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### Messages - Victor Ivrii

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1
##### Test-2 / Re: Test-2 problem-1 confusion regarding boundary conditions
« on: March 07, 2021, 04:12:41 AM »
Solution is allowed to be discontinuous.

2
##### Chapter 3 / Re: 3.1 Heaviside step function
« on: February 09, 2021, 05:11:06 AM »
Everything is correct. You need to look carefully at limits in the integrals

3
##### Chapter 3 / Re: 3.2 Theorem1
« on: February 09, 2021, 05:10:26 AM »
$\int_0^\infty$. I fixed it

4
##### Chapter 2 / Re: Example 6 from Week 3 Lec 2
« on: February 02, 2021, 04:31:46 PM »
Indeed

5
##### Quiz 1 / Re: Quiz 1 - Variant 2E - Problem 2
« on: January 29, 2021, 11:45:32 AM »
Try to avoid high-riser notations. Several years ago I was a referee for a paper which used notations like this $\widehat{\dot{\widetilde{\mathcal{D}}}}$ and sometimes this little pesky dot was missing

6
##### Quiz 1 / Re: Quiz 1 - Variant 2E - Problem 1
« on: January 29, 2021, 11:41:45 AM »
Indeed, this equation has $u=0$ as a solution but the notion of "homogeneou"s does not apply to nonlinear.

7
##### Quiz 1 / Re: LEC9101 Quiz#1 B Question#2
« on: January 28, 2021, 08:01:17 PM »
not $sin(x)$ but $\sin(x)$ (it is \sin, \cos, \ln, \sum, \int and so on for "math operators")

8
##### Chapter 1 / Re: Math Notation Question
« on: January 24, 2021, 07:56:05 AM »
it depends: if $u=u(x)$ you can use either. If $u=u(x,y)$ then only partial derivative would be correct.

9
##### Chapter 3 / Re: Rouché's Theorem
« on: December 12, 2020, 06:25:05 PM »
You need to indicate that there are no zeroes on $\gamma$

10
##### Chapter 7 / Re: Drawing phase portrait with complex eigenvalues question
« on: December 01, 2020, 08:46:13 PM »
"How ellipses wouls look like" means the directions and relative size of their semi-axis. See frame 4 of MAT244_W8L3 handout

11
##### Chapter 4 / Re: Section 4.4 Past Final Exam Question
« on: November 22, 2020, 05:19:33 AM »
There could be misprints

12
##### Chapter 4 / Re: General Question related to something in Section 4.1
« on: November 13, 2020, 12:54:44 PM »
For higher order equations it is covered in MAT244-LEC0201-W6L2 (see modules). It is mandatory material.

13
##### Chapter 4 / Re: 4.2 question 28
« on: October 25, 2020, 01:13:50 PM »
you need to write it, if you hope for any answer

14
##### Test 1 / Re: 2020TT1 Deferred Sitting #1
« on: October 15, 2020, 08:17:16 AM »
Indeed, instead of $\log(\pm w)$ with $+2\pi mi$ we write $\log(w)$  with $+\pi mi$ since $\log (-w)=\log(w)+\i i$

15
##### Test 1 / Re: Abel's Theorem
« on: October 15, 2020, 08:11:23 AM »
Of some fundamental set (remember a constant factor!)

Pages: [1] 2 3 ... 152

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text:
# American Institute of Mathematical Sciences

2011, 18: 97-111. doi: 10.3934/era.2011.18.97

## Simple loops on 2-bridge spheres in 2-bridge link complements

 1 Department of Mathematics, Pusan National University, San-30 Jangjeon-Dong, Geumjung-Gu, Pusan, 609-735, South Korea 2 Department of Mathematics,, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan

