lower bound technique for length reducing automata the class of growing context sensitive languages gcsl was proposed as a naturally defined subclass of context sensitive languages whose membership problem is solvable in polynomial time growing context sensitive languages and their deterministic counterpart called church rosser languages crl complement the chomsky hierarchy in a natural way as the classes filling the gap between context free languages and context sensitive languages they possess characterizations by a natural machine model length reducing two pushdown automata lrtpda we introduce a lower bound technique for lrtpdas using this technique we prove the conjecture of mcnaughton narendran and otto that the set of palindromes is not in crl as a consequence we obtain that cfl cocfl as well as ucfl coucfl are not included in crl where ucfl denotes the class of unambiguous context free languages this solves an open problem posed by beaudry holzer niemann and otto another corollary is that crl is a strict subset of gcsl cogcsl

level set approach to mean curvature flow in arbitrary codimension we develop a level set theory for the mean curvature evolution of surfaces with arbitrary co dimension thus generalizing the previous work 8 15 on hypersurfaces the main idea is to surround the evolving surface of co dimension k in r by a family of hypersurfaces the level sets of a function evolving with normal velocity equal to the sum of the d k smallest principal curvatures the existence and the uniqueness of a weak level set solution is easily established using mainly the results of 8 and the theory of viscosity solutions for second order nonlinear parabolic equations the level set solutions coincide with the classical solutions whenever the latter exist the proof of this connection uses a careful analysis of the squared distance from the surfaces it is also shown that varifold solutions constructed by brakke 7 are included in the level set solutions the idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of de giorgi an introduction to the theory of barriers and his connection to the level set solutions is also provided

partially overdetermined elliptic boundary value problems we consider semilinear elliptic dirichlet problems in bounded domains overdetermined with a neumann condition on a proper part of the boundary under different kinds of assumptions we show that these problems admit a solution only if the domain is a ball when these assumptions are not fulfilled we discuss possible counterexamples to symmetry we also consider neumann problems overdetermined with a dirichlet condition on a proper part of the boundary and the case of partially overdetermined problems on exterior domains

existence of minima for nonconvex functionals in spaces of functions depending on the distance from the boundary we prove that some nonconvex functionals admit a unique minimum in a functional space of functions which depend only on the distance from the boundary of the plane domain where they are defined the domains considered are disks and regular polygons we prove that the sequence of minima of the functional on the polygons converges to the unique minimum on the circumscribed disk as the number of sides tends to infinity our method also allows us to determine the explicit form of the minima

a deterministic control based approach motion by curvature the level set formulation of motion by mean curvature is a degenerate parabolic equation we show that its solution can be interpreted as the value function of a deterministic two person game more precisely we give a family of discrete time two person games whose value functions converge in the continuous time limit to the solution of the motion by curvature pde for a convex domain the boundary s first arrival time solves a degenerate elliptic equation this corresponds in our game theoretic setting to a minimum exit time problem for a nonconvex domain the two person game still makes sense we draw a connection between its minimum exit time and the evolution of curves with velocity equal to the positive part of the curvature these results are unexpected because the value function of a deterministic control problem is normally the solution of a first order hamilton jacobi equation our situation is different because the usual first order calculation is singular 2005 wiley periodicals inc

pointwise boundary differentiability for the infinity laplace equations abstract in this paper we study the boundary differentiability for infinity harmonic functions at some general corner points we prove the slope estimate and provide a necessary and sufficient condition of the boundary differentiability for infinity harmonic function at less than corner points or greater than corner points as defined in the introduction the boundary differentiable conclusions of the homogeneous infinity laplace equations also apply to the inhomogeneous case

