Exact constants of approximation for differentiable periodic functions

For all odd r we construct a linear operator B_r,r(f) which maps the set of 2-periodic functionsf(t) X^(r) (X^(r)=C^(r) or L₁ ^(r)) into a set of trigonometric polynomials of order not higher than n-1 such that where X is the C or L₁ metric, E_n(f)_X and (f, )_X are the best approximation by means of trigonometric polynomials of order not higher than n-1 and the modulus of continuity of the functionf in the X metric, respectively; K_r are the known Favard constants.Translated from Matematicheskie Zametki, Vol. 14, No. 1, pp. 21–30, July, 1973.In conclusion, the author wishes to express his deep gratitude to N. P. Korneichuk under whose guidance this paper was written.     

On lower bounds in radial basis approximation

We investigate the approximation by manifolds _n() generated by linear combinations of n radial basis functions on R^d of the form (|–a|), where is the thin-plate spline type function. We obtain exact asymptotic estimates for the approximation of Sobolev classes W^r(B^d) in the space L(B^d) on the unit ball B^d. AMS subject classification 41A25, 41A63, 65D07, 41A15     

A New Law of the Iterated Logarithm in Rd with Application to Matrix-Normalized Sums of Random Vectors

Let (X _n , n1) be a sequence of independent centered random vectors in R ^d . We study the law of the iterated logarithm lim sup _n(2 log log B _n )^–1/2 B ^–1/2 _n S _n =1 a.s., where B _n is the covariance matrix of S _n = ⁿ _i=1 X _i , n1. Application to matrix-normalized sums of independent random vectors is given.     

Application of Topological Degree to the Study of Bifurcation in von Kármán Equations

The aim of this paper is to illustrate the use of topological degree for the study of bifurcation in von Kármán equations with two real positive parameters and for a thin elastic disk lying on the elastic base under the action of a compressing force, which may be written in the form of an operator equation F(x, , ) = 0 in some real Banach spaces X and Y. The bifurcation problem that we study is a mathematical model for a certain physical phenomenon and it is very important in the mechanics of elastic constructions. We reduce the bifurcation problem in the solution set of equation F(x, , ) = 0 at a point (0, ₀, ₀) X × IR ₊ ² to the bifurcation problem in the solution set of a certain equation in IR ⁿ at a point (0, ₀, ₀) IR ⁿ × IR ₊ ², where n = dim Ker F _x (0, ₀, ₀) and F _x (0, ₀, ₀): X Y is a Fréchet derivative of F with respect to x at (0, ₀, ₀). To solve the bifurcation problem obtained as a result of reduction, we apply homotopy and degree theory.     

Strong approximation of additive functionals

LetX ₁,X ₂,...be a sequence of i.i.d. random variables and putS ₀=0,S _n =X ₁+...+X _n . A strong approximation type result is given forA _N = _i=1 ^N f(S _i ) whereF(x),xR is a real valued function. A similar result is given for ₀ ^t g(B(s))ds. Some weak convergence type implications are also discussed.1991 Mathematics Subject Classification: Primary: 60F15, Secondary: 60J15.Supported by the Hungarian National Foundation for Scientific Research, Grant No. 1905.Supported by an NSERC Canada Grant at Carleton University.Supported by a PSC CUNY Grant No.662349.     

Approximation of Distributions by Parametric Sets

Let P be a probability distribution on a separable metric space (S,d). We study the following problem of approximation of a distribution P by a set from a given class A2 ^S : W(A,P) _S (d(x,A))P(dx)min _AA , where is a nondecreasing function. A special case where A is a parametric class A={A():T} is considered in detail. Our main interest is to obtain convergence results for sequences {A ^* _n }, where A ^* _n is an optimal set for a measure P _n satisfying P _n P, as n.     

Exact convergence rates in strong approximation laws for large increments of partial sums

Summary Consider partial sumsS _n of an i.i.d. sequenceX ₁ X ₂, ..., of centered random variables having a finite moment generating function in a neighborhood of zero. The asymptotic behaviour of is investigated, where 1b _n n denotes an integer sequence such thatb _n /logn asn. In particular, ifb _n =o(log ^p n) asn for somep>1, the exact convergence rate ofU _n /b _{n n} =1 +0 (1) is determined, where _n depends uponb _n and the distribution ofX ₁. In addition, a weak limit law forU _n is derived. Finally, it is shown how strong invariance takes over if b _n (loglogn)²/log³ n=.     

