global regularity for a class of quasi linear local and nonlocal elliptic equations on extension domains abstract we consider the solvability of quasi linear elliptic equations with either local or nonlocal neumann robin or wentzell boundary conditions defined in the generalized sense on a bounded w 1 p extension domain whose boundary is an upper d set for an appropriate d 0 then we extend the fine regularity theory for weak solutions of the elliptic equations with the above boundary conditions known for bounded lipschitz domains to bounded w 1 p extension domains whose boundaries are upper d sets by showing that such weak solutions are globally holder continuous consequently we generalize substantially the class of bounded domains where weak solutions of boundary value problems of type neumann robin or wentzell may be uniformly continuous up to the boundary

some potential theory for reflecting brownian motion in holder and lipschitz domains bounds are found on the transition densities and green functions for brownian motion with normal reflection in holder and lipschitz domains for lipschitz domains reflecting brownian motion and boundary local time are constructed a harnack inequality valid up to the boundary is proved a probabilistic solution to the neumann problem is given and the kuramochi boundary is identified

an efficient algorithm for computing bisimulation equivalence we propose an efficient algorithmic solution to the problem of determining a bisimulation relation on a finite structure working both on the explicit and on the implicit symbolic representation as far as the explicit case is concerned starting from a set theoretic point of view we propose an algorithm that optimizes the solution to the relational coarsest partition problem given by paige and tarjan siam j comput 16 6 1987 973 its use in model checking packages is also discussed and tested for well structured graphs our algorithm reaches a linear worst case behaviour the proposed algorithm is then re elaborated to produce a symbolic version

quasi linear boundary value problems with generalized nonlocal boundary conditions abstract we investigate a quasi linear boundary value problem of the form div u p 2 u 0 involving a general boundary map and mixed neumann boundary conditions on a bounded lipschitz domain we show existence uniqueness and holder continuity of the weak solution of this mixed boundary value problem and obtain maximum principles for this class of mixed equations as a consequence we obtain uniform continuity up to the boundary to solutions associated with a class of electrical models described by maxwell s equations with nonlocal boundary conditions an extension to boundary value problems with generalized nonlocal robin boundary conditions is also achieved

an asynchronous spacetime discontinuous galerkin finite element method for time domain electromagnetics abstract we present an asynchronous spacetime discontinuous galerkin asdg method for time domain electromagnetics in which space and time are directly discretized by using differential forms we express maxwell s equations and consequently their discontinuous galerkin discretization for arbitrary domains in spacetime the elements are discretized with electric and magnetic basis functions that are discontinuous across all inter element boundaries and can have arbitrary high and per element spacetime orders when restricted to unstructured grids that satisfy a specific causality constraint the method has a local and asynchronous solution procedure with linear solution complexity in terms of the number of elements we numerically investigate the convergence properties of the method for 1d to 3d uniform grids for energy dissipation an error relative to the exact solution and von neumann dissipation and dispersion errors two dimensional simulations demonstrate the effectiveness of the method in resolving sharp wave fronts

mixed boundary value problems for maxwell s equations we study the maxwell system with mixed boundary conditions in a lipschitz domain in 3 it is assumed that two disjoint relatively open subsets e h of such that e h e h have been fixed and one prescribes the tangential components of the electric and magnetic fields on e and h respectively under suitable geometric assumptions on e and h we prove that this boundary value problem is well posed when l p estimates for the nontangential maximal function are sought with p near 2 a higher dimensional version of this result is established as well in the language of differential forms this extends earlier work by r brown and by the author and collaborators

a high order non conforming discontinuous galerkin method for time domain electromagnetics in this paper we discuss the formulation stability and validation of a high order non dissipative discontinuous galerkin dg method for solving maxwell s equations on non conforming simplex meshes the proposed method combines a centered approximation for the numerical fluxes at inter element boundaries with either a second order or a fourth order leap frog time integration scheme moreover the interpolation degree is defined at the element level and the mesh is refined locally in a non conforming way resulting in arbitrary level hanging nodes the method is proved to be stable and conserves a discrete counterpart of the electromagnetic energy for metallic cavities numerical experiments with high order elements show the potential of the method

dissipative terms and local time stepping improvements in a spatial high order discontinuous galerkin scheme for the time domain maxwell s equations in this paper we present some improvements in terms of accuracy and speed up for a particular well adapted discontinuous galerkin method devoted to the time domain maxwell equations first to reduce spurious modes on very distorted meshes the addition of dissipative terms as penalization in the numerical scheme is studied and compared on examples second in order to increase the efficiency of the method a multi class local time stepping strategy is presented and its validation and advantages are highlighted on different examples

