The almost contact metric structure on a real hypersurface M in complex projective space allows to define on M, for any nonnull real number k and any operator C, two tensor fields of type (1,2) denoted, respectively, by C_F^(k) and C_T^(k) . We will classify real hypersurfaces in complex projective space for which the h-operator satisfies that either h_F^(k) (respectively, h_T^(k) ) is either symmetric or skewsymmetric.
This paper introduces deep neural network operators based on a special class of sigmoidal functions, using a constructive approximation approach. We begin by constructing a neural network operator for the univariate case with two hidden layers and inductively extend it to an operator with an arbitrary but finite number of hidden layers. The interpolation properties are established, along with error estimation in terms of the modulus of continuity. We establish Bernstein-type inequality for the derivative of a two-hidden-layer neural network operator. Further, using the Peetre K-functional, we prove a converse theorem of approximation. The results obtained in the univariate case are then extended to a multivariate case. These results show the denseness of the constructed operators in the space of continuous functions C[a, b] for both univariate and C( ) , where is a hyperrectangle in the multivariate case. This study also provides an error analysis comparing the approximation capabilities between deep and shallow neural network operators. Numerical experiments for both univariate and multivariate cases validate the theoretical results for these operators, highlighting their efficiency.
This article develops a differential—geometric framework for the Hamilton–Jacobi theory of autonomous contact Hamiltonian systems, with emphasis on dissipative dynamics on the extended phase space T^*Q×ℝ , endowed with the contact form η . Solutions of the contact Hamilton–Jacobi equation are interpreted, in the sense of Lie, as integral submanifolds of the exterior differential system generated by (η ,dH) . Using Cartan reduction theory, we identify the characteristic distribution of this system with the evolution vector field ℰ_H , canonically associated with the contact structure, and we exploit its algebra of first integrals to obtain local normal forms. Under natural transversality and regularity assumptions, we construct n= Q functionally independent first integrals of ℰ_H that are pairwise in involution (in a sense specified in the paper). These integrals determine a local Legendrian fibration of the zero-energy hypersurface ℋ={H=0} , yielding complete solutions of the dissipative Hamilton–Jacobi equation and, consequently, integration by quadratures along the fibers. The approach highlights a proposal for complete integrability in contact geometry, and places it in perspective through a comparison with other formulations in the literature.
We prove a uniqueness theorem for Delaunay surfaces among connected constant mean curvature surfaces in three-dimensional space forms that meet a one-parameter family of geodesic spheres orthogonally along arcs. Our result requires no embeddedness assumption and no topological hypothesis other than connectedness.
We establish the existence of a positive and non-trivial periodic solution for the following class of delay differential equations θ”(t)+θ (t)(1-θ (t)-f(θ (t-r)))=0, where f is a bistable-type nonlinearity. Precisely, our result states that if |f'(ν )| is sufficiently large, there exists a finite collection of disjoint intervals with the following property: if the delay is within one of these intervals, then the equation admits a positive, nonconstant, even 2r-periodic solution. Here, ν is the unique equilibrium of the equation that is greater than 0 but less than 1.
We introduce and study semi-slant Riemannian maps defined on CR submanifolds of Kähler manifolds. The horizontal distribution is assumed to admit an orthogonal decomposition into a J̅ -invariant component and a slant component with constant slant angle θ∈ (0,π/2) . Under a natural compatibility condition between the complex structure of the target manifold and the normalized horizontal almost complex structure 𝒥_ℋ , we characterize when the integral manifolds of the range distribution Range π _* carry a Kähler structure. We derive a harmonicity criterion by decomposing the tension field with respect to the invariant and slant components of the horizontal distribution, where the trace over the slant horizontal distribution is expressed in a φ -adapted form involving the normalization factor ^2θ . We also clarify how the harmonicity condition must be reformulated at the anti-invariant endpoint θ =π/2 . Furthermore, we establish φ -sectional curvature formulas, partial Ricci-type curvature identities, and a scalar curvature-type decomposition along Range π _* , in which the slant angle enters explicitly through ^2θ and ^4θ factors. Several examples, including a proper and a nonlinear case, illustrate the theory.
We estimate the Hausdorff dimension of a subset of the unit sphere where a positive solution of a semilinear elliptic equation with source term of negative exponent in the unit ball grows faster than a prescribed order. Also, the sharpness of our estimate is shown by proving the existence of a positive solution growing faster than a prescribed order on a given subset of the unit sphere.
In this paper, we establish a new Fueter theorem and its inverse for generalized partial-slice monogenic functions. First, a new version Fueter theorem is first proved for an extended function class, from which a novel result is derived for the specific case of generalized partial-slice monogenic functions. Subsequently, the corresponding inverse Fueter theorems are formulated. The classical theoretical framework is significantly extended.
In this paper, drawing on the methodological approach of Grosse-Erdmann (Studia Math 139(1):47–68, 2000) and leveraging the characteristics of double sequence spaces, we have achieved a characterization of the dynamical behavior of double shift operators ( B^(l),B^(u)) on the double Fréchet sequence spaces, and obtained some sufficient conditions and necessary conditions for (B^(l), B^(u)) to be hypercyclic, weakly mixing, topological mixing and chaotic the sense of Devaney.
