soliton dynamics in computational anatomy computational anatomy ca has introduced the idea of anatomical structures being transformed by geodesic deformations on groups of diffeomorphisms among these geometric structures landmarks and image outlines in ca are shown to be singular solutions of a partial differential equation that is called the geodesic epdiff equation a recently discovered momentum map for singular solutions of epdiff yields their canonical hamiltonian formulation which in turn provides a complete parameterization of the landmarks by their canonical positions and momenta the momentum map provides an isomorphism between landmarks and outlines for images and singular soliton solutions of the epdiff equation this isomorphism suggests a new dynamical paradigm for ca as well as new data representation

kinematical symmetries of three dimensional incompressible flows abstract the motion of an incompressible fluid in lagrangian coordinates involves infinitely many symmetries generated by the left lie algebra of group of volume preserving diffeomorphisms of the three dimensional domain occupied by the fluid utilizing a 1 3 dimensional hamiltonian setting an explicit realization of this symmetry algebra is constructed recursively a dynamical connection is used to split the symmetries into reparametrization of trajectories and one parameter family of volume preserving diffeomorphisms of the fluid domain algebraic structures of symmetries and hamiltonian structures of their generators are inherited from the same construction a comparison with the properties of two dimensional flows is included

semidirect products and reduction in mechanics this paper shows how to reduce a hamiltonian system on the cotangent bundle of a lie group to a hamiltonian system in the dual of the lie algebra of a semidirect product the procedure simplifies unifies and extends work of greene guillemin holm holmes kupershmidt marsden morrison ratiu sternberg and others the heavy top compressible fluids magnetohydrodynamics elasticity the maxwell vlasov equations and multifluid plasmas are presented as examples starting with lagrangian variables our method explains in a direct way why semidirect products occur so frequently in examples it also provides a framework for the systematic introduction of clebsch or canonical variables

a differential complex for poisson manifolds construction du complexe canonique d une variete de poisson homologie canonique des varietes symplectiques application a l homologie de hochschild des algebres non commutatives

on the cauchy problem for the two component camassa holm system in this paper we establish the local well posedness for the two component camassa holm system in a range of the besov spaces we also derive a wave breaking mechanism for strong solutions in addition we determine the exact blow up rate of such solutions to the system

decay rates for inverses of band matrices spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices the rate of decay of a 1 can be bounded in terms of the essential spectrum of aa for general a and in terms of the essential spectrum of a for positive definite a in the positive definite case the bound can be attained these results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices we also establish decay rates for certain generalized inverses

application of the parallel diagonal dominant algorithm for the incompressible navier stokes equations abstract the accuracy and the applicability of the parallel diagonal dominant pdd algorithm are explored for highly scalable computation of the incompressible navier stokes equations which are integrated using a fully implicit fractional step method in parallel computational environments the pdd algorithm is known to be applicable only for an evenly diagonal dominant matrix in the present study however it is shown mathematically that the pdd algorithm is utilizable even for non diagonal dominant matrices derived from discretization of incompressible momentum equations the order of accuracy and the error characteristics are investigated in detail in terms of the courant friedrichs lewy cfl number and the grid spacing by conducting simulations of decaying vortices in both two and three dimensions flow in a lid driven cavity and flow over a circular cylinder in order to reduce communication cost which is one of bottlenecks in parallel computation an aggregative data communication method is combined with the pdd algorithm parallel performance of the present pdd based method is investigated by measuring the speedup efficiency overhead and serial fraction

lichnerowicz jacobi cohomology in this paper we extend the notion of lichnerowicz poisson cohomology to jacobi manifolds we study the relation of the so called lichnerowicz jacobi cohomology with the basic de rham cohomology and the cohomology of the lie algebra of functions relative to the representation defined by the hamiltonian vector fields a natural pairing with the canonical homology is constructed the relation between the lichnerowicz poisson cohomology of a quantizable poisson manifold and the lichnerowicz jacobi cohomology of the total space of a prequantization bundle is obtained particular cases of cosymplectic contact and locally conformal symplectic manifolds are discussed finally the lichnerowicz jacobi cohomology of a non transitive example is studied

thickenings and conformal gravity a twistor correspondence is given for complex conformal space times with vanishing bach and eastwood dighton tensors when the weyl curvature is algebraically general these equations are precisely the conformal version of einstein s vacuum equations with cosmological constant this gives a fully curved version of the linearized correspondence of baston and mason b m

on dual pairs in dirac geometry in this note we discuss weak dual pairs in dirac geometry we show that this notion appears naturally when studying the problem of pushing forward a dirac structure along a surjective submersion and we prove a dirac theoretic version of libermann s theorem from poisson geometry our main result is an explicit construction of self dual pairs for dirac structures this theorem not only recovers the global construction of symplectic realizations from crainic and m rcu j symplectic geom 9 4 435 444 2011 but allows for a more conceptual understanding of it yielding a simpler and more natural proof as an application of the main theorem we present a different approach to the recent normal form theorem around dirac transversals from bursztyn et al j fur die reine und angewandte mathematik crelles j doi 10 1515 crelle 2017 0014 2017

