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Sun Chemical Selects OpenText to Improve Customer Service and Experience with Machine Learning
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OpenText Business Center for SAP® Digitizes Sales Order Processes to Increase Operational Efficiency
Waterloo, ON – 2018-11-19 – OpenText™ (NASDAQ: OTEX, TSX: OTEX), a global leader in Enterprise Information Management (EIM), today announced that Sun Chemical, a member of the DIC group, has selected OpenText™ Business Center for SAP® Solutions to transform their customer experience and service. With OpenText Business Center, Sun Chemical can automate data-heavy customer service processes, freeing more time for high-value customer engagement. Advanced machine learning will reduce setup effort and drive greater automation and efficiency over time.
"Customer service representatives spend an average of 30 to 40 percent of their time on data entry," said Carrie Proffit, Director of Customer Services at Sun Chemical. "Applying technology from OpenText will allow us to automate the time-intensive sales order process, and with machine learning capability, to continuously improve efficiency. More importantly, this will enable our team to spend more of their time delivering a truly exceptional level of customer service."
Sun Chemical is a recognized leader in printing inks, coatings and supplies for the packaging, publication, commercial, industrial, and digital markets. Together with DIC, Sun Chemical has annual sales of more than $7.5 billion and over 20,000 employees located in 63 countries. The deployment of OpenText Business Center for SAP Solutions will help Sun Chemical customer service representatives to focus on higher value sales and customer-facing service, rather than manual data entry, which is both time consuming and prone to human error.
OpenText Business Center for SAP Solutions provides tools necessary to move from manual, document-centric processes to fully automated digital processes more quickly and effectively. The solution captures and validates information from multiple incoming source to initiate the SAP process. Deep integration with SAP facilitates accuracy of data, while advanced machine learning minimizes setup effort and drives higher automation rates over time. Digitizing processes with OpenText Business Center assists our customers in preventing costly re-work due to input errors and helps keep their sales and service employees where they belong, working with customers.
"We're committed to improving performance on the essentials of business," said Simone Edwards, Global Business Process Analyst at Sun Chemical. "The fact that we will be able to customize and adapt the solution to our unique business needs, with the OpenText Professional Services team available to help with deployment, made OpenText the ideal choice."
To learn more about OpenText Business Center for SAP Solutions, visit: https://www.opentext.com/products-and-solutions/products/opentext-suite-for-sap/opentext-business-center-for-sap-solutions.
To learn more about OpenText Professional Services, visit: https://www.opentext.com/products-and-solutions/services.
About OpenText
OpenText, The Information Company™, enables organizations to gain insight through market leading information management solutions, on-premises or in the cloud. For more information about OpenText (NASDAQ: OTEX, TSX: OTEX) visit opentext.com.
OpenText CEO Mark Barrenechea's blog
Certain statements in this press release may contain words considered forward-looking statements or information under applicable securities laws. These statements are based on OpenText's current expectations, estimates, forecasts and projections about the operating environment, economies and markets in which the company operates. These statements are subject to important assumptions, risks and uncertainties that are difficult to predict, and the actual outcome may be materially different. OpenText's assumptions, although considered reasonable by the company at the date of this press release, may prove to be inaccurate and consequently its actual results could differ materially from the expectations set out herein. For additional information with respect to risks and other factors which could occur, see OpenText's Annual Report on Form 10-K, Quarterly Reports on Form 10-Q and other securities filings with the SEC and other securities regulators. Unless otherwise required by applicable securities laws, OpenText disclaims any intention or obligations to update or revise any forward-looking statements, whether as a result of new information, future events or otherwise.
Copyright © 2018 Open Text. All rights reserved. OpenText is a trademark or registered trademark of Open Text. The list of trademarks is not exhaustive of other trademarks. Registered trademarks, product names, company names, brands and service names mentioned herein are property of Open Text or other respective owners.
Austin DeArman
publicrelations@opentext.com


Ronne Ostby is an experienced social marketing professional, well-known and respected within the discipline. She has a passion for studying and applying social marketing and behavior change concepts and theories to society's most critical health and well-being concerns. She has worked for nearly 15 years in the fields of social marketing and communications with proven expertise in behavioral theory innovation, strategy development, research and evaluation, creative planning and project leadership.At ICF she leads social marketing strategy design and formative audience-based research and manages integrated communications efforts.
Prior to joining ICF, Ms. Ostby led strategy development through in-depth consumer research for the White House National Youth Anti-Drug Media Campaign. Prior to joining the Executive Office of the President, Ronne was a senior director in AED's Center for Social Marketing and Behavior Change. She led social marketing research, program development, and evaluation efforts, working with numerous U.S. federal agencies, including the Centers for Disease Control and Prevention (CDC) and National Institutes of Health (NIH), and several national nonprofit foundations. She has particular experience with issues affecting teens and tweens, including drug and tobacco use prevention and obesity prevention. She is an adjunct faculty member at American University, teaching Social Marketing and other graduate-level courses, and serves as Associate Editor for Social Marketing Quarterly.


Light up your Beavers pride in your home or office with this beautiful Tiffany table lamp. With classic-style, this lamp boasts a hand-cut Oregon State-themed stained glass shade and an elegant cast metal base. It will warm up any room with the soft glow of Oregon State Beavers spirit!
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Everyone in the real estate industry agrees that in today's market, be it hot or cold, for first-time buyers or luxury vacation homes, a Realtor must have a website to be competitive. But that's about the only thing there's agreement on. The sometimes conflicting information that is available gives the impression that how to go about developing an effective web presence is open to interpretation. Do a little research and it quickly becomes evident there are as many opinions about what defines a "good" website (and plenty of self-proclaimed experts to offer them) as there are pages on the Internet.
Dakno's experience developing countless fully customized websites has taught us there is only one factor that truly sets a real estate website apart: leads. When we develop a website with a client, no matter its size or scope, we evaluate every one of the hundreds of decisions made during the process by asking if it will increase the chances of capturing a lead. If the answer to this simple question is no, we'll tell you so, even if it isn't necessarily what you'd like to hear.
We conduct ourselves, both as individuals and as a company, in keeping with this simple truth: people don't like to be manipulated. We understand that if instead you provide them all the information they need to make an informed decision, they will appreciate your openness and reward you with their business now, and perhaps even more importantly, in the future.
We also understand that trust is earned, not given, which is why we take the time to explain to clients the reasons behind the things we do when designing, developing and promoting websites. Expertise is what sets a professional apart, and by sharing our understanding of the sometimes overwhelmingly complicated world of the Internet, we provide the facts you need to confidently decide which direction to take your website and business.
Too often, in the interest of time (or because it's just that much easier), service providers will take the "just trust me" route. As the buying public, we usually feel uneasy with this arrangement, but since we're in a position of need, and because we often lack the knowledge and understanding needed to represent our own best interests, we go along and take the expert's word for it. After all, they do it for a living, right?
Well, we all know how that story ends. And maybe the most aggravating thing when this happens is that infuriating feeling you get when you eventually "know better" and look back and see that you've been had. At Dakno we will never use a client's ignorance against them. You will always be told why we make our recommendations and have the option to accept our guidance or to go your own way. At the end of the day the customer may not always be right, but they do sign the checks, so they certainly get want they want!
Why Should Another Realtor's Success Matters to You?
Dakno Admin, the control panel we developed that allows you to enhance, grow and manage your Dakno website, is the result of many years of using this approach to real estate website development. The custom programming of numerous top-producing Realtors from around the United States and Canada we have worked with has been distilled down and refined to include the most sought after, valued, and proven functionality that makes a website a powerful selling tool.
And because our understanding of effective website design is constantly being refined, so is our product line and services. Dakno Admin, like all the best business systems, is a work in progress, and as improvements are made due to insights we gain from working with our many savvy and driven partners, the resulting enhancements are passed along to you automatically and free of charge.
At a certain point a company's record of success begins to speak for itself, but while we are proud of our accomplishments, at Dakno we tend to be more concerned with looking ahead than back. Of course evaluating what you have done is crucial in business, which is why we are big believers in constantly analyzing a website's ongoing performance, looking for ways to improve and refine it.
By constantly seeking to understand why an individual website succeeds at generating leads, we learn the techniques that allow us to replicate that success for all of our clients. We apply this approach to every aspect of our company in an ongoing effort to improve the quality of our work. Please take the time to browse our website to get a better understanding of the ways we strive to earn your trust first, and your business as a result. And as always, don't hesitate to contact us if you have a question that isn't addressed, would like clarification on a topic, or are ready to get started. Thank you for your valuable time.


