Cp

The spacec _p is the class of operators on a Hilbert space for which thec _p norm |T| _p =[trace(T*T) ^p/2]^1/p is finite. We prove many of the known results concerningc _p in an elementary fashion, together with the result (new for 1<p<2) thatc _p is as uniformly convex a Banach space asl _p. In spite of the remarkable parallel of norm inequalities in the spacesc _p andl _p, we show thatp ≠ 2, noc _p built on an infinite dimensional Hilbert space is equivalent to any subspace of anyl _p orL _p space. The author was supported by National Science Foundation Grant GP-5707.     

A stochastic Ramsey theorem

We establish a stochastic extension of Ramsey's theorem. Any Markov chain generates a filtration relative to which one may define a notion of stopping times. A stochastic colouring is any k-valued (k<∞) colour function defined on all pairs consisting of a bounded stopping time and a finite partial history of the chain truncated before this stopping time. For any bounded stopping time θ and any infinite history ω of the Markov chain, let ω|θ denote the finite partial history up to and including the time θ(ω). Given k=2, for every ?>0, we prove that there is an increasing sequence θ₁<θ₂<? of bounded stopping times having the property that, with probability greater than 1−?, the history ω is such that the values assigned to all pairs (ω|θi,θj), with i<j, are the same. Just as with the classical Ramsey theorem, we also obtain an analogous finitary stochastic Ramsey theorem. Furthermore, with appropriate finiteness assumptions, the time one must wait for the last stopping time (in the finitary case) is uniformly bounded, independently of the probability transitions. We generalise the results to any finite number k of colours.     

Primarite deL p (l r , 1<p,r<∞

In this article we show that if 1<p, r<∞, the spaceL _p (l _r ) is primary. If we let (h _k ) be the Haar system inL _p and (e _i ) the usual base ofl _r , we give sufficient conditions on a subsequence of (h _k ⊗e _i ) _{k ≧1,i≧1} for it to generate a space isomorphic toL _p (l _r ). We deduce the primarity ofL _p (l _r ).      

Abelian groups admitting a Fréchet-Urysohn pseudocompact topological group topology

We show that every Abelian group G with r₀(G)=|G|=|G|^ω admits a pseudocompact Hausdorff topological group topology T such that the space (G,T) is Fréchet-Urysohn. We also show that a bounded torsion Abelian group G of exponent n admits a pseudocompact Hausdorff topological group topology making G a Fréchet-Urysohn space if for every prime divisor p of n and every integer k≥0, the Ulm-Kaplansky invariant f_p,k of G satisfies (f_p,k)^ω=f_p,k provided that f_p,k is infinite and f_p,k>f_p,i for each i>k.Our approach is based on an appropriate dense embedding of a group G into a Σ-product of circle groups or finite cyclic groups.     

The equality S1 = D = R

The new result of this paper is that for θ( x ; a )‐stable (a weakening of “T is stable”) we have S1[θ( x ; a )] = D[θ( x ; a ), L, ∞]. S1 is Hrushovski's rank. This is an improvement of a result of Kim and Pillay, who for simple theories under the (strong) assumption that either of the ranks be finite obtained the same identity. Only the first equality is new, the second equality is a result of Shelah from the seventies. We derive it by studying localizations of several rank functions, we get the following Main Theorem. Suppose that μ is regular satisfying μ ≥ |T|⁺, p is a finite type, and Δ is a set of formulas closed under Boolean operations. If either (a) R[p, Δ, μ⁺] < ∞ or (b) p is Δ‐stable and μ satisfies “for every sequence {μ_i : i < |Δ| + ?₀} of cardinals μ_i < μ we have that holds”, then S[p, Δ, μ⁺] = D[p, Δ, μ⁺] = R[p, Δ, μ⁺]. The S rank above is a localized version of Hrushovski's S1 rank. This rank, as well as our systematic use of local stability, allows us to get a more conceptual proof of the equality of D and R, which is an old result of Shelah. A particular (asymptotic) case of the theorem offers a new sufficient condition for the equality of S1 and D[·, L, ∞]. We also manage, due to a more general approach, to avoid some combinatorial difficulties present in Shelah's original exposition.     

Espaces de Banach stables

We define the notion of “stable Banach space” by a simple condition on the norm. We prove that ifE is a stable Banach space, then every subspace ofL ^p(E) (1≦p<∞) is stable. Our main result asserts that every infinite dimensional stable Banach space containsl ^p, for somep, 1≦p<∞. This is a generalization of a theorem due to D. Aldous: every infinite dimensional subspace ofL ¹ containsl ^p, for somep in the interval [1, 2].     

