Stephen Montgomery-Smith

University of Missouri | Mizzou · Department of Mathematics

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 b...

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...

This paper is concerned with difficulties encountered by engineers when they attempt to predict the orientation of fibers in the creation of injection molded plastic parts. It is known that Jeffery's equation, which was designed to model a single fiber in an infinite fluid, breaks down very badly when applied to this situation. In a previous paper,...

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 previous closur...

Following the work of Lloyd Shapley on the Shapley value, and tangentially the work of Guillermo Owen, we offer an alternative non-probabilistic formulation of part of the work of Robert J. Weber in his 1978 paper "Probabilistic values for games." Specifically, we focus upon efficient but not symmetric allocations of value for cooperative games. We...

The Levenberg-Marquardt (LM) algorithm is the most commonly used training algorithm for moderate-sized feed forward artificial neural networks (ANNs) due to its high convergence rate and reasonably good accuracy. It conventionally employs a Jacobian-based approximation to the Hessian matrix, since exact evaluation of the Hessian matrix is generally...

Microbial populations adapt to their environment by acquiring advantageous mutations, but in the early twentieth century, questions about how these organisms acquire mutations arose. The experiment of Salvador Luria and Max Delbr\"uck that won them a Nobel Prize in 1969 confirmed that mutations don't occur out of necessity, but instead can occur ma...

This paper calculates probability distributions modeling the Luria-Delbr\"uck experiment. We show that by thinking purely in terms of generating functions, and using a 'backwards in time' paradigm, that formulas describing various situations can be easily obtained. This includes a generating function for Haldane's probability distribution due to Yc...

The effects of the initial conditions on the oscillatory flow in oscillating heat pipes (OHPs) are investigated numerically. The initial vapor temperature, pressure, and liquid displacement are taken into consideration. Three test cases are solved with different initial condition settings. The liquid displacement and vapor temperature tendencies ar...

Microbial cultures swiftly adapt to lethal agents such as antibiotics or viruses by acquiring resistance mutations. Does this remarkable adaptability require a Lamarckian explanation, whereby the agent specifically directs resistance mutations? Soon after the question arose, Luria and Delbrück devised a clever experiment, the fluctuation test, that...

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 tim...

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 math...

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 equation $\dot q = \frac{\partial H}{\partial p}(q,p), \quad \dot p = - \frac{\partial H}{\partial q}(q,p) - \frac...

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 H. Let 1<p<infinity and suppose that x in B(H) is an operator such that the commutator [A, x] is contained in the Schatten class S_p. It is proved by the last two authors, that then al...

This paper seeks to provide clues as to why experimental evidence for the alignment of slender fibres in semi-dilute suspensions under shear flows does not match theoretical predictions. This paper posits that the hydrodynamic interactions between the different fibres that might be responsible for the deviation from theory, can at least partially b...

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 previous closur...

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...

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 ass...

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...

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 flu...

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 th...

This paper extends the work of Bird, Warner, Stewart, Sørensen, 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 numer...

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 ligand...

The kinetics of the fiber orientation during processing of shortfiber 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 co...

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...

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 f...

Fiber orientation kinematic models of non‐dilute suspensions have relied on the Folgar and Tucker (1984) model for diffusion for over two decades. Recent research, however, has exposed the propensity of this fiber collision model to over‐predict the rate of alignment. To promote the advancement of light‐weight, high strength composites, a new fund...

This paper presents a method to numerically solve partial differential equations such as the Jeffery&apos;s equation, which calculates the orientation of fiber in a slow moving fluid. Our method relies on spherical harmonics. This method is equivalent to using the higher order moment tensors with the linear closure, but using tensors of very high o...

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 becau...

An operator defined by averaging of the Fourier-Haar coefficients of a function is studied. A criterion for the boundedness of such an operator acting in a pair of rearrangement-invariant spaces is derived.

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....

We examine a Bayesian approach to estimating the number of classes in a population, in the situation that we are able to take many inde- pendent samples from an infinite population.

We introduce unbiased estimators for the Shannon entropy and the class number, in the situation that we are able to take sequences of inde- pendent samples of arbitrary length.

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 Tema...

We consider whether L = lim sup n!1 nkT 1 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 (T ) = f1g and T 6= I , then lim inf n!1 nkT k 1=e.

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 != I, then liminf_{n to infty} n ||T^{n+1}-T^n|| >= 1/e. The con...

Let X 1 ,X 2 ,⋯,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≤q<∞. This extends work of G. Johnson and W. B. Schech...

For each 1 < p < infinity, 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 |e_k|] le E[s_k vee |d_k|] then for all positive integers n...

