The paper extends a classical result on the convergence of the Krawtchouk polynomials to the Hermite polynomials. We provide a uniform asymptotic expansion in terms of Hermite polynomials and obtain explicit expressions for a few first terms of this expansion. The research is motivated by the study of ergodic sums of the Pascal adic transformation. Bibliography: 10 titles.
We obtain central limit type theorems for the total number of edges in the generalized random graphs with random vertex weights under different moment conditions on the distributions of the weights. © 2016 Springer Science+Business Media New York.
A polynomial with exactly two critical values is called a generalized Chebyshev polynomial (or Shabat polynomial). A polynomial with exactly three critical values is called a Zolotarev polynomial. Two Chebyshev polynomials f and g are called Z-homotopic if there exists a family pα, α(Formula presented.) [0, 1], where p0 = f, p1 = g, and pα is a Zolotarev polynomial if α(Formula presented.) (0, 1). As each Chebyshev polynomial defines a plane tree (and vice versa), Z-homotopy can be defined for plane trees. In this work, we prove some necessary geometric conditions for the existence of Z-homotopy of plane trees, describe Z-homotopy for trees with five and six edges, and study one interesting example in the class of trees with seven edges. © 2015 Springer Science+Business Media New York
Рассмотрен класс градиентно-подобных диффеоморфизмов, обладающих аттрактором и репеллером, разделенными окружностью на двумерной сфере. Для любого диффеоморфизма данного класса построена устойчивая дуга, соединяющая его с системой источник-сток.
In this paper we conduct a comparative analysis of the powers of the two-sample Kolmogorov–Smirnov and Anderson–Darling tests under various alternatives using simulation. We consider two examples. In the first example the alternatives to the standard normal distribution are the distributions of the so-called contaminated normal model. We study the influence of a small contamination with a positive shift on the powers of the test. In the second example the alternatives are the logistic and the Laplace distributions, which are symmetric and differ in shape from the normal distribution having a larger kurtosis coefficient and heavier tails.
We compute the class Wn−4(Formula presented.), which is Poincaré dual to the first Stiefel–Whitney class for the variety (Formula presented.) in terms of the natural cell decomposition of (Formula presented.) © 2015 Springer Science+Business Media New York.