Improvement of Bruck's completion theorem

Improving Bruck's Completion-Theorem for nets, we show that a net of order k and degree k + 1 – can be extended to an affine plane, if 3k > 8³ – 18² + 8 + 4. As applications we obtain the following two theorems: A maximal partial t-spread in PG(2t + 1, q), q not a square, with deficiency > 0 satisfies 8³ – 18² + 8 + 4 3q ². There exists an absolute constant c such that every linear space with constant point degree n + 1 and minimum line degree n + 1 – a can be embedded in a protective plane of order n provided that n > ca ³.     

On Piecewise-Constant Approximation of Continuous Functions of n Variables in Integral Metrics

We consider the approximation by piecewise-constant functions for classes of functions of many variables defined by moduli of continuity of the form (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ), where _i ( _i ) are ordinary moduli of continuity that depend on one variable. In the case where _i ( _i ) are convex upward, we obtain exact error estimates in the following cases: (i) in the integral metric L ₂ for (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ); (ii) in the integral metric L _p (p 1) for (₁, ..., _n ) = c ₁₁ + ... + c _{n n} ; (iii) in the integral metric L _{(2, ..., 2, 2r)} (r = 2, 3, ...) for (₁, ..., _n ) = ₁(₁) + ... + _{n – 1}( _{n – 1}) + c _{n n} .     

Large deviations for squares of Bessel and Ornstein-Uhlenbeck processes

Let (X_t⁽⁾,t0) be the BESQ process starting at x. We are interested in large deviations as for the family {^–1X_t⁽⁾,tT}, – or, more generally, for the family of squared radial OU process. The main properties of this family allow us to develop three different approaches: an exponential martingale method, a Cramér–type theorem, thanks to a remarkable additivity property, and a Wentzell–Freidlin method, with the help of McKean results on the controlled equation. We also derive large deviations for Bessel bridges.Mathematics Subject Classification (2000): 60F10, 60J60     

The integral modulus of continuity of the Dirichlet kernel and the conjugate Dirichlet kernel

Asymptotic estimates for the integral modulus of continuity of order s of the Dirichlet kernel and the conjugate Dirichlet kernel are obtained. For example, if k/2, then _s (D _k ,)=2 ^s +1/²sin ^s k/2 log(1+k/s)+O(2 ^s sin ^s k/2)holds uniformly with respect to all the parameters.Translated from Matematicheskie Zametki, Vol. 54, No. 3, pp. 98–105, September, 1993.     

Problem of correctness of the best approximation in the space of continuous functions

Let W^rH _w be the subclass of those functions of C^r[a, b], for which (f ^(r),)(), where () is a given modulus of continuity, and P_n be the space of algebraic polynomials of degree at most n and _n(f) be the polynomial of best approximation for f(x) on [a, b]. Estimates for and moduli of continuity of the operators of best approximation on W^rH _w are established. For example, if ()=, then Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 351–360, March, 1978.The author thanks S. B. Stechkin for the formulation of the problem and assistance with the article.     

Boundedness in the mean of orthonormalized polynomials

For the polynomials {p_n(t)} ₀ , orthonormalized on [–1, 1] with weightp(t) = (1–t) (1+t) _v=1 ^m , we obtain necessary and sufficient conditions for boundedness of the sequences of norms: 1) 2) and 3) with the conditions that on [–1, 1] and (H,)^–1 L²(0, 2), where(H,) is the modulus of continuity in C(–1, 1) of function H.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 759–770, May, 1973.     

Stability and uniqueness of the solution of the inverse kinematic problem of seismology in higher dimensions

The following inverse kinematic problem of seismology is considered. In the compact domain M of dimension ,2 with the metric, we consider the problem of constructing a new metricdu=nds according to the known formula where ,M and K_, is the geodesic in the metric du, connecting the points , . One proves uniqueness and one obtains a stability estimate, where the refraction indices n₁, n₂ are the solutions of the inverse kinematic problem, constructed relative to the functions ₁, ₂, respectively, is the differential form on M×Mwhere =₂–₁,.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 84, pp. 3–6, 1979.     

