Article
Mathematics
Xiaoqi Huang, Cheng Zhang
Summary: This paper investigates the eigenvalues and eigenfunctions of Schrödinger operators with singular potentials on general n-dimensional Riemannian manifolds. We establish pointwise Weyl laws for potentials in the Kato class and those in Ln(M), extending previous results in the 3-dimensional case. Our proof relies on heat kernel bounds and eigenfunction estimates.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics, Applied
Roger C. Baker, Changhao Chen, Igor E. Shparlinski
Summary: We obtain the exact value of the Hausdorff dimension of the set of coefficients of Gauss sums and provide a new upper bound on the Hausdorff dimension of the set of polynomial coefficients and integer sequences. Our method gives the exact value under specific conditions.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Aida Abiad, Jozefien D'haeseleer, Willem H. Haemers, Robin Simoens
Summary: We present three infinite families of graphs in the Johnson and Grassmann schemes that cannot be uniquely determined by their spectrum. We demonstrate this by constructing graphs that have the same spectrum but are non-isomorphic to these graphs.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2023)
Article
Multidisciplinary Sciences
Shitao Li, Chang Wan, Ali Asghar Talebi, Masomeh Mojahedfar
Summary: Vague graphs (VGs), belonging to the fuzzy graphs (FGs) family, are powerful tools for modeling ambiguity and uncertainty in decision-making problems, particularly in identifying influential individuals in various relations. Vague fuzzy graph structures (VFGSs) are a generalization of VGs that can handle uncertainty associated with inconsistent and indeterminate information, and are useful tools for various domains in computer science and decision-making. Energy and Laplacian energy on VFGS are important concepts that can be applied to decision-making problems.
Article
Mathematics
Yoshihisa Miyanishi
Summary: This article deduces the eigenvalue asymptotics of the Neumann-Poincare operators in three dimensions and discusses the Weyl's law for eigenvalue problems of these operators.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics, Applied
Akashdeep Dey
Summary: This article discusses the properties of a certain type of closed, singular, minimal hypersurfaces in a closed Riemannian manifold. The study shows that under certain volume and eigenvalue bounds, there exists a hypersurface that satisfies these conditions, and the sequence can weakly converge to this hypersurface. Moreover, the study also proves that if a singular, minimal hypersurface satisfies a certain condition, the index of the associated varifold is the same as the index of its regular part.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Xiaohong Li, Yongqin Zhang, Jianfeng Wang, Guang Li, Da Huang
Summary: In this paper, the wreath product of two graphs is considered, resulting in larger graphs than other graph products. A formula for the adjacency spectrum of the wreath product of a complete graph and a circulant graph is derived. The (normalized) Laplacian spectra are obtained, and the adjacency spectra are computed for circulant graphs such as the Mobius ladder graph, the crown graph, and the Andrasfai graph. The paper also generalizes some known results and proposes a problem for further study.
RICERCHE DI MATEMATICA
(2023)
Article
Mathematics, Applied
Changhao Chen, Igor E. Shparlinski
Summary: This paper proves that a set in [0, 1)(d) contains a dense g(δ) set and has a positive Hausdorff dimension. Similar statements also hold for the generalized Gaussian sums.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics
Changxing Miao, Jiye Yuan, Tengfei Zhao, Alex Barron
Summary: In this paper, we study the maximal estimate for Weyl sums on torus Td with d >= 2, and consider its variants along rational lines and on the generic torus. The applications include obtaining new upper bounds on the Hausdorff dimension of sets associated with large values of Weyl sums, which reflect the compound phenomenon between square root cancellation and constructive interference.
JOURNAL OF FUNCTIONAL ANALYSIS
(2023)
Article
Mathematics, Applied
Tanvi Jain
Summary: For every 2n x 2n real positive definite matrix A, there exists a real symplectic matrix M such that M^TAM has a specific diagonal form, known as the symplectic eigenvalues of A. In this paper, we discussed some properties of symplectic eigenvalues, such as analogues of Wielandt's extremal principle and multiplicative Lidskii's inequalities.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2021)
Article
Mathematics
Leonardo Colzani
Summary: This study investigates the speed of convergence in numerical integration using Weyl sums over Kronecker sequences in the torus.
MONATSHEFTE FUR MATHEMATIK
(2023)
Article
Mathematics
Eren Mehmet Kiral, Maki Nakasuji
Summary: In this study, we stratify the SL3 big cell Kloosterman sets using the reduced word decomposition of the Weyl group element, inspired by the Bott-Samelson factorization. As a result, the SL3 long word Kloosterman sum is decomposed into finer parts, expressed as a finite sum of a product of two classical Kloosterman sums. These fine Kloosterman sums are important for the Bruggeman-Kuznetsov trace formula on the congruence subgroup Gamma(0)(N) subset of SL3(Z). Another finding is a new explicit formula, which expresses the triple divisor sum function in terms of a double Dirichlet series of exponential sums, generalizing Ramanujan's formula.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics
Saieed Akbari, Jalal Askari, Kinkar Chandra Das
Summary: This paper studies the properties of the Seidel eigenvalues of graph G and presents some lower and upper bounds for the Seidel energy.
LINEAR & MULTILINEAR ALGEBRA
(2022)
Correction
Mathematics, Applied
Seyed Ahmad Mojallal, Pierre Hansen
Summary: This corrigendum addresses and corrects errors in Theorems 3.1 and 4.1, as well as Corollaries 4.2 and 4.3, in the paper published in [Linear Algebra Appl. 595 (2020) 1-12].
LINEAR ALGEBRA AND ITS APPLICATIONS
(2021)
Article
Mathematics, Applied
Zoran Stanic
Summary: This passage discusses the association between oriented graphs and signed graphs, proving that they are mutually associated when the underlying graph is bipartite. Based on this result, it is shown that in the bipartite case, the skew spectrum of G' can be obtained from the spectrum of an associated signed graph G, and vice versa. In the non-bipartite case, it is proved that the skew spectrum of G' can be obtained from the spectrum of a signed graph associated with the bipartite double of G. Thus, the theory of skew spectra of oriented graphs is shown to have a strong relationship with the theory of spectra of signed graphs, and some problems concerning oriented graphs can be considered within the framework of signed graphs.
LINEAR ALGEBRA AND ITS APPLICATIONS
(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)