# Preprints for Stephen Montgomery-Smith

These are most of my preprints. If you want other preprints, please email me at stephen@missouri.edu, but chances are I don't have it in electronic form. Some of the tex files are plain tex, some are latex, some are amstex. I would like to thank the NSF for their support of the research that is contained in these papers.

• List of Publications on AMS Server (you need access to MathSciNet to use this).
• Preprints on arxiv.org.
• Hamiltonians representing equations of motion with damping due to friction. Electron. J. Diff. Equ., Vol. 2014 (2014), No. 89, pp. 1-10. Suppose that $H(q,p)$ is a Hamiltonian on a manifold $M$, and $\tilde L(q,\dot q)$, the Rayleigh dissipation function, satisfies the same hypotheses as a Lagrangian on the manifold $M$. We provide a Hamiltonian framework that gives the equations $\dot p = \frac{\partial H}{\partial p}$, $\dot q = - \frac{\partial H}{\partial q} - \frac{\partial \tilde L}{\partial \dot q}$. The method is to embed $M$ into a larger framework where the motion drives a wave equation on the negative half line, where the energy in the wave represents heat being carried away from the motion. We obtain a version of Nöther's Theorem that is valid for dissipative systems. We also show that this framework fits the widely held view of how Hamiltonian dynamics can lead to the "arrow of time." (pdf, actual article.)
• (With Hannah Morgan) Obtaining Laws of Thermodynamics for Ideal Gases using Elastic Collisions. Preprint. The purpose of this note is to see to what extent ideal gas laws can be obtained from simple Newtonian mechanics, specifically elastic collisions. We present simple one-dimensional situations that seem to validate the laws. The first section describes a numerical simulation that demonstrates the second law of thermodynamics. The second section mathematically demonstrates the adiabatic law of expansion of ideal gases. (pdf.)
• Non-linear Instability of periodic orbits of suspensions of thin fibers in fluids. Preprint. It is known that Jeffery's equation predicts that fibers with Jeffery's parameter less than one will exhibit periodic behavior when subjected to shear flows. Yet this behavior is not seen in suspensions containing many fibers. This paper explores the extent to which coupling Jeffery's equation with the viscosity of the suspension causes instability that breaks up this periodic behavior. A simple one-dimensional model is presented, which suggests that there is at least some non-linear instability, so that this may at least partially account for why periodic behavior is not observed. One interesting observation is that this instability only grows linearly if only two dimensions are considered, whereas the instability can have exponential growth if the third dimension is considered. (pdf.)
• (with Weijun Huang) A numerical method to model dynamic behavior of thin inextensible elastic rods in three dimensions. Journal of Computational and Nonlinear Dynamics, 9,1, (2014). Static equations for thin inextensible elastic rods, or elastica as they are sometimes called, have been studied since before the time of Euler. In this paper, we examine how to model the dynamic behavior of elastica. We present a fairly high speed, robust numerical scheme that uses (i) a space discretization that uses cubic splines, and (ii) a time discretization that preserves a discrete version of the Hamiltonian. A good choice of numerical scheme is important, because these equations are very stiff, that is, most explicit numerical schemes will become unstable very quickly. The authors conducted this research anticipating describing the dynamic Kirchhoff problem, that is, the behavior of general springs that have natural curvature, and for which the equations take into account torsion of the rod. (pdf, actual article.)
• (with Martijn Caspers, Denis Potapov, Fedor Sukochev) The best constants for operator Lipschitz functions on Schatten classes. Preprint. Suppose that $f$ is a Lipschitz function on the real numbers with Lipschitz constant smaller or equal to 1. Let $A$ be a bounded self-adjoint operator on a Hilbert space $\mathcal H$. Let $p \in (1,\infty)$ and suppose that $x$ in $B(\mathcal H)$ is an operator such that the commutator $[A,x]$ is contained in the Schatten class $\mathcal S_p$. It is proved by the last two authors, that then also $[f(A),x]$ is contained in $\mathcal S_p$ and there exists a constant $C_p$ independent of $x$ and $f$ such that ${\|[f(A),x]\|}_p \le C_p {\|[A,x]\|}_p$. The main result of this paper is to give a sharp estimate for $C_p$ in terms of $p$. Namely, we show that $C_p \sim p^2/(p-1)$. In particular, this gives the best estimates for operator Lipschitz inequalities. We treat this result in a more general setting. This involves commutators of $n$ self-adjoint operators, for which we prove the analogous result. The case described here in the abstract follows as a special case. (pdf.)