О метрических свойст вах особых множеств г оломорфных функций нескольких п еременных

LetG be a domain inC ⁿ ,EG, mes E=0 for (r)=r ^2n–1(r), where (r) is a nondecreasing non-negative function (r>0). Iff(z) is holomorphic inGE and (,f, GE)(), C=const, thenf(z) is holomorphic inG.The impossibility of the relaxation of the stipulations on () and(r) is also established.The statement above is a corollary to a more general result about the representation of a holomorphic function from a certain class in the form of an integral with respect to -measure, extended over the set of singular points of the function.     

p-Sets for random walks

Summary Let (,,P) be a probability space and let {itX _n ()} _n=1 be a sequence of i.i.d. random vectors whose state space isZ ^m for some positive integerm, where Z denotes the integers. Forn = 1, 2,... letS _n () be the random walk defined by . ForxZ ^m andU ^m, them-dimensional torus, let . Finally let be the characteristic function of the X's.In this paper we show that, under mild restrictions, there exists a set withP{ ₀ } = 1 such that for ₀ we have for all aU ^m,le0.As a consequence of this theorem, we obtain two corollaries. One is concerned with occupancy sets form-dimensional random walks, and the other is a mean ergodic theorem.Research supported by N.S.F. Grant # MCS 77-26809     

On the complexity of following the central path of linear programs by linear extrapolation II

A class of algorithms is proposed for solving linear programming problems (withm inequality constraints) by following the central path using linear extrapolation with a special adaptive choice of steplengths. The latter is based on explicit results concerning the convergence behaviour of Newton's method to compute points on the central pathx(r), r>0, and this allows to estimate the complexity, i.e. the total numberN = N(R, ) of steps needed to go from an initial pointx(R) to a final pointx(), R>>0, by an integral of the local weighted curvature of the (primal—dual) path. Here, the central curve is parametrized with the logarithmic penalty parameterr0. It is shown that for large classes of problems the complexity integral, i.e. the number of stepsN, is not greater than constm log(R/), where < 1/2 e.g. = 1/4 or = 3/8 (note that = 1/2 gives the complexity of zero order methods). We also provide a lower bound for the complexity showing that for some problems the above estimation can hold only for 1/3.As a byproduct, many analytical and structural properties of the primal—dual central path are obtained: there are, for instance, close relations between the weighted curvature and the logarithmic derivatives of the slack variables; the dependence of these quantities on the parameterr is described. Also, related results hold for a family of weighted trajectories, into which the central path can be embedded.On leave from the Institute of Mathematics, Eötvös University Budapest, H-1080 Budapest, Hungary.     

Vitesse de convergence du théorème de la limite centrale pour des champs faiblement dépendants

Résumé Nous étudions la vitesse de convergence du théorème de la limite centrale pour des champs de ^d, faiblement dépendants: m-dépendant ou -fortement mélangeant. Dès que le champ est dans L ²⁺, >0, la vitesse de convergence obtenue est _n ^{– (1)} avec un facteur (log _n ) ^a qui intervient quand est à décroissance exponentielle et dans le cas m-dépendant quand 1. Le cas où est à décroissance puissance est aussi étudié. Ces résultats ne font intervenir ni la stationarité, ni la géométrie des domaines sur lesquels le T.L.C. est étudié.     

On the problem of two linearized wells

We study the behaviour of sequences of elastic deformationsy ^{n n} whose gradients approach two linearized wells, and give an application to magnetostriction.This article was processed by the author using the style filepljour1m from Springer-Verlag.     

Weak solutions of the Stokes problem in weighted Sobolev spaces

For n2 we consider the Stokes problem in ⁿ, -u + p=f, -divu=g, in weighted Soboiev spaces H ₆ ^m,r , where the weights are proportional to (1+|x|). We prove the existence of weak solutions for any K, whereK is a discrete set of critical values. Furthermore, we characterize the solutions of the homogeneous problem.This research was supported by the DFG research group Equations of Hydrodynamics, Universities of Bayreuth and Paderborn.     

On the representation of integers as sums of triangular numbers

Summary In this survey article we discuss the problem of determining the number of representations of an integer as sums of triangular numbers. This study yields several interesting results. Ifn 0 is a non-negative integer, then thenth triangular number isT _n =n(n + 1)/2. Letk be a positive integer. We denote by _k (n) the number of representations ofn as a sum ofk triangular numbers. Here we use the theory of modular forms to calculate _k (n). The case wherek = 24 is particularly interesting. It turns out that, ifn 3 is odd, then the number of points on the 24 dimensional Leech lattice of norm 2n is 2¹²(2¹² – 1) ₂₄(n – 3). Furthermore the formula for ₂₄(n) involves the Ramanujan(n)-function. As a consequence, we get elementary congruences for(n). In a similar vein, whenp is a prime, we demonstrate ₂₄(p ^k – 3) as a Dirichlet convolution of ₁₁(n) and(n). It is also of interest to know that this study produces formulas for the number of lattice points insidek-dimensional spheres.     

