In this paper, the authors first elaborate on the concept of (α, m)-(β, F)-convex functions, as a unification of (α, m)-convex and F-convex functions, then establish a series of integral inequalities of the Hermite–Hadamard type for (α, m)-(β, F)-convex functions, and further deduce some novel integral inequalities for the Hermite–Hadamard type of several generalized convex functions.
In this paper, the authors introduce a new class of generalized convex functions, called GTH-convex functions. They investigate their fundamental properties and establish several new integral inequalities of Hermite–Hadamard type for this class of functions.
This paper investigates the second-order discrete Hamiltonian system Δ[P(k)Δu(k−1)]−L(k)u(k)+∇W(k,u(k))=0 ${\Delta}\left[\mathbb{P}\left(k\right){\Delta}u\left(k-1\right)\right]-\mathbb{L}\left(k\right)u\left(k\right)+\nabla \mathbb{W}\left(k,u\left(k\right)\right)=0$ , where L(k) $\mathbb{L}\left(k\right)$ is a matrix-valued function that is positive definite for all |k| sufficiently large. By employing variational methods, we obtain a new sufficient condition for the existence of infinitely many nontrivial homoclinic solutions when W(k,x) $\mathbb{W}\left(k,x\right)$ is locally defined subquadratic with respect to x (that is W(k,x) $\mathbb{W}\left(k,x\right)$ is only locally defined near the origin with respect to x). The obtained results extend and improve many known theorems in the recent literature.
Luo, Xu and Yu proposed an extremal problem on group connectivity of graphs as follows: for an abelian group A with |A| ≥ 3 and an integer n ≥ 3, find ex(n, A), where ex(n, A) is the maximum number such that every simple graph with n vertices and at most ex(n, A) edges is not A-connected. In this paper, we determine the values ex(n, Z 2 × Z 2) for 3 ≤ n ≤ 10.
In this paper, we attempt to deal with two types of set-valued quadratic ρ-functional inequalities, which are related to quadratic type set-valued functional equation. We also discuss the Hyers–Ulam stability of such set-valued quadratic ρ-functional inequalities by applying the fixed point approach.
We obtain some necessary and sufficient conditions for the boundedness of the p-adic multilinear Hardy–Littlewood operators Hφ,n,m ${\mathcal{H}}_{\varphi ,n,m}$ on the p-adic Morrey–Herz spaces ∏i=1mMK̇ℓ,qi(⋅),ωiαi(⋅),λiQpn $\prod _{i=1}^{m}{M \dot {K}}_{\ell ,{q}_{i}\left(\cdot \right),{\omega }_{i}}^{{\alpha }_{i}\left(\cdot \right),{\lambda }_{i}}\left({\mathbb{Q}}_{p}^{n}\right)$ , the p-adic central Morrey spaces ∏i=1mḂωiqi(⋅),λiQpn $\prod _{i=1}^{m}{\dot {B}}_{{\omega }_{i}}^{{q}_{i}\left(\cdot \right),{\lambda }_{i}}\left({\mathbb{Q}}_{p}^{n}\right)$ , and the p-adic local central Morrey spaces ∏i=1mḂωi,locqi(⋅),λiQpn $\prod _{i=1}^{m}{\dot {B}}_{{\omega }_{i},\mathrm{l}\mathrm{o}\mathrm{c}}^{{q}_{i}\left(\cdot \right),{\lambda }_{i}}\left({\mathbb{Q}}_{p}^{n}\right)$ with power weights.
This study presents a new computational technique for solving weakly singular Fredholm integral equations of the first kind, based on a regularization–discretization strategy that combines Tikhonov regularization with Legendre spectral projection methods. To incorporate the singular part of the kernel, we divide the integration into two parts and employ a straightforward variable transformation using two Jacobi weight functions. The presented numerical method effectively transforms the solution of weakly singular integral equations of the first kind into the solution of a linear system of algebraic equations. A thorough theoretical analysis of the proposed technique is presented. Finally, a series of numerical tests were conducted to demonstrate the validity and efficiency of our approach.
Let M = M 1 ∪F M 2 be an amalgamation of two irreducible 3-manifolds M 1 and M 2 along a compact connected surface F (not necessarily closed). In the paper, we introduce the extended curve complex C̃(F) $\tilde {C}\left(F\right)$ of a compact connected surface F, whose vertices are the isotopy classes of non-trivial simple closed curves on F. In case that F is bi-compressible in the amalgamated 3-manifold M = M 1 ∪F M 2 and in case that F is compressible only in one of M 1 and M 2, we give some sufficient conditions in terms of distance between some vertex subsets of C̃(F) $\tilde {C}\left(F\right)$ for M to be irreducible respectively.
