Capillary surface discontinuities above reentrant corners

A particular configuration of a vertical capillary tube for which S is the equilibrium interface between two fluids in the presence of a downward pointing gravitational field was investigated. S is the graph a function u whose domain is the (horizontal) cross section gamma of the tube. The mean curvature of S is proportional to its height above a fixed reference plane and lambda is a prescribed constant and may be taken between zero and pi/2. Domains gamma for which us is a bounded function but does not extend continuously to d gamma are sought. Simple domains are found and the behavior of u in those domains is studied. An important comparison principle that has been used in the literature to derive many of the results in capillarity is reviewed. It allows one to deduce the approximate shape of a capillary surface by constructing comparison surfaces with mean curvature and contact angle close to those of the (unknown) solution surface. In the context of nonparametric problems the comparison principle leads to height estimates above and below for the function u. An example from the literature where these height estimates have been used successfully is described. The promised domains for which the bounded u does not extend continuously to the boundary are constructed. The point on the boundary at which u has a jump discontinuity will be the vertext of a re-entrant corner having any interior angle theta pi. Using the comparison principle the behavior of u near this point is studied

Roots under convolution of sequences

AbstractThe convolution a * b of the sequences a = a0, a1, a2, ⋯ ∼ and b is the sequence with elements ∑0n akbn − k. One sets 1, 1, 1, ⋯ equal to σ. Given that a * a with a ≥ 0 is close to σ * σ, how close is a to σ? More generally, one asks how close a is to σ if the p-th convolution power, a*P with a ≥ 0, is close to σ*P. Power series and complex analysis form a natural tool to estimate the ‘summed deviation’ ρ = σ * (a — σ) in terms of b = a * a — σ * σ or b = a*P − σ*P. Optimal estimates are found under the condition ∑k=0n bk2 = %plane1D;512;(n2β + 1) whenever −½ < β < p − 1. It is not known what the optimal estimates are for the special case bn = %plane1D;512;(nβ)

A representation of mixed derivatives with an application to the edge-of-the-wedge theorem

AbstractThe authors prove a lemma which expresses the mixed derivatives of a function in ℝn in terms of its directional derivatives of the same order in an angle. The lemma is used to derive an edge-ofthe-wedge theorem for ℂn with an explicit domain of analytic continuation. Other applications will be given in subsequent papers

Fekete Potentials and Polynomials for Continua

AbstractFor planar continua, upper and lower bounds are given for the growth of the associated Fekete potentials, polynomials and energies. The main result is that for continua K of capacity 1 whose outer boundary is an analytic Jordan curve, the family of Fekete polynomials is bounded on K. Our work makes use of precise results of Pommerenke on the growth of the discriminant and on the distribution of the Fekete points. We also use potential theory, including the exterior Green function with pole at infinity. The Lipschitz character of this function determines the separation of the Fekete points

OPE Convergence in Conformal Field Theory

We clarify questions related to the convergence of the OPE and conformal block decomposition in unitary Conformal Field Theories (for any number of spacetime dimensions). In particular, we explain why these expansions are convergent in a finite region. We also show that the convergence is exponentially fast, in the sense that the operators of dimension above Delta contribute to correlation functions at most exp(-a Delta). Here the constant a>0 depends on the positions of operator insertions and we compute it explicitly.Comment: 26 pages, 6 figures; v2: a clarifying note and two refs added; v3: note added concerning an extra constant factor in the main error estimate, misprint correcte

Average prime-pair counting formula

Taking $r>0$, let $\pi_{2r}(x)$ denote the number of prime pairs $(p,\,p+2r)$ with $p\le x$. The prime-pair conjecture of Hardy and Littlewood (1923) asserts that $\pi_{2r}(x)\sim 2C_{2r}\,{\rm li}_2(x)$ with an explicit constant $C_{2r}>0$. There seems to be no good conjecture for the remainders \om_{2r}(x)=\pi_{2r(x)-2C_{2r}\,{\rm li}_2(x) that corresponds to Riemann's formula for $\pi(x)-{\rm li}(x)$. However, there is a heuristic approximate formula for averages of the remainders \om_{2r}(x) which is supported by numerical results

Functions holomorphic along holomorphic vector fields

The main result of the paper is the following generalization of Forelli's theorem: Suppose F is a holomorphic vector field with singular point at p, such that F is linearizable at p and the matrix is diagonalizable with the eigenvalues whose ratios are positive reals. Then any function $\phi$ that has an asymptotic Taylor expansion at p and is holomorphic along the complex integral curves of F is holomorphic in a neighborhood of p. We also present an example to show that the requirement for ratios of the eigenvalues to be positive reals is necessary