cosmological post newtonian expansions to arbitrary order we prove the existence of a large class of one parameter families of solutions to the einstein euler equations that depend on the singular parameter epsilon v t c 0 epsilon epsilon 0 where c is the speed of light and v t is a typical speed of the gravitating fluid these solutions are shown to exist on a common spacetime slab m cong 0 t times mathbb t 3 and converge as epsilon searrow 0 to a solution of the cosmological poisson euler equations of newtonian gravity moreover we establish that these solutions can be expanded in the parameter epsilon to any specified order with expansion coefficients that satisfy epsilon independent nonlocal symmetric hyperbolic equations

an electromagnetic field algorithm for 2d implicit plasma simulation abstract a new robust algorithm is presented for the implicit calculation of the electromagnetic fields used in the full particle and hybrid modeling of 2d simulation plasmas the algorithm allows for calculations at time steps t well in excess of the plasma period and for mesh scales far exceeding the debye length with electron inertial terms retained the implicit fields suppress the numerical instability associated with plasma waves still the at remain constrained by an electron courant limit the algorithm is considerably simpler than earlier implicit schemes and more complete in its treatment of field errors in its present form the algorithm is limited to plasmas moving and accelerating in a plane across a single component of magnetic field an extension to include all the field components is suggested however in accordance with the implicit moment method estimated electric and magnetic fields are obtained by solving maxwell s equations self consistently for a set of implicit sources estimated with the aid of an auxiliary set of lower fluid moment equations for component fluxes and density the fluid pressure terms are treated explicitly and the spatial differencing of the auxiliary moments is centered to facilitate the solution of the resultant field equations solution for the single magnetic field component is obtained by one elliptic equation inversion readily managed by a vectorized solver package a subsequent irrotational old e field correction is found to be crucial for the maintainence of anticipated quasi neutrality a concomitant rotational correction is needed for physical solutions in steep density gradient problems we show that both corrections can be obtained simultaneously by referencing the deviations between the true currents flowing and the currents predicted to flow in the plasma at the end of a cycle the current correction is shown to be equivalent to the first and usually sufficient step of an iterative procedure leading to an exact solution for the fields in addition we demonstrate that electrostatic solutions can be obtained from the implicit algorithm by setting the speed of light to very large multiples of its physical value comparisons are made with earlier moment and direct method approaches and the scheme is related to previous classical hybrid models demonstrative applications are discussed

a curvilinear fully implicit conservative electromagnetic pic algorithm in multiple dimensions we extend a recently proposed fully implicit pic algorithm for the vlasov darwin model in multiple dimensions chen and chacon 2015 1 to curvilinear geometry as in the cartesian case the approach is based on a potential formulation a and overcomes many difficulties of traditional semi implicit darwin pic algorithms conservation theorems for local charge and global energy are derived in curvilinear representation and then enforced discretely by a careful choice of the discretization of field and particle equations additionally the algorithm conserves canonical momentum in any ignorable direction and preserves the coulomb gauge a 0 exactly an asymptotically well posed fluid preconditioner allows efficient use of large cell sizes which are determined by accuracy considerations not stability and can be orders of magnitude larger than required in a standard explicit electromagnetic pic simulation we demonstrate the accuracy and efficiency properties of the algorithm with numerical experiments in mapped meshes in 1d 3v and 2d 3v

some unconditionally stable time stepping methods for the 3d maxwell s equations almost all the difficulties that arise in the numerical solution of maxwell s equations are due to material interfaces in case that their geometrical features are much smaller than a typical wavelength one would like to use small space steps with large time steps the first time stepping method which combines a very low cost per time step with unconditional stability was the adi fdtd method introduced in 1999 the present discussion starts with this method and with an even more recent crank nicolson based split step method with similar properties we then explore how these methods can be made even more efficient by combining them with techniques that increase their temporal accuracies

a boundary value problem for the stationary vlasov poisson equations the plane diode the stationary vlasov poisson boundary value problem in a spatially one dimensional domain is studied the equations describe the flow of electrons in a plane diode existence is proved when the boundary condition the cathode emission distribution is a bounded function which decays super linearly or a dirac mass uniqueness is proved for physically realistic boundary conditions which are decreasing functions of the velocity variable it is shown that uniqueness does not always hold for the dirac mass boundary conditions

