Let Ω⊂ℂ^n be a bounded starlike circular domain with 0∈Ω . Let F: Ω→ℂ^n be a holomorphic mapping such that F has the power series expansion F(z)=z+∑ _l=k+1^∞D^lF(0)(z^l)/l! for z near the origin, where k is a positive integer. In this paper, we first establish the Fekete and Szegö inequalities for F associated with the class of quasi-convex mappings of type B on Ω . Next, as an application, we obtain the sharp bounds of the generalized determinants formed over the related terms of homogeneous expansion of F.
An odd coloring of a graph G is a proper coloring such that for every non-isolated vertex v, there exists a color that appears an odd number of times on its neighbors. Recently, Dai, Ouyang, and Pirot introduced a strengthening of the odd coloring. For positive integers h and k, a proper vertex coloring of G is an h-odd k-coloring if for every vertex v, there are min{d(v), h} colors each of which appears an odd number of times in N(v). The h-odd chromatic number of G, denoted χ ^h_o(G) is the smallest integer k such that G has an h-odd k-coloring. In this paper, we show that χ ^2_o(G)≤ 6 for any graph G with mad(G)<18/7 , where mad(G)=max{2|E(H)|/|V(H)|: H⊆ G, |V(H)|>0} . Moreover, the threshold 18/7 is best possible with respect to replacing the strict inequality by a non-strict one.
The main purpose of this paper is to investigate a diffusive system incorporating nonlocal delay under Dirichlet boundary condition. The existence, stability, and multiplicity of spatially nonhomogeneous steady-state solution and periodic solutions are investigated by using Lyapunov–Schmidt reduction. Moreover, we illustrate our general results by an application to the food-limited population model with one-dimensional spatial domain.
While considerable work has been devoted to q-supercongruences modulo cyclotomic polynomials Φ _n(q) , results concerning higher powers of the q-integer [n] remain scarce. To address this gap, we present a proof of the Guo-Schlosser conjecture modulo [n]_q^2^4Φ _n(q^2) in this paper. Our proof employs Watson’s transformation and Bailey’s transformation to introduce auxiliary parameters into the congruence, followed by an analysis involving specific parameter specializations at every primitive m-th root of unity, where m divides n. Additionally, we establish a new q-supercongruence modulo [n]^3 that extends the one formulated by Wei.
This paper is concerned with the asymptotically autonomous stability of pullback random attractors for fractional nonclassical diffusion equations with fractional Laplace-multiplier noise. In order to overcome the difficulties caused by the lack of compact Sobolev embedding on unbounded domains and weak dissipative structure of the equation, we first prove the existence, uniqueness and backward compactness of a special kind of pullback random attractor using the method of spectral decomposition in bounded domains and the uniform tail-estimates of solutions outside bounded domains over the infinite time interval. The measurability of this class of attractors is established by proving that two classes of defined attractors are equal with respect to two different universes. More importantly, we prove backward convergence from time-fibers of the non-autonomous attractor to the autonomous attractor.
Optimal boundary trace embeddings for Orlicz–Sobolev spaces associated with arbitrary Young functions were previously established by Cianchi using symmetrization and rearrangement techniques. In this paper, we develop a different and more direct approach under additional structural assumptions on the Young function. We explicitly construct the trace Young function from the Sobolev conjugate and prove the corresponding continuous and compact trace embeddings on bounded Lipschitz domains. We further show, without relying on symmetrization or rearrangement arguments, that the resulting target space coincides, up to equivalence of norms, with the optimal Orlicz trace space identified by Cianchi. As an application, we study a Neumann problem with Orlicz growth and establish the existence of solutions via monotone operator theory when the data are independent of the solution, and via the mountain pass theorem in the fully nonlinear case.
In this paper we analyze the Lipschitz boundedness of the following type of operators T_N^ℒ f(x)=∑ _j=N_1^N_2 v_j (e^-a_j+1ℒ f(x)-e^-a_jℒ f(x) ), x∈ℝ^n and its maximal operator T^*,ℒ= sup _N | T^ℒ_N f(x)| , where {e^-tℒ}_t>0 is the heat semigroup of the operator ℒ=-Δ +V with Δ being the classical Laplacian, the nonnegative potential V belonging to the reverse Hölder class RH_q with q>n/2 and n≥ 3 , N=(N_1, N_2)∈ℤ^2 with N_1
We introduce a class of selective max-residual operators for approximating common fixed points of finite families of cutters in real Hilbert spaces. By adaptively selecting at each iteration the component with the maximal residual, these operators preserve the fundamental properties of the underlying cutters. We establish weak and strong convergence theorems under standard assumptions. The theoretical framework is applied to develop computationally efficient algorithms for the split feasibility problem with multiple output sets and the multiple-sets split feasibility problem, eliminating the need for prior knowledge of the norms of the linear operators involved.
In this work, we develop a stochastic SIRI epidemic model with relapse, where the environmental stochasticity is driven by Black-Karasinski process. First, the existence and uniqueness of global solution of the model is studied. By constructing some desirable Lyapunov functions, we derive two critical values ℛ_0^E and ℛ_0^S related to the basic reproduction number ℛ_0 of the deterministic system. It turns out that (i) the disease will go extinct exponentially when ℛ_0^E<1 ; (ii) if ℛ_0^S>1 , the stochastic model admits a stationary distribution (i.e., a reflection of the long-term prevalence of disease); (iii) ℛ_0^E=ℛ_0^S=ℛ_0 if there is no environmental noise in disease transmission. Then we derive an approximate expression of local probability density function of the stationary distribution. Finally, several numerical examples are provided to demonstrate the theoretical findings. Our results reveal that relapse and environmental noise will facilitate disease pandemic.
