Diagonal Entries Research Articles

THE NON-COPRIME GRAPHS OF UPPER UNITRIANGULAR MATRIX GROUPS OVER THE RING OF INTEGER MODULO WITH PRIME ORDER AND THEIR TOPOLOGICAL INDICES

In its application graph theory is widely applied in various fields of science, including scheduling, transportation, industry, and structural chemistry, such as topological indexes. The study of graph theory is also widely applied as a form of representation of algebraic structures, including groups. One form of graph representation that has been studied is non-coprime graphs. The upper unitriangular matrix group is a form of group that can be represented in graph form. This group consists of upper unitriangular matrices, which are a special form of upper triangular matrix with entries in a ring R and all main diagonal entries have a value of one. In this research, we look for the form of a non-coprime graph from the upper unitriangular matrix group over a ring of prime modulo integers and several topological indexes, namely the Harmonic index, Wiener index, Harary index, and First Zagreb index. The findings of this research indicate that the structure of the graph and the general formula for the Harmonic index, Wiener index, Harary index, and First Zagreb index were successfully obtained.

Complex Eigenvalue Analysis of a Linear Pencil: An Experimental Study

This paper is concerned with a finite-dimensional example of a linear pencil which leads to a class of non-self-adjoint matrices. We consider the linear pencil $H_c-\lambda L$, where $H_c$ is a tri-diagonal matrix with a constant parameter $c$ on the main diagonal and off-diagonal entries equal to one, and $L$ is a diagonal matrix whose elements decrease linearly from one to minus one. In general, the spectra of operator polynomials may contain non-real eigenvalues as well as real eigenvalues. Nevertheless, they exhibit certain patterns. Our aim in this research is to carry out a variety of numerical investigation on the eigenvalues so as to understand the eigenvalue behaviour of such pencils from different points of view. In accordance with our numerical findings, a series of conjectures are offered and various heuristics has been discussed.

The Markov chain embedding problem in a one-jump setting

Abstract The embedding problem of Markov chains examines whether a stochastic matrix $\mathbf{P} $ can arise as the transition matrix from time 0 to time 1 of a continuous-time Markov chain. When the chain is homogeneous, it checks if $ \mathbf{P}=\exp{\mathbf{Q}}$ for a rate matrix $ \mathbf{Q}$ with zero row sums and non-negative off-diagonal elements, called a Markov generator. It is known that a Markov generator may not always exist or be unique. This paper addresses finding $ \mathbf{Q}$ , assuming that the process has at most one jump per unit time interval, and focuses on the problem of aligning the conditional one-jump transition matrix from time 0 to time 1 with $ \mathbf{P}$ . We derive a formula for this matrix in terms of $ \mathbf{Q}$ and establish that for any $ \mathbf{P}$ with non-zero diagonal entries, a unique $ \mathbf{Q}$ , called the ${\unicode{x1D7D9}}$ -generator, exists. We compare the ${\unicode{x1D7D9}}$ -generator with the one-jump rate matrix from Jarrow, Lando, and Turnbull (1997), showing which is a better approximate Markov generator of $ \mathbf{P}$ in some practical cases.

Transition of the semiclassical resonance widths across a tangential crossing energy-level

We consider a 1D 2×2 matrix-valued operator (1.1) with two semiclassical Schrödinger operators on the diagonal entries and small interactions on the off-diagonal ones. When the two potentials cross at a turning point with contact order n, the corresponding two classical trajectories at the crossing level intersect at one point in the phase space with contact order 2n. Below this level, they have no intersection, which suggests exponentially small widths of resonances (see e.g., [1,2]), while above this level, on the contrary, they intersect at two points, which implies a polynomial order of the widths as proved in [3]. We prove that the transition of the resonance widths near the crossing level is described in terms of a generalized Airy function. This result generalizes [4] to the tangential crossing and [3] to the crossing at a turning point. Our approach is based on the computation of the microlocal transfer matrix at the crossing point between the incoming and outgoing microlocal solutions.

Randomized methods for computing joint eigenvalues, with applications to multiparameter eigenvalue problems and root finding

AbstractIt is well known that a family of $${n}\times {n}$$ n × n commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the n joint eigenvalues of the family. In this work, we consider the task of numerically computing approximations to such joint eigenvalues for a family of (nearly) commuting matrices. This task arises, for example, in solvers for multiparameter eigenvalue problems and systems of multivariate polynomials, which are our main motivations. We propose and analyze a simple approach that computes eigenvalues as one-sided or two-sided Rayleigh quotients from eigenvectors of a random linear combination of the matrices in the family. We provide some analysis and numerous numerical examples, showing that such randomized approaches can compute semisimple joint eigenvalues accurately and lead to improved performance of existing solvers.