on optimal message vector length for block single parallel partition algorithm in a three dimensional adi solver it has long been recognized that many direct parallel tridiagonal solvers are only efficient for solving a single tridiagonal equation of large sizes and they become inefficient when naively used in a three dimensional adi solver in order to improve the parallel efficiency of an adi solver using a direct parallel solver we implement the single parallel partition spp algorithm in conjunction with message vectorization which aggregates several communication messages into one to reduce the communication costs the measured performances show that the longest allowable message vector length mvl is not necessarily the best choice to understand this observation and optimize the performance we propose an improved model that takes the cache effect into consideration the optimal mvl for achieving the best performance is shown to depend on number of processors and grid sizes similar dependence of the optimal mvl is also found for the popular block pipelined method

a parallel poisson generator using parallel prefix in this paper we use the renewal theory to develop a poisson random number algorithm without restart a parallel poisson random number generator is designed based on this algorithm and prefix computation this generator iteratively produces m poisson random numbers with mean in average time complexity o m n f n p on erew pram where f n p is the time for computing an n element parallel prefix on p processors in each iteration assuming that parallel uniform random numbers can be generated at the rate of one number per unit time per processor if n is selected near m it achieves linear speedup when p is small and the average time complexity is o log m when p is o m

extensions of banach lie poisson spaces abstract the extension of banach lie poisson spaces is studied and linked to the extension of a special class of banach lie algebras the case of w algebras is given particular attention semidirect products and the extension of the restricted banach lie poisson space by the banach lie poisson space of compact operators are given as examples

band asymptotics in two dimensions let x be a rank one symmetric space e g s rp cp and let a be its laplace beltrami operator it is known that the spectrum of a is discrete and consists of a sequence of eigenvalues a k k a k 0 1 2 a 1 1 each occurring with multiplicity n b n l k o k 1 2 n being the dimension of x and a and b being positive integers which depend only on x let q be a zeroth order self adjoint pseudodifferential operator if we perturb a by adding q to it the eigenvalues 1 1 break up into bands

parallel minimal norm method for tridiagonal linear systems based on the parallel minimal norm method an algorithm is derived to solve tridiagonal linear systems with a high degree of parallelism no conditions need to be posed with respect to the system experiments indicate that the numerical stability of the algorithm is similar to gaussian elimination with partial pivoting

persistence exponents for fluctuating interfaces numerical and analytic results for the exponent ensuremath theta describing the decay of the first return probability of an interface to its initial height are obtained for a large class of linear langevin equations the models are parametrized by the dynamic roughness exponent ensuremath beta with 0l ensuremath beta l1 for ensuremath beta frac 1 2 the time evolution is markovian using simulations of solid on solid models of the discretized continuum equations as well as of the associated zero dimensional stationary gaussian process we address two problems the return of an initially flat interface and the return to an initial state with fully developed steady state roughness the two problems are shown to be governed by different exponents for the steady state case we point out the equivalence to fractional brownian motion which has a return exponent ensuremath theta s 1 ensuremath ensuremath beta the exponent ensuremath theta 0 for the flat initial condition appears to be nontrivial we prove that ensuremath theta 0 ensuremath rightarrow ensuremath infty for ensuremath beta ensuremath rightarrow 0 ensuremath theta 0 g ensuremath theta s for ensuremath beta l frac 1 2 and ensuremath theta 0 l ensuremath theta s for ensuremath beta g frac 1 2 and calculate ensuremath theta 0 s perturbatively to first order in an expansion around the markovian case ensuremath beta frac 1 2 using the exact result ensuremath theta s 1 ensuremath ensuremath beta accurate upper and lower bounds on ensuremath theta 0 can be derived which show in particular that ensuremath theta 0 g 1 ensuremath ensuremath beta 2 ensuremath beta for small ensuremath beta

coadjoint orbits vortices and clebsch variables for incompressible fluids this paper is a study of incompressible fluids especially their clebsch variables and vortices using symplectic geometry and the lie poisson structure on the dual of a lie algebra following ideas of arnold and others it is shown that euler s equations are lie poisson equations associated to the group of volume preserving diffeomorphisms the dual of the lie algebra is seen to be the space of vortices and kelvin s circulation theorem is interpreted as preservation of coadjoint orbits in this context clebsch variables can be understood as momentum maps the motion of n point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit and the standard canonical variables for them are a special kind of clebsch variables point vortices with cores vortex patches and vortex filaments can be understood in a similar way this leads to an explanation of the geometry behind the hald beale majda convergence theorems for vorticity algorithms symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the kdv and the schrodinger equations respectively

