On polynomial congruences

The work of the first author was supported by the Russian Foundation for Fundamental Research, Grant 93-011-240.     

On quadratic congruences

It is proved that under appropriate assumptions, the solutions f of quadratic congruences Q(f) ≡ 0 (mod n), n ≤ x, with (f|n) = ±1, are distributed asymptotically equally as x → ∞. Bibliography: 5 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 195–200.     

On polynomial congruences

We deal with functions which fulfil the condition \({\Delta_h^{n+1} \varphi(x)\in\mathbb{Z}}\) for all x, h taken from some linear space V. We derive necessary and sufficient conditions for such a function to be decent in the following sense: there exist functions \({f\colon V\rightarrow \mathbb{R},\ g\colon V \rightarrow \mathbb{Z}}\) such that \({\varphi = f + g}\) and \({\Delta_h^{n+1}f(x)=0}\) for all \({x, h\in V}\).     

On multiplicative congruences

Let ${\varepsilon}$ be a fixed positive quantity, m be a large integer, x _j denote integer variables. We prove that for any positive integers N ₁, N ₂, N ₃ with ${N_1N_2N_3 > m^{1+\varepsilon}, }$ the set $$\{x_1x_2x_3 \quad ({\rm mod}\,m): \quad x_j\in [1,N_j]\}$$ contains almost all the residue classes modulo m (i.e., its cardinality is equal to m + o(m)). We further show that if m is cubefree, then for any positive integers N ₁, N ₂, N ₃, N ₄ with ${ N_1N_2N_3N_4 > m^{1+\varepsilon}, }$ the set $$\{x_1x_2x_3x_4 \quad ({\rm mod}\,m): \quad x_j\in [1,N_j]\}$$ also contains almost all the residue classes modulo m. Let p be a large prime parameter and let ${p > N > p^{63/76+\varepsilon}.}$ We prove that for any nonzero integer constant k and any integer ${\lambda\not\equiv 0 \,\, ({\rm mod}\,p)}$ the congruence $$p_1p_2(p_3+k)\equiv \lambda \quad ({\rm mod}\, p) $$ admits (1 + o(1))π(N)³/p solutions in prime numbers p ₁, p ₂, p ₃ ≤ N.     

On congruences of Jacobi forms

We consider congruences and filtrations of Jacobi forms. More specifically, we extend Tate's theory of theta cycles to Jacobi forms, which allows us to prove a criterion for an analog of Atkin's -operator applied to a Jacobi form to be nonzero modulo a prime.

     

On right congruences of semigroups having no proper essential right congruences

A right congruence ?? in a semigroup S is essential if for any right congruence ?? we have ??????=?? (the identity relation) implies ??=??. Clearly, the universal relation, ??, is an essential right congruence. We say ?? is proper if ??????. In this paper we get a necessary and sufficient condition for a semigroup with an identity element?1 and having no proper essential right congruences to have a distributive lattice of right congruences.     

On congruences of-normal semigroups-normal semigroups

Communicated by G. Lallement     

On the sectional irregularity of congruences

Partially supported by CICYT PB90-0637     

On the congruences of Jacobi forms

In this paper, we study congruence properties of coefficient of Jacobi forms. The result for elliptic modular form case was studied by Sturm (Lecture Notes in Mathematics, Springer, Berlin Heidelberg New York 1987).     

On rational isotropic congruences of lines

The aim of this paper is to show a way to find an explicit parametrization of rational isotropic congruences of lines in Euclidean three-space It will be shown that also the focal surfaces of these congruences admit a rational parametrization. Furthermore, the close relation of isotropic congruences of lines to minimal surfaces will be used to find the related polynomial minimal surfaces.     

On some congruences of power algebras

In a natural way we can “lift” any operation defined on a set A to an operation on the set of all non-empty subsets of A and obtain from any algebra (A, Ω) its power algebra of subsets. In this paper we investigate extended power algebras (power algebras of non-empty subsets with one additional semilattice operation) of modes (entropic and idempotent algebras). We describe some congruence relations on these algebras such that their quotients are idempotent. Such congruences determine some class of non-trivial subvarieties of the variety of all semilattice ordered modes (modals).     

On small solutions to quadratic congruences

We estimate the deviation of the number of solutions of the congruence

     

On congruences in closed post classes

It is shown herein that on almost all closed Post classes only trivial congruences are possible. Closed Post classes having nontrivial congruences are indicated, and all congruences of these classes are described. For k3 closed classes having an infinity of congruences are indicated.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 725–734, May, 1973.The author is grateful to S. V. Yablonskii for valuable comments and assistance in the research.