Regular homomorphisms of generalized projective planes

A generalized projective plane is an incidence structure together with a relation distant on the set of points and also on the set of lines, such that any two distant points A,B (lines a,b) have a unique common line (A,B) (common point (a,b)) and three further axioms hold. Every commutative ring with 1 supplies a model. A homomorphism of into an incidence structure is called regular if the following condition and its dual are valid: A distant B and c IA,B implies c=(A,B). We shall prove the following two theorems. Let be a generalized projective plane satisfying a richness condition called (U). Let M I m. If and are regular homomorphisms of such that X = M X = M for each point X of the line m then A = B A = B for any two points A,B. If is a projective plane over a commutative ring such that (U) holds then the surjective regular homomorphisms of are induced by the ideals of the ring; in particular, the image of under a regular homomorphism is again a projective plane over a ring, and preserves distant.     

On the exactness of a theorem of G.A. Fomin

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On convergence of multiparameter strong submartingales in Banach lattices

- , ., B - — , B- {M _n ,nN ^d } n. .     

Асимптоический Анализ Распределения Статистики Диксона

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Sur la répartition des valeurs de la fonction d'Euler

Let (x) denote the number of those integers n with (n) x, where denotes the Euler function. Improving on a well-known estimate of Bateman (1972), we show that (x)-Ax R(x), where A=(2)(3)/(6) and R(x) is essentially of the size of the best available estimate for the remainder term in the prime number theorem.     

Hardy spaces of predictable martingales

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This research was partially supported by the Hungarian Scientific Foundation.     

Pointwise convergence of expansions with respect to certain product systems

- , - _n A_n-,. , . , ¦_n¦=1 (n=1,2, ), . , — — , .     

Quasiclassical asymptotic behavior of the scattering cross section for asymptotically homogeneous potentials

One considers the total scattering cross section on the potential gV(x), x^m, m3, for large values of the coupling constant g and of the wave number k. One assumes that V(x)(x/|1x|)|x|^–, 2>m+1, as ¦x¦. It is shown that for gk^–1 , g^3–ak^2(a–2) the scattering cross section is equal asymptotically to _a(gk^–1), x=(m–1)(–1)^–1. Here the coefficient _a is determined only by the function and the number . Under the additional conditions >0, V>0, the indicated asymptotic behavior holds in the large domain gk^–1 , gk^a–z c(gk^–1), >0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 105–136, 1986.     

Configuration of subgroups in the special linear group over a skew field with infinite center

Let T be a skew field with infinite center, let be the special linear group over T of degree 3, and let be the subgroup of diagonal matrices with unit Dieudonee determinant. It is proved that for each intermediate subgroup H, H , there exists a net of order n such that ( H N().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 175, pp. 5–12, 1989.In conclusion, the author would like to thank his instructor Z. I. Borevich, as well as N. A. Vavilov, for their assistance.     

Improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set

We will establish the following improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set: For each k and d, 0 k d, define f(d,k) = d+1 if k = 0 and f(d,k) = max{d+1,2d–2k+2} if 1 k d.Theorem 1. Let S be a compact, connected, locally starshaped set in R^d, S not convex. Then for a k with 0 k d, dim ker S k if and only if every f(d, k) lnc points of S are clearly visible from a common k-dimensional subset of S.Theorem 2. Let S be a nonempty compact set in R^d. Then for a k with 0 k d, dim ker S k if and only if every f (d, k) boundary points of S are clearly visible from a common k-dimensional subset of S. In each case, the number f(d, k) is best possible for every d and k.     

О числе Делителей “Сдвинутых” Простых чисел-Близнецов

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Existence for the Cahn-Hilliard phase separation model with a nondifferentiable energy

Summary The Cahn-Hilliard model for phase separation in a binary alloy leads to the equations (I) u_t=w, (II) w= (u)– u with an associated energy functional F(u)=f [(u)+ +¦u¦²/2] dx. In this paper we discuss the existence theory for initial bounday value problems arising from modifications to the Cahn-Hilliard model due to the addition of the non-differentiable term ¦u¦dx to the energy F(u).     

Cohomology of solvable lie algebras and solvmanifolds

The cohomology H^* (G/,) of the de Rham complex *(G/) of a compact solvmanifold G/ with deformed differential d = d + , where is a closed 1 -form, is studied. Such cohomologies naturally arise in Morse-Novikov theory. It is shown that, for any completely solvable Lie group G containing a cocompact lattice G, the cohomology H^*(G/, ) is isomorphic to the cohomology H^*( ) of the tangent Lie algebra of the group G with coefficients in the one-dimensional representation : defined by () = (). Moreover, the cohomology H ^*(G/,) is nontrivial if and only if -[] belongs to a finite subset of H ¹(G/,) defined in terms of the Lie algebra .Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 67–79.Original Russian Text Copyright © 2005 by D. V. Millionshchikov.This revised version was published online in April 2005 with a corrected issue number.     

Critical points of essential norms of singular integral operators in weighted spaces

We show that for any simple piecewise Ljapunov contour there exists a power weight such that the essential norm |S | in the spaceL ₂(, ) does not depend on the angles of the contour and it is given by formula (2). All such weights are described. For the union =₁₂ of two simple piecewise Lyapunov curves we prove that the essential norm |S | inL ₂() is minimal if both ₁ and ₂ are smooth in some neighborhoods of the common points. It is the case when the norm |S | in the spaceL ₂() as well as inL ₂(, ) does not depend on the values of the angles and it can be calculated by formula (5).     

On generalized gamma functions related to the Laurent coefficients of the Riemann zeta function

Summary We study a class of generalized gamma functions _k (z) which relate to the generalized Euler constants _k (basically the Laurent coefficients of(s)) as (z) does to the Euler constant. A new series expansion for _k is derived, and the constant term in the asymptotic expansion for log _k (z) is studied in detail. These and related constants are numerically computed for 1 k 15.     

Existence of compatible families of proper regular conditional probabilities

Summary Let (, , ) be a perfect probability space with countably generated, and let IB be a family of sub--fields of . Under a countability condition on the family IB, I show that there exists a family {}_IB of regular conditional probabilities which are everywhere compatible. Under a more stringent condition on IB, I show that the can furthermore be chosen to be everywhere proper. It follows that in the Dobrushin-Lanford-Ruelle formulation of the statistical mechanics of classical lattice systems, every (perfect) probability measure is a Gibbs measure for some specification.Research supported in part by NSF PHY-78-23952NSF Predoctoral Fellow (1976–79) and Danforth Fellow (1979–81).     

Concretely Locally Presentable and Locally Generated Categories

We give concrete versions of the characterizations of locally -presentable (resp. -generated) categories as categories of models of (resp. weakly) -ary limit-theories and of (resp. weakly) -ary essentially algebraic theories. By concrete we mean that starting with a category of -structures, the theories obtained are extensions of the original ones, and the equivalences of categories are concrete isomorphisms. -presentable and -generated objects are also investigated from this viewpoint.     

On the uniform (C, 1)-summability of Fourier series and integrability of series with respect to multiplicative orthonormal systems

H (G), f(g)H (G) , (, 1)- OHMC G. , OHMC, A. H. . , . , OHMC, lim supp _n=, , ,n .. . , 117 234 . . -      

On term by term dyadic differentiability of Walsh series

. . . . : {ja _j },j=1,2,... — , f(x) , , f ^[1](x) — f .     

Orthogonal systems invariant under an automorphism

, , - ( ) . , - , ( ) .