Volume 23, Issue 1

Given a finite group G G with identity e e , the TI-power graph (trivial intersection power graph) defined on G G , denoted by Γ ( G ) \Gamma \left(G) , is an undirected graph with vertex set G G where distinct vertices a a and b b are adjacent if ⟨ a ⟩ ∩ ⟨ b ⟩ = { e } \langle a\rangle \cap \langle b\rangle =\left\{e\right\} . We classify all finite groups whose TI-power graph is claw-free, K 1 , 4 {K}_{1,4} -free, C 4 {C}_{4} -free, and P 4 {P}_{4} -free. In addition, we classify the finite groups whose TI-power graph is a threshold graph, a cograph, a chordal graph, and a split graph.

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.

For a simple graph Γ \Gamma , Γ \Gamma is said to be s s -regular, provided that the automorphism group of Γ \Gamma regularly acts on the set consisting of s s -arcs of Γ \Gamma . Given a positive integer n n , the question on finding all s s -regular graphs of order n n and degree 3 has received considerable attention. An s s -regular graph with degree 3 is so-called a cubic symmetric graph. Let p p be a prime. We show that if Γ \Gamma is a cubic symmetric graph of order 88 p 88p , then p ∈ { 5 , 11 , 23 } p\in \left\{5,11,23\right\} ; if Γ \Gamma is a cubic symmetric graph of order 88 p 2 88{p}^{2} , then p = 11 p=11 . Moreover, we classify all cubic symmetric graphs of order 88 p 88p and 88 p 2 88{p}^{2} .

This study makes some preliminary observations towards an extension of current work on graphs defined on groups to simplicial complexes. I define a variety of simplicial complexes on a group, which are preserved by automorphisms of the group, and in many cases have a relation to familiar graphs on the group. The ones which seem to reach deepest into the graph structure are two forms of independence complex , and some results on the class of groups for which these two complexes coincide are given. Other examples are treated more briefly.

For n ≥ 4 n\ge 4 , let OPD n {{\mathcal{OPD}}}_{n} be the orientation-preserving and order-decreasing transformation semigroup on the finite chain X n = { 1 < … < n } {X}_{n}=\left\{1\lt \ldots \lt \hspace{0.30em}n\right\} . First, we determine the set of two-sided zero-divisors of OPD n {{\mathcal{OPD}}}_{n} , and its cardinality. Then, we let Γ ( OPD n ) \Gamma \left({{\mathcal{OPD}}}_{n}) be the graph whose vertices are the two-sided zero-divisors of OPD n {{\mathcal{OPD}}}_{n} excluding the zero element θ \theta and distinct two vertices α \alpha and β \beta joined by an edge in case α β = θ = β α \alpha \beta =\theta =\beta \alpha . In this study, we prove that Γ ( OPD n ) \Gamma \left({{\mathcal{OPD}}}_{n}) is a connected graph, and we find the diameter, girth, domination number, minimum degree, and maximum degree of Γ ( OPD n ) \Gamma \left({{\mathcal{OPD}}}_{n}) . Moreover, we give a lower bound for clique number of Γ ( OPD n ) \Gamma \left({{\mathcal{OPD}}}_{n}) and we prove that Γ ( OPD n ) \Gamma \left({{\mathcal{OPD}}}_{n}) is an imperfect graph.

The research on the permanents of graph matrices is one of the contemporary research topic in algebraic combinatorics. Brualdi and Goldwasser characterized the upper and lower bounds of permanents of Laplacian matrices of trees. In this article, we determined the second and third minimal permanents of the Laplacian matrices of trees, and the second maximal permanent of the Laplacian matrices of trees is given. The corresponding extremal graphs are characterized. Furthermore, we determined bounds of permanents of the Laplacian matrices of non-caterpillar trees with given graph parameters. Moreover, the corresponding extremal graphs are determined.

The aim of this article is to present a local generalized fixed-point theorem for set-valued mappings in b-metric spaces, which brings together the framework of different (such as Nadler’s and Kannan’s) fixed-point theorems. For general set-valued graph contractions, a global fixed-point theorem and qualitative results concerning the fixed-point sets are obtained. We also provide an answer to the open question regarding Ulam-Hyers stability of the fixed-point inclusion proposed in [Petruşel et al., Multi-valued graph contraction principle with applications , Optimization 69 (2020), 1541–1556].

Around one third of CO 2 {{\rm{CO}}}_{2} emissions in the atmosphere are linked to vehicular traffic. Pollutant agents have an impact on the environment, in particular, the increased presence of particulate matter (PM) creates negative effects on human health. This article examines how autonomy could positively reduce the emission of air pollutants due to traffic. The methodology involves the analyses of PM emissions as a function of traffic conditions, especially in the presence of autonomous vehicles (AVs) dampening traffic waves. The starting point is traffic measurements that, gathered from real experiments involving a fleet of vehicles moving on a ring track, exhibit the presence of stop-and-go waves that are dampened by control strategies implemented on a unique AV. Using a system of ordinary differential equations modeling the principal chemical reactions in the atmosphere, it is proved that wave dampening implies a significant decrease in PM emissions at ground level. The horizontal diffusion of the pollutants is estimated by partial differential equations combined with the model for chemical reactions. The obtained outcomes show advantages given by the improvements in traffic flows and the mitigation effect of green barriers.

The L p {L}_{p} Gauss image problem amounts to solving a class of Monge-Ampère type equations on the sphere. In this article, we discuss the regularity of solutions to the L p {L}_{p} Gauss image problem.

In the field of applied sciences, systems are frequently modeled using mathematical frameworks that include systems of nonlinear algebraic equations. Identifying their roots, whether real or complex, is of critical importance. The widespread use of complex numbers in science and engineering highlights the importance of accurately determining the complex roots of equations. This article presents a study in which the complex roots of a system of equations are identified through an approach that utilizes homotopy continuation, with the curve being traced using a hyperspherical path tracking technique. Furthermore, this article details five case studies on electrical networks where this method is applied to solve systems of equations containing imaginary coefficients to find mesh currents. The path tracking shows the behavior of system equation in each case study. Finally, an analysis of the precision of the solutions obtained in these case studies is provided, demonstrating an accuracy of up to 15 SDs in a single iteration during the refinement stage.