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Faculty of Mathematics and Informatics
Academic Staff A-Z
Faculty Structure
Vice-deans
Department of Student Еducation
Academic staff, project and science work
Teachers in English
Information Services Laboratory
Doctoral degree programmes
Mathematical Guide
Tournaments and competitions for students
Programmes by speciality
Research structures and people
Students in research
Events and Cooperation
Annual of the FMI
Information system Authors
IS Authors
Internal - funded by SU
Projects Information system (in Bulgarian)
Research calls information
Research Memberships
FMI Colloquium
Conferences organized by FMI
Links with research organizations
Faculty - Industry Relations
Research data bases – ISI, Scopus, …
Начало Academic Staff A-Z Stefan Petrov Ivanov
Stefan Petrov Ivanov
ivanovsp@fmi.uni-sofia.bg
FzF-05
Publications and Quotes
Extremals of the Sobolev inequality and the quaternionic contact Yamabe problem, (with D. Vassilev), World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. xviii+219 pp.
Quaternionic contact Einstein structures and quaternionic contact Yamabe problem, (with I. Minchev and D. Vassilev), Memoirs Amer. Math. Soc. (2014), 231, no. 1086.
Survey papers
The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds, (with Dimiter Vassilev), Nonlinear Analysis - Theory 126 (2015), 262-323. arXiv:1504.03259.
Quaternionic contact hypersurfaces in hyper-K\"ahler manifolds, (with Ivan Minchev, Dimiter Vassilev), to appear in Annali di matematica Pura ed Applicata. arXiv:1406.4256.
Quaternionic contact Einstein manifolds , (with Ivan Minchev and Dimiter Vassilev), to appear in Math. Research Letters. arXiv:1306.0474.
Connections on non-symmetric (generalized) Riemannian manifold and gravity, (with Milan Zlatanovic), to appear in Class. Quntum Gravity. arXiv:1503.05217.
The Obata sphere theorems on a quaternionic contact manifold of dimension bigger than seven , (with Alexander Petkov and Dimiter Vassilev), to appear in J. Spectral Theory. arXiv:1303.0409.
The quaternionic Heisenberg group and Heterotic String Solutions with non-constant dilaton in dimensions 7 and 5 , (with Marisa Fernandez, Luis Ugarte, Dimiter Vassilev), Comm. Math. Phys. 339 (2015), no. 1, 199–219. arXiv:arXiv:1410.4130.
Non-Kaehler Heterotic String Solutions with non-zero fluxes and non-constant dilaton , (with Marisa Fernandez, Luis Ugarte, Dimiter vassilev), J. High Energy Physics (JHEP) 06 (2014) 073; arXiv:arXiv:1402.6107.
Vanishing theorems on $(l|k)$-strong Kaehler manifolds with torsion, (with George Papadopoulos), Adv. Math. 237 (2013), 147-164. arXiv: arXiv:1202.6470 .
An Obata type result for the first eigenvalue of the sub-Laplacian on a CR manifold with a divergence free torsion , (with Dimiter Vassilev), J. Geom 103, 3 (2012), 475-504. arXiv: arXiv:1203.5812.
An Obata-type theorem on a three-dimensional CR manifold,(with Dimiter Vassilev), Glasgow Math. J., 56 (2014), 283-294. arXiv: arXiv:1208.1040.
The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold in dimension seven (with Alexander Petkov and Dimiter Vassilev), Nonlinear Analysis-Theory, 93 (2013), 51-61. arXiv:arXiv:1210.6932.
The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold , (with Alexander Petkov and Dimiter Vassilev), J. Geom. Analysis, 24 (2014), no. 2, 595-612. arXiv:arXiv:1112.0779.
The optimal constant in the $L^2$ Folland-Stein inequality on the quaternionic Heisenberg group, (with I. Minchev and D. Vassilev), Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XI (2012), 635-652; arXiv:arXiv:1009.2978.
Quaternionic Kaehler and Spin(7) metrics arising from quaternionic contact Einstein structures, (with L. C. de Andres, M. Fernandez, J. A. Santisteban, L. Ugarte and D. Vassilev), Annali di matematica Pura ed Applicata, 193, 1 (2014), 261-290. arXiv:arXiv:1009.2745.
Bianchi type A hyper-symplectic and hyper-K\"ahler metrics in 4D , (with Luis C. de Andre's, Marisa Ferna'ndez, Jose A. Santisteban, Luis Ugarte and Dimiter Vassilev), Class. Quant. Grav. 29 (2012), 025003. arXiv:arXiv:1102.1518.
Compact supersymmetric solutions of the heterotic equations of motion in dimensions 7 and 8, (with M. Ferna'ndez, L. Ugarte and R. Villacampa), Adv. Theor. Math. Phys., (2011) 15, 245-284. arXiv:0806.4356.
HKT manifolds with holonomy SL(n,H), (with A. Petkov), International Mathematics Research Notices, 2012, 16, 3779--3799; arXiv:arXiv:1010.5052.
The twistor space of a quaternionic contact manifold, (with J. Davidov and I. Minchev), Quart. J. Math. 63 (2012), no. 4, 873-890. arXiv:arXiv:1010.4994.
Heterotic supersymmetry, anomaly cancellation and equations of motion, Phys. Lett. B685 (2010), 190-196. arXiv:0908.2927.
Quaternionic contact manifolds with a closed fundamental 4-form, (with D. Vassilev), Bull. London Math. Soc., 42 (6) (2010), 1021–1030. arXiv:0810.3888,.
Conformal quaternionic contact curvature and the local sphere theorem, (with D. Vassilev), Journal de Mathe'matiques Pures et Applique'es (JMPA) 93 (2010), 277-307. arXiv:0707.1289.
Extremals for the Sobolev inequality on the seven dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem, (with I. Minchev and D. Vassilev), J. Eur. Math. Soc. (JEMS) 12 (2010), 1041–1067. math.DG/0703044
Compact supersymmetric solutions of the heterotic equations of motion in dimension 5, (with M. Ferna'ndez, L. Ugarte and R. Villacampa), Nuclear Physics B820 (2009), 483-502. arXiv:0811.2137.
Non-Kaehler Heterotic String Compactifications with non-zero fluxes and constant dilaton, (with M. Ferna'ndez, L. Ugarte and R. Villacampa), Comm. Math. Phys. 288 (2009), 677-697. arXiv:0804.1648
Conformal Paracontact curvature and the local flatness theorem, (with D. Vassilev and S. Zamkovoy), Geom. Dedicata 144 (2010), 79-100. arXiv:0707.3773
Curvature decomposition of $G_2$ manifolds (with R. Cleyton), J. Geom. Phys. 58 (2008), 1429-1449. math.DG/0702289.
Nearly hypo structures and compact Nearly K\"ahler 6-manifolds with conical singularities (with M. Fernandez, V. Munos and L. Ugarte), J. London Math. Soc. 78 (2008), 580-604. math.DG/0602160.
Conformal equivalence between certain geometries in dimension 6 and 7, (with R. Cleyton), Math. Res. Lett. 15 (2008), 631-641. math.DG/0607487.
Twistor and Reflector Spaces of Almost Para-Quaternionic Manifolds, (with I. Minchev and S. Zamkovoy), to appear in "Handbook of pseudo-Riemannian Geometry and Supersymmetry" to be published by the European Mathematical Society EMS in the series IRMA Lectures in Mathematics and Theoretical Physics. math.DG/0511657.
SU(3)-structures on submanifolds of a Spin(7)-manifold, (with F. Martin Cabrera), Diff. Geom. Appl. 26 (2008), 113-132. math.DG/0510406.
On the geometry of closed $G_2$-structures, (with R. Cleyton), Commun. Math. Phys., 270 (2007), 53--67. math.DG/0306362.
Locally conformal parallel $G_2$ and $Spin(7)$ manifolds, (with M. Parton and P. Piccinni), Math. Res. Lett., 13 (2006), 167--177. math.DG/0509038.
Hyper-ParaHermitian manifolds with torsion, (with V. Tsanov and S. Zamkovoy), J. Geom. Phys. 56, (2006), 670-690. math.DG/0405585.
Para-Hermitian and Para-Quaternionic manifolds, (with S. Zamkovoy), Diff. Geom. Appl. 23 (2005), 205-234. math.DG/0310415.
$SU(3)$-instantons and $G_2, Spin(7)$ heterotic string solitons, (with P. Ivanov), Commun. Math. Phys. 259 (2005), 79-102. math.DG/0312094.
Connection with torsion, parallel spinors and geometry of Spin(7) manifolds, Math. Res. Lett. 11 (2004), 171--186. math.DG/0111216.
Quaternionic K\"ahler and hyperK\"ahler manifolds with torsion and twistor spaces, (with I. Minchev), J. reine angew. Math. 567 (2004), 215-233, math.DG/0112157.
Deformations of generalized calibrations and compact non-Kahler manifolds with vanishing first Chern class, (with J. Gutowski and G. Papadopoulos), Asian J. Math. 7 (2003), 39-80, math.DG/0205012.
Almost contact manifolds, connections with torsion, and parallel spinors, (with Th. Friedrich), J. reine angew. Math. 559 (2003), 217-236, math.DG/0111131.
Killing spinor equations in dimension 7 and geometry of integrable $G_2$ manifolds, (with Th. Friedrich), J. Geom. Phys. 48 (2003), 1-11, math.DG/0112201
Weyl structure with positive Ricci tensor, (with B. Alexandrov), Diff. Geom. Appl. 18 (2003), 343-350, math.DG/9902033.
Parallel spinors and connections with skew-symmetric torsion in string theory, (with Th. Friedrich), Asian J. Math. 6 (2002), 303-336, math.DG/0102142.
Geometry of Quaternionic Kaehler connections with torsion, J. Geom. Phys. 41 (2002), 235-257, math.DG/0003214.
A no-go theorem for string warped compactification, (with G.Papadopoulos), Phys. Lett. B 497 (2001) 309-316, hep-th/0008232.
Vanishing theorems and String Backgrounds, (with G.Papadopoulos), Class. Quantum Gravity 18 (2001), 1089-1110, math.DG/0010038.
Harmonic spinors of Dirac operator of connection with torsion in dimension 4, (with P.Dalakov), Class. Quantum Gravity 18 (2001), 253-265, [PS]
Vanishing theorems on Hermitian manifolds, (with B. Alexandrov), Diff. Geom. Appl. 14 (3) (2001), 251-265, math.DG/9901090.
The Dolbeault operator on Hermitian spin surfaces, (with. B. Alexandrov and G. Grantcharov), Ann. Inst. Fourier (Grenoble) 51 1 (2001), 221-235, math.DG/9902005.
Harmonic and holomorphic 1-forms on compact balanced Hermitian manifold (with G.Ganchev), Diff. Geom. Appl. 14 (1) (2001), 79-93. [PS]
Holomorphic and Killing vector fields on compact balanced Hermitian manifolds, (with G.Ganchev), Int. J. Math. 11 (2000), 15-28. [PS]
Dirac operators on Hermitian spin surfaces (with B. Alexandrov), Ann. Glob. Anal. Geom. 18 (2000), 529-539. [PS]
Einstein-Weyl structures on certain compact conformal manifolds, Quart. J. Math. (Oxford) (2) 50 (1999), 457-462. [PS]
Orthogonal complex structures on certain Riemannian 6-manifolds, (with V. Apostolov and G. Grantcharov), Diff. Geom. Appl. 11 (1999) 279-296. [PS]
Riemannian manifold in which the skew-symmetric curvature operator has pointwise constant eigenvalues, (with I.Petrova), Geom. Dedicata 70 (1998), 269-282. [PS]
Curvature properties of twistor spaces of quaternionic Kaehler manifolds, (with B. Alexandrov and G. Grantcharov), J. Geom. 62 (1998), 1-12. [PS]
An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifold admitting parallel one form, (with B. Alexandrov and G. Grantcharov), J. Geom. Phys. 28 (1998), 263-270. [PS]
Hermitian structures on twistor spaces, (with V. Apostolov and G. Grantcharov), Ann. Glob. Anal. Geom. 16 (1998), 291-308. [PS]
Compact Hermitian surfaces of constant antiholomorphic sectional curvature, (with V.Apostolov and G.Ganchev), Proc. Amer. Math. Soc. 125 (1997), 3705-3714. [PS]
Einstein-Hermitian surfaces and Hermitian Einstein-Weyl structures in dimension 4, (with P.Gauduchon), Mathematische Zeitschrift 226 (1997), 317-326. [PS]
On the fundamental theorem for non-degenerate complex affine hypersurface immersions, Monatshefte Mathematics 123 (1997), 321-336. [PS]
Riemannian manifolds in which certain curvature operator has constant eigenvalues along each circle, (with I.Petrova), Ann. Glob. Anal. Geom. 15 (1997), 157-171. [PS]
Formality of 7-dimensional 3-Sasakian manifolds, (with Marisa Fernandez and Vicente Munoz). arXiv:1511.08930.
Solution of the qc Yamabe equation on a 3-Sasakian manifold and the quaternionic Heisenberg group, (with Ivan Minchev, Dimiter Vassilev). arXiv:1504.03142.
Sasaki-like almost contact complex Riemannian manifolds, (with H. Manev and M. Manev). arXiv:1402.5426.
Explicit Quaternionic Contact Structures and Metrics with Special Holonomy, (with L. C. de Andres, M. Fernandez, J. A. Santisteban, L. Ugarte and D. Vassilev), arXiv:0903.1398.
Twistors of Almost Quaternionic Manifolds, (with I. Minchev and S. Zamkovoy), math.DG/0511525.
Complex product structures on some simple Lie groups, (with V. Tsanov), math.DG/0405584.
https://debian.fmi.uni-sofia.bg/~ivanovsp/
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redpajama_set_name: RedPajamaCommonCrawl