jordan mappings of semiprime rings an additive mapping 8 of a ring r into a 2 torsion free ring r is called a jordan homomorphism if 6 ab z a 8 b 6 b a for all a be r a well known result of i n herstein 4 states that every jordan homomorphism onto a prime ring is either a homomorphism or an antihomomorphism suppose that r contains ideals u and v with null intersection let cp r u be a homomorphism and ii r v be an antihomomorphism a mapping 6 40 is a so called direct sum of mappings rp and 11 obviously b is a jordan homomorphism according to this construction we see that herstein s result 4 does not hold in semip me rings moreover w e baxter and w s martindale el showed by an example that a jordan homomorphism 0 of a ring r onto a semiprime ring r is not necessarily a direct sum of a homomorphism and an antihomomorphism but they proved that there always exists an essential ideal e of r such that the restriction of 8 to e is a direct sum of a homomorphism cp e r and an antihomomorphism y5 e r 1 theorem 2 71 in this paper theorem 2 3 we extend this result by showing that e can be choosen so that it is a sum of ideals u and v of r such that cp vanishes on v and tc vanishes on u even more for each x e r we have b ux e u e x for all u e u and b ux f x e o for all u e v this result also contains an affirmative answer to the question of baxter and martindale whether there is a way to choose the ideal e so that 8 e is an associative subring of r as we shall see in our case 6 e is the essential associative ideal of r an additive mapping 0 of a ring r into a ring r which satisfies b aba a 6 b 6 a for all a b e r will be called a jordan triple homomorphism an easy computation shows that every jordan homomorphism is also a

on local solutions of the initial value problem for the vlasov maxwell equation the initial value problem of the vlasov maxwell equation has a unique solution in a time interval 0 t for each initial data in some function space t is estimated by the size of the initial data the solution is classical if the initial data is smooth

jordan isomorphisms of upper triangular matrix rings abstract let r be a 2 torsionfree ring with identity 1 and let t n r n 2 be the ring of all upper triangular n n matrices over r we describe additive jordan isomorphisms of t n r onto an arbitrary ring and generalize several results on this line

divergence preservation in the adi algorithms for electromagnetics the recent advances in alternating direct implicit adi methods promise important new capability for time domain plasma simulations namely the elimination of numerical stability limits on the time step but the utility of these methods in simulations with charge and current sources such as in electromagnetic particle in cell empic computations has been uncertain as the methods introduced so far do not have the property of divergence preservation this property is related to charge conservation and self consistency and is critical for accurate and robust empic simulation this paper contains a complete study of these adi methods in the presence of charge and current sources it is shown that there are four significantly distinct cases with four more related by duality of those only one preserves divergence and thus is guaranteed to be stable in the presence of moving charged particles computational verification of this property is accomplished by implementation in existing 3d empic simulation software of the other three cases two are verified unstable as expected and one remains stable despite the lack of divergence preservation this other stable algorithm is shown to be related to the divergence preserving case by a similarity transformation effectively providing the complement of the divergence preserving field in the finite difference energy quantity

propagation of moments and regularity for the 3 dimensional vlasov poisson system we prove that if the initial data has moments inv higher than three then the solution of vlasov poisson has also moments inv higher than three we deduce from this different regularity results on the local density the force field or the solution itself also we give a new uniqueness result and new regularity results for solutions satisfying only the energy andl bounds our proofs are based on a new representation formula and logarithmic estimates for the force field

approximately bisimilar symbolic models for nonlinear control systems control systems are usually modeled by differential equations describing how physical phenomena can be influenced by certain control parameters or inputs although these models are very powerful when dealing with physical phenomena they are less suited to describe software and hardware interfacing with the physical world for this reason there is a growing interest in describing control systems through symbolic models that are abstract descriptions of the continuous dynamics where each symbol corresponds to an aggregate of states in the continuous model since these symbolic models are of the same nature of the models used in computer science to describe software and hardware they provide a unified language to study problems of control in which software and hardware interact with the physical world furthermore the use of symbolic models enables one to leverage techniques from supervisory control and algorithms from game theory for controller synthesis purposes in this paper we show that every incrementally globally asymptotically stable nonlinear control system is approximately equivalent bisimilar to a symbolic model the approximation error is a design parameter in the construction of the symbolic model and can be rendered as small as desired furthermore if the state space of the control system is bounded the obtained symbolic model is finite for digital control systems and under the stronger assumption of incremental input to state stability symbolic models can be constructed through a suitable quantization of the inputs

