This paper establishes necessary and sufficient conditions for the Finsler metrization of sprays with non-vanishing scalar curvature, by studying symplectic forms, the Berwald connection, and symmetry-based criteria. We derive global and local metrizability conditions for sprays with non-vanishing scalar flag curvature, extending to isotropic and constant curvature cases. Examples on manifolds including S^2 , ℝ^2 × S^1 , ℝ× T^2 , and warped product spaces illustrate these conditions, highlighting the interplay of curvature with non-trivial topologies. These results refine and generalize existing frameworks, advancing applications in Finsler geometry, differential geometry, and mathematical physics.
We extend several known results by completely characterizing the boundedness and compactness of the generalized Volterra integration operator on weighted Fock spaces F_α ^2 . Equivalent and more tractable formulations of these conditions are also obtained. Furthermore, we establish a characterization of the order boundedness of the operator from F_α ^2 to F_α ^q .
We introduce slow metric mean dimensions, a new family of invariants for dynamical systems with subexponential growth complexity. For the exponential scale a_χ (n)=e^χ n , these invariants are shown to coincide with classical metric mean dimensions. We establish fundamental inequalities relating their measure-theoretic, Katok, and topological versions.Under an additional homogeneity assumption on the invariant measure, we obtain an equality linking the topological slow metric mean dimension to a pointwise Bowen-ball growth rate.
This paper is devoted to Wilker–Huygens type inequalities for generalized trigonometric and hyperbolic functions. Building upon earlier work by Neuman, we address several limitations and inaccuracies in the existing literature and present three key advances. First, we correct the flawed condition in Neuman’s work for hyperbolic function inequalities. Second, we extend significantly the parameter range by introducing a refined parameter p̃ for trigonometric functions and p^2/(1+2p) for hyperbolic functions, covering both positive and negative power cases. Third, we adopt a monotonicity analysis of the constructed function Φ (x) (combined with properties of Gaussian hypergeometric functions) to establish necessary and sufficient conditions for the inequalities, which fills gaps in previous results.
We are interested in the following critical biharmonic Schrödinger equation {[ Δ ^2 u+λ u=g(u)+|u|^4^*-2u in ℝ^N,; ∫ _ℝ^N|u|^2 dx=c, ]. where 5≤ N≤ 7 , 4^*:=2N/N-4 , c>0 and λ∈ℝ appears as a Lagrange multiplier. The novelty of this paper is that, under a class of general mass-supercritical conditions on g(u), we obtain the existence of ground state solutions and derive an asymptotic behavior of the ground state energy as c→ +∞ . The key ingredient of our proof relies on an alternative criterion and some subtle energy estimation technique. Some recent results are generalized and improved significantly.
In this study, a weighted generalization of the Fourier transform using a weight function w with respect to a function g is defined and some fundamental properties are examined. Then, the existence of the weighted Fourier transform is proven, and the transforms of some example functions are calculated to facilitate understanding of the transform, and then the inverse of the weighted Fourier transform is given. Furthermore, a new weighted convolution operation is defined, and a weighted convolution theorem is proven. Finally, the weighted Fourier transform of the weighted derivative and the fractional weighted derivative of a function f with a weighted w respect to another function g is investigated. Additionally, the certain theorems are verified with examples, and our study is enriched with numerical graphs.
The purpose of this article is to continue our studies of single and multiple (q-)hypergeometric functions. We shall thus export the so-called q-case of multiple hypergeometric functions, Appell’s transformation formula, first Lauricella function transformation formula, Euler-Pfaff formulas for triple functions, transformation formula between the first and third Appell functions, integral representation and difference equations. First we prove Euler-Pfaff and reduction formulas for triple q-Saran hypergeometric functions including an equivalence relation for them. Then we shall prove both q-Euler and q-Laplace integral representations, as well as formal q-integral representations with the third q-real number. As usual, q-Euler integral representations are proved by the q-Beta integral. The q-Laplace integral representations contain confluent q-hypergeometric functions, which were previously defined. These formulas are proved by using the q-integral expression for the q-Gamma function. Because of the confluence, powers of (1-q) occur in several formulas and two new triple, confluent q-hypergeometric functions are used. Furthermore, systems of q-difference equations for some q-Saran functions are stated whose proofs are obvious. Some of Sarans formulas are corrected, and in some cases new triple hypergeometric formulas are inserted. Finally, two transformations for the first q-Lauricella function are proved, one of which requires the use of a q-real number.
