Exponential generating functions and complexity of Lie varieties

Suppose that % MathType!End!2!1! is a variety of Lie algebras, and letc _n( % MathType!End!2!1!) be the dimension of the linear span of all multilinear words onn distinct letters in the free algebraF( % MathType!End!2!1!,X) of the variety % MathType!End!2!1!. We consider an exponential generating function % MathType!End!2!1!, called the complexity function. The complexity function is an entire function of a complex variable provided the variety of Lie algebras is nontrivial. In this paper we introduce the notion of complexity for Lie varieties in terms of the growth of complexity functions; also we describe what the complexity means for the codimension growth of the variety. Our main goal is to specify the complexity of a product of two Lie varieties in terms of the complexities of multiplicands. The main observation here is thatC( % MathType!End!2!1!),z) behaves like a composition of three functionsC( % MathType!End!2!1!),z), exp(z), andC( % MathType!End!2!1!),z). Partially supported by grant RFFI 96-01-00146; the author is grateful to the University of Bielefeld for hospitality, where he was DAAD-fellow.     

Exact asymptotic behaviour of the codimensions of some P.I. algebras

Letc _n (A) denote the codimensions of a P.I. algebraA, and assumec _n (A) has a polynomial growth: . Then, necessarily,q∈ℚ [D3]. If 1∈A, we show that , wheree=2.71…. In the non-unitary case, for any 0<q∈ℚ, we constructA, with a suitablek, such that . In memory of S. A. Amitsur, our teacher and friend Partially supported by Grant MM404/94 of Ministry of Education and Science, Bulgaria and by a Bulgarian-American Grant of NSF. Partially supported by NSF grant DMS-9101488.     

On directional entropy functions

Given aZ ²-process, the measure theoretic directional entropy function,h( % MathType!End!2!1!), is defined on % MathType!End!2!1!. We relate the directional entropy of aZ ²-process to itsR ² suspension. We find a sufficient condition for the continuity of directional entropy function. In particular, this shows that the directional entropy is continuous for aZ ²-action generated by a cellular automaton; this finally answers a question of Milnor [Mil]. We show that the unit vectors whose directional entropy is zero form aG _δ subset ofS ¹. We study examples to investigate some properties of directional entropy functions. This research is supported in part by BSRI and KOSEF 95-0701-03-3.     

Complexities of finite families of polynomials,Weyl systems,and constructions in combinatorial number theory

We introduce two notions of complexity of a system of polynomials p ₁,..., p _r ∈ ℤ[n] and apply them to characterize the limits of the expressions of the form where T is a skew-product transformation of a torus and are measurable sets. The dynamical results obtained allow us to construct subsets of integers with specific combinatorial properties related to the polynomial Szemerédi theorem. Bergelson and Leibman were supported by NSF grants DMS-0345350 and DMS-0600042.     

Self-similar solutions for a convection-diffusion equation with absorption inR N

We prove the existence of a positive and smooth solution for the following semilinear elliptic problem: % MathType!End!2!1! for anya∈R ^N , 1<p<1+2/N andq=(p+1)/2. This solution decays exponentially as |x|→+∞. Moreover, if |a| is sufficiently small, this positive and rapidly decaying solution is unique. The existence of a positive, self-similar solution % MathType!End!2!1! follows for the following convection-diffusion equation with absorption: % MathType!End!2!1!. It is also a very singular solution. This solution decays as |x|→+∞ for anyt>0 fixed. Because of the nonvariational nature of the elliptic problem, a fixed point method is used for proving the existence result. The uniqueness is proved applying the Implicit Function Theorem. The work of the first author has been partially supported by Grant 1273/00003/88 of the University of the Basque Country. The work of the second author has been supported by Grant PB 86-0112-C02-00 of the Dirección General de Investigación Científica y Técnica.     

Forbidden paths and cycles in ordered graphs and matrices

Assume % MathType!End!2!1! and let Ω⊂R ^N(N≥4) be a smooth bounded domain, 0∈Ω. We study the semilinear elliptic problem: % MathType!End!2!1!. By investigating the effect of the coefficientQ, we establish the existence of nontrivial solutions for any λ>0 and multiple positive solutions with λ,μ>0 small.     

