Article
Mathematics, Applied
Grigoriy Blekherman, Santanu S. Dey, Kevin Shu, Shengding Sun
Summary: This study examines a computational approach for enforcing PSD-ness on a collection of submatrices in large-scale positive semidefinite (PSD) programs. It introduces new insights on the eigenvalues of Sn,k matrices and improves known upper bounds on Frobenius distance. The research also demonstrates the tightness of relaxation for the case of k = n-1 and presents a structure theorem on nonsingular matrices in Sn,k.
SIAM JOURNAL ON OPTIMIZATION
(2022)
Article
Physics, Multidisciplinary
Lluis A. Belanche-Munoz, Malgorzata Wiejacha
Summary: Kernel methods have been crucial in data science for modeling and visualizing complex problems over the past two decades. However, the selection of kernel functions and the reasons behind their varying performances are still poorly understood. Additionally, the computational costs associated with kernel-based methods call for careful kernel design and parameter selection, as standard model selection methods are highly inefficient. This paper examines these issues from an entropic perspective, focusing on kernelized relevance vector machines (RVMs) and identifying desirable properties of kernel matrices that improve generalization power and model fitting ability. A heuristic approach is also provided to achieve optimal modeling results with limited computational resources.
Article
Mathematics, Applied
Mostafa Hayajneh, Saja Hayajneh, Fuad Kittaneh
Summary: For positive semidefinite matrices A and B, norm inequalities are proven for specific t values using unitarily invariant norms. These inequalities are sharper than previous ones derived by Alakhrass and closely related to an unsolved question posed by Bourin. In fact, these inequalities lead to an affirmative solution for Bourin's question at t=1/4 and 3/4, as demonstrated by Hayajneh and Kittaneh in 2021.
RESULTS IN MATHEMATICS
(2023)
Article
Mathematics
Bat-Od Battseren
Summary: This article demonstrates that group exactness is a von Neumann equivalence invariant, extending the previous knowledge that group exactness is stable under measure equivalence and W*-equivalence.
JOURNAL OF FUNCTIONAL ANALYSIS
(2023)
Article
Mathematics
Shavkat Ayupov, Karimbergen Kudaybergenov
Summary: This paper provides a complete description of ring isomorphisms between algebras of measurable operators affiliated with von Neumann algebras of type II1.
JOURNAL OF FUNCTIONAL ANALYSIS
(2021)
Article
Mathematics, Applied
Jinghao Huang, Karimbergen Kudaybergenov, Fedor Sukochev
Summary: This paper proves that if A is an EW*-algebra and its bounded part Ab is a W*-algebra without finite type I direct summands, then any ring derivation from A into LS(Ab) is an inner derivation. An example that satisfies the condition M⊆A is also provided.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2023)
Article
Mathematics, Applied
Mohd Arif Raza, Aisha Jabeen, Abdul Nadim Khan, Husain Alhazmi
Summary: The manuscript provides a detailed characterization of Lie type derivation on factor von Neumann algebra of zero product and projection product, revealing that they have a standard form.
Article
Physics, Mathematical
Lewis Bowen, Ben Hayes, Yuqing Frank Lin
Summary: The classical Multiplicative Ergodic Theorem of Oseledets is generalized here to cocycles taking values in a semi-finite von Neumann algebra, allowing for a continuous Lyapunov distribution.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2021)
Article
Mathematics, Applied
Ulrik Enstad
Summary: This study investigates the converse theorem for group representations restricted to lattices using the dimension theory of Hilbert modules. It computes the center-valued von Neumann dimension and applies the results to characterize the existence of multiwindow superframes and Riesz sequences.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Angel Rodriguez Palacios
Summary: By studying projections of complete normed complex algebras under different conditions, a series of results about the properties of the algebras and satisfying inequalities have been obtained.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Computer Science, Software Engineering
Michal Kocvara
Summary: The decomposition of large matrix inequalities can reduce the problem size of semidefinite optimization problems and improve efficiency by introducing new matrix variables with rank one that can be replaced by vector variables. This approach leads to significant improvement in efficiency and has demonstrated linear growth in complexity in numerical examples. The connection to the standard theory of domain decomposition and the outcomes of discrete Steklov-Poincare operators further illustrate the benefits of this method.
MATHEMATICAL PROGRAMMING
(2021)
Article
Mathematics
Gilles Pisier
Summary: It is shown that a QWEP von Neumann algebra has the weak* positive approximation property if and only if it is seemingly injective in a certain sense. However, there are examples of algebras that are not seemingly injective, such as B(H)** and certain finite examples defined using ultraproducts. Moreover, it suffices for an algebra to be seemingly injective to have a specific factorization of the identity through B(H) with certain positive properties.
ACTA MATHEMATICA SCIENTIA
(2021)
Article
Mathematics
Martijn Caspers, Mario Klisse, Adam Skalski, Gerrit Vos, Mateusz Wasilewski
Summary: This article introduces the relative Haagerup approximation property for a unital, expected inclusion of arbitrary von Neumann algebras. It shows that if the smaller algebra is finite, then the notion only depends on the inclusion itself and not on the choice of the conditional expectation. Several variations of the definition are shown to be equivalent in this case, and the approximating maps can be chosen to be unital and preserving the reference state. The concept is applied to amalgamated free products of von Neumann algebras and used to deduce the stability of the standard Haagerup property under taking free products with amalgamation over finite-dimensional subalgebras. The general results are illustrated by examples from q deformed Hecke-von Neumann algebras and von Neumann algebras of quantum orthogonal groups.
