Riemann's Fragment on Limit Values of Elliptic Modular Functions

We give an interpretation different from that of Dedekind to work n° 28 of the complete works of Riemann Fragmente über die Grenzfälle der Elliptischen Modulfunctionen.We prove a theorem of inversion of radial limit and sum in a series of functions. This allows us to justify all of Riemann's reasoning in the fragment to obtain the limit values of modular elliptic functions. In particular we prove the statement of Riemann that for every rational number x we have where denotes the periodic function with period 1, such that (x) = x when |x| < 1/2, and (n + ) = 0 for every n Z.This assertion of Riemann was criticized by Dedekind. We also give the transformation formulae of the logarithms of the classical theta-function ₃(0), giving an alternative form to that obtained by B. C. Berndt [1].     

Vilenkin群上的加权Hardy空间中的乘子

本文主要研究了Vilenkin群上的加权Hardy空间的原子分解,证明了一个加权H^p乘子定理并给出其应用。     

Multipliers for the symmetry groups of p-adic spacetime

The consequences for particle classification of the Volovich hypothesis that spacetime geometry is non-archimedean at the Planck scale are explored. The multiplier groups and universal topological central extensions of the p-adic Poincaré and Galilean groups are determined. The text was submitted by the author in English.     

p-adic Measures of Algebraic Independence for the Values of Ramanujan Functions and Klein Modular Functions

In this paper, we give the p-adic measures of algebraic independence for the values of Ramanujan functions and Klein modular functions at algebraic points.     

Modular Transformations of Ramanujan's Fifth and Seventh Order Mock Theta Functions

In his last letter to Hardy, Ramanujan defined 17 functions F(q), where |q| < 1. He called them mock theta functions, because as q radially approaches any point e ^2ir (r rational), there is a theta function F _r(q) with F(q) – F _r(q) = O(1). In this paper we obtain the transformations of Ramanujan's fifth and seventh order mock theta functions under the modular group generators + 1 and –1/, where q = e ⁱ. The transformation formulas are more complex than those of ordinary theta functions. A definition of the order of a mock theta function is also given.     

Multipliers and Generalized Derivatives of Performance Functions

It is known that multipliers in mathematical programming have to do with the interpretation of generalized derivatives for the performance function of some perturbed problem. In economics, this fact is known under the term of shadow price. Here, we point out a precise relationship with the subdifferential of the performance function in the contingent sense and in the Fréchet sense, which are the simplest notions of nonsmooth analysis.     

Coefficients of Products of Powers of Eta Functions

We give an elementary proof of a conjecture of Cooper, Hirschhorn and Lewis concerning the coefficients of certain products of powers of Dedekind eta functions.     

Decomposable Multipliers on a Banach Algebra

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Modular Equations for Ramanujan's Cubic Continued Fraction

In this paper we present two new identities providing relations between Ramanujan's cubic continued fraction G(q) and the two continued fractions G(q⁵) and G(q⁷).     

Strongly Modular Subgroup in Finite Group

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Modular Group Algebras with Almost Maximal Lie Nilpotency Indices

Let be a field of positive characteristic and the group algebra of a group . It is known that, if is Lie nilpotent, then its upper (and lower) Lie nilpotency index is at most , where is the order of the commutator subgroup. The authors previously determined those groups for which this index is maximal and here they determine the groups for which it is `almost maximal', that is, it takes the next highest possible value, namely .Presented by V. Dl a b.Dedicated to Professor Vjacheslav Rudko on his 65th birthday.The research was supported by OTKA No. T 037202, No. T 038059 and Italian National Research Project “Group Theory and Application.”     

二阶线性微分方程的乘子可积性

在偏微分方程Riemann解法和微分方程裂变思想的启发下,引入了微分方程乘子函数(解)和乘子解法的概念,系统地讨论了二阶线性微分方程的乘子可积性.得到了二阶线性微分方程乘子可积的条件以及Riceati方程可积的充分必要条件,并分别给出了二阶线性微分方程和Riccati方程在乘子解下的通积分.     

Mean-square Almost Periodic Random Functions

In this paper, we present a basic theory of mean-square almost periodicity, apply the theory in random differential equation, and obtain mean-square almost periodic solution of some types stochastic differential equation.     

Certain Modular Functions Similar to the Dedekind eta Function

In this paper, we study certain modular functions η_ψ(z) similar to the Dedekind eta function η(z). The functions are given by an analogue of Borcherds type liftings. It turns out that the functions η_ψ(z) have some good properties, similarly as in the case of the Dedekind eta function.     

On the Genesis of the Lagrange Multipliers

The genesis of the Lagrange multipliers is analyzed in this work. Particularly, the author shows that this mathematical approach was introduced by Lagrange in the framework of statics in order to determine the general equations of equilibrium for problems with constraints. Indeed, the multipliers allowed Lagrange to treat the questions of maxima and minima in differential calculus and in calculus of variations in the same way as problems of statics: if the equilibrium of a point or a system of points is required, there is an analogy between statics and differential calculus; if the equilibrium of a rigid body is required, there is an analogy between statics and calculus of variations.     

均方概周期随机函数

In this paper, we present a basic theory of mean-square almost periodicity, apply the theory in random differential equation, and obtain mean-square almost periodic solution of some types stochastic differential equation.     

Banach代数A=A上左(右)乘子的Fredholm定理

本文得到了某些不可换Banach代数上左(右)乘子的Fredholm定理.作为应用,我们刻划了紧群上的Fredholm左(右)乘子.     

Exceptional Congruences for Powers of the Partition Function

In Journal of London Math. Soc. 31 (1956), 350–359, Morris Newman studied vector spaces of functions arising from lifts to ₀(p) of certain eta-products on the group ₀(pQ), Q = p ⁿ. In this paper, the author considers vector spaces of modular functions obtained as lifts of more general eta-products from ₀(pQ) to ₀(p), (Q, p) = 1. Specifically considered are functions arising as lifts of functions of the form

Metacyclic groups and modular forms

In the paper we find the metacyclic groups of the form 〈a, b:a ^m=e, b ^s=e, b ⁻¹ ab=a ^r〉, wherem=3, 4, 5, 7, 11, 23, such that the modular forms associated with all elements of these groups by some faithful representation are multiplicative η-products. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 163–173, February, 2000.