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We model the folding of ordinary paper via piecewise isometries R2→R3. The collection of crease lines and vertices in the unfolded paper is called the crease pattern. Our results generalize the previously known necessity conditions from the more restrictive case of folding paper flat (into R2); if the crease pattern is foldable, then the product (in a non-intuitive order) of the associated rotational matrices is the identity matrix. This condition holds locally in a multiple vertex crease pattern and can be adapted to a global condition. Sufficiency conditions are significantly harder, and are not known except in the two-dimensional single-vertex case. [Copyright &y& Elsevier]

Abstract: The Szeged index extends the Wiener index for cyclic graphs by counting the number of vertices on both sides of each edge and sum these counts. Klavzar et al. [S. Klavzar, A. Rajapakse, I. Gutman, The Szeged and the Wiener index of graphs, Appl. Math. Lett. 9 (5) (1996) 45–49] provided an exact formula for computing Szeged index of product of graphs. In this paper, we apply a matrix method to obtain exact formulae for computing the Szeged index of join and composition of graphs. The join and composition of the vertex PI index of graphs are also computed. [Copyright &y& Elsevier]

Abstract: A tree which has exactly one vertex of degree greater than two is said to be starlike. In spite of seemingly simple structure of these trees, not much is known about their spectral properties. In this paper, we introduce a generalization of the notion of cospectrality called -cospectrality which turns out to be useful in constructing cospectral graphs. Based on this, we construct cospectral mates for some starlike trees. We also present a set of necessary and sufficient conditions for divisibility of the characteristic polynomial of a starlike tree by the characteristic polynomial of a path. [Copyright &y& Elsevier]

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]

A linear group is called unisingular if every element of it has eigenvalue 1. In this paper we develop some general machinery for the study of unisingular irreducible linear groups. A motivation for the study of such groups comes from several sources, including algebraic geometry, Galois theory, finite group theory and representation theory. In particular, a certain aspect of the theory of abelian varieties requires the knowledge of unisingular irreducible subgroups of the symplectic groups over the field of two elements, and in this paper we concentrate on this special case of the general problem. A more special but important question is that of the existence of such subgroups in the symplectic groups of particular degrees. We answer this question for almost all degrees 2 n < 250 , specifically, the question remains open only 7 values of n. [ABSTRACT FROM AUTHOR]

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]

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]

Abstract In this paper, we generalize classical von Neumann symmetrization of two-person zero-sum games to general linear games. We use this symmetrization to show that for a given general linear game there exists a symmetric linear game whose solution yields a solution to the underlying linear game. We define symmetric linear games of type gRPS (generalized Rock–Paper–Scissors) and prove that a symmetric linear game has a pure strategy equilibrium if and only if it is not a gRPS game. From this we deduce that a completely mixed symmetric linear game is gRPS. [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]