Based on an equivalent derivative non-linear Schrödinger equation, we derive some periodic and non-periodic two-parameter solutions of the deformed continuous Heisenberg spin (DCHS) equation. The ill-posedness of these solutions is demonstrated through Fourier integral estimates in the Sobolev space H S 2 s {H}_{{{\rm{S}}}^{2}}^{s} (for the periodic solution in H S 2 s ( T ) {H}_{{{\rm{S}}}^{2}}^{s}\left({\mathbb{T}}) and the non-periodic solution in H S 2 s ( R ) {H}_{{{\rm{S}}}^{2}}^{s}\left({\mathbb{R}}) , respectively). When α ≠ 0 \alpha \ne 0 , the range of the weak ill-posedness index is 1 < s < 3 2 1\lt s\lt \frac{3}{2} for both periodic and non-periodic solutions. However, the periodic solution exhibits a strong ill-posedness index in the range of 3 2 < s < 7 2 \frac{3}{2}\lt s\lt \frac{7}{2} , whereas for the non-periodic solution, the range is 1 < s < 2 1\lt s\lt 2 . These findings extend our previous work on the DCHS model to include the case of periodic solutions and explore a different fractional Sobolev space.

In this article, we study the value distribution of meromorphic functions concerning Hayman’s alternative. We extend and improve some results due to Zhu [ A far-reaching form of Hayman’s inequality; fixed points of meromorphic functions and their derivatives , Kexue Tongbao (Chinese) 31 (1986), no. 11, 801–804], Hua and Chuang [ On a conjecture of Hayman , Acta Math. Sinica (N.S.) 7 (1991), no. 2, 119–126], and Charak and Singh [ A value distribution result related to Hayman’s alternative , Commun. Korean Math. Soc. 34 (2019), no. 2, 495–506].

This article introduces a generalization of the concept of inverse graphs applicable to both graphs and mixed graphs. Given a graph G G with adjacency matrix A ( G ) A\left(G) , the inverse graph G − 1 {G}^{-1} is defined such that its adjacency matrix is similar to the inverse of A ( G ) A\left(G) through a diagonal matrix with entries of ± 1 \pm 1 . While this diagonal matrix may or may not exist for graphs with nonsingular adjacency matrices, our study extends the concept to include mixed graphs as well. It has been proven that for certain unicyclic graphs, such a diagonal matrix does not exist. Motivated by this, we generalized the definition of inverse graphs to include mixed graphs, allowing us to find inverse mixed graphs for a class previously shown to lack one.

We investigate the two-species chemotaxis predator-prey system given by the following system: u t = Δ u − χ ∇ ⋅ ( u ∇ w ) + u ( λ 1 − μ 1 u r 1 − 1 + a v ) , x ∈ Ω , t > 0 , v t = Δ v + ξ ∇ ⋅ ( v ∇ z ) + v ( λ 2 − μ 2 v r 2 − 1 − b u ) , x ∈ Ω , t > 0 , 0 = Δ w − w + v , x ∈ Ω , t > 0 , 0 = Δ z − z + u , x ∈ Ω , t > 0 , \left\{\begin{array}{ll}{u}_{t}=\Delta u-\chi \nabla \cdot \left(u\nabla w)+u\left({\lambda }_{1}-{\mu }_{1}{u}^{{r}_{1}-1}+av),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ {v}_{t}=\Delta v+\xi \nabla \cdot \left(v\nabla z)+v\left({\lambda }_{2}-{\mu }_{2}{v}^{{r}_{2}-1}-bu),& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ 0=\Delta w-w+v,& x\in \Omega ,\hspace{0.33em}t\gt 0,\\ 0=\Delta z-z+u,& x\in \Omega ,\hspace{0.33em}t\gt 0,\end{array}\right. in a bounded domain Ω ⊂ R N ( N ≥ 1 ) \Omega \subset {{\mathbb{R}}}^{N}\left(N\ge 1) with smooth boundary, where parameters χ , ξ , λ i , μ i > 0 \chi ,\xi ,{\lambda }_{i},{\mu }_{i}\gt 0 , and r i > 1 ( i = 1 , 2 ) {r}_{i}\gt 1\hspace{0.33em}\left(i=1,2) . Under appropriate conditions, utilizing suitable a priori estimates, we demonstrate that if ( N − 2 ) + N < max ( r 1 − 1 ) ( r 2 − 1 ) , 4 N 2 , ( r 1 − 1 ) 2 N , ( r 2 − 1 ) 2 N \frac{{\left(N-2)}_{+}}{N}\lt \max \left\{\phantom{\rule[-0.95em]{}{0ex}},\left({r}_{1}-1)\left({r}_{2}-1),\frac{4}{{N}^{2}},\left({r}_{1}-1)\frac{2}{N},\left({r}_{2}-1)\frac{2}{N}\right\} , then the system admits a unique, uniformly bounded global classical solution. This finding extends the results of several previous studies.

In this work, we settle a conjecture about the number of set-orbits of permutation group acting on the power set.

This study investigates a number of integrals possessing products of different indices of the univariate Lommel function, s μ , ν ( a ) {s}_{\mu ,\nu }\left(a) , with various elementary and special functions. As a consequence, connections between the function and Chebyshev polynomials of the first kind are established.

We study the Euler-Bernoulli equations with time delay: u t t + Δ 2 u − g 1 ∗ Δ 2 u + g 2 ∗ Δ u + μ 1 u t ( x , t ) ∣ u t ( x , t ) ∣ m − 2 + μ 2 u t ( x , t − τ ) ∣ u t ( x , t − τ ) ∣ m − 2 = f ( u ) , {u}_{tt}+{\Delta }^{2}u-{g}_{1}\ast {\Delta }^{2}u+{g}_{2}\ast \Delta u+{\mu }_{1}{u}_{t}\left(x,t){| {u}_{t}\left(x,t)| }^{m-2}+{\mu }_{2}{u}_{t}\left(x,t-\tau ){| {u}_{t}\left(x,t-\tau )| }^{m-2}=f\left(u), where τ \tau represents the time delay. We exhibit the blow-up behavior of solutions with both positive and nonpositive initial energy for the Euler-Bernoulli equations involving time delay.

