test fields on compact space times in this paper some basic aspects of lorentzian field theory on compact lorentz manifolds are studied all compact space times are acausal i e possess closed timelike curves this makes them a useful testbed in analyzing some new notions of causality that will be introduced for more general acausal space times in addition studying compact space times in their own right raises a wide range of fascinating mathematical problems some of which will be explored it will be shown that it is reasonable to expect lorentzian field theory on a compact space time to provide information on the topology of the underlying manifold if this is true then this information is likely to be orthogonal or complementary to the information obtained through the study of euclidean field theory

gamma convergence of gradient flows with applications to ginzburg landau we present a method to prove convergence of gradient flows of families of energies that converge to a limiting energy it provides lower bound criteria to obtain the convergence that correspond to a sort of c1 order convergence of functionals we then apply this method to establish the limiting dynamical law of a finite number of vortices for the heat flow of the ginzburg landau energy in dimension 2 retrieving in a different way the existing results for the case without magnetic field and obtaining new results for the case with magnetic field 2004 wiley periodicals inc

collisions and phase vortex interactions in dissipative ginzburg landau dynamics in this paper we describe a natural framework for the vortex dynamics in the parabolic complex ginzburg landau equation in r this general setting does not rely on any assumption of well preparedness and has the advantage to be valid even after collision times we analyze carefully collisions leading to annihilation a new phenomenon is identified the phase vortex interaction related to persistence of low frequency oscillations and leading to an unexpected drift in the motion of vortices

limiting behavior of the ginzburg landau functional we continue our study of the functional ee u u12 u 2 14e2 1 u 2 2dx for u h1 u r2 where u is a bounded open subset of r2 compactness results for the scaled jacobian of ue are proved under the assumption that ee ue is bounded uniformly by a function of e in addition the gamma limit of ee ue ln e 2 is shown to be e v 12 v 22 v m where v is the limit of j ue ln e j ue ue due and m is the total variation of a radon measure these results are applied to the ginzburg landau functionalfe u a hext u12 au 2 14e2 1 u 2 2 12 a hext dx with external magnetic field hext h ln e the gamma limit of fe ln e 2 is calculated to bef v a h 12 v a 22 v m a h 22 where v is as before and a is the limit of ae ln e

gamma limit of the nonself dual chern simons higgs energy abstract we consider the gamma limit of the abelian chern simons higgs energy g csh 1 2 u a e u e 2 2 4 curl a e h ex 2 u e 2 1 e 2 u e 2 1 u e 2 2 d x on a bounded simply connected two dimensional domain under the e 0 limit as a first step we study the gamma limit of e csh 1 2 u u e 2 1 e 2 u e 2 1 u e 2 2 d x under two different scalings e csh log e and e csh log e 2 we apply the log e 2 scaling result to the full chern simons higgs energy g csh and as a consequence we are able to compute the first critical field h 1 h 1 u for the nucleation of a vortex the method entails estimating in certain weak topologies the jacobian j u e det u e in terms of the chern simons higgs energy e csh

a product estimate for ginzburg landau and corollaries abstract we prove a new inequality for the jacobian or vorticity associated to the ginzburg landau energy in any dimension it allows to retrieve existing lower bounds on the energy to extend them to the case of unbounded vorticity and to get a few other corollaries it also provides a new estimate on the time variation for time dependent families which has applications for the study of ginzburg landau dynamics

geometric quantization and the bogoliubov transformation this paper treats some simple applications of geometric quantization theory to problems involving the interaction of quantum fields with external classical potentials the main new result is a representation of the transition amplitudes for a boson field in terms of functional integrals over the classical phase space the space of solutions of the classical field equations an analogous representation for fermion fields is also constructed

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

complex manifold methods in quantum field theory in curved space time a real analytic space time n can be embedded as a real slice in a complex riemannian manifold m g a theorem due to woodhouse int j theor phys 16 671 1977 stating that real slices are necessarily totally geodesic submanifolds of m g is used to find all real slices of the complexified sphere csnr of dimension n and radius r and of complexified robertson walker space time cr the real slices n of csnr are n dimensional spaces of constant curvature r 2 of all possible signatures cr is determined by a holomorphic radius function r x the x constant hypersurfaces being isometric to csnr x in general all real slices of cr are robertson walker type spaces of various signatures the interesting case that a robertson walker space time with radius function r t intersects a euclidean real slice in an n dimensional hypersurface can only occur if r t is time reflection symmetric finally propagators to the scalar wave operator m2 on real slices n1 of cs4r are considered and their anal

