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 . . -      

Mittelwert der Eulerschen φ-Funktion und des Quadrates der Dirichletschen Teilerfunktion in algebraischen Zahlkörpern

LetK be an algebraic number field, and for every integer K let () andd(), respectively, denote the number of relatively prime residue classes and the number of divisors of the principal ideal (). Asymptotic equalities are proved for the sums () and d ²(), where runs through certain finite sets of integers ofK.     

Angular limits of holomorphic functions which satisfy an integrability condition

Let (–1,1), let 2/(1–)p<, letp denote the Hölder conjugate ofp, and let be an open arc of the unit circle. It is shown that, iff is a holomorphic function on the unit disc such that: (i) (1–|z|)log⁺|f(z)| isL ^p -integrable on the sector {r:0f has an infinite asymptotic value has -finite (2–(1+)p)-dimensional Hausdorff, measure, thenf has finite angular limits on a subset of of positive linear measure. In fact, a stronger conclusion will be established.     

A priori estimates of the solution of the dirichlet problem for the Monge-ampére equation in weight spaces

One proves that a priori boundedness of the norm of the solution of the problem det(U_xx)=f(x,u,u_x)>>0,u¦=0. The magnitudes of the exponents,() depends on whether the arguments u p occur or not in f (x,u,p).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 74–90, 1983.     

γ-Subdifferential and γ-convexity of functions on the real line

In this paper the -subdifferential and -convexity of real-valued functions on the real line are introduced. By means of the -subdifferential, a new necessary condition for global minima (or maxima) is formulated which many local minima (or maxima) cannot satisfy. The -convexity is used to state sufficient conditions for global minima. The class of -convex functions is relatively large. For example, there are -convex functions which are not continuous anywhere. Nevertheless, a -local minimum of a -convex function is always a global minimum. Furthermore, if a -convex function attains its global minimum, then it does so near the boundary of its domain.This research was supported by the Alexander von Humboldt Foundation.     

The intersection of complete geodesics on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> with positive curvature

Let M be a Riemannian manifold. A complete geodesic on M means that :(-,+)M is a normalized geodesic. In this paper, we prove that on (S²,g) with positive curvature, any two complete geodesics must intersect an infinite number of times, and a complete geodesic must self-intersect an infinite number of times. Mathematics Subject Classification (2000) 53C40 (53C22)     

Controllability of linear systems,differential geometry curves in grassmannians and generalized grassmannians,and riccati equations

We study the linear system =Ax+Bu from a differential geometric point of view. It is well-known that controllability of the system is related to the one-parameter family of operators e^t B. We use this to give a proof of the classical controllability conditions in terms of the differential geometry of certain curves in ⁿ. We then view (t)=Im(e^t B) as a curve in appropriate Grassmannian and see that, in local coordinates, is an integral curve of the flow induced by a matrix Riccati equation. We obtain qualitative geometric conditions on that are equivalent to the controllability of the system. To get quantitiative results, we lift to a curve l' in a splitting space, a generalized Grassmannian, which has the advantage of being a reductive homogeneous space of the general linear group, GL(ⁿ). Explicit and simple expressions concerning the geometry of are computed in terms of the Lie algebra of GL(ⁿ), and these are related to the controllability of the system.James Wolper was a visiting professor in the Department of Mathematics at Texas Tech University while much of this research was conducted. He would like to express appreciation for the hospitality he received during his visit.     

О лагранжевом интерполировании с узлами в корнях многочленов сонина-маркова

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Convex independent sets and 7-holes in restricted planar point sets

For a finite setA of points in the plane, letq(A) denote the ratio of the maximum distance of any pair of points ofA to the minimum distance of any pair of points ofA. Fork>0 letc (k) denote the largest integerc such that any setA ofk points in general position in the plane, satisfying for fixed , contains at leastc convex independent points. We determine the exact asymptotic behavior ofc (k), proving that there are two positive constants=(), such thatk ^1/3c (k)k ^1/3. To establish the upper bound ofc (k) we construct a set, which also solves (affirmatively) the problem of Alonet al. [1] about the existence of a setA ofk points in general position without a 7-hole (i.e., vertices of a convex 7-gon containing no other points fromA), satisfying . The construction uses Horton sets, which generalize sets without 7-holes constructed by Horton and which have some interesting properties.     

Explicitly solvable model of Mössbauer scattering

An explicitly solvable model of Mössbauer scattering of rays by a nucleus bound in a harmonic-oscillator potential is constructed. The probability of elastic scattering, which is proportional to the Debye—Waller factor, is calculated in the framework of the explicitly solvable scattering problem. It is assumed that the rms deviation x of the nucleus and the photon wave numberk satisfykxE /E , whereE andE are typical energy levels of the photon and the oscillator states.St Petersburg State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 3, pp. 439–430, June, 1993.     