Разработка трудноизвлекаемых запасов угля мобильными горнодобывающими комплексами без постоянного присутствия людей в зоне ведения горных работ

Today, millions of tons of coil remain in the subsoil and are written off as losses. These include hard-to-recover reserves unfit for mining with traditional high-performance means of integrated mechanization. One of the all-purpose technical solutions to the problem of developing such reserves can be technologies using robotic mobile mining equipment based on a heading-and-winning machine and a self-propelled car. Flowcharts for coal deposit development have been designed for seams unfit for mining with integrated mechanization means (medium thick seams, thick seams, flat and steeply inclined seams). These include open-pit and underground mining of marginal reserves from the side of the pit wall and limited reserves from the daytime surface performed by mobile mechanization means. During the operation of these means, the following process operations are repeated: coal breaking, loading into a self-propelled car, transportation along a preset trajectory, unloading into a reloader or other transport vehicle, or a warehouse on the bench. These operations can be robotized, which will increase both the safety and efficiency of coal mining. In order to perform coal-mining operations without constant attendance of maintenance staff, the existing mechanization means must be retrofitted with machine vision and a remote control system. Moreover, a robotized control system for the set of equipment must be developed to resolve the following problems. For the heading-and-winning machine: transportation (feed) along a straight line; transportation along the preset curve (formation of a diagonal entry); coal breaking at the preset thickness; coal breaking within coal-rock boundaries or along a preset cross-sectional contour; loading of broken coal into a self-propelled car; and control of the car loading level. For a self-propelled car: transportation along a preset trajectory from the loading point to the unloading point; control and management of the completeness of loading (unloading).

A linear-algebraic derivation of the Lorentz transformation

In order to explain the Lorentz transformation, it is advantageous to use its eigencoordinates rather than Cartesian coordinates. In this manner, the matrix can be diagonalized. Moreover, the diagonal entries have a direct physical meaning—permitting their determination without the need of algebraic equations. This yields a proof that is easier to remember and reproduce.

Robust Direction-of-Arrival Estimation in the Presence of Outliers and Noise Nonuniformity

In direction-of-arrival (DOA) estimation with sensor arrays, the background noise is usually modeled to be uncorrelated uniform white noise, such that the related algorithms can be greatly simplified by making use of the property of the noise covariance matrix being a diagonal matrix with identical diagonal entries. However, this model can be easily violated by the nonuniformity of sensor noise and the presence of outliers that may arise from unexpected impulsive noise. To tackle this problem, we first introduce an exploratory factor analysis (EFA) model for DOA estimation in nonuniform noise. Then, to deal with the outliers, a generalized extreme Studentized deviate (ESD) test is applied for outlier detection and trimming. Based on the trimmed data matrix, a modified EFA model, which belongs to weighted least-squares (WLS) fitting problems, is presented. Furthermore, a monotonic convergent iterative reweighted least-squares (IRLS) algorithm, called the iterative majorization approach, is introduced to solve the WLS problem. Simulation results show that the proposed algorithm offers improved robustness against nonuniform noise and observation outliers over traditional algorithms.

Stability and stabilization of discrete-time linear compartmental switched systems via Markov chains

The stabilizing switching signal design of discrete-time linear compartmental switched systems (DT-LCSSs) has been heretofore unsolved. It has been proven that a DT-LCSS is stabilizable if and only if it is stabilizable by a periodic switching signal. However, it still needs to be determined whether the period of a stabilizing switching signal can be confined within a bound. Moreover, the existing design method for stabilizing periodic switching signals requires the diagonal entries of system matrices of all subsystems to be strictly positive. In this study, we propose a novel approach to solve this problem completely. We construct a discrete-time Markov chain for a given DT-LCSS, termed the associated Markov chain, and prove the equivalence of stability and stabilizability between the DT-LCSS and the associated Markov chain. Based on this, verifiable necessary and sufficient conditions for stability and stabilizability are derived. Especially, the period of a stabilizing switching signal for an n-dimensional DT-LCSS can always be chosen within the bound n2−n+1. We propose a state-independent stabilizing switching signal design method for general stabilizable DT-LCSSs. We also prove the equivalence between stabilizability by state-independent switching laws and stabilizability by state-dependent switching laws. A state-dependent global stabilizing switching signal design method is also proposed. Additionally, the proposed results are applied to the consensus analysis of discrete-time leader–follower multi-agent systems with switching communication digraphs. The effectiveness of the theoretical results is demonstrated by examples.