initial layers and zero relaxation limits of euler maxwell equations in this paper we consider zero relaxation limits for periodic smooth solutions of euler maxwell systems for well prepared initial data we propose an approximate solution based on a new asymptotic expansion up to any order for ill prepared initial data we construct initial layer corrections in an explicit way in both cases the asymptotic expansions are valid in time intervals independent of the relaxation time and their convergence is justified by establishing uniform energy estimates

high order unconditionally stable fc ad solvers for general smooth domains ii elliptic parabolic and hyperbolic pdes theoretical considerations a new pde solver was introduced recently in part i of this two paper sequence on the basis of two main concepts the well known alternating direction implicit adi approach on one hand and a certain fourier continuation fc method for the resolution of the gibbs phenomenon on the other unlike previous alternating direction methods of order higher than one which only deliver unconditional stability for rectangular domains the new high order fc ad fourier continuation alternating direction algorithm yields unconditional stability for general domains at an o nlog n cost per time step for an n point spatial discretization grid in the present contribution we provide an overall theoretical discussion of the fc ad approach and we extend the fc ad methodology to linear hyperbolic pdes in particular we study the convergence properties of the newly introduced fc gram fourier continuation method for both approximation of general functions and solution of the alternating direction odes we also present for parabolic pdes on general domains and thus for our associated elliptic solvers a stability criterion which when satisfied ensures unconditional stability of the fc ad algorithm use of this criterion in conjunction with numerical evaluation of a series of singular values of the alternating direction discrete one dimensional operators suggests clearly that the fifth order accurate class of parabolic and elliptic fc ad solvers we propose is indeed unconditionally stable for all smooth spatial domains and for arbitrarily fine discretizations to illustrate the fc ad methodology in the hyperbolic pde context finally we present an example concerning the wave equation demonstrating sixth order spatial and fourth order temporal accuracy as well as a complete absence of the debilitating dispersion error also known as pollution error that arises as finite difference and finite element solvers are applied to solution of wave propagation problems

the classical limit of the relativistic vlasov maxwell system solutions of the relativistic vlasov maxwell system of partial differential equations are considered in three space dimensions the speed of light c appears as a parameter in this system for smooth cauchy data classical solutions are shown to exist on a time interval that is independent ofc then using an integral representation for the electric and magnetic fields due to glassey and strauss 6 conditions are given under which solutions of the relativistic vlasov maxwell system converge in pointwise sense to solutions of the non relativistic vlasov poisson system at the asymptotic rate of 1 c asc tends to infinity

quasi lacunae of maxwell s equations classical lacunae in the solutions of hyperbolic differential equations and systems in the spaces of odd dimension are a manifestation of the huygens principle if the source terms are compactly supported in space and time then at any finite location in space the solution becomes identically zero after a finite interval of time in other words the propagating waves have sharp aft fronts for maxwell s equations though even if the currents that drive the field are compactly supported in time they may still lead to the accumulation of charges in that case the solution won t have the lacunae per se we show however that the notion of classical lacunae can be generalized and that even when the steady state charges are present the waves still have sharp aft fronts yet behind those aft fronts there is a nonzero electrostatic solution rather than one identically zero

fast integral equation methods for rothe s method applied to the isotropic heat equation we present an efficient integral equation approach to solve the forced heat equation u t x email protected x f x u t in a two dimensional multiply connected domain with dirichlet boundary conditions instead of using an integral equation formulation based on the heat kernel we discretize in time first this approach known as rothe s method leads to a non homogeneous modified helmholtz equation that is solved at each time step we formulate the solution to this equation as a volume potential plus a double layer potential and both of these potentials are calculated with available tools accelerated by the fast multipole method for a total of n points in the discretization of the boundary and the domain the total computational cost per time step is o n we demonstrate our approach on the heat equation and the allen cahn equation

higher regularity of the tangential fields in the relativistic vlasov maxwell system abstract it is shown that the tangential electric and magnetic fields in the glassey strauss representation formulas are in fact bounded in l loc t l x 2 for some 0

jordan isomorphisms of triangular matrix algebras over a connected commutative ring abstract let c be a 2 torsionfree commutative ring with identity 1 and let t r c r 2 be the algebra of all upper triangular r r r 2 matrices over c then c contains no idempotents except 0 and 1 if and only if every jordan isomorphism of t r c onto an arbitrary algebra over c is either an isomorphism or an anti isomorphism

classical solutions and the glassey strauss theorem for the 3d vlasov maxwell system r glassey and w strauss have proved in arch rational mech anal 92 1986 59 90 that c1 solutions to the relativistic vlasov maxwell system in three space dimensions do not develop singularities as long as the support of the distribution function in the momentum variable remains bounded the present paper simplifies their proof

