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In this paper, we solve a classical counting problem for non-degenerate forms of symplectic and hermitian type defined on a vector space: given a subspace π , we find the number of non-singular subspaces that are trivially intersecting with π and span a non-singular subspace with π. Lower bounds for the quantity of such pairs where π is non-singular were first studied in "Glasby, Niemeyer, Praeger (Finite Fields Appl., 2022)", which was later improved in "Glasby, Ihringer, Mattheus (Des. Codes Cryptogr., 2023)" and generalised in "Glasby, Niemeyer, Praeger (Linear Algebra Appl., 2022)". In this paper, we derive explicit formulae, which allow us to give the exact proportion and improve the known lower bounds. [ABSTRACT FROM AUTHOR]

In this paper, we examine almost (strictly) semimonotone matrices which are also either Z -matrices or matrices having Property (+ +) and show that both parts of Wendler's conjecture [22] (2019) hold for these matrices. Another important result of this paper is the characterization of copositive matrices beyond the matrices with non-positive off-diagonal elements. We discuss some interesting properties related to the structure of almost (strictly) semimonotone matrices and also present results pertaining to the existence and multiplicity of solutions to the linear complementarity problem associated with an almost strictly semimonotone matrix. [ABSTRACT FROM AUTHOR]

Let { A n } n be a sequence of square matrices such that size (A n) = d n → ∞ as n → ∞. We say that { A n } n has an asymptotic spectral distribution described by a Lebesgue measurable function f : D ⊂ R k → C if, for every continuous function F : C → C with bounded support, lim n → ∞ ⁡ 1 d n ∑ i = 1 d n F (λ i (A n)) = 1 μ k (D) ∫ D F (f (x)) d x , where μ k is the Lebesgue measure in R k and λ 1 (A n) , ... , λ d n (A n) are the eigenvalues of A n. In the last decades, the asymptotic spectral distribution of increasing size matrices has become a subject of investigation by several authors. Special attention has been devoted to matrices A n arising from the discretization of differential equations, whose size d n diverges to ∞ along with the mesh-fineness parameter n. However, despite the popularity that the topic has reached nowadays, the role of the so-called test functions F appearing in the above limit relation has been inexplicably neglected so far. In particular, a natural question such as "Which is the largest set of test functions F for which the above limit relation is satisfied?" is still unanswered. In the present paper, we provide a definitive answer to this question by identifying the largest set of test functions F for which the above limit relation is satisfied. We also present some applications of this result to the analysis of spectral clustering, including new asymptotic estimates on the number of outliers. A special attention is devoted to the so-called "inner outliers" of Hermitian block Toeplitz matrices. We conclude the paper with an interpretation of the main result in the context of the vague convergence of probability measures. [ABSTRACT FROM AUTHOR]

In a recent paper (Groenewald et al. 2021 [9]) we considered an unbounded Toeplitz-like operator T Ω generated by a rational matrix function Ω that has poles on the unit circle T of the complex plane. A Wiener-Hopf type factorization was proved and this factorization was used to determine some Fredholm properties of the operator T Ω , including the Fredholm index. Due to the lower triangular structure (rather than diagonal) of the middle term in the Wiener-Hopf type factorization and the lack of uniqueness, it is not straightforward to determine the dimension of the kernel of T Ω from this factorization, and hence of the co-kernel, even when T Ω is Fredholm. In the current paper we provide a formula for the dimension of the kernel of T Ω under an additional assumption on the Wiener-Hopf type factorization. In the case that Ω is a 2 × 2 matrix function, a characterization of the kernel of the middle factor of the Wiener-Hopf type factorization is given and in many cases a formula for the dimension of the kernel is obtained. The characterization of the kernel of the middle factor for the 2 × 2 case is partially extended to the case of matrix functions of arbitrary size. [ABSTRACT FROM AUTHOR]

In this paper, I will answer a question that Albrecht asked me when preparing a paper on spectral instabilities: Is the ε -pseudospectrum of a Toeplitz operator T (a) , say with a continuous generating function a , always contained in the ε -pseudospectrum of the compactly perturbed operator T (a) + K ? [ABSTRACT FROM AUTHOR]

