A Note on the Singular-Value Decomposition of Circulant Jacket Matrices
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Quantum Cluster Algebras and Quantum Nilpotent Algebras
Quantum cluster algebras and quantum nilpotent algebras K. R. Goodearl ∗ and M. T. Yakimov † ∗Department of Mathematics, University of California, Santa Barbara, CA 93106, U.S.A., and †Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. Proceedings of the National Academy of Sciences of the United States of America 111 (2014) 9696–9703 A major direction in the theory of cluster algebras is to construct construction does not rely on any initial combinatorics of the (quantum) cluster algebra structures on the (quantized) coordinate algebras. On the contrary, the construction itself produces rings of various families of varieties arising in Lie theory. We prove intricate combinatorial data for prime elements in chains of that all algebras in a very large axiomatically defined class of noncom- subalgebras. When this is applied to special cases, we re- mutative algebras possess canonical quantum cluster algebra struc- cover the Weyl group combinatorics which played a key role tures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results in categorification earlier [7, 9, 10]. Because of this, we expect for a large class of Poisson nilpotent algebras. Many important fam- that our construction will be helpful in building a unified cat- ilies of coordinate rings are subsumed in the class we are covering, egorification of quantum nilpotent algebras. Finally, we also which leads to a broad range of application of the general results prove similar results for (commutative) cluster algebras using to the above mentioned types of problems. As a consequence, we Poisson prime elements. -
Introduction to Cluster Algebras. Chapters
Introduction to Cluster Algebras Chapters 1–3 (preliminary version) Sergey Fomin Lauren Williams Andrei Zelevinsky arXiv:1608.05735v4 [math.CO] 30 Aug 2021 Preface This is a preliminary draft of Chapters 1–3 of our forthcoming textbook Introduction to cluster algebras, joint with Andrei Zelevinsky (1953–2013). Other chapters have been posted as arXiv:1707.07190 (Chapters 4–5), • arXiv:2008.09189 (Chapter 6), and • arXiv:2106.02160 (Chapter 7). • We expect to post additional chapters in the not so distant future. This book grew from the ten lectures given by Andrei at the NSF CBMS conference on Cluster Algebras and Applications at North Carolina State University in June 2006. The material of his lectures is much expanded but we still follow the original plan aimed at giving an accessible introduction to the subject for a general mathematical audience. Since its inception in [23], the theory of cluster algebras has been actively developed in many directions. We do not attempt to give a comprehensive treatment of the many connections and applications of this young theory. Our choice of topics reflects our personal taste; much of the book is based on the work done by Andrei and ourselves. Comments and suggestions are welcome. Sergey Fomin Lauren Williams Partially supported by the NSF grants DMS-1049513, DMS-1361789, and DMS-1664722. 2020 Mathematics Subject Classification. Primary 13F60. © 2016–2021 by Sergey Fomin, Lauren Williams, and Andrei Zelevinsky Contents Chapter 1. Total positivity 1 §1.1. Totally positive matrices 1 §1.2. The Grassmannian of 2-planes in m-space 4 §1.3. -
A Matrix Handbook for Statisticians
A MATRIX HANDBOOK FOR STATISTICIANS George A. F. Seber Department of Statistics University of Auckland Auckland, New Zealand BICENTENNIAL BICENTENNIAL WILEY-INTERSCIENCE A John Wiley & Sons, Inc., Publication This Page Intentionally Left Blank A MATRIX HANDBOOK FOR STATISTICIANS THE WlLEY BICENTENNIAL-KNOWLEDGE FOR GENERATIONS Gachgeneration has its unique needs and aspirations. When Charles Wiley first opened his small printing shop in lower Manhattan in 1807, it was a generation of boundless potential searching for an identity. And we were there, helping to define a new American literary tradition. Over half a century later, in the midst of the Second Industrial