Showing papers on "Space (mathematics) published in 1997"
TL;DR: A model quantum register made of replicas (cells) of a given finite-dimensional quantum system S realizes a noiseless quantum code in which information can be stored, in principle, for an arbitrarily long time without being affected by errors.
Abstract: In this paper we study a model quantum register $R$ made of $N$ replicas (cells) of a given finite-dimensional quantum system S. Assuming that all cells are coupled with a common environment with equal strength we show that, for $N$ large enough, in the Hilbert space of $R$ there exists a linear subspace ${C}_{N}$ which is dynamically decoupled from the environment. The states in ${C}_{N}$ evolve unitarily and are therefore decoherence-dissipation free. The space ${C}_{N}$ realizes a noiseless quantum code in which information can be stored, in principle, for an arbitrarily long time without being affected by errors.
1,075 citations
TL;DR: In this paper, a new action principle was proposed to be associated with a non-commutative space, where the universal formula for the spectral action is a spinor on the Hilbert space.
Abstract: We propose a new action principle to be associated with a noncommutative space \(\). The universal formula for the spectral action is \(\) where \(\) is a spinor on the Hilbert space, \(\) is a scale and \(\) a positive function. When this principle is applied to the noncommutative space defined by the spectrum of the standard model one obtains the standard model action coupled to Einstein plus Weyl gravity. There are relations between the gauge coupling constants identical to those of SU(5) as well as the Higgs self-coupling, to be taken at a fixed high energy scale.
838 citations
01 Mar 1997
TL;DR: The β-tungsten structure as discussed by the authors has a surface area that is approximately 0.3% less than that of Kelvin's structure, which is a counter-example of a structure analogous to that of some clathrate compounds.
Abstract: Kelvin's conjecture, that a b.c.c. arrangement of his minimal tetrakaidecahedron divides space into equal cells of minimum surface area, has stood for over one hundred years. We have found a counter-example, in the form of a structure analogous to that of some clathrate compounds and also related to the β-tungsten structure. Its surface area is approximately 0.3% less than that of Kelvin's structure.
453 citations
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TL;DR: In this article, a local geometric description of how charged matter arises in type IIA, M-theory, of Ftheory compactifications on Calabi-Yau manifolds is given.
351 citations
TL;DR: In this paper, the prefactork0 of the fractal aggregate scaling relationship was determined for both diffusion limited and diffusion limited cluster aggregation processes in spatial dimensions of 2, 3, 4, and 5.
322 citations
TL;DR: In this paper, a quantitative theory of Gabor expansions f (x ) = √ k, n c k, n e 2 πinαx g (x − kβ ).
296 citations
TL;DR: In this article, the fine properties of functions in, the space of functions with bounded deformation, were analyzed, and it was shown that functions are approximately differentiable in almost every point of their domain.
Abstract: The paper is concerned with the fine properties of functions in , the space of functions with bounded deformation. We analyse the set of Lebesgue points and the set where these functions have one-sided approximate limits. Moreover, following the analogy with , we decompose the symmetric distributional derivative into an absolutely continuous part , a jump part , and a Cantor part . The main result of the paper is a structure theorem for functions, showing that these parts of the derivative can be recovered from the corresponding ones of the one-dimensional sections. Moreover, we prove that functions are approximately differentiable in almost every point of their domain.
283 citations
Book•
01 Jun 1997
TL;DR: Using the theory of linear relations in Pontryagin spaces, this paper extended the nonpositive case of reproducing kernel spaces associated with contractions in Hilbert spaces to the non-positive case.
Abstract: Using the theory of linear relations in Pontryagin spaces we extend to the nonpositive case the theory of reproducing kernel spaces associated with contractions in Hilbert spaces.
227 citations
TL;DR: In this article, generalized membrane solutions of D = 11 supergravity, for which the transverse space is a toric hyper-Kahler manifold, are shown to have IIB duals representing the intersection of parallel 3-brane with 5-branes whose orientations are determined by their Sl(2, Z) charge vectors.
167 citations
TL;DR: In this article, the authors describe movement as a pattern of discrete transitions from one space to another, and describe movement patterns as a set of transitions from a given perspective to a new perspective.
Abstract: As we move through buildings, we experience not only continuous changes of perspective but also discrete transitions from one space to another. To describe movement as a pattern of such transitions...