Let K be an Artin-Schreier extension of the rational function field k≔Fq(T) $k{:=}{\mathbb{F}}_{q}\left(T\right)$ , where Fq ${\mathbb{F}}_{q}$ is a finite field of order q and q is a power of p. We study the behavior of the divisor class groups of all constant field extensions Kn≔KFqn ${K}_{n}{:=}K{\mathbb{F}}_{{q}^{n}}$ of K for every positive integer n. We explicitly find the closed formulae for the divisor class numbers of all Artin-Schreier extensions K n with genus up to four. We express the divisor class number formulae for K n in terms of the number of places of Fqn(T) ${\mathbb{F}}_{{q}^{n}}\left(T\right)$ with degree m which are unramified in K n and have inertia degree r, where m and r depend on the genus of K n. Then we classified all Artin-Schreier extensions Kn=KFqn ${K}_{n}=K{\mathbb{F}}_{{q}^{n}}$ with genus up to four such that their p-class group ranks stay zero in every tower of constant field extensions of K. In particular, we point out that all the divisor class numbers of K n are congruent to 1 modulo p. Tables 1 through 4 present the divisor class numbers of all Artin-Schreier extensions with genus up to 4.
We study a class of Markov decision processes on Borel state spaces motivated by situations where both the horizon and the notion of performance are uncertain. The process may terminate at a random time, generating a survival profile that effectively plays the role of a time-dependent discount factor. To capture imprecise performance specifications, rewards are modeled as trapezoidal fuzzy numbers, and policies are compared through weighted-vertex ranking functionals. Our results show that, under Lyapunov-type drift conditions and standard continuity/compactness assumptions, the ranked control problem admits Bellman optimality equations and optimal Markov policies. Technically, we reduce the original model to an equivalent discounted problem on an augmented state space, which leads to a contraction property, a fixed-point representation of the value function, and a convergent value-iteration scheme. A continuous-state neurostimulation example illustrates the implications in a concrete setting and outlines a practical discretization-based implementation.
Let G be a finite group, and let cd (G) denote the set of all irreducible character degrees of G. The set ρ(G) represents all prime divisors of the integers in cd (G). For a prime p and a positive integer n, let n p denote the p part of n. The degree prime-power graph of G is defined as a graph whose vertex set is given by V(G)=pep(G)∣p∈ρ(G) $V\left(G\right)=\left\{{p}^{{e}_{p}\left(G\right)}\mid p\in \rho \left(G\right)\right\}$ , where pep(G) ${p}^{{e}_{p}\left(G\right)}$ is the maximum value of the p-part of the degrees of the irreducible characters of G. An edge exists between distinct vertices x, y ∈ V(G) if their product xy divides some integer in cd (G). The authors have previously shown that certain non-abelian simple groups can be uniquely determined by their orders and degree prime-power graphs. In this paper, they continue this line of research, demonstrating that the simple unitary group PSU 3(p) can be uniquely identified by its order and degree prime-power graph. Furthermore, they establish that every simple group of Lie type S over a finite field GF(p f) with |S| < p 10 can also be uniquely identified by its order and degree prime-power graph.
To effectively manage biological populations in ecosystems, this paper introduces a delayed feedback mechanism into a Leslie-Gower prey harvesting model, resulting in a more realistic dynamic system. First, using fixed-point theory, differential inequalities, and suitably constructed functions, we establish the existence and uniqueness, non-negativity, and boundedness of solutions. Subsequently, stability criteria and Hopf bifurcation conditions are derived via the Routh-Hurwitz criterion, stability theory, and bifurcation principles for delay differential equations. Crucially, we analytically estimate the critical time-delay length that triggers stability switching. In response to the sensitivity of ecological parameters to disturbances, three control strategies are designed: delayed feedback control, hybrid control, and speed feedback control. These effectively postpone or advance the Hopf bifurcation, thereby enlarging or shrinking the stability region. Numerical simulations clearly demonstrate that delay alters stability and bifurcation dynamics, while also confirming the effectiveness of the controllers. This study provides a novel theoretical framework and technical approaches for real-time regulation and sustainable management of complex ecosystems.
In this paper we establish some Holder's type generalizations of Opial's inequalities for two absolutely continuous functions. Applications related to the trapezoid weighted inequalities and to Fejer's inequality for convex functions are also provided. Some Gr & uuml;ss' type inequalities for p -norms are pointed out. Certain examples for norm inequalities in Hilbert spaces and approximations of Finite Fourier transform are given as well. Among others we show that, if h, k : a , b -> C are absolutely continuous on [a,b] with h ' is an element of L-p [a,b], k ' is an element of L-q[a,b], k(a) = k(b) = 0 and p, q > 1 with 1/p + 1/q = 1 , then integral(b)(a) |h '(t)k(t)|dt <= (integral(b)(a) K(t)|h '(t )|(P) dt)(1/p) (integral(b)(a)|a + b/2 - t||k '(t)|(q) dt)(1/q) <= integral(b)(a) [1/p K(t)|h '(t )|(P) dt + 1/q|a + b/2 - t||k '(t)|(q) ]dt, where K is defined by K(t) & colone; 1/2 (b - a) - |a + b/2 - t|, t is an element of [a,b] .