Solution of integral equations in fractional Sobolev spaces

We consider linear integral equations and Urysohn equations with constant integration limits. Sufficient conditions are given for the solutions of these equations to be in Sobolev spacesW ₂ (0,1), 0 2. Finite-difference schemes are constructed for approximate solution of the original equation by special averaging of the right-hand side kernel. The rate of convergence of the approximate solution to the averaged exact solution is shown to beO(h|ln h|(1/2,)⁺(3/2,)).Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 3–19, 1987.     

Voronoi Diagrams of Real Algebraic Sets

A collection of n (possibly singular) semi-algebraic sets in ^d of dimension d–1, each defined by polynomials of maximal degree , has ((n) ^d ) first-order Voronoi cells (for any fixed d). In the nonhypersurface case, where the maximal dimension of the semi-algebraic sets is m d–2, the number of first-order Voronoi cells is bounded above by O(n ^{m+1 d} ) (for nonsingular semi-algebraic sets) or by O((n) ^d ) (in general). The complexity of the entire kth-order Voronoi diagram of a generic collection of n non-singular real algebraic sets in R ^d of maximal dimension m<d and maximal degree is O(n ^{min(d+k,2d)2(m+1)d} ).     

Pyramids in the complex projective plane

We study the critical points of the diameter functional on the n-fold Cartesian product of the complex projective plane C P ² with the Fubini-Study metric. Such critical points arise in the calculation of a metric invariant called the filling radius, and are akin to the critical points of the distance function. We study a special family of such critical points, P _kC P ¹C P ², k=1,2... We show that P _k is a local minimum of by verifying the positivity of the Hessian of (a smooth approximation to) at P _k. For this purpose, we use Shirokov's law of cosines and the holonomy of the normal bundle of C P ¹C P ². We also exhibit a critical point of , given by a subset which is not contained in any totally geodesic submanifold of C P ².     

Blow-up rates for parabolic systems

Let ⁿ be a bounded domain andB _R be a ball in ⁿ of radiusR. We consider two parabolic systems: ut=u +f(), i= +g(u) in × (0,T) withu=v=0 on × (0,T) andu _t =u, v _t =v inB _r × (0,T) withe/v=f (v), e/v=g(u) onB _R × (0,T). Whenf(v) andg(u) are power law or exponential functions, we establish estimates on the blow-up rates for nonnegative solutions of the systems.     

Flat points of two-dimensional Brownian motion

Summary SupposeZ(·) is a two-dimensional Brownian motion. It is shown that a.s. there existt ₀ and >0 such thatZ(t ₀) is an extremal point of the convex hull of {Z(t)|t ₀–tt₀} and also an extremal point of the convex hull of {Z(t)|t ₀tt₀+} and, moreover, the tangent lines to the convex hulls atZ(t ₀) form a non-zero angle.The result is related to the following unsolved problem of S.J. Taylor. Do there exist a.s.t ₀ and >0 such that the intersection of the convex hulls of {Z(t)|t ₀–tt₀} and {Z(t)|t ₀tt₀+} contains onlyZ(t ₀)?This research was partially supported by Grant-in-Aid for Scientific Research (No. 400101540202), Ministry of Education, Science and Culture     

The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms

Given a vector of real numbers=(₁,... _d ) ^d , the Jacobi-Perron algorithm and related algorithms, such as Brun's algorithm and Selmer's algorithm, produce a sequence of (d+1)×(d+1) convergent matrices {C⁽ⁿ⁾():n1} whose rows provide Diophantine approximations to . Such algorithms are specified by two mapsT:[0, 1] ^d [0, 1] ^d and A:[0,1] ^d GL(d+1,), which compute convergent matrices C⁽ⁿ⁾())...A(T())A(). The quality of the Diophantine approximations these algorithms find can be measured in two ways. The best approximation exponent is the upper bound of those values of for which there is some row of the convergent matrices such that for infinitely many values ofn that row of C⁽ⁿ⁾() has . The uniform approximation exponent is the upper bound of those values of such that for all sufficiently large values ofn and all rows of C⁽ⁿ⁾⁽⁾ one has . The paper applies Oseledec's multiplicative ergodic theorem to show that for a large class of such algorithms and take constant values and on a set of Lebesgue measure one. It establishes the formula where are the two largest Lyapunov exponents attached by Oseledec's multiplicative ergodic theorem to the skew-product (T, A,d), whered is aT-invariant measure, absolutely continuous with respect to Lebesgue measure. We conjecture that holds for a large class of such algorithms. These results apply to thed-dimensional Jacobi-Perron algorithm and Selmer's algorithm. We show that; experimental evidence of Baldwin (1992) indicates (nonrigorously) that. We conjecture that holds for alld2.