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

Amenability of algebras of approximable operators

We give a necessary and sufficient condition for amenability of the Banach algebra of approximable operators on a Banach space. We further investigate the relationship between amenability of this algebra and factorization of operators, strengthening known results and developing new techniques to determine whether or not a given Banach space carries an amenable algebra of approximable operators. Using these techniques, we are able to show, among other things, the non-amenability of the algebra of approximable operators on Tsirelson's space.Comment: 20 pages, to appear in Israel Journal of Mathematic

Stability of chemical reaction fronts in solids:Analytical and numerical approaches

Localized chemical reactions in deformable solids are considered. A chemical transformation is accompanied by the transformation strain and emerging mechanical stresses, which affect the kinetics of the chemical reaction front to the reaction arrest. A chemo-mechanical coupling via the chemical affinity tensor is used, in which the stresses affect the reaction rate. The emphasis is made on the stability of the propagating reaction front in the vicinity of the blocked state. There are two major novel contributions. First, it is shown that for a planar reaction front, the diffusion of the gaseous-type reactant does not influence the stability of the reaction front – the stability is governed only by the mechanical properties of solid reactants and stresses induced by the transformation strain and the external loading, which corresponds to the mathematically analogous phase transition problem. Second, the comparison of two computational approaches to model the reaction front propagation is performed – the standard finite-element method with a remeshing technique to resolve the moving interface is compared to the cut-finite-element-based approach, which allows the interface to cut through the elements and to move independently of the finite-element mesh. For stability problems considered in the present paper, the previously-developed implementation of the cut-element approach has been extended with the additional post-processing procedure that obtains more accurate stresses and strains, relying on the fact that the structured grid is used in the implementation. The approaches are compared using a range of chemo-mechanical problems with stable and unstable reaction fronts.</p

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

Measurement of the Luminosity in the ZEUS Experiment at HERA II

The luminosity in the ZEUS detector was measured using photons from electron bremsstrahlung. In 2001 the HERA collider was upgraded for operation at higher luminosity. At the same time the luminosity-measuring system of the ZEUS experiment was modified to tackle the expected higher photon rate and synchrotron radiation. The existing lead-scintillator calorimeter was equipped with radiation hard scintillator tiles and shielded against synchrotron radiation. In addition, a magnetic spectrometer was installed to measure the luminosity independently using photons converted in the beam-pipe exit window. The redundancy provided a reliable and robust luminosity determination with a systematic uncertainty of 1.7%. The experimental setup, the techniques used for luminosity determination and the estimate of the systematic uncertainty are reported.Comment: 25 pages, 11 figure

