Preprints for Stephen Montgomery-Smith

These are most of my preprints. If you want other preprints, please email me at stephen@math.missouri.edu, or check the preprint server at Los Alamos, but chances are I don't have it in electronic form. I would like to thank the NSF for their support of the research that is contained in these papers.

Many of the abstracts listed below contain mathematical symbols. I have found that these symbols do not render well on old versions of Netscape, and a few of the symbols do not render well on Internet Explorer. However they all seem to render very well on more recent versions of Netscape and Mozilla.

• List of Publications. (tex, dvi, ps, pdf, html).
• List of Publications on AMS Server (you need a pawprint account at the University of Missouri to use this).
• Preprints on the Preprint Server at Los Alamos.
• (with Yong Gan and Zhen Chen) A Numerical Scheme for Coupled CFD and CSD Simulation of 3-D Fluid-Membrane Interactions. Preprint. For coupled computational fluid dynamics (CFD) and computational solid dynamics (CSD) simulation of the three-dimensional fluid-membrane interaction, a numerical scheme is developed by improving the material point method (MPM). In order to compute the stress state at any membrane point, a plane stress assumption is made in the local tangent plane consisting of membrane points, and a simple procedure is proposed to find the effective point connectivity information for determining the orientation of the local tangent plane. With an iterative algorithm, the existing MPM is improved for formulating fluid points so that fluid dynamics problems with strong shocks could be better simulated. The use of an Eulerian background mesh for solving the momentum equations enables the MPM to automatically handle fluid-membrane interactions without requiring special treatment. Three examples are used to demonstrate the features of the proposed numerical scheme, and to compare them with the analytical and FEM solutions. It appears from the comparison that the proposed procedure is robust and efficient to simulate the three-dimensional fluid-membrane interactions. (pdf, figures (zipped).)
• (with T. Schürmann) Unbiased Estimators for Entropy and Class Number. Preprint. We introduce unbiased estimators for the Shannon entropy and the class number, in the situation that we are able to take sequences of independent samples of arbitrary length. (tex, dvi, ps, pdf.)
• On a Bayesian Approach to Estimating Class Number. Preprint. We examine a Bayesian approach to estimating the number of classes in a population, in the situation that we are able to take many independent samples from an infinite population. (tex, dvi, ps, pdf.)
• (with S. Geiss and E. Saksman) On singular integral and martingale transforms. Preprint. Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD(p)-constant of a Banach space X equals the norm of the real (or the imaginary) part of the Beurling-Ahlfors singular integral operator, acting on the X-valued L^p-space on the plane. Moreover, replacing equality by a linear equivalence, this is found to be the typical property of even multipliers. A corresponding result for odd multipliers and the Hilbert transform is given. (tex, dvi, ps, pdf.)
• Conditions implying regularity of the three dimensional Navier-Stokes equation. Applications of Mathematics 50, (2005), 451-464. We obtain logarithmic improvements for conditions for regularity of the Navier-Stokes equation, similar to those of Prodi-Serrin or Beale-Kato-Majda. Some of the proofs make use of a stochastic approach involving Feynman-Kac like inequalities. As part of the our methods, we give a different approach to a priori estimates of Foias, Guillope and Temam. (tex, dvi, ps, pdf.)
• (with Nigel Kalton, Krzysztof Oleszkiewicz and Yuri Tomilov) Power-bounded operators and related norm estimates. Journal of London Math. Soc. 70, (2004), 463-478. We consider whether L = lim sup_n→∞ n ∥Tⁿ⁺¹-Tⁿ∥ < ∞ implies that the operator T is power bounded. We show that this is so if L < 1/e, but it does not necessarily hold if L = 1/e. As part of our methods, we include an improvement of a result of Esterle, showing that if σ(T) = {1} and T ≠ I, then lim inf_n→∞ n ∥Tⁿ⁺¹-Tⁿ∥ ≥ 1/e. The constant 1/e is sharp. Finally we describe a way to create many generalizations of Esterle's result, and also give many conditions on an operator which imply that its norm is equal to its spectral radius. (tex, dvi, ps, pdf.)
• (with Shih-Chi Shen) An Extension to the Tangent Sequence Martingale Inequality. For each 1 < p < ∞, there exists a positive constant c_p, depending only on p, such that the following holds. Let (d_k), (e_k) be real-valued martingale difference sequences. If for for all bounded nonnegative predictable sequences (s_k) and all positive integers k we have E[s_k∨|e_k|] ≤ E[s_k∨|d_k|] then we have ∥∑ e_k∥_p ≤ c_p ∥∑ d_k∥_p. (tex, dvi, ps, pdf.)