They are still out there. Who is "they"? They are the audit departments that are convinced it is perfectly fine to live in a past where internal audit tics and ties, fills in the blanks, completes ICQs, points out everything the auditee is doing wrong, and then goes on their merry way, secure in the belief that they have fought the good fight and have done everything they can possibly do within the limits of their responsibilities, their charter, and (sadly) their skills.
The sheer lack of vision exhibited by some audit departments still flabbergasts me. Here's the latest case in point.
This is an internal auditor talking about the roles and responsibilities of internal audit. Take a little time to wrap your head around this statement. Walk around it. Observe it from all angles. Examine all the nooks and crannies of its shortsightedness. Try to fathom the wrong-headedness — the utter and complete misunderstanding of what the profession of internal auditing is trying to accomplish — contained in this one profession-destroying sentence.
You want to know why there are C-suites, executive managers, and boards who feel internal audit is not an important part of their organizations? The answer is in those 19 words.
Do not get me wrong. I sincerely believe that part of internal audit's role is, indeed, cost savings. But our value goes far beyond that.
Sure covers a lot more than "cost savings," doesn't it?
The audit department that limits itself to cost savings (not to mention limiting activities to production, inventories, and purchasing processes as in the comment we are discussing) will find itself outsourced quicker than the steno pool.
Ask yourself (ask your department, ask your audit department employees, ask your audit department leaders) what internal audit should be providing to its customers. And if the answer does not reflect all or a significant part of what is now included in The IIA's Mission of Internal Audit, Core Principles for the Professional Practice of Internal Auditing, and Definition of Internal Auditing, then it is time for a big change.


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Year: - Any -20212020201920182017201620152014
Toggle Summary September 8, 2015 Capnia Acquires Neonatal Pulmonary Resuscitation Solutions From NeoForce Group, Inc.
Acquisition Strengthens Capnia's Presence in Neonatology NeoForce President and Chief Executive Officer to Join Capnia as General Manager of Neonatology REDWOOD CITY, Calif. , Sept. 8, 2015 (GLOBE NEWSWIRE) -- Capnia, Inc. (NASDAQ:CAPN), focused on the development and commercialization of novel
Toggle Summary June 29, 2015 Capnia Added to Russell Microcap(R) Index
REDWOOD CITY, Calif. , June 29, 2015 (GLOBE NEWSWIRE) -- Capnia, Inc. (Nasdaq:CAPN), focused on the development of novel products based on its proprietary technologies for precision metering of gas flow, today announced it was added to the Russell Microcap® Index when the Russell Investment Group
Toggle Summary January 6, 2015 Capnia and Clinvest® Partner to Develop Cluster Headache Therapeutic
REDWOOD CITY, Calif. , Jan. 6, 2015 / PRNewswire / -- Capnia, Inc. (NASDAQ: CAPN), focused on the development of novel diagnostics and therapeutics based on its proprietary technologies for precision metering of gas flow, and Clinvest®, a research organization dedicated to the advancement of
Toggle Summary December 27, 2016 Capnia and Essentialis Announce Merger to Create Rare Disease Therapeutics Company
Combined Company to Focus on Advancing Novel, Late-Stage Asset for the Treatment of Prader-Willi Syndrome, a Devastating, Orphan Disease Enrollment in a Phase II/III Clinical Trial Expected to Begin in Second Half of 2017 $8 Million Concurrent Financing; Sufficient to Fund Development Plan Through
Toggle Summary July 27, 2015 Capnia Announces $10 Million At-the-Market Firm Commitment Stock Purchase Agreement With Aspire Capital
Company Advancing Promising Treatment for Cluster Headache, Trigeminal Neuralgia and Allergic Rhinitis REDWOOD CITY, Calif. , July 27, 2015 (GLOBE NEWSWIRE) -- Capnia, Inc. (NASDAQ:CAPN), focused on the development and commercialization of novel products based on its proprietary technologies for
Toggle Summary June 26, 2015 Capnia Announces Commencement of Series B Warrant Exchange Offer
REDWOOD CITY, Calif. , June 26, 2015 (GLOBE NEWSWIRE) -- Capnia, Inc. (Nasdaq:CAPN), focused on the development of novel products based on its proprietary technologies for precision metering of gas flow, today announced that its previously filed Registration Statement on Form S-4 in connection with
Toggle Summary July 22, 2015 Capnia Announces Enrollment of First Patient in Clinical Trial for Treatment of Cluster Headache
REDWOOD CITY, Calif. , July 22, 2015 (GLOBE NEWSWIRE) -- Capnia, Inc. (NASDAQ:CAPN), focused on the development and commercialization of novel products based on its proprietary technologies for precision metering of gas flow, today announced enrollment of the first patient in an
Toggle Summary June 7, 2016 Capnia Announces Exclusive Distribution Agreement for CoSense® End-Tidal Carbon Monoxide Monitor in India
-Agreement represents continued expansion of CoSense outside the US- REDWOOD CITY, Calif. , June 07, 2016 (GLOBE NEWSWIRE) -- Capnia, Inc. (NASDAQ:CAPN), a diversified healthcare company that develops and commercializes innovative diagnostics, devices and therapeutics addressing unmet medical
Toggle Summary March 16, 2016 Capnia Announces Exclusive Distribution Agreement for Sale of CoSense® in China
Five-year agreement represents significant expansion of CoSense outside the US REDWOOD CITY, Calif. , March 16, 2016 (GLOBE NEWSWIRE) -- Capnia, Inc. (NASDAQ:CAPN), a diversified healthcare company that develops and commercializes innovative diagnostics, devices and therapeutics addressing unmet
Toggle Summary February 16, 2016 Capnia Announces Expiration of Series B Warrants
REDWOOD CITY, Calif. and PALO ALTO, Calif. , Feb. 16, 2016 (GLOBE NEWSWIRE) -- Capnia, Inc. (NASDAQ:CAPN), a diversified healthcare company that develops innovative diagnostics, devices and therapeutics addressing unmet medical needs, announced the expiration of the Company's Series B Warrants as


WCIO SearchPoint, Data Specifications Manuals, Statistical Error Tables, DCO Product Applicability Chart, PEO Applicable Code Chart by State, and Data Dictionary have been updated with approved changes dated September 26, 2018. Please see the Summary of Changes document found within the Data Specifications Manual by individual data type, i.e., WCPOLS, WCSTAT.
WCIO SearchPoint, Data Specifications Manuals, Statistical Error Tables, DCO Product Applicability Chart, PEO Applicable Code Chart by State, various DCO Contact Reports and Data Dictionary have been updated with approved changes dated May 23, 2018. Please see the Summary of Changes document found within the Data Specifications Manual by individual data type, i.e., WCPOLS, WCSTAT.
WCIO SearchPoint, Data Specifications Manuals, Data Reporting Handbook, Statistical Error Tables, DCO Product Applicability Chart, various DCO Contact Information and Data Dictionary have been updated with approved changes dated September 27, 2017. Please see the Summary of Changes document found within the Data Specifications Manual by individual data type, i.e., WCPOLS, WCSTAT.
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In 2012, just under 50,000 cases of malicious cyber attacks were reported to the United States Computer Emergency Team, a department within Home land Security dedicated to computer based threats. In 2015, damage costs from cyber crime soared to three trillion dollars internationally.
This year's DECA State Competition was a whirlwind of a showing from the GMHS DECA team. Our school brought 31 members of our chapter to States, which is the highest attendance in our chapter's history, and had 10 people get onto the stage during the final ceremony.


TMS stands for transcranial magnetic stimulation. It is used to treat depression by stimulating the brain non-invasively using electromagnetic fields, similar to those produced by an MRI machine. During TMS Therapy, a magnetic field is administered in very short pulses to the part of the brain that research has demonstrated to be associated with depression. The typical initial course of treatment is 18 minutes daily over 4-6 weeks.
A typical course of TMS Therapy is 5 times per week for 18 minute sessions over 4-6 weeks.


text:
\section{introduction}

Neutrino physics becomes one the most interesting fields of
elementary particle physics, especially in the wake of the recent
experimental achievements in studying of solar and reactor
neutrinos (see, e.g., Refs.~\cite{Aha04,Ali05}). One of the most
intriguing puzzles in neutrino physics in the neutrino
oscillations problem. Neutrinos of one type can be converted into
another type if there is a mixing between different neutrino
eigenstates. Presently three major opportunities for neutrino
oscillations are discussed in literature. The first one is
neutrino flavor oscillations. The example of this oscillations
type are $\nu_e\to\nu_{\mu,\tau}$ or $\nu_\mu\to\nu_\tau$
conversions that are the possible explanations of the solar and
atmospheric neutrino problems. The second type of neutrino
oscillations is the transitions between neutrino helicity states
within one flavor, or neutrino spin oscillations. The last
opportunity is the combination of the two first neutrino
oscillations types, i.e. the transition when both flavor and
helicity states can change. This type of neutrino oscillations is
called neutrino spin-flavor oscillations. It can be important in
solving of the solar neutrino problem. In this paper we restrict
ourselves to studying of neutrino spin oscillations.

