Search

The electrophoretic deposition of a mixture of copper-manganese spinel oxide matrix with perovskite oxide dispersion phase and subsequent two-step sintering was investigated as a means of obtaining a new spinel/perovskite composite coating for application in metallic interconnects for solid oxide cell (SOC) technologies. The effect of a number of factors was examined, including electrophoretic deposition parameters such as solvent type, time, voltage, and dispersant content as well as different heat treatment conditions on the quality of the matrix of the coating material. A solution of ethanol-acetone (25/75 v/v) and 2.0 g L−1 of iodine dispersant with a deposition time of 60 s and a voltage of 60 V provided the thickest, most uniform coating as deposited. SEM observations showed that a dense coating of Cu–Mn spinel was obtained after 8 h of reduction at 1000 °C followed by 6 h of oxidation at 850 °C. Selected preparation parameters were applied to deposit composite coatings with a Cu–Mn spinel matrix and the addition of LNF, LSC and LSCF perovskites. [Display omitted] • EOH25/75ACT was determined to be the most suitable mixture for the deposition. • Mass gain of the deposit generally correlates with particle mobility. • Reduction in Ar–10%H 2 at 1000 °C and re-synthesis in air at 850 °C is optimal. • Morphology of composite coatings depend on type of perovskite additive. [ABSTRACT FROM AUTHOR]

For a matrix A which satisfies Crouzeix's conjecture, we construct several classes of matrices from A for which the conjecture will also hold. We discover a new link between cyclicity and Crouzeix's conjecture, which shows that Crouzeix's Conjecture holds in full generality if and only if it holds for the differentiation operator on a class of analytic functions. We pose several open questions, which if proved, will prove Crouzeix's conjecture. We also begin an investigation into Crouzeix's conjecture for symmetric matrices and in the case of 3 × 3 matrices, we show Crouzeix's conjecture holds for symmetric matrices if and only if it holds for analytic truncated Toeplitz operators. [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]

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 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]

This paper aims to quantitatively analyze the relationship between forces acting on the tool tip and tool movement during drilling operations. The study encompasses axial and lateral vibrations superimposed on the nominal tool movement, arising from rigid body motion (rotational and axial velocities). Specifically, only forces attributed to the cutting process are considered, excluding considerations of indentation forces around the chisel edge. The research adopts a generalized approach, spanning from tool measurements to establishing the force model. The investigation involves measuring cutting forces and correlating them with the varying rake and inclination angles of the drill's cutting edges. An analytical model is proposed to describe the distribution of all local force components along drill edges, considering the evolution of forces and geometry. The dynamic coefficient matrix is evaluated by using the identified cutting coefficient and tool geometry. Validation of the proposed methodology is demonstrated through drilling experiments on Ti6Al4V alloy, utilizing three solid carbide drills with distinct geometries. The proposed procedure allows complete identification of the dynamic characteristics from the measurements taken at the entrance stage of hole drilling operation. Moreover, the influence of tool geometry on cutting coefficients and dynamic coefficient matrices are discussed. • A novel method is developed to achieve minimized experimental effort to identify cutting coefficients in drilling process. • Experiments with industrially relevant tools validated the accurate predictions of transient drilling axial force and torque. • The method is extended to inversely obtain the dynamic coefficient matrix, for the first time, from experimentally measured forces. • Effects of geometry and dynamic interactions were demonstrated using three solid carbide drills with distinct geometries. [ABSTRACT FROM AUTHOR]

By adding a scalar parameter to the classical Lyapunov matrix inequality, the underlying structure turns to be richer: Partial order of stability sets is introduced. This is then used to improve estimates of trajectories associated with differential inclusions. Technically, the spectrum of all matrices, satisfying a given Lyapunov inequality, lies within a special disk in the right-half plane: Under inversion such a disk is mapped onto itself. As a by-product, it is shown that these disks are a natural tool to understanding the Matrix-Sign-Function iteration scheme, used in matrix computations. Hyper-Lyapunov inclusions are formulated through Matrix-Quadratic-Form Inequalities and so are the analogous Hyper-Stein sets of matrices whose spectrum lies within sub -unit disks. [ABSTRACT FROM AUTHOR]

