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853
Covariant Theory of Asymptotic Symmetries, Conservation Laws and Central Charges
, 2001
"... Under suitable assumptions on the boundary conditions, it is shown that there is a bijective correspondence between non trivial asymptotic reducibility parameters and non trivial asymptotically conserved n 2 forms in the context of Lagrangian gauge theories. The asymptotic reducibility parameters ar ..."
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Cited by 70 (13 self)
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Under suitable assumptions on the boundary conditions, it is shown that there is a bijective correspondence between non trivial asymptotic reducibility parameters and non trivial asymptotically conserved n 2 forms in the context of Lagrangian gauge theories. The asymptotic reducibility parameters are the parameters of gauge transformations that vanish suciently fast when evaluated at the background. A universal formula for asymptotically conserved n 2 forms in terms of the reducibility parameters is derived. Sucient conditions for niteness of the charges built out of the asymptotically conserved n 2 forms and for the existence of a Lie algebra g among equivalence classes of asymptotic reducibility parameters are given. The representation of g in terms of the charges may be centrally extended. An explicit and covariant formula for the central charges is constructed. They are shown to be 2cocycles on the Lie algebra g. The general considerations and formulas are applied to electrodynamics, YangMills theory and Einstein gravity where they reproduce familiar results.
Characteristi cohomology of differential systems (II): Conservation laws for a class of parabolic systems
 277080320 Email address: bryantfmath. duke. edu INSTITUTE FOR ADVANCED STUDY
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Cited by 54 (3 self)
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
Canonical structure of classical field theory in the polymomentum phase space
 Rep. Math. Phys
, 1998
"... Canonical structure of classical field theory in n dimensions is studied within the covariant polymomentum Hamiltonian formulation of De Donder–Weyl (DW). The bivertical (n + 1)form, called polysymplectic, is put forward as a generalization of the symplectic form in mechanics. Although not given i ..."
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Cited by 49 (8 self)
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Canonical structure of classical field theory in n dimensions is studied within the covariant polymomentum Hamiltonian formulation of De Donder–Weyl (DW). The bivertical (n + 1)form, called polysymplectic, is put forward as a generalization of the symplectic form in mechanics. Although not given in intrinsic geometric terms differently than a certain coset it gives rise to an invariantly defined map between horizontal forms playing the role of dynamical variables and the socalled vertical multivectors generalizing Hamiltonian vector fields. The analogue of the Poisson bracket on forms is defined which leads to the structure of Zgraded Lie algebra on the socalled Hamiltonian forms for which the map above exists. A generalized Poisson structure appears in the form of what we call a “higherorder ” and a right Gerstenhaber algebra. The equations of motion of forms are formulated in terms of the Poisson bracket with the DW Hamiltonian nform H ˜ vol ( ˜ vol is the spacetime volume form, H is the DW Hamiltonian function) which is found to be related to the operation of the total exterior differentiation of forms. A few applications and a relation to the standard Hamiltonian formalism in field theory are briefly discussed. 1
Symplectic Integration Of Constrained Hamiltonian Systems
"... . A Hamiltonian system in potential form (H(q; p) = p t M \Gamma1 p=2 + F (q)) subject to smooth constraints on q can be viewed as a Hamiltonian system on a manifold, but numerical computations must be performed in R n . In this paper, methods which reduce "Hamiltonian differentialalgebraic eq ..."
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Cited by 46 (10 self)
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. A Hamiltonian system in potential form (H(q; p) = p t M \Gamma1 p=2 + F (q)) subject to smooth constraints on q can be viewed as a Hamiltonian system on a manifold, but numerical computations must be performed in R n . In this paper, methods which reduce "Hamiltonian differentialalgebraic equations" to ODEs in Euclidean space are examined. The authors study the construction of canonical parameterizations or local charts as well as methods based on the construction of ODE systems in the space in which the constraint manifold is embedded which preserve the constraint manifold as an invariant manifold. In each case, a Hamiltonian system of ordinary differential equations is produced. The stability of the constraintinvariants and the behavior of the original Hamiltonian along solutions are investigated both numerically and analytically. Key words. differentialalgebraic equations, constrained Hamiltonian systems, canonical discretization schemes, symplectic methods AMS(MOS) subj...