It was realized more than twenty years ago that external fields
drastically change the process of vacuum neutrino oscillations. In
Refs.~\cite{Wol78,MikSmi85eng} it was established that neutrino
weak interactions with background matter result in the resonant
enhancement of neutrino flavor oscillations. The electromagnetic
interactions are of great importance in studying of neutrino spin
and spin-flavor oscillations. For example, the neutrino
interaction with an external electromagnetic field provides one of
the mechanisms for the mixing between different helicity
eigenstates. It is also interesting to study the influence of
gravitational fields on neutrino oscillations (neutrino does not
seem to participate in strong interactions). Despite the
gravitational interaction is relatively weak compared to
electromagnetic and weak interactions there are rather strong
gravitational fields in the Universe. Thus neutrino oscillations
in gravitational fields are of interest in astroparticle physics
and cosmology.

The influence of the gravitational interaction on neutrino
oscillations has been studied in many publications (see, e.g.,
Refs.~\cite{CasMon94,AhlBur96,PirRoyWud96,Ahl97,CarFul97,KosMew04,SinMobPap04,DvoGriStu05}).
First we mention Refs.~\cite{PirRoyWud96,CarFul97} where the
formalism for the description of neutrino oscillations in
gravitational fields in presence of matter and external
electromagnetic fields was worked out. One of the approaches to
the description of the spin dynamics of a neutrino in a
gravitational field consists in decomposing of the Dirac
Hamiltonian (in presence of an external gravitational field) and
establishing terms that mix different \emph{chirality} components
of the neutrino wave function. This method was used in
Ref.~\cite{CasMon94}. To describe the \emph{helicity} evolution in
a gravitational field one can use the evolution equation in
Heisenberg representation with the Hamiltonian in Foldy-Wouthuysen
representation. This approach was implemented in
Ref.~\cite{SinMobPap04}. Neutrino spin oscillations and spin light
of a neutrino in weak gravitational fields were studied in our
work~\cite{DvoGriStu05}. Gravity induced neutrino flavor
oscillations were examined in Refs.~\cite{AhlBur96,Ahl97}.

There is, however, another approach for the treatment of the
spinning particle dynamics in an external gravitational field. The
system of equations for the description of the spinning particle
motion in gravitational fields was derived by A.~Papapetrou in
Ref.~\cite{Pap51}. Since then plenty of works where Papapetrou
equations were analysed have been published. The major
difficulties in solving Papapetrou equations are that these
equations are non-linear in the particle spin as well as the
motion of a spinning particle deviates from the geodesical. Thus
to describe the motion of a \emph{finite size} spinning particle
in a gravitational field one has to take into account the
contribution of the spin term into the particle motion law. It was
established in Ref.~\cite{KhrPom98} that Papapetrou equations can
be significantly simplified when one uses the quasi-classical
approach and linear in spin approximation of these equations for a
\emph{point} particle.

In this paper we study neutrino spin oscillations in gravitational
fields within the quasi-classical approach. Note that the
quasi-classical treatment of neutrino spin oscillations in moving
and polarized matter under the influence of electromagnetic fields
was considered in Refs.~\cite{EgoLobStu00,LobStu01,DvoStu02JHEP}.
In Sec.~\ref{General} we formulate the main equations necessary
for the description of the neutrino motion and spin evolution. The
limit of a weak gravitational field is examined in Sec.~\ref{WGF}.
On the basis of the general equation presented in
Sec.~\ref{General} we obtain the effective Hamiltonian for
neutrino spin oscillations. This effective Hamiltonian is valid
for arbitrary neutrino velocity. We also compare the neutrino spin
evolution equation with the analogous one derived in our previous
work \cite{DvoGriStu05} and find out that the equation obtained in
the present work correctly accounts for the neutrino velocity
dependence. Then we receive neutrino transition probability when a
particle propagates in the gravitational field of a rotating
massive object. In Sec.~\ref{SM} we apply the general technique to
the description of neutrino spin oscillations in the Schwarzschild
metric. We obtain the neutrino spin evolution equation which is
valid for the neutrino motion even in the vicinity of a black
hole. The effective Hamiltonian and the transition probability are
also derived. We examine neutrino oscillations process on
different circular orbits and analyze the frequencies of the spin
transitions. In Sec.~\ref{CONCL} we discuss our results. The
calculation of covariant derivatives of the vierbein vectors
(Appendix~\ref{VF}) and the basic elements of $SL(2,C)$ group
(Appendix~\ref{LorGroup}) are also presented.

\section{Motion of a spinning point particle in
gravitational fields}\label{General}

The motion of a spinning particle in gravitational fields was
described in Ref.~\cite{Pap51}. The evolution of the spin tensor
of a particle $S^{\mu\nu}$ and particle's momentum $p^\mu$ is
described by the following equations,
\begin{align}
  \label{PapS}
  \frac{DS^{\mu\nu}}{D\tau} & =
  p^{\mu}v^{\nu}-p^{\nu}v^{\mu}, \\
  \label{Papp}
  \frac{Dp^{\mu}}{D\tau} & =-\frac{1}{2}
  R^{\mu}{}_{\nu\rho\sigma}v^\nu S^{\rho\sigma},
\end{align}
where $v^\mu$ is the unit tangent vector to the center-of-mass
world line, $\tau$ is the parameter (not necessarily the proper
time) which changes along the world line, $D/D\tau$ denotes the
covariant derivative along the world line and
$R^{\mu}{}_{\nu\rho\sigma}$ is the Riemann tensor. The spin vector
is introduced in the following way (see, e.g., Ref.~\cite{Wal72}),
\begin{equation*}
  S_\rho=\frac{1}{2m}\sqrt{-g}
  \varepsilon_{\mu\nu\lambda\rho}p^\mu S^{\nu\lambda},
\end{equation*}
where $\varepsilon_{\mu\nu\lambda\rho}$ is the completely
antisymmetric tensor density, $g=\det(g_{\mu\nu})$ and $m^2=p_\mu
p^{\mu}$ is the constant of the particle's motion.

When we study the motion of point particles in a gravitational
field, Eqs.~\eqref{PapS} and \eqref{Papp} can be significantly
simplified. It was shown in Ref.~\cite{Wei72p121} that using only
the Principle of General Covariance the equations for the
description of particle's spin vector $S^{\mu}$ and four-velocity
$U^\mu=dx^\mu/d\tau$ evolution take the form,
\begin{align}
  \label{WeiS}
  \frac{DS^\mu}{D\tau} & =0, \\
  \label{WeiU}
  \frac{DU^\mu}{D\tau} & =0.
\end{align}
Here the covariant derivatives are taken with respect to the
proper time $\tau$. It follows from Eqs.~\eqref{WeiS} and
\eqref{WeiU} that particle's spin and four-velocity are parallel
transported along its world-line. It should be noted that usual
properties of the spin vector $S^\mu$ in Minkowski space-time,
namely $S^\mu U_\mu=0$ and $S^\mu S_\mu$ is the constant value,
remain unchanged when a particle moves in curved space-time. One
can verify these properties using Eqs.~\eqref{WeiS} and
\eqref{WeiU}. The basic Eqs.~\eqref{WeiS} and \eqref{WeiU} can be
also rewritten with help of the Christoffel symbols of the second
kind $\Gamma^{\mu}{}_{\alpha\beta}$,
\begin{align}
  \label{SGamma}
  \frac{dS^\mu}{d\tau} & =
  -\Gamma^{\mu}{}_{\alpha\beta}U^{\alpha}S^{\beta}, \\
  \label{UGamma}
  \frac{dU^\mu}{d\tau} & =
  -\Gamma^{\mu}{}_{\alpha\beta}U^{\alpha}U^{\beta}.
\end{align}

Eq.~\eqref{SGamma} describes the spin evolution in a general
coordinate system. However the particle's properties are
determined by the spin vector components measured with respect to
the particle's rest frame. The particle's rest-frame-spin
precession in gravitational fields was discussed in
Ref.~\cite{KhrPom98}. In order to proceed in our analysis of
particle's spin evolution, we should rewrite Eqs.~\eqref{SGamma}
and \eqref{UGamma} in a locally minkowskian frame. One can
implement this coordinates transformation with help of the
vierbein vectors $e^a{}_\mu$. We recall the basic vierbein vectors
properties,
\begin{align}\label{vierbeinprop}
  g_{\mu\nu} & =e^a{}_\mu e^b{}_\nu\eta_{ab},
  \quad
  \delta^\mu{}_\nu=e_a{}^\mu e^a{}_\nu,
  \\
  \notag
  \eta_{ab} & =e_a{}^\mu e_b{}^\mu g_{\mu\nu},
  \quad
  \delta^a{}_b=e^a{}_\mu e_b{}^\mu,
\end{align}
where $e_a{}^\mu=\eta_{ab}g^{\mu\nu}e^b{}_\nu$ is the inverse
vierbein and $\eta_{ab}=\mathrm{diag}(+1,-1,-1,-1)$ is the metric
tensor in a locally minkowskian frame.