Quantum geometrodynamics revived: II. Hilbert space of positive definite metrics

Abstract This paper represents the second in a series of works aimed at reinvigorating the quantum geometrodynamics program. Our approach introduces a lattice regularization of the hypersurface deformation algebra, such that each lattice site carries a set of canonical variables given by the components of the spatial metric and the corresponding conjugate momenta. In order to quantize this theory, we describe a representation of the canonical commutation relations that enforces the positivity of the operators q_{ab} s^a s^b for all choices of s. Moreover, symmetry of q_{ab} and p^{ab} is ensured. This reflects the physical requirement that the spatial metric should be a positive definite, symmetric tensor. To achieve this end, we resort to the Cholesky decomposition of the spatial metric into upper triangular matrices with positive diagonal entries. Moreover, our Hilbert space also carries a representation of the vielbein fields and naturally separates the physical and gauge degrees of freedom. Finally, we introduce a generalization of the Weyl quantization for our representation. We want to emphasize that our proposed methodology is amenable to applications in other fields of physics, particularly
in scenarios where the configuration space is restricted by complicated relationships among the degrees of freedom.

Exact relations between the conductivities and their connection to the chemical composition of QCD matter

We present exact relations between the diffusion coefficients or conductivities, κqq′/T=σqq′, of strongly interacting matter. We show that once the diagonal entries are known in two different charge representations, the off-diagonal coefficients are functions of the diagonal entries once isospin symmetry applies. As an important example, we infer the conductivities on the basis of available calculations from lattice quantum chromodynamics and argue that these computations suffer under the approximations made to achieve them. Further, we argue that the representation of the conductivities with respect to the conserved quark flavors may deliver more insight into the chemical composition of strongly interacting matter. Published by the American Physical Society 2024

Entries of the bottleneck matrices of maximal outerplanar graphs

In this paper, we consider the entries of the bottleneck matrices of maximal outerplanar graphs. We recall patterns from [Kirkland SJ, Neumann M, Shader BL. Characteristic vertices of weighted trees via Perron values. Linear Multilinear Algebra. 1996;40:311–325.] involving the entries of bottleneck matrices for trees. We first establish similar patterns involving the diagonal entries of bottleneck matrices for maximal outerplanar graphs. We then investigate patterns involving the off-diagonal entries of bottleneck matrices for maximal outerplanar graphs and establish similar results to those found in [Kirkland SJ, Neumann M, Shader BL. Characteristic vertices of weighted trees via Perron values. Linear Multilinear Algebra. 1996;40:311–325.] for trees. We close with examples illustrating our results.

The controllability and structural controllability of Laplacian dynamics

In this paper, classic controllability and structural controllability under two protocols are investigated. For classic controllability, the multiplicity of eigenvalue zero of general Laplacian matrix L ∗ is shown to be determined by the sum of the numbers of zero circles, identical nodes and opposite pairs; however, it is always simple for the Laplacian L with diagonal entries in absolute form. For a fixed structurally balanced topology, the controllable subspace is proved to be invariant even if the antagonistic weights are selected differently under the corresponding protocol with L. For a graph expanded from a star graph rooted from a single leader, the dimension of controllable subspace is two under the protocol associated with L ∗ . In addition, the system is structurally controllable under both protocols if and only if the topology without unaccessible nodes is connected. As a reinforcing case of structural controllability, strong structural controllability requires the system to be controllable for any choice of weights. The connection between parent nodes and child nodes affects strong structural controllability because it determines the linear relationship of the control information from parent nodes. This discovery is a major factor in establishing the sufficient conditions on strong structural controllability for multi-agent systems under both protocols, rather than for complex networks, about latter results are already abundant. Finally, we present a sufficient and necessary condition to identify the strong structural controllability. Moreover, a method to construct a strongly structurally controllable graph is proposed.

Positive vectors, pairwise comparison matrices and directed Hamiltonian cycles

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.

The volume of an isocanted cube is a determinant*

In any dimension d ≥ 2 , we give exact volume formulas of two mutually polar dual convex d-polytopes. The primal body is called isocanted cube of dimension d, depending on two real parameters 0 < a < ℓ . The limit case a = 0 yields a d-cube of edge-length ℓ. We prove that the volume of such a body is the determinant of the matrix of order d having diagonal entries equal to ℓ and a elsewhere. We also compute the volume of the polar dual body, getting a rational expression in ℓ , a , homogeneous of degree − d with rational coefficients. Isocanted cubes are origin-symmetric zonotopes. Zonoids (defined as the limits of families of zonotopes) satisfy the Mahler conjecture; in particular, zonotopes do. Nonetheless, we confirm (by elementary methods) that the Mahler conjecture holds for isocanted cubes.

