Mean-Field Conditions for Percolation on Finite Graphs

Let {G _n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G _n | = n. Denote by p ^t (v, v) the return probability after t steps of the non-backtracking random walk on G _n . We show that if p ^t (v, v) has quasi-random properties, then critical bond-percolation on G _n behaves as it would on a random graph. More precisely, if $\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}$ then the size of the largest component in p-bond-percolation with ${p =\frac{1+O(n^{-1/3})}{d-1}}Let {G _n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G _n | = n. Denote by p ^t (v, v) the return probability after t steps of the non-backtracking random walk on G _n . We show that if p ^t (v, v) has quasi-random properties, then critical bond-percolation on G _n behaves as it would on a random graph. More precisely, if

lim sup  n n1/3 ?t = 1n1/3 tpt(v,v) < ￥,\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}      2. A Note About Critical Percolation on Finite Graphs      Gady Kozma  Asaf Nachmias 《Journal of Theoretical Probability》2011,24(4):1087-1096 In this note we study the geometry of the largest component C1\mathcal {C}_{1} of critical percolation on a finite graph G which satisfies the finite triangle condition, defined by Borgs et al. (Random Struct. Algorithms 27:137–184, 2005). There it is shown that this component is of size n 2/3, and here we show that its diameter is n 1/3 and that the simple random walk takes n steps to mix on it. By Borgs et al. (Ann. Probab. 33:1886–1944, 2005), our results apply to critical percolation on several high-dimensional finite graphs such as the finite torus \mathbbZnd\mathbb{Z}_{n}^{d} (with d large and n→∞) and the Hamming cube {0,1} n .      3. A randomized embedding algorithm for trees      Benny Sudakov  Jan Vondrák 《Combinatorica》2010,30(4):445-470 In this paper, we propose a simple and natural randomized algorithm to embed a tree T in a given graph G. The algorithm can be viewed as a “self-avoiding tree-indexed random walk“. The order of the tree T can be as large as a constant fraction of the order of the graph G, and the maximum degree of T can be close to the minimum degree of G. We show that our algorithm works in a variety of interesting settings. For example, we prove that any graph of minimum degree d without 4-cycles contains every tree of order εd 2 and maximum degree at most d-2εd-2. As there exist d-regular graphs without 4-cycles and with O(d 2) vertices, this result is optimal up to constant factors. We prove similar nearly tight results for graphs of given girth and graphs with no complete bipartite subgraph K s,t .      4. Self-similar extremal processes      E. I. Pancheva 《Journal of Mathematical Sciences》1998,92(3):3911-3920 Given an extremal process X: [0,∞)→[0,∞)d with lower curve C and associated point process N={(tk, Xk):k≥0}, tk distinct and Xk independent, given a sequence ζ n =(τ n , ξ n ), n≥1, of time-space changes (max-automorphisms of [0,∞)d+1), we study the limit behavior of the sequence of extremal processes Yn(t)=ξ n -1 ○ X ○ τn(t)=Cn(t) V max {ξ n -1 ○ Xk: tk ≤ τn(t){ ⇒ Y under a regularity condition on the norming sequence ζn and asymptotic negligibility of the max-increments of Yn. The limit class consists of self-similar (with respect to a group ηα=(σα, Lα), α>0, of time-space changes) extremal processes. By self-similarity here we mean the property Lα ○ Y(t) = d Y ○ αα(t) for all α>0. The univariate marginals of Y are max-self-decomposable. If additionally the initial extremal process X is assumed to have homogeneous max-increments, then the limit process is max-stable with homogeneous max-increments. Supported by the Bulgarian Ministry of Education and Sciences (grant No. MM 234/1996). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.      