r max a general polynomial time algorithm for near optimal reinforcement learning r max is a very simple model based reinforcement learning algorithm which can attain near optimal average reward in polynomial time in r max the agent always maintains a complete but possibly inaccurate model of its environment and acts based on the optimal policy derived from this model the model is initialized in an optimistic fashion all actions in all states return the maximal possible reward hence the name during execution it is updated based on the agent s observations r max improves upon several previous algorithms 1 it is simpler and more general than kearns and singh s e3 algorithm covering zero sum stochastic games 2 it has a built in mechanism for resolving the exploration vs exploitation dilemma 3 it formally justifies the optimism under uncertainty bias used in many rl algorithms 4 it is simpler more general and more efficient than brafman and tennenholtz s lsg algorithm for learning in single controller stochastic games 5 it generalizes the algorithm by monderer and tennenholtz for learning in repeated games 6 it is the only algorithm for learning in repeated games to date which is provably efficient considerably improving and simplifying previous algorithms by banos and by megiddo

nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

on the ideal theory of graphs we study algebras defined by finite sets g m1 mq of monomials of a polynomial ring r there are two basic algebras i k g k m1 mq the k subalgebra of r spanned by the mi and ii the quotient ring r i g where i g m1 mq they come together in the construction of the rees algebra r i g of the ideal i g the emphasis is almost entirely on sets of squarefree monomials of degree two and their attached graphs the main results are assertions about the cohen macaulay behaviour of the koszul homology of i g and how normality or cohen macaulayness of one of the algebras can be read off the properties of the graph or in the other algebra

nonexistence of global solutions to critical semilinear wave equations in exterior domain in high dimensions abstract the aim of this paper is to study the long time behavior of solutions to the initial boundary value problem of critical semilinear wave equations with small data in exterior domain in high dimensions n 5 we prove that solutions cannot exist globally in time the novelty is that we first use a cutoff function to transform the exterior problem to a cauchy problem then we divide the time into two intervals in one time interval we establish an energy estimate to control the error terms while in the other interval the asymptotic behavior of two test functions is used furthermore we obtain the upper bound of the lifespan by following the idea of zhou and han 2014

a probabilistic plan recognition algorithm based on plan tree grammars we present the phatt algorithm for plan recognition unlike previous approaches to plan recognition phatt is based on a model of plan execution we show that this clarifies several difficult issues in plan recognition including the execution of multiple interleaved root goals partially ordered plans and failing to observe actions we present the phatt algorithm s theoretical basis and an implementation based on tree structures we also investigate the algorithm s complexity both analytically and empirically finally we present phatt s integrated constraint reasoning for parametrized actions and temporal constraints

lifespan of solutions to semilinear damping wave equations in de sitter spacetime abstract this paper studies blowup phenomena for two kind of initial value problem of semilinear damping wave equations in de sitter spacetime the first problem comes from the latter problem of conjecture of strauss in de sitter spacetime and the second problem arises from the latter problem of conjecture of glassey in de sitter spacetime we give blowup and lifespan of solutions on those two problems comparing with non damping case established in yan 2018 one can see that the damping term with a positive bounded time dependent coefficient does not effect blowup of solutions meanwhile we establish a new ode blowup result concerning damping term

existence and nonexistence of global solutions of some system of semilinear wave equations abstract an initial boundary value problem for systems of semilinear wave equations in a bounded domain is considered we prove the global existence uniqueness and blow up of solutions by energy methods and give some estimates for the lifespan of solutions

the lifespan of radially symmetric solutions to nonlinear systems of odd dimensional wave equations abstract the sharp upper bound of the lifespan of the solution to p q systems in high space dimensions has been obtained in the lifespan of solutions to nonlinear systems of high dimensional wave equation preprint for the sub critical case they also proved the lower bound of the lifespan with a small error term the present paper remove this error term and give the sharp lifespan under the assumption of the radial symmetricity of the solution

blow up for a weakly coupled system of semilinear damped wave equations in the scattering case with power nonlinearities abstract in this work we study the blow up of solutions of a weakly coupled system of damped semilinear wave equations in the scattering case with power nonlinearities we apply an iteration method to study both the subcritical case and the critical case in the subcritical case our approach is based on lower bounds for the space averages of the components of local solutions in the critical case we use the slicing method and a couple of auxiliary functions recently introduced by wakasa yordanov to modify the definition of the functionals with the introduction of weight terms in particular we find as critical curve for the pair p q of the exponents in the nonlinear terms the same one as for the weakly coupled system of semilinear wave equations with power nonlinearities

