The asymptotic behavior of Christoffel functions is established at points of discontinuity of the first kind.

For a system of smooth Jordan curves asymptotics for Christoffel functions is established almost everywhere for measures belonging to Szegő’s class. 1 The result Let μ be a finite Borel-measure on the plane with compact support consisting of infinitely many points. The Christoffel functions associated with μ are defined as λn(z, μ) = inf Pn(z)=1 ∫ |Pn|dμ, where the infimum is taken for all poly...

The primary current distribution and the resistance of a cell containing a slotted electrode were calculated using numerical methods coupled with the Schwarz-Christoffel transformation. Results are presented and compared to asymptotic solutions. An approximate analytic expression for the cell resistance is presented. Primary current and potential distributions apply when the surface overpotenti...

Sturmian sequences are well-known as the ones having minimal complexity over a 2-letter alphabet. They are also the balanced sequences over a 2-letter alphabet and the sequences describing discrete lines. They are famous and have been extensively studied since the 18th century. One of the generalization of these sequences are the episturmian sequences, introduced by A. de Luca [dL97a] and studi...

E. B. Christoffel, Gesammelte mathematische Abhandlungen, Unter Mitwirkung von A. KRAZER und G. FABER, herausgegeben von L. MAURER. Zwei Bande. B. G. Teubner, Leipzig, 1910. WHEN one turns over the pages of the collected works of a mathematician such as this one, arranged in chronological order, and notes the varied fields in which the author worked, he feels an impulse to follow the methods of...

Discrete geometry redefines notions borrowed from Euclidean geometry creating a need for new algorithmical tools. The notion of convexity does not translate trivially, and detecting if a discrete region of the plane is convex requires a deeper analysis. To the many different approaches of digital convexity, we propose the combinatorics on words point of view, unnoticed until recently in the pat...

A modified Schwarz-Christoffel transformation (SCT) is used to obtain guidedand radiated-wave characteristics of a singlycurved rectangular patch antenna. The method is to map a straight channel into an arbitrarily-curved channel. This simplifies the problem to that of a planar rectangular patch antenna. Applying conventional SCT to the problem confronts two difficulties: the region under inves...

We verify the following three basic results on the free loop space LM . (1) We show that the set of all points, where the fundamental form on LM is nondegenerate, is an open subset. (2) The connections of a Fréchet bundle over LM can be extended to S1-central extensions and, in particular, there exist natural connections on the string structures. (3) The notion of Christoffel symbols and the cu...

1.1. Geometry. A covariant derivative on a manifold M is an operator ∇XY on vector fields X and Y satisfying for any smooth function f : (i) ∇fXY = f∇XY ; and (ii) ∇X(fY ) = f∇XY + (∇Xf)Y . If g is a Riemannian metric on M , then there is associated with g a unique covariant derivative ∇ characterized by: (iii) ∇XY −∇YX = [X,Y ]; and (iv) ∇X ( g(Y,Z) ) = g(∇XY, Z) + g(Y,∇XZ). We define the Chri...