a stochastic differential game for the inhomogeneous laplace equation given a bounded c 2 domain gr m and functions g 2 c g r and h 2 c g r 0 let u denote the unique viscosity solution to the equation 2 u h in g with boundary data g we provide a representation for u as the value of a two player zero sum stochastic differential game ams 2000 subject classifications 91a15 91a23 35j70 49l20 g r 0 be given we study a two player zero sum stochastic differential game sdg defined in terms of an m dimensional state process that is driven by a one dimensional brow nian motion played until the state exits the domain the functions g and h serve as terminal and respectively running payoffs the players controls enter in a diffusion coefficient and in an unbounded drift coefficient of the state process the dynamics are degenerate in that it is possible for the players to completely switch off the brownian motion we show that the game has value and characterize the value function as the unique viscosity solution u uniqueness of solutions is known from 9 of the equation 2 u h in g

some convergence results for howard s algorithm this paper deals with convergence results of howard s algorithm for the resolution of min a in mathcal a b a x c a 0 where b a is a matrix c a is a vector and mathcal a is a compact set we show a global superlinear convergence result under a monotonicity assumption on the matrices b a extensions of howard s algorithm for a max min problem of the form max b in mathcal b min a in mathcal a b a b x c a b 0 are also proposed in the particular case of an obstacle problem of the form min ax b x g 0 where a is an n times n matrix satisfying a monotonicity assumption we show the convergence of howard s algorithm in no more than n iterations instead of the usual 2 n bound still in the case of an obstacle problem we establish the equivalence between howard s algorithm and a primal dual active set algorithm m hintermuller k ito and k kunisch siam j optim 13 2002 pp 865 888 the algorithms are illustrated on the discretization of nonlinear pdes arising in the context of mathematical finance american option and merton s portfolio problem of front propagation problems and for the double obstacle problem

on the dirichlet and serrin problems for the inhomogeneous infinity laplacian in convex domains regularity and geometric results given an open bounded subset of mathbb r n which is convex and satisfies an interior sphere condition we consider the pde delta infty u 1 in subject to the homogeneous boundary condition u 0 on we prove that the unique solution to this dirichlet problem is power concave precisely 3 4 concave and it is of class c1 we then investigate the overdetermined serrin type problem formerly considered in buttazzo and kawohl int math res not pp 237 247 2011 obtained by adding the extra boundary condition nabla u a on by using a suitable p function we prove that if satisfies the same assumptions as above and in addition contains a ball which touches at two diametral points then the existence of a solution to this serrin type problem implies that necessarily the cut locus and the high ridge of coincide in turn in dimension n 2 this entails that must be a stadium like domain and in particular it must be a ball in case its boundary is of class c2

the construction of three dimensional invariant cubature formulae abstract a method of constructing three dimensional cubature formulae of high degree for three dimensional regions is presented the formulae possess full polyhedral symmetry this means that every knot of the formula belongs to an orbit invariant with respect to the symmetries of the polyhedron and that all points of an orbit are knots of the formula with the same weight

the obstacle problem for the p laplacian via optimal stopping of tug of war games we present a probabilistic approach to the obstacle problem for the p laplace operator the solutions are approximated by running processes determined by tug of war games plus noise and letting the step size go to zero not unlike the case when brownian motion is approximated by random walks rather than stopping the process when the boundary is reached the value function is obtained by maximizing over all possible stopping times that are smaller than the exit time of the domain

boundary differentiability of infinity harmonic functions abstract we prove that an infinity harmonic function u x c is differentiable at a boundary point x 0 if both and the boundary data g x are differentiable at x 0 this work improved a former result of changyou wang and yifeng yu 2012 10 they obtained the boundary differentiability of u x with the c 1 assumption on and g x

c1 regularity for infinity harmonic functions in two dimensions a continuous function open image in new window is said to be infinity harmonic if it satisfies the pde

overdetermined problems with possibly degenerate ellipticity a geometric approach given an open bounded connected subset of n we consider the overdetermined boundary value problem obtained by adding both zero dirichlet and constant neumann boundary data to the elliptic equation div a u u 1 in we prove that if this problem admits a solution in a suitable weak sense then is a ball this is obtained under fairly general assumptions on and a in particular a may be degenerate and no growth condition is required our method of proof is quite simple it relies on a maximum principle for a suitable p function combined with some geometric arguments involving the mean curvature of

