Bulletin of the Malaysian Mathematical Sciences Society最新文献

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.

In this paper, we study the radial epiderivative notion for nonconvex functions, which extends the (classical) directional derivative concept. The paper presents new definition and new properties for this notion and establishes relationships between the radial epiderivative, the Clarke’s directional derivative, the Rockafellar’s subderivative and the directional derivative. The radial epiderivative notion is used to establish new regularity conditions without convexity conditions. The paper presents explicit formulations for computing the radial epiderivatives in terms of weak subgradients and vice versa. We also present an iterative algorithm for approximate computing of radial epiderivatives and show that the algorithm terminates in a finite number of iterations. The paper analyzes necessary and sufficient conditions for global optimums in nonconvex optimization via the radial epiderivatives. We formulate a necessary and sufficient condition for a global descent direction for radially epidifferentiable nonconvex functions. All the properties and theorems presented in this paper are illustrated and interpreted on examples.

In this paper, we introduce a combinatorial proof approach for a theorem regarding the upper Banach density of a subset S of ({mathbb {N}}) and present a finite IP-set revision of the theorem originally established by Li, Tu, and Ye.

We investigate the blowup conditions to the Cauchy problem for a semilinear wave equation with scale-invariant damping, mass and general nonlinear memory term (see Eq. (1.1) in the Introduction). We first establish a local (in time) existence result for this problem by Banach’s fixed point theorem, where Palmieri’s decay estimates on the solution to the corresponding linear homogeneous equation play an essential role in the proof. We then formulate a blowup result for energy solutions by applying the iteration argument together with the test function method.

In this paper, we consider several improved regularity criteria to the 3D micropolar fluid equations. In particular, we prove regularity criteria that only require control of the middle eigenvalue of strain tensor in critical Besov spaces, which can be regarded as improvement and extension of results very recently obtained.

Let G be a graph and ({mathcal {F}}) be a family of connected graphs. A subset S of G is called an ({mathcal {F}})-isolating set if (G-N[S]) contains no member in ({mathcal {F}}) as a subgraph, and the minimum cardinality of an ({mathcal {F}})-isolating set of graph G is called the ({mathcal {F}})-isolation number of graph G, denoted by (iota (G,{mathcal {F}})). For simplicity, let (iota (G,{K_{1,k+1}})=iota _k(G)). Thus, (iota _1(G)) is the cardinality of a smallest set S such that (G-N[S]) consists of (K_1) and (K_2) only. In this paper, we prove that for any claw-free cubic graph G of order n, (iota _1(G)le frac{n}{4}). The bound is sharp.

Existence of nodal (i.e., sign changing) solutions and constant-sign solutions for quasilinear elliptic equations involving convection–absorption terms are presented. A location principle for nodal solutions is obtained by means of constant-sign solutions whose existence is also derived. The proof is chiefly based on sub-supersolutions technique together with monotone operator theory.

Abstract

In this paper, we establish several characterizations of planar symmetric maps, such as preserving Euclidean circles, preserving k-Apollonius circles for some (kin (0,infty )) , and preserving geometric moduli of all pairs of disjoint continua in the extended complex plane.

Abstract

We provide novel lower bounds on the Betti numbers of Vietoris–Rips complexes of hypercube graphs of all dimensions and at all scales. In more detail, let (Q_n) be the vertex set of (2^n) vertices in the n-dimensional hypercube graph, equipped with the shortest path metric. Let (textrm{VR}(Q_n;r)) be its Vietoris–Rips complex at scale parameter (r ge 0) , which has (Q_n) as its vertex set, and all subsets of diameter at most r as its simplices. For integers (r<r') the inclusion (textrm{VR}(Q_n;r)hookrightarrow textrm{VR}(Q_n;r')) is nullhomotopic, meaning no persistent homology bars have length longer than one, and we therefore focus attention on the individual spaces (textrm{VR}(Q_n;r)) . We provide lower bounds on the ranks of homology groups of (textrm{VR}(Q_n;r)) . For example, using cross-polytopal generators, we prove that the rank of (H_{2^r-1}(textrm{VR}(Q_n;r))) is at least (2^{n-(r+1)}left( {begin{array}{c}n r+1end{array}}right) ) . We also prove a version of homology propagation: if (qge 1) and if p is the smallest integer for which (textrm{rank}H_q(textrm{VR}(Q_p;r))ne 0) , then (textrm{rank}H_q(textrm{VR}(Q_n;r)) ge sum _{i=p}^n 2^{i-p} left( {begin{array}{c}i-1 p-1end{array}}right) cdot textrm{rank}H_q(textrm{VR}(Q_p;r))) for all (n ge p) . When (rle 3) , this result and variants thereof provide tight lower bounds on the rank of (H_q(textrm{VR}(Q_n;r))) for all n, and for each (r ge 4) we produce novel lower bounds on the ranks of homology groups. Furthermore, we show that for each (rge 2) , the homology groups of (textrm{VR}(Q_n;r)) for (n ge 2r+1) contain propagated homology not induced by the initial cross-polytopal generators.

In this paper, we use the Mordukhovich/limiting subdifferential to establish approximate optimality conditions/approximate duality theorems/approximate saddle point theorems for multiobjective optimization problems with infinite constraints. The main results obtained in this paper are new and extend some corresponding known results. Some examples are given for the illustration of our results.