Related papers: Notes on the Weierstrass Elliptic Function

We show that the graph of the classical Weierstrass function $\sum_{n=0}^\infty \lambda^n \cos (2\pi b^n x)$ has Hausdorff dimension $2+\log\lambda/\log b$, for every integer $b\ge 2$ and every $\lambda\in (1/b,1)$. Replacing $\cos(2\pi x)$…

We prove a Carleman estimate for elliptic second order partial differential operators with Lipschitz continuous coefficients. The Carleman estimate is valid for any complex-valued function $u\in W^{2,2}$ with support in a punctured ball of…

A new variational approach to solve the problem of estimating the (possibly discontinuous) coefficient functions $p$, $q$ and $f$ in elliptic equations of the form $-\nabla \cdot (p(x)\nabla u) + \lambda q(x) u = f$, $x \in \Omega \subset…

Let $(M,g)$ be a closed Riemannian manifold of dimension $n \geq 3$ and let $f\in C^{\infty}(M)$, such that the operator $P_f:= \Delta_g+f$ is positive. If $g$ is flat near some point $p$ and $f$ vanishes around $p$, we can define the mass…

The Wigner function W(q,p) is formulated as a phase-space path integral, whereby its sign oscillations can be seen to follow from interference between the geometrical phases of the paths. The approach has similarities to the path-centroid…

The general analytic solution to the functional equation $$ \phi_1(x+y)= { { \biggl|\matrix{\phi_2(x)&\phi_2(y)\cr\phi_3(x)&\phi_3(y)\cr}\biggr|} \over { \biggl|\matrix{\phi_4(x)&\phi_4(y)\cr\phi_5(x)&\phi_5(y)\cr}\biggr|} } $$ is…

A floating hemisphere under forced harmonic oscillation at very high and very low frequencies is considered. The problem is reduced to an elliptic one, that is, the Laplace operator in the exterior domain with standard Dirichlet and Neumann…

For a given second order elliptic operation $\mathcal{L}$ in a domain $\Omega\subset{\mathbb{R}}^\mathbf{N}$, $\mathbf{N}\ $, and a compact set $\mathbf{K}\subset\Omega$, order $\mathbf{N}$-$2$-Ahlfors-David regular, we define the space…

The paper provides an explicit description of the structure of the domain of the Friedrichs extension of a second order semibounded elliptic wedge operator, initially defined on smooth functions or sections with compact support away from…

Using a physically motivated stress energy tensor, we prove weak and strong monotonicity formulas for solutions to the semilinear elliptic system $\Delta u=\nabla W(u)$ with $W$ nonnegative. In particular, we extend a recent two dimensional…

We discuss the theory of generalized Weierstrass $\sigma$ and $\wp$ functions defined on a trigonal curve of genus four, following earlier work on the genus three case. The specific example of the "purely trigonal" (or "cyclic trigonal")…

We obtain KSS, Strichartz and certain weighted Strichartz estimate for the wave equation on $(\R^d, \mathfrak{g})$, $d \geq 3$, when metric $\mathfrak{g}$ is non-trapping and approaches the Euclidean metric like $ x ^{- \rho}$ with…

In the present paper we study the geodesic motion of test particles and light rays in the spacetime of a static charged black hole in $f(R)$ gravity. The complete set of analytic solutions of the geodesic equations in the spacetime of this…

Consider the Schr\"odinger operator $\mathcal{L}=-\Delta+V$ in $\mathbb{R}^n, n\ge 3,$ where $V$ is a nonnegative potential satisfying a reverse H\"older condition of the type \begin{equation*} \left( \frac{1}{|B|}\int_B…

We present the complete set of analytical solutions of the geodesic equation in Taub-NUT space-times in terms of the Weierstrass elliptic function. We systematically study the underlying polynomials and characterize the motion of test…

Given a local ring $(R,\mathfrak{m})$ and an elliptic curve $E(R/\mathfrak{m})$, we define elliptic loops as the points of $\mathbb{P}^2(R)$ projecting to $E$ under the canonical modulo-$\mathfrak{m}$ reduction, endowed with an operation…

We determine the Weierstrass semigroup at one and two totally ramified places in a Kummer extension defined by the affine equation $y^{m}=\prod_{i=1}^{r} (x-\alpha_i)^{\lambda_i}$ over $K$, the algebraic closure of $\mathbb{F}_q$, where…

The elliptic gamma function is a generalization of the Euler gamma function and is associated to an elliptic curve. Its trigonometric and rational degenerations are the Jackson q-gamma function and the Euler gamma function, respectively.…

This paper deals with the exact solutions of a nonlinear coupled coupled wave equation. The (G'/G)-expansion method has been applied to derive kink solutions and singular wave solutions. The restrictions on the coefficients of the governing…

The elliptic flow ($v_{2}$) at $\sqrt{s_{\rm NN}} = $ 11.5, 39, and 200 GeV and triangular flow ($v_{3}$) at $\sqrt{s_{\rm NN}} = $ 200 GeV of identified particles ($\pi^{\pm}, K^{\pm}, K^{0}_{S}, p, \bar{p}, \phi, \Lambda$ and…