• Rearrangement Invariant Norms of Symmetric Sequence Norms of Independent Sequences of Random Variables. Israel Journal of Mathematics, 131, (2002), 51-60. Let X₁, X₂,...,X_n be a sequence of independent random variables, let M be a rearrangement invariant space on the underlying probability space, and let N be a symmetric sequence space. This paper gives an approximate formula for the quantity ∥ ∥(X_i)∥_N∥_M whenever L_q embeds into M for some finite q. This extends work of Johnson and Schechtman who tackled the case when N = l_p, and recent work of Gordon, Litvak, Schütt and Werner who obtained similar results for Orlicz spaces. (tex, dvi, ps, pdf.)
• (with Milan Pokorný) A counterexample to the smoothness of the solution to an equation arising in fluid mechanics. Commentationes Mathematicae Universitatis Carolinae, 43, 1, (2002), 61-75. We analyze the equation coming from the Eulerian-Lagrangian description of fluids. We discuss a couple of ways to extend this notion to viscous fluids. The main focus of this paper is to discuss the first way, due to Constantin. We show that this description can only work for short times, after which the "back to coordinates map" may have no smooth inverse. Then we briefly discuss a second way that uses Brownian motion. We use this to provide a plausibility argument for the global regularity for the Navier-Stokes equation. (tex, dvi, ps, pdf, html.)
• (with Nakhlé Asmar) Decomposition of analytic measures on groups and measure spaces. Studia Math, 146, (2001), 261-284. This paper provides a new approach to proving generalizations of the F.&M. Riesz Theorem, for example, the result of Helson and Lowdenslager, the result of Forelli (and de Leeuw and Glicksberg), and more recent results of Yamagushi. We study actions of a locally compact abelian group with ordered dual onto a space of measures, and consider those measures that are analytic, that is, the spectrum of the action on the measure is contained within the positive elements of the dual of the group. The classical results tell us that the singular and absolutely continuous parts of the measure (with respect to a suitable measure) are also analytic. The approach taken in this paper is to adopt the transference principle developed by the authors and Saeki in another paper, and apply it to martingale inequalities of Burkholder and Garling. In this way, we obtain a decomposition of the measures, and obtain the above mentioned results as corollaries. (tex, dvi, ps, pdf, html.)
• (with David Greaves) Unforgeable Marker Sequences. (Computer Science) A binary number of n bits consists of an ordered sequence of n digits taken from the set {0,1}. A sequence is said to be an unforgeable marker if all subsequences of consecutive digits starting at the left-hand end are dissimilar from the sequence of the same length which ends at the right-hand end. Unforgeable marker sequences are so called because, when misaligned in a shift-register or other view port of the correct length, there is no possibility of adjacent random digits impersonating the true sequence. Such sequences are used for frame alignment purposes in serial data communications systems. (tex, dvi, ps, pdf, html.) Since we wrote this paper, we found out that these sequences had been studied by others, as bifix-free words, or as autocorrelations. We refer the reader to On-Line Encyclopedia of Integer Sequences A003000 and http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/autocorrelation.
• Finite time blow up for a Navier-Stokes like equation. Proc. A.M.S., 129, (2001), 3017-3023. We consider an equation similar to the Navier-Stokes equation. We show that there is initial data that exists in every Triebel-Lizorkin or Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the solution is in no Triebel-Lizorkin or Besov space (and hence in no Lebesgue or Sobolev space). The purpose is to show the limitations of the so called semigroup method for the Navier-Stokes equation. We also consider the possibility of existence of solutions with initial data in the Besov space B_∞^-1,∞. We give initial data in this space for which there is no reasonable solution for the Navier-Stokes like equation. (tex, dvi, ps, pdf, html.)
• (Pawel Hitczenko) Measuring the magnitude of sums of independent random variables. Annals of Probability, 29, (2001), 447-466. This paper considers how to measure the magnitude of the sum of independent random variables in several ways. We give a formula for the tail distribution for sequences that satisfy the so called Lévy property. We then give a connection between the tail distribution and the pth moment, and between the pth moment and the rearrangement invariant norms. (tex, dvi, ps, pdf, html.)
• (with Alexander Pruss) A comparison inequality for sums of independent random variables. J.M.A.A., 254, (2001), 35-42. We give a comparison inequality that allows one to estimate the tail probabilities of sums of independent Banach space valued random variables in terms of those of independent identically distributed random variables. More precisely, let X₁,...,X_n be independent Banach-valued random variables. Let I be a random variable independent of X₁,...,X_n and uniformly distributed over {1,...,n}. Put Z₁ = X_I, and let Z₂,...,Z_n be independent identically distributed copies of Z₁. Then, P(∥X₁+...+X_n∥ ≥ λ) ≤ c P(∥Z₁+...+Z_n∥ ≥ λ/c) for all λ > 0, where c is an absolute positive constant. (tex, dvi, ps, pdf, actual article.)
