Showing papers on "Space (mathematics) published in 2001"

On the Existence of Globally Defined Weak Solutions to the Navier—Stokes Equations

Abstract: We prove the existence of globally defined weak solutions to the Navier—Stokes equations of compressible isentropic flows in three space dimensions on condition that the adiabatic constant satisfies $ \gamma > 3/2 $ .

Computational line geometry

Abstract: Fundamentals.- Models of Line Space.- Linear Complexes.- Approximation in Line Space.- Ruled Surfaces.- Developable Surfaces.- Line Congruences and Line Complexes.- Linear Line Mappings #x2014 Computational Kinematics.

Another look at Sobolev spaces

Abstract: The standard seminorm in the space $W^{s,p}$, with $s$<$1$, does not converge, when $s$ approaches $1$, to the corresponding $W^{1,p}$ seminorm We prove that continuity is restored provided we multiply the $W^{s,p}$ seminorm by an appropriate factor

Stringy gravity, interacting tensionless strings and massless higher spins

Abstract: Consequences of a strong version of the AdS/CFT correspondence for extremely stringy physics are examined. In particular, properties of N = 4 supersymmetric Yang-Mills theory are used to extract results about interacting tensionless strings and massless higher spin fields in an AdS5 × S5 background. Furthermore, the thermodynamics of this model signals the presence of a Hawking-Page phase transition between AdS5 space and a “black hole”-like high temperature configuration even in the extreme string limit.

Dynamic texture recognition

Abstract: Dynamic textures are sequences of images that exhibit some form of temporal stationarity, such as waves, steam, and foliage. We pose the problem of recognizing and classifying dynamic textures in the space of dynamical systems where each dynamic texture is uniquely represented. Since the space is non-linear, a distance between models must be defined We examine three different distances in the space of autoregressive models and assess their power.

The Quantum Hall Fluid and Non-Commutative Chern Simons Theory

Abstract: The first part of this paper is a review of the author's work with S. Bahcall which gave an elementary derivation of the Chern Simons description of the Quantum Hall effect for filling fraction $1/n$. The notation has been modernized to conform with standard gauge theory conventions. In the second part arguments are given to support the claim that abelian non-commutative Chern Simons theory at level $n$ is exactly equivalent to the Laughlin theory at filling fraction $1/n$. The theory may also be formulated as a matrix theory similar to that describing D0-branes in string theory. Finally it can also be thought of as the quantum theory of mappings between two non-commutative spaces, the first being the target space and the second being the base space.

Existence of blow-up solutions in the energy space for the critical generalized KdV equation

Abstract: From these conservation laws, H appears as an energy space, so that it is a natural space in which to study the solutions. Note that p = 2 is a special case for equation (2). Indeed, from the integrability theory (see Lax [14]), we have for suitable u0 (u0 and its derivatives with fast decay at infinity) an infinite number of conservation laws. The general question is to understand the dynamics induced by such equations. Local existence in time of solutions of (2) in the energy space is now well understood; see Kato [10], Ginibre and Tsutsumi [8] for the H theory (s > 32 ), Kenig, Ponce and Vega [11] for the L theory in the case of equation (1) and sharp H theory for (2), and Bourgain [3] and [4] for the periodic case.

The Space of K\"ahler metrics (II)

Abstract: This paper, the second of a series, deals with the function space of all smooth K\"ahler metrics in any given closed complex manifold $M$ in a fixed cohomology class. The previous result of the second author \cite{chen991} showed that the space is a path length space and it is geodesically convex in the sense that any two points are joined by a unique path, which is always length minimizing and of class C^{1,1}. This already confirms one of Donaldson's conjecture completely and verifies another one partially. In the present paper, we show first of all, that the space is, as expected, a path length space of non-positive curvature in the sense of A. D. Alexanderov. The second result is related to the theory of extremal K\"ahler metrics, namely that the gradient flow of the K energy is strictly length decreasing on all paths except those induced by a path of holomorphic automorphisms of $M$. This result, in particular, implies that extremal K\"ahler metric is unique up to holomorphic transformations, provided that Donaldson's conjecture on the regularity of geodesic is true.

