Coxeter energy of graphs

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Methods and results from the representation theory of quivers and finite dimensional algebras have led to many interactions with other areas of mathematics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further develop such interactions and to stimulate progress in the representation theory of algebras.

Following the spectral graph theory, a graph coloring technique, and algebraic methods in graph theory, we study the category ${\cal U}{\cal B} igr_n$ of loop-free edge-bipartite (signed) graphs $\Delta$, with $n\geq 2$ vertices, up to $\mathbb{Z}$-congruences $\sim_\mathbb{Z}$ and $\approx_\mathbb{Z}$ (defined in the paper), by means of the Gram matrix $ \check G_\Delta \in \mathbb{M}_n(\mathbb{Z})$, the Coxeter--Gram matrix ${\rm Cox}_\Delta= - \check G_\Delta\cdot \check G_\Delta^{-tr} \in \mathbb{M}_n(\mathbb{Z})$, the spectrum $ {\bf specc}_\Delta$ of $\mbox{\rm Cox}_\Delta$, the Coxeter polynomial $\mbox{\rm cox}_\Delta(t)\in \mathbb{Z}[t]$, the Coxeter number ${\bf c}_\Delta \geq 2$, and the $\mathbb{Z}$-bilinear Gram form $b_\Delta: \mathbb{Z}^n\times \mathbb{Z}^n \to \mathbb{Z}$ of $\Delta$. Our main inspiration for the study comes from the representation theory of posets, groups and algebras, Lie theory, and Diophantine geometry problems. One of our aims is to compute the set ${\cal C}{\cal G} p...

Lie groups and algebraic groups are important in many major areas of mathematics and mathematical physics. We find them in diverse roles, notably as groups of automorphisms of geometric structures, as symmetries of differential systems, or as basic tools in the theory of automorphic forms. The author looks at their development, highlighting the evolution from the almost purely local theory at the start to the global theory that we know today. Starting from Lie's theory of local analytic transformation groups and early work on Lie algebras, he follows the process of globalization in its two main frameworks: differential geometry and topology on one hand, algebraic geometry on the other. Chapters II to IV are devoted to the former, Chapters V to VIII, to the latter. The essays in the first part of the book survey various proofs of the full reducibility of linear representations of \\mathbf{SL}_2{(\\mathbb{C})}, the contributions of H. Weyl to representations and invariant theory for semisimple Lie groups, and conclude with a chapter on E. Cartan's theory of symmetric spaces and Lie groups in the large. The second part of the book first outlines various contributions to linear algebraic groups in the 19th century, due mainly to E. Study, E. Picard, and above all, L. Maurer. After being abandoned for nearly fifty years, the theory was revived by C. Chevalley and E. Kolchin, and then further developed by many others. This is the focus of Chapter VI. The book concludes with two chapters on the work of Chevalley on Lie groups and Lie algebras and of Kolchin on algebraic groups and the Galois theory of differential fields, which put their contributions to algebraic groups in a broader context. Professor Borel brings a unique perspective to this study. As an important developer of some of the modern elements of both the differential geometric and the algebraic geometric sides of the theory, he has a particularly deep understanding of the underlying mathematics. His lifelong involvement and his historical research in the subject area give him a special appreciation of the story of its development.

This book brings together the essential ideas and methods behind applications of variational theory in theoretical physics and chemistry. The emphasis is on understanding physical and computational applications of variational methodology rather than on rigorous mathematical formalism. The text begins with an historical survey of familiar variational principles in classical mechanics and optimization theory, then proceeds to develop the variational principles and formalism behind current computational methodology for bound and continuum quantum states of interacting electrons in atoms, molecules, and condensed matter. It covers multiple-scattering theory, including a detailed presentation of contemporary methodology for electron-impact rotational and vibrational excitation of molecules. The book ends with an introduction to the variational theory of relativistic fields. Ideal for graduate students and researchers in any field that uses variational methodology, this book is particularly suitable as a backup reference for lecture courses in mathematical methods in physics and theoretical chemistry.

This is a call for contributions to a special issue of Journal of Physics A: Mathematical and Theoretical dedicated to cluster algebras in mathematical physics. Over the ten years since their introduction by Fomin and Zelevinsky, the theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links that have been discovered with a wide range of subjects in mathematics and, increasingly, theoretical and mathematical physics. The main motivation of this special issue is to gather together reviews, recent developments and open problems, mainly from a mathematical physics viewpoint, into a single comprehensive issue. We expect that such a special issue will become a valuable reference for the broad scientific community working in mathematical and theoretical physics. The issue will consist of invited review articles and contributed papers containing new results on the interplays of cluster algebras with mathematical physics.Editorial policyThe Guest Editors for this issue are Philippe Di Francesco, Michael Gekhtman, Atsuo Kuniba and Masahito Yamazaki.The areas and topics for this issue include, but are not limited to: discrete integrable systems arising from cluster mutationscluster structure on Poisson varietiescluster algebras and soliton interactionscluster positivity conjectureY-systems in the thermodynamic Bethe ansatz and Zamolodchikov's periodicity conjectureT-system of transfer matrices of integrable lattice modelsdilogarithm identities in conformal field theorywall crossing in 4d N = 2 supersymmetric gauge theories4d N = 1 quiver gauge theories described by networksscattering amplitudes of 4d N = 4 theories3d N = 2 gauge theories described by flat connections on 3-manifoldsintegrability of dimer/Ising models on graphs.All contributions will be refereed and processed according to the usual procedure of the journal.Guidelines for preparation of contributions The deadline for contributed papers is 31 March 2014. This deadline will allow the special issue to appear at the end of 2014.There is no strict regulation on article size, but as a guide the preferable size is 15–30 pages for contributed papers and 40–60 pages for reviews. Further advice on publishing your work in Journal of Physics A may be found at iopscience.iop.org/jphysa.Contributions to the special issue should be submitted by web upload via ScholarOne Manuscripts, quoting ‘JPhysA special issue on cluster algebras in mathematical physics’. Submissions should ideally be in standard LaTeX form. Please see the website for further information on electronic submissions.All contributions should be accompanied by a read-me file or covering letter giving the postal and e-mail addresses for correspondence. The Publishing Office should be notified of any subsequent change of address.The special issue will be published in the print and online versions of the journal.

