Showing papers on "Space (mathematics) published in 2007"

Toward accurate dynamic time warping in linear time and space

Abstract: Dynamic Time Warping (DTW) has a quadratic time and space complexity that limits its use to small time series. In this paper we introduce FastDTW, an approximation of DTW that has a linear time and space complexity. FastDTW uses a multilevel approach that recursively projects a solution from a coarser resolution and refines the projected solution. We prove the linear time and space complexity of FastDTW both theoretically and empirically. We also analyze the accuracy of FastDTW by comparing it to two other types of existing approximate DTW algorithms: constraints (such as Sakoe-Chiba Bands) and abstraction. Our results show a large improvement in accuracy over existing methods.

Automatic discovery of metastable states for the construction of Markov models of macromolecular conformational dynamics.

Abstract: To meet the challenge of modeling the conformational dynamics of biological macromolecules over long time scales, much recent effort has been devoted to constructing stochastic kinetic models, often in the form of discrete-state Markov models, from short molecular dynamics simulations. To construct useful models that faithfully represent dynamics at the time scales of interest, it is necessary to decompose configuration space into a set of kinetically metastable states. Previous attempts to define these states have relied upon either prior knowledge of the slow degrees of freedom or on the application of conformational clustering techniques which assume that conformationally distinct clusters are also kinetically distinct. Here, we present a first version of an automatic algorithm for the discovery of kinetically metastable states that is generally applicable to solvated macromolecules. Given molecular dynamics trajectories initiated from a well-defined starting distribution, the algorithm discovers long lived, kinetically metastable states through successive iterations of partitioning and aggregating conformation space into kinetically related regions. The authors apply this method to three peptides in explicit solvent—terminally blocked alanine, the 21-residue helical Fs peptide, and the engineered 12-residue β-hairpin trpzip2—to assess its ability to generate physically meaningful states and faithful kinetic models.

Holomorphic Functions in the Plane and n-dimensional Space

Abstract: Numbers.- Functions.- Integration and integral theorems.- Series expansions and local behavior.

Minkowski space structure of the Higgs potential in 2HDM

Abstract: The Higgs potential of 2HDM keeps its generic form under the group of transformation $GL(2,C)$, which is larger than the usually considered reparametrization group $SU(2)$. This reparametrization symmetry induces the Minkowski space structure in the orbit space of 2HDM. Exploiting this property, we present a geometric analysis of the number and properties of stationary points of the most general 2HDM potential. In particular, we prove that charge-breaking and neutral vacua never coexist in 2HDM and establish conditions when the most general explicitly $CP$-conserving Higgs potential has spontaneously $CP$-violating minima. We also define the prototypical model of a given 2HDM, which has six free parameters less than the original one but still contains all the essential physics. Our analysis avoids manipulation with high-order algebraic equations.

Sobolev Active Contours

Abstract: All previous geometric active contour models that have been formulated as gradient flows of various energies use the same L 2-type inner product to define the notion of gradient. Recent work has shown that this inner product induces a pathological Riemannian metric on the space of smooth curves. However, there are also undesirable features associated with the gradient flows that this inner product induces. In this paper, we reformulate the generic geometric active contour model by redefining the notion of gradient in accordance with Sobolev-type inner products. We call the resulting flows Sobolev active contours. Sobolev metrics induce favorable regularity properties in their gradient flows. In addition, Sobolev active contours favor global translations, but are not restricted to such motions; they are also less susceptible to certain types of local minima in contrast to traditional active contours. These properties are particularly useful in tracking applications. We demonstrate the general methodology by reformulating some standard edge-based and region-based active contour models as Sobolev active contours and show the substantial improvements gained in segmentation.

Statistics of critical points of Gaussian fields on large-dimensional spaces.

Abstract: We calculate the average number of critical points of a Gaussian field on a high-dimensional space as a function of their energy and their index. Our results give a complete picture of the organization of critical points and are of relevance to glassy and disordered systems and landscape scenarios coming from the anthropic approach to string theory.

Modeling trees with a space colonization algorithm

Abstract: We extend the open leaf venation model by Runions et al. [RFL*05] to three dimensions and show that it generates surprisingly realistic tree structures. Model parameters correspond to visually relevant tree characteristics identified in landscaping, offering convenient control of tree shape and structure.