Revolution, it was a generation focused on building the future. Once again, we were there, supplying the critical scientific, technical, and engineering knowledge that helped frame the world. Throughout the 20th Century, and into the new millennium, nations began to reach out beyond their own borders and a new international community was born. Wiley was there, expanding its operations around the world to enable a global exchange of ideas, opinions, and know-how. For 200 years, Wiley has been an integral part of each generation's journey, enabling the flow of information and understanding necessary to meet their needs and fulfill their aspirations. Today, bold new technologies are changing the way we live and learn. Wiley will be there, providing you the must-have knowledge you need to imagine new worlds, new possibilities, and new opportunities. Generations come and go, but you can always count on Wiley to provide you the knowledge you need, when and where you need it! n WILLIAM J. -
Group Matrix Ring Codes and Constructions of Self-Dual Codes
Group Matrix Ring Codes and Constructions of Self-Dual Codes S. T. Dougherty University of Scranton Scranton, PA, 18518, USA Adrian Korban Department of Mathematical and Physical Sciences University of Chester Thornton Science Park, Pool Ln, Chester CH2 4NU, England Serap S¸ahinkaya Tarsus University, Faculty of Engineering Department of Natural and Mathematical Sciences Mersin, Turkey Deniz Ustun Tarsus University, Faculty of Engineering Department of Computer Engineering Mersin, Turkey February 2, 2021 arXiv:2102.00475v1 [cs.IT] 31 Jan 2021 Abstract In this work, we study codes generated by elements that come from group matrix rings. We present a matrix construction which we use to generate codes in two different ambient spaces: the matrix ring Mk(R) and the ring R, where R is the commutative Frobenius ring. We show that codes over the ring Mk(R) are one sided ideals in the group matrix k ring Mk(R)G and the corresponding codes over the ring R are G - codes of length kn. Additionally, we give a generator matrix for self- dual codes, which consist of the mentioned above matrix construction. 1 We employ this generator matrix to search for binary self-dual codes with parameters [72, 36, 12] and find new singly-even and doubly-even codes of this type. In particular, we construct 16 new Type I and 4 new Type II binary [72, 36, 12] self-dual codes. 1 Introduction Self-dual codes are one of the most widely studied and interesting class of codes. They have been shown to have strong connections to unimodular lattices, invariant theory, and designs. -
Chapter 2 Solving Linear Equations
Chapter 2 Solving Linear Equations Po-Ning Chen, Professor Department of Computer and Electrical Engineering National Chiao Tung University Hsin Chu, Taiwan 30010, R.O.C. 2.1 Vectors and linear equations 2-1 • What is this course Linear Algebra about? Algebra (代數) The part of mathematics in which letters and other general sym- bols are used to represent numbers and quantities in formulas and equations. Linear Algebra (線性代數) To combine these algebraic symbols (e.g., vectors) in a linear fashion. So, we will not combine these algebraic symbols in a nonlinear fashion in this course! x x1x2 =1 x 1 Example of nonlinear equations for = x : 2 x1/x2 =1 x x x x 1 1 + 2 =4 Example of linear equations for = x : 2 x1 − x2 =1 2.1 Vectors and linear equations 2-2 The linear equations can always be represented as matrix operation, i.e., Ax = b. Hence, the central problem of linear algorithm is to solve a system of linear equations. x − 2y =1 1 −2 x 1 Example of linear equations. ⇔ = 2 3x +2y =11 32 y 11 • A linear equation problem can also be viewed as a linear combination prob- lem for column vectors (as referred by the textbook as the column picture view). In contrast, the original linear equations (the red-color one above) is referred by the textbook as the row picture view. 1 −2 1 Column picture view: x + y = 3 2 11 1 −2 We want to find the scalar coefficients of column vectors and to form 3 2 1 another vector . -
A Constructive Arbitrary-Degree Kronecker Product Decomposition of Tensors
