{C}}\cap {S_{\mathbb{R}}})$,  $\sharp (S_{\mathbb{C},l}\cup {S_{\mathbb{R},l}})=\sharp (S_{\mathbb{C},l}) +\sharp (S_{\mathbb{R},l})-
\sharp (S_{\mathbb{C},l}\cap S_{\mathbb{R},l})$ and $h^1(\mathcal {I}_{S_{\mathbb{C}}\cup {S_{\mathbb{R}}}}(d))
= h^1(l,\mathcal {I}_{S_{\mathbb{C},l}\cup {S_{\mathbb{R},l}}})$, Grassmann's formula gives
that $\langle \nu _d(S_{\mathbb{C}})\rangle \cap \langle \nu _d({S_{\mathbb{R}}})\rangle$
is generated by $\langle \nu _d(S_{\mathbb{C},l})\rangle \cap \langle \nu _d(S_{\mathbb{R},l})\rangle$.
Since $P\notin \langle \nu _d(S')\rangle$ for any $S'\subsetneq S_{\mathbb{C}}$, we get
$S_{\mathbb{C}} = S_{\mathbb{C},l} \cup (S_{\mathbb{C}}\cap {S_{\mathbb{R}}})_{\hat{l}}$, i.e. $S_{\hat{l}} = {S_{\mathbb{R}}}\setminus S_{\mathbb{C},l}$. Hence
we can consider case (\ref{a}), and (\ref{SChat}) proves property (\ref{ai}) also for the case (1.2) that we are treating. The
point $P_l$ that we need to get (aii) can be identified with the point $O'$ defined in (\ref{Opset}) while (aiii) comes from
our hypotheses.

This gives all cases (\ref{a}) of Theorem \ref{i1}.

\vspace{0.1cm}
\quad (1.2.3) Assume that both $\sharp  (S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{r} \ge d+2$ and $\sharp (S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{l} \geq d+2$.
\\
We need to prove that we are in case
(\ref{c}).
Recall that $S_{\mathbb{C},{\hat{\Gamma}}} = S_{\mathbb{R},{\hat{\Gamma}}}$
and that $h^1(\mathcal {I}_{S_{\mathbb{C}, {\hat{\Gamma}}}\cup \Gamma}(d))=0$. The latter
equality implies, as in Remark \ref{d1}, that $\langle \{P\}\cup \nu _d(S_{\mathbb{C},{\Gamma}})\rangle
\cap \langle \nu _d(\Gamma)\rangle$ is a single real point $O_1$,
that $S_{\mathbb{C}, \Gamma}$ evinces $r_{\mathbb{C}}(O_1)$ and that ${S_{\mathbb{R},\Gamma}}$ evinces $O_1$. Now $O_1$ plays the role of $O_{\Gamma}$ of case (\ref{ciii}) in Theorem \ref{i1}.

Since
$\langle \nu _d(l)\rangle \cap \langle \nu_d(r)\rangle =\emptyset$ and $O_1\in \langle \nu _d(\Gamma)\rangle$,
the sets $\langle \{O_1\}\cup \nu _d(l)\rangle \cap \langle \nu _d(l)\rangle$ (resp. $\langle \{O_1\}\cup \nu _d(r)\rangle \cap
\langle \nu _d(r)\rangle$) are formed by a unique point $O_2$ (resp. $O_3$).
Remark \ref{d1} gives that $O_i\in \mathbb {P}^N(\mathbb {R})$, $i=1,2$, $S_{\mathbb{C},l}$ evinces $r_{\mathbb{C}} (O_2)$,
$S_{\mathbb{R},l}$ evinces $r_{\mathbb{R}} (O_2)$,  $S_{\mathbb{C},r}$ evinces $r_{\mathbb{C}} (O_3)$ and  ${S_{\mathbb{R},r}}$ evinces $r_{\mathbb{R}} (O_3)$.
The hypotheses of the case (1.2.3) coincide with (\ref{cii}) of the statement of the theorem, while (\ref{SChat}) gives
also property (\ref{ci}). Moreover $O_2$ and $O_3$ defined above coincide with $O_l$ and $O_r$ in (\ref{civ}) of Theorem \ref{i1}, therefore we have also proved case (\ref{c}) of Theorem \ref{i1}.

