Showing papers on "Space (mathematics) published in 1992"

The Production of Space

Abstract: The book is a search for a reconciliation between mental space (the space of the philosophers) and real space (the physical and social spheres in which we all live). In the course of his exploration, Henri Lefebvre moves from metaphysical and ideological considerations of the meaning of space to its experience in the everyday life of home and city. He seeks, in other words, to bridge the gap between the realms of theory and practice, between the mental and the social, and between philosophy and reality. In doing so, he ranges through art, literature, architecture and economics, and further provides a powerful antidote to the sterile and obfuscatory methods and theories characteristic of much recent continental philosophy.

Quaternion kinematic and dynamic differential equations

Abstract: Many useful identities pertaining to quaternion multiplications are generalized. Among them multiplicative commutativity is the most powerful. Since quaternion space includes the 3D vector space, the physical quantities related to rotations, such as angular displacement, velocity, acceleration, and momentum, are shown to be vector quaternions, and their expressions in quaternion space are derived. These kinematic and dynamic differential equations are further shown to be invertible due to the fact that they are written in quaternion space, and the highest order term of the rotation parameters can be expressed explicitly in closed form. >

Quantifying the neighborhood preservation of self-organizing feature maps

Abstract: It is shown that a topographic product P, first introduced in nonlinear dynamics, is an appropriate measure of the preservation or violation of neighborhood relations. It is sensitive to large-scale violations of the neighborhood ordering, but does not account for neighborhood ordering distortions caused by varying areal magnification factors. A vanishing value of the topographic product indicates a perfect neighborhood preservation; negative (positive) values indicate a too small (too large) output space dimensionality. In a simple example of maps from a 2D input space onto 1D, 2D, and 3D output spaces, it is demonstrated how the topographic product picks the correct output space dimensionality. In a second example, 19D speech data are mapped onto various output spaces and it is found that a 3D output space (instead of 2D) seems to be optimally suited to the data. This is an agreement with a recent speech recognition experiment on the same data set. >

Reduction of some classical non-holonomic systems with symmetry

Abstract: Two types of nonholonomic systems with symmetry are treated: (i) the configuration space is a total space of a G-principal bundle and the constraints are given by a connection; (ii) the configuration space is G itself and the constraints are given by left-invariant forms. The proofs are based on the method of quasicoordinates. In passing, a derivation of the Maurer-Cartan equations for Lie groups is obtained. Simple examples are given to illustrate the algorithmical character of the main results.

Stable pairs, linear systems and the Verlinde formula

Abstract: We study the moduli problem of pairs consisting of a rank 2 vector bundle and a nonzero section over a fixed smooth curve. The stability condition involves a parameter; as it varies, we show that the moduli space undergoes a sequence of flips in the sense of Mori. As applications, we prove several results about moduli spaces of rank 2 bundles, including the Harder-Narasimhan formula and the SU(2) Verlinde formula. Indeed, we prove a general result on the space of sections of powers of the ideal sheaf of a curve in projective space, which includes the Verlinde formula.

On the definition of a probabilistic normed space

Abstract: In this paper we give a new definition of a probabilistic normed space. This definition, which is based on a characterization of normed spaces by means of a betweenness relation, includes the earlier definition of A. N. Serstnev as a special case and leads naturally to the definition of the principal class of probabilistic normed spaces, the Menger spaces.

Renormalization group analysis of the τ-lepton decay within QCD

Abstract: A technique to sum up the regular corrections appearing under the analytical continuation from the space like momentum region to the time like one is proposed. A perturbative part of the inclusive semileptonic decay width of the τ-lepton in analyzed in detail.

