the complexity of mean payoff games on graphs abstract we study the complexity of finding the values and optimal strategies of mean payoff games on graphs a family of perfect information games introduced by ehrenfeucht and mycielski and considered by gurvich karzanov and khachiyan we describe a pseudo polynomial time algorithm for the solution of such games the decision problem for which is in np conp finally we describe a polynomial reduction from mean payoff games to the simple stochastic games studied by condon these games are also known to be in np conp but no polynomial or pseudo polynomial time algorithm is known for them

the speed of reaction diffusion wavefronts in nonsteady media the evolution of the speed of wavefronts for reaction diffusion equations with time varying parameters is analysed we make use of singular perturbative analysis to study the temporal evolution of the speed for pushed fronts the analogy with hamilton jacobi dynamics allows us to consider the problem for pulled fronts which is described by kolmogorov petrovskii piskunov kpp reaction kinetics both analytical studies are in good agreement with the results of numerical solutions

stationary strategies for recursive games we study two person zero sum recursive matrix games framing them in the more general context of nonleavable games our aim is to extend a result of orkin orkin m 1972 recursive matrix games j appl probab 9 813 820 by proving that a uniformly e optimal stationary strategy is available to player i player ii in any recursive game such that the set where the value of the game is strictly greater less than its utility is finite the proof exploits some new connections between nonleavable and leavable games in particular we find sufficient conditions for the existence of e optimal stationary strategies for player i in a leavable game and we use these as a basis for constructing uniformly e optimal stationary strategies for the recursive games with which we are concerned

combinatorial structure and randomized subexponential algorithms for infinite games the complexity of solving infinite games including parity mean payoff and simple stochastic is an important open problem in verification automata and complexity theory in this paper we develop an abstract setting for studying and solving such games based on function optimization over certain discrete structures we introduce new classes of recursively local global rlg and partial recursively local global prlg functions and show that strategy evaluation functions for simple stochastic mean payoff and parity games belong to these classes in this setting we suggest randomized subexponential algorithms appropriate for rlg and prlg function optimization we show that the subexponential algorithms for combinatorial linear programming due to kalai and matousek sharir welzl can be adapted for optimizing the rlg and prlg functions

zero forcing sets and controllability of dynamical systems defined on graphs in this technical note controllability of systems defined on graphs is discussed we consider the problem of controllability of the network for a family of matrices carrying the structure of an underlying directed graph a one to one correspondence between the set of leaders rendering the network controllable and zero forcing sets is established to illustrate the proposed results special cases including path cycle and complete graphs are discussed moreover as shown for graphs with a tree structure the proposed results of the present technical note together with the existing results on the zero forcing sets lead to a minimal leader selection scheme in particular cases

singular limit of a p laplacian reaction diffusion equation with a spatially inhomogeneous reaction term we study singular limit of a p laplacian reaction diffusion equation with a spatially inhomogeneous reaction term the coefficient of the reaction term is much larger than the diffusion coefficient and sharp interfaces appear between two phases we show by matched asymptotic expansions that the limit equation interface equation is a mean curvature flow with drift terms similar to the case p 2

between controllable and uncontrollable the distance between a system a b and the set of uncontrollable systems is the minimum of the smallest singular value of i a b with respect to

aggregating pattern dynamics in a chemotaxis model including growth abstract a population model including diffusion chemotaxis and growth is studied assuming that the diffusion rate and the chemotactic rate are both very small compared with the growth rate we derive a new equation to describe the time evolution of the aggregating region of biological individuals and show the conditions for the existence and stability of radially symmetric equilibrium solutions of the equation which indicate the aggregation of individuals

influence of obstacles on the propagation speeds of nonlinear waves driven by curvature abstract we consider nonlinear waves driven by curvature in the presence of periodically distributed obstacles when the obstacles are small matano et al 2006 proved rigorously that the propagation speed of a wave mainly depends on the opening angle of the obstacles in this paper we first explain in a formal way that when the obstacles are small the propagation behavior of a wave near the obstacles has four different stages in one spatial period they can be described by traveling waves self similar solutions or curvature flows without driving force when the obstacles are large by numerical simulation we show that obstacles always decrease the propagation of waves the propagation speed is decreasing in the opening angles and the sizes of the obstacles but it is not monotone in the homogenization degree of the obstacles the periods of obstacles in a row or the isthmus widths between two rows