integrable systems on semidirect product lie groups we study integrable systems on the semidirect product of a lie group and its lie algebra as the representation space of the adjoint action regarding the tangent bundle of a lie group as phase space endowed with this semidirect product lie group structure we construct a class of symplectic submanifolds equipped with a dirac bracket on which integrable systems in the adler kostant symes sense are naturally built through collective dynamics in doing so we address other issues such as factorization poisson lie structures and dressing actions we show that the procedure becomes recursive for some particular hamilton functions giving rise to a tower of nested integrable systems

banach lie poisson spaces and reduction the category of banach lie poisson spaces is introduced and studied it is shown that the category of w algebras can be considered as one of its subcategories examples and applications of banach lie poisson spaces to quantization and integration of hamiltonian systems are given the relationship between classical and quantum reduction is discussed

unified architecture for divide and conquer based tridiagonal system solvers the solution of tridiagonal systems is a topic of great interest in many areas of numerical analysis several algorithms have recently been proposed for solving triadiagonal systems based on the divide and conquer dc strategy in this work we propose a unified parallel architecture for dc algorithms which present the data flows of the successive doubling recursive doubling and parallel cyclic reduction methods the architecture is based in the perfect unshuffle permutation which transforms these data flows into a constant geometry one the partition of the data arises in a natural manner giving way to a systolic data flow with a wired control section we conclude that the constant geometry cyclic reduction architecture is the most appropriate one for solving tridiagonal systems and from the point of view of integration in vlsi technology is the one which uses the least amount of area and the smallest number of pins

geodesic flows on semidirect product lie groups geometry of singular measure valued solutions the epdiff equation or dispersionless camassa holm equation in 1d is a well known example of geodesic motion on the diff group of smooth invertible maps diffeomorphisms its recent two component extension governs geodesic motion on the semidirect product mathrm diff circleds mathcal f where mathcal f denotes the space of scalar functions this paper generalizes the second construction to consider geodesic motion on mathrm diff circleds mathfrak g where mathfrak g denotes the space of scalar functions that take values on a certain lie algebra for example mathfrak g mathcal f otimes mathfrak so 3 measure valued delta like solutions are shown to be momentum maps possessing a dual pair structure thereby extending previous results for the epdiff equation the collective hamiltonians are shown to fit into the kaluza klein theory of particles in a yang mills field and these formulations are shown to apply also at the continuum pde level in the continuum description the kaluza klein approach produces the kelvin circulation theorem

probability distribution of persistent spins in an ising chain we study the probability distribution q n t of n t the fraction of spins unflipped up to time t in an ising chain with ferromagnetic interactions the distribution shows a peak at n nmax and in general is non gaussian and asymmetric in nature however for n nmax it shows a gaussian decay data collapse can be obtained when q n t l versus n nmax l is plotted with 0 45 and 0 6 interestingly nmax t shows different behaviour compared to n t p t the persistence probability which follows the well known behaviour p t t a quantitative estimate of the asymmetry and non gaussian nature of q n t is made by calculating its skewness and kurtosis

application specific architecture for fast transforms based on the successive doubling method the successive doubling method is an efficient procedure for the design of fast algorithms for orthogonal transforms of length n r sup n where the radix r is a power of 2 a partitioned systolic architecture is presented for the two standard radix successive doubling algorithms decimation in time dit and decimation in frequency dif the index space of the data is projected onto the index space associated with a column of processors interconnected using a perfect unshuffle dit or shuffle dif interconnection network defined by permutations of the order log sub 2 r the result is a partitioned systolic array with q processors q r sup i 0

the density manifold and configuration space quantization on developpe la structure geometrique differentielle d une variete de frechet de densites ceci fournit un cadre geometrique a la quantification reliee a la mecanique stochastique de nelson on etudie les structures riemanniennes et symplectiques de la variete densite et on deduit l equation de schrodinger a partir d un principe variationnel

rigidity of hamiltonian actions on poisson manifolds abstract this paper is about the rigidity of compact group actions in the poisson context the main result is that hamiltonian actions of compact semisimple type are rigid we prove it via a nash moser normal form theorem for closed subgroups of sci type this nash moser normal form has other applications to stability results that we will explore in a future paper we also review some classical rigidity results for differentiable actions of compact lie groups and export it to the case of symplectic actions of compact lie groups on symplectic manifolds

stability of poisson equilibria and hamiltonian relative equilibria by energy methods we develop a general stability theory for equilibrium points of poisson dynamical systems and relative equilibria of hamiltonian systems with symmetries including several generalisations of the energy casimir and energy momentum methods using a topological generalisation of lyapunov s result that an extremal critical point of a conserved quantity is stable we show that a poisson equilibrium is stable if it is an isolated point in the intersection of a level set of a conserved function with a subset of the phase space that is related to the topology of the symplectic leaf space at that point this criterion is applied to generalise the energy momentum method to hamiltonian systems which are invariant under non compact symmetry groups for which the coadjoint orbit space is not hausdorff we also show that a g stable relative equilibrium satisfies the stronger condition of being a stable where a is a specific group theoretically defined subset of g which contains the momentum isotropy subgroup of the relative equilibrium the results are illustrated by an application to the stability of a rigid body in an ideal irrotational fluid