We study a matrix analog of the Erdős-Falconer distance problems in vector spaces over finite fields. We provide lower bounds for the cardinality of a subset of the matrix algebra over a finite field such that its distance set is the whole matrix algebra. There arises an interesting analysis of certain quadratic matrix Gauss sums. [ABSTRACT FROM AUTHOR]

In this paper the construction of C k basis functions on meshes with hanging nodes for hp -finite element methods is proposed. The construction is limited to axis-parallel, rectangular meshes, which can, however, be of any dimension. The basis functions are composed by using hierarchical tensor product shape functions which are defined on patches of so-called constraining nodes and combine univariate Hermite interpolation polynomials and Gegenbauer polynomials. The paper discusses how to integrate the constructed C k basis functions into the assembling process of stiffness matrices and load vectors by using connectivity matrices. Numerical examples illustrate the applicability of the proposed concept. [ABSTRACT FROM AUTHOR]

In this paper we construct new families of extremal copositive matrices in arbitrary dimension by an algorithmic procedure. Extremal copositive matrices are organized in relatively open subsets of real-algebraic varieties, and knowing a particular such matrix A allows in principle to obtain the variety in which A is embedded by solving the corresponding system of algebraic equations. We show that if A is a matrix associated to a so-called COP-irreducible graph with stability number equal 3, then by a trigonometric transformation these algebraic equations become linear and can be solved by linear algebra methods. We develop an algorithm to construct and solve the corresponding linear systems and give examples where the variety contains singularities at the initial matrix A. For the cycle graph C 7 we completely characterize the part of the variety which consists of copositive matrices. [ABSTRACT FROM AUTHOR]

In this paper we generalize the transition probability matrix for a random walk on a finite network by defining the transition probabilities through a symmetric M -matrix. Usually, the walker jumps from a vertex to a neighbor according to the probabilities given by the adjacency matrix. Moreover, we can find in the literature the relation between random walks and the normalized laplacian or the combinatorial laplacian that are singular and symmetric M -matrices. Our model takes into consideration not only the probability of transitioning given by the adjacency matrix but also some added probability that depends on a node property. This also includes the probability of remaining in each node, when the M -matrix is not singular. The nodes importance is taking into account by considering the lower eigenvalue and its associated eigenfunction for the given M -matrix. We give expressions for the mean first passage time and Kemeny's constant for such a random walks in terms of 1-inverses of the considered M -matrix. [ABSTRACT FROM AUTHOR]

In this paper we will look at the well known interlacing problem, but here we consider the result for Hermitian matrices in the Minkowski space, an indefinite inner product space with one negative square. More specific, we consider the n × n matrix A = [ J u − u ⁎ a ] with a ∈ R , J = J ⁎ and u ∈ C n − 1. Then A is H -selfadjoint with respect to the matrix H = I n − 1 ⊕ (− 1). The canonical form for the pair (A , H) plays an important role and the sign characteristic coupled to the pair is also discussed. [ABSTRACT FROM AUTHOR]

A well known method to solve the Polynomial Eigenvalue Problem (PEP) is via linearization. That is, transforming the PEP into a generalized linear eigenvalue problem with the same spectral information and solving such linear problem with some of the eigenvalue algorithms available in the literature. Linearizations of matrix polynomials are usually defined using unimodular transformations. In this paper we establish a connection between the standard definition of linearization for matrix polynomials introduced by Gohberg, Lancaster and Rodman and the notion of polynomial system matrix introduced by Rosenbrock. This connection gives new techniques to show that a matrix pencil is a linearization of the corresponding matrix polynomial arising in a PEP. [ABSTRACT FROM AUTHOR]

