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Averaging bounds for lattices and linear codes
 IEEE Trans. Information Theory
, 1997
"... Abstract — General random coding theorems for lattices are derived from the Minkowski–Hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out. A new version of the Minkowski–Hlawka theorem itself is obtained as the limit, for p!1,ofa ..."
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Cited by 97 (1 self)
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,ofasimple lemma for linear codes over GF (p) used with plevel amplitude modulation. The relation between the combinatorial packing of solid bodies and the informationtheoretic “soft packing ” with arbitrarily small, but positive, overlap is illuminated. The “softpacking” results are new. When specialized
Symplectic bifurcations of plane curves and isotropic liftings
 Quarterly J. Math
"... We study the classification of varieties in the Marsden–Weinstein reduction and their liftability. In particular the complete symplectic classification of the Bruce–Gaffney plane curve singularites is provided and is applied to obtain naturally the Lagrangian openings. 1. ..."
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Cited by 19 (8 self)
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We study the classification of varieties in the Marsden–Weinstein reduction and their liftability. In particular the complete symplectic classification of the Bruce–Gaffney plane curve singularites is provided and is applied to obtain naturally the Lagrangian openings. 1.
Minimum Rank and Maximum Eigenvalue Multiplicity of Symmetric Tree Sign Patterns
, 2005
"... The set of real matrices described by a sign pattern (a square matrix whose entries are elements of {+, −, 0}) has been studied extensively. A simple graph has been associated with the set of symmetric matrices having a zerononzero pattern of offdiagonal entries described by the graph. In this p ..."
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Cited by 12 (6 self)
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The set of real matrices described by a sign pattern (a square matrix whose entries are elements of {+, −, 0}) has been studied extensively. A simple graph has been associated with the set of symmetric matrices having a zerononzero pattern of offdiagonal entries described by the graph. In this paper, we present a unified approach to the study of the set of symmetric matrices described by a sign pattern and the set of matrices associated with a graph allowing loops, with the presence or absence of loops describing the zerononzero pattern of the diagonal. We call any family of matrices having a common graph a cohort. For a cohort whose graph is a tree, we provide an algorithm for the calculation of the maximum of the multiplicities of eigenvalues of any matrix in the cohort. For a symmetric tree sign pattern or tree that allows loops, this algorithm allows exact computation of maximum multiplicity and minimum rank, and can be used to obtain a symmetric integer matrix realizing minimum rank.
Continuous dependence estimates for nonlinear fractional convectiondiffusion equations
 SIAM J. MATH. ANAL
, 2012
"... We develop a general framework for finding error estimates for convectiondiffusion equations with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators that are generators of pure jump Lévy processes (e.g. the f ..."
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Cited by 8 (4 self)
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We develop a general framework for finding error estimates for convectiondiffusion equations with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators that are generators of pure jump Lévy processes (e.g. the fractional Laplacian). As an application, we derive continuous dependence estimates on the nonlinearities and on the Lévy measure of the diffusion term. Estimates of the rates of convergence for general nonlinear nonlocal vanishing viscosity approximations of scalar conservation laws then follow as a corollary. Our results both cover, and extend to new equations, a large part of the known error estimates in the literature.
ON COUNTABLE YON NEUMANN REGULAR RINGS
"... This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DMLCZ: The Czech Digital ..."
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This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DMLCZ: The Czech Digital
A Numerical Perspective on HartreeFockBogoliubov Theory
, 2012
"... The method ofchoice fordescribing attractive quantum systems isHartreeFockBogoliubov (HFB) theory. This is a nonlinear model which allows for the description of pairing effects, the main explanation for the superconductivity of certain materials at very low temperature. This paper is the first stu ..."