5. On the limit of large girth graph sequences      Gábor Elek 《Combinatorica》2010,30(5):553-563 Let d≥2 be given and let μ be an involution-invariant probability measure on the space of trees T ∈ T d with maximum degrees at most d. Then μ arises as the local limit of some sequence {G n } n=1∞ of graphs with all degrees at most d. This answers Question 3.3 of Bollobás and Riordan [4].      6. Regularity of the density for the total weighted occupation measure of super-Brownain motion      Rong Li Liu 《数学学报(英文版)》2011,27(8):1557-1572 Let (X t ) be a super-Brownian motion in a bounded domain D in ℝ d . The random measure Y D (·) = ∫0∞ X t (·)dt is called the total weighted occupation time of (X t ). We consider the regularity properties for the densities of a class of Y D . When d = 1, the densities have continuous modifications. When d ≥ 2, the densities are locally unbounded on any open subset of D with positive Y D (dx)-measure.      7. The upper traceable number of a graph      Futaba Okamoto  Ping Zhang  Varaporn Saenpholphat 《Czechoslovak Mathematical Journal》2008,58(1):271-287 For a nontrivial connected graph G of order n and a linear ordering s: v 1, v 2, …, v n of vertices of G, define . The traceable number t(G) of a graph G is t(G) = min{d(s)} and the upper traceable number t +(G) of G is t +(G) = max{d(s)}, where the minimum and maximum are taken over all linear orderings s of vertices of G. We study upper traceable numbers of several classes of graphs and the relationship between the traceable number and upper traceable number of a graph. All connected graphs G for which t +(G) − t(G) = 1 are characterized and a formula for the upper traceable number of a tree is established. Research supported by Srinakharinwirot University, the Thailand Research Fund and the Commission on Higher Education, Thailand under the grant number MRG 5080075.      8. Dynamical borel-cantelli lemma for hyperbolic spaces      François Maucourant 《Israel Journal of Mathematics》2006,152(1):143-155 We prove that almost every (resp. almost no) geodesic rays in a finite volume hyperbolic manifold of real dimensionn intersects for arbitrary large timest a decreasing family of balls of radiusr t, provided the integral ∫ 0 ∞ r t n −1 dt diverges (resp. converges).      9. On Optimal Orientations of Cartesian Products of Trees      Khee Meng Koh  E. G. Tay 《Graphs and Combinatorics》2001,17(1):79-97 . Let d(D) (resp., d(G)) denote the diameter and r(D) (resp., r(G)) the radius of a digraph D (resp., graph G). Let G×H denote the cartesian product of two graphs G and H. An orientation D of G is said to be (r, d)-invariant if r(D)=r(G) and d(D)=d(G). Let {T i }, i=1,…,n, where n≥2, be a family of trees. In this paper, we show that the graph ∏ i =1 n T i admits an (r, d)-invariant orientation provided that d(T 1)≥d(T 2)≥4 for n=2, and d(T 1)≥5 and d(T 2)≥4 for n≥3. Received: July 30, 1997 Final version received: April 20, 1998      10. Fragmentation Processes with an Initial Mass Converging to Infinity      Bénédicte Haas 《Journal of Theoretical Probability》2007,20(4):721-758 We consider a family of fragmentation processes where the rate at which a particle splits is proportional to a function of its mass. Let F 1(m)(t),F 2(m)(t),… denote the decreasing rearrangement of the masses present at time t in a such process, starting from an initial mass m. Let then m→∞. Under an assumption of regular variation type on the dynamics of the fragmentation, we prove that the sequence (F 2(m),F 3(m),…) converges in distribution, with respect to the Skorohod topology, to a fragmentation with immigration process. This holds jointly with the convergence of m−F 1(m) to a stable subordinator. A continuum random tree counterpart of this result is also given: the continuum random tree describing the genealogy of a self-similar fragmentation satisfying the required assumption and starting from a mass converging to ∞ will converge to a tree with a spine coding a fragmentation with immigration. Research supported in part by EPSRC GR/T26368.      