Quadratically hyponormal weighted shifts with two equal weights

We present an extensive analysis ofpositively quadratically hyponormal weighted shiftsW with ₀ = ₁ = 1. Our main result states that such weighted shifts abound! Specifically, by focusing on recursively generated weighted shifts of the form, we establish that the planar set is positively quadratically hyponormal} is a closed convex, set with nonempty interior. In addition, we are able to describe in detail the boundary of Research partially supported by NSF grants DMS-9401455 and DMS-9800931Research partially supported by KOSEF grant 971-0102-006-2 and by TGRC-KOSEF     

Branch rings, thinned rings, tree enveloping rings

We develop the theory of “branch algebras”, which are infinite-dimensional associative algebras that are isomorphic, up to taking subrings of finite codimension, to a matrix ring over themselves. The main examples come from groups acting on trees. In particular, for every field % MathType!End!2!1! we contruct a % MathType!End!2!1! which

Asymptotics for Amitsur's Capelli-type polynomials and verbally prime PI-algebras

We consider associativePI-algebras over a field of characteristic zero. The main goal of the paper is to prove that the codimensions of a verbally prime algebra [11] are asymptotically equal to the codimensions of theT-ideal generated by some Amitsur's Capelli-type polynomialsE _M,L ^* [1]. We recall that two sequencesa _n,b _nare asymptotically equal, and we writea _n ≃b _n,if and only if lim _n→∞(a _n/b _n)=1.In this paper we prove that % MathType!End!2!1!, whereG is the Grassmann algebra. These results extend to all verbally primePI-algebras a theorem of A. Giambruno and M. Zaicev [9] giving the asymptotic equality % MathType!End!2!1! between the codimensions of the matrix algebraM _k(F) and the Capelli polynomials. The second author is partially supported by grants RFFI 04-01-00739a, E02-2.0-26.     

Improved complexity results on solving real-number linear feasibility problems

We present complexity results on solving real-number standard linear programs LP(A,b,c), where the constraint matrix the right-hand-side vector and the objective coefficient vector are real. In particular, we present a two-layered interior-point method and show that LP(A,b,0), i.e., the linear feasibility problem A x = b and x ≥ 0, can be solved in in O(n ^2.5 c(A)) interior-point method iterations. Here 0 is the vector of all zeros and c(A) is the condition measure of matrix A defined in [25]. This complexity iteration bound is reduced by a factor n from that for general LP(A, b, c) in [25]. We also prove that the iteration bound will be further reduced to O(n ^1.5 c(A)) for LP(A, 0, 0), i.e., for the homogeneous linear feasibility problem. These results are surprising since the classical view has been that linear feasibility would be as hard as linear programming. This author was supported in part by NSF Grants DMS-9703490 and DMS-0306611     

Canonical maps to twisted rings

If A is a strongly noetherian graded algebra generated in degree one, then there is a canonically constructed graded ring homomorphism from A to a twisted homogeneous coordinate ring , which is surjective in large degree. This result is a key step in the study of projectively simple rings. The proof relies on some results concerning the growth of graded rings which are of independent interest. D. Rogalski was partially supported by NSF grant DMS-0202479. J. J. Zhang was partially supported by NSF grant DMS-0245420 and Leverhulme Research Interchange Grant F/00158/X (UK).     

Strong monotonicity of operator functions

LetA, B be bounded selfadjoint operators on a Hilbert space. We will give a formula to get the maximum subspace such that is invariant forA andB, and . We will use this to show strong monotonicity or strong convexity of operator functions. We will see that when 0≤A≤B, andB−A is of finite rank,A ^t ≤B ^t for somet>1 if and only if the null space ofB−A is invariant forA.     