Received  April 2011 Revised  June 2011 Published  August 2011

The purpose of this note is to announce complete answers to the following questions. (1) For an essential simple loop on a 2-bridge sphere in a 2-bridge link complement, when is it null-homotopic in the link complement? (2) For two distinct essential simple loops on a 2-bridge sphere in a 2-bridge link complement, when are they homotopic in the link complement? We also announce applications of these results to character varieties and McShane's identity.
Citation: Donghi Lee, Makoto Sakuma. Simple loops on 2-bridge spheres in 2-bridge link complements. Electronic Research Announcements, 2011, 18: 97-111. doi: 10.3934/era.2011.18.97
##### References:
 [1] C. Adams, Hyperbolic 3-manifolds with two generators, Comm. Anal. Geom., 4 (1996), 181-206.  Google Scholar [2] H. Akiyoshi, H. Miyachi and M. Sakuma, A refinement of McShane's identity for quasifuchsian punctured torus groups, In the Tradition of Ahlfors and Bers, III, Contemporary Math., 355, Amer. Math. Soc., Providence, RI, (2004), 21-40.  Google Scholar [3] H. Akiyoshi, H. Miyachi and M. Sakuma, Variations of McShane's identity for punctured surface groups, Proceedings of the Workshop "Spaces of Kleinian Groups," London Math. Soc. Lecture Note Series, 329, Cambridge Univ. Press, Cambridge, (2006), 151-185.  Google Scholar [4] H. Akiyoahi, M. Sakuma, M. Wada and Y. Yamashita, "Punctured Torus Groups and 2-Bridge Knot Groups (I)," Lecture Notes in Mathematics, 1909, Springer, Berlin, 2007.  Google Scholar [5] K. I. Appel and P. E. Schupp, The conjugacy problem for the group of any tame alternating knot is solvable, Proc. Amer. Math. Soc., 33 (1972), 329-336. doi: 10.1090/S0002-9939-1972-0294460-X.  Google Scholar [6] B. H. Bowditch, A proof of McShane's identity via Markoff triples, Bull. London Math. Soc., 28 (1996), 73-78. doi: 10.1112/blms/28.1.73.  Google Scholar [7] B. H. Bowditch, Markoff triples and quasi-Fuchsian groups, Proc. London Math. Soc. (3), 77 (1998), 697-736. doi: 10.1112/S0024611598000604.  Google Scholar [8] B. H. Bowditch, A variation of McShane's identity for once-punctured torus bundles, Topology, 36 (1997), 325-334. doi: 10.1016/0040-9383(96)00017-1.  Google Scholar [9] C. Gordon, "Problems," Workshop on Heegaard Splittings, 401-411, Geom. Topol. Monogr., 12, Geom. Topol. Publ., Coventry, 2007.  Google Scholar [10] K. Johnsgard, The conjugacy problem for the groups of alternating prime tame links is polynomial-time, Trans. Amer. Math. Soc., 349 (1997), 857-901. doi: 10.1090/S0002-9947-97-01617-6.  Google Scholar [11] D. Lee and M. Sakuma, Epimorphisms between 2-bridge link groups: Homotopically trivial simple loops on 2-bridge spheres,, Proc. London Math. Soc., ().   Google Scholar [12] D. Lee and M. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (I),, \arXiv{1010.2232}., ().   Google Scholar [13] D. Lee and M. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (II),, \arXiv{1103.0856}., ().   Google Scholar [14] D. Lee and M. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (III),, preliminary notes., ().   Google Scholar [15] D. Lee and M. Sakuma, A variation of McShane's identity for 2-bridge links,, in preparation., ().   Google Scholar [16] R. C. Lyndon and P. E. Schupp, "Combinatorial Group Theory," Ergebnisse der Mathematik und ihrer Grenzgebiete, Band, 89, Springer-Verlag, Berlin-New York, 1977.  Google Scholar [17] G. McShane, "A Remarkable Identity for Lengths of Curves," Ph.D. Thesis, University of Warwick, 1991. Google Scholar [18] G. McShane, Simple geodesics and a series constant over Teichmuller space, Invent. Math., 132 (1998), 607-632. doi: 10.1007/s002220050235.  Google Scholar [19] M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math., 167 (2007), 179-222. doi: 10.1007/s00222-006-0013-2.  Google Scholar [20] T. Ohtsuki, R. Riley and M. Sakuma, Epimorphisms between 2-bridge link groups, Teh Zieschang Gedenkschrift, Geom. Topol. Monogr., 14, Geom. Topol. Publ., Coventry, (2008), 417-450.  Google Scholar [21] J.-P. Préaux, Conjugacy problems in groups of oriented geometrizable 3-manifolds, Topology, 45 (2006), 171-208. doi: 10.1016/j.top.2005.06.002.  Google Scholar [22] R. Riley, Parabolic representations of knot groups. I, Proc. London Math. Soc. (3), 24 (1972), 217-242.  Google Scholar [23] M. Sakuma, Variations of McShane's identity for the Riley slice and 2-bridge links, In "Hyperbolic Spaces and Related Topics" (Japanese) (Kyoto, 1998), Sūrikaisekikenkyūsho Kōkyūroku, 1104 (1999), 103-108.  Google Scholar [24] Z. Sela, The conjugacy problem for knot groups, Topology, 32 (1993), 363-369. doi: 10.1016/0040-9383(93)90026-R.  Google Scholar [25] S. P. Tan, Private communication, May, 2011. Google Scholar [26] S. P. Tan, Y. L. Wong and Y. Zhang, The $\SL(2,\CC)$ character variety of a one-holed torus, Electon. Res. Announc. Amer. Math. Soc., 11 (2005), 103-110. doi: 10.1090/S1079-6762-05-00153-8.  Google Scholar [27] S. P. Tan, Y. L. Wong and Y. Zhang, Generalizations of McShane's identity to hyperbolic cone-surfaces, J. Differential Geom., 72 (2006), 73-112.  Google Scholar [28] S. P. Tan, Y. L. Wong and Y. Zhang, Necessary and sufficient conditions for McShane's identity and variations, Geom. Dedicata, 119 (2006), 199-217. doi: 10.1007/s10711-006-9069-9.  Google Scholar [29] S. P. Tan, Y. L. Wong and Y. Zhang, Generalized Markoff maps and McShane's identity, Adv. Math., 217 (2008), 761-813. doi: 10.1016/j.aim.2007.09.004.  Google Scholar [30] S. P. Tan, Y. L. Wong and Y. Zhang, End invariants for $SL(2,\CC)$ characters of the one-holed torus, Amer. J. Math., 130 (2008), 385-412. doi: 10.1353/ajm.2008.0010.  Google Scholar [31] S. P. Tan, Y. L. Wong and Y. Zhang, McShane's identity for classical Schottky groups, Pacific J. Math., 237 (2008), 183-200. doi: 10.2140/pjm.2008.237.183.  Google Scholar [32] C. M. Weinbaum, The word and conjugacy problems for the knot group of any tame, prime, alternating knot, Proc. Amer. Math. Soc., 30 (1971), 22-26. doi: 10.1090/S0002-9939-1971-0279169-X.  Google Scholar