small data blow up for semilinear klein gordon equations energy heuristics predict global existence from small data for semi linear klein gordon systems when the nonlinearity has order p 1 2 n we complement these heuristics and previous rigorous existence results by demonstrating blow up from small data for p 1 2 n

improved kato s lemma on ordinary differential inequality and its application to semilinear wave equations abstract we are interested in the upper bound of the lifespan of solutions of semilinear wave equations for the sub critical case in high dimensions it has been believed that the basic tools of its analysis are kato s lemma on ordinary differential inequalities and the rescaling argument in the functional method but there is a small lack of delicate analysis and no published paper about this here we give a simple alternative proof by means of improved kato s lemma without any rescaling argument

nonexistence of global solutions of a nonlinear hyperbolic system consider the initial value problem utt au ivip vtt av u q x e r t t o u x 0 f x v x 0 h x ut x 0 g x vt x 0 x with 1 0 we show that there exists a bound b n oo such that if 1 pq b n all nontrivial solutions with compact support blow up in finite time

lifespan of semilinear wave equation with scale invariant dissipation and mass and sub strauss power nonlinearity abstract in this paper we study the blow up of solutions for semilinear wave equations with scale invariant dissipation and mass in the case in which the model is somehow wave like a strauss type critical exponent is determined as the upper bound for the exponent in the nonlinearity in the main theorems two blow up results are obtained for the subcritical case and for the critical case respectively in both cases an upper bound lifespan estimate is given

lifespan estimates of solutions to quasilinear wave equations with scattering damping abstract the formation of singularity of solutions to the quasilinear wave equations with scattering damping and divergence form nonlinearities is investigated the upper bound lifespan estimates of solutions to the problems with variable coefficient in the critical case and constant coefficient in the sub critical case are established by applying the rescaled test function method and iterative method

weighted strichartz estimates and global existence for semilinear wave equations in this paper we prove a sharp global existence theorem in all dimensions for nonlinear wave equations with power type nonlinearities the proof is based on a weighted strichartz estimate involving powers of the lorentz distance

an example of blow up for the complex kdv equation and existence beyond the blow up the kdv equation with smooth complex initial data that blows up in a finite amount of time is studied first examples are given using the elliptic solutions of airault mckean and moser second the kdv flow of spectral coordinates is described showing that blow up occurs when eigenvalues run to infinity in finite time but eigenvalue collisions are harmless and do not cause a singularity eigenvalue collisions and blow up are studied numerically finally the solutions are continued through the singularity by continuing the coordinate flows on the appropriate riemann surface

an elementary proof of strauss conjecture abstract in this paper we consider the cauchy problem for semilinear wave equation in high dimensions first we use a morawetz energy estimate which is obtained by integrating on the light cone to get a weighted l 2 l 2 estimate of the solution and then give an elementary proof of the weighted strichartz estimate in georgiev et al 3 hence the strauss conjecture we also obtain a variant of the weighted strichartz estimates and give the sharp estimate of the lifespan for the semilinear wave equation with subcritical nonlinearity

an elementary proof of the blow up for semilinear wave equation in high space dimensions abstract this paper concerns the blow up of solutions to u tt u u p in high dimensions for n 4 and 1 p p 0 n where p 0 n is a critical exponent we proved that the solutions blow up in finite time by estimating the solutions near the wave front using elementary inequalities

finite time blow up for critical wave equations in high dimensions we prove that solutions to the critical wave equation 1 1 with dimension n 4 can not be global if the initial values are positive somewhere and nonnegative this completes the solution to the famous strauss conjecture about semilinear wave equations of the form u t2u u p 0 the rest of the cases the lower dimensional case n 3 and the sub or super critical cases were settled many years earlier by the work of several authors

global existence for the dissipative wave equations with space time dependent potential abstract this research paper aims at studying the long time behavior of solutions for wave equations with focusing nonlinearity we consider p c r n p n 2 n 2 and separable potential a t x x t with x 1 x and t 1 t satisfying conditions 3 4 which are given below the main result is that if the exponent p lies between p c r 1 4 1 2 n 1 2 p n 2 n 2 the solutions are global for sufficiently small data also the precise decay estimates for the energy l 2 and l p 1 norms of solutions in the region of global solutions have been found