global weak solutions of the vlasov maxwell system with boundary conditions boundaries occur naturally in physucal systems which satisfy the vlasov maxwell system assume perfect conductor boundary conditions for maxwell and either specular reflection or partial absorption for vlasov then weak solutions with finite energy exist for all time

computation of maximal safe sets for switching systems the problem of determining maximal safe sets and hybrid controllers is computationally intractable because of the mathematical generality of hybrid system models given the practical and theoretical relevance of the problem finding implementable procedures that could at least approximate the maximal safe set is important to this end we begin by restricting our attention to a special class of hybrid systems switching systems we exploit the structural properties of the graph describing the discrete part of a switching system to develop an efficient procedure for the computation of the safe set this procedure requires the computation of a maximal controlled invariant set we then restrict our attention to linear discrete time systems for which there is a wealth of results available in the literature for the determination of maximal controlled invariant sets however even for this class of systems the computation may not converge in a finite number of steps we then propose to compute inner approximations that are controlled invariant and for which a procedure that terminates in a finite number of steps can be obtained a tight bound on the error can be given by comparing the inner approximation with the classical outer approximation of the maximal controlled invariant set our procedure is applied to the idle speed regulation problem in engine control to demonstrate its efficiency

equivalence of switching linear systems by bisimulation a general notion of hybrid bisimulation is proposed for the class of switching linear systems connections between the notions of bisimulation based equivalence state space equivalence algebraic and input output equivalence are investigated an algebraic characterization of hybrid bisimulation and an algorithmic procedure converging in a finite number of steps to the maximal hybrid bisimulation are derived hybrid state space reduction is performed by hybrid bisimulation between the hybrid system and itself by specializing the results obtained on bisimulation also characterizations of simulation and abstraction are derived connections between observability bisimulation based reduction and simulation based abstraction are studied

a new continuation criterion for the relativistic vlasov maxwell system the global existence of solutions to the relativistic vlasov maxwell system given sufficiently regular finite energy initial data is a longstanding open problem the main result of glassey and strauss arch ration mech anal 92 59 90 1986 shows that a solution f e b remains c 1 as long as the momentum support of f remains bounded alternate proofs were later given by bouchut et al arch ration mech anal 170 1 15 2003 and klainerman and staffilani commun pure appl anal 1 103 125 2002 we show that only the boundedness of the momentum support of f after projecting to any two dimensional plane is needed for f e b to remain c 1

a domain decomposition method for pseudo spectral electromagnetic simulations of plasmas abstract pseudo spectral electromagnetic solvers i e representing the fields in fourier space have extraordinary precision in particular haber et al presented in 1973 a pseudo spectral solver that integrates analytically the solution over a finite time step under the usual assumption that the source is constant over that time step yet pseudo spectral solvers have not been widely used due in part to the difficulty for efficient parallelization owing to global communications associated with global ffts on the entire computational domains a method for the parallelization of electromagnetic pseudo spectral solvers is proposed and tested on single electromagnetic pulses and on particle in cell simulations of the wakefield formation in a laser plasma accelerator the method takes advantage of the properties of the discrete fourier transform the linearity of maxwell s equations and the finite speed of light for limiting the communications of data within guard regions between neighboring computational domains although this requires a small approximation test results show that no significant error is made on the test cases that have been presented the proposed method opens the way to solvers combining the favorable parallel scaling of standard finite difference methods with the accuracy advantages of pseudo spectral methods

the newtonian limit for asymptotically flat solutions of the vlasov einstein system it is shown that there exist families of asymptotically flat solutions of the einstein equations coupled to the vlasov equation describing a collisionless gas which have a newtonian limit these are sufficiently general to confirm that for this matter model as many families of this type exist as would be expected on the basis of physical intuition a central role in the proof is played by energy estimates in unweighted sobolev spaces for a wave equation satisfied by the second fundamental form of a maximal foliation

the newtonian limit for perfect fluids we prove that there exists a class of non stationary solutions to the einstein euler equations which have a newtonian limit the proof of this result is based on a symmetric hyperbolic formulation of the einstein euler equations which contains a singular parameter vt c where vt is a characteristic velocity scale associated with the fluid and c is the speed of light the symmetric hyperbolic formulation allows us to derive e independent energy estimates on weighted sobolev spaces these estimates are the main tool used to analyze the behavior of solutions in the limit 0