Deformations of filiform Lie algebras and symplectic structures

We study symplectic structures on filiform Lie algebras, which are niplotent Lie algebras with the maximal length of the descending central sequence. Let g be a symplectic filiform Lie algebra and dim g = 2k ≥ 12. Then g is isomorphic to some ℕ-filtered deformation either of m₀(2k) (defined by the structure relations [e ₁, e _i ] = e _i+1, i = 2,…, 2k − 1) or of V _2k , the quotient of the positive part of the Witt algebra W ₊ by the ideal of elements of degree greater than 2k. We classify ℕ-filtered deformations of V _n : [e _i , e _j ] = (j − i)e _i+1 + Σ _l≥1 c _ij ^l e _i+j+l . For dim g = n ≥ 16, the moduli space ℳ_n of these deformations is the weighted projective space . For even n, the subspace of symplectic Lie algebras is determined by a single linear equation. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 252, pp. 194–216.     

A characterization of the class of credibility matrices corresponding to a certain class of discrete distributions

Recently (see De Vylder & Goovaerts (1984), this issue) so called credibility matrices have been introduced and studied in the framework of general properties of matrices, such as non-negativity, total positivity etc. In the present note we characterize a class of credibility matrices generated by the normed sequence of functions (p_l, p_l,…, p_n) on K = [0, b] where $\text{p}{}_{\mathrm{i}}\text{(θ) =?(i)g(θ)h}{}^{\mathrm{i}}\text{(θ)}$, i=0, …, n, θ ? K, and where ?, g, h are nonnegative (eventually depending on n, n may be finite or infinite). For simplicity we suppose h to be monotonic and continuous.     

Asymptotically optimal empirical bayes procedures for selecting good

Let ∏₁,…,∏_k denote k independent populations, where a random observation from population ∏ _i has a uniform distribution over the interval (0,θ _i ) and θ _i is a realization of a random variable having an unknown prior distribution G _i . Population ∏ _i is said to be a good population if θ _i ≥θ₀, where θ₀ is a given, positive number. This paper provides a sequence of empirical Bayes procedures for selecting the good populationsamong ∏₁,…,∏ _k . It is shown that these procedures are asymptotically optimal and that the order of associated convergence rates is O(n^-r/4) for some r, 0<r<2, where n is the number of accumulated past observations

at hand     

On symmetric basic sequences in lorentz sequence spaces II

It is shown that if {y _n} is a block of type I of a symmetric basis {x _n} in a Banach spaceX, then {y _n} is equivalent to {x _n} if and only if the closed linear span [y _n] of {y _n} is complemented inX. The result is used to study the symmetric basic sequences of the dual space of a Lorentz sequence spaced(a, p). Let {x _n,f _n} be the unit vector basis ofd(a, p), for 1≤p<+∞. It is shown that every infinite-dimensional subspace ofd(a, p) (respectively, [f _n] has a complemented subspace isomorphic tol _p (respectively,l _q, 1/p+1/q=1 when 1<p<+∞ andc ₀ whenp=1) and numerous other results on complemented subspaces ofd(a, p) and [f _n] are obtained. We also obtain necessary and sufficient conditions such that [f _n] have exactly two non-equivalent symmetric basic sequences. Finally, we exhibit a Banach spaceX with symmetric basis {x _n} such that every symmetric block basic sequence of {x _n} spans a complemented subspace inX butX is not isomorphic to eitherc ₀ orl _p, 1≤p<+∞.     

The chromatic numbers of random hypergraphs

For a pair of integers 1≤γ<r, the γ-chromatic number of an r-uniform hypergraph H=(V, E) is the minimal k, for which there exists a partition of V into subsets T₁,…,T_k such that |e∩T_i|≤γ for every e∈E. In this paper we determine the asymptotic behavior of the γ-chromatic number of the random r-uniform hypergraph H_r(n, p) for all possible values of γ and for all values of p down to p=Θ(n^−r+1). © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 381–403, 1998     

Subspaces ofl p N of small codimension

In this paper the structure of subspaces and quotients ofl _p ^N of dimension very close toN is studied, for 1≤p≤∞. In particular, the maximal dimensionk=k(p, m, N) so that an arbitrarym-dimensional subspaceX ofl _p ^N contains a good copy ofl _p ^k , is investigated form=N−o(N). In several cases the obtained results are sharp.     