an almost confluent congruential language which is not church rosser congruential it is fairly easy to show that every regular set is an almost confluent congruential language accl and it is known 3 that every regular set is a church rosser congruential language crcl whether there exists an accl which is not a crcl seems to remain an open question in this note we present one such accl

a stochastic representation for mean curvature type geometric flows a smooth solution t t 0 t rd of a parabolic geometric flow is characterized as the reachability set of a stochastic target problem in this control problem the controller tries to steer the state process into a given deterministic set tc with probability one the reachability set v t for the target problem is the set of all initial data x from which the state process xx t tc for some control process this representation is proved by studying the squared distance function to t for the codimension k mean curvature flow the state process is dx t 2 pdw t where w t is a d dimensional brownian motion and the control p is any projection matrix onto a d k dimensional plane smooth solutions of the inverse mean curvature flow and a discussion of non smooth solutions are also given

a tour of the theory of absolutely minimizing functions a detailed analysis of the class of absolutely minimizing functions in euclidean spaces and the relationship to the infinity laplace equation

a morphological scheme for mean curvature motion and applications to anisotropic diffusion and motion of level sets this paper introduces a discrete scheme for mean curvature motion using a morphological image processing approach an axiomatic approach of image processing and the mean curvature partial differential equation pde are briefly presented then the properties of the proposed scheme are studied in particular consistency and convergence are proved the applications of mean curvature motion in image denoising and form evolution are developed and experiences are presented

the undecidability of the preperfectness of thue systems abstract since the word problem for thue systems was proved to be undecidable by post and markov in the 1940 s there has been some interest in developing conditions under which the word problem is decidable at present it is possible to decide whether a finite thue system is almost confluent or church rosser but testing for preperfectness has been an open problem for years here we prove this to be undecidable

a deterministic control based approach to fully nonlinear parabolic and elliptic equations we show that a broad class of fully nonlinear second order parabolic or elliptic pde s can be realized as the hamilton jacobi bellman equations of deterministic twoperson games more precisely given the pde we identify a deterministic discretetime two person game whose value function converges in the continuous time limit to the viscosity solution of the desired equation our game is roughly speaking a deterministic analogue of the stochastic representation recently introduced by cheridito soner touzi and victoir comm pure appl math 60 2006 1081 1110 in the parabolic setting with no u dependence it amounts to a semidiscrete numerical scheme whose timestep is a min max our result is interesting because the usual control based interpretations of second order pde s involve stochastic rather than deterministic control

slope estimate and boundary differentiability for inhomogeneous infinity laplace equation on convex domains abstract we study the boundary differentiability for inhomogeneous infinity laplace equations on convex domains with the inhomogeneous term f c l and differentiable boundary data g at a flat point the boundary point where the blow up of the domain is the half space u is differentiable due to a previous result of the second author in hong 2014 at a corner point the boundary point where the blow up of the domain is not the half space we establish a slope estimate for u and provide a necessary and sufficient condition for the boundary differentiability of u in this paper

optimal spherical designs and numerical integration on the sphere so called spherical t designs in three dimensional euclidean space r3 are considered it is shown that the maximal possible value of t for designs which are permuted transitively under a finite group of euclidean transformations is 9 this means that we can find a set of 60 points on the surface of the unit sphere in r3 which form an orbit under the icosahedral group such that the average of the function values at the 60 points and the average over the spherical surface are identical for all spherical harmonics of degree less than or equal to t 9 while there is no such orbit for t 10 among all 9 designs which are orbits of the icosahedral group exactly one is optimal with respect to approximation of integrals of harmonics of tenth and eleventh order this set is given in section 2 of this paper it is very well suited for numerically evaluating integrals of continuous functions over the sphere a problem occurring very often in mathematics and science as e g physics astronomy etc