The language of space

Abstract: Space as Language Space and the Human Dimension Mechanisms of Perceiving Space Four Ways of Perceiving Space Space and Distance Proxemics The Territory Space and Time Recording Space.

Three-point functions for M^N/S^N orbifolds with N=4 supersymmetry

Abstract: The D1-D5 system is believed to have an `orbifold point' in its moduli space where its low energy theory is a N=4 supersymmetric sigma model with target space M^N/S^N, where M is T^4 or K3. We study correlation functions of chiral operators in CFTs arising from such a theory. We construct a basic class of chiral operators from twist fields of the symmetric group and the generators of the superconformal algebra. We find explicitly the 3-point functions for these chiral fields at large N; these expressions are `universal' in that they are independent of the choice of M. We observe that the result is a significantly simpler expression than the corresponding expression for the bosonic theory based on the same orbifold target space.

On embeddings between classical Lorentz spaces

Abstract: Let p ∈ (0, ∞) , let v be a weight on (0, ∞) and let Λp(v) be the classical Lorentz space, determiined by the norm ∥f∥Λp(v) := (∫∞0(f*(t))pv(t)dt) 1/p. When p ∈ (1, ∞), this space is known to be a Banach space if and only if v is non-increasing, while it is only equivalent to a Banach space if and only if Λp(v) = Γp(v), where ∥f∥Γp(v) := (∫∞0(f**(t))pv(t) t/p. We may thus conclude that, for p ∈ (1, ∞), the space Λp(v) is equivalent to a Banach space if and only if the norm of a function f in it can be expressed in terms of f**. We study the question whether an analogous assertion holds when p = 1. Motivated by this problem, we consider general embeddings between four types of classical and weak Lorentz spaces, namely, Λp(v), Λp,∞(v), Γp(v), Γp,∞(v), where Λp,∞(v) and Γp∞(v) are certain weak analogues of the spaces Λp(v) and Γp(v), respectively. We present a unified approach to these embeddings, based on rearrangement techniques. We survey all the known results and prove new ones. Our main results concern the embedding Γp,∞(v) → Λq(w) which had not been characterized so far. We apply our results to the characterization of associate spaces of classical and weak Lorentz spaces and we give a characterization of fundamental functions for which the endpoint Lorentz space and the endpoint Marcinkiewicz space coincide.

The state space and physical interpretation of self-similar spherically symmetric perfect-fluid models

Abstract: The purpose of this paper is to further investigate the solution space of self-similar spherically symmetric perfect-fluid models and gain a deeper understanding of the physical aspects of these solutions We achieve this by combining the state space description of the homothetic approach with the use of the physically interesting quantities arising in the comoving approach We focus on three types of models First, we consider models that are natural inhomogeneous generalizations of the Friedmann universe; such models are asymptotically Friedmann in their past and evolve fluctuations in the energy density at later times Secondly, we consider so-called quasi-static models This class includes models that undergo self-similar gravitational collapse and is important for studying the formation of naked singularities If naked singularities do form, they have profound implications for the predictability of general relativity as a theory Thirdly, we consider a new class of asymptotically Minkowski self-similar spacetimes, emphasizing that some of them are associated with the self-similar solutions associated with the critical behaviour observed in recent gravitational collapse calculations

Gibbs states on the symbolic space over an infinite alphabet

Abstract: We consider subshifts of finite type on the symbolic space generated by incidence matrices over a countably infinite alphabet. We extend the definition of topological pressure to this context and, as our main result, we construct a new class of Gibbs states of Holder continuous potentials on these symbol spaces. We establish some basic stochastic properties of these Gibbs states: exponential decay of correlations, central limit theorem and an a.s. invariance principle. This is accomplished via detailed studies of the associated Perron-Frobenius operator and its conjugate operator.