Reduction of Systems of Nonlinear Partial Differential Equations to Simplified Involutive Forms
, 1996
"... We describe the rif algorithm which uses a finite number of differentiations and algebraic operations to simplify analytic nonlinear systems of partial differential equations to what we call reduced involutive form. This form includes the integrability conditions of the system and satisfies a consta ..."
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Cited by 43 (14 self)
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We describe the rif algorithm which uses a finite number of differentiations and algebraic operations to simplify analytic nonlinear systems of partial differential equations to what we call reduced involutive form. This form includes the integrability conditions of the system and satisfies a constant rank condition. The algorithm is useful for classifying initial value problems for determined pde systems and can yield dramatic simplifications of complex overdetermined nonlinear pde systems. Such overdetermined systems arise in analysis of physical pdes for reductions to odes using the Nonclassical Method, the search for exact solutions of Einstein's field equations and the determination of discrete symmetries of differential equations. Application of the algorithm to the associated nonlinear overdetermined system of 856 pdes arising when the Nonclassical Method is applied to a cubic nonlinear Schrodinger system yields new results. Our algorithm combines features of geometric involutiv...
LieButcher theory for RungeKutta methods
 BIT
, 1995
"... . RungeKutta methods are formulated via coordinate independent operations on manifolds. It is shown that there is an intimate connection between Lie series and Lie groups on one hand and Butcher's celebrated theory of order conditions on the other. In Butcher's theory the elementary differentials ..."
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Cited by 43 (15 self)
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. RungeKutta methods are formulated via coordinate independent operations on manifolds. It is shown that there is an intimate connection between Lie series and Lie groups on one hand and Butcher's celebrated theory of order conditions on the other. In Butcher's theory the elementary differentials are represented as trees. In the present formulation they appear as commutators between vector fields. This leads to a theory for the order conditions, which can be developed in a completely coordinate free manner. Even if this theory is developed in a language that is not widely used in applied mathematics, it is structurally simple. The recursion for the order conditions rests mainly on three lemmas, each with very short proofs. The techniques used in the analysis are prepared for studying RKlike methods on general Lie groups and homogeneous manifolds, but these themes are not studied in detail within the present paper. AMS subject classification: 65L06. Key words: Butcher theory, Run...
Integrable Evolution Equations on Associative Algebras
, 1997
"... This paper surveys the classi cation of integrable evolution equations whose eld variables take values in an associative algebra, which includes matrix, Clifford, and group algebra valued systems. A variety of new examples of integrable systems possessing higher order symmetries are presented. Symm ..."
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Cited by 40 (7 self)
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This paper surveys the classi cation of integrable evolution equations whose eld variables take values in an associative algebra, which includes matrix, Clifford, and group algebra valued systems. A variety of new examples of integrable systems possessing higher order symmetries are presented. Symmetry reductions lead to associative algebravalued version of the Painleve transcendent equations. The basic theory of Hamiltonian structures for associative algebravalued systems is developed and the biHamiltonian structures for several examples are found.
Active Contours for Tracking Distributions
 IEEE Trans. Image Processing
, 2002
"... A new approach to tracking using geometric active contours is presented. The class of objects to be tracked is assumed to be characterized by a probability distribution over some variable, such as intensity, colour, or texture. The goal of the algorithm is to find the region within the current im ..."
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Cited by 38 (4 self)
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A new approach to tracking using geometric active contours is presented. The class of objects to be tracked is assumed to be characterized by a probability distribution over some variable, such as intensity, colour, or texture. The goal of the algorithm is to find the region within the current image, such that the sample distribution of the interior of the region most closely matches the model distribution. Several criteria for matching distributions are examined, and the curve evolution equations are derived in each case. A particular flow is shown to perform well in two experiments.
Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications
 Eur. J. Appl. Math
, 2002
"... This paper gives a general treatment and proof of the direct conservation law method presented in Part I (see [3]). In particular, the treatment here applies to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard CauchyKovalevskaya ..."
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Cited by 35 (5 self)
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This paper gives a general treatment and proof of the direct conservation law method presented in Part I (see [3]). In particular, the treatment here applies to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard CauchyKovalevskaya form. A summary of the general method and its effective computational implementation is also given. 1