Let G be a graph and d i be the degree of a vertex v i in G. For a symmetric real function f (x , y) , one can get an edge-weighted graph in such a way that for each edge v i v j of G , the weight of v i v j is assigned by f (d i , d j). Hence, we have a weighted adjacency matrix A f (G) of G , in which the ij -entry is equal to f (d i , d j) if v i v j ∈ E (G) and 0 otherwise. In this paper, we use a unified approach to deal with the spectral properties of A f (G) for f (x , y) to be the functions of graphical or topological function-indices. Firstly, we obtain uniform interlacing inequalities for the weighted adjacency eigenvalues. For the edge-weight functions defined by almost a half of popularly used topological indices, it can be shown that our inequalities cannot be improved. Secondly, we establish a uniform equivalent condition for a connected graph G to have m distinct weighted adjacency eigenvalues. As an application, a combinatorial characterization for a graph to have two and three distinct weighted adjacency eigenvalues is presented, respectively. Moreover, bipartite graphs and unicyclic graphs with three distinct weighted adjacency eigenvalues are characterized. This paper attempts to unify the spectral study for weighted adjacency matrices of graphs with degree-based edge-weights. [ABSTRACT FROM AUTHOR]

In this paper we consider the twisted Hamiltonian extended affine Lie algebra (THEALA). We classify the irreducible integrable modules for these Lie algebras with finite dimensional weight spaces when the finite dimensional center acts non-trivially. This Lie algebra has a triangular decomposition, which is different from the natural triangular decomposition of twisted full toroidal Lie algebra. Any irreducible integrable module of it is a highest weight module with respect to the given triangular decomposition. In this paper we describe the highest weight space in detail. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of source recovery in a dynamical system utilizing space-time samples. This is a specific issue within the broader field of dynamical sampling, which involves collecting samples from solutions to a differential equation across both space and time with the aim of recovering critical data, such as initial values, the sources, the driving operator, or other relevant details. Our focus in this study is the recovery of unknown, stationary sources across both space and time, leveraging space-time samples. This research may have significant applications; for instance, it could provide a model for strategically placing devices to measure the number of pollutants emanating from factory smokestacks and dispersing across a specific area. Space-time samples could be collected using measuring devices placed at various spatial locations and activated at different times. We present necessary and sufficient conditions for the positioning of these measuring devices to successfully resolve this dynamical sampling problem. This paper provides both a theoretical foundation for the recovery of sources in dynamical systems and potential practical applications. [ABSTRACT FROM AUTHOR]

In this work we complement the description of the extreme rays of the 6 × 6 copositive cone with some topological structure. In a previous paper we decomposed the set of extreme elements of this cone into a disjoint union of subsets of algebraic varieties of different dimension. In this paper we link this classification to the recently introduced combinatorial characteristic called extended minimal zero support set. We determine those subsets which are essential, i.e., which are not embedded in the boundary of other subsets. This allows to drastically decrease the number of cases one has to consider when investigating different properties of the 6 × 6 copositive cone. As an application, we construct an example of a copositive 6 × 6 matrix with all ones on the diagonal which does not belong to the Parrilo inner sum of squares relaxation K 6 (1). [ABSTRACT FROM AUTHOR]

When the conjugate gradient (CG) method for solving linear algebraic systems was formulated about 70 years ago by Lanczos, Hestenes, and Stiefel, it was considered an iterative process possessing a mathematical finite termination property. With the deep insight of the original authors, CG was placed into a very rich mathematical context, including links with Gauss quadrature and continued fractions. The optimality property of CG was described via a normalized weighted polynomial least squares approximation to zero. This highly nonlinear problem explains the adaptation of CG iterates to the given data. Karush and Hayes immediately considered CG in infinite dimensional Hilbert spaces and investigated its superlinear convergence. Since then, the view of CG, as well as other Krylov subspace methods developed in the meantime, has changed. Today these methods are considered primarily as computational tools, and their behavior is typically characterized using linear upper bounds, or heuristics based on clustering of eigenvalues. Such simplifications limit the mathematical understanding of Krylov subspace methods, and also negatively affect their practical application. This paper offers a different perspective. Focusing on CG and the generalized minimal residual (GMRES) method, it presents mathematically important as well as practically relevant phenomena that uncover their behavior through a discussion of computed examples. These examples provide an easily accessible approach that enables understanding of the methods, while pointers to more detailed analyses in the literature are given. This approach allows readers to choose the level of depth and thoroughness appropriate for their intentions. Some of the points made in this paper illustrate well known facts. Others challenge mainstream views and explain existing misunderstandings. Several points refer to recent results leading to open problems. We consider CG and GMRES crucially important for the mathematical understanding, further development, and practical applications also of other Krylov subspace methods. The paper additionally addresses the motivation of preconditioning. [ABSTRACT FROM AUTHOR]

Given a finite field F q and a positive integer n , a flag is a sequence of nested F q -subspaces of a vector space F q n and a flag code is a nonempty collection of flags. The projected codes of a flag code are the constant dimension codes containing all the subspaces of prescribed dimensions that form the flags in the flag code. In this paper we address the notion of equivalence for flag codes and explore in which situations such an equivalence can be reduced to the equivalence of the corresponding projected codes. In addition, this study leads to new results concerning the automorphism group of certain families of flag codes, some of them also introduced in this paper. [ABSTRACT FROM AUTHOR]

