Results 1  10
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179
E(8) Gauge Theory and a Derivation of Ktheory from Mtheory
"... The partition function of RamondRamond pform fields in Type IIA supergravity on a tenmanifold X contains subtle phase factors that are associated with Tduality, selfduality, and the relation of the RR fields to Ktheory. The analogous partition function of Mtheory on X × S 1 contains subtle pha ..."
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Cited by 109 (9 self)
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The partition function of RamondRamond pform fields in Type IIA supergravity on a tenmanifold X contains subtle phase factors that are associated with Tduality, selfduality, and the relation of the RR fields to Ktheory. The analogous partition function of Mtheory on X × S 1 contains subtle phases that are similarly associated with E8 gauge theory. We analyze the detailed phase factors on the two sides and show that they agree, thereby testing Mtheory/Type IIA duality as well as the Ktheory formalism in an interesting way. We also show that certain Dbrane states wrapped on nontrivial homology cycles are actually unstable, that (−1) FL symmetry in Type IIA superstring theory depends in general on a cancellation between a fermion anomaly and an anomaly of RR fields, and that Type IIA superstring theory with no wrapped branes is welldefined only on a spacetime with W7 = 0. On leave from Institute for Advanced Study, Princeton, NJ 08540.
Existence of conformal metrics with constant Qcurvature
, 2004
"... Given a compact four dimensional manifold, we prove existence of conformal metrics with constant Qcurvature under generic assumptions. The problem amounts to solving a fourthorder nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbou ..."
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Cited by 67 (4 self)
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Given a compact four dimensional manifold, we prove existence of conformal metrics with constant Qcurvature under generic assumptions. The problem amounts to solving a fourthorder nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and minimax schemes, jointly with the compactness result of [31].
Central extensions of infinitedimensional Lie groups
 ANNALES DE L’INST. FOURIER 52:5
, 2004
"... In the present paper we study abelian extensions of connected Lie groups G modeled on locally convex spaces by smooth Gmodules A. We parametrize the extension classes by a suitable cohomology group H 2 s (G,A) defined by locally smooth cochains and construct an exact sequence that describes the di ..."
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Cited by 57 (6 self)
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In the present paper we study abelian extensions of connected Lie groups G modeled on locally convex spaces by smooth Gmodules A. We parametrize the extension classes by a suitable cohomology group H 2 s (G,A) defined by locally smooth cochains and construct an exact sequence that describes the difference between H 2 s (G,A) and the corresponding continuous Lie algebra cohomology space H 2 c (g,a). The obstructions for the integrability of a Lie algebra extensions to a Lie group extension are described in terms of period and flux homomorphisms. We also characterize the extensions with global smooth sections resp. those given by global smooth cocycles. Finally we apply the general theory to extensions of several types of diffeomorphism groups.
Computational Topology: Ambient Isotopic Approximation of 2Manifolds
 THEORETICAL COMPUTER SCIENCE
, 2001
"... A fundamental issue in theoretical computer science is that of establishing unambiguous formal criteria for algorithmic output. This paper does so within the domain of computeraided geometric modeling. For practical geometric modeling algorithms, it is often desirable to create piecewise linear app ..."
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Cited by 40 (19 self)
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A fundamental issue in theoretical computer science is that of establishing unambiguous formal criteria for algorithmic output. This paper does so within the domain of computeraided geometric modeling. For practical geometric modeling algorithms, it is often desirable to create piecewise linear approximations to compact manifolds embedded in and it is usually desirable for these two representations to be "topologically equivalent". Though this has traditionally been taken to mean that the two representations are homeomorphic, such a notion of equivalence suffers from a variety of technical and philosophical difficulties; we adopt the stronger notion of ambient isotopy. It is shown here, that for any C², compact, 2manifold without boundary, which is embedded in R³, there exists a piecewise linear ambient isotopic approximation. Furthermore, this isotopy has compact support, with specific bounds upon the size of this compact neighborhood. These bounds may be useful for practical application in computer graphics and engineering design simulations. The proof given relies upon properties of the medial axis, which is explained in this paper.
Minimal surface representations of virtual knots and
"... Equivalence classes of virtual knot diagrams are in a one to one correspondence with decorated immersions of S 1 into orientable, closed surfaces modulo stable handle equivalence and Reidemeister moves. Each virtual knot diagram corresponds to an immersion of S 1 with over/under markings in a unique ..."
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Cited by 39 (28 self)
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Equivalence classes of virtual knot diagrams are in a one to one correspondence with decorated immersions of S 1 into orientable, closed surfaces modulo stable handle equivalence and Reidemeister moves. Each virtual knot diagram corresponds to an immersion of S 1 with over/under markings in a unique minimal surface. If a virtual knot diagram is equivalent to a classical knot diagram then this minimal surface is a sphere. We use minimal surfaces and a generalized version of the bracket polynomial for surfaces to determine when a virtual knot diagram is nontrivial and nonclassical. 1 1
Computing shortest nontrivial cycles on orientable surfaces of bounded genus in almost linear time
 In SOCG ’06: Proc. 22nd Symp. Comput. Geom
, 2006
"... We present an algorithm that computes a shortest noncontractible and a shortest nonseparating cycle on an orientable combinatorial surface of bounded genus in O(n log n) time, where n denotes the complexity of the surface. This solves a central open problem in computational topology, improving upon ..."
