Volume 22, Issue 1

The atom-bond sum-connectivity (ABS) index of a graph G G with edges e 1 , … , e m {e}_{1},\ldots ,{e}_{m} is the sum of the numbers 1 − 2 ( d e i + 2 ) − 1 \sqrt{1-2{\left({d}_{{e}_{i}}+2)}^{-1}} over 1 ≤ i ≤ m 1\le i\le m , where d e i {d}_{{e}_{i}} is the number of edges adjacent to e i {e}_{i} . In this article, we study the maximum values of the ABS index over graphs with given parameters. More specifically, we determine the maximum ABS index of connected graphs of a given order with a fixed (i) minimum degree, (ii) maximum degree, (iii) chromatic number, (iv) independence number, or (v) number of pendent vertices. We also characterize the graphs attaining the maximum ABS values in all of these classes.

We find new upper bounds for the global cyclicity index, a variant of the Kirchhoff index, and discuss the wide family of graphs for which the bounds are attained.

In this manuscript, we delve into the exploration of the first and second Zagreb connection indices of both polyomino chains and random polyomino chains. Our methodology relies on the utilization of Markov chain theory. Within this framework, the article thoroughly examines precise formulas and investigates extreme values. Leveraging the derived formulas, we further explore and elucidate the long-term behavior exhibited by random polyomino chains.

Multiplicative sum Zagreb index is a modified version of the famous Zagreb indices. For a graph G G , the multiplicative sum Zagreb index is defined as Π 1 * ( G ) = ∏ u v ∈ E ( G ) ( d G ( u ) + d G ( v ) ) {\Pi }_{1}^{* }\left(G)={\prod }_{uv\in E\left(G)}\left({d}_{G}\left(u)+{d}_{G}\left(v)) , where E ( G ) E\left(G) is the edge set of G G and d G ( u ) {d}_{G}\left(u) stands for the degree of vertex u u in G G . In this article, we determine the extremal multiplicative sum Zagreb indices among all n n -vertex molecular trees, molecular unicyclic graphs, molecular bicyclic graphs and molecular tricyclic graphs.

The number of j j -matchings in a graph H H is denote by m ( H , j ) m\left(H,j) . If for two graphs H 1 {H}_{1} and H 2 {H}_{2} , m ( H 1 , j ) ≥ m ( H 2 , j ) m\left({H}_{1},j)\ge m\left({H}_{2},j) for all j j , then we write H 1 ≽ H 2 {H}_{1}\succcurlyeq {H}_{2} . If H 1 ≽ H 2 {H}_{1}\succcurlyeq {H}_{2} , and m ( H 1 , i ) > m ( H 2 , i ) m\left({H}_{1},i)\gt m\left({H}_{2},i) for some i i , then we write H 1 ≻ H 2 {H}_{1}\hspace{0.33em}\succ \hspace{0.33em}{H}_{2} . In this article, by utilizing several new graph transformations, we determine the least element with respect to the quasi-order ≽ \succcurlyeq among all unicyclic graphs with fixed order and number of vertices of degree two. As consequences, we characterize the graphs with minimum matching energy and with minimum Hosoya index in the set of all unicyclic graphs with fixed order and number of vertices of degree two.

We establish weighted Hermite-Hadamard-type inequalities for some classes of differentiable functions without assuming any symmetry property on the weight function. Next, we apply our obtained results to the approximation of some classes of weighted integrals.

This work investigates the asymptotic behavior of energy solutions to the focusing nonlinear Schrödinger equation of Choquard type i ∂ t u + Δ u + ∣ u ∣ p − 2 ( I α * ∣ u ∣ p ) u = 0 , p = 1 + 2 + α N − 2 , N ≥ 3 . i{\partial }_{t}u+\Delta u+{| u| }^{p-2}\left({I}_{\alpha }* {| u| }^{p})u=0,\hspace{1.0em}p=1+\frac{2+\alpha }{N-2},\hspace{1.0em}N\ge 3. Indeed, in the energy-critical spherically symmetric regime, one proves a global existence and scattering versus finite time blow-up dichotomy. Precisely, if the data have an energy less than the ground state one, two cases are possible. If the kinetic energy of the radial data is less than the ground state one, then the solution is global and scatters. Otherwise, if the data have a finite variance or is spherically symmetric and have a finite mass, then the solution is nonglobal. The main difficulty is to deal with the nonlocal source term. The argument is the concentration-compactness-rigidity method introduced by Kenig and Merle ( Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case , Invent. Math. 166 (2006), no. 3, 645–675). This note naturally complements the work by Saanouni ( Scattering theory for a class of defocusing energy-critical Choquard equations , J. Evol. Equ. 21 (2021), 1551–1571), where the scattering of the defocusing energy-critical generalized Hartree equation was obtained.

Based on the concept of ( p , q ) \left(p,q) -compact operator for p ∈ [ 1 , ∞ ] p\in \left[1,\infty ] and q ∈ [ 1 , p * ] q\in \left[1,{p}^{* }] , we introduce and study the notion of ( p , q ) \left(p,q) -compact holomorphic mapping between Banach spaces. We prove that the space formed by such mappings is a surjective p q ∕ ( p + q ) pq/\left(p+q) -Banach bounded-holomorphic ideal that can be generated by composition with the ideal of ( p , q ) \left(p,q) -compact operators. In addition, we study Mujica’s linearization of such mappings, its relation with the ( u * v * + t v * + t u * ) ∕ t u * v * \left({u}^{* }{v}^{* }+t{v}^{* }+t{u}^{* })/t{u}^{* }{v}^{* } -Banach bounded-holomorphic composition ideal of the ( t , u , v ) \left(t,u,v) -nuclear holomorphic mappings for t , u , v ∈ [ 1 , ∞ ] t,u,v\in \left[1,\infty ] , its holomorphic transposition via the injective hull of the ideal of ( p , q * , 1 ) \left(p,{q}^{* },1) -nuclear operators, the Möbius invariance of ( p , q ) \left(p,q) -compact holomorphic mappings on D {\mathbb{D}} , and its full compact factorization through a compact holomorphic mapping, a ( p , q ) \left(p,q) -compact operator, and a compact operator.

In this study, we discuss on the problem of minimizing the differences of two non-positive valued increasing, co-radiant and quasi-concave (ICRQC) functions defined on X X (where X X is a real locally convex topological vector space). For this purpose, we first gave different characterizations of the upper support set’s minimal elements of non-positive co-radiant functions. Then, we presented sufficient and necessary conditions for the global minimizers of the differences of two non-positive ICRQC functions.

In this article, we study some Hermite-Hadamard-type inequalities for strongly h h -convex functions on co-ordinates in R n {{\mathbb{R}}}^{n} , which extend some known results. Some mappings connected with these inequalities and related applications are also obtained.

This article is devoted to the global existence and extinction behavior of the weak solution to a fast diffusion p p -Laplace equation with logarithmic nonlinearity and special medium void. By applying energy estimates approach, Hardy-Littlewood-Sobolev inequality, and some ordinary differential inequalities, the global existence result is proved and the sufficient conditions on the occurrence of the extinction and nonextinction phenomena are given.

Fejér’s integral inequality is a weighted version of the Hermite-Hadamard inequality that holds for the class of convex functions. To derive his inequality, Fejér [ Über die Fourierreihen, II , Math. Naturwiss, Anz. Ungar. Akad. Wiss. 24 (1906), 369–390] assumed that the weight function is symmetric w.r.t. the midpoint of the interval. In this study, without assuming any symmetry condition on the weight function, Fejér’s inequality is extended to the class of sub-biharmonic functions, namely, the set of functions f ∈ C 4 ( I ) f\in {C}^{4}\left({\mathbb{I}}) satisfying f ″ ″ ≤ 0 {f}^{^{\prime\prime} ^{\prime\prime} }\le 0 , where I {\mathbb{I}} is an interval of R {\mathbb{R}} . In the special case when the weight function is symmetric w.r.t. the midpoint of some interval, and the function f f is convex and sub-biharmonic, an interesting refinement of Fejér’s inequality is deduced. Moreover, this inequality is extended to a new class of functions defined on the plane, that we call the set of sub-biharmonic functions on the coordinates. Assuming in addition that the function is convex on the coordinates, a refinement of an inequality due to Dragomir [ On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane , Taiwan. J. Math. 5 (2001), 775–788] is obtained.

Reflexivity is characterized by the weak-compactness of the unit ball. The weak-compactness of bounded, closed, and convex sets is characterized through sup-attaining functionals in view of the famous James’ theorem. The aim of this mathematical note is to provide the construction of bounded, closed, and convex subsets for which there exists a functional attaining its supremum on such a set but not its infimum. This construction leads to a characterization of reflexivity in the category of normed spaces and a characterization of full norm-attainment also in the category of normed spaces.

This study presents a regularization method for the two membranes problem with non-homogeneous boundary conditions. We establish both convergence results and a priori estimates for this method. Using duality theory from convex analysis, we identify the dual problem related to the two membranes problem. Based on this dual formulation, we give a posteriori error estimates for both the continuous and discrete versions of the problem. These a posteriori error estimates are crucial for the practical implementation of the regularized problem.

We prove that a finite pointed abelian group generates a finitely axiomatizable variety that has a finite quasivariety lattice. As a consequence, we obtain that a quasivariety generated by a finite pointed abelian group has a finite basis of quasi-identities. The problems arising from the results obtained are also discussed.

The article explores research findings akin to Amitsur’s theorem, asserting that any derivation within a matrix ring can be expressed as the sum of an inner derivation and a hereditary derivation. In most results related to rings and semirings, Birkenmeier’s semicentral idempotents play a crucial role. This article is intended for PhD students, postdocs, and researchers.

Fractional partial differential equations (FPDEs) have gained significant attention in various scientific and engineering fields due to their ability to describe complex phenomena with memory and long-range interactions. Solving FPDEs analytically can be challenging, leading to a growing need for efficient numerical methods. This review article presents the recent analytical and numerical methods for solving FPDEs, where the fractional derivatives are assumed in Riemann-Liouville’s sense, Caputo’s sense, Atangana-Baleanu’s sense, and others. The primary objective of this study is to provide an overview of numerical techniques commonly used for FPDEs, focusing on appropriate choices of fractional derivatives and initial conditions. This article also briefly illustrates some FPDEs with exact solutions. It highlights various approaches utilized for solving these equations analytically and numerically, considering different fractional derivative concepts. The presented methods aim to expand the scope of analytical and numerical solutions available for time-FPDEs and improve the accuracy and efficiency of the techniques employed.

This article surveys some results in the study of Iannelli [ Su un modello di Oberbeck-Boussinesq relativo al moto di un fluido viscoso soggetto a riscaldamento , Fisica Matematica, Istituto Lombardo (rend. Sc.) A 121 (1987), 145–191], in which the motion of a viscous, compressible fluid in a two-dimensional domain, subject to heating at the walls, is studied. A global existence and uniqueness theorem for the time-dependent problem is given, and also, under more stringent assumptions, an existence and uniqueness theorem in the stationary case is given. A theorem on the asymptotic behavior for t → ∞ t\to \infty of the time-dependent solutions is proved.