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Cited by 2 (0 self)
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The method ofchoice fordescribing attractive quantum systems isHartreeFockBogoliubov (HFB) theory. This is a nonlinear model which allows for the description of pairing effects, the main explanation for the superconductivity of certain materials at very low temperature. This paper is the first study of HartreeFockBogoliubov theory from the point of view of numerical analysis. We start by discussing its proper discretization and then analyze the convergence of the simple fixed point (Roothaan) algorithm. Following works by Cancès, Le Bris and Levitt for electrons in atoms and molecules, we show that this algorithm either converges to a solution of the equation, or oscillates between two states, none of them being a solution to the HFB equations. We also adapt the Optimal Damping Algorithm of Cancès and Le Bris to the HFB setting and we analyze it. The last part of the paper is devoted to numerical experiments. We consider a purely gravitational system and numerically discover that pairing always occurs. We then examine a simplified model for nucleons, with an effective interaction similar to what is often used in nuclear physics. In both cases we discuss the importance of using a damping algorithm. c ○ 2012 by the authors. This paper may be reproduced, in its entirety, for noncommercial purposes. Contents 1
FORMULAS FOR PRIMITIVE IDEMPOTENTS IN FROBENIUS ALGEBRAS AND AN APPLICATION TO DECOMPOSITION MAPS
"... Abstract. Inthefirstpartofthispaperwepresentexplicitformulasforprimitive idempotents in arbitrary Frobenius algebras using the entries of representing matrices coming from projective indecomposable modules with respect to a certain choice of basis. The proofs use a generalisation of the wellknown F ..."
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Abstract. Inthefirstpartofthispaperwepresentexplicitformulasforprimitive idempotents in arbitrary Frobenius algebras using the entries of representing matrices coming from projective indecomposable modules with respect to a certain choice of basis. The proofs use a generalisation of the wellknown FrobeniusSchur relations for semisimple algebras. The second part of this paper considers Ofree Oalgebras of finite Orank over a discrete valuation ring O and their decomposition maps under modular reduction modulo the maximal ideal of O, thereby studying the modular representation theory of such algebras. Using the formulas from the first part we derive general criteria for such a decomposition map to be an isomorphism that preserves the classes of simple modules involving explicitly known matrix representations on projective indecomposable modules. Finally, we show how this approach could eventually be used to attack a conjecture by Gordon James in the formulation of Meinolf Geck for IwahoriHecke algebras, provided the necessary matrix representations on projective indecomposable modules could be constructed explicitly. 1.
Documenta Math. 147 Variations on a Theme of Groups Splitting by a Quadratic Extension and GrothendieckSerre Conjecture for Group Schemes F4 with Trivial g3 Invariant
, 2009
"... Abstract. Westudystructurepropertiesofreductivegroupschemes defined over a local ring and splitting over its étale quadratic extension. As an application we prove Serre–Grothendieck conjecture on rationally trivial torsors over a local regular ring containing a field of characteristic 0 for group sc ..."
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Abstract. Westudystructurepropertiesofreductivegroupschemes defined over a local ring and splitting over its étale quadratic extension. As an application we prove Serre–Grothendieck conjecture on rationally trivial torsors over a local regular ring containing a field of characteristic 0 for group schemes of type F4 with trivial g3 invariant.
DepartmentofComputerScience SeriesofPublicationsA ReportA19951 MappingBayesianNetworkstoStochasticNeural Networks:AFoundationforHybrid BayesianNeuralSystems
"... ability(MAP)valueassignmentsforasetofdiscreteattributes,giventheconstraint ISSN12388645,ISBN9514572114 Helsinki,December1995,93pages thatsomeoftheattributesarepermanentlyxedtosomevaluesapriori.Forbuild Inthiswork,weareinterestedintheproblemofndingmaximumaposterioriprob Abstract ingasystemcapab ..."
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ability(MAP)valueassignmentsforasetofdiscreteattributes,giventheconstraint ISSN12388645,ISBN9514572114 Helsinki,December1995,93pages thatsomeoftheattributesarepermanentlyxedtosomevaluesapriori.Forbuild Inthiswork,weareinterestedintheproblemofndingmaximumaposterioriprob Abstract ingasystemcapableofthistypeofuncertainreasoninginpractice,weneedrstto massivelyparallelplatformforsearchingthemodelstatespace.Themainapplic forsolvingthesetwosubtasks.TheBayesiannetworkcomponentcanbeusedfor available.Theneuralnetworkcomponentprovidesthenacomputationallyecient, ilitydistributionquicklyandreliably,assumingthatsuitableexpertknowledgeis constructingacompact,highlevelrepresentationfortheproblemdomainprobab establishanecientsearchmechanismforndingMAPcongurationswithinthe constructedmodel.WeproposeahybridBayesiannetworkneuralnetworksystem constructanaccurateabstractrepresentationoftheproblemdomain,andthento
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