Additive operators preserving rank-additivity on symmetry matrix spaces

We characterize the additive operators preserving rank-additivity on symmetry matrix spaces. LetS _n(F) be the space of alln×n symmetry matrices over a fieldF with 2,3 ∈F ^*, thenT is an additive injective operator preserving rank-additivity onS _n(F) if and only if there exists an invertible matrixU∈M _n(F) and an injective field homomorphism ? ofF to itself such thatT(X)=cUX ^?U^T, ?X=(x_ij)∈S_n(F) wherec∈F ^*,X ^?=(?(x _ij)). As applications, we determine the additive operators preserving minus-order onS _n(F) over the fieldF.     

On polynomial approximations to f a (Z)(z-a)−1 with complex a and some applications to certain non-hermitian matrices

We are concerned with the problem 1 $$\mathop {min}\limits_{p \in P_n } \mathop {max}\limits_{z \in [ - 1,1]} |w(z)(f_a (z) - p(z))|,a \in C/[ - 1,1],n = 0 \cdots $$ of best polynomial approximation of degree n to f_a(z)=(z?a)^?1 on the unit interval. Here P_n denotes the class of complex polynomials of degree at most n, and ω belongs to a certain classical family of weight functions. For real a the solution of this approximation problem is known. In this paper, we obtain the best approximations for purely imaginary a. For general a, close approximations to the optimal polynomials are derived by solving the approximation problem expli citly for a certain subclass of P_n. We then use these polynomials to devise an iterative method for the solution of linear systems Ax=b with coefficient matrices of the form A=cI+dT where T=T^H and c, d ∈C. Finally, as a further appication of our results, we derive bounds for the decay rates of the inverses of banded matrices A=cI+dT.     

Følner sequences and Hilbert’s Irreducibility Theorem over Q

Letf(X; T ₁, ...,T _n) be an irreducible polynomial overQ. LetB be the set ofb teZ ⁿ such thatf(X;b) is of lesser degree or reducible overQ. Let ?={F _j}{F _j } _j?1 ^∞ be a Følner sequence inZ ⁿ — that is, a sequence of finite nonempty subsetsF _j ?Z ⁿ such that for eachvteZ ⁿ , $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap (F_j + \upsilon )} \right|}}{{\left| {F_j } \right|}} = 1$ Suppose ? satisfies the extra condition that forW a properQ-subvariety ofP ⁿ ?A ⁿ and ?>0, there is a neighborhoodU ofW(R) in the real topology such that $\mathop {lim sup}\limits_{j \to \infty } \frac{{\left| {F_j \cap U} \right|}}{{\left| {F_j } \right|}}< \varepsilon $ whereZ ⁿ is identified withA ⁿ (Z). We prove $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap B} \right|}}{{\left| {F_j } \right|}} = 0$ .     

Discrete approximating operators on function algebras

We give a new presentation and various extensions of one theorem of Somorjai. For any sequence of operatorsL _n , given byL _n f=∑ _k=1 ⁿ f(z _n,k )l _n,k withz _{n, k} ∈T andl _{n, k} ∈A(T), there exists a functionf∈A(T) such thatL _n f does not converge tof.     

Order-constrained uniform approximations to the exponential based on restricted rationals

LetR _n/m(z∶γ)=P _n(z∶γ)/(1?γz) ^m be a rational approximation to exp (z),z ∈C, of ordern for all real positiveγ. In this paper we show there exists exactly one value ofγ in each of min(n+1,m) interpolation intervals such that the uniform error overR ^? is at a local minimum.     

Über Bruwirsche Reihen mit nicht negativen Koeffizienten

LetK be the space of entire functions $$f(z) = \sum\limits_{n = 0}^\infty {a_n (1 + z/n)^n + a_\infty \exp (z),z \in \mathbb{C}} ,$$ withf(0)=1 and nonnegative coefficientsa _n. Then it is shown thatK is a Choquetsimplex and can be characterized in three ways: by its set of extrem points, by differential inequalities or as closed convex hull of a set of normed polynomials having only negative zeros and with harmonic mean of these zeros less or equal ?n, n being the degree of the polynomial. Additional properties ofK are derived by using the above characterizations.     

