Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.     

A general Hobby-Rice Theorem and cake cutting

LetX be a Borel subset of a separable Banach spaceE. Letμ be a non-atomic,σ-finite, Borel measure onX. LetG ⊆L ₁ (X, Σ,μ) bem-dimensional. Theorem:There is an l ∈ E* and real numbers −∞=x ₀<x ₁<x ₂<…<x _n<x _n+1=∞with n≤m, such that for all g ∈ G,      

Finitary reconstruction of a measure preserving transformation

This paper considers thefinitary reconstruction of an ergodic measure preserving transformationT of a complete separable metric spaceX from a single trajectoryx, Tx, …, or more generally, from a suitable reconstruction sequence x=x ₁,x ₂, … withx _i∈X. Ann-sample reconstruction is a functionT _n: X ⁿ⁺¹ →X; the map (·;x ₁, …,x _n)is treated as an estimate ofT(·) based on then initial elements of x. Given a reference probability measureμ ₀ and constantM>1, functionsT ₁,T ₂, … are defined, and it is shown that for everyμ with 1/M≤dμ/dμ ₀≤M, everyμ-preserving transformationT, and every reconstruction sequence x forT, the estimates (·;x ₁, …,x _nconverge toT in the weak topology. For the family of interval exchange transformations of [0, 1] a simple family of estimates is described and shown to be consistent both pointwise and in the strong topology. However, it is also shown that no finitary estimation scheme is consistent in the strong topology for the family of all ergodic Lebesgue measure preserving transformations of the unit interval, even if x is assumed to be a generic trajectory ofT. Supported in part by NSF Grant DMS-9501926.     

A criterion for the primeness of ideals generated by polynomials with separated variables

Proving primeness of an idealI=〈f ₁, …,f _m〉 in a polynomial ringR=K[X ₁, …,X _n]ofn indeterminates over an algebraically closed fieldK is a difficult task in general. Although there are straightforward algorithms that decide whetherI is prime or not, they are prohibitively lengthy if the number of indeterminates or the degrees of thef _iare large. In this paper we will give an easy criterion for the primeness ofI if thef _iare polynomials with separated variables, i.e. no mixed monomials occur in thef _i. The work on this paper was done while the author was a MINERVA fellow at Tel Aviv University.     

On calculation of moments of ladder heights

Let X,X ₁,X ₂, … be independent identically distributed random variables, F(x) = P{X < x}, S ₀ = 0, and S _n =Σ _i=1 ⁿ X _i . We consider the random variables, ladder heights Z ₊ and Z ₋ that are respectively the first positive sum and the first negative sum in the random walk {S _n }, n = 0, 1, 2, …. We calculate the first three (four in the case EX = 0) moments of random variables Z ₊ and Z ₋ in the qualitatively different cases EX > 0, EX < 0, and EX = 0. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 159–179, April–June, 2006.     

On the range of random walk

Let {S _n , n=0, 1, 2, …} be a random walk (S _n being thenth partial sum of a sequence of independent, identically distributed, random variables) with values inE _d , thed-dimensional integer lattice. Letf _n =Prob {S ₁ ≠ 0, …,S _n −1 ≠ 0,S _n =0 |S ₀=0}. The random walk is said to be transient if and strongly transient if . LetR _n =cardinality of the set {S ₀,S ₁, …,S _n }. It is shown that for a strongly transient random walk with p<1, the distribution of [R _n −np]/σ √n converges to the normal distribution with mean 0 and variance 1 asn tends to infinity, where σ is an appropriate positive constant. The other main result concerns the “capacity” of {S ₀, …,S _n }. For a finite setA inE _d , let C(A=Σ _x∈A ) Prob {S _n ∉A, n≧1 |S ₀=x} be the capacity ofA. A strong law forC{S ₀, …,S _n } is proved for a transient random walk, and some related questions are also considered. This research was partially supported by the National Science Foundation.     