on the existence and nonexistence of global solutions for certain semilinear exterior problems with nontrivial robin boundary conditions abstract we consider three types of semilinear equations elliptic parabolic and hyperbolic posed in the n dimensional exterior domain r n d where n 2 and d is the closed unit ball in r n a nontrivial robin boundary condition is imposed on the boundary of d using a test function approach with judicious choices of the test functions we show that the considered problems share a common critical behavior we discuss separately the cases n 2 and n 3 moreover in the case n 3 the dependence of the critical exponent on initial data is discussed to the best our knowledge the study of the critical behavior in an exterior domain with a nontrivial robin boundary condition has never been studied in the literature

blow up and critical exponents for nonlinear hyperbolic equations we prove nonexistence results for the cauchy problem for the abstract hyperbolic equation in a banach space x utt f u t 0 u 0 u0 ut 0 u1 where f x r is a c1 function several applications to the second and higher order hyperbolic equations with local and nonlocal nonlinearities are presented we also describe an approach to kato s and john s critical exponents for the semilinear equations ut u b x t u p p 1 which are responsible for phenomena of stability unstability blow up and asymptotic behaviour

nash q learning for general sum stochastic games we extend q learning to a noncooperative multiagent context using the framework of general sum stochastic games a learning agent maintains q functions over joint actions and performs updates based on assuming nash equilibrium behavior over the current q values this learning protocol provably converges given certain restrictions on the stage games defined by q values that arise during learning experiments with a pair of two player grid games suggest that such restrictions on the game structure are not necessarily required stage games encountered during learning in both grid environments violate the conditions however learning consistently converges in the first grid game which has a unique equilibrium q function but sometimes fails to converge in the second which has three different equilibrium q functions in a comparison of offline learning performance in both games we find agents are more likely to reach a joint optimal path with nash q learning than with a single agent q learning method when at least one agent adopts nash q learning the performance of both agents is better than using single agent q learning we have also implemented an online version of nash q learning that balances exploration with exploitation yielding improved performance

a shift in the strauss exponent for semilinear wave equations with a not effective damping abstract in this note we study the global existence of small data solutions to the cauchy problem for the semilinear wave equation with a not effective scale invariant damping term namely v t t v 2 1 t v t v p v 0 x v 0 x v t 0 x v 1 x where p 1 n 2 we prove blow up in finite time in the subcritical range p 1 p 2 n and existence theorems for p p 2 n n 2 3 in this way we find the critical exponent for small data solutions to this problem our results lead to the conjecture p 2 n p 0 n 2 for n 2 where p 0 n is the strauss exponent for the classical semilinear wave equation with power nonlinearity

posterior predictive p values extending work of rubin this paper explores a bayesian counterpart of the classical p value namely a tail area probability of a test statistic under a null hypothesis the bayesian formulation using posterior predictive replications of the data allows a test statistic to depend on both data and unknown nuisance parameters and thus permits a direct measure of the discrepancy between sample and population quantities the tail area probability for a test statistic is then found under the joint posterior distribution of replicate data and the nuisance parameters both conditional on the null hypothesis this posterior predictive p value can also be viewed as the posterior mean of a classical p value averaging over the posterior distribution of nuisance parameters under the null hypothesis and thus it provides one general method for dealing with nuisance parameters two classical examples including the behrens fisher problem are used to illustrate the posterior predictive p value and some of its interesting properties which also reveal a new bayesian interpretation for some classical p values an application to multiple imputation inference is also presented a frequency evaluation shows that in general if the replication is defined by new nuisance parameters and new data then the type i frequentist error of an level posterior predictive test is often close to but less than and will never exceed 2

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

on existence and scattering with minimal regularity for semilinear wave equations abstract we prove existence and scattering results for semilinear wave equations with low regularity data we also determine the minimal regularity that is needed to ensure local existence and well posedness and we give counterexamples to well posedness more specifically we show that equations of the type u u p with initial data u u t in r n 1 r n have a local solution if p n and we construct counterexamples if p n the existence results rely on mixed norm space time estimates of strichartz type

existence and blow up of small amplitude nonlinear waves with a sign changing potential abstract we study the nonlinear wave equation with a sign changing potential in any space dimension if the potential is small and rapidly decaying then the existence of small amplitude solutions is driven by the nonlinear term if the potential induces growth in the linearized problem however solutions that start out small may blow up in finite time

combined effects of two nonlinearities in lifespan of small solutions to semi linear wave equations this paper investigates the combined effects of two distinctive power type nonlinear terms with parameters p q 1 in the lifespan of small solutions to semi linear wave equations we determine the full region of p q to admit global existence of small solutions at least for spatial dimensions n 2 3 moreover for many p q when there is no global existence we obtain sharp lower bound of the lifespan which is of the same order as the upper bound of the lifespan