Let A {\mathcal{A}} be a complex unital Banach algebra and ( R 1 , R 2 {R}_{1},{R}_{2} ) be a pair of semiregularities in A {\mathcal{A}} . In order to investigate the boundary of spectra in the axiomatic theory of spectrum, this article defines the concept of spectrum boundary domination (SBD) of semiregularities and gives a criterion for a pair ( R 1 , R 2 {R}_{1},{R}_{2} ) possessing SBD property. Furthermore, the conditions such that σ R 1 ( a ) = σ R 2 ( a ) {\sigma }_{{R}_{1}}\left(a)={\sigma }_{{R}_{2}}\left(a) are described for a ∈ A a\in {\mathcal{A}} under the assumption that the pair ( R 1 , R 2 {R}_{1},{R}_{2} ) possesses SBD property. In addition, the transfer of SBD property through a Banach algebra homomorphism is discussed.

Probability distribution has proven its usefulness in almost every discipline of human endeavors. A novel extension of Bur X distribution is developed in this study employing the record-based transmuted mapping technique, which can be used to fit skewed and complex data. We referred to this novel distribution as a record-based transmuted Burr X model. We established the shape of the probability density function and hazard function. Numerous statistical and mathematical properties are provided, including quantile function, moment, and ordered statistics of the proposed model. Further, we obtain the estimation of the model parameters using the maximum likelihood estimation method, and four sets of Monte Carlo simulation studies are carried out to evaluate the efficiency of these estimates. Finally, the practical applicability of the developed model is demonstrated by analyzing three data sets, comparing its performance with several well-known distributions. The results highlight the flexibility and accuracy of the model, establishing it as a powerful and reliable tool for advanced statistical modeling in environmental and survival research.

The aim of this article is to develop a modified predictor–corrector scheme for solving the system of nonlinear Ψ \Psi -Caputo fractional differential equations with order 0 < α < 1 0\lt \alpha \lt 1 . By using the graded mesh and considering the predictor–corrector scheme for Ψ \Psi -Caputo fractional derivative, the numerical solutions of nonlinear Ψ \Psi -Caputo fractional-order systems are derived. Moreover, the error estimations of predictor–corrector scheme with graded mesh are also investigated. Particularly, the accuracy of numerical solutions depended on the function Ψ \Psi and the partition size on graded mesh. Numerical examples for linear and nonlinear fractional differential systems with various kernels and meshes are considered to explain the value and effectiveness of the proposed schemes.

Infectious diseases still remain one of the major causes of death worldwide, despite the fact that various treatments, such as antibiotics, antiviral drugs, and vaccines for some diseases, are more available to people. Factors such as drug resistance, lack of access to health care, and environmental changes contribute to their persistence and spread. Motivated by this fact, in this study, the stochastic susceptible-infectious-recovered (SIR) epidemiological model with treatment and non-linear incidence rate is extended, by introducing coloured noise, to model which takes into account the seasonal nature of the disease, as well as the fact that the disease is constantly changing through mutations, which leads to the appearance of new disease strains. For the model formulated in this way, we first prove the existence and uniqueness of the global positive solution. Then, we provide conditions under which the disease persists in the population, as well as sufficient conditions for the disease to die out. The theoretical results of the current study are validated by numerical simulations. For that purpose, we use data on the spread of the Ebola epidemic in Sierra Leone and the coronavirus disease 2019 (COVID-19) pandemic in Pakistan. Both theoretical and numerical results can lead us to conclusion that our model represents the solid research base for further investigation in the field of epidemiological modelling.

In this work, we study some classes of modules closed under submodules, quotients, and projective covers, even if the left projective cover of an arbitrary left module not always exists. We obtain a characterization of artinian principal ideal rings when the class of left R R -modules closed under submodules and projective covers and the class of left R R -modules closed under quotients and projective covers coincide. Also, for commutative rings, we characterize noetherian rings in which every cyclic module is quasi-injective.

We provide a recursive formula for the dimension of the algebraic sum of finitely many subspaces in a finite-dimensional vector space over an arbitrary field.

Consider an autonomous ordinary differential equation in R n {{\mathbb{R}}}^{n} , which has a heteroclinic loop. Assume that the heteroclinic loop consists of two degenerate heteroclinic orbits γ 1 {\gamma }_{1} , γ 2 {\gamma }_{2} and two saddle points with different Morse indices. The degenerate heteroclinic orbit in the sense that variational equation along the heteroclinic orbit γ i {\gamma }_{i} has d i {d}_{i} ( d i > 1 {d}_{i}\gt 1 , i = 1 i=1 , 2) linearly independent bounded solutions. By the different Morse indices and d i {d}_{i} , the heteroclinic loop is a heterodimensional loop, at the same time, it has high codimension in this situation. Applying Lin’s method to the heteroclinic loop, we derived the bifurcation function. The zeros of this function correspond to the conditions under which periodic or homoclinic orbits can bifurcate from the high-codimension heteroclinic loop in the perturbed system.

The aim of this study is to explore the complete lifts of almost Norden structures on tangent bundles of Walker four-manifolds. Furthermore, we examine the integrability conditions of the complete lifts J C {J}^{C} of the proper almost complex structure and identify the conditions for the isotropic Kähler-Norden property of the structure J C {J}^{C} on ( T ( M 4 ) , J C , g C ) \left(T\left({M}_{4}),{J}^{C},{g}^{C}) . Finally, we investigate the holomorphic conditions of the Norden (Kähler-Norden) metrics g C {g}^{C} and discuss Quasi-Kähler-Norden metrics g C {g}^{C} on the tangent bundle ( T ( M 4 ) , φ C , g C ) \left(T\left({M}_{4}),{\varphi }^{C},{g}^{C}) of Walker four-manifolds.