163 citations
Journal Article•
Patent•
29 Jul 1997TL;DR: In this paper, a patient is automatically accurately positioned relative to a fixed reference of a treatment/diagnostic device by an optical system which operates a patient positioning assembly to bring fiducials or skin markers on the patient into coincidence with impigement points of laser beams projected in a fixed pattern relative to the device.
Abstract: A patient is automatically accurately positioned relative to a fixed reference of a treatment/diagnostic device by an optical system which operates a patient positioning assembly to bring fiducials or skin markers on the patient into coincidence with impigement points of laser beams projected in a fixed pattern relative to the device. Cameras record images of the fiducials and laser impingement points from which alignment error and velocity error in pixel space are determined. The velocity error in pixel space is converted to a velocity error in room space by the inverse of an Image Jacobian. The Image Jacobian is initially derived using rough values for system parameters and is continuously updated and refined using the calculated errors in pixel space derived from the camera images and errors in room space derived from position encoders on the treatment/diagnostic device.
TL;DR: In this paper, the authors clarified the relationship between immersions of surfaces and solutions of the inhomogeneous Dirac equation and the main idea leading to the description of a surface M 2 by a spinor field is the observation that the restriction to M 2 of any parallel spinor phi on R^3 is (with respect to the inner geometry of M 2) a non-trivial spinor fields on M^2 of constant length which is a solution of the inner geometrical equation and vice versa.
Abstract: The aim of the present paper is to clarify the relationship between immersions of surfaces and solutions of the inhomogeneous Dirac equation. The main idea leading to the description of a surface M^2 by a spinor field is the observation that the restriction to M^2 of any parallel spinor phi on R^3 is (with respect to the inner geometry of M^2) a non-trivial spinor field on M^2 of constant length which is a solution of the inhomogeneous Dirac equation and vice versa.
TL;DR: In this article, the authors studied geometrical and topological aspects of the generalised dimensional reduction of supergravities in D=11 and D=10 dimensions, which give rise to massive theories in lower dimensions.
Abstract: We study some geometrical and topological aspects of the generalised dimensional reduction of supergravities in D=11 and D=10 dimensions, which give rise to massive theories in lower dimensions. In these reductions, a global symmetry is used in order to allow some of the fields to have a non-trivial dependence on the compactifying coordinates. Global consistency in the internal space imposes topological restrictions on the parameters of the compactification as well as the structure of the space itself. Examples that we consider include the generalised reduction of the type IIA and type IIB theories on a circle, and also the massive ten-dimensional theory obtained by the generalised reduction of D=11 supergravity.
TL;DR: A unified view in which space is based on the physical properties of manipulability, locomotion, and size of space is proposed, which has implications for various theoretical and methodological questions concerning the design and use of spatial information tools.
Abstract: The way people conceptualize space is an important consideration for the design of GIS, because a better match with people's thinking is expected to lead to easier-to-use information systems. Everyday space, the basis to GIS, has been characterized in the literature as being either small-scale (from table-top to room-size spaces) or large-scale (inside-of-building spaces to city-size spaces). While this dichotomy of space is grounded in the view from psychology that people's perception of space, spatial cognition, and spatial behaviour are experience-based, it is in contrast to current GIS, which enable us to interact with large-scale spaces as though they were small-scale or manipulable. We analyse different approaches to characterizing spaces and propose a unified view in which space is based on the physical properties of manipulability, locomotion, and size of space. Within the structure of our framework, we distinguish six types of spaces: manipulable object space (smaller than the human body), non-manipulable object space (greater than the human body, but less than the size of a building), environmental space (from inside-of-building spaces to city-size spaces), geographic space (state, country, and continent-size spaces), panoramic space (spaces perceived via scanning the landscape), and map space. Such a categorization is an important part of Naive Geography, a set of theories on how people intuitively or spontaneously conceptualize geographic space and time, because it has implications for various theoretical and methodological questions concerning the design and use of spatial information tools. Of particular concern is the design of effective spatial information tools that lead to better communication.
Book•
27 Nov 1997
TL;DR: In this paper, Copies of c 0 and 1 in L p (?, X) spaces are given, and the space L?(?, X) is used for tabulation of results.
Abstract: Preliminaries.- Copies of c 0 and ?1 in L p (?, X).- C(K, X) spaces.- L p (?, X) spaces.- The space L ?(?, X).- Tabulation of results.- Some related open problems.