• (with Evgueni Semenov) Embeddings of rearrangement invariant spaces that are not strictly singular. Positivity, 4, (2000), 397-404. We give partial answers to the following conjecture: the natural embedding of a rearrangement invariant space E into L₁([0,1]) is strictly singular if and only if G does not embed into E continuously, where G is the closure of the simple functions in the the Orlicz space L_Φ with Φ(x) = exp(x²)-1. (tex, dvi, ps, pdf. The tex file requires kluwer style files available here (kluwer.tgz), or in other formats at http://www.wkap.nl/kaphtml.htm/STYLEFILES.)
• Global regularity of the Navier-Stokes equation on thin three dimensional domains with periodic boundary conditions. Electronic J. Differential Equations, 1999, (1999), no. 11, 1-19. This paper gives another version of results due to Raugel and Sell, and similar results due to Moise, Temam and Ziane, that state the following: the solution of the Navier-Stokes equation on a thin 3 dimensional domain with periodic boundary conditions has global regularity, as long as there is some control on the size of the initial data and the forcing term, where the control is larger than that obtainable via "small data" estimates. The approach taken is to consider the three dimensional equation as a perturbation of the equation when the vector field does not depend upon the coordinate in the thin direction. (tex, dvi, ps, pdf).
• (with Stephen Clark, Yuri Latushkin and Tim Randolph) Stability radius and internal versus external stability in Banach spaces: an evolution semigroup approach. S.I.A.M. J. of Control Optim, 38, (2000), 1757-1793. In this paper the theory of evolution semigroups is developed and used to provide a framework to study the stability of general linear control systems. These include time-varying systems modeled with unbounded state-space operators acting on Banach spaces. This approach allows one to apply the classical theory of strongly continuous semigroups to time-varying systems. In particular, the complex stability radius may be expressed explicitly in terms of the generator of a (evolution) semigroup. Examples are given to show that classical formulas for the stability radius of an autonomous Hilbert-space system fail in more general settings. Upper and lower bounds on the stability radius are provided for these general systems. In addition, it is shown that the theory of evolution semigroups allows for a straightforward operator-theoretic analysis of internal stability as determined by classical frequency-domain and input-output operators, even for nonautonomous Banach-space systems. (tex, dvi, ps, pdf, actual article. The tex file also requires siamltex.cls and siam10.clo.)
• Concrete representation of martingales. Electronic J. Probability, 3, (1998), Paper 15. Let (f_n) be a mean zero vector valued martingale sequence. Then there exist vector valued functions (d_n) from [0,1]ⁿ such that ∫₀¹ d_n(x₁,...,x_n) dx_n = 0 for almost all x₁,...,x_n-1, and such that the law of (f_n) is the same as the law of (∑_k=1ⁿ d_n(x₁,...,x_n)). Similar results for tangent sequences and sequences satisfying condition (C.I.) are presented. We also present a weaker version of a result of McConnell that provides a Skorohod like representation for vector valued martingales. (tex, dvi, ps, pdf).
• (with Pawel Hitczenko) A note on sums of independent random variables. Advances in Stochastic Inequalities, Ed.: T. Hill and C. Houdre, Contemporary Mathematics 234, A.M.S., Providence R.I., 1999. In this note a two sided bound on the tail probability of sums of independent, and either symmetric or nonnegative, random variables is obtained. We utilize a recent result by Latala on bounds of moments of such sums. We also give a new proof of Latala's result for nonnegative random variables, and improve one of the constants in his inequality. (tex, dvi, ps, pdf. The tex file also requires ams-p.sty, ams-spec.sty and conm-p.sty.)
• (with Al Baernstein) Some conjectures about integral means of ∂f and ∂¯f. Complex Analysis and Differential Equations, edited by C.Kiselman, Acta Universitatis Upsaliensis C., Volume 64 (1999), 92-109. We discuss some conjectural inequalities concerning a problem from the calculus of variations, namely that rank 1 convex functions are quasi-convex. An affirmative answer would also give the best constants for the Beurling-Ahlfors operator that appears in the theory of quasi-conformal mappings on the plane. (tex, dvi, ps, pdf).
• (with Nakhlé Asmar and Sadahiro Saeki) Transference in Spaces of Measures. J. Functional Analysis 165, (1999), 1-23. The transference theory for L^p spaces of Calderon, Coifman, and Weiss is a powerful tool with many applications to singular integrals, ergodic theory, and spectral theory of operators. Transference methods afford a unified approach to many problems in diverse areas, which before were proved by a variety of methods. The purpose of this paper is to bring about a similar approach to the study of measures. Specifically, deep results in classical harmonic analysis and ergodic theory, due to Bochner, de Leeuw-Glicksberg, Forelli, and others, are all extensions of the classical F.&M. Riesz Theorem. We will show that all these extensions are obtainable via our new transference principle for spaces of measures. (tex, dvi, ps, pdf, html, actual article).