Supersymmetric states of N=4 Yang-Mills from giant gravitons

Abstract: Mikhailov has constructed an infinite family of 1 BPS D3-branes in AdS5 × S 5 . We regulate Mikhailov's solution space by focussing on finite dimensional submani- folds. Our submanifolds are topologically complex projective spaces with symplectic form cohomologically equal to 2πN times the Fubini-Study Kahler class. Upon quan- tization and removing the regulator we find the Hilbert Space of N noninteracting Bose particles in a 3d Harmonic oscillator, a result previously conjectured by Beasley. This Hilbert Space is isomorphic to the classical chiral ring of 1 BPS states in N = 4 Yang-Mills theory. We view our result as evidence that the spectrum of 1 BPS states in N = 4 Yang Mills theory, which is known to jump discontinuously from zero to infinitesimal coupling, receives no further renormalization at finite values of the 't Hooft coupling.

Function spaces of variable smoothness and integrability

Abstract: In this article we introduce Triebel--Lizorkin spaces with variable smoothness and integrability. Our new scale covers spaces with variable exponent as well as spaces of variable smoothness that have been studied in recent years. Vector-valued maximal inequalities do not work in the generality which we pursue, and an alternate approach is thus developed. Applying it, we give molecular and atomic decomposition results and show that our space is well-defined, i.e., independent of the choice of basis functions. As in the classical case, a unified scale of spaces permits clearer results in cases where smoothness and integrability interact, such as Sobolev embedding and trace theorems. As an application of our decomposition, we prove optimal trace theorems in the variable indices case.

Finite elements for symmetric tensors in three dimension

Abstract: We construct finite element subspaces of the space of symmetric tensors with square-integrable divergence on a three-dimensional domain. These spaces can be used to approximate the stress field in the classical Hellinger--Reissner mixed formulation of the elasticty equations, when standard discontinous finite element spaces are used to approximate the displacement field. These finite element spaces are defined with respect to an arbitrary simplicial triangulation of the domain, and there is one for each positive value of the polynomial degree used for the displacements. For each degree, these provide a stable finite element discretization. The construction of the spaces is closely tied to discretizations of the elasticity complex, and can be viewed as the three-dimensional analogue of the triangular element family for plane elasticity previously proposed by Arnold and Winther.

Non-supersymmetric attractor flow in symmetric spaces

Abstract: We derive extremal black hole solutions for a variety of four dimensional models which, after Kaluza-Klein reduction, admit a description in terms of 3D gravity coupled to a sigma model with symmetric target space. The solutions are in correspondence with certain nilpotent generators of the isometry group. In particular, we provide the exact solution for a non-BPS black hole with generic charges and asymptotic moduli in = 2 supergravity coupled to one vector multiplet. Multi-centered solutions can also be generated with this technique. It is shown that the non-supersymmetric solutions lack the intricate moduli space of bound configurations that are typical of the supersymmetric case.

On the moduli space of semilocal strings and lumps

Abstract: We study BPS non-abelian semilocal vortices in U(Nc) gauge theory with Nf flavors, Nf > Nc, in the Higgs phase. The moduli space for arbitrary winding number is described using the moduli matrix formalism. We find a relation between the moduli spaces of the semilocal vortices in a Seiberg-like dual pairs of theories, U(Nc) and U(Nf-Nc). They are two alternative regularizations of a "parent" non-Hausdorff space, which tend to the same moduli space of sigma-model lumps in the infinite gauge coupling limits. We examine the normalizability of the zero-modes and find the somewhat surprising phenomenon that the number of normalizable zero-modes, dynamical fields in the effective action, depends on the point of the moduli space we are considering. We find, in the lump limit, an effective action on the vortex worldsheet, which we compare to that found by Shifman and Yung.

A Cross-Match of 2MASS and SDSS: Newly-Found L and T Dwarfs and an Estimate of the Space Densitfy of T Dwarfs

Abstract: We report new L and T dwarfs found in a cross-match of the SDSS Data Release 1 and 2MASS. Our simultaneous search of the two databases effectively allows us to relax the criteria for object detection in either survey and to explore the combined databases to a greater completeness level. We find two new T dwarfs in addition to the 13 already known in the SDSS DR1 footprint. We also identify 22 new candidate and bona-fide L dwarfs, including a new young L2 dwarf and a peculiar L2 dwarf with unusually blue near-IR colors: potentially the result of mildly sub-solar metallicity. These discoveries underscore the utility of simultaneous database cross-correlation in searching for rare objects. Our cross-match completes the census of T dwarfs within the joint SDSS and 2MASS flux limits to the 97% level. Hence, we are able to accurately infer the space density of T dwarfs. We employ Monte Carlo tools to simulate the observed population of SDSS DR1 T dwarfs with 2MASS counterparts and find that the space density of T0-T8 dwarf systems is 0.0070 (-0.0030; +0.0032) per cubic parsec (95% confidence interval), i.e., about one per 140 cubic parsecs. Compared to predictions for the T dwarf space density that depend on various assumptions for the sub-stellar mass function, this result is most consistent with models that assume a flat sub-stellar mass function dN/dM ~ M^0. No >T8 dwarfs were discovered in the present cross-match, though less than one was expected in the limited area (2099 sq. degrees) of SDSS DR1.

