In this paper, we consider the blow-up behavior of linearly implicit L1 method for fractional ordinary differential equations. Based on Nakagawa's criteria, a suitable adaptive step strategy is introduced. The existence of global numerical solution is proved. Moreover, we show that the finite blow-up behaviors are replicated for any positive solution. Finally, some numerical examples are given to test the main result.
The Sturm-Liouville wavelet transform (SLWT) is a novel addition to the class of Sturm-Liouville transforms, which has gained a respectable status in the realm of time-frequency signal analysis within a short span of time. Given that the study of time-frequency analysis is of both theoretically interesting and practically useful, the aim of this paper is to explore a class of quantitative uncertainty principles (UP) associated with SLWT, including Faris local-type UP, Shannon-type UP, Donoho-Stark-type UP and Benedicks-type UP.
We consider three-dimensional Lorentzian Egorov and epsilon-spaces and show that the existence of proper biharmonic helices in these manifolds is highly constrained and closely tied to the causal character of the Frenet frame. In particular, we prove that no proper biharmonic helices exist in Egorov spaces unless the space is flat and the normal vector is spacelike. Furthermore, we establish that no proper biharmonic helices exist in threedimensional epsilon-spaces.
In the present article, we propose a numerical resolution of the Caputo-Fabrizio temporal fractional wave equation with non-homogenous Dirichlet and Neumann functional conditions. We derive and analyze the semi- and fully discrete approximations using the introduced finite difference scheme for the time Caputo-Fabrizio derivative and the finite element scheme for the spacial derivative. Result of the existence and uniqueness of the solution is discussed, stability and error estimates are established. To support the theoretical studies, a numerical example is given.
We obtain necessary and sufficient conditions for an increasing nonnegative sequence omega n, n is an element of N, to be a Weyl multiplier for a.e. unconditional convergence for general Haar or Franklin systems corresponding to an admissible sequence T.
A basis of a real symmetric bilinear space is called an isotropic basis if all its elements are isotropic. In this paper, we provide both necessary and sufficient conditions for the existence of such an isotropic basis. We present one geometric method and two linear algebraic methods for constructing isotropic bases. Additionally, we address a question arising from the properties of symmetric bilinear forms. As a consequence, we explore various properties of the vector space spanned by the preimage set of a point under a real-valued continuous function. We also demonstrate some applications of these properties within the context of real symmetric bilinear spaces.
In this paper, we examine the geometric properties of linear operators associated with normalized Jackson and Hahn-Exton q-Bessel functions that arise through suitable transformations and q-extension of the Hohlov integral operator. These operators are investigated in the framework of the function from the class M ,theta and sufficient conditions for functions in the class M ,theta Additionally, we explore inclusion properties by using the Taylor coefficients of z2 phi 1(a, b; c; q, z) and the normalized Jackson and Hahn-Exton q-Bessel functions. The primary objective is to derive sufficient conditions under which the convolution operators will be in different subclasses of q-starlike and q-convex functions.
This paper provides the solution to the problem of Gibbsian description of Gaussian random fields based on the Gibbs scheme currently being developed in the theory of lattice random fields.
This paper provides a comprehensive derivation of geodesic equations in a two-dimensional Finsler space characterized by a Matsumoto-type metric. It further explores the geometric applications of these equations on various surfaces, including cylinders, spheres, pseudo-spheres, and catenoids. Using illustrative examples and graphs, we analyze the geometric properties of geodesics for different parametric forms of these surfaces. The results enhance our understanding of geodesic behavior in Finsler geometry and shed light on curvature and geometric structures in diverse contexts.
In this paper, we consider the blow-up of explicit L1 scheme for time-fractional partial differential equations. Firstly, we discretize the mentioned equation by the explicit L1 scheme and obtain the corresponding matrix form. Secondly, we introduce the concept of discrete energy. Based on Nakagawa's criteria, a suitable adaptive time-stepping strategy is given by the discrete energy. Thirdly, with the help of lower discrete energy, the finite blow-up behaviors of numerical solution are studied. Finally, some numerical examples for verifying the theoretical results are provided.
The purpose of this study is to obtain a decomposition of the solution to a backward stochastic differential equation used in the dual problem of mathematical finance. Some explicitly solvable equations considered. We convert the equation into a system of recurrent relations. By solving this system and proving convergence of the series the solution to the equation can be determined. In this study, Adomian's method was applied to solve the backward stochastic differential equation. An explicit solution was obtained for some examples.
