Showing papers on "Space (mathematics) published in 1996"

The Production of Space

Abstract: ‘The Production of Space’, in: Frans Jacobi, Imagine, Space Poetry, Copenhagen, 1996, unpaginated.

Separable dual-space Gaussian pseudopotentials

Abstract: We present pseudopotential coefficients for the first two rows of the Periodic Table. The pseudopotential is of an analytic form that gives optimal efficiency in numerical calculations using plane waves as a basis set. At most, seven coefficients are necessary to specify its analytic form. It is separable and has optimal decay properties in both real and Fourier space. Because of this property, the application of the nonlocal part of the pseudopotential to a wave function can be done efficiently on a grid in real space. Real space integration is much faster for large systems than ordinary multiplication in Fourier space, since it shows only quadratic scaling with respect to the size of the system. We systematically verify the high accuracy of these pseudopotentials by extensive atomic and molecular test calculations. \textcopyright{} 1996 The American Physical Society.

Logarithmic sobolev inequalities for finite markov chains

Abstract: This is an expository paper on the use of logarithmic Sobolev inequalities for bounding rates of convergence of Markov chains on finite state spaces to their stationary distributions. Logarithmic Sobolev inequalities complement eigenvalue techniques and work for nonreversible chains in continuous time. Some aspects of the theory simplify considerably with finite state spaces and we are able to give a self-contained development. Examples of applications include the study of a Metropolis chain for the binomial distribution, sharp results for natural chains on the box of side n in d dimensions and improved rates for exclusion processes. We also show that for most r-regular graphs the log-Sobolev constant is of smaller order than the spectral gap. The log-Sobolev constant of the asymmetric two-point space is computed exactly as well as the log-Sobolev constant of the complete graph on n points.

Riemannian symmetric superspaces and their origin in random‐matrix theory

Abstract: Gaussian random‐matrix ensembles defined over the tangent spaces of the large families of Cartan’s symmetric spaces are considered. Such ensembles play a central role in mesoscopic physics, as they describe the universal ergodic limit of disordered and chaotic single‐particle systems. The generating function for the spectral correlations of each ensemble is reduced to an integral over a Riemannian symmetric superspace in the limit of large matrix dimension. Such a space is defined as a pair (G/H,M r ), where G/H is a complex‐analytic graded manifold homogeneous with respect to the action of a complex Lie supergroup G, and M r is a maximal Riemannian submanifold of the support of G/H.

Randomness is linear in space

Abstract: We show that any randomized algorithm that runs in space S and time T and uses poly(S) random bits can be simulated using only O(S) random bits in space Sand time T+ poly(S). A deterministic simulation in space S follows. Of independent interest is our main technical tool : a procedure which extracts randomness from a defective random source using a small additional number of truly random bits.

