[en] When f(z) is given by a known power series expansion, it is possible to construct the power series expansion for f(z; p) = e/sup -p/z f(z). We define p/sub opt/ to be the value of p for which the expansion for f(z; p) converges most rapidly. When f(z) is an entire function of order 1, we show that p/sub opt/ is uniquely defined and may be characterized in terms of the set of singularities z/sub i/ = 1/sigma/sub i/ of an associated function h(z). Specifically, it is the center of the smallest circle in the complex plane which contains all points sigma/sub i/

[en] In this paper, nonlocal symmetries for the bilinear semi-discrete Kadomtsev-Petviashvili (KP) and bilinear semi-discrete B-type KP (BKP) equations are derived. By expanding these nonlocal symmetries in power series of each of two parameters, we have derived bilinear negative semi-discrete KP and BKP hierarchies. An interesting observation is that bilinear positive and negative semi-discrete KP hierarchies may be derived from the same nonlocal symmetry for the semi-discrete KP equation.

[en] Von Neumann's method of generating random variables with the exponential distribution and Forsythe's method for obtaining distributions with densities of the form e/sup -G//sup( x/) are generalized to apply to certain power series representations. The flexibility of the power series methods is illustrated by algorithms for the Cauchy and geometric distributions

[en] We find order estimates for the modulus of variation and the averaged modulus of the sum of a lacunary trigonometric series in terms of its coefficients. These interesting global characteristics of a function and their applications have been studied in papers by Chanturiya, Dolzhenko, Sevast'yanov, Sendov, Popov, and others. Since the sum of a lacunary trigonometric series has frequently been used in the theory of functions to provide an example of a function having one property or another, it is useful to know as much as possible about such a function, especially such global characteristics as the modulus of variation and the averaged modulus. We also give necessary and sufficient conditions for the sum of a lacunary trigonometric series to belong to certain classes of functions defined in terms of these characteristics

[en] We study the series expansion of the tau function of the BKP hierarchy applying the addition formulae of the BKP hierarchy. Any formal power series can be expanded in terms of Schur functions. It is known that, under the condition $\mathrm{\tau <!--\; ??\; -->}(x)\ne <!--\; ???\; -->0$, a formal power series $\mathrm{\tau <!--\; ??\; -->}(x)$ is a solution of the KP hierarchy if and only if its coefficients of Schur function expansion are given by the so called Giambelli type formula. A similar result is known for the BKP hierarchy with respect to Schur’s Q-function expansion under a similar condition. In this paper we generalize this result to the case of $\mathrm{\tau <!--\; ??\; -->}(0)=0$. (paper)

[en] Certain special classes of division algebras over the field of Laurent power series with arbitrary residue field are studied. We call algebras in these classes split and well-split algebras. These classes are shown to contain the group of tame division algebras. For the class of well-split division algebras we prove a decomposition theorem which is a generalization of the well-known decomposition theorems of Jacob and Wadsworth for tame division algebras. For both classes we introduce the notion of a δ-map and develop the technique of δ-maps for division algebras in these classes. Using this technique we prove decomposition theorems, reprove several old well-known results of Saltman, and prove Artin's conjecture on the period and index in the local case: the exponent of a division algebra A over a C₂-field F is equal to the index of A if F=F₁((t)), where F₁ is a C₁-field. In addition we obtain several results on split division algebras, which, we hope, will help in further research of wild division algebras.

[en] In this paper we present a new approach to proving some exponential inequalities involving the sinc function. Power series expansions are used to generate new polynomial inequalities that are sufficient to prove the given exponential inequalities.

[en] We study the behaviour of the quantities a_n(q) and b_n(q), that is, the number of nth power residues in the reduced and complete residue systems modulo a composite number q, respectively, where n≥2 is an arbitrary fixed number. In particular, we prove asymptotic formulae for the sum functions A_n(x) and B_n(x) of these quantities.

[en] Let p be an odd prime. The ''first factor'' h (p) of the class number of the field of pth roots of unity has been the subject of many investigations beginning with Kummer (1861). In the present paper it is shown how the theory of a function introduced by T. A. Pierce (1917) can be used to find the prime factors of h

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