DOCSLIB.ORG

A Fast Block Hankel Solver Based on an Inversion

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

A FAST BLOCK HANKEL SOLVER BASED ON AN INVERSION FORMULA FOR BLOCK LOEWNER MATRICES (1) Peter Kravanja Marc Van Barel 3 2 Abstract We prop ose a new O p n algorithm for solving complex np np linear systems that have blo ck Hankel structure where the blo cks are square matrices of size p p Via FFTs the blo ck Hankel system is transformed into a blo ck Lo ewner system An inversion formula enables us to calculate the inverse of the blo ck Lo ewner matrix explicitely The parameters that o ccur in this inversion formula are calculated by solving two rational inter p olation problems on the unit circle We have implemented our algorithm in Fortran Numerical examples are included Intro duction Let n and p b e p ositive integers Consider a sequence H H in 0 2n2 n1 pp C such that the np np blo ck Hankel matrix H H is regular k +l k l=0 n p 1 Let b C We consider the problem of computing x H b n1 I transforms the The np np blo ck exchange matrix E p k n1l k l=0 blo ck Hankel system H x b into the blo ck To eplitz system T x b where T HE and x E x For a given value of p the classical algorithms for solving a blo ck Hankel or a blo ck To eplitz system exploit the structure and 3 require less arithmetic op erations compared to O n for general linear systems 2 The socalled fast algorithms require O n arithmetic op erations while the 2 sup erfast ones need only O n log n op erations by using a divide and conquer strategy The ow of these metho ds is determined by the exact singularity of the square leading principal blo ck submatrices of H or T The fast metho ds compute the solutions corresp onding to successive nonsingular leading principal submatrices sections However in niteprecision arithmetic not only singular Received (1) Katholieke Universiteit Leuven Department of Computer Science Celestij nenlaan A B Heverlee Belgium Email napkravanjananetornlgov and marccskuleuvenacbe P Kravanja M Van Barel A fast block Hankel solver but also illconditioned sections should b e avoided The algorithms that have b een develop ed for this purp ose are called lo okahead algorithms They lo ok ahead from one wellconditioned section to the next one and jump over the illconditioned sections that lie in b etween For a more detailed overview concerning scalar Hankel and To eplitz systems we refer the interested reader to the intro duction of our pap er and the references cited therein For the blo ck To eplitz and the blo ck Hankel case the reader may lo ok at the intro duction and the references of Go o d lo okahead strategies are dicult to design Only recently a totally dierent approach was considered This approach transforms the blo ck Hankel or To eplitz matrix into a generalized Cauchy matrix Such a matrix can b e factorized using pivoting without destroying the structure Hence the Cauchy system can b e solved without using lo okahead and the solution can b e transformed back to the solution of the original system We refer the interested reader to and the references cited therein In we presented such a transformation approach based on an inversion formula for Lo ewner matrices In this pap er we will generalize this approach to the blo ck case The outline of this pap er is as follows In Section we will show how the blo ck Hankel system H x b can b e transformed into a blo ck Lo ewner 0 0 2 1 system Lx b in O p n log n ops An explicit formula for L enables us 0 1 0 to calculate x as L b This inversion formula for blo ck Lo ewner matrices is discussed in Section It involves certain p p matrices that can b e computed by solving two rational interp olation problems on the unit circle In Section 3 2 we present an O p n algorithm to solve these interp olation problems We conclude with numerical examples in Section Transformation into a blo ck Lo ewner system Let y y z z b e n mutually distinct complex numb ers and 1 n 1 n dene y y y and z z z Let Ly z b e the class of np np 1 n 1 n blo ck matrices i o n h n pp C D k l j C C D D C Ly z 1 n 1 n y z k l k l=1 The elements of Ly z are called blo ck Lo ewner matrices They b ear the name of Karl Lo ewner who studied them for p in the context of rational interp olation and monotone matrix functions pp pp The set Ly z is a C mo dule and a submo dule of the C mo dule of all the complex np np blo ck matrices having p p blo cks Since addition of a constant matrix to all the n p p