Computational and Applied Mathematics最新文献

Computing the optimal low rank approximations of quaternion matrices is the key target in many quaternion matrix related problems including color images inpainting and recognition, which can be reconstructed by some dominant singular values of quaternion matrices. However, the singular value decomposition of large-scale quaternion matrices requires expensive storage and computational costs. In this paper, we propose an incremental quaternion singular value decomposition (IQSVD) method for a class of quaternion matrices, where the number of columns far exceeds the number of rows, to improve computing efficiency. What’s more, based on IQSVD, we consider the low rank quaternion matrix completion problem and design a proximal linearized minimization algorithm with convergence guarantee to solve it. Numerical experiments on synthetic data and real-world videos illustrate the efficiency of IQSVD and the proposed proximal linearized minimization algorithm involved IQSVD.

This work deals with the problem of multilinear principal component analysis (MPCA) and multilinear discriminant analysis (MDA), that solve for a tensor to tensor projection (TTP) using Einstein product. MPCA and MDA are considered as a higher-order extension of principal component analysis (PCA ) and linear discriminant analysis (LDA), respectively. MPCA seeks to find a low-dimensional representation that captures most of the variation present in the original data tensor. Whereas MDA seeks to find discriminative features that maximize the separation between classes, while preserving the multilinear structure. Specifically, we are interested in finding a projective tensor that maps the original data tensor onto a new lower-dimensional subspace. In this paper, we propose to solve the MPCA problem by employing the global Lanczos procedure via Einstein product for a fourth-order tensor, while solving the MDA problem by combining Newton method and global tensorial Lanczos method. The numerical experiments illustrate the use of these algorithms for face recognition problems, compression and classification.

Let G be a connected graph with adjacency matrix A(G) and distance matrix D(G). Let ({{,textrm{dist},}}(u,v)) denote the distance between the pair of vertices (u,vin V(G)), then the transmission ({{,textrm{trs},}}(u)) of vertex u is defined as (sum _{vin V(G)}{{,textrm{dist},}}(u,v)). Let ({{,textrm{trs},}}(G)) be the diagonal matrix whose diagonal elements are the transmissions of the vertices of G. And, let (deg (G)) be the diagonal matrix whose diagonal elements are the degrees of the vertices of G. In this paper we investigate the Smith normal form (SNF) and the spectrum of the matrices (D^{deg }_+(G):=deg (G)+D(G)), (D^{deg }(G):=deg (G)-D(G)), (A^{{{,textrm{trs},}}}_+(G):={{,textrm{trs},}}(G)+A(G)) and (A^{{{,textrm{trs},}}}(G):={{,textrm{trs},}}(G)-A(G)). In particular, we explore how good the SNF and the spectrum of these matrices are for determining graphs up to isomorphism. We found that the SNF of (A^{{{,textrm{trs},}}}) has an interesting behaviour when compared with other classical matrices. We note that the SNF of (A^{{{,textrm{trs},}}}) can be used to compute the structure of the sandpile group of certain graphs. We compute the SNF of (D^{deg }_+), (D^{deg }), (A^{{{,textrm{trs},}}}_+) and (A^{{{,textrm{trs},}}}) for several graph families. We prove that the SNF of (D^{deg }_+), (D^{deg }), (A^{{{,textrm{trs},}}}_+) and (A^{{{,textrm{trs},}}}) determine complete graphs. Finally, we derive some results about the spectrum of (D^{deg }) and (A^{{{,textrm{trs},}}}).

The connections between parameter surfaces enable the development of various geometric designs. However, these surfaces are generally connected along their boundaries in a rectangular domain. This study investigated the methods for connecting surfaces along a curve. To this end, we introduced two-variable degenerate functions and utilized their algebraic properties to characterize the form of the parameter surfaces for practical surface construction. The results were used to deform the surfaces along the curve. For application, we presented the examples of deformations using Bézier surfaces and extended them to general surfaces.

Consider a commutative ring with unity denoted as (mathscr {R}), and let (W(mathscr {R})) represent the set of non-unit elements in (mathscr {R}). The coannihilator graph of (mathscr {R}), denoted as (AG'(mathscr {R})), is a graph defined on the vertex set (W(mathscr {R})^*). This graph captures the relationships among non-unit elements. Specifically, two distinct vertices, x and y, are connected in (AG'(mathscr {R})) if and only if either (x notin xymathscr {R}) or (y notin xymathscr {R}), where (wmathscr {R}) denotes the principal ideal generated by (w in mathscr {R}). In the context of this paper, the primary objective is to systematically classify finite rings (mathscr {R}) based on distinct characteristics of their coannihilator graph. The focus is particularly on cases where the coannihilator graph exhibits a genus or crosscap of two. Additionally, the research endeavors to provide a comprehensive characterization of finite rings (mathscr {R}) for which the connihilator graph (AG'(mathscr {R})) attains an outerplanarity index of two.