optimal canonization of all substrings of a string abstract any word can be decomposed uniquely into lexicographically nonincreasing factors each one of which is a lyndon word this paper addresses the relationship between the lyndon decomposition of a word x and a canonical rotation of x i e a rotation w of x that is lexicographically smallest among all rotations of x the main combinatorial result is a characterization of the lyndon factor of x with which w must start as an application faster on line algorithms for finding the canonical rotation s of x are developed by nontrivial extension of known lyndon factorization strategies unlike their predecessors the new algorithms lend themselves to incremental variants that compute in linear time the canonical rotations of all prefixes of x the fastest such variant represents the main algorithmic contribution of the paper it performs within the same 3 x character comparisons bound as that of the fastest previous on line algorithms for the canonization of a single string this leads to the canonization of all substrings of a string in optimal quadratic time within less than 3 x 2 character comparisons and using linear auxiliary space

continuant polynomials and worst case behavior of hopcroft s minimization algorithm this paper is concerned with the analysis of the worst case behavior of hopcroft s algorithm for minimizing deterministic finite state automata we extend a result of castiglione restivo and sciortino they show that hopcroft s algorithm has a worst case behavior for the automata recognizing fibonacci words in a previous paper we have proved that this holds for all standard sturmian words having an ultimately periodic directive sequence the directive sequence for fibonacci words is 1 1 we prove here that the same conclusion holds for all standard sturmian words having a directive sequence with bounded elements more precisely we obtain in fact a characterization of those directive sequences for which hopcroft s algorithm has worst case running time these are the directive sequences d 1 d 2 d 3 for which the sequence of geometric means d 1d 2 d n 1 n is bounded as a consequence we easily show that there exist directive sequences for which the worst case for the running time is not attained

explicit finite element methods for symmetric hyperbolic equations a family of explicit space time finite element methods for the initial boundary value problem for linear symmetric hyperbolic systems of equations is described and analyzed the method generalizes the discontinuous galerkin method and as is typical for this method obtains error estimates of order o h n 1 2 for approximations by polynomials of degree le n

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

causal path local time stepping in the discontinuous galerkin method for maxwell s equations we introduce a novel local time stepping technique for marching in time algorithms the technique is denoted as causal path local time stepping cplts and it is applied for two time integration techniques fourth order low storage explicit runge kutta lserk4 and second order leap frog lf2 the cplts method is applied to evolve email protected s curl equations using a discontinuous galerkin dg scheme for the spatial discretization numerical results for lf2 and lserk4 are compared with analytical solutions and the email protected s lf2 technique the results show that the cplts technique improves the dispersive and dissipative properties of lf2 lts scheme

an efficient parallel algorithm for the single function coarsest partition problem we describe an efficient parallel algorithm to solve the single function coarsest partition problem the algorithm runs in o log n time using o n log log n operations on the arbitrary crcw pram the previous best known algorithms run in o log2 n time using o n log2n operations on the crew pram and o log n time using o n log n operations on the arbitrary crcw pram our solution is based on efficient algorithms for solving several subproblems that are of independent interest in particular we present efficient parallel algorithms to find a minimal starting point of a circular string with respect to lexicographic ordering and to sort lexicographically a list of strings of different lengths

regularity and well posedness of some quasi linear elliptic and parabolic problems with nonlinear general wentzell boundary conditions on nonsmooth domains abstract we show that on a bounded domain r n with lipschitz continuous boundary if 2 n n 2 p n 1 then weak solutions of the quasi linear elliptic equation p u u p 2 u 1 x u f in with the general wentzell boundary conditions p u u p 2 u u p 2 u 2 x u g weakly on are globally holder continuous on here p and p denote the p laplace operator and the p laplace beltrami operator on the boundary respectively the functions 1 x for x and 2 x for x are continuous on r and satisfy a certain growth condition we also obtain that a realization of the operator p u 1 x u in c with the boundary conditions p u 1 x u p u u p 2 u 2 x u 0 on generates a strongly continuous semigroup of nonlinear operators on c