This article studies the regularity of weak solutions to the 3D stationary tropical climate model. We prove that when ( U , V , θ ) \left(U,V,\theta ) belongs to the homogeneous Morrey space M ˙ 2 , p ( R 3 ) {\dot{M}}^{2,p}\left({{\mathbb{R}}}^{3}) with p > 3 p\gt 3 , then ( U , V , θ ) ∈ C ∞ ( R 3 ) \left(U,V,\theta )\in {C}^{\infty }\left({{\mathbb{R}}}^{3}) .

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.

In this article, we investigate the relationship between freely quasiconformal mappings and locally weakly quasisymmetric mappings in quasiconvex and complete metric spaces. We first prove the equivalence of freely quasiconformal mappings, ring properties, and locally weakly quasisymmetric mappings. Finally, we prove that the composition of two locally weakly quasisymmetric mappings in metric spaces is locally weakly quasisymmetric mapping.

Let G G be a simple connected graph on n n vertices. The degree of a vertex v ∈ V ( G ) v\in V\left(G) , denoted by d v {d}_{v} , is the number of edges incident with v v and the distance between any two vertices u , v ∈ V ( G ) u,v\in V\left(G) , denoted by d u v {d}_{uv} , is defined as the length of the shortest path from u u to v v . The distance matrix of G G , denoted by, D ( G ) D\left(G) , is defined as D ( G ) = d u v D\left(G)={d}_{uv} . We now define and investigate the degree distance matrix of a connected graph G G , defined as M D D ( G ) = ( ( d u + d v ) d u v ) u , v ∈ V ( G ) {M}_{DD}\left(G)={\left(\left({d}_{u}+{d}_{v}){d}_{uv})}_{u,v\in V\left(G)} . In this article, first, we derive bounds for the largest eigenvalue of the degree distance matrix. Then, we establish some bounds for the energy of the degree distance matrix of G G .

Two conic sections C 1 {C}_{1} and C 2 {C}_{2} in the Euclidean plane that pass through two given points can generally have two further points of intersection. It is shown how these can be constructed using compass and ruler. The idea is to construct the degenerate conics in the pencil of the two conics C 1 {C}_{1} and C 2 {C}_{2} . Their intersections are then the four intersection points of C 1 {C}_{1} and C 2 {C}_{2} . The same idea is then used to reduce a general quartic equation to a cubic equation and to solve it. This is performed by interpreting the solutions of the quartic as intersections of two complex conic sections. The degenerate complex conics in their pencil can then be found through a cubic equation.

The main purpose of this article is to investigate a topology based on implication and upsets in a bounded BCK-algebra. First, we introduce a special kind of sets associated with implication and upsets in a bounded BCK-algebra, and some basic properties of these sets are derived. Furthermore, a topology related to implication and upsets is constructed by means of the sets. Moreover, we discuss some topological properties of the topology such as compactness and continuity. In particular, it is proved that a Glivenko BCK-algebra with the topology for implication is a left topological BCK-algebra. Also, a bounded product BCK-algebra with such topology for product is a para-topological BCK-algebra. Finally, the relationship between the topology and the quotient topology on a bounded quotient BCK-algebra is revealed.

The objective of this study is to establish a novel category of multivalent analytic functions by utilizing generalized multiplier transformations. The subclass is further examined for coefficient estimations, growth and distortion theorems, extreme points, and the radius of starlikeness and convexity.

We investigate the existence of local minimizers for the nonlinear Schrödinger (NLS) equation with localized nonlinearity on noncompact metric graphs. In the absence of ground states, we prove that normalized local minimizers of the NLS equation do exist under suitable topological and metric assumptions of the graphs. In particular, we provide a criterion for the existence of local minimizers for the NLS equation in this article. Our results rely on the variational method and an application of Gagliardo-Nirenberg inequalities.

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.

In this article, we study the semilinear biharmonic problem with a critical growth: Δ 2 u = μ Q ( x ) ∣ u ∣ p − 2 u + V ( x ) ∣ u ∣ 2 * * − 2 u in Ω , u = Δ u = 0 on ∂ Ω , \left\{\begin{array}{ll}{\Delta }^{2}u=\mu Q\left(x){| u| }^{p-2}u+V\left(x){| u| }^{{2}^{* * }-2}u\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=\Delta u=0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Ω ⊂ R N ( N > 4 ) \Omega \subset {{\mathbb{R}}}^{N}\hspace{0.25em}\left(N\gt 4) is a smooth bounded domain, μ > 0 \mu \gt 0 , 1 < p < 2 1\lt p\lt 2 , and 2 * * = 2 N ⁄ ( N − 4 ) {2}^{* * }=2N/\left(N-4) is the critical Sobolev exponent. By using the Nehari method and the critical point theory, at least k k nontrivial solutions of the above equations for μ \mu sufficiently small and under some appropriate assumptions on Q Q and V V are obtained.

In this article, we investigate the relations between various types of weighted measure-theoretic pressures and different versions of weighted topological pressures. We show that various types of weighted measure-theoretic pressures for an ergodic measure with respect to a potential function are equal to the sum of measure-theoretic entropy of this measure and the integral of the potential function.

In 2001, S. S. Dragomir introduced a generalized class of convexity, the so-called ( m , M , ψ ) \left(m,M,\psi ) -convex functions, which covers many other classes of convexity. In this article, we prove some useful characterizations of this generalized class of convex functions. We obtain majorization-type inequalities for ( m , M , ψ ) \left(m,M,\psi ) -convex functions, providing also applications to new estimates for some well-known mean inequalities.

Spivey found a remarkable recurrence relation for Bell numbers, which was generalized to that for Bell polynomials by Gould-Quaintance. The aim of this article is to generalize their recurrence relation for Bell polynomials to that for the probabilistic Dowling polynomials associated with Y Y and also that for the probabilistic r r -Dowling polynomials associated with Y Y . Here Y Y is a random variable whose moment generating function exists in a neighborhood of the origin.