The components of the spin and four-velocity with respect to a
locally minkowskian frame are,
\begin{equation*}
  s^a=e^a{}_\mu S^\mu,
  \quad
  u^a=e^a{}_\mu U^\mu.
\end{equation*}
The equations for the description of the $s^a$ and $u^a$ evolution
have the form (see also Ref.~\cite{KhrPom98}),
\begin{align}
  \label{sG}
  \frac{ds^a}{dt} & =
  \frac{1}{\gamma}G^{ab}s_b, \\
  \label{uG}
  \frac{du^a}{dt} & =
  \frac{1}{\gamma}G^{ab}u_b,
\end{align}
where $G^{ab}=\eta^{ac}\eta^{bd}\gamma_{cde}u^e=-G^{ba}$ is the
analog of the electromagnetic field tensor and
$\gamma_{abc}=\eta_{ad}e^d{}_{\mu;\nu}e_b{}^{\mu}e_c{}^{\nu}$ are
the Ricii rotation coefficients. In Eqs.~\eqref{sG} and \eqref{uG}
we keep the usual notation $\gamma=U^0=dt/d\tau$, however it
should be noted that $\gamma\not= u^0$. The derivatives in the
right-handed sides of these equations are taken with respect to
the laboratory time.

To make the coordinate transformation into the particle's rest
frame one uses a boost within a locally minkowskian frame. The
relation between the spin vector $s^a$
and the three dimensional spin in the particle's rest frame
$\bm{\zeta}$ is given by following expression,
\begin{equation}\label{szeta}
  s^a=
  \left(
    (\bm{\zeta}\mathbf{u}),\bm{\zeta}+
    \frac{\mathbf{u}(\bm{\zeta}\mathbf{u})}{1+u^0}
  \right).
\end{equation}
Here $u^0$ and $\mathbf{u}$ are the time and space components of
the four-velocity $u^a$. We remind that $u^a$ is the four velocity
vector in the vierbein frame.

Using Eqs.~\eqref{sG} and \eqref{szeta} we can readily derive the
equation for the description of the three dimensional spin vector
evolution,
\begin{equation}\label{zetaeq}
  \frac{d\bm{\zeta}}{dt}=
  \frac{2}{\gamma}[\bm{\zeta}\times\mathbf{G}],
\end{equation}
where
\begin{equation}\label{GEB}
  \mathbf{G}=\frac{1}{2}
  \left(
    \mathbf{B}+
    \frac{1}{1+u^0}[\mathbf{E}\times\mathbf{u}]
  \right).
\end{equation}
In deriving of Eqs.~\eqref{zetaeq} and \eqref{GEB} we use the fact
that any antisymmetric tensor in four dimensional (minkowskian)
space-time can be expressed in terms of the two three dimensional
vectors (analogs of electric and magnetic fields), i.e.
$G_{ab}=(\mathbf{E},\mathbf{B})$, where $G_{0i}=E_i$ and
$G_{ij}=-\varepsilon_{ijk}B_k$. Eqs.~\eqref{zetaeq} and
\eqref{GEB} describe particle's spin precession in arbitrary
gravitational field. These equations are linear in spin vector.
One can, however, discuss further (non-linear in spin) terms
contributing to the particle's spin evolution (see
Ref.~\cite{KhrPom98}).

At the end of this section me mention that Eqs.~\eqref{sG} and
\eqref{uG} [or alternatively Eqs.~\eqref{zetaeq} and \eqref{GEB}]
are similar to the equations for the description of the spin and
four-velocity evolution of a charged particle with gyromagnetic
ratio $\mathfrak{g}=2$ interacting with an external
electromagnetic field (see also Ref.~\cite{KhrPom98}). The spin
evolution of a charged particle with $\mathfrak{g}=2$ was studied
in Refs.~\cite{Zwa65,LobPav99}. It was shown that the spin
dynamics was completely determined by the particle's motion law,
i.e. by the solution of the Lorentz equation \eqref{uG}.

\section{Neutrino spin evolution in weak gravitational fields}
\label{WGF}

In this section we study the particle spin precession in a weak
gravitational field. We describe the neutrino spin oscillations
and discuss the obtained results as well as we consider analogous
approach elaborated in our previous work.

Weak gravitational fields can be found, for instance, at great
distances from a finite massive object under study. In this case
we can always choose the quasi-minkowskian coordinate system,
\begin{equation}\label{weakgf}
  g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},
\end{equation}
where $\eta_{\mu\nu}=\mathrm{diag}(+1,-1,-1,-1)$ is the Minkowski
metric tensor. The metric perturbation $h_{\mu\nu}$ has to vanish
at the infinity. One of the possible examples of the metric given
in Eq.~\eqref{weakgf} is the gravitational field created by a
massive rotating object at great distances. Using post-Neutonian
approximation we obtain for the components of the tensor
$h_{\mu\nu}$ (see, e.g., Ref.~\cite{Wei72p212}),
\begin{gather}
  \label{hmunuPN}
  h_{00}=2\varphi,
  \quad
  h_{ij}=2\varphi\delta_{ij},
  \quad
  h_{0i}=-h_i,
  \\
  \label{phih}
  \varphi=-\frac{M}{r},
  \quad
  \mathbf{h}=\frac{2}{r^3}
  [\mathbf{r}\times\mathbf{J}],
\end{gather}
where $M$ is the mass of the object, $\mathbf{J}$ is its total
angular momentum and $r$ is the distance from the object.

First let us consider the particle's spin evolution when an
observer is in the particle's rest frame. For this purpose we can
use directly Eq.~\eqref{SGamma}. In our case the four dimensional
spin vector is reduced to the three dimensional one, i.e.
$S^\mu\to(0,\bm{\zeta})$. Eq.~\eqref{SGamma}, written for the
spatial components of the spin vector, takes the following form,
\begin{equation}\label{zetaeqrfG}
  \frac{d\zeta_i}{dt}=
  -\Gamma^{i}{}_{0j}\zeta_{j}.
\end{equation}
Here we assume that $U^0=1$ and $\mathbf{U}=0$ in particle's rest
frame. Note that we do not distinguish between upper and lower
indexes when we use three dimensional vectors. Christoffel symbols
$\Gamma^{i}{}_{0j}$ can be calculated with help of
Eq.~\eqref{hmunuPN},
\begin{equation}\label{Gi0j}
  \Gamma^{i}{}_{0j}=
  \frac{1}{2}
  \left(
    \frac{\partial h_i}{\partial x_j}-
    \frac{\partial h_j}{\partial x_i}
  \right)=
  -\frac{1}{2}\varepsilon_{ijk}
  [\bm{\nabla}\times\mathbf{h}]_k.
\end{equation}
Using Eqs.~\eqref{zetaeqrfG} and \eqref{Gi0j} we obtain the
equation for the particle's spin evolution in its rest frame,
\begin{equation}\label{zetaeqr}
  \frac{d\bm{\zeta}}{dt}=
  \frac{1}{2}[\bm{\zeta}\times[\bm{\nabla}\times\mathbf{h}]].
\end{equation}
One can see that Eq.~\eqref{zetaeqr} coincides (to within the
sign) with the analogous equation derived in our previous work
(see Ref.~\cite{DvoGriStu05}) if we set there $\gamma=1$ and
$\bm{\beta}=0$.

Now let us discuss the particle's spin precession in a weak
gravitational field, with an observer being placed in the
laboratory frame, i.e. a particle has the non-zero velocity with
respect to him. In this case we should use Eqs.~\eqref{zetaeq} and
\eqref{GEB} to describe particle's spin evolution. The Ricci
rotation coefficients were calculated in Ref.~\cite{KhrPom98} in
the weak gravitational field approximation. They can be expressed
in the following way,
\begin{equation}\label{Ricciwf}
  \gamma_{abc}=\frac{1}{2}
  \left(
    \frac{\partial h_{bc}}{\partial x^a}-
    \frac{\partial h_{ac}}{\partial x^b}
  \right),
\end{equation}
where $h_{ab}$ and $x^a$ are the components of the tensor
$h_{\mu\nu}$ and vector $x^\mu$ in the vierbein frame. Note that
we do not distinguish the vierbein and world indexes in the weak
gravitational field approximation.

Using Eqs.~\eqref{hmunuPN} and \eqref{Ricciwf} as well as the
definition of the tensor $G_{ab}$ we receive the expressions for
the "electric" and "magnetic" fields,
\begin{equation*}
  \mathbf{E}=-\gamma\bm{\nabla}\varphi+
  \frac{\gamma}{2}v_i\bm{\nabla}h_i,
  \quad
  \mathbf{B}=\gamma[\mathbf{v}\times\bm{\nabla}]\varphi+
  \frac{\gamma}{2}[\bm{\nabla}\times\mathbf{h}].
\end{equation*}
One can readily find the vector $\mathbf{G}$, which determines the
particle's spin precession, in the explicit form,
\begin{align}\label{Gwf}
  \mathbf{G}= & \frac{\gamma}{2}
  \bigg(
    \frac{1}{2}[\bm{\nabla}\times\mathbf{h}]+
    \frac{2\gamma+1}{\gamma+1}[\mathbf{v}\times\bm{\nabla}]\varphi
    \\
    \notag
    & -
    \frac{\gamma}{2(\gamma+1)}v_i[\mathbf{v}\times\bm{\nabla}]h_i
  \bigg),
\end{align}
where $\mathbf{v}=\dot{\mathbf{r}}$ is the particle's velocity and
$\gamma=(1-v^2)^{-1/2}$. Note that if we set $\mathbf{v}=0$ in
Eq.~\eqref{Gwf}, we obtain the expression consistent with
Eq.~\eqref{zetaeqr} obtained directly from Eq.~\eqref{SGamma}. It
proves the validity of the used technique.

Eqs.~\eqref{zetaeq} and \eqref{Gwf} govern particle's spin
evolution in a weak gravitational field. These equations are valid
for arbitrary particle velocities. Using Eqs.~\eqref{zetaeq} and
\eqref{Gwf} we can describe neutrino spin oscillations in the weak
gravitational field. For example, it is possible to study the
particle's spin dynamics of a neutrino emitted in the vicinity of
a rotating black hole and then propagating faraway from the
massive object (see Ref.~\cite{DvoGriStu05}). Supposing that a
neutrino has the velocity directed along the $\mathbf{e}_z$ base
vector we obtain from Eq.~\eqref{Gwf} the expression for the
effective Hamiltonian (see Ref.~\cite{EgoLobStu00})
\begin{equation}\label{Heffwf}
  H_\mathrm{eff}=-\frac{1}{\gamma}(\bm{\sigma}\mathbf{G}),
\end{equation}
where $\bm{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ are the
Pauli matrices.