• (with Babatunde O. Agboola and David A. Jack) Effectiveness of Recent Fiber-interaction Diffusion Models for Orientation and the Part Stiffness Predictions in Injection Molded Short-fiber Reinforced Composites. Composites: Part A 43 (2012) 1959-1970. Two fiber interaction models for predicting the fiber orientation and resulting stiffness of a short-fiber reinforced thermoplastic composite are investigated, the isotropic rotary diffusion of Folgar and Tucker (1984) and the anisotropic rotary diffusion of Phelps and Tucker (2009). This study employs several fiber orientation tensor closure approximations for both diffusion models and results are compared to those from the numerically exact spherical harmonic approach. Results are presented for variations in the fiber orientation and the processed part stiffness. A significant difference was observed between the stiffness predicted by both rotary diffusion models. It is worth noting that not all closures behave the same between the diffusion models, thus encouraging further studies to refine and validate the new fiber interaction models and solution approaches. A study of the predicted flexural modulus is presented, and results suggest that flexural modulus experiments may aid in further refining the fiber interaction models. (pdf.)
• (with Dongdong Zhang, Douglas E. Smith and David Jack) Rheological study on multiple fiber suspensions for fiber reinforced composite materials processing. Proceedings of the ASME 2011 International Mechanical Engineering Congress & Exposition IMECE 2011 Novermber 11-17, 2011, Denver, Colorado, USA. This paper studies the rheological properties of a semi-dilute fiber suspension for short fiber reinforced composite materials processing. For industrial applications, the volume fraction of short fibers could be large for semi-dilute and concentrated fiber suspensions. Therefore, fiber-fiber interactions consisting of hydrodynamic interactions and direct mechanical contacts could affect fiber orientations and thus the rate of fiber alignment in the manufacturing processing. In this paper, we study the semi-dilute fiber suspensions, i.e. the gap between fibers becomes closer, and hydrodynamic interactions becomes stronger, but the physical/mechanical contacts are still rare. We develop a three-dimensional finite element approach for simulating the motions of multiple fibers in low-Reynolds-number flows typical of polymer melt flow. We extend our earlier single fiber model to consider hydrodynamic interactions between fibers. This approach computes the hydrodynamic forces and torques on fibers by solving governing equations of motion in fluid. The hydrodynamic forces and torques result from two scenarios: gross fluid motion and hydrodynamic interactions from other fibers. Our approach seeks fibers' velocities that zero the hydrodynamic torques and forces acting on the fibers by the surrounding fluid. Fiber motions are then computed using a Runge-Kutta approach to update fiber positions and orientations as a function of time. This method is quite general and allows for solving multiple fiber suspensions in complex fluids. Examples with fibers having various starting positions and orientations are considered and compared with Jeffery's single fiber solution (1922). Meanwhile, we study the effect of the presence of a bounded wall on fiber motions, which is ignored in Jeffery's original work. The possible reasons why fiber motions observed in experiments align slower than those predicted by Jeffery's theory are discussed in this paper. (pdf.)
• (with Dongdong Zhang, Douglas E. Smith and David Jack) Numerical evaluation of single fiber motion for short-fiber-reinforced composite materials processing. Journal of Manufacturing Science and Engineering, 133(5), 2011, doi:10.1115/1.4004831. This paper presents a computational approach for simulating the motion of a single fiber suspended within a viscous fluid. We develop a Finite Element Method (FEM) for modeling the dynamics of a single rigid fiber suspended in a moving fluid. Our approach seeks solutions using the Newton-Raphson method for the fiber's linear and angular velocities such that the net hydrodynamic forces and torques acting on the fiber are zero. Fiber motion is then computed with a Runge-Kutta method to update the fiber position and orientation as a function of time. Low-Reynolds-number viscous flows are considered since these best represent the flow conditions for a polymer melt within a mold cavity. This approach is first used to verify Jeffery's orbit (1922) and addresses such issues as the role of a fiber's geometry on the dynamics of a single fiber, which were not addressed in Jeffery's original work. The method is quite general and allows for fiber shapes that include, but are not limited to, ellipsoidal fibers (such as that studied in Jeffery's original work), cylindrical fibers and bead-chain fibers. The relationships between equivalent aspect ratio and geometric aspect ratio of cylindrical and other axisymmetric fibers are derived in this paper. (pdf, actual article.)