\begin{enumerate}
\setcounter{enumi}{\themycounter}

\item\label{2}  Now assume the existence of a conic $C\subset \mathbb {P}^m$ such that
\begin{equation}\label{CRC}
\deg (S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_C \ge 2d+2.
\end{equation}

\end{enumerate}

Since $\sharp (S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_C\le 3d-1-2d-2 \le d-1$, we
have $h^1(\mathcal {I}_{(S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{\hat{C}}}(d-2))=0$. By Lemma \ref{c2}
we have
\begin{equation}\label{SCC}
S_{\mathbb{C}, \hat{C}} = {S_{\mathbb{R}, \hat{C}}},
\end{equation}
the set $\langle \{P\} \cup \nu_d(S_{\mathbb{C}, \hat{C}})\rangle  \cap
\langle \nu _d(\langle C \rangle)\rangle$ is a single point:
\begin{equation}\label{Pp}
P':=\langle \{P\} \cup \nu_d(S_{\mathbb{C}, \hat{C}})\rangle  \cap
\langle \nu _d(\langle C \rangle)\rangle
\end{equation}
and $S_{\mathbb{C},C}$ evinces
$r_{\mathbb{C}} (P')$. Moreover, if $C$ is defined over $\mathbb {R}$, then $P'\in \mathbb {P}^N(\mathbb {R})$ and $S_{\mathbb{R},C}$ evinces $r_{\mathbb{R}} (P')$. Hence $\sharp (S_{\mathbb{C},C})< \sharp (S_{\mathbb{C}}\cap {S_{\mathbb{R}}})$.

\vspace{0.1cm}
\quad (2.1) Assume that $C$ is smooth.  Therefore (\ref{SCC}) proves (\ref{bi}) of the statement of the theorem in the case where $C$ is smooth. Since the reduced case is proved above (immediately after the displayed formula (\ref{SChat})), we have concluded the proof of (\ref{bi}).

Moreover the hypothesis (\ref{CRC}) coincides with (\ref{biii}) of the statement of the theorem since $\sharp (S_{\mathbb{C},C})$ is obviously strictly smaller than $ \sharp (S_{\mathbb{R},C})$. This concludes (\ref{biii}).

The fact that $\sharp (S_{\mathbb{C},C})< \sharp (S_{\mathbb{R},C})$ also implies that
$\sharp (S_{\mathbb{R},C}) \ge 5$. Since each point of ${S_{\mathbb{R}}}$ is real, $C$ is real. Remark \ref{d1} gives
that $ \nu_d({S_{\mathbb{R},C}})$ evinces $r_{\mathbb{R}} (P')$. Since ${S_{\mathbb{R},C}} \subset C$, ${S_{\mathbb{R},C}}$ also evinces
the real symmetric tensor rank of $P'$ with respect to the degree $2d$ rational normal
curve $\nu _d(C)$.
The point $P'$ defined in (\ref{Pp}) plays the role of the point $P_C$ appearing in (\ref{bii}) of the statement of the theorem. Therefore, we have just proved  (\ref{bii}) of Theorem \ref{i1}.

We treat the case (2.2) below for sake of completeness, but we can observe that this concludes the proof of Theorem \ref{i1}.

\vspace{0.1cm}
\quad (2.2) Assume that $C$ is reducible, say $C =L_1\cup L_2$ with $L_1$ and $L_2$ lines
and $\sharp ((S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{L_1}) \ge \sharp ((S_{\mathbb{C}}\cap {S_{\mathbb{R}}})_{L_2})$.
 If $\sharp ((S_{\mathbb{C}}\cup {S_{\mathbb{R}}})\cap (L_2\setminus L_2\cap L_1)) \le d$, then we proved in step  (\ref{1}) that we are
in case (\ref{a}) with respect to the line $L_1$. Hence we may assume $\sharp ((S_{\mathbb{C}}\cup {S_{\mathbb{R}}})\cap (L_2\setminus L_2\cap L_1)) \ge d+1$. Thus even condition  (\ref{biv}) is satisfied as already remarked above after the displayed formula (\ref{SChat}).
\endproof

\vspace{0.3cm}

\def\fine{

We analyzed in step (a) all cases arising
if there is a line $L\subset C$ such that $\sharp (L\cap C) \ge d+2$. Hence we may assume
that there is no such line. In particular $C$ is not a double line. If $C$ is smooth, then $C$ is defined over $\mathbb {R}$, because $\sharp ( {S_{\mathbb{R},C}})\ge d+2 \ge 5$ and each point of ${S_{\mathbb{R}}}$ is real. If $C$ is reducible, say $C = C_1\cup C_2$, the Since $\sharp (S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_C\le 3d-1-2d-2 \le d-1$, we
have $h^1(\mathcal {I}_{(S_{\mathbb{C}}\cup {S_{\mathbb{R}}})_{\hat{C}}}(d-2))=0$.