• (with Evgueni Semenov) Rearrangements and Operators. 25 Years of Voronezh Winter Mathematical School, Proceedings in honor of S. Krein, A.M.S. Let m = (m_i,j) be an n by n matrix. Pick a permutation π of {1,2,...,n} at random. Kwapien and Schütt considered the problem of finding E(∥(m_i,π(i))∥_p^q)^1/q. In this paper, we generalize their results to rearrangement invariant spaces. We also consider the property of D and D* convexity for rearrangement invariant spaces. (tex, dvi, ps, pdf. The tex file also requires ams-p.sty, ams-spec.sty, amsppt.sti, amsppt.sty and gen-p.sty).
• Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equations. Duke Math J. 91 (1998), 393-408. Let u(x,t) be the solution of the Schroedinger or wave equation with L₂ initial data. We provide counterexamples to plausible conjectures involving the decay in t of the BMO norm of u(t,-). The proofs make use of random methods, in particular, Brownian motion. (tex, dvi, ps, pdf.) Since this paper was written, the unsolved problem remaining in this paper has been solved by Keel and Tao. You may find a copy of their paper at either of their web sites. (Keel, Tao).
• (with Loukas Grafakos and Olexei Motrunich) A sharp estimate for the Hardy-Littlewood maximal function. Studia Math, 134, (1999), 57-67. The best constant in the usual L^p norm inequality for the centered Hardy-Littlewood maximal function on R¹ is obtained for the class of all "peak-shaped" functions. A positive function on the line is called "peak-shaped" if it is positive and convex except at one point. The techniques we use include convexity and an adaptation of the standard Euler-Langrange variational method. (tex, dvi, ps, pdf.)
• (with Loukas Grafakos) Best constants for uncentered maximal functions. Bul. London Math. Soc., 29, (1997), 60-64. We precisely evaluate the operator norm of the uncentered Hardy-Littlewood maximal function on L^p(R¹), showing that it is the unique positive root of the polynomial (p-1)x ^p-px ^p-1-1. Consequently, we compute the operator norm of the "strong" maximal function on L^p(Rⁿ), and we observe that the operator norm of the uncentered Hardy-Littlewood maximal function over balls on L^p(Rⁿ) grows exponentially as n → ∞. (tex, dvi, ps, pdf.)
• (with Alexander Koldobsky) Inequalities of correlation type for symmetric stable random vectors. Stat. and Probab. Letters, 28, (1996), 485-490. We point out a certain class of functions f and g for which random variables f(X₁,...,X_m) and g(X_m+1,...,X_k) are non-negatively correlated for any symmetric jointly stable random variables X_i. We also show another result that is related to the correlation problem for Gaussian measures of symmetric convex sets. (tex, dvi, ps, pdf.)
• (with Nakhlé Asmar and Annela Kelly) Vector-valued weakly analytic measures. Hokkaido Math. J., 27, (1998), 457-473. A celebrated result of Forelli extends the classical F.&M. Riesz Theorem to representations on spaces of Baire measures on a locally compact Hausdorff topological space. We extend these results to representations on vector valued measures, using methods previously developed by two of the authors. The results contained herein complement a result of Ryan. Our paper is not based upon Forelli's result or methods. (tex, dvi, ps, pdf.)
• (with Nakhlé Asmar) A transference theorem for ergodic H¹. Quarterly J. of Math. 48, (1997), 417-430. We extend the basic transference theorem for convolution operators on L^p spaces of Coifman and Weiss to H¹ spaces. (tex, dvi, ps, pdf.)
• (with Nakhlé Asmar) Hardy martingales and Jensen's Inequality. Bull. Australian Math. Soc., 55, (1997), 185-195. Hardy martingales were introduced by Garling and used to study analytic functions on the N-dimensional torus T^N, where analyticity is defined using a lexicographic order on the dual group Z^N. We show how, by using basic properties of orders on Z^N, we can apply Garling's method in the study of analytic functions on an arbitrary compact abelian group with an arbitrary order on its dual group. We illustrate our approach by giving a new and simple proof of a famous generalized Jensen's Inequality due to Helson and Lowdenslager. (tex, dvi, ps, pdf.)
• (with Nakhlé Asmar) Analytic measures and Bochner measurability. Bull. Sc. Math., 122, (1998), 39-66. Let Σ be a σ-algebra over Ω, and let M(Σ) denote the Banach space of complex measures. Consider a representation T_t for t∈R acting on M(Σ). We show that under certain, very weak hypotheses, that if for a given μ∈M(Σ) and all A∈Σ the map t→T_tμ(A) is in H^∞(R), then it follows that the map t→T_tμ is Bochner measurable. The proof is based upon the idea of the Analytic Radon Nikodym Property. Straightforward applications yield a new and simpler proof of Forelli's main result concerning analytic measures (Analytic and quasi-invariant measures, Acta Math., 118 (1967), 33--59). (tex, dvi, ps, pdf.)