P.I.G-rings and the contractability of primes

LetR=F{x ₁, …, x_k} be a prime affine p.i. ring andS a multiplicative closed set in the center ofR, Z(R). The structure ofG-rings of the formR _s is completely determined. In particular it is proved thatZ(R _s)—the normalization ofZ(R _s) —is a prüfer ring, 1≦k.d(R _s)≦p.i.d(R _s) and the inequalities can be strict. We also obtain a related result concerning the contractability ofq, a prime ideal ofZ(R) fromR. More precisely, letQ be a prime ideal ofR maximal to satisfyQϒZ(R)=q. Then k.dZ(R)/q=k.dR/Q, h(q)=h(Q) andh(q)+k.dZ(R)/q=k.dz(R). The last condition is a necessary butnot sufficient condition for contractability ofq fromR.     

On order preserving contractions

Let (Ω,Σ,μ) be a measure space and letP be an operator onL ₂(Ω,Σ,μ) with ‖P‖≦1,Pf≧0 a.e. wheneverf≧0. If the subspaceK is defined byK={x| ||P ⁿ x||=||P ^*n x||=||x||,n=1,2,...} thenK=L ₂(Ω,Σ₁,μ), where Σ₁ ⊂ Σ and onK the operatorP is “essentially” a measure preserving transformation. Thus the eigenvalues ofP of modulus one, form a group under multiplication. This last result was proved by Rota for finiteμ here finiteness is not assumed) and is a generalization of a theorem of Frobenius and Perron on positive matrices. The research reported in this document has been sponsored in part by Air Force Office of Scientific Research, OAR through the European Office, Aerospace Research, United States Air Force.     

Central polynomials and matrix invariants

LetK be a field, charK=0 andM _n (K) the algebra ofn×n matrices overK. If λ=(λ₁,…,λ _m ) andμ=(μ ₁,…,μ _m ) are partitions ofn ² let wherex ₁,…,x _n ²,y ₁,…,y _n ² are noncommuting indeterminates andS _n ² is the symmetric group of degreen ². The polynomialsF ^{λ, μ} , when evaluated inM _n (K), take central values and we study the problem of classifying those partitions λ,μ for whichF ^{λ, μ} is a central polynomial (not a polynomial identity) forM _n (K). We give a formula that allows us to evaluateF ^{λ, μ} inM(K) in general and we prove that if λ andμ are not both derived in a suitable way from the partition δ=(1, 3,…, 2n−3, 2n−1), thenF ^{λ, μ} is a polynomial identity forM _n (K). As an application, we exhibit a new class of central polynomials forM _n (K). In memory of Shimshon Amitsur Research supported by a grant from MURST of Italy.     

Topological realizations of families of ergodic automorphisms, multitowers and orbit equivalence

We study minimal topological realizations of families of ergodic measure preserving automorphisms (e.m.p.a.'s). Our main result is the following theorem. Theorem: Let {T_p:p∈I} be an arbitrary finite or countable collection of e.m.p.a.'s on nonatomic Lebesgue probability spaces (Y _p v _p ). Let S be a Cantor minimal system such that the cardinality of the set ε _S of all ergodic S-invariant Borel probability measures is at least the cardinality of I. Then for any collection {μ _p :pεI} of distinct measures from ε _S there is a Cantor minimal system S′ in the topological orbit equivalence class of S such that, as a measure preserving system, (S ¹,μ _p ) is isomorphic to T_p for every p∈I. Moreover, S′ can be chosen strongly orbit equivalent to S if and only if all finite topological factors of S are measure-theoretic factors of T_p for all p∈I. This result shows, in particular, that there are no restrictions at all for the topological realizations of countable families of e.m.p.a.'s in Cantor minimal systems. Namely, for any finite or countable collection {T ₁,T₂,…} of e.m.p.a.'s of nonatomic Lebesgue probability spaces, there is a Cantor minimal systemS, whose collection {μ1,μ2…} of ergodic Borel probability measures is in one-to-one correspondence with {T ₁,T₂,…}, and such that (S,μ _i ) is isomorphic toT _i for alli. Furthermore, since realizations are taking place within orbit equivalence classes of a given Cantor minimal system, our results generalize the strong orbit realization theorem and the orbit realization theorem of [18]. Those theorems are now special cases of our result where the collections {T _p}, {T _p }{μ _p } consist of just one element each. Research of I.K. was supported by NSF grant DMS 0140068.     