Let us consider neutrino spin oscillations when a neutrino
interacts with the gravitational field having the properties given
in Eq.~\eqref{phih}. Using Eqs.~\eqref{phih} and \eqref{Gwf} we
receive for the vector $\mathbf{G}$ the following expression,
\begin{widetext}
\begin{equation}
  \label{GJphi}
  \mathbf{G}=\frac{\gamma}{2}
  \left\{
    \frac{1}{r^5}
    \left[
      r^2\mathbf{J}-3\mathbf{r}(\mathbf{J}\mathbf{r})
    \right]-
    \frac{\gamma}{\gamma+1}
    \left[
      \frac{1}{r^3}
      \left[
        v^2\mathbf{J}-\mathbf{v}(\mathbf{v}\mathbf{J})
      \right]-
      \frac{3}{r^5}
      [\mathbf{v}\times\mathbf{r}]
      (\mathbf{v}[\mathbf{r}\times\mathbf{J}])
    \right]+
    \frac{2\gamma+1}{\gamma+1}\frac{M}{r^3}
    [\mathbf{v}\times\mathbf{r}]
  \right\}.
\end{equation}
\end{widetext}
It can be seen that this expression for the vector $\mathbf{G}$
agrees with the analogous formula obtained in our prevoius work
\cite{DvoGriStu05} only for slow neutrinos having $v\ll 1$ (see
also above).  This discrepancy results from the incorrect account
of the spin-orbital interaction in Ref.~\cite{DvoGriStu05}.
However, if we study only the radial neutrino motion, the main
contribution to the neutrino spin evolution comes from the
interaction with the angular momentum of the massive object and we
reach an agreement with our recent work.

We consider therefore a neutrino propagating along the radius,
i.e. $\mathbf{v}=v(\mathbf{r}/r)$. In this case we can rewrite
Eq.~\eqref{GJphi} in the form,
\begin{equation}\label{GJradial}
  \mathbf{G}=\frac{\gamma}{2r^3}
  \left\{
    \frac{1}{\gamma}\mathbf{J}-
    \mathbf{n}(\mathbf{J}\mathbf{n})
    \left(
      2+\frac{1}{\gamma}
    \right)
  \right\},
\end{equation}
where $\mathbf{n}$ is the unit vector along the neutrino velocity.
One can see that the vector $\mathbf{G}$ given in
Eq.~\eqref{GJradial} coincides with that derived in our previous
work (see Ref.~\cite{DvoGriStu05}). Thus all results concerning
the neutrino spin light obtained in Ref.~\cite{DvoGriStu05} are
valid \emph{only} for the radial neutrino motion.

Now let us write down the effective Hamiltonian for neutrino spin
oscillations in our case. We can always choose the coordinate
system so that,
\begin{gather}
  \label{nJ}
  \mathbf{n}=(0,0,1),
  \quad
  \mathbf{J}=(J_1,0,J_3),
  \\
  \notag
  J_1=J\sin\vartheta,
  \quad
  J_3=J\cos\vartheta,
\end{gather}
where $\vartheta$ is the angle between vectors $\mathbf{n}$ and
$\mathbf{J}$. Using Eqs.~\eqref{Heffwf}, \eqref{GJradial} and
\eqref{nJ} we obtain the expression for the neutrino spin
oscillations effective Hamiltonian,
\begin{equation}\label{HeffJ}
  H_\mathrm{eff}=\frac{J}{r^3}
  \begin{pmatrix}
    \cos\vartheta & \sin\vartheta/(2\gamma) \\
    \sin\vartheta/(2\gamma) & -\cos\vartheta
  \end{pmatrix}.
\end{equation}
Here $\mathbf{r}=\mathbf{r}_0+\mathbf{v}t$, where $\mathbf{r}_0$
is the initial neutrino coordinate. It follows from
Eq.~\eqref{HeffJ} that transitions between different polarization
states of a neutrino are suppressed by the factor $1/\gamma$ which
is small for ultrarelativistic neutrinos. The most intensive
neutrino spin oscillations take place when a neutrino is
propagating in the direction perpendicular to the vector
$\mathbf{J}$ ($\vartheta=\pi/2$). There are no oscillations at all
when a neutrino is propagating along the vector $\mathbf{J}$
($\vartheta=0$).

The transition probability for the $\nu_L\leftrightarrow\nu_R$
oscillations can be obtained with help of the effective
Hamiltonian given in Eq.~\eqref{HeffJ}. Let us suppose that
$\nu_R(0)=0$ and $\nu_L(0)=1$, then we receive the expression for
the probability to find a right-handed neutrino in this neutrino
beam,
\begin{equation}\label{Probabr}
  P(r)=
  \frac{\sin^2\vartheta}{4\gamma^2\cos^2\vartheta+\sin^2\vartheta}
  \sin^2[\Phi(r)],
\end{equation}
where
\begin{align}\label{Phir}
  \Phi(r)= & \frac{J}{4\sqrt{\gamma^2-1}}
  \left(
    \frac{1}{r_0^2}-\frac{1}{r^2}
  \right)
  \\
  \notag
  & \times
  \sqrt{4\gamma^2\cos^2\vartheta+\sin^2\vartheta}.
\end{align}
Here $r_0$ is the distance between a neutrino and the center of
the massive object at $t=0$. Eqs.~\eqref{Probabr} and \eqref{Phir}
coincide with the similar expression for the transition
probability derived in our previous work \cite{DvoGriStu05}.

\section{Neutrino spin oscillations in Schwarzschild metric}
\label{SM}

General equations derived in Sec.~\ref{General} allow one to
describe particle's spin evolution not only in weak gravitational
fields. In this section we discuss the neutrino spin evolution and
oscillations when a particle interacts with a non-rotating black
hole in the case when a neutrino propagates even in the vicinity
of a black hole.

When we study the gravitational field of a non-rotating black
hole, the interval is known to be expressed with help of the
Schwarzschild metric,
\begin{equation}\label{schwarz}
  d\tau^2=A^{2}dt^2-A^{-2}dr^2-
  r^2(d\theta^2+\sin^2\theta d\phi^2),
\end{equation}
where
\begin{equation}\label{Aschwarz}
  A=\sqrt{1-\frac{r_g}{r}},
\end{equation}
and $r_g$ is the Schwarzschild radius. In Eq.~\eqref{schwarz} we
use the spherical coordinate system for the world (not for the
vierbein) coordinates. We can decompose the four-velocity $U^\mu$
with help of the spherical coordinates,
\begin{equation}\label{Umuspher}
  U^\mu=(U^0,U_r,U_\theta,U_\phi).
\end{equation}

In order to construct the "electric" and "magnetic" fields we
should choose the appropriate vierbein vectors. One can verify
that the following vectors,
\begin{align}
  \label{e0}
  e^{0}{}_{\mu} & =(A,0,0,0), \\
  e^{1}{}_{\mu} & =(0,A^{-1},0,0), \\
  e^{2}{}_{\mu} & =(0,0,r,0), \\
  \label{e3}
  e^{3}{}_{\mu} & =(0,0,0,r\sin\theta),
\end{align}
satisfy the general properties of the vierbein vectors given in
Eq.~\eqref{vierbeinprop}.

Using Eqs.~\eqref{Umuspher}-\eqref{e3} we can find vierbein
components of the vector $u^a$,
\begin{equation*}
  u^a=(\gamma A,U_r A^{-1},U_\theta r,U_\phi r\sin\theta).
\end{equation*}
With help of Eqs.~\eqref{cde0}-\eqref{Gabconv} as well as using
the definition of the tensor $G_{ab}$ one can write down the
components of the "electric" and "magnetic" fields,
\begin{align}\label{EBschwarz}
  \mathbf{E} & =
  \left(
    -\gamma\frac{r_g}{2r^2},0,0
  \right),
  \\
  \notag
  \mathbf{B} & =(U_\phi\cos\theta,-U_\phi A\sin\theta,U_\theta A).
\end{align}
Neutrino spin precession is determined by the vector
$\bm{\Omega}=\mathbf{G}/\gamma$. The components of this vector can
be found on the basis of Eqs.~\eqref{GEB} and \eqref{EBschwarz},
\begin{align}
  \label{Omega1}
  \Omega_1 & =\frac{1}{2}v_\phi\cos\theta, \\
  \label{Omega2}
  \Omega_2 & =v_\phi\sin\theta\frac{1}{2}
  \left(
    -A+\frac{\gamma}{(1+\gamma A)}
    \frac{r_g}{2r}
  \right), \\
  \label{Omega3}
  \Omega_3 & =v_\theta\frac{1}{2}
  \left(
    A-\frac{\gamma}{(1+\gamma A)}
    \frac{r_g}{2r}
  \right),
\end{align}
where $\mathbf{v}=(v_r,v_\theta,v_\phi)$ are the components of the
world (not the vierbein) velocity. In deriving of
Eqs.~\eqref{Omega1}-\eqref{Omega3} we use the fact that
$\mathbf{U}=\gamma\mathbf{v}$.

Let us discuss a neutrino orbiting a black hole. For simplicity we
consider only circular orbits with the radius $R$. We may restrict
ourselves to the studying of the orbits lying only in the
equatorial plane ($\theta=\pi/2$ or equivalently $v_\theta=0$)
because of the spherical symmetry of the gravitational field. In
this case $\Omega_1=0$ and $\Omega_3=0$ in Eqs.~\eqref{Omega1} and
\eqref{Omega3}. The expressions for the neutrino angular velocity
and $\gamma$ are presented in Ref.~\cite{Wei72p185},
\begin{equation}\label{vphico}
  v_\phi=\frac{d\phi}{dt}=\sqrt{\frac{r_g}{2R^3}},
\end{equation}
and
\begin{equation}\label{gammaco}
  \gamma^{-1}=\frac{d\tau}{dt}=\sqrt{1-\frac{3r_g}{2R}}.
\end{equation}
It should be noted that the vierbein four velocity now takes the
form, $u^a=(\gamma A, 0, 0, \gamma v_\phi r)$. One can verify that
$u^a u_a=1$ with help of Eqs.~\eqref{Aschwarz}, \eqref{vphico} and
\eqref{gammaco}. We also mention that Eq.~\eqref{uG} is also
identically satisfied. Indeed $G_{ab}u^b=0$ since $E_1 u^0=B_2
u^3$ and hence $du^a/d\tau=0$. Therefore a neutrino has constant
four velocity with respect to the vierbein frame. Using
Eqs.~\eqref{Aschwarz}, \eqref{Omega2} and \eqref{gammaco} we can
rewrite $\Omega_2$ in the more simple form,
\begin{equation}\label{Omaga2sf}
  \Omega_2=-\gamma^{-1}\frac{v_\phi}{2}.
\end{equation}
This equation has very clear physical meaning. It is well known
that the spin precession of a charged particle with
$\mathfrak{g}=2$ in the external electromagnetic field is
completely determined by the particle's motion law (see
Sec.~\ref{General} or Ref.~\cite{Zwa65}). Therefore
Eq.~\eqref{Omaga2sf} is nothing else as the Lorentz transformation
of the spin rotation frequency from the rest frame to the
laboratory frame.