equivalence of dynamical systems by bisimulation a general notion of bisimulation is defined for linear input state output systems using analogies with the theory of concurrent processes a characterization of bisimulation and an algorithm for computing the maximal bisimulation relation is derived using geometric control theory bisimulation is shown to be a notion which unifies the concepts of state space equivalence and state space reduction and which allows to study equivalence of systems with nonminimal state space dimension the notion of bisimulation is especially powerful for nondeterministic dynamical systems and leads in this case to a notion of equivalence which is finer than equality of external behavior for abstractions of systems it is shown how the results specialize to previously obtained results by other authors extensions of the main results to the nonlinear case are provided

fast integral equation methods for the modified helmholtz equation we present integral equation methods for the solution to the two dimensional modified helmholtz equation u x email protected email protected x 0 in bounded or unbounded multiply connected domains we consider both dirichlet and neumann problems we derive well conditioned fredholm integral equations of the second kind which are discretized using high order hybrid gauss trapezoid rules our fast multipole based iterative solution procedure requires only o n operations where n is the number of nodes in the discretization of the boundary we demonstrate the performance of our methods on several numerical examples and we show that they have both the ability to handle highly complex geometry and the potential to solve large scale problems

functional identities of degree 2 in triangular rings abstract let r be a triangular ring the problem of describing the form of additive maps f 1 f 2 g 1 g 2 r r satisfying functional identity f 1 x y f 2 y x xg 2 y yg 1 x 0 for all x y r is considered as an application generalized inner biderivations and commuting additive maps of certain triangular rings are determined

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

discontinuous galerkin methods for the vlasov maxwell equations discontinuous galerkin methods are developed for solving the vlasov maxwell system methods that are designed to be systematically as accurate as one wants with provable conservation of mass and possibly total energy such properties in general are hard to achieve within other numerical method frameworks for simulating the vlasov maxwell system the proposed scheme employs discontinuous galerkin discretizations for both the vlasov and the maxwell equations resulting in a consistent description of the distribution function and electromagnetic fields it is proven up to some boundary effects that charge is conserved and the total energy can be preserved with suitable choices of the numerical flux for the maxwell equations and the underlying approximation spaces error estimates are established for several flux choices the scheme is tested on the streaming weibel instability the order of accuracy and conservation properties of the proposed method are verified

higher order a stable schemes for the wave equation using a successive convolution approach in several recent works we developed a new second order a stable approach to wave propagation problems based on the method of lines transpose mol t formulation combined with alternating direction implicit adi schemes because our method is based on an integral solution of the adi splitting of the mol t formulation we are able to easily embed non cartesian boundaries and include point sources with exact spatial resolution further we developed an efficient o n convolution algorithm for rapid evaluation of the solution which makes our method competitive with explicit finite difference e g finite difference time domain solvers in terms of both accuracy and time to solution even for courant numbers slightly larger than 1 we have demonstrated the utility of this method by applying it to a range of problems with complex geometry including cavities with cusps in this work we present several important modifications to our recently developed wave solver we obtain a family of wave solvers

jordan homomorphisms of upper triangular matrix rings abstract in this paper we investigate jordan homomorphisms of upper triangular matrix rings and give a sufficient condition under which they are necessarily homomorphisms or anti homomorphisms

local existence and uniqueness theory of the vlasov maxwell system on developpe une theorie de l existence et de l unicite locale par rapport au temps pour le systeme de vlasov maxwell qui n utilise pas la theorie des semigroupes de kato

post newtonian expansions for perfect fluids we prove the existence of a large class of dynamical solutions to the einstein euler equations that have a first post newtonian expansion the results here are based on the elliptic hyperbolic formulation of the einstein euler equations used in 15 which contains a singular parameter epsilon v t c where vt is a characteristic velocity associated with the fluid and c is the speed of light as in 15 energy estimates on weighted sobolev spaces are used to analyze the behavior of solutions to the einstein euler equations in the limit epsilon searrow 0 and to demonstrate the validity of the first post newtonian expansion as an approximation

the boost problem in general relativity we show that any asymptotically flat initial data for the einstein field equations have a development which includes complete spacelike surfaces boosted relative to the initial surface furthermore the asymptotic fall off is preserved along these boosted surfaces and there exists a global system of harmonic coordinates on such a development we also extend former results on global solutions of the constraint equations by virtue of this extension the constraint and evolution parts of the problem fit together exactly several theorems are given which concern the behaviour in the large of general classes of linear and quasilinear differential systems this paper contains in addition a systematic exposition of the functional spaces employed