matrices C D leads to the same k l blo ck Lo ewner matrix its dimension is n The set of all the complex based on an inversion formula for block Loewner matrices np np blo ck Hankel matrices having p p blo cks also forms a submo dule of dimension n Blo ck Hankel and blo ck Lo ewner matrices are even more closely related In the scalar case p a theorem of Fiedler asserts that every Hankel matrix can b e transformed into a Lo ewner matrix and vice versa In this section we will formulate this theorem for the blo ck case and discuss 2 our O p n log n implementation of the transformation First we have to deal with some preliminaries concerning Vandermonde matrices Let t t b e n complex numb ers and dene t t t The 1 n 1 n Vandermonde matrix with no des t t is given by 1 n n1 t t 1 1 V t V t t 1 n n1 t t n n Let f z b e the monic p olynomial of degree n that has zeros t t t 1 n f z z t z t t 1 n and dene Y f z z t j n tj k k 6=j Note that f z is a monic p olynomial of degree n for j n Dene tj the n n matrix W t by the equation f z t1 f z t2 z W t n1 z f z tn This means that the j th row of W t contains the co ecients of f z when tj n1 written in terms of the standard monomial basis z z Then T W t V t diag f t f t D t t1 1 tn n The Vandermonde matrix V t is regular if and only if its no des t t are 1 n mutually distinct In that case implies that W t is regular and also that 1 T 1 V t W t D t Let V y z b e the n n Vandermonde matrix with no des y y and 1 n z z and similarly for W y z 1 n Recall that the Kronecker pro duct b etween two matrices A a k l k l=1 C and B C is dened as the blo ck matrix A B a B C k l k l=1 P Kravanja M Van Barel A fast block Hankel solver T Theorem The matrix L W y I H W z I is a blo ck Lo ewner p p matrix in Ly z whose p p matrices C C D D are given by up 1 n 1 n pp to an arbitrary additive matrix C C 1 H 0 H 1 C n W y z I p D 1 H 2n2 D n Proof The scalar case p has b een proven by Fiedler Theorem The generalization to the blo ck case is straightforward The Kronecker pro duct A B is regular if A and B are b oth regular This implies that W y I and W z I are regular and thus L is regular p p A judicious choice of the p oints y and z enables us to transform the blo ck 0 0 2 Hankel system H x b into a blo ck Lo ewner system Lx b in O p n log n k 1 ops Let exp in and supp ose from now on that y for k n1 k n That is let y Let exp in and supp ose n1 from now on that z y for k n That is let z k k Note that 2 2n2 3 2n1 y and z Let b e the n n Fourier matrix n n1 p V n n H amount to an inverse discrete Matrixvector pro ducts involving n n Fourier transform DFT and can b e evaluated via the celebrated inverse fast Fourier transform FFT in O n log n ops Finally let D and D b e n n the n n diagonal matrices n1 n1 D diag and D diag n n Theorem The solution to the blo ck Hankel system H x b is given by p n1 H 0 n D D I x x n n p n p 0 1 0 0 where x L b with b n D I b n p n 1 T 0 Proof Theorem implies that x H b is given by x W z I x p 0 1 0 0 where x L b with b W y I b In the pro of of Theorem it p is shown that p p T n1 H n D and W z n D D W y n n n n n based on an inversion formula for block Loewner matrices T T As W z I W z I the result follows p p It follows that 0 x x r r 0 x x p p+r p+r n1 H n D D n n n 0 x x (n1)p+r (n1)p+r and 0 b b r r 0 b b p p+r p+r n D n n 0 b b (n1)p+r (n1)p+r 0 0 for r p Thus the transformations b b and x x can b e done in O pn log n oating p oint op erations Let P b e the p ermutation matrix dened by y 1 y n P z 1 2n1 z n let b e the n n Fourier matrix 2n 2n1 p V 2n n 2n1 and let D diag 2n Theorem The p p matrices C C D D of the blo ck Lo ewner 1 n 1 n pp matrix L are given by up to an arbitrary additive matrix C C 1 H 0 H 1 p C n T n P D I 2n 2n p D 1 H 2n2 D n p T Proof In Theorem it is shown that W y z n P D 2n 2n The result then follows immediately from Theorem P Kravanja M Van Barel A fast block Hankel solver The previous theorem implies that C 1 k l H 0 k l H 1 k l p C n k l T n P D 2n 2n D 1 k l H 2n2 k l k l D n k l 2 for k l p Thus H can b e transformed into L in O p n log n ops Note As we already mentioned in the pro of of Theorem in Theorem it is shown that p p T n1 H W y n D and W z n D D n n n n n p p T n and W z n are unitary The Kronecker pro duct of two Thus W y unitary matrices is a unitary matrix As T p p L n W y I H W z I p p n n 1 1 1 it follows that kLk nkH k and kL
Recommended publications
  • Quantum Cluster Algebras and Quantum Nilpotent Algebras