• (with David Jack and Douglas E. Smith) The Fast Exact Closure for Jeffery's Equation with Diffusion. Journal of Non-Newtonian Fluid Mechanics, Volume 166, Issues 7-8, April 2011, Pages 343-353 . Jeffery's equation with diffusion is widely used to predict the motion of concentrated fiber suspensions in flows with low Reynold's numbers. Unfortunately, the evaluation of the fiber orientation distribution can require excessive computation, which is often avoided by solving the related second order moment tensor equation. This approach requires a `closure' that approximates the distribution function's fourth order moment tensor from its second order moment tensor. This paper presents the Fast Exact Closure (FEC) which uses conversion tensors to obtain a pair of related ordinary differential equations; avoiding approximations of the higher order moment tensors altogether. The FEC is exact in that when diffusion is absent it exactly solves Jeffery's equation. Numerical examples are provided with both Folgar-Tucker (1984) diffusion and the recent anisotropic rotary diffusion of Phelps and Tucker (2009). Computations demonstrate that the FEC exhibits improved accuracy with computational speeds equivalent to or better than existing closure approximations. (pdf, actual article.)
• (with Z.C. Feng, J.K. Chen and Yuwen Zhang) Temperature and Heat Flux Estimation from Sampled Transient Sensor Measurements. International Journal of Thermal Sciences, 49 (2010) 2385-2390. Laplace transform is used to solve the problem of heat conduction over a finite slab. The temperature and heat flux on the two surfaces of a slab are related by the transfer functions. These relationships can be used to calculate the front surface heat input (temperature and heat flux) from the back surface measurements (temperature and/or heat flux) when the front surface measurements are not feasible to obtain. This paper demonstrates that the front surface inputs can be obtained from the sensor data without resorting to inverse Laplace transform. Through Hadamard Factorization Theorem, the transfer functions are represented as infinite products of simple polynomials. Consequently, the relationships between the front and back surfaces are translated to the time-domain without inverse Laplace transforms. These time-domain relationships are used to obtain approximate solutions through iterative procedures. We select a numerical method that can smooth the data to filter out noise and at the same time obtain the time derivatives of the data. The smoothed data and time derivatives are then used to calculate the front surface inputs. (pdf, actual article.)
• (with Frank Schmidt) Statistical Methods for Estimating Complexity from Competition Experiments between Two Populations. Journal of Theoretical Biology, Volume 264, Issue 3, 7 June 2010, Pages 1043-1046. Often a screening or selection experiment targets a cell or tissue, which presents many possible molecular targets and identifies a correspondingly large number of ligands. We describe a statistical method to extract an estimate of the complexity or richness of the set of molecular targets from competition experiments between distinguishable ligands, including aptamers derived from combinatorial experiments (SELEX or phage display). In simulations, the nonparametric statistic provides a robust estimate of complexity from a 100x100 matrix of competition experiments, which is clearly feasible in high-throughput format. The statistic and method are potentially applicable to other ligand binding situations. (pdf, actual article.)
