a geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities action symplecticity signature and complex instability are fundamental concepts in hamiltonian dynamics which can be characterized in terms of the symplectic structure in this paper hamiltonian pdes on unbounded domains are characterized in terms of a multisymplectic structure where a distinct differential two form is assigned to each space direction and time this leads to a new geometric formulation of the conservation of wave action for linear and nonlinear hamiltonian pdes and via stokes s theorem a conservation law for symplecticity each symplectic structure is used to define a signature invariant on the eigenspace of a normal mode the first invariant in this family is classical krein signature or energy sign when the energy is time independent and the other spatial signatures are energy flux signs leading to a classification of instabilities that includes information about directional spatial spreading of an instability the theory is applied to several examples the boussinesq equation the water wave equations linearized about an arbitrary stokes s wave rotating shallow water flow and flow past a compliant surface some implications for non conservative systems are also discussed

numerical methods for the discretization of random fields by means of the karhunen loeve expansion the computational efficiency of random field representations with the karhunen loeve kl expansion relies on the solution of a fredholm integral eigenvalue problem this contribution compares different methods that solve this problem focus is put on methods that apply to arbitrary shaped domains and arbitrary autocovariance functions these include the nystrom method as well as collocation and galerkin projection methods among the galerkin methods we investigate the finite element method fem and propose the application of the finite cell method fcm this method is based on an extension to the fem but avoids mesh generation on domains of complex geometric shape the fcm was originally presented in parvizian et al 2007 17 for the solution of elliptic boundary value problems as an alternative to the l2l2 projection of the covariance function used in the galerkin method h1 2h1 2 projection and discrete projection are investigated it is shown that the expansion optimal linear estimation eole method proposed in li and der kiureghian 1993 18 constitutes a special case of the nystrom method it is found that the eole method is most efficient for the numerical solution of the kl expansion the fem and the fcm are more efficient than the eole method in evaluating a realization of the random field and therefore are suitable for problems in which the time spent in the evaluation of random field realizations has a major contribution to the overall runtime e g in finite element reliability analysis

analytic classification of a class of cuspidal foliations in c3 0 abstract in this article we study the analytic classification of certain types of quasi homogeneous cuspidal holomorphic foliations in c 3 0 via the essential holonomy defined over one of the components of the exceptional divisor that appears in the reduction of the singularities of the foliation

multisymplectic formulation of fluid dynamics using the inverse map we construct multisymplectic formulations of fluid dynamics using the inverse of the lagrangian path map this inverse map the back to labels map gives the initial lagrangian label of the fluid particle that currently occupies each eulerian position explicitly enforcing the condition that the fluid particles carry their labels with the flow in hamilton s principle leads to our multisymplectic formulation we use the multisymplectic one form to obtain conservation laws for energy momentum and an infinite set of conservation laws arising from the particle relabelling symmetry and leading to kelvin s circulation theorem we discuss how multisymplectic numerical integrators naturally arise in this approach

the stochastic finite element method past present and future a powerful tool in computational stochastic mechanics is the stochastic finite element method sfem sfem is an extension of the classical deterministic fe approach to the stochastic framework i e to the solution of static and dynamic problems with stochastic mechanical geometric and or loading properties the considerable attention that sfem received over the last decade can be mainly attributed to the spectacular growth of computing power rendering possible the efficient treatment of large scale problems this article aims at providing a state of the art review of past and recent developments in the sfem area and indicating future directions as well as some open issues to be examined by the computational mechanics community in the future

analytical and formal classifications of quasi homogeneous foliations in c2 0 we prove a result of classification for germs of formal and convergent quasi homogeneous foliations in c 2 with fixed separatrix basically we prove that the analytical and formal class of such a foliation depend respectively only on the analytical and formal class of its representation of projective holonomy

on the rate of convergence of optimal solutions of monte carlo approximations of stochastic programs in this paper we discuss monte carlo simulation based approximations of a stochastic programming problem we show that if the corresponding random functions are convex piecewise linear and the distribution is discrete then an optimal solution of the approximating problem provides an exact optimal solution of the true problem with probability one for sufficiently large sample size moreover by using the theory of large deviations we show that the probability of such an event approaches one exponentially fast with increase of the sample size in particular this happens in the case of linear two or multi stage stochastic programming with recourse if the corresponding distributions are discrete the obtained results suggest that in such cases monte carlo simulation based methods could be very efficient we present some numerical examples to illustrate the ideas involved

weak existence for the semigeostrophic equations formulated as a coupled monge ampere transport problem hoskins s semigeostrophic equations are reformulated as a coupled monge ampere transport problem b j hoskins quart j royal met soc 97 1971 pp 139 153 existence of global weak solutions is obtained for this formulation