Miura opers and critical points of master functions

Critical points of a master function associated to a simple Lie algebra come in families called the populations [11]. We prove that a population is isomorphic to the flag variety of the Langlands dual Lie algebra . The proof is based on the correspondence between critical points and differential operators called the Miura opers. For a Miura oper D, associated with a critical point of a population, we show that all solutions of the differential equation DY=0 can be written explicitly in terms of critical points composing the population. Supported in part by NSF grant DMS-0140460 Supported in part by NSF grant DMS-0244579     

Analytic Hausdorff gaps II: The density zero ideal

We prove two results about the quotient over the asymptotic density zero ideal. First, it is forcing equivalent to % MathType!End!2!1!, where % MathType!End!2!1! is the homogeneous probability measure algebra of characterc. Second, if it has analytic Hausdorff gaps, then they look considerably different from proviously known gaps of this form. Partially supported by NSERC.     

On the Classification of Cartan Actions

We study higher-rank Cartan actions on compact manifolds preserving an ergodic measure with full support. In particular, we classify actions by with k ≥ 3 whose one-parameter groups act transitively as well as nondegenerate totally nonsymplectic -actions for k ≥ 3. The first author is supported in part by NSF grants DMS-0140513. The second author is supported in part by NSF grant DMS-0203735. Received: July 2005 Revision: March 2006 Accepted: March 2006     

Singly generatedC *-algebras

We consider aC ^*-algebraA generated byk self-adjoint elements. We prove that, for , the algebraM _n (A) is singly generated, i.e., generated by one non-self-adjoint element. We present an example of algebraA for which the property thatM _n (A) is singly generated implies the relation . Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 8, pp. 1136–1141, August, 1999.     

Almost orthogonal submatrices of an orthogonal matrix

Lett≥1 and letn, M be natural numbers,n<M. Leta=(a _i,j ) be ann xM matrix whose rows are orthonormal. Suppose that the ℓ₂-norms of the columns ofA are uniformly bounded. Namely, for allj Using majorizing measure estimates we prove that for every ε>0 there exists, a setI ⊃ {1,…,M} of cardinality at most such that the matrix , whereA _I =(a _i,j ) _j∈I , acts as a (1+ε)-isomorphism from ℓ ₂ ⁿ into . Research supported in part by a grant of the US-Israel BSF. Part of this research was performed when the author held a postdoctoral position at MSRI. Research at MSRI was supported in part by NSF grant DMS-9022140.     

Harmonic measure of curves in the disk

A powerful tool for studying the growth of analytic and harmonic functions is Hall's Lemma, which states that there is a constantC>0 so that the harmonic measure of a subsetE of the closed unit disk evaluated at 0 satisfies whereE _rad is the radial projection ofE onto . FitzGerald, Rodin and Warschawski proved that ifE is a continuum in whose radial projection has length at most π then (*) is true withC=1, and they asked how large the length, |E _rad|, can be in order for their result to be valid. We prove that (*) holds withC=1 for every continuum satisfying and θc cannot be replaced by a larger number. Fuchs asked for the largest constantC so that (*) holds for allE. We show that for every continuum , (*) holds withC=C _2π≅.977126698498665669…, whereC _2π is the harmonic measure of the two long sides of a 3∶1 rectangle evaluated at the center. There are Jordan curves for which equality holds in (*) withC=C _2π. The authors are supported in part by NSF grants DMS-9302823 and DMS-9401027, and while at MSRI by NSF grant DMS-9022140.     

Three-Color Ramsey Numbers For Paths

We prove—for sufficiently large n—the following conjecture of Faudree and Schelp:

Asymptotics of degrees of someS n-sub regular representations

The numbers % MathType!End!2!1!, λ ⊢n appear in the enumeration of various objects, as well as coefficients inS _nrepresentations associated with products of higher commutators. We study their asymptotics asn→∞ and show that if (λ₁, λ₂, …)≈(α ₁,α ₂, …)n, if (λ′₁, λ′₂, …)≈(β ₁,β ₂, …)n and ifγ=1− Σ _k⩽1(α _k⩽1+β _k⩽1), then % MathType!End!2!1!. Work partially supported by N.S.F. Grant No. DMS 94-01197.