r matrix theory formal casimirs and the periodic toda lattice the nonunitary r matrix theory and the associated bi and trihamiltonian schemes are considered the language of poisson pencils and of their formal casimirs is applied in this framework to characterize the bihamiltonian chains of integrals of motion pointing out the role of the schur polynomials in these constructions this formalism is subsequently applied to the periodic toda lattice some different algebraic settings and lax formulations proposed in the literature for this system are analyzed in detail and their full equivalence is exploited in particular the equivalence between the loop algebra approach and the method of differential difference operators is illustrated moreover two alternative lax formulations are considered and appropriate reduction algorithms are found in both cases allowing us to derive the multihamiltonian formalism from r matrix theory the systems of integrals for the periodic toda lattice known after flaschka and henon and their functional relations are recovered through systematic application of the previously outlined schemes

a note on multi dimensional camassa holm type systems on the torus we present a 2n component nonlinear evolutionary pde which includes the n dimensional versions of the camassa holm and the hunter saxton systems as well as their partially averaged variations our goal is to apply arnold s geometric formalism arnold 1966 ann inst fourier grenoble 16 319 61 ebin and marsden 1970 ann math 92 102 63 to this general equation in order to obtain results on well posedness conservation laws or stability of its solutions following the line of arguments of kohlmann 2011 j phys a math theor 44 465205 we present geometric aspects of a two dimensional periodic b equation on the diffeomorphism group of the torus in this context

the hamiltonian structure of classical chromohydrodynamics noncanonical hamiltonian structures are presented both for yang mills vlasov plasmas and for ideal fluids interacting with yang mills fields the hamiltonian structure for the yang mills vlasov system passes over to that for the yang mills fluid in the cold plasma limit the resulting hamiltonian structure is shown to correspond to a lie algebra

a parallel alternating direction implicit preconditioning method abstract the alternating direction implicit adi iterative method is an efficient iterative method to solve systems of linear equations due to its extremely fast convergence the adi method has also been used successfully as a preconditioner in some other iterative methods such as the preconditioned conjugate gradient in this paper a parallel algorithm for the adi preconditioning is proposed in this algorithm several steps of the adi iteration are computed simultaneously this means that several tridiagonal systems that are traditionally solved sequentially are now solved concurrently the high performance of this algorithm is achieved by increasing the degree of parallelism and reducing memory contention the algorithm can easily be implemented in a multiprocessor architecture experiments have been conducted on the myrias sps 2 computer with 64 processors and good performance of this algorithm is observed

tangent lifts of poisson and related structures the derivation dt on the exterior algebra of forms on a manifold m with values in the exterior algebra of forms on the tangent bundle tm is extended to multivector fields these tangent lifts are studied with application to the theory of poisson structures their symplectic foliations canonical vector fields and poisson lie groups

reduction theory for symmetry breaking with applications to nematic systems we formulate euler poincare and lagrange poincare equations for systems with broken symmetry we specialize the general theory to present explicit equations of motion for nematic systems ranging from single nematic molecules to biaxial liquid crystals the geometric construction applies to order parameter spaces consisting of either unsigned unit vectors directors or symmetric matrices alignment tensors on the hamiltonian side we provide the corresponding poisson brackets in both lie poisson and hamilton poincare formulations the explicit form of the helicity invariant for uniaxial nematics is also presented together with a whole class of invariant quantities casimirs for two dimensional incompressible flows

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

input output decoupling of hamiltonian systems the linear case in this note we give necessary and sufficient conditions for a linear hamiltonian system to be input output decouplable by hamiltonian feedback i e feedback that preserves the hamiltonian structure in a second paper we treat the same problem for nonlinear hamiltonian systems

poisson integrators for lie poisson structures on mathbb r 3 this paper is concerned with the study of poisson integrators we are interested in lie poisson systems on first we focus on poisson integrators for constant poisson systems and the transformations used for transforming lie poisson structures to constant poisson structures then we construct local poisson integrators for lie poisson systems on finally we present the results of numerical experiments for two lie poisson systems and compare our poisson integrators with other known methods

explicitly solvable non linear optimal control problems the method of collective hamiltonians is applied to the study of optimal control problems on manifolds this approach enables us to single out exactly solvable situations various examples are considered necessary results from hamiltonian formalism are given including an industrial technique for the construction of completely integrable systems

particle like structure of lie algebras if a lie algebra structure x1d524 on a vector space is the sum of a family of mutually compatible lie algebra structures x1d524 i s we say that x1d524 is simply assembled from the x1d524 i s repeating this procedure with a number of lie algebras themselves simply assembled from the x1d524 i s one obtains a lie algebra assembled in two steps from x1d524 i s and so on we describe the process of modular disassembling of a lie algebra into a unimodular and a non unimodular part we then study two inverse questions which lie algebras can be assembled from a given family of lie algebras and from which lie algebras can a given lie algebra be assembled we develop some basic assembling and disassembling techniques that constitute the elements of a new approach to the general theory of lie algebras the main result of our theory is that any finite dimensional lie algebra over an algebraically closed field of characteristic zero or over r can be assembled in a finite number of steps from two elementary constituents which we call dyons and

