Related papers: Formulas for the approximation of the complete Ell…

We express the leading electromagnetic corrections in K-> pi pi amplitudes as integrals over the virtual photon squared-momentum Q^2. The high Q^2 behavior is obtained via the operator product expansion. The low Q^2 behavior is calculated…

For a congruence subgroup $\Gamma$, we define the notion of $\Gamma$-equivalence on binary quadratic forms which is the same as proper equivalence if $\Gamma = \mathrm{SL}_2(\mathbb Z)$. We develop a theory on $\Gamma$-equivalence such as…

Division polynomials associated to an elliptic curve $E/K$ are polynomials $\phi_n, \psi_n^2$ that arise from the sequence of points $\{nP\}_{n \in \mathbb{N}}$ on this curve. If one wishes to study $\mathbb{Z}$--linear combination of…

We consider elliptic curves defined by an equation of the form $y^2=x^3+f(t)$, where $f\in k[t]$ has coefficients in a perfect field $k$ of characteristic not $2$ or $3$. By performing $2$ and $3$-descent, we obtain, under suitable…

In this paper, we propose an efficient exponential integrator finite element method for solving a class of semilinear parabolic equations in rectangular domains. The proposed method first performs the spatial discretization of the model…

We derive an analytic representation of the ten-particle, two-loop double-box integral as an elliptic integral over weight-three polylogarithms. To obtain this form, we first derive a four-fold, rational (Feynman-)parametric representation…

We prove a sharp H\"older estimate for solutions of linear two-dimensional, divergence form elliptic equations with measurable coefficients, such that the matrix of the coefficients is symmetric and has {\em unit determinant}. Our result…

We examine exponential sums of the form $\sum_{n \le X} w(n) e^{2\pi i\alpha n^k}$, for $k=1,2$, where $\alpha$ satisfies a generalized Diophantine approximation and where $w$ are different arithmetic functions that might be multiplicative,…

Elliptic functions are largely studied and standardized mathematical objects. The two usual approaches are due to Jacobi and Weierstrass. From a contour integral which allowed us to unify many summation formulae (Euler-MacLaurin, Poisson,…

Let $\Pi$ be a polar space of type $\textsf{D}_{n}$. Denote by ${\mathcal G}_{\delta}(\Pi)$, $\delta\in \{+,-\}$ the associated half-spin Grassmannians and write $\Gamma_{\delta}(\Pi)$ for the corresponding half-spin Grassmann graphs. In…

Accurate and efficient methods to evaluate cosmological distances are an important tool in modern precision cosmology. In a flat $\Lambda$CDM cosmology, the luminosity distance can be expressed in terms of elliptic integrals. We derive an…

Asymptotic approximations to the zeros of Hermite and Laguerre polynomials are given, together with methods for obtaining the coefficients in the expansions. These approximations can be used as a standalone method of computation of Gaussian…

We study the renormalization of the DeltaS=1 effective weak Hamiltonian with overlap fermions. The mixing coefficients among dimension-six operators are computed at one loop in perturbation theory. As a consequence of the chiral symmetry at…

The beta integral is applied to accelerate the hypergeometric function $2 F 1\left\{1, B; C ; w\right\}$ to derive new infinite series for constants such as $\pi$ and values of the gamma function. A compendium of new infinite series is…

We present an elliptic version of Selberg's integral formula.

Semiclassical approximations often involve the use of stationary phase approximations. This method can be applied when $\hbar$ is small in comparison to relevant actions or action differences in the corresponding classical system. In many…

Let $\mathcal{K}\left( x\right) $ be the complete elliptic integral of the first kind and \begin{equation*} \mathcal{G}_{p}\left( x\right) =e^{\mathcal{K}\left( \sqrt{x} \right) }-\frac{p}{\sqrt{1-x}} \end{equation*} for $p\in \mathbb{R}$…

This note is a continuation of the work \cite{CaoXiangYan2014}. We study the following quasilinear elliptic equations \[ -\Delta_{p}u-\frac{\mu}{|x|^{p}}|u|^{p-2}u=Q(x)|u|^{\frac{Np}{N-p}-2}u,\quad\, x\in\mathbb{R}^{N}, \] where…

Sharp upper and lower estimates are obtained of the approximation numbers of a Sobolev embedding and an integral operator of Volterra type. These lead to asymptotic formulae for the approximation numbers and certain other s-numbers.

We prove estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds. These equations are not arbitrary, but arise naturally in the study of conformal geometry.