We consider the long-time global dynamics of non-autonomous Oregonator systems. This system is a coupled system of three reaction-diffusion equations, that arises from the Belousov-Zhabotinskii reaction. We first present some sufficient conditions for the existence of pullback and uniform exponential attractors for non-autonomous dynamical system. Then, we apply abstract results to prove the existence of a pullback exponential attractor for Oregonator systems affected by time-dependent forces and a uniform exponential attractor for Oregonator systems driven by quasi-periodic external forces.

In this article, we prove the following result. Let n ≥ 3 n\ge 3 be some fixed integer and let R R be a prime ring with char ( R ) ≠ ( n + 1 ) ! 2 n − 2 {\rm{char}}\left(R)\ne \left(n+1)\!{2}^{n-2} . Suppose there exists an additive mapping D : R → R D:R\to R satisfying the relation 2 n − 2 D ( x n ) = ∑ i = 0 n − 2 n − 2 i x i D ( x 2 ) x n − 2 − i + ( 2 n − 2 − 1 ) ( D ( x ) x n − 1 + x n − 1 D ( x ) ) + ∑ i = 1 n − 2 ∑ k = 2 i ( 2 k − 1 − 1 ) n − k − 2 i − k + ∑ k = 2 n − 1 − i ( 2 k − 1 − 1 ) n − k − 2 n − i − k − 1 x i D ( x ) x n − 1 − i \begin{array}{rcl}{2}^{n-2}D\left({x}^{n})& =& \left(\mathop{\displaystyle \sum }\limits_{i=0}^{n-2}\left(\genfrac{}{}{0.0pt}{}{n-2}{i}\right){x}^{i}D\left({x}^{2}){x}^{n-2-i}\right)+\left({2}^{n-2}-1)\left(D\left(x){x}^{n-1}+{x}^{n-1}D\left(x))\\ & & +\mathop{\displaystyle \sum }\limits_{i=1}^{n-2}\left(\mathop{\displaystyle \sum }\limits_{k=2}^{i}\left({2}^{k-1}-1)\left(\genfrac{}{}{0.0pt}{}{n-k-2}{i-k}\right)+\mathop{\displaystyle \sum }\limits_{k=2}^{n-1-i}\left({2}^{k-1}-1)\left(\genfrac{}{}{0.0pt}{}{n-k-2}{n-i-k-1}\right)\right){x}^{i}D\left(x){x}^{n-1-i}\end{array} for all x ∈ R . x\in R. In this case, D D is a derivation. This result is related to a classical result of Herstein, which states that any Jordan derivation on a prime ring with char ( R ) ≠ 2 {\rm{char}}\left(R)\ne 2 is a derivation.

In this article, we prove that the product of two algebraic numbers of degrees 4 and 6 over Q {\mathbb{Q}} cannot be of degree 8. This completes the classification of so-called product-feasible triplets ( a , b , c ) ∈ N 3 \left(a,b,c)\in {{\mathbb{N}}}^{3} with a ≤ b ≤ c a\le b\le c and b ≤ 7 b\le 7 . The triplet ( a , b , c ) \left(a,b,c) is called product-feasible if there are algebraic numbers α , β \alpha ,\beta , and γ \gamma of degrees a , b a,b , and c c over Q {\mathbb{Q}} , respectively, such that α β = γ \alpha \beta =\gamma . In the proof, we use a proposition that describes all monic quartic irreducible polynomials in Q [ x ] {\mathbb{Q}}\left[x] with four roots of equal moduli and is of independent interest. We also prove a more general statement, which asserts that for any integers n ≥ 2 n\ge 2 and k ≥ 1 k\ge 1 , the triplet ( a , b , c ) = ( n , ( n − 1 ) k , n k ) \left(a,b,c)=\left(n,\left(n-1)k,nk) is product-feasible if and only if n n is a prime number. The choice ( n , k ) = ( 4 , 2 ) \left(n,k)=\left(4,2) recovers the case ( a , b , c ) = ( 4 , 6 , 8 ) \left(a,b,c)=\left(4,6,8) as well.

In this study, we solve the system of additive functional equations: h ( x + y ) = h ( x ) + h ( y ) , g ( x + y ) = f ( x ) + f ( y ) , 2 f x + y 2 = g ( x ) + g ( y ) , \left\{\begin{array}{l}h\left(x+y)=h\left(x)+h(y),\\ g\left(x+y)=f\left(x)+f(y),\\ 2f\left(\frac{x+y}{2}\right)=g\left(x)+g(y),\end{array}\right. and we investigate the stability of (homomorphism, derivation)-systems in Banach algebras.

This article introduces new Young-type inequalities, leveraging the Kantorovich constant, by refining the original inequality. In addition, we present a range of norm-based inequalities applicable to positive semidefinite matrices, such as the Hilbert-Schmidt norm and the trace norm. The importance of these results lies in their dual significance: they hold inherent value on their own, and they also extend and build upon numerous established results within the existing literature.

This article focuses on local and global existence and uniqueness for the strong solution to the Boussinesq system in R n {{\mathbb{R}}}^{n} ( n ≥ 3 n\ge 3 ) with full viscosity in Besov spaces. Under the hypotheses 1 < p < ∞ 1\lt p\lt \infty and − min { n ∕ p , 2 − n ∕ p } < s ≤ n ∕ p -\min \left\{n/p,2-n/p\right\}\lt s\le n/p , and the initial condition ( θ 0 , u 0 ) ∈ B ˙ p , 1 s − 1 × B ˙ p , 1 n ∕ p − 1 \left({\theta }_{0},{u}_{0})\in {\dot{B}}_{p,1}^{s-1}\times {\dot{B}}_{p,1}^{n/p-1} , the Boussinesq system is proved to have a unique local strong solution. Under the hypotheses n ≤ p < ∞ n\le p\lt \infty and − n ∕ p < s ≤ n ∕ p -n/p\lt s\le n/p , or especially n ≤ p < 2 n n\le p\lt 2n and − n ∕ p < s < n ∕ p − 1 -n/p\lt s\lt n/p-1 , and the initial condition ( θ 0 , u 0 ) ∈ ( B ˙ p , 1 s − 1 ∩ L n ∕ 3 ) × ( B ˙ p , 1 n ∕ p − 1 ∩ L n ) \left({\theta }_{0},{u}_{0})\in \left({\dot{B}}_{p,1}^{s-1}\cap {L}^{n/3})\times \left({\dot{B}}_{p,1}^{n/p-1}\cap {L}^{n}) with sufficiently small norms ‖ θ 0 ‖ L n ∕ 3 {\Vert {\theta }_{0}\Vert }_{{L}^{n/3}} and ‖ u 0 ‖ L n {\Vert {u}_{0}\Vert }_{{L}^{n}} , the Boussinesq system is proved to have a unique global strong solution.

The symmetric stochastic inverse eigenvalue problem (SSIEP) asks which lists of real numbers occur as the spectra of symmetric stochastic matrices. When the cardinality of a list is 4, Kaddoura and Mourad provided a sufficient condition for SSIEP by a mapping and convexity technique. They also conjectured that the sufficient condition is the necessary condition. This study presents the same sufficient condition for SSIEP, but we do it in terms of the list elements. In this way, we provide a different but more straightforward construction of symmetric stochastic matrices for SSIEP compared to those of Kaddoura and Mourad.

A conjecture of Mallows and Sloane conveys the dominance of Hilbert series for finding basic invariants of finite linear groups if the Hilbert series of the invariant ring is of a certain explicit canonical form. However, the conjecture does not hold in general by a well-known counterexample of Stanley. In this article, we give a constraint on lower bounds for the degrees of homogeneous system of parameters of rings of invariants of finite linear groups depending on the universal denominator of Hilbert series defined by Derksen. We consider the conjecture with the universal denominator on abelian groups and provide some criteria guaranteeing the existence of homogeneous system of parameters of certain degrees. In this case, Stanley’s counterexample could be avoided, and the homogeneous system of parameters is optimal.

It is shown that the expression as A 2 = α A + β P {A}^{2}=\alpha A+\beta P for generalized quadratic matrices is not unique by numerical examples. Then it is proven that the uniqueness of expression for generalized quadratic matrices is concerned not only with the properties of A A but also with the rank of P P . Furthermore, the sufficient and necessary conditions for the uniqueness of generalized quadratic matrices’expression are obtained. Finally, some related discussions about generalized quadratic matrices are also given.

The main purpose of this article is to study the calculating problem of the fourth power mean of the two-term exponential sums and provide an accurate calculating formula for utilizing analytical methods and character sums’ properties. In the meantime, a result of the fourth power mean of Gauss sums is improved.

In this article, we investigate the following Schrödinger equation: − Δ u − μ ∣ x ∣ 2 u = g ( u ) in R N , -\Delta u-\frac{\mu }{{| x| }^{2}}u=g\left(u)\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N}, where N ≥ 3 N\ge 3 , μ ∣ x ∣ 2 \frac{\mu }{{| x| }^{2}} is called the Hardy potential and g g satisfies Berestycki-Lions conditions. If 0 < μ < ( N − 2 ) 2 4 0\lt \mu \lt \frac{{\left(N-2)}^{2}}{4} , we will take symmetric mountain pass approaches to prove the existence of infinitely many solutions of this problem.

A signed graph is a simple graph in which every edge has a positive or negative sign. In this article, we employ several algebraic techniques to compute the determinant of a signed graph in terms of the spectrum of a vertex-deleted subgraph. Particular cases, including vertex-deleted subgraphs without repeated eigenvalues or singular vertex-deleted subgraphs are considered. As applications, an algorithm for the determinant of a signed graph with pendant edges is established, the determinant of a bicyclic graph and the determinant of a chain graph are computed. In the end, the uniqueness of the polynomial reconstruction for chain graphs is proved.

In this article, we prove two results on the value distribution of meromorphic functions. Using the theorem of Yamanoi, the first result gives a precise estimation of the relationship between the characteristic function of a meromorphic function and its k k th derivative in a concise form. This result extends and improves some results of Shan, Singh, Gopalakrishna, Edrei, Weitsman, Yang, Wu and Wu, etc. The second result answers a conjecture posed by C. C. Yang. This conjecture turned to be false by a counter-example, but it will be true with an additional condition.

Let ℜ \Re be a commutative ring with unity and ℑ \Im be a left ℜ \Re -module. We define the secondary-like spectrum of ℑ \Im to be the set of all secondary submodules K K of ℑ \Im such that the annihilator of the socle of K K is the radical of the annihilator of K K , and we denote it by Spec L ( ℑ ) {{\rm{Spec}}}^{L}\left(\Im ) . In this study, we introduce a topology on Spec L ( ℑ ) {{\rm{Spec}}}^{L}\left(\Im ) having the Zariski topology on the second spectrum Spec s ( ℑ ) {{\rm{Spec}}}^{s}\left(\Im ) as a subspace topology and study several topological structures of this topology.