recursive markov decision processes and recursive stochastic games we introduce recursive markov decision processes rmdps and recursive simple stochastic games rssgs which are classes of finitely presented countable state mdps and zero sum turn based perfect information stochastic games they extend standard finite state mdps and stochastic games with a recursion feature we study the decidability and computational complexity of these games under termination objectives for the two players one player s goal is to maximize the probability of termination at a given exit while the other player s goal is to minimize this probability in the quantitative termination problems given an rmdp or rssg and probability p we wish to decide whether the value of such a termination game is at least p or at most p in the qualitative termination problem we wish to decide whether the value is 1 the important 1 exit subclasses of these models 1 rmdps and 1 rssgs correspond in a precise sense to controlled and game versions of classic stochastic models including multitype branching processes and stochastic context free grammars where the objective of the players is to maximize or minimize the probability of termination extinction we provide a number of upper and lower bounds for qualitative and quantitative termination problems for rmdps and rssgs we show both problems are undecidable for multi exit rmdps but are decidable for 1 rmdps and 1 rssgs specifically the quantitative termination problem is decidable in pspace for both 1 rmdps and 1 rssgs and is at least as hard as the square root sum problem a well known open problem in numerical computation we show that the qualitative termination problem for 1 rmdps i e a controlled version of branching processes can be solved in polynomial time both for maximizing and minimizing 1 rmdps the qualitative problem for 1 rssgs is in np conp and is at least as hard as the quantitative termination problem for condon s finite state simple stochastic games whose complexity remains a well known open problem finally we show that even for 1 rmdps more general qualitative and quantitative model checking problems with respect to linear time temporal properties are undecidable even for a fixed property

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

the singular limit of the allen cahn equation and the fitzhugh nagumo system abstract we consider an allen cahn type equation of the form u t u e 2 f e x t u where e is a small parameter and f e x t u f u e g e x t u a bistable nonlinearity associated with a double well potential whose well depths can be slightly unbalanced given a rather general initial data u 0 that is independent of e we perform a rigorous analysis of both the generation and the motion of interface more precisely we show that the solution develops a steep transition layer within the time scale of order e 2 ln e and that the layer obeys the law of motion that coincides with the formal asymptotic limit within an error margin of order e this is an optimal estimate that has not been known before for solutions with general initial data even in the case where g e 0 next we consider systems of reaction diffusion equations of the form u t u e 2 f e u v v t d v h u v which include the fitzhugh nagumo system as a special case given a rather general initial data u 0 v 0 we show that the component u develops a steep transition layer and that all the above mentioned results remain true for the u component of these systems

speed of reaction diffusion fronts in spatially heterogeneous media the front speed problem for nonuniform reaction rate and diffusion coefficient is studied by using singular perturbation analysis the geometric approach of hamilton jacobi dynamics and the local speed approach exact and perturbed expressions for the front speed are obtained in the limit of large times for linear and fractal heterogeneities the analytic results have been compared with numerical results exhibiting a good agreement finally we reach a general expression for the speed of the front in the case of smooth and weak heterogeneities

complexity of node coverage games modern software systems may exhibit a nondeterministic behavior due to many unpredictable factors in this work we propose the node coverage game a two player turn based game played on a finite game graph as a formalization of the problem to test such systems each node in the graph represents a functional equivalence class of the software under test sut one player the tester wants to maximize the node coverage measured by the number of nodes visited when exploring the game graphs while his opponent the sut wants to minimize it an optimal test would maximize the cover and it is an interesting problem to find the maximal number of nodes that the tester can guarantee to visit irrespective of the responses of the sut we show that the decision problem of whether the guarantee is less than a given number is np complete we also discuss two extensions of our result and present a testing procedure based on our result when the sut is not so hostile we establish that the decision problem of maximal node coverage guarantee of non deterministic game graphs is np complete we present two variations of node coverage game with np complete decision problem complexity we also provide an adaptive procedure that employs an np procedure to generate test cases with maximal coverage guarantee and reevaluate the mcg and regenerate test cases when the sut is not so hostile

on explosions of solutions to a system of partial differential equations modelling chemotaxis a system of partial differential equations modelling chemotactic aggregation is analysed keller segel model conditions on the system of paramaters are given implying global existence of smooth solutions in two space dimensions and radially symmetric situations explosion of the bacteria concentration in finite time is shown for a class of initial values