Sampling theory in Fourier and signal analysis : foundations

Abstract: 1. An introduction to sampling theory 1.1 General introduction 1.2 Introduction - continued 1.3 The seventeenth to the mid twentieth century - a brief review 1.4 Interpolation and sampling from the seventeenth century to the mid twentieth century - a brief review 1.5 Introduction - concluding remarks 2. Background in Fourier analysis 2.1 The Fourier Series 2.2 The Fourier transform 2.3 Poisson's summation formula 2.4 Tempered distributions - some basic facts 3. Hilbert spaces, bases and frames 3.1 Bases for Banach and Hilbert spaces 3.2 Riesz bases and unconditional bases 3.3 Frames 3.4 Reproducing kernel Hilbert spaces 3.5 Direct sums of Hilbert spaces 3.6 Sampling and reproducing kernels 4. Finite sampling 4.1 A general setting for finite sampling 4.2 Sampling on the sphere 5. From finite to infinite sampling series 5.1 The change to infinite sampling series 5.2 The Theorem of Hinsen and Kloosters 6. Bernstein and Paley-Weiner spaces 6.1 Convolution and the cardinal series 6.2 Sampling and entire functions of polynomial growth 6.3 Paley-Weiner spaces 6.4 The cardinal series for Paley-Weiner spaces 6.5 The space ReH1 6.6 The ordinary Paley-Weiner space and its reproducing kernel 6.7 A convergence principle for general Paley-Weiner spaces 7. More about Paley-Weiner spaces 7.1 Paley-Weiner theorems - a review 7.2 Bases for Paley-Weiner spaces 7.3 Operators on the Paley-Weiner space 7.4 Oscillatory properties of Paley-Weiner functions 8. Kramer's lemma 8.1 Kramer's Lemma 8.2 The Walsh sampling therem 9. Contour integral methods 9.1 The Paley-Weiner theorem 9.2 Some formulae of analysis and their equivalence 9.3 A general sampling theorem 10. Ireggular sampling 10.1 Sets of stable sampling, of interpolation and of uniqueness 10.2 Irregular sampling at minimal rate 10.3 Frames and over-sampling 11. Errors and aliasing 11.1 Errors 11.2 The time jitter error 11.3 The aliasing error 12. Multi-channel sampling 12.1 Single channel sampling 12.3 Two channels 13. Multi-band sampling 13.1 Regular sampling 13.3 An algorithm for the optimal regular sampling rate 13.4 Selectively tiled band regions 13.5 Harmonic signals 13.6 Band-ass sampling 14. Multi-dimensional sampling 14.1 Remarks on multi-dimensional Fourier analysis 14.2 The rectangular case 14.3 Regular multi-dimensional sampling 15. Sampling and eigenvalue problems 15.1 Preliminary facts 15.2 Direct and inverse Sturm-Liouville problems 15.3 Further types of eigenvalue problem - some examples 16. Campbell's generalised sampling theorem 16.1 L.L. Campbell's generalisation of the sampling theorem 16.2 Band-limited functions 16.3 Non band-limited functions - an example 17. Modelling, uncertainty and stable sampling 17.1 Remarks on signal modelling 17.2 Energy concentration 17.3 Prolate Spheroidal Wave functions 17.4 The uncertainty principle of signal theory 17.5 The Nyquist-Landau minimal sampling rate

Étale Cohomology of Rigid Analytic Varieties and Adic Spaces

Abstract: Summary of the results on the etale cohomology of rigid analytic varieties - Adic spaces - The etale site of a rigid analytic variety and an adic space - Comparison theorems - Base change theorems - Cohomology with compact support - Finiteness - Poincare Duality - Partially proper sites of rigid analytic varieties and analytic adic spaces

Cosmic Topology

Abstract: General relativity does not allow one to specify the topology of space, leaving the possibility that space is multi-- rather than simply--connected We review the main mathematical properties of multi--connected spaces, and the different tools to classify them and to analyse their properties Following the mathematical classification, we describe the different possible muticonnected spaces which may be used to construct universe models We briefly discuss some implications of multi--connectedness for quantum cosmology, and its consequences concerning quantum field theory in the early universe We consider in details the properties of the cosmological models where space is multi--connected, with emphasis towards observable effects We then review the analyses of observational results obtained in this context, to search for a possible signature of multi--connectedness, or to constrain the models They may concern the distribution of images of cosmic objects like galaxies, clusters, quasars,, or more global effects, mainly those concerning the Cosmic Microwave Background, and the present limits resulting from them

Banach spaces of analytic functions

Abstract: In this paper, we explore certain Banach spaces of analytic functions. In particular, we study the space A -I, demonstrating some of its basic properties including non-separability. We ask the question: Given a class C of analytic functions on the unit disk ID> and a sequence { Zn} of points in the disk, is there an non-zero analytic function f E C with f(zn) = 0 for all n? Finally, we explore the Mz invariant subspaces of A-t, demonstrating that they may possess the codimension-2 property. This paper is part of the requirements for honors in mathematics. The signatures below, by the advisor, a departmental reader, and a representative of the departmental honors committee, demonstrate that Michael T. Nimchek has met all the requirements needed to receive honors in mathematics.

Linear estimation in Krein spaces. I. Theory

Abstract: The authors develop a self-contained theory for linear estimation in Krein spaces. The derivation is based on simple concepts such as projections and matrix factorizations and leads to an interesting connection between Krein space projection and the recursive computation of the stationary points of certain second-order (or quadratic) forms. The authors use the innovations process to obtain a general recursive linear estimation algorithm. When specialized to a state-space structure, the algorithm yields a Krein space generalization of the celebrated Kalman filter with applications in several areas such as H/sup /spl infin//-filtering and control, game problems, risk sensitive control, and adaptive filtering.