We suppose that initially a neutrino is left-handed, i.e. its
initial spin vector is antiparallel to the particle's velocity.
According to Eqs.~\eqref{zetaeq} and \eqref{Omega1}-\eqref{Omega3}
the neutrino spin rotates around the second axis. Therefore using
Eq.~\eqref{zetaeq} we can construct the effective Hamiltonian for
the neutrino spin oscillations in the gravitational field of a
non-rotating black hole,
\begin{equation}\label{HeffBH}
  H_\mathrm{eff}=
  \begin{pmatrix}
    0 & -i\Omega_2 \\
    i\Omega_2 & 0
  \end{pmatrix}.
\end{equation}
It is interesting to note that in Eq.~\eqref{HeffBH} the parameter
$|\Omega_2|{\not=}0$ if $\gamma^{-1}{\not=}0$ (see below). One can
see it directly in Eq.~\eqref{Omaga2sf}. Using Eq.~\eqref{HeffBH}
we write the expression for the neutrino transition probability,
\begin{equation}\label{probBH}
  P(t)=\sin^2(\Omega_2 t).
\end{equation}
We can see that there is a full mixing in our case and neutrino
transition probability can achieve a unit value.

It is possible to describe the neutrino spin evolution using
formalism elaborated in Ref.~\cite{LobPav99} where the the
approach for the studying of Lorentz and Bargmann-Michel-Telegdi
equations was described. It follows from the results of that work
that the spin dynamics is completely determined by the particle's
motion law when $\mathfrak{g}=2$. Eq.~\eqref{uG} can be rewritten
in the equivalent form,
\begin{equation*}
  \frac{d}{d\tau}\undertilde{\Lambda}=
  \undertilde{F}\undertilde{\Lambda},
  \quad
  \undertilde{F}=\frac{i}{2}\bm{\sigma}(\mathbf{B}-i\mathbf{E}),
\end{equation*}
and the operator $\undertilde{\Lambda}$ implements the shift of
the particle's velocity along the trajectory. We present the main
properties of the $SL(2,C)$ group in Appendix~\ref{LorGroup}. If
one has found the operator $\undertilde{\Lambda}$, the resolvent
of the Eq.~\eqref{sG} takes the form ($\mathfrak{g}=2$),
\begin{equation}\label{utRdef}
  \undertilde{R}(\tau,\tau_0)=
  \undertilde{L}^{-1}(\tau)\undertilde{\Lambda}\undertilde{L}(\tau_0),
\end{equation}
where the matrix $\undertilde{L}$ implements the boost from the
rest frame to the laboratory frame.

If we consider the neutrino motion in the equatorial plane from
the very beginning, the operator $\undertilde{\Lambda}$ is
expressed in the following way,
\begin{equation}\label{utLambda}
  \undertilde{\Lambda}=\exp(\undertilde{F}\tau)=
  \begin{pmatrix}
    \cos\alpha t & \alpha_{+}\sin\alpha t \\
    \alpha_{-}\sin\alpha t & \cos\alpha t
  \end{pmatrix},
\end{equation}
where $\alpha=\sqrt{B^2-E^2}/(2\gamma)$ and
\begin{equation*}
  \alpha_{\pm}=\frac{E\pm B}{\sqrt{B^2-E^2}}.
\end{equation*}
Note that using Eqs.~\eqref{Aschwarz}, \eqref{EBschwarz},
\eqref{vphico} and \eqref{gammaco} we can show that
$\alpha=v_\phi/(2\gamma)$.
It can be verified that
$\undertilde{\Lambda}\undertilde{u}\undertilde{\Lambda}^\dagger=
\undertilde{u}$, where
\begin{equation*}
  \undertilde{u}=
  \begin{pmatrix}
    u^0+u^3 & 0 \\
    0 & u^0-u^3
  \end{pmatrix},
\end{equation*}
i.e. a neutrino moves along a strait line (in the veirbein frame).
This fact also agrees with the result obtained above in present
paper. The operator $\undertilde{L}$ does not depend on time since
the neutrino's four velocity is constant. This matrix is expressed
in the following way (see Appendix~\ref{LorGroup}),
\begin{equation}\label{utL}
  \undertilde{L}=
  \begin{pmatrix}
    \sqrt{u^0+u^3} & 0 \\
    0 & \sqrt{u^0-u^3}
  \end{pmatrix}.
\end{equation}
Using Eqs.~\eqref{utRdef}-\eqref{utL} we receive the resolvent
$\undertilde{R}(t)$ in the form (here we assume that $t_0=0$),
\begin{equation}\label{utR}
  \undertilde{R}(t)=
  \begin{pmatrix}
    \cos\alpha t & \sin\alpha t \\
    -\sin\alpha t & \cos\alpha t
  \end{pmatrix}.
\end{equation}
In order to find the spin rotation axis one adopts the unit vector
$\mathbf{k}$ along this axis, so that
$[\undertilde{R},\bm{\sigma}\mathbf{k}]=0$. Here use the fact that
$\undertilde{R}\in SO(3)$ and
$\undertilde{R}^{\dagger}=\undertilde{R}^{-1}$. It can be easily
derived with help of Eq.~\eqref{utR} that
$[\undertilde{R},\sigma_y]=0$. Thus the spin precession axis is
the second axis in the complete agreement with the above obtained
results.

One can also plot the frequency of neutrino spin oscillations
versus the radius of the orbit. It is possible to see on
Fig.~\ref{freq} that $|\Omega_2|=0$ at $R=1.5 r_g$ and
$|\Omega_2|\to 0$ at $R\to\infty$.
\begin{figure}
  \centering
  \includegraphics[scale=.45]{frequency.eps}
  \caption{Neutrino spin oscillations frequency versus the radius
  of the neutrino orbit.}\label{freq}
\end{figure}
One can conclude that $|\Omega_2|$ has its maximal value (equal to
$6.25\times 10^{-2} r_g^{-1}$) at $R=2 r_g$ [see also
Eq.~\eqref{Omaga2sf}]. Let us evaluate the number of revolutions
$N$, that a neutrino should make, necessary for the total spin
flip. Using Eqs.~\eqref{vphico} and \eqref{Omaga2sf} we can easily
find for $N$,
\begin{equation*}
  N=\frac{v_\phi}{2|\Omega_2|}=
  \left.
    \gamma
  \right|_{R=2 r_g}
  =2.
\end{equation*}
It is interesting to evaluate the characteristic period of the
neutrino spin oscillations (oscillations length). For $M=10
M_\odot$ at $R=2 r_g$ we get for $T=\pi/|\Omega_2|\approx
4.94\times 10^{-3}\thinspace\text{s}$.

We can discuss another orbit, namely $R=1.5 r_g$. It follows from
Eq.~\eqref{Omaga2sf} (see also Fig.~\ref{freq}) that the frequency
of the transitions between two polarization states vanishes. It
should be noted that $d\tau=0$ at $R=1.5 r_g$ [see
Eq.~\eqref{gammaco}]. Therefore this orbit corresponds to massless
particles. We reveal that the property of a massless particle in
the Minkowsky space-time to keep unchanged its spin direction with
respect to the momentum remains unaffected in the Schwarzschild
metric. It is interesting to note that the absence of the spin
rotation for massless particles is valid only for the
Schwarzschild metric. It was shown in
Refs.~\cite{MohMukPra02,DebMukDad06} that for a background metric
with at least one off-diagonal space component, e.g. the Kerr
metric, in the limit $m_\nu \to 0$ still there are transitions
between neutrino and antineutrino, i.e. neutrino spin flip. This
phenomenon is used in Refs.~\cite{MohMukPra02,Muk05} to explain
the neutrino-antineutrino asymmetry problem.

It is also worth examining the behaviour of $\Omega_2$ for large
orbits. If a neutrino moves on a remote orbit, the gravitational
field is weak. Thus we again do not distinguish between world and
vierbein indexes. We define the coordinate system so that the
angular velocity is expressed in the following way,
$\bm{\omega}_\mathrm{frame}=v_{\phi}\mathbf{e}_z$, where
$\mathbf{e}_z$ is the unit vector. Using Eq.~\eqref{Omaga2sf} in
the weak field limit we obtain for the spin rotation frequency,
\begin{equation*}
  \bm{\omega}_\mathrm{abs}=-2\Omega_2\mathbf{e}_z=v_\phi
  \left(
    1-\frac{3}{4}\frac{r_g}{R}+\dots
  \right)\mathbf{e}_z,
\end{equation*}
since the second vierbein coordinate axis is nothing else as the
$\theta$-axis and for the orbit in the equatorial plane we have
$\mathbf{e}_\theta=-\mathbf{e}_z$. Note that the effective
Hamiltonian in Eq.~\eqref{HeffBH} determines the evolution of the
neutrino rest-frame-spin vector. However an observer is in the
laboratory frame ($t$ is time in the laboratory frame). Thus
$\Omega_2$ is the oscillations frequency measured in the
laboratory frame. In order to obtain the spin rotation frequency
measured by an observer in the co-moving frame we should subtract
$\bm{\omega}_\mathrm{frame}$ from $\bm{\omega}_\mathrm{abs}$ since
$\bm{\omega}_\mathrm{frame}$ is the angular velocity of the
orbital motion, i.e. the angular velocity of the rest frame
rotation with respect to the laboratory frame. Finally we get for
$\bm{\omega}_\mathrm{rel}=\bm{\omega}_\mathrm{abs}-
\bm{\omega}_\mathrm{frame}$ the following expression,
\begin{equation*}
  \bm{\omega}_\mathrm{rel}=
  -\frac{3}{4}\frac{r_g}{R} v_\phi \mathbf{e}_z,
\end{equation*}
which agrees with the classical result for the spin rotation
frequency in the co-moving frame (see, e.g., Ref.~\cite{Sch60} or
Eq.~\eqref{GJphi} in the limit $\gamma\to 1$).

