Weighted Maximum Over Minimum Modulus of Polynomials, Applied to Ray Sequences of Padé Approximants

Let a≥ 0 , ɛ >0 . We use potential theory to obtain a sharp lower bound for the linear Lebesgue measure of the set Here P is an arbitrary polynomial of degree ≤ n . We then apply this to diagonal and ray Padé sequences for functions analytic (or meromorphic) in the unit ball. For example, we show that the diagonal \left{ [n/n]\right} _n=1 ^∞ sequence provides good approximation on almost one-eighth of the circles centre 0 , and the \left{ [2n/n]\right} _n=1 ^∞ sequence on almost one-quarter of such circles. July 18, 2000. Date revised: . Date accepted: April 19, 2001.     

Approximation by p -Faber polynomials in the weighted Smirnov class Ep( G,ω ) and the Bieberbach polynomials

Let G\subset C be a finite domain with a regular Jordan boundary L . In this work, the approximation properties of a p -Faber polynomial series of functions in the weighted Smirnov class E ^p (G,ω) are studied and the rate of polynomial approximation, for f∈ E ^p ( G,ω) by the weighted integral modulus of continuity, is estimated. Some application of this result to the uniform convergence of the Bieberbach polynomials π _n in a closed domain \overline G with a smooth boundary L is given. February 25, 1999. Date revised: October 20, 1999. Date accepted: May 26, 2000.     

Rates of Best Uniform Rational Approximation of Analytic Functions by Ray Sequences of Rational Functions

In this paper, problems related to the approximation of a holomorphic function f on a compact subset E of the complex plane C by rational functions from the class of all rational functions of order (n,m) are considered. Let ρ _n,m = ρ _n,m (f;E) be the distance of f in the uniform metric on E from the class . We obtain results characterizing the rate of convergence to zero of the sequence of the best rational approximation { ρ _n,m(n) } _n=0 ^∞ , m(n)/n → θ ∈ (0,1] as n → ∞ . In particular, we give an upper estimate for the liminf _{n →∞} ρ _n,m(n) ^1/(n+m(n)) in terms of the solution to a certain minimum energy problem with respect to the logarithmic potential. The proofs of the results obtained are based on the methods of the theory of Hankel operators. June 16, 1997. Date revised: December 1, 1997. Date accepted: December 1, 1997. Communicated by Ronald A. DeVore.     

On projectively normal embeddings of generalk-gonal curves

Fix integersg, k andt witht>0,k≥3 andtk<g/2−1. LetX be a generalk-gonal curve of genusg andR∈Pic ^k (X) the uniqueg _k ¹ onX. SetL:=K _X⊗(R ^*)^⊗t.L is very ample. Leth _L:X→P(H ⁰(X, L)^*) be the associated embedding. Here we prove thath _L(X) is projectively normal. Ifk≥4 andtk<g/2−2 the curveh _L(X) is scheme-theoretically cut out by quadrics. The author was partially supported by MURST and GNSAGA of CNR (Italy).     

On the transversal Helly numbers of disjoint and overlapping disks

A family of disks is said to have the property T(k) if any k members of the family have a common line transversal. We call a family of unit diameter disks t-disjoint if the distances between the centers are greater than t. We consider for each natural number k≧ 3 the infimum t_k of the distances t for which any finite family of t-disjoint unit diameter disks with the property T(k) has a line transversal. We determine exact values of t₃ and t₄, and give general lower and upper bounds on the sequence t_k, showing that t_k = O(1/k) as k → ∞. In honour of Helge Tverberg’s seventieth birthday Received: 9 June 2005     

Piecewise Monotone Pointwise Approximation

Let r, k, s be three integers such that , or We prove the following: Proposition. Let Y:={y _i } _i=1 ^s be a fixed collection of distinct points y _i ∈ (-1,1) and Π (x):= (x-y ₁ ). ... .(x-y _s ). Let I:=[-1,1]. If f ∈ C ^(r) (I) and f'(x)Π(x) ≥ 0, x ∈ I, then for each integer n ≥ k+r-1 there is an algebraic polynomial P _n =P _n (x) of degree ≤ n such that P _n '(x) Π (x) ≥ 0 and $$ \vert f(x)-P_n(x) \vert \le B\left(\frac{1}{n^2}+\frac{1}{n}\sqrt{1-x^2}\right)^r \omega_k \left(f^{(r)};\frac{1}{n^2}+\frac{1}{n}\sqrt{1-x^2}\right) \legno{(1)}$$ for all x∈ I, where ω _k (f ^(r) ;t) is the modulus of smoothness of the k -th order of the function f ^(r) and B is a constant depending only on r , k , and Y. If s=1, the constant B does not depend on Y except in the case (r=1, k=3). In addition it is shown that (1) does not hold for r=1, k>3. March 20, 1995. Dates revised: March 11, 1996; December 20, 1996; and August 7, 1997.     

