Showing papers on "Space (mathematics) published in 1998"
Book•
09 Oct 1998
TL;DR: In this article, the Hahn-Banach Extension Theorem (HBMT) is used to describe the properties of normed spaces and linear operators between normed space.
Abstract: 1 Basic Concepts.- 1.1 Preliminaries.- 1.2 Norms.- 1.3 First Properties of Normed Spaces.- 1.4 Linear Operators Between Normed Spaces.- 1.5 Baire Category.- 1.6 Three Fundamental Theorems.- 1.7 Quotient Spaces.- 1.8 Direct Sums.- 1.9 The Hahn-Banach Extension Theorems.- 1.10 Dual Spaces.- 1.11 The Second Dual and Reflexivity.- 1.12 Separability.- 1.13 Characterizations of Reflexivity.- 2 The Weak and Weak Topologies.- 2.1 Topology and Nets.- 2.2 Vector Topologies.- 2.3 Metrizable Vector Topologies.- 2.4 Topologies Induced by Families of Functions.- 2.5 The Weak Topology.- 2.6 The Weak Topology.- 2.7 The Bounded Weak Topology.- 2.8 Weak Compactness.- 2.9 James's Weak Compactness Theorem.- 2.10 Extreme Points.- 2.11 Support Points and Subreflexivity.- 3 Linear Operators.- 3.1 Adjoint Operators.- 3.2 Projections and Complemented Subspaces.- 3.3 Banach Algebras and Spectra.- 3.4 Compact Operators.- 3.5 Weakly Compact Operators.- 4 Schauder Bases.- 4.1 First Properties of Schauder Bases.- 4.2 Unconditional Bases.- 4.3 Equivalent Bases.- 4.4 Bases and Duality.- 4.5 James's Space J.- 5 Rotundity and Smoothness.- 5.1 Rotundity.- 5.2 Uniform Rotundity.- 5.3 Generalizations of Uniform Rotundity.- 5.4 Smoothness.- 5.5 Uniform Smoothness.- 5.6 Generalizations of Uniform Smoothness.- A Prerequisites.- B Metric Spaces.- D Ultranets.- References.- List of Symbols.
1,099 citations
TL;DR: Several space-time statistical models are constructed based on both classical empirical studies of clustering and some more speculative hypotheses, and the goodness-of-fit of the models, as measured by AIC values, is discussed for two high quality data sets, in different tectonic regions as mentioned in this paper.
Abstract: Several space-time statistical models are constructed based on both classical empirical studies of clustering and some more speculative hypotheses. Then we discuss the discrimination between models incorporating contrasting assumptions concerning the form of the space-time clusters. We also examine further practical extensions of the model to situations where the background seismicity is spatially non-homogeneous, and the clusters are non-isotropic. The goodness-of-fit of the models, as measured by AIC values, is discussed for two high quality data sets, in different tectonic regions. AIC also allows the details of the clustering structure in space to be clarified. A simulation algorithm for the models is provided, and used to confirm the numerical accuracy of the likelihood calculations. The simulated data sets show the similar spatial distributions to the real ones, but differ from them in some features of space-time clustering. These differences may provide useful indicators of directions for further study.
1,060 citations
TL;DR: In this article, a boundary of the anti-deSitter space analogous to a cut-off on the Liouville coordinate of the two-dimensional string theory is introduced.
Abstract: We suggest a means of obtaining certain Green's functions in 3+1-dimensional ${\cal N} = 4$ supersymmetric Yang-Mills theory with a large number of colors via non-critical string theory. The non-critical string theory is related to critical string theory in anti-deSitter background. We introduce a boundary of the anti-deSitter space analogous to a cut-off on the Liouville coordinate of the two-dimensional string theory. Correlation functions of operators in the gauge theory are related to the dependence of the supergravity action on the boundary conditions. From the quadratic terms in supergravity we read off the anomalous dimensions. For operators that couple to massless string states it has been established through absorption calculations that the anomalous dimensions vanish, and we rederive this result. The operators that couple to massive string states at level $n$ acquire anomalous dimensions that grow as $2\left (n g_{YM} \sqrt {2 N} )^{1/2}$ for large `t Hooft coupling. This is a new prediction about the strong coupling behavior of large $N$ SYM theory.