hodge decomposition with degenerate weights and the gross pitaevskii energy abstract we solve a hodge decomposition problem with respect to a weight function that vanishes on the boundary this problem is present in several works on the gross pitaevskii energy and we use our solution to show that a convergence result that links these works is still valid for more general data than previously considered

a simple proof of geodesical completeness for compact space times of zero curvature any riemannian metric on a compact manifold is geodesically complete by contrast it is widely known that incomplete compact lorentzian manifolds exist in this article it will be proven that a compact manifold with a smooth lorentz metric must be complete if the metric is flat everywhere this question of completeness which has significant implications for the classification of compact lorentzian space forms has remained unresolved until quite recently when it was finally established by carriere as a special case of an even more general result here a simple alternative proof of this special case of carriere s theorem is given the proof requires minimal mathematical machinery but still involves the use of some intriguing connections between the topology and global geometry of a compact flat space time

numerical analyses of operator splitting algorithms for the two dimensional advection diffusion equation abstract holly and preissmann s hp scheme has been the basis of a large number of operator splitting algorithms for the solution of the advection diffusion equation however these algorithms including hp are first order accurate in time due to splitting errors error analyses of these algorithms incorporating splitting error and errors resulting from numerical solutions of the split advection and diffusion equations are lacking in this paper error analysis of a second order accurate adaptation of the hp scheme ahp is presented for the two dimensional advection diffusion equation a modified ahp scheme mahp is suggested to remove the ad hoc nature of boundary conditions during the diffusion step of computations in both hp and ahp as boundary conditions specified for the advection diffusion equation are not applicable to the split equations second order accurate boundary conditions for the split advection and diffusion equations are derived an analysis of numerical dispersion and dissipation associated with the numerical procedure for the advection equation is presented the analysis establishes a criterion so that computational errors are small in two dimensional advection dominated transport problems several numerical examples are presented to verify the numerical analyses presented in the paper in addition a review of the current status of operator splitting algorithms for the advection diffusion equation is presented the objective of the review is to identify the issues that have not been addressed in the previous studies

structure of lorentzian tori with a killing vector field all lorentzian tori with a non discrete group of isometries are characterized and explicitly obtained they can lie into three cases a flat b conformally flat but non flat and c geodesically incomplete a detailed study of many of their properties including results on the logical dependence of the three kinds of causal completeness on geodesic connectedness and on prescribed curvature is carried out the incomplete case is specially analyzed and several known examples and results in the literature are generalized from a unified point of view

flat compact hermite lorentz manifolds in dimension 4 we give a classification up to finite cover of flat compact complete hermite lorentz manifolds up to complex dimension 4

asymptotics for the ginzburg landau equation in arbitrary dimensions let be a bounded simply connected regular domain of rn n 2 for 0 e 1 let ue c be a smooth solution of the ginzburg landau equation in with dirichlet boundary condition ge i e formula we are interested in the asymptotic behavior of ue as e goes to zero under the assumption that ee ue m0 log e and some conditions on ge which allow singularities of dimension n 3 on

three dimensional affine crystallographic groups those groups r which act properly discontinuously and aillnely on ii with compact fundamental domain are classified first it is shown that such a group f contains a solvable subgroup of finite index thus establishing a conjecture of auslander in dimension three then unimodular simply transitive altine actions on ir are classified this leads to a classification of attine crystallographic groups acting on ir3 a characterization of which abstract groups admit such an action is given moreover it is proved that every isomorphism between virtually solvable atline crystallographic groups respectively simply transitive afline groups is induced by conjugation by a polynomial automorphism of the affrne space a characterization is given of which closed 3 manifolds can be represented as quotients of ir by groups of afftne transformations a closed 3 manifold m admits a complete atline structure if and only if m has a finite covering homeomorphic or homotopy equivaient to a 2 torus bundle over the circle

an operator splitting algorithm for coupled one dimensional advection diffusion reaction equations abstract an operator splitting algorithm for a system of one dimensional advection diffusion reaction equations describing the transport of non conservative pollutants is presented in this paper the algorithm is a strang type splitting procedure incorporating contributions from the inhomogeneous terms by the duhamel s principle the associated homogeneous equations are split into advection diffusion and reaction equations and solved by a backward method of characteristic a finite element method and an explicit runge kutta method respectively the boundary conditions applicable to the split equations are derived numerical analyses of the algorithm consisting of the stability the accuracy and the convergence of the solution procedure are presented the composite algorithm is second order accurate in time and space and conditionally stable the numerical characteristics of the algorithm are demonstrated by several examples

