Bulletin of the Malaysian Mathematical Sciences Society最新文献

The usual theory of negative type (and p-negative type) is heavily dependent on an embedding result of Schoenberg, which states that a metric space isometrically embeds in some Hilbert space if and only if it has 2-negative type. A generalisation of this embedding result to the setting of bi-lipschitz embeddings was given by Linial, London and Rabinovich. In this article we use this newer embedding result to define the concept of distorted p-negative type and extend much of the known theory of p-negative type to the setting of bi-lipschitz embeddings. In particular we show that a metric space ((X,d_{X})) has p-negative type with distortion C ((0le p<infty ), (1le C<infty )) if and only if ((X,d_{X}^{p/2})) admits a bi-lipschitz embedding into some Hilbert space with distortion at most C. Analogues of strict p-negative type and polygonal equalities in this new setting are given and systematically studied. Finally, we provide explicit examples of these concepts in the bi-lipschitz setting for the bipartite graphs (K_{m,n}).

In this paper, with the help of potential function, we extend the classical Brezis–Lieb lemma on Euclidean space to graphs, which can be applied to the following Kirchhoff equation

on a connected locally finite graph (G=(mathbb {V}, mathbb {E})), where (b, lambda >0), (p>2) and V(x) is a potential function defined on (mathbb {V}). The purpose of this paper is four-fold. First of all, using the idea of the filtration Nehari manifold technique and a compactness result based on generalized Brezis–Lieb lemma on graphs, we prove that there admits a positive solution (u_{lambda , b} in E_lambda ) with positive energy for (b in (0, b^*)) when (2<p<4). In the sequel, when (p geqslant 4), a positive ground state solution (w_{lambda , b} in E_lambda ) is also obtained by using standard variational methods. What’s more, we explore various asymptotic behaviors of (u_{lambda , b}, w_{lambda , b} in E_lambda ) by separately controlling the parameters (lambda rightarrow infty ) and (b rightarrow 0^{+}), as well as jointly controlling both parameters. Finally, we utilize iteration to obtain the (L^{infty })-norm estimates of the solution.

In this paper, we study the discrete Kirchhoff–Choquard equation

where (a,,b>0), (alpha in (0,3)) are constants and (R_{alpha }) is the Green’s function of the discrete fractional Laplacian that behaves as the Riesz potential. Under some suitable assumptions on V and f, we prove the existence of nontrivial solutions and ground state solutions respectively by variational methods.

This article presents a self-adjustable branch-and-bound algorithm for globally solving a class of linear multiplicative programming problems (LMP). In this algorithm, a self-adjustable branching rule is introduced and it can continuously update the upper bound for the optimal value of LMP by selecting suitable branching point under certain conditions, which differs from the standard bisection rule. The proposed algorithm further integrates the linear relaxation program and the self-adjustable branching rule. The dependability and robustness of the proposed algorithm are demonstrated by establishing the global convergence. Furthermore, the computational complexity of the proposed algorithm is estimated. Finally, numerical results validate the effectiveness of the self-adjustable branching rule and demonstrate the feasibility of the proposed algorithm.

Let (nge 2) be an integer. The Grimaldi graph G(n) is defined by taking the elements of the set ({ 0, ldots , n-1 }) as vertices. Two distinct vertices x and y are adjacent in G(n) if and only if (gcd (x+y, n) =1). In this paper, we examine the Betti numbers of the edge ideals of these graphs and their complements.

Holditch’s theorem is a classical geometrical result on the areas of a given closed curve and another one, its Holditch curve, which is constructed as the locus of a fixed point dividing a chord of constant length that moves with its endpoints over the given curve and that returns back to its original position after some full revolution. Holditch curves have already been studied from the parametric point of view, although numerical methods and approximations are often necessary for their computation. In this paper, implicit equations of Holditch curves of algebraic curves are studied. The implicit equations can be simply found from the computation of a resultant of two polynomials. With the same techniques, Holditch curves of two initial algebraic curves are also considered. Moreover, the use of implicit equations allows to find new and explicit parameterizations of non-trivial Holditch curves, such as in the case of having an ellipse as an initial curve.

In this paper, we establish uniform asymptotic formulas for the rank and crank statistics of cubic partitions. This partly improves upon the asymptotic results established by Kim–Kim–Nam in 2016.

The Ramsey numbers (R(T_n,W_8)) are determined for each tree graph (T_n) of order (nge 7) and maximum degree (Delta (T_n)) equal to either (n-4) or (n-5). These numbers indicate strong support for the conjecture, due to Chen, Zhang and Zhang and to Hafidh and Baskoro, that (R(T_n,W_m) = 2n-1) for each tree graph (T_n) of order (nge m-1) with (Delta (T_n)le n-m+2) when (mge 4) is even.

Let (mathcal {H}) be the class of harmonic functions (f=h+overline{g}) in the unit disk (mathbb {D}:={zin mathbb {C}:|z|<1}), where h and g are analytic in (mathbb {D}). In 2020, N. Ghosh and V. Allu introduced the class (mathcal {P}_{mathcal {H}}^0(M)) of normalized harmonic mappings defined by (mathcal {P}_{mathcal {H}}^0(M)={f=h+overline{g}in mathcal {H}: text {Re}(zh''(z))>-M+|zg''(z)|;text {with};M>0, g'(0)=0, zin mathbb {D}}). In this paper, we investigate various geometric properties such as starlikeness, convexity, convex combination and convolution for functions in the class (mathcal {P}_{mathcal {H}}^0(M)). Furthermore, we determine the sharp Bohr–Rogosinski radius, improved Bohr radius and refined Bohr radius for the class (mathcal {P}_{mathcal {H}}^0(M)).

In the continuous setting, Morrey spaces have been studied extensively, especially since the late 1960s. Meanwhile, Morrey sequence spaces, which are also known as discrete Morrey spaces, have only been developed by Gunawan et al. since 2018. In this article, we extend some known results on their inclusion properties and their (lack of) uniform nonsquareness to mixed Morrey double-sequence spaces, i.e. Morrey double-sequence spaces equipped with a mixed norm. As in the calculation of three geometric constants of Morrey spaces by Gunawan et al. in 2019, we also compute three geometric constants, namely Von Neumann-Jordan constant, James constant, and Dunkl-Williams constant for mixed Morrey double-sequence spaces. These constants measure uniformly nonsquareness of any Banach space. Through the values of the three constants, we reveal that mixed Morrey double-sequence spaces are not uniformly nonsquare. A relation between mixed Morrey double-sequence spaces and mixed Morrey spaces is also discussed.