The equations governing the conditions of mechanical equilibrium in fluid membranes subject to bending are revisited thanks to the principle of virtual work. The note proposes systematic tools to obtain the shape equation and the line condition instead of Christoffel symbols and the complex calculations they entail. The method seems adequate to investigate all problems involving surface energies.

From Sturmian and Christoffel words we derive a strictly increasing function ∆ : [0,∞) → R. This function is continuous at every irrational point, while at rational points, left-continuous but not right-continuous. Moreover, it assumes algebraic integers at rationals, and transcendental numbers at irrationals. We also see that the differentiation of ∆ distinguishes some irrationality measures o...

Abstract. We obtain expressions for the Christoffel–Darboux kernel of antisymmetric multivariable orthogonal polynomials as determinants and pfaffians. These kernels include correlation functions of orthogonal polynomial ensembles (with β = 2). In subsequent work, our results are applied in combinatorics (enumeration of marked shifted tableaux) and number theory (representation of integers as s...

Bleher and Kuijlaars, and Daems and Kuijlaars showed that the correlation functions of the eigenvalues of a random matrix from unitary ensemble with external source can be expressed in terms of the ChristoffelDarboux kernel for multiple orthogonal polynomials. We obtain a representation of this Christoffel-Darboux kernel in terms of the usual orthogonal polynomials.

We consider the corresponding Christoffel-Minkowski problem for curvature measures. The existence of star-shaped (n − k)-convex bodies with prescribed k-th curvature measures (k > 0) has been a longstanding problem. This is settled in this paper through the establishment of a crucial C a priori estimate for the corresponding curvature equation on S.