Multiplicative <Emphasis Type="Italic">A</Emphasis><Subscript>∞</Subscript>-structure in terms of spectral sequences of fibrations

In the present paper, the technique of spectral sequences with A _∞ -structures in their terms is developed for differential algebras with filtrations. Applications of this technique to the multiplicative spectral sequences of fibrations are given. We show that if the base of fibration is connected and simply connected, then the structure graded A _∞ -algebra in the second term of the spectral sequence of a fibration is the tensor product of the cohomology A _∞ -algebra of the base and the cohomology A _∞ -algebra of the fibre of this fibration.     

Structure relations in special <Emphasis Type="Italic">A</Emphasis><Subscript>∞</Subscript>-bialgebras

We compute the structure relations in special A _∞-bialgebras whose operations are limited to those defining the underlying A _∞-(co)algebra substructure. Such bialgebras appear as the homology of certain loop spaces. Whereas structure relations in general A _∞-bialgebras depend upon the combinatorics of permutahedra, only Stasheff’s associahedra are required here. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 43, Topology and Its Applications, 2006.     

On Large Deviations of Multivariate Heavy-Tailed Random Walks

Let {S _n ;n=1,2,…} be a random walk in R ^d and E(S ₁)=(μ ₁,…,μ _d ). Let a _j >μ _j for j=1,…,d and A=(a ₁,∞)×⋅⋅⋅×(a _d ,∞). We are interested in the probability P(S _n /n∈A) for large n in the case where the components of S ₁ are heavy tailed. An objective is to associate an exact power with the aforementioned probability. We also derive sharper asymptotic bounds for the probability and show that in essence, the occurrence of the event {S _n /n∈A} is caused by large single increments of the components in a specific way.      

An optimal bound on the tail distribution of the number of recurrences of an event in product spaces

 Let X ₁ ,X ₂ ,... be independent random variables and a a positive real number. For the sake of illustration, suppose A is the event that |X _i+1 +...+X _j |≥a for some integers 0≤i<j<∞. For each k≥2 we upper-bound the probability that A occurs k or more times, i.e. that A occurs on k or more disjoint intervals, in terms of P(A), the probability that A occurs at least once. More generally, let X=(X ₁ ,X ₂ ,...)Ω=Π _{j ≥1}Ω _j be a random element in a product probability space (Ω,ℬ,P=⊗ _{j ≥1} P _j ). We are interested in events AB that are (at most contable) unions of finite-dimensional cylinders. We term such sets sequentially searchable. Let L(A) denote the (random) number of disjoint intervals (i,j] such that the value of X _(i,j] =(X _i+1 ,...,X _j ) ensures that XA. By definition, for sequentially searchable A, P(A)≡P(L(A)≥1)=P(𝒩_{−ln (P(Ac))} ≥1), where 𝒩_γ denotes a Poisson random variable with some parameter γ>0. Without further assumptions we prove that, if 0<P(A)<1, then P(L(A)≥k)<P(𝒩_{−ln (P(Ac))} ≥k) for all integers k≥2. An application to sums of independent Banach space random elements in l ^∞ is given showing how to extend our theorem to situations having dependent components. Received: 8 June 2001 / Revised version: 30 October 2002 Published online: 15 April 2003 RID="*" ID="*" Supported by NSF Grant DMS-99-72417. RID="†" ID="†" Supported by the Swedish Research Council. Mathematics Subject Classification (2000): Primary 60E15, 60G50 Key words or phrases: Tail probability inequalities – Hoffmann-Jo rgensen inequality – Poisson bounds – Number of event recurrences – Number of entrance times – Product spaces     

A critically loaded multirate link with trunk reservation

We consider a loss system model of interest in telecommunications. There is a single service facility with N servers and no waiting room. There are K types of customers, with type ί customers requiring A _ί servers simultaneously. Arrival processes are Poisson and service times are exponential. An arriving type ί customer is accepted only if there are Rί(⩾A_ί ) idle servers. We examine the asymptotic behavior of the above system in the regime known as critical loading where both N and the offered load are large and almost equal. We also assume that R ₁,..., R _K-1 remain bounded, while R _K ^N ←∞ and R _K ^N /√N ← 0 as N ← ∞. Our main result is that the K dimensional “queue length” process converges, under the appropriate normalization, to a particular K dimensional diffusion. We show that a related system with preemption has the same limit process. For the associated optimization problem where accepted customers pay, we show that our trunk reservation policy is asymptotically optimal when the parameters satisfy a certain relation. This revised version was published online in June 2006 with corrections to the Cover Date.     