permanent regimes for the vlasov maxwell equations with specular boundary conditions the subject matter of this paper concerns the existence of permanent regimes i e stationary or time periodic solutions for the vlasov maxwell system in a bounded domain we are looking for equilibrium configurations by imposing specular boundary conditions the main difficulty is the treatment of such boundary conditions our analysis relies on perturbative techniques based on uniform a priori estimates

asymptotic behavior of weak solutions for the relativistic vlasov maxwell equations with large light speed abstract we study here the behavior of time periodic weak solutions for the relativistic vlasov maxwell boundary value problem in a three dimensional bounded domain with strictly star shaped boundary when the light speed becomes infinite we prove the convergence toward a time periodic weak solution for the classical vlasov poisson equations

strichartz estimates and moment bounds for the relativistic vlasov maxwell system we consider the relativistic vlasov maxwell system with data of unrestricted size and without compact support in momentum space in the two dimensional and the two and a half dimensional cases glassey schaeffer proved commun math phys 185 257 284 1997 arch ration mech anal 141 331 354 1998 arch ration mech anal 141 355 374 1998 that for regular initial data with compact momentum support this system has unique global in time classical solutions in this work we do not assume compact momentum support for the initial data and instead require only that the data have polynomial decay in momentum space in the two dimensional and the two and a half dimensional cases we prove the global existence uniqueness and regularity for solutions arising from this class of initial data to this end we use strichartz estimates and prove that suitable moments of the solution remain bounded moreover we obtain a slight improvement of the temporal growth of the l infty x norms of the electromagnetic fields compared to glassey and schaeffer commun math phys 185 257 284 1997 arch ration mech anal 141 355 374 1998 in the three dimensional case we apply strichartz estimates and moment bounds to show that a regular solution can be extended as long as p 0 theta f l q x l 1 p remains bounded for theta frac 2 q 2 q leqq infty this improves previous results of pallard indiana univ math j 54 5 1395 1409 2005 commun math sci 13 2 347 354 2015

multiplicative lie n derivations of triangular rings abstract we introduce the notion of a multiplicative lie n derivation of a ring generalizing the notion of a lie triple derivation the main goal of the paper is to consider the question of when do all multiplicative lie n derivations of a triangular ring t have the so called standard form the main result is applied to the classical examples of triangular rings nest algebras and block upper triangular matrix rings

on the definition of post newtonian approximations a definition of post newtonian approximations is presented where the whole formalism is derived from a minimal set of axioms this establishes a link between the existing precise formulation of the newtonian limit of general relativity and the post newtonian equations which are used in practical calculations the breakdown of higher post newtonian approximations is examined within this framework it is shown that the choice of harmonic gauge leads to equations which do not admit asymptotically flat solutions at the second post newtonian level if one starts with a generic newtonian solution the most simple choice of gauge gives equations which are solvable at the 2pn level but which in general have no solutions in the case of the third post newtonian approximation

a particle in cell method for the simulation of plasmas based on an unconditionally stable field solver abstract we propose a new particle in cell pic method for the simulation of plasmas based on a recently developed unconditionally stable solver for the wave equation this method is not subject to a cfl restriction limiting the ratio of the time step size to the spatial step size typical of explicit methods while maintaining computational cost and code complexity comparable to such explicit schemes we describe the implementation in one and two dimensions for both electrostatic and electromagnetic cases and present the results of several standard test problems showing good agreement with theory with time step sizes much larger than allowed by typical cfl restrictions

corrigendum to jordan homomorphisms of upper triangular matrix rings linear algebra appl 439 12 2013 4063 4069 abstract in this paper we will present a new proof of the main theorem in linear algebra appl 439 2013 4063 4069

jordan homomorphisms of upper triangular matrix rings over a prime ring abstract the aim of the paper is to prove that under a mild assumption every jordan homomorphism from an upper triangular matrix ring over a unital ring onto another upper triangular matrix ring over a unital prime ring of characteristic not 2 is either a homomorphism or an anti homomorphism this result is an extension of a classical theorem of herstein