In this article, we first put forward the concept of deferred f -double natural density for double sequences, where f is an unbounded modulus. Then, we combine f -density with deferred statistical convergence for double sequences and investigate deferred f -statistical convergence and strongly deferred Cesàro summability with respect to modulus f . Moreover, we extend these concepts to deferred f -statistical convergence for double sequences of random variables in the Wijsman sense and prove some inclusions. Finally, we consider the concepts of deferred f -statistical convergence of order α \alpha and strongly deferred f -summability of order α \alpha for double sequences and obtain some conclusions.

Let R R be a commutative ring with identity and j {\mathscr{j}} an ideal of R R . An ideal I I of R R is said to be a j {\mathscr{j}} -ideal if I ⊈ j I\hspace{0.33em} \nsubseteq \hspace{0.33em}{\mathscr{j}} . We define R R to be a j {\mathscr{j}} -Noetherian ring if each j {\mathscr{j}} -ideal of R R is finitely generated. In this work, we study some properties of j {\mathscr{j}} -Noetherian rings. More precisely, we investigate j {\mathscr{j}} -Noetherian rings via the Cohen-type theorem, the flat extension, decomposable ring, the trivial extension ring, the amalgamated duplication, the polynomial ring extension, and the power series ring extension.

In this article, the complete convergence and the Kolmogorov strong law of large numbers for weighted sums of ( α , β ) \left(\alpha ,\beta ) -mixing random variables are presented. An application to simple linear errors-in-variables model is provided. Simulation studies are also carried out to support the theoretical results.

Asymptotic formulae are established for the number of natural numbers m m with largest square-free divisor not exceeding m ϑ {m}^{{\vartheta }} , for any fixed positive parameter ϑ {\vartheta } . Related counting functions are also considered.

In this study, we are concerned with the existence and exponential stability issue of a delayed differential neoclassical growth model with discontinuous control strategy. By employing the Filippov’s theory and dichotomy theory, together with the Lyapunov functional method, novel criteria on existence and exponential stability are established for the addressed model. The established theoretical results extend and supplement the related results in the existing literature. Moreover, a simulation example is presented to verify the practicability of the proposed results.

A new distributionally robust ratio optimization model is proposed under the known first and second moments of the uncertain distributions. In this article, both standard deviation (SD) and conditional value-at-risk (CVaR) are used to measure the risk, avoiding both fat-tail and volatility. The new model can be reduced to a simple distributionally robust model under assumptions on the measurements of reward, CVaR and SD. Furthermore, it can be rewritten as a tractable semi-definite programming problem by the duality theorem under partially known information of the uncertain parameters. Finally, the model is tested on portfolio problems and verified from numerical results that it can give a reasonable decision under only the first and second moments.

In this study, we consider a viscoelastic Shear beam model with no rotary inertia. Specifically, we study ρ 1 φ t t − κ ( φ x + ψ ) x + ( g ∗ φ x x ) ( t ) = 0 , − b ψ x x + κ ( φ x + ψ ) = 0 , \begin{array}{rcl}{\rho }_{1}{\varphi }_{tt}-\kappa {\left({\varphi }_{x}+\psi )}_{x}+\left(g\ast {\varphi }_{xx})\left(t)& =& 0,\\ -b{\psi }_{xx}+\kappa \left({\varphi }_{x}+\psi )& =& 0,\end{array} where the convolution memory function g g belongs to a class of L 1 ( 0 , ∞ ) {L}^{1}\left(0,\infty ) functions that satisfies g ′ ( t ) ≤ − ξ ( t ) ϒ ( g ( t ) ) , ∀ t ≥ 0 , g^{\prime} \left(t)\le -\xi \left(t)\Upsilon \left(g\left(t)),\hspace{1.0em}\forall t\ge 0, where ξ \xi is a positive nonincreasing differentiable function and ϒ \Upsilon is an increasing and convex function near the origin. Using just this general assumptions on the behavior of g g at infinity, we provide optimal and explicit general energy decay rates from which we recover the exponential and polynomial rates when ϒ ( s ) = s p \Upsilon \left(s)={s}^{p} and p p covers the full admissible range [ 1 , 2 ) \left[1,2) . Given this degree of generality, our results improve some of earlier related results in the literature.

Let Δ = A N B A M A B B \Delta =\left(\begin{array}{cc}A& {}_{A}N_{B}\\ {}_{B}M_{A}& B\end{array}\right) be a Morita ring, where M ⊗ A N = 0 = N ⊗ B M M{\otimes }_{A}N=0=N{\otimes }_{B}M . Let X X be left A A -module and Y Y be left B B -module. We prove that ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) \left(X,M{\otimes }_{A}X,1,0)\oplus \left(N{\otimes }_{B}Y,Y,0,1) is a silting module if and only if X X is a silting A A -module, Y Y is a silting B B -module, M ⊗ A X M{\otimes }_{A}X is generated by Y Y , and N ⊗ B Y N{\otimes }_{B}Y is generated by X X . As a consequence, we obtain that if M A {M}_{A} and N B {N}_{B} are flat, then ( X , M ⊗ A X , 1 , 0 ) ⊕ ( N ⊗ B Y , Y , 0 , 1 ) \left(X,M{\otimes }_{A}X,1,0)\oplus \left(N{\otimes }_{B}Y,Y,0,1) is a tilting Δ \Delta -module if and only if X X is a tilting A A -module, Y Y is a tilting B B -module, M ⊗ A X M{\otimes }_{A}X is generated by Y Y , and N ⊗ B Y N{\otimes }_{B}Y is generated by X X .

We study linear differential equations whose coefficients consist of products of powers of natural logarithm and general continuous functions. We derive conditions that guarantee the non-oscillation of all non-trivial solutions of the treated type of equations. The conditions are formulated as a non-oscillation criterion, which is the counterpart of a previously obtained oscillation theorem. Therefore, from the presented main result, it follows that the analysed equations are conditionally oscillatory. The used method is based on averaging techniques for the combination of the generalized adapted Prüfer angle and the modified Riccati transformation. This article is finished by new corollaries and examples.

In this study, we give a synthetic construction of a complete system of mutually unbiased bases in C 3 {{\mathbb{C}}}^{3} .

In this article, we study the Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three, of hyperbolic type, using Bielecki norm. Sufficient conditions are established to ensure Hyers-Ulam and Hyers-Ulam-Rassias stability for these equations. These types of equations appear in various applications in engineering, biology, chemistry, economics (price fluctuation – Black-Scholes equation), etc.

In this work, we define the notions of Wronskian and simplified Wronskian for Stieltjes derivatives and study some of their properties in a similar manner to the context of time scales or the usual derivative. Later, we use these tools to investigate second-order linear differential equations with Stieltjes derivatives to find linearly independent solutions, as well as to derive the variation of parameters method for problems with g g -continuous coefficients. This theory is later illustrated with some examples such as the study of the one-dimensional linear Helmholtz equation with piecewise-constant coefficients.

The transition between strong and weak Allee effects in prey provides a simple regime shift in ecology. In this article, we study a discrete predator-prey system with Holling type II functional response and Allee effect. First, the number of fixed points of the system, local stability, and global stability is discussed. The population changes of predator and prey under strong or weak Allee effects are proved using the nullclines and direction field, respectively. Second, using the bifurcation theory, the bifurcation conditions for the system to undergo transcritical bifurcation and Neimark-Sacker bifurcation at the equilibrium point are obtained. Finally, the dynamic behavior of the system is analyzed by numerical simulation of bifurcation diagram, phase diagram, and maximum Lyapunov exponent diagram. The results show that the system will produce complex dynamic phenomena such as periodic state, quasi-periodic state, and chaos.

In this article, we study a p -Laplacian Schrödinger-Poisson system involving a parameter q ≠ 0 q\ne 0 in bounded domains. By using the Nehari manifold and the fibering method, we obtain the non-existence and multiplicity of nontrivial solutions. On one hand, there exists q * > 0 {q}^{* }\gt 0 such that only the trivial solution is admitted for q ∈ ( q * , + ∞ ) . q\in \left({q}^{* },+\infty ). On the other hand, there are two positive solutions existing for q ∈ ( 0 , q 0 * + ε ) q\in \left(0,{q}_{0}^{* }+\varepsilon ) , where ε > 0 \varepsilon \gt 0 and q 0 * + ε < q * . {q}_{0}^{* }+\varepsilon \lt {q}^{* }. In particular, q * {q}^{* } and q 0 * {q}_{0}^{* } correspond to the supremum for the nonlinear generalized Rayleigh quotients, respectively. The specific form of the nonlinear generalized Rayleigh quotients is calculated. Moreover, it is worth mentioning that we also obtain the qualitative properties associated with the energy level of the solutions.

For a given sequence a = ( a 1 , … , a n ) ∈ R n a=\left({a}_{1},\ldots ,{a}_{n})\in {{\mathbb{R}}}^{n} , our aim is to obtain an estimate of E n ≔ a 1 + a n 2 − 1 n ∑ i = 1 n a i {E}_{n}:= \left|\hspace{-0.33em},\frac{{a}_{1}+{a}_{n}}{2}-\frac{1}{n}{\sum }_{i=1}^{n}{a}_{i},\hspace{-0.33em}\right| . Several classes of sequences are studied. For each class, an estimate of E n {E}_{n} is obtained. We also introduce the class of convex matrices, which is a discrete version of the class of convex functions on the coordinates. For this set of matrices, new discrete Hermite-Hadamard-type inequalities are proved. Our obtained results are extensions of known results from the continuous case to the discrete case.

We investigate third-order strongly nonlinear differential equations of the type ( Φ ( k ( t ) u ″ ( t ) ) ) ′ = f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) , a.e. on [ 0 , T ] , \left(\Phi \left(k\left(t){u}^{^{\prime\prime} }\left(t)))^{\prime} =f\left(t,u\left(t),u^{\prime} \left(t),{u}^{^{\prime\prime} }\left(t)),\hspace{1em}\hspace{0.1em}\text{a.e. on}\hspace{0.1em}\hspace{0.33em}\left[0,T], where Φ \Phi is a strictly increasing homeomorphism, and the non-negative function k k may vanish on a set of measure zero. Using the upper and lower solution method, we prove existence results for some boundary value problems associated with the aforementioned equation. Moreover, we also consider second-order integro-differential equations like ( Φ ( k ( t ) v ′ ( t ) ) ) ′ = f t , ∫ 0 t v ( s ) d s , v ( t ) , v ′ ( t ) , a.e. on [ 0 , T ] , \left(\Phi \left(k\left(t)v^{\prime} \left(t)))^{\prime} =f\left(t,\underset{0}{\overset{t}{\int }}v\left(s){\rm{d}}s,v\left(t),v^{\prime} \left(t)\right),\hspace{1em}\hspace{0.1em}\text{a.e. on}\hspace{0.1em}\hspace{0.33em}\left[0,T], for which we provide existence results for various types of boundary conditions, including periodic, Sturm-Liouville, and Neumann-type conditions.