Quantum Field Theory on Noncommutative Spaces

Abstract: A pedagogical and self-contained introduction to noncommutative quantum field theory is presented, with emphasis on those properties that are intimately tied to string theory and gravity. Topics covered include the Weyl-Wigner correspondence, noncommutative Feynman diagrams, UV/IR mixing, noncommutative Yang-Mills theory on infinite space and on the torus, Morita equivalences of noncommutative gauge theories, twisted reduced models, and an in-depth study of the gauge group of noncommutative Yang-Mills theory. Some of the more mathematical ideas and techniques of noncommutative geometry are also briefly explained.

Split string field theory I

Abstract: We describe projection operators in the matter sector of Witten's cubic string field theory using modes on the right and left halves of the string. These projection operators represent a step towards an analytic solution of the equations of motion of the full string field theory, and can be used to construct Dp-brane solutions of the string field theory when the BRST operator Q is taken to be pure ghost, as suggested in the recent conjecture by Rastelli, Sen and Zwiebach. We show that a family of solutions related to the sliver state are rank one projection operators on the appropriate space of half-string functionals, and we construct higher rank projection operators corresponding to configurations of multiple D-branes.

Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces

Abstract: We prove the compactness of the imbedding of the Sobolev space $W^{1,2}_0(\Omega)$ into $L^2(\Omega)$ for any relatively compact open subset $\Omega$ of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approximated by the Laplacian induced from the DC-structure on the Alexandrov space. We also prove the existence of the locally Holder continuous heat kernel.

Relativistic theory for time and frequency transfer to order c -3

Abstract: This paper is motivated by the current development of several space missions (e.g. ACES on International Space Station) that will use Earth-orbit laser cooled atomic clocks, providing a time-keeping accuracy of the order of 5 10-17 in fractional frequency. We show that to such accuracy, the theory of frequency transfer between Earth and Space must be extended from the currently known relativistic order (which has been needed in previous space experiments such as GP-A) to the next relativistic correction of order . We find that the frequency transfer includes the first and second-order Doppler contributions, the Einstein gravitational red-shift and, at the order , a mixture of these effects. As for the time transfer, it contains the standard Shapiro time delay, and we present an expression also including the first and second-order Sagnac corrections. Higher-order relativistic corrections, at least , are numerically negligible for time and frequency transfers in these experiments, being for instance of order 10-20 in fractional frequency. Particular attention is paid to the problem of the frequency transfer in the two-way experimental configuration. In this case we find a simple theoretical expression which extends the previous formula (Vessot et al. [CITE]) to the next order . In the Appendix we present the detailed proofs of all the formulas which will be needed in such experiments.

Space bounds for resolution

Abstract: We introduce a new way to measure the space needed in resolution refutations of CNF formulas in propositional logic. With the former definition (1994, B. H. Kleine and T. Lettman, "Aussangenlogik: Deduktion und Algorithmen, Teubner, Stuttgart) the space required for the resolution of any unsatisfiable formula in CNF is linear in the number of clauses. The new definition allows a much finer analysis of the space in the refutation, ranging from constant to linear space. Moreover, the new definition allows us to relate the space needed in a resolution proof of a formula to other well-studied complexity measures. It coincides with the complexity of a pebble game in the resolution graphs of a formula and, as we show, has relationships to the size of the refutation. We also give upper and lower bounds on the space needed for the resolution of unsatisfiable formulas. We show that Tseitin formulas associated to a certain kind of expander graphs of n nodes need resolution space n-c for some constant c. Measured on the number of clauses, this result is the best possible. We also show that the formulas expressing the general pigeonhole principle with n holes and more than n pigeons need space n+1 independent of the number of pigeons. Since a matching space upper bound of n+1 for these formulas exists, the obtained bound is exact. We also point to a possible connection between resolution space and resolution width, another measure for the complexity of resolution refutations. 2001 Elsevier Science.