an efficient derivation of the aronsson equation for 1 p the equation which characterizes minima of the functional u u du p dx subject to fixed values of u on u is p u 0 here p is the well known p laplacian when p the corresponding functional is u du 2 l u a new feature arises in that minima are no longer unique unless u is allowed to vary leading to the idea of absolute minimizers aronsson showed that then the appropriate equation is u 0 that is u is infinity harmonic as explained below jensen showed that infinity harmonic functions understood in the viscosity sense are precisely the absolute minimizers here we advance results of barron jensen and wang concerning more general functionals u f x u du l u by giving a simplified derivation of the corresponding necessary condition under weaker hypotheses

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

the principle of least action and fundamental solutions of mass spring and n body two point boundary value problems two point boundary value problems for conservative systems are studied in the context of the least action principle one obtains a fundamental solution whereby two point boundary value problems are converted to initial value problems via an idempotent convolution of the fundamental solution with a cost function related to the terminal data the classical mass spring problem is included as a simple example the n body problem under gravitation is also studied there the least action principle optimal control problem is converted to a differential game where an opposing player maximizes over an indexed set of quadratics to yield the gravitational potential solutions are obtained as indexed sets of solutions of riccati equations

rate of convergence of the mean curvature flow we study the flow mt of a smooth strictly convex hypersurface by its mean curvature in n 1 the surface remains smooth and convex shrinking monotonically until it disappears at a critical time t and point x which is due to huisken this is equivalent to saying that the corresponding rescaled mean curvature flow converges to a sphere sn of radius n in this paper we will study the rate of exponential convergence of a rescaled flow we will present here a method that tells us that the rate of the exponential decay is at least 2 n we can define the arrival time u of a smooth strictly convex n dimensional hypersurface as it moves with normal velocity equal to its mean curvature via u x t if x mt for x int m0 huisken proved that for n 2 u x is c2 near x the case n 1 has been treated by kohn and serfaty 11 they proved c3 regularity of u as a consequence of the obtained rate of convergence of the mean curvature flow we prove that u is not necessarily c3 near x for n 2 we also show that the obtained rate of convergence 2 n which arises from linearizing a mean curvature flow is the optimal one at least for n 2 2007 wiley periodicals inc

church rosser thue systems and formal languages since about 1971 much research has been done on thue systems that have properties that ensure viable and efficient computation the strongest of these is the church rosser property which states that two equivalent strings can each be brought to a unique canonical form by a sequence of length reducing rules in this paper three ways in which formal languages can be defined by thue systems with this property are studied and some general results about the three families of languages so determined are studied

the euler equation and absolute minimizers of l functionals the aronsson euler equation for the functional on w g 1 m i e w 1 with boundary data g is this equation has been derived for smooth absolute minimizers i e a function which minimizes f on every subdomain we prove in this paper that for m 1 n 1 or n 1 m 1 an absolute minimizer of f exists in w g 1 m and for m 1 n 1 any absolute minimizer of f must be a viscosity solution of the aronsson euler equation

finite difference methods for the infinity laplace and p laplace equations abstract we build convergent discretizations and semi implicit solvers for the infinity laplacian and the game theoretical p laplacian the discretizations simplify and generalize earlier ones we prove convergence of the solution of the wide stencil finite difference schemes to the unique viscosity solution of the underlying equation we build a semi implicit solver which solves the laplace equation as each step it is fast in the sense that the number of iterations is independent of the problem size this is an improvement over previous explicit solvers which are slow due to the cfl condition

finite element approximation of level set motion by powers of the mean curvature in this paper we study the level set formulations of certain geometric evolution equations from a numerical point of view specifically we consider the flow by powers greater than one of the mean curvature and the inverse mean curvature flow since the corresponding equations in level set form are quasilinear degenerate and especially possibly singular a regularization method is used in the literature to approximate these equations to overcome the singularities of the equations motivated by the paper 29 which studies the finite element approximation of inverse mean curvature flow we prove error estimates for the finite element approximation of the regularized equations for the flow by powers of the mean curvature we validate the rates with numerical examples additionally the regularization error in the rotational symmetric case for both flows is analyzed numerically all calculations are performed in the 2d case