The (k , r) -fan, denoted by F k , r , is the graph consisting of k copies of the complete graph K r which intersect in a single vertex. Desai et al. [7] proved that E X s p (n , F k , r) ⊆ E X (n , F k , r) for sufficiently large n , where E X s p (n , F k , r) and E X (n , F k , r) are the sets of n -vertex F k , r -free graphs with maximum spectral radius and maximum size, respectively. In this paper, the set E X s p (n , F k , r) is uniquely determined for n large enough. Let H s , t 1 , ... , t k be the graph consisting of s triangles and k odd cycles of lengths t 1 , ... , t k ≥ 5 intersecting in exactly one common vertex, denoted by H s , k for short. Li and Peng [12] showed that E X s p (n , H s , k) ⊆ E X (n , H s , k) for n large enough. In this paper, the set E X s p (n , H s , k) is uniquely characterized for sufficiently large n. [ABSTRACT FROM AUTHOR]

Let G n , α be the set of all connected graphs of order n with independence number α. A graph is called the Q -minimizer graph (A -minimizer graph) if it attains the minimum signless Laplacian spectral radius (adjacency spectral radius) over all graphs in G n , α. In this paper, we first show that the Q -minimizer graph must be a tree for α ≥ ⌈ n 2 ⌉ , and then we derive seven propositions about the Q -minimizer graph. Moreover, when n − α is a constant, the structure of the Q -minimizer graph is characterized. The method of getting Q -minimizer graph in this paper is different from that of getting A -minimizer graph. As applications, we determine the Q -minimizer graphs for α = n − 1 , n − 2 , n − 3 and n − 4 , respectively. The results of α = n − 1 , n − 2 , n − 3 are consistent with that in Li and Shu (2010) [15] and the result of α = n − 4 is new. Interestingly, the Q -minimizer graph in G n , n − 4 is unique, which is exactly one of the A -minimizer graphs in G n , n − 4. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence solutions to nonlinear Fučík problems of the form (1) A x = α x + − β x − + g (x) , where A is a symmetric n × n matrix, α , β are real numbers, and g : R n → R n is continuous. The nonlinear problem (1) is motivated by application to nonlinear oscillating systems such as the Tacoma Narrows Bridge. The paper begins by developing a qualitative picture of Fučík spectrum associated with the matrix equation A x = α x + − β x −. In this setting, we present two characterizations: first, we show that under appropriate assumptions the Fučík spectrum consists of curves bifurcating from points (λ , λ) ∈ R 2 , where λ is an eigenvalue of A ; second, we give more global variational characterization of the Fučík curves. In both cases, we present various qualitative properties of the Fučík curves. The paper finishes by presenting two existence theorems for the nonlinear Fučík problem under mild assumptions on the nonlinear term g. [ABSTRACT FROM AUTHOR]

In this paper, we determine the set S of straight lines L 0 that have intersections with four given distinct lines L 1 , ... , L 4 in R 3. If any two of the four given lines are skew, i.e., not co-planar, Bielinski and Lapinska used techniques in projective geometry to show that there are either zero, one, or two elements in the set S. Using linear algebra techniques, we determine S and show that there are no, one, two or infinitely many elements L 0 in S , where the last case was overlooked in the earlier paper. For the sake of completeness, we provide a comprehensive determination of all the elements L 0 in S if at least two of the four given lines are co-planar. In this scenario, there may also be zero, one, two, or infinitely many solutions. [ABSTRACT FROM AUTHOR]

A sign pattern matrix is a matrix whose entries are from the set { + , − , 0 }. For a sign pattern matrix A , the qualitative class of A , denoted Q (A) , is the set of all real matrices whose entries have signs given by the corresponding entries of A. An n × n sign pattern matrix A requires all distinct eigenvalues if every real matrix in Q (A) has n distinct eigenvalues. Li and Harris (2002) [13] characterized the 2 × 2 and 3 × 3 irreducible sign pattern matrices that require all distinct eigenvalues, and established some useful general results on n × n sign patterns that require all distinct eigenvalues. In this paper, we characterize 4 × 4 irreducible sign patterns that require four distinct eigenvalues. This is done by characterizing 4 × 4 irreducible sign patterns that require four distinct real eigenvalues, that require four distinct nonreal real eigenvalues, or that require two distinct real eigenvalues and a pair of conjugate nonreal eigenvalues. The last case turns out to be much more involved. Some interesting open problems are presented. Three important tools that are used in the paper are the following: the discriminant of a polynomial; the fact that if a square sign pattern matrix A requires all distinct eigenvalues then A requires a fixed number of real eigenvalues; and the known result that if A is a " k -cycle" sign pattern then for each B ∈ Q (A) , the k nonzero eigenvalues of B are evenly distributed on a circle in the complex plane centered at the origin. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the adjacency matrices of finite undirected simple graphs over a finite prime field F p. We apply symmetric (row and column) elementary transformations to the adjacency matrix over F p in order to get a direct sum decomposition by other adjacency matrices. In this paper, we give a complete description of the direct sum decomposition of the adjacency matrix of any graph over F p for any odd prime p. Our key tool is quadratic residues of F p. [ABSTRACT FROM AUTHOR]