Optimal Tight Frames and Quantum Measurement

Abstract: Tight frames and rank-one quantum measurements are shown to be intimately related. In fact, the family of normalized tight frames for the space in which a quantum mechanical system lies is precisely the family of rank-one generalized quantum measurements (POVMs) on that space. Using this relationship, frame-theoretical analogues of various quantum-mechanical concepts and results are developed. The analogue of a least-squares quantum measurement is a tight frame that is closest in a least-squares sense to a given set of vectors. The least-squares tight frame is found for both the case in which the scaling of the frame is specified (constrained least-squares frame (CLSF)) and the case in which the scaling is free (unconstrained least-squares frame (ULSF)). The well-known canonical frame is shown to be proportional to the ULSF and to coincide with the CLSF with a certain scaling. Finally, the canonical frame vectors corresponding to a geometrically uniform vector set are shown to be geometrically uniform and to have the same symmetries as the original vector set.

Ergodicity of the 2D Navier--Stokes Equations¶with Random Forcing

Abstract: We consider the Navier-Stokes equation on a two dimensional torus with a random force, acting at discrete times and analytic in space, for arbitrarily small viscosity coefficient. We prove the existence and uniqueness of the invariant measure for this system as well as exponential mixing in time.

On the three-dimensional spaces which admit a continuous group of motions

Abstract: namely the law by which we measure infinitesimal arclengths in the space Sn, from which the law of measure for finite arclengths follows. We consider n independent real variables x1, x2, . . . , xn and assume that the coefficients aik of the quadratic differential form (1) as well as their first and second derivatives are real, finite and continuous functions of the x for the entire range of values which we consider. We also assume that the discriminant of expression (1) is always nonzero and that the coefficients aik fulfill the well known inequalities which make this differential form positivedefinite. It is well known how the law for measuring angles and the entire geometry of the space Sn is determined by equation (1). If two spaces Sn, S′ n can be put into a one-to-one correspondence in such a way that the line elements are the same, the two spaces will be called isometric and the two geometries will be identical. When the line elements of the two spaces only differ by a constant factor or can be reduced to this relationship by a transformation of coordinates, the two spaces will be called similar, and we will consider them as belonging to the same type. Their geometries are essentially identical; the only thing which changes from one to the other is the unit of linear measure. An isometry of a space Sn into itself will be called a motion of this space. We will consider the spaces which admit continuous motions into themselves, namely, such that in the corresponding equations of the transformation appear some arbitrary parameters. The set of all these motions for a given Sn clearly forms a group. Simple geometrical considerations show that the number of parameters of this group is necessarily finite, which is in fact easily demonstrated analytically as we will see. If r is the number of these parameters in the complete group of motions, in every case this group will consist of a Original title: Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti, Memorie di Matematica e di Fisica della Societa Italiana delle Scienze, Serie Terza, Tomo XI, pp. 267–352 (1898). Printed with the kind permission of the Accademia Nazionale delle Scienze, detta dei XL, in Rome, the current copyright owner. Translated by Robert Jantzen, Department of Mathematical Sciences, Villanova University, Villanova, Pa 19085, USA. This paper was also reprinted in: Opere [The Collected Works of Luigi Bianchi], Rome, Edizione Cremonese, 1952, vol. 9, pp. 17-109.

Space of State Vectors in PT Symmetrical Quantum Mechanics

Abstract: Space of states of PT symmetrical quantum mechanics is examined. Requirement that eigenstates with different eigenvalues must be orthogonal leads to the conclusion that eigenfunctions belong to the space with an indefinite metric. The self consistent expressions for the probability amplitude and average value of operator are suggested. Further specification of space of state vectors yield the superselection rule, redefining notion of the superposition principle. The expression for the probability current density, satisfying equation of continuity and vanishing for the bound state, is proposed.

De Sitter Space from Warped Supergravity Solutions

Abstract: The solutions of 10 and 11 dimensional supergravity that are warped products of de Sitter space with a non-compact ‘internal’ space are investigated. A convenient form of the metric is found and it is shown that in each case the internal space is asymptotic to a cone over a product of spheres. A consistent truncation gives gauged supergravities with non-compact gauge groups. The BPS domain wall solutions of the non-compact gauged supergravities are lifted to warped solutions in 10 or 11 dimensions.