This is a call for contributions to a special issue of Journal of Physics A: Mathematical and Theoretical dedicated to cluster algebras in mathematical physics. Over the ten years since their introduction by Fomin and Zelevinsky, the theory of cluster algebras has witnessed a spectacular growth, first and foremost due to the many links that have been discovered with a wide range of subjects in mathematics and, increasingly, theoretical and mathematical physics. The main motivation of this special issue is to gather together reviews, recent developments and open problems, mainly from a mathematical physics viewpoint, into a single comprehensive issue. We expect that such a special issue will become a valuable reference for the broad scientific community working in mathematical and theoretical physics. The issue will consist of invited review articles and contributed papers containing new results on the interplays of cluster algebras with mathematical physics.Editorial policyThe Guest Editors for this issue are Philippe Di Francesco, Michael Gekhtman, Atsuo Kuniba and Masahito Yamazaki.The areas and topics for this issue include, but are not limited to: discrete integrable systems arising from cluster mutationscluster structure on Poisson varietiescluster algebras and soliton interactionscluster positivity conjectureY-systems in the thermodynamic Bethe ansatz and Zamolodchikov's periodicity conjectureT-system of transfer matrices of integrable lattice modelsdilogarithm identities in conformal field theorywall crossing in 4d N = 2 supersymmetric gauge theories4d N = 1 quiver gauge theories described by networksscattering amplitudes of 4d N = 4 theories3d N = 2 gauge theories described by flat connections on 3-manifoldsintegrability of dimer/Ising models on graphs.All contributions will be refereed and processed according to the usual procedure of the journal.Guidelines for preparation of contributions The deadline for contributed papers is 31 March 2014. This deadline will allow the special issue to appear at the end of 2014.There is no strict regulation on article size, but as a guide the preferable size is 15–30 pages for contributed papers and 40–60 pages for reviews. Further advice on publishing your work in Journal of Physics A may be found at iopscience.iop.org/jphysa.Contributions to the special issue should be submitted by web upload via ScholarOne Manuscripts, quoting ‘JPhysA special issue on cluster algebras in mathematical physics’. Submissions should ideally be in standard LaTeX form. Please see the website for further information on electronic submissions.All contributions should be accompanied by a read-me file or covering letter giving the postal and e-mail addresses for correspondence. The Publishing Office should be notified of any subsequent change of address.The special issue will be published in the print and online versions of the journal.

The Weyl algebra is one of the most important objects in mathematical physics and representation theory of Lie groups and Kac-Moody algebras.

This is a report on advances in the representation theory of finite dimensional algebras in the years 1984 – 1990. During these years, the German research council (DFG) has sponsered a Forschungsschwerpunkt devoted to the representation theory of finite groups and finite dimensional algebras; it started in 1984 and will be finished by 1991.KeywordsEndomorphism RingIndecomposable ModuleFinite Dimensional AlgebraHall AlgebraHereditary AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Orthogonal polynomials, special functions and mathematical physics

The basic structure and representation theory of finite-dimensional simple Lie algebras over \(\mathbb{C} \), as developed by W. Killing, E. Cartan and H. Weyl, is a well-known and frequently used branch of mathematics. It is a striking fact that very significant new results in this classical area of mathematics have come to light during the past decade. These new results have originated from the development of the theory of quantum groups. The purpose of this article is to survey briefly the classical results on the representation theory of simple Lie algebras and then to describe the recent developments primarily due to G. Lusztig, M. Kashiwara and P. Littelmann.

This annual conference was established at the National Nuclear Research University – Moscow Engineering Institute in the year 2012. The scope was initially on mathematical physics and mathematical modelling, while since this year theoretical physics was also included. The Conference was organized by three leading departments of the MEPhI in the field of theoretical and mathematical physics.Each year, more than 100 participants from different countries including Belorussia, Czech Republic, France, Great Britain, Japan, Moldova, Russia, Slovakia, Spain, Sweden and the USA take part in the Conference with oral talks.The aim of this conference is to bring together researchers with interests in: Analytical Theory of Differential Equations, Interaction of Radiation With Matter and Physics of Extreme Light, Nuclear Physics and Theory of Elementary Particles, Optimization Problems, Solid Mechanics, Soft Matter and Interdisciplinary Physics, Fluid Mechanics, Mathematical Modelling, Time Series.Nikolay A. KudryashovOleg V. NagornovSergey V. Popruzhenko

Age-Related Differences in Resting-State Functional Connectivity in Women

The XXVIIIth International Colloquium on Group-Theoretical Methods in Physics (ICGTMP), also known as the GROUP 28 conference, took place in Newcastle upon Tyne from 26–30 July 2010. There were 128 attendees from 22 countries – physicists, mathematicians, chemists and engineers who use mathematical and numerical methods based on geometry and symmetry in their work. The conference was organised jointly by Northumbria and Durham Universities. The conference website is http://group28.northumbria.ac.uk.

This book provides a complete and concise review of the main methods used in the study of algebraic structures of symmetries for use in models of theoretical and mathematical physics, namely groups...