Recently, a spectral Favard theorem was presented for bounded banded lower Hessenberg matrices that possess a positive bidiagonal factorization. The paper establishes conditions, expressed in terms of continued fractions, under which an oscillatory tetradiagonal Hessenberg matrix can have such a positive bidiagonal factorization. Oscillatory tetradiagonal Toeplitz matrices are examined as a case study of matrices that admit a positive bidiagonal factorization. Furthermore, the paper proves that oscillatory banded Hessenberg matrices are organized in rays, where the origin of the ray does not have a positive bidiagonal factorization, but all the interior points of the ray do have such a positive bidiagonal factorization. [ABSTRACT FROM AUTHOR]

The notion of automatic selfadjointness of all ideals in a multiplicative semigroup of the bounded linear operators on a separable Hilbert space B (H) arose in a 2015 discussion with Heydar Radjavi who pointed out that B (H) and the finite rank operators F (H) possessed this unitary invariant property which category we named SI semigroups (for automatic selfadjoint ideal semigroups). Equivalent to the SI property is the solvability, for each A in the semigroup, of the bilinear operator equation A ⁎ = X A Y which we believe is a new connection relating the semigroup theory with the theory of operator equations. We found in our earlier works in the subject that even at the basic level of singly generated semigroups, the investigation of SI semigroups led to interesting algebraic and analytic phenomena when generated by rank one operators, normal operators, partial and power partial isometries, subnormal-hyponormal-essentially normal operators, and weighted shift operators; and generated by commuting families of normal operators. In this paper, we focus on a separate M n (C) treatment for singly generated SI semigroups that requires studying the solvability of the bilinear matrix equation A ⁎ = X A Y in a multiplicative semigroup of finite matrices. This separate focus is needed because the techniques employed in our earlier works we could not adapt to finite matrices. In this paper we find that for certain classes of generators, being a partial isometry is equivalent to generating an SI semigroup. Such classes are: degree 2 nilpotent matrices, weighted shifts, and non-normal Jordan matrices. For the key tools used to establish these equivalences, we developed a number of necessary conditions for singly generated semigroups to be SI for the very general classes: nonselfadjoint matrices, nonzero nilpotent matrices, nonselfadjoint invertible matrices, and Jordan blocks. We also show, for a nonselfadjoint matrix generator in an SI semigroup, the matrix being a partial isometry is equivalent to having norm one. And as an aside, we also prove necessary generator conditions for the SI property when generated by matrices with nonnegative entries. [ABSTRACT FROM AUTHOR]

This paper continues the previous work (Quantum Inf. Process 11 (2019)) by two authors of the present paper about a spectral mapping property of chiral symmetric unitary operators. In physics, they treat non-unitary time-evolution operators to consider quantum walks in open systems. In this paper, we generalize the above result to include a chiral symmetric non-unitary operator whose coin operator only has two eigenvalues. As a result, the spectra of such non-unitary operators are included in the (possibly non-unit) circle and the real axis in the complex plane. We also give some examples of our abstract results, such as non-unitary quantum walks defined by Mochizuki et al. Furthermore, the main theorem applies not only to quantum walks but also to the Ihara zeta function and correlated random walks on regular graphs. [ABSTRACT FROM AUTHOR]

The inverse eigenvalue problem studies the possible spectra among matrices whose off-diagonal entries have their zero-nonzero patterns described by the adjacency of a graph G. In this paper, we refer to the i -nullity pair of a matrix A as (null (A) , null (A (i)) , where A (i) is the matrix obtained from A by removing the i -th row and column. The inverse i -nullity pair problem is considered for complete graphs, cycles, and trees. The strong nullity interlacing property is introduced, and the corresponding supergraph lemma and decontraction lemma are developed as new tools for constructing matrices with a given nullity pair. [ABSTRACT FROM AUTHOR]