Introduction Throughout this paper, N denotes a xed but arbitrary positive integer, T denotes the circle group, and T N denotes the product of N copies of T. The normalized Lebesgue measure on T N will be symbolized by P . For a measurable function f , we let kfk 1 = sup y>0 y f (y) where f (y) = P fx 2 T N : jf(x)j > yg . The integers will be deno...

. A Banach space X is called an HT space if the Hilbert transform is bounded from L p (X) into L p (X), where 1 < p < 1. We introduce the notion of an ACF Banach space, that is, a Banach space X for which we have an abstract M. Riesz Theorem for conjugate functions in L p (X), 1 < p < 1. Berkson, Gillespie, and Muhly [5] showed that X 2 HT =) X 2 A...

this paper we study properties of weakly analytic vector-valued measures. To motivate the discussion, let us start with some relevant results for scalar-valued measures. A fundamental result in harmonic analysis, the F. and M. Riesz Theorem, states that if a complex Borel measure on the unit circle T (in symbols, 2 M(T)) is analytic that is, Z e in...

Let G be a locally compact abelian group whose dual group contains a Haar measurable order P . Using the order P we dene the conjugate function operator on L p (G), 1 p < 1, as was done by Helson [7]. 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 u...

. Let X = P "nxn be a Rademacher series with vector-valued coecients. We obtain an approximate formula for the distribution of the random variable jjXjj in terms of its mean and a certain quantity derived from the K-functional of interpolation theory. Several applications of the formula are given. 1. Results In [6] the second-named author calculate...

ecoupling inequalities allow one to compare expressions of the rst kind with expressions of the second kind. Such results permit the almost direct transfer of results for sums of independent random variables to the case of generalized Ustatistics. The reason being that conditionally on fX (2) i g; :::; fX (k) i g, the second sum above is a sum inde...

Let X; Y and Z be Banach spaces, and let Q p (Y; Z) (1 p < 1) denote the space of p-summing operators from Y to Z. We show that, if X is a $1 -space, then a bounded linear operator T : X ^

. A new proof of a result of Lutz Weis is given, that states that the stability of a positive strongly continuous semigroup (e tA ) t0 on Lp 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 (A). Consider a strongly continuous semigroup (e tA ) t0...

Introduction Let us consider an autonomous dierential equation v 0 = Av in a Banach space E, where A is a generator of C 0 -semigroup fe tA g t0 . Denote, as usual, s(A) = supfRe : 2 (A)g and !(A) = inff! : ke tA k Me !t g. A classical result of A. M. Lyapunov (see, e.g., [9]) shows that for any bounded operator A 2 B(E) the spectrum (A) of A is re...

The spectrum of the kinematic dynamo operator for an ideally conducting uid and the spectrum of the corresponding group acting in the space of continuous divergence free vector elds 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 determ...

. We point out a certain class of functions f and g for which random variables f(X1 ; : : : ; Xm ) and g(Xm+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. 1.

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...

We give necessary and sucient conditions for the Hardy operator to be bounded on a rearrangement invariant quasi-Banach space in terms of its Boyd indices. MAIN RESULTS A rearrangement invariant space X on R is a set of measurable functions (modulo functions equal almost everywhere) with a complete quasi-norm k k X such that the following holds: i)...

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 smoo...

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,…,Xn be independent Banach-valued random variables. Let I be a random variable independent of X1,…,Xn and uni...

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...

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 Levy property. We then give a connection between the tail distribution and the pth moment, and between the pth moment and the rearrangement invariant norms...

. 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 nite time, the solution is in no Triebel-Lizorkin or Besov space (and hence in no Lebesgue or Sobolev space). The purpose is to sh...

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 Φ with Φ(x) = exp(x2)-1.

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 me...

. 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 autonomous and nonautonomous systems modeled with unbounded state-space operators acting on Banach spaces. This approach allows one to apply the classical theory of strongly continuo...

We survey some recent literature concerning the spectrum of evolutionary semigroups. 1 Introduction Let us consider an absract Cauchy problem: d dt v(t) = A(t)v(t) (t t 0 ); v(t 0 ) = v 0 : (1) Here A(t) is a given unbounded operator with dense domain from a Banach space X to itself, v 0 is in the domain of A(t 0 ), and we are looking for a solutio...

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 three-dimensional domain with periodic boundary conditions has global regularity, as long as there is some control on the size of the initial data and t...

this paper is to bring about a similar approach to spaces of measures. Our main transference result is motivated by the extensions of the classical F.&M. Riesz Theorem due to Bochner [3], Helson-Lowdenslager [10, 11], de Leeuw-Glicksberg [6], Forelli [9], and others. It might seem that these extensions should all be obtainable via transference meth...