In his seminal monograph, S. I. Adian established a lower bound on the growth of the free Burnside group B(2, n) for odd n >= 665. Building on Adian's methods, we extend this result to free Burnside groups B(m, n) of arbitrary rank m >= 2. As a consequence, we obtain growth estimates for a broad class of finitely generated groups.
In this paper, we formulate the conformable Fourier transform on time scales, drawing motivation from the structure of the conformable bilateral Laplace transform. Some of the elementary properties are proved, including shifting, transform of derivative, conjugation, transform of Hilger delta function, and transform of integral.
This article focuses on the determination of appropriate lower bounds for a general term defined as the sum of two specific integrals. This term has the property of depending on four functions, one of which is associated with the two integrals involved. Two theorems are established: one with mono-tonicity and sign assumptions on the functions considered, and another, more technical, with special primitive-like inequality assumptions on these functions. The connections, advantages and limitations of these assumptions are discussed in detail.
The main objective of this paper is to study some new local fractional Hilbert-type inequalities with a general kernel. We apply our main results to non-homogeneous kernels. In addition, we obtain the best possible constants.
This study continues previous research on the approximation of functions by means of singular integrals. We begin by introducing the truncated Picard singular integral. Subsequently, using this integral along with the classical Picard--Cauchy and Gauss-Weierstrass singular integrals, we establish the orders of approximation for functions belonging to a generalized Zygmund space, both in the Lp-norm and in the corresponding norm of the generalized Zygmund space.
The Eneström-Kakeya theorem states that if $P(z)=\sum_{\ell =0}^n a_\ell z^\ell$ is a polynomial of degree $n$ with real coefficients satisfying $0\leq a_0\leq a_1\leq \cdots\leq a_n$, then all zeros of $P$ lie in $|z|\leq 1$ in the complex plane. Motivated by recent results concerning an Eneström-Kakeya "type" condition on the real and imaginary parts of complex coefficients, we give similar results with hypotheses concerning the real and imaginary parts of the coefficients of a quaternionic polynomial. We give bounds on the moduli of quaternionic zeros of such polynomials.
Consider the following problem: given a positive integer, what is the minimum number of positive integer powers having unlike exponents greater than one such that their sum is equal to the given number? We deal with this open question by presenting some experimental results, indicating some inequalities and relations, presenting some new integer sequences, obtaining a bivariate generating function, and eventually proposing a conjecture.
All physical phenomena in the four-dimensional spacetime are invariant under the Poincaré group. The Standard Model of fundamental interactions, Electroweak theory, and Quantum Chromodynamics are required to be invariant under Poincaré group. Any possible extension of the Poincaré group hints to the existence of a new physics beyond the Standard Model. In particular, the supersymmetric extension of the Poincaré group predicts the existence of new particles that are supersymmetric partners of the elementary particles of the Standard Model: leptons, quarks, W and Z bosons and gluons. In a recently suggested high-spin extension of the Poincaré group, new massless particles of increasing spins are predicted to exist. In that respect we are interested in investigating a massless representation of the Poincaré algebra that has high-spin states. The massless states are described by the helicity operator, which has only two polarisations equal to the components of spin along the direction of motion, as it takes place for photons and gravitons. This means that not all of the 2s+1 spin magnetic quantum states exist and the spin operator is not defined anymore. In order to eliminate the spin operator from a massless representation and ensure that only helicity operator is included into the representations, Schwinger suggested that new non-commuting coordinates should be defined. We investigate the uncertainty relations that follow from non-commutativity of these new coordinates. It is the average wavelength of a massless particle that sets the scale of the coordinate uncertainty.
This article is based on the construction procedure of bivariate hyperbolic box spline functions. Generally, box splines are considered as the multivariate generalizations of univariate B-splines. Both B-splines and box splines are refinable functions. Two different kinds of box splines like the polynomial box splines and the trigonometric box splines along with their usefulness are well studied in literature. However, another variant of box splines named as the class of hyperbolic box spline functions, has not gained much attention. This article focuses on the construction of bivariate hyperbolic box spline functions from univariate hyperbolic B-spline functions through directional convolution method. Also, the importance and usefulness of such functions are discussed.