Limit Theorems for the Maximum Terms of a Sequence of Random Variables with Marginal Geometric Distributions

Let X _n1 ^* , ... X _nn ^* be a sequence of n independent random variables which have a geometric distribution with the parameter p _n = 1/n, and M _n ^* = \max\{X _n1 ^* , ... X _nn ^* }. Let Z ₁, Z₂, Z₃, ... be a sequence of independent random variables with the uniform distribution over the set N _n = {1, 2, ... n}. For each j N _n let us denote X _nj = min{k : Z_k = j}, M _n = max{X_n1, ... X_nn}, and let S _n be the 2nd largest among X _n1, X_n2, ... X_nn. Using the methodology of verifying D(u_n) and D'(u_n) mixing conditions we prove herein that the maximum M _n has the same type I limiting distribution as the maximum M _n ^* and estimate the rate of convergence. The limiting bivariate distribution of (S_n, M_n) is also obtained. Let _n, _n N_n, , and T _n = min{M(A_n), M(B_n)}. We determine herein the limiting distribution of random variable T _n in the case _n , _n/_n > 0, as n .     

Multivalued Stochastic Differential Equations: Convergence of a Numerical Scheme

In this paper we show the strong mean square convergence of a numerical scheme for a R ^d -multivalued stochastic differential equation: dX _t +A(X _t )dtb(t,X _t )dt+(t,X _t )dW _t and obtain the rate of convergence O(( log(1/)^1/2) when the diffusion coefficient is bounded. By introducing a discrete Skorokhod problem, we establish L ^p -estimates (p2) for the solutions and prove the convergence by using a deterministic result. Numerical experiments for the rate of convergence are presented.     

Large Deviations of Local Times of Lévy Processes

For X(t) a real-valued symmetric Lévy process, its characteristic function is E(e ^iX(t))=exp(–t()). Assume that is regularly varying at infinity with index 1<2. Let L ^x _t denote the local time of X(t) and L* _t =sup _xR L ^x _t . Estimates are obtained for P(L ⁰ _t y) and P(L* _t y) as y and t fixed.     

Interpolating them-th power ofx at the zeros of then-th Chebyshev-polynomial yields an almost best Chebyshev-approximation

LetX={x ₁,x ₂,..., _n }I=[–1, 1] and . ForfC ₁(I) definef* byf–p _f =f*, wherep _f denotes the interpolation-polynomial off with respect toX. We state some properties of the operatorf f*. In particular, we treat the case whereX consists of the zeros of the Chebyshev polynomialT _n (x) and obtain x ^m –p _x ^m8eE _n–1(x ^m ), whereE _n–1(f) denotes the sup-norm distance fromf to the polynomials of degree less thann. Finally we state a lower estimate forE _n (f) that omits theassumptionf ⁽ⁿ⁺¹⁾>0 in a similar estimate of Meinardus.     

Numerical studies of the Möbius power series

We define the Möbius power series throughf(z)= _n-1 z ⁿ ,g(z)= _n=1 (n)z ⁿ /n where (n) is the usual Möbius function. This paper presents some heuristic estimates describing the behavior off(z) andg(z) when |z| is close to 1 together with representations in terms of elementary functions for real values ofz. Function tables are also given together with zeros and a few other special values.     

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.     

Relative stability and the strong law of large numbers

Summary LetX ₁,X ₂,..., be i.i.d. random variables andS _n=X ₁+X ₂+. +X _n. In this paper we simplify Rogozin's condition forS _n/B _n ±1for someB _n+, which generalises Hinin's condition for relative stability ofS _n. We also consider convergence of subsequences ofS _n/B _n. As an application of our methods, we extend a result of Chow and Robbins to show thatS _n/B _n±1 a.s. for someB _n + if and only if 0<¦EX¦E¦X¦<+ .     

