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Xiao-Jun Yang

Publications and source records attributed to Xiao-Jun Yang.

At least 19 recordsLinked to original sources
arXiv2022

On the Riemann-Hardy hypothesis for the Ramanujan zeta function

The Ramanujan zeta function was in $1916$ proposed by an Indian mathematician Srinivasa Ramanujan. As an analogue of the Riemann hypothesis, an English mathematician Godfrey Harold Hardy proposed in $1940$ that the real part of all complex zeros of the Ramanujan zeta function is $6$. This is the well-known Riemann-Hardy hypothesis for the Ramanujan zeta function. This article is devoted to the proof of this hypothesis derived from the Ramanujan-Rankin function. Owing to the integral representation of the Ramanujan-De Bruijn function, we establish its series. We also reduce its product using the Hadamard's factorization theorem. By a class with its series and product representations, we conclude that the real part of all zeros for Ramanujan-De Bruijn function is zero. we also obtain its products of Conrey and Ghosh and Hadamard-type for the Ramanujan-Rankin function. Based on the obtained result, we prove that the Riemann-Hardy hypothesis is true.

math.GM↗
arXiv2022

On the de Bruijn-Newman constant: a new approach

The conjecture of Newman, proposed in 1976 by Newman, states that all zeros of $Ξ_\aleph \left( λ\right)$ are real for $\aleph \in \mathbb{R}$. Its equivalent statement is that $\mathbb{M}_\aleph \left( τ\right)$ has purely imaginary zeros for $\aleph \in \mathbb{R}$. It is well known that $\mathbb{M}_\aleph \left( τ\right)$ is an even entire function of order one. This article addresses the product representation for $\mathbb{M}_\aleph \left( τ\right)$ by the works of Hadamard and Csordas, Norfolk and Varga. We establish a new class of $\mathbb{M}_\aleph \left( τ\right)$ by its series and product. Based on the obtained result, we prove that it has only purely imaginary zeros for $\aleph \in \mathbb{R}$. This implies that the conjecture of Newman is true.

math.GM↗
arXiv2022

A new class of the entire function of order one: a case study

In this article, a new class of the entire function of order one, expressed by the series and product representations with the real positive coefficients and complex zeros, is investigated for the first time. The entire function on the critical line deduces an even entire function of order one. It is proved that the real part of the complex zeros is equal to the critical line. An equivalent representation theorem is obtained to set up the sufficient conditions for the critical line for the entire function. As a typical example, the critical line for the special hyperbolic cosine function obtained by the present theorem agrees with the result of Euler. We also discover the new products of the hyperbolic cosine and sinc functions.

math.GM↗
arXiv2022

On all real zeros for a new class of the even entire function

In this article we propose a new class of the even entire function connected with the product and series with the real coefficients. We address a sufficient condition for all real zeros for it. As a typical example, we give an answer to the problem of Lagarias and Montague. We suggest the open problems for the class of the even entire function.

math.GM↗
arXiv2021

A scaling law chaotic system

In this article, we propose an anomalous chaotic system of the scaling-law ordinary differential equations involving the Mandelbrot scaling law. This chaotic behavior shows the "Wukong" effect. The comparison among the Lorenz and scaling-law attractors is discussed in detail. We also suggest the conjecture for the fixed point theory for the fractal SL attractor. The scaling-law chaos may be open a new door in the study of the chaos theory.

math.DS↗
arXiv2021

An insight on the fractal power law flow: from a Hausdorff vector calculus perspective

In the article we suggest the Hausdorff vector calculus based on the Chen Hausdorff calculus for the first time. The Gauss-Ostrogradsky-like, Stokes-like, and Green-like theorems, and Green-like identities are obtained in the framework of the Hausdorff vector calculus. The formula is proposed as a mathematical tool to describe the real world problems for the fractal power-law flow equations with the anomalous diffusion equation. A conjecture for the fractal power-law flow equations analogous to the Smales 15th Problem (one of the Millennium Prize Problems for the Navier--Stokes equations) is also addressed.

math.GM↗
arXiv2020

The scaling-law flows: An attempt at scaling-law vector calculus

In this paper, the scaling-law vector calculus, which is related to the connection between the vector calculus and the scaling law in fractal geometry, is addressed based on the Leibniz derivative and Stieltjes integral for the first time. The Gauss-Ostrogradsky-like theorem, Stokes-like theorem, Green-like theorem, and Green-like identities are considered in the sense of the scaling-law vector calculus. The Navier-Stokes-like equations are obtained in detail. The obtained result is as a potentially mathematical tool proposed to develop an important way of approaching this challenge for the scaling-law flows.