• (with Nakhlé Asmar) On a weak type (1,1) inequality for a maximal conjugate function. Studia Math. 125, (1997), 13-21. In a celebrated paper, Burkholder, Gundy, and Silverstein used Brownian motion to derive a maximal function characterization of H^p spaces for 0 < p < ∞. In this paper, we show that their method extends to higher dimensions and yields a dimension-free weak type (1,1) estimate for a conjugate function on the N-dimensional torus. (tex, dvi, ps, pdf.)
• (with Nakhlé Asmar) Hahn's Embedding Theorem for orders and harmonic analysis on groups with ordered duals. Colloq. Math. 70, (1996), 235-252. Let G be a locally compact abelian group whose dual group Γ contains a Haar measurable order P. Using the order P we define the conjugate function operator on L^p(G), 1 ≤ p < ∞, as was done by Helson. We will show how to use Hahn's Embedding Theorem for orders and the ergodic Hilbert transform to study the conjugate function. Our approach enables us to define a filtration of the Borel σ-algebra on G, which in turn will allow us to introduce tools from martingale theory into the analysis on groups with ordered duals. We illustrate our methods by describing a concrete way to construct the conjugate function in L^p(G). This construction is in terms of an unconditionally convergent conjugate series whose individual terms are constructed from specific ergodic Hilbert transforms. We also present a study of the square function associated with the conjugate series. (tex, dvi, ps, pdf.)
• (with Nakhlé Asmar and Brian Kelly) A Note on UMD Spaces and Transference in Vector-valued Function Spaces. Proc. Edin. Math. Soc. 39, (1996), 485-490. We introduce the notion of an ACF space, that is, a space for which a generalized version of M. Riesz's theorem for conjugate functions with values in the Banach space is bounded. We use transference to prove that spaces for which the Hilbert transform is bounded, i.e. X∈HT, are ACF spaces. We then show that Bourgain's proof of X∈HT⇒X∈UMD is a consequence of this result. (tex, dvi, ps, pdf.)
• Boyd Indices of Orlicz-Lorentz Spaces. Function Spaces, The Second Conference, Ed: K. Jarosz, 321-334, Marcel Dekker, New York, 1995. Orlicz-Lorentz spaces provide a common generalization of Orlicz spaces and Lorentz spaces. In this paper, we investigate their Boyd indices. Bounds on the Boyd indices in terms of the Matuszewska-Orlicz indices of the defining functions are given. Also, we give an example to show that the Boyd indices and Zippin indices of an Orlicz-Lorentz space need not be equal, answering a question of Maligranda. Finally, we show how the Boyd indices are related to whether an Orlicz-Lorentz space is p-convex or q-concave. (tex, dvi, ps, pdf. The LaTeX file also requires the file homegrownmarcdek.sty.)
• The Hardy Operator and Boyd Indices. Interaction between Probability, Harmonic Analysis and Functional Analysis, Ed: N. Kalton, S.J. Montgomery-Smith, E. Saab, Lecture Notes in Pure and Appl. Math, 175, Marcel Dekker, New York, 1995. We give necessary and sufficient conditions for the Hardy operator to be bounded on a rearrangement invariant quasi-Banach space in terms of its Boyd indices. (tex, dvi, ps, pdf. The LaTeX file also requires the file homegrownmarcdek.sty.)
• (with Pawel Hitczenko and Krzysztof Oleszkiewicz) Moment inequalities for sums of certain independent symmetric random variables. Studia Math. 123, (1997), 15-42. This paper gives upper and lower bounds for moments of sums of independent random variables (X_k) which satisfy the condition that P(|X_k| ≥ t) = exp(-N_k(t)), where N_k are concave functions. As a consequence we obtain precise information about the tail probabilities of linear combinations of independent random variables for which N(t) = |t|^r for some fixed 0 < r ≤ 1. This complements work of Gluskin and Kwapien who have done the same for convex functions N. (tex, dvi, ps, pdf.)
• (with Pawel Hitczenko) Tangent Sequences in Orlicz and Rearrangement Invariant Spaces. Proc. Camb. Phil. Soc. 119, (1996), 91-101. Let (f_n) and (g_n) be two sequences of random variables adapted to an increasing sequence of σ-algebras (F_n) such that the conditional distributions of f_n and g_n given F_n-1 coincide, and such that the sequence (g_n) is conditionally independent. Then it is known that ∥∑ f_n∥_p ≤ C ∥∑ g_n∥_p where the constant C is independent of p. The aim of this paper is to extend this result to certain classes of Orlicz and rearrangement invariant spaces. This paper includes fairly general techniques for obtaining rearrangement invariant inequalities from Orlicz norm inequalities. (tex, dvi, ps, pdf.)