At the end of this section we briefly examine the applicability of
the quasi-classical approach to the description of the neutrino
spin evolution in the gravitational field of a black hole. Basing
on the similarity between particle spin precession in
gravitational and electromagnetic fields we can apply the results
of Ref.~\cite{Ter90} where the quasi-classical approximation for
the electron spin precession in an external magnetic field was
considered. It was found in that paper that one can use the
quasi-classical limit if the corresponding inequality (we rewrite
it for our purposes) is satisfied,
\begin{equation}\label{qqlcond}
  \frac{\hbar}{2\mathcal{E}}
  \left|
    \frac{d\bm{\zeta}}{dt'}
  \right|
  \ll 1
\end{equation}
where $\mathcal{E}$ is the neutrino energy in the vierbein frame,
the derivative is also taken with respect to the vierbein time
$t'$. We again study the case of circular orbits. For the neutrino
energy one finds, $\mathcal{E}=m_\nu u^0=m_\nu \gamma A$, where
$m_\nu$ is the neutrino mass. Using Eq.~\eqref{e0} we obtain that
$dt'=A dt$. Let us discuss an electron neutrino with mass
$m_{\nu_e}\approx 2\thinspace\text{eV}$ rotating around a black
hole with $M=10 M_\odot$ on the orbit with radius $R=2r_g$. Then
Eq.~\eqref{qqlcond} reads,
\begin{equation*}
  \frac{\hbar}{2\mathcal{E}}
  \left|
    \frac{d\bm{\zeta}}{dt'}
  \right|=\frac{\hbar}{2m_\nu \gamma^2 A^2}v_\phi\approx
  4.19\times 10^{-13}\ll 1.
\end{equation*}
We can see that the condition of the validity of the
quasi-classical approach is satisfied almost in all reasonable
astrophysical objects.

\section{Conclusion}\label{CONCL}

In conclusion we note that neutrino spin oscillations in
gravitational fields within the quasi-classical approach have been
studied. We have started from the particle spin evolution equation
in the vierbein frame. On the the basis of this approach we have
investigated neutrino spin oscillations in a weak gravitational
field created by a massive rotating object. The three dimensional
spin evolution equation has been obtained. We have improved
analogous calculation performed in our previous work
\cite{DvoGriStu05}. The neutrino spin evolution equation derived
in the present work is valid for arbitrary neutrino velocities and
correctly takes into account the neutrino velocity dependence.
Then we have written down the neutrino oscillations effective
Hamiltonian [Eq.~\eqref{HeffJ}] and transition probability
[Eqs.~\eqref{Probabr} and \eqref{Phir}] when a neutrino propagates
along the radial direction. The neutrino spin conversion rate has
been analyzed for different neutrino velocity directions. Then we
have examined neutrino spin oscillations in the gravitational
field of a non-rotating black hole. We have studied a neutrino
moving on circular orbits, with our calculations being valid for
the neutrino motion in the vicinity of a black hole. We have
derived the effective Hamiltonian [Eq.~\eqref{HeffBH}] and
transition probability [Eq.~\eqref{probBH}] in this case. It has
been demonstrated that neutrino spin oscillations occur in the
Schwarzschield metric. We have also analyzed the dependence of the
neutrino spin oscillations frequency on the radius of the orbit.
The validity of the quasi-classical approach to the description of
neutrino spin oscillations in a gravitational field has been
analyzed in our work.

It should be noted that the method used in the present paper for
the studying of neutrino spin oscillations is valid not only for
neutrinos but also for any spin-$1/2$ particles. The basic
equations \eqref{zetaeq} and \eqref{GEB} were demonstrated in
Ref.~\cite{SilTer05} to be applicable to a massive Dirac particle
in a weak static gravitational field. Therefore we can study,
e.g., electron spin evolution with help of this approach. It is
known that rather frequently massive gravitational objects have
inherent magnetic fields and are surrounded by the moving matter
(accretion disks). Thus, if we discuss the particle spin
oscillations process comprehensively, i.e. take into account all
factors, the cases of neutrinos and electrons are different
because these particles have diverse types of the interaction with
magnetic field and with the background matter. In principle
massless particles can be also included into consideration (see
Sec.~\ref{SM}). Despite of the fact that the mass of a particle is
absent in Eqs.~\eqref{zetaeq} and \eqref{GEB}, the law of motion
(i.e. the function $u^a(\mathbf{r},t)$, which is involved in the
spin evolution equations) of massless particles differs from that
of massive particles. However in the case of massless particles
the helicity and chirality are the same and one can apply the
methods used in the recent work~\cite{Muk05}.

According to Eqs.~\eqref{Omega1}-\eqref{Omega3} there is no
neutrino spin flip in case of radial neutrino motion in the
Schwarzschield metric. This our result agrees with the deduction
of Ref.~\cite{PirRoyWud96} where the radial propagation of
neutrinos from active galactic nuclei was studied. However one can
hardly agree with the claims of Ref.~\cite{CarFul97} that a
spherically symmetric, static Schwarzschild space-time cannot
cause the particle spin flip. That statement confront both our
results and the analysis of Ref.~\cite{SilTer05} in which it was
shown by means of direct calculations that in the Schwarzschield
metric the spin of a Dirac fermion can precess and change its
direction with respect to the particle momentum. The calculations
in that paper were based on Dirac equation in curved space. That
result also agrees with the statement of our paper that the
gravitational field of a non-rotating black hole can cause
neutrino spin oscillations.

Note that the results of our paper can be applied to the
description of the neutrino spin evolution in various
astrophysical media. It is known that relic massive neutrinos can
cluster into a halo around a galaxy contributing to cold dark
matter. Despite of the fact that the average size of a halo ranges
between $10^{2}$ and $10^{3}\thinspace\text{kpc}$ for a typical
galaxy like the Milky Way (see, e.g., Ref.~\cite{RinWon04}), there
is a possibility for a neutrino to be gravitationally captured on
orbits close to the central massive object. In this case strong
gravitational field can essentially contribute to the neutrino
spin oscillations process.

\begin{acknowledgments}
This research was supported by grant of Russian Science Support
Foundation. The author is very thankful to Andrey Lobanov (MSU)
and Andrey Zorin (JINR) for helpful discussions as well as to
Takuya Morozumi for making the facilities of Hiroshima University
available for him and for hospitality. The referee's comments are
also appreciated.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
Maggie's Plan: Parenthood and Love...
Maggie's Plan: Parenthood and Love Triangles
June 10, 2016 by Anja Savcic
Directed by Rebecca Miller, "Maggie's Plan" is a quirky offbeat comedy set in New York City. The location and cast, Greta Gerwig in particular, bring a familiar Woody Allen feel and aesthetic to the screen. A romantic comedy with a stellar cast, this film follows the tangled events of parenthood and the many complications of love.
Miller wastes no time diving into plot. The first scene shows Maggie expressing anxiety over her ticking biological clock. She hatches a plan to artificially inseminate herself and decides to use the sperm of a fellow acquaintance and pickle entrepreneur, Guy (Travis Fimmel). She falls short when a married man, professor John Harding (Ethan Hawke), enters her life. John's rocky marriage leads him to share his unfinished novel and vulnerabilities with Maggie and soon the two are indulging in an illicit affair. Fast-forwarding a couple years, Maggie and John are seen living together and raising a child. Maggie is unhappy with her relationship with John, a now struggling novelist, and guilt over breaking up his first marriage gnaws at her. We then meet Georgette (Julianne Moore), John's now ex-wife, and watch as Maggie orchestrates a plan to get the couple back together. Sure enough Maggie's plan leaves her juggling the emotions of herself and others.
Falling in love on the cusp of a life changing decision is a loaded topic, however the film shifts away from these complexities to favour those in the second half: divorce, jealousy and manipulation. In this way, the sight of Maggie's initial plan, the stakes of parenthood, is lost. While the many twists and turns in the storyline are entertaining, the events take up a lot of the film's content which mean less time for character development. In particular, given more time, John and Maggie's romance could have become fiercer; this also would have increased the stakes for the second part of the movie when the 'three years later' relationship shift occurs. It seems that the film's running time didn't allow for the intricacies of a love triangle and the responsibility/consequences of motherhood to come to light.
In terms of characters, the introduction of best friends, Maggie (Greta Gerwig) and Tony (Bill Hader) was refreshing. Portraying a strong plutonic relationship meant their scenes strayed far away from any romantic tension. In turn, scenes with supporting characters, Tony and Felicia (Maya Rudolph), were what I enjoyed the most. The couple's obvious chemistry lead to flawless comedic scenes filled with playful and sarcastic humour. Fimmel's short stint in this film also brought a special kind of awkwardness to the movie while Gerwig charmed the screen with her own offbeat quirks.
Overall, this was a fun film to watch. It was beautiful aesthetically and all the recognizable actors on screen left me vying to learn more about each and every one of them. Perhaps with a longer running time it could have been the perfect film, but I suppose like Maggie's plan, things don't always work out exactly the way you'd hoped.
Anja Savcic
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Since getting signed to Made Men Music Group in 2013, Tekno Miles has always wowed his fans all over the world with his chart topping singles.With songs like "Pana", "Rara" "Yawa" , "Be", "Samantha", "Go", "Mama" to his discography, Tekno Miles adds another one which is titled "Yur Luv"The song titled "Yur Luv" has been highly anticipated and raved about in the previous weeks.
Finally, the self produced song by Tekno gets available for the listening pleasure of fans.This is Tekno's opening for the year 2018. Well, we hope the record reaches the peak as envisaged by fans.


Creating your own content, escaping from Cuba, and speaking your truth. This week Actress/Writer and Filmmaker Leslie Lluvet dropped by the studio to shine her light. The ups and down of the industry, how everything has led up to this point, and the beauty in the lessons learned, you name it its here. There's a little bit of everything for everyone in this episode, so don't miss it.