Abstract 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.

Abstract 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.

Abstract Let X X be a compact Riemann surface of genus g ≥ 2 g\ge 2 . The geometry of the moduli space ℳ ( Spin ( 8 , C ) ) {\mathcal{ {\mathcal M} }}\left({\rm{Spin}}\left(8,{\mathbb{C}})) of Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) -Higgs bundles over X X is of great interest both in algebraic geometry and mathematical physics. Consequently, several works have studied subvarieties of fixed points of this moduli space, especially those arising from the automorphism induced by the action of triality. In this work, fixed points of automorphisms of ℳ ( Spin ( 8 , C ) ) {\mathcal{ {\mathcal M} }}\left({\rm{Spin}}\left(8,{\mathbb{C}})) induced by outer automorphisms of Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) are studied. This is done by giving explicit descriptions of the spectral data of these fixed points induced by the Hitchin fibration, under certain technical conditions that must be required. Specifically, it is proved that stable Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) -Higgs bundles that admit nontrivial automorphisms reduce their structure group to a subgroup isomorphic to SL ( 2 , C ) 4 = SL ( 2 , C ) × SL ( 2 , C ) × SL ( 2 , C ) × SL ( 2 , C ) {\rm{SL}}{\left(2,{\mathbb{C}})}^{4}={\rm{SL}}\left(2,{\mathbb{C}})\times {\rm{SL}}\left(2,{\mathbb{C}})\times {\rm{SL}}\left(2,{\mathbb{C}})\times {\rm{SL}}\left(2,{\mathbb{C}}) . Subsequently, the spectral data of these reductions are described and the manner in which the outer automorphisms of Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) act on them is analyzed. The final application of these results allows a description of the mentioned fixed points through their spectral data.

Abstract We deal with the existence and multiplicity of positive solutions for differential systems depending on two parameters, λ 1 , λ 2 {\lambda }_{1},{\lambda }_{2} , subjected to periodic boundary conditions. We establish the existence of a continuous curve Γ \Gamma that separates the first quadrant into two disjoint unbounded open sets O 1 {{\mathcal{O}}}_{1} and O 2 {{\mathcal{O}}}_{2} . Specifically, we prove that the periodic system has no positive solutions if ( λ 1 , λ 2 ) ∈ O 1 \left({\lambda }_{1},{\lambda }_{2})\in {{\mathcal{O}}}_{1} , at least one positive solution if ( λ 1 , λ 2 ) ∈ Γ \left({\lambda }_{1},{\lambda }_{2})\in \Gamma , and at least two positive solutions if ( λ 1 , λ 2 ) ∈ O 2 \left({\lambda }_{1},{\lambda }_{2})\in {{\mathcal{O}}}_{2} . Our approach relies on the fixed point index theory and the method of lower and upper solutions.

Abstract Recall that weakly type B semigroups are generalized inverse semigroups on semi-abundant semigroups. The main aim of this article is to prove that the translational hull of a weakly type B semigroup is still a semigroup of the same type. Some characterizations and properties on the translational hull of weakly type B semigroups are obtained. As an application, some characterizations of the translational hull on weakly type B semigroups with E-properties are given.

Abstract Let G G be a simple graph of order n n with eigenvalues λ 1 ≥ λ 2 ≥ … ≥ λ n . {\lambda }_{1}\ge {\lambda }_{2}\ge \ldots \ge {\lambda }_{n}. The resolvent energy of G G is a spectrum-based graph invariant defined as ER ( G ) = ∑ i = 1 n ( n − λ i ) − 1 . {\rm{ER}}(G)={\sum }_{i=1}^{n}{(n-{\lambda }_{i})}^{-1}. In this work, we propose some new bounds for ER ( G ) {\rm{ER}}(G) . As a direct consequence of these bounds, we present some ( n , m ) (n,m) -type results for triangle-free graphs.

Abstract In this article, we introduce a new type of simplicial homology group called the chaotic simplicial homology group. We construct the structure of this group and obtain some types of conditional foldings and unfoldings restricted to the elements of the chaotic simplicial homology group. Also, we deduce the chaotic simplicial homology groups under the limit of foldings. We describe the variant and invariant of the chaotic simplicial homology group under the folding of a chaotic simplicial complex into itself. We use the chain of a commutative diagram on the simplicial complexes to induce the chain of a commutative diagram of simplicial homology groups.

Abstract In this article, we give a representation of multi-Cauchy mappings on groupoids as an equation and then establish the (Hyers and Găvruţa) stability of such mappings on groupoids. In the case that the range is a subset of Banach space, the stability result will be different.

Abstract Let p p be a prime that divides the order of a finite group G G , and let P P be a Sylow p p -subgroup of G G . Assume that d d is the smallest number of generators of P P and define ℳ d ( P ) = { P 1 , P 2 , … , P d } {{\mathcal{ {\mathcal M} }}}_{d}\left(P)=\left\{{P}_{1},{P}_{2},\ldots ,{P}_{d}\right\} as a collection of maximal subgroups of P P such that ⋂ i = 1 d P i = Φ ( P ) {\bigcap }_{i=1}^{d}{P}_{i}=\Phi \left(P) , the Frattini subgroup of P P . This article focuses on the exploration of the structure of a finite group G G for which every element of ℳ d ( P ) {{\mathcal{ {\mathcal M} }}}_{d}\left(P) is either c # {c}^{\#} -normal or S S -quasinormally embedded in G G . Our results improve and generalize many known results.

Abstract Let f f be a transcendental entire function of hyperorder strictly less than 1, and let c c be a nonzero finite complex number. We prove that if f ( z ) f\left(z) and f ( z + c ) f\left(z+c) partially share 0, 1 ignoring multiplicity (i.e., E ¯ ( 0 , f ( z ) ) ⊆ E ¯ ( 0 , f ( z + c ) ) \overline{E}\left(0,f\left(z))\subseteq \overline{E}\left(0,f\left(z+c)) and E ¯ ( 1 , f ( z ) ) ⊆ E ¯ ( 1 , f ( z + c ) ) \overline{E}\left(1,f\left(z))\subseteq \overline{E}\left(1,f\left(z+c)) ), then f ( z ) ≡ f ( z + c ) f\left(z)\equiv f\left(z+c) . This result is a generalization and improvement of the previous theorem due to Li and Yi.