show all references

##### References:
 [1] C. Adams, Hyperbolic 3-manifolds with two generators, Comm. Anal. Geom., 4 (1996), 181-206.  Google Scholar [2] H. Akiyoshi, H. Miyachi and M. Sakuma, A refinement of McShane's identity for quasifuchsian punctured torus groups, In the Tradition of Ahlfors and Bers, III, Contemporary Math., 355, Amer. Math. Soc., Providence, RI, (2004), 21-40.  Google Scholar [3] H. Akiyoshi, H. Miyachi and M. Sakuma, Variations of McShane's identity for punctured surface groups, Proceedings of the Workshop "Spaces of Kleinian Groups," London Math. Soc. Lecture Note Series, 329, Cambridge Univ. Press, Cambridge, (2006), 151-185.  Google Scholar [4] H. Akiyoahi, M. Sakuma, M. Wada and Y. Yamashita, "Punctured Torus Groups and 2-Bridge Knot Groups (I)," Lecture Notes in Mathematics, 1909, Springer, Berlin, 2007.  Google Scholar [5] K. I. Appel and P. E. Schupp, The conjugacy problem for the group of any tame alternating knot is solvable, Proc. Amer. Math. Soc., 33 (1972), 329-336. doi: 10.1090/S0002-9939-1972-0294460-X.  Google Scholar [6] B. H. Bowditch, A proof of McShane's identity via Markoff triples, Bull. London Math. Soc., 28 (1996), 73-78. doi: 10.1112/blms/28.1.73.  Google Scholar [7] B. H. Bowditch, Markoff triples and quasi-Fuchsian groups, Proc. London Math. Soc. (3), 77 (1998), 697-736. doi: 10.1112/S0024611598000604.  Google Scholar [8] B. H. Bowditch, A variation of McShane's identity for once-punctured torus bundles, Topology, 36 (1997), 325-334. doi: 10.1016/0040-9383(96)00017-1.  Google Scholar [9] C. Gordon, "Problems," Workshop on Heegaard Splittings, 401-411, Geom. Topol. Monogr., 12, Geom. Topol. Publ., Coventry, 2007.  Google Scholar [10] K. Johnsgard, The conjugacy problem for the groups of alternating prime tame links is polynomial-time, Trans. Amer. Math. Soc., 349 (1997), 857-901. doi: 10.1090/S0002-9947-97-01617-6.  Google Scholar [11] D. Lee and M. Sakuma, Epimorphisms between 2-bridge link groups: Homotopically trivial simple loops on 2-bridge spheres,, Proc. London Math. Soc., ().   Google Scholar [12] D. Lee and M. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (I),, \arXiv{1010.2232}., ().   Google Scholar [13] D. Lee and M. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (II),, \arXiv{1103.0856}., ().   Google Scholar [14] D. Lee and M. Sakuma, Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (III),, preliminary notes., ().   Google Scholar [15] D. Lee and M. Sakuma, A variation of McShane's identity for 2-bridge links,, in preparation., ().   Google Scholar [16] R. C. Lyndon and P. E. Schupp, "Combinatorial Group Theory," Ergebnisse der Mathematik und ihrer Grenzgebiete, Band, 89, Springer-Verlag, Berlin-New York, 1977.  Google Scholar [17] G. McShane, "A Remarkable Identity for Lengths of Curves," Ph.D. Thesis, University of Warwick, 1991. Google Scholar [18] G. McShane, Simple geodesics and a series constant over Teichmuller space, Invent. Math., 132 (1998), 607-632. doi: 10.1007/s002220050235.  Google Scholar [19] M. Mirzakhani, Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math., 167 (2007), 179-222. doi: 10.1007/s00222-006-0013-2.  Google Scholar [20] T. Ohtsuki, R. Riley and M. Sakuma, Epimorphisms between 2-bridge link groups, Teh Zieschang Gedenkschrift, Geom. Topol. Monogr., 14, Geom. Topol. Publ., Coventry, (2008), 417-450.  Google Scholar [21] J.-P. Préaux, Conjugacy problems in groups of oriented geometrizable 3-manifolds, Topology, 45 (2006), 171-208. doi: 10.1016/j.top.2005.06.002.  Google Scholar [22] R. Riley, Parabolic representations of knot groups. I, Proc. London Math. Soc. (3), 24 (1972), 217-242.  Google Scholar [23] M. Sakuma, Variations of McShane's identity for the Riley slice and 2-bridge links, In "Hyperbolic Spaces and Related Topics" (Japanese) (Kyoto, 1998), Sūrikaisekikenkyūsho Kōkyūroku, 1104 (1999), 103-108.  Google Scholar [24] Z. Sela, The conjugacy problem for knot groups, Topology, 32 (1993), 363-369. doi: 10.1016/0040-9383(93)90026-R.  Google Scholar [25] S. P. Tan, Private communication, May, 2011. Google Scholar [26] S. P. Tan, Y. L. Wong and Y. Zhang, The $\SL(2,\CC)$ character variety of a one-holed torus, Electon. Res. Announc. Amer. Math. Soc., 11 (2005), 103-110. doi: 10.1090/S1079-6762-05-00153-8.  Google Scholar [27] S. P. Tan, Y. L. Wong and Y. Zhang, Generalizations of McShane's identity to hyperbolic cone-surfaces, J. Differential Geom., 72 (2006), 73-112.  Google Scholar [28] S. P. Tan, Y. L. Wong and Y. Zhang, Necessary and sufficient conditions for McShane's identity and variations, Geom. Dedicata, 119 (2006), 199-217. doi: 10.1007/s10711-006-9069-9.  Google Scholar [29] S. P. Tan, Y. L. Wong and Y. Zhang, Generalized Markoff maps and McShane's identity, Adv. Math., 217 (2008), 761-813. doi: 10.1016/j.aim.2007.09.004.  Google Scholar [30] S. P. Tan, Y. L. Wong and Y. Zhang, End invariants for $SL(2,\CC)$ characters of the one-holed torus, Amer. J. Math., 130 (2008), 385-412. doi: 10.1353/ajm.2008.0010.  Google Scholar [31] S. P. Tan, Y. L. Wong and Y. Zhang, McShane's identity for classical Schottky groups, Pacific J. Math., 237 (2008), 183-200. doi: 10.2140/pjm.2008.237.183.  Google Scholar [32] C. M. Weinbaum, The word and conjugacy problems for the knot group of any tame, prime, alternating knot, Proc. Amer. Math. Soc., 30 (1971), 22-26. doi: 10.1090/S0002-9939-1971-0279169-X.  Google Scholar
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This is the first volume of a modern introduction to quantum field theory which addresses both mathematicians and physicists, at levels ranging from advanced undergraduate students to professional scientists. The book bridges the acknowledged gap between the different languages used by mathematicians and physicists. For students of mathematics the author shows that detailed knowledge of the physical background helps to motivate the mathematical subjects and to discover interesting interrelationships between quite different mathematical topics. For students of physics, fairly advanced mathematics is presented, which goes beyond the usual curriculum in physics.

An Introduction to Riemann Surfaces

Author: Terrence Napier,Mohan Ramachandran

Publisher: Springer Science & Business Media

ISBN: 9780817646936

Category: Mathematics

Page: 560

View: 7608

This textbook presents a unified approach to compact and noncompact Riemann surfaces from the point of view of the so-called L2 $\bar{\delta}$-method. This method is a powerful technique from the theory of several complex variables, and provides for a unique approach to the fundamentally different characteristics of compact and noncompact Riemann surfaces. The inclusion of continuing exercises running throughout the book, which lead to generalizations of the main theorems, as well as the exercises included in each chapter make this text ideal for a one- or two-semester graduate course.

Compactifying Moduli Spaces

Publisher: Birkhäuser

ISBN: 3034809212

Category: Mathematics

Page: 135

View: 6601

This book focusses on a large class of objects in moduli theory and provides different perspectives from which compactifications of moduli spaces may be investigated. Three contributions give an insight on particular aspects of moduli problems. In the first of them, various ways to construct and compactify moduli spaces are presented. In the second, some questions on the boundary of moduli spaces of surfaces are addressed. Finally, the theory of stable quotients is explained, which yields meaningful compactifications of moduli spaces of maps. Both advanced graduate students and researchers in algebraic geometry will find this book a valuable read.

Krichever–Novikov Type Algebras

Theory and Applications

Author: Martin Schlichenmaier

Publisher: Walter de Gruyter GmbH & Co KG

ISBN: 3110381478

Category: Mathematics

Page: 375

View: 1183

Krichever and Novikov introduced certain classes of infinite dimensional Lie algebras to extend the Virasoro algebra and its related algebras to Riemann surfaces of higher genus. The author of this book generalized and extended them to a more general setting needed by the applications. Examples of applications are Conformal Field Theory, Wess-Zumino-Novikov-Witten models, moduli space problems, integrable systems, Lax operator algebras, and deformation theory of Lie algebra. Furthermore they constitute an important class of infinite dimensional Lie algebras which due to their geometric origin are still manageable. This book gives an introduction for the newcomer to this exciting field of ongoing research in mathematics and will be a valuable source of reference for the experienced researcher. Beside the basic constructions and results also applications are presented.

Riemann Surfaces by Way of Complex Analytic Geometry

Author: Dror Varolin

Publisher: American Mathematical Soc.