slope estimate and boundary differentiability of infinity harmonic functions on convex domains abstract we study the boundary differentiability of infinity harmonic functions with given differentiable boundary data on convex domains at a flat point the boundary point where the blow up of the domain is the half space the infinity harmonic function u is differentiable due to a previous result of the first author in hong 2013 at a corner point the boundary point where the blow up of the domain is not the half space an example shows that u is not necessarily differentiable in this paper we establish a slope estimate for u at corner points and provide a necessary and sufficient condition for the differentiability of u at corner points

regularity of the level set flow we showed earlier that the level set function of a monotonic advancing front is twice differentiable everywhere with bounded second derivative and satisfies the equation classically we show here that the second derivative is continuous if and only if the flow has a single singular time where it becomes extinct and the singular set consists of a closed c1 manifold with cylindrical singularities 2017 wiley periodicals inc

counterexample to c1 boundary regularity of infinity harmonic functions abstract in this short note we construct a planar infinity harmonic function u in a c 1 domain with smooth boundary data g however d u is not continuous on the boundary changyou wang and yifeng yu 2012 proved the c 1 boundary regularity of planar infinity harmonic functions provided that and g are c 2 they asked the question whether the result holds when and g are assumed to be c 1 our counterexample answered this question negatively our construction of the counterexample is strongly inspired by the counterexample of dongsheng li and lihe wang 2006 for uniformly elliptic equations

superposition principle on the viscosity solutions of infinity laplace equations abstract we consider the sum of the solutions of two infinity laplace equations in disjoint variables we prove that the superposed function is a viscosity solution of the infinity laplace equation in the extension domains with the sum of inhomogeneous terms if one of the solutions is in the sense of viscosity and the other is in the classical sense we also construct a counterexample to show that the conclusion may not be true if both of the solutions are merely in the viscosity sense

a semi lagrangian scheme for the curve shortening flow in codimension 2 we consider the model problem where a curve in r 3 moves according to the mean curvature flow the curve shortening flow we construct a semi lagrangian scheme based on the feynman kac representation formula of the solutions of the related level set geometric equation the first step is to obtain an approximation of the associated codimension 1 problem formulated by ambrosio and soner where the squared distance from the initial curve is used as initial condition since the e sublevel of this evolution contains the curve the next step is to extract the curve itself by following an optimal trajectory inside each e sublevel we show that this procedure is robust and accurate as long as the fattening phenomenon does not occur moreover it can still single out the physically meaningful solution when it occurs

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

inhomogeneous infinity laplace equation abstract we present the theory of the viscosity solutions of the inhomogeneous infinity laplace equation x i u x j u x i x j 2 u f in domains in r n we show existence and uniqueness of a viscosity solution of the dirichlet problem under the intrinsic condition f does not change its sign we also discover a characteristic property which we call the comparison with standard functions property of the viscosity sub and super solutions of the equation with constant right hand side applying these results and properties we prove the stability of the inhomogeneous infinity laplace equation with nonvanishing right hand side which states the uniform convergence of the viscosity solutions of the perturbed equations to that of the original inhomogeneous equation when both the right hand side and boundary data are perturbed in the end we prove the stability of the well known homogeneous infinity laplace equation x i u x j u x i x j 2 u 0 which states the viscosity solutions of the perturbed equations converge uniformly to the unique viscosity solution of the homogeneous equation when its right hand side and boundary data are perturbed simultaneously

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

characterization of stadium like domains via boundary value problems for the infinity laplacian abstract we give a complete characterization as stadium like domains of convex subsets of r n where a solution exists to serrin type overdetermined boundary value problems in which the operator is either the infinity laplacian or its normalized version in case of the not normalized operator our results extend those obtained in a previous work where the problem was solved under some geometrical restrictions on while in the case of the normalized operator they are new in the normalized case we also show that stadium like domains are precisely the unique convex sets in r n where the solution to a dirichlet problem is of class c 1 1