In this article, we consider the following quasilinear Schrödinger system: − ε Δ u + u + k 2 ε [ Δ ∣ u ∣ 2 ] u = 2 α α + β ∣ u ∣ α − 2 u ∣ v ∣ β , x ∈ R N , − ε Δ v + v + k 2 ε [ Δ ∣ v ∣ 2 ] v = 2 β α + β ∣ u ∣ α ∣ v ∣ β − 2 v , x ∈ R N , \left\{\begin{array}{ll}-\varepsilon \Delta u+u+\frac{k}{2}\varepsilon \left[\Delta \hspace{-0.25em}{| u| }^{2}]u=\frac{2\alpha }{\alpha +\beta }{| u| }^{\alpha -2}u{| v| }^{\beta },& x\in {{\mathbb{R}}}^{N},\\ -\varepsilon \Delta v+v+\frac{k}{2}\varepsilon \left[\Delta \hspace{-0.25em}{| v| }^{2}]v=\frac{2\beta }{\alpha +\beta }{| u| }^{\alpha }{| v| }^{\beta -2}v,& x\in {{\mathbb{R}}}^{N},\end{array}\right. where ε > 0 , k < 0 \varepsilon \gt 0,k\lt 0 are real constants, N ≥ 3 N\ge 3 , α , β \alpha ,\beta are integers multiple of constant 2. By using the Mountain Pass Theorem in a suitable Orlicz space proposed by Abbas Moameni [ Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in R N {{\mathbb{R}}}^{N} , J. Differential Equations 229 (2006), 570–587], we proved the existence of soliton solution ( u ε , v ε ) \left({u}_{\varepsilon },{v}_{\varepsilon }) for the above system, and ( u ε ( x ) , v ε ( x ) ) → ( 0 , 0 ) ({u}_{\varepsilon }\left(x),{v}_{\varepsilon }\left(x))\to \left(0,0) as ∣ ε ∣ → 0 | \varepsilon | \to 0 .

In this article, we establish Hermite-Hadamard-type inequalities for the two classes of functions X ± λ ( Ω ) = { f ∈ C 2 ( Ω ) : Δ f ± λ f ≥ 0 } {X}_{\pm \lambda }\left(\Omega )=\{f\in {C}^{2}\left(\Omega ):\Delta f\pm \lambda f\ge 0\} , where λ > 0 \lambda \gt 0 and Ω \Omega is an open subset of R 2 {{\mathbb{R}}}^{2} . We also obtain a characterization of the set X − λ ( Ω ) {X}_{-\lambda }\left(\Omega ) . Notice that in the one-dimensional case, if Ω = I \Omega =I (an open interval of R {\mathbb{R}} ) and λ = ρ 2 \lambda ={\rho }^{2} , ρ > 0 \rho \gt 0 , then X λ ( Ω ) {X}_{\lambda }\left(\Omega ) (resp. X − λ ( Ω ) {X}_{-\lambda }\left(\Omega ) ) reduces to the class of functions f ∈ C 2 ( I ) f\in {C}^{2}\left(I) such that f f is trigonometrically ρ \rho -convex (resp. hyperbolic ρ \rho -convex) on I I .

In this note, we prove that the toroidal pseudo-differential operator is bounded from L ∞ ( T n ) {L}^{\infty }\left({{\mathbb{T}}}^{n}) to BMO ( T n ) {\rm{BMO}}\left({{\mathbb{T}}}^{n}) if the symbol belongs to the toroidal Hörmander class S ρ , δ n ( ρ − 1 ) ∕ 2 ( T n × Z n ) {S}_{\rho ,\delta }^{n\left(\rho -1)/2}\left({{\mathbb{T}}}^{n}\times {{\mathbb{Z}}}^{n}) with 0 < ρ ≤ 1 0\lt \rho \le 1 and 0 ≤ δ < 1 0\le \delta \lt 1 . As a corollary, we obtain a result of toroidal pseudo-differential operators on L p {L}^{p} when 2 < p < ∞ 2\lt p\lt \infty for symbols in the class S ρ , δ m ( T n × Z n ) {S}_{\rho ,\delta }^{m}\left({{\mathbb{T}}}^{n}\times {{\mathbb{Z}}}^{n}) with m ≤ n ( ρ − 1 ) 1 2 − 1 p + n p min { 0 , ρ − δ } m\le n\left(\rho -1)\left(\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2}-\frac{1}{p}\right)+\frac{n}{p}\min \left\{0,\rho -\delta \right\} .

We consider the tensor products of square matrices of different sizes and introduce the stretching maps, which can be viewed as a generalized matricization. Stretching maps conserve algebraic properties of the tensor product, but are not necessarily injective. Dropping the injectivity condition allows us to construct examples of stretching maps with additional symmetry properties. Furthermore, this leads to the averaging of the tensor product and possibly could be used to compress the data.

In this article, we obtain a very sharp version of some singular perturbation results going back to Dancer and Hess [ Behaviour of a semilinear periodic-parabolic problem when a parameter is small , Lecture Notes in Mathematics, Vol. 1450, Springer-Verlag, Berlin, 1990, pp. 12–19] and Daners and López-Gómez [ The singular perturbation problem for the periodic-parabolic logistic equation with indefinite weight functions , J. Dynam. Differential Equations 6 (1994), 659–670] valid for a general class of semilinear periodic-parabolic problems of logistic type under general boundary conditions of mixed type. The results of Dancer and Hess [ Behaviour of a semilinear periodic-parabolic problem when a parameter is small , Lecture Notes in Mathematics, Vol. 1450, Springer-Verlag, Berlin, 1990, pp. 12–19] and [ The singular perturbation problem for the periodic-parabolic logistic equation with indefinite weight functions , J. Dynam. Differential Equations 6 (1994), 659–670] were found, respectively, for Neumann and Dirichlet boundary conditions with L = − Δ {\mathfrak{L}}=-\Delta . In this article, L {\mathfrak{L}} stands for a general second-order elliptic operator.

In this manuscript, we study some algebraic and differential properties of matrix orthogonal polynomials with respect to the Laguerre-Sobolev right sesquilinear form defined by ⟨ p , q ⟩ S ≔ ∫ 0 ∞ p * ( x ) W L A ( x ) q ( x ) d x + M ∫ 0 ∞ ( p ′ ( x ) ) * W ( x ) q ′ ( x ) d x , {\langle p,q\rangle }_{{\bf{S}}}:= \underset{0}{\overset{\infty }{\int }}{p}^{* }\left(x){{\bf{W}}}_{{\bf{L}}}^{{\bf{A}}}\left(x)q\left(x){\rm{d}}x+{\bf{M}}\underset{0}{\overset{\infty }{\int }}{(p^{\prime} \left(x))}^{* }{\bf{W}}\left(x)q^{\prime} \left(x){\rm{d}}x, where W L A ( x ) = e − λ x x A {{\bf{W}}}_{{\bf{L}}}^{{\bf{A}}}\left(x)={e}^{-\lambda x}{x}^{{\bf{A}}} is the Laguerre matrix weight, W {\bf{W}} is some matrix weight, p p and q q are the matrix polynomials, M {\bf{M}} and A {\bf{A}} are the matrices such that M {\bf{M}} is non-singular and A {\bf{A}} satisfies a spectral condition, and λ \lambda is a complex number with positive real part.

This article deals with the asymptotic behavior of fractional lattice systems with time-varying delays in weighted space. First, we establish some sufficient conditions for the existence and uniqueness of solutions. Subsequently, we demonstrate the existence of pullback attractors for the considered fractional lattice systems.

In this article, we mainly study the geometric properties of spherical surface of a curve on a hypersurface Σ \Sigma in four-dimensional Euclidean space. We define a family of tangent height functions of a curve on Σ \Sigma as the main tool for research and combine the relevant knowledge of singularity theory. It is shown that there are three types of singularities of spherical surface, that is, in the local sense, the spherical surface is respectively diffeomorphic to the cuspidal edge, the swallowtail, and the cuspidal beaks. In addition, we give two examples of the spherical surface.

In this study, we consider a Kirchhoff-type second-order impulsive differential system with the Dirichlet boundary condition and obtain the existence and multiplicity of solutions to the impulsive problem via variational methods.

This article can be considered as a continuation of Petrović and Milošević [ The truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay , Filomat 35 (2021), no. 7, 2457–2484], where the authors established the L q {L}^{q} -convergence of the truncated Euler-Maruyama (EM) method for neutral stochastic differential equations with time-dependent delay under the Khasminskii-type condition. However, the convergence rate of the method has not been studied there, which is the main goal of this article. Also, there are some restrictions on the truncated coefficients of the considered equations, and these restrictions sometimes might force the step size to be so small that the application of the truncated EM method would be limited. Therefore, the convergence rate without these restrictions will be considered in this article. Moreover, one of the sufficient conditions for obtaining the main result of this article, which is related to Lipschitz constants for the neutral term and delay function, is weakened. In that way, some of the results of the cited article are generalized. The main result of this article is proved by employing two conditions related to the increments to the coefficients and the neutral term of the equations under consideration, among other conditions. The main theoretical result is illustrated by an example.

In this article, we study a mathematical model for a one-dimensional suspension bridge problem with nonlinear damping. The model takes into consideration the vibration of the bridge deck in the vertical plane and main cable from which the bridge deck is suspended by the suspenders. We use the multiplier method to establish explicit and generalized decay results, without imposing restrictive growth assumption near the origin on the damping terms. Our results substantially improve, extend, and generalize some earlier related results in the literature.

This article proves several weak limit theorems for the joint version of extreme order statistics and partial sums of independently and identically distributed random variables. The results are also extended to almost sure limit version.

In this study, we discuss new Hardy-type inequalities for operators involving iteration and provide explicit characterizations of these inequalities. As an application of our results, we consider the problem of the boundedness of the multidimensional Hardy operator from a Lebesgue space to a Morrey-type space.

In this study, a multipoint boundary value problem for Volterra-Fredholm integro-differential equations is considered. The addition of a new function converts the system of Volterra-Fredholm integro-differential equations to a system of Fredholm integro-differential equations. In contrast to the original problem, the dimension of a Fredholm integro-differential equation is determined by the number of matrices in the degenerate kernel of the Volterra integral. A numerical algorithm of Dzhumabaev parameterization method for addressing a multipoint boundary value problem for Volterra-Fredholm integro-differential equations is proposed. The main advantage of the proposed method is splitting the problem into auxiliary Cauchy problems for ordinary differential equations and a system of algebraic equations with respect to the parameters. The conditions for the unique solvability of the multipoint boundary value problem for Fredholm integro-differential equations are established. Finally, various numerical examples are provided to demonstrate the efficiency and correctness of the suggested technique.

