Orthonormal systems satisfying an inequality of S. M. Nikol'skii

N- (p, q) (1 pN-, L _p - L _q -. , , , L L _q - , , .     

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.     

On aU-set for multiple Walsh series

m- (m2) , - , .     

Approximate first-order and second-order directional derivatives of a marginal function in convex optimization

Given a convex functionf: ^p × ^q (–, +], the marginal function is defined on ^p by (x)=inf{f(x, y)|y ^q }. Our purpose in this paper is to express the approximate first-order and second-order directional derivatives of atx ₀ in terms of those off at (x ₀,y ₀), wherey ₀ is any element for which (x ₀)=f(x ₀,y ₀).The author is indebted to one referee for pointing out an inaccuracy in an earlier version of Theorem 4.1.     

OnU ϕ-sets of trigonometric integrals

, (t) >0 E(–, +),E<, , ¦f(t)¦ (t) xE, f(t)=0 (–, +).     

Almost-periodic solutions for a second order abstract Cauchy problem

Summary Let M be a number field of degree m with ring of integers \bZ_M. Let F\in\bZ_M[X,Y] be a form of degree n such that F(X,1) has distinct roots. Let\break G\in\bZ[X,Y] be an arbitrary polynomial of degree k. Assuming that k\le n-2m\pl 1 if all roots of F^{(i)}(X,1) (1\le i\le n) are complex and k\le n-4m\pl 1 otherwise, we provide an efficient algorithm for finding all solutions X,Y\in\bZ_M, \max\b(\overline{|X|},\overline{|Y|}\,\b)\ki C of the inequality \overline{\b|F(X,Y)\b|\!}\,\le c \cdot \overline{\b|G(X,Y)\b|\!}\,. We provide numerical examples with m=3 and C=10^{100}.     

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, .     

Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain in R ², we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ^–,>0 with nonnegative functions qK(). Some estimates on the solution are also given.     

Epsilon efficiency

This paper considers the extension of -optimality for scalar problems to vector maximization problems, or efficiency problems, which havem objective functions defined on a set .It is shown that the natural extension of the scalar -optimality concepts [viz, given >0, given a solution setS, ifxS there exists an efficient solutiony with f(x)–f(y), and given an efficient solutiony, there exists anxS with f(x)–f(y)] do not hold for some methods used. Six concepts of -efficient sets are introduced and examined, to a very limited extent, in the context of five methods used for generating efficient points or near efficient points.In doing so, a distinction is drawn between methods in which the surrogate optimizations are carried out exactly, and those where terminal -optimal solutions are obtained.The author would like to thank the referees whose thoroughness was extremely helpful for the revised paper.     

Large transversals to small families of unit disks

Summary It is proved that if the nonempty intersection of bounded closed convex sets AnB is contained in (A + F)U(B+F) and one of the following holds true: (i) the space X is less-than-three dimensional, (ii) AUB is convex, (iii) F is a one-point set, then AnBCA+F or AnBCB+F (Theorems 2 and 3). Moreover, under some hypotheses the characterization of A and B such that AnB is a summand of AUB is given (Theorem 3).     

Die Verteilung der schlichten Funktionen in einem Funktionenraum

We consider the set of regular functions . We construct a Borel measure and a class of outer measures _h onH. With these and _h we show that: (HS)=0 and _h (HS)=0, (S is the set of normed univalent functions). From _h (HS)=0 follows—forh=t —that the Hausdorff—Billingsley-dimension ofHS is zero.     

Estimation de sommes multiples de fonctions arithmétiques

We estimate some sums of the shape S(X ¹,..., X ^m ):=_{1 d1 X1...1 dm Xm} f(d ₁,..., d _m )when m N and f is a nonnegative arithmetical function. We relate them to the behaviour of the associated Dirichlet series F(s ₁,..., s _m ) = _{d1 = 1} ... _{dm = 1} f(d ₁,..., d _m )/d ₁ ^s1 ... d _m ^sm.The main aim of this work is to develop analytic tools to count the rational points of bounded height on toric varieties.     

Moduli of smoothness and best approximation in the spacesL p , 0<p<1

L ^p , 0<<1, .     

Martingale Hardy spaces,BMO and VMO spaces with nonlinearly ordered stochastic basis

VMO. , ⁺, «» VMO . « » .     

Uniform Continuity of a Family of Weakly Regular Set Functions on Topological Spaces

Conditions for the uniform continuity of a family of weakly regular set functions defined on an algebra of subsets of a -topological space (T,) and taking values in an arbitrary topological space are found.     

On the equivalence of renormalizations in standard and dimensional regularizations of 2D four-fermion interactions

We discuss the problem of equivalence between the standard (integer-dimensional d=2) and the d=2+ dimensional renormalization schemes for the complete U_N-symmetrical four-fermion interaction model. To ensure the multiplicative renormalizability of the theory, we need three charges in the first case; in the second, we need an infinite series of independent charges g {g_n, n=0, 1, ...}. After the usual MS-renormalization. there exists a UV-finite renormalization of fields. Charges gg'(g) exist such that the renormalized Green's functions in the limit 0 depend only on the three lower charges g'_n(g) with n=0, 1, 2. rather than on the whole set. This ensures the possibility of establishing the equivalence of the two renormalization schemes. The results of calculations in the MS scheme up to two loops for the -functions, and up to three loops for the anomalous field dimension are presented. These are presented together with the derivation of the projection technique relations, which allows one to express the higher renormalized composite operators of the 4F-interaction via the lower ones in the limit 0.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 1, pp. 27–46, April, 1996.Translated by L. O. Chekhov.     

On the solution of a generalized Schrödinger-type equation

Lu(t)+(u,F)g(t)=f(t), tS. , ( F, g). .

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The authors wish to thank Professor Yu. A. Rozanov for his help and discussions.     

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

A conditional dilatation equation

Summary The following theorem holds true. Theorem. Let X be a normed real vector space of dimension 3 and let k > 0 be a fixed real number. Suppose that f: X X and g: X × X are functions satisfying x – y = k f(x) – f(y) = g(x, y)(x – y) for all x, y X. Then there exist elements and t X such that f(x) = x + t for all x X and such that g(x, y) = for all x, y X with x – y = k.     

The interpolation ofℓ r sequences by hp functions

The sequence spaceH ^P (z)={{f (z_h)}:f H ^p} is defined for a fixed sequence Z={z_k} of different points of the open unit disk and the Hardy class HP of analytic functions in the disk. For an arbitrary p[1, ) is constructed a point sequence Z= {z_k} such that ¹h ^p(z), but ^r h^p (Z) for r > 1. It follows from a well-known result of L. Carleson that the inclusions ^r h (Z) for all r[1,] are equivalent.Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 503–508, April, 1977.