In 1876 H. J. S. Smith defined an LCM matrix as follows: let S = { x 1 , x 2 , ... , x n } be a set of positive integers with x 1 < x 2 < ⋯ < x n. The LCM matrix [ S ] on the set S is the n × n matrix with lcm (x i , x j) as its ij entry. During the last 30 years singularity of LCM matrices has interested many authors. In 1992 Bourque and Ligh ended up conjecturing that if the GCD closedness of the set S (which means that gcd ⁡ (x i , x j) ∈ S for all i , j ∈ { 1 , 2 , ... , n }), suffices to guarantee the invertibility of the matrix [ S ]. However, a few years later this conjecture was proven false first by Haukkanen et al. and then by Hong. It turned out that the conjecture holds only on GCD closed sets with at most 7 elements but not in general for larger sets. However, the given counterexamples did not give much insight on why does the conjecture fail exactly in the case when n = 8. This situation was later improved in a couple of articles, where a new lattice theoretic approach was introduced (the method is based on the fact that because the set S is assumed to be GCD closed, the structure (S , |) actually forms a meet semilattice). For example, it has been shown that in the case when the set S has 8 elements and the matrix [ S ] is singular, there is only one option for the semilattice structure of (S , |) , namely the cube structure. Since the cases n ≤ 8 have been thoroughly studied in various articles, the next natural step is to apply the methods to the case n = 9. This was done by Altınışık and Altıntaş as they consider the different lattice structures of (S , |) with nine elements that can result in a singular LCM matrix [ S ]. However, their investigation leaves two open questions, and the main purpose of this presentation is to provide solutions to them. We shall also give a new lattice theoretic proof for a result referred to as Sun's conjecture, which was originally proven by Hong via number theoretic approach. [ABSTRACT FROM AUTHOR]

In this paper, we primarily focus on the eigenvalues of the adjacency matrix and Seidel matrix of chain graphs, referred to as eigenvalues and Seidel eigenvalues of these graphs, respectively. Firstly, we utilize the characteristic polynomial of the adjacency matrix of a chain graph to construct infinite pairs of non-isomorphic cospectral chain graphs. Next, we determine the inertia of the Seidel matrix of a chain graph and establish an interval that does not contain the Seidel eigenvalues of chain graphs. Lastly, we characterize chain graphs with distinct Seidel eigenvalues. [ABSTRACT FROM AUTHOR]

We consider invertible, row diagonally dominant real matrices and give inequalities on their minors and diagonal entries of their inverses. A very special case is that all diagonal entries of an inverse, of a row stochastic, row diagonally dominant and invertible matrix, are at least 1, with strict inequality at least when the dominance is strict. This was conjectured in international trade theory in economics and motivated the present work (though much more is proven). Some of the results generalize previously known facts for M -matrices. [ABSTRACT FROM AUTHOR]

We study bivariate orthogonal polynomials associated with an inner product that satisfies a symmetry property such that it is invariant when both variables are interchanged. Under that hypothesis, the structure of the polynomial vectors of the orthogonal polynomial systems is described by using centrosymmetric matrices. Also, we prove that the coefficient matrices of the three-term relations, one for each variable, are connected by a reverse operation over matrices. Finally, several particular cases and examples are analysed. [ABSTRACT FROM AUTHOR]

Although eigenvectors belong to the core of linear algebra, relatively few closed-form expressions exist, which we bundle and discuss here. A particular goal is their interpretation for graph-related matrices, such as the adjacency matrix of an undirected, possibly weighted graph. [ABSTRACT FROM AUTHOR]

The Matrix Waring problem is if we can write every matrix as a sum of k -th powers. Here, we look at the same problem for triangular matrix algebra T n (F q) consisting of upper triangular matrices over a finite field. We prove that for all integers k , n ≥ 1 , there exists a constant C (k , n) , such that for all q > C (k , n) , every matrix in T n (F q) is a sum of three k -th powers. Moreover, if −1 is k -th power in F q , then for all q > C (k , n) , every matrix in T n (F q) is a sum of two k -th powers. We make use of Lang-Weil estimates about the number of solutions of equations over finite fields to achieve the desired results. [ABSTRACT FROM AUTHOR]