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

a shorter proof that palindromes are not a church rosser language with extensions to almost confluent and preperfect thue systems in 2002 jurdzinski and lorys settled a long standing conjecture that palindromes are not a church rosser language their proof involved a difficult analysis of computation graphs associated with 2 pushdown stack automata we present a shorter and easier proof in terms of 1 tape turing machines we also discuss how the proof generalises to almost confluent thue systems and the differing powers of church rosser almost confluent and preperfect thue systems in relation to palindromes

tug of war and the infinity laplacian we consider a class of zero sum two player stochastic games called tug of war and use them to prove that every bounded real valued lipschitz function f on a subset y of a length space x admits a unique absolutely minimal am extension to x i e a unique lipschitz extension u x for which lip u u lip u u for all open u x y

reconstruction of the clarke subdifferential by the lasry lions regularizations abstract we prove that the clarke subdifferential of a locally lipschitz function with a growth condition defined on a hilbert space can be represented by the derivatives of its lasry lions regularizations we complete a result of j benoist showing a similar representation for the clarke subdifferential of a lipschitz function in an arbitrary banach space by the clarke subdifferentials of its lasry lions regularizations

cubature for the sphere and the discrete spherical harmonic transform using a result of bannai and damerell it is shown that a cubature formula with n points of degree 2s 4 for the surface of the n dimensional sphere u n cannot achieve the classical lower bound of dim mathcal p s where mathcal p s is the space of all polynomials in n variables of at most degree s restricted to u n this implies that for n 2 there does not exist a cubature based discrete n dimensional spherical harmonic transform for degree s 2 with the same number of points as spectral coefficients

lattice integration on the 7 sphere abstract recently the 7 sphere s7 has been the subject of increased interest for mathematicians and theoretical physicists in this paper we present integration algorithms of several different orders up to 19 which require no more than a few thousand points in this respect our formulae are highly superior to standard methods e g gaussian integration or the use of spherical t designs

uniqueness of lipschitz extensions minimizing the sup norm of the gradient in this paper we examine the problem of minimizing the sup norm of the gradient of a function with prescribed boundary values geometrically this can be interpreted as finding a minimal lipschitz extension due to the weak convexity of the functional associated to this problem solutions are generally nonunique by adopting g aronsson s notion of absolutely minimizing we are able to prove uniqueness by characterizing minimizers as the unique solutions of an associated partial differential equation in fact we actually prove a weak maximum principle for this partial differential equation which in some sense is the euler equation for the minimization problem this is significantly difficult because the partial differential equation is both fully nonlinear and has very degenerate ellipticity to overcome this difficulty we use the weak solutions of m g crandall and p l lions also known as viscosity solutions in conjunction with some arguments using integration by parts

a sharp upper bound for the torsional rigidity of rods by means of web functions using web functions we approximate the dirichlet integral which represents the torsional rigidity of a cylindrical rod with planar convex cross section to this end we use a suitably defined piercing function which enables us to obtain bounds for both the approximate and the exact value of the torsional rigidity when varies we show that the ratio between these two values is always larger than we prove that this is a sharp lower bound and that it is not attained several extremal cases are also analyzed and studied by numerical methods

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

convergent difference schemes for nonlinear parabolic equations and mean curvature motion explicit finite difference schemes are given for a collection of parabolic equations which may have all of the following complex features degeneracy quasilinearity full nonlinearity and singularities in particular the equation of motion by mean curvature is included the schemes are monotone and consistent so that convergence is guaranteed by the general theory of approximation of viscosity solutions of fully nonlinear problems in addition an intriguing new type of nonlocal problem is analyzed which is related to the schemes and another very different sort of approximation is presented as well