ADVANCES IN MATHEMATICS
(2023)
Article
Physics, Multidisciplinary
Igor Klep, Victor Magron, Jurij Volcic
Summary: Motivated by recent progress in quantum information theory, this article aims to optimize trace polynomials and introduces a novel Positivstellensatz for certifying positivity. A hierarchy of semidefinite relaxations converging monotonically to the optimum of a trace polynomial subject to tracial constraints is provided. The Gelfand-Naimark-Segal construction is applied to extract optimizers, and the techniques used in this paper are inspired by real algebraic geometry, operator theory, and noncommutative algebra.
ANNALES HENRI POINCARE
(2022)
Article
Mathematics
Iulia-Elena Chiru, Septimiu Crivei
Summary: We provide a constructive sufficient condition for determining whether a matrix over a commutative ring is von Neumann regular, and we prove that this condition is also necessary over local rings. Specifically, we show that a matrix A over a local commutative ring is von Neumann regular if and only if A has an invertible rho(A) x rho(A)-submatrix if and only if the determinantal rank rho(A) and the McCoy rank of A coincide. We also derive consequences for (products of local) commutative rings, as well as determine the number of von Neumann regular matrices over finite rings of residue classes and group algebras.
LINEAR & MULTILINEAR ALGEBRA
(2023)
Article
Mathematics, Applied
Muhammad Syifa'ul Mufid, Ebrahim Patel, Sergei Sergeev
Summary: This paper presents an approach to solve maxmin-omega linear systems by performing normalization and generating a principal order matrix. The possible solution indices can be identified using the principal order matrix and the parameter omega, and the fully active solutions can be obtained from these indices. Other solutions can be found by applying a relaxation to the fully active solutions. This approach can be seen as a generalization of solving max-plus or min-plus linear systems. The paper also highlights the unusual feature of maxmin-omega linear systems having a finite number of solutions when the solution is non-unique.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
E. Mainar, J. M. Pena, B. Rubio
Summary: A bidiagonal decomposition of quantum Hilbert matrices is obtained and the total positivity of these matrices is proved. This factorization is used for accurate algebraic computations and the numerical errors caused by imprecise computer arithmetic or perturbed input data are analyzed. Numerical experiments demonstrate the accuracy of the proposed methods.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Zhong-Zhi Bai
Summary: This study explores the algebraic structures and computational properties of Wasserstein-1 metric matrices. It shows that these matrices can be expressed using the Neumann series of nilpotent matrices and can be accurately and stably computed by solving unit bidiagonal triangular systems of linear equations.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Bogdan Nica
Summary: This study investigates the relationship between the independence number and chromatic number in a graph of non-singular matrices over a finite field, and obtains an upper bound for the former and a lower bound for the latter.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Dijian Wang, Yaoping Hou, Deqiong Li
Summary: In this paper, a Turán-like problem in signed graphs is studied. The properties of signed graphs are proven in the context of the problem.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Tyler Chen, Thomas Trogdon
Summary: This study focuses on the stability of the Lanczos algorithm when applied to problems with eigenvector empirical spectral distribution close to a reference measure characterized by well-behaved orthogonal polynomials. The analysis reveals that the Lanczos algorithm is forward stable on many large random matrix models, even in finite precision arithmetic, which indicates that random matrices differ significantly from general matrices and caution must be exercised when using them to test numerical algorithms.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Constantin Costara
Summary: This passage discusses linear mappings on matrices and the relationship between subsets of the spectrum, providing corresponding characterization conditions.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Amir Hossein Ghodrati, Mohammad Ali Hosseinzadeh
Summary: This paper presents tight upper bounds for all signless Laplacian eigenvalues of a graph with prescribed order and minimum degree, improving upon previously known bounds. Additionally, the relationship between the number of signless Laplacian eigenvalues falling within specific intervals and various graph parameters such as independence, clique, chromatic, edge covering, and matching numbers is explored.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Ya-Lei Jin, Jie Zhang, Xiao-Dong Zhang
Summary: This paper investigates the relationship between the spectral radius of a symmetric matrix and its principal submatrices, and uses these relationships to obtain upper bounds of the spectral radius of graphs.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Davide Bolognini, Paolo Sentinelli
Summary: We introduce immanant varieties associated with simple characters of a finite group and discuss the features of one-dimensional characters and trivial characters.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
A. S. Gordienko
Summary: We introduce the concept of a graded group action on a graded algebra, or equivalently, a group action by graded pseudoautomorphisms. We study the properties of groups of graded pseudoautomorphisms and prove several important theorems and conjectures regarding graded algebras with a group action.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Jiaqi Gu, Shenghao Feng, Yimin Wei
Summary: We propose a tensor product structure compatible with the hypergraph structure and define the algebraic connectivity of the hypergraph in this product, establishing its relationship with vertex connectivity. We introduce connectivity optimization problems into the hypergraph and solve them using algebraic connectivity. Additionally, we apply the Laplacian eigenmap algorithm to the hypergraph under our tensor product.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)
Article
Mathematics, Applied
Samuel Lichtenberg, Abiy Tasissa
Summary: This paper explores a dual basis approach to Classical Multidimensional Scaling (CMDS) and provides explicit formulas for the dual basis vectors. It also characterizes the spectrum of an essential matrix in the dual basis framework. Connections to a related problem in metric nearness are made.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2024)