The generalized Weyl algebras (GWA) are a class of associative algebras, which play an important role in the classification of simple modules for the Lie algebra 𝔰𝔩_2 , its quantum version U_q(𝔰𝔩_2) and many analogues. In this paper, we give a complete description of derivations for an arbitrary degree-1 GWA using skew-derivations of the base ring of the GWA.
The purpose of the article is to give a complete characterization of the hypercyclic linear fractional composition operators as well as their multiples acting on the derivative Hardy spaces S^p and on the weighted Dirichlet spaces 𝒟_α ^p . This answers affirmatively two problems posed by Colonna and Martínez-Avendaño and generalizes several results in literature.
We first establish a complete variational theory for triharmonic maps between pseudo-Riemannian manifolds. As a central application, we then characterize the stability of triharmonic CMC hypersurfaces in pseudo-Riemannian space forms. Our analysis reveals a fundamental and decisive divergence from the Riemannian setting.
For a normalized univalent function f(z) = z + ∑ _n=1^∞ a_n z^n defined in the unit disc 𝔻 , the coefficients γ _n determined by the expansion log (f(z)/z ) = 2 ∑ _n=1^∞γ _n z^n are called the logarithmic coefficients of f . In this paper, sharp bounds for the quantity |γ _3 |- |γ _2 | is established, where γ _2 and γ _3 denote the second and third logarithmic coefficients, respectively, for functions belonging to the following subclasses: starlike functions of order α ; close-to-convex functions satisfying Re( 1 + z f”(z)/f'(z) ) > α , functions satisfying Re (1 + z f”(z)/f'(z) ) < 1 + α /2 ; and functions of bounded turning with Ref'(z) > α . For the class of bounded turning functions, the estimate of | γ _2| -| γ _1| is also discussed.
In this article, robust optimality for nonsmooth multiobjective programming problems and some remarks on Mond–Weir-type duality involving image space analysis (ISA) are employed. First, we study robust optimality conditions via a separation scheme for nonsmooth multiobjective optimization problems under data uncertainty (for brevity, (UP)) and discuss some fundamental characterizations for a class of regular weak separation functions as well as ISA. Second, in terms of augmented Lagrangian functions and image space analysis, we derive robust necessary and sufficient optimality conditions for the uncertain multiobjective programming problem. Finally, we shall show that the Mond–Weir-type duality is a Lagrange-type duality, addressing an interesting question that had been raised earlier in another related research paper under suitable assumptions on the 𝒞- differentiation of the objective and constraint functions. Some illustrative examples to demonstrate the obtained main results are provided, too.
In this paper, we consider a type of square functions in difference form, which is a modified one of sharp-edged square functions introduced by Wilson. By adding Lipschitz conditions on the constituent kernels of the square functions, we show that they are bounded on L^p(ℝ^d) for all p∈ (1,∞ ) .
We investigate traveling wave solutions for a system of coupled nonlinear Schrödinger equations arising from two-component Bose-Einstein condensates with spin-orbit and Raman couplings. The presence of spin-orbit coupling breaks the Galilean invariance, making the existence of traveling waves a nontrivial problem. By minimizing the action functional on the associated Nehari manifold and using the concentration-compactness principle, we establish the existence of non-radial traveling solitary waves in ℝ^d ( d=1,2,3 ). Furthermore, we prove that as the frequency ω→ +∞ , the rescaled profiles of the boosted ground states or ground states converge strongly in the energy space (H^1(ℝ^d,ℂ))^2 to the ground states of a stationary system without spin-orbit and Raman couplings.
For a locally compact group G, the Chabauty space 𝒮𝒰ℬ( G) of closed subgroups is a compact Hausdorff space. Although 𝒮𝒰ℬ( H) is always a closed subspace of 𝒮𝒰ℬ( G) for every closed subgroup H of G, it is not necessarily open. Understanding when 𝒮𝒰ℬ( H) is open provides insight into the topological and algebraic structure of H and its interaction with G. In this paper, we investigate closed subgroups with open Chabauty subspaces and introduce the class 𝒯( G) of such subgroups. Our first main theorem establishes a criterion relating open morphisms and Chabauty openness: for a continuous morphism φ :G→ H with G, H compact groups, the induced map φ _*:𝒮𝒰ℬ( G) →𝒮𝒰ℬ( H) , defined by L↦φ (L) , is open if and only if φ is open, provided that H_0⊆φ (G) . We also study isolated points of 𝒮𝒰ℬ( G) when G is compact. We prove that a closed subgroup H is isolated in 𝒮𝒰ℬ( G) precisely when it is strongly finitely generated and belongs to 𝒯( G) . This result generalizes previous work on discrete and profinite groups and yields a complete characterization in compact groups.
In this paper, we study numerical invariants associated with a homogeneous submodule of the Hardy module over the bidisk. We focus on the submodule generated by the polynomial (z-w)(2) and obtain explicit formulas for the corresponding invariants. As an application, we verify the monotonicity property in this concrete setting. Our results provide a detailed example illustrating the behavior of these invariants beyond the linear case.