In this paper, we present a second order parameter-uniform numerical method for a singularly perturbed Volterra delay-integro-differential equation on a Bakhvalov-type mesh. The equation is discretized by using the variable two-step backward differentiation formula of the first derivative term and the trapezoidal formula of the integral term. The stability and convergence of the numerical method in the discrete maximum norm are proved. Finally, the theoretical results are verified by some numerical experiments.

Let (Gamma ) be a simple finite graph with vertex set (V(Gamma )) and edge set (E(Gamma )). Let (mathcal {R}) be an equivalence relation on (V(Gamma )). The (mathcal {R})-super (Gamma ) graph (Gamma ^{mathcal {R}}) is a simple graph with vertex set (V(Gamma )) and two distinct vertices are adjacent if either they are in the same (mathcal {R})-equivalence class or there are elements in their respective (mathcal {R})-equivalence classes that are adjacent in the original graph (Gamma ). We first show that (Gamma ^{mathcal {R}}) is a generalized join of some complete graphs and using this we obtain the adjacency and Laplacian spectrum of conjugacy super commuting graphs and order super commuting graphs of dihedral group (D_{2n}; (nge 3)), generalized quaternion group (Q_{4m} ;(mge 2)) and the nonabelian group (mathbb {Z}_p rtimes mathbb {Z}_q) of order pq, where p and q are distinct primes with (q|(p-1)).

For a connected graph G, let A(G) be the adjacency matrix of G and D(G) be the diagonal matrix of the degrees of the vertices in G. The (A_{alpha })-matrix of G is defined as

The largest eigenvalue of (A_{alpha }(G)) is called the (A_{alpha })-spectral radius of G. In this article, we characterize the graphs with maximum (A_{alpha })-spectral radius among the class of unicyclic and bicyclic graphs of order n with fixed girth g. Also, we identify the unique graphs with maximum (A_{alpha })-spectral radius among the class of unicyclic and bicyclic graphs of order n with k pendant vertices.

Rough sets (RSs) and fuzzy sets (FSs) are designed to tackle the uncertainty in the data. By taking into account the control or reference parameters, the linear Diophantine fuzzy set (LD-FS) is a novel approach to decision making (DM), broadens the previously dominant theories of the intuitionistic fuzzy set (IFS), Pythagorean fuzzy set (PyFS), and q-rung orthopair fuzzy set (q-ROFS), and allows for a more flexible representation of uncertain data. A promising avenue for RS theory is to investigate RSs within the context of LD-FS, where LD-FSs are approximated by an intuitionistic fuzzy relation (IFR). The major goal of this article is to create a novel method of roughness for LD-FSs employing an IFR over dual universes. The notions of lower and upper approximations of an LD-FS are established by using an IFR, and some axiomatic systems are carefully investigated in detail. Moreover, a link between LD-FRSs and linear Diophantine fuzzy topology (LDF-topology) has been established. Eventually, based on lower and upper approximations of an LD-FS, several similarity relations are investigated. Meanwhile, we apply the recommended model of LD-FRSs over dual universes for solving the DM problem. Furthermore, a real-life case study is given to demonstrate the practicality and feasibility of our designed approach. Finally, we conduct a detailed comparative analysis with certain existing methods to explore the effectiveness and superiority of the established technique.

This paper introduces two new identification methods for linear quadratic (LQ) ordinal potential differential games (OPDGs). Potential games are notable for their benefits, such as the computability and guaranteed existence of Nash Equilibria. While previous research has analyzed ordinal potential static games, their applicability to various engineering applications remains limited. Despite the earlier introduction of OPDGs, a systematic method for identifying a potential game for a given LQ differential game has not yet been developed. To address this gap, we propose two identification methods to provide the quadratic potential cost function for a given LQ differential game. Both methods are based on linear matrix inequalities (LMIs). The first method aims to minimize the condition number of the potential cost function’s parameters, offering a faster and more precise technique compared to earlier solutions. In addition, we present an evaluation of the feasibility of the structural requirements of the system. The second method, with a less rigid formulation, can identify LQ OPDGs in cases where the first method fails. These novel identification methods are verified through simulations, demonstrating their advantages and potential in designing and analyzing cooperative control systems.