new properties and examples of incomplete lorentzian tori a new procedure to construct incomplete metrics on a two torus is introduced relations between curvature and completeness are explored and examples showing a wide independence among them are given incompleteness appears as a usual property of the lorentzian tori and the density of the incomplete metrics in the set of all the lorentzian ones is discussed this article can be seen as a partial study of compact space times and the conformal lorentzian moduli space mod t2 as claimed by yurtsever j math phys 31 3064 1990 and as a continuation of the authors previous article j math phys 34 3768 1993

canonical variation of a lorentzian metric abstract given a lorentzian manifold m g l and a timelike unitary vector field e we can construct the riemannian metric g r g l 2 being the metrically equivalent one form to e we relate the curvature of both metrics especially in the case of e being killing or closed and we use the relations obtained to give some results about m g l

rectifiability of flat chains we prove without using federer s structure theorem that a finite mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0 dimensional slices are rectifiable this implies that every flat chain of finite mass and finite size is rectifiable it also leads to a simple necessary and sufficient condition on the coefficient group in order for every finite mass flat

ginzburg landau functionals and renormalized energy a revised γ convergence approach abstract we give short and self contained proofs of convergence results for ginzburg landau energy functionals in two dimensions in the logarithmic energetic regime in particular we derive the renormalized energy by convergence

duality and conformal structure in four dimensions two metrics that are conformally related define the same hodge dual operator on the space of two forms the converse namely that two metrics that have the same hodge dual are conformally related is established this is true for metrics of arbitrary nondegenerate signature for euclidean signature a stronger result namely that the conformal class of the metric is completely determined by choosing a dual operator on two forms satisfying certain conditions is proved

eulerian lagrangian solution of the vertically averaged groundwater transport equation eulerian lagrangian methods elms have been developed to reduce smearing and oscillations in numerically generated advection dominated transport solutions improvements in front propagation characteristics are gained but new sources of errors are introduced in the tracking step of elms the eulerian lagrangian least squares collocation method ellesco is applied to the vertically averaged transport equation and the ellesco method is used to demonstrate how different types of discretization errors affect elm solutions the accuracy of elms is partially dependent on the accuracy of a tracking step the tracking integration can be made as accurate as desired by dynamically adjusting the number of second order runge kutta integretion steps per finite difference transport step however the accuracy of the final position is dependent on accurate velocities inaccurate velocities can lead to poorly located concentration fronts and mass balance errors although ellesco can extend the spatial and temporal discretization limits of classical methods concentration fronts must be spread over one and a half elements to avoid oscillations in the solution time truncation errors associated with concentration dependent source terms lead to mass balance errors and source terms should be computed with the crank nicolson approximation to avoid excessive truncation error the size of the finite difference transport time step and the magnitude of the local velocity determine the distance between the head and foot of the particle back tracks this distance referred to as the lagrangian spatial resolution must be less than the characteristic length of areal sources or distortions will occur in the concentration front the limit on the lagrangian spatial resolution is an effective limit on the transport finite difference time step size

a higher order numerical scheme for scalar transport abstract the computational principles of a numerical scheme for the solution of the two dimensional scalar transport equation are presented the scheme is designed for use in transient flow situations where accurate simulation of the advective process is important advective transport is computed by the method of characteristics in which the scalar field is represented by a hermitian polynomial complete through the third degree in both coordinate directions while diffusion is computed by central differencing the superior accuracy of the new method is demonstrated by analysing its propagation characteristics and by comparing its performance on standard test problems with that of some well known lower order methods finally the method s applicability is demonstrated in several examples involving tracer releases into channel flows where possible the results of these simulations are compared with analytical solutions

lower bounds for the energy of unit vector fields and applications we prove lower bounds for the dirichlet energy of a unit vector field defined in a perforated domain of r2with nonzero degree on the outer boundary in terms of the total diameter of the holes we use this to derive lower bounds and then compactness results for sequences ue of minimizers or almost minimizers of ginzburg landau functionals with coupling constant 1 e2tending to

