Functions and Analysis

2312 Submissions

[5] viXra:2312.0167 [pdf] submitted on 2023-12-31 20:26:57

Complete Integers: Extending Integers to Allow Real Powers Have Discontinuities in Zero

Authors: Davide Peressoni
Comments: 8 Pages.

We will define a superset of integers (the complete integers), which contains the dual of integers along parity (e.g. the odd zero, the even one, ...). Then we will see how they form a ring and how they can be used as exponents for real numbers powers, in order to write functions which have a discontinuity in zero (the function itself or one of its derivates), as for example |x| and sgn(x).
Category: Functions and Analysis

[4] viXra:2312.0132 [pdf] submitted on 2023-12-26 03:24:49

Homogenization of the First Initial-Boundary Value Problem for Periodic Hyperbolic Systems. Principal Term of Approximation

Authors: Yulia Meshkova
Comments: 17 Pages.

Let $mathcal{O}subset mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In $ L_2(mathcal{O};mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $A_{D,varepsilon}$ with the Dirichlet boundary condition. Here $varepsilon >0$ is a small parameter. The coefficients of the operator $A_{D,varepsilon}$ are periodic and depend on $mathbf{x}/varepsilon$. The principal terms of approximations for the operator cosine and sine functions are given in the $(H^2ightarrow L_2)$- and $(H^1ightarrow L_2)$-operator norms, respectively. The error estimates are of the precise order $O(varepsilon)$ for a fixed time. The results in operator terms are derived from the quantitative homogenization estimate for approximation of the solution of the initial-boundary value problem for the equation $(partial _t^2+A_{D,varepsilon})mathbf{u}_varepsilon =mathbf{F}$.
Category: Functions and Analysis

[3] viXra:2312.0125 [pdf] submitted on 2023-12-23 13:07:24

Quadratic Phase Quaternion Domain Fourier Transform

Authors: Eckhard Hitzer
Comments: 12 Pages. Published in: Bin Sheng, Lei Bi, J inman Kim, Nadia Magnenat Thalmann, Daniel Thalmann (eds), Advances in Computer Graphics. CGI 2023. LNCS, vol 14498. Springer, Cham, First Online: 24 Dec. 2023. https://doi.org/10.1007/978-3-031-50078-7_21

Based on the quaternion domain Fourier transform (QDFT)of 2016 and the quadratic-phase Fourier transform of 2018, we introduce the quadratic-phase quaternion domain Fourier transform (QPQDFT) and study some of its properties, like its representation in terms of the QDFT, linearity, Riemann-Lebesgue lemma, shift and modulation, scaling, inversion, Parseval type identity, Plancherel theorem, directional uncertainty principle, and the (direction-independent) uncertainty principle. The generalization thus achieved includes the special cases of QDFT, a quaternion domain (QD) fractional Fourier transform, and a QD linear canonical transform.
Category: Functions and Analysis

[2] viXra:2312.0112 [pdf] submitted on 2023-12-21 23:18:31

On Wilker-Type Inequalities

Authors: Yi-Chieh Huang, Li-Chang Hung
Comments: 6 Pages.

In this paper, we present elementary proofs of Wilker-type inequalities involving trigonometric and hyperbolic functions. In addition, we propose some conjectures which extend and generalize the Wilker-type inequalities.
Category: Functions and Analysis

[1] viXra:2312.0111 [pdf] submitted on 2023-12-21 23:17:21

On Generalized li-Yau Inequalities

Authors: Li-Chang Hung
Comments: 11 Pages.

We generalize the Li-Yau inequality for second derivatives and we also establish Li-Yau type inequality for fourth derivatives. Our derivation relies on the representation formula for the heat equation.
Category: Functions and Analysis