On the subspaces ofL p (p>2) spanned by sequences of independent random variables

Let 2<p<∞. The Banach space spanned by a sequence of independent random variables inL ^p , each of mean zero, is shown to be isomorphic tol ²,l ^p ,l ²⊕l ^p , or a new spaceX _p , and the linear topological properties ofX _p are investigated. It is proved thatX _p is isomorphic to a complemented subspace ofL ^p and another uncomplemented subspace ofL ^p , whence there exists an uncomplemented subspace ofl ^p isomorphic tol ^p . It is also proved thatX _p is not isomorphic to the previously knownℒ _p spaces. The work for this research was partially supported by the National Science Foundation GP-12997.     

On the value distribution of positive definite quadratic forms

Denote by 0 = λ ₀ < λ ₁ ≤ λ ₂ ≤ . . . the infinite sequence given by the values of a positive definite irrational quadratic form in k variables at integer points. For l ≥ 2 and an (l −1)-dimensional interval I = I ₂×. . .×I _l we consider the l-level correlation function K^(l)_I(R){K^{(l)}_I(R)} which counts the number of tuples (i ₁, . . . , i _l ) such that l_i1,?,l_il ￡ R²{\lambda_{i_1},\ldots,\lambda_{i_l}\leq R^2} and l_ij-l_i1 ? I_j{\lambda_{i_{j}}-\lambda_{i_{1}}\in I_j} for 2 ≤ j ≤ l. We study the asymptotic behavior of K^(l)_I(R){K^{(l)}_I(R)} as R tends to infinity. If k ≥ 4 we prove K^(l)_I(R) ~ c_l(Q) vol(I)R^lk-2(l-1){K^{(l)}_I(R)\sim c_l(Q)\,{\rm vol}(I)R^{lk-2(l-1)}} for arbitrary l, where c _l (Q) is an explicitly determined constant. This remains true for k = 3 under the restriction l ≤ 3.     

The second centralizer of a Bernoulli shift is just its powers

If ℐ is a collection of measure preserving transformations of a probability space, byC(ℐ), the centralizer of ℐ, we mean the group of all measure preserving transformationsS such thatTS=ST for allT ∈ ℐ. We show here that ifT is a Bernoulli shift, thenC(C(T))={T ⁱ |i ∈ Z}. The proof is carried out by constructing an action of Z², {T ₁ ⁱ °T ₂ ⁱ |i, j ∈ Z}, whereT ₁ is a Bernoulli shift of arbitrary entropy, but for anyj ≠ 0,C({T ₁,T ₂ ^i} ={T ₁ ⁱ °T ₂ ^k l, k ∈ Z}. The construction is a two-dimensional analogue of Ornstein’s “rank one mixing” transformation.     

Fractional Poisson equations and ergodic theorems for fractional coboundaries

For a given contractionT in a Banach spaceX and 0<α<1, we define the contractionT _α=Σ _j=1 ^∞ a _j T ^j , where {a _j } are the coefficients in the power series expansion (1-t)^α=1-Σ _j=1 ^∞ a _j t ^j in the open unit disk, which satisfya _j >0 anda _j >0 and Σ _j=1 ^∞ a _j =1. The operator calculus justifies the notation(I−T) ^α :=I−T _α (e.g., (I−T _1/2)²=I−T). A vectory∈X is called an, α-fractional coboundary for T if there is anx∈X such that(I−T) ^α x=y, i.e.,y is a coboundary forT _α . The fractional Poisson equation forT is the Poisson equation forT _α . We show that if(I−T)X is not closed, then(I−T) ^α X strictly contains(I−T)X (but has the same closure). ForT mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove thaty∈X is an α-fractional coboundary if and only if Σ _k=1 ^∞ T ^k y/k ^1-α converges in norm, and conclude that lim _n ‖(1/n ^1-α)Σ _k=1 ⁿ T ^k y‖=0 for suchy. For a Dunford-Schwartz operatorT onL ₁ of a probability space, we consider also a.e. convergence. We prove that iff∈(I−T) ^α L ₁ for some 0<α<1, then the one-sided Hilbert transform Σ _k=1 ^∞ T ^k f/k converges a.e. For 1<p<∞, we prove that iff∈(I−T) ^α L _p with α>1−1/p=1/q, then Σ _k=1 ^∞ T ^k f/k ^1/p converges a.e., and thus (1/n ^1/p ) Σ _k=1 ⁿ T ^k f converges a.e. to zero. Whenf∈(I−T) ^1/q L _p (the case α=1/q), we prove that (1/n ^1/p (logn)^1/q )Σ _k=1 ⁿ T ^k f converges a.e. to zero.     