SOME LIMIT PROPERTIES OF LOCAL TIME FOR RANDOM WALK

Let X,X1,X2 be i. i. d. random variables with EX^2＋δ〈∞ （for some δ〉0）. Consider a one dimensional random walk S={Sn}n≥0, starting from S0 =0. Let ζ＊ （n）=supx∈zζ（x,n）,ζ（x,n） =#{0≤k≤n：[Sk]=x}. A strong approximation of ζ（n） by the local time for Wiener process is presented and the limsup type and liminf-type laws of iterated logarithm of the maximum local time ζ＊（n） are obtained. Furthermore,the precise asymptoties in the law of iterated logarithm of ζ＊（n） is proved.     

Noncommutative classical invariant theory

In this thesis, we consider some aspects ofnoncommutative classical invariant theory, i.e., noncommutative invariants ofthe classical group SL(2, k). We develop asymbolic method for invariants and covariants, and we use the method to compute some invariant algebras. The subspaceĨ _d ^m of the noncommutative invariant algebraĨ _d consisting of homogeneous elements of degreem has the structure of a module over thesymmetric group S _m . We find the explicit decomposition into irreducible modules. As a consequence, we obtain theHilbert series of the commutative classical invariant algebras. TheCayley—Sylvester theorem and theHermite reciprocity law are studied in some detail. We consider a new power series H(Ĩ _d,t) whose coefficients are the number of irreducibleS _m -modules in the decomposition ofĨ _d ^m , and show that it is rational. Finally, we develop some analogues of all this for covariants.     

相关序的进一步研究

Let (X1,X2,…,Xn) and (Y1,Y2,…Yn) be real random vectors with the same marginal distributions,if (X1,X2,…,Xn)≤c(Y1,Y2,…Yn), it is showed in this paper that ∑i=1^n Xi≤cx∑i=1^n Yi and max1≤k≤n∑i=1^k Xi≤icx max1≤k≤n∑i=1^k Yi hold. Based on this fact,a more general comparison theorem is obtained.     

M?bius transform, moment-angle complexes and Halperin–Carlsson conjecture

The motivation for this paper comes from the Halperin–Carlsson conjecture for (real) moment-angle complexes. We first give an algebraic combinatorics formula for the M?bius transform of an abstract simplicial complex K on [m]={1,…,m} in terms of the Betti numbers of the Stanley–Reisner face ring k(K) of K over a field k. We then employ a way of compressing K to provide the lower bound on the sum of those Betti numbers using our formula. Next we consider a class of generalized moment-angle complexes Z_K^{(\mathbb D, \mathbb S)}\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})}, including the moment-angle complex Z_K\mathcal{Z}_{K} and the real moment-angle complex \mathbbRZ_K\mathbb{R}\mathcal {Z}_{K} as special examples. We show that H^*(Z_K^{(\mathbb D, \mathbb S)};k)H^{*}(\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})};\mathbf{k}) has the same graded k-module structure as Tor  ^k[v](k(K),k). Finally we show that the Halperin–Carlsson conjecture holds for Z_K\mathcal{Z}_{K} (resp. \mathbb RZ_K\mathbb{ R}\mathcal{Z}_{K}) under the restriction of the natural T ^m -action on Z_K\mathcal{Z}_{K} (resp. (ℤ₂) ^m -action on \mathbb RZ_K\mathbb{ R}\mathcal{Z}_{K}).     

The additive subgroup generated by a polynomial

SupposeR is a prime ring with the centerZ and the extended centroidC. Letp(x ₁, …,x _n) be a polynomial overC in noncommuting variablesx ₁, …,x _n. LetI be a nonzero ideal ofR andA be the additive subgroup ofRC generated by {p(a ₁, …,a _n):a ₁, …,a _n ∈I}. Then eitherp(x ₁, …,x _n) is central valued orA contains a noncentral Lie ideal ofR except in the only one case whereR is the ring of all 2 × 2 matrices over GF(2), the integers mod 2.     

A finiteness condition for semigroups which generalizes Green-Rees' theorem

LetS be a finitely generated semigroup. ThenS is finite if every finitely generated subgroup ofS is finite and, for some integerm≥1, for everym-tuplex ₁,x ₂,…x _m of elements ofS there exist an integeri: 1≤i≤m and an integer ρ>1 such that:x _i +1…x _m (x ₁ x ₂…x _m )^ρ=x _i +1…x _m x ₁…x _m . The proof of the result is a direct generalisation of the original one by Green and Rees for the casem=1.     

