This article demonstrates the broader flexibility of the techniques developed in previous studies, showing that they can be adapted to cover a wider range of Hilbert-type integral inequalities. By introducing a framework based on the incomplete gamma function, we derive two new generalizations that extend existing results while maintaining connections with classical inequalities. Several examples illustrate these extensions, o ering new perspectives and expanding the scope of potential applications.
This article is devoted to the study of a new three-dimensional Hardy-Hilbert integral inequality. It is innovative mainly for its generality, characterized by the presence of many adjustable parameters and complex power-sum interactions of the variables. Several new three-dimensional integral inequalities are derived from the main result, showing how it can be applied in di erent analytical frameworks. The proofs are presented in detail, ensuring a rigorous theoretical foundation.
In this article we prove a common fixed point result for interpolative Kannan type contraction mappings in a nonempty, closed and bounded subset with respect to n linearly independent vectors in an n-Banach space. On the other hand, we introduce interpolative Dass and Gupta rational type contraction mappings on a nonempty, closed and bounded subset with respect to n linearly independent vectors in an n-Banach space. In particular, we discuss the existence and uniqueness of a fixed point of such a mapping on a nonempty, closed and bounded subset with respect to n linearly independent vectors in an n-Banach space.
Text classification is the task of assigning textual data to predefined categories, playing a crucial role in natural language processing. In recent years, deep learning models have demonstrated superior performance over traditional machine learning approaches in text classification tasks. This paper presents a supervised deep learning approach for hierarchical Arabic text classification. To facilitate this study, we developed WiHArD, a novel hierarchical Arabic text dataset, where each text is systematically labeled according to a structured category hierarchy. We then propose a deep learning model that integrates BERT-based feature extraction with a neural network classifier. BERT encodes textual inputs into dense vector representations, while the neural network learns to accurately classify texts within the hierarchical structure. Our comparative study demonstrates that the proposed BERT-ANN model achieves significant improvements in hierarchical classification performance, outperforming the existing HMATC model. These findings highlight the e ectiveness of deep learning-based approaches in advancing Arabic text classification.
Our idea is to use the global diffeomorphism group G (M) to study certain properties of differentiable manifolds endowed with global vector fields. We define a partial parallelizable manifold (p.p.m.) as a couple (M, ρp), where ρp is a global partial frame (g.p.f.) of the tangent bundle T M of M. In this text there are studied properties of the p.p.m. Certain specific invariants for these manifolds, relative to the group G (M) of the global diffeomorphisms of M, are constructed. We also give some applications and examples. We obtain that a G (M) -invariant [M × ℝp] of partial parallelizable manifold (M, ρp) is an equivalence class defined by action group G (M) on M.
We extend a refinement of the Hermite-Hadamard inequality to other convex functions, thus some integral of these convex functions can be estimated by series. We also generalize part of this refinement by introducing one more parameter, then the Stolarsky mean can be refined and more general integrals of convex functions can be estimated.
In this article, new Hardy-Hilbert-type integral inequalities are established. Our main result is based on a special inhomogeneous two-parameter kernel function. It is of the ratio power form, and has the property of involving a product term which perturbs the standard homogeneity property. We then use this result to derive new weighted integral norm inequalities and other Hardy-Hilbert-type integral inequalities. They are also defined with inhomogeneous kernel functions, but with innovative power and logarithmic forms. Some of them are obtained by treating an adjustable parameter as a variable and integrating with respect to it, which remains an original technique of proof. The article concludes with an at-tempt to unify some new and old Hardy-Hilbert-type integral inequalities. Due to the mathematical complexity, the optimality of the final result remains an open question, giving some new perspectives to a classical topic.
This paper addresses the existence of solutions for a boundary value problem characterized by a fractional differential equation involving a multi-term formulation of the Riemann-Liouville-Caputo derivative. The multi-term approach allows for more accurate modelling of complex systems with diverse memory and hereditary properties. To explore the existence and uniqueness of solutions, fixed point theorems are employed. Additionally, two numerical examples are provided to illustrate the theoretical results.
Motivated by a question on the ranges of the commutators of dilated floor functions in [10], together with a related problem in [3], we investigate the precise ranges of certain generalized polynomials dependent on a real parameter. Our analysis requires non-trivial tools, including Kronecker’s approximation theorem. The results highlight sharp distinctions between irrational parameters and sub-unitary and supra-unitary rational parameters. We also propose several conjectures for the irrational and supra-unitary rational cases, supported by extensive computations in Wolfram Mathematica.
Some refinements of the celebrated Aczél-Popoviciu and Bellman inequalities in both discrete and Lebesgue integral forms are provided. We also express Hölder and Aczél-Popoviciu inequalities in a monotonous sequence S ( m ), and the special case S (0) ≥ S (+ ∞ ) is our refinement.