• (with Babatunde O. Agboola, David A. Jack and Douglas E. Smith) Investigation of the effectiveness and efficiency of the exact closure: comparison with industrial closures and spherical harmonic solutions. Proceedings of the ASME 2010 International Mechanical Engineering Congress & Exposition IMECE 2010 November 12-18, Vancouver, British Columbia, Canada. For stiffness predictions of short fiber reinforced polymer composites, it is essential to understand the orientation during processing. This is often performed through the equation of change of the fiber orientation tensor to simulate the fiber orientation during processing. Unfortunately this approach, while computationally efficient, requires the next higher ordered orientation tensor, thus requiring the use of a closure approximation. Many efforts have been made to develop closures to approximate the fourth-order orientation tensor in terms of the second order orientation tensor. Recently, Montgomery-Smith et al (2010) developed a pair of exact closures, one for systems with dilute suspensions and a second for dense suspensions, where the later works well for a variety of diffusion models. In this paper we compare the fiber orientation results of the Fast Exact Closure (FEC) for dense suspensions to that of the Spherical Harmonic solution, which although considered to be numerically exact does not readily lend itself to implementations in current industrial processing CFD codes. This paper focuses on a series of comparisons of material stiffness predictions between the FEC, current fitted closure models, and the spherical harmonics solution for a thin plate subjected to pure shear. Results for the select flows considered show the similarities between the current class of orthotropic fitted closures and that of the FEC. Although the results are similar between the fitted closures and the FEC, it is important to recognize that the Fast Exact Closure is formed without a fitting process. Consequently, the results are anticipated, in general, to be more robust in implementation. (pdf.)
• (with David A. Jack and Douglas E. Smith) Fast solutions for the fiber orientation of concentrated suspensions of short-fiber composites using the exact closure method. Proceedings of the ASME 2010 International Mechanical Engineering Congress & Exposition IMECE 2010 November 12-18, Vancouver, British Columbia, Canada. The kinetics of the fiber orientation during processing of short-fiber composites governs both the processing characteristics and the cured part performance. The flow kinetics of the polymer melt dictates the fiber orientation kinetics, and in turn the underlying fiber orientation dictates the bulk flow characteristics. It is beyond computational comprehension to model the equation of motion of the full fiber orientation probability distribution function. Instead, typical industrial simulations rely on the computationally efficient equation of motion of the second-order orientation tensor (also known as the second-order moment of the orientation distribution function) to model the characteristics of the fiber orientation within a polymer suspension. Unfortunately, typical implementation forms of any order orientation tensor equation of motion requires the next higher, even ordered, orientation tensor, thus necessitating a closure of the higher order expression. The recently developed Fast Exact Closure avoids the classical closure problem by solving a set of related second-order tensor equations of motion, and yields the exact solution for pure Jeffery's motion as the diffusion goes to zero. Typical closures are obtained through a fitting process, and are often obtained by fitting for orientation states obtained from solutions of the full orientation distribution function, thus tying the closure to the flows from which it was fit. With the recent understandings of the limitations of the Folgar and Tucker (1984) model of fiber interactions during processing, it has become clear the importance of developing a closure that is independent of any choice of fitting data. The Fast Exact Closure presents an alternative in that it is constructed independent of any fitting process. Results demonstrate that when diffusion exists, the solution is not only physical, but solutions for flows experiencing Folgar-Tucker diffusion are shown to exhibit an equal to or greater accuracy than solutions relying on closures developed via a curve fitting approach. (pdf.)
• (with Dongdong Zhang, Douglas E. Smith and David A. Jack) Numerical evaluation of single fiber motion for short fiber composites materials processing. Proceedings of the ASME 2010 International Mechanical Engineering Congress & Exposition IMECE 2010 November 12-18, Vancouver, British Columbia, Canada. This paper presents a numerical approach for calculating the single fiber motion in a viscous flow. This approach addresses such issues as the role of axis ratio and fiber shape on the dynamics of a single fiber, which was not addressed in Jeffery's original work. We develop a Finite Element Method (FEM) for modeling the dynamics of a single rigid fiber suspended in a moving fluid. Low Reynolds number viscous flows are considered since these best represent the flow conditions for a polymer melt within a mold cavity. Our approach seeks the fiber angular velocities that zero the hydrodynamic torques acting on the fiber using the Newton-Raphson method. Fiber motion is then computed with a Runge-Kutta method to update the position, i.e. the angle of the fiber as a function of time. This method is quite general and allows for fiber shapes that include, but are not limited to, ellipsoidal fibers (such as that studied in Jeffery's original work), cylindrical fibers and beads-chain fibers. The relationships between equivalent axis ratios and geometrical axis ratios for cylindrical and beads-chain fibers are derived in this paper. (pdf.)
