Search ResultsShowing 1-4 of 4

1. arXiv:1909.11322 (Published 2019-09-25)

   A Few Surprising Integrals

   Malin Palö Forsström, Jeffrey E. Steif
   Comments: 5 pages, some overlap with arXiv:1812.08455v2 [math.PR]

   Categories: math.PR

   Subjects: 60E07, 60G52

   Keywords: surprising integrals, color model, fact independent, quantities
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   Using formulas for certain quantities involving stable vectors, due to I. Molchanov, and in some cases utilizing the so-called divide and color model, we prove that certain families of integrals which, ostensibly, depend on a parameter are in fact independent of this parameter.

2. arXiv:1808.00306 (Published 2018-08-01)

   Equilibrium fluctuations for a chain of anharmonic oscillators in the Euler scaling limit

   Stefano Olla, Lu Xu
   Categories: math.PR, cond-mat.stat-mech, math-ph, math.MP

   Keywords: euler scaling limit, anharmonic oscillators, equilibrium fluctuations, macroscopic behavior, quantities
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   We study the macroscopic behavior of the fluctuations in equilibrium for the conserved quantities of an anharmonic chain of oscillators under hyperbolic scaling of space and time. Under a stochastic perturbation of the dynamics conservative of such quantities, we prove that these fluctuations evolve macroscopically following the linearized Euler system of equations.

3. arXiv:1610.02295 (Published 2016-10-03)

   Explicit measures for the homogeneous transform

   Hubert Holin
   Comments: 14 pages, 3 figures (one of which is a composite)

   Categories: math.FA, math.PR

   Subjects: 62E15, G.3

   Keywords: homogeneous transform, explicit measures, practical applications outside, quantities, proportions
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   The homogeneous transform has many practical applications outside the realm of mathematics, for instance to represent the proportions of several chemical substances. We aim here to present results about the transformation of measures, which could be used to take into account the uncertainties of the quantities to be homogeneously transformed.

4. arXiv:1306.3779 (Published 2013-06-17, updated 2015-07-16)

   Bounds on restricted isometry constants of random matrices

   Mihailo Stojnic
   Comments: acknowledgement footnote added

   Categories: math.OC, cs.IT, math.IT, math.PR

   Keywords: random matrices, isometry properties, quantities, lower restricted isometry constants, powerful tail estimates
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   In this paper we look at isometry properties of random matrices. During the last decade these properties gained a lot attention in a field called compressed sensing in first place due to their initial use in \cite{CRT,CT}. Namely, in \cite{CRT,CT} these quantities were used as a critical tool in providing a rigorous analysis of $\ell_1$ optimization's ability to solve an under-determined system of linear equations with sparse solutions. In such a framework a particular type of isometry, called restricted isometry, plays a key role. One then typically introduces a couple of quantities, called upper and lower restricted isometry constants to characterize the isometry properties of random matrices. Those constants are then usually viewed as mathematical objects of interest and their a precise characterization is desirable. The first estimates of these quantities within compressed sensing were given in \cite{CRT,CT}. As the need for precisely estimating them grew further a finer improvements of these initial estimates were obtained in e.g. \cite{BCTsharp09,BT10}. These are typically obtained through a combination of union-bounding strategy and powerful tail estimates of extreme eigenvalues of Wishart (Gaussian) matrices (see, e.g. \cite{Edelman88}). In this paper we attempt to circumvent such an approach and provide an alternative way to obtain similar estimates.