The paper considers the convergence of the complex block Jacobi diagonalization methods under the large set of the generalized serial pivot strategies. The global convergence of the block methods for Hermitian, normal and J -Hermitian matrices is proven. In order to obtain the convergence results for the block methods that solve other eigenvalue problems, such as the generalized eigenvalue problem, we consider the convergence of a general block iterative process which uses the complex block Jacobi annihilators and operators. [ABSTRACT FROM AUTHOR]

Let F be a field and q (x) a quadratic polynomial in F [ x ] with q (0) ≠ 0. We denote by T ∞ (F) the algebra of all infinite upper triangular matrices over the field F. A matrix A ∈ T ∞ (F) is called a quadratic matrix with respect to q (x) if q (A) = 0. In this paper, we first investigate the subgroup in T ∞ (F) generated by all quadratic matrices with respect to q (x) and then present some applications. [ABSTRACT FROM AUTHOR]

A complex unit gain graph, or T -gain graph, is a triple Φ = (G , T , φ) comprised of a simple graph G as the underlying graph of Φ, the set of unit complex numbers T = { z ∈ C : | z | = 1 } , and a gain function φ : E → → T with the property that φ (e i j) = φ (e j i) − 1. A cactus graph is a connected graph in which any two cycles have at most one vertex in common. In this paper, we firstly show that there does not exist a complex unit gain graph with nullity n (G) − 2 m (G) + 2 c (G) − 1 , where n (G) , m (G) and c (G) are the order, matching number, and cyclomatic number of G. Next, we provide a lower bound on the nullity for connected complex unit gain graphs and an upper bound on the nullity for complex unit gain bipartite graphs. Finally, we characterize all non-singular complex unit gain bipartite cactus graphs, which generalizes a result in Wong et al. (2022) [30]. [ABSTRACT FROM AUTHOR]

In this paper, we study the representation of an infinite-dimensional Lie algebra C related to the q-analog Virasoro-like Lie algebra. We give the necessary and sufficient conditions for the highest weight irreducible module V (ϕ) of C to be a Harish-Chandra module. We prove that the Verma C -module V ¯ (ϕ) is either irreducible or has the corresponding irreducible highest weight C -module V (ϕ) that is a Harish-Chandra module. We also give the maximal proper submodule of the Verma module V ¯ (ϕ) and the e -character of the irreducible highest weight C -module V (ϕ) when the highest weight ϕ satisfies some natural conditions. Furthermore, we give the classification of the Harish-Chandra C -modules with nontrivial central charge. [ABSTRACT FROM AUTHOR]

In this paper we look at the families of random walks arising from FI-graphs. One may think of these objects as families of nested graphs, each equipped with a natural action by a symmetric group S n , such that these actions are compatible and transitive. Families of graphs of this form were introduced by the authors in [9] , while a systematic study of random walks on these families were considered in [10]. In the present work, we illustrate that these random walks never exhibit the so-called product condition, and therefore also never display total variation cutoff as defined by Aldous and Diaconis [1]. In particular, we provide a large family of algebro-combinatorially motivated examples of collections of Markov chains which satisfy some well-known algebraic heuristics for cutoff, while not actually having the property. [ABSTRACT FROM AUTHOR]

It is known that the generating function associated with the enumeration of non-backtracking walks on finite graphs is a rational matrix-valued function of the parameter; such function is also closely related to graph-theoretical results such as Ihara's theorem and the zeta function on graphs. In Grindrod et al. [13] , the radius of convergence of the generating function was studied for simple (i.e., undirected, unweighted and with no loops) graphs, and shown to depend on the number of cycles in the graph. In this paper, we use technologies from the theory of polynomial and rational matrices to greatly extend these results by studying the radius of convergence of the corresponding generating function for general, possibly directed and/or weighted, graphs. We give an analogous characterization of the radius of convergence for directed (unweighted or weighted) graphs, showing that it depends on the number of cycles in the undirectization of the graph. We also consider backtrack-downweighted walks on unweighted digraphs, and we prove a version of Ihara's theorem in that case. Finally, for weighted directed graphs, we provide for the first time an exact formula for the radius of convergence, improving a previous result that exhibited a lower bound, and we also prove a version of Ihara's theorem. [ABSTRACT FROM AUTHOR]

For a connected graph G with order n , let e (G) be the number of its distinct eigenvalues and d be the diameter. We denote by m G (μ) the eigenvalue multiplicity of μ in G. It is well known that e (G) ≥ d + 1 , which shows m G (μ) ≤ n − d for any real number μ. A graph is called m i n i m a l if e (G) = d + 1. In 2013, Wong et al. characterize all minimal graphs with m G (0) = n − d. In this paper, by applying the star complement theory, we prove that if G is not a path and m G (μ) = n − d , then μ ∈ { 0 , − 1 }. Furthermore, we completely characterize all minimal graphs with m G (− 1) = n − d. [ABSTRACT FROM AUTHOR]