• Stability and Dichotomy of Positive Semigroups on L_p. Proc. A.M.S. 8, (1996), 2433-2437. A new proof of a result of Lutz Weis is given, that states that the stability of a positive strongly continuous semigroup (e^tA)_t≥0 on L_p may be determined by the quantity s(A). We also give an example to show that the dichotomy of the semigroup may not always be determined by the spectrum σ(A). (tex, dvi, ps, pdf.)
• (with Carmen Chicone and Yuri Latushkin) The Annular Hull Theorems for the Kinematic Dynamo Operator for an Ideally Conducting Fluid. Indiana J. 45, (1996), 361-379. The group generated by the kinematic dynamo operator in the space of continuous divergence-free sections of the tangent bundle of a smooth manifold is studied. As shown in previous work, if the underlying Eulerian flow is aperiodic, then the spectrum of this group is obtained from the spectrum of its generator by exponentiation, but this result does not hold for flows with an open set of periodic trajectories. In the present paper, we consider Eulerian vector fields with periodic trajectories and prove the following annular hull theorems: The spectrum of the group belongs to the annular hull of the exponent of the spectrum of the kinematic dynamo operator, that is to the union of all circles centered at the origin and intersecting this set. Also, the annular hull of the spectrum of the group on the space of divergence free vector fields coincides with the smallest annulus, containing the spectrum of the group on the space of all continuous vector fields. As a corollary, the spectral abscissa of the generator coincides with the growth bound for the group. (tex, dvi, ps, pdf.)
• (with Yuri Latushkin and Tim Randolph) Evolutionary semigroups, spectral mapping theorems, linear skew-product flows, exponential dichotomy. J. Diff. Eq. 125, (1996), 73-116. We study evolutionary semigroups generated by a strongly continuous semi-cocycle over a locally compact metric space acting on Banach fibers. This setting simultaneously covers evolutionary semigroups arising from nonautonomuous abstract Cauchy problems and C₀-semigroups, and linear skew-product flows. The spectral mapping theorem for these semigroups is proved. The hyperbolicity of the semigroup is related to the exponential dichotomy of the corresponding linear skew-product flow. To this end a Banach algebra of weighted composition operators is studied. The results are applied in the study of: "roughness" of the dichotomy, dichotomy and solutions of nonhomogeneous equations, Green's function for a linear skew-product flow, "pointwise" dichotomy versus "global" dichotomy, and evolutionary semigroups along trajectories of the flow. (tex, dvi, ps, pdf.)
• (with Carmen Chicone and Yuri Latushkin) The Spectrum of the Kinematic Dynamo Operator for an Ideally Conducting Fluid. Commun. Math. Phys. 173, (1995), 379-400. The spectrum of the kinematic dynamo operator for an ideally conducting fluid and the spectrum of the corresponding group acting in the space of continuous divergence free vector fields on a compact Riemannian manifold are described. We prove that the spectrum of the kinematic dynamo operator is exactly one vertical strip whose boundaries can be determined in terms of the Lyapunov-Oseledets exponents with respect to all ergodic measures for the Eulerian flow. Also, we prove that the spectrum of the corresponding group is obtained from the spectrum of its generator by exponentiation. In particular, the growth bound for the group coincides with the spectral bound for the generator. (tex, dvi, ps, pdf.)
• (with Yuri Latushkin) Evolutionary Semigroups and Lyapunov Theorems in Banach Spaces. J. Func. Anal. 127, (1995), 173-197. We present a spectral mapping theorem for continuous semigroups of operators on any Banach space E. The condition for the hyperbolicity of a semigroup on E is given in terms of the generator of an evolutionary semigroup acting in the space of E-valued functions. The evolutionary semigroup generated by the propagator of a nonautonomous differential equation in E is also studied. A "discrete" technique for the investigating of the evolutionary semigroup is developed and applied to describe the hyperbolicity (exponential dichotomy) of the nonautonomuos equation. (tex, dvi, ps, pdf.)
• (with Yuri Latushkin) Lyapunov theorems for Banach spaces. Bull. Amer. Math. Soc. (N.S.) 31 (1994) 44-49. We present a spectral mapping theorem for semigroups on any Banach space E. From this, we obtain a characterization of exponential dichotomy for nonautonomous differential equations for E-valued functions. This characterization is given in terms of the spectrum of the generator of the semigroup of evolutionary operators. (tex, dvi, ps, pdf.)
• Comparison of Sums of independent Identically Distributed Random Variables. Prob. and Math. Stat. 14, (1993), 281-285. Let S_k be the k-th partial sum of Banach space valued independent identically distributed random variables. In this paper, we compare the tail distribution of ∥S_k∥ with that of ∥S_j∥, and deduce some tail distribution maximal inequalities. Theorem: There is universal constant c such that for j < k we have Pr(∥S_j∥ > t) ≤ c Pr(∥S_k∥ > t/c). (tex, dvi, ps, pdf.)