aks hierarchy and bi hamiltonian geometry of gelfand zakharevich type a bi hamiltonian system is a system of differential equations which can be written in hamiltonian form in two distinct ways the applications of gelfand zakharevich bi hamiltonian structure which is an extension of a poisson nijenhuis structure on phase space has been extensively explored by falqui magri and pedroni in the context of separation of variables it is well known that the integrable hamiltonian systems defined by the adler kostant symes aks scheme contains bi hamiltonian structure in this paper we unveil the connection between adler kostant symes formalism applied to loop algebra and the gelfand zakharevich bi hamiltonian structure by superposition the results of fordy and kulish in the aks scheme we also study the commuting flows of the aks hierachy and its connection to the zakharov shabat hierarchy

the curvature of semidirect product groups associated with two component hunter saxton systems in this paper we study two component versions of the periodic hunter saxton equation and its variant considering both equations as a geodesic flow on the semidirect product of the circle diffeomorphism group with a space of scalar functions on we show that both equations are locally well posed the main result of this paper is that the sectional curvature associated with the 2hs is constant and positive and that 2 hs allows for a large subspace of positive sectional curvature the issues of this paper are related to some of the results for 2ch and 2dp presented in escher et al 2011 j geom phys 61 436 52

homology of algebras of families of pseudodifferential operators abstract we compute the hochschild cyclic and periodic cyclic homology groups of algebras of families of laurent complete symbols on manifolds with corners we show in particular that the spectral sequence associated with hochschild homology degenerates at e 2 and converges to hochschild homology as a byproduct we identify the space of residue traces on fibrations by manifolds with corners in the process we prove some structural results about algebras of complete symbols on manifolds with corners

efficient tridiagonal solvers on multicomputers three parallel algorithms namely the parallel partition lu ppt algorithm the parallel partition hybrid pph algorithm and the parallel diagonal dominant pdd algorithm are proposed for solving tridiagonal linear systems on multicomputers these algorithms are based on the divide and conquer parallel computation model the ppt and pph algorithms support both pivoting and nonpivoting the ppt algorithm is good when the number of processors is small otherwise the pph algorithm is better when the system is diagonal dominant the pdd algorithm is highly parallel and provides an approximate solution which equals the exact solution within machine accuracy computation and communication complexities of the three algorithms are presented all three methods have been implemented on a 64 node ncube 1 multicomputer the analytic results closely match the results measured from the ncube 1 machine

hamiltonian formalism for general relativistic adiabatic fluids abstract we derive the hamiltonian structures of three theories non relativistic special relativistic and general relativistic adiabatic fluids each in the eulerian representation in riemannian space or lorentzian spacetime all by the same procedure using standard variational principles the evolution in each case is generated by a hamiltonian that is equivalent to that obtained from a canonical analysis for the gravitational variables the poisson bracket has the usual canonical symplectic structure however for the fluid variables the three theories all share the same lie poisson bracket when expressed in the appropriate spaces of physical variables constructed here this shared lie poisson bracket is associated to the dual of the semidirect product lie algebra of vector fields acting on differential forms an immediate consequence of this shared structure is that each of these theories possesses an infinite family of conservation laws the so called casimirs that belong to the kernel of the lie poisson bracket the role of these casimirs in the study of lyapunov stability or dynamic stability for fluid equilibria is discussed the relationship of this approach to other approaches in the literature is also discussed

the hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces the hyperbolic moduli space of flat connections and the isomorphism of symplectic multiplicity spaces

computing discrete fourier transform on a rectangular data array the authors generalize the weighted redundancy transform wrt algorithm for computing the multidimensional discrete fourier transform dft in the case in which the sample size blocklength is not the same on every axis the proposed algorithm like the wrt algorithm is based on the one dimensional fast fourier transform fft and compared to the traditional ways of computing the multidimensional dft offers substantial savings in the number of one dimensional fft procedure calls while the algorithm is applicable to transforms of any dimensions only the two dimensional case is explored in detail

lie algebroid foliations and script e1 m dirac structures we prove some general results about the relation between the 1 cocycles of an arbitrary lie algebroid a over m and the leaves of the lie algebroid foliation on m associated with a using these results we show that a 1 m dirac structure l induces on every leaf f of its characteristic foliation a 1 f dirac structure lf which comes from a precontact structure or from a locally conformal presymplectic structure on f in addition we prove that a dirac structure on m can be obtained from l and we discuss the relation between the leaves of the characteristic foliations of l and