    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

    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.
  • Seminar VII for the Course GROUP THEORY in PHYSICS Micael Flohr

    Seminar VII for the Course GROUP THEORY in PHYSICS Micael Flohr

    Seminar VII for the course GROUP THEORY IN PHYSICS Mic~ael Flohr The classical Lie groups 25. January 2005 MATRIX LIE GROUPS Most Lie groups one ever encouters in physics are realized as matrix Lie groups and thus as subgroups of GL(n, R) or GL(n, C). This is the group of invertibel n × n matrices with coefficients in R or C, respectively. This is a Lie group, since it forms as an open subset of the vector space of n × n matrices a manifold. Matrix multiplication is certainly a differentiable map, as is taking the inverse via Cramer’s rule. The only condition defining the open 2 subset is that the determinat must not be zero, which implies that dimKGL(n, K) = n is the same as the one of the vector space Mn(K). However, GL(n, R) is not connected, because we cannot move continuously from a matrix with determinant less than zero to one with determinant larger than zero. It is worth mentioning that gl(n, K) is the vector space of all n × n matrices over the field K, equipped with the standard commutator as Lie bracket. We can describe most other Lie groups as subgroups of GL(n, K) for either K = R or K = C. There are two ways to do so. Firstly, one can give restricting equations to the coefficients of the matrices. Secondly, one can find subgroups of the automorphisms of V =∼ Kn, which conserve a given structure on Kn. In the following, we give some examples for this: SL(n, K).
  • Inertia of the Matrix [(Pi + Pj) ]

    Inertia of the Matrix [(Pi + Pj) ]

    isid/ms/2013/12 October 20, 2013 http://www.isid.ac.in/estatmath/eprints r Inertia of the matrix [(pi + pj) ] Rajendra Bhatia and Tanvi Jain Indian Statistical Institute, Delhi Centre 7, SJSS Marg, New Delhi{110 016, India r INERTIA OF THE MATRIX [(pi + pj) ] RAJENDRA BHATIA* AND TANVI JAIN** Abstract. Let p1; : : : ; pn be positive real numbers. It is well r known that for every r < 0 the matrix [(pi + pj) ] is positive def- inite. Our main theorem gives a count of the number of positive and negative eigenvalues of this matrix when r > 0: Connections with some other matrices that arise in Loewner's theory of oper- ator monotone functions and in the theory of spline interpolation are discussed. 1. Introduction Let p1; p2; : : : ; pn be distinct positive real numbers. The n×n matrix 1 C = [ ] is known as the Cauchy matrix. The special case pi = i pi+pj 1 gives the Hilbert matrix H = [ i+j ]: Both matrices have been studied by several authors in diverse contexts and are much used as test matrices in numerical analysis. The Cauchy matrix is known to be positive definite. It possessesh ai ◦r 1 stronger property: for each r > 0 the entrywise power C = r (pi+pj ) is positive definite. (See [4] for a proof.) The object of this paper is to study positivity properties of the related family of matrices r Pr = [(pi + pj) ]; r ≥ 0: (1) The inertia of a Hermitian matrix A is the triple In(A) = (π(A); ζ(A); ν(A)) ; in which π(A); ζ(A) and ν(A) stand for the number of positive, zero, and negative eigenvalues of A; respectively.
  • Chapter 2 Solving Linear Equations