Uniform and mean approximation by certain weighted polynomials,with applications

LetW(x):= exp(-{tiQ(x})), where, for example, Q(x) is even and convex onR, and Q(x)/logx → ∞ asx → ∞. A result of Mhaskar and Saff asserts that ifa _n =a _n (W) is the positive root of the equation $$n = ({2 \mathord{\left/ {\vphantom {2 \pi }} \right. \kern-\nulldelimiterspace} \pi })\int_0^1 {{{a_n xQ'(a_n x)} \mathord{\left/ {\vphantom {{a_n xQ'(a_n x)} {\sqrt {1 - x^2 } }}} \right. \kern-\nulldelimiterspace} {\sqrt {1 - x^2 } }}dx,}$$ then, given any polynomialP _n(x) of degree at mostn, the sup norm ofP _n(x)W(a _n x) overR is attained on [-1, 1]. In addition, any sequence of weighted polynomials {p _n (x)W(a _n x)} ₁ ^∞ that is uniformly bounded onR will converge to 0, for ¦x¦>1. In this paper we show that under certain conditions onW, a function g(x) continuous inR can be approximated in the uniform norm by such a sequence {p _n (x)W(a _n x)} ₁ ^∞ if and only if g(x)=0 for ¦x¦? 1. We also prove anL _p analogue for 0W(x)=exp(?|x| ^α ), when α >1. Further applications of our results are upper bounds for Christoffel functions, and asymptotic behavior of the largest zeros of orthogonal polynomials. A final application is an approximation theorem that will be used in a forthcoming proof of Freud's conjecture for |x| ^p exp(?|x| ^α ),α > 0,p > ?1.     

An inequality for polynomials with a prescribed zero

Let p_n(z)=∑_(k-0)~n a_kz~k be a polynomial of degree n such that |p_n(z)|≤M for |z|≤1. It is well.known that for 0≤u相似文献   

Continuous additive functions and difference equations of infinite order

Пустьq∈(1, 2) иL=(q?1)^?1. Дляz∈[0,L] обозначимδ(z) функцию, для которойδ(z)=1, еслиz≧1/q иδ(z)=0, еслиz<1/q. Пустьy(z) определяется из урав ненияz= =δ(z)q ^?1+y(z)q ^?1, и регулярное представление \(\mathop \Sigma \limits_{n = 1}^\infty \varepsilon _n \left( x \right)q^{ - n} \) аргументах определя ется из следующих соотношен ий: $$x = x_0 , \varepsilon _n \left( x \right) = \delta \left( {x_n } \right), x_{n + 1} = y\left( {x_n } \right).$$ ФункцияF: [0,L]→C называе тся аддитивной, если о на представляется в вид е $$F\left( x \right) = \mathop \Sigma \limits_{n = 1}^\infty \varepsilon _n \left( x \right)a_n ,$$ где ε ¦a _n ¦<∞. «Бесконеч ное» представление 1=εl _i q ^?1 числа 1 определяется с ледующим образом: еслие _n (1)=1 для б есконечно многихп, т оl _n =ε _n (1) (n=1, 2, ...); если ? максим альный индекс, для которогоε _s (1)=1, то $$l_{ks + 1} = \left\{ \begin{gathered} \varepsilon _i \left( 1 \right) \left( {k = 0, 1, 2, ...; i = 1, ..., s - 1} \right) \hfill \\ 0 \left( {i = 0; k = 1, 2, ...} \right). \hfill \\ \end{gathered} \right.$$ В более ранней работе, опубликованной в это м журнале, авторы доказали, что а ддитивная функция является неп рерывной на отрезке [0,L] тогда и только тогда, когда ра венство $$a_n = \mathop \Sigma \limits_{i = 1}^\infty l_i a_{n + 1} $$ выполняется для всехn∈N. В настоящей работе ра ссматриваются непре рывные функции для которых в ыполняются дополнительные усло вия видаa _n =O(q ^??n ) (0a _n ≧0. Анализируются их свя зи с корнями функцииG(z)=1 +ε l _i z ⁱ . Доказы вается, что непрерывн ая аддитивная функция и ли вляется линейной, или нигде не дифференцир уема на отрезке [0,L].     