On bounded solutions of abstract differential equations

Summary We construct nontrivial bounded solutions of an abstract evolution equationu'(t)=Au(t) (—∞<t<∞) whereA generates a (C ₀) semigroup of operators {T _t;t≥0} such thatT _t converges strongly to zero ast→∞.

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Riassunto Si costruiscono soluzioni limitate non triviali di una equazioneu'(t)=Au(t) (—∞<t<∞) doveA è il generatore di un semigruppo di operatori {T _t;t≥0} tale cheT _t converge fortemente verso 0 pert→∞.

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Supported by National Science Foundation grant GP-12722.     

On tame operators between non-archimedean power series spaces

Let p ∈ {1, ∞}. We show that any continuous linear operator T from A1 (a) to Ap (b) is tame, i.e., there exists a positive integer c such that sup x||Tx||k/|x|ck ∞ for every k ∈ N. Next we prove that a similar result holds for operators from A∞(a) to Ap(b) if and only if the set Mb,a of all finite limit points of the double sequence (bi /aj ) i,j∈N is bounded. Finally we show that the range of every tame operator from A∞(a) to A∞(b) has a Schauder basis.     

Functorial polar functions

W _∞ denotes the category of archimedean ℓ-groups with designated weak unit and complete ℓ-homomorphisms that preserve the weak unit. CmpT _2,∞ denotes the category of compact Hausdorff spaces with continuous skeletal maps. This work introduces the concept of a functorial polar function on W _∞ and its dual a functorial covering function on CmpT _2,∞.     

Large deviations probabilities for random walks in the absence of finite expectations of jumps

 Let be independent identically distributed random variables with regularly varying distribution tails:

A ∞-simplicial objects andA ∞-topological groups

The notion ofA _∞-topological group is introduced. It is proved that, if a space is obtained by deformation retraction of a topological groupG, then it has the structure of anA _∞-topological group, and theA _∞-homotopy equivalence holds. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 913–919, December, 1999.     

Noncommutative interpolation and Poisson transforms

General results of interpolation (e.g., Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebraF ^∞ (resp., noncommutative disc algebraA _n) with consequences to the interpolation by bounded operator-valued analytic functions in the unit ball of ℂⁿ are obtained. Noncommutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric representations of the quotient algebraF ^∞/J on Hilbert spaces whereJ is anyw ^*-closed, 2-sided ideal ofF ^∞, are obtained and used to construct aw ^*-continuous,F ^∞/J-functional calculus associated to row contractionsT=[T ₁,…,T _n] whenf(T₁, …, T_n)=0 for anyf∈J. Other properties of the dual algebraF ^∞/J are considered. The second author was partially supported by NSF DMS-9531954.     

The weak and strong asymptotic equivalence relations and the generalized inverse<Superscript>*</Superscript>

We discuss the relationship between the weak and strong asymptotic equivalence relations and the generalized inverse in the class A \mathcal {A} of all nondecreasing unbounded positive functions on a half-axis [a,+∞) (a > 0). As a main result, we prove a proper characterization of the functional class R _∞ ∩ A \mathcal {A} , where R _∞ is the class of all rapidly varying functions. Also, we prove a characterization of the functional class PI ^* ∩ A \mathcal {A} .     

Approximation of (ψ, β)-differentiable functions by Poisson integrals in the uniform metric

We obtain asymptotic equalities for upper bounds of approximations of functions from the class C _β,∞^ψ by Poisson integrals in the metric of the space C.     

Large-deviation probability and the local dimension of sets

Let X ₁ , X ₂ , ..., X_n be n independent identically distributed real random variables and S_n = Σ _n=1 ⁿ X_i. We obtain precise asymptotics forP (S_n ∈ nA) for rather arbitrary Borel sets A₁ in terms of the density of the dominating points in A. Our result extends classical theorems in the field of large deviations for independent samples. We also obtain asymptotics forP (S_n ∈ γ_nA), with γ_n/n → ∞. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I.     

