scholarly journals Honeycombs from Hermitian Matrix Pairs

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Glenn Appleby ◽  
Tamsen Whitehead
Keyword(s):  
Crystal Structure ◽  
Representation Theory ◽  
Linear Algebra ◽  
Hermitian Matrix ◽  
Hermitian Matrices ◽  
Orthonormal Bases ◽  
Crystal Graphs ◽  
The Matrix ◽  
International Audience ◽  
Matrix Pair

International audience Knutson and Tao's work on the Horn conjectures used combinatorial invariants called hives and honeycombs to relate spectra of sums of Hermitian matrices to Littlewood-Richardson coefficients and problems in representation theory, but these relationships remained implicit. Here, let $M$ and $N$ be two $n ×n$ Hermitian matrices. We will show how to determine a hive $\mathcal{H}(M, N)={H_ijk}$ using linear algebra constructions from this matrix pair. With this construction, one may also define an explicit Littlewood-Richardson filling (enumerated by the Littlewood-Richardson coefficient $c_μν ^λ$ associated to the matrix pair). We then relate rotations of orthonormal bases of eigenvectors of $M$ and $N$ to deformations of honeycombs (and hives), which we interpret in terms of the structure of crystal graphs and Littelmann's path operators. We find that the crystal structure is determined \emphmore simply from the perspective of rotations than that of path operators.

Structural/Property Relationships for Optical Materials

1993 ◽  
Vol 329 ◽  
Author(s):  
Vivien D.
Keyword(s):  
Crystal Structure ◽  
Optical Properties ◽  
Electronic Structure ◽  
Chemical Composition ◽  
Field Strength ◽  
Optical Materials ◽  
Site Occupancy ◽  
Laser Materials ◽  
Crystal Field Strength ◽  
The Matrix

AbstractIn this paper the relationships between the crystal structure, chemical composition and electronic structure of laser materials, and their optical properties are discussed. A brief description is given of the different laser activators and of the influence of the matrix on laser characteristics in terms of crystal field strength, symmetry, covalency and phonon frequencies. The last part of the paper lays emphasis on the means to optimize the matrix-activator properties such as control of the oxidation state and site occupancy of the activator and influence of its concentration.

A general inverse matrix series relation and associated polynomials – II

2021 ◽  
Vol 71 (2) ◽  
pp. 301-316
Author(s):  
Reshma Sanjhira
Keyword(s):  
Matrix Polynomial ◽  
Combinatorial Identities ◽  
Inverse Matrix ◽  
Matrix Polynomials ◽  
Special Cases ◽  
Series Representations ◽  
The Matrix ◽  
Associated Polynomials ◽  
Matrix Pair ◽  
Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

scholarly journals Eigenbranes in Jackiw-Teitelboim gravity

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Andreas Blommaert ◽  
Thomas G. Mertens ◽  
Henri Verschelde
Keyword(s):  
Finite Volume ◽  
Chaotic System ◽  
Path Integral ◽  
Hermitian Matrices ◽  
The Matrix ◽  
Integral Contour

Abstract It was proven recently that JT gravity can be defined as an ensemble of L × L Hermitian matrices. We point out that the eigenvalues of the matrix correspond in JT gravity to FZZT-type boundaries on which spacetimes can end. We then investigate an ensemble of matrices with 1 ≪ N ≪ L eigenvalues held fixed. This corresponds to a version of JT gravity which includes N FZZT type boundaries in the path integral contour and which is found to emulate a discrete quantum chaotic system. In particular this version of JT gravity can capture the behavior of finite-volume holographic correlators at late times, including erratic oscillations.

MRHS solver based on linear algebra and exhaustive search

2018 ◽  
Vol 12 (3) ◽  
pp. 143-157 ◽  
Author(s):  
Håvard Raddum ◽  
Pavol Zajac
Keyword(s):  
Linear Algebra ◽  
Matrix Representation ◽  
Equation System ◽  
Exhaustive Search ◽  
Binary Matrix ◽  
Algebraic Attacks ◽  
The Matrix ◽  
Speed Up ◽  
Symmetric Key

Abstract We show how to build a binary matrix from the MRHS representation of a symmetric-key cipher. The matrix contains the cipher represented as an equation system and can be used to assess a cipher’s resistance against algebraic attacks. We give an algorithm for solving the system and compute its complexity. The complexity is normally close to exhaustive search on the variables representing the user-selected key. Finally, we show that for some variants of LowMC, the joined MRHS matrix representation can be used to speed up regular encryption in addition to exhaustive key search.