convergence in time periodic quasilinear parabolic equations in one space dimension abstract we consider time periodic quasilinear parabolic equations in the domain t x r 2 0 x r t t 0 where the right boundary r t of the spatial interval is a positive function which might be periodic or asymptotically periodic or a function tending to infinity or infinity we show that in the first case that is r t is a periodic function any bounded solution of the equations converges as t to a periodic one in the other three cases any positive bounded solution converges as t to a nonnegative periodic one using such a result we study the long time dynamics of the initial boundary value problem on the half line as well as the stefan free boundary problem of a general heterogeneous reaction diffusion equation also we use the convergence result to study the long time dynamics of the initial boundary value problem for a time periodic mean curvature flow equation

determinacy and optimal strategies in infinite state stochastic reachability games we consider perfect information reachability stochastic games for 2 players on countable graphs such a game is strongly determined if whenever we fix an inequality and a threshold p either player max has a strategy which forces the value of the game to satisfy p against any strategy of player min or min has a strategy which forces the opposite against any strategy of max one of our results shows that whenever one of the players has an optimal strategy in every state of a game then this game is strongly determined this significantly generalises e g recent results on finitely branching reachability games for strong determinacy our methods are substantially different based on which player has the optimal strategy because the roles of the players are not symmetric we also do not restrict the branching of the games and where we provide an extension of results for finitely branching games we had to overcome significant complications and employ new methods as well the other result is finding a subclass of stochastic games where player max has an optimal strategy in each state the subclass is defined by the property that if v is an accumulation point of the set of all values of a game then v 0 these results complement recent results classifying the existence of an optimal strategy for player min and our general strong determinacy theorem applies here as well we also apply our results for max in the context of recently studied one counter stochastic games this work extends a workshop version of this paper which appeared in gandalf 2011 in particular we prove a conjecture raised in that paper for the class of all reachability games

fast reaction slow diffusion and curve shortening the reaction diffusion problem u 1 varepsilon delta u varepsilon 1 v n u quad u x 0 varepsilon g x quad partial n u 0 text on partial omega for a vector u x t varepsilon is considered in a domain omega in r m an asymptotic solution is constructed for varepsilon small it shows that at each x u tends quickly to a minimum of v u when v has several minima when u tends to a piecewise constant function boundary layer expansions are constructed around the resulting surfaces of discontinuity or fronts each front is found to move along its normal with a constant velocity determined by the discontinuity v in v across it when v 0 the front s normal velocity is varepsilon kappa where kappa is its mean curvature the motion of fronts in this manner is studied for arcs in the plane which are normal to partial omega at their endpoints and for fronts that are closed curves it is shown a front can shrink to a point in

the complexity of stochastic games abstract we consider the complexity of stochastic games simple games of chance played by two players we show that the problem of deciding which player has the greatest chance of winning the game is in the class np co np

front propagation and phase field theory the connection between the weak theories for a class of geometric equations and the asymptotics of appropriately rescaled reaction diffusion equations is rigorously established two different scalings are studied in the first the limiting geometric equation is a first order equation in the second it is a generalization of the mean curvature equation intrinsic definitions for the geometric equations are obtained and uniqueness under a geometric condition on the initial surface is proved in particular in the case of the mean curvature equation this condition is satisfied by surfaces that are strictly starshaped that have positive mean curvature or that satisfy a condition that interpolates between the positive mean curvature and the starshape conditions

qualitative analysis of concurrent mean payoff games we consider concurrent games played by two players on a finite state graph where in every round the players simultaneously choose a move and the current state along with the joint moves determine the successor state we study the most fundamental objective for concurrent games namely mean payoff or limit average objective where a reward is associated to each transition and the goal of player 1 is to maximize the long run average of the rewards and the objective of player 2 is strictly the opposite i e the games are zero sum the path constraint for player 1 could be qualitative i e the mean payoff is the maximal reward or arbitrarily close to it or quantitative i e a given threshold between the minimal and maximal reward we consider the computation of the almost sure resp positive winning sets where player 1 can ensure that the path constraint is satisfied with probability 1 resp positive probability almost sure winning with qualitative constraint exactly corresponds to the question of whether there exists a strategy to ensure that the payoff is the maximal reward of the game our main results for qualitative path constraints are as follows 1 we establish qualitative determinacy results that show that for every state either player 1 has a strategy to ensure almost sure resp positive winning against all player 2 strategies or player 2 has a spoiling strategy to falsify almost sure resp positive winning against all player 1 strategies 2 we present optimal strategy complexity results that precisely characterize the classes of strategies required for almost sure and positive winning for both players and 3 we present quadratic time algorithms to compute the almost sure and the positive winning sets matching the best known bound of the algorithms for much simpler problems such as reachability objectives for quantitative constraints we show that a polynomial time solution for the almost sure or the positive winning set would imply a solution to a long standing open problem of solving the value problem of turn based deterministic mean payoff games that is not known to be solvable in polynomial time