11. Adaptive policies for time-varying stochastic systems under discounted criterion      Nadine Hilgert  J. Adolfo Minjárez-Sosa 《Mathematical Methods of Operations Research》2001,54(3):491-505 We consider a class of time-varying stochastic control systems, with Borel state and action spaces, and possibly unbounded costs. The processes evolve according to a discrete-time equation x n + 1=G n (x n , a n , ξn), n=0, 1, … , where the ξn are i.i.d. ℜk-valued random vectors whose common density is unknown, and the G n are given functions converging, in a restricted way, to some function G ∞ as n→∞. Assuming observability of ξn, we construct an adaptive policy which is asymptotically discounted cost optimal for the limiting control system x n+1=G ∞ (x n , a n , ξn).      12. Good points for diophantine approximation      Daniel Berend  Artūras Dubickas 《Proceedings Mathematical Sciences》2009,119(4):423-429 Given a sequence (x n ) n=1∞ of real numbers in the interval [0, 1) and a sequence (δ n ) n=1∞ of positive numbers tending to zero, we consider the size of the set of numbers in [0, 1] which can be ‘well approximated’ by terms of the first sequence, namely, those y ∈ [0, 1] for which the inequality |y − x n | < δ n holds for infinitely many positive integers n. We show that the set of ‘well approximable’ points by a sequence (x n ) n=1∞, which is dense in [0, 1], is ‘quite large’ no matter how fast the sequence (δ n ) n=1∞ converges to zero. On the other hand, for any sequence of positive numbers (δ n ) n=1∞ tending to zero, there is a well distributed sequence (x n ) n=1∞ in the interval [0, 1] such that the set of ‘well approximable’ points y is ‘quite small’.      13. Gauss-Bonnet Formula, Finiteness Condition, and Characterizations of Graphs Embedded in Surfaces      Beifang Chen  Guantao Chen 《Graphs and Combinatorics》2008,24(3):159-183 Let G be an infinite graph embedded in a closed 2-manifold, such that each open face of the embedding is homeomorphic to an open disk and is bounded by finite number of edges. For each vertex x of G, define the combinatorial curvature as that of [8], where d(x) is the degree of x, F(x) is the multiset of all open faces σ in the embedding such that the closure contains x (the multiplicity of σ is the number of times that x is visited along ∂σ), and |σ| is the number of sides of edges bounding the face σ. In this paper, we first show that if the absolute total curvature ∑ x∈V(G) |Φ G (x)| is finite, then G has only finite number of vertices of non-vanishing curvature. Next we present a Gauss-Bonnet formula for embedded infinite graphs with finite number of accumulation points. At last, for a finite simple graph G with 3 ≤ d G (x) < ∞ and Φ G (x) > 0 for every x ∈ V(G), we have (i) if G is embedded in a projective plane and #(V(G)) = n ≥ 1722, then G is isomorphic to the projective wheel P n ; (ii) if G is embedded in a sphere and #(V(G)) = n ≥ 3444, then G is isomorphic to the sphere annulus either A n or B n ; and (iii) if d G (x) = 5 for all x ∈ V(G), then there are only 49 possible embedded plane graphs and 16 possible embedded projective plane graphs. Guantao Chen: The second author was partially supported by NSF DMS-0070514 and NSA-H98230-04-1-0300.      14. Asymptotic representations of solutions of nonautonomous ordinary differential equations with regularly varying nonlinearities      V. M. Evtukhov  A. M. Samoilenko 《Differential Equations》2011,47(5):627-649 We obtain asymptotic representations as t ↑ ω, ω ≤ + ∞, for all possible types of P ω(Y 0, λ 0)-solutions (where Y 0 is zero or ±∞ and −∞ ≤ λ0 ≤ +∞) of nonlinear differential equations y (n) = α 0 p(t)φ(y), where α 0 ∈ {−1, 1}, p: [a, ω[→]0,+∞[ is a continuous function, and φ is a continuous regularly varying function in a one-sided neighborhood of Y 0.      