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

nonexistence of global solutions to semilinear wave equations in high dimensions on considere le probleme de cauchy pour l equation d onde semi lineaire 2 t 2 u x t u x t p p 1 u x o f x u t x o g x avec f g c o r n

on abstract strichartz estimates and the strauss conjecture for nontrapping obstacles we establish the strauss conjecture concerning small data global existence for nonlinear wave equations in the setting of exterior domains to compact obstacles for space dimensions n 3 and 4 the obstacle is assumed to be nontrapping and the solution is assumed to satisfy either dirichlet or neumann conditions along the boundary of the obstacle the key step in the proof is establishing certain abstract strichartz estimates for the linear wave equation on exterior domains

on the solvability of one multidimensional version of the first darboux problem for some nonlinear wave equations in the present paper for wave equations with power nonlinearity we investigate the problem of the existence or nonexistence of global solutions of a multidimensional version of the first darboux problem in the conic domain

global nonexistence results for a class of hyperbolic systems abstract our concern in this paper is to prove blow up results to the non autonomous nonlinear system of wave equations u t t u a t x v p v t t v b t x u q t 0 x r n in any space dimension we show that a curve f p q 0 depending on the space dimension on the exponents p q and on the behavior of the functions a t x and b t x exists such that all nontrivial solutions to the above system blow up in a finite time whenever f p q 0 our method of proof uses some estimates developed by galaktionov and pohozaev in 11 for a single non autonomous wave equation enabling us to obtain a system of ordinary differential inequalities from which the desired result is derived our result generalizes some important results such as the ones in del santo et al 1996 12 and galaktionov and pohozaev 2003 11 the advantage here is that our result applies to a wide variety of problems

p values for composite null models abstract the problem of investigating compatibility of an assumed model with the data is investigated in the situation when the assumed model has unknown parameters the most frequently used measures of compatibility are p values based on statistics t for which large values are deemed to indicate incompatibility of the data and the model when the null model has unknown parameters p values are not uniquely defined the proposals for computing a p value in such a situation include the plug in and similar p values on the frequentist side and the predictive and posterior predictive p values on the bayesian side we propose two alternatives the conditional predictive p value and the partial posterior predictive p value and indicate their advantages from both bayesian and frequentist perspectives

long time existence for small amplitude semilinear wave equations we prove that a certain class of semilinear wave equations has global solutions if the initial data is small these existence results rely on mixed norm angular radial space time estimates more specifically we consider power nonlinearities u f p u where f p u u p and 2 t denotes the d alembertian on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i in 1979 john investigated this equation for small initial data in dimension n 3 he proved that if p 1 2 then there is a global solution whereas for p n dimensions and the critical power p c above which global solutions exist should be the positive root of n 1 p 2 c n 1 p c 2 0 sideris proved that for all n the solutions can blow up if p p c recently zhou proved that for n 4 one has global solutions if p p c 2 here we prove that for n 8 one has global solutions if p p c furthermore we proved that for all n one has global solutions if p p c and the initial data are spherically symmetric in the radial case we use l p estimates derived from the explicit form of the fundamental solution using inequalities from classical analysis in the nonradial case we write the fundamental solution in polar coordinates and use fourier integral and special function estimates to handle the angular part

the sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions the final open part of strauss conjecture on semilinear wave equations was the blow up theorem for the critical case in high dimensions this problem was solved by yordanov and zhang 2006 18 or zhou 2007 21 independently but the estimate for the lifespan the maximal existence time of solutions was not clarified in both papers in this paper we refine their theorems and introduce a new iteration argument to get the sharp upper bound of the lifespan as a result with the sharp lower bound by li and zhou 1995 10 the lifespan t e of solutions of utt u u2 in r4 0 with the initial data u x 0 ef x ut x 0 eg x of a small parameter e 0 compactly supported smooth functions f and g has an estimate exp ce 2 t e exp ce 2 where c and c are positive constants depending only on f and g this upper bound has been known to be the last open optimality of the general theory for fully nonlinear wave equations

finite time blow up to critical semilinear wave equation outside the ball in 3 d abstract in this paper we study the initial boundary value problem of critical semilinear wave equation with small initial data in 3 d blow up result will be established no matter how small the data are we find a radial and positive solution of the corresponding linear problem then our blow up result follows from the idea of zhou and han 2014 what is more we get the upper bound of the lifespan estimate