Freud's conjecture and approximation of reciprocals of weights by polynomials

LetW(x) be a function that is nonnegative inR, positive on a set of positive measure, and such that all power moments ofW ² (x) are finite. Let {p _n (W ²;x)} ₀ ^∞ denote the sequence of orthonormal polynomials with respect to the weightW ², and let {α _n } ₁ ^∞ and {β _n } ₁ ^∞ denote the coefficients in the recurrence relation $$xp_n (W^2 ,x) = \alpha _{n + 1} p_{n + 1} (W^2 ,x) + \beta _n p_n (W^2 ,x) + \alpha _n p_{n - 1} (W^2 ,x).$$ We obtain a sufficient condition, involving mean approximation ofW ^?1 by reciprocals of polynomials, for $$\mathop {\lim }\limits_{n \to \infty } {{\alpha _n } \mathord{\left/ {\vphantom {{\alpha _n } {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }} = \tfrac{1}{2}and\mathop {\lim }\limits_{n \to \infty } {{\beta _n } \mathord{\left/ {\vphantom {{\beta _n } {c_{n + 1} }}} \right. \kern-\nulldelimiterspace} {c_{n + 1} }} = 0,$$ wherec _{n 1} ^∞ is a certain increasing sequence of positive numbers. In particular, we obtain a sufficient condition for Freud's conjecture associated with weights onR.     

Sum-free sets in abelian groups

An IP system is a functionn taking finite subsets ofN to a commutative, additive group Ω satisfyingn(α∪β)=n(α)+n(β) whenever α∩β=ø. In an extension of their Szemerédi theorem for finitely many commuting measure preserving transformations, Furstenberg and Katznelson showed that ifS _i ,1≤i≤k, are IP systems into a commutative (possibly infinitely generated) group Ω of measure preserving transformations of a probability space (X, B, μ, andA∈B with μ(A)>0, then for some ø≠α one has μ(? _i=1 ^k S _i({α})A>0). We extend this to so-called FVIP systems, which are polynomial analogs of IP systems, thereby generalizing as well joint work by the author and V. Bergelson concerning special FVIP systems of the formS(α)=T(p(n(α))), wherep:Z ^t →Z ^d is a polynomial vanishing at zero,T is a measure preservingZ ^d action andn is an IP system intoZ ^t . The primary novelty here is potential infinite generation of the underlying group action, however there are new applications inZ ^d as well, for example multiple recurrence along a wide class ofgeneralized polynomials (very roughly, functions built out of regular polynomials by iterated use of the greatest integer function).     

Classification theory for non-elementary classes I: The number of uncountable models ofψ ∈L ω_1, ω. Part A

Assuming that 2^Nn < 2^Nn+1 forn < ω, we prove that everyψ ∈L _{ω_1, ω} has many non-isomorphic models of powerN _n for somen>0or has models in all cardinalities. We can conclude that every such Ψ has at least 2 _N 1 non-isomorphic uncountable models. As for the more vague problem of classification, restricting ourselves to the atomic models of some countableT (we can reduce general cases to this) we find a cutting line named “excellent”. Excellent classes are well understood and are parallel to totally transcendental theories, have models in all cardinals, have the amalgamation property, and satisfy the Los conjecture. For non-excellent classes we have a non-structure theorem, e.g., if they have an uncountable model then they have many non-isomorphic ones in someN _n (provided {ie212-7}).     

On holomorphic isometric embeddings of the unit n-ball into products of two unit m-balls

We study holomorphic isometric embeddings of the complex unit n-ball into products of two complex unit m-balls with respect to their Bergman metrics up to normalization constants (the isometric constant). There are two trivial holomorphic isometric embeddings for m ?? n, given by F ₁(z)?=?(0, I _n;m (z)) with the isometric constant equal to (m?+?1)/(n?+?1) and F ₂(z)?=?(I _n;m (z), I _n;m (z)) with the isometric constant equal to 2(m?+?1)/(n?+?1). Here ${I_{n;m}:\mathbb{C}^n \longrightarrow \mathbb{C}^m}$ is the canonical embedding. We prove that when m < 2n, these are the only holomorphic isometric embeddings up to unitary transformations.     