Decoupling control is a commonly employed technique for achieving high precision in multiple-input multiple-output (MIMO) motion control systems. A static decoupling matrix, which can be determined using geometric construction, is widely used due to its practicality and simplicity. However, inaccurate geometric parameters will lead to a coarse decoupling matrix, and result in interactions among the system axes and performance deterioration. To tackle this challenge, various attempts have been undertaken to calibrate the decoupling matrix. Data-driven on-line approaches have gained considerable attention for their ability to calibrate the decoupling matrix without interrupting the normal operation of the system. This paper presents a data-driven approach to calibrate the decoupling matrix for MIMO and linear time invariant (LTI) systems. Through some reasonable assumptions, a calibrated static decoupling matrix can be derived to improve the performance of the system. Moreover, considering the inevitable presence of measurement noise, the consistency of the proposed method has been analyzed. As a result, the instrument variable is introduced in the improved method to eliminate the impact of the measurement noise. Finally, the effectiveness and practicality of the proposed method are demonstrated through both numerical simulations and experiments carried out on an ultraprecision wafer stage. • The performance of MIMO systems declines if mechanical parameters are inaccurate. • Traditional calibration of these parameters is complicated. • Optimizing the decoupling matrix directly with measurement data. • A calibrated static decoupling matrix with consistency is obtained. [ABSTRACT FROM AUTHOR]

The Q -index of a graph G is the largest eigenvalue of its Q -matrix Q (G) = D (G) + A (G) , where D (G) and A (G) are the diagonal matrix of vertex degrees and the adjacency matrix of G , respectively. Let 3 K 3 denote the graph consisting of three vertex-disjoint triangles. A graph is called 3 K 3 -free if it does not contain 3 K 3 as a subgraph. In this paper, we present a sharp upper bound on the Q -index of 3 K 3 -free graphs of order n ≥ 453 , and characterize the unique extremal graph which attains the bound. [ABSTRACT FROM AUTHOR]

The generalized arithmetic–geometric matrix of a simple connected graph G = (V , E) is defined to be the | V | × | V | matrix whose i j -entry is d i α + d j α 2 d i α d j α if v i v j ∈ E , and 0 otherwise, where α is an arbitrary real number and d i is the degree of v i ∈ V. This matrix is a general form of the arithmetic–geometric matrix (α = 1) and the extended adjacency matrix (α = 2), both of which have been well studied in spectral graph theory and chemical graph theory. In this paper, we focus on the spectral radius ρ a g α of the generalized arithmetic–geometric matrix of connected graphs. Some chemical applications of ρ a g α are explored, and some extremal results on ρ a g α are obtained; in particular, the connected graphs (including trees) having the maximum and minimum ρ a g α are determined. [ABSTRACT FROM AUTHOR]

The determinants of { ± 1 } -matrices are calculated via the oriented hypergraphic Laplacian and summing over incidence generalizations of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on their hyperedge containment. Every non-edge-monic family is shown to contribute a net value of 0 to the Laplacian, while each edge-monic family is shown to sum to the absolute value of the determinant of the original incidence matrix. Simple symmetries are identified as well as their relationship to Hadamard's maximum determinant problem. Finally, the entries of the incidence matrix are reclaimed using only the signs of an adjacency-minimal set of cycle-covers from an edge-monic family. [ABSTRACT FROM AUTHOR]