TL;DR: The non-Arrhenius temperature-dependence of the average relaxation time of glass-forming liquids is studied in this article, where the authors review the classical phenomenological models for this phenomenon -the free volume model and the entropy model -and critique against these models.
Abstract: A major mystery of glass-forming liquids is the non-Arrhenius temperature-dependence of the average relaxation time. This paper briefly reviews the classical phenomenological models for this phenomenon - the free-volume model and the entropy model - and critiques against these models. We then discuss a recent model [Dyre, Olsen, and Christensen, Phys. Rev. B 53, 2171 (1996)] according to which the activation energy for the average relaxation time is determined by the work done in shoving aside the surrounding liquid to create space needed for a flow event. In this model the non-Arrhenius temperature-dependence is a consequence of the fact that the instantaneous (infinite-frequency) shear modulus increases upon cooling.
12 May 1997
TL;DR: The design of K projection patterns for a structured light system with L distinct planes of light is shown to be equivalent to the placement of L points in a K dimensional space subject to certain constraints.
Abstract: A methodology for the optimal design of projection patterns for stereometric structured light systems is presented. The similarity as well as the difference between the design of projection patterns and the design of optimal signals for digital communication are discussed. The design of K projection patterns for a structured light system with L distinct planes of light is shown to be equivalent to the placement of L points in a K dimensional space subject to certain constraints. optimal design in the MSE sense is defined, but shown to lead to an intractable multi-parameter global optimization problem. Intuitively appealing suboptimal solutions derived from the family of K dimensional space-filling Hilbert curves are obtained. Preliminary experimental results are presented.
TL;DR: In this article, the on-top pair density P r, r gives the probability that one electron will be found on top of another at position r. The results show that the local spin density LSD and generalized. gradient GGA approximations for exchange and correlation predict this quantity with remarkable accuracy.
Abstract: The on-top pair density P r, r gives the probability that one electron will be found on top . of another at position r. We find that the local spin density LSD and generalized . gradient GGA approximations for exchange and correlation predict this quantity with remarkable accuracy. We show how this fact and the usual sum-rule arguments explain the success of these approximations for real atoms, molecules, and solids, where the electron spin densities do not vary slowly over space. Self-consistent LSD or GGA . calculations make realistic predictions for the total energy E, the total density n r , and .
TL;DR: The framework needed to apply modern high accuracy numerical methods from computational gas dynamics to this extended system of convection-diffusion equations with stiff source terms is developed.
TL;DR: In this paper, the authors prove some version of Morrison's conjecture on the cone of divisors for Calabi-Yau fiber spaces with non-trivial base pace whose total space is 3-dimensional.
Abstract: We prove some version of Morrison's conjecture on the cone of divisors for Calabi-Yau fiber spaces with non-trivial base pace whose total space is 3-dimensional.
TL;DR: In this article, the authors use the HyperKahler quotient of flat space to obtain some monopole moduli space metrics in explicit form and discuss their topology, completeness and isometries.
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 HyperKahler metrics which, depending on the “mass parameter” being positive or negative, tend to either the asymptotic metric on the moduli space of many SU(2) monopoles, or to previously unknown metrics. These new metrics are singular or complete depending on the particular choice of the level set of the moment map. The singular 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.
TL;DR: In this paper, a coset-space unification model for families based on $E_7/SU(5) \times U(1)^3) was studied, and it was shown that qualitative structure of quark and lepton mass matrices in this model describes very well the observation.
Abstract: We study a coset-space unification model for families based on $E_7/SU(5) \times U(1)^3$. We find that qualitative structure of quark and lepton mass matrices in this model describes very well the observation. We stress, in particular, that the large mixing angle, $\sin^22\theta_{
u_\mu
u_\tau} \simeq 1$, required for the atmospheric neutrino oscillation reported by the SuperKamiokande collaboration, is naturally obtained, which is a consequence of unparallel family structure in the present coset-space unification.
TL;DR: In this article, the authors consider the form of the eigenvalue counting function ρ for Laplacians on p.c.f. self-similar sets and show that ρ(x)x−ds/# does not converge as x!¢.