In this paper, we provide a formula to compute the permanent of block matrices depending on entries of each block. As a consequence, a generalized Lieb permanent inequality on positive semi-definite block matrices is given. [ABSTRACT FROM AUTHOR]

For a simple graph on n vertices, any of its signless Laplacian eigenvalues is in the interval [ 0 , 2 n − 2 ]. In this paper, we give relationships between the number of signless Laplacian eigenvalues in specific intervals in [ 0 , 2 n − 2 ] and graph invariants including matching number and diameter. [ABSTRACT FROM AUTHOR]

Let D be a Noetherian infinite integral domain, denote by M 2 (D) and by s l 2 (D) the 2 × 2 matrix algebra and the Lie algebra of the traceless matrices in M 2 (D) , respectively. In this paper we study the weak polynomial identities for the natural grading by the cyclic group Z 2 of order 2 on M 2 (D) and on s l 2 (D). We describe a finite basis of the graded polynomial identities for the pair (M 2 (D) , s l 2 (D)). Moreover we prove that the ideal of the graded identities for this pair satisfies the Specht property, that is every ideal of graded identities of pairs (associative algebra, Lie algebra), satisfying the graded identities for (M 2 (D) , s l 2 (D)) , is finitely generated. The polynomial identities for M 2 (D) are known if D is any field of characteristic different from 2. The identities for the Lie algebra s l 2 (D) are known when D is an infinite field. The identities for the pair we consider were first described by Razmyslov when D is a field of characteristic 0, and afterwards by the second author when D is an infinite field. The graded identities for the pair (M 2 (D) , g l 2 (D)) were also described, by Krasilnikov and the second author. In order to obtain these results we use certain graded analogues of the generic matrices, and also techniques developed by G. Higman concerning partially well ordered sets. [ABSTRACT FROM AUTHOR]

In recent years, higher-order trace formulas of operator functions have attracted considerable attention to a large part of the perturbation theory community. In this direction, we prove estimates for traces of higher-order derivatives of multivariate operator functions with associated scalar functions arising from multivariate analytic function space and, as a consequence, derive higher-order spectral shift measures for pairs of tuples of commuting contractions under Hilbert-Schmidt perturbations. These results substantially extend the main results of [26] , where the estimates were proved for traces of first and second-order derivatives of multivariate operator functions. In the context of the existence of higher-order spectral shift measures, our results extend the relative results of [6,20] from a single-variable to a multivariate case under Hilbert-Schmidt perturbations. Our results rely crucially on heavy uses of explicit expressions of higher-order derivatives of operator functions and estimates of the divided difference of multivariate analytic functions, which are developed in this paper, along with the spectral theorem of tuple of commuting normal operators. In conclusion, we explore the significance of our results and provide relevant examples. [ABSTRACT FROM AUTHOR]

In this paper, we give the rank of the walk matrix of the Dynkin graph A n , and prove that its Smith normal form is diag ( 1 , ... , 1 ︸ ⌈ n 2 ⌉ , 0 , ... , 0). [ABSTRACT FROM AUTHOR]

A graph G = (V G , E G) is said to be a cograph if the path P 4 does not appear among its induced subgraphs. The vicinal preorder ≺ on the vertex set V G is defined in terms of inclusions between neighborhoods. The minimum number ∇ (G) of ≺-chains required to cover G is called the Dilworth number of G. In this paper it is proved that for a cograph G , the multiplicity of every Seidel eigenvalue λ ≠ ± 1 does not exceed ∇ (G). This bound turns out to be tight and can be further improved for threshold graphs. Moreover, it is shown that cographs with at least two vertices have no Seidel eigenvalues in the interval (− 1 , 1). [ABSTRACT FROM AUTHOR]

In this paper, we obtain a combinatorial expression for the Perron root and eigenvectors of a non-negative integer matrix using techniques from symbolic dynamics. We associate such a matrix with a multigraph and consider the edge shift corresponding to it. This gives rise to a collection of forbidden words F which correspond to the non-existence of an edge between two vertices, and a collection of repeated words R with multiplicities which correspond to multiple edges between two vertices. In general, for given collections F of forbidden words and R of repeated words with pre-assigned multiplicities, we construct a generalized language as a multiset. A combinatorial expression that enumerates the number of words of fixed length in this generalized language gives the Perron root and eigenvectors of the adjacency matrix. We also obtain conditions under which such a generalized language is a language of an edge shift. [ABSTRACT FROM AUTHOR]

