相关论文: A transcendence result on elliptic functions using…
We show that the iterative logarithm of each non-linear entire function is differentially transcendental over the ring of entire functions, and we give a sufficient criterion for such an iterative logarithm to be differentially…
It is proved that for all but a finite set of the square-free integers $d$ the value of transcendental function $\exp~(2\pi i ~x+\log\log y)$ is an algebraic number for the algebraic arguments $x$ and $y$ lying in a real quadratic field of…
Given a bounded Lipschitz domain $\omega\subset\mathbb{R}^{d-1}$ and a lower semicontinuous function $W:\mathbb{R}^N\to\mathbb{R}_+\cup\{+\infty\}$ that vanishes on a finite set and that is bounded from below by a positive constant at…
The Weierstrassian $\wp, \zeta$ and $\sigma $ functions are generalized to ${\bf R}^{n}$. The $n=3$ and $n=4$ cases have already been used in gravitational and Yang-Mills instanton solutions which may be interpreted as explicit realizations…
We consider Riemann's Xi function $\xi(s)$ which is evaluated at $s = \frac{1}{2} + \sigma + i \omega$, given by $\xi(\frac{1}{2} + \sigma + i \omega)= E_{p\omega}(\omega)$, where $\sigma, \omega$ are real and compute its inverse Fourier…
A transcendental function usually returns transcendental values at algebraic points. The (algebraic) exceptions form the so-called \emph{exceptional set}, as for instance the unitary set $\{0\}$ for the function $f(z) = e^z \,$, according…
The theory of Schwarzschild geodesics is revisited. Basing on a result by Weierstrass and Biermann, we derive a formula describing all non radial, timelike and null trajectories in terms of Weierstrass elliptic functions. Quite remarkably,…
It is expected that Schanuel's Conjecture contains all ``reasonable" statements that can be made on the values of {\em the exponential function}. In particular it implies the Lindemann-Weierstrass Theorem and the Conjecture on algebraic…
It is known that the elliptic function solutions of the nonlinear Schr\"odinger equation are reduced to the algebraic differential relation in terms of the Weierstrass sigma function, $\displaystyle{…
We show how to calculate particular values of the Gamma function for specific rational arguments in the interval (0,1) without using the Elliptic K-function. Instead we use transcendental constants or periods defined by hyperelliptic…
We investigate the first-order system `$s\,' = c^3, \, c\,' = - s^3; \, s(0) = 0, \, c(0) = 1$'. Its solutions have the property that $s \, c$, $s^2$ and $c^2$ extend to simply-poled elliptic functions, which we explicitly identify in terms…
In this paper, we present counterexamples showing that for any $p\in (1,\infty)$, $p\neq 2$, there is a non-divergence form uniformly elliptic operator with piecewise constant coefficients in $\mathbb{R}^2$ (constant on each quadrant in…
Complete analytic solutions to quasi-continuous-wave four-wave mixing in nonlinear optical fibres are presented in terms of Weierstrass elliptic $\wp$, $\zeta$, and $\sigma$ functions, providing the full complex envelopes for all four waves…
For a real analytic periodic function $\phi:\mathbb{R}\to\mathbb{R}^d$, an integer $b \ge 2$ and $\lambda\in(1/b,1)$, we prove that the box dimension and the Hausdorff dimension of the graph of the Weierstrass function…
The Weierstrass function is a classic example of a continuous nowhere differentiable function, defined as a sum of high-frequency complex exponentials. In this paper, we follow a suggestion of M.V. Berry and study the convergence properties…
We exhibit a non-varying phenomenon for the counting problem of cylinders, weighted by their area, passing through two marked (regular) Weierstrass points of a translation surface in a hyperelliptic connected component…
Infinite products expansions of the Weierstrass elliptic function \ $\wp(z) = \wp(z,1,\tau)$\ and $n$-order transformations allow us to provide some modular relations.
While the distribution of the non-trivial zeros of the Riemann zeta function constitutes a central theme in Mathematics, nothing is known about the algebraic nature of these non-trivial zeros. In this article, we study the transcendental…
In the theory of elliptic functions and elliptic curves, the Weierstrass $zeta$ function (which is essentially an antiderivative of the Weierstrass $\wp$ function) plays a prominent role. Although it is not an elliptic function, Eisenstein…
A new algebraic method to find two special types of exact traveling wave solutions and the solitary type solutions to some conformable fractional partial differential equations is proposed. The two special types of solutions given by the…