The aim of this article is to study how the greatest common divisor of the degree and codegree of an irreducible character of a finite group influences its structure. We study a finite group G G with gcd ( χ ( 1 ) , χ c ( 1 ) ) {\rm{\gcd }}\left(\chi \left(1),{\chi }^{c}\left(1)) a prime for almost all irreducible characters χ \chi of G G , and obtain the following two conclusions: (1) There does not exist any finite group G G such that gcd ( χ ( 1 ) , χ c ( 1 ) ) {\rm{\gcd }}\left(\chi \left(1),{\chi }^{c}\left(1)) is a prime, for each χ ∈ Irr ( G ) ♯ \chi \in {\rm{Irr}}{\left(G)}^{\sharp } , where Irr ( G ) ♯ {\rm{Irr}}{\left(G)}^{\sharp } is the set of non-principal irreducible characters of G G . (2) Let G G be a finite group, if gcd ( χ ( 1 ) , χ c ( 1 ) ) {\rm{\gcd }}\left(\chi \left(1),{\chi }^{c}\left(1)) is a prime, for each χ ∈ Irr ( G ) \ Lin ( G ) \chi \left\in {\rm{Irr}}\left(G)\backslash {\rm{Lin}}\left(G) , then G G is solvable, where Lin ( G ) {\rm{Lin}}\left(G) is the set of all linear irreducible characters of G G .

The multivariate Gaussian random fields with matrix-based scaling laws are widely used for inference in statistics and many applied areas. In such contexts, interests are often Hölder regularities of spatial surfaces in any given direction. This article analyzes the almost sure sample function behavior for operator fractional Brownian motion, including multivariate fractional Brownian motion. We obtain the estimations of small ball probability and the strongly locally nondeterministic for operator fractional Brownian motion in any given direction. By applying these estimates, we obtain Chung type laws of the iterated logarithm for operator fractional Brownian motion. Our results show that the precise Hölder regularities of these spatial surfaces are completely determined by the real parts of the eigenvalues of self-similarity exponent and the covariance matrix at time point 1.

The aim of this study is to develop the theory of prime ideals in ordered semigroups. First, to ensure the existence of prime ideals, we study a class of ordered semigroups which will be denoted by S I P {{\mathbb{S}}}_{IP} . And then we introduce the hull-kernel topology for the prime ideals P ( S ) {\mathcal{P}}\left(S) and the topological properties like separation axioms, compactness and connectedness are studied. Finally, we focus on the subspace ℳ ( S , I ) {\mathcal{ {\mathcal M} }}\left(S,I) , minimal prime ideals containing the ideal I I in an ordered semigroup S S . We investigate topological properties of this subspace and connections between this subspace and the ordered semigroup S S .

The theory of generalized metric spaces is an active topic in general topology. In this article, we utilize the concepts of ideal convergence and networks to discuss the metrization problem and the mutual classification problem between spaces and mappings in topological spaces. We define ℐ {\mathcal{ {\mathcal I} }} - s n sn -metrizable spaces, obtain several characterizations of ℐ {\mathcal{ {\mathcal I} }} - s n sn -metrizable spaces, and establish some mapping relations between ℐ {\mathcal{ {\mathcal I} }} - s n sn -metrizable spaces and semi-metric spaces. These not only generalize some theorems in generalized metric theory, but also find further applications of ideal convergence in general topology.

In this study, the strong law of large numbers and the convergence rate for weighted sums of non-identically distributed widely orthant dependent random variables are established. As applications, the strong consistency for weighted estimator in nonparametric regression model and the rate of strong consistency for least-squares estimator in multiple linear regression model are obtained. Some numerical simulations are also provided to verify the validity of the theoretical results.

This note focuses on establishing a fractional version akin to the Schweitzer inequality, specifically tailored to accommodate the left-sided Riemann-Liouville fractional integral operator. The Schweitzer inequality is a fundamental mathematical expression, and extending it to the fractional realm holds significance in advancing our understanding and applications of fractional calculus.

In this work, we first define two special sets of real numbers, and then, we construct a half-discrete kernel function where the variables are defined in the whole plane, and the parameters in the kernel function are limited to the newly constructed special sets. Estimate the kernel function in the whole plane by converting it to the first quadrant, and then, a class of new Hilbert-type inequality is established. Additionally, it is proved that the constant factor of the newly established inequality is the best possible. Furthermore, assigning special values to the parameters and using rational fraction expansion of cosecant function, some special results are presented at the end of this article.

In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate are proved on both the coarse grid and fine grid. It is shown that the two-grid method can achieve asymptotically optimal approximation as long as the coarse grid size H H and the fine grid size h h satisfy h = O ( H ( 2 k + 1 ) ⁄ ( k + 1 ) ) h={\mathcal{O}}\left({H}^{\left(2k+1)/\left(k+1)}) ( k ≥ 1 k\ge 1 ), where k k is the degree of the approximating space for the primary variable. Numerical experiment is presented to demonstrate the accuracy and the efficiency of the proposed method.

Stability estimates are proposed for two variants of Markov processes defined by stochastic difference equations: uncontrolled and controlled. Processes of this type are widely used in applications where their “governing distributions” are known only approximately, for example, as statistical estimates obtained from real data. Therefore, the problem of estimating deviations of output characteristics arises. The Kantorovich metric is used to measure the variations of probability distributions that govern the processes. In the uncontrolled case, the Kantorovich distance between the stationary distributions of the initial process and its perturbation is evaluated. On the other hand, the control processes being compared are endowed with an expected total discounted cost, and the inequality for the corresponding stability index is obtained. The stability index measures the increase in costs when using the control policy optimal for the “approximating process.”

Three classes of trigonometric integrals involving an integer parameter are evaluated by the contour integration and the residue theorem. The resulting formulae are expressed in terms of Riemann zeta function and Dirichlet beta function. Several remarkable integral identities are presented.

On a real hypersurface M M in complex projective space, we can define two tensor fields of type (1, 2), A F ( k ) {A}_{F}^{\left(k)} and A T ( k ) {A}_{T}^{\left(k)} , associated with the shape operator A A of the real hypersurface, for any nonnull real number k k . Following Tachibana [ Analytic tensor and its generalization , Tohoku Math. J. 12 (1960), 208–221], we study the purity and hybridness of such tensors with respect to A A .

In this article, we study the following weighted integral system: u ( x ) = ∫ R + n + 1 y n + 1 β f ( u ( y ) , v ( y ) ) ∣ x − y ∣ λ d y , x ∈ R + n + 1 , v ( x ) = ∫ R + n + 1 y n + 1 β g ( u ( y ) , v ( y ) ) ∣ x − y ∣ λ d y , x ∈ R + n + 1 . \left\{\begin{array}{l}u\left(x)=\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}_{+}^{n+1}}\frac{{y}_{n+1}^{\beta }f\left(u(y),v(y))}{{| x-y| }^{\lambda }}{\rm{d}}y,\hspace{1em}x\in {{\mathbb{R}}}_{+}^{n+1},\hspace{1.0em}\\ v\left(x)=\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}_{+}^{n+1}}\frac{{y}_{n+1}^{\beta }g\left(u(y),v(y))}{{| x-y| }^{\lambda }}{\rm{d}}y,\hspace{1em}x\in {{\mathbb{R}}}_{+}^{n+1}.\hspace{1.0em}\end{array}\right. Under nature structure conditions on f f and g g , we classify the positive solutions using the method of moving spheres.

We consider solving a severely ill-posed problem for determining the surface temperature and heat flux distribution of the nonhomogeneous sideways heat equation by a quasi-reversibility regularization method. An analytical solution is deduced based on the Fourier transform, and then, the logarithmic-type error estimate for the regularized solution is achieved. Finally, three numerical examples, including smooth and non-smooth functions, validate the feasibility and effectiveness of the proposed approach.

In this article, we introduce and study a new class of higher-order fractional q -difference equations involving Riemann-Liouville q -derivatives with dual hybrid terms, supplemented with nonlocal multipoint q -integral boundary conditions. The existence of a unique solution to the given problem is shown by applying Banach’s fixed point theorem. We also present existing criteria for solutions to the problem at hand with the aid of Krasnoselskii’s fixed point theorem and Leray-Schauder’s nonlinear alternative. Illustrative examples are given to demonstrate the application of the obtained results.

Let A ⊆ B A\subseteq B be an extension of commutative rings with identity, ( S , ≤ ) \left(S,\le ) a nonzero strictly ordered monoid, and S * = S \ { 0 } {S}^{* }\left=S\backslash \left\{0\right\} . Let A + 〚 B S * , ≤ 〛 = { f ∈ 〚 B S , ≤ 〛 ∣ f ( 0 ) ∈ A } A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] =\{f\in [\kern-2pt[ {B}^{S,\le }]\kern-2pt] \hspace{0.33em}| \hspace{0.33em}f\left(0)\in A\} . In this study, we determine when the ring A + 〚 B S * , ≤ 〛 A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring. We prove that when S S is a strict monoid, if A + 〚 B S * , ≤ 〛 A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring, then A A is a Noetherian ring, B B is a finitely generated A A -module, and S S is finitely generated. We also show that if B B is a finitely generated A A -module over a Noetherian ring A A and ( S , ≤ ) \left(S,\le ) is a positive strictly ordered monoid, which is finitely generated, then A + 〚 B S * , ≤ 〛 A+[\kern-2pt[ {B}^{{S}^{* },\le }]\kern-2pt] is a Noetherian ring.

In this article, we consider the asymptotic behaviour of the modified Mellin transform Z ( s ) {\mathcal{Z}}\left(s) , s = σ + i t s=\sigma +it , of the Riemann zeta-function using weak convergence of probability measures in the space of analytic functions defined by means of shifts Z ( s + i φ ( τ ) ) {\mathcal{Z}}\left(s+i\varphi \left(\tau )) , where φ ( τ ) \varphi \left(\tau ) is a real increasing to + ∞ +\infty differentiable function with monotonically decreasing derivative satisfying a certain estimate connected to the second moment of Z ( s ) {\mathcal{Z}}\left(s) . We prove in this case that the limit measure is concentrated at the point g 0 ( s ) ≡ 0 {g}_{0}\left(s)\equiv 0 . This result is applied to the approximation of g 0 ( s ) {g}_{0}\left(s) by shifts Z ( s + i φ ( τ ) ) {\mathcal{Z}}\left(s+i\varphi \left(\tau )) .

Counting perfect matchings is an interesting and challenging combinatorial task. It has important applications in statistical physics and chemistry. As the general problem is #P-complete, it is usually tackled by randomized heuristics and approximation schemes. Let G G and H H be two graphs. Denote by G × H G\times H the Cartesian product of graphs G G and H H . The number of perfect matchings of some types of Cartesian product G × H G\times H is investigated by the Pfaffian method, where G ≅ P 2 G\cong {P}_{2} , P 3 {P}_{3} , P 4 {P}_{4} or C 4 {C}_{4} , and H ≅ T H\cong T or H H is a non-bipartite graph with a unique cycle. In this article, we construct a Pfaffian orientation of the graph G × P 2 G\times {P}_{2} , where G G is a non-bipartite graph with k k odd cycles. We obtain the counting formula on the number of perfect matchings of G × P 2 G\times {P}_{2} .

By using the weight coefficients and the techniques of real analysis, a new extended Mulholland’s inequality with multi-parameters involving one partial sum is given. The equivalent statements of the best value related to several parameters are provided. The equivalent forms, some inequalities in particular parameters, and the operator expressions are obtained.