ISBN: 0821853694

Category: Mathematics

Page: 236

View: 2913

This book establishes the basic function theory and complex geometry of Riemann surfaces, both open and compact. Many of the methods used in the book are adaptations and simplifications of methods from the theories of several complex variables and complex analytic geometry and would serve as excellent training for mathematicians wanting to work in complex analytic geometry. After three introductory chapters, the book embarks on its central, and certainly most novel, goal of studying Hermitian holomorphic line bundles and their sections. Among other things, finite-dimensionality of spaces of sections of holomorphic line bundles of compact Riemann surfaces and the triviality of holomorphic line bundles over Riemann surfaces are proved, with various applications. Perhaps the main result of the book is Hormander's Theorem on the square-integrable solution of the Cauchy-Riemann equations. The crowning application is the proof of the Kodaira and Narasimhan Embedding Theorems for compact and open Riemann surfaces. The intended reader has had first courses in real and complex analysis, as well as advanced calculus and basic differential topology (though the latter subject is not crucial). As such, the book should appeal to a broad portion of the mathematical and scientific community. This book is the first to give a textbook exposition of Riemann surface theory from the viewpoint of positive Hermitian line bundles and Hormander $\bar \partial$ estimates. It is more analytical and PDE oriented than prior texts in the field, and is an excellent introduction to the methods used currently in complex geometry, as exemplified in J. P. Demailly's online but otherwise unpublished book Complex analytic and differential geometry.'' I used it for a one quarter course on Riemann surfaces and found it to be clearly written and self-contained. It not only fills a significant gap in the large textbook literature on Riemann surfaces but is also rather indispensible for those who would like to teach the subject from a differential geometric and PDE viewpoint. --Steven Zelditch

Introduction to Algebraic Curves

Author: Phillip Griffiths

Publisher: American Mathematical Soc.

ISBN: 0821845306

Category: Courbes algébriques

Page: 221

View: 1252

This book gives the first systematic exposition of geometric analysis on Riemannian symmetric spaces and its relationship to the representation theory of Lie groups. The book starts with modern integral geometry for double fibrations and treats several examples in detail. After discussing the theory of Radon transforms and Fourier transforms on symmetric spaces, inversion formulas, and range theorems, Helgason examines applications to invariant differential equations on symmetric spaces, existence theorems, and explicit solution formulas, particularly potential theory and wave equations. The canonical multitemporal wave equation on a symmetric space is included. The book concludes with a chapter on eigenspace representations--that is, representations on solution spaces of invariant differential equations. Known for his high-quality expositions, Helgason received the 1988 Steele Prize for his earlier books Differential Geometry, Lie Groups and Symmetric Spaces and Groups and Geometric Analysis. Containing exercises (with solutions) and references to further results, this revised edition would be suitable for advanced graduate courses in modern integral geometry, analysis on Lie groups, and representation theory of Lie groups.

Geometry of Characteristic Classes

Author: Shigeyuki Morita

Publisher: American Mathematical Soc.

ISBN: 0821821393

Category: Mathematics

Page: 185

View: 1293

Characteristic classes are central to the modern study of the topology and geometry of manifolds. They were first introduced in topology, where, for instance, they could be used to define obstructions to the existence of certain fiber bundles. Characteristic classes were later defined using connections on vector bundles, thus revealing their geometric side. The purpose of this book is to introduce the reader to the three theories of characteristic classes that were developed in the late 1960s. They include characteristic classes of flat bundles, characteristic classes of foliations, and characteristic classes of surface bundles. The book is intended for graduate students and research mathematicians working in various areas of geometry and topology.

Algebraic Analysis

Papers Dedicated to Professor Mikio Sato on the Occasion of His Sixtieth Birthday

Author: Masaki Kashiwara,Takahiro Kawai

ISBN: 1483267946

Category: Mathematics

Page: 500

View: 9584

Algebraic Analysis: Papers Dedicated to Professor Mikio Sato on the Occasion of his 60th Birthday, Volume II is a collection of research papers on algebraic analysis and related topics in honor to Professor Mikio Sato’s 60th birthday. This volume is divided into 29 chapters and starts with research works concerning the fundamentals of KP equations, strings, Schottky problem, and the applications of transformation theory for nonlinear integrable systems to linear prediction problems and isospectral deformations,. The subsequent chapters contain papers on the approach to nonlinear integrable systems, the Hodge numbers, the stochastic different equation for the multi-dimensional weakly stationary process, and a method of harmonic analysis on semisimple symmetric spaces. These topics are followed by studies on the quantization of extended vortices, moduli space for Fuchsian groups, microfunctions for boundary value problems, and the issues of multi-dimensional integrable systems. The remaining chapters explore the practical aspects of pseudodifferential operators in hyperfunction theory, the elliptic solitons, and Carlson’s theorem for holomorphic functions. This book will prove useful to mathematicians and advance mathematics students.

Algebraic Geometry

Author: Robin Hartshorne

Publisher: Springer Science & Business Media

ISBN: 1475738498

Category: Mathematics

Page: 496

View: 2232

An introduction to abstract algebraic geometry, with the only prerequisites being results from commutative algebra, which are stated as needed, and some elementary topology. More than 400 exercises distributed throughout the book offer specific examples as well as more specialised topics not treated in the main text, while three appendices present brief accounts of some areas of current research. This book can thus be used as textbook for an introductory course in algebraic geometry following a basic graduate course in algebra. Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. He is the author of "Residues and Duality", "Foundations of Projective Geometry", "Ample Subvarieties of Algebraic Varieties", and numerous research titles.


text:
IFC 4.3.0.1 (IFC4X3) development

8.8.3.31 IfcRightCircularCone

8.8.3.31.1 Semantic definition

The IfcRightCircularCone is a Construction Solid Geometry (CSG) 3D primitive. It is a solid with a circular base and a point called apex as the top. The tapers from the base to the top. The axis from the center of the circular base to the apex is perpendicular to the base. The inherited Position attribute defines the IfcAxisPlacement3D and provides the location and orientation of the cone:

• SELF\IfcCsgPrimitive3D.Position: The location and orientation of the axis system for the primitive.
• SELF\IfcCsgPrimitive3D.Position.Location: The center of the circular area being the bottom face of the cone.
• SELF\IfcCsgPrimitive3D.Position.Position[3]: The z-axis of the inherited placement coordinate system provides the center axis of the IfcRightCircularCone, and the apex is at the Height value applied to the positive direction of the z-axis. The BottomRadius defines the circular base at the xy-plane of the placement coordinate system.

Figure 8.8.3.31.A illustrates geometric parameters of the cone. The cone is positioned within its own placement coordinate system. The origin is the center of the bottom circular disk, that lies in the XY plane. The apex lies on the positive z axis at [0, 0, Height].

Texture use definition

On the circular side, textures are aligned facing upright with origin at the back (+Y direction) revolving counter-clockwise. Textures are stretched or repeated to the extent of the base circumference according to RepeatS. Textures are compressed linearly going upwards towards the top point according to RepeatT.

On the bottom face, textures are aligned facing front-to-back, with the center of the circle aligned to the center of the texture.

Figure 8.8.3.31.B illustrates default texture mapping with a clamped texture (RepeatS=False and RepeatT=False). The image on the left shows the texture where the S axis points to the right and the T axis points up. The image on the right shows the texture applied to the geometry where the X axis points back to the right, the Y axis points back to the left, and the Z axis points up.

8.8.3.31.4 Formal representation

ENTITY IfcRightCircularCone
SUBTYPE OF (IfcCsgPrimitive3D);
Height : IfcPositiveLengthMeasure;
END_ENTITY;


text:
# Encoder '18 - Solution Outlines

While I get to the organizers for editorials, here are the short solution outlines which you can refer to for solving the problem.