concerning the strauss conjecture on asymptotically euclidean manifolds abstract in this paper we verify the strauss conjecture for semilinear wave equations on asymptotically euclidean manifolds when n 3 4 we also give an almost sharp lifespan for the subcritical case 2 p p c when n 3 the main ingredients include a keel smith sogge type estimate with 0 1 2 and weighted strichartz estimates of order two

on the complexity of nash equilibria and other fixed points we reexamine what it means to compute nash equilibria and more generally what it means to compute a fixed point of a given brouwer function and we investigate the complexity of the associated problems specifically we study the complexity of the following problem given a finite game gamma with 3 or more players and given epsilon 0 compute an approximation within epsilon of some actual nash equilibrium we show that approximation of an actual nash equilibrium even to within any nontrivial constant additive factor epsilon 1 2 in just one desired coordinate is at least as hard as the long standing square root sum problem as well as a more general arithmetic circuit decision problem that characterizes p time in a unit cost model of computation with arbitrary precision rational arithmetic thus placing the approximation problem in p or even np would resolve major open problems in the complexity of numerical computation we show similar results for market equilibria it is hard to estimate with any nontrivial accuracy the equilibrium prices in an exchange economy with a unique equilibrium where the economy is given by explicit algebraic formulas for the excess demand functions we define a class fixp which captures search problems that can be cast as fixed point computation problems for functions represented by algebraic circuits straight line programs over basis max min with rational constants we show that the exact or approximate computation of nash equilibria for 3 or more players is complete for fixp the price equilibrium problem for exchange economies with algebraic demand functions is another fixp complete problem we show that the piecewise linear fragment of fixp equals ppad many other problems in game theory economics and probability theory can be cast as fixed point problems for such algebraic functions we discuss several important such problems computing the value of shapley s stochastic games and the simpler games of condon extinction probabilities of branching processes probabilities of stochastic context free grammars and termination probabilities of recursive markov chains we show that for some of them the approximation or even exact computation problem can be placed in ppad while for others they are at least as hard as the square root sum and arithmetic circuit decision problems

on semilinear tricomi equations with critical exponents or in two space dimensions abstract this paper is a complement of our recent works on the semilinear tricomi equations in 9 and 10 for the semilinear tricomi equation t 2 u t u u p with initial data u 0 t u 0 u 0 u 1 where t 0 x r n n 3 p 1 and u i c 0 r n i 0 1 we have shown in 9 and 10 that there exists a critical exponent p crit n 1 such that the solution u in general blows up in finite time when 1 p p crit n and there is a global small solution for p p crit n in the present paper firstly we prove that the solution of t 2 u t u u p will generally blow up for the critical exponent p p crit n and n 2 secondly we establish the global existence of small data solution to t 2 u t u u p for p p c r i t n and n 2 thus we have given a systematic study on the blowup or global existence of small data solution u to the equation t 2 u t u u p for n 2

belief and truth in hypothesised behaviours there is a long history in game theory on the topic of bayesian or rational learning in which each player maintains beliefs over a set of alternative behaviours or types for the other players this idea has gained increasing interest in the artificial intelligence ai community where it is used as a method to control a single agent in a system composed of multiple agents with unknown behaviours the idea is to hypothesise a set of types each specifying a possible behaviour for the other agents and to plan our own actions with respect to those types which we believe are most likely given the observed actions of the agents the game theory literature studies this idea primarily in the context of equilibrium attainment in contrast many ai applications have a focus on task completion and payoff maximisation with this perspective in mind we identify and address a spectrum of questions pertaining to belief and truth in hypothesised types we formulate three basic ways to incorporate evidence into posterior beliefs and show when the resulting beliefs are correct and when they may fail to be correct moreover we demonstrate that prior beliefs can have a significant impact on our ability to maximise payoffs in the long term and that they can be computed automatically with consistent performance effects furthermore we analyse the conditions under which we are able complete our task optimally despite inaccuracies in the hypothesised types finally we show how the correctness of hypothesised types can be ascertained during the interaction via an automated statistical analysis