Contra-continuous functions and strongly S-closed spaces

Abstract: In 1989 Ganster and Reilly [6] introduced and studied the notion of LC-continuous functions via the concept of locally closed sets. In this paper we consider a stronger form of LC-continuity called contra-continuity. We call a function f:(X,τ)→(Y,σ) contra-continuous if the preimage of every open set is closed. A space (X,τ) is called strongly S-closed if it has a finite dense subset or equivalently if every cover of (X,τ) by closed sets has a finite subcover. We prove that contra-continuous images of strongly S-closed spaces are compact as well as that contra-continuous, β-continuous images of S-closed spaces are also compact. We show that every strongly S-closed space satisfies FCC and hence is nearly compact.

Convergence of evolutionary algorithms in general search spaces

Abstract: This paper provides conditions under which evolutionary algorithms with an elitist selection rule will converge to the global optimum of some function whose domain may be an arbitrary space. These results generalize the previously developed convergence theory for binary and Euclidean search spaces to general search spaces.

Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents

Abstract: © Annales de l’institut Fourier, 1996, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Constructible sets in real geometry

Abstract: I. A First Look at Semialgebraic Geometry.- 1. Real Closed Fields and Transfer Principles.- 2. What is Semialgebraic Geometry?.- 3. Real Spaces.- 4. Examples.- II. Real Algebra.- 1. The Real Spectrum of a Ring.- 2. Specializations, Zero Sets and Real Ideals.- 3. Real Valuations.- 4. Real Going-Up and Real Going-Down.- 5. Abstract Semialgebraic Functions.- 6. Cylindrical Decomposition.- 7. Real Strict Localization.- Notes.- III. Spaces of Signs.- 1. The Axioms.- 2. Forms.- 3. SAP-Spaces and Fans.- 4. Local Spaces of Signs.- 5. The Space of Signs of a Ring.- 6. Subspaces.- Notes.- IV. Spaces of Orderings.- 1. The Axioms Revisited.- 2. Basic Constructions.- 3. Spaces of Finite Type.- 4. Spaces of Finite Chain Length.- 5. Finite Type = Finite Chain Length.- 6. Local-Global Principles.- 7. Representation Theorem and Invariants.- Notes.- V. The Main Results.- 1. Stability Formulae.- 2. Complexity of Constructible Sets.- 3. Separation.- 4. Real Divisors.- 5. The Artin-Lang Property.- Notes.- VI. Spaces of Signs of Rings.- 1. Fans and Valuations.- 2. Field Extensions: Upper Bounds.- 3. Field Extensions: Lower Bounds.- 4. Algebras.- 5. Algebras Finitely Generated over Fields.- 6. Archimedean Rings.- 7. Coming Back to Geometry.- Notes.- VII. Real Algebra of Excellent Rings.- 1. Regular Homomorphisms.- 2. Excellent Rings.- 3. Extension of Orderings Under Completion.- 4. Curve Selection Lemma.- 5. Dimension, Valuations and Fans.- 6. Closures of Constructible Sets.- 7. Real Going-down for Regular Homomorphisms.- 8. Connected Components of Constructible Sets.- Notes.- VIII. Real Analytic Geometry.- 1. Semianalytic Sets.- 2. Semianalytic Set Germs.- 3. Cylindrical Decomposition of Germs.- 4. Rings of Global Analytic Functions.- 5. Hilbert's 17th Problem and Real Nullstellensatz.- 6. Minimal Generation of Global Semianalytic Sets.- 7. Topology of Global Semianalytic Sets.- 8. Germs at Compact Sets.- Notes.

Real oscillations of virtual neutrinos.

Abstract: We study the conditions for neutrino oscillations in a field-theoretical approach by taking into account that only the neutrino production and detection processes, which are localized in space around the coordinates {ital x{searrow}}{sub {ital P}} and {ital x{searrow}}{sub {ital D}}, respectively, can be manipulated. In this sense the neutrinos whose oscillations are investigated appear as virtual lines connecting production with detection in the total Feynman graph and all neutrino fields or states to be found in the discussion are mass eigenfields or eigenstates. We perform a thorough examination of the integral over the spatial components of the inner neutrino momentum and show that in the asymptotic limit {ital L}={vert_bar}{ital x{searrow}}{sub {ital D}}{minus}{ital x{searrow}}{sub {ital P}}{vert_bar}{r_arrow}{infinity} the virtual neutrinos become {open_quote}{open_quote}real{close_quote}{close_quote} and under certain conditions the usual picture of neutrino oscillations emerges without ambiguities. {copyright} {ital 1996 The American Physical Society.}