existence of value and randomized strategies in zero sum discrete time stochastic dynamic games two players with conflicting objectives are simultaneously controlling a discrete time stochastic system the goal of this paper is to analyze such zero sum discrete time stochastic systems when the two players are allowed to use randomized strategies previous results have been restricted to systems with finite or compact state spaces such restrictions are usually untenable from the point of view of applications since many applications frequently use either the integers or mathbb r n as a state space our results are proved for complete separable metric spaces which are very useful for applications all previously known results emerge as special cases of our results in addition a variety of conjectures and open problems are resolved regarding the existence of a value function its properties such as borel measurability or continuity and the existence for either or both players of optimal or varepsilon optimal stationary strategies

concurrent reachability games we consider concurrent two player games with reachability objectives in such games at each round player 1 and player 2 independently and simultaneously choose moves and the two choices determine the next state of the game the objective of player 1 is to reach a set of target states the objective of player 2 is to prevent this these are zero sum games and the reachability objective is one of the most basic objectives determining the set of states from which player 1 can win the game is a fundamental problem in control theory and system verification there are three types of winning states according to the degree of certainty with which player 1 can reach the target from type 1 states player 1 has a deterministic strategy to always reach the target from type 2 states player 1 has a randomized strategy to reach the target with probability 1 from type 3 states player 1 has for every real e 0 a randomized strategy to reach the target with probability greater than email protected we show that for finite state spaces all three sets of winning states can be computed in polynomial time type 1 states in linear time and type 2 and type 3 states in quadratic time the algorithms to compute the three sets of winning states also enable the construction of the winning and spoiling strategies

singular limit of a reaction diffusion equation with a spatially inhomogeneous reaction term we study reaction diffusion equations with a spatially inhomogeneous reaction term if the coefficient of these reaction term is much larger than the diffusion coefficient a sharp interface appears between two different phases we show that the equation of motion of such an interface involves a drift term despite the absence of drift in the original diffusion equations in particular we show that the same rich spatial patterns observed for a chemotaxis growth model can be realized by a system without a drift term

the observability radius of networks this paper studies the observability radius of network systems which measures the robustness of a network to perturbations of the edges we consider linear networks where the dynamics are described by a weighted adjacency matrix and dedicated sensors are positioned at a subset of nodes we allow for perturbations of certain edge weights with the objective of preventing observability of some modes of the network dynamics to comply with the network setting our work considers perturbations with a desired sparsity structure thus extending the classic literature on the observability radius of linear systems the paper proposes two sets of results first we propose an optimization framework to determine a perturbation with smallest frobenius norm that renders a desired mode unobservable from the existing sensor nodes second we study the expected observability radius of networks with given structure and random edge weights we provide fundamental robustness bounds dependent on the connectivity properties of the network and we analytically characterize optimal perturbations of line and star networks showing that line networks are inherently more robust than star networks

generation and propagation of interfaces for reaction diffusion equations abstract this paper is concerned with the asymptotic behavior as e 0 of the solution ue of the reaction diffusion equation in r n r u t 1 2 0 where is the derivative of a bistable potential we show that if the initial data u 0 has values in both domains of attraction of the potential then an interface will develop in a short time o e 2 ln e we also show that if the wells of the potential are of equal depth then this interface will propagate with normal velocity equal to its mean curvature our result is valid as long as the interface remains smooth if the initial interface is compact and n 2 then the interface will disappear in a finite time but not if n 1 in case the depths of the wells are not equal then in order to obtain get reasonable results we must work on the scaled time s t e slower time scale in this scale we show that the interface moves with a constant speed proportional to the difference of the depths of the two wells along the normal towards the domain of the deeper well this result is valid for all s 0 and does not actually depend on the regularity of the interface we also extend the above results to the homogeneous second initial boundary value problem in case the depths of the wells are equal and the initial interface is orthogonal to the boundary we prove that the interface moves with normal velocity equal to its mean curvature provided that there is a family of hypersurfaces which moves according to their mean curvature and intersect the boundary orthogonally