Abstract: In this paper we consider the form of the eigenvalue counting function ρ for Laplacians on p.c.f. selfsimilar sets, a class of self-similar fractal spaces. It is known that on a p.c.f. self-similar set K the function ρ(x) O(xds/#) as x!¢, for some d s " 0. We show that if there exist localized eigenfunctions (that is, a non-zero eigenfunction which vanishes on some open subset of the space) and K satisfies some additional conditions (‘ the lattice case ’) then ρ(x)x−ds/# does not converge as x!¢. We next establish a number of sufficient conditions for the existence of a localized eigenfunction in terms of the symmetries of the space K. In particular, we show that any nested fractal with more than two essential fixed points has localized eigenfunctions.
TL;DR: It is shown that the category of symmetrically compact -continuity spaces with continuous maps has many of the key properties required of a category of domains and that it captures, in a natural way, the traditional examples.
TL;DR: It is concluded that the algebraic approach to fitness landscape analysis can be extended to recombination spaces and provides an effective way to analyze the relative hardness of a landscape for a given recombination operator.
Abstract: A new mathematical representation is proposed for the configuration space structure induced by recombination, which we call “P-structure.” It consists of a mapping of pairs of objects to the power set of all objects in the search space. The mapping assigns to each pair of parental “genotypes” the set of all recombinant genotypes obtainable from the parental ones. It is shown that this construction allows a Fourier decomposition of fitness landscapes into a superposition of “elementary landscapes.” This decomposition is analogous to the Fourier decomposition of fitness landscapes on mutation spaces. The elementary landscapes are obtained as eigenfunctions of a Laplacian operator defined for P-structures. For binary string recombination, the elementary landscapes are exactly the p-spin functions (Walsh functions), that is, the same as the elementary landscapes of the string point mutation spaces (i.e., the hypercube). This supports the notion of a strong homomorphism between string mutation and recombination spaces. However, the effective nearest neighbor correlations on these elementary landscapes differ between mutation and recombination and among different recombination operators. On average, the nearest neighbor correlation is higher for one-point recombination than for uniform recombination. For one-point recombination, the correlations are higher for elementary landscapes with fewer interacting sites as well as for sites that have closer linkage, confirming the qualitative predictions of the Schema Theorem. We conclude that the algebraic approach to fitness landscape analysis can be extended to recombination spaces and provides an effective way to analyze the relative hardness of a landscape for a given recombination operator.
Journal Article•
TL;DR: The Besicovitch and Weyl pseudometrics on the space AŸ of biinfinite sequences measure the density of differences in either the central or arbitrary segments of given sequences.
Abstract: The Besicovitch and Weyl pseudometrics on the space AŸ of biinfinite sequences measure the density of differences in either the central or arbitrary segments of given sequences. The Besicovitch and Weyl spaces are obtained from A by factoring through the equivalence of zero distance. Cellular automata are considered as dynamical systems on the Besicovitch and Weyl spaces and their topological and dynamical properties are compared with those they possess in the Cantor space.
TL;DR: In this article, an optimal Penrose-like inequality for the mass of any asymptotically flat Riemannian 3-manifold having an inner minimal 2-sphere and nonnegative scalar curvature was shown.
Abstract: We prove an optimal Penrose-like inequality for the mass of any asymptotically flat Riemannian 3-manifold having an inner minimal 2-sphere and nonnegative scalar curvature. Our result shows that the mass is bounded from below by an expression involving the area of the minimal sphere (as in the original Penrose conjecture) and some nomalized Sobolev ratio. As expected, the equality case is achieved if and only if the metric is that of a standard spacelike slice in the Schwarzschild space.
TL;DR: In this article, the authors derived a simple way to express the redshift distortions in galaxy redshift surveys with arbitrary geometry in closed form, which provides an almost ideal way to measure the value of $\beta=\Omega_0^{0.6}/b$ in wide area surveys, since all pairs in the survey can be used for the analysis.
Abstract: Using a novel two-dimensional coordinate system, we have derived a particularly simple way to express the redshift distortions in galaxy redshift surveys with arbitrary geometry in closed form. This method provides an almost ideal way to measure the value of $\beta=\Omega_0^{0.6}/b$ in wide area surveys, since all pairs in the survey can be used for the analysis. In the limit of small angles, this result straightforwardly reduces to the plane-parallel approximation. This expansion can also be used together with more sophisticated methods such as for the calculation of Karhunen-Loeve eigenvectors in redshift space for an arbitrary survey geometry. Therefore, these results should provide for more precise methods in which to measure the large scale power spectrum and the value of $\beta$ simultaneously.