DMRG Approach to Fast Linear Algebra in the TT-Format

2011 ◽  
Vol 11 (3) ◽  
pp. 382-393 ◽  
Author(s):  
Ivan Oseledets
Keyword(s):  
Linear Systems ◽  
Linear Algebra ◽  
Direct Method ◽  
Fast Algorithms ◽  
Approximate Solutions ◽  
Vector Product ◽  
The Matrix ◽  
Minimization Scheme

AbstractIn this paper, the concept of the DMRG minimization scheme is extended to several important operations in the TT-format, like the matrix-by-vector product and the conversion from the canonical format to the TT-format. Fast algorithms are implemented and a stabilization scheme based on randomization is proposed. The comparison with the direct method is performed on a sequence of matrices and vectors coming as approximate solutions of linear systems in the TT-format. A generated example is provided to show that randomization is really needed in some cases. The matrices and vectors used are available from the author or at http://spring.inm.ras.ru/osel

scholarly journals An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-28 ◽  
Author(s):  
Feng Yin ◽  
Guang-Xin Huang
Keyword(s):  
Iterative Algorithm ◽  
Special Kind ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Sylvester Matrix ◽  
Special Cases ◽  
The Matrix ◽  
Solution Pair ◽  
Sylvester Matrix Equations ◽  
Matrix Pair

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations(AXB-CYD,EXF-GYH)=(M,N), which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matricesXandY. When the matrix equations are consistent, for any initial generalized reflexive matrix pair[X1,Y1], the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair[X̂,Ŷ]to a given matrix pair[X0,Y0]in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair[X̃*,Ỹ*]of a new corresponding generalized coupled Sylvester matrix equation pair(AX̃B-CỸD,EX̃F-GỸH)=(M̃,Ñ), whereM̃=M-AX0B+CY0D,Ñ=N-EX0F+GY0H. Several numerical examples are given to show the effectiveness of the presented iterative algorithm.

Orthonormal Isotropic Vector Bases

1998 ◽  
Author(s):  
Namik Ciblak ◽  
Harvey Lipkin
Keyword(s):  
Recursive Algorithm ◽  
Computer Applications ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Orthonormal Bases ◽  
Isotropic Vector ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Deviatoric Stresses ◽  
Isotropic Vectors

Abstract Orthonormal bases of isotropic vectors for indefinite square matrices are proposed and solved. A necessary and sufficient condition is that the matrix must have zero trace. A recursive algorithm is presented for computer applications. The isotropic vectors of 3 × 3 matrices are solved explicitly. Deviatoric stresses in continuum mechanics, the existence of isotropic vectors (particularly in screw space), and stiffness synthesis by springs are shown to be related to the isotropic vector problem.

Applications

2004 ◽  
Author(s):  
Wesley Petersen ◽  
Peter Arbenz
Keyword(s):  
Linear Algebra ◽  
Sparse Matrix ◽  
Sparse Matrices ◽  
Simple Procedure ◽  
Complex Numbers ◽  
Multiprocessor Computer ◽  
Basic Linear Algebra Subprograms ◽  
The Matrix ◽  
Shared Memory Multiprocessor ◽  
Real Complex

Linear algebra is often the kernel of most numerical computations. It deals with vectors and matrices and simple operations like addition and multiplication on these objects. Vectors are one-dimensional arrays of say n real or complex numbers x0, x1, . . . , xn−1. We denote such a vector by x and think of it as a column vector, On a sequential computer, these numbers occupy n consecutive memory locations. This is also true, at least conceptually, on a shared memory multiprocessor computer. On distributed memory multicomputers, the primary issue is how to distribute vectors on the memory of the processors involved in the computation. Matrices are two-dimensional arrays of the form The n · m real (complex) matrix elements aij are stored in n · m (respectively 2 · n ·m if complex datatype is available) consecutive memory locations. This is achieved by either stacking the columns on top of each other or by appending row after row. The former is called column-major, the latter row-major order. The actual procedure depends on the programming language. In Fortran, matrices are stored in column-major order, in C in row-major order. There is no principal difference, but for writing efficient programs one has to respect how matrices are laid out. To be consistent with the libraries that we will use that are mostly written in Fortran, we will explicitly program in column-major order. Thus, the matrix element aij of the m×n matrix A is located i+j · m memory locations after a00. Therefore, in our C codes we will write a[i+j*m]. Notice that there is no such simple procedure for determining the memory location of an element of a sparse matrix. In Section 2.3, we outline data descriptors to handle sparse matrices. In this and later chapters we deal with one of the simplest operations one wants to do with vectors and matrices: the so-called saxpy operation (2.3). In Tables 2.1 and 2.2 are listed some of the acronyms and conventions for the basic linear algebra subprograms discussed in this book.

scholarly journals Casselman’s Basis of Iwahori Vectors and Kazhdan–Lusztig Polynomials

2019 ◽  
Vol 71 (6) ◽  
pp. 1351-1366
Author(s):  
Daniel Bump ◽  
Maki Nakasuji
Keyword(s):  
Representation Theory ◽  
Functional Equations ◽  
Transition Matrix ◽  
Principal Series ◽  
Surprising Result ◽  
Transition Matrices ◽  
Natural Basis ◽  
Series Representations ◽  
The Matrix ◽  
Lusztig Polynomials

AbstractA problem in representation theory of $p$-adic groups is the computation of the Casselman basis of Iwahori fixed vectors in the spherical principal series representations, which are dual to the intertwining integrals. We shall express the transition matrix $(m_{u,v})$ of the Casselman basis to another natural basis in terms of certain polynomials that are deformations of the Kazhdan–Lusztig R-polynomials. As an application we will obtain certain new functional equations for these transition matrices under the algebraic involution sending the residue cardinality $q$ to $q^{-1}$. We will also obtain a new proof of a surprising result of Nakasuji and Naruse that relates the matrix $(m_{u,v})$ to its inverse.

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