A CONSTRUCTIVE ARBITRARY-DEGREE KRONECKER PRODUCT DECOMPOSITION OF TENSORS KIM BATSELIER AND NGAI WONG∗ Abstract. We propose the tensor Kronecker product singular value decomposition (TKPSVD) that decomposes a real k-way tensor A into a linear combination of tensor Kronecker products PR (d) (1) with an arbitrary number of d factors A = j=1 σj Aj ⊗ · · · ⊗ Aj . We generalize the matrix (i) Kronecker product to tensors such that each factor Aj in the TKPSVD is a k-way tensor. The algorithm relies on reshaping and permuting the original tensor into a d-way tensor, after which a polyadic decomposition with orthogonal rank-1 terms is computed. We prove that for many (1) (d) different structured tensors, the Kronecker product factors Aj ;:::; Aj are guaranteed to inherit this structure. In addition, we introduce the new notion of general symmetric tensors, which includes many different structures such as symmetric, persymmetric, centrosymmetric, Toeplitz and Hankel tensors. Key words. Kronecker product, structured tensors, tensor decomposition, TTr1SVD, general- ized symmetric tensors, Toeplitz tensor, Hankel tensor AMS subject classifications. 15A69, 15B05, 15A18, 15A23, 15B57 1. Introduction. Consider the singular value decomposition (SVD) of the fol- lowing 16 × 9 matrix A~ 0 1:108 −0:267 −1:192 −0:267 −1:192 −1:281 −1:192 −1:281 1:102 1 B 0:417 −1:487 −0:004 −1:487 −0:004 −1:418 −0:004 −1:418 −0:228C B C B−0:127 1:100 −1:461 1:100 −1:461 0:729 −1:461 0:729 0:940 C B C B−0:748 −0:243 0:387 −0:243 0:387 −1:241 0:387 −1:241 −1:853C B C B 0:417 -
Introduction to Cluster Algebras Sergey Fomin
Joint Introductory Workshop MSRI, Berkeley, August 2012 Introduction to cluster algebras Sergey Fomin (University of Michigan) Main references Cluster algebras I–IV: J. Amer. Math. Soc. 15 (2002), with A. Zelevinsky; Invent. Math. 154 (2003), with A. Zelevinsky; Duke Math. J. 126 (2005), with A. Berenstein & A. Zelevinsky; Compos. Math. 143 (2007), with A. Zelevinsky. Y -systems and generalized associahedra, Ann. of Math. 158 (2003), with A. Zelevinsky. Total positivity and cluster algebras, Proc. ICM, vol. 2, Hyderabad, 2010. 2 Cluster Algebras Portal hhttp://www.math.lsa.umich.edu/˜fomin/cluster.htmli Links to: • >400 papers on the arXiv; • a separate listing for lecture notes and surveys; • conferences, seminars, courses, thematic programs, etc. 3 Plan 1. Basic notions 2. Basic structural results 3. Periodicity and Grassmannians 4. Cluster algebras in full generality Tutorial session (G. Musiker) 4 FAQ • Who is your target audience? • Can I avoid the calculations? • Why don’t you just use a blackboard and chalk? • Where can I get the slides for your lectures? • Why do I keep seeing different definitions for the same terms? 5 PART 1: BASIC NOTIONS Motivations and applications Cluster algebras: a class of commutative rings equipped with a particular kind of combinatorial structure. Motivation: algebraic/combinatorial study of total positivity and dual canonical bases in semisimple algebraic groups (G. Lusztig). Some contexts where cluster-algebraic structures arise: • Lie theory and quantum groups; • quiver representations; • Poisson geometry and Teichm¨uller theory; • discrete integrable systems. 6 Total positivity A real matrix is totally positive (resp., totally nonnegative) if all its minors are positive (resp., nonnegative). -
ON the PROPERTIES of the EXCHANGE GRAPH of a CLUSTER ALGEBRA Michael Gekhtman, Michael Shapiro, and Alek Vainshtein 1. Main Defi
Math. Res. Lett. 15 (2008), no. 2, 321–330 c International Press 2008 ON THE PROPERTIES OF THE EXCHANGE GRAPH OF A CLUSTER ALGEBRA Michael Gekhtman, Michael Shapiro, and Alek Vainshtein Abstract. We prove a conjecture about the vertices and edges of the exchange graph of a cluster algebra A in two cases: when A is of geometric type and when A is arbitrary and its exchange matrix is nondegenerate. In the second case we also prove that the exchange graph does not depend on the coefficients of A. Both conjectures were formulated recently by Fomin and Zelevinsky. 