Deformation of certain quadratic algebras and the corresponding quantum semigroups

LetV be a finite-dimensional vector space. Given a decompositionV⊗V=⊕ _i=1,…n I _i , definen quadratic algebrasQ(V, J _(m)) whereJ _(m)=⊕ _i≠m I _i . There is also a quantum semigroupM(V; I ₁, …,I _n ) which acts on all these quadratic algebras. The decomposition determines as well a family of associative subalgebras of End (V ^⊗k ), which we denote byA _k =A _k (I ₁,…,I _n ),k≥2. In the classical case, whenV⊗V decomposes into the symmetric and skewsymmetric tensors,A _k coincides with the image of the representation of the group algebra of the symmetric groupS _k in End(V ^⊗k ). LetI _i,h be deformations of the subspacesI _i . In this paper we give a criteria for flatness of the corresponding deformations of the quadratic algebrasQ(V, J _(m),h ) and the quantum semigroupM(V;I _1,h ,…,I _n,h ). It says that the deformations will be flat if the algebrasA _k (I ₁, …,I _n ) are semisimple and under the deformation their dimension does not change. Usually, the decomposition intoI _i is defined by a given semisimple operatorS onV⊗V, for whichI _i are its eigensubspaces, and the deformationsI _i,h are defined by a deformationS _h ofS. We consider the cases whenS _h is a deformation of Hecke or Birman-Wenzl symmetry, and also the case whenS _h is the Yang-Baxter operator which appears by a representation of the Drinfeld-Jimbo quantum group. Applying the flatness criteria we prove that in all these cases we obtain flat deformations of the quadratic algebras and the corresponding quantum semigroups. Partially supported by a grant from the Israel Science Foundation administered by the Israel Academy of Sciences.     

Ring of invariants of general linear group over local ring \mathbb{Z}_{p^m }

Let \mathbbZ_p^m \mathbb{Z}_{p^m } be the ring of integers modulo p ^m , where p is a prime and m ⩾ 1. The general linear group GL _n ( \mathbbZ_p^m \mathbb{Z}_{p^m } ) acts naturally on the polynomial algebra A _n := \mathbbZ_p^m \mathbb{Z}_{p^m } [x ₁, …, x _n ]. Denote by A_n^{GL₂ (\mathbbZ_pm )} A_n^{GL_2 (\mathbb{Z}_{p^m } )} the corresponding ring of invariants. The purpose of the present paper is to calculate this invariant ring. Our results also generalize the classical Dickson’s theorem.     

Bounding the number of connected components of a real algebraic set

For every polynomial mapf=(f ₁,…,f _k): ℝ ⁿ →ℝ ^k , we consider the number of connected components of its zero set,B(Z _f) and two natural “measures of the complexity off,” that is the triple(n, k, d), d being equal to max(degree off _i), and thek-tuple (Δ₁,...,Δ₄), Δ _k being the Newton polyhedron off _i respectively. Our aim is to boundB(Z _f) by recursive functions of these measures of complexity. In particular, with respect to (n, k, d) we shall improve the well-known Milnor-Thom’s bound μ _d (n)=d(2d−1) ⁿ⁻¹. Considered as a polynomial ind, μ _d (n) has leading coefficient equal to 2 ⁿ⁻¹. We obtain a bound depending onn, d, andk such that ifn is sufficiently larger thank, then it improves μ _d (n) for everyd. In particular, it is asymptotically equal to 1/2(k+1)n ^k−1 dⁿ, ifk is fixed andn tends to infinity. The two bounds are obtained by a similar technique involving a slight modification of Milnor-Thom's argument, Smith's theory, and information about the sum of Betti numbers of complex complete intersections.     

Additive latin transversals

We prove that for every odd primep, everyk≤p and every two subsets A={a ₁, …,a _k } andB={b ₁, …,b _k } of cardinalityk each ofZ _p , there is a permutationπ ∈S _k such that the sumsa _i +b _π(i) (inZ _p ) are pairwise distinct. This partially settles a question of Snevily. The proof is algebraic, and implies several related results as well. Research supported in part by a State of New Jersey grant and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.