numerical integration on the sphere and its effect on the material symmetry of constitutive equations a comparative study in the present paper we consider a class of constitutive models based on numerical integration on the unit sphere the directional behaviour of the quadrature schemes and its effect on the symmetry properties of these constitutive models are studied by subjecting the set of integration points on the sphere to arbitrary rigid rotations we investigate a number of recently proposed integration schemes in application to a full network model of rubber elasticity and to an exponential model for soft tissues in order to assess and compare these schemes statistical methods are presented and applied the analysis discloses a number of integration schemes that offer a good compromise between the numerical error and the number of integration points however as a general result it turns out that numerical integration is prone to introduce strong anisotropy into originally isotropic constitutive equations in particular for highly non linear integrand functions the consequences for application of the investigated class of constitutive models in finite element calculations are highlighted in a benchmark like numerical example copyright 2009 john wiley sons ltd

viscosity solutions with shocks a solution of single nonlinear first order equations may develop jump discontinuities even if initial data is smooth typical examples include a crude model equation describing some bunching phenomena observed in epitaxial growth of crystals as well as conservation laws where jump discontinuities are called shocks conventional theory of viscosity solutions does not apply we introduce a notion of proper viscosity solutions to track whole evolutions for such equations in multi dimensional spaces we establish several versions of comparison principles we also study the vanishing viscosity method to construct a unique global proper solution at least when the evolution is monotone in time or the initial data is monotone in some sense under additional technical assumptions in fact we prove that the graph of approximate solutions converges to that of a proper solution in the hausdorff distance topology such a convergence is also established for conservation laws with monotone data in particular local uniform convergence outside shocks is proved 2001 john wiley sons inc

nonlocal tug of war and the infinity fractional laplacian motivated by the tug of war game studied by peres et al in 2009 we consider a nonlocal version of the game that goes as follows at every step two players pick respectively a direction and then instead of flipping a coin in order to decide which direction to choose and then moving a fixed amount e 0 as is done in the classical case it is an s stable levy process that chooses at the same time both the direction and the distance to travel starting from this game we heuristically derive a deterministic nonlocal integrodifferential equation that we call the infinity fractional laplacian we study existence uniqueness and regularity both for the dirichlet problem and for a double obstacle problem both problems having a natural interpretation as tug of war games 2011 wiley periodicals inc

fattening and comparison principle for level set equations of mean curvature type in this paper we give several applications of the discrete game approach to partial differential equations pdes we first present a rigorous game theoretic proof of fattening phenomenon for motion by curvature with figure eight shaped initial curves without using parabolic pde theory the proof is based on a comparison between the game value and its inverse accompanied by the example of figure eight our second result shows for the stationary equation of mean curvature type in an arbitrary region omega that fattening of positive curvature flow with initial surface partial omega causes loss of the weak comparison principle which partially answers an open question posed by kohn and serfaty in 2006 in addition we prove the existence of solutions of the stationary problem and its game approximation in the absence of comparison principles but under regularity conditions of the flow the main difference between our games and those in other papers is that we take the domain perturbation into consideration

growing context sensitive languages and church rosser languages the growing context sensitive languages gcsl are characterized by a nondeterministic machine model the so called shrinking two pushdown automaton stpda then the deterministic version of this automaton sdtpda is shown to characterize the class of generalized church rosser languages gcrl finally we prove that each growing context sensitive language is accepted in polynomial time by some one way auxiliary pushdown automaton with a logarithmic space bound ow auxpda log poly as a consequence the class of generalized church rosser languages and the class of context free languages are incomparable under set inclusion which verifies a conjecture of mc naughton et al mno88

on near optimal trajectories for a game associated with the laplacian a two player stochastic differential game representation has recently been obtained for solutions of the equation u h in a mathcal c 2 domain with dirichlet boundary condition where h is continuous and takes values in mathbb r setminus 0 under appropriate assumptions including smoothness of u we identify a family of diffusion processes that may arise as the vanishing limit law of the state process when both players play optimally we also identify the limit law of the state process under a sequence of near saddle points