15. Trace formulas for a class of Toeplitz-like operators      Harry Dym 《Israel Journal of Mathematics》1977,27(1):21-48 LetP T denote projection onto the space of entire functions of exponential type ≦T which are square summable on the line relative to a measuredΔ and letG denote multiplication by a suitably restricted complex valued function,g. For a reasonably large class of measuresdΔ, which includes Lebesgue measuredγ, it is shown that trace {(P TGPT)n−PTGnPT} tends boundedly to a limit asT↑∞ and that the limit isindependent of the choice ofdΔ within the permitted class. This extends the range of validity of a formula due to Mark Kac who evaluated this limit in the special casedΔ=dγ using a different formalism.      16. The Global Existence of Small Solutions to the Oldroyd-B Model      Wenjing ZHAO 《数学年刊B辑(英文版)》2011,32(2):215-222 The Cauchy problem to the Oldroyd-B model is studied. In particular, it is shown that if the smooth solution (u, τ) to this system blows up at a finite time T*, then ∫0 T* ‖▿u(t)‖ L ∞dt = ∞. Furthermore, the global existence of smooth solution to this system is given with small initial data.      17. Brownian bridge on hyperbolic spaces and on homogeneous trees      Philippe Bougerol  Thierry Jeulin 《Probability Theory and Related Fields》1999,115(1):95-120 Let B be the Brownian motion on a noncompact non Euclidean rank one symmetric space H. A typical examples is an hyperbolic space H n , n > 2. For ν > 0, the Brownian bridge B (ν) of length ν on H is the process B t , 0 ≤t≤ν, conditioned by B 0 = B ν = o, where o is an origin in H. It is proved that the process converges weakly to the Brownian excursion when ν→ + ∞ (the Brownian excursion is the radial part of the Brownian Bridge on ℝ3). The same result holds for the simple random walk on an homogeneous tree. Received: 4 December 1998 / Revised version: 22 January 1999      18. Joint ergodicity and mixing      Daniel Berend 《Journal d'Analyse Mathématique》1985,45(1):255-284 The study of jointly ergodic measure preserving transformations of probability spaces, begun in [1], is continued, and notions of joint weak and strong mixing are introduced. Various properties of ergodic and mixing transformations are shown to admit analogues for several transformations. The case of endomorphisms of compact abelian groups is particularly emphasized. The main result is that, given such commuting endomorphisms σ1σ2,...,σ, ofG, the sequence ((1/N)Σ n=0 N−1 σ 1 n f 1·σ 2 n f 2· ··· · σ s n f sconverges inL 2(G) for everyf 1,f 2,…,f s∈L ∞(G). If, moreover, the endomorphisms are jointly ergodic, i.e., if the limit of any sequence as above is Π i=1 s ∫ G f 1 d μ, where μ is the Haar measure, then the convergence holds also μ-a.e.      19. Possible limit laws for entrance times of an ergodic aperiodic dynamical system      Y. Lacroix 《Israel Journal of Mathematics》2002,132(1):253-263 LetG denote the set of decreasingG: ℝ→ℝ withGэ1 on ]−∞,0], and ƒ 0 ∞ G(t)dt⩽1. LetX be a compact metric space, andT: X→X a continuous map. Let μ denone aT-invariant ergodic probability measure onX, and assume (X, T, μ) to be aperiodic. LetU⊂X be such that μ(U)>0. Let τ U (x)=inf{k⩾1:T k xεU}, and defineG U (t)=1/u(U)u({xεU:u(U)τU(x)>t),tεℝ We prove that for μ-a.e.x∈X, there exists a sequence (U n ) n≥1 of neighbourhoods ofx such that {x}=∩ n U n , and for anyG ∈G, there exists a subsequence (n k ) k≥1 withG U n k ↑U weakly. We also construct a uniquely ergodic Toeplitz flowO(x ∞,S, μ), the orbit closure of a Toeplitz sequencex ∞, such that the above conclusion still holds, with moreover the requirement that eachU n be a cylinder set. In memory of Anzelm Iwanik      20. Largeness of the set of finite products in a semigroup      Chase Adams III  Neil Hindman  Dona Strauss 《Semigroup Forum》2008,76(2):276-296 We investigate when the set of finite products of distinct terms of a sequence 〈x n 〉 n=1∞ in a semigroup (S,⋅) is large in any of several standard notions of largeness. These include piecewise syndetic, central, syndetic, central*, and IP*. In the case of a “nice” sequence in (S,⋅)=(ℕ,+) one has that FS(〈x n 〉 n=1∞) has any or all of the first three properties if and only if {x n+1−∑ t=1 n x t :n∈ℕ} is bounded from above. N. Hindman acknowledges support received from the National Science Foundation via Grant DMS-0554803.