751 citations
TL;DR: The Metropolis-Hastings algorithm is a method of constructing a reversible Markov transition kernel with a specified invariant distribution as discussed by the authors, which is used to construct reversible transition kernels.
Abstract: The Metropolis-Hastings algorithm is a method of constructing a reversible Markov transition kernel with a specified invariant distribution. This note describes necessary and sufficient conditions on the candidate generation kernel and the acceptance probability function for the resulting transition kernel and invariant distribution to satisfy the detailed balance conditions. A simple general formulation is used that covers a range of special cases treated separately in the literature. In addition, results on a useful partial ordering of finite state space reversible transition kernels are extended to general state spaces and used to compare the performance of two approaches to using mixtures in Metropolis-Hastings kernels.
411 citations
TL;DR: In this paper, a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces is presented.
Abstract: We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. G. Sprindzuk in 1964. We also prove several related hypotheses of Baker and Sprindzuk formulated in 1970s. The core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence of unipotent flows on the space of lattices.
381 citations
TL;DR: In this article, a brane model for supersymmetric non-linear sigma models with target space such as CPn−1 and the Grassmann manifolds is proposed and the quantum properties of these models are realized in M-theory by taking the strong Type IIA coupling limit.
247 citations
Posted Content•
TL;DR: In this article, a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces is presented.
Abstract: We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. Sprindzhuk in 1964. We also prove several related hypotheses of A. Baker and V. Sprindzhuk formulated in 1970s. The core of the proof is a theorem which generalizes and sharpens earlier results on non-divergence of unipotent flows on the space of lattices.
244 citations
TL;DR: In this article, all SU (2) caloron solutions with topological charge one and arbitrary Polyakov loop at spatial infinity (with trace 2cos(2 πω )), using the Nahm duality transformation and ADHM, were determined.
184 citations
Proceedings Article•
01 Jul 1998
TL;DR: This paper poses the mapping problem as a statistical maximum likelihood problem, and devises an efficient algorithm for search in likelihood space that integrates two phases: a topological and a metric mapping phase.
Abstract: The problem of concurrent mapping and localization has received considerable attention in the mobile robotics community. Existing approaches can largely be grouped into two distinct paradigms: topological and metric. This paper proposes a method that integrates both. It poses the mapping problem as a statistical maximum likelihood problem, and devises an efficient algorithm for search in likelihood space. It presents an novel mapping algorithm that integrates two phases: a topological and a metric mapping phase. The topological mapping phase solves a global position alignment problem between potentially indistinguishable, significant places. The subsequent metric mapping phase produces a fine-grained metric map of the environment in floating-point resolution. The approach is demonstrated empirically to scale up to large, cyclic, and highly ambiguous environments.
180 citations
Posted Content•
TL;DR: In this paper, the authors show that the Kochen-Specker theorem is equivalent to the statement that a certain presheaf defined on the category of self-adjoint operators has no global elements.
Abstract: The Kochen-Specker theorem asserts the impossibility of assigning values to quantum quantities in a way that preserves functional relations between them. We construct a new type of valuation which is defined on all operators, and which respects an appropriate version of the functional composition principle. The truth-values assigned to propositions are (i) contextual; and (ii) multi-valued, where the space of contexts and the multi-valued logic for each context come naturally from the topos theory of presheaves.
The first step in our theory is to demonstrate that the Kochen-Specker theorem is equivalent to the statement that a certain presheaf defined on the category of self-adjoint operators has no global elements. We then show how the use of ideas drawn from the theory of presheaves leads to the definition of a generalized valuation in quantum theory whose values are sieves of operators. In particular, we show how each quantum state leads to such a generalized valuation.
