In this paper, we investigate the Pre-Schwarzian norm estimates for classes of log-harmonic mappings associated with the tanh function. Motivated by earlier studies on sine starlike and sine close-to-starlike log-harmonic mappings, we introduce the class of tanh log-harmonic mappings and determine their radius of starlikeness. Furthermore, we establish sharp upper bounds for the Pre-Schwarzian norm of locally univalent log-harmonic mappings of the form f (z) = h(z)g(z), where h(z) belongs to the class SB & lowast;. As an application, we derive explicit estimates for a special subclass of log-harmonic mappings of the form f(z) = h(z)h '(z).
. This paper provides an analysis of spacetimes characterized by the pseudo W8curvature tensor, with particular focus on their geometric and physical properties within the framework of Einstein's field equations. The study investigates pseudo W8-flat spacetimes as potential Einstein manifolds and examines the associated covariant properties of the energy-momentum tensor, including scenarios with a cosmological constant. It also explores perfect fluid models with vanishing pseudo W8-curvature to understand the behavior of energy density and isotropic pressure, particularly under the timelike convergence condition and its relation to the strong energy condition. Dust-type fluid spacetimes are considered to analyze vacuum solutions with non-zero energy density and their geometric structure. Additionally, the paper constructs explicit spacetime models to illustrate the geometric and physical implications of the vanishing pseudo W8-curvature condition.
. This study seeks to formulate and establish multiple fixed point and common fixed point theorems for a novel class of contractions, called C-type contractions, within the context of S-metric spaces. Furthermore, we provide illustrative examples to substantiate the theoretical findings. The results of this study generalize and extend several previous results in the literature. We use these fixed point results to establish the existence and uniqueness of solutions for a class of conformable differential equations, representing a recent development in fractional calculus.
. In this paper, we focus on constructing analytical asymptotic approximations for solutions of second-order matrix difference operators using the Liouville-Green method. General analytical approximations are provided based on the Discrete Levinson's Fundamental Theorem and Kooman's Theorem. Furthermore, we use the discrete Gronwall inequality to establish error bounds for the asymptotic approximations.
. This paper presents a novel fourth-order two-step iterative method for solving nonlinear equations f (x) = 0, derived by optimizing a key parameter in a combination of established third-order methods. The proposed method enhances efficiency and accuracy, outperforming existing iterative techniques. Extensive numerical experiments on diverse polynomial and transcendental functions demonstrate superior convergence speed and precision. Additionally, the method shows significant promise for applications in polynomiography and broader mathematical analysis, as evidenced by its robust performance in visualization and root-finding tasks.
Let X be a finite set and let T(X) denote the full transformation semigroup on X. For a fixed nonempty subset Y of X, we define S(X, Y ) as the subsemigroup of T(X) given by S(X, Y ) = {alpha E T(X) : Y alpha C_ Y }. In this paper, we identify all maximal subsemigroups of S(X, Y ) when Y is a proper subset of X. Additionally, we show that within this semigroup, the maximal subsemigroups and the maximal regular subsemigroups coincide if and only if Y = 1.
The purpose of the present work is to study the geometric characteristics of m-th root Finsler metrics under a conformal generalized Kropina transformation. We derive expressions for the fundamental tensor and the spray coefficients of the transformed metrics, and demonstrate that these geometric quantities become rational functions in the directional arguments under the conformal transformation. Further, we study the necessary and sufficient conditions for the conformal generalized Kropina transformation of an m-th root metric to be locally dually flat and derive conditions under which the conformal factor is a homothety. Additionally, we provide conditions under which the transformed metric is Einstein.
This study investigates the binormal curvature flows of timelike curves in the Minkowski space R2,1. We analyze the binormal velocity as a function of the curvature. In addition, we derive the time evolution equations for the Frenet frame associated with the curve in R2,1. We also establish the time evolution equations for its curvatures. Moreover, we construct timelike surfaces generated by the binormal flows of a family of such curves and present their geometric properties, including the mean and Gaussian curvatures. Finally, we introduce several new applications of the timelike surfaces constructed from these flows in R2,1.
. Let M and N be even dimensional connected closed manifolds. For self-indexing Morse functions f : M -* R and g : N -* R, we denote by C(f, g) the fiber product of f and g. The purpose of this paper is twofold: firstly, we prove a formula which describes chi(C(f, g)), where chi denotes the Euler characteristic. Using the formula, we compute chi(C(f, g)) for various f and g. Secondly, we compute H & lowast;(C(f, g); Z) when M and N are connected closed surfaces.
We establish the Hardy-Littlewood-Sobolev inequality for the quaternion Heisenberg group.
In this study, we introduce a new family of Leonardo quaternions defined by means of quantum integers, which we call q-generalized Leonardo quaternions. The aim is to extend the previously introduced Leonardo quaternions and generalized Leonardo quaternions. Furthermore, we derive several formulas and identities for this new family, including a Binet-like formula, exponential and Poisson generating functions, summation formulas, and Catalan-like, Cassini-like, and d'Ocagne-like identities.