critical curve for p q systems of nonlinear wave equations in three space dimensions abstract the existence of the critical curve for p q systems for nonlinear wave equations was already established by d del santo v georgiev and e mitidieri 1997 global existence of the solutions and formation of singularities for a class of hyperbolic systems in geometric optics and related topics f colombini and n lerner eds progress in nonlinear differential equations and their applications vol 32 pp 117 139 birkhauser basel except for the critical case our main purpose is to prove a blow up theorem for which the nonlinearity p q is just on the critical curve in three space dimensions moreover the lower and upper bounds of the lifespan of solutions are precisely estimated including the sub critical case

blow up theorem for semilinear wave equations with non zero initial position abstract one of the features of solutions of semilinear wave equations can be found in blow up results for non compactly supported data in spite of finite propagation speed of the linear wave we have no global in time solution for any power nonlinearity if the spatial decay of the initial data is weak this was first observed by asakura 1986 2 finding out a critical decay to ensure the global existence of the solution but the blow up result is available only for zero initial position having positive speed in this paper the blow up theorem for non zero initial position by uesaka 2009 22 is extended to higher dimensional case and the assumption on the nonlinear term is relaxed to include an example u p 1 u moreover the critical decay of the initial position is clarified by example

a competition between fujita and strauss type exponents for blow up of semi linear wave equations with scale invariant damping and mass abstract we obtain a blow up result for solutions to a semi linear wave equation with scale invariant dissipation and mass and power non linearity in the case in which the model has a wave like behavior we perform a change of variables that transforms our starting equation in a strictly hyperbolic semi linear wave equation with time dependent speed of propagation applying kato s lemma we prove a blow up result for solutions to the transformed equation under some assumptions on the initial data the limit case that is when the exponent p is exactly equal to the upper bound of the range of admissible values of p yielding blow up needs special considerations in this critical case an explicit integral representation formula for solutions of the corresponding linear cauchy problem in 1d is derived finally carrying out the inverse change of variables we get a non existence result for global in time solutions to the original model

testing a point null hypothesis the irreconcilability of p values and evidence abstract the problem of testing a point null hypothesis or a small interval null hypothesis is considered of interest is the relationship between the p value or observed significance level and conditional and bayesian measures of evidence against the null hypothesis although one might presume that a small p value indicates the presence of strong evidence against the null such is not necessarily the case expanding on earlier work especially edwards lindman and savage 1963 and dickey 1977 it is shown that actual evidence against a null as measured say by posterior probability or comparative likelihood can differ by an order of magnitude from the p value for instance data that yield a p value of 05 when testing a normal mean result in a posterior probability of the null of at least 30 for any objective prior distribution objective here means that equal prior weight is given the two hypotheses and that the prior is symmetric and nonincreasing away from the null other definiti

regular articlegeneralized strichartz inequalities for the wave equation we make a synthetic exposition of the generalized strichartz inequalities for the wave equation obtained in 6 together with the limiting cases recently obtained in 13 with as simple proofs as possible the proofs combine stationary phase estimates dyadic decompositions the hardy littlewood inequality or the holder inequality in time and abstract duality and interpolation arguments

the first darboux problem for wave equations with a nonlinear positive source term abstract we consider the first darboux problem for nonlinear wave equations with positive power nonlinearity source term depending on the power of nonlinearity we investigate the problem on a global existence and blow up of solutions of the first darboux problem the question of local solvability of the problem is also considered

on the solvability of the cauchy characteristic problem for a nonlinear equation with iterated wave operator in the principal part we consider one multidimensional version of the cauchy characteristic problem in the light cone of the future for a hyperbolic equation with power nonlinearity with iterated wave operator in the principal part depending on the exponent of nonlinearity and spatial dimension of equation we investigate the problem on the nonexistence of global solutions of the cauchy characteristic problem the question on the local solvability of that problem is also considered

global solvability of the cauchy characteristic problem for one class of nonlinear second order hyperbolic systems the cauchy characteristic problem in the light cone of the future for one class of nonlinear hyperbolic systems of the second order is considered the existence and uniqueness of global solution of this problem is proved

level rings and algebras with straightening laws abstract stanley 13 defined the concept of level rings intermediate between cohen macaulay and gorenstein which is of interest from viewpoints of both commutative algebra and combinatorics via the theory of canonical modules the study of level rings turns out to be one of the most important foundations of the theory of algebras with straightening laws the purpose of this paper is to state some basic results on level rings which are either stanley reisner rings or algebras with straightening laws