Based on the monotone iteration method and Leray–Schauder degree theory, we obtain the Ambrosetti-Prodi type results for multiparameter Neumann systems with mean curvature operator in Minkowski space {[ -(u'/√(1-u'^2))'=f(x,u,v)+r+h(x), x∈ (0,1),; -(v'/√(1-v'^2))'=g(x,u,v)+s+l(x), x∈ (0,1),; u'(0)=u'(1)=0, v'(0)=v'(1)=0, ]. where f,g∈ C([0,1]×ℝ×ℝ), r,s∈ℝ are parameters, h,l∈ C[0,1], and ∫ _0^1h(x)dx=0,∫ _0^1l(x)dx=0.
Iterates of mappings contracting perimeters without a periodic point of a prime period two are graphic contractions except for finitely many. The reverse conclusion does not hold, there are graphic contractions without iterates that are mappings contracting perimeters of triangles. We prove that a mapping contracting perimeters of triangles cannot have a periodic point of a prime period greater than two, and prove the existence of a periodic point using the Saturated principle of graphic contraction. The theory is substantiated with numerous examples.
We study the well-known Rössler system ẋ=-y-z, ẏ=x+a y, ż=b-c z+x z. First, we give a global qualitative description of the flow of the completely degenerate case a=b=c=0 restricted to each invariant surface H=h of its first integral, including the behaviour at infinity via Poincaré compactification. Second, we use first-order averaging to prove the existence of periodic orbits for sufficiently small parameters (in a perturbation of the integrable case) and provide leading-order approximations of their initial conditions.
In this paper, we consider coupled nonsymmetric algebraic Riccati equations (CNAREs) arising in jump linear quadratic differential system. Firstly, we show that the minimal positive matrix m-tuple solutions of the CNAREs exist under some specific assumptions. Secondly, we review the basic fixed-point iterative method for solving the CNAREs, and prove that the fixed-point iteration is linearly or sublinearly convergent. Thirdly, we propose Newton’s method to solve the CNAREs, and show that Newton’s iteration is quadratically or linearly convergent. We figure out that our convergence analysis about the fixed-point iteration corrects and refines the analysis given in a recent paper (Zhang et al. in J Appl Math Comput 68:4119–4133, 2022), and our proposed Newton’s method is definitely the standard Newton’s method which differs from that proposed in a recent paper (Liu et al. in Comput Appl Math 39(2):1–17, 2020). Besides, we also devise a mixed algorithm by executing several steps of fixed-point iterations and then switching to Newton’s iterations and give its convergence results. Finally, we also give some examples to demonstrate the validity and applicability of the proposed methods.
For a graph G, the total k-cut complex Δ ^t_k(G) , introduced by Bayer et al. (Disc Math 346(7), 2023) is the simplicial complex whose facets are the complements of independent vertex sets of size k in G. It is known that Δ ^t_k(G) is vertex decomposable for all k if and only if G is chordal. In this paper, we study the Alexander dual of Δ ^t_k(G) for some specific complexes. We prove that the Alexander dual of the total cut complexes of powers of a path graph is pure vertex decomposable. To establish this result, we characterize the minimal hitting sets of the family of independent vertices of size k in G. As a consequence, combined with a result of Bayer et al., we confirm two conjectures of Fröberg which state that the Stanley–Reisner rings of the total cut complexes of paths and squared paths are Cohen–Macaulay and have linear resolutions. Furthermore, for any cycle C_n of length n, we show that, unlike Δ ^t_k(C_n) itself, its Alexander dual Δ ^t_k(C_n)^∨ is pure vertex decomposable, implying that the Stanley–Reisner ring of Δ ^t_k(C_n) has a linear resolution. We investigate vertex decomposability for the total cut complexes of certain relative augmented graphs obtained from a vertex decomposable graph. We determine the homotopy type of Δ ^t_k(C_n)^∨ , as well as that of the Alexander dual of the total cut complexes of powers of a path. Finally, we prove pure vertex decomposability of the Alexander dual of the total cut complexes of certain complete multipartite graphs and then characterize higher powers of cycles for which the Alexander dual of their total cut complexes are pure vertex decomposable.