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.

In this note, we study a global existence and blow-up of the positive solutions to the initial-boundary value problem of the nonlinear pseudo-parabolic equation for the Baouendi-Grushin operator. The approach is based on the concavity argument and the Poincaré inequality related to the Baouendi-Grushin operator from [Suragan and Yessirkegenov, Sharp remainder of the Poincaré inequality for Baouendi–Grushin vector fields , Asian-Eur. J. Math. 16 (2023), 2350041], inspired by the recent work [Ruzhansky et al., Global existence and blow-up of solutions to porous medium equation and pseudo-parabolic equation, I. Stratified group , Manuscripta Math. 171 (2023), 377–395].

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.

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.

The Piatetski-Shapiro sequences are sequences of the form ( ⌊ n c ⌋ ) n = 1 ∞ {\left(\lfloor {n}^{c}\rfloor )}_{n=1}^{\infty } and the Beatty sequence is the sequence of integers ( ⌊ α n + β ⌋ ) n = 1 ∞ {(\lfloor \alpha n+\beta \rfloor )}_{n=1}^{\infty } . We prove that there are infinitely many Carmichael numbers composed of entirely the primes from the intersection of a Piatetski-Shapiro sequence and a Beatty sequence for c ∈ 1 , 19137 18746 c\in \left(\phantom{\rule[-0.75em]{}{0ex}},1,\frac{19137}{18746}\right) , α > 1 \alpha \gt 1 irrational and of finite type by investigating the Piatetski-Shapiro primes in arithmetic progressions in a Beatty sequence. Moreover, we also discuss the intersection of a Piatetski-Shapiro sequence and multiple Beatty sequences in arithmetic progressions.

Let F q {{\mathbb{F}}}_{q} be the finite field of characteristic p p and F q * = F q \ { 0 } {{\mathbb{F}}}_{q}^{* }\left={{\mathbb{F}}}_{q}\backslash \left\{0\right\} . In this article, we use Smith normal form of exponent matrices to present exact formulas for the numbers of rational points on suitable affine algebraic varieties defined by the following systems of equations over F q {{\mathbb{F}}}_{q} : a 1 x 1 e 11 … x m 1 e 1 m 1 + … + a m 1 x 1 e m 1 , 1 … x m 1 e m 1 , m 1 = b 1 , a m 1 + 1 x 1 e m 1 + 1,1 … x m 2 e m 1 + 1 , m 2 + … + a m 2 x 1 e m 2 , 1 … x m 2 e m 2 , m 2 = b 2 \left\{\begin{array}{l}{a}_{1}{x}_{1}^{{e}_{11}}\ldots {x}_{{m}_{1}}^{{e}_{1{m}_{1}}}+\ldots +{a}_{{m}_{1}}{x}_{1}^{{e}_{{m}_{1},1}}\ldots {x}_{{m}_{1}}^{{e}_{{m}_{1},{m}_{1}}}={b}_{1},\\ {a}_{{m}_{1}+1}{x}_{1}^{{e}_{{m}_{1}+\mathrm{1,1}}}\ldots {x}_{{m}_{2}}^{{e}_{{m}_{1}+1,{m}_{2}}}+\ldots +{a}_{{m}_{2}}{x}_{1}^{{e}_{{m}_{2},1}}\ldots {x}_{{m}_{2}}^{{e}_{{m}_{2},{m}_{2}}}={b}_{2}\end{array}\right. and c 1 x 1 d 11 … x n 1 d 1 n 1 + … + c n 1 x 1 d n 1 , 1 … x n 1 d n 1 , n 1 = l 1 , c n 1 + 1 x 1 d n 1 + 1,1 … x n 2 d n 1 + 1 , n 2 + … + c n 2 x 1 d n 2 , 1 … x n 2 d n 2 , n 2 = l 2 , c n 2 + 1 x 1 d n 2 + 1,1 … x n 3 d n 2 + 1 , n 3 + … + c n 3 x 1 d n 3 , 1 … x n 3 d n 3 , n 3 = l 3 \left\{\begin{array}{l}{c}_{1}{x}_{1}^{{d}_{11}}\ldots {x}_{{n}_{1}}^{{d}_{1{n}_{1}}}+\ldots +{c}_{{n}_{1}}{x}_{1}^{{d}_{{n}_{1},1}}\ldots {x}_{{n}_{1}}^{{d}_{{n}_{1},{n}_{1}}}={l}_{1},\\ {c}_{{n}_{1}+1}{x}_{1}^{{d}_{{n}_{1}+\mathrm{1,1}}}\ldots {x}_{{n}_{2}}^{{d}_{{n}_{1}+1,{n}_{2}}}+\ldots +{c}_{{n}_{2}}{x}_{1}^{{d}_{{n}_{2},1}}\ldots {x}_{{n}_{2}}^{{d}_{{n}_{2},{n}_{2}}}={l}_{2},\\ {c}_{{n}_{2}+1}{x}_{1}^{{d}_{{n}_{2}+\mathrm{1,1}}}\ldots {x}_{{n}_{3}}^{{d}_{{n}_{2}+1,{n}_{3}}}+\ldots +{c}_{{n}_{3}}{x}_{1}^{{d}_{{n}_{3},1}}\ldots {x}_{{n}_{3}}^{{d}_{{n}_{3},{n}_{3}}}={l}_{3}\end{array}\right.\hspace{1.15em} when the determinants of exponent matrices are coprime to q − 1 q-1 , where e i j , d i ′ j ′ ∈ Z + (the set of positive integers) , a i , c i ′ ∈ F q * , 1 ≤ i , j ≤ m 2 , 1 ≤ i ′ , j ′ ≤ n 3 , {e}_{ij},{d}_{i^{\prime} j^{\prime} }\in {{\mathbb{Z}}}^{+}\hspace{0.1em}\text{(the set of positive integers)}\hspace{0.1em},{a}_{i},{c}_{i^{\prime} }\in {{\mathbb{F}}}_{q}^{* },1\le i,j\le {m}_{2},1\le i^{\prime} ,j^{\prime} \le {n}_{3}, and b 1 , b 2 , l 1 , l 2 , l 3 ∈ F q {b}_{1},{b}_{2},{l}_{1},{l}_{2},{l}_{3}\in {{\mathbb{F}}}_{q} . These formulas extend the theorem obtained by Wang and Sun ( An explicit formula of solution of some special equations over a finite field , Chinese Ann. Math. Ser. A 26 (2005), 391–396, https://www.cqvip.com/doc/journal/977048790. (in Chinese)). Our results also give a partial answer to an open problem of Hu et al. raised in ( The number of rational points of a family of hypersurfaces over finite fields , J. Number Theory 156 (2015), 135–153, doi: https://doi.org/10.1016/j.jnt.2015.04.006).