phase transitions and generalized motion by mean curvature we study the limiting behavior of solutions to appropriately rescaled versions of the allen cahn equation a simplified model for dynamic phase transitions we rigorously establish the existence in the limit of a phase antiphase interface evolving according to mean curvature motion this assertion is valid for all positive time the motion interpreted in the generalized sense of evans spruck and chen giga goto after the onset of geometric singularities

the effect of a singular perturbation on nonconvex variational problems we study the effect of a singular perturbation on certain nonconvex variational problems the goal is to characterize the limit of minimizers as some perturbation parameter 0 the technique utilizes the notion of convergence of variational problems developed by de giorgi the essential idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem in so doing one characterizes the limit of minimizers as the solution of a new variational problem for the cases considered here these new problems have a simple geometric nature involving minimal surfaces and geodesics

on the convergence rate of a reaction diffusion equation with chemotaxis to a viscosity solution abstract the singular limit of an advection reaction diffusion equation arising from a chemotaxis growth model equation is investigated for a very specific initial data the convergence to the viscosity solution of a limit free boundary problem is known henry et al 1999 15 in this work we not only allow rather general initial data but also construct refined barriers in some special cases these barriers provide an o e ln e estimate of the location and the thickness measured in space time of the transition layers

local minimisers and singular perturbations we construct local minimisers to certain variational problems the method is quite general and relies on the theory of convergence the approach is demonstrated through the model problem it is shown that in certain nonconvex domains n and for e small there exist nonconstant local minimisers u e satisfying u e 1 except in a thin transition layer the location of the layer is determined through the requirement that in the limit u e u 0 the hypersurface separating the states u 0 1 and u 0 1 locally minimises surface area generalisations are discussed with for example vector valued u and anisotropic perturbations replacing u 2

an analysis of stochastic shortest path problems we consider a stochastic version of the classical shortest path problem whereby for each node of a graph we must choose a probability distribution over the set of successor nodes so as to reach a certain destination node with minimum expected cost the costs of transition between successive nodes can be positive as well as negative we prove natural generalizations of the standard results for the deterministic shortest path problem and we extend the corresponding theory for undiscounted finite state markovian decision problems by removing the usual restriction that costs are either all nonnegative or all nonpositive

on the structured distance to uncontrollability this article is concerned with the structured distance to uncontrollability of a linear time invariant system and relates this concept to a variation of the value the developed framework is applied to derive computational expressions for the class of real perturbations as well as for hermitian symmetric and skew symmetric perturbations in a relatively simple manner examples demonstrate that the structured distance can differ from the standard unstructured distance to uncontrollability by an arbitrary amount it is also shown how systems of higher order can be addressed

total least squares for affinely structured matrices and the noisy realization problem structured rank deficient matrices arise in many applications in signal processing system identification and control theory the author discusses the structured total least squares stls problem which is the problem of approximating affinely structured matrices i e matrices affine in the parameters by similarly structured rank deficient ones while minimizing an l sub 2 error criterion it is shown that the optimality conditions lead to a nonlinear generalized singular value decomposition which can be solved via an algorithm that is inspired by inverse iteration next the author concentrates on the so called l sub 2 optimal noisy realization problem which is equivalent with approximating a given data sequence by the impulse response of a finite dimensional time invariant linear system of a given order this can be solved as a structured total least squares problem it is shown with some simple counter examples that classical algorithms such as the steiglitz mcbride 1965 iterative quadratic maximum likelihood and cadzow s 1988 iteration do not converge to the optimal l sub 2 solution despite misleading claims in the literature