Locally supersymmetric D=3 non-linear sigma models

Abstract: We study non-linear sigma models with N local supersymmetries in three space-time dimensions. For N=1 and 2 the target space of these models is Riemannian or Kahler, respectively. All N>2 theories are associated with Einstein spaces. For N=3 the target space is quaternionic, while for N=4 it generally decomposes into two separate quaternionic spaces, associated with inequivalent supermultiplets. For N=5,6,8 there is a unique (symmetric) space for any given number of supermultiplets. Beyond that there are only theories based on a single supermultiplet for N=9,10,12 and 16, associated with coset spaces with the exceptional isometry groups $F_{4(-20)}$, $E_{6(-14)}$, $E_{7(-5)}$ and $E_{16(+8)}$, respectively. For $N=3$ and $N\geq5$ the $D=2$ theories obtained by dimensional reduction are two-loop finite.

Countable sections for locally compact group actions

Abstract: It has been shown by J. Feldman, P. Hahn and C. C. Moore that every non-singular action of a second countable locally compact group has a countable (in fact so-called lacunary) complete measurable section. This is extended here to the purely Borel theoretic category, consisting of a Borel action of such a group on an analytic Borel space (without any measure). Characterizations of when an arbitrary Borel equivalence relation admits a countable complete Borel section are also established.

Effect of topology on the thermodynamic limit for a string gas.

Abstract: We discuss the thermodynamic limit for a gas of strings at high energy densities. This is defined by studying the statistical properties of the gas in a compact space and taking the size of the space to infinity keeping the energy density finite. We obtain a behavior that is different from earlier treatments where the gas is considered at the same energy density but living in a noncompact space. In particular we show that the gas is not dominated by a single energetic string above the Hagedorn energy density, but instead the number of energetic strings is ln{ital R}/ {radical}{alpha}{prime} where {ital R} is the radius of the universe and {alpha}{prime} the slope parameter. The reason for the thermodynamic behavior being sensitive to topology is the existence of winding modes that can sense the large-scale structure of the space.

The Superparticle and the Lorentz Group

Abstract: We present a unified group-theoretical framework for superparticle theories. This explains the origin of the ``twistor-like'' variables that have been used in trading the superparticle's $\kappa$-symmetry for worldline supersymmetry. We show that these twistor-like variables naturally parametrise the coset space ${\cal G}/{\cal H}$, where $\cal G$ is the Lorentz group $SO^\uparrow(1,d-1)$ and $\cal H$ is its maximal subgroup. This space is a compact manifold, the sphere $S^{d-2}$. Our group-theoretical construction gives the proper covariantisation of a fixed light-cone frame and clarifies the relation between target-space and worldline supersymmetries.

The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces

Abstract: The paper “The Dirac operator on homogeneous spaces and its spectrum on 3-dimensional lens spaces”, Arch. Math. 59, 65–79 (1992), contains some misprints which I would like to correct in this erratum. One important point: In Theorem 5 on p. 79 (and also on p. 78, l. 6) replace [−(m + 1)/N ] < i by −[(m − 2)/N ] ≤ i. Similarly, replace [(1 − m − N ′)/N ] < i in Theorem 5 by −[(m− 2 + N ′)/N ] ≤ i. Some fairly obvious misprints: • p. 66, l. 12: Replace [g, b̄0 ·Θ(Λ)] by [g,Θ(Λ)]. • p. 68, l. 5: Replace (dg0 · X̄)so([g0, b̄0]) = d dt [g0e tX , b̄0(Θ · Λ(t))]|t=0 by (dg0 · X̄)so([g0, 1so]) = d dt [g0e tX ,Θ · Λ(t)]|t=0. • p. 68, l. 20: Replace ∇ dt [e tX , b̄0 · (Θ · Λ(t))]|t=0 by ∇ dt [e tX ,Θ · Λ(t)]|t=0. • p. 69, l. 14: Add |t=0. • p. 71, l. 6: Replace Aπγ(g)v by Aπγ(g)v. • p. 72, l. 2: Replace E3 = 1 T ( i 0 0 −i )

Low‐velocity scattering of vortices in a modified Abelian Higgs model

Abstract: The moduli space metric for hyperbolic vortices is constructed, and their slow motion scattering is calculated in terms of geodesics in this space.