• Perturbations of the coupled Jeffery-Stokes equations. Journal of Fluid Mechanics, volume 681, (2011), pp. 622-638. This paper seeks to provide clues as to why experimental evidence for the alignment of slender fibers in semi-dilute suspensions under shear flows does not match theoretical predictions. This paper posits that the hydrodynamic interactions between the different fibers that might be responsible for the deviation from theory, can at least partially be modeled by the coupling between Jeffery's equation and Stokes' equation. It is proposed that if the initial data is slightly non-uniform, in that the probability distribution of the orientation has small spacial variations, then there is feedback via Stokes' equation that causes these non-uniformities to grow significantly in short amounts of time, so that the standard uncoupled Jeffery's equation becomes a poor predictor when the volume ratio of fibers to fluid is not extremely low. This paper provides numerical evidence, involving spectral analysis of the linearization of the perturbation equation, to support this theory. (pdf, actual article.) Since the paper was published, a mistake was found. The correction is here.
• (with Wei He, David Jack and Douglas E. Smith) Exact tensor closures for the three dimensional Jeffery's Equation. Journal of Fluid Mechanics, volume 680, (2011), pp. 321-335. This paper presents an exact formula for calculating the fourth moment tensor from the second moment tensor for the three dimensional Jeffery's equation. Although this approach falls within the category of a moment tensor closure, it does not rely upon an approximation, either analytic or curve fit, of the fourth moment tensor as do the quadratic, hybrid or current orthotropic closures. This closure is orthotropic in the sense of Cintra and Tucker, or equivalently, a natural closure in the sense of Verleye and Dupret. The existence of these explicit formulae has been asserted by previous authors, but as far as the authors know, the explicit forms have yet to be published. The formulae involve elliptic integrals, and are valid whenever the fiber orientation tensor was isotropic at some point in time. Finally, this paper presents the Fast Exact Closure (FEC), a fast and in principle exact method for solving Jeffery's equation, which does not require approximate closures, nor the computation of the elliptic integrals. (pdf, actual article.)
• (with S. Dostoglou and R.R. Gastler) Negative longitudinal correlation for isotropic flows. Preprint. Examples of physically reasonable homogeneous, isotropic, three-dimensional divergence-free vector fields with longitudinal correlation negative on some interval are presented. The negativity of the longitudinal correlation persists in the Galerkin approximation of the hydrodynamic equations at least for some time. Both the outline of the mathematical arguments and the numerical implementation are included. (pdf.)
• (with David Jack and Douglas E. Smith) Modeling Orientational Diffusion in Short Fiber Composite Processing Simulations. Preprint. Numerical simulations of fiber orientation in short-fiber composites have relied on the Folgar and Tucker (1984) model for diffusion for over twenty years. Unfortunately, it has recently been shown that this fiber collision model tends to over-predict the rate of alignment; exposing the need for a new fundamental approach to more accurately capture fiber interactions within the melt flow. Here we present our initial work in the development of an objective directional diffusion model and a variable lambda model for fiber collisions where we modify Jeffery's model (1922) to incorporate local directionally dependent effects assumed proportional to the probability of fiber-fiber collisions. We show that our directional diffusion model performs well in extensional flows, whereas its usefulness appears limited in shearing flows. Conversely, preliminary results from the variable lambda model in both elongational and shearing flows are quite promising and will be the focus of future investigations. (pdf.)
• (with Xiaofang Jin, Jessica Rose Newton and George P. Smith) A generalized kinetic model for amine modification of proteins with application to phage display. BioTechniques 46:175-182 (March 2009) doi 10.2144/000113074. Amine modification of filamentous virions (phage particles) is widely used in phage display technology to couple small groups such as biotin or fluorescent dyes to the major coat protein pVIII. We have developed a generalized kinetic model for protein amine modification and applied it to the modification of pVIII with biotin and the near-infrared fluorophor Alexa Fluor 680. Empirically optimized kinetic parameters for the two modification reactions allow the modification level to be predicted for a wide range of virions and modifying reagent concentrations. Virions with 0.03 biotins per pVIII subunit have 50% of the maximal binding capacity for a streptavidin conjugate. (pdf.)