1.$\text{KCHESS-}$Create a hash of positions which are near to king- 3*3 matrix. And after that check whether all the positions are attacking or not.
2.PNTWLL- The last paint will definitely exist on the wall. So we just have to find the increasing sequence from reverse and count it.Moreover to keep the colors distinct, we insert the color id in a set stl and at the end print the set size. Another approach can be using stack as suggested by ____(guess who :p) using stack is possible, where we can just push Ai to stack, until you encounter a element greater than it. Once you find Aj, such that Aj> top of stack, pop top of stack until its empty or Aj< top of stack.
3.SLAEL- Thanks to @Arjun Arul for pointed a noteworthy solution - Find and store all indices of elements >K. Say we stored them in array A. Now find longest length between two such values such that there is exactly 1 element >K in between, which, precisely, is A[i+2]-A[i]. With little modification to above, we cna handle case of multiple maximums in a range- simply dont add the maximum if its equal to top of stack or is equal to most recently inserted maxima.
4. OLDFAR- We are proposing a dynamic programming solution for this problem. For a matrix “mat” with dimensions N*M

a. dp[i][j] represents the maximum sum that can be obtained by traversing a ‘L’ shaped path formed from the coordinates, (i,j), (x,j) and (x,M) where (i<= x<=N) , for the submatrix with top-left corner at i,j and bottom right corner at N,M.
b. revCum[i][j] represents the sum, mat[i][j]+mat[i][j+1]+mat[i][j+2]...............mat[i][M]

c. dp[i][j] can be calculated as- dp[i][j] = mat[i][j] + max(dp[i+1][j],revCum[i][j+1])(Base cases can be handled accordingly)
d. After the dp and revCum have been calculated, the matrix can then be traversed and answer can be calculated using
ans=max(ans, revCum[i][j]+dp[i+1][j])

5.CTOUR- The problem can be solved using LCA and Kadane’s Algorithm on trees.

Preprocessing steps:

Profit from root to all the nodes - profit[n].
For calculating LCA in log(n).

Using kadane’s algorithm (top down), calculate the max profit for each node, if we are at the node and go towards the root and come back - up[n].

Using kadane’s algorithm (bottom-up) calculate the max profit for each node, if we are at the node and go towards downwards and come back - down[n].

Calculation steps:

Every query can be answered in O(log(n)).
There are three cases.

a. If LCA(A,B) != A or B. Then we have only one choice- going from A to LCA(A,B), then towards root to some extent and then to B. The total amount we can gain is profit[A] + profit** - 2*profit[LCA(A,B)] + 2*up[LCA(A,B)].
b. If LCA(A,B) = A. Then we have two choices- going upwards from A and back, or going downwards from B and back. The total amount we can gain is profit** - profit[A] + 2*max(up[A],down**).
c. If LCA(A,B) = B. Then we have two choices- going upwards from B and back, or going downwards from A and back. The total amount we can gain is profit[A] - profit** + 2*max(up**,down[A]).

2 Likes

____(guess who :p) = @vijju123 xD

1 Like

When will you guys publish the combined ranklist for Div1 and Div2?

Please see the announcement on the contest page.

An interesting question arising from KCHESS would be to see if the knights can keep a non-checkmated king confined to an area of the board. If the king can get all but one of the knights behind him by a couple of squares, he can certainly never be driven back to the body of knights.

    #include<bits/stdc++.h>
using namespace std;
int main(){
int n,m,t;
cin>>t;
for(int i=0;i<t;i++)
{
cin>>n>>m;
int count=0;
long int h[n],c[n],max;
for(int j=0;j<n;j++)
cin>>h[j];
for(int j=0;j<n;j++)
cin>>c[j];
max=h[n-1];
int pos=n-1;
count++;
for(int j=n-1;j>=0;j--)
{
if(h[j]>max&&c[pos]!=c[j])
{
count++;
max=h[j];
pos=j;
}

}
cout<<count<<endl;
}
return 0;
}


text:
# Probability of genetics [closed]

The color of a persons eyes is determined by a single pair of genes.   If they are both blue-eyed genes, then the person will have blue eyes.   If they are both brown-eyed genes, then the person will have brown eyes.   Also, if one of them is a blue-eyed gene and the other a brown-eyed gene, then the person will have brown eyes.

A newborn child independently receives one gene from each of his parents and the gene it receives from a parent is equally likely to be either of the two eye genes of that parent.

Suppose that Joe and both his parents have brown eyes, but Joe's sister and Joe's wife Claudia have blue eyes.

(a) What is the probability that Joe possesses a blue-eyed gene?

(b) What is the probability that Joe and Claudia’s first child will have blue eyes?

(c) If Joe and Claudia’s first child has brown eyes, what is the probability that their next child will also have brown eyes? (Hint: This is not the same as one minus the answer to part (b)).

How to solve part(c)? Suppose we let E1=(the first child has brown eyes),E2=(the next child has brown eyes) How to calculate Pr(E2|E1)?

## closed as off-topic by TZakrevskiy, Graham Kemp, jameselmore, user147263, graydadSep 15 '15 at 2:54

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – TZakrevskiy, jameselmore, graydad
If this question can be reworded to fit the rules in the help center, please edit the question.

• What did you try to solve this question? Did it work? Where do you get stuck? – TZakrevskiy Sep 14 '15 at 23:55

To get you started:

Let A be the allele for brown eyes and a the allele for blue eyes. If an individual has genotype AA, Aa, or aA --one allele from each parent--then their phenotype is brown eyes. The only genotype with phenotype blue eyes is genotype aa.

Now :

• What is the genotype of Joe's wife? aa, as that is the only genotype for phenotype blue eyes

• Both parents have phenotype brown eyes and thus their genotypes are either AA, Aa or aA. But as Joe's sister has genotype aa, what must be the genotype of both parents? Both of Joe's parents must have genotype Aa or aA.

• Thus with equal probability, Joe's genotype is AA, Aa, aA. In other words, with probability $2/6 = 1/3$ Joe has passes on an a allele, the allele for blue eyes

Can you take it from here?

I strongly suggest drawing a genetic diagram, showing the genotype for each individual.

• Thank you. Following your tips, I try to solve (b) in this way: – Emily Sep 15 '15 at 0:14
• Thank you. Following your tips, I try to solve (b) in this way: If Joe has the genotype of Aa or aA, and gives the a allele to his child, with Claudia's genotype being aa, then their first child can have blue eyes. Since the probability of Joe's genotype being Aa or aA is 2/3 and the probability of giving the a allele from the genotype Aa or aA is 1/2. The probability of Joe and Claudia's first child to have blue eyes is 2/3*1/2=1/3. Does it work? – Emily Sep 15 '15 at 0:29
• Yes, that's fine. – Simon S Sep 15 '15 at 1:46
• I notice that you mentioned in your answer that the probability of Joe having an a allele is 2/6=1/3, but I think it is 2/3. – Emily Sep 15 '15 at 2:08
• I notice that you mentioned in your answer that the probability of Joe having an a allele is 2/6=1/3, but I think it is 2/3, because among the three possible genotype for Joe (AA,Aa,aA), when Joe's genotype is Aa or aA, Joe has an a allele, that is 2 out of 3. – Emily Sep 15 '15 at 2:15


text:
# greatest least bound and least upper bound proof

• November 4th 2009, 12:21 PM
tae1466
greatest least bound and least upper bound proof
I've been struggling with this one.

