A Generalization of a Theorem of Liao

Let X be a metrizable space and let φ：R× X → X be a continuous flow on X. For any given {φt}-invariant Borel probability measure, this paper presents a {φt}-invariant Borel subset of X satisfying the requirements of the classical ergodic theorem for the contiImous flow （X, {φt}）. The set is more restrictive than the ones in the literature, but it might be more useful and convenient, particularly for non-uniformly hyperbolic systems and skew-product flows.     

Extension of Abel's Lemma with <Emphasis Type="Italic">q</Emphasis>-Series Implications

In previous work arising from the study of Ramanujan's Lost Notebook, a new Abel type lemma was proved. In this paper, we discuss extensions of this lemma and use it to prove many q-series identities. The first author was partially supported by NSF grant DMS–0200047. The second author was partially supported by FCT, Portugal, through program POCTI. 2000 Mathematics Subject Classification:Primary—33D15; Secondary—05A30     

Kähler–Ricci Flow With Small Initial Energy

In this paper, we prove that the K?hler–Ricci flow converges to a K?hler–Einstein metric when E ₁ energy is small. We also prove that E ₁ is bounded from below if and only if the K-energy is bounded from below in the canonical class. The first named author is partially supported by a NSF grant, while the third author was partially supported by a NSF supplement grant.     

Curvature and uniformization

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu-e ^2u=K_o(z) on general open surfaces. A few other topics are discussed, including boundary behavior of the conformal factore ^2u giving the Poincaré metric when the Riemann surface has smoothly bounded compact closure, and also a curvature equation proof of Koebe's disk theorem. Research supported in part by NSF Grant DMS-9971975 and also at MSRI by NSF grant DMS-9701755. Research supported in part by NSF Grant DMS-9877077     

On the product formulas for <Emphasis Type="Italic">C</Emphasis>-semigroups

In this paper we obtain a Chernoff-type approximation theorem for C-semigroups, from which we derive the Trotter product formula and the Post-Widder inversion formula for C-semigroups. The first author was supported by the NSF of China (Grant No. 10501032) and the NSFC-RFBR Programm (Grant No. 108011120015), and the second author by the NSF of China (Grant No. 10671079).     

A computational study of a gradient-based log-barrier algorithm for a class of large-scale SDPs

 The authors of this paper recently introduced a transformation [4] that converts a class of semidefinite programs (SDPs) into nonlinear optimization problems free of matrix-valued constraints and variables. This transformation enables the application of nonlinear optimization techniques to the solution of certain SDPs that are too large for conventional interior-point methods to handle efficiently. Based on the transformation, we proposed a globally convergent, first-order (i.e., gradient-based) log-barrier algorithm for solving a class of linear SDPs. In this paper, we discuss an efficient implementation of the proposed algorithm and report computational results on semidefinite relaxations of three types of combinatorial optimization problems. Our results demonstrate that the proposed algorithm is indeed capable of solving large-scale SDPs and is particularly effective for problems with a large number of constraints. Received: June 22, 2001 / Accepted: January 20, 2002 Published online: December 9, 2002 RID="†" ID="†"Computational results reported in this paper were obtained on an SGI Origin2000 computer at Rice University acquired in part with support from NSF Grant DMS-9872009. RID="⋆" ID="⋆"This author was supported in part by NSF Grants CCR-9902010, INT-9910084 and CCR-0203426 RID="⋆⋆" ID="⋆⋆"This author was supported in part by NSF Grants CCR-9902010, INT-9910084 and CCR-0203113 RID="⋆⋆⋆" ID="⋆⋆⋆"This author was supported in part by DOE Grant DE-FG03-97ER25331, DOE/LANL Contract 03891-99-23 and NSF Grant DMS-9973339. Key Words. semidefinite program – semidefinite relaxation – nonlinear programming – interior-point methods – limited memory quasi-Newton methods. Mathematics Subject Classification (1991): 90C06, 90C27, 90C30.     

System of Generalized Vector Quasi-Equilibrium Problems in Locally FC-Spaces

A new class of locally finite continuous topological spaces （for short, locally FC-spaces） and a class of system of generalized vector quasi-equilibrium problems are introduced. By applying a generalized Himmelberg type fixed point theorem for a set-valued mapping with KKM-property due to the author, a collectively fixed point and an equilibrium existence theorem of generalized game are first proved in locally FC-spaces. By applying our equilibrium existence theorem of generalized game, some new existence theorems of equilibrium points for the system of generalized vector quasi-equilibrium problems are proved in locally FC-spaces. These theorems improve, unify and generalize many known results in the literatures.     