TL;DR: In this paper, a construction in Matrix theory of longitudinal 5-branes whose geometry in transverse space corresponds to a 4-sphere is presented, and the physical properties of these branes correspond with those expected from M-theory; in particular, these objects have an energy and a leading long distance interaction with gravitons which agrees with 11D supergravity at leading order in N.
TL;DR: In this article, the authors considered the (2+1)-dimensional Kerr-de Sitter space and its statistical entropy computation, and they showed that this space has only one (cosmological) event horizon, and there is a phase transition between the stable horizon and the evaporating horizon at a point M 2 = 1 3 J 2 /l 2 together with a lower bound of the horizon temperature.
TL;DR: In this article, it was shown that any two points in Teichmuller space are joined by an earthquake path, and that any earthquake path R → T (S) extends to a proper holomorphic mapping of a simply connected domain D into Technical space, where R ⊆ D ⊂ C. These complex earthquakes relate Weil-Petersson geometry, projective structures, pleated surfaces and quasifuchsian groups.
Abstract: It is known that any two points in Teichmuller space are joined by an earthquake path. In this paper we show any earthquake path R → T (S) extends to a proper holomorphic mapping of a simplyconnected domain D into Teichmuller space, where R ⊂ D ⊂ C. These complex earthquakes relate Weil-Petersson geometry, projective structures, pleated surfaces and quasifuchsian groups. Using complex earthquakes, we prove grafting is a homeomorphism for all 1-dimensional Teichmuller spaces, and we construct bending coordinates on Bers slices and their generalizations. In the appendix we use projective surfaces to show the closure of quasifuchsian space is not a topological manifold.
TL;DR: The authors characterise word hyperbolic groups as those groups which act properly discontinuously and cocompactly on the space of distinct triples of a compact metrisable space, which is equivalent to a convergence group for which every point of the space is a conical limit point.
Abstract: We characterise word hyperbolic groups as those groups which act properly discontinuously and cocompactly on the space of distinct triples of a compact metrisable space. This is, in turn, equivalent to a convergence group for which every point of the space is a conical limit point.
TL;DR: In this article, a purely group-theoretical derivation of the continuous wavelet transform (CWT) on the (n-1)-sphere Sn-1 is presented.
Abstract: We present a purely group-theoretical derivation of the continuous wavelet transform (CWT) on the (n-1)-sphere Sn-1. based on the construction of general coherent states associated to square integrable group representations. The parameter space of the CWT, X similar to SO(n)xR*(+), is embedded into the generalized Lorentz group SO0(n,1) via the Iwasawa decomposition, so that X similar or equal to SO0(n,1)IN, where N similar or equal to Rn-1. Then the CWT on Sn-1 is derived from a suitable unitary representation of SO0(n,1) acting in the space L-2(Sn-1,d mu) of finite energy signals on Sn-1, which turns out to be square integrable over X. We find a necessary condition for the admissibility of a wavelet, in the form of a zero mean condition, which entails all the usual filtering properties of the CWT. Next the Euclidean limit of this CWT on Sn-1 is obtained by redoing the construction on a sphere of radius R and performing a group contraction for R-->infinity, from which one recovers the usual CWT on flat Euclidean space. Finally, we discuss the extension of this construction to the two-sheeted hyperboloid Hn-1SO0(n-1,1)/SO(n-1) and some other Riemannian symmetric spaces. (C) 1998 American Institute of Physics. [S0022-2488(98)00308-9].
TL;DR: In this paper, the authors studied strong anomalous diffusion, where the superdiffusion behavior is not completely characherized by a unique exponent, i.e., it can be generated by nontrivial chaotic dynamics, e.g. Lagrangian motion in time-dependent incompressible velocity fields, symplectic maps and intermittent maps.