Let A be a partially ordered set. Suppose X⊆Y⊆A.

1. Assuming that all the least upper bounds and greatest lower bounds exist, prove that glb(Y) ≤ glb(X) ≤ lub(X) ≤ lub (Y)

2. Find two subsets X and Y of Real Numbers for which X is a proper subset of Y and yet glb(Y) = glb(X) and lub(X) = lub(y)
• November 4th 2009, 04:44 PM
Drexel28
Quote:

Originally Posted by tae1466
I've been struggling with this one.

Let A be a partially ordered set. Suppose X⊆Y⊆A.

1. Assuming that all the least upper bounds and greatest lower bounds exist, prove that glb(Y) ≤ glb(X) ≤ lub(X) ≤ lub (Y)

2. Find two subsets X and Y of Real Numbers for which X is a proper subset of Y and yet glb(Y) = glb(X) and lub(X) = lub(y)

1. I'll do part of this. See if you can do the rest.

Problem: Let $X\subseteq Y$. Prove that $\sup\left(X\right)\le\sup\left(Y\right)$

Proof: We know that since $X\subseteq Y$ that $x\in X\implies x\in Y$. But $\forall x\in Y\quad x\le\sup\left(Y\right)$, so $\forall x\in X\quad x\le \sup\left(Y\right)$. Therefore $\sup\left(Y\right)$ is an upper bound for $X$. And by definition $\sup\left(X\right)$ is at least as small as any upper bound of $X$. The conclusion follows.

2. What about $X=(0,1)$ and $Y=[0,1]$?


text:
## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

$\displaystyle \frac{1}{9}$
Apply rule $\quad a^{m}\cdot a^{n}=a^{m+n}$: $3^{-6}\cdot 3^{4}=3^{-6+4}=3^{-2}$ Apply rule $\displaystyle \quad a^{-n}=\frac{1}{a^{n}}$: $=\displaystyle \frac{1}{3^{2}}$ $=\displaystyle \frac{1}{9}$


text:
Hide

# Problem JSimply Sudoku

Sudoku puzzles come in all different shapes and difficulty levels. Traditionally a Sudoku puzzle is a $9\times 9$ grid. Initially, some of the cells are filled in with numbers and some are empty. The goal is to fill in each cell with a number in the range $1$$9 subject to the following restrictions: • Each digit 1$$9$ must appear once in each row

• Each digit $1$$9 must appear once in each column • Each digit 1$$9$ must appear once in each $3\times 3$ sub-grid

The difficulty of a Sudoku puzzle can vary widely. The easiest puzzles can be solved with the following two simple techniques:

Single Value Rule:

search for squares which only have one possible value that can be put there.

Unique Location Rule:

within any row, column or sub-grid search for a value that can only be placed in one of the nine locations.

Consider the partially solved Sudoku puzzle shown in Figure 1. The Single Value Rule applies to grid square A$7$ where $8$ is the only value that can be placed there. The Unique Location Rule can be used to put a $5$ in square B$3$ as it is the only location in row $3$ where a $5$ can be placed.

The easiest Sudoku puzzles can be solved with only these two rules; harder puzzles use techniques like swordfish, x-wings and BUGs.

For this problem you will be given a Sudoku puzzle and must determine if it is an easy puzzle, i.e., whether it can be solved by just using the Single Value and Unique Location rules.

## Input

Input consists of a single Sudoku puzzle given over nine lines. Each line contains $9$ values in the range $0$ to $9$, where a $0$ indicates a blank in the puzzle.

## Output

Output the word Easy followed by the solved Sudoku puzzle if it is an easy puzzle. The puzzle should be printed on nine lines with a single space separating values on a line. If the puzzle is not easy output Not easy followed by as much of the Sudoku puzzle as can be solved using the two rules described above. Use the same format for the partial solution as for the complete solution using a ‘.’ instead of a digit for a unfilled square.

Sample Input 1 Sample Output 1
2 6 0 5 1 0 3 0 0
3 0 0 0 6 0 0 0 2
0 1 5 0 7 3 9 0 4
0 0 9 0 0 0 5 0 0
0 0 2 6 0 1 4 0 0
0 0 6 0 0 0 7 0 0
6 0 1 9 4 0 2 3 0
9 0 0 0 2 0 0 0 5
0 0 8 0 3 5 0 4 9

Easy
2 6 4 5 1 9 3 7 8
3 9 7 8 6 4 1 5 2
8 1 5 2 7 3 9 6 4
1 3 9 4 8 7 5 2 6
5 7 2 6 9 1 4 8 3
4 8 6 3 5 2 7 9 1
6 5 1 9 4 8 2 3 7
9 4 3 7 2 6 8 1 5
7 2 8 1 3 5 6 4 9

Sample Input 2 Sample Output 2
0 0 0 0 0 0 7 0 1
0 0 0 0 0 1 2 3 5
0 0 1 8 0 0 0 0 6
0 0 0 0 2 5 0 9 3
9 0 0 0 0 0 0 0 2
3 1 0 6 7 0 0 0 0
2 0 0 0 0 3 8 0 0
1 3 8 9 0 0 0 0 0
4 0 6 0 0 0 0 0 0

Not easy
. . 3 . . . 7 8 1
. . . . . 1 2 3 5
. . 1 8 3 . 9 4 6
. . . . 2 5 . 9 3
9 . . 3 . . . 7 2
3 1 2 6 7 9 4 5 8
2 . . . . 3 8 . .
1 3 8 9 . . 5 . .
4 . 6 . . . 3 . .


text:
# Show Thoughts

When asked to ignore stimuli in the visual modality and attend to the auditory modality, increased activity in the auditory temporal cortex and decreased activity in the visual occipital cortex can be observed (and vice versa).

Task-irrelevant visual cues do not affect visual orienting (visual spatial attention). Task-irrelevant auditory cues, however, seem to do so.

Semantical congruence can influence multisensory integration.

In one of their experiments, Warren et al. had their subjects localize visual or auditory components of visual-auditory stimuli (videos of people speaking and the corresponding sound). Stimuli were made compelling' by playing video and audio in sync anduncompelling' by introducing a temporal offset.

They found that their subjects performed as under a unity assumptions'' when told they would perceive cross-sensory stimuli, and when the stimuli were compelling' and under a lowunity assumption'' when they were told there could be separate auditory or visual stimuli and/or the stimuli were made uncompelling'.