1. Main definitions and results A cluster algebra is an axiomatically defined commutative ring equipped with a distinguished set of generators (cluster variables). These generators are subdivided into overlapping subsets (clusters) of the same cardinality that are connected via sequences of birational transformations of a particular kind, called cluster transfor- mations. Transfomations of this kind can be observed in many areas of mathematics (Pl¨ucker relations, Somos sequences and Hirota equations, to name just a few exam- ples). Cluster algebras were initially introduced in [FZ1] to study total positivity and (dual) canonical bases in semisimple algebraic groups. The rapid development of the cluster algebra theory revealed relations between cluster algebras and Grassmannians, quiver representations, generalized associahedra, Teichm¨uller theory, Poisson geome- try and many other branches of mathematics, see [Ze] and references therein. In the present paper we prove some conjectures on the general structure of cluster algebras formulated by Fomin and Zelevinsky in [FZ3]. To state our results, we recall the definition of a cluster algebra; for details see [FZ1, FZ4]. -
Spectral Clustering and Multidimensional Scaling: a Unified View
Spectral clustering and multidimensional scaling: a unified view Fran¸coisBavaud Section d’Informatique et de M´ethodes Math´ematiques Facult´edes Lettres, Universit´ede Lausanne CH-1015 Lausanne, Switzerland (To appear in the proccedings of the IFCS 2006 Conference: “Data Science and Classification”, Ljubljana, Slovenia, July 25 - 29, 2006) Abstract. Spectral clustering is a procedure aimed at partitionning a weighted graph into minimally interacting components. The resulting eigen-structure is de- termined by a reversible Markov chain, or equivalently by a symmetric transition matrix F . On the other hand, multidimensional scaling procedures (and factorial correspondence analysis in particular) consist in the spectral decomposition of a kernel matrix K. This paper shows how F and K can be related to each other through a linear or even non-linear transformation leaving the eigen-vectors invari- ant. As illustrated by examples, this circumstance permits to define a transition matrix from a similarity matrix between n objects, to define Euclidean distances between the vertices of a weighted graph, and to elucidate the “flow-induced” nature of spatial auto-covariances. 1 Introduction and main results Scalar products between features define similarities between objects, and re- versible Markov chains define weighted graphs describing a stationary flow. It is natural to expect flows and similarities to be related: somehow, the exchange of flows between objects should enhance their similarity, and tran- sitions should preferentially occur between similar states. This paper formalizes the above intuition by demonstrating in a general framework that the symmetric matrices K and F possess an identical eigen- structure, where K (kernel, equation (2)) is a measure of similarity, and F (symmetrized transition. -
Orbital-Dependent Exchange-Correlation Functionals in Density-Functional Theory Realized by the FLAPW Method
Orbital-dependent exchange-correlation functionals in density-functional theory realized by the FLAPW method Von der Fakultät für Mathematik, Informatik und Naturwissenschaen der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaen genehmigte Dissertation vorgelegt von Dipl.-Phys. Markus Betzinger aus Fröndenberg Berichter: Prof. Dr. rer. nat. Stefan Blügel Prof. Dr. rer. nat. Andreas Görling Prof. Dr. rer. nat. Carsten Honerkamp Tag der mündlichen Prüfung: Ô¥.Ôò.