A Note on the Behavior of the Randomized Kaczmarz Algorithm of Strohmer and Vershynin

In a recent paper by Strohmer and Vershynin (J. Fourier Anal. Appl. 15:262–278, 2009) a “randomized Kaczmarz algorithm” is proposed for solving consistent systems of linear equations {〈a ⁱ ,x〉=b _i } _i=1 ^m . In that algorithm the next equation to be used in an iterative Kaczmarz process is selected with a probability proportional to ‖a ⁱ ‖². The paper illustrates the superiority of this selection method for the reconstruction of a bandlimited function from its nonuniformly spaced sampling values. In this note we point out that the reported success of the algorithm of Strohmer and Vershynin in their numerical simulation depends on the specific choices that are made in translating the underlying problem, whose geometrical nature is “find a common point of a set of hyperplanes”, into a system of algebraic equations. If this translation is carefully done, as in the numerical simulation provided by Strohmer and Vershynin for the reconstruction of a bandlimited function from its nonuniformly spaced sampling values, then indeed good performance may result. However, there will always be legitimate algebraic representations of the underlying problem (so that the set of solutions of the system of algebraic equations is exactly the set of points in the intersection of the hyperplanes), for which the selection method of Strohmer and Vershynin will perform in an inferior manner.     

A zero-one law for random subgroups of some totally disconnected groups

Let A be a locally compact group topologically generated by d elements and let k > d. Consider the action, by precomposition, of Γ = Aut(F _k ) on the set of marked, k-generated, dense subgroups $ {D_{k,A}}: = \left\{ {\eta \in {\text{Hom}}\left( {{F_k},A} \right)\left| {\overline {\left\langle {\phi \left( {{F_k}} \right)} \right\rangle } = A} \right.} \right\} Let A be a locally compact group topologically generated by d elements and let k > d. Consider the action, by precomposition, of Γ = Aut(F _k ) on the set of marked, k-generated, dense subgroups D_k,A: = { h ? \textHom( F_k,A )| [`( á f( F_k ) ñ )] = A } {D_{k,A}}: = \left\{ {\eta \in {\text{Hom}}\left( {{F_k},A} \right)\left| {\overline {\left\langle {\phi \left( {{F_k}} \right)} \right\rangle } = A} \right.} \right\} . We prove the ergodicity of this action for the following two families of simple, totally disconnected, locally compact groups:

•	A = PSL2(K) where K is a non-Archimedean local field (of characteristic ≠ 2);
•	A = Aut0(T q+1)—the group of orientation-preserving automorphisms of a q + 1 regular tree, for q \geqslant 2.q \geqslant 2.

In contrast, a recent result of Minsky’s shows that the same action fails to be ergodic for A = PSL₂(C) and, when k is even, also for A = PSL₂(R). Therefore, if k \geqslant 4 k \geqslant 4 is even and K is a local field (with char(K) ≠ 2), the action of Aut(F _k ) on D_{k,\textPS\textL2(K)} {D_{k,{\text{PS}}{{\text{L}}_2}(K)}} is ergodic if and only if K is non-Archimedean. Ergodicity implies that every “measurable property” either holds or fails to hold for almost every k-generated dense subgroup of A.     

Markov decision processes with exponentially representable discounting

We generalize the geometric discount of finite discounted cost Markov Decision Processes to “exponentially representable”discount functions, prove existence of optimal policies which are stationary from some time N onward, and provide an algorithm for their computation. Outside this class, optimal “N-stationary” policies in general do not exist.     

Molecular optimization of multiply-functionalized mesoporous films with ion conduction properties

Sequential processing of multiply functionalized mesoporous films is shown to yield materials that are compositionally and structurally heterogeneous on mesoscopic and molecular length scales, both of which must be controlled to optimize macroscopic ion-conduction properties. Cubic mesoporous silica films prepared from strongly acidic solutions were subsequently functionalized under highly alkaline conditions to incorporate hydrophilic aluminosilica surface moieties, followed by nonaqueous conditions to introduce perfluorosulfonic-acid surface groups. Such sequential combination of individually incompatible steps yielded stable mesoporous films with high surface hydrophilicities and strong acid functionalities that exhibited high proton conductivities (ca. 9 × 10(-2) S/cm) at elevated temperatures (120 °C). Molecular, mesoscopic, and macroscopic properties of the multiply functionalized films were monitored and correlated at each stage of the syntheses by nuclear magnetic resonance (NMR) spectroscopy, small-angle X-ray scattering (SAXS), transmission electron microscopy (TEM), elemental analysis, adsorption, and ion conductivity measurements. In particular, variable-temperature solid-state two-dimensional (2D) (27)Al{(1)H}, (29)Si{(1)H}, (27)Al{(19)F}, and (29)Si{(19)F} HETeronuclear chemical-shift CORrelation (HETCOR) NMR spectra reveal separate surface adsorption and grafting sites for the different functional surface species within the mesopore channels. The hydrophilic aluminosilica and acidic fluoro-group loadings and interaction sites are demonstrated to be strongly affected by the different synthesis and functionalization treatments, which must be separately and collectively optimized to maximize the proton conductivities.     

Limitations to Fréchet’s metric embedding method

Fréchet’s classical isometric embedding argument has evolved to become a major tool in the study of metric spaces. An important example of a Fréchet embedding is Bourgain's embedding [4]. The authors have recently shown [2] that for every ε>0, anyn-point metric space contains a subset of size at leastn ^1−ε which embeds into ℓ₂ with distortion . The embedding used in [2] is non-Fréchet, and the purpose of this note is to show that this is not coincidental. Specifically, for every ε>0, we construct arbitrarily largen-point metric spaces, such that the distortion of any Fréchet embedding into ℓ_p on subsets of size at leastn ^1/2+ε is Ω((logn)^1/p ). Supported in part by a grant from the Israeli National Science Foundation. Supported in part by a grant from the Israeli National Science Foundation. Supported in part by the Landau Center.     

Spatial periodic forcing can displace patterns it is intended to control

Spatial periodic forcing of pattern-forming systems is an important, but lightly studied, method of controlling patterns. It can be used to control the amplitude and wave number of one-dimensional periodic patterns, to stabilize unstable patterns, and to induce them below instability onset. We show that, although in one spatial dimension the forcing acts to reinforce the patterns, in two dimensions it acts to destabilize or displace them by inducing two-dimensional rectangular and oblique patterns.     

A characterization of vector measure games inpNA

We give a complete characterization of games inpNA of the formf ^o μ (where μ is a vector of finite number of non-atomic probability measures, andf is a real valued function on the range of μ withf(0)=0). Specifically, we show thatf ^o μ is inpNA iff “f is continuous at μ” (the definition of the latter is given in the paper). This work was supported by National Science Foundation Grant SOC 75-21820-A01 at the Institute for Mathematical Studies in the Social Sciences, Stanford University.