In 2010, Nikiforov conjectured that for ℓ ≥ 2 and n sufficiently large, S n , ℓ − 1 1 is the unique graph with the maximum spectral radius over all n -vertex C 2 ℓ -free graphs. In 2022, Cioabǎ, Desai and Tait solved this conjecture. The theta graph Θ t , ℓ consists of two vertices joined by t vertex-disjoint paths, each of length ℓ. Particularly, Θ 2 , ℓ ≅ C 2 ℓ. In this paper, we characterize the unique extremal graph which attains the maximum spectral radius among all Θ t , ℓ -free graphs of order n , where t , ℓ ≥ 3 and n is sufficiently large. [ABSTRACT FROM AUTHOR]

We study the individual behavior of the eigenvalues of the laplacian matrices of the cyclic graph of order n , where one edge has weight α ∈ C , with Re (α) < 0 , and all the others have weights 1. This paper is a sequel of a previous one where we considered Re (α) ∈ [ 0 , 1 ] (Grudsky et al., 2022 [12]). We prove that for Re (α) < 0 and n > Re (α − 1) / Re (α) , one eigenvalue is negative while the others belong to [ 0 , 4 ] and are distributed as the function x ↦ 4 sin 2 ⁡ (x / 2). Additionally, we prove that as n tends to ∞, the outlier eigenvalue converges exponentially to 4 Re (α) 2 / (2 Re (α) − 1). We give exact formulas for half of the inner eigenvalues, while for the others we justify the convergence of Newton's method and fixed-point iteration method. We find asymptotic expansions, as n tends to ∞, both for the eigenvalues belonging to [ 0 , 4 ] and the outlier. We also compute the eigenvectors and their norms. [ABSTRACT FROM AUTHOR]

Solutions to most nonlinear ordinary differential equations (ODEs) rely on numerical solvers, but this gives little insight into the nature of the trajectories and is relatively expense to compute. In this paper, we derive analytic solutions to a class of nonlinear, homogeneous ODEs with linear and quadratic terms on the right-hand side. We formulate a power series expansion of each state variable, whose function depends on the eigenvalues of the linearized system, and solve for the coefficients using some linear algebra and combinatorics. Various experiments exhibit quickly decaying coefficients, such that a good approximation to the true solution consists of just a few terms. [ABSTRACT FROM AUTHOR]

In this paper we consider an orthonormal basis, generated by a tensor product of Fourier basis functions, half period cosine basis functions, and the Chebyshev basis functions. We deal with the approximation problem in high dimensions related to this basis and design a fast algorithm to multiply with the underlying matrix, consisting of rows of the non-equidistant Fourier matrix, the non-equidistant cosine matrix and the non-equidistant Chebyshev matrix, and its transposed. Using this, we derive the ANOVA (analysis of variance) decomposition for functions with partially periodic boundary conditions through using the Fourier basis in some dimensions and the half period cosine basis or the Chebyshev basis in others. We consider sensitivity analysis in this setting, in order to find an adapted basis for the underlying approximation problem. More precisely, we find the underlying index set of the multidimensional series expansion. Additionally, we test this ANOVA approximation with mixed basis at numerical experiments, and refer to the advantage of interpretable results. [ABSTRACT FROM AUTHOR]

In this paper we obtain some general results about mixed second order de Branges–Rovnyak spaces defined by an operator A such that both A and A ⁎ are 2-hypercontractions. As applications of these results, we study mixed second order sub-Bergman spaces since the analytic Toeplitz operator T b on weighted Bergman space A α 2 for α ≥ 0 on the unit disk is such that both T b and T b ⁎ are 2-hypercontractions. [ABSTRACT FROM AUTHOR]

The main aim of the present paper is to establish inversion formulas of Gohberg-Semencul type for Toeplitz-plus-Hankel matrices. In particular, it is shown how the inverse of such a structured matrix A n of order n is computed by means of their first two and last two columns or rows under the additional assumption that a certain 2 × 2 matrix is nonsingular. Moreover, a formula for the inverse of the Toeplitz-plus-Hankel matrix A n − 2 of order n − 2 in the center of A n is established and sufficient invertibility criteria for both A n and A n − 2 are obtained. Hereby a main tool is to use known inversion formulas involving not only rows or columns of the inverse but also solutions of equations the right hand sides of which depend on entries of the Toeplitz-plus-Hankel matrix itself. [ABSTRACT FROM AUTHOR]

With this paper we continue the investigations started in [6] and concerned with stability conditions for collocation-quadrature methods based on the zeros of classical Jacobi polynomials, not only Chebyshev polynomials. While in [6] we only proved the necessity of certain conditions, here we will show also their sufficiency in particular cases. [ABSTRACT FROM AUTHOR]