Cognitive factors can influence multisensory processing.


text:
## B. Rider, VIGRE Postdoc

 Office Location: 133 Physics Bldg Office Phone: (919) 660-2816 Email Address: Web Page: http://www.math.duke.edu/~rider

Office Hours:

Wednesdays 5-7
Education:

B.S Mathematics, MIT 1995 Ph.D. Mathematics, Courant Institute 2000
Research Interests: Probability/ Mathematical Physics

I'm interested in asymptotic problems related to physics, or Large Deviation type questions. Right now I think mostly about Random Matrix models. The grand hope is to do something interesting related to local structure without relying on the orthogonal polynomial technology. Towards this I try to focus on the non-Hermitian and beta not equal to two settings.

Representative Publications

1. Brownian Motion with Restoring Drift: Micro-canonical Ensemble and Thermodynamic Limit, Communications in Math Physics, (231) 463-480, (2002). [ps]
2. On the $\infty$-Volume Limit of Focussing Cubic Schr{\"o}dinger, Communications on Pure and Applied Mathematics, (55) 1231-1248 (2002). [ps]
3. A Limit Theorem at the edge of a non-Hermitian Random Matrix Ensemble, Journal of Physics A: Mathematical and General (36) 3401-3409 (2003). [ps]
4. with Shmuel Friedland and Ofer Zeitouni, Concentration of Permanent Estimators for Certain Large Matrices, Annals of Applied Probability (Accepted, 0) [ps]
5. Deviations from the Circular Law, Probability Theory and Related Fields (Accepted, 0) [ps]


text:
# What is $\gcd(0,a)$, where $a$ is a positive integer?

I have tried $\gcd(0,8)$ in a lot of online gcd (or hcf) calculators, but some say $\gcd(0,8)=0$, some other gives $\gcd(0,8)=8$ and some others give $\gcd(0,8)=1$. So really which one of these is correct and why there are different conventions?

• I haven't encountered the convention of gcd(0,8) = 1. It depends on how you define the phrase "a divides b" Mar 18 '11 at 2:49
• Mar 18 '11 at 2:53
• @The Chaz: They are really the same things but with different names. see en.wikipedia.org/wiki/Greatest_common_divisor Mar 18 '11 at 4:04
• Should be a, because anything is a divisor of 0. Nov 26 '20 at 7:48

Let's recall the definition of  "$\rm a$ divides $\rm b$"  in a ring $\rm\,Z,\,$ often written as $\rm\ a\mid b\ \ in\ Z.$

$\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \rm\ a\mid b\ \ in\ Z\ \iff\ a\,c = b\ \$ for some $\rm\ c\in Z$

Recall also the definition of $\rm\ gcd(a,b),\,$ namely

$(1)\rm\qquad\quad \rm gcd(a,b)\mid a,b\qquad\qquad\qquad\$ the gcd is a common divisor

$(2)\rm\qquad\quad\! \rm c\mid a,b\ \ \ \Longrightarrow\ \ c\mid gcd(a,b)\quad$ the gcd is a greatest common divisor

$\ \ \ \$ i.e. $\rm\quad\ c\mid a,b\ \iff\ c\mid gcd(a,b)\quad\,$ expressed in $\iff$ form  [put $\rm\ c = gcd(a,b)\$ for $(1)$]

Notice $\rm\quad\, c\mid a,0\ \iff\ c\mid a\,\$ so $\rm\ gcd(a,0)\ =\ a\$ by the prior "iff" form of the gcd definition.

Note that $\rm\ gcd(0,8) \ne 0\,$ since $\rm\ gcd(0,8) = 0\ \Rightarrow\ 0\mid 8\$ contra $\rm\ 0\mid x\ \iff\ x = 0.$

Note that $\rm\ gcd(0,8) \ne 1\,$ else $\rm\ 8\mid 0,8\ \Rightarrow\ 8\mid gcd(0,8) = 1\ \Rightarrow\ 1/8 \in \mathbb Z.$

Therefore it makes no sense to define $\rm\ gcd(0,8)\$to be $\,0\,$ or $\,1\,$ since $\,0\,$ is not a common divisor of $\,0,8\,$ and $\,1\,$ is not the greatest common divisor.

The $\iff$ gcd definition is universal - it may be employed in any domain or cancellative monoid, with the convention that the gcd is defined only up to a unit factor. This $\iff$ definition is very convenient in proofs since it enables efficient simultaneous proof of both implication directions. $\$ For example, below is a proof of this particular form for the fundamental GCD distributive law $\rm\ (ab,ac)\ =\ a\ (b,c)\$ slightly generalized (your problem is simply $\rm\ c=0\$ in the special case $\rm\ (a,\ \ ac)\ =\,\ a\ (1,c)\ =\ a\,$).

Theorem $\rm\quad (a,b)\ =\ (ac,bc)/c\quad$ if $\rm\ (ac,bc)\$ exists.

Proof $\rm\quad d\mid a,b\ \iff\ dc\mid ac,bc\ \iff\ dc\mid (ac,bc)\ \iff\ d|(ac,bc)/c$

See here for further discussion of this property and its relationship with Euclid's Lemma.

Recall also how this universal approach simplifies the proof of the basic GCD * LCM law:

Theorem $\rm\;\; \ (a,b) = ab/[a,b] \;\;$ if $\;\rm\ [a,b] \;$ exists.

Proof $\rm\quad d|\,a,b \;\iff\; a,b\,|\,ab/d \;\iff\; [a,b]\,|\,ab/d \;\iff\; d\,|\,ab/[a,b] \quad\;\;$

For much further discussion see my many posts on GCDs.

Another way to look at it is by the divisibility lattice, where gcd is the greatest lower bound. So 5 is the greatest lower bound of 10 and 15 in the lattice.

The counter-intuitive thing about this lattice is that the 'bottom' (the absolute lowest element) is 1 (1 divides everything), but the highest element, the one above everybody, is 0 (everybody divides 0).

So $\gcd(0, x)$ is the same as ${\rm glb}(0, x)$ and should be $x$, because $x$ is the lower bound of the two: they are not 'apart' and 0 is '$>'$ $x$ (that is the counter-intuitive part).

In fact, the top answer can be generalized slightly: if $$a \mid b$$, then $$\gcd(a,b)=a$$ (and this holds in any algebraic structure where divisibility makes sense, e.g. a commutative, cancellative monoid).

To see why, well, it's clear that $$a$$ is a common divisor of $$a$$ and $$b$$, and if $$\alpha$$ is any common divisor of $$a$$ and $$b$$, then, of course, $$\alpha \mid a$$. Thus, $$a=\gcd(a,b)$$.

• Indeed, even more generally, it is a special case of the distributive law - see my answer. As for commutative monoids, one usually requires them to be cancellative in order to obtain a rich theory. Mar 18 '11 at 18:26

It might be partly a matter of convention. However, I believe that stating that $$\gcd(8,0) = 8$$ is safer. In fact, $$\frac{0}{8} = 0$$, with no remainder. The proof of the division, indeed is that "Dividend = divider $$\times$$ quotient plus remainder". In our case, 0 (dividend) = 8 (divisor) x 0 (quotient). No remainder. Now, why should 8 be the GCD? Because, while the same method of proof can be used for all numbers, proving that $$0$$ has infinite divisors, the greatest common divisor cannot be greater than $$8$$, and for the reason given above, is $$8$$.