a positivity preserving high order semi lagrangian discontinuous galerkin scheme for the vlasov poisson equations the vlasov poisson equations describe the evolution of a collisionless plasma represented through a probability density function pdf that self interacts via an electrostatic force one of the main difficulties in numerically solving this system is the severe time step restriction that arises from parts of the pdf associated with moderate to large velocities the dominant approach in the plasma physics community for removing these time step restrictions is the so called particle in cell pic method which discretizes the distribution function into a set of macro particles while the electric field is represented on a mesh several alternatives to this approach exist including fully lagrangian fully eulerian and so called semi lagrangian methods the focus of this work is the semi lagrangian approach which begins with a grid based eulerian representation of both the pdf and the electric field then evolves the pdf via lagrangian dynamics and finally projects this evolved field back onto the original eulerian mesh in particular we develop in this work a method that discretizes the 1 1 vlasov poisson system via a high order discontinuous galerkin dg method in phase space and an operator split semi lagrangian method in time second order accuracy in time is relatively easy to achieve via strang operator splitting with additional work using higher order splitting and a higher order method of characteristics we also demonstrate how to push this scheme to fourth order accuracy in time we show how to resolve all of the lagrangian dynamics in such a way that mass is exactly conserved positivity is maintained and high order accuracy is achieved the poisson equation is solved to high order via the smallest stencil local discontinuous galerkin ldg approach we test the proposed scheme on several standard test cases

three new results on continuation criteria for the 3d relativistic vlasov maxwell system abstract we consider continuation criteria for the three dimensional relativistic vlasov maxwell system when the particle density f t x p is compactly supported at t 0 we prove p 0 18 5 r 1 f l t l x r l p 1 1 where 1 r 2 and 0 is arbitrarily small is a continuation criteria our continuation criteria is an improvement in the 1 r 2 range to the previously best known criteria p 0 4 r 1 f l t l x r l 1 p 1 due to kunze 7 we also consider continuation criteria when f 0 x p has noncompact support in this regime luk strain 9 proved that p 0 f l x 1 l p 1 1 is a continuation criteria for 5 we improve this result to 3 finally we build on another result by luk strain 8 the authors proved boundedness of momentum support on a fixed two dimensional plane is a sufficient continuation criteria we prove the same result even if the plane varies continuously in time

finite speed propagation of the solutions for the relativistic vlasov maxwell system in this paper we investigate the continuous dependence with respect to the initial data of the solutions for the 1d and 1 5d relativistic vlasov maxwell system more precisely we prove that these solutions propagate with finite speed we formulate our results in the framework of mild solutions i e the particle densities are solutions by characteristics and the electro magnetic fields are lipschitz continuous functions

a critical comparison of eulerian grid based vlasov solvers a common problem with direct vlasov solvers is ensuring that the distribution function remains positive a related problem is to guarantee that the numerical scheme does not introduce false oscillations in velocity space in this paper we use a variety of schemes to assess the importance of these issues and to determine an optimal strategy for eulerian split approaches to vlasov solvers from these tests we conclude that maintaining positivity is less important than correctly dissipating the fine scale structure which arises naturally in the solution to many vlasov problems furthermore we show that there are distinct advantages to using high order schemes i e third order rather than second a natural choice which satisfies all of these requirements is the piecewise parabolic method ppm which is applied here to vlasov s equation for the first time

a converse lyapunov theorem for a class of dynamical systems which undergo switching the authors investigate the stability of a system in which the dynamics at any instant in time will follow one of a fixed set of vector fields they allow switching between members of the family of vector fields to be completely random the focus of this paper is to prove a converse lyapunov theorem for this class of systems

bisimilar control affine systems the notion of bisimulation plays a very important role in theoretical computer science where it provides several notions of equivalence between models of computation these equivalences are in turn used to simplify verification and synthesis for these models as well as to enable compositional reasoning in systems theory a similar notion is also of interest in order to develop modular verification and design tools for purely continuous or hybrid control systems in this paper we introduce two notions of bisimulation for nonlinear systems we present differential geometric characterizations of these notions and show that bisimilar systems of different dimensions are obtained by factoring out certain invariant distributions furthermore we also show that all bisimilar systems of different dimension are of this form

yet another criterion for global existence in the 3d relativistic vlasov maxwell system abstract we prove that solutions of the 3d relativistic vlasov maxwell system can be extended as long as the quantity 1 t x max 1 r 3 d p 1 p 2 1 1 v f t x p is bounded in l x 2

a multi level multi domain method for particle in cell plasma simulations a novel adaptive technique for electromagnetic particle in cell pic plasma simulations is presented here two main issues are identified as regards the development of the algorithm first the choice of the size of the particle shape function in progressively refined grids with the decision to avoid both time dependent shape functions and cumbersome particle to grid interpolation techniques and second the necessity to comply with the strict stability constraints of the explicit pic algorithm the adaptive implementation presented responds to these demands with the introduction of a multi level multi domain mlmd system where a cloud of self similar domains is fully simulated with both fields and particles and the use of an implicit moment pic method as baseline algorithm for the adaptive evolution information is exchanged between the levels with the projection of the field information from the refined to the coarser levels and the interpolation of the boundary conditions for the refined levels from the coarser level fields particles are bound to their level of origin and are prevented from transitioning to coarser levels but are repopulated at the refined grid boundaries with a splitting technique the presented algorithm is tested against a series of simulation challenges

