Finiteness of Ricci flatN=2 supersymmetric σ-models

We study the ultraviolet behavior of two dimensional supersymmetric non-linear -models with target space an arbitrary Kähler manifoldM, so that the models areN=2 supersymmetric. We point out that these models have an additional fermionic axial symmetry if and only if the metric onM is Ricci flat. We show that the preservation of this symmetry in perturbation theory implies that both bare and renormalized metrics onM are Ricci flat. Combining this result with the constraint ofN=2 supersymmetry requiring that all counter-terms to the metric beyond one-loop order be cohomologically trivial, we argue thatN=2 models defined on Ricci flat Kähler manifolds are on-shell ultraviolet finite to all orders of perturbation theory.     

On the average length of lexicographic paths

A bound is given for the average length of a lexicographic path, a definition that is motivated by degeneracies encountered when using the randomized simplex method.     

Selection theorems,based on generalized variation and oscillation

In this paper we prove selection theorems for everywhere and almost everywhere convergent subsequences, based on the notion of uniformly limited oscillation, ofA, Λ-oscillation and ofA, Λ-variation, whereA is a system of intervals and Λ a sequence of reals. By this, we generalize and strengthen the selection theorems of Schrader (for oscillation) and of Waterman (for Λ-variation).     

Stability of the eigenvalues of certain types of singular integral equations

In the paper one finds necessary and sufficient conditions for the existence of nontrivial solutions of the equation (A+C)f=f for all compact (or small in norm) perturbations C; for the unperturbed operators A one considers the operators of dual and conjugate to dual integral equations with kernels depending on the difference of arguments, Wiener-Hopf operators, and singular integral operators. The answer is given in terms of the symbol S (A) of the operator A.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 127–136, 1984.     

Generalized Alternating Polynomials, some Properties and Numerical Applications

In the present paper we introduce the generalized alternatingpolynomials , the coefficients of which are defined together with the parameter W_n by the linear system where T_n(x) = cos (n arc cos x), is a set of n+2 distinct points in the interval [–1, 1], and fis a continuous function on [–1, 1], For the set of nodesx_k = cos [k/(n+1)] the w_n-polynomials coincide with the polynomialsof equiamplitude alternation introduced by de La Vall?e-Poussinand discussed in the literature earlier (Eterman, Malozemov,Meinardus, Cheney & Rivlin, Phillips & Taylor, Brutman,and others). It is shown that the generalized alternating polynomials arerelated to the polynomials of interpolation through the Lanczoseconomization process. Some approximation properties of w_n -polynomialsand W_n -parameters are studied. The application of w_n -polynomialsto function approximation and to the estimate of remainder termsfor quadrature formulas is discussed.