On the law of the iterated logarithm for subsequences for a stable subordinator

Let {X(t): 0≤t<∞) be a stable subordinator with α∈(0,1). For increasing sequences t_k we give normalizing constants a_k such thatliminf _k→∞ a _k ⁻¹ X(t_k) is a.s constant. We also derive a.s. upper bounds. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models Part I, Eger, Hungary, 1994.     

Best Meromorphic Approximation of Markov Functions on the Unit Circle

Let E \subset(-1,1) be a compact set, let μ be a positive Borel measure with support \supp μ =E , and let H _p (G), 1≤ p ≤∈fty, be the Hardy space of analytic functions on the open unit disk G with circumference Γ={z \colon |z|=1} . Let Δ _n,p be the error in best approximation of the Markov function \frac{1}{2π i} ∈t_E \frac{d μ(x)}{z-x} in the space L _p (Γ) by meromorphic functions that can be represented in the form h=P/Q , where P ∈ H _p (G), Q is a polynomial of degree at most n , Q\not \equiv 0 . We investigate the rate of decrease of Δ _n,p , 1≤ p ≤∈fty , and its connection with n -widths. The convergence of the best meromorphic approximants and the limiting distribution of poles of the best approximants are described in the case when 1<p≤∈fty and the measure μ with support E=[a,b] satisfies the Szegő condition ∈t_a^b \frac{\log(d μ/ d x)}{\sqrt{(x-a)(b-x)}} dx >- ∈fty. July 27, 2000. Final version received: May 19, 2001.     

Roots of the affine Cremona group

Let k ^[n] = k[x ₁,…, x _n ] be the polynomial algebra in n variables and let \mathbbAⁿ = \textSpec  \boldk^{[ n ]} {\mathbb{A}^n} = {\text{Spec}}\;{{\bold{k}}^{\left[ n \right]}} . In this note we show that the root vectors of \textAu\textt^*( \mathbbAⁿ ) {\text{Au}}{{\text{t}}^*}\left( {{\mathbb{A}^n}} \right) , the subgroup of volume preserving automorphisms in the affine Cremona group \textAut( \mathbbAⁿ ) {\text{Aut}}\left( {{\mathbb{A}^n}} \right) , with respect to the diagonal torus are exactly the locally nilpotent derivations x ^α (∂/∂x _i ), where x ^α is any monomial not depending on x _i . This answers a question posed by Popov.     

On tense modules for extended algebras over rational fields

Let k(x) be the field of fractions of the polynomial algebra k[x] over the field k. We prove that, for an arbitrary finite dimensional k-algebra Λ, any finitely generated Λ ⊗_k k(x)-module M such that its minimal projective presentation admits no non-trivial selfextension is of the form M ≅ N^k(x), for some finitely generated Λ-module N. Some consequences are derived for tilting modules over the rational algebra Λ ⊗_k k(x) and for some generic modules for Λ. Received: 24 November 2003; revised: 11 February 2005     

On Stable Local Bases for Bivariate Polynomial Spline Spaces

Stable locally supported bases are constructed for the spaces \cal S _d ^r (\triangle) of polynomial splines of degree d≥ 3r+2 and smoothness r defined on triangulations \triangle , as well as for various superspline subspaces. In addition, we show that for r≥ 1 , in general, it is impossible to construct bases which are simultaneously stable and locally linearly independent. February 2, 2000. Date revised: November 27, 2000. Date accepted: March 7, 2001.     

On the sum of a prime and a k-th power of prime in short intervals

Let H_k\mathcal{H}_{k} denote the set {n∣2|n, n\not o 1 (mod p)n\not\equiv 1\ (\mathrm{mod}\ p) ∀ p>2 with p−1|k}. We prove that when X^{\frac1120(1-\frac12k) +e}\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{2k}\right) +\varepsilon}\leqq H\leqq X, almost all integers n ? \allowbreak H_k ?(X, X+H]n\in\allowbreak {\mathcal{H}_{k} \cap (X, X+H]} can be represented as the sum of a prime and a k-th power of prime for k≧3. Moreover, when X^{\frac1120(1-\frac1k) +e}\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{k}\right) +\varepsilon}\leqq H\leqq X, almost all integers n∈(X,X+H] can be represented as the sum of a prime and a k-th power of integer for k≧3.     

Asymptotic Distribution of Nodes for Near-Optimal Polynomial Interpolation on Certain Curves in R 2

Let E\subset \Bbb R ^s be compact and let d _n ^E denote the dimension of the space of polynomials of degree at most n in s variables restricted to E . We introduce the notion of an asymptotic interpolation measure (AIM). Such a measure, if it exists , describes the asymptotic behavior of any scheme τ _n ={ \bf x _k,n } _k=1 ^dnE , n=1,2,\ldots , of nodes for multivariate polynomial interpolation for which the norms of the corresponding interpolation operators do not grow geometrically large with n . We demonstrate the existence of AIMs for the finite union of compact subsets of certain algebraic curves in R ² . It turns out that the theory of logarithmic potentials with external fields plays a useful role in the investigation. Furthermore, for the sets mentioned above, we give a computationally simple construction for ``good' interpolation schemes. November 9, 2000. Date revised: August 4, 2001. Date accepted: September 14, 2001.     