Abstract: The superdiffusion behavior, i.e. $ \sim t^{2
u}$, with $
u > 1/2$, in general is not completely characherized by a unique exponent. We study some systems exhibiting strong anomalous diffusion, i.e. $ \sim t^{q
u(q)}$ where $
u(2)>1/2$ and $q
u(q)$ is not a linear function of $q$. This feature is different from the weak superdiffusion regime, i.e. $
u(q)=const > 1/2$, as in random shear flows. The strong anomalous diffusion can be generated by nontrivial chaotic dynamics, e.g. Lagrangian motion in $2d$ time-dependent incompressible velocity fields, $2d$ symplectic maps and $1d$ intermittent maps. Typically the function $q
u(q)$ is piecewise linear. This corresponds to two mechanisms: a weak anomalous diffusion for the typical events and a ballistic transport for the rare excursions. In order to have strong anomalous diffusion one needs a violation of the hypothesis of the central limit theorem, this happens only in a very narrow region of the control parameters space.
Posted Content•
TL;DR: In this paper, it was shown that the space of all quasi-regular Sasakian-Einstein orbifolds has a natural multiplication on it, and necessary and sufficient conditions were given for the ''product'' of two SAIs to be a smooth SAIs.
Abstract: We discuss Sasakian-Einstein geometry under a quasi-regularity assumption. It is shown that the space of all quasi-regular Sasakian-Einstein orbifolds has a natural multiplication on it. Furthermore, necessary and sufficient conditions are given for the `product' of two Sasakian-Einstein manifolds to be a smooth Sasakian-Einstein manifold. Using spectral sequence arguments we work out the cohomology ring in many cases of interest. This type of geometry has recently become of interest in the physics of supersymmetric conformal field theories.
Journal Article•
TL;DR: In this article, a random walk model is proposed for space fractional diffusion, where the fundamental solutions of these generalized diffusion equations are shown to provide certain probability density functions, in space or time, which are related to the relevant class of stable distributions.
Abstract: FRACTIONAL calculus allows one to generalize the linear (one-dimensional) diffusion equation by replacing either the first time-derivative or the second space-derivative by a derivative of a fractional order. The fundamental solutions of these generalized diffusion equations are shown to provide certain probability density functions, in space or time, which are related to the relevant class of stable distributions. For the space fractional diffusion, a random-walk model is also proposed.
TL;DR: In this paper, the authors introduced a method of estimating the space analyticity radius of solutions for the Navier-Stokes and related equations in terms of L p and L ∞ norms of the initial data.
TL;DR: This paper discusses explicit embedded integration methods with large stability domains of order 3 and 4 that realize a corresponding explicit Runge-Kutta method with the help of the theory of composition methods.
Abstract: This paper discusses explicit embedded integration methods with large stability domains of order 3 and 4. The high order produces accurate results, the large stability domains allow some reasonable stiffness, the explicitness enables the method to treat very large problems, often space discretization of parabolic PDEs, and the embedded formulas permit an efficient stepsize control. The construction of these methods is achieved in two steps: firstly we compute stability polynomials of a given order with optimal stability domains, i.e., possessing a Chebyshev alternation; secondly we realize a corresponding explicit Runge-Kutta method with the help of the theory of composition methods.
Book•
01 Jan 1998
TL;DR: In this article, the algebra of sets is used for linear functional analysis, where sets of functions and Lebesgue integration of continuous functions are studied. But the results are restricted to the Euclidean space.
Abstract: Contents- Introduction- Linear Functional Analysis- Sets- The algebra of sets- Sets of numbers- Rn and its subsets- Relations, equivalence classes and Zorn's lemma- Theorem-proving- Bibliographical remarks- Exercises- Sets of functions and Lebesgue integration- Continuous functions- Meansure of sets in Rn- Lebesgue integration and the space Lp(_)- Bibliographical remarks - Exercises- Vector spaces, normed and inner product spaces
TL;DR: In this paper, the results of large-basis shell-model calculations for nuclei with $A=7\ensuremath{-}11$ are presented, where the effective interactions used in the study were derived microscopically from the Reid93 potential and take into account the Coulomb potential as well as the charge dependence of $T=1$ partial waves.