high order unconditionally stable fc ad solvers for general smooth domains i basic elements we introduce a new methodology for the numerical solution of partial differential equations in general spatial domains our algorithms are based on the use of the well known alternating direction implicit adi approach in conjunction with a certain fourier continuation fc method for the resolution of the gibbs phenomenon unlike previous alternating direction methods of order higher than one which can only deliver unconditional stability for rectangular domains the present high order algorithms possess the desirable property of unconditional stability for general domains the computational time required by our algorithms to advance a solution by one time step in turn grows in an essentially linear manner with the number of spatial discretization points used in this paper we demonstrate the fc ad methodology through a variety of examples concerning the heat and laplace equations in two and three dimensional domains with smooth boundaries applications of the fc ad methodology to hyperbolic pdes together with a theoretical discussion of the method will be put forth in a subsequent contribution the numerical examples presented in this text demonstrate the unconditional stability and high order convergence of the proposed algorithms as well the very significant improvements they can provide in one of our examples we demonstrate a one thousand improvement factor over the computing times required by some of the most efficient alternative general domain solvers

bisimilar linear systems the notion of bisimulation in theoretical computer science is one of the main complexity reduction methods for the analysis and synthesis of labeled transition systems bisimulations are special quotients of the state space that preserve many important properties expressible in temporal logics and in particular reachability in this paper the framework of bisimilar transition systems is applied to various transition systems that are generated by linear control systems given a discrete time or continuous time linear system and a finite observation map we characterize linear quotient maps that result in quotient transition systems that are bisimilar to the original system interestingly the characterizations for discrete time systems are more restrictive than for continuous time systems due to the existence of an atomic time step we show that computing the coarsest bisimulation which results in maximum complexity reduction corresponds to computing the maximal controlled or reachability invariant subspace inside the kernel of the observations map these results establish strong connections between complexity reduction concepts in control theory and computer science

global weak solutions of vlasov maxwell systems we study here the vlasov maxwell system in its classical and relativistic form we prove the stability of solutions in weak topologies and deduce from this stability result the global existence of a weak solution with large initial data the main tools consist of a new regularity result for velocity averages of solutions of some general linear transport equations and the method of renormalization introduced by the authors in previous work

method of lines transpose an implicit solution to the wave equation we present a new method for solving the wave equation implicitly our approach is to discretize the wave equation in time following the method of lines transpose sometimes referred to as the transverse method of lines or rothe s method we differ from conventional methods that follow this approach in that we solve the resulting system of partial differential equations using boundary integral methods our algorithm extends to higher spatial dimensions using an alternating direction implicit adi framework thus we develop a boundary integral solution that is competitive with explicit finite difference methods both in terms of accuracy and speed however it provides more flexibility in the treatment of source functions and complex boundaries we provide the analytical details of our one dimensional method herein along with a proof of the convergence of our schemes in free space and on a bounded domain we find that the method is unconditionally stable and achieves second order accuracy a caveat of the analysis is the derivation of a unique and novel optimal quadrature method which can be viewed as a lax type correction

reliability of perturbation theory in general relativity the relation between perturbation theory and exact solutions in general relativity is tackled by investigating the existence and properties of smooth one parameter families of solutions on the one hand the coefficients of the taylor expansion in the parameter of any given smooth family of solutions necessarily satisfy the hierarchy of perturbation equations on the other hand it is the converse question does any solution of the perturbation equations come from taylor expanding some family of exact solutions which is of importance for the mathematical justification of the use of perturbation theory this converse question is called the one of the reliability of perturbation theory using and completing recent results on the characteristic initial value problem the local reliability of perturbation theory for general relativity in vacuum is proven very generally this result is then generalized to the einstein yang mills equations and therefore in particular to the einstein maxwell ones

regularity of the moments of the solution of a transport equation let u u x v satisfy the transport equation u v xu f x rn v rnwhere f belongs to some space of type lp dx d v where is a positive bounded measure on rn we study the resulting regularity of the moment u x v d v in terms of fractional sobolev spaces for example counter examples are given in order to test the optimality of our results