• (with Victor de la Peña) Decoupling Inequalities for the Tail Probabilities of Multivariate U-statistics. Annals Prob. 23, (1995), 806-816. In this paper the following result, which allows one to decouple U-Statistics in tail probability, is proved in full generality. Theorem 1. Let X_i be a sequence of independent random variables taking values in a measure space S, and let f_i1...ik be measurable functions from S^k to a Banach space B. Let (X_i^(j)) be independent copies of (X_i). The following inequality holds for all t ≥ 0 and all n ≥ 2
          P(∥∑_{1≤i1≠...≠ik≤n} f_i1...ik(X_i1,...,X_ik)∥ ≥ t) ≤ C_k P(C_k∥∑_{1≤i1≠...≠ik≤n} f_i1...ik(X_i1⁽¹⁾,...,X_ik^(k))∥ ≥ t).
  Furthermore, the reverse inequality also holds in the case that the functions {f_i1...ik} satisfy the symmetry condition
          f_i1...ik(X_i1,...,X_ik) = f_{iπ(1)...iπ(k)}(X_iπ(1),...,X_iπ(k))
  for all permutations π of {1,...,k}. Note that the expression i₁≠...≠i_k means that i_r ≠ i_s for r ≠ s. Also, C_k is a constant that depends only on k. (tex, dvi, ps, pdf.)
• (with Victor de la Peña) Bounds on the tail probability of U-statistics and quadratic forms. Bull. Amer. Math. Soc. (N.S.) 31 (1994) 223-227. The authors announce a general tail estimate, called a decoupling inequality, for a symmetrized sum of non-linear k-correlations of n > k independent random variables. (tex, dvi, ps, pdf.)
• (with Victor de la Peña and Jerzy Szulga) Contraction and decoupling inequalities for multilinear forms and u-statistics. Annals Prob., 22, (1994), 1745-1765. We prove decoupling inequalities for random polynomials in independent random variables with coefficients in vector space. We use various means of comparison, including rearrangement invariant norms (e.g., Orlicz and Lorentz norms), tail distributions, tightness, hypercontractivity, etc. (tex, dvi, ps, pdf.)
• The Distribution of Non-Commutative Rademacher Series. Math. Ann. 302, (1995), 395-416. We give a formula for the tail of the distribution of the non-commutative Rademacher series, which generalizes the result that is already available in the commutative case. As a result, we are able to calculate the norm of these series in many rearrangement invariant spaces, generalizing work of Pisier and Rodin and Semyonov. (tex, dvi, ps, pdf.)
• (with Stephen Dilworth) The distribution of vector-valued Rademacher series. Annals Prob. 21, (1993), 2046-2052. Let X = ∑ ε_nx_n be a Rademacher series with vector-valued coefficients. We obtain an approximate formula for the distribution of the random variable ∥X∥ in terms of its mean and a certain quantity derived from the K-functional of interpolation theory. Several applications of the formula are given. (tex, dvi, ps, pdf.)
• (with Nigel Kalton) Set-functions and factorization. Arch. Math. 61, (1993), 183-200. If ϕ is a submeasure satisfying an appropriate lower estimate we give a quantitative result on the total mass of a measure μ satisfying 0 ≤ μ ≤ ϕ. We give a dual result for supermeasures and then use these results to investigate convexity on non-locally convex quasi-Banach lattices. We then show how to use these results to extend some factorization theorems due to Pisier to the setting of quasi-Banach spaces. We conclude by showing that if X is a quasi-Banach space of cotype two then any operator T:C(Ω)→)X is 2-absolutely summing and factors through a Hilbert space and discussing general factorization theorems for cotype two spaces. (tex, dvi, ps, pdf. The tex file also requires vanilla.sty.)
• (with Nakhlé Asmar) On the distribution of Sidon series. Arkiv Mat. 31, (1993), 13-26. Let B denote an arbitrary Banach space, G a compact abelian group with Haar measure μ and dual group Γ. Let E be a Sidon subset of Γ with Sidon constant S(E). Let r_n denote the n-th Rademacher function on [0, 1]. We show that there is a constant c, depending only on S(E), such that, for all α > 0:
          c^-1 P[|∑ a_nr_n| ≥ cα] ≤ μ[|∑ a_nγ_n| ≥ α] ≤ c P[|∑ a_nr_n| ≥ c^-1α]
  where a₁,...,a_N are arbitrary elements of B, and γ₁,...,γ_N are arbitrary elements of E. We prove a similar result for Sidon subsets of dual objects of compact groups, and apply our results to obtain new lower bounds for the distribution functions of scalar-valued Sidon series. We also note that either one of the above inequalities, even in the scalar case, characterizes Sidon sets. (tex, dvi, ps, pdf.)