The energy of a simple graph G is defined as the sum of the absolute values of eigenvalues of the adjacency matrix of G. For a complex matrix M the spread of M is the maximum absolute value of the differences between any two eigenvalues of M. Thus if λ 1 , ... , λ n are the eigenvalues of M , then the spread of M is max 1 ≤ i , j ≤ n ⁡ | λ i − λ j |. The spread of a graph G is defined as the spread of its adjacency matrix and is denoted by s (G). The inertia of G is an integer triple (n + , n − , n 0) specifying the numbers of positive, negative and zero eigenvalues of the adjacency matrix of G. In this paper we find some bounds for energy of graphs in terms of some parameters of graphs such as rank, inertia and spread of graphs. We find some bounds for spread of graphs and matrices that improve the previous bounds. In particular, we show that if G is a graph with m edges and inertia (n + , n − , n 0) , then s (G) ≥ 2 m (n + + n −) n + n − and the equality holds if and only if G = r K s ∪ t K 1 or G = r K p , ... , p ︸ q ∪ t K 1 or G = r 1 K a 1 , b 1 ∪ ⋯ ∪ r h K a h , b h , for some non-negative integers r , s , t , p , q and r 1 , a 1 , b 1 , ... , r h , a h , b h such that a 1 b 1 = ⋯ = a h b h , p ≥ 2 and q ≥ 3. [ABSTRACT FROM AUTHOR]

We continue the study of truncated matrix-valued moment problems begun in [12]. Let q ∈ N. Suppose that (X , X) is a measurable space and E is a finite-dimensional vector space of measurable mappings of X into H q , the Hermitian q × q matrices. A linear functional Λ on E is called a moment functional if there exists a positive H q -valued measure μ on (X , X) such that Λ (F) = ∫ X 〈 F , d μ 〉 for F ∈ E. In this paper a number of special topics on the truncated matricial moment problem are treated. We restate a result from [11] to obtain a matricial version of the flat extension theorem. Assuming that X is a compact space and all elements of E are continuous on X we characterize moment functionals in terms of positivity and obtain an ordered maximal mass representing measure for each moment functional. The set of masses of representing measures at a fixed point and some related sets are studied. The class of commutative matrix moment functionals is investigated. We generalize the apolar scalar product for homogeneous polynomials to the matrix case and apply this to the matricial truncated moment problem. [ABSTRACT FROM AUTHOR]

The eccentricity matrix ɛ (G) of a strongly connected digraph G is defined as ɛ (G) i j = d (v i , v j) , if d (v i , v j) = min { e + (v i) , e − (v j) } , 0 , otherwise. , where e + (v i) = max { d (v i , v j) ∣ v j ∈ V (G) } is the out-eccentricity of the vertex v i of G , and e − (v j) = max { d (v i , v j) ∣ v i ∈ V (G) } is the in-eccentricity of the vertex v j of G. The eigenvalue of ɛ (G) with the largest modulus is called the eccentricity spectral radius of G. In this paper, we obtain lower bounds for the eccentricity spectral radius among all join digraphs with a fixed dichromatic number. We also give upper bounds for the eccentricity spectral radius of some special join digraphs with a fixed dichromatic number. [ABSTRACT FROM AUTHOR]

A linear Lie algebra is splittable if it contains the semisimple and nilpotent parts of each element. It is early known that a solvable linear Lie algebra g is splittable if and only if g = a + n , where a is an abelian subalgebra of g composed of semisimple elements and n is the ideal of all nilpotent matrices of g. In this paper, using elementary linear algebra we give a direct proof of the theorem and related results. Besides, we determine the structure of linear Lie algebras composed of semisimple or nilpotent elements. [ABSTRACT FROM AUTHOR]

The Gau-Wu number is an important matrix invariant describing the geometry of the numerical range. In this work, the group of non-singular linear preservers of the Gau-Wu number is completely characterized. [ABSTRACT FROM AUTHOR]

Let f (x , y) > 0 be a real symmetric function. For a connected graph G , the weight of edge v i v j is equal to the value f (d i , d j) , where d i is the degree of vertex v i. The degree-based weighted adjacency matrix is defined as A f (G) , in which the (i , j) -entry is equal to f (d i , d j) if v i v j is an edge of G and 0 otherwise. In this paper, we first give some bounds of the weighted adjacency eigenvalue λ 1 (A f (G)) in terms of λ 1 (A f (H)) , where H is obtained from G by some kinds of graph operations, including deleting vertices, deleting an edge and subdividing an edge, and examples are given to show that bounds are tight. Second, we obtain some bounds for the largest weighted adjacency eigenvalue λ 1 (A f (G)) of irregular weighted 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]