Improved algorithms for path, matching, and packing problems

Abstract: Improved randomized and deterministic algorithms are presented for PATH, MATCHING, and PACKING problems. Our randomized algorithms are based on the divide-and-conquer technique, and improve previous best algorithms for these problems. For example, for the k-PATH problem, our randomized algorithm runs in time O(4kk3.42m) and space O(nklogk + m), improving the previous best randomized algorithm for the problem that runs in time O(5.44kkm) and space O(2kkn + m). To achieve improved deterministic algorithms, we study a number of previously proposed de-randomization schemes, and also develop a new derandomization scheme. These studies result in a number of deterministic algorithms: one of time O(4k+o(k)m) for the k-PATH problem, one of time O(2.803kk nlog2n) for the 3-D MATCHING problem, and one of time O(43k+o(k)n) for the 3-SET PACKING problem. All these significantly improve previous best algorithms for the problems.

Blow Up of Solutions to Semilinear Wave Equations with Critical Exponent in High Dimensions

Abstract: In this paper, the author considers the Cauchy problem for semilinear wave equations with critical exponent in n ≥ 4 space dimensions. Under some positivity conditions on the initial data, it is proved that there can be no global solutions no matter how small the initial data are.

Decompositions of Besov–Morrey spaces and Triebel–Lizorkin–Morrey spaces

Abstract: The aim of this paper is to define the Besov–Morrey spaces and the Triebel– Lizorkin–Morrey spaces and to present a decomposition of functions belonging to these spaces. Our results contain an answer to the conjecture proposed by Mazzucato.

A trilinear method for finding null points in a 3D vector space

Abstract: Null points are important locations in vector fields, such as a magnetic field. A new technique (a trilinear method for finding null points) is presented for finding null points over a large grid of points, such as those derived from a numerical experiment. The method was designed so that the null points found would agree with any fieldlines traced using the commonly used trilinear interpolation. It is split into three parts: reduction, analysis and positioning, which, when combined, provide an efficient means of locating null points to a user-defined sub-grid accuracy. We compare the results of the trilinear method with that of a method based on the Poincare index, and discuss the accuracy and limitations of both methods.

Existence and Regularity of Weak Solutions to the Quasi-Geostrophic Equations in the Spaces L p or $$\dot{H}^{-1/2}$$

Abstract: In this paper we study the 2D quasi-geostrophic equation with and without dissipation. We give global existence results of weak solutions for an initial data in the space Lp or \(\dot{H}^{-1/2}\) . In the dissipative case, when the initial data is in Lp, p > 2, we give a regularity result of these solutions.

Global well-posedness of the KP-I initial-value problem in the energy space

Abstract: We prove that the KP-I initial value problem is globally well-posed in the natural energy space of the equation.

Accelerating Cosmologies in the Einstein-Gauss-Bonnet Theory with Dilaton

Abstract: We study cosmological solutions in the low-energy effective heterotic string theory, which is the Einstein gravity with Gauss-Bonnet term and the dilaton. We show that the field equations are cast into an autonomous system for flat internal and external spaces, and derive all the fixed points in the system. We also examine the time evolution of the solutions and whether the solutions can give (transient) accelerated expansion of our four-dimensional space in the Einstein frame.

Fock space, quantum fields, and κ-Poincaré symmetries

Abstract: We study the quantization of a linear scalar field, whose symmetries are described by the {kappa}-Poincare Hopf algebra, via deformed Fock space construction. The one-particle sector of the theory exhibits a natural (Planckian) cutoff for the modes of the field. At the ''multiparticle'' level the nontrivial coalgebra structure of {kappa}-Poincare leads to a deformed bosonization in the construction of Fock space states. These physical states carry energy-momentum charges which are divergenceless and obey a deformed dispersion relation.