Volume 24, Issue 1

Accretive partial transpose (APT) matrices have been recently defined, as a natural extension of positive partial transpose (PPT) matrices. In this paper, we discuss further properties of APT matrices in a way that extends some of those properties known for PPT matrices. Among many results, we show that if A , B , X are n  ×  n complex matrices such that A , B are sectorial with sector angle α for some α ∈ [0, π /2), and if f : (0, ∞ ) → (0, ∞ ) is a certain operator monotone function such that cos 2 ( α ) f ( A ) X X * cos 2 ( α ) f ( B ) $\left[\begin{matrix}\hfill {\mathrm{cos}}^{2}\left(\alpha \right)f\left(A\right)\hfill & \hfill X\hfill \\ \hfill {X}^{{\ast}}\hfill & \hfill {\mathrm{cos}}^{2}\left(\alpha \right)f\left(B\right)\hfill \end{matrix}\right]$ is APT, Then f ( A ) ∇ t f ( B ) X X * f ( A ∇ t B ) $\left[\begin{matrix}\hfill f\left(A\right){\nabla }_{t}f\left(B\right)\hfill & \hfill X\hfill \\ \hfill {X}^{{\ast}}\hfill & \hfill f\left(A{\nabla }_{t}B\right)\hfill \end{matrix}\right]$ is APT for any 0 ≤ t ≤ 1, where A ∇ t B is the weighted arithmetic mean of A and B .

For an edge-colored graph, the Maximum Colored Cut problem is to find a bipartition maximizing the number of colors in edges going across the bipartition. This problem is a generalization of the classical Max-Cut problem. Let G be an edge-colored graph with p colors, and let mcc ( G ) be the maximum number of colors in a cut of G . In this work, we show that (1) if G is a complete graph containing no properly colored K 4 − ${K}_{4}^{-}$ s, where K 4 − ${K}_{4}^{-}$ is the graph obtained from the complete graph on four vertices by deleting an edge, then mcc( G ) ≥ 2 p /3; (2) if G is a complete k -partitie graph ( k ≥ 3) containing no properly colored four-cycles, then mcc( G ) ≥ p − 1.

This study examines the spatial gradient estimation of solutions to a particular class of variational inequality problems that arise from the structure of nonlinear degenerate parabolic operators. Specifically, under the assumption that 2 ≤ p ≤ 2 ( 1 + 2 n ) $2\le p\le 2\left(1+2/n \right.\right)$ (where n denotes the spatial dimension), we derive an estimate for the spatial gradient of the solution within a local cylindrical domain, referred to as the L p energy estimate, which is complemented by an additional L 2 estimate.

In this paper, we consider a boundary problem for a p -biharmonic equation Δ Δ u p − 2 Δ u = a x u p − 2 u ⁡ ln u + b x u p − 2 u ${\Delta}\left({\left\vert {\Delta}u\right\vert }^{p-2}{\Delta}u\right)=a\left(x\right){\left\vert u\right\vert }^{p-2}u\mathrm{ln}\left\vert u\right\vert +b\left(x\right){\left\vert u\right\vert }^{p-2}u$ with a sign-changing logarithmic nonlinearity a x u p − 2 u ⁡ ln u $a\left(x\right){\left\vert u\right\vert }^{p-2}u\mathrm{ln}\left\vert u\right\vert $ and a polynomial nonlinearity b x u p − 2 u $b\left(x\right){\left\vert u\right\vert }^{p-2}u$ in a bounded domain. We derive a new result regarding the existence of solutions and illustrate how the weight functions a x $a\left(x\right)$ and b x $b\left(x\right)$ can affect the existence and multiplicity of solutions. Furthermore, the sign-changing weight function a x $a\left(x\right)$ , associated with the logarithmic nonlinearity a x u p − 2 u ⁡ ln u $a\left(x\right){\left\vert u\right\vert }^{p-2}u\mathrm{ln}\left\vert u\right\vert $ , plays a pivotal role in determining the existence and multiplicity of these solutions.

In this paper, the solvability of some initial-boundary value problems is considered for a nonlocal analogue of the degenerate parabolic equation. The inverse problems are studied for the case where it is necessary to find not only a solution to the equation but also the function in its right-hand side. The theorems on the existence and uniqueness of the solution to the considered problems are proved.

We demonstrate an analytical solution for the dynamic transversal vibrations of non-uniform beams on variable foundations. The exact expressions of eigenvalues and eigenfunctions are derived. The proposed dynamic model of the non-uniform beam with clamped conditions reduces to the dynamic model of the uniform beam with the same boundary conditions. It is shown that this reduction is associated with the isospectral properties of a non-uniform beam with clamped-clamped ends. The analytical solution is used to study the behavior of the beam. The analytical results obtained are illustrated by numerical examples.

In this article, we consider the following singular double phase problem with variable exponents and convolution term of the form: − T p ( x ) , q ( x ) α ( x ) ( u ) = λ s ( x ) u − γ ( x ) + ∫ Ω | u ( x ) | h ( x ) | x − y | μ ( x , y ) | u ( y ) | h ( x ) − 2 u ( y ) in Ω , u = 0 on ∂ Ω , $$\begin{cases}-{\mathcal{T}}_{p\left(x\right),q\left(x\right)}^{\alpha \left(x\right)}\left(u\right)=\lambda s\left(x\right){u}^{-\gamma \left(x\right)}+\left({\int }_{{\Omega}}\frac{\vert u\left(x\right){\vert }^{h\left(x\right)}}{\vert x-y{\vert }^{\mu \left(x,y\right)}}\right)\vert u\left(y\right){\vert }^{h\left(x\right)-2}u\left(y\right) \text{in} {\Omega},\hfill \\ u=0 \text{on} \partial {\Omega},\hfill \end{cases}$$ where the operator T p ( x ) , q ( x ) α ( x ) ( u ) : = div ∇ u p ( x ) − 2 ∇ u + α ( x ) ∇ u q ( x ) − 2 ∇ u ${\mathcal{T}}_{p\left(x\right),q\left(x\right)}^{\alpha \left(x\right)}\left(u\right) : =\text{div}\left({\left\vert \nabla u\right\vert }^{p\left(x\right)-2}\nabla u+\alpha \left(x\right){\left\vert \nabla u\right\vert }^{q\left(x\right)-2}\nabla u\right)$ is the double phase operator with variable exponents, Ω ⊂ R N ${\Omega}\subset {\mathbb{R}}^{N}$ is a bounded domain with smooth boundary ∂Ω, 0 ≤ α (⋅) ∈ L ∞ (Ω), λ is a positive real parameter. The functions s ( x ) ∈ C ( Ω ̄ ) $s\left(x\right)\in C\left(\bar{{\Omega}}\right)$ are positive with compact support in Ω, h : R N → R $h : {\mathbb{R}}^{N}\to \mathbb{R}$ and μ : R N × R N → R $\mu : {\mathbb{R}}^{N}{\times}{\mathbb{R}}^{N}\to \mathbb{R}$ are continuous functions. Under the suitable conditions, the existence of at least one weak solution is obtained for the above problem by using the Nehari manifold approach. The novelty of this paper is that this problem includes singular term and convolution term. Moreover, the emergence of p ( x ) and q ( x ) Laplacian operator makes the study of this problem more complicated and interesting.