architecture oriented regular algorithms for discrete sine and cosine transforms we propose fast algorithms for a class of trigonometric transforms these algorithms have a unified structure and simple data exchange similar to the constant geometry pease 1968 algorithm derived from the cooley tukey 1965 fft we can easily extend many of the parallel fft approaches for these algorithms the idea of the method is to localize the nonregularities peculiar to fast algorithms for other trigonometric transforms into the nodes of the cooley tukey fft type computational graph only the basic operation in the nodes of the computational graph will be different for different transforms thus a simple programmable processor element for executing the node functions can be the basis for parallel constructs

poisson structures and integrable systems a poisson coalgebra is used to construct integrable hamiltonian systems we consider a poisson structure given by the bivector which does not form a lie algebra with respect to the poisson bracket xa xb p pab x we prove that this coalgebra may be used to generate integrable hamiltonian systems as an example we give the poisson tensor and we show that it is linked with the calogero system

levi decomposition for smooth poisson structures we prove the existence of a local smooth levi decomposition for smooth poisson structures and lie algebroids near a singular point this levi decomposition is a kind of normal form or partial linearization which was established in the formal case by wade wad97 and in the analytic case by the second author zun03 in particular in the case of smooth poisson structures with a compact semisimple linear part we recover conn s smooth linearization theorem con85 and in the case of smooth lie algebroids with a compact semisimple isotropy lie algebra our levi decomposition result gives a positive answer to a conjecture of weinstein wei00 on the smooth linearization of such lie algebroids in the appendix of this paper we show an abstract nashmoser normal form theorem which generalizes our levi decomposition result and which may be helpful in the study of other smooth normal form problems

fedosov products and quantum momentum maps the purpose of this paper is to study various aspects of star products on a symplectic manifold related to the fedosov method by introducing the notion of quantum exponential maps we give a characterization of fedosov connections as an application a geometric realization is obtained for the equivalence between an arbitrary product and a fedosov one every fedosov product is shown to be a vey product consequently we find that every product is equivalent to a vey product a classical result of lichnerowicz quantization of a hamiltonian g space and in particular quantum momentum maps are studied lagrangian submanifolds are also studied under a deformation quantization

singular solutions of a modified two component camassa holm equation the camassa holm ch equation is a well known integrable equation describing the velocity dynamics of shallow water waves this equation exhibits spontaneous emergence of singular solutions peakons from smooth initial conditions the ch equation has been recently extended to a two component integrable system ch2 which includes both velocity and density variables in the dynamics although possessing peakon solutions in the velocity the ch2 equation does not admit singular solutions in the density profile we modify the ch2 system to allow a dependence on the average density as well as the pointwise density the modified ch2 system mch2 does admit peakon solutions in the velocity and average density we analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data numerical results for the mch2 system are given and compared with the pure ch2 case these numerics show that the modification in the mch2 system to introduce the average density has little short time effect on the emergent dynamical properties however an analytical and numerical study of pairwise peakon interactions for the mch2 system shows a different asymptotic feature namely besides the expected soliton scattering behavior seen in overtaking and head on peakon collisions the mch2 system also allows the phase shift of the peakon collision to diverge in certain parameter regimes

a recursive doubling algorithm for solution of tridiagonal systems on hypercube multiprocessors abstract the recursive doubling algorithm as developed by stone can be used to solve a tridiagonal linear system of size n on a parallel computer with n processors using o log n parallel arithmetic steps in this paper we give a limited processor version of the recursive doubling algorithm for the solution of tridiagonal linear systems using o n p log p parallel arithmetic steps on a parallel computer with p n processors the main technique relies on fast parallel prefix algorithms which can be efficiently mapped on the hypercube architecture using the binary reflected gray code for p n this algorithm achieves linear speedup and constant efficiency over its sequential implementation as well as over the sequential lu decomposition algorithm these results are confirmed by numerical experiments obtained on an intel ipsc d5 hypercube multiprocessor

self averaging distribution of pseudocritical temperatures and finite size scaling in critical disordered systems we evaluate by monte carlo simulations various singular thermodynamic quantities x for ensembles of quenched random ising and ashkin teller models the measurements are taken at t c and we study how the distributions p x and in particular their relative squared width r x over the ensemble depend on the system size l the ashkin teller model was studied in the regime where bond randomness is irrelevant and we found weak self averaging r x ensuremath sim l ensuremath alpha ensuremath nu ensuremath rightarrow 0 where ensuremath alpha l0 and ensuremath nu are the exponents of the pure model fixed point governing the transition for the site dilute ising model on a cubic lattice known to be governed by a random fixed point we find that r x tends to a constant as predicted by aharony and harris we tested whether this constant is universal however this constant is different for canonical and grand canonical disorder we identify the pseudocritical temperature of each sample i t c i l as the temperature at which the susceptibility reaches its maximal value ensuremath chi mathrm max the distribution of these sample dependent t c i l was investigated we found that its variance scales as ensuremath delta t c l 2 ensuremath sim l ensuremath 2 ensuremath nu our previously proposed finite size scaling ansatz for disordered systems was tested and found to hold we did observe deviations from a single function which imply that sample dependent scaling functions are needed these deviations are however relatively small and hence to obtain a fixed statistical error it may be more computationally efficient to measure ensuremath chi mathrm max than the commonly used ensuremath chi t c ensuremath infty