A Note About Critical Percolation on Finite Graphs

In this note we study the geometry of the largest component C₁\mathcal {C}_{1} of critical percolation on a finite graph G which satisfies the finite triangle condition, defined by Borgs et al. (Random Struct. Algorithms 27:137–184, 2005). There it is shown that this component is of size n ^2/3, and here we show that its diameter is n ^1/3 and that the simple random walk takes n steps to mix on it. By Borgs et al. (Ann. Probab. 33:1886–1944, 2005), our results apply to critical percolation on several high-dimensional finite graphs such as the finite torus \mathbbZ_n^d\mathbb{Z}_{n}^{d} (with d large and n→∞) and the Hamming cube {0,1} ⁿ .     

A randomized embedding algorithm for trees

In this paper, we propose a simple and natural randomized algorithm to embed a tree T in a given graph G. The algorithm can be viewed as a “self-avoiding tree-indexed random walk“. The order of the tree T can be as large as a constant fraction of the order of the graph G, and the maximum degree of T can be close to the minimum degree of G. We show that our algorithm works in a variety of interesting settings. For example, we prove that any graph of minimum degree d without 4-cycles contains every tree of order εd ² and maximum degree at most d-2εd-2. As there exist d-regular graphs without 4-cycles and with O(d ²) vertices, this result is optimal up to constant factors. We prove similar nearly tight results for graphs of given girth and graphs with no complete bipartite subgraph K _s,t .     

Self-similar extremal processes

Given an extremal process X: [0,∞)→[0,∞)^d with lower curve C and associated point process N={(t_k, X_k):k≥0}, t_k distinct and X_k independent, given a sequence ζ _n =(τ _n , ξ _n ), n≥1, of time-space changes (max-automorphisms of [0,∞)^d+1), we study the limit behavior of the sequence of extremal processes Y_n(t)=ξ _n ^-1 ○ X ○ τ_n(t)=C_n(t) V max {ξ _n ^-1 ○ X_k: t_k ≤ τ_n(t){ ⇒ Y under a regularity condition on the norming sequence ζ_n and asymptotic negligibility of the max-increments of Y_n. The limit class consists of self-similar (with respect to a group η_α=(σ_α, L_α), α>0, of time-space changes) extremal processes. By self-similarity here we mean the property L_α ○ Y(t) ₌ ^d Y ○ α_α(t) for all α>0. The univariate marginals of Y are max-self-decomposable. If additionally the initial extremal process X is assumed to have homogeneous max-increments, then the limit process is max-stable with homogeneous max-increments. Supported by the Bulgarian Ministry of Education and Sciences (grant No. MM 234/1996). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

On the limit of large girth graph sequences

Let d≥2 be given and let μ be an involution-invariant probability measure on the space of trees T ∈ T _d with maximum degrees at most d. Then μ arises as the local limit of some sequence {G _n } _n=1^∞ of graphs with all degrees at most d. This answers Question 3.3 of Bollobás and Riordan [4].     

Regularity of the density for the total weighted occupation measure of super-Brownain motion

Let (X _t ) be a super-Brownian motion in a bounded domain D in ℝ ^d . The random measure Y ^D (·) = ∫₀^∞ X _t (·)dt is called the total weighted occupation time of (X _t ). We consider the regularity properties for the densities of a class of Y ^D . When d = 1, the densities have continuous modifications. When d ≥ 2, the densities are locally unbounded on any open subset of D with positive Y ^D (dx)-measure.     

The upper traceable number of a graph

For a nontrivial connected graph G of order n and a linear ordering s: v ₁, v ₂, …, v _n of vertices of G, define . The traceable number t(G) of a graph G is t(G) = min{d(s)} and the upper traceable number t ⁺(G) of G is t ⁺(G) = max{d(s)}, where the minimum and maximum are taken over all linear orderings s of vertices of G. We study upper traceable numbers of several classes of graphs and the relationship between the traceable number and upper traceable number of a graph. All connected graphs G for which t ⁺(G) − t(G) = 1 are characterized and a formula for the upper traceable number of a tree is established. Research supported by Srinakharinwirot University, the Thailand Research Fund and the Commission on Higher Education, Thailand under the grant number MRG 5080075.     

Dynamical borel-cantelli lemma for hyperbolic spaces

We prove that almost every (resp. almost no) geodesic rays in a finite volume hyperbolic manifold of real dimensionn intersects for arbitrary large timest a decreasing family of balls of radiusr _t, provided the integral ∫ ₀ ^∞ r _t ⁿ −1 dt diverges (resp. converges).     