topological invariants and equidesingularization for holomorphic vector fields on etudie l invariance du nombre de milnor et ses applications on demontre un theoreme d equidesingularisation

growth at infinity and index of polynomial maps abstract let f k n k n be a polynomial map such that f 1 0 is compact where k r or c then we give a condition implying that there is a uniform bound for the ojasiewicz exponent at infinity in certain deformations of f this fact gives a result about the invariance of the global index of f

multi symplectic formulation of near local hamiltonian balanced models we transform near local hamiltonian balanced models hbms describing nearly geostrophic fluid motion with constant coriolis parameter into multi symplectic ms systems this allows us to determine conservation of lagrangian momentum energy and potential vorticity for salmon s l1 dynamics a similar approach works for other near local balanced models such as the model the ms approach also enables us to determine a class of systems that have a contact structure similar to that of the semigeostrophic model the contact structure yields a contact transformation that makes the problem of front formation tractable the new class includes the first local model with a variable coriolis parameter that preserves all of the most useful geometric features of the semigeostrophic model

a c2 continuous approximation to the mohr coulomb yield surface abstract in spite of the development of more sophisticated constitutive models for soil the mohr coulomb yield criterion remains a popular choice for geotechnical analysis due to its simplicity and ease of use by practising engineers the implementation of the criterion in finite element programs however presents some numerical difficulties due to the gradient discontinuities which occur at both the edges and the tip of the hexagonal yield surface pyramid furthermore some implicit techniques utilising consistent tangent stiffness formulations are unable to achieve full quadratic convergence as the yield criteria is not c2 continuous this paper extends the previous work of abbo and sloan 1995 through the introduction of c2 continuous rounding of the mohr coulomb yield surface in the octahedral plane this approximation when combined with the hyperbolic approximation in the meridional plane abbo and sloan 1995 describes a yield surface that is c2 continuous at all stress states the new smooth yield surface can be made to approximate the mohr coulomb yield function as closely as required by adjusting only two parameters and is suitable for consistent tangent stiffness formulations

hyper kahler geometry and semi geostrophic theory we use the formalism of monge ampere operators to study the geometric properties of the monge ampere equations arising in semi geostrophic sg theory and related models of geophysical fluid dynamics we show how kahler and hyper kahler structures arise and the legendre duality arising in sg theory is generalized to other models of nearly geostrophic flows

numerical methods for hamiltonian pdes the paper provides an introduction and survey of conservative discretization methods for hamiltonian partial differential equations the emphasis is on variational symplectic and multi symplectic methods the derivation of methods as well as some of their fundamental geometric properties are discussed basic principles are illustrated by means of examples from wave and fluid dynamics

multisymplectic conservation laws for differential and differential difference equations many well known partial differential equations can be written as multisymplectic systems such systems have a structural conservation law from which scalar conservation laws can be derived these c

vorticity and symplecticity in lagrangian fluid dynamics the relationship between potential vorticity pv and the symplectic form is explored for the shallow water equations governing lagrangian particle paths starting with the symplectic form the pv is found by the pull back operation to the reference space at first sight the encoding of pv in the symplectic form appears to be independent of the particle relabelling symmetry the analysis is carried a step further in two ways using the conservation of symplecticity as a starting point the fluxes of symplecticity arise as differential forms and a complete pull back of the flux forms leads to a geometric description of pv conservation secondly symmetry methods are used to give a rigorous connection between particle relabelling symplecticity and pv conservation generalizations of these issues to semi geostrophic flow and three dimensional lagrangian fluid flows and the connection with ertel s theorem are also discussed

efficient markov chain monte carlo for combined subset simulation and nonlinear finite element analysis abstract typical probabilistic problems in an engineering context include rare event probability estimation for physical models where spatial autocorrelation of material property parameters is significant subset simulation a markov chain monte carlo technique can be used to estimate rare event probabilities in physical models more efficiently than monte carlo simulation this efficiency gain is important when the sampling operation is computationally demanding as is the case in the solution of stochastic partial differential equations in high dimensional spaces where polynomial chaos or other direct integration techniques become intractable sampling methods may be the only way to compute integral functions in probabilistic analysis in this paper subset simulation is applied to probability of failure estimation in nonlinear elasto plastic finite element problems further a derivation of confidence intervals for subset simulation relative errors is presented this new technique allows for vastly improved efficiency in the computation of error estimates for subset simulation significantly the numerical studies presented indicate that for the tested finite element problems metropolis hastings sampling can outperform componentwise metropolis hastings and gibbs sampling this result is relevant to the design of efficient subset simulation methodologies