Ramanujan listed several q-series identities in his lost notebook. The most well known q-series identities are the Rogers-Ramanujan type identities which are first discovered by Rogers and then rediscovered by Ramanujan. In this paper, we give partition-theoretic interpretations of some of the Rogers-Ramanujan type identities using overpartition and colour partition of positive integers, and prove infinite families of congruences modulo powers of 2.
. This paper introduces the notion of graded 0t-prime ideals as a natural bridge between graded prime ideals and graded quasi-primary ideals within the framework of Ggraded commutative rings, where G is a group. We investigate their fundamental properties and characterizations, focusing on their behavior under graded ring constructions such as localization, factor rings, and trivial extensions. Several equivalences and transfer properties are established, offering a comprehensive understanding of graded 0t-prime ideals. Illustrative examples highlight their distinctions from other generalized ideals and emphasize their intermediate nature. Furthermore, we characterize graded rings in which every graded ideal is a graded 0t-prime ideal.
. In this paper, the q-derivative operator for matrix functions is defined, and q-calculus is examined in a broader context. The q-matrix exponential, the q-binomial of two matrices, and q-trigonometric and q-hyperbolic matrix functions are introduced. The q-derivatives of these newly defined functions are calculated in detail, and the fundamental properties of these structures are investigated. This study provides new insights into the interactions between q-calculus and matrix analysis.
Let D denote the open unit disc centered at the origin in the complex plane, and let S denote the usual class of normalized univalent functions defined on D. For c > 0 and for f is an element of S define, M-c,M-h[f](z) = f(z) + c(h (*) f)(z)/1 + c + (Sic)f(z) -c(h (*) f)(z)/1 + c , where h : D -> C is an analytic function with h(0) = 0, h '(0) not equal 0 and satisfying R((1 z)(2)h '(z)) > 0 for z is an element of D. Here '*' denotes the Hadamard product (or convolution) of two analytic functions. We find conditions on f such that M-c,M-h[f ] is harmonical convex; and conditions on h and f such that M-c,M-h[f ] is stably harmonically convex. We also investigate properties of the classical Cesa`ro means and de la Vallee Poussin means of M-c,M-h[I], where I(z) = z/(1 - z). Furthermore, we present, among other results, a subordination relation between the Cesa`ro means and the de la Vallee Poussin means of a harmonic function.
In this paper, we consider the problem of reconstructing the heat distribution for a nonlinear space-fractional diffusion equation from the final data with Gaussian white noise. As is commonly acknowledged, the problem is severely ill-posed according to Hadamard's definition. Consequently, we propose the Fourier truncation method to regularize the problem. With different assumptions on the exact solution, we obtain an estimation of the expectation of the error between the regularized solution and the exact solution. Finally, we provide an example to illustrate our theoretical results.
This paper establishes new weighted Hardy-type inequalities on arbitrary time scales, aiming to unify and extend classical results across both continuous and discrete settings. The primary objective is to develop general dynamic inequalities using tools from delta calculus, including the chain rule, integration by parts, Holder's inequality, Jensen's inequality, and the properties of convex and submultiplicative functions. Several novel inequalities are derived under suitable assumptions, improving known results and introducing entirely new findings within the framework of time scales. These contributions not only enhance the theoretical foundation of integral inequalities but also provide useful applications in the analysis of dynamic equations and discrete systems.
In this paper, we investigate the existence and non-existence of non-negative solutions for a non-homogeneous fractional p-Laplacian problem ((-triangle)(s)(p)u= |u|(p & lowast;s-2)u +lambda f(x) in ohm, u = 0 in R-N\ohm, where ohm is a smooth bounded domain in R-N (N > sp), with s is an element of (0, 1), lambda> 0 and 1 < p < +infinity, where p(s)(& lowast;) := (N)(p) (N-sp) is the fractional critical exponent, and where f is a function with changing sign.
We introduce a new polynomial invariant for oriented dichromatic links, called the projection image polynomial, derived from the dibiquandle colorings of link diagrams. This invariant simultaneously extends the dibiquandle counting invariant and the image enhancement polynomial by incorporating the two independent biquandle projections of a dibiquandle. We establish that the projection image polynomial is preserved under oriented dichromatic Reidemeister moves and provide an explicit correspondence between colorings and homomorphisms of the fundamental dibiquandle. Through several examples, we show that this polynomial distinguishes dichromatic links which share identical counting invariants and image enhancement polynomial. Furthermore, we prove its algebraic properties, including multiplicativity under disjoint union, a normalization law for connected sum, and duality relations under mirror image and orientation reversal. Finally, we outline a natural generalization of this framework to oriented n-colored links via the notion of an n-multibiquandle, which leads to the associated n-projection image polynomial. This multicolored extension unifies the biquandle (n = 1) and dibiquandle (n = 2) cases within a common algebraic framework and provides a foundation for subsequent module-theoretic and categorified refinements.
