scholarly journals An Inverse Matrix Formula in the Right-Quantum Algebra

10.37236/747 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Matjaž Konvalinka
Keyword(s):  
Linear Algebra ◽  
Quantum Algebra ◽  
Matrix Inverse ◽  
Bijective Proof ◽  
The Matrix ◽  
Matrix Formula ◽  
Fundamental Transformation ◽  
The Right ◽  
Quantum Matrix ◽  
Master Theorem

The right-quantum algebra was introduced recently by Garoufalidis, Lê and Zeilberger in their quantum generalization of the MacMahon master theorem. A bijective proof of this identity due to Konvalinka and Pak, and also the recent proof of the right-quantum Sylvester's determinant identity, make heavy use of a bijection related to the first fundamental transformation on words introduced by Foata. This paper makes explicit the connection between this transformation and right-quantum linear algebra identities; we give a new, bijective proof of the right-quantum matrix inverse theorem, we show that similar techniques prove the right-quantum Jacobi ratio theorem, and we use the matrix inverse formula to find a generalization of the (right-quantum) MacMahon master theorem.

scholarly journals Re-nnd SOLUTIONS OF THE MATRIX EQUATION AXB=C

2008 ◽  
Vol 84 (1) ◽  
pp. 63-72 ◽  
Author(s):  
DRAGANA S. CVETKOVIĆ-ILIĆ
Keyword(s):  
Inverse Problem ◽  
Linear Algebra ◽  
Matrix Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Explicit Solutions ◽  
Matrix Inverse ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Special Case

AbstractIn this article we consider Re-nnd solutions of the equation AXB=C with respect to X, where A,B,C are given matrices. We give necessary and sufficient conditions for the existence of Re-nnd solutions and present a general form of such solutions. As a special case when A=I we obtain the results from a paper of Groß (‘Explicit solutions to the matrix inverse problem AX=B’, Linear Algebra Appl.289 (1999), 131–134).

scholarly journals Querying a Matrix through Matrix-Vector Products

2021 ◽  
Vol 17 (4) ◽  
pp. 1-19
Author(s):  
Xiaoming Sun ◽  
David P. Woodruff ◽  
Guang Yang ◽  
Jialin Zhang
Keyword(s):  
Linear Algebra ◽  
Incidence Matrix ◽  
Communication Complexity ◽  
Property Testing ◽  
Distributed Computation ◽  
Maximum Eigenvalue ◽  
Graph Problems ◽  
The Matrix ◽  
The Right ◽  
Matrix Vector

We consider algorithms with access to an unknown matrix M ε F n×d via matrix-vector products , namely, the algorithm chooses vectors v 1 , ⃛ , v q , and observes Mv 1 , ⃛ , Mv q . Here the v i can be randomized as well as chosen adaptively as a function of Mv 1 , ⃛ , Mv i-1 . Motivated by applications of sketching in distributed computation, linear algebra, and streaming models, as well as connections to areas such as communication complexity and property testing, we initiate the study of the number q of queries needed to solve various fundamental problems. We study problems in three broad categories, including linear algebra, statistics problems, and graph problems. For example, we consider the number of queries required to approximate the rank, trace, maximum eigenvalue, and norms of a matrix M; to compute the AND/OR/Parity of each column or row of M, to decide whether there are identical columns or rows in M or whether M is symmetric, diagonal, or unitary; or to compute whether a graph defined by M is connected or triangle-free. We also show separations for algorithms that are allowed to obtain matrix-vector products only by querying vectors on the right, versus algorithms that can query vectors on both the left and the right. We also show separations depending on the underlying field the matrix-vector product occurs in. For graph problems, we show separations depending on the form of the matrix (bipartite adjacency versus signed edge-vertex incidence matrix) to represent the graph. Surprisingly, very few works discuss this fundamental model, and we believe a thorough investigation of problems in this model would be beneficial to a number of different application areas.