Global inequalities for curves and surfaces in three-space

Summary The aim of the paper is to give upper bounds for the total curvature of smooth curves and surfaces embedded in euclidean space, in terms of other global geometric characters; in particular, for a plane curve , we prove the inequality K() < (2 + f()d()/2), where d() is the geometric degree of and f() is the number of its inflection points. In the case of a surface S, a bound is given in terms of the genus g(S), the number of components of the parabolic points on S and the geometry of its apparent contour.     

Über geschlossene affine Zwangläufe in der Ebene

We consider integral coverings y:{1,2,..,} of an affine plane which occur when is moved under a continuous periodic affine motion(t):. One can distinguish normal points × , i.e. is constant in a certain neighborhood of x, and singular points. If (x) is the number of times x passes through its orbit (t)x all normal points x have (x)=1, and the set of all singular points consists of a number of isolated points and lines. If (x) is the tangent rotation number of the orbit of x all singular points lie on the moving pole curve.     

On the γ-Interior and γ-Closure of a Set

For a set X, let : exp X exp X satisfy A B whenever A B X. In [4], -open subsets of X, -interior iA and -closure cA of A X have been defined. The purpose of the present paper is to show that, under suitable conditions on , explicit formulas furnish iA and cA.     

Infinitesimal,first order bendings of smooth,convex surfaces of revolution subject to conic,sleeve-like constraints along the boundary

We prove that on a closed, smooth, convex surface of revolution , whose poles are not flattening points, there exists only a countable set of parallels _n. Each of these parallels cuts surface into two parts so that one of the parts, , admits nontrivial, infinitesimal bendings in the process of which all the points of its boundary _n are displaced on a preassigned, conic sleeve K that is coaxial with the surface. The sequence of such parallels _n converges to parallel ^*, which has the following properties: 1) the tangent cone to surface along ^* is orthogonal to sleeve K; 2) surface , cut off from surface by parallel ^*, has rigidity of first order in the indicated class of bendings.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 3–8, 1990.     

Random Walks On Finite Convex Sets Of Lattice Points

This paper examines the convergence of nearest-neighbor random walks on convex subsets of the lattices^d. The main result shows that for fixedd, O(²) steps are sufficient for a walk to get random, where is the diameter of the set. Toward this end a new definition of convexity is introduced for subsets of lattices, which has many important properties of the concept of convexity in Euclidean spaces.     

The compact support property for solutions to the heat equation with noise

Summary We consider all solutions of a martingale problem associated with the stochastic pde and show thatu(t,·) has compact support for allt0 ifu(0,·) does and if <1. This extends a result of T. Shiga who derived this compact support property for 1/2 and complements a result of C. Mueller who proved this property fails if 1.The author's research was supported by an NSF grant and an NSERC operating grantThe author's research was supported by an NSERC operating grant     

On the radicals of structural matrix rings

The relationship between the radical of a ringR and a structural matrix ring overR has been determined for some radicals. We continue these investigations, amongst others, determining exactly which radicals have the property (M(,R))=M( _s ,(R))+M( _a ,⁺(R))for any structural matrix ringM(,R) and finding (M(,R)) for any hereditary subidempotent radical .     

Symmetrization of functions in Sobolev spaces and the isoperimetric inequality

A positive measurable function f on R^d can be symmetrized to a function f^* depending only on the distance r, and with the same distribution function as f. If the distribution derivatives of f are Radon measures then we have the inequality f^*f, where f is the total mass of the gradient. This inequality is a generalisation of the classical isoperimetric inequality for sets. Furthermore, and this is important for applications, if f belongs to the Sobolev space H¹,P then f^* belongs to H¹,P and f^*_pf_p.     

Turán's extremal problem in random graphs: Forbidding odd cycles

For 0<1 and graphsG andH, we writeGH if any -proportion of the edges ofG span at least one copy ofH inG. As customary, we writeC ^k for a cycle of lengthk. We show that, for every fixed integerl1 and real >0, there exists a real constantC=C(l, ), such that almost every random graphG _{n, p} withp=p(n)Cn ^–1+1/2l satisfiesG _n,p1/2+ C ^2l+1. In particular, for any fixedl1 and >0, this result implies the existence of very sparse graphsG withG _1/2+ C ^2l+1.The first author was partially supported by NSERC. The second author was partially supported by FAPESP (Proc. 93/0603-1) and by CNP_q (Proc. 300334/93-1). The third author was partially sopported by KBN grant 2 1087 91 01.     

On the isotopy of Legendrian knots

Let ₀ and ₁ be Legendrian knots which are isotopic as usual knots, and which have the same obvious invariants rot and link. It seems to be an open question whether ₀ and ₁ are isotopic as Legendrian knots. In the paper we give a positive answer to this question for the (rather restricted) class of Legendrian knots with nonintersecting fronts.