On Optimal Orientations of Cartesian Products of Trees

. Let d(D) (resp., d(G)) denote the diameter and r(D) (resp., r(G)) the radius of a digraph D (resp., graph G). Let G×H denote the cartesian product of two graphs G and H. An orientation D of G is said to be (r, d)-invariant if r(D)=r(G) and d(D)=d(G). Let {T _i }, i=1,…,n, where n≥2, be a family of trees. In this paper, we show that the graph ∏ _{i =1} ⁿ T _i admits an (r, d)-invariant orientation provided that d(T ₁)≥d(T ₂)≥4 for n=2, and d(T ₁)≥5 and d(T ₂)≥4 for n≥3. Received: July 30, 1997 Final version received: April 20, 1998     

Fragmentation Processes with an Initial Mass Converging to Infinity

We consider a family of fragmentation processes where the rate at which a particle splits is proportional to a function of its mass. Let F ₁^(m)(t),F ₂^(m)(t),… denote the decreasing rearrangement of the masses present at time t in a such process, starting from an initial mass m. Let then m→∞. Under an assumption of regular variation type on the dynamics of the fragmentation, we prove that the sequence (F ₂^(m),F ₃^(m),…) converges in distribution, with respect to the Skorohod topology, to a fragmentation with immigration process. This holds jointly with the convergence of m−F ₁^(m) to a stable subordinator. A continuum random tree counterpart of this result is also given: the continuum random tree describing the genealogy of a self-similar fragmentation satisfying the required assumption and starting from a mass converging to ∞ will converge to a tree with a spine coding a fragmentation with immigration. Research supported in part by EPSRC GR/T26368.     

Adaptive policies for time-varying stochastic systems under discounted criterion

We consider a class of time-varying stochastic control systems, with Borel state and action spaces, and possibly unbounded costs. The processes evolve according to a discrete-time equation x _{n + 1}=G _n (x _n , a _n , ξ_n), n=0, 1, … , where the ξ_n are i.i.d. ℜ^k-valued random vectors whose common density is unknown, and the G _n are given functions converging, in a restricted way, to some function G _∞ as n→∞. Assuming observability of ξ_n, we construct an adaptive policy which is asymptotically discounted cost optimal for the limiting control system x _n+1=G _∞ (x _n , a _n , ξ_n).     

Good points for diophantine approximation

Given a sequence (x _n ) _n=1^∞ of real numbers in the interval [0, 1) and a sequence (δ _n ) _n=1^∞ of positive numbers tending to zero, we consider the size of the set of numbers in [0, 1] which can be ‘well approximated’ by terms of the first sequence, namely, those y ∈ [0, 1] for which the inequality |y − x _n | < δ _n holds for infinitely many positive integers n. We show that the set of ‘well approximable’ points by a sequence (x _n ) _n=1^∞, which is dense in [0, 1], is ‘quite large’ no matter how fast the sequence (δ _n ) _n=1^∞ converges to zero. On the other hand, for any sequence of positive numbers (δ _n ) _n=1^∞ tending to zero, there is a well distributed sequence (x _n ) _n=1^∞ in the interval [0, 1] such that the set of ‘well approximable’ points y is ‘quite small’.     

Gauss-Bonnet Formula, Finiteness Condition, and Characterizations of Graphs Embedded in Surfaces

Let G be an infinite graph embedded in a closed 2-manifold, such that each open face of the embedding is homeomorphic to an open disk and is bounded by finite number of edges. For each vertex x of G, define the combinatorial curvature

Asymptotic representations of solutions of nonautonomous ordinary differential equations with regularly varying nonlinearities

We obtain asymptotic representations as t ↑ ω, ω ≤ + ∞, for all possible types of P _ω(Y ₀, λ ₀)-solutions (where Y ₀ is zero or ±∞ and −∞ ≤ λ₀ ≤ +∞) of nonlinear differential equations y ⁽ⁿ⁾ = α ₀ p(t)φ(y), where α ₀ ∈ {−1, 1}, p: [a, ω[→]0,+∞[ is a continuous function, and φ is a continuous regularly varying function in a one-sided neighborhood of Y ₀.     