• Comparison of Orlicz-Lorentz spaces. Stud. Math. 103 (2), (1992), 161-189. Orlicz-Lorentz spaces provide a common generalization of Orlicz spaces and Lorentz spaces. They have been studied by many authors, including Mastylo, Maligranda, and Kaminska. In this paper, we consider the problem of comparing the Orlicz-Lorentz norms, and establish necessary and sufficient conditions for them to be equivalent. As a corollary, we give necessary and sufficient conditions for a Lorentz-Sharpley space to be equivalent to an Orlicz space, extending results of Lorentz and Raynaud. We also give an example of a rearrangement invariant space that is not an Orlicz-Lorentz space. (tex, dvi, ps, pdf.)
• Orlicz-Lorentz Spaces. Proceedings of the Orlicz Memorial Conference, (Ed. P. Kranz and I. Labuda), Oxford, Mississippi (1991). This is an article summerizing some of my work on Orlicz-Lorentz Spaces. (tex, dvi, ps, pdf.)
• (with Paulette Saab) p-summing operators on injective tensor products of spaces. B. Royal Soc. Edin. 120A, (1992), 283-296. Let X, Y and Z be Banach spaces, and let Π_p(Y,Z) (1 ≤ p < ∞) denote the space of p-summing operators from Y to Z. We show that, if X is a £_∞-space, then a bounded linear operator T:X⊗_εY→Z is 1-summing if and only if a naturally associated operator T^#:X→Π₁(Y,Z) is 1-summing. This result need not be true if X is not a £_∞-space. For p > 1, several examples are given with X = C[0,1] to show that T^# can be p-summing without T being p-summing. Indeed, there is an operator T on C[0,1]⊗_εl₁ whose associated operator T^# is 2-summing, but for all N∈N, there exists an N-dimensional subspace U of C[0,1]⊗_εl₁ such that T restricted to U is equivalent to the identity operator on l_∞^N. Finally, we show that there is a compact Hausdorff space K and a bounded linear operator T:C(K)⊗_εl₁→l₂ for which T^#:C(K)→Π₁(l₁,l₂) is not 2-summing. (tex, dvi, ps, pdf.)
• (with D.J.H. Garling) Complemented subspaces of spaces obtained by interpolation. J. L.M.S. (2) 44 (1991), 503-513. If Z is a quotient of a subspace of a separable Banach space X, and V is any separable Banach space, then there is a Banach couple (A₀,A₁) such that A₀ and A₁ are isometric to X⊕V, and any intermediate space obtained using the real or complex interpolation method contains a complemented subspace isomorphic to Z. Thus many properties of Banach spaces, including having non-trivial cotype, having the Radon-Nikodym property, and having the analytic unconditional martingale difference sequence property, do not pass to intermediate spaces. (tex, dvi, ps, pdf.)
• The p^1/p in Pisier's Factorization Theorem. Proceedings of Conference on Geometry of Spaces at Strobl, Ed: P.F.X. Müller and W. Schachermayer, L.M.S. 1990. We show that the constants in Pisier's factorization theorem for (p,1)-summing operators from C(Ω) cannot be improved. (tex, dvi, ps, pdf.)
• (with Michel Talagrand) The Rademacher cotype of operators from l_∞^N. Proc. A.M.S. 112 (1991), 187-194. We show that for any operator T:l_∞^N→Y, where Y is a Banach space, that its cotype 2 constant, K₂(T), is related to its (2,1)-summing norm, π_2,1(T), by K₂(T) ≤ c log log N π_2,1(T). Thus, we can show that there is an operator T:C(K)→Y that has cotype 2, but is not 2-summing. (tex, dvi, ps, pdf.)
• The Gaussian Cotype of Operators from C(K). Israel Journal of Math. 68 (1989), 123-128. We show that the canonical embedding C(K) to L_Φ(μ) has Gaussian cotype p, where μ is a Radon probabilty measure on K, and Φ is an Orlicz function equivalent to t^p(log t)^p/2 for large t. (tex, dvi, ps, pdf.)
• The Distribution of Rademacher Sums. Proc. A.M.S. 109 (1990), 517-522. We find upper and lower bounds for Pr(∑ ±x_n > t), where x₁, x₂,... are real numbers. We express the answer in terms of the K-interpolation norm from the theory of interpolation of Banach spaces. (tex, dvi, ps, pdf.)
• Ph.D. Thesis. The Cotype of Operators from C(K), 1988 (Cambridge) If you want to see tortuous mathematical writing, I can recommend my own Ph.D. thesis. Unfortunately, at the time I wrote this, I did not know much about how to communicate effectively via the written word. Well, maybe I still don't but I am better than this. (tex, dvi, ps, pdf.)