the strauss conjecture on kerr black hole backgrounds we examine solutions to semilinear wave equations on black hole backgrounds and give a proof of an analog of the strauss conjecture on the schwarzschild and kerr with small angular momentum black hole backgrounds the key estimates are a class of weighted strichartz estimates which are used near infinity where the metrics can be viewed as small perturbations of the minkowski metric and a localized energy estimate on the black hole background which handles the behavior in the remaining compact set

almost global solutions of semilinear wave equations with the critical exponent in high dimensions abstract we are interested in the almost global in time existence of classical solutions in the general theory for nonlinear wave equations all the three such cases are known to be sharp due to blow up results in the critical case for model equations however it is known that we have a possibility to get the global in time existence for two of them in low space dimensions if the nonlinear term is of derivatives of the unknown function and satisfies the so called null condition or non positive condition but another one for the quadratic term in four space dimensions is out of the case as the nonlinear term should include a square of the unknown function itself in this paper we get one more example guaranteeing the sharpness of the almost global in time existence in four space dimensions it is also the first example of the blow up of classical solutions for non single and indefinitely signed term in high dimensions such a term arises from the neglect of derivative loss factors in duhamel s formula for positive and single nonlinear term this fact may help us to describe a criterion to get the global in time existence in this critical situation

the lifespan of solutions to nonlinear systems of a high dimensional wave equation abstract in this work we study the lifespan of solutions to a p q system in the higher dimensional case n 4 a suitable local existence curve in the p q plane is found the curve characterizes the local solutions in sobolev space h s with s 0 further some lower and upper bounds of the lifespan of classical solutions are found too this work is an extension of work geometric optics and related topics progress in nonlinear differential equations and their applications vol 32 birkhauser basel 1997 pp 117 140 where a suitable global existence small data curve is studied in the subcritical case we give almost precise results for the lower bounds of the lifespan by using a suitable weighted strichartz estimate for the higher dimensional wave equation

new blow up results for nonlinear boundary value problems in exterior domains abstract we consider exterior boundary value problems with nontrivial boundary conditions including three types of semilinear equations heat equation wave equation and damped wave equation using a test function approach with a choice of a suitable family of test functions we obtain the critical exponents in the sense of fujita for the considered problems our obtained results close some open questions from the literature

weighted decay estimates for the wave equation abstract in this work we study weighted sobolev spaces in r n generated by the lie algebra of vector fields 1 x 2 1 2 x j j 1 n interpolation properties and sobolev embeddings are obtained on the basis of a suitable localization in r n as an application we derive weighted l q estimates for the solution of the homogeneous wave equation for the inhomogeneous wave equation we generalize the weighted strichartz estimate established by v georgiev 1997 amer j math 119 1291 1319 and establish global existence results for the supercritical semilinear wave equation with non compact small initial data in these weighted sobolev spaces

critical criteria of fujita type for a system of inhomogeneous wave inequalities in exterior domains abstract we consider blow up results for a system of inhomogeneous wave inequalities in exterior domains we will handle three type boundary conditions dirichlet type neumann type and mixed boundary conditions we use a unified approach to show the optimal criteria of fujita type for each case our study yields naturally optimal nonexistence results for the corresponding stationary wave system and equation we provide many new results and close some open questions

bisimulation through probabilistic testing abstract we propose a language for testing concurrent processes and examine its strength in terms of the processes that are distinguished by a test by using probabilistic transition systems as the underlying semantic model we show how a testing algorithm can distinguish with a probability arbitrarily close to one between processes that are not bisimulation equivalent we also show a similar result in a slightly stronger form for a new process relation called 2 3 bisimulation which lies strictly between that of simulation and bisimulation finally the ultimately strength of the testing language is shown to identify a new process relation called probabilistic bisimulation which is strictly stronger than bisimulation

blow up of solutions to critical semilinear wave equations with variable coefficients abstract we verify the critical case p p 0 n of strauss conjecture 30 concerning the blow up of solutions to semilinear wave equations with variable coefficients in r n where n 2 the perturbations of laplace operator are assumed to be smooth and decay exponentially fast at infinity we also obtain a sharp lifespan upper bound for solutions with compactly supported data when p p 0 n the unified approach to blow up problems in all dimensions combines several classical ideas in order to generalize and simplify the method of zhou 43 and zhou han 45 exponential eigenfunctions of the laplacian 37 are used to construct the test function q for linear wave equation with variable coefficients and john s method of iterations 13 is augmented with the slicing method of agemi kurokawa and takamura 1 for lower bounds in the critical case