This study explores the power vector inequalities for a pair of operators ( B , C ) \left(B,C) in a Hilbert space. By utilizing a Mitrinović-Pečarić-Fink-type inequality for inner products and norms, we derive various power vector inequalities. Specifically, we consider the cases where ( B , C ) \left(B,C) is equal to ( A , A * ) \left(A,{A}^{* }) or ( Re ( A ) , Im ( A ) ) (\hspace{0.1em}\text{Re}\hspace{0.1em}\left(A),\hspace{0.1em}\text{Im}\hspace{0.1em}\left(A)) for an operator A A in B ( H ) B\left(H) , where H H is a Hilbert space. This leads to the derivation of vector, norm, and numerical radius inequalities for a single operator. Furthermore, we obtain power inequalities for the s s - r r -norm and s s - r r -numerical radius of the operator pair ( B , C ) ∈ B ( H ) \left(B,C)\in B\left(H) , which generalizes the Euclidean norm and Euclidean numerical radius. Finally, we apply these results to derive the corresponding inequalities for a single operator A ∈ B ( H ) A\in B\left(H) .

In this article, we study common zeros of the iterated derivatives of the Eisenstein series for Γ 0 + ( N ) {\Gamma }_{0}^{+}\left(N) of level N = 1 , 2 , and 3 , N=1,2, and 3, which are quasi-modular forms. More precisely, we investigate the common zeros of quasi-modular forms and prove that all the zeros of the iterated derivatives of the Eisenstein series θ m E k ( N ) {\theta }^{m}{E}_{k}^{\left(N)} of weight k = 2 , 4 , 6 k=2,4,6 for Γ 0 + ( N ) {\Gamma }_{0}^{+}\left(N) of level N = 2 , 3 N=2,3 are simple by generalizing the results of Meher and Gun-Oesterlé for SL 2 ( Z ) {{\rm{SL}}}_{2}\left({\mathbb{Z}}) .

This article aims to investigate the calculation problem of the fourth-power mean of the specific Kloosterman sums by utilizing analytic methods and the properties of classical Gauss sums. It provides a captivating calculation formula for this purpose. Additionally, an effective method for studying high-power mean of Kloosterman sums is introduced in this research.

In this article, we focus on exploring the fuzzy version of the Hyers-Ulam-Rassias stability of n n -ring homomorphisms and n n -ring derivations in the context of fuzzy Banach algebras. Our investigation utilizes the direct method.

In this article, we extend inequalities concerning the polar derivative of a polynomial to integral mean for the class of polynomials with s-fold zero at the origin and the remaining zeros inside some closed disk of radius k k for k ≥ 1 k\ge 1 and k ≤ 1 k\le 1 , and we thereby obtain bounds that depend on some coefficients of the underlying polynomial. Our results not only extend some known polynomial inequalities but also reduce some interesting results in particular cases. Furthermore, we provide numerical examples and graphical representations to demonstrate the superior precision of our results compared to previously established results.

In this work, we present necessary and sufficient conditions for the boundedness of the commutators generated by multilinear fractional maximal operators on the products of Morrey spaces when the symbol belongs to Lipschitz spaces.

In this article, we work on vector optimization problems in linear topological spaces. Our vector optimization problems have weakened convex inequality constraints and weakened affine equality constraints. Our inequalities are given by partial orders that are induced by pointed convex cones. We prove a Farkas–Minkowski-type theorem of alternative and obtain some optimality conditions through the discussions of vector saddle points and scalar saddle points.

There are many entropy inequalities in geometry, some of them can be seen as the Minkowski inequalities in the form of entropy, which play important roles in convex geometry. In this article, let P P and Q Q be convex bodies, we introduce the mixed curvature entropy M f ( P , Q ) {M}_{f}\left(P,Q) and the curvature entropy E f ( P , Q ) {E}_{f}\left(P,Q) with respect to a continuous function f f . Moreover, we establish the log \log -Minkowski inequalities of the mixed curvature entropy and the curvature entropy for two classes of three-dimensional convex bodies.

Let G G be a graph with n ( G ) n\left(G) vertices and e ( G ) e\left(G) edges, and S k ( G ) {S}_{k}\left(G) be the sum of the k k largest Laplacian eigenvalues of G G . Brouwer conjectured that S k ( G ) ≤ e ( G ) + k + 1 2 {S}_{k}\left(G)\le e\left(G)+\left(\phantom{\rule[-0.75em]{}{0ex}},\genfrac{}{}{0.0pt}{}{k+1}{2}\right) for 1 ≤ k ≤ n ( G ) 1\le k\le n\left(G) . In this article, we obtain upper bounds of S k ( G ) {S}_{k}\left(G) in terms of the graphs that contain the friendship graph or the book graph as a subgraph. Further, we show that Brouwer’s conjecture holds for Halin graphs and certain classes of graphs.

In this article, a fourth-order finite-difference ghost-point method for the Poisson equation on regular Cartesian mesh is presented. The method can be considered the high-order extension of the second-order ghost method introduced earlier by the authors. Three different discretizations are considered, which differ in the stencil that discretizes the Laplacian and the source term. It is shown that only two of them provide a stable method. The accuracy of such stable methods is numerically verified on several test problems.

This article investigates the geometric and topologic of warped product submanifolds in Riemannian warped product Q ε m × R {{\mathbb{Q}}}_{\varepsilon }^{m}\times {\mathbb{R}} . In this respect, we obtain the first Chen inequality that involves extrinsic invariants like the length of the warping functions and the mean curvature. This inequality involves two intrinsic invariants (sectional curvature and δ \delta -invariant). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the Ricci curvature gradient. We aim to apply this theory to many structures and obtain Dirichlet eigenvalues for problem applications. Some new results regarding the vanishing mean curvature are presented as a partial solution, and this can be considered for the well-known problem given by Chern.

The main purpose of this article is to study the problem of calculating the 2 m th power mean of the generalized quadratic Gauss sums, and using the analytic method and an interesting combinatorial identity to give a sharp asymptotic formula for the 2 m th power mean. Thus, a new simple proof of this existing result [N. Bag, A. Rojas-León, and W. P. Zhang, On some conjectures on generalized quadratic Gauss sums and related problems , Finite Fields Appl. 86 (2023), 10213] is provided.

In this article, we investigate the Euler- α \alpha equations in a three-dimensional bounded domain. On the one hand, we prove in the Euler setting that the equations are locally well-posed with initial data in H s ( s ≥ 3 ) {H}^{s}\left(s\ge 3) . On the other hand, the relationship between the H s {H}^{s} -norm of the velocity field and the parameter α \alpha is clarified.

Let ( A , C , ℬ ) \left({\mathcal{A}},{\mathcal{C}},{\mathcal{ {\mathcal B} }}) be a recollement of exact categories. An explicit procedure about gluing complete hereditary cotorsion pairs from A {\mathcal{A}} and ℬ {\mathcal{ {\mathcal B} }} to C {\mathcal{C}} has been established by Hu et al. to provide a new method on the construction of recollements of triangulated categories. In this article, we study when the validity of Enochs conjecture for the left-hand classes of those complete hereditary cotorsion pairs is preserved in the aforementioned gluing procedure. Applications are given to cotorsion pairs induced by the class of projective objects or Gorenstein projective objects over comma categories.

In this article, we first prove that the Hankel determinant of order three of the polynomial sequence { P n ( x ) = ∑ k ≥ 0 P ( n , k ) x k } n ≥ 0 {\left\{{P}_{n}\left(x)={\sum }_{k\ge 0}P\left(n,k){x}^{k}\right\}}_{n\ge 0} is weakly (Hurwitz) stable, where P ( n , k ) P\left(n,k) satisfies the recurrence relation P ( n , k ) = ( a 1 n + a 2 ) P ( n − 1 , k ) + ( b 1 n + b 2 ) P ( n − 1 , k − 1 ) , P\left(n,k)=\left({a}_{1}n+{a}_{2})P\left(n-1,k)+\left({b}_{1}n+{b}_{2})P\left(n-1,k-1), with P ( n , k ) = 0 P\left(n,k)=0 wherever k ∉ { 0 , 1 , … , n } . k\notin \left\{0,1,\ldots ,n\right\}. The stability of a polynomial is closely associated with the interlacing property, which is based on the Hermite-Biehler theorem. We also show the interlacing property of the polynomial sequence ( U n ( x ) ) n ≥ 0 , {\left({U}_{n}\left(x))}_{n\ge 0}, which satisfies the following recurrence relation: U n ( x ) = ( α n x + β n ) U n − 1 ( x ) + ( u n x 2 + v n x ) U n − 1 ′ ( x ) {U}_{n}\left(x)=\left({\alpha }_{n}x+{\beta }_{n}){U}_{n-1}\left(x)+\left({u}_{n}{x}^{2}+{v}_{n}x){U}_{n-1}^{^{\prime} }\left(x) based on the Hermite-Biehler theorem. As applications, we obtain the weak (Hurwitz) stability of the Hankel determinant of order three for the row polynomials of the (unsigned) Stirling numbers of the first kind, the Whitney numbers of the first kind, and show the interlacing property of Eulerian polynomials, Bell polynomials, and Dowling polynomials.

In this article, we consider the asymptotic behavior of solutions for the Kirchhoff-type reaction–diffusion equations driven by a nonlinear colored noise defined on unbounded domains. We prove the existence and uniqueness of pullback random attractors by the uniformly estimate.

With the help of the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we mainly give a description of the existence and the forms of the transcendental entire solutions of several product-type complex partial differential equations systems ( a 1 f z 1 + a 2 f z 2 ) ( a 3 g z 1 + a 4 g z 2 ) = e h 1 , ( a 1 g z 1 + a 2 g z 2 ) ( a 3 f z 1 + a 4 f z 2 ) = e h 2 , \left\{\begin{array}{l}({a}_{1}{f}_{{z}_{1}}+{a}_{2}{f}_{{z}_{2}})({a}_{3}{g}_{{z}_{1}}+{a}_{4}{g}_{{z}_{2}})={e}^{{h}_{1}},\\ ({a}_{1}{g}_{{z}_{1}}+{a}_{2}{g}_{{z}_{2}})({a}_{3}{f}_{{z}_{1}}+{a}_{4}{f}_{{z}_{2}})={e}^{{h}_{2}},\end{array}\right. and ( a 1 f z 1 + a 2 g z 1 ) ( a 3 f z 2 + a 4 g z 2 ) = e h 1 , ( a 1 f z 2 + a 2 g z 2 ) ( a 3 f z 1 + a 4 g z 1 ) = e h 2 , \left\{\begin{array}{l}({a}_{1}{f}_{{z}_{1}}+{a}_{2}{g}_{{z}_{1}})({a}_{3}{f}_{{z}_{2}}+{a}_{4}{g}_{{z}_{2}})={e}^{{h}_{1}},\\ ({a}_{1}{f}_{{z}_{2}}+{a}_{2}{g}_{{z}_{2}})({a}_{3}{f}_{{z}_{1}}+{a}_{4}{g}_{{z}_{1}})={e}^{{h}_{2}},\end{array}\right. where f z t = ∂ f t ∂ z t {f}_{{z}_{t}}=\frac{\partial {f}_{t}}{\partial {z}_{t}} , g z t = ∂ g t ∂ z t {g}_{{z}_{t}}=\frac{\partial {g}_{t}}{\partial {z}_{t}} , t = 1 , 2 t=1,2 ; a j ∈ C , j = 1 , 2 , 3 , 4 {a}_{j}\in {\mathbb{C}},\hspace{0.33em}j=1,2,3,4 , and h 1 {h}_{1} and h 2 {h}_{2} are the polynomials in C 2 {{\mathbb{C}}}^{2} . More importantly, we list some examples to explain that our results about the forms of solutions of such systems are precise to some extent.