on a camassa holm type equation with two dependent variables we consider a generalization of the camassa holm ch equation with two dependent variables called ch2 introduced in a paper by liu and zhang liu s q and zhang y 2005 j geom phys 54 427 53 we briefly provide an alternative derivation of it based on the theory of hamiltonian structures on the dual of a lie algebra the lie algebra involved here is the same algebra as underlies the nls hierarchy we study the structural properties of the hierarchy defined by the ch2 equation within the bi hamiltonian theory of integrable pdes and provide its lax representation then we explicitly discuss how to construct classes of solutions both of peakon and of algebro geometrical type finally we sketch the construction of a class of singular solutions defined by setting to zero one of the two dependent variables

inverse jacobi multiplier as a link between conservative systems and poisson structures some aspects of the relationship between conservativeness of a dynamical system namely the preservation of a finite measure and the existence of a poisson structure for that system are analyzed from the local point of view due to the flow box theorem we restrict ourselves to neighborhoods of singularities in this sense we characterize poisson structures around the typical zero hopf singularity in dimension 3 under the assumption of having a local analytic first integral with non vanishing first jet by connecting with the classical poincar e center problem from the global point of view we connect the property of being strictly conservative the invariant measure must be positive with the existence of a poisson structure depending on the phase space dimension finally weak conservativeness in dimension two is introduced by the extension of inverse jacobi multipliers as weak solutions of its defining partial differential equation and some of its applications are developed examples including lotka volterra systems quadratic isochronous centers and non smooth oscillators are provided

weak geodesic flow on a semidirect product and global solutions to the periodic hunter saxton system abstract we give explicit solutions for the two component hunter saxton system on the unit circle moreover we show how global weak solutions can be naturally constructed using the geometric interpretation of this system as a re expression of the geodesic flow on the semidirect product of a suitable subgroup of the diffeomorphism group of the circle with the space of smooth functions on the circle these spatially and temporally periodic solutions turn out to be conservative

on the geometry of lie algebras and poisson tensors a geometric programme to analyse the structure of lie algebras is presented with special emphasis on the geometry of linear poisson tensors the notion of decomposable poisson tensors is introduced and an algorithm to construct all solvable lie algebras is presented poisson liouville structures are also introduced to discuss a new class of lie algebras which include as a subclass semi simple lie algebras a decomposition theorem for poisson tensors is proved for a class of poisson manifolds including linear ones simple lie algebras are also discussed from this viewpoint and lower dimensional real lie algebras are analysed

new solutions of the jacobi equations for three dimensional poisson structures a systematic investigation of the skew symmetric solutions of the three dimensional jacobi equations is presented as a result three disjoint and complementary new families of solutions are characterized such families are very general thus unifying many different and well known poisson structures seemingly unrelated which now appear embraced as particular cases of a more general solution this unification is not only conceptual but allows the development of algorithms for the explicit determination of important properties such as the symplectic structure the casimir invariants and the darboux canonical form which are known only for a limited sample of poisson structures these common procedures are thus simultaneously valid for all the particular cases which can now be analyzed in a unified and more economic framework instead of using a case by case approach in addition the methods developed are valid globally in phase space thus ameliorating the usual scope of darboux reduction which is only of

lie algebroids holonomy and characteristic classes abstract we extend the notion of connection in order to study singular geometric structures namely we consider a notion of connection on a lie algebroid which is a natural extension of the usual concept of a covariant connection it allows us to define holonomy of the orbit foliation of a lie algebroid and prove a stability theorem we also introduce secondary or exotic characteristic classes thus providing invariants which generalize the modular class of a lie algebroid

non trivial exponents in the zero temperature dynamics of the 1d ising and potts models we consider the glauber dynamics of the q state potts model in one dimension at zero temperature starting with a random initial configuration we measure the density r t of spins which have never flipped from the beginning of the simulation until time t we find that for large t the density r t has a power law decay left r t sim t theta right where the exponent theta varies with q our simulations lead to theta simeq 37 for q 2 theta simeq 53 for q 3 and theta longrightarrow 1 as q longrightarrow infty

hamiltonian structure of dynamical systems which solve linear programming problems abstract a new class of completely integrable hamiltonian systems which solve linear programming problems is introduced the relations with reduced toda flows double bracket equations interior point methods are established an interpretation in terms of total actions on kahler manifolds is given a complete description of the phase portrait including the explicit description of action angle variables is provided

reduction of some classical non holonomic systems with symmetry 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