Given a pair of real symmetric matrices A , B ∈ R n × n with nonzero patterns determined by the edges of any pair of chosen graphs on n vertices, we consider an inverse eigenvalue problem for the structured matrix C = [ A B I O ] ∈ R 2 n × 2 n. We conjecture that C can attain any spectrum that is closed under conjugation. We use a structured Jacobian method to prove this conjecture for A and B of orders at most 4 or when the graph of A has a Hamilton path, and prove a weaker version of this conjecture for any pair of graphs with a restriction on the multiplicities of eigenvalues of C. [ABSTRACT FROM AUTHOR]

In the Analytic Hierarchy Process (AHP) the efficient vectors for a pairwise comparison matrix (PC matrix) are based on the principle of Pareto optimal decisions. To infer the efficiency of a vector for a PC matrix we construct a directed Hamiltonian cycle of a certain digraph associated with the PC matrix and the vector. We describe advantages of using this process over using the strong connectivity of the digraph. As an application of our process we find efficient vectors for a PC matrix, A, obtained from a consistent one by perturbing three entries above the main diagonal and the corresponding reciprocal entries, in a way that there is a square submatrix of A of order 2 containing three of the perturbed entries and not containing a diagonal entry of A. For completeness, we include examples showing conditions under which, when deleting a certain entry of an efficient vector for the square matrix A of order n, we have a non-efficient vector for the corresponding square principal submatrix of order n-1 of A. [ABSTRACT FROM AUTHOR]

In this article, taking a Fiedler's result on the spectrum of a matrix formed from two symmetric matrices as a motivation, we deduce a more general result on the eigenvalues of a matrix, which form from n symmetric matrices. As an important application, we obtain the adjacency spectra, Laplacian spectra and signless Laplacian spectra of a graph with a particular almost equitable partition. [ABSTRACT FROM AUTHOR]

We study products of functions evaluated at self-adjoint polynomials in deterministic matrices and independent Wigner matrices; we compute the deterministic approximations of such products and control the fluctuations. We focus on minimizing the assumption of smoothness on those functions while optimizing the error term with respect to N , the size of the matrices. As an application, we build on the idea that the long-time Heisenberg evolution associated to Wigner matrices generates asymptotic freeness as first shown in [9]. More precisely given P a self-adjoint non-commutative polynomial and Y N a d -tuple of independent Wigner matrices, we prove that the quantum evolution associated to the operator P (Y N) yields asymptotic freeness for large times. [ABSTRACT FROM AUTHOR]

In this research, we propose a novel and fast computational technique for solving a class of space-time fractional-order linear and non-linear partial differential equations. Caputo-type fractional derivatives are considered. The proposed method is based on the operational and pseudo-operational matrices for the fractional-order Lagrange polynomials. To carry out the method, first, we find the integer and fractional-order operational and pseudo-operational matrix of integration. The collocation technique and obtained operational and pseudo-operational matrices are then used to generate a system of algebraic equations by reducing the given space-time fractional differential problem. The resultant algebraic system is then easily solved by Newton's iterative methods. The upper bound of the fractional-order operational matrix of integration is also provided, which confirms the convergence of fractional-order Lagrange polynomial's approximation. Finally, some numerical experiments are conducted to demonstrate the applicability and usefulness of the suggested numerical scheme. [ABSTRACT FROM AUTHOR]