Trace formulas for a class of Toeplitz-like operators

LetP _T denote projection onto the space of entire functions of exponential type ≦T which are square summable on the line relative to a measuredΔ and letG denote multiplication by a suitably restricted complex valued function,g. For a reasonably large class of measuresdΔ, which includes Lebesgue measuredγ, it is shown that trace {(P _TGP_T)ⁿ−P_TGⁿP_T} tends boundedly to a limit asT↑∞ and that the limit isindependent of the choice ofdΔ within the permitted class. This extends the range of validity of a formula due to Mark Kac who evaluated this limit in the special casedΔ=dγ using a different formalism.     

The Global Existence of Small Solutions to the Oldroyd-B Model

The Cauchy problem to the Oldroyd-B model is studied. In particular, it is shown that if the smooth solution (u, τ) to this system blows up at a finite time T*, then ∫₀ ^T* ‖▿u(t)‖ _L ^∞dt = ∞. Furthermore, the global existence of smooth solution to this system is given with small initial data.     

Brownian bridge on hyperbolic spaces and on homogeneous trees

Let B be the Brownian motion on a noncompact non Euclidean rank one symmetric space H. A typical examples is an hyperbolic space H _n , n > 2. For ν > 0, the Brownian bridge B ^(ν) of length ν on H is the process B _t , 0 ≤t≤ν, conditioned by B ₀ = B _ν = o, where o is an origin in H. It is proved that the process converges weakly to the Brownian excursion when ν→ + ∞ (the Brownian excursion is the radial part of the Brownian Bridge on ℝ³). The same result holds for the simple random walk on an homogeneous tree. Received: 4 December 1998 / Revised version: 22 January 1999     

Joint ergodicity and mixing

The study of jointly ergodic measure preserving transformations of probability spaces, begun in [1], is continued, and notions of joint weak and strong mixing are introduced. Various properties of ergodic and mixing transformations are shown to admit analogues for several transformations. The case of endomorphisms of compact abelian groups is particularly emphasized. The main result is that, given such commuting endomorphisms σ₁σ₂,...,σ, ofG, the sequence ((1/N)Σ _n=0 ^N−1 σ ₁ ⁿ f ¹·σ ₂ ⁿ f ²· ··· · σ _s ⁿ f ^sconverges inL ²(G) for everyf ₁,f ₂,…,f _s∈L ^∞(G). If, moreover, the endomorphisms are jointly ergodic, i.e., if the limit of any sequence as above is Π _i=1 ^s ∫ _G f ₁ d μ, where μ is the Haar measure, then the convergence holds also μ-a.e.     

Possible limit laws for entrance times of an ergodic aperiodic dynamical system

LetG denote the set of decreasingG: ℝ→ℝ withGэ1 on ]−∞,0], and ƒ ₀ ^∞ G(t)dt⩽1. LetX be a compact metric space, andT: X→X a continuous map. Let μ denone aT-invariant ergodic probability measure onX, and assume (X, T, μ) to be aperiodic. LetU⊂X be such that μ(U)>0. Let τ _U (x)=inf{k⩾1:T ^k xεU}, and defineG _U (t)=1/u(U)u({xεU:u(U)τU(x)>t),tεℝ We prove that for μ-a.e.x∈X, there exists a sequence (U _n ) _n≥1 of neighbourhoods ofx such that {x}=∩ _n U _n , and for anyG ∈G, there exists a subsequence (n _k ) _k≥1 withG _{U n k} ↑U weakly. We also construct a uniquely ergodic Toeplitz flowO(x ^∞,S, μ), the orbit closure of a Toeplitz sequencex ^∞, such that the above conclusion still holds, with moreover the requirement that eachU _n be a cylinder set. In memory of Anzelm Iwanik     

Largeness of the set of finite products in a semigroup

We investigate when the set of finite products of distinct terms of a sequence 〈x _n 〉 _n=1^∞ in a semigroup (S,⋅) is large in any of several standard notions of largeness. These include piecewise syndetic, central, syndetic, central*, and IP*. In the case of a “nice” sequence in (S,⋅)=(ℕ,+) one has that FS(〈x _n 〉 _n=1^∞) has any or all of the first three properties if and only if {x _n+1−∑ _t=1 ⁿ x _t :n∈ℕ} is bounded from above. N. Hindman acknowledges support received from the National Science Foundation via Grant DMS-0554803.