\begin{lemma}\label{c1}
Fix a line $l\subset \mathbb {P}^m$ defined
over $\mathbb {R}$, sets ${S_{\mathbb{C}}}_1\subset l(\mathbb {C})$, ${S_{\mathbb{R}}}_1\subset l(\mathbb {R})$,
$E\subset \mathbb {P}^m(\mathbb {R})\setminus l(\mathbb {R})$ and
$P_l\in \langle l\rangle (\mathbb {R})$ such that ${S_{\mathbb{C}}}_1\ne {S_{\mathbb{R}}}_1$, $\sharp (E) \le (d-1)/2$, ${S_{\mathbb{C}}}_1$ evinces $Crs (P_l)$
with respect to the rational normal curve $\nu _d(l)$ and ${S_{\mathbb{R}}}_1$ evinces $r_{\mathbb{R}} (P_l)$ with
respect to the rational normal curve curve $\nu _d(l)$ and that $\sharp ({S_{\mathbb{C}}}_1)<\sharp ({S_{\mathbb{R}}}_1)$. Fix
$P\in \langle \nu _d(E)\cup \{P_l\}\rangle (\mathbb {R})$ such that $P\notin \langle G\rangle$
for any $G\subsetneq \nu _d (E)\cup \{P_l\}$. Set $S_{\mathbb{C}}:= E\cup {S_{\mathbb{C}}}_1$ and ${S_{\mathbb{R}}}:= E\cup {S_{\mathbb{R}}}_1$.
Assume $\sharp (S_{\mathbb{C}})+\sharp ({S_{\mathbb{R}}}) \le 2d+2$ and $\sharp ({S_{\mathbb{C}}}_1)<\sharp ({S_{\mathbb{R}}}_1)$.

Then $S_{\mathbb{C}}$ evinces $r_{\mathbb{C}} (P)$ and ${S_{\mathbb{R}}}$ evinces $r_{\mathbb{R}} (P)$.
\end{lemma}

\begin{proof}
We first prove the lemma for $S_{\mathbb{C}}$. Assume that it is not true and take $F\subset \mathbb {P}^m(\mathbb {C})$ evincing $Crs (P)$. Since $\sharp (F) < \sharp (S_{\mathbb{C}}) \le d$, we
have $\sharp (S_{\mathbb{C}}\cup F) \le 2d-1$. Since $h^1(\mathcal {I}_{S_{\mathbb{C}}\cup F}(d)) >0$, we
get the existence of a line $r\subset \mathbb {P}^m$ such that $\sharp ((S_{\mathbb{C}}\cup F)) \ge d+2$.
Since $h^1(\mathcal {I}_{S_{\mathbb{C}}\cup F}(d)) >0$ (\cite{bb}, Lemma 1), there is a line $r\subset \mathbb {P}^m$
such that $\sharp (r\cap (S_{\mathbb{C}}\cup F)) \ge d+2$. Since $\sharp (S_{\mathbb{C}}\cup F\setminus (S_{\mathbb{C}}\cup F)\cap r))
\le d-1$, we have $h^1(\mathcal {I}_{S_{\mathbb{C}}\cup F\setminus (S_{\mathbb{C}}\cup F)\cap r)}(d-1))=0$.
Hence $F\setminus F\cap r = S_{\mathbb{C}}\setminus S_{\mathbb{C}}\cap r$. Since $\sharp (F)<\sharp (S_{\mathbb{C}})$, we
have $\sharp (F\cap r) < \sharp (S_{\mathbb{C}}\cap r)$. Hence $\sharp (S_{\mathbb{C}}\cap r) > (d+2)/2$. First assume $r\ne l$. Since
$\sharp (r\cap l) \le 1$, we have
$\sharp (E) \ge d/2$, a contradiction. Now assume $r=l$. Since $P\in \langle
\nu _d(F)\rangle$ and $P\notin \langle
\nu _d(F)\rangle$, the set $\langle \nu _d(F\setminus F\cap l)\cup \{P\}\langle \cap
\langle \nu _d(F\cap l)\rangle$ is a single point (call it $P_C)$. For
the same reason we have $\{P_l\} = \langle \nu _d(S_{\hat{l}})\cup \{P\}\langle \cap
\langle \nu _d(S_{\mathbb{C},l})\rangle$. Since $F\setminus F\cap l = S_{\hat{l}}$,
we have $P_C=P_l$. Since ${S_{\mathbb{C}}}_1$ evinces $r_{\mathbb{C}} (P_l)$, we get $\sharp (F\cap l)\ge \sharp
(S_{\mathbb{C}}\cap F)$. Hence $\sharp (F)\ge \sharp (S_{\mathbb{C}})$, a contradiction.