In this article, we construct filling disks for meromorphic functions of order zero and that way we prove the existence of Borel directions of these functions. In the latter part of this article, we demonstrate the existence of filling disks using the Borel direction of meromorphic functions.

In this article, we investigate the long-time behavior for the ill-posed problems ∂ 2 u ∂ t 2 + ∂ u ∂ t + λ u − Δ u − Δ ∂ u ∂ t − Δ ∂ 2 u ∂ t 2 = f ( t , u ( x , t − ρ ( t ) ) ) + g ( t , x ) , in ( τ , + ∞ ) × R N , \frac{{\partial }^{2}u}{\partial {t}^{2}}+\frac{\partial u}{\partial t}+\lambda u-\Delta u-\Delta \frac{\partial u}{\partial t}-\Delta \frac{{\partial }^{2}u}{\partial {t}^{2}}=f\left(t,u\left(x,t-\rho \left(t)))+g\left(t,x),\hspace{1.0em}{\rm{in}}\hspace{0.33em}\left(\tau ,+\infty )\times {{\mathbb{R}}}^{N}, with some hereditary characteristics. First, we establish the existence of solutions for the second-order non-autonomous evolution equation by the standard Faedo-Galerkin methods, but without any Lipschitz conditions on the nonlinear term f ( ⋅ ) f\left(\cdot ) . Then, by proving the D {\mathfrak{D}} -pullback asymptotically upper-semicompact property for the multivalued process { U ( t , τ ) } \left\{U\left(t,\tau )\right\} , we establish the existence of pullback attractors A C H 1 ( R N ) , H 1 ( R N ) {{\mathcal{A}}}_{{C}_{{H}^{1}\left({{\mathbb{R}}}^{N}),{H}^{1}\left({{\mathbb{R}}}^{N})}} in the Banach spaces C H 1 ( R N ) , H 1 ( R N ) {C}_{{H}^{1}\left({{\mathbb{R}}}^{N}),{H}^{1}\left({{\mathbb{R}}}^{N})} for the multi-valued process generated by a class of second-order non-autonomous evolution equations with delays and ill-posedness.

In this article, we show that there is a one-to-one correspondence between the eigenfunctions associated with the perturbed Laplacian operator Δ q {\Delta }_{q} on a Schrödinger infinite network { X , t , q } \{X,t,q\} with weight function q ( a ) q\left(a) and the eigenfunctions associated with classical Laplacian operator Δ \Delta on the infinite network { X , t } . \{X,t\}.

This article aims to delve deeper into the weighted grand variable Herz-Morrey spaces, and try to establish the boundedness of fractional sublinear operators and their multilinear commutators within this framework. The results are still new even in the unweighted setting, extending previous research. As applications, the corresponding boundedness estimates for the commutators of maximal operator and Riesz potential operator are established.

Recently, Shen et al. showed that the SI 2 -topology on a T 0 {T}_{0} -space can be described completely in terms of SI 2 -convergence, and the SI 2 -convergence is topological whenever the given space is SI 2 -continuous. In this article, we give a characterization of T 0 {T}_{0} -spaces for the SI 2 -convergence being topological by introducing the notion of strongly I 2 -continuous spaces, which are strictly weaker than SI 2 -continuous spaces but are more closely related to the SI 2 -convergence. Moreover, as a common generalization of the irr-convergence and the S {\mathcal{S}} -convergence, we introduce the concept of SI 2 * {}_{2}^{* } -convergence in T 0 {T}_{0} -spaces and the related concept of SI 2 * {}_{2}^{* } -continuous spaces. It is proved that if a T 0 {T}_{0} -space X X is SI 2 * {}_{2}^{* } -continuous, then the SI 2 * {}_{2}^{* } -convergence in X X is topological.

We investigate the existence of blowing-up solutions of the following almost-critical problem: − Δ u + V ( x ) u = u p − ε , u > 0 in Ω , u = 0 on ∂ Ω , -\Delta u+V\left(x)u={u}^{p-\varepsilon },\hspace{1.0em}u\gt 0\hspace{0.25em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,u=0\hspace{0.25em}\hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.25em}\partial \Omega , where Ω \Omega is a bounded regular domain in R n {{\mathbb{R}}}^{n} , n ≥ 4 n\ge 4 , ε \varepsilon is a small positive parameter, p + 1 = ( 2 n ) ∕ ( n − 2 ) p+1=\left(2n)/\left(n-2) is the critical Soblolev exponent, and the potential V V is a smooth positive function. We find solutions that exhibit bubbles clustered inside as ε \varepsilon goes to zero. To the best of our knowledge, this is the first existence result for interior non-simple blowing-up positive solutions to Dirichlet problems in general domains. Our results are proven through delicate asymptotic estimates of the gradient of the associated Euler-Lagrange functional.

We find all integer polynomials of degree d d that take the values ± 1 \pm 1 at exactly d d integer arguments, and determine the irreducibility of these polynomials by means of an elementary approach.

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.

The averaging method is applied to the study of optimal control problems for systems of integro-differential equations with rapidly oscillating coefficients and a small parameter. The original problem is associated with an averaged optimal control problem, formulated for a system of ordinary differential equations, which significantly simplifies the analysis. It is proven that as the small parameter tends to zero, the quality criterion, optimal control, and optimal trajectory of the original problem converge to those of the averaged problem.