The hybrid rapid decay property for the pair (G, H), where H is a subgroup of G, was introduced by Chatterji and Zarka for discrete groups as a relative version of the classical rapid decay property. In this paper, we extend several results from the discrete setting to compactly generated locally compact groups. We first establish equivalent characterizations of hybrid rapid decay property for the pair (G, H). We then prove that if H is compact and (G, H) has hybrid rapid decay property, then G has the rapid decay property, while if G has rapid decay property and H is a discrete amenable cocompact subgroup of G, then (G, H) has hybrid rapid decay property. We also study the behavior of hybrid rapid decay property along infinite ascending chains of closed subgroups of finite index, obtaining permanence results for the pairs (G,H_i) and (H_j,H_i) . Finally, we show that if H is a normal compact subgroup of G, then (G, H) has hybrid rapid decay property if and only if the quotient group G/H has rapid decay property.
In this paper, we investigate the order boundedness of the generalized Stević–Sharma operators T_u,v,φ^m,nf(z)=u(z)f^(m)(φ (z))+v(z)f^(n)(φ (z)) between the weighted Bergman spaces A_ω^p induced by doubling weights. We also characterize the boundedness, essential norm, and compactness of T_u,v,φ^m,n acting from A_ω^p spaces into Zygmund-type spaces.
In this paper, we describe the general form of relatively uniformly continuous linear separating maps between unital f-algebras and commutative von Neumann regular algebras. As applications, we first show that, under a suitable regularity condition, every relatively uniformly continuous separating linear map from a unital f-algebra into a commutative von Neumann regular algebra is a weighted linear map. Second, we establish a Banach–Stone-type theorem for relatively uniformly continuous biseparating linear maps between C(X) and C(Y), where X and Y are complete metric spaces. This fits into the type of results by Jarosz [11], Araujo et al. [1] and Boulabiar et al. 5.
In this study, we establish a generalized framework for Hermite–Hadamard-type inequalities for subharmonic functions on ℝ^n, n≥ 2. Moving beyond traditional convex analysis, we utilize potential-theoretic methods (specifically, Dirichlet solutions and Poisson kernels), to derive integral inequalities over annular and rectangular domains. We provide a unified measure-theoretic interpretation that clarifies and extend recent results, demonstrating that the classical Hermite–Hadamard inequality is a specific one-dimensional manifestation of broader potential-theoretic principles. By employing approximation techniques, we relax standard regularity requirements, extending the applicability of these results to general subharmonic functions.
In this paper, we first deal with a class of neutral evolution equation with delay in ordered Banach space X d/dt(z(t)-cz(t-δ ))+A(z(t)-cz(t-δ ))=f(t, z(t), z(t-τ )), t∈ℝ, where | c| <1 , the constants δ ,τ >0 are defined as time lags, A:𝒟(A)⊂ X→ X is a closed linear operator, -A generates a positive and compact C_0 -semigroup T(t)(t⩾ 0) , and f:ℝ× X^2→ X is continuous function which is ω -periodic in t. Based on the characteristics of positive operator semigroup and monotone iterative technique, we provide important ordered conditions on the nonlinear terms f to guarantee that equation has positive and asymptotically stable ω -periodic mild solutions. Finally, an example of our main results is presented.
In this paper, we introduce the notions of (F,ψ ,φ ,α ,1-α ) -type contractive mappings and (F,ψ ,φ ,α ,1-α ) -type convex contractive mappings in metric spaces by employing C-class functions together with altering distance functions. Within this framework, we establish existence and uniqueness results for fixed points of continuous self-mappings defined on complete metric spaces. The proposed formulation provides a unified setting in which several known contractive conditions can be recovered through suitable choices of the auxiliary functions.
Abstract We study codimension one distributions on the projective three-space, focusing on cases where the tangent sheaf of the distribution is nonsplit and unstable. We relate the order of instability to the degree of the induced subfoliation by curves, showing that the order of instability is bounded. Moreover, we classify the tangent sheaf of the codimension one distributions that admit a subfoliation by curves of degree 1. In other words, assuming the sheaf is nonsplit, we classify the situations in which the tangent sheaf attains the maximal possible order of instability.
This paper introduces twisted sequential warped products. This new class of warped products extends solutions to Einstein’s field equations. We derive explicit formulas for covariant derivatives, curvature tensors, Ricci curvature, and scalar curvature. We characterize Einstein conditions and analyze geodesics. We also study conformal vector fields, including Killing and concircular types. Furthermore, we examine gradient almost Ricci solitons and the concircular curvature tensor. Explicit examples of such manifolds are presented, and applications to cosmology, black hole physics, and quantum field theory are discussed. These findings deepen the understanding of warped product geometry in general relativity. They also provide a foundation for further research.
In this paper, we consider generalized knot theories and we prove an Alexander-type theorem in this setting. More precisely, we prove that every oriented normal generalized link is the closure of a quasitoric normal generalized braid. In doing so, we compute a generating set for the pure normal generalized braid group. Further, we prove that the set of quasitoric normal generalized braids forms a subgroup of the normal generalized braid group.
This study presents a family of non-symmetric, explicit six-point block methods of sixth order for solving second-order ordinary and partial differential equations. The construction focuses on reducing phase and amplification errors to improve performance in oscillatory and dissipative problems. The order, phase-error behavior, and stability properties of the methods are analyzed, and their stability regions are illustrated. Numerical experiments on benchmark ODE and PDE problems confirm that the proposed schemes deliver higher accuracy. Comparisons with 12-step explicit symplectic Runge–Kutta methods and trigonometric-fitted schemes further show that the new methods are computationally faster.
