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Demenkov E.,Saint Petersburg State University | Kulikov A.S.,Steklov Institute of Mathematics
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2011

A Boolean function f: double-struck F2n → double-struck F 2 is called an affine disperser of dimension d, if f is not constant on any affine subspace of double-struck F2 n of dimension at least d. Recently Ben-Sasson and Kopparty gave an explicit construction of an affine disperser for sublinear d. The main motivation for studying such functions comes from extracting randomness from structured sources of imperfect randomness. In this paper, we show another application: we give a very simple proof of a 3n-o(n) lower bound on the circuit complexity (over the full binary basis) of affine dispersers for sublinear dimension. The same lower bound 3n-o(n) (but for a completely different function) was given by Blum in 1984 and is still the best known. The main technique is to substitute variables by linear functions. This way the function is restricted to an affine subspace of F2n. An affine disperser for sublinear dimension then guarantees that one can make n-o(n) such substitutions before the function degenerates. It remains to show that each such substitution eliminates at least 3 gates from a circuit. © 2011 Springer-Verlag GmbH.

Livshitz E.D.,Moscow State University | Temlyakov V.N.,University of South Carolina | Temlyakov V.N.,Steklov Institute of Mathematics
IEEE Transactions on Information Theory | Year: 2014

We study sparse approximation by greedy algorithms. Our contribution is twofold. First, we prove exact recovery with high probability of random K-sparse signals within (K(1+ε) iterations of the orthogonal matching pursuit (OMP). This result shows that in a probabilistic sense, the OMP is almost optimal for exact recovery. Second, we prove the Lebesgue-type inequalities for the weak Chebyshev greedy algorithm, a generalization of the weak orthogonal matching pursuit to the case of a Banach space. The main novelty of these results is a Banach space setting instead of a Hilbert space setting. However, even in the case of a Hilbert space, our results add some new elements to known results on the Lebesgue-type inequalities for the restricted isometry property dictionaries. Our technique is a development of the recent technique created by Zhang. © 2014 IEEE.

Kiselev A.P.,Steklov Institute of Mathematics | Parker D.F.,University of Edinburgh
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | Year: 2010

Just as uni-directional Rayleigh waves at the traction-free surface of a transversely isotropic elastic half-space and Stoneley waves at the interface between two such media may have arbitrary waveform and may be represented in terms of a single function harmonic in a half-plane, it is shown that surface-guided waves travelling simultaneously in all directions parallel to the surface may be represented, at each instant, in terms of a single function satisfying Laplace's equation in a three-dimensional half-space. That harmonic function is determined so that its normal derivative at the surface equals the normal displacement of the surface (or interface). It is shown, moreover, that the time evolution of that normal displacement may be any solution to the membrane equation with wave speed being equal to that of classical, uni-directional, time-harmonic Rayleigh or Stoneley waves. A similar representation is also shown to exist for Schölte waves at a fluid-solid interface, in the non-evanescent case. Thus, every surface- or interfaceguided disturbance in media having rotational symmetry about the surface normal is governed by the membrane equation with appropriate wave speed, provided that the combination of materials allows uni-directional, time-harmonic waves that are nonevanescent. Conversely, each solution to the membrane equation may be used to construct a representation of either a Rayleigh wave, a Stoneley wave or a (non-evanescent) Schölte wave. In each case, the disturbance at all depths may be represented at each instant in terms of a single function harmonic in a half-space. © 2010 The Royal Society.

Malyshev C.,Steklov Institute of Mathematics
Annals of Physics | Year: 2014

The behavior of the shear modulus caused by proliferation of dipoles of non-singular screw dislocations with finite-sized core is considered. The representation of two-dimensional Coulomb gas with smoothed-out coupling is used, and the stress-stress correlation function is calculated. A convolution integral expressed in terms of the modified Bessel function K0 is derived in order to obtain the shear modulus in approximation of interacting dipoles. Implications are demonstrated for the shear modulus near the melting transition which are due to the singularityless character of the dislocations. © 2014 Elsevier Inc.

