Almost-Euclidean subspaces of ℓ1N\ell_1^N via tensor products: a simple approach to randomness reduction

It has been known since 1970's that the N-dimensional $\ell_1$-space contains nearly Euclidean subspaces whose dimension is $\Omega(N)$. However, proofs of existence of such subspaces were probabilistic, hence non-constructive, which made the results not-quite-suitable for subsequently discovered applications to high-dimensional nearest neighbor search, error-correcting codes over the reals, compressive sensing and other computational problems. In this paper we present a "low-tech" scheme which, for any $a > 0$, allows to exhibit nearly Euclidean $\Omega(N)$-dimensional subspaces of $\ell_1^N$ while using only $N^a$ random bits. Our results extend and complement (particularly) recent work by Guruswami-Lee-Wigderson. Characteristic features of our approach include (1) simplicity (we use only tensor products) and (2) yielding "almost Euclidean" subspaces with arbitrarily small distortions.Comment: 11 pages; title change, abstract and references added, other minor change

Relativistic MHD with Adaptive Mesh Refinement

This paper presents a new computer code to solve the general relativistic magnetohydrodynamics (GRMHD) equations using distributed parallel adaptive mesh refinement (AMR). The fluid equations are solved using a finite difference Convex ENO method (CENO) in 3+1 dimensions, and the AMR is Berger-Oliger. Hyperbolic divergence cleaning is used to control the $\nabla\cdot {\bf B}=0$ constraint. We present results from three flat space tests, and examine the accretion of a fluid onto a Schwarzschild black hole, reproducing the Michel solution. The AMR simulations substantially improve performance while reproducing the resolution equivalent unigrid simulation results. Finally, we discuss strong scaling results for parallel unigrid and AMR runs.Comment: 24 pages, 14 figures, 3 table

Nonlinear spectral calculus and super-expanders

Nonlinear spectral gaps with respect to uniformly convex normed spaces are shown to satisfy a spectral calculus inequality that establishes their decay along Cesaro averages. Nonlinear spectral gaps of graphs are also shown to behave sub-multiplicatively under zigzag products. These results yield a combinatorial construction of super-expanders, i.e., a sequence of 3-regular graphs that does not admit a coarse embedding into any uniformly convex normed space.Comment: Typos fixed based on referee comments. Some of the results of this paper were announced in arXiv:0910.2041. The corresponding parts of arXiv:0910.2041 are subsumed by the current pape

Reliability and Validity of a Flume-Based Maximal Oxygen Uptake Swimming Test

A mode-specific swimming protocol to assess maximal aerobic uptake (VO2maxsw) is vital to accurately evaluate swimming performance. A need exists for reliable and valid swimming protocols that assess VO2maxsw in a flume environment. The purpose was to assess: (a) reliability and (b) “performance” validity of a VO2maxsw flume protocol using the 457-m freestyle pool performance swim (PS) test as the criterion. Nineteen males (n = 9) and females (n = 10) (age, 28.5 ± 8.3 years.; height, 174.7 ± 8.2 cm; mass, 72.9 ± 12.5 kg; %body fat, 21.4 ± 5.9) performed two flume VO2maxsw tests (VO2maxswA and VO2maxswB) and one PS test [457 m (469.4 ± 94.7 s)]. For test–retest reliability (Trials A vs. B), moderately strong relationships were established for VO2maxsw (mL·kg−1·min−1)(r= 0.628, p = 0.002), O2pulse (mL O2·beat−1)(r = 0.502, p = 0.014), VEmax (L·min−1) (r = 0.671, p = 0.001), final test time (sec) (0.608, p = 0.004), and immediate post-test blood lactate (IPE (BLa)) (0.716, p = 0.001). For performance validity, moderately strong relationships (p \u3c 0.05) were found between VO2maxswA (r =−0.648, p = 0.005), O2pulse (r= −0.623, p = 0.008), VEmax (r = −0.509 p = 0.037), and 457-m swim times. The swimming flume protocol examined is a reliable and valid assessment of VO2maxsw., and offers an alternative for military, open water, or those seeking complementary forms of training to improve swimming performance

Spectral density and Sobolev inequalities for pure and mixed states

We prove some general Sobolev-type and related inequalities for positive operators A of given ultracontractive spectral decay, without assuming e^{-tA} is submarkovian. These inequalities hold on functions, or pure states, as usual, but also on mixed states, or density operators in the quantum mechanical sense. This provides universal bounds of Faber-Krahn type on domains, that apply to their whole Dirichlet spectrum distribution, not only the first eigenvalue. Another application is given to relate the Novikov-Shubin numbers of coverings of finite simplicial complexes to the vanishing of the torsion of some l^{p,2}-cohomology

Intertwining relations for one-dimensional diffusions and application to functional inequalities

International audienceFollowing the recent work [13] fulfilled in the discrete case, we pro- vide in this paper new intertwining relations for semigroups of one-dimensional diffusions. Various applications of these results are investigated, among them the famous variational formula of the spectral gap derived by Chen and Wang [15] together with a new criterion ensuring that the logarithmic Sobolev inequality holds. We complete this work by revisiting some classical examples, for which new estimates on the optimal constants are derived

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1

Telephone-administered psychotherapy for depression in MS patients: moderating role of social support

Depression is common in individuals with multiple sclerosis (MS). While psychotherapy is an effective treatment for depression, not all individuals benefit. We examined whether baseline social support might differentially affect treatment outcome in 127 participants with MS and depression randomized to either Telephone-administered Cognitive-Behavioral Therapy (T-CBT) or Telephone-administered Emotion-Focused Therapy (T-EFT). We predicted that those with low social support would improve more in T-EFT, since this approach emphasizes the therapeutic relationship, while participants with strong social networks and presumably more emotional resources might fare better in the more structured and demanding T-CBT. We found that both level of received support and satisfaction with that support at baseline did moderate treatment outcome. Individuals with high social support showed a greater reduction in depressive symptoms in the T-CBT as predicted, but participants with low social support showed a similar reduction in both treatments. This suggests that for participants with high social support, CBT may be a more beneficial treatment for depression compared with EFT

Quantum-information entropies for highly excited states of single-particle systems with power-type potentials

The asymptotics of the Boltzmann-Shannon information entropy as well as the Renyi entropy for the quantum probability density of a single-particle system with a confining (i.e., bounded below) power-type potential V(x)=x^2k with k∈N and x∈R, is investigated in the position and momentum spaces within the semiclassical (WKB) approximation. It is found that for highly excited states both physical entropies, as well as their sum, have a logarithmic dependence on its quantum number not only when k=1 (harmonic oscillator), but also for any fixed k. As a by-product, the extremal case k→∞ (the infinite well potential) is also rigorously analyzed. It is shown that not only the position-space entropy has the same constant value for all quantum states, which is a known result, but also that the momentum-space entropy is constant for highly excited states