persistence in an antiferromagnetic ising system with conserved magnetisation abstract we obtain the persistence exponents for an antiferromagnetic ising system in which the magnetisation is kept constant this system belongs to model c system with non conserved order parameter with a conserved density and is expected to have persistence exponents different from that of model a system with no conservation but independent of the conserved density our numerical results for both local persistence at zero temperature and global persistence at the critical temperature however indicate that the exponents are dependent on the conserved magnetisation in both two and three dimensions this non universal feature is attributed to the presence of the conserved field and is special to the persistence phenomena

reduction of constrained mechanical systems and stability of relative equilibria a mechanical system with perfect constraints can be described under some mild assumptions as a constrained hamiltonian system m h d w m thephase space is a symplectic manifold h thehamiltonian a smooth function onm d theconstraint submanifold a submanifold ofm andw theprojection bundle a vector sub bundle oft d m the reduced tangent bundle alongd we prove that when these data satisfy some suitable conditions the time evolution of the system is governed by a well defined differential equation ond we define constrained hamiltonian systems with symmetry and prove a reduction theorem application of that theorem is illustrated on the example of a convex heavy body rolling without slipping on a horizontal plane two other simple examples show that constrained mechanical systems with symmetry may have an attractive or repulsive set of relative equilibria

poisson brackets and clebsch representations for magnetohydrodynamics multifluid plasmas and elasticity abstract poisson brackets are constructed by the same mathematical procedure for three physical theories ideal magnetohydrodynamics multifluid plasmas and elasticity each of these brackets is given a simple lie algebraic interpretation moreover each bracket is induced to physical space by use of a canonical poisson bracket in the space of clebsch potentials which are constructed for each physical theory by the standard procedure of constrained lagrangians

recursion operators and bi hamiltonian formulations of the landau lifshitz equation hierarchies using a polynomial pencil of lax pairs we give a new bi hamiltonian formulation for the landau lifshitz equation hierarchy and find a new recursion operator related to it

perturbation theory for evaluation algorithms of arithmetic expressions the paper presents the theoretical foundation of a forward error analysis of numerical algorithms under data perturbations rounding error in arithmetic floating point operations and approximations in built in functions the error analysis is based on the linearization method that has been proposed by many authors in various forms fundamental tools of the forward error analysis are systems of linear absolute and relative a priori and a posteriori error equations and associated condition numbers constituting optimal bounds of possible accumulated or total errors derivations representations and properties of these condition numbers are studied in detail the condition numbers enable simple general quantitative definitions of numerical stability backward analysis welland ill conditioning of a problem and an algorithm the well known illustration of algorithms and their linear error equations by graphs is extended to a method of deriving condition numbers and associated bounds for many algorithms the associated condition numbers can be determined analytically a priori and be computed numerically a posteriori the theoretical results of the paper have been applied to a series of concrete algorithms including gaussian elimination and have proved to be very effective means of both a priori and a posteriori error analysis introduction evaluation algorithms are defined by finite sequences f fo f of input operations evaluations of built in functions and arithmetic operations for determining sequences u uo u of input data intermediate and final results in the form 1 ut f u t 0 n it is presupposed in the following that fo is a constant function and the function values ft x depend on xo xt1 but not on xt x for t 1 i n under perturbations an evaluation algorithm yields approximations vt of ut that can be written in the form 2 vt 1 e ft v t o n the so called local errors et are the relative errors of the data input function evaluation or rounding in the arithmetic floating point operation in step t of the algorithm we shall assume that the local errors are bounded by je1j yt7 for t o n where yt are suitable nonnegative weights and q is an accuracy constant received june 12 1979 revised november 26 1979 and august 27 1980 1980 mathematics subject classification primary 65g05

quadratic poisson structures an algebraic characterization of homogeneous quadratic poisson structures is obtained and some methods to construct examples are given a sufficient condition for the commutativity of a homogeneous quadratic poisson structure and the lie poisson structure on the dual of a lie algebra is given quadratic poisson structures on lie algebras are discussed a new poisson structure is introduced and its application is shown

hamiltonian systems admitting a runge lenz vector and an optimal extension of bertrand s theorem to curved manifolds bertrand s theorem asserts that any spherically symmetric natural hamiltonian system in euclidean 3 space which possesses stable circular orbits and whose bounded trajectories are all periodic is either a harmonic oscillator or a kepler system in this paper we extend this classical result to curved spaces by proving that any hamiltonian on a spherically symmetric riemannian 3 manifold which satisfies the same conditions as in bertrand s theorem is superintegrable and given by an intrinsic oscillator or kepler system as a byproduct we obtain a wide panoply of new superintegrable hamiltonian systems the demonstration relies on perlick s classification of bertrand spacetimes and on the construction of a suitable globally defined generalization of the runge lenz vector

on blow up of solutions to the two component π camassa holm system abstract we investigate cauchy problem of the two component camassa holm system which arises from geodesic flow on a semidirect product lie group of the circle the precise blow up scenarios of strong solutions are derived for the system then several criteria to guarantee blow up of strong solutions are presented finally the exact blow up rate is determined