Primitive Inner Illuminating Systems for Convex Bodies

A set X of boundary points of a (possibly unbounded) convex body KE ^d illuminating K from within is called primitive if no proper subset of X still illuminates K from within. We prove that for such a primitive set X of an unbounded, convex set KE ^d (distinct from a cone) one has X=2 if d=2, X6 if d=3, and that there is no upper bound for X if d4.     

A convolution metric in the space of measures and ε-isometries in Lp

For >0, 2,4,6,... on the set of those Borel measures on , such that x^pd(x)<, one introduces a metric, characterizing the nearness of the convolutions x^p* and x^p*. From the convergence of a sequence of probability measures in this metric there follows its convergence in the Kantorovich-Rubinshtein metric. From here one derives theorems on the approximation of -isometries: if H is a finite-dimensional subspace in L_p, then there exists a continuous function _H(), such that for any linear -isometric operator THL_p there exists a linear isometry UH L_p, such that T–U<_H().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 151–156, 1987.     

On Sets of Planes in Projective Spaces Intersecting Mutually in One Point

Let be a projective space. In this paper we consider sets of planes of such that any two planes of intersect in exactly one point. Our investigation will lead to a classification of these sets in most cases. There are the following two main results:- If is a set of planes of a projective space intersecting mutually in one point, then the set of intersection points spans a subspace of dimension 6. There are up to isomorphism only three sets where this dimension is 6. These sets are related to the Fano plane.- If is a set of planes of PG(d,q) intersecting mutually in one point, and if q3, 3(q²+q+1), then is either contained in a Klein quadric in PG(5,q), or is a dual partial spread in PG(4,q), or all elements of pass through a common point.     

Existence of solutions for the Navier-Stokes equations,having an infinite dissipation of energy,in a class of domains with noncompact boundaries

In the domain ³ with two exits at infinity, ₁ = {x:x<g ₁(x ₃),x ₃ > 2} and ₂ = {x:0<x ₃<g ₃(x), x>2}, one investigates the stationary system of Navier-Stokes equations under the boundary condition. One proves existence and uniqueness theorems for the solution of this problem with an infinite Dirichlet integral, under a given flow of the velocity vector across the cross sections of the exits at infinity. As an example one considers the case wheng _i(t) 1.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 180–202, 1981.     

k-regular factors and semi-k-regular factors in bipartite graphs

LetG be a graph, andk1 an integer. LetU be a subset ofV(G), and letF be a spanning subgraph ofG such that deg _F (x)=k for allx V(G)–U. If deg _F (x)k for allxU, thenF is called an upper semi-k-regular factor with defect setU, and if deg _F (x)k for allxU, thenF is called a lower semi-k-regular factor with defect setU. Now letG=(X, Y;E(G)) be a bipartite graph with bipartition (X,Y) such that X=Yk+2. We prove the following two results.(1) Suppose that for each subsetU ₁X such that U ₁=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.(2) Suppose that for each subsetU ₁X such that U ₁=X–1/k+1,G has a lower semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=X–1/k+1,G has a lower semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.     

Critical determinant of the region ∣x∣p+∣y∣p<1 (supplement to the paper “on the application of the electronic computer to the proof of a conjecture of Minkowski from the geometry of numbers,” Zap. Nauchn. Sem. Lomi,71, 163–180, 1977)

Supplement to the paper cited in the title. One presents the results of the calculations on an electronic computer, by the method developed in that paper, proving Minkowski's conjecture regarding the critical determinant of the region x^p+y^p<1 for all real numbers p>2.035 and for 1.01p1.99.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 82, pp. 29–32, 1979.     

Cauchy problem for a semilinear wave equation. II

The existence and the uniqueness of a weak solution of the Cauchy problem, for a second-order semilinear pseudodifferential hyperbolic equation on a smooth Riemannian manifold (of dimension n3) without boundary, is proved. In particular, for the equation ü–u+u^p–1=0 one has unique solvability for the Cauchy problem when 1(n+2)/(n–2).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 182, pp. 38–85, 1990.     