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# [texhax] need certain amount of room on page

Fri Jan 30 15:56:03 CET 2009

On Thu, Jan 29, 2009 at 05:18:54PM -0700, D. R. Evans wrote:
> In plain TeX I want to do the following:
>   if there is X amount of remaining space on the current page:
>     do A
>   else
>     do B

If B = "start new page and A", then you can do something like plain
TeX's \beginsection:

\outer\def\beginsection#1\par{\vskip\z@ plus.3\vsize\penalty-250
\vskip\z@ plus-.3\vsize\bigskip\vskip\parskip
...
}

Oleg


text:
# American Institute of Mathematical Sciences

July  2007, 1(3): 425-442. doi: 10.3934/jmd.2007.1.425

## Global rigidity of certain Abelian actions by toral automorphisms

 1 IMERL-Facultad de Ingeniería, Universidad de la República, ulio Herrera y Reissig 565, CC 30, 11300 Montevideo, Uruguay

Received  June 2006 Revised  March 2007 Published  April 2007

We prove global rigidity results for some linear abelian actions on tori. The type of actions we deal with includes in particular maximal rank semisimple actions on $\mathbb T^N$.
Citation: Federico Rodriguez Hertz. Global rigidity of certain Abelian actions by toral automorphisms. Journal of Modern Dynamics, 2007, 1 (3) : 425-442. doi: 10.3934/jmd.2007.1.425
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2021 Impact Factor: 0.641


text:
# Wiki Contributions

I haven’t tested extensively but first impression is that this is indeed the case. Would be interesting to see if Sydney is similar but I think there’s a limit on number of messages per conversation or something?

When you did this do you let ChatGPT play both sides or were you playing one side? I think it is much better if it gets to play both sides.

I tried this with chatGPT to see just how big the difference was.

ChatGPT is pretty terrible at FEN in both games (Zack and Erik). In Erik’s game it insisted on giving me a position after 13 moves even though 25 moves had happened. When I pointed this out it told me that because there were no captures or pawn moves between moves 13 and 25 the FEN stayed the same…

However it is able to give sensible continuations of >20 ply to checkmate for both positions provided you instruct it not to give commentary and to only provide the moves. The second you allow it to comment on moves it spouts nonsense and starts making illegal moves. I also sometimes had to point out that it was black to play.

In Zack’s game ChatGPT has black set a trap for white (14… Ne5) and has white fall into it (15. Qxd4). After this 15… Nf3+ wins the Queen with a discovered attack from the bishop on g7.

Example continuation:

14... Ne5 15. Qxd4 Nf3+ 16. gxf3 Bxd4 17. Nxd4 Qxd4 18. Be3 Qxb2 19. Rec1 Bh3 20. Rab1 Qf6 21. Bd1 Rfc8 22. Rxc8+ Rxc8 23. Rxb7 Qa1 24. Bxa7 Qxd1#

One other continuation it gave got to checkmate on move 51 without any illegal moves or obvious blunders! (Other than whites initial blunder to fall into the trap)

In Erik’s game ChatGPT manages to play 29 ply of near perfect game for both players:

25... d5 26. g5 d4 27. cxd4 exd4 28. Qd2 Bb3+ 29. Ke1 Nd7 30. Nf5 Ne5 31. Be2 d3 32. Bd1 Bxd1 33. Kxd1 Nxf3 34. Qc3 f6 35. Qc4+ Kh8 36. Qe6 Qa5 37. Kc1 Qc3+ 38. Kb1 Rb8+ 39. Ka2 Qb2#

Stockfish prefers 26… dxc4+ and later on keeps wanting Bc5 for black plus black takes slightly to complete the checkmate than optimal but overall this is very accurate for both players.

Might be worth playing a game against chatGPT while telling it not to give any commentary?

Despite being a GPT-3 instance DALL-E appears to be able to draw an adequate " SolidGoldMagikarp" (if you allow for its usual lack of ability to spell). I tried a couple of alternative prompts without any anomalous results.

FWIW this matches my own experience with one as a company car pretty exactly.

(On mine (UK, right hand drive) the wipers can be activated manually by pressing the button on the end of the left stalk. This also brings up an on-screen menu for selecting constant wiping)

Get out of our houses before we are driven to expend effort killing them, and similarly for all the other places ants conflict with humans (stinging, eating crops, ..)

Ant mafia: "Lovely house you've got there, wouldn't it be a shame if it got all filled up with ants?"

I can't tell you whether this is right or not but this is what ChatGPT thinks:

"Is it possible to put a latex table in a footnote?"

"Yes, it is possible to include a LaTeX table in a footnote. You can do this by using the footnote or footnotetext commands provided by the footmisc package.

Here is an example of how to use these commands to include a table in a footnote:

\usepackage{footmisc}

 ...

 \begin{table}[h]

\centering

\begin{tabular}{c c c}

A & B & C \\

1 & 2 & 3 \\

4 & 5 & 6

\end{tabular}

\caption{Table caption}

\end{table}

 ...

 \footnote{

\begin{table}[h]

\centering

\begin{tabular}{c c c}

D & E & F \\

7 & 8 & 9 \\

10 & 11 & 12

\end{tabular}

\caption{Table caption in footnote}

\end{table}

}

Note that when including a table in a footnote, you should use the table environment as you would normally, but place it inside the footnote or footnotetext command. You should also make sure to use the [h] option for the table environment to prevent the table from floating.

I hope this helps! Let me know if you have any other questions."

I think the article undersells the problems of ChatGPT's hallucinations. One example from the article where ChatGPT is said to win is a recipe for risotto. However, I wouldn't follow a risotto recipe for ChatGPT just because I can't be confident it hasn't hallucinated some portion of the recipe but would happily follow one from Google, even if the format is a bit more annoying. Same issue with calculating load bearing capacity for a beam only more serious!

Having said that, it does seem like there are definitely specific areas where ChatGPT will be more useful. Coding is a good example as verifying the code is usually straightforward and/or would need to be done anyway. In many cases ChatGPT for an overview followed by Google for more detail/verification is probably a good way to go - I think this would be a good idea for the load bearing wall example.

One thing I've found useful is to make sure I identify to the supplier what specifically I need about the product I'm ordering - sometimes they have something similar in stock which meets my requirements.

One thing I think makes a big difference to me is whether I feel like the provider is taking a collaborative or adversarial stance.

1. I don't usually skip ads on Youtube content but if the channel is often clickbaity/misrepresenting content then I will
2. The printer/ink thing feels very out to get me. The alternative model of printer subscription (e.g. hp) feels alot more collaborative so I don't feel the need to ensure that every page I print is as filled with ink as possible so as to get the "best" deal.
3. If the premium charged on foods in an amusement park/movie theatre feels excessive then I will feel less bad about eating my own food.