In this article, we consider some types of derivations in Banach algebras. In detail, we investigate the question of whether the superstability can be achieved under some conditions for some types of derivations, such as Jordan derivations, generalized Lie 2-derivations, and generalized Lie derivations.

Graph theory (GT) is a highly captivating area of study within applied mathematics. Over the last 50 years, the number of mathematicians contributing to research in this field has consistently risen. This surge in interest can largely be attributed to its vast potential for practical applications. It has found uses in various disciplines such as engineering, physical sciences, genetics, computer science, sociology, operations research and economics. Driven and encouraged by these points, we introduce and analyze the convergence theorems for the modified UO-iterative sequences applied to G G -nonexpansive mapping in Banach spaces via a graph. In addition, the modified UO-iterative algorithm’s applications to image deblurring and signal enhancement are demonstrated.

The method of brackets is a procedure to evaluate definite integrals over a half-line. It consists of a small number of rules. This article illustrates the method by evaluating an integral by several variations of the method. The integrand is the product of a Bessel function and an exponential integral function.

The main focus of this study is to discuss the existence of positive ω \omega -periodic mild solutions for evolution equation with delay in an ordered Banach space E E . Under the ordered conditions of the growth exponent of the nonlinearity g g with respect to the semigroup T ( t ) ( t ≥ 0 ) {\mathcal{T}}\left(t)\hspace{.25em}\left(t\ge 0) or the first eigenvalue of the operator A {\mathcal{A}} , the existence results of mild solutions and positive mild solutions for evolution equation are obtained by applying the Poincaré mapping and monotone iterative method, and adding appropriate conditions to the nonlinearity term g g without assuming the existence of upper and lower solutions. Finally, we give an example to verify the applicability of our abstract results.

In this article, we provide a characterization of tilings, sub-tilings, and spectral sets on the p p -adic space Q p d {{\mathbb{Q}}}_{p}^{d} . Our methods are based on p p -adic distributions and p p -adic Fourier analysis. We also obtained a sufficient condition and a necessary condition on tiling of functions by translation on Q p d {{\mathbb{Q}}}_{p}^{d} .

The aim of the article is to better understand the higher octahedral axiom of n n -angulated categories. We discuss several possible additional axioms for pre- n n -angulated categories and prove that they are all equivalent to the higher octahedral axiom.

In this article, we provide a comprehensive study on the continuity and essential norm of an operator defined by an infinite tridiagonal matrix, specifically when it operates from a weighted Orlicz sequence space or a weighted l ∞ {l}^{\infty } space into another space of similar nature. Our findings include significant characterizations regarding the compactness of this operator across various contexts of weighted Orlicz and l ∞ {l}^{\infty } sequence spaces.

A family of commuting contraction semigroups ( P n ( t ) ) n ∈ N {\left({P}_{n}\left(t))}_{n\in {\mathbb{N}}} , defined on l 1 ( N ) {l}^{1}\left({\mathbb{N}}) , is presented. For this family, the product semigroup ∏ n = 1 ∞ P n ( t ) {\prod }_{n=1}^{\infty }{P}_{n}\left(t) exists and has bounded generator. The infinite product of the corresponding family of adjoint semigroups ( P n ∗ ( t ) ) n ∈ N {\left({P}_{n}^{\ast }\left(t))}_{n\in {\mathbb{N}}} , defined on l ∞ ( N ) {l}^{\infty }\left({\mathbb{N}}) , also exists and its generator is bounded. Explicit formulae for these generators are also given.

The purpose of this article is to establish the existence of the pullback attractors for the non-autonomous magneto-micropolar fluid equations in 2D bounded domains. To this end, the asymptotic compactness of the processes generated by the solutions is required, which is proved by verifying the flattening property (also known as the “Condition C”) of the corresponding processes.

We define the weighted Orlicz-Lorentz-Morrey and weak weighted Orlicz-Lorentz-Morrey spaces to generalize the Orlicz spaces, the weighted Lorentz spaces, the Orlicz-Lorentz spaces, and the Orlicz-Morrey spaces. Furthermore, necessary and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator, generalized fractional integral, and maximal operators on the weighted Orlicz-Lorentz-Morrey and weak Orlicz-Lorentz-Morrey spaces are given, based on the exploration of properties of Young functions, B p {B}_{p} weights, and B p , ∞ {B}_{p,\infty } weights. Specifying the weights and the Young functions, we recover the existing results and we obtain new results in the new and old settings.

In this work, a boundary value problems for a system of nonlinear ordinary differential equations that incorporates impulsive actions is considered. This formulation is significant for modeling real-world phenomena in which abrupt changes occur at specific time instants. The study established sufficient conditions for the existence of isolated solutions to the proposed boundary value problems. This is crucial to ensure that the mathematical models accurately reflect the behavior of systems subject to impulsive actions. Algorithms were developed to find solutions to the boundary value problems. These algorithms leverage the parameterization method, which is effective in handling the discontinuities introduced by impulsive actions. The research includes a numerical implementation of the proposed algorithms, demonstrating their practicality and effectiveness in solving the boundary value problems with impulsive actions. The findings have implications in various fields, including mechanics, electrical engineering, and biology, where systems often experience sudden changes due to external influences. In general, the research contributes to the understanding and solution of nonlinear boundary value problems affected by impulsive actions, providing a framework for further exploration and application in scientific and engineering contexts.