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Cited by 35 (0 self)
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We present an algorithm that computes a shortest noncontractible and a shortest nonseparating cycle on an orientable combinatorial surface of bounded genus in O(n log n) time, where n denotes the complexity of the surface. This solves a central open problem in computational topology, improving upon the currentbest O(n 3/2)time algorithm by Cabello and Mohar (ESA 2005). Our algorithm uses universalcover constructions to find short cycles and makes extensive use of existing tools from the field. 1
Linear SelfAssemblies: Equilibria, Entropy and Convergence Rates
 IN SIXTH INTERNATIONAL CONFERENCE ON DIFFERENCE EQUATIONS AND APPLICATIONS
, 2001
"... Selfassembly is a ubiquitous process by which objects autonomously assemble into complexes. In the context of computation, selfassembly is important to both DNA computing and amorphous computing. Thus a well developed mathematical theory of selfassembly will be useful in these and other domains. ..."
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Cited by 35 (2 self)
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Selfassembly is a ubiquitous process by which objects autonomously assemble into complexes. In the context of computation, selfassembly is important to both DNA computing and amorphous computing. Thus a well developed mathematical theory of selfassembly will be useful in these and other domains. As a first problem in selfassembly, we will explore the creation of linear polymers. Polymers are chains of molecular units. For examples, a molecule of DNA is a polymer made from the bases adenine, guanine, cytosine and thymine; and proteins are polymers formed from the twenty amino acids. In our model the dynamics of a linear polymerization system is determined by difference and di#erential equations with initial conditions, and we investigate the equilibrium behavior of such a system. A good characterization of the equilibria of these systems will allow us to predict which computations can be carried out with adequate yields and what quantities of initial substrates these computations require. The rates at which such systems approach equilibria are
Exact Cellular Decompositions in Terms of Critical Points of Morse Functions
 In IEEE International Conference on Robotics and Automation
, 2000
"... Exact cellular decompositions are structures that globally encode the topology of a robot's free space, while locally describing the free space's geometry. These structures have been widely used for path planning between two points, but can be used for mapping and coverage of robot free sp ..."
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Cited by 27 (11 self)
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Exact cellular decompositions are structures that globally encode the topology of a robot's free space, while locally describing the free space's geometry. These structures have been widely used for path planning between two points, but can be used for mapping and coverage of robot free spaces. In this paper, we define exact cellular decompositions where critical points of Morse functions indicate the location of cell boundaries. Morse functions are those whose critical points are nondegenerate. Between critical points, the structure of a space is effectively the same, so simple control strategies to achieve tasks, such as coverage, are feasible within each cell. In this paper, we derive a general framework for defining decompositions in terms of critical points and then give examples, each corresponding to a different task. All of the results in this paper are derived in an mdimensional Euclidean space, but the examples depicted in the figures are two and threedimensional for ease ...
GromovWitten invariants and pseudo symplectic capacities
, 2001
"... We define the concept of pseudo symplectic capacities that is a mild generalization of that of the symplectic capacities. In particular a typical pseudo symplectic capacity, whose special case is a variant of the HoferZehnder symplectic capacity, is constructed and estimated in terms of the Gromov ..."
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Cited by 26 (7 self)
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We define the concept of pseudo symplectic capacities that is a mild generalization of that of the symplectic capacities. In particular a typical pseudo symplectic capacity, whose special case is a variant of the HoferZehnder symplectic capacity, is constructed and estimated in terms of the GromovWitten invariants. An example is also given to illustrate that using the pseudo symplectic capacity may get better estimation result than doing the HoferZehnder capacity. Among various potential applications of these estimations three typical applications are given. The first one is to derive some general nonsqueezing theorems that generalize and unite many past versions. The second is to give an alternate generalization of LalondeMcDuff theorem on length minimizing Hamiltonian paths to a general closed symplectic manifold. In the third application we give the new symplectic packing obstructions for a wider class of symplectic manifolds.
Inequalities between Entropy and Index of Coincidence derived from Information Diagrams
 IEEE Trans. Inform. Theory
, 2001
"... To any discrete probability distribution P we can associate its entropy H(P) = − � pi ln pi and its index of coincidence IC(P) = � p 2 i. The main result of the paper is the determination of the precise range of the map P � (IC(P), H(P)). The range looks much like that of the map P � (Pmax, H(P ..."
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Cited by 25 (11 self)
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To any discrete probability distribution P we can associate its entropy H(P) = − � pi ln pi and its index of coincidence IC(P) = � p 2 i. The main result of the paper is the determination of the precise range of the map P � (IC(P), H(P)). The range looks much like that of the map P � (Pmax, H(P)) where Pmax is the maximal point probability, cf. research from 1965 (Kovalevskij [18]) to 1994 (Feder and Merhav [7]). The earlier results, which actually focus on the probability of error 1 − Pmax rather than Pmax, can be conceived as limiting cases of results obtained by methods here presented. Ranges of maps as those indicated are called Information Diagrams. The main result gives rise to precise lower as well as upper bounds for the entropy function. Some of these bounds are essential for the exact solution of certain problems of universal coding and prediction for Bernoulli sources. Other applications concern Shannon theory (relations betweeen various measures of divergence), statistical decision theory and rate distortion theory. Two methods are developed. One is topological, another involves convex analysis and is based on a “lemma of replacement ” which is of independent interest in relation to problems of optimization of mixed type (concave/convex optimization).