The Passage Time Distribution for a Birth-and-Death Chain: Strong Stationary Duality Gives a First Stochastic Proof

A well-known theorem usually attributed to Keilson states that, for an irreducible continuous-time birth-and-death chain on the nonnegative integers and any d, the passage time from state 0 to state d is distributed as a sum of d independent exponential random variables. Until now, no probabilistic proof of the theorem has been known. In this paper we use the theory of strong stationary duality to give a stochastic proof of a similar result for discrete-time birth-and-death chains and geometric random variables, and the continuous-time result (which can also be given a direct stochastic proof) then follows immediately. In both cases we link the parameters of the distributions to eigenvalue information about the chain. We also discuss how the continuous-time result leads to a proof of the Ray–Knight theorem. Intimately related to the passage-time theorem is a theorem of Fill that any fastest strong stationary time T for an ergodic birth-and-death chain on {0,…,d} in continuous time with generator G, started in state 0, is distributed as a sum of d independent exponential random variables whose rate parameters are the nonzero eigenvalues of −G. Our approach yields the first (sample-path) construction of such a T for which individual such exponentials summing to T can be explicitly identified. Research of J.A. Fill was supported by NSF grant DMS–0406104 and by The Johns Hopkins University’s Acheson J. Duncan Fund for the Advancement of Research in Statistics.     

Distances between measures from 1-dimensional projections as implied by continuity of the inverse radon transform

Summary Distances between measures on IR ^d are determined from distances between their 1-dimensional projections. The method employed involves considering the 1-dimensional projections to be the Radon transform of the measures. Crucial to the main theorem is a continuity result for the inverse Radon transform. Focus is restricted to the Prohorov, dual bounded Lipschitz and d _k metrics which metrize weak convergence of probability measures. These metrics are related to each other and to the Sobolev norms. The d _k results extend from measures to generalized functions.Partially supported by NSF Grant No. MCS-81-01895Partially supported by NSF Grant No. MCS-82-01627 and support from the Mellon Foundation     

Mixed and Isoperimetric Estimates on the Log-Sobolev Constants of Graphs and Markov Chains

Two types of lower bounds are obtained on the log-Sobolev constants of graphs and Markov chains. The first is a mixture of spectral gap and logarithmic isoperimetric constant, the second involves the Gaussian isoperimetric constant. The sharpness of both types of bounds is tested on some examples. Product generalizations of some of these results are also briefly given. Received March 1, 1999/Revised July 17, 2000 RID="*" ID="*" Research supported in part by the NSF Grant No. DMS–9803239. The author greatly enjoyed the hospitality of CIMAT, Gto, Mexico, where part of this work was done.     

The Characterization of Finite Simple Groups, L3（3^2m-1）（m≥2）, by Their Element Orders

In this paper the following theorem is proved： Every group L3（q） for q = 3^（2m-1）（m≥2） is characterized by its set of element orders.     

Extreme value distribution of a recursive-type detector in a linear model

We study a CUSUM–type monitoring scheme designed to sequentially detect changes in the regression parameter of an underlying linear model. The test statistic used is based on recursive residuals. Main aim of this paper is to derive the limiting extreme value distribution under the null hypothesis of structural stability. The model assumptions are flexible enough to include rather general classes of error sequences such as augmented GARCH(1,1) processes. The result is underlined by an illustrative simulation study. Research partially supported by NSF grants DMS–0604670 and DMS–065242.     

A Quenched CLT for Super-Brownian Motion with Random Immigration

A quenched central limit theorem is derived for the super-Brownian motion with super-Brownian immigration, in dimension d≥4. At the critical dimension d=4, the quenched and annealed fluctuations are of the same order but are not equal. W. Hong was supported by the Program for New Century Excellent Talents in University (NCET) and NSFC (Grant No. 10121101). O. Zeitouni was partially supported by NSF grant DMS-0503775.     

