相关论文: Ptolemy Constants as Described by Eccentricity

Let $V$ be a vector space over a finite field $k=\mathbb{F} _q$ of dimension $n$. For a polynomial $P:V\to k$ we define the bias of $P$ to be $$b_1(P)=\frac {|\sum _{v\in V}\psi (P(V))|}{q^n}$$ where $\psi :k\to \mathbb{C} ^\star$ is a…

Starting from the classical division polynomials we construct homogeneous polynomials $\alpha_n$, $\beta_n$, $\gamma_n$ such that for $P = (x:y:z)$ on an elliptic curve in Weierstrass form over an arbitrary ring we have $nP =…

The problem of maximizing the $L^p$ norms of chords connecting points on a closed curve separated by arclength $u$ arises in electrostatic and quantum--mechanical problems. It is known that among all closed curves of fixed length, the…

The best constant in the usual Lp norm inequality for the centered Hardy-Littlewood maximal function on R1 is obtained for the class of all ``peak-shaped'' functions. A positive function on the line is called ``peak-shaped'' if it is…

This paper develops a new divergence that generalizes relative entropy and can be used to compare probability measures without a requirement of absolute continuity. We establish properties of the divergence, and in particular derive and…

We provide an exact infinite power series solution that describes the trajectory of a nonlinear simple pendulum undergoing librating and rotating motion for all time. Although the series coefficients were previously given in [V. Fair\'en,…

The symmetric convex hull of random points that are independent and distributed according to the cone probability measure on the $\ell_p$-unit sphere of $\mathbb R^n$ for some $1\leq p < \infty$ is considered. We prove that these random…

We propose a simple derivation of an upper bound for the perimeter of an ellipse. The procedure, which relies on the use of elliptic integrals, consists in introducing, via inequalities and convexity properties, specific integrals which can…

We count algebraic points of bounded height and degree on the graphs of certain functions analytic on the unit disk, obtaining a bound which is polynomial in the degree and in the logarithm of the multiplicative height. We combine this work…

In this paper, among other results, we improve the best known estimates for the constants of the generalized Bohnenblust-Hille inequality. These enhancements are then used to improve the best known constants of the Hardy--Littlewood…

Let $L$ be an analytic Jordan curve and let $\{p_n(z)\}_{n=0}^\infty$ be the sequence of polynomials that are orthonormal with respect to the area measure over the interior of $L$. A well-known result of Carleman states that \label{eq12}…

We consider an i.i.d. supercritical bond percolation on Z^d , every edge is open with a probability p > p\_c (d), where p\_c (d) denotes the critical parameter for this percolation. We know that there exists almost surely a unique infinite…

Let $p(z)$ be a monic polynomial of degree $n$, with complex coefficients, and let $q(z)$ be its monic factor. We prove an asymptotically sharp inequality of the form $\|q\|_{E} \le C^n \|p\|_E$, where $\|\cdot\|_E$ denotes the sup norm on…

Einstein's perihelion advance formula can be given a geometric interpretation in terms of the curvature of the ellipse. The formula can be obtained by splitting the constant term of an auxiliary polar equation for an elliptical orbit into…

Ihara and Birch obtained a formula expressing the sum of powers of the traces of elliptic curves over a fixed finite field of characteristic $p$ in terms of the traces of Hecke operators for $\mathrm{SL}_2(\mathbb{Z})$. Generalizing the…

The optimal one-sided parametric polynomial approximants of a circular arc are considered. More precisely, the approximant must be entirely in or out of the underlying circle of an arc. The natural restriction to an arc's approximants…

We characterize the boundary at infinity of a complex hyperbolic space as a compact Ptolemy space that satisfies four incidence axioms.

We provide numerical bounds on the Crouzeix ratiofor KLS matrices $A$ which have a line segment on the boundary of the numerical range. The Crouzeix ratio is the supremum over all polynomials $p$ of the spectral norm of $p(A)$ dividedby the…

We study a question with connections to linear algebra, real algebraic geometry, combinatorics, and complex analysis. Let $p(x,y)$ be a polynomial of degree $d$ with $N$ positive coefficients and no negative coefficients, such that $p=1$…

We bound the variance and other moments of a random vector based on the range of its realizations, thus generalizing inequalities of Popoviciu (1935) and Bhatia and Davis (2000) concerning measures on the line to several dimensions. This is…