construction and decomposition of reflecting diffusions on lipschitz domains with holder cusps we consider a d dimensional euclidean domain d whose boundary is lipschitz continuous but admits locally finite number of outward or inward holder cusp points using a method of stampacchia and moser for pde we first construct a conservative diffusion process on the euclidean closure of d possessing a strong feller resolvent and associated with a second order uniformly elliptic differential operator of divergence form with measurable coefficients a ij the sample path of the constructed diffusion can be uniquely decomposed as a sum of a martingale additive functional and an additive functional locally of zero energy the second additive functional will be proved to be of bounded variation with a skorohod type expression whenever a ij is weakly differentiable and the holder exponent at each outward cusp boundary point is greater than 1 2 regardless the dimension d

standard sturmian words and automata minimization algorithms the study of some close connections between the combinatorial properties of words and the performance of the automata minimization process constitutes the main focus of this paper these relationships have been in fact the basis of the study of the tightness and the extremal cases of hopcroft s algorithm that is up to now the most efficient minimization method for deterministic finite state automata recently increasing attention has been paid to another minimization method that unlike the approach proposed by hopcroft is not based on refinement of the set of states of the automaton but on automata operations such as determinization and reverse and is also applicable to non deterministic finite automata however even for deterministic automata it was proved that the method incurs almost surely in an explosion of the number of the states very recently some polynomial variants of brzozowski s method have been introduced in this paper by using some combinatorial properties of words we analyze the performance of one of such algorithms when applied to a particular infinite family of automata called standard word automata constructed by using standard sturmian words in particular n log n is the worst case time complexity when the algorithm is applied to the infinite family of word automata associated to fibonacci words representing also the worst case of hopcroft s minimization algorithm

on extremal cases of hopcroft s algorithm in this paper we consider the problem of minimization of deterministic finite automata dfa with reference to hopcroft s algorithm hopcroft s algorithm has several degrees of freedom so there can exist different executions that can lead to different sequences of refinements of the set of the states up to the final partition we find an infinite family of binary automata for which such a process is unique whatever strategy is chosen some recent papers cf berstel and carton 2004 3 castiglione et al 2008 6 and berstel et al 2009 1 have been devoted to find families of automata for which hopcroft s algorithm has its worst execution time they are unary automata associated with circular words however automata minimization can be achieved also in linear time when the alphabet has only one letter cf paige et al 1985 14 but such a method does not seem to extend to larger alphabet so in this paper we face the tightness of hopcroft s algorithm when the alphabet contains more than one letter in particular we define an infinite family of binary automata representing the worst case of hopcroft s algorithm for each execution they are automata associated with particular trees and we deepen the connection between the refinement process of hopcroft s algorithm and the combinatorial properties of such trees

riemann solutions and spacetime discontinuous galerkin method for linear elastodynamic contact abstract we derive riemann solutions for stick slip and separation contact modes in a linear elastic material with an isotropic coulomb friction relation and explore their numerical implementation the riemann solutions preserve the characteristic structure of the underlying elastodynamic system and imply dynamic contact conditions that are distinct from the quasi static conditions used in some numerical models nonphysical discontinuities in the standard coulomb model at stick slip transitions can cause contact mode chatter in numerical simulations we restate the coulomb relation to remove these artificial discontinuities and eliminate the need for algorithmic remedies discontinuous response at abrupt separation to contact transitions is physically reasonable and we propose a regularization scheme to address this case we implement the riemann contact solutions within an adaptive spacetime discontinuous galerkin sdg code and report numerical results that demonstrate the model s efficacy

a spacetime discontinuous galerkin method for hyperbolic heat conduction non fourier conduction models remedy the paradox of infinite signal speed in the traditional parabolic heat equation for applications involving very short length or time scales hyperbolic conduction models better represent the physical thermal transport processes this paper reviews the maxwell cattaneo vernotte modification of the fourier conduction law and describes its implementation within a spacetime discontinuous galerkin sdg finite element method that admits jumps in the primary variables across element boundaries with arbitrary orientation in space and time a causal advancing front meshing procedure enables a patch wise solution procedure with linear complexity in the number of spacetime elements an h adaptive scheme and a special sdg shock capturing operator accurately resolve sharp solution features in both space and time numerical results for one spatial dimension demonstrate the convergence properties of the sdg method as well as the effectiveness of the shock capturing method simulations in two spatial dimensions demonstrate the proposed method s ability to accurately resolve continuous and discontinuous thermal waves in problems where rapid and localized heating of the conducting medium takes place