Self-consistent-field theory for confined polyelectrolyte chains

Abstract: The Poisson-Boltzmann theory of the forces between two charged macroscopic surfaces has been generalized to a case where the intervening space contains a uni-univalent electrolyte as well as long polyelectrolyte chains

Quadratic algebras and dynamics in curved spaces. II. The Kepler problem

Abstract: The symmetry aspects of the Kepler problem in a space of constant negative curvature are considered. It is shown that the algebra of the hidden symmetry reduces to the quadratic Racah algebraQR(3), and this makes it possible to express the coefficients of the overlapping of the wave functions in the spherical and parabolic coordinates in terms of Wilson-Racah polynomials. It is shown that the dynamical symmetry algebra that generates the spectrum is the quadratic Jacobi algebraQJ(3). Its ladder operators permit explicit construction of wave functions in the coordinate representation with the ground state as the starting point.

SU(1,1) Lie algebraic approach to linear dissipative processes in quantum optics

Abstract: An SU(1,1) Lie algebraic formulation is presented for investigating the linear dissipative processes in quantum optical systems. The Liouville space formulation, thermo field dynamics, and the disentanglement theorem of SU(1,1) Lie algebra play essential roles in this formulation. In the Liouville space, the time‐evolution equation for the state vector of a system is solved algebraically by using the decomposition formulas of SU(1,1) Lie algebra and the thermal state condition of thermo field dynamics. The presented formulation is used for investigating a dissipative nonlinear oscillator, the quantum mechanical model of phase modulation, and the photon echo in the localized electron–phonon system. This algebraic formulation gives a systematic treatment for investigating the phenomena in quantum optical systems.

Quantum affine symmetry in vertex models

Abstract: We study the higher spin anologs of the six vertex model on the basis of its symmetry under the quantum affine algebra $U_q(\slth)$. Using the method developed recently for the XXZ spin chain, we formulate the space of states, transfer matrix, vacuum, creation/annihilation operators of particles, and local operators, purely in the language of representation theory. We find that, regardless of the level of the representation involved, the particles have spin $1/2$, and that the $n$-particle space has an RSOS-type structure rather than a simple tensor product of the $1$-particle space. This agrees with the picture proposed earlier by Reshetikhin.

Canonical Representation of Set Functions

Abstract: The representation of a cooperative transferable utility game as a linear combination of unanimity games may be viewed as an isomorphism between not-necessarily additive set functions on the players space and additive ones on the coalitions space. We extend the unanimity-basis representation to general (infinite) spaces of players, study spaces of games of games which satisfy certain properties and provide some conditions for sigma-additivity of the resulting additive set function (on the space of coalitions). These results also allow us to extend some representations of the Choquet integral from finite to infinite spaces.

The quantum group GLh(2)

Abstract: Among all quantum group structures on the space Mat(2) of 2 by 2 matrices, how many have a central quantum determinant, so that one can define quantum SL(2) out of quantum GL(2)? Up to isomorphism, there are two such structures, GLq(2) and GLh(2). The former is well known, the latter is described in this letter.

Competitive Equilibrium in Sobolev Spaces Without Bounds on Short Sales

Abstract: Following Chichilnisky and Chichilnisky-Kalman we establish existence and optimality of competitive equilibrium when commodity spaces are infinite dimensional Sobolev spaces, including Hilbert spaces such as weighted L2 which have Linf as dense subspaces. We allow general consumption sets with or without lower bounds, thus including securities markets with infinitely many assets and unbounded short sales and economies with production. We give non-arbitrage conditions on endowments and preferences which suffice for the existence of an equilibrium. Prices are in the same space as commodities. Equilibrium allocations are approximated by allocations in other frequently used spaces such as C(R) and Linf.

Inertial manifolds and inertial sets for the phase-field equations

Abstract: The phase-field system is a mathematical model of phase transition, coupling temperature with a continuous order parameter which describes degree of solidification. The flow induced by this system is shown to be smoothing in H1×L2 and a global attractor is shown to exist. Furthermore, in low-dimensional space, the flow is essentially finite dimensional in the sense that a strongly attracting finite-dimensional manifold (or set) exists.