• An s-stage one-step method is developed for highly nonlinear problems. • Theorems for convergence and stability presented for authenticity. • Highly accurate solution attained as compared to the existing results. • Computational order of convergence investigated in temporal and spatial direction. The behavior of nonlinear dynamical systems arising in mathematical physics through numerical tools is a challenging task for researchers. In this context, an efficient semi-spectral method is proposed and applied to observe the robust solutions for the mathematical physics problems. Firstly, the space variable is approximated by the Vieta-Lucas polynomials and then the s -stage one-step method is applied to discretize the temporal variable which transfers the problem in the form C n + 1 = C n + Δ t ϕ (x , t , C n , F (u n)). Novel operational matrices of integer order are developed to replace the spatial derivative terms presented in the discussed problem. Related theorems are included in the study to validate the approach mathematically. The proposed semi-spectral schemes convert the considered nonlinear problem to a system of linear algebraic equations which is easier to tackle. We also accomplish an investigation on the error bound and convergence to confirm the mathematical formulation of the computational algorithm. To show the accuracy and effectiveness of the suggested computational method numerous test problems, such as the advection-diffusion problem, generalized Burger-Huxley, sine-Gordon, and modified KdV–Burgers' equations are considered. An inclusive comparative examination demonstrates the currently suggested computational method in terms of credibility, accuracy, and reliability. Moreover, the coupling of the spectral method with the fourth-order Runge-Kutta method seems outstanding to handle the nonlinear problem to examine the precise smooth and non-smooth solutions of physical problems. The computational order of convergence (COC) is computed numerically through numerous simulations of the proposed schemes. It is found that the proposed schemes are in exponential order of convergence in the spatial direction and the COC in the temporal direction validates the studies in the literature. [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]

For an arbitrary q ≥ 2 , we find an upper bound for the dimension of a subalgebra of the full matrix algebra M n (K) over an arbitrary field K satisfying the identity [ [ x 1 , y 1 ] , z 1 ] ⋅ [ [ x 2 , y 2 ] , z 2 ] ⋅ ⋯ ⋅ [ [ x q , y q ] , z q ] = 0 , and we show that this upper bound is sharp by presenting an example in block triangular form of a subalgebra of M n (K) with dimension equal to the obtained upper bound. We apply this result to Lie solvable algebras of index 2, i.e., algebras satisfying the identity [ [ x 1 , y 1 ] , [ x 2 , y 2 ] ] = 0. To be precise, for n ≤ 4 , we find the sharp upper bound for the dimension of a Lie solvable subalgebra of M n (K) of index 2, and for n > 4 , we obtain the relatively tight (at least for small values of n > 4) interval [ 2 + ⌊ 3 n 2 8 ⌋ , 2 + ⌊ 5 n 2 12 ⌋ ] for the maximum dimension of a Lie solvable subalgebra of M n (K) of index 2, the exact value of which is not known. [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]

The Four point condition (4PC henceforth) is a well known condition characterising distances in trees T. Let w , x , y , z be four vertices in T and let d x , y denote the distance between vertices x , y in T. The 4PC condition says that among the three terms d w , x + d y , z , d w , y + d x , z and d w , z + d x , y the maximum value equals the second maximum value. We define an ( n 2 ) × ( n 2 ) sized matrix Max4PC T from a tree T where the rows and columns are indexed by size-2 subsets. The entry of Max4PC T corresponding to the row indexed by { w , x } and column { y , z } is the maximum value among the three terms d w , x + d y , z , d w , y + d x , z and d w , z + d x , y. In this work, we determine basic properties of this matrix like rank, give an algorithm that outputs a family of bases, and find the determinant of Max4PC T when restricted to our basis. We further determine the inertia and the Smith Normal Form (SNF) of Max4PC T. [ABSTRACT FROM AUTHOR]

We study the image of SU (n) under the diagonal product map. Using only elementary tools of linear algebra, analysis and general topology, we find the analytical formula for the boundary of this image and find all special unitary matrices whose diagonal product lies on that boundary. [ABSTRACT FROM AUTHOR]

Haagerup's proof of the non commutative little Grothendieck inequality raises some questions on the commutative little inequality, and it offers a new result on scalar matrices with non negative entries. The theory of completely bounded maps may be used to show that the commutative Grothendieck inequality follows from the little commutative inequality, and that this passage may be given a geometric form as a relation between a pair of compact convex sets of positive matrices, which, in turn, characterizes the little constant k G C . [ABSTRACT FROM AUTHOR]

