Bulletin of the Malaysian Mathematical Sciences Society最新文献

An interval in which a given graph has no eigenvalues is called a gap interval. We show that for any real number R greater than (frac{1}{2}(-1+sqrt{2})), there exist infinitely many threshold graphs with gap interval (0, R). We provide a new recurrence relation for computing the characteristic polynomial of the threshold graphs and based on it, we conclude that the sequence of the least positive (resp. largest negative) eigenvalues of a certain sequence of threshold graphs is convergent. In some particular cases, we compute the limit points.

In this paper, we explore some properties of the dual Wills functional, which are part of the dual Brunn–Minkowski theory. We give the upper and lower bounds for the dual Wills functional in terms of the 1-th dual volume of star bodies. Moreover, an inequality that is associated with the section of convex bodies for isotropic measures is presented.

The problem of injective coloring in graphs can be revisited through two different approaches: coloring the two-step graphs and vertex partitioning of graphs into open packing sets, each of which is equivalent to the injective coloring problem itself. Taking these facts into account, we observe that the injective coloring lies between graph coloring and domination theory. We make use of these three points of view in this paper so as to investigate the injective coloring of some well-known graph products. We bound the injective chromatic number of direct and lexicographic product graphs from below and above. In particular, we completely determine this parameter for the direct product of two cycles. We also give a closed formula for the corona product of two graphs.

Abstract

This article is concerned with the stability, ergodicity and mixing of invariant measures of a class of stochastic lattice plate equations with nonlinear damping driven by a family of nonlinear white noise. The polynomial growth drift term has an arbitrary order growth rate, and the diffusion term is a family of locally Lipschitz continuous functions. By modifying and improving several energy estimates of the solutions uniformly for initial data when time is large enough, we prove that the noise intensity union of all invariant measures of the stochastic equation is tight on (ell ^2times ell ^2) . Then, we show that the weak limit of every sequence of invariant measures in this union must be an invariant measure of the corresponding limiting equation under the locally Lipschitz assumptions on the drift and diffusion terms. Under some globally Lipschitz conditions on the drift and diffusion terms, we also prove that every invariant measure of the stochastic equation must be ergodic and exponentially mixing in the pointwise and Wasserstein metric sense.

In this research, the constant type of neutral delay Volterra integro-differential equations (NDVIDEs) are currently being resolved by applying the proposed technique in numerical analysis namely, two-point two off-step point block multistep method (2OBM4). This new technique is being applied in solving NDVIDE, identified as a hybrid block multistep method, developed using Taylor series interpolating polynomials. To complete the algorithm, two alternative numerical approaches are introduced to resolve the integral and differential parts of the problems. Note that the differentiation is approximated by the divided difference formula while the integration is interpolated using composite Simpson’s rule. The proposed method has been analysed thoroughly in terms of its order, consistency, zero stability and convergence. The suitable stability region for 2OBM4 in solving NDVIDE has been constructed and the stability region is built based on the stability polynomial obtained. Consequently, numerical results are presented to demonstrate the effectiveness of the proposed 2OBM4.

Abstract

In this paper, we consider the following discrete nonlinear Schrödinger equation $$begin{aligned} left{ begin{array}{l} -Delta u_{n}+V_{n}u_{n}-omega u_{n}=f_{n}(u_{n}), quad n in mathbb {Z}, lim _{|n| rightarrow infty } u_{n}=0, end{array}right. end{aligned}$$ with non-periodic potentials. We obtain the existence of nontrivial solutions of this equation with various nonlinearities by using Morse theory. In this way, we need to compute the critical groups of the corresponding action functional of the equation. To the best of our knowledge, there is no result focusing on this issue in the literature.

In this paper, making use of the coefficient inequality for subordinate functions on the unit disc (mathbb {U}) in (mathbb {C}), we establish some refinements of the Fekete and Szegö inequalities for a class of holomorphic mappings related to spirallike mappings on bounded starlike circular domains in (mathbb {C}^n). The results presented here would generalize some known results.

In this paper, we deal with the mean value theorem for tempered fractional operators, some properties with monotone functions and the Taylor theorem for Caputo tempered fractional derivative. Furthermore, we present a geometric interpretation of the Riemann-Liouville tempered fractional integral as a shadow on the wall.

Let H and K be infinite dimensional separable complex Hilbert spaces and B(K, H) the algebra of all bounded linear operators from K into H. Let (Ain B(H)) and (Bin B(K)). We denote by (M_C) the operator acting on (Hoplus K) of the form (M_C=left( begin{array}{cc}A&{}C 0&{}B end{array}right) ). In this paper, we give necessary and sufficient conditions for (M_C) to be an upper semi-Fredholm operator with (n(M_C)>0) and (hbox {ind}(M_C)<0) for some left invertible operator (Cin B(K,H)). Meanwhile, we discover the relationship between (n(M_C)) and n(A) during the exploration. And we also describe all left invertible operators (Cin B(K,H)) such that (M_C) is an upper semi-Fredholm operator with (n(M_C)>0) and (hbox {ind}(M_C)<0).

In this paper, we consider a discrete Hamiltonian system on nonnegative integers, and using Sylvester’s inertia indices theory, we construct maximal subspaces on which the Hermitian form has a certain sign. After constructing nested ellipsoids, we introduce a lower bound for the number of linearly independent summable-square solutions of the discrete equation. Finally, we provide a limit-point criterion.