the fundamental group of a compact flat lorentz space form is virtually polycyclic a flat lorentz space form is a geodesically complete lorentzian manifold of zero curvature it is well known see auslander markus 3 that such a space m may be represented as a quotient r where r is an dimensional minkowski space n equals the dimension of m and is a group of lorentz isometries acting properly discontinuously and freely on r in particular the universal covering of m is isometric to r and the fundamental group m is isomorphic to theorem let m be a compact flat lorentz space form then m is virtually poly cyclic recall that a group is virtually polycyclic if it can be built by iterated extensions from finitely many finite groups and cyclic groups this result affirms a conjecture of milnor 13 in a special case for discussion of this conjecture as well as another special case we refer to fried goldman 8 the importance of this result is that it reduces the classification of compact flat lorentz space forms to fairly elementary problems concerning lie algebras and lattices in solvable lie groups we hope to pursue this classification in a future publication for a description of this reduction and the classification in dimension 3 see fried goldman 8 in dimension 4 the classification is worked out in fried 7 one immediate consequence of the structure theory developed in fried goldman 8 1 and kamishima 12 is the following corollary let m be the compact flat lorentz space form then m has a finite covering which is diffeomorphic to a solvmanifold the outline of this paper is as follows in the first section we collect some basic facts about the group e n 1 1 of isometries of dimensional minkowski space in the second section we prove the theorem in the special

γ convergence analysis of systems of edge dislocations the self energy regime this paper deals with the elastic energy induced by systems of straight edge dislocations in the framework of linearized plane elasticity the dislocations are introduced as point topological defects of the displacement gradient fields following the core radius approach we introduce a parameter varepsilon 0 representing the lattice spacing of the crystal we remove a disc of radius varepsilon around each dislocation and compute the elastic energy stored outside the union of such discs namely outside the core region then we analyze the asymptotic behaviour of the elastic energy as varepsilon rightarrow 0 in terms of convergence we focus on the self energy regime of order log frac 1 varepsilon we show that configurations with logarithmic diverging energy converge up to a subsequence to a finite number of multiple dislocations and we compute the corresponding limit

ginzburg landau vortex dynamics driven by an applied boundary current in this paper we study the time dependent ginzburg landau equations on a smooth bounded domain c 2 subject to an electrical current applied on the boundary the dynamics with an applied current are nondissipative but via the identification of a special structure in an interaction energy we are able to derive a precise upper bound for the energy growth we then turn to the study of the dynamics of the vortices of the solutions in the limit e 0 we first consider the original time scale in which the vortices do not move and the solutions undergo a phase relaxation then we study an accelerated time scale in which the vortices move according to a derived dynamical law in the dynamical law we identify a novel lorentz force term induced by the applied boundary current 2010 wiley periodicals inc

a berger green type inequality for compact lorentzian manifolds we give a lorentzian metric on the null congruence associated with a timelike conformal vector field a liouville type theorem is proved and a boundedness for the volume of the null congruence analogous to a well known berger green theorem in the riemannian case will be derived by studying conjugate points along null geodesics as a consequence several classification results on certain compact lorentzian manifolds without conjugate points on its null geodesics are obtained finally several properties of null geodesics of a natural lorentzian metric on each odd dimensional sphere have been found

metastability and dynamics of discrete topological singularities in two dimensions a γ convergence approach this paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions based on convergence we consider discrete systems described by scalar functions defined on a square lattice and governed by periodic interaction potentials our main motivation comes from xy spin systems described by the phase parameter and screw dislocations described by the displacement function for these systems we introduce a discrete notion of vorticity as the lattice spacing tends to zero we derive the first order limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices as a byproduct of this analysis we show that such systems exhibit increasingly many metastable configurations of singularities therefore we propose a variational approach to the depinning and dynamics of discrete vortices based on minimizing movements we show that letting first the lattice spacing and then the time step of the minimizing movements tend to zero the vortices move according with the gradient flow of the renormalized energy as in the continuous ginzburg landau framework

an investigation of eulerian lagrangian methods for solving heterogeneous advection dominated transport problems numerical simulation of solute transport in heterogeneous porous media is greatly complicated by the large velocity and concentration gradients induced by spatial variations in hydraulic conductivity eulerian lagrangian methods for solving the transport equation can give accurate solutions to heterogeneous problems if their interpolation algorithms are properly selected this paper compares the performance of four eulerian lagrangian solvers that rely on linear quadratic cubic spline and taut spline interpolators in each case a tensor product decomposition is used to reduce the general n dimensional interpolation problem to a sequence of n one dimensional problems comparisons of a set of test problems indicate that the linear and taut spline interpolators are dispersive while the quadratic and cubic spline interpolators are oscillatory the cubic and taut spline interpolators give consistently better accuracy than the more conventional linear and quadratic alternatives simulation experiments in two and three dimensional heterogeneous media indicate that the taut spline interpolator which is applied here for the first time to a solute transport problem is able to yield accurate essentially nonoscillatory solutions for high grid peclet numbers the cubic spline interpolator requires significantly less computational effort to achieve performance comparable to the other methods