• (with David Jack and Douglas E. Smith) A Systematic Approach to Obtaining Numerical Solutions of Jeffery's Type Equations using Spherical Harmonics. Composites Part A, Volume 41, Issue 7, July 2010, Pages 827-835. This paper extends the work of Bird, Warner, Stewart, Sorensen, Larson, Ottinger, Vukadinovic, and Forest et al., who have applied Spherical Harmonics to numerically solve certain types of partial differential equations on the two-dimensional sphere. We present a systematic approach and implementation for solving such equations with efficient numerical solutions. In particular we are able to solve a wide variety of fiber orientation equations considered before by Jeffery, Folgar and Tucker, and Koch, and include several recently introduced fiber orientation collision models. The main tools used to compute the coefficients for the Spherical Harmonic-based expansion are Rodrigues' formula and the ladder operators. We show that solutions of the Folgar-Tucker model using our new algorithm retains the accuracy of full simulations of the fiber orientation distribution function with computational efforts that are only slightly more than the Advani-Tucker orientation/moment tensor solutions commonly used in industrial applications. The spherical harmonic approach requires a computational effort of just three times that of the orientation tensor approach employing the orthotropic closure of VerWeyst, but with less than 1/1000th the computational effort of numerical solutions of the full orientation distribution function obtained using control volume methods. (pdf, actual article.)
• (with Yong Gan and Zhen Chen) Improved material point method for simulating the zona failure response in piezo-assisted intracytoplasmic sperm injection. CMES: Computer Modeling in Engineering & Sciences, Vol. 73, No. 1, pp. 45-76, 2011. The material point method (MPM), which is an extension from computational fluid dynamics (CFD) to computational solid dynamics (CSD), is improved for the coupled CFD and CSD simulation of the zona failure response in piezo-assisted intracytoplasmic sperm injection (piezo-ICSI). To evaluate the stresses at any zona material point, a plane stress assumption is made in the local tangent plane of the membrane point, and a simple procedure is proposed to find the effective point connectivity for the orientation of the local tangent plane. With an iterative algorithm in each time step, the original MPM is improved to better simulate fluid dynamics problems involving strong shocks. The use of an Eulerian mesh for solving the momentum equations enables the MPM to automatically handle fluid-membrane interactions without requiring the interface-tracking module. Several examples are used to demonstrate the robustness and efficiency of the proposed numerical scheme for simulating three-dimensional fluid-membrane interactions. Finally, the proposed procedure is applied to the shock-induced zona failure analysis for the piezo-ICSI experiment. (pdf, actual paper.)
• (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. Transactions of the American Math Society, 362, (2010), 553-575. 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-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 $L^p_X(\mathbb R^2)$ with $p\in (1,\infty).$ 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 $L = \limsup_{n\to\infty} n {\|T^{n+1}-T^n\|} < \infty$ 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 improve a result of Esterle, showing that if $\sigma(T) = \{1\}$ and $T \ne I$, then $\liminf_{n\to\infty} n {\|T^{n+1}-T^n\|} \ge 1/e$. 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 $1 < p < \infty$, 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 $E[s_k \vee d_k] \le E[s_k \vee e_k]$ then we have ${\|\sum d_k\|}_p \le c_p {\|\sum e_k\|}_p$. (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 $X_1$, $X_2,\dots,$ $X_n$ 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 $1 \le q < \infty$. This extends work of Johnson and Schechtman who tackled the case when $N = \ell_p$, 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 $B_\infty^{-1,\infty}$. 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 $p$th moment, and between the $p$th 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 $X_1,\dots,X_n$ be independent Banach-valued random variables. Let $I$ be a random variable independent of $X_1,\dots,X_n$ and uniformly distributed over $\{ 1,\dots,n \}$. Put $\tilde X_1=X_I$, and let $\tilde X_2,\dots,\tilde X_n$ be independent identically distributed copies of $\tilde X_1$. Then, $P(\|X_1+\dots+X_n\| \ge \lambda)\le cP(\|\tilde X_1+\dots+\tilde X_n\|\ge \lambda/c)$ for all $\lambda\ge 0$, where $c$ is an absolute 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 $L_1([0,1])$ 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 Orlicz space $L_\Phi$ with $\Phi(x) = \exp(x^2)-1$. (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 $\int_0^1 d_n(x_1,\dots,x_n)\,dx_n = 0$ for almost all $x_1,\dots,x_{n-1}$, and such that the law of $(f_n)$ is the same as the law of $(\sum_{k=1}^n d_k(x_1,\dots,x_k))$. 