Complex Chi-Squares: Matrix Notation and Calculation

1972 ◽  
Vol 30 (3) ◽  
pp. 743-746 ◽  
Author(s):  
Edward F. Gocka
Keyword(s):  
Computational Algorithms ◽  
Behavioral Sciences ◽  
Matrix Notation ◽  
Standard Formula ◽  
The Matrix ◽  
Matrix Formula

A matrix formula available for the calculation of complex chi-squares allows several computational variations, each of which requires fewer steps than the standard formula. However, neither the matrix formula nor the associated computational algorithms have been given adequate exposure in statistical texts for the behavioral sciences. This paper reintroduces the formula, expands the notation, and shows how several computational variations can be derived.

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 Improvement of the Impact Properties of Composite Laminates by Means of Nano-Modification of the Matrix—A Review

2018 ◽  
Vol 8 (12) ◽  
pp. 2406 ◽  
Author(s):  
Hamed Saghafi ◽  
Mohamad Fotouhi ◽  
Giangiacomo Minak
Keyword(s):  
Composite Laminates ◽  
Cost Effective ◽  
Careful Consideration ◽  
Impact Performance ◽  
Compression After Impact ◽  
The Matrix ◽  
The Right ◽  
2D And 3D ◽  
The Impact ◽  
Selection Of

This paper reviews recent works on the application of nanofibers and nanoparticle reinforcements to enhance the interlaminar fracture toughness, to reduce the impact induced damage and to improve the compression after impact performance of fiber reinforced composites with brittle thermosetting resins. The nanofibers have been mainly used as mats embedded between plies of laminated composites, whereas the nanoparticles have been used in 0D, 1D, 2D, and 3D dimensional patterns to reinforce the matrix and consequently the composite. The reinforcement mechanisms are presented, and a comparison is done between the different papers in the literature. This review shows that in order to have an efficient reinforcement effect, careful consideration is required in the manufacturing, materials selection and reinforcement content and percentage. The selection of the right parameters can provide a tough and impact resistant composite with cost effective reinforcements.

The inflammatory macrophage: a story of Jekyll and Hyde

2002 ◽  
Vol 104 (1) ◽  
pp. 27-38 ◽  
Author(s):  
Jeremy S. DUFFIELD
Keyword(s):  
Apoptotic Cells ◽  
Tissue Healing ◽  
Immune Mediated ◽  
Selective Depletion ◽  
The Matrix ◽  
Matrix Components ◽  
The Right ◽  
Extracellular Structures ◽  
Two Populations ◽  
Inflammatory Macrophage

Recent investigations have highlighted new roles for the macrophage (Mϕ) in the biology of inflammation. Selective depletion of Mϕs from inflamed sites has confirmed their predominant role in immune-mediated damage. The components of this injury have been dissected. Mϕs mediate death of stromal, parenchymal and other immune cells by engaging the death programme, resulting in apoptosis. In addition, Mϕs induce destruction of matrix and extracellular structures both directly and indirectly by inducing stromal cells to release matrix metalloproteinases. However, there is another side to the inflammatory Mϕ. Evidence is provided that Mϕs at the same sites possess the ability to aid cell proliferation, secrete and stabilize new matrix components and induce resident cells to secrete matrix components themselves. Mϕ phagocytosis of apoptotic cells brings about a change from the cell-killing matrix-degrading cell to the matrix-generating cell-proliferating tissue-healing cell. Just as both Mϕ types are necessary at the inflamed site, the right balance of these two populations is required for healing and resolution. Evidence of excessive inflammation as a manifestation of impaired phagocytosis of apoptotic cells emphasizes that defects in the transition from one Mϕ type to another may account for the uncontrolled excessive inflammation seen in disease. Recent insights into the mechanisms by which apoptotic cells signal the change of function to the Mϕ offer the prospect of novel targets for manipulation of Mϕs in the inflamed tissue.

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