Divergence of diagonal Padé approximants and the asymptotic behavior of orthogonal polynomials associated with nonpositive measures

In this paper we prove the existence of real- and complex-valued measuresμ on the interval [?1,1] with the property that the diagonal Padé approximants [n/n],n=1,2,..., to the functionf(z)=∫dμ(x)/(x?z) neither converge at any fixed pointz∈C～[?1,1] nor converge in capacity on any open (nonempty) setS inC～[?1,1]. This result is derived from a theorem on the asymptotic behavior of orthogonal polynomials. It will be shown that it is possible to construct measuresμ. on [?1,1] such that for any arbitrarily prescribed asymptotic behavior there exist subsequences of the associated orthogonal polynomialsQ _n that have this behavior.     

On the radii of convexity and starlikeness for some special classes of analytic functions in a disk

We introduce the class O _α, 0≤α≤1, of functions w=?(z), ?(0)=0, ?′(0)=0,..., ? ₍₀₎ ^(n?1) =0, f ⁽ⁿ⁾(0)=(n-l)! analytic in the disk |z|<1 and satisfying the condition $$\operatorname{Re} \left( {\frac{{1 - 2z^n \cos \Theta + z^{2n} }}{{z^{n - 1} }}f'(z)} \right) > \alpha , 0 \leqslant \Theta \leqslant \pi , n = 1,2,3,... .$$ We establish the radius of convexity in the class Oα and the radius of starlikeness in the class Uα of functions σ(z)=z?′(z), ?(z)?O _α.     

Interpolation with lagrange polynomials a simple proof of Markov inequality and some of its generalizations

The following theorem is provedTheorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying inthe interval[-1,1] and△'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}.If polynomial pP_n satisfies the inequalitythen for each k=1,n and any x[-1,1]its k-th derivative satisfies the inequality丨p~(k)(x)丨≤max{丨q~((k))(x)丨,丨1/k(x~2-1)q~(k+1)(x)+xq~((k))(x)丨}.This estimate leads to the Markov inequality for the higher order derivatives ofpolynomials if we set q=T_n,where Tn is Chebyshev polynomial least deviated from zero.Some other results are established which gives evidence to the conjecture that under theconditions of Theorem 1 the inequality ‖p~((k))‖≤‖q~(k)‖holds.     

Siegel’s Lemma and Sum-Distinct Sets

Let L(x)=a ₁ x ₁+a ₂ x ₂+???+a _n x _n , n≥2, be a linear form with integer coefficients a ₁,a ₂,…,a _n which are not all zero. A basic problem is to determine nonzero integer vectors x such that L(x)=0, and the maximum norm ‖x‖ is relatively small compared with the size of the coefficients a ₁,a ₂,…,a _n . The main result of this paper asserts that there exist linearly independent vectors x ₁,…,x _n?1∈? ⁿ such that L(x _i )=0, i=1,…,n?1, and $$\|{\mathbf{x}}_{1}\|\cdots\|{\mathbf{x}}_{n-1}\|<\frac{\|{\mathbf{a}}\|}{\sigma_{n}},$$ where a=(a ₁,a ₂,…,a _n ) and $$\sigma_{n}=\frac{2}{\pi}\int_{0}^{\infty}\left(\frac{\sin t}{t}\right)^{n}\,dt.$$ This result also implies a new lower bound on the greatest element of a sum-distinct set of positive integers (Erdös–Moser problem). The main tools are the Minkowski theorem on successive minima and the Busemann theorem from convex geometry.     

Self-Affine Sets and Graph-Directed Systems

Abstract. A self-affine set in R ⁿ is a compact set T with A(T)= ∪ _{d∈ D} (T+d) where A is an expanding n× n matrix with integer entries and D ={d ₁ , d ₂ ,···, d _N } ? Z ⁿ is an N -digit set. For the case N = | det(A)| the set T has been studied in great detail in the context of self-affine tiles. Our main interest in this paper is to consider the case N > | det(A)| , but the theorems and proofs apply to all the N . The self-affine sets arise naturally in fractal geometry and, moreover, they are the support of the scaling functions in wavelet theory. The main difficulty in studying such sets is that the pieces T+d, d∈ D, overlap and it is harder to trace the iteration. For this we construct a new graph-directed system to determine whether such a set T will have a nonvoid interior, and to use the system to calculate the dimension of T or its boundary (if T ^o ). By using this setup we also show that the Lebesgue measure of such T is a rational number, in contrast to the case where, for a self-affine tile, it is an integer.