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 $\partial f$ and $\bar\partial f$. 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 $\pi$ of $\{1,2,\dots,n\}$ at random. Kwapien and Schütt considered the problem of finding $E\left(\left\|(m_{i,\pi(i)})\right\|_p^q\right)^{1/q}$. 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 Schrödinger or wave equation with $L_2$ initial data. We provide counterexamples to plausible conjectures involving the decay in $t$ of the BMO norm of $u(t,\cdot)$. 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 $\mathbb R^1$ 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 $L^p(\mathbb R^1)$, 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 $L^p(\mathbb R^n)$, and we observe that the operator norm of the uncentered Hardy-Littlewood maximal function over balls on $L^p(\mathbb R^n)$ grows exponentially as $n\to\infty$. (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 $f(X_1,\dots,X_m)$ and $g(X_{m+1},\dots,X_k)$ are non-negatively correlated for any symmetric jointly stable random variables $Xi$. 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 $H^1$ 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 $\mathbb T^N$, where analyticity is defined using a lexicographic order on the dual group $\mathbb Z^N$. We show how, by using basic properties of orders on $\mathbb Z^N$, 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 $\Sigma$ be a $\sigma$-algebra over $\Omega$, and let $M(\Sigma)$ denote the Banach space of complex measures. Consider a representation $T_t$ for $t\in\mathbb R$ acting on $M(\Sigma)$. We show that under certain, very weak hypotheses, that if for a given $\mu\in M(\Sigma)$ and all $A\in\Sigma$ the map $t\mapsto T_t\mu(A)$ is in $H^\infty(\mathbb R)$, then it follows that the map $t\mapsto T_ t\mu$ 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 $0 < p < \infty$. 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 $\Gamma$ contains a Haar measurable order $P$. Using the order $P$ we define the conjugate function operator on $L^p(G)$, $1 \le p < \infty$, 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 $\sigma$-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 $\text{ACF}$ 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. $X\in \text{HT}$, are $\text{ACF}$ spaces. We then show that Bourgain's proof of $X \in \text{HT} \Rightarrow X \in \text{UMD}$ 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 $0 < r \le 1$. 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 $\sigma$-algebras $(\mathcal F_n)$ such that the conditional distributions of $f_n$ and $g_n$ given $\mathcal F_n$ coincide, and such that the sequence $(g_n)$ is conditionally independent. Then it is known that $\left\|\sum f_n\right\|_p \le C \left\| \sum g_n \right\|_p$ 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 $(e^{tA})_{t\ge 0}$ 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 $\sigma(A)$. (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 $j \lt k$ we have $\Pr(\|S_j\| > t) \le c \Pr (\|S_k\| > t/c)$. (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 $\mathcal S$, and let $f_{i_1,\dots,i_k}$ be measurable functions from $\mathcal S^k$ to a Banach space $B$. Let $(X_i^{(j)})$ be independent copies of $(X_i)$. The following inequality holds for all $t \ge 0$ and all $n \ge 2$ $$P\left(\left\|\sum_{1 \le i_1 \ne \cdots \ne i_k \le n} f_{i_1,\dots,i_k}(X_{i_1},\dots,X_{i_k})\right\| \ge t\right) \le C_k P\left(\left\|\sum_{1 \le i_1 \ne \cdots \ne i_k \le n} f_{i_1,\dots,i_k}(X_{i_1}^{(1)},\dots,X_{i_k}^{(k)})\right\| \ge t\right) .$$ Furthermore, the reverse inequality also holds in the case that the functions $\{f_{i_1,\dots,i_k}\}$ satisfy the symmetry condition $$f_{i_1,\dots,i_k}(X_{i_1},\dots,X_{i_k}) = f_{i_{\pi(1)},\dots,i_{\pi(k)}}(X_{i_{\pi(1)}},\dots,X_{i_{\pi(k)}})$$ for all permutations $\pi$ of $\{1,\dots,k\}$. Note that the expression $i_1 \ne \cdots \ne i_k$ means that $i_r \ne i_s$ for $r \ne s$. 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 $X = \sum \epsilon_n x_n$ 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 $\phi$ is a submeasure satisfying an appropriate lower estimate we give a quantitative result on the total mass of a measure $\mu$ satisfying $0\le \mu \le \phi$. 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 $T:C(\Omega) \to X$ 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 $\mu$ and dual group $\Gamma$. Let $E$ be a Sidon subset of $\Gamma$ 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 $\alpha>0$: $$c^{-1} P\left[\left|\sum a_n r_n\right| \ge c \alpha\right] \le \mu\left[\left|\sum a_n \gamma_n \right| \ge \alpha \right] \le c P\left[\left|\sum a_n r_n\right| \ge c^{-1} \alpha\right]$$ where $a_1,\dots,a_N$ are arbitrary elements of $B$, and $\gamma_1,\dots,\gamma_N$ 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.)
