arXiv subjects

Stephen Montgomery-Smith

Publications and source records attributed to Stephen Montgomery-Smith.

27 records · Page 2Linked to original sources

arXiv2002

A counterexample to the smoothness of the solution to an equation arising in fluid mechanics

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.

math.AP↗

arXiv2001

Rearrangement Invariant Norms of Symmetric Sequence Norms of Independent Sequences of Random Variables

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 le q < infty. This extends work of Johnson and Schechtman who tackled the case when N = l_p, and recent work of Gordon, Litvak, Schuett and Werner who obtained similar results for Orlicz spaces.

math.PR↗

arXiv2001

An Extension to the Tangent Sequence Martingale Inequality

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 we have || sum_{k=1}^n e_k ||_p le c_p || \sum_{k=1}^n d_k ||_p .

math.PR↗

arXiv2000

Decomposition of analytic measures on groups and measure spaces

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.

math.FA↗

arXiv2000

Finite time blow up for a Navier-Stokes like equation

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 $\dot B^{-1,\infty}_\infty$. We give initial data in this space for which there is no reasonable solution for the Navier-Stokes like equation.

math.AP↗

arXiv1999

A note on sums of independent random variables

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

math.PR↗

arXiv1999

Measuring the magnitude of sums of independent random variables

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.

math.PR↗

arXiv1998

A comparison inequality for sums of independent random variables

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,...,X_n be independent Banach-valued random variables. Let I be a random variable independent of X_1,...,X_n and uniformly distributed over {1,...,n}. Put Z_1 = X_I, and let Z_2,...,Z_n be independent identically distributed copies of Z_1. Then, P(||X_1+...+X_n|| > t) < c P(||Z_1+...+Z_n|| > t/c), for all t>0, where c is an absolute constant.

math.PR↗

arXiv1993

Lyapunov theorems for Banach spaces

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.

math.FA↗