The present paper is focused of some concepts of dichotomy with growth rates (h-dichotomy, weak h-dichotomy) for linear discrete systems. These concepts use two types of dichotomy projections sequences (invariant and strongly invariant) and generalize some well-known dichotomy concepts. More precisely, necessary and sufficient conditions of Datko type are given using both invariant and strongly invariant projections sequences for h-dichotomy and weak h-dichotomy.
This article solves an open problem that was previously stated by providing an exact evaluation of a logarithmic integral. Furthermore, the result is generalized by introducing a new adjustable parameter. Based on this extension, other integral formulas are derived. Applications are presented for integral inequalities of Hölder and Hardy-Hilbert types, in which the introduced parameter plays a pivotal role in establishing the bounds under suitable integrability conditions.
In this article, we derive new integral formulas involving a ratio function, a maximum function, and three adjustable parameters. Two of these parameters control the maximum function in di erent ways. The arctangent function plays a central role in the resulting expressions. These formulas are then used to construct new and varied types of integral inequalities. In particular, we present weighted Hölder-type integral inequalities, as well as new Hardy-Hilbert-type integral inequalities. Their novelty lies mainly in the inclusion of the maximum function and the two parameters governing it. Detailed proofs are given.
Let A be a local Artinian ring with residue field k(A). Let X be a curve over A and let be X′ = X ×spec A spec k(A) the fiber of X over k(A). Consider ℒ an invertible sheaf on X and ℒ ′ = ϕ*ℒ ∈ Pic(X′), where ϕ : X′ → X is the natural map. Let C and C′ be the algebraic geometric codes constructed using the groups of cohomology Γ(X, ℒ) and Γ(X′, ℒ ′) respectively. In this note, we first give the complete relation between Γ(X, ℒ) and Γ(X′, ℒ ′) without any condition and finally, we provide relations between C ⊗ A k(A) and C′ using exact sequences and dimensional theory. Therefore we extend, some results of Walker [18, 20] giving the characterization of WAG codes over rings.
Building on the simplified mathematical model derived from the dynamics of the hypothalamic-pituitary-adrenal (HPA) axis, which describes the interactions between ACTH and cortisol through a system of differential equations with distributed delays, this paper focuses on the criticality analysis of the Hopf bifurcation. This analysis is conducted using the method of multiple time scales, which is applicable to systems with arbitrary distributed delays and indicates whether the Hopf bifurcation leads to a stable or unstable limit cycle. Furthermore, the approach presented in this paper can be applied to a broader class of systems that exhibit similar structural properties, extending its relevance beyond the specific biological context considered here.
Hasanalizade [5] studied Deaconescu’s conjecture for positive composite integer n . A positive composite integer n ≥ 4 is said to be a Deaconescu number if S 2 ( n ) | ϕ ( n ) − 1. In this paper, we improve Hasanalizade’s result by proving that a Deaconescu number n must have at least seventeen distinct prime divisors, i.e., ω ( n ) ≥ 17 and must be strictly larger than 5.86 · 10 22 . Further, we prove that if any Deaconescu number n has all prime divisors greater than or equal to 11, then ω ( n ) ≥ p *, where p * is the smallest prime divisor of n and if n ∈ D 3 then all the prime divisors of n must be congruent to 2 modulo 3 and ω ( n ) ≥ 48.
In this article, we establish new and flexible integral inequalities that have the property of involving four adaptable functions. Some of them generalize existing results in the literature. They can also be reformulated in the context of probability theory. Several special examples are discussed in detail. Additional integral inequalities based on the notion of convexity are also proposed.
In this study, we discuss the existence and the uniqueness of the solution to Caputo-Hadamard Cauchy problems for a system of fractional differential equations, by using Perov’s fixed point theorem. Finally, two examples are provided to illustrate our results.
Does 20 have a friend? Or is it a solitary number? A folklore conjecture asserts that 20 has no friends i.e. it is a solitary number. In this article, we prove that, a friend N of 20 is of the form N=2·5^2a· m^2, with (3,m)=(7,m)=1 and it has at least six distinct prime divisors. Furthermore, we show that Ω(N)≥ 2ω(N)+6a-5 and if Ω(m)≤ K then N< 10· 6^(2^K-2a+3-1)^2, where Ω(n) and ω(n) denote the total number of prime divisors and the number of distinct prime divisors of the integer n respectively. In addition, we deduce that, not all exponents of odd prime divisors of friend N of 20 are congruent to -1 modulo f, where f is the order of 5 in (ℤ/pℤ)^× such that 3| f and p is a prime congruent to 1 modulo 6. Also, we prove necessary upper bounds for all prime divisors of friends of 20 in terms of the number of divisors of the friend. In addition, we prove that, if P is the largest prime divisor of N then P<N^1/4.
Some work has been done in the study of non-geodesic III-harmonic curves in some model spaces. In this paper, we study III-harmonic curves in SL2ℝ˜\widetilde {{\rm{S}}{{\rm{L}}_2}\mathbb{R}} space. We give necessary and su cient conditions for helices to be III-harmonic. Also, we characterize III-harmonic curves in terms of their curvature and torsion.