Now we prove the lemma for ${S_{\mathbb{R}}}$, using that it is true for $S_{\mathbb{C}}$.
Obviously ${S_{\mathbb{R}}}\subset \mathbb {P}^m(\mathbb {R})$ and $\sharp ({S_{\mathbb{R}}}) = \sharp ({S_{\mathbb{R}}}_1)+\sharp (E)$. We have
that $\sharp ({S_{\mathbb{R}}}_1)\le d$ (\cite{co}, ) and hence $\sharp ({S_{\mathbb{R}}}) \le (3d-1)/2$. By the real case of \cite{l}, Exercise 3.2.2.2, ${S_{\mathbb{R}}}_1$ evinces $r_{\mathbb{R}} (P_l)$. Assume that ${S_{\mathbb{R}}}$ does not
evinces $r_{\mathbb{R}} (P)$ and take $G\subset \mathbb {P}^m(\mathbb {R})$ such that $\sharp (G)< \sharp ({S_{\mathbb{R}}})$ and $G$ evinces $r_{\mathbb{R}} (P)$. First assume  $h^1(\mathcal {I}_{G\cup S_{\mathbb{C}}}(d)) >0$. Since
$\sharp (S_{\mathbb{C}})+\sharp (G) \le 2d$, there is a line $C_1$ such
that $\sharp (C_1\cap (S_{\mathbb{C}}\cup G)) \ge d+2$. Since
$\sharp (S_{\mathbb{C}}\cup G)-\sharp ((S_{\mathbb{C}}\cup G)\cap C_1)
\le 2d-d-2 \le d$, we have $h^1(\mathcal {I}_{(S_{\mathbb{C}}\cup G)\setminus (S_{\mathbb{C}}\cup G)\cap C_1)} (d-1)) =0$. Lemma \ref{c2} gives $G\setminus G\cap C_1 = S_{\mathbb{C}}\setminus {S_{\mathbb{C},C}}_1$. Hence
each point of $S_{\mathbb{C}}\setminus C_1$ is real. First assume
$C_1=l$. Hence $G\setminus G\cap l = E$. Since $S_{\mathbb{C}}$ evinces $r_{\mathbb{C}} (P)$, we
have $\sharp (G\cap l) \ge \sharp ({S_{\mathbb{C}}}_1)$ and in particular, $G\cap l \ne \emptyset$. Since
$G$ evinces $r_{\mathbb{R}} (P)$, the set $\langle \nu _d(G\cap l)\rangle \cap \langle \{P\} \cup E\rangle$
is a single point, $P_C$, and $G\cap l$ evinces $r_{\mathbb{R}} (P_C)$. Since
$h^1(\mathcal {I}_{(S_{\mathbb{C}}\cup G)\setminus (S_{\mathbb{C}}\cup G)\cap C_1)} (d-1)) =0$ and $C_1=l$, we have
$\langle \nu _d(l)\rangle \cap \langle \nu _d(E)\rangle =\emptyset$. Hence
$\langle \nu _d(l)\rangle \cap \langle \nu _d(E)\cup \{P\}\rangle$ is at most one point.
Since $P_l\in \langle \nu _d(l)\rangle \cap \langle \nu _d(E)\cup \{P\}\rangle$, we get
$P_l=P_C$. Hence $G\cap l$ evinces $r_{\mathbb{R}} (P_l)$. Hence $\sharp (G\cap l) =\sharp ({S_{\mathbb{R}}}_1)$.
Hence $\sharp (G)=\sharp ({S_{\mathbb{R}}})$, a contradiction.
Now assume $C_1\ne l$. Since $\sharp (C_1\cap l)\le 1$, and each point
of $E = S_{\hat{l}}$ is real, at most one point
of $S_{\mathbb{C}}$ is not real. Since $\sigma (S_{\mathbb{C}})=S_{\mathbb{C}}$, we get that
each point of $S_{\mathbb{C}}$ is real. Hence $Crs (P) =r_{\mathbb{R}} (P)$. Since
${S_{\mathbb{C}}}_1$ evinces $r_{\mathbb{C}} (P_l)$ and each point of $S_{\mathbb{C}}$ is real, we also get
$r_{\mathbb{R}} (P_l)=r_{\mathbb{C}} (P_l)$, i.e. $\sharp ({S_{\mathbb{C}}}_1)=\sharp ({S_{\mathbb{R}}}_1)$, i.e. $\sharp (S_{\mathbb{C}})=\sharp ({S_{\mathbb{R}}})$, a contradiction.
\end{proof}