quantitative fair simulation games simulation is an attractive alternative for language inclusion for automata as it is an under approximation of language inclusion but usually has much lower complexity for non deterministic automata while language inclusion is pspace complete simulation can be computed in polynomial time simulation has also been extended in two orthogonal directions namely 1 fair simulation for simulation over specified set of infinite runs and 2 quantitative simulation for simulation between weighted automata again while fair trace inclusion is pspace complete fair simulation can be computed in polynomial time for weighted automata the quantitative language inclusion problem is undecidable for mean payoff automata and the decidability is open for discounted sum automata whereas the quantitative simulation reduce to mean payoff games and discounted sum games which admit pseudo polynomial time algorithms in this work we study quantitative simulation for weighted automata with buchi acceptance conditions i e we generalize fair simulation from non weighted automata to weighted automata we show that imposing buchi acceptance conditions on weighted automata changes many fundamental properties of the simulation games for example whereas for mean payoff and discounted sum games the players do not need memory to play optimally we show in contrast that for simulation games with buchi acceptance conditions i for mean payoff objectives optimal strategies for both players require infinite memory in general and ii for discounted sum objectives optimal strategies need not exist for both players while the simulation games with buchi acceptance conditions are more complicated e g due to infinite memory requirements for mean payoff objectives as compared to their counterpart without buchi acceptance conditions we still present pseudo polynomial time algorithms to solve simulation games with buchi acceptance conditions for both weighted mean payoff and weighted discounted sum automata

real controllability stabilizability radius of lti systems in this note we present a simple and readily computable formula for the calculation of the real controllability stabilizability and observability detectability radii for linear time invariant systems

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

alternating time temporal logic temporal logic comes in two varieties linear time temporal logic assumes implicit universal quantification over all paths that are generated by the execution of a system branching time temporal logic allows explicit existential and universal quantification over all paths we introduce a third more general variety of temporal logic alternating time temporal logic offers selective quantification over those paths that are possible outcomes of games such as the game in which the system and the environment alternate moves while linear time and branching time logics are natural specification languages for closed systems alternating time logics are natural specification languages for open systems for example by preceding the temporal operator eventually with a selective path quantifier we can specify that in the game between the system and the environment the system has a strategy to reach a certain state the problems of receptiveness realizability and controllability can be formulated as model checking problems for alternating time formulas depending on whether or not we admit arbitrary nesting of selective path quantifiers and temporal operators we obtain the two alternating time temporal logics atl and atl atl and atl are interpreted over concurrent game structures every state transition of a concurrent game structure results from a choice of moves one for each player the players represent individual components and the environment of an open system concurrent game structures can capture various forms of synchronous composition for open systems and if augmented with fairness constraints also asynchronous composition over structures without fairness constraints the model checking complexity of atl is linear in the size of the game structure and length of the formula and the symbolic model checking algorithm for ctl extends with few modifications to atl over structures with weak fairness constraints atl model checking requires the solution of 1 pair rabin games and can be done in polynomial time over structures with strong fairness constraints atl model checking requires the solution of games with boolean combinations of buchi conditions and can be done in pspace in the case of atl the model checking problem is closely related to the synthesis problem for linear time formulas and requires doubly exponential time

traveling waves of a curvature flow in almost periodic media abstract in a plane media with almost periodic vertical striations we study a curvature flow and construct two kinds of traveling waves one having a straight line like profile and the other having a v shaped profile for each of the first kind traveling waves its profile is the graph of a function whose derivative is almost periodic for each of the second kind traveling waves its profile is like a pulsating cone whose two tails approach asymptotically the profiles of the first kind traveling waves also we consider a homogenization problem and provide an explicit formula for the homogenized traveling speed

properties of numerical algorithms related to computing controllability the numerical properties of some methods for computing controllability are used in an expository way to motivate a wider understanding of numerical computations in particular the numerical rank of a matrix the numerical stability of algorithms the sensitivity of problems and the scaling of problems are discussed a numerically stable algorithm is given for computing controllability but it is pointed out that a measure of the distance of the given system from the nearest uncontrollable system would be more useful and this appears to be an open computational problem

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

singular limits of spatially inhomogeneous convection reaction diffusion equations we study convection reaction diffusion equations which have spatially inhomogeneous steady states when the coefficients of the reaction terms are much larger than the diffusion coefficients sharp interfaces appear between two stable inhomogeneous steady states by using the method of matched asymptotic expansions we derive the equations of the motion of such interfaces which depend on the mean curvature of the interfaces and depend on the inhomogeneous coefficients locally

motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic allen cahn equation we consider the spatially inhomogeneous and anisotropic reaction diffusion equation ut m x 1 div m x ap x u e 2 f u involving a small parameter e 0 and a bistable nonlinear term whose stable equilibria are 0 and 1 we use a finsler metric related to the anisotropic diffusion term and work in relative geometry we prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces more precisely we show that within the time scale of order e2 ln e the unique weak solution u e develops a steep transition layer that separates the regions u e 0 and u e 1 then on a much slower time scale the layer starts to propagate consequently as e 0 the solution u e converges almost everywhere a e to 0 in t and 1 in t where t and t are sub domains of separated by an interface t whose motion is driven by its anisotropic mean curvature we also prove that the thickness of the transition layer is of order e