Triangular Dirichlet Kernels and Growth of L p Lebesgue Constants

Let P be a polygon in ℤ² and consider the mapping of an L¹(\mathbbT²)L^{1}(\mathbb{T}^{2}) function into the partial sum of its Fourier series determined by the dilate of P by the integer N. If the image space is endowed with the L ^p norm, 1<p<∞, then the operator norm will be given by the L ^p norm of ∑_(m,n)∈NP e ^{2π i(mx+ny)}. The size of this operator norm is shown to be O(N ^2(1−1/p)) when the polygon is a triangle. The estimate is independent of the shape of the triangle. For a k sided polygon the corresponding estimate is O(kN ^2(1−1/p)).     

ℓ<Superscript>1</Superscript>-spreading models in mixed Tsirelson spacemodels in mixed Tsirelson space

Suppose that (F _n ) _n=1 ^∞ is a sequence of regular families of finite subsets of ℝ and (θ _n ) _n=1 ^∞ is a nonincreasing null sequence in (0,1). The mixed Tsirelson spaceT[(θ _n ,F _n ) _n=1 ^∞ ] is the completion ofc ₀₀ with respect to the implicitly defined norm , where the last supremum is taken over all sequences (E _i ) _i=1 ^k in [ℕ]^<∞ such that maxE _i<minE _i +1 and . Necessary and sufficient conditions are obtained for the existence of higher order ℓ¹-spreading models in every subspace generated by a subsequence of the unit vector basis ofT[(θ _n ,F _n ) _n=1 ^∞ ].     

Optimal broadcast on parallel locality models

In this paper matching upper and lower bounds for broadcast on general purpose parallel computation models that exploit network locality are proven. These models try to capture both the general purpose properties of models like the PRAM or BSP on the one hand, and to exploit network locality of special purpose models like meshes, hypercubes, etc., on the other hand. They do so by charging a cost l(|i−j|) for a communication between processors i and j, where l is a suitably chosen latency function.An upper bound T(p)=∑_i=0^loglogp2ⁱ·l(p^1/2i) on the runtime of a broadcast on a p processor H-PRAM is given, for an arbitrary latency function l(k).The main contribution of the paper is a matching lower bound, holding for all latency functions in the range from l(k)=Ω(logk/loglogk) to l(k)=O(log²k). This is not a severe restriction since for latency functions l(k)=O(logk/log¹⁺log(k)) with arbitrary >0, the runtime of the algorithm matches the trivial lower bound Ω(logp) and for l(k)=Θ(log¹⁺k) or l(k)=Θ(k), the runtime matches the other trivial lower bound Ω(l(p)). Both upper and lower bounds apply for other parallel locality models like Y-PRAM, D-BSP and E-BSP, too.     

Tychonoff expansions with prescribed resolvability properties

The recent literature offers examples, specific and hand-crafted, of Tychonoff spaces (in ZFC) which respond negatively to these questions, due respectively to Ceder and Pearson (1967) [3] and to Comfort and García-Ferreira (2001) [5]: (1) Is every ω-resolvable space maximally resolvable? (2) Is every maximally resolvable space extraresolvable? Now using the method of KID expansion, the authors show that every suitably restricted Tychonoff topological space (X,T) admits a larger Tychonoff topology (that is, an “expansion”) witnessing such failure. Specifically the authors show in ZFC that if (X,T) is a maximally resolvable Tychonoff space with S(X,T)?Δ(X,T)=κ, then (X,T) has Tychonoff expansions U=Ui (1?i?5), with Δ(X,Ui)=Δ(X,T) and S(X,Ui)?Δ(X,Ui), such that (X,Ui) is: (i=1) ω-resolvable but not maximally resolvable; (i=2) [if κ^′ is regular, with S(X,T)?κ^′?κ] τ-resolvable for all τ<κ^′, but not κ^′-resolvable; (i=3) maximally resolvable, but not extraresolvable; (i=4) extraresolvable, but not maximally resolvable; (i=5) maximally resolvable and extraresolvable, but not strongly extraresolvable.