nlin.CD↗
arXiv2017

A new fractional derivative involving the normalized sinc function without singular kernel

In this paper, a new fractional derivative involving the normalized sinc function without singular kernel is proposed. The Laplace transform is used to find the analytical solution of the anomalous heat-diffusion problems. The comparative results between classical and fractional-order operators are presented. The results are significant in the analysis of one-dimensional anomalous heat-transfer problems.

math.CA↗
arXiv2016

Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems

In the present paper, we address a class of the fractional derivatives of constant and variable orders for the first time. Fractional-order relaxation equations of constants and variable orders in the sense of Caputo type are modeled from mathematical view of point. The comparative results of the anomalous relaxation among the various fractional derivatives are also given. They are very efficient in description of the complex phenomenon arising in heat transfer.

physics.class-ph↗
arXiv2016

A new fractional operator of variable order: application in the description of anomalous diffusion

In this paper, a new fractional operator of variable order with the use of the monotonic increasing function is proposed in sense of Caputo type. The properties in term of the Laplace and Fourier transforms are analyzed and the results for the anomalous diffusion equations of variable order are discussed. The new formulation is efficient in modeling a class of concentrations in the complex transport process.

cond-mat.stat-mech↗
arXiv2013

Superconductivity, charge- or spin-density wave, and metal-nonmetal transition in BaTi$_{2}$(Sb$_{1-x}$Bi$_{x}$)$_{2}$O

We have performed an isovalent substitution study in a layered titanium oxypnictide system BaTi$_{2}$(Sb$_{1-x}$Bi$_{x}$)$_{2}$O (0$\leq x\leq$ 0.40) by the measurements of x-ray diffraction, electrical resistivity and magnetic susceptibility. The parent compound BaTi$_{2}$Sb$_{2}$O is confirmed to exhibit superconductivity at 1.5 K as well as charge- or spin-density wave (CDW/SDW) ordering below 55 K. With the partial substitution of Sb by Bi, the lattice parameters $a$, $c$ and $c/a$ all increase monotonically, indicating negative chemical pressure and lattice distortion on the (super)conducting Ti$_2$Sb$_2$O-layers. The Bi doping elevates the superconducting transition temperature to its maximum $T_c$=3.7 K at $x=$0.17, and then $T_c$ decreases gradually with additional Bi doping. A metal-to-nonmetal transition takes place around $x$=0.3, and superconductivity at $\sim$1K exists at the nonmetal side. The CDW/SDW anomaly, in comparison, is rapidly suppressed by the Bi doping, and vanishes for $x\geq$0.17. The results are discussed in terms of negative chemical pressure and disorder effect.

cond-mat.supr-con↗
arXiv2012

A generalized model for Yang-Fourier transforms in fractal space

Local fractional calculus deals with everywhere continuous but nowhere differentiable functions in fractal space. The Yang-Fourier transform based on the local fractional calculus is a generalization of Fourier transform in fractal space. In this paper, local fractional continuous non-differentiable functions in fractal space are studied, and the generalized model for the Yang-Fourier transforms derived from the local fractional calculus are introduced. A generalized model for the Yang-Fourier transforms in fractal space and some results are proposed in detail.

math-ph↗
arXiv2012

Generalized Local Fractional Taylor's Formula with Local Fractional Derivative

In the present paper, a generalized local Taylor formula with the local fractional derivatives (LFDs) is proposed based on the local fractional calculus (LFC). From the fractal geometry point of view, the theory of local fractional integrals and derivatives has been dealt with fractal and continuously non-differentiable functions, and has been successfully applied in engineering problems. It points out the proof of the generalized local fractional Taylor formula, and is devoted to the applications of the generalized local fractional Taylor formula to the generalized local fractional series and the approximation of functions. Finally, it is shown that local fractional Taylor series of the Mittag-Leffler type function is discussed.

math-ph↗
arXiv2011

A short introduction to local fractional complex analysis

This paper presents a short introduction to local fractional complex analysis. The generalized local fractional complex integral formulas, Yang-Taylor series and local fractional Laurent's series of complex functions in complex fractal space, and generalized residue theorems are investigated.

math-ph↗