qualitative determinacy and decidability of stochastic games with signals we consider two person zero sum stochastic games with signals a standard model of stochastic games with imperfect information the only source of information for the players consists of the signals they receive they cannot directly observe the state of the game nor the actions played by their opponent nor their own actions we are interested in the existence of almost surely winning or positively winning strategies under reachability safety buchi or co buchi winning objectives and the computation of these strategies when the game has finitely many states and actions we prove two qualitative determinacy results first in a reachability game either player 1 can achieve almost surely the reachability objective or player 2 can achieve surely the dual safety objective or both players have positively winning strategies second in a buchi game if player 1 cannot achieve almost surely the buchi objective then player 2 can ensure positively the dual co buchi objective we prove that players only need strategies with finite memory the number of memory states needed to win with finite memory strategies ranges from one corresponding to memoryless strategies to doubly exponential with matching upper and lower bounds together with the qualitative determinacy results we also provide fix point algorithms for deciding which player has an almost surely winning or a positively winning strategy and for computing an associated finite memory strategy complexity ranges from exptime to 2exptime with matching lower bounds our fix point algorithms also enjoy a better complexity in the cases where one of the players is better informed than their opponent our results hold even when players do not necessarily observe their own actions the adequate class of strategies in this case is mixed or general strategies they are equivalent behavioral strategies are too restrictive to guarantee determinacy it may happen that one of the players has a winning general strategy but none of them has a winning behavioral strategy on the other hand if a player can observe their actions then general mixed and behavioral strategies are equivalent finite memory strategies are sufficient for determinacy to hold provided that randomized memory updates are allowed

necessity of internal and boundary bulk balance law for existence of interfaces for an elliptic system with nonlinear boundary condition abstract for a system of stationary solutions to a reaction diffusion equations with small diffusion coefficient and nonlinear flux boundary condition we prove that the bulk balance law not only on the domain but on its boundary as well is a necessary condition for formation of internal and boundary layers

measuring how far a controllable system is from an uncontrollable one the main goal of this note is to explore some of the properties of a controllable system which in a precise sense to be defined is near an uncontrollable pair to do this we first discuss the notion of the distance between a controllable system and the nearest uncontrollable one in both real and complex cases 2 5 when this distance is relatively small we show to what extent this distance corresponds to certain properties of the uncontrollability matrix and the reachability grammian

a comprehensive method for reachability analysis of uncertain nonlinear hybrid systems reachability analysis of nonlinear uncertain hybrid systems i e continuous discrete dynamical systems whose continuous dynamics guard sets and reset functions are defined by nonlinear functions can be decomposed in three algorithmic steps computing the reachable set when the system is in a given operation mode computing the discrete transitions i e detecting and localizing when and where the continuous flowpipe intersects the guard sets and aggregating the multiple trajectories that result from an uncertain transition once the whole flow pipe has transitioned so that the algorithm can resume this paper proposes a comprehensive method that provides a nicely integrated solution to the hybrid reachability problem at the core of the method is the concept of mspb i e geometrical object obtained as the minkowski sum of a parallelotope and an axes aligned box mspb are a way to control the over approximation of the taylor s interval integration method as they happen to be a specific type of zonotope they articulate perfectly with the zonotope bounding method that we propose to enclose in an optimal way the set of flowpipe trajectories generated by the transition process the method is evaluated both theoretically by analyzing its complexity and empirically by applying it to well chosen hybrid nonlinear examples

a pseudo polynomial algorithm for mean payoff stochastic games with perfect information and few random positions abstract we consider two person zero sum stochastic mean payoff games with perfect information or bwr games given by a digraph g v e with local rewards r e z and three types of positions black v b white v w and random v r forming a partition of v it is a long standing open question whether a polynomial time algorithm for bwr games exists or not even when v r 0 in fact a pseudo polynomial algorithm for bwr games would already imply their polynomial solvability in this paper 1 we show that bwr games with a constant number of random positions can be solved in pseudo polynomial time more precisely in any bwr game with v r o 1 a saddle point in uniformly optimal pure stationary strategies can be found in time polynomial in v w v b the maximum absolute local reward and the common denominator of the transition probabilities