Table of contents

In this paper we give a classification of affine surfaces which are quasihomogeneous with respect to an algebraic group. In particular, we also obtain a classification of affine surfaces which are homogeneous with respect to an algebraic group.

In this paper we disprove a conjecture of C. L. Siegel on the uniform boundedness of the number of integral points on hyperelliptic curves of given genus and defined over a function field.

The relationship between the structure of a simple Lie algebra of finite characteristic and the structure of the group of its automorphisms is investigated. The results obtained are used to classify simple Lie algebras of characteristic

p > 5

for which the largest reduced subgroup in the scheme of automorphisms is a maximal subscheme. An analogous classification theorem is proved for "simple" group schemes, i.e. schemes every normal divisor of which lying in the reduced subscheme is the kernel of some purely nonseparable isogeny. For characteristics 2 and 3, families of counterexamples are constructed to all results obtained for

p > 5

.

The paper contains a characterization, by means of their biprimary subgroups of even order, of the following finite simple groups: the special projective linear groups PSL(2, q) (q > 3), the Suzuki groups Sz(q), and Janko's group J of order 175,560.

In this work we improve Philip Hall's estimate for the number of cyclic subgroups in a finite -group. From our result it follows that if a -group is not absolutely regular and not a group of maximal class, then 1) the number of solutions of the equation in is equal to , where is a nonnegative integer; 2) if , then the number of solutions of the equation in is divisible by . This permits us to strengthen important theorems of Hall and Norman Blackburn on the existence of normal subgroups of prime exponent. The latter results in turn permit us to give a factorization of -groups with absolutely regular Frattini subgroup. Another application is a theorem on the number of subgroups of maximal class in a -group.

A simple construction is given for proving the Poincaré duality theorem for generalized manifolds, which also applies to generalized manifolds without locally constant "orientation" sheaves (for example, to manifolds with "boundary"). It appears that some other well-known duality relations in generalized manifolds are either special cases of Poincaré duality, or simple consequences of it.

Algebraic K-theory can be constructed by means of the homotopy groups of the abstract simplicial structure on the group of invertible matrices GL(A) of the ring A. This structure may be naturally taken as two-sidedly invariant. Of basic interest is the multiplication in the functor so obtained, which for different rings A assumes different aspects.

We introduce a new method of approximation of nonperiodic functions by algebraic polynomials. In particular, by this method we establish necessary and sufficient conditions for a function on the interval [-1,1] to satisfy Hölder's condition in the L_p metric.

Let {T(X)} be a uniformly bounded one-parameter strongly continuous group of operators on a Banach space. Suppose its corresponding ergodic projection exists and is zero. Then we can define a certain countable family of parametric norms (norms depending on a positive parameter t) whose rate of decay with respect to t at a given element involves the "ergodic" properties of that element. They ate called the ergodic moduli of a given (positive integral) order. This article is basically devoted to explaining the properties of ergodic moduli and to finding one- and two-sided estimates for them in terms of other parametric norms. An interesting analogy can be drawn between the properties of ergodic moduli and those of the smoothness moduli of functions as studied in approximation theory.

Conditions depending on the properties of the polynomials and are found for the correct solvability of the boundary value problem

The algebra generated by one-dimensional singular integral operators with piecewise continuous coefficients is studied. Operators acting on the space L_p with a weight are considered. The contour is assumed to consist of closed and open arcs. The structure of the symbols of the operators considered is elucidated. It is found that the symbol is a matrix-function of the second order depending both on p and on the weight. Criteria that the operator be Fredholm and a formula for its index are established.