On non-bipartite graphs with strong reciprocal eigenvalue property

Let G be a simple connected graph and A(G) be the adjacency matrix of G. A diagonal matrix with diagonal entries ±1 is called a signature matrix. If A(G) is nonsingular and X=SA(G)−1S−1 is entrywise nonnegative for some signature matrix S, then X can be viewed as the adjacency matrix of a unique weighted graph. It is called the inverse of G, denoted by G+. A graph G is said to have the reciprocal eigenvalue property (property(R)) if A(G) is nonsingular, and 1λ is an eigenvalue of A(G) whenever λ is an eigenvalue of A(G). Further, if λ and 1λ have the same multiplicity for each eigenvalue λ, then G is said to have the strong reciprocal eigenvalue property (property (SR)). It is known that for a tree T, the following conditions are equivalent: a) T+ is isomorphic to T, b) T has property (R), c) T has property (SR) and d) T is a corona tree (it is a tree which is obtained from another tree by adding a new pendant at each vertex).Studies on the inverses, property (R) and property (SR) of bipartite graphs are available in the literature. However, their studies for the non-bipartite graphs are rarely done. In this article, we study the inverse and property (SR) for non-bipartite graphs. We first introduce an operation, which helps us to study the inverses of non-bipartite graphs. As a consequence, we supply a class of non-bipartite graphs for which the inverse graph G+ exists and G+ is isomorphic to G. It follows that each graph G in this class has property (SR).

Differentiable microstructures design via anisotropic thermal diffusion

We propose a novel method to design differentiable microstructures. Central to our algorithm is a new representation of the mapping from the parameters to microstructures, formulated as the anisotropic thermal diffusion. A metric field governs the anisotropic diffusion. The metric associated with each point is represented as a 2 × 2 symmetric positive definite matrix that becomes the design variable. To alleviate the difficulties caused by symmetric positive definite constraints, we perform the singular value decomposition of the metric matrix so that the design variable includes a rotation angle and a diagonal matrix. Then, the positive definiteness is converted to requiring the two diagonal entries of the diagonal matrix to be positive, which is easier to deal with. The effectiveness of our algorithm is demonstrated through evaluations and comparisons over various examples.

A note on heat kernel of graphs

Consider a simple undirected connected graph G, with D(G) and A(G) representing its degree and adjacency matrices, respectively. Furthermore, L(G)=D(G)−A(G) is the Laplacian matrix of G, and Ht=exp⁡(−tL(G)) is the heat kernel (HK) of G, with t>0 denoting the time variable. For a vertex u∈V(G), the uth element of the diagonal of the HK is defined as Ht(u,u)=(exp⁡(−tL(G)))uu=∑k=0∞((−tL(G))k)uuk!, and HE(G)=∑i=1ne−tλi=∑u=1nHt(u,u) is the HK trace of G, where λ1,λ2,⋯,λn denote the eigenvalues of L(G). This study provides new computational formulas for the HK diagonal entries of graphs using an almost equitable partition and the Schur complement technique. We also provide bounds for the HK trace of the graphs.

Dominant eigenvalue and universal winners of digraphs

Let A be a nonnegative irreducible matrix of order n and Ai(t) be the matrix obtained by increasing its ith diagonal entry by a positive number t. An index i∈[n]={1,…,n} is called a universal winner for A, if, letting ρ(⋅) denote the spectral radius, ρ(Ai(t))≥ρ(Aj(t)) for j∈[n] and for t∈(0,∞).Let G be a strongly connected digraph with vertex set [n]. Let WG be the set of all nonnegative matrices whose underlying graph structure is G. We say a vertex u structurally dominates another vertex v in G, if ρ(Au(t))≥ρ(Av(t)) for all t∈(0,∞) and for all A∈WG. We characterize the class of digraphs G that do not have a vertex that structurally dominates all other vertices. We say two vertices u and v structurally tie if they structurally dominate each other in G. We supply an equivalent graph theoretic condition for the structural tying of two vertices in G. Let S⊆[n] be nonempty. We characterize the class of strongly connected digraphs G with vertex set [n] such that S is the set of universal winners for each A∈WG. We also characterize the strongly connected digraphs whose vertex set can be partitioned into subsets P1,P2,…,Pk such that vertices inside a part Pi structurally tie with each other and vertices of Pi structurally dominate vertices of Pj strictly (without tying) for i<j.

Triple perturbed consistent matrix and the efficiency of its principal right eigenvector

Let A be a pairwise comparison matrix obtained from a consistent one by perturbing three entries above the main diagonal, x,y,z, and the corresponding reciprocal entries, in a way that there is a submatrix of size 2 containing the three perturbed entries and not containing a diagonal entry. In this paper we describe the relations among x,y,z with which A always has its principal right eigenvector efficient. Previously, and only for a few cases of this problem, R. Fernandes and S. Furtado (2022) proved the efficiency of the principal right eigenvector of A. In this paper, we continue to use the strong connectivity of a certain digraph associated with A and its principal right eigenvector to characterize the vector efficiency. For completeness, we show that the existence of a sink in this digraph is equivalent to the inefficiency of the principal right eigenvector of A.