Mathematics

2006, No. 6/1

T.Y. Andreeva

АN ASYMPTOTIC FORMULA FOR THE NUMBER OF NATURAL NUMBERS REPRESENTATION BY A QUADRATIC FORM

In the paper an asymptotic formula for the number of representation of natural numbers by a special quadratic form of the four variables is presented. The coefficients can increase and the approximation error is less then that in Kloosterman’s formula.

G.V. Voskresenskaya

MODULAR FORMS WITH MULTIPLICATIVE COEFFICIENTS AND REPRESENTATIONS OF GROUPS OF THE ORDER 24

In the paper various exact representations of all nonabelian groups of the order 24 which associate with the elements of these groups multiplicative η?products are considered. For each group we write in detail all irreducible representations, classes of conjugate elements and tables by which the connections between elements and modular forms are presented.

F.M. Korkmasov

ON ESTIMATION OF THE LEBESQUE FUNCTION OF THE DISCRETE FOURIER–JACOBI SUMS

We consider the system of the classical Jacobi polynomials of degree at most N which generate an orthogonal system on the discrete set of the zeros of the Jacobi polynomial of degree N. Given an arbitrary continuous function f (t) on the interval [-1, 1], we construct the discrete Fourier–Jacobi sums S^α,_n,^β_N(f ,t) over the orthonormal system introduced above. We prove that the Lebesgue function L^α,_n,^β_N(t) of the discrete Fourier–Jacobi sums for - 1/2  < α, β < 1/2, n ≤N - 1, n ≥ 1 on the interval [ -1+ ε, 1 - ε] (ε > 0) have the O(ln n) order; on the intervals [-1, -1+ ε] and [1 - ε, 1], accordingly, the O(n^β+1/2) order and the O(n^α+1/2) order.

K.V. Lykov

ON COMPLEMENTABILITY OF SUBSPACES GENERATED BY COMPRESSIONS AND TRANSLATIONS IN A SYMMETRIC SPACE

In the paper properties of symmetric space infinite dimensional subspaces generated by compressions and translations of a function are presented. Relation to multiplicator space is considered. Also we give a characterization of L_p – spaces among all symmetric spaces.

V.N. Orlov

AN EXISTENCE CRITERION FOR MOVING SINGULAR POINTS OF THE RICCATI DIFFERENTIAL EQUATION

In the paper a problem concerning unique solutions of non-linear differential equations with moving singular points is studied. An existence criterion of moving singular points of the Riccati differential equation solution in complex area is given.

S.M. Ratseev

INCREASE OF MANIFOLDS OF THE LEIBNIZ ALGEBRAS

Let K be a field of any characteristic. In the paper a growth of manifolds of the Leibniz algebras over K with a nilpotent commutator is studied. It is proved that in case of the Leibniz algebras the varieties of intermediate increase are absent since char K ≠ 2.

K.E. Tikhomirov

ON MEASURE OF A UNION OF POINT SETS k TIMES COVERED BY k-FOLD EXTENDED INTERVALS

In the paper the problem of estimating a measure of a union of point sets covered k times by k-fold extended intervals is formulated. It was formulated in 1992 (Proceedings of Israel Mathematical Conference). It is proved that this problem is equivalent to a combinatorial matrix problem. Moreover, the positive answer on it allows to strengthen the Marcinkiewicz theorem related to the structure of closed sets of reals and to establish new properties of integrable functions.