    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

    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
  • Fast Approximation Algorithms for Cauchy Matrices, Polynomials and Rational Functions

    Fast Approximation Algorithms for Cauchy Matrices, Polynomials and Rational Functions

    City University of New York (CUNY) CUNY Academic Works Computer Science Technical Reports CUNY Academic Works 2013 TR-2013011: Fast Approximation Algorithms for Cauchy Matrices, Polynomials and Rational Functions Victor Y. Pan How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/gc_cs_tr/386 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] Fast Approximation Algorithms for Cauchy Matrices, Polynomials and Rational Functions ? Victor Y. Pan Department of Mathematics and Computer Science Lehman College and the Graduate Center of the City University of New York Bronx, NY 10468 USA [email protected], home page: http://comet.lehman.cuny.edu/vpan/ Abstract. The papers [MRT05], [CGS07], [XXG12], and [XXCBa] have combined the advanced FMM techniques with transformations of matrix structures (traced back to [P90]) in order to devise numerically stable algorithms that approximate the solutions of Toeplitz, Hankel, Toeplitz- like, and Hankel-like linear systems of equations in nearly linear arith- metic time, versus classical cubic time and quadratic time of the previous advanced algorithms. We show that the power of these approximation al- gorithms can be extended to yield similar results for computations with other matrices that have displacement structure, which includes Van- dermonde and Cauchy matrices, as well as to polynomial and rational evaluation and interpolation. The resulting decrease of the running time of the known approximation algorithms is again by order of magnitude, from quadratic to nearly linear.
  • An Accurate Computational Solution of Totally Positive Cauchy Linear Systems

    An Accurate Computational Solution of Totally Positive Cauchy Linear Systems

    AN ACCURATE COMPUTATIONAL SOLUTION OF TOTALLY POSITIVE CAUCHY LINEAR SYSTEMS Abdoulaye Cissé Andrew Anda Computer Science Department Computer Science Department St. Cloud State University St. Cloud State University St. Cloud, MN 56301 St. Cloud, MN 56301 [email protected] [email protected] Abstract ≈ ’ ∆ 1 The Cauchy linear systems C(x, y)V = b where C(x, y) = ∆ , x = (xi ) , y = ( yi ) , « xi − y j V = (vi ) , b = (bi ) can be solved very accurately regardless of condition number using Björck-Pereyra-type methods. We explain this phenomenon by the following facts. First, the Björck-Pereyra methods, written in matrix form, are essentially an accurate and efficient bidiagonal decomposition of the inverse of the matrix, as obtained by Neville Elimination with no pivoting. Second, each nontrivial entry of this bidiagonal decomposition is a product of two quotients of (initial) minors. We use this method to derive and implement a new method of accurately and efficiently solving totally positive Cauchy linear systems in time complexity O(n2). Keywords: Accuracy, Björck-Pereyra-type methods, Cauchy linear systems, floating point architectures, generalized Vandermonde linear systems, Neville elimination, totally positive, LDU decomposition. Note: In this paper we assume that the nodes xi and yj are pair wise distinct and C is totally positive 1 Background Definition 1.1 A Cauchy matrix is defined as: ≈ ’ ∆ 1 C(x, y) = ∆ , for i, j = 1,2,..., n (1.1) « xi − y j where the nodes xi and yj are assumed pair wise distinct. C(x,y) is totally positive (TP) if 2 2 0 < y1 < y2 < < yn−1 < yn < x1 < x2 < < xn−1 < xn .
  • Displacement Rank of the Drazin Inverse Huaian Diaoa, Yimin Weib;1, Sanzheng Qiaoc;∗;2