Abstract: Results of large-basis shell-model calculations for nuclei with $A=7\ensuremath{-}11$ are presented. The effective interactions used in the study were derived microscopically from the Reid93 potential and take into account the Coulomb potential as well as the charge dependence of $T=1$ partial waves. For $A=7$, a $6\ensuremath{\Elzxh}\ensuremath{\Omega}$ model space was used, while for the rest of the studied nuclides, the calculations were performed in a $4\ensuremath{\Elzxh}\ensuremath{\Omega}$ model space. It is demonstrated that the shell model combined with microscopic effective interactions derived from modern nucleon-nucleon potentials is capable of providing good agreement with the experimental properties of the ground state as well as with those of the low-lying excited states.
TL;DR: In this article, the authors used the two-point correlation function to calculate the clustering properties of the recently completed SSRS2 survey and found that early types are more clustered than late types, but that in the absence of rich clusters, the relative bias between early and late types in real space, is not as strong as previously estimated.
Abstract: We use the two-point correlation function to calculate the clustering properties of the recently completed SSRS2 survey. The redshift space correlation function for the magnitude-limited SSRS2 is given by xi(s)=(s/5.85 h-1 Mpc)^{-1.60} for separations between 2 L*) are more clustered than sub-L* galaxies and that the luminosity segregation is scale-independent. We find that early types are more clustered than late types, but that in the absence of rich clusters, the relative bias between early and late types in real space, is not as strong as previously estimated. Furthermore, both morphologies present a luminosity-dependent bias, with the early types showing a slightly stronger dependence on luminosity. We also find that red galaxies are significantly more clustered than blue ones, with a mean relative bias stronger than that seen for morphology. Finally, we find that the relative bias between optical and iras galaxies in real space is b_o/b_I $\sim$ 1.4.
TL;DR: In this paper, the authors constructed a synthetic parallel transportation along a geodesic in Alexandrov space with curvature bounded below, and proved an analog of the second variation formula for this case.
Abstract: In this paper we construct a "synthetic" parallel transportation along a geodesic in Alexandrov space with curvature bounded below, and prove an analog of the second variation formula for this case. A closely related construction has been made for Alexandrov space with bilaterally bounded curvature by Igor Nikolaev (see [N]).¶Naturally, as we have a more general situation, the constructed transportation does not have such good properties as in the case of bilaterally bounded curvature. In particular, we cannot prove the uniqueness in any good sense. Nevertheless the constructed transportation is enough for the most important applications such as Synge's lemma and Frankel's theorem. Recently by using this parallel transportation together with techniques of harmonic functions on Alexandrov space, we have proved an isoperimetric inequality of Gromov's type.¶Author is indebted to Stephanie Alexander, Yuri Burago and Grisha Perelman for their willingness to understand, interest and important remarks.
TL;DR: In this paper, it was shown that a universal Harsanyi type space does exist even when spaces are defined in pure measure theoretic terms, and that coherent hierarchies of beliefs do not necessarily describe types.
TL;DR: In this article, the authors generalize results of Ford and Roman which place lower bounds on the renormalized energy density of a quantum field averaged against a choice of sampling function.
Abstract: We generalize results of Ford and Roman which place lower bounds---known as quantum inequalities---on the renormalized energy density of a quantum field averaged against a choice of sampling function. Ford and Roman derived their results for a specific non-compactly supported sampling function; here we use a different argument to obtain quantum inequalities for a class of smooth, even and non-negative sampling functions which are either compactly supported or decay rapidly at infinity. Our results hold in $d$-dimensional Minkowski space $(dg~2)$ for the free real scalar field of mass $mg~0.$ We discuss various features of our bounds in 2 and 4 dimensions. In particular, for massless field theory in two-dimensional Minkowski space, we show that our quantum inequality is weaker than Flanagan's optimal bound by a factor of $\frac{3}{2}.$
TL;DR: The transformations of all the Schrodinger operators with point interactions in dimension one under space reflection P, time reversal T and (Weyl) scaling Wλ are presented in this paper.
Abstract: The transformations of all the Schrodinger operators with point interactions in dimension one under space reflection P, time reversal T and (Weyl) scaling Wλ are presented. In particular, those operators which are invariant (possibly up to a scale) are selected. Some recent papers on related topics are commented upon.