Malyshev C.,Steklov Institute of Mathematics
Journal of Physics A: Mathematical and Theoretical | Year: 2011

A field theory is developed for a thermodynamical description of an array of parallel non-singular screw dislocations in an elastic cylinder. The partition function of the system is considered in the functional integral form. Self-energy of the dislocation cores is chosen in the form suggested by the gauge-translational model of non-singular screw dislocation. It is shown that the system of dislocations is equivalent to the two-dimensional Coulomb gas. The coupling potential is prevented from a short-distance divergency since the core energies are taken into account. Two-point correlation functions of the stress components are obtained. Renormalization of the shear modulus caused by the presence of dislocations is studied in the approximation of non-interacting dislocation dipoles. It is demonstrated that the finite size of the dislocation cores results in a modification of the renormalization law. © 2011 IOP Publishing Ltd.

Sagitov S.,Chalmers University of Technology | Jagers P.,Chalmers University of Technology | Vatutin V.,Steklov Institute of Mathematics
Theoretical Population Biology | Year: 2010

We establish convergence to the Kingman coalescent for the genealogy of a geographically-or otherwise-structured version of the Wright-Fisher population model with fast migration. The new feature is that migration probabilities may change in a random fashion. This brings a novel formula for the coalescent effective population size (EPS). We call it a quenched EPS to emphasize the key feature of our model - random environment. The quenched EPS is compared with an annealed (mean-field) EPS which describes the case of constant migration probabilities obtained by averaging the random migration probabilities over possible environments. © 2010 Elsevier Inc.

Faddeev L.D.,Steklov Institute of Mathematics
International Journal of Modern Physics A | Year: 2010

In the terminology of theoretical physics, the term "ghost" is used to identify an object that has no real physical meaning. The name "FaddeevPopov ghosts" is given to the fictitious fields that were originally introduced in the construction of a manifestly Lorentz covariant quantization of the YangMills field. Later, these objects acquired more widespread application, including in string theory. The necessity of ghosts is associated with gauge invariance. In gauge invariant theories, one usually has to deal with local fields, whose number exceeds that of physical degrees of freedom. For example in electrodynamics, in order to maintain manifest Lorentz invariance, one uses a four component vector potential Aμ(x), whereas the photon has only two polarizations. Thus, one needs a suitable mechanism in order to get rid of the unphysical degrees of freedom. Introducing fictitious fields, the ghosts, is one way of achieving this goal. © 2010 World Scientific Publishing Company.

Matiyasevich Y.,Steklov Institute of Mathematics
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2012

In mathematics sometimes methods from one area can be fruitfully applied for getting results in another area, occasionally looking very remote from the other area. A well-known example is given by analytic geometry that enables us, besides proving "elementary" geometrical theorems, to establish otherwise untractable results like unsolvability of the problems of angle trisection and doubling the cube by compass and straightedge and to reduce calculation of the kissing numbers of spheres to verification of a first-order formula about real numbers (and that could be done, in principle, by Tarski algorithm). © 2012 Springer-Verlag.

Kojevnikov A.,Steklov Institute of Mathematics | Kulikov A.S.,Steklov Institute of Mathematics
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2010

In this note, we use lower bounds on Boolean multiplicative complexity to prove lower bounds on Boolean circuit complexity. We give a very simple proof of a 7n/3-c lower bound on the circuit complexity of a large class of functions representable by high degree polynomials over GF(2). The key idea of the proof is a circuit complexity measure assigning different weights to XOR and AND gates. © 2010 Springer-Verlag Berlin Heidelberg.

Matiyasevich Y.,Steklov Institute of Mathematics
Combinatorics Probability and Computing | Year: 2010

The paper presents yet another way to reformulate the Four Colour Conjecture as a statement concerning conditional probabilities of certain events involving planar graphs. © 2010 Cambridge University Press.

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