critical blowup for systems of semilinear wave equations in low space dimensions abstract we consider the cauchy problem for systems of semilinear wave equations in two and three space dimensions with small initial data del santo et al geometric optics and related topics f colombini and n lerner eds progress in nonlinear differential equations and their applications vol 32 pp 117 140 birkhauser boston 1997 have studied the existence and nonexistence of global classical solutions of the cauchy problem except for the critical case in this paper we study the critical case and we show the nonexistence of global classical solutions and also give the upper bounds of the life span

on singularity of a nonlinear variational sine gordon equation abstract we study a sine gordon type of nonlinear variational wave equation whose wave speed is a sinusoidal function of the wave amplitude this equation arises naturally from long waves on a dipole chain in the continuum limit which provides a crude model for some polymers using characteristic methods we describe a blow up result for the one dimensional nonlinear variational sine gordon equation which shows that smooth solutions breakdown in finite time

simplicial complexes and macaulay s inverse systems let be a simplicial complex on v x 1 x n with stanley reisner ideal i delta subseteq r k x 1 ldots x n the goal of this paper is to investigate the class of artinian algebras a a delta a 1 ldots a n r i delta x 1 a 1 ldots x n a n where each a i 2 by utilizing the technique of macaulay s inverse systems we can explicitly describe the socle of a in terms of as a consequence we determine the simplicial complexes that we will call levelable for which there exists a tuple a 1 a n such that a a 1 a n is a level algebra

bayesianly justifiable and relevant frequency calculations for the applied statistician a common reaction among applied statisticians is that the bayesian statistician s energies in an applied problem must be directed at the a priori elicitation of one model specification from which an optimal design and all inferences follow automatically by applying bayes s theorem to calculate conditional distributions of unknowns given knowns i feel however that the applied bayesian statistician s tool kit should be more extensive and include tools that may be usefully labeled frequency calculations three types of bayesianly justifiable and relevant frequency calculations are presented using examples to convey their use for the applied statistician

small data global existence for the semilinear wave equation with space time dependent damping abstract in this paper we consider the critical exponent problem for the semilinear wave equation with space time dependent damping when the damping is effective it is expected that the critical exponent agrees with that of only the space dependent coefficient case we shall prove that there exists a unique global solution for small data if the power of nonlinearity is larger than the expected exponent moreover we do not assume that the data are compactly supported however it is still open whether there exists a blow up solution if the power of nonlinearity is smaller than the expected exponent

small data blowup for systems of semilinear wave equations with different propagation speeds in three space dimensions abstract we consider the cauchy problem for a system of semilinear wave equations with small initial data and critical nonlinearity as for a class of systems of quasilinear wave equations with critical nonlinearity the small data global existence has been well developed for the case when the propagation speeds are distinct in contrast with the quasilinear case we show that the critical small data blowup occurs for the semilinear case even if the propagation speeds are different from each other

prediction optimization and learning in repeated games consider a two player discounted repeated game in which each player optimizes with respect to a prior belief about his opponent s repeated game strategy one would like to argue that if beliefs are cautious then each player s best response will be in the support loosely speaking of his opponent s belief and that therefore players will learn as the game unfolds to predict the continuation path of play if this conjecture were true a convergence result due to kalai and lehrer would imply that the continuation path of the repeated game would asymptotically resemble that of a nash equilibrium one would thus have constructed a theory in which nash equilibrium behavior is a necessary long run consequence of optimization by cautious players this paper points out an obstacle to such a theory loosely put in many repeated games if players optimize with respect to beliefs that satisfy a diversity condition termed neutrality then each player will choose a strategy that his opponent was certain would not be played

life span of small solutions to a system of wave equations abstract we study the cauchy problem with small initial data for a system of semilinear wave equations u v q v t u p in n dimensional space when n 2 we prove that blow up can occur for arbitrarily small data if p q lies below a curve in the p q plane on the other hand we show a global existence result for n 3 which asserts that a portion of the curve is in fact the borderline between global in time existence and finite time blow up we also estimate the maximal existence time and get its upper bound which is sharp at least for n p q 2 2 2 and 3 2 2

single point blow up for a semilinear initial value problem abstract the initial value problem on r r is considered u t t x u xx t x u t x u t r 0 u 0 x x where 0 and is a fixed large number it is known that for some initial values the solution u t x exists only up to some finite time t and that u t as t t for the specific initial value k where 0 xx 0 r 0 k is sufficiently large it is shown that if x 0 then lim t t u t x and lim t t u x t x exist and are finite in other words blow up occurs only at the point x 0