}

\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}

redpajama_set_name: RedPajamaArXiv


text:
Beatles music to be free after 2013
By James Rivington published 24 July 07
Music older than 50 years released to public domain
If you create your own music, by the time the Earth orbits the sun 50 times, your neighbour will be able to pinch it for free
The UK government has rejected plans to extend copyright ownership on sound recordings from 50 to 70 years. This means that all music recorded will be released into the public domain after just 50 years.
The real world effect this will have is that ownership of The Beatles ' debut album 'Please Please Me', for example, which spent 30 weeks at the top of the album chart in 1963, will be taken away from the band and record label in 2013. So in six years time, that album will belong to you just as much as it does them.
John Kennedy, chairman and chief executive of IFPI ( International Federation of the Phonographic Industry ), responded in absolute dismay.
"The UK is a world-beating source of great music," he said. "So it is frustrating that on the issue of copyright term the government has shown scant respect for British artists and the UK recording industry.
Copyright conundrum
"Some of the greatest works of British music will soon be taken away from the artists who performed them and the companies that invested in them. Extending copyright term would promote vital investment in young talent and new music, all of which will help to secure the UK's future as an exciting music market."
In May, a Select Committee recommended that the government press the case with the European Commission for extending sound recording copyright term. But the government has elected not to do this, sparking anger from musicians.
Roger Daltry of legendary rock band The Who said: "Thousands of musicians have no pensions and rely on royalties to support themselves. These people helped to create one of Britain's most successful industries, poured money into the British economy and enriched people's lives. They are not asking for a handout, just a fair reward for their creative endeavours."
James Rivington
James was part of the TechRadar editorial team for eight years up until 2015 and now works in a senior position for TR's parent company Future.
See more Hi-fi & radio news


text:
Asterix: Asterix in Britain: Album 8 (#8, Astérix)
Paperback published 2004-09-30 in United Kingdom by Hachette
Publisher: Hachette
Published: 30 September, 2004
Australian Libraries (Trove)
Editions: 8 other editions of this product
Part of Astérix (Series)
Others in this series:
1 Asterix the Gaul
2 Asterix and the Golden Sickle
3 Asterix and the Goths
4 Asterix the Gladiator
5 Asterix and the Banquet
6 Asterix and Cleopatra
7 Asterix and the Big Fight
8 Asterix in Britain
9 Asterix and the Normans
10 Asterix the Legionary: Book 10
11 Asterix and the Chieftain's Shield
12 Asterix at the Olympic Games
13 Asterix and the Cauldron
14 Asterix in Spain
15 Asterix and the Roman Agent
16 Asterix in Switzerland
17 The Mansions of the Gods
18 Asterix and the Laurel Wreath
19 Asterix and the Soothsayer
20 Asterix in Corsica
21 Asterix and Caesar's Gift
22 Asterix and the Great Crossing
23 Obelix and Co
24 Asterix in Belgium
25 Asterix and the Great Divide
26 Asterix and the Black Gold
27 Asterix and Son
28 Asterix and the Magic Carpet
30 Asterix and Obelix All at Sea
31 Asterix and the Actress
32 Asterix and the Class Act
33 Asterix and the Falling Sky
34 Asterix and Obelix's Birthday
Paperback published September 2004 in United Kingdom
Gaul was divided into three parts. No, four parts - for one small village of indomitable Gauls still held out against the Roman invaders. BOOK 8 The Romans have invaded Britain but one village still holds out against them. At his British cousin Anticlimax's request, Asterix and Obelix set off to help with a barrel full of magic potion.