We present a novel identity for norms related to gyroaddition of four elements which satisfy some orthogonal conditions in the M & ouml;bius gyrovector space. As an application, we give a concrete and simple procedure to compute the norm of the M & ouml;bius subtraction between two orthogonal gyrolinear combinations consisting of arbitrary finite number of terms, as a counterpart to the classical identity of the norm of the ordinary subtraction between two ordinary orthogonal linear combinations in a real Hilbert space.
In this paper, we establish some new results on Bernstein and Tur & aacute;n-type inequalities for polynomials on the unit circle on the complex plane by using an improved Schwarz Lemma at the boundary recently proved by Mercer. The obtained results strengthen several recent findings by various authors known in the literature. Moreover, some concrete numerical examples are presented, showing that in some situations, the bounds derived from our results can be considerably sharper than the ones previously known.
We are concerned with the multiplicity of solutions for the singular superlinear Kirchhoff equation involving indefinite potentials -(1 + b integral(R3) vertical bar del u vertical bar(2)dx)Delta u + V(x)u = lambda g(x)u(-gamma) + h(x)u(p) in R-3, when 0 < g is an element of L-gamma 0 (R-3), the potential h is an element of L-infinity(R-3) may change its sign, V is a positive continuous function, and lambda > 0 is a real parameter. The main difficulties come from the non-differentiability of the energy functional and the fact that the Nehari extremal value is not available. We overcome these difficulties by analyzing the structure of the Nehari manifold and redefine the range for the existence of solutions, which does not depend on the Nehari extremal value. As a result, two different solutions of that problem are obtained with negative and positive energy, respectively.
Let F be a field of characteristic p > 2 and D-2p be the dihedral group of order 2 p. In this paper, we consider the 2-dimensional indecomposable representation V-2(+), V-2(-) of D-2p and the ring of vector invariants F[mV(2)(+) ] (D)(2p) , F[mV(2)(-)] (D)(2p) for any positive integer m. We give a generating set for F[mV(2)(+) ] (D)(2p) , F[mV(2)(-)] (D)(2p) and a generating set for the invariant ring of the decomposable representation of D-2p whose summand is the 2-dimensional indecomposable representation of D-2p. Moreover, we give a class of relations of the generators of the ring of vector invariant F[mV(2)](Cp) of the 2- dimensional indecomposable representation V-2 of the cyclic group C-p of order p.
We show that if there is a nonconstant function f is an element of C-1,C-alpha(M) satisfying the Obata equation on a Finsler n-manifold (M, F), then (M, F) is isometric to a Finsler n-sphere. In particular, for a Randers metric, F is determined analytically by the standard sphere metric and a homothetic field, and the S-curvature is constant under the Busemann-Hausdorff volume form. As applications, we prove Lichnerowicz-Obata type rigidity theorem: for a closed Finsler n-manifold with the weighted Ricci curvature Ric(N) >= (N - 1)k > 0, the first eigenvalue of the Finsler Laplacian is bounded below by Nk with equality if and only if (M, F) is isometric to a Finsler n-sphere.
The eccentric atom-bond sum-connectivity index (ABSC e ) of a graph G is defined as A B S C e ( G ) = & sum; u v is an element of E ( G ) e u + e v - 2 e u + e v $ABS{C}_{e}\left(G ight)={\sum }_{uv\in E\left(G ight)}\sqrt{\frac{{e}_{u}+{e}_{v}-2}{{e}_{u}+{e}_{v}}}$ , where e u and e v represent the eccentricities of u and v, respectively. In this paper, we investigate the eccentric atom-bond sum-connectivity index of trees and unicyclic graphs on n vertices with diameter d. We also characterize the extremal graphs.
In this article, we consider the recently derived mathematical model of the full von K & aacute;rm & aacute;n beam with temperature and microtemperature effects. The nonlinear governing equations were derived by using Hamilton principle in the framework of Euler-Bernoulli beam theory. The new aspect we propose here is to introduce the second sound law in the temperatures and the microtemperatures which turns to the Gurtin-Pipkin's one. Hence the derived equations are physically more realistic since they overcome the property of infinite propagation speed and take into account the thermal memory. Even more so, the case of Fourier's law, Cattaneo's law and the Coleman-Gurtin's law can be recovered from the derived system by considering a rescaled kernel in place of the original kernels through a proper singular limit procedure. Based on semigroups theory, we establish existence and uniqueness of weak and strong solutions to the derived problem under Gurtin-Pipkin's law. By adding a frictional damping function acting on the transversal component and by using the multiplier method, we show that solutions decay exponentially. These results are obtained under condition which establishes the null solution as the only equilibrium of the derived model.