The classical Rouché Theorem from complex function theory has been generalized to a result involving Fredholm operator valued functions (see [9] and [6]). The setting in question is (generally) a non-commutative one, so actually there are two versions of the generalization. In this paper, a unifying enhanced variant is presented involving the notion of the spectral radius. Its proof is based on a maximum principle for the spectral radius. An example is given showing that the enhanced Rouché Theorem presented here is a genuine improvement over the earlier version of the result. [ABSTRACT FROM AUTHOR]

Inspired by the paper of Groenewald, Kaashoek and Ran (2017) [19] , we present an operator-theoretic approach to provide further insight and simpler computational formulas for the Wiener-Hopf indices of a rational matrix valued function taking unimodular values on the unit circle. [ABSTRACT FROM AUTHOR]

Given p ∈ (1 , ∞) and the power weight ω (x) = | x + i | β on R + = (0 , ∞) with β ∈ (− 1 / p , 1 − 1 / p) , let L N p (R + , ω) be the weighted Lebesgue space of functions f : R + → C N. The paper deals with index calculation for the Fredholm on L N p (R + , ω) operators of the form A = ∑ j = 1 m a j W (b j) , where a j are continuous on R + matrix coefficients that slowly oscillate at 0 and ∞, and W (b j) are Wiener-Hopf operators with piecewise continuous matrix symbols b j with finite sets of discontinuities on R , which are Mellin multipliers on L N p (R + , ω). To this end we apply a modification of Duduchava's decomposition scheme, results on Fourier and Mellin pseudodifferential operators, and results on convolution type operators with piecewise slowly oscillating data. Fredholm symbols constructed for the Banach algebra generated by the operators of the considered form are N × N matrix functions of specific structure. [ABSTRACT FROM AUTHOR]

We investigate the convex polytope Ω n (123 ‾) of doubly stochastic matrices spanned by the n × n permutation matrices that avoid an increasing pattern of length 3, the 123 ‾ -permutation matrices. We determine some of its facets and other faces, including faces of small dimension, and their connection to facets and faces of the polytope Ω n of all n × n doubly stochastic matrices. The paper concludes with some relations concerning the frequencies of the positions of the 1's in n × n 123 ‾ -permutation matrices, and the vertex-edge graph of Ω n (123 ‾). [ABSTRACT FROM AUTHOR]

Let A be a positive bounded operator on a Hilbert space H and let ‖ T ‖ A , w A (T) , and m A (T) denote the A -operator seminorm, the A -numerical radius, and the A -minimum modulus of an operator T in the semi-Hilbertian space (H , ‖ ⋅ ‖ A) , respectively. In this paper, we present new improvements of certain A -Cauchy–Schwarz type inequalities and as applications of our results, we provide refinements of some A -numerical radius inequalities for semi-Hilbertian space operators. It is shown, among other inequalities, that w A (T) ≤ (1 − 1 2 inf λ ∈ C ⁡ m A 2 (I − λ T)) ‖ T ‖ A , where I is the identity operator on H. A refinement of the triangle inequality for semi-Hilbertian space operators is also given. [ABSTRACT FROM AUTHOR]

In this paper, we present some weighted Hölder-type inequalities for N −tuple of n × n complex matrices, with applications that include refined bounds for the norm of the sum of two matrices, the numerical radius of a matrix and the numerical radius for the product of two matrices. The relation with the existing literature will be discussed, where some results are proven to be improvements of celebrated results in the literature. Further, numerical examples are given to support the significance of other results. [ABSTRACT FROM AUTHOR]

Haemers' conjecture on the Seidel energy of graphs, states that the energy of the Seidel matrix of any graph of order n is at least 2 n − 2 , where the equality holds for the complete graph. In this paper, we give a short simple proof of this conjecture. [ABSTRACT FROM AUTHOR]

In the paper we give some basic properties of the superderivations of Lie superalgebras. Under certain condition, for solvable Lie superalgebras with given nilradicals, we give estimates for upper bounds to the dimensions of complementary subspaces to the nilradicals. Moreover, under the condition that an analogue of Lie's theorem is true, the description of solvable Lie superalgebras of maximal rank is obtained. Namely, we prove that an arbitrary solvable Lie superalgebra of maximal rank under the mentioned condition is isomorphic to the maximal solvable extension of nilradical of maximal rank. Finally, along with effective method of construction of solvable Lie superalgebras of maximal rank we present the description of special type of maximal solvable extension of nilpotent Lie superalgebras. [ABSTRACT FROM AUTHOR]