• (with Nakhlé Asmar) Littlewood-Paley theory on solenoids. Colloquium Mathematicum 65, (1993), 69-82. (actual article.)
• 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 $\Pi_p(X,Y)$ $(1 \le p < \infty)$ denote the space of $p$-summing operators from $Y$ to $Z$. We show that, if $X$ is a $\mathcal L_\infty$-space, then a bounded linear operator $T:X\otimes_\epsilon Y\to Z$ is 1-summing if and only if a naturally associated operator $T^\#:X\to \Pi_1(Y,Z)$ is 1-summing. This result need not be true if $X$ is not a $\mathcal L_\infty$-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 $C[0,1]\otimes_\epsilon \ell_1$ whose associated operator $T^\#$ is 2-summing, but for all $N\in\mathbb N$, there exists an $N$-dimensional subspace $U$ of $C[0,1]\otimes_\epsilon \ell_1$ such that $T$ restricted to $U$ is equivalent to the identity operator on $\ell_\infty^N$. Finally, we show that there is a compact Hausdorff space $K$ and a bounded linear operator $T:C(K) \otimes_\epsilon \ell_1 \to \ell_2$ for which $T^\#:C(K) \to \Pi_1(\ell_1,\ell_2)$ 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 $(A_0,A_1)$ such that $A_0$ and $A_1$ are isometric to $X\oplus V$, 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 $C(\Omega)$ cannot be improved. (tex, dvi, ps, pdf.)
• (with Michel Talagrand) The Rademacher cotype of operators from $\ell_\infty^N$. Proc. A.M.S. 112 (1991), 187-194. We show that for any operator $T:\ell_\infty^N \to Y$, where $Y$ is a Banach space, that its cotype 2 constant, $K_2(T)$, is related to its $(2,1)$-summing norm, $\pi_{2,1}(T)$, by $K_2(T) \le c \log \log N \pi_{2,1}(T)$. Thus, we can show that there is an operator $T:C(K)\to Y$ 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 $L_\Phi(\mu)$ has Gaussian cotype $p$, where $\mu$ is a Radon probabilty measure on $K$, and $\Phi$ 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 $\Pr\left(\sum \pm x_n > t\right)$, where $x_1, x_2, \dots$ 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.)
• (with Raimund Ober) Bilinear Transformation of Infinite-Dimensional State-Space Systems and Balanced Realizations of Nonrational Transfer Functions. SIAM J. Control Optim. 28(2) (1990), 438-465. The bilinear transform maps the open right half plane to the open unit disk and is therefore a suitable tool for carrying over results for continuous-time systems to discrete-time systems and vice versa. Corresponding state-space formulae are widely used and well understood for the case of finite-dimensional systems. In this paper infinite-dimensional generalizations of these formulae are studied for a general class of infinite-dimensional state-space systems. In particular, it is shown that reachability and observability are carried over and that the reachability and observability gramians are preserved under this transformation. Young showed that a wide class of nonrational discrete-time transfer functions admit a balanced state-space representation. It is shown that this result carries over to the continuous-time situation via the bilinear transformation. (actual article.)
• 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.)