a space time discontinuous galerkin method for linearized elastodynamics with element wise momentum balance abstract we present a new space time discontinuous galerkin finite element method for linearized elastodynamics that delivers exact balance of linear and angular momentum over every space time element the method is formulated for use with fully unstructured space time grids and uses displacement basis functions that are discontinuous across all inter element boundaries we introduce a new space time formulation of continuum elastodynamics that uses differential forms and the exterior calculus on manifolds to generate a system of space time field equations and jump conditions then we invoke a bubnov galerkin weighted residuals procedure to formulate the finite element method we describe an implementation on patch wise causal meshes that features linear complexity in the number of elements and special per pixel accurate visualization numerical examples confirm an a priori error estimate and demonstrate the method s shock capturing capabilities

generalizations of suffix arrays to multi dimensional matrices we propose multi dimensional index data structures that generalize suffix arrays to square matrices and cubic matrices giancarlo proposed a two dimensional index data structure the lsuffix tree that generalizes suffix trees to square matrices however the construction algorithm for lsuffix trees maintains complicated data structures and uses a large amount of space we present simple construction algorithms for multi dimensional suffix arrays by applying a new partitioning technique to lexicographic sorting our contributions are the first efficient algorithms for constructing two dimensional and three dimensional suffix arrays directly

explicit local time stepping methods for maxwell s equations explicit local time stepping methods are derived for time dependent maxwell equations in conducting and non conducting media by using smaller time steps precisely where smaller elements in the mesh are located these methods overcome the bottleneck caused by local mesh refinement in explicit time integrators when combined with a finite element discretisation in space with an essentially diagonal mass matrix the resulting discrete time marching schemes are fully explicit and thus inherently parallel in a non conducting source free medium they also conserve a discrete energy which provides a rigorous criterion for stability starting from the standard leap frog scheme local time stepping methods of arbitrarily high accuracy are derived for non conducting media numerical experiments with a discontinuous galerkin discretisation in space validate the theory and illustrate the usefulness of the proposed time integration schemes

a class of quasi linear parabolic and elliptic equations with nonlocal robin boundary conditions abstract let p 1 n r n a bounded w 1 p extension domain and let be an upper d ahlfors measure on with d n p n we show in the first part that for every p 2 n n 2 n 1 n a realization of the p laplace operator with nonlinear generalized nonlocal robin boundary conditions generates a nonlinear strongly continuous submarkovian semigroup on l 2 and hence the associated first order cauchy problem is well posed on l q for every q 1 in the second part we investigate existence uniqueness and regularity of weak solutions to the associated quasi linear elliptic equation more precisely global a priori estimates of weak solutions are obtained

a linear time solution to the single function coarsest partition problem abstract the problem of finding the coarsest partition of a set s with respect to another partition of s one or more functions on s has several applications one of which is the state minimization of finite state automata in 1971 hopcroft presented an algorithm to solve the many function coarsest partition problem for sets of n elements in o n log n time and o n space in 1974 aho hopcroft and ullman presented an o n log n algorithm that solves the special case of this problem for only one function both these algorithms use a negative strategy that repeatedly refines the original partition until a solution is found we present a new algorithm to solve the single function coarsest partition problem in o n time and space using a different constructive approach our algorithm can be applied to the automated manufacturing of woven fabric

a space time discontinuous galerkin trefftz method for time dependent maxwell s equations we consider the discretization of electromagnetic wave propagation problems by a discontinuous galerkin method based on trefftz polynomials this method fits into an abstract framework for space time discontinuous galerkin methods for which we can prove consistency stability and energy dissipation without the need to completely specify the approximation spaces in detail any method of such a general form results in an implicit time stepping scheme with some basic stability properties for the local approximation on every space time element we then consider trefftz polynomials i e the subspace of polynomials that satisfy maxwell s equations exactly on the respective element we present an explicit construction of a basis for the local trefftz spaces in two and three dimensions and summarize some of their basic properties using local properties of the trefftz polynomials we can establish the well posedness of the resulting discontinuous galerkin trefftz method consistency stability and energy dis

electrical conductivity reconstruction using nonlocal boundary conditions in this paper we study existence uniqueness and solution estimates to the mixed problem nabla cdot sigma nabla u 0 with dirichlet to neumann map boundary conditions and neumann boundary conditions we then show how this can be used in the reconstruction of sigma given the relationship between u and its normal derivative on the boundary portion where we do not apply the dirichlet to neumann map a numerical reconstruction scheme is also derived