    Displacement Rank of the Drazin Inverse Huaian Diaoa, Yimin Weib;1, Sanzheng Qiaoc;∗;2

    Available online at www.sciencedirect.com Journal of Computational and Applied Mathematics 167 (2004) 147–161 www.elsevier.com/locate/cam Displacement rank of the Drazin inverse Huaian Diaoa, Yimin Weib;1, Sanzheng Qiaoc;∗;2 aInstitute of Mathematics, Fudan University, Shanghai 200433, PR China bDepartment of Mathematics, Fudan University, Shanghai 200433, PR China cDepartment of Computing and Software, McMaster University, Hamilton, Ont., Canada L8S 4L7 Received 20 February 2003; received in revised form 4 August 2003 Abstract In this paper, we study the displacement rank of the Drazin inverse. Both Sylvester displacement and the generalized displacement are discussed. We present upper bounds for the ranks of the displacements of the Drazin inverse. The general results are applied to the group inverse of a structured matrix such as close-to- Toeplitz, generalized Cauchy, Toeplitz-plus-Hankel, and Bezoutians. c 2003 Elsevier B.V. All rights reserved. MSC: 15A09; 65F20 Keywords: Drazin inverse; Displacement; Structured matrix; Jordan canonical form 1. Introduction Displacement gives a quantitative way of identifying the structure of a matrix. Consider an n × n Toeplitz matrix t0 t−1 ··· t−n+1 . t .. .. 1 T = ; . .. .. . t−1 tn−1 ··· t1 t0 ∗ Corresponding author. Tel.: +1-905-525-9140; fax: +1-905-524-0340. E-mail addresses: [email protected] (Y. Wei), [email protected] (S. Qiao). 1 Supported by National Natural Science Foundation of China under Grant 19901006. 2 Partially supported by Natural Science and Engineering Research Council of Canada. 0377-0427/$ - see front matter c 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2003.09.050 148 H.
  • Introduction to Cluster Algebras Sergey Fomin

    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).
  • Arxiv:2104.06123V2 [Nlin.SI] 19 Jul 2021

    Arxiv:2104.06123V2 [Nlin.SI] 19 Jul 2021

    LINEAR INTEGRAL EQUATIONS AND TWO-DIMENSIONAL TODA SYSTEMS YUE YIN AND WEI FU Abstract. The direct linearisation framework is presented for the two-dimensional Toda equations associated with the infinite-dimensional (1) (2) (1) (2) Z+ Lie algebras A∞, B∞ and C∞, as well as the Kac–Moody algebras Ar , A2r , Cr and Dr+1 for arbitrary integers r ∈ , from the aspect of a set of linear integral equations in a certain form. Such a scheme not only provides a unified perspective to understand the underlying integrability structure, but also induces the direct linearising type solution potentially leading to the universal solution space, for each class of the two-dimensional Toda system. As particular applications of this framework to the two-dimensional Toda lattices, we rediscover the Lax pairs and the adjoint Lax pairs and simultaneously construct the generalised Cauchy matrix solutions. 1. Introduction In the modern theory of integrable systems, the notion of integrability of nonlinear equations often refers to the property that a differential/difference equation is exactly solvable under an initial-boundary condition, which in many cases allows us to construct explicit solutions. Motivated by this, many mathematical methods were invented and developed to search for explicit solutions of nonlinear models, for instance, the inverse scattering transform, the Darboux transform, Hirota’s bilinear method, as well as the algebro-geometric method, etc., see e.g. the monographs [1,11,16,26]. These techniques not only explained the nonlinear phenomena such as solitary waves and periodic waves in nature mathematically, but also motivated the discoveryof a huge number of nonlinear equations that possess “nice” algebraic and geometric properties.
  • ON the PROPERTIES of the EXCHANGE GRAPH of a CLUSTER ALGEBRA Michael Gekhtman, Michael Shapiro, and Alek Vainshtein 1. Main Defi

    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].