A theorem is proved on the degree of rational approximation of sequences of analytic functions given by Cauchy-type integrals of the formThe theorem is formulated in terms connected with the equilibrium distribution of the charge on the plates of a capacitor (E, F) under the assumption that an external field φ = lim n _ t3O (2«)~I log|Φ Π | -Ι acts on the plate F, and this plate satisfies a certain symmetry condition in the field φ. The theorem is used to solve the problem of the degree of rational approximation of the function e~x on [0, +00).Bibliography: 44 titles.1= /' 8 {ζ,ί)άμ{1), zet\E,

A geometric braid consists of n curves stretching between two parallel planes. A group structure may be defined on a geometric braid yielding an algebraic braid. Braids provide information on the structure of knots and links, and have many practical applications, for example in fluid mechanics and astrophysics. A well-known problem in braid theory consists of finding minimal crossing number braids. No efficient algorithm which solves this problem using group theory seems to be possible for n > 3. Here we investigate several different approaches to obtaining minimal configurations: we employ three different relaxation techniques and compare them with each other and with an algebraic heuristic algorithm, in terms of minimization (of energy and crossing number) and time efficiency. By energy we mean total string length of the braid. It is found that more than half of the crossings of a sufficiently large braid (in terms of crossing number and number of strings) are redundant. We analyse the different methods and say in what circumstances which method is to be favoured and conclude that minimum braid energy and minimum braid crossing number are substantially different measures of topological complexity for braids.

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