Regressions and monotone chains: A ramsey-type extremal problem for partial orders

Aregression is a functiong from a partially ordered set to itself such thatg(x)≦x for allz. Amonotone k-chain is a chain ofk elementsx ₁<x ₂ <...<x _k such thatg(x ₁)≦g(x ₂)≦...≦g(x _k ). If a partial order has sufficiently many elements compared to the size of its largest antichain, every regression on it will have a monotone (k + 1)-chain. Fixingw, letf(w, k) be the smallest number such that every regression on every partial order with size leastf(w, k) but no antichain larger thanw has a monotone (k + 1)-chain. We show thatf(w, k)=(w+1) ^k . Dedicated to Paul Erdős on his seventieth birthday Research supported in part by the National Science Foundation under ISP-80-11451.     

The chromatic number of the product of two ℵ1-chromatic graphs can be countable

We prove (in ZFC) that for every infinite cardinal ϰ there are two graphsG ₀,G ₁ with χ(G ₀)=χ(G ₁)=ϰ⁺ and χ(G ₀×G ₁)=ϰ. We also prove a result from the other direction. If χ(G ₀)≧≧ℵ₀ and χ(G ₁)=k<ω, then χ(G ₀×G ₁)=k.     

The signed <Emphasis Type="Italic">k</Emphasis>-domination number of directed graphs

Let k ≥ 1 be an integer, and let D = (V; A) be a finite simple digraph, for which d _D ⁻ ≥ k − 1 for all v ɛ V. A function f: V → {−1; 1} is called a signed k-dominating function (SkDF) if f(N ⁻[v]) ≥ k for each vertex v ɛ V. The weight w(f) of f is defined by $ \sum\nolimits_{v \in V} {f(v)} $ \sum\nolimits_{v \in V} {f(v)} . The signed k-domination number for a digraph D is γ _kS (D) = min {w(f|f) is an SkDF of D. In this paper, we initiate the study of signed k-domination in digraphs. In particular, we present some sharp lower bounds for γ _kS (D) in terms of the order, the maximum and minimum outdegree and indegree, and the chromatic number. Some of our results are extensions of well-known lower bounds of the classical signed domination numbers of graphs and digraphs.     

Automorphism group and representation of a twisted multi-loop algebra

Let G be the complexification of the real Lie algebra so（3） and A = C[t1^±1, t2^±1] be the Lau-ent polynomial algebra with commuting variables. Let L：（t1, t2, 1） = G c .A be the twisted multi-loop Lie algebra. Recently we have studied the universal central extension, derivations and its vertex operator representations. In the present paper we study the automorphism group and bosonic representations ofL（t1, t2, 1）.     

On the Best Approximation by Ridge Functions in the Uniform Norm

We consider the best approximation of some function classes by the manifold M _n consisting of sums of n arbitrary ridge functions. It is proved that the deviation of the Sobolev class W _p ^r,d from the manifold M _n in the space L _q for any 2≤ q≤ p≤∈fty behaves asymptotically as n ^-r/(d-1) . In particular, we obtain this asymptotic estimate for the uniform norm p=q=∈fty . January 10, 2000. Date revised: March 1, 2001. Date accepted: March 12, 2001.     

Exponential Convergence for a System of Conuclear Space-Valued Diffusions with Mean-Field Interaction

We consider an interacting system of n diffusion processes X ⁿ _j (t): t∈[0,1] , j=1,2,. . ., n , taking values in a conuclear space Φ' . Let ζ ⁿ _t =(1/n)Σ ⁿ _j=1 δ _Xnj(t) be the empirical process. It has been proved that ζ ⁿ , as n→∞ , converges to a deterministic measure-valued process which is the unique solution of a nonlinear differential equation. In this paper we show that, under suitable conditions, ζ ⁿ converges to ζ at an exponential rate. Accepted 20 October 1997     

Precise rates in the law of the iterated logarithm under dependence assumptions

Let {X, X1, X2,...} be a strictly stationaryφ-mixing sequence which satisfies EX = 0,EX^2（log2{X}）^2〈∞and φ（n）=O（1/log n）^Tfor some T〉2.Let Sn=∑k=1^nXk and an=O（√n/（log2n）^γ for some γ〉1/2.We prove that limε→√2√ε^2-2∑n=3^∞1/nP（｜Sn｜≥ε√ESn^2log2n＋an）=√2.The results of Gut and Spataru （2000） are special cases of ours.