òýÔÔ Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar. is document was typeset with LATEX. Figures were created using Gnuplot, InkScape, and VESTA. Das Unverständlichste am Universum ist im Grunde, dass wir es verstehen können. Albert Einstein Abstract In this thesis, we extended the applicability of the full-potential linearized augmented-plane- wave (FLAPW) method, one of the most precise, versatile and generally applicable elec- tronic structure methods for solids working within the framework of density-functional the- ory (DFT), to orbital-dependent functionals for the exchange-correlation (xc) energy. In con- trast to the commonly applied local-density approximation (LDA) and generalized gradient approximation (GGA) for the xc energy, orbital-dependent functionals depend directly on the Kohn-Sham (KS) orbitals and only indirectly on the density. Two dierent schemes that deal with orbital-dependent functionals, the KS and the gen- eralized Kohn-Sham (gKS) formalism, have been realized. While the KS scheme requires a local multiplicative xc potential, the gKS scheme allows for a non-local potential in the one- particle Schrödinger equations. Hybrid functionals, combining some amount of the orbital-dependent exact exchange en- ergy with local or semi-local functionals of the density, are implemented within the gKS scheme. -
Matrix Algebra and Control
Appendix A Matrix Algebra and Control Boldface lower case letters, e.g., a or b, denote vectors, boldface capital letters, e.g., A, M, denote matrices. A vector is a column matrix. Containing m elements (entries) it is referred to as an m-vector. The number of rows and columns of a matrix A is nand m, respectively. Then, A is an (n, m)-matrix or n x m-matrix (dimension n x m). The matrix A is called positive or non-negative if A>, 0 or A :2:, 0 , respectively, i.e., if the elements are real, positive and non-negative, respectively. A.1 Matrix Multiplication Two matrices A and B may only be multiplied, C = AB , if they are conformable. A has size n x m, B m x r, C n x r. Two matrices are conformable for multiplication if the number m of columns of the first matrix A equals the number m of rows of the second matrix B. Kronecker matrix products do not require conformable multiplicands. The elements or entries of the matrices are related as follows Cij = 2::;;'=1 AivBvj 'Vi = 1 ... n, j = 1 ... r . The jth column vector C,j of the matrix C as denoted in Eq.(A.I3) can be calculated from the columns Av and the entries BVj by the following relation; the jth row C j ' from rows Bv. and Ajv: column C j = L A,vBvj , row Cj ' = (CT),j = LAjvBv, (A. 1) /1=1 11=1 A matrix product, e.g., AB = (c: c:) (~b ~b) = 0 , may be zero although neither multipli cand A nor multiplicator B is zero. -
Preconditioners for Symmetrized Toeplitz and Multilevel Toeplitz Matrices∗
PRECONDITIONERS FOR SYMMETRIZED TOEPLITZ AND MULTILEVEL TOEPLITZ MATRICES∗ J. PESTANAy Abstract. When solving linear systems with nonsymmetric Toeplitz or multilevel Toeplitz matrices using Krylov subspace methods, the coefficient matrix may be symmetrized. The precondi- tioned MINRES method can then be applied to this symmetrized system, which allows rigorous upper bounds on the number of MINRES iterations to be obtained. However, effective preconditioners for symmetrized (multilevel) Toeplitz matrices are lacking. Here, we propose novel ideal preconditioners, and investigate the spectra of the preconditioned matrices. We show how these preconditioners can be approximated and demonstrate their effectiveness via numerical experiments. Key words. Toeplitz matrix, multilevel Toeplitz matrix, symmetrization, preconditioning, Krylov subspace method AMS subject classifications. 65F08, 65F10, 15B05, 35R11 1. Introduction. Linear systems (1.1) Anx = b; n×n n where An 2 R is a Toeplitz or multilevel Toeplitz matrix, and b 2 R arise in a range of applications. These include the discretization of partial differential and integral equations, time series analysis, and signal and image processing [7, 27]. Additionally, demand for fast numerical methods for fractional diffusion problems| which have recently received significant attention|has renewed interest in the solution of Toeplitz and Toeplitz-like systems [10, 26, 31, 32, 46]. Preconditioned iterative methods are often used to solve systems of the form (1.1). When An is Hermitian, CG [18] and MINRES [29] can be applied, and their descrip- tive convergence rate bounds guide the construction of effective preconditioners [7, 27]. On the other hand, convergence rates of preconditioned iterative methods for non- symmetric Toeplitz matrices are difficult to describe.