This paper aims to study a Kaup-Newell type matrix eigenvalue problem with four potentials, based on a specific matrix Lie algebra, and construct an associated soliton hierarchy of combined derivative nonlinear Schrödinger (NLS) equations, within the zero curvature formulation. The Liouville integrability of the resulting soliton hierarchy is shown by exploring its hereditary recursion operator and bi-Hamiltonian formulation. The first nonlinear example provides an integrable model consisting of combined derivative NLS equations with two arbitrary constants. [ABSTRACT FROM AUTHOR]

In this paper, we study the q -distance matrix for a distance-regular graph and show that the q -distance matrix of a distance-regular graph with classical parameters (D , q , α , β) has exactly three distinct eigenvalues, of which one is zero. Moreover, we study distance-regular graphs whose q -distance matrix has exactly one positive eigenvalue. [ABSTRACT FROM AUTHOR]

Except for the empty graph, we show that the orthogonal matrix X of the adjacency matrix A determines that adjacency matrix completely, but not always uniquely. The proof relies on interesting properties of the Hadamard product Ξ = X ∘ X. As a consequence of the theory, we show that irregular co-eigenvector graphs exist only if the number of nodes N ≥ 6. Co-eigenvector graphs possess the same orthogonal eigenvector matrix X , but different eigenvalues of the adjacency matrix. Co-eigenvector graphs are the dual of co-spectral graphs, that share all eigenvalues of the adjacency matrix, but possess a different orthogonal eigenvector matrix. We deduce general properties of co-eigenvector graph and start to enumerate all co-eigenvector graphs on N = 6 and N = 7 nodes. Finally, we list many open problems. [ABSTRACT FROM AUTHOR]

The first order eigenvalue perturbation theory for structure-preserving perturbations of real or complex symplectic matrices started in [21] is continued and completed. While in [21] all eigenvalues were considered that qualitatively behave similarly under structure-preserving and structure-ignoring perturbations, the focus of this paper is on one particular case when the behavior under structure-preserving perturbations differs significantly from the one under structure-ignoring perturbations. This case occurs for the exceptional eigenvalues ±1 and is caused by restrictions in the possible Jordan structure of symplectic matrices. The main result gives asymptotic expansions of specific perturbed eigenvalues that are not yet covered by the results from [21]. While the main theorem requires that the rank of the perturbation is equal to the rank of its first order term, it is also discussed what can be said if this hypothesis is not satisfied. [ABSTRACT FROM AUTHOR]

We consider cyclic algebras, Milnor's symbol algebras, and certain graded algebra structures on them. We classify these gradings with respect to both isomorphism and equivalence relations. Some of them induce gradings on matrix algebras, which we also classify. As an application, we obtain the classification of all group gradings on the algebra M p (F) , where p is a prime number and F is an arbitrary field. [ABSTRACT FROM AUTHOR]

Let F be a field of characteristic zero and let m ≥ 2 be an integer. In this paper, we prove that if a group grading on U T m (F) admits a graded involution then this grading is a coarsening of a Z ⌊ m 2 ⌋ -grading on U T m (F) and the graded involution is equivalent to the reflection or symplectic involution on U T m (F) , this grading is called the finest grading on U T m (F). Furthermore, if m ≤ 4 the algebra U T m (F) with the finest grading satisfies no non-trivial monomial identities. For the finest grading, a finite basis for the (Z ⌊ m 2 ⌋ , ⁎) -identities is exhibited with the reflection and symplectic involutions and the asymptotic growth of the (Z ⌊ m 2 ⌋ , ⁎) -codimensions is determined. As a consequence, we prove that for any G -grading on U T m (F) and any graded involution the (G , ⁎) -exponent is m. Finally, we study the algebra U T 3 (F). For this algebra, there are, up to equivalence, two non-trivial gradings that admit a graded involution: the canonical Z -grading and the Z 2 -grading induced by (0 , 1 , 0). We determine a basis for the (Z 2 , ⁎) -identities and we compute the codimension sequence for the (Z 2 , ⁎) -graded identities for U T 3 (F). [ABSTRACT FROM AUTHOR]