This paper establishes a new framework for Hermite-Hadamard type inequalities by employing Gateaux derivatives and the Bochner integral. By extending the classical scalar results to real Banach spaces, we provide sharp error bounds for trapezoidal and midpoint type functional approximations. Under the assumption that the magnitude of the directional curvature |φ ^'| is convex, we derive deterministic error estimates with an optimal constant of 1/8. Furthermore, we demonstrate the practical utility of these results through applications in nonlinear optimization search paths, quadratic growth characterization, and the stability analysis of energy potentials in nonlinear elasticity. These results remain valid in infinite-dimensional Banach spaces and therefore extend the classical Hermite-Hadamard framework beyond the usual real-variable setting. An illustrative example on the space ℓ ^2 demonstrates how the proposed approach provides effective error bounds in infinite-dimensional optimization problems.
In this paper, we study the eigenvalue problem for the third-order differential operator with a positive delta point interaction under the periodic boundary condition. For a constant α >0 , we consider the operator H=id^3/dx^3+αδ (x) in L^2(-1/2,1/2) , where δ (· ) is the Dirac’s delta function supported at the origin. We impose a function y∈dom(H) the periodic boundary conditions y^(j)(1/2)=y^(j)(-1/2), j=0,1,2 . Then, its spectrum σ (H) is discrete. We show that there exists exactly one eigenvalue λ _n in the interval ((2nπ )^3,(2(n+1)π )^3) for each n∈ℤ .
A dissociation set of a graph is a vertex subset whose induced subgraph has maximum degree at most one. The dissociation number of a graph is the cardinality of its largest dissociation set. In this paper, we establish both lower and upper bounds on the dissociation number of unicyclic graphs, parameterized by invariants such as maximum degree and diameter. We also characterize the structures of the extremal graphs attaining the upper bounds and the lower bounds.
In this paper, we study generalized Berwald square metrics F=α +2β +β ^2/α where α =√(a_ij(x)y^iy^j) is a Riemannian metric and β =b_i(x)y^i is a one-form on a manifold M. Let F be a generalized Berwald square metric with isotropic S-curvature. We show that F is a generalized Douglas–Weyl metric if and only if it is R-quadratic if and only if it is R-reversible. We also prove that F is Ricci-quadratic if and only if it is Ricci-reversible. Finally, we show that every weakly Einstein square metric is Ricci-reversible if and only if it is Ricci-quadratic.
We consider C^3 unimodal maps with a non-flat critical point of order ℓ , all of whose periodic points are hyperbolic repelling. We prove that for any such map f that is only finitely renormalizable, and for any t > 0 for which the Poincaré series 𝒫(c,t) is finite, the series ∑ _n=0^∞ |Df^n(f(c))|^-t/ℓ converges. The proof proceeds by showing that such maps satisfy the backward contracting property.
The aim of this paper is to study the quasilinear elliptic equation {[ -Δ u-Δ (u^2)u-λ u=Q(x)h(u), x∈ℝ^N,; ∫ _ℝ^N|u|^2dx=a, ]. where N≥ 2 , λ∈ℝ , a>0 is a given mass and Q∈𝒞(ℝ^N,[0,+∞ )) is a nonlinear function. By a suitable change of variables, the existence of ground state normalized solutions with the non-autonomous nonlinearity are established via a dual method, which has rarely been considered for quasilinear elliptic problems. Our results are the supplement of quasilinear elliptic equations with prescribed mass.