Bounding the consistency strength of a five element linear basis

In [13] it was demonstrated that the Proper Forcing Axiom implies that there is a five element basis for the class of uncountable linear orders. The assumptions needed in the proof have consistency strength of at least infinitely many Woodin cardinals. In this paper we reduce the upper bound on the consistency strength of such a basis to something less than a Mahlo cardinal, a hypothesis which can hold in the constructible universe L. A crucial notion in the proof is the saturation of an Aronszajn tree, a statement which may be of broader interest. We show that if all Aronszajn trees are saturated and PFA(ω ₁) holds, then there is a five element basis for the uncountable linear orders. We show that PFA(ω ₂) implies that all Aronszajn trees are saturated and that it is consistent to have PFA(ω ₁) plus every Aronszajn tree is saturated relative to the consistency of a reflecting Mahlo cardinal. Finally we show that a hypothesis weaker than the existence of a Mahlo cardinal is sufficient to force the existence of a five element basis for the uncountable linear orders. The first author acknowledges a fellowship granted by the French ministry of research. The research of the second author was partially supported by the Centre de Rercerca Matemàtica of the Universitat Autònoma de Barcelona, and by NSF Grant DMS-0401603. The second author would also like to thank the third and fourth authors for bringing him to Boise and Paris respectively for further discussions. The third author was supported by NSF grants DMS-0401893 and DMS-0200671. The second and fourth authors would like to thank CIRM in Luminy for hosting them during a petit group de travaille, and to thank the others participants, Ralf Schindler and Ernest Schimmerling, for discussions on this topic.     

On the unsteady Poiseuille flow in a pipe

We show that in an unsteady Poiseuille flow of a Navier–Stokes fluid in an infinite straight pipe of constant cross-section, σ, the flow rate, F(t), and the axial pressure drop, q(t), are related, at each time t, by a linear Volterra integral equation of the second type, where the kernel depends only upon t and σ. One significant consequence of this result is that it allows us to prove that the inverse parabolic problem of finding a Poiseuille flow corresponding to a given F(t) is equivalent to the resolution of the classical initial-boundary value problem for the heat equation. G. P. Galdi: Partially supported by the NSF grant DMS–0404834. K. Pileckas: Supported by EC FP6 MCToK program SPADE2, MTKD–CT–2004–014508 A. L. Silvestre: Supported by FCT-Project POCI/MAT/61792/2004     

Harmonic Functions,Entropy, and a Characterization of the Hyperbolic Space

Let (M ⁿ ,g) be a compact Riemannian manifold with Ric ≥−(n−1). It is well known that the bottom of spectrum λ ₀ of its universal covering satisfies λ ₀≤(n−1)²/4. We prove that equality holds iff M is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy. The author was partially supported by NSF Grant 0505645.     

Hankel and Toeplitz–Schur multipliers

We study the problem of characterizing Hankel–Schur multipliers and Toeplitz–Schur multipliers of Schatten–von Neumann class for . We obtain various sharp necessary conditions and sufficient conditions for a Hankel matrix to be a Schur multiplier of . We also give a characterization of the Hankel–Schur multipliers of whos e symbols have lacunary power series. Then the results on Hankel–Schur multipliers are used to obtain a characterization of the Toeplitz–Schur multipliers of . Finally, we return to Hankel–Schur multipliers and obtain new results in the case when the symbol of the Hankel matrix is a complex measure on the unit circle. Received: 16 February 2001 / revised version: 2 December 2001 / Published online: 27 June 2002 The first author is partially supported by Grant 99-01-00103 of Russian Foundation of Fundamental Studies and by Grant 326.53 of Integration. The second author is partially supported by NSF grant DMS 9970561.     

The X-Ray Transform for a Generic Family of Curves and Weights

We study the weighted integral transform on a compact manifold with boundary over a smooth family of curves Γ. We prove generic injectivity and a stability estimate under the condition that the conormal bundle of Γ covers T ^* M. Second author was partly supported by NSF Grant DMS-0400869; third author was partly supported by NSF and a Walker Family Endowed Professorship.     

On the structure of stationary sets

We isolate several classes of stationary sets of [k]ωand investigate implications among them. Under a large cardinal assumption, we prove a structure theorem for stationary sets.     

Wavelet sets in ℝ n

A congruency theorem is proven for an ordered pair of groups of homeomorphisms of a metric space satisfying an abstract dilation-translation relationship. A corollary is the existence of wavelet sets, and hence of single-function wavelets, for arbitrary expansive matrix dilations on L ² (ℝ ⁿ). Moreover, for any expansive matrix dilation, it is proven that there are sufficiently many wavelet sets to generate the Borel structure ofℝ ⁿ. The second author is supported in part by NSF Grant DMS-9401544. The third author was a Graduate Research Assistant at Workshop in Linear Analysis and Probability, Texas A&M University.