In this article, a diffusive predator-prey model with memory effect and predation-driven Allee effect is considered. Through eigenvalue analysis, the local asymptotic stability of positive constant steady-state solutions is analyzed, and it is found that memory delay affects the stability of positive constant steady-state solutions and induces Hopf bifurcation. The properties of Hopf bifurcating periodic solutions have also been analyzed through the central manifold theorem and the normal form method. Finally, our theoretical analysis results were validated through numerical simulations. It was found that both memory delay and predation-driven Allee effect would cause the positive constant steady-state solution of the model to become unstable, accompanied by the emergence of spatially inhomogeneous periodic solutions. Increasing the memory period will cause periodic oscillations in the spatial distribution of the population. In addition, there would also be high-dimensional bifurcation such as Hopf–Hopf bifurcation, making the spatiotemporal changes of the population more complex.

The orthogonal decomposition (OD) of special linear Lie algebras over the complex numbers was first explored in the early 1980s and has gained renewed interest in the past decade due to its applications in quantum information theory. In this study, we investigate the OD of special linear Lie algebras over commutative rings with identity. Additionally, we present examples of specific rings that facilitate the existence of such ODs for these Lie algebras.

In this work, we study the δ \delta -chromatic number of a graph, which is the chromatic number of the δ \delta -complement of a graph. We give a structure of the δ \delta -complements and sharp bounds on the δ \delta -chromatic numbers of the Cartesian products of graphs. Furthermore, we compute the δ \delta -chromatic numbers of various classes of Cartesian product graphs, including the Cartesian products among cycles, paths, and stars.

In this article, we compute binomial convolution sums of divisor functions associated with the Dirichlet character modulo 8, which is the remaining primitive Dirichlet character modulo powers of 2 yet to be considered. To do this, we provide an explicit expression of a general modular form of even weight for Γ 0 ( 32 ) {\Gamma }_{0}\left(32) in terms of its basis, and use an identity of Huard, Ou, Spearman, and Williams. As an application, we also compute the number of representations by quadratic forms under parity conditions.

Let L = − △ + V L=-\bigtriangleup +V be the Schrödinger operator on R n {{\mathbb{R}}}^{n} , where V ≠ 0 V\ne 0 is a non-negative function satisfying the reverse Hölder class R H q 1 R{H}_{{q}_{1}} for some q 1 > n ⁄ 2 {q}_{1}\gt n/2 . △ \bigtriangleup is the Laplacian on R n {{\mathbb{R}}}^{n} . Assume that b b is a member of the Campanato space Λ ν θ ( ρ ) {\Lambda }_{\nu }^{\theta }\left(\rho ) and that the fractional integral operator associated with L L is ℐ β L {{\mathcal{ {\mathcal I} }}}_{\beta }^{L} . We study the boundedness of the commutators [ b , ℐ β L ] \left[b,{{\mathcal{ {\mathcal I} }}}_{\beta }^{L}] with b ∈ Λ ν θ ( ρ ) b\in {\Lambda }_{\nu }^{\theta }\left(\rho ) on local generalized mixed Morrey spaces. Generalized mixed Morrey spaces M p → , φ α , V {M}_{\overrightarrow{p},\varphi }^{\alpha ,V} , vanishing generalized mixed Morrey spaces V M p → , φ α , V V{M}_{\overrightarrow{p},\varphi }^{\alpha ,V} , and L M p → , φ α , V , { x 0 } L{M}_{\overrightarrow{p},\varphi }^{\alpha ,V,\left\{{x}_{0}\right\}} are related to the Schrödinger operator, in that order. We demonstrate that the commutator operator [ b , ℐ β L ] \left[b,{{\mathcal{ {\mathcal I} }}}_{\beta }^{L}] is satisfied when b b belongs to Λ ν θ ( ρ ) {\Lambda }_{\nu }^{\theta }\left(\rho ) with θ > 0 \theta \gt 0 , 0 < ν < 1 0\lt \nu \lt 1 , and ( φ 1 , φ 2 ) \left({\varphi }_{1},{\varphi }_{2}) satisfying certain requirements are bounded from L M p → , φ 1 α , V , { x 0 } L{M}_{\overrightarrow{p},{\varphi }_{1}}^{\alpha ,V,\left\{{x}_{0}\right\}} to L M q → , φ 2 α , V , { x 0 } L{M}_{\overrightarrow{q},{\varphi }_{2}}^{\alpha ,V,\left\{{x}_{0}\right\}} ; from M p → , φ 1 α , V {M}_{\overrightarrow{p},{\varphi }_{1}}^{\alpha ,V} to M q → , φ 2 α , V {M}_{\overrightarrow{q},{\varphi }_{2}}^{\alpha ,V} , and from V M p → , φ 1 α , V V{M}_{\overrightarrow{p},{\varphi }_{1}}^{\alpha ,V} to V M q → , φ 2 α , V V{M}_{\overrightarrow{q},{\varphi }_{2}}^{\alpha ,V} , ∑ i = 1 n 1 ⁄ p i − ∑ i = 1 n 1 ⁄ q i = β + ν {\sum }_{i=1}^{n}1/{p}_{i}-{\sum }_{i=1}^{n}1/{q}_{i}=\beta +\nu .

In this article, we study the mathematical properties of the solution u = ( u 1 , … , u k ) {\bf{u}}=({u}^{1},\ldots ,{u}^{k}) to the degenerate parabolic system u t = ∇ ⋅ ( ∣ ∇ u ∣ p − 2 ∇ u ) , ( p > 2 ) . {{\bf{u}}}_{t}=\nabla \hspace{0.25em}\cdot \hspace{0.25em}({| \nabla {\bf{u}}| }^{p-2}\nabla {\bf{u}}),\hspace{1.0em}(p\gt 2). More precisely, we show the existence and uniqueness of solution u {\bf{u}} and investigate a priori L ∞ {L}^{\infty } boundedness of the gradient of the solution. Assuming that the solution decays quickly at infinity, we also prove that the component u l {u}^{l} , ( 1 ≤ l ≤ k ) (1\le l\le k) , converges to the function c l ℬ {c}^{l}{\mathcal{ {\mathcal B} }} in space as t → ∞ t\to \infty . Here, the function ℬ {\mathcal{ {\mathcal B} }} is the fundamental or Barenblatt solution of p p -Laplacian equation, and the constant c l {c}^{l} is determined by the L 1 {L}^{1} -mass of u l {u}^{l} . The proof is based on the existence of entropy functional. As an application of the asymptotic large-time behaviour, we establish a Harnack-type inequality, which makes the size of the spatial average controlled by the value of the solution at one point.

In this study, we present a stochastic rumor model. The stability of the disease-free equilibrium state and instability of the free equilibrium E 0 {E}_{0} of stochastic epidemics model are considered with the help of Lyapunov functions. Sufficient conditions of persistence and extinction of rumor are given. Numerical simulations verify the analytical results.

This study presents a novel family of bivariate 2D- q q Hermite polynomials. We derive explicit forms and q q -partial differential equations and investigate numerical aspects associated with these polynomials. Furthermore, we deduce the monomiality principle and other key properties of the bivariate 2D- q q Hermite polynomials, including q q -recurrence relations and q q -difference equations. In addition, the graphical representations for the bivariate 2D- q q Hermite polynomials and q q -Hermite polynomials are explored for different values of indices.

Using the direct method, we prove the Hyers-Ulam-Rassias stability of the following functional equation: ϕ ( x + y , z + w ) + ϕ ( x − y , z − w ) − 2 ϕ ( x , z ) − 2 ϕ ( x , w ) = 0 \phi \left(x+y,z+w)+\phi \left(x-y,z-w)-2\phi \left(x,z)-2\phi \left(x,w)=0 in ρ \rho -complete convex modular spaces satisfying Fatou property or Δ 2 {\Delta }_{2} -condition.

In this article, the authors introduce Qi’s normalized remainder of the Maclaurin series expansion of Qi’s normalized remainder for the cosine function. By virtue of a monotonicity rule for the quotient of two series and with the aid of an increasing monotonicity of a sequence involving the quotient of two consecutive non-zero Bernoulli numbers, they prove the logarithmic convexity of Qi’s normalized remainder. In view of a higher order derivative formula for the quotient of two functions, they expand the logarithm of Qi’s normalized remainder into a Maclaurin series whose coefficients are expressed in terms of determinants of a class of specific Hessenberg matrices. In light of a monotonicity rule for the quotient of two series, they present the monotonicity of the ratio between two normalized remainders. Finally, the authors connect two of their main results with the generalized hypergeometric functions.

In the context of a simple undirected graph G G , a k k -prime labeling refers to assigning distinct integers from the set { k , k + 1 , … , ∣ V ( G ) ∣ + k − 1 } \left\{k,k+1,\ldots ,| V\left(G)| +k-1\right\} to its vertices, such that adjacent vertices in G G are labeled with numbers that are relatively prime to each other. If G G has a k k -prime labeling, we say that G G is a k k -prime graph ( k -PG). In this article, we characterize when a graph up to order 6 is a k -PG and characterize when a graph of order 7 is a k -PG whenever k k and k + 1 k+1 are not divisible by 5. Also, we find a lower bound for the independence number of a k -PG. Finally, we study when a cycle is a k -PG.

A sequence α \alpha of nonnegative integers is said to be graphic if it is the degree sequence of a simple graph G G , and such a graph G G is called a realization of α \alpha . In this article, we generalize Gale and Ryser’s theorem and give the sufficient condition and necessary condition for a triple to be realized by a tripartite graph. Not only that, we also give another stronger monotonous degree condition.

A graph is said to be symmetric if its automorphism group is transitive on its arcs. Guo et al. ( Pentavalent symmetric graphs of order 12 p , Electron. J. Combin. 18 (2011), no. 1, #P233, DOI: https://doi.org/10.37236/720) and Ling ( Classifying pentavalent symmetric graphs of order 24p , Bull. Iranian Math. Soc. 43 (2017), no. 6, 1855–1866) Li and Ling ( Symmetric graphs and interconnection networks , Future Gener. Comput. Syst. 83 (2018), no. 1, 461–467, DOI: https://doi.org/10.1016/j.future.2017.05.016) determined all pentavalent symmetric graphs of order 12 p 12p , 24 p 24p , and 36 p 36p . In this article, we shall generalize these results by determining all connected pentavalent symmetric graphs of order 12 p q 12pq , where p > q p\gt q are distinct primes.