Limit algebras of differential forms in non-commutative geometry

Given a C*-normed algebra A which is either a Banach *-algebra or a Frechet *-algebra, we study the algebras Ω_∞ A and Ω_ε A obtained by taking respectively the projective limit and the inductive limit of Banach *-algebras obtained by completing the universal graded differential algebra Ω*A of abstract non-commutative differential forms over A. Various quantized integrals on Ω_∞ A induced by a K-cycle on A are considered. The GNS-representation of Ω_∞ A defined by a d-dimensional non-commutative volume integral on a d ⁺-summable K-cycle on A is realized as the representation induced by the left action of A on Ω*A. This supplements the representation A on the space of forms discussed by Connes (Ch. VI.1, Prop. 5, p. 550 of [C]).     

The Karp complexity of unstable classes

A class K of structures is controlled if, for all cardinals λ, the relation of L _∞,λ-equivalence partitions K into a set of equivalence classes (as opposed to a proper class). We prove that the class of doubly transitive linear orders is controlled, while any pseudo-elementary class with the ω-independence property is not controlled. Received: 23 September 1998 / Revised version: 6 July 1999 / Published online: 21 December 2000     

Intégration Symplectique des Variétés de Poisson Régulières

Asymplectic integration of a Poisson manifold (M, Λ) is a symplectic groupoid (Γ,η) whichrealizes the given Poisson manifold, i.e. such that the space of units Γ₀ with the induced Poisson structure Λ₀ is isomorphic to (M, Λ). This notion was introduced by A. Weinstein in [28] in order to quantize Poisson manifolds by quantizing their symplectic integration. Any Poisson manifold can be integrated by alocal symplectic groupoid ([4], [13]) but already for regular Poisson manifolds there are obstructions to global integrability ([2], [6], [11], [17], [28]). The aim of this paper is to summarize all the known obstructions and present a sufficient topological condition for integrability of regular Poisson manifolds; we will indeed describe a concrete procedure for this integration. Further our criterion will provide necessary and sufficient if we require Γ to be Hausdorff, which is a suitable condition to proceed to Weinstein’s program of quantization. These integrability results may be interpreted as a generalization of the Cartan-Smith proof of Lie’s third theorem in the infinite dimensional case.

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Recherche supportée par D.G.I.C.Y.T. Espagne (Proyecto PB90-0765) et Xunta de Galicia (Proxecto XUGA20704B90)     

Singular semi-linear equations inL 1(R)

Letg be a positive continuous function onR which tends to zero at −∞ and which is not integrable overR. The boundary-value problem −u″+g(u)=f, u′(±∞)=0, is considered forf∈L ¹(R). We show that this problem can have a solution if and only ifg is integrable at −∞ and if this is so then the problem is solvable precisely when ∫ _−∞ ^∞ . Some extensions of this result are also given. Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and by the National Science Foundation, Grant MPS 75-05501.     

Hyperinvariant subspaces for some <Emphasis Type="Italic">B</Emphasis>–circular operators

We show that if A is a Hilbert–space operator, then the set of all projections onto hyperinvariant subspaces of A, which is contained in the von Neumann algebra υN(A) that is generated by A, is independent of the representation of υ N(A), thought of as an abstract W^*–algebra. We modify a technique of Foias, Ko, Jung and Pearcy to get a method for finding nontrivial hyperinvariant subspaces of certain operators in finite von Neumann algebras. We introduce the B–circular operators as a special case of Speicher's B–Gaussian operators in free probability theory, and we prove several results about a B–circular operator z, including formulas for the B–valued Cauchy– and R–transforms of z^*z. We show that a large class of L^∞([0,1])–circular operators in finite von Neumann algebras have nontrivial hyperinvariant subspaces, and that another large class of them can be embedded in the free group factor L(F₃). These results generalize some of what is known about the quasinilpotent DT–operator. Supported in part by NSF Grant DMS-0300336. with an Appendix by Gabriel Tucci     

The 2-primary torsion on elliptic curves in the Z p -extensions of Q

Let E be an elliptic curve over Q and p a prime number. Denote by Q_p,∞ the Z_p-extension of Q. In this paper, we show that if p≠3, then where E(Q_p,∞)₍₂₎ is the 2-primary part of the group E(Q_p,∞) of Q_p,∞-rational points on E. More precisely, in case p=2, we completely classify E(Q_2,∞)₍₂₎ in terms of E(Q)₍₂₎; in case p≥5 (or in case p=3 and E(Q)₍₂₎≠{O}), we show that E(Q_p,∞)₍₂₎=E(Q)₍₂₎.