: 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 R 1 0 d n (x 1 ; : : : ; x n ) dx n = 0 for almost all x 1 ; : : : ; x nGamma1 , and such that the law of (f n ) is the same as the law of ( P n k=1 d k (x 1 ; : : : ; x k )). Similar results for tangent sequences a...

. 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 Lata/la on bounds on moments of such sums. We also give a new proof of Lata/la's result for nonnegative random variables, and improve one of the constants in his inequality...

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 f...

This article is devoted to the development of these methods. Let 1 q 1, and ` be a one-to-one correspondence of Sn to f1; 2; : : : ; n!g.

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 Lata{\l}a on bounds on moments of such sums. We also give a new proof of Lata{\l}a's result for nonnegative random variables, and improve one of the constants in his inequali...

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, ·). The proofs make use of random methods, in particular, Brownian motion.

We discuss some conjectural inequalities that are related to singular integrals, martingales, quasiconformal mappings, and the calculus of variations. Specifically, we present evidence for a conjecture of Iwaniec concerning the best constant for the Beurling-Ahlfors Operator.

The transference theory for Lp 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 b...

. Using the theory of evolution semigroups, this paper investigates the stability radius for a wide class of linear nonautonomous systems in Banach spaces. A spectral-mapping theorem for evolution semigroups acting on vector-valued functions on [0; 1) is proven first. This allows the stability radius to be expressed in terms of the spectrum of the...

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 C 0 -semigroups, and linear skew-product flows. The spectral mapping theorem for these semig...

The best constant in the usual Lp 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 E...

Let $u(x,t)$ be the solution of the Schr\"odinger 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. (Since this paper was written, the unsolved problem remaining in this pa...

Consider an n by n matrix x_ij, and consider the quantity || x_{i,pi(i)} ||_X where X is a symmetric sequence space as a random variable where the permutation pi is chosen randomly. This was considered by Kwapien and Schutt, and we extend their results to rearrangement invariant spaces. We also consider the property of D and D^* convexity for rearr...

Abstract This paper gives upper and lower bounds for moments,of sums of independent random variables (Xk) which satisfy the condition that P (jXjk t) = exp( Nk(t)), where Nk 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) =jtj ,...

We extend ideas of Garling to consider the so called Hardy martingales in a more general setting of H^p theory of compact abelian groups with ordered dual. As a consequence, we obtain a new proof of a result of Helson and Lowdenslager which generalizes Jensen's Inequality for H^1 functions.

In this paper we present a decoupling inequality that shows that multivariate $U$-statistics can be studied as sums of (conditionally) independent random variables. This result has important implications in several areas of probability and statistics including the study of random graphs and multiple stochastic integration. More precisely, we get th...

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 $t\in\Bbb R$ 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(\Bb...

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 n...

We precisely evaluate the operator norm of the uncentered Hardy-Littlewood maximal function on $L^p(\Bbb R^1)$. We also compute the operator norm of the uncentered Hardy-Littlewood maximal function over rectangles on $L^p(\Bbb R^n)$, and we show that the operator norm of the uncentered Hardy-Littlewood maximal function over balls on $L^p(\Bbb R^n)$...

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\. $X\in\text{HT}$, are ACF spaces. We then show that Bourgain's proof of $X\in\t...

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 $\sigma(A)$.

Let (f_n) and (g_n) be two sequences of random variables adapted to an increasing sequence of $\sigma$-algebras $({\cal F}_n)$ such that the conditional distributions of f_n and g_n given ${\cal F}_{n-1}$ coincide, and such that the sequence (g_n) is conditionally independent. Then it is known that $\normo{\sum f_k}_p \le C \normo{\sum g_k}_p$, $1...

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 and so forth.

In a celebrated paper, Burkholder, Gundy, and Silverstein used Brownian motion to derive a maximal function characterization of H^p spaces for 0 < p < infinity. 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.

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 Pr(||S_j|| > t) <= c Pr(||S_k|| > t/c).

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 $f_{i_1...i_k}$ be measurable functions from $S^k$ to a Banach space $B$. Let $(X_i^{(j)})$ be independent c...

to appear in Probability theory. Here {Xi} is a sequence of independent random variables taking values in a measurable space (S, S), and {fi1...ik} is a sequence of measurable functions from Sk into a Banach Space (B, � · �). Special cases of this type of random variable appear, for example, in Statistics in the form of U-statistics and quadratic f...

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 differe...

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. Comment: 6 pages

(S f)^ = 1 b f, where, whenever A is a set, 1A is the indicator function of the set A. If 1 < p < 1 and S is bounded from L2(G) \Lp(G) into Lp(G), we use the same symbol to denote the bounded extension of S to all Lp(G). We say that the decomposition ( j)j2I has the LP (Littlewood-Paley) property if for every p 2 (1,1) there are constants p and p s...