Riemannian Manifolds of Conullity Two

Abstract: Definition and early development local structure of semi-symmetric spaces explicit treatment of foliated semi-symmetric spaces curvature homogeneous semi-symmetric spaces asymptotic distributions and algebraic rank three-dimensional Riemannian manifolds of conullity two asymptotically foliated semi-symmetric spaces elliptic semi-symmetric spaces complete foliated semi-symmetric spaces application - local rigidity problems for hypersurfaces with type number two in IR4 three-dimensional Riemannian manifolds of relative conullity two appendix - more about curvature homogeneous spaces.

Noncollinear intra-atomic magnetism

Abstract: An intra-atomic noncollinear magnetization density has been calculated for the case of ferromagnetic fcc Pu, by means of a newly implemented general-local-spin-density-approximation method which treats the magnetization density as a continuous vector quantity. The presence of noncollinearity is a general effect, not specific to Pu, which is shown to rise due to the interplay of the local exchange and the spin-orbit coupling. The form of the noncollinear part of the magnetization density is very sensitive to the space group symmetry as is demonstrated by calculations with the average spin moment along [001] and [111], respectively.

Irreducible components of the space of holomorphic foliations of degree two in cp(n), n 3

Abstract: In this paper we will prove that the space of holomorphic fo- liations of codimension 1 and degree 2 in CP(n), n > 3, has six irreducible components.

On Riemann-Roch Formulas for Multiplicities

Abstract: A Theorem due to Guillemin and Sternberg about geometric quantization of Hamiltonian actions of compact Lie groups $G$ on compact Kaehler manifolds says that the dimension of the $G$-invariant subspace is equal to the Riemann-Roch number of the symplectically reduced space. Combined with the shifting-trick, this gives explicit formulas for the multiplicities of the various irreducible components. One of the assumptions of the Theorem is that the reduction is regular, so that the reduced space is a smooth symplectic manifold. In this paper, we prove a generalization of this result to the case where the reduced space may have orbifold singularities. Our proof uses localization techniques from equivariant cohomology, and relies in particular on recent work of Jeffrey-Kirwan and Guillemin. Since there are no complex geometry arguments involved, the result also extends to non Kaehlerian settings.

HyperK\"{a}hler Quotient Construction of BPS Monopole Moduli Spaces

Abstract: We use the HyperKahler quotient of flat space to obtain some monopole moduli space metrics in explicit form. Using this new description, we discuss their topology, completeness and isometries. We construct the moduli space metrics in the limit when some monopoles become massless, which corresponds to non-maximal symmetry breaking of the gauge group. We also introduce a new family of HyperK"{a}hler metrics which, depending on the ``mass parameter'' being positive or negative, give rise to either the asymptotic metric on the moduli space of many SU(2) monopoles, or to previously unknown metrics. These new metrics are complete if one carries out the quotient of a non-zero level set of the moment map, but develop singularities when the zero-set is considered. These latter metrics are of relevance to the moduli spaces of vacua of three dimensional gauge theories for higher rank gauge groups. Finally, we make a few comments concerning the existence of closed or bound orbits on some of these manifolds and the integrability of the geodesic flow.

Wick calculus in Gaussian analysis

Abstract: We define an extension of the distribution spaces conventionally used in Gaussian analysis. This space, characterized by analytic properties of S-transforms, allows for a calculus based on the Wick product. We indicate some of its features.

Quantum Immanants and Higher Capelli Identities

Abstract: We consider remarkable central elements of the universal enveloping algebra of the general linear algebra which we call quantum immanants. We express them in terms of generators $E_{ij}$ and as differential operators on the space of matrices. These expressions are a direct generalization of the classical Capelli identities. They result in many nontrivial properties of quantum immanants.

Phase diagram of the two-dimensional complex Ginzburg-Landau equation

Abstract: After a brief introduction to the complex Ginzburgh-Landau equation, some of its important features in two space dimensions are reviewed. A comprehensive study of the various phases observed numerically in large systems over the whole parameter space is then presented. The nature of the transitions between these phases is investigated and some theoretical problems linked to the phase diagram are discussed.