Factorization of Operators Through Orlicz Spaces

[EN] We study factorization of operators between quasi-Banach spaces. We prove the equivalence between certain vector norm inequalities and the factorization of operators through Orlicz spaces. As a consequence, we obtain the Maurey-Rosenthal factorization of operators into L-p-spaces. We give several applications. In particular, we prove a variant of Maurey's Extension Theorem.The research of the first author was supported by the National Science Centre (NCN), Poland, Grant No. 2011/01/B/ST1/06243. The research of the second author was supported by Ministerio de Economia y Competitividad, Spain, under project #MTM2012-36740-C02-02Mastylo, M.; Sánchez Pérez, EA. (2017). Factorization of Operators Through Orlicz Spaces. Bulletin of the Malaysian Mathematical Sciences Society. 40(4):1653-1675. https://doi.org/10.1007/s40840-015-0158-5S16531675404Calderón, A.P.: Intermediate spaces and interpolation, the complex method. Stud. Math. 24, 113–190 (1964)Davis, W.J., Garling, D.J.H., Tomczak-Jaegermann, N.: The complex convexity of quasi-normed linear spaces. J. Funct. Anal. 55, 110–150 (1984)Defant, A.: Variants of the Maurey–Rosenthal theorem for quasi Köthe function spaces. Positivity 5, 153–175 (2001)Defant, A., Mastyło, M., Michels, C.: Orlicz norm estimates for eigenvalues of matrices. Isr. J. Math. 132, 45–59 (2002)Defant, A., Sánchez Pérez, E.A.: Maurey–Rosenthal factorization of positive operators and convexity. J. Math. Anal. Appl. 297, 771–790 (2004)Defant, A., Sánchez Pérez, E.A.: Domination of operators on function spaces. Math. Proc. Camb. Phil. Soc. 146, 57–66 (2009)Diestel, J.: Sequences and Series in Banach Spaces. Springer, Berlin (1984)Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)Dilworth, S.J.: Special Banach lattices and their applications. In: Handbook of the Geometry of Banach Spaces, vol. 1. Elsevier, Amsterdam (2001)Figiel, T., Pisier, G.: Séries alétoires dans les espaces uniformément convexes ou uniformément lisses. Comptes Rendus de l’Académie des Sciences, Paris, Série A 279, 611–614 (1974)Kalton, N.J., Montgomery-Smith, S.J.: Set-functions and factorization. Arch. Math. (Basel) 61(2), 183–200 (1993)Kamińska, A., Mastyło, M.: Abstract duality Sawyer formula and its applications. Monatsh. Math. 151(3), 223–245 (2007)Kantorovich, L.V., Akilov, G.P.: Functional Analysis, 2nd edn. Pergamon Press, Oxford (1982)Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)Lozanovskii, G.Ya.: On some Banach lattices IV, Sibirsk. Mat. Z. 14, 140–155 (1973) (in Russian); English transl.: Siberian. Math. J. 14, 97–108 (1973)Lozanovskii, G.Ya.:Transformations of ideal Banach spaces by means of concave functions. In: Qualitative and Approximate Methods for the Investigation of Operator Equations, Yaroslavl, vol. 3, pp. 122–147 (1978) (Russian)Mastyło, M., Szwedek, R.: Interpolative constructions and factorization of operators. J. Math. Anal. Appl. 401, 198–208 (2013)Nikišin, E.M.: Resonance theorems and superlinear operators. Usp. Mat. Nauk 25, 129–191 (1970) (Russian)Okada, S., Ricker, W.J., Sánchez Pérez, E.A.: Optimal Domain and Integral Extension of Operators acting in Function Spaces. Operator Theory: Adv. Appl., vol. 180. Birkhäuser, Basel (2008)Pisier, G.: Factorization of linear operators and geometry of Banach spaces. CBMS Regional Conference Series in Mathematics, vol. 60. Published for the Conference Board of the Mathematical Sciences, Washington, DC (1986)Reisner, S.: On two theorems of Lozanovskii concerning intermediate Banach lattices, geometric aspects of functional analysis (1986/87). Lecture Notes in Math., vol. 1317, pp. 67–83. Springer, Berlin (1988)Wojtaszczyk, P.: Banach Spaces for Analysts. Cambridge University Press, Cambridge (1991

Stability of chemical reaction fronts in solids : analytical and numerical approaches

Localized chemical reactions in deformable solids are considered. A chemical transformation is accompanied by the transformation strain and emerging mechanical stresses, which affect the kinetics of the chemical reaction front to the reaction arrest. A chemo-mechanical coupling via the chemical affinity tensor is used, in which the stresses affect the reaction rate. The emphasis is made on the stability of the propagating reaction front in the vicinity of the blocked state. There are two major novel contributions. First, it is shown that for a planar reaction front, the diffusion of the gaseous-type reactant does not influence the stability of the reaction front – the stability is governed only by the mechanical properties of solid reactants and stresses induced by the transformation strain and the external loading, which corresponds to the mathematically analogous phase transition problem. Second, the comparison of two computational approaches to model the reaction front propagation is performed – the standard finite-element method with a remeshing technique to resolve the moving interface is compared to the cut-finite-element-based approach, which allows the interface to cut through the elements and to move independently of the finite-element mesh. For stability problems considered in the present paper, the previously-developed implementation of the cut-element approach has been extended with the additional post-processing procedure that obtains more accurate stresses and strains, relying on the fact that the structured grid is used in the implementation. The approaches are compared using a range of chemo-mechanical problems with stable and unstable reaction fronts