Tate–Vogel Completions of Half-Exact Functors

We provide a general method to construct the Tate–Vogel homology theory for a general half-exact functor with one variable, aiming at a good generalization of Cohen–Macaulay approximations of modules over commutative Gorenstein rings. For a half exact functor F, using the left and right satellites (S _n and S ⁿ ), we define F (X)=lim S _n S ⁿ F(X) and F (X)=lim S _n S ⁿ F(X), and call F and F the Tate–Vogel completions of F. We provide several properties of F and F , and their relations with the G-dimension and the projective dimension of the functor F. A comparison theorem of Tate–Vogel completions with ordinary Tate–Vogel homologies is proved. If F is a half exact functor over the category of R-modules, where R is a commutative Noetherian local ring inspired by Martsinkovsky's works, we can define the invariants (F) and (F) of F. If F=Ext _R ⁱ (M, ), then they coincide with Martsinkovsky's -invariants and Auslander's delta invariants. Our advantage is that we can consider these invariants for any half exact functors. We also compute these invariants for the local cohomology functors.     

The asymptotic distribution of magnitude trimmed sums for distributions in the Feller class

LetX, X ₁,X ₂,... be i.i.d. with common distribution functionF. Rather than study limit behavior of the sum,S _n =X ₁++X _n , under constant normalizations, we consider the sum with ther _n summands largest in magnitude removed from the sumS _n , wherer _n andr _n n ^–10. This is known as an intermediate magnitude trimmed sum. LetF be such that lim inf_t lim inf _t EX ² I(|X|t/)t ² P((|X|>t)>0. The collection of suchF is known as the Feller class, a large class of distributions which includes all domains of attraction (in particular the stable laws). Pruitt(¹³) showed that asymptotic normality always holds for the trimmed sums ifF is in the Feller class and ifF is symmetric. Here, for anyF in the Feller class, we obtain complete results including the form of the possible limit laws and their convergence criteria, thus generalizing Pruitt's result to the asymmetric setting.This paper forms a portion of the author's Ph.D. dissertation under the supervision of Daniel C. Weiner.     

Interacting Brownian particles and the Wigner law

Summary In this paper, we study interacting diffusing particles governed by the stochastic differential equationsdX _j (t)= _n dB _j (t) –D _jØ_n(X ₁,...,X _n)dt,j=1, 2,...,n. Here theB _jare independent Brownian motions in ^d , and Ø _n (X ₁,...,X _n)= _{n ij} V(X _iX _j) + _ni U(X ₁). The potentialV has a singularity at 0 strong enough to keep the particles apart, and the potentialU serves to keep the particles from escaping to infinity. Our interest is in the behaviour as the number of particles increases without limit, which we study through the empirical measure process. We prove tightness of these processes in the case ofd=1,V(x)=–log|x|,U(x)=x ²/2 where it is possible to prove uniqueness of the limiting evolution and deduce that a limiting measure-valued process exists. This process is deterministic, and converges to the Wigner law ast. Some information on the rates of convergence is derived, and the case of a Cauchy initial distribution is analysed completely.Supported by SERC grant number GR/H 00444     

Measure ratio of equidecomposable sets

Let n 2. There are Lebesgue measurable sets A and B in ³ such that (B)/(A)=r and A _n B if and only if 2/n r n/2.     

An improved asymptotic analysis of the expected number of pivot steps required by the simplex algorithm

Leta ₁ ...,a _m be i.i.d. points uniformly on the unit sphere in ⁿ ,m n 3, and letX:= {x ⁿ |a _i ^T x1} be the random polyhedron generated bya ₁, ...,a _m . Furthermore, for linearly independent vectorsu, in ⁿ , letS _u , (X) be the number of shadow vertices ofX inspan(u,). The paper provides an asymptotic expansion of the expectation value¯S _n,m := _in4 ¹ E(S _u, ) for fixedn andm .¯S _n,m equals the expected number of pivot steps that the shadow vertex algorithm — a parametric variant of the simplex algorithm — requires in order to solve linear programming problems of type max u ^T ,xX, if the algorithm will be started with anX-vertex solving the problem max ^T ,x X. Our analysis is closely related to Borgwardt's probabilistic analysis of the simplex algorithm. We obtain a refined asymptotic analysis of the expected number of pivot steps required by the shadow vertex algorithm for uniformly on the sphere distributed data.     

The Higher K-Theory of Real Curves

If X is a smooth curve defined over the real numbers , we show that K _n (X) is the sum of a divisible group and a finite elementary Abelian 2-group when n 2. We determine the torsion subgroup of K _n (X), which is a finite sum of copies of and 2, only depending on the topological invariants of X() and X(), and show that (for n 2) these torsion subgroups are periodic of order 8.