Exact estimates for partially monotone approximation

f(x) — , - [–1,1], (f, ) — , as— f, . . (- ) (x _i,x _{i+ 1}) (i=0, 1, ...,s–1; =–1,x _s,=1), f(x) . , n=0,1,... _n() , [– 1,1] signf(x) sign _n(x) 0, ¦f(x)– _n(x)¦ C(s) (f, 1/n+1, C(s) s. , - , « » .     

Mixed-norm estimates for a class of integral operators

(, ) — R ^m ×R ⁿ . f R ^m ×R ⁿ f_p,q, f L _p (R ^m) x y, L^q(Rⁿ). ׃ _q,r cƒ _p,r , ׃ R ^m ×R ⁿ , , , q r . , ( ¦¦) K ₀ (y); p, g r , K ₀.     

Galerkin methods for parabolic equations with nonlinear boundary conditions

A variety of Galerkin methods are studied for the parabolic equationu _t =(a(x) u),x ⁿ ,t (O,T], subject to the nonlinear boundary conditionu _v =g(x,t,u),x,t (O,T] and the usual initial condition. Optimal order error estimates are derived both inL ² () andH ¹ () norms for all methods treated, including several that produce linear computational procedures.The authors were partially supported by The National Science Foundation during the preparation of this paper.     

О предельном распределении для сумм независимых слууайных велиуин на бикомпактных коммутативных топологиуеских группах

,      

On the summability of Fourier series with the method of lacunary arithmetic means

. f- ,S _n (f)— . {n _k }, n _k+1/n _k >1+ck ^– ,— , 0<1/2, f ₀, .     

A quantitative refinement of Rado's theorem

The fundamental result of the paper is the following. Theorem: Let be a k-quasiconformal Jordan curve and let be another Jordan curve (not necessarily quasiconformal). Assume that f maps conformallyext ontoext , f()=, f()>0. We assume that there exists a homeomorphism between and such that Then there exist numbers =(k)>0 and A=A(k), such that f(())– A, .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 103–112, 1987.     

Events correlated with respect to every subposet of a fixed poset

Posets are said to be (positively) correlated with respect to a third posetR onX (we writeA _R B) ifP(AR) P(AR B). HereP(CR) is the probability that a randomly chosen linear extension ofR is also a linear extension ofC. We classify posetsR onX such that(x, y) _s (u, v) holds for all posetsS onX which are subposets ofR, wherex, y, u, v are distinct elements ofX. On the way to proving this result, we show when a correlation inequality due to Shepp holds strictly.     

НАИЛУЧШЕЕ ПРИБЛИЖЕ НИЕ УГЛОМ И ПРИБЛИЖЕНИЕ УГЛОМ ИЗ СИНГУЛЯРНЫХ ИНТЕГРАЛОВ ФУНКЦИЙf∈L p (R n ),2<P<∞

. .     

On the Riemann derivatives ofC sP-integrable functions

[4] , .     

Uniform boundedness of a family of set functions

Let be a ring of sets, X a normed space, : X ( ) a bounded family of triangular functions. The following generalized Nikodym theorem is established: the family {} is uniformly bounded on if and only if it is bounded on every sequence of pairwise disjoint sets of which the union is a part of some set in . An analogous criterion is established also for semiadditive functions. In addition, it is shown that uniform boundedness of a family of triangular functions is preserved in passing from a ring to the -ring it generates.Translated from Matematicheskie Zametki, Vol. 23, No. 6, pp. 855–861, June, 1978.     

On the optimal mapping of distributions

We consider the problem of mappingXY, whereX andY have given distributions, so as to minimize the expected value of X–Y². This is equivalent to finding the joint distribution of the random variable (X, Y), with specified marginal distributions forX andY, such that the expected value of X–Y² is minimized. We give a sufficient condition for the minimizing joint distribution and supply numerical results for two special cases.     

Fields with vanishing K2. Torsion in H1(X,K2) and Ch2(x)

This paper describes fields F of nonzero characteristic with the property that for all finite extensions E/F K₂E=0. We consider a somewhat wider class of fields which includes finite and separably closed fields. For smooth projective varieties X over such a field we show that the groups H¹(X, K₂){} and H²(X_et, (2)), NH³(X_et, (2)) and Ch²(X){} are isomorphic. These results are applied to describe the groups SK₁ of a smooth affine curve over such a field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 108–118, 1982.