In this article, we first propose a concept of almost periodic functions on arbitrary time scales, which is defined by trigonometric polynomial approximations with respect to supremum norm, and study some basic properties of these kinds of functions. Then, on almost periodic time scales, we introduce the concepts of the mean value and Fourier series of almost periodic functions and give some related results. Finally, we give the definitions of almost periodic functions in the sense of Bohr and in the sense of Bochner on time scales, respectively, and prove the equivalence of the above three definitions on almost periodic time scales.

In this article, we mainly study the solution of higher order difference equations. We solve some questions posed by Chen and Xu [ Uniqueness on entire functions and their nth order exact differences with two shared values , Open Math. 18 (2020), no. 1, 211–215] and give a negative answer to some conjectures posed by Gao et al. [ Uniqueness of meromorphic functions sharing values with their nth order exact differences , Anal. Math. 45 (2019), no. 2, 321–334] and Ahamed [ Class of meromorphic functions partially shared values with their differences or shifts , Kyungpook Math. J. 61 (2021), no. 4, 745–763]. Meanwhile, we extend some results due to Adud and Chakraborty [ Some results related to differential-difference counterpart of the Brück conjecture , Commun. Korean Math. Soc. 39 (2024), no. 1, 117–125].

An n n -dimensional Riemannian manifold ( N n , g ) ({N}^{n},g) isometrically immersed in Euclidean space R n + 1 {R}^{n+1} with unit normal ζ \zeta and shape operator S S , for a fixed constant unit vector a → \overrightarrow{{\bf{a}}} in R n + 1 {R}^{n+1} induces a vector field v {\bf{v}} and a smooth function ρ \rho on N n {N}^{n} called its code vector and coding function, respectively. The sextuple ( N n , g , ζ , S , v , ρ ) ({N}^{n},g,\zeta ,S,{\bf{v}},\rho ) is called a coded hypersurface of Euclidean space R n + 1 {R}^{n+1} . In the first result of this article, we show that a compact and simply connected coded hypersurface ( N n , g , ζ , S , v , ρ ) ({N}^{n},g,\zeta ,S,{\bf{v}},\rho ) of R n + 1 {R}^{n+1} has integral of Ricci curvature Ric ( v , v ) {\rm{Ric}}({\bf{v}},{\bf{v}}) with a suitable lower bound if and only if it is isometric to the sphere S b n {S}_{b}^{n} of constant curvature b b . In the second result, we show that a complete and simply connected coded hypersurface ( N n , g , ζ , S , v , ρ ) ({N}^{n},g,\zeta ,S,{\bf{v}},\rho ) of R n + 1 {R}^{n+1} of positive Ricci curvature, with shape operator S S invariant under its code vector field v {\bf{v}} and coding function ρ \rho satisfying the static perfect fluid equation is isometric to S b n {S}_{b}^{n} and the converse also holds. Finally, we show that a complete and simply connected coded hypersurface ( N n , g , ζ , S , v , ρ ) ({N}^{n},g,\zeta ,S,{\bf{v}},\rho ) of R n + 1 {R}^{n+1} has a point p ∈ N n p\in {N}^{n} with Ric ( v , v ) ( p ) > 0 {\rm{Ric}}({\bf{v}},{\bf{v}})(p)\gt 0 and has mean curvature H H constant along integral curves of code vector v {\bf{v}} satisfies H 2 ≥ c {H}^{2}\ge c for a positive constant c c , and the coding function ρ \rho satisfies Δ ρ = − n c ρ \Delta \rho =-nc\rho if and only if it is isometric to S c n {S}_{c}^{n} .

For p ∈ { 2 , 3 } p\in \left\{2,3\right\} and an even integer k k , let W k − 2 − ( p ) {W}_{k-2}^{-}\left(p) be the space of period polynomials of weight k − 2 k-2 on Γ 0 + ( p ) {\Gamma }_{0}^{+}\left(p) with eigenvalue − 1 -1 under the Fricke involution. We determine the dimension formula for W k − 2 − ( p ) {W}_{k-2}^{-}\left(p) and construct an explicit basis for it using period functions for weakly holomorphic modular forms. Furthermore, for a quadratic form Q Q , we define the function F − ( z , Q ) {F}^{-}\left(z,Q) on the complex upper half-plane as a generating function of the cycle integrals of the canonical basis elements for the space of weakly holomorphic modular forms of weight k k and eigenvalue − 1 -1 under the Fricke involution on Γ 0 ( p ) {\Gamma }_{0}\left(p) . We also show that F − ( z , Q ) {F}^{-}\left(z,Q) is a modular integral on Γ 0 + ( p ) {\Gamma }_{0}^{+}\left(p) . Our approach focuses on calculating cycle integrals within Γ 0 ( p ) {\Gamma }_{0}\left(p) rather than Γ 0 + ( p ) {\Gamma }_{0}^{+}\left(p) , which allows us to overcome certain technical challenges. This study extends earlier work by Choi and Kim ( Rational period functions and cycle integrals in higher level cases , J. Math. Anal. Appl. 427 (2015), no. 2, 741–758) which focused on eigenvalue +1, providing new insights by examining eigenvalue − 1 -1 cases in the theory of rational period functions and cycle integrals in this setting.

This article is devoted to present the degree to which a mapping defined from L X {L}^{X} to M , M, which is an ( L , M ) \left(L,M) -fuzzy bornology in the sense of Liang et al. In this respect, the degree to which L L -subset is bounded according to the given bornological structure is described. In addition, the notions of boundedness degree and the boundedness preserving degree of a mapping defined between bornological spaces are introduced. The characterizations of the provided notions are studied.

For a general vector of all unknown vectors in a constrained linear mixed model (CLMM), this study compared the dispersion matrices of the best linear unbiased predictors with any symmetric matrix for determining the optimality of predictors among others. Using the methodology of block matrix inertias and ranks as well as elementary block matrix operations, some equalities and inequalities are derived for comparisons. Additionally, the comparison findings for the CLMM reduce to special instances of the general vector of all unknown vectors. We also provide some comparison findings between the CLMM and its unconstrained form. Finally, we give a numerical example to illustrate our theoretical findings.

Let S ⊆ N p S\subseteq {{\mathbb{N}}}^{p} be a semigroup, any P ⊆ S P\subseteq S is an ideal of S S if P + S ⊆ P P+S\subseteq P , and an I ( S ) I\left(S) -semigroup is the affine semigroup P ∪ { 0 } P\cup \left\{0\right\} , with P P an ideal of S S . We characterise the I ( S ) I\left(S) -semigroups and the ones that also are C {\mathcal{C}} -semigroups. Moreover, some algorithms are provided to compute all the I ( S ) I\left(S) -semigroups satisfying some properties. From a family of ideals of S S , we introduce the affine semigroups with maximal embedding dimension, characterising them and describing some families.

An explicit solution for linear control systems on Lie groups is presented using the variation of parameters method. This solution is formulated as a series of Lie brackets, and the convergence of these series is analyzed. Additionally, a concept of equivalence between two elements within this class is introduced.

This article considers existence of solution for a class of fractional Schrödinger-Poisson system. By using the Nehari method and the variational method, we obtain a uniqueness result for positive solutions.

This article aims to investigate the Zariski topology on the set of prime ideals of a weak commutative neutrosophic extended triplet group (NETG) N N , denoted by Prim ( N ) {\rm{Prim}}\left(N) . First, by giving an equivalent characterization of idempotent weak commutative NETGs, we show that a topological space X X is an S S SS -space if and only if X X is homeomorphic to the space Prim ( N ) {\rm{Prim}}\left(N) of some weak commutative NETG. In addition, we prove that there exists an adjunction between the dual category of weak commutative NETGs and the category of S S SS -spaces. Finally, we further study the categorical relation between idempotent weak commutative NETGs and that of S S SS -spaces, which leads to a conclusion that the category of idempotent weak commutative NETGs is equivalent to that of commutative idempotent semigroups.

This study concerns a Krasnoselskii-type fixed point theorem for the sum of two operators A , B A,B in a Banach space E E , where B B is a Reich-type contractive mapping and A A is a k-set contractive mapping. We introduce a class of operators θ : X × X → [ 1 , + ∞ ) \theta :X\times X\to \left[1,+\infty ) satisfying some axioms and use it as a new metric to prove a fixed point theorem in the spirit of Azam et al. [ Reich-Krasnoselskii-type fixed point results with applications in integral equations , J. Inequal. Appl. 2023 (2023), 131].

This article studies the structure of rings R R over which the 2 × 2 2\times 2 upper triangular matrix rings with the same diagonal are right duo, denoted by D 2 ( R ) {D}_{2}\left(R) . We prove that for any right regular element d d of such a ring R R , d R dR contains the ideal of R R generated by all commutators. It is also proved that for a domain R R , D 2 ( R ) {D}_{2}\left(R) is right duo if and only if R R is either commutative or a division ring. Moreover, it is proved that if R R is a ring of characteristic 2 such that D 2 ( R ) {D}_{2}\left(R) is right duo, then R R has an ascending chain of nil ideals N R i ⊆ N R i + 1 {N}_{{R}_{i}}\subseteq {N}_{{R}_{i+1}} ( i = 0 , 1 , … i=0,1,\ldots ) such that N R i + 1 ⁄ N R i {N}_{{R}_{i+1}}/{N}_{{R}_{i}} is contained in the center of R ⁄ N R i R/{N}_{{R}_{i}} . Furthermore, we give a simpler proof to the famous result that if R R is a simple noncommutative ring then R R coincides with its subring generated by all commutators (by Herstein). Finally, we show that if D 2 ( R ) {D}_{2}\left(R) is right duo over a ring R R , then S a ⊆ a S Sa\subseteq aS for any a ∈ R a\in R , where S S is any of the following: (i) the prime radical, (ii) the Jacobson radical, (iii) the group of all units in R R , and (iv) the set of one-sided zero-divisors.

By making use of Bolle’s method, we show that the following problem has infinitely many solutions: x ¨ + V ′ ( x ) = 0 , x ( 0 ) cos α − x ˙ ( 0 ) sin α = x 0 , x ( 1 ) cos β − x ˙ ( 1 ) sin β = x 1 , \begin{array}{rcl}\ddot{x}+{V}^{^{\prime} }\left(x)& =& 0,\\ x\left(0)\cos \alpha -\dot{x}\left(0)\sin \alpha & =& {x}_{0},\\ x\left(1)\cos \beta -\dot{x}\left(1)\sin \beta & =& {x}_{1},\end{array} where α , β ∈ ( 0 , π ) x 0 , x 1 ∈ R n \alpha ,\beta \in \left(0,\pi ){x}_{0},{x}_{1}\in {{\rm{R}}}^{n} are given and V ∈ C 2 ( R n , R ) V\in {C}^{2}\left({{\rm{R}}}^{n},{\rm{R}}) is even and is super-quadratic at infinity.