In this article, we investigate the following nonlinear Kirchhoff equation with Sobolev critical growth: − a + b ∫ R 3 ∣ ∇ u ∣ 2 d x Δ u + λ u = μ f ( u ) + ∣ u ∣ 4 u , in R 3 , u > 0 , ∫ R 3 ∣ u ∣ 2 d x = m 2 , in R 3 , ( P m ) \left\{\begin{array}{l}-\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{3}}{| \nabla u| }^{2}{\rm{d}}x\right)\Delta u+\lambda u=\mu f\left(u)+{| u| }^{4}u,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\\ u\gt 0,\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{3}}{| u| }^{2}{\rm{d}}x={m}^{2},\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3},\end{array}\right.\hspace{2.0em}\hspace{2.0em}\left({P}_{m}) where μ \mu is a positive parameter, a , b > 0 a,b\gt 0 , and the frequency λ \lambda appears as a positive Lagrange multiplier. The nonlinearity f f is more general and satisfies Sobolev subcritical conditions. With the assistance of the Pohožaev constraints and the Sobolev subcritical approximation method, we have achieved a couple of the normalized ground-state solutions to ( P m ) \left({P}_{m}) and the asymptotic behavior of the ground-state is also studied.

Let F ( s ) F\left(s) be a function from the extended Selberg class. We consider decompositions F ( s ) = f ( h ( s ) ) F\left(s)=f\left(h\left(s)) , where f f and h h are meromorphic functions. Among other things, we show that F F is prime if and only if the greatest common divisor of the orders of all zeros and the pole of F F is 1.

For subharmonic functions s s in R n , n ≥ 2 , {{\mathbb{R}}}^{n},n\ge 2, there is an associated Radon measure μ \mu that is used to represent s s locally as an integral up to an additive harmonic function. We prove that the total measure μ ( R 2 ) \mu ({{\mathbb{R}}}^{2}) is finite if and only if the subharmonic function s s has harmonic majorant outside a compact set. However, this relationship does not hold in higher dimensions.

The study of ( h , g ; α − m ) \left(h,g;\hspace{1.42271pt}\alpha -m) -convex functions extends the classical concept of convexity to more generalized forms, which provide flexible tools for analysis. The aim of this article is to yield the generalization of the Fejér type inequalities for various classes of convex functions, such as ( h , g ; α − m ) \left(h,g;\hspace{1.42271pt}\alpha -m) -convex and ( h , g ; m ) \left(h,g;\hspace{1.42271pt}m) -convex functions.

This article aims to provide a series of characterizations of the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) mapping for spectral norm regularized convex optimization problems. By establishing the variational properties of the spectral norm function, we directly prove that the KKT mapping is isolated calm if and only if the strict Robinson constraint qualification (SRCQ) and the second-order sufficient condition (SOSC) hold. Furthermore, we obtain the crucial result that the SRCQ for the primal/dual problem and the SOSC for the dual/primal problem are equivalent. The obtained results can derive more equivalent conditions of the robust isolated calmness of the KKT mapping, thereby enriching the stability theory of spectral norm regularized optimization problems and enhancing the usability of isolated calmness in algorithm applications.

In this article, we focus on a class of p p -Kirchhoff-type equations that include logarithmic and concave terms. By applying the variational method, we establish the existence and multiplicity of positive solutions.

In the article, we obtain that, for algebraically independent over Q {\mathbb{Q}} parameters α 1 , … , α r {\alpha }_{1},\ldots ,{\alpha }_{r} , there are infinitely many shifts ( ζ ( s + i τ , α 1 ) , … , ζ ( s + i τ , α r ) ) \left(\zeta \left(s+i\tau ,{\alpha }_{1}),\ldots ,\zeta \left(s+i\tau ,{\alpha }_{r})) of Hurwitz zeta-functions with τ ∈ [ T , T + H ] \tau \in \left[T,T+H] , T 27 ⁄ 82 ⩽ H ⩽ T 1 ⁄ 2 {T}^{27/82}\leqslant H\leqslant {T}^{1/2} , that approximate any r r -tuple of analytic functions on the strip { s ∈ C : 1 ⁄ 2 < σ < 1 } \left\{s\in {\mathbb{C}}:1/2\lt \sigma \lt 1\right\} . More precisely, the latter set of shifts has a positive density. For the proof, a probabilistic approach is applied.

Let S S be a semigroup. In this study, we first introduce the Green’s digraphs and Green’s graphs related to the Green’s relations L {\mathscr{L}} , R {\mathscr{R}} , and J {\mathscr{J}} of S S . Further, the connectedness and completeness of the Green’s graphs are discussed. For a finite semigroup S S , we show that each of the Green’s graphs of S S has a transitive orientation. Moreover, we obtain that these Green’s graphs are perfect. Finally, the structures of the Green’s graphs are characterized using the generalized lexicographic product.

We establish novel Hermite-Hadamard-type inequalities for the product of two strongly h h -convex functions defined on balls and ellipsoids in multidimensional Euclidean spaces. Additionally, we investigate mappings associated with these inequalities and explore their applications. Our results generalize several existing findings in the literature.

In this article, we consider a class of Kirchhoff-type equations: − a + b ∫ Ω ∣ ∇ u ∣ 2 d x Δ u = f ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω . \left\{\begin{array}{ll}-\left(a+b\mathop{\displaystyle \int }\limits_{\Omega }{| \nabla u| }^{2}{\rm{d}}x\right)\Delta u=f\left(x,u),\hspace{1.0em}& x\in \Omega ,\\ u=0,\hspace{1.0em}& x\in \partial \Omega .\end{array}\right. It is a generalization of the classical wave equation and is often used to model wave propagation in various physical media. The nonlocal term b ( ∫ Ω ∣ ∇ u ∣ 2 d x ) Δ u b\left({\int }_{\Omega }{| \nabla u| }^{2}{\rm{d}}x)\Delta u in the equation causes the variational functional of the equation to have fundamentally different properties from the case of b = 0 b=0 . As far as we know, there are relatively few conclusions regarding infinitely many solutions of this equation. Under weaker assumptions, we obtain that the equation has infinitely many high-energy solutions. And our results extend the conclusions of Mao-Zhang (2009) and Zhang-Perera (2006).

In this article, we continue the study of the nonvanishing minors property initiated by Garcia, Karaali, and Katz, for the compressed Fourier matrix attached to a subgroup H H of the multiplicative group of a finite field F q {{\mathbb{F}}}_{q} and a character χ \chi defined over H H . Here, we provide a characterization of this aforementioned property for symmetries arising from an index-3 subgroup H H and a nontrivial character χ \chi .