critical exponents for semilinear equations of mixed elliptic hyperbolic and degenerate types for semilinear gellerstedt equations with tricomi goursat or dirichlet boundary conditions we prove poho aev type identities and derive nonexistence results that exploit an invariance of the linear part with respect to certain nonhomogeneous dilations a critical exponent phenomenon of power type in the nonlinearity is exhibited in these mixed elliptic hyperbolic or degenerate settings where the power is 1 less than the critical exponent in a relevant sobolev embedding c 2003 wiley periodicals inc

conservation laws for equations of mixed elliptic hyperbolic and degenerate types for partial differential equations of mixed elliptic hyperbolic and degenerate types which are the euler lagrange equations for an associated lagrangian invariance with respect to changes in independent and dependent variables is investigated as are results in the classification of continuous one parameter symmetry groups for the variational and divergence symmetries conservation laws are derived via the method of multipliers the conservation laws resulting from anisotropic dilations are applied to prove uniqueness theorems for linear and nonlinear problems and the invariance under dilations of the linear part is used to derive critical exponent phenomena and to obtain localized energy estimates for supercritical problems

global existence for the one dimensional second order semilinear hyperbolic equations abstract the aim of the paper is to study necessary and sufficient conditions for the existence of the global solution of the one dimensional semilinear equation appearing in the boundary value problems of gas dynamics we investigate the cauchy problem for such equation in the domain where the operator is weakly hyperbolic we obtain the necessary condition for the existence of the self similar solutions for the semilinear gellerstedt type equation the approach used in the paper is based on the fundamental solution of the linear gellerstedt operator and the l p l q estimates

the self similar solutions of the one dimensional semilinear tricomi type equations abstract in this article we investigate the issue of existence of global in time solutions of semilinear tricomi type equations we give conditions that relate the nonlinearity the speed of propagation and the order of singularities of initial data these conditions guarantee existence of global in time solutions in particular we prove existence of solutions invariant under dilation by solving the cauchy problem with initial data which are homogeneous functions

multistate components assignment problem with optimal network reliability subject to assignment budget abstract the typical assignment problem for finding the optimal assignment of a set of components to a set of locations in a system has been widely studied in practical applications however this problem mainly focuses on maximizing the total profit or minimizing the total cost without considering component s failure in practice each component should be multistate due to failure partially failure or maintenance that is each component has several capacities with a probability distribution and may fail when a set of multistate components is assigned to a system the system can be treated as a stochastic flow network the network reliability is the probability that d units of homogenous commodity can be transmitted through the network successfully the multistate components assignment problem to maximize the network reliability is never discussed therefore this paper focuses on solving this problem under an assignment budget constraint in which each component has an assignment cost the network reliability under a components assignment can be evaluated in terms of minimal paths and state space decomposition subsequently an optimization method based on genetic algorithm is proposed the experimental results show that the proposed algorithm can be executed in a reasonable time

spatial boundary problem with the dirichlet neumann condition for a singular elliptic equation abstract in the present work we find an explicit solution of a boundary problem with the dirichlet neumann condition for elliptic equation with singular coefficients in a quarter of ball for this aim the method of green s function has been used since found green s function contains a hypergeometric function of appell we had to deal with decomposition formulas formulas of differentiation and some adjacent relations for this hypergeometric function in order to get explicit solution of the formulated problem

a model of pair aggregation inspired by the familiar phenomenon of the cherry entanglement we compute the clustering properties of a new model in which pairs of elements located on the sites of a lattice and tiling it completely attach to form aggregates of various sizes avoiding double occupancy the distance between the two elements of each pair is chosen at random with a prescribed maximum lenght l the resulting clustering behaviour crucially depends on the dimension of the embedding lattice and on l we devise a transfer matrix method to solve the model in 1 dimension results in 2 dimensions are obtained by monte carlo simulations in 1 dimension we define a characteristic cluster size it depends exponentially on l in 2 dimensions for a big enough l we find scaling for the probability of occurrence of clusters of size n versus n possible generalizations of this model are qualitatively discussed

one armed bandit problems with covariates as does woodroofe we consider a bayesian sequential allocation between two treatments that incorporates a covariate the goal is to maximize the total discounted expected reward from an infinite population of patients although our model is more general than woodroofe s we are able to duplicate his main result the myopic rule is asymptotically optimal

adaptive treatment allocation and the multi armed bandit problem a class of simple adaptive allocation rules is proposed for the problem often called the multi armed bandit problem of sampling x 1 cdots x n sequentially from k populations with densities belonging to an exponential family in order to maximize the expected value of the sum s n x 1 cdots x n these allocation rules are based on certain upper confidence bounds which are developed from boundary crossing theory for the k population parameters the rules are shown to be asymptotically optimal as n rightarrow infty from both bayesian and frequentist points of view monte carlo studies show that they also perform very well for moderate values of the horizon n

fundamental solutions of the wave equation in robertson walker spaces abstract in this article we construct the fundamental solutions for the wave equation in the robertson walker spaces arising in the de sitter model of the universe we then use these fundamental solutions to represent solutions of the cauchy problem for the equation with and without a source term

response surface bandits in this paper we define a response surface bandit as the sequential design problem that maximizes an expected bandit utility but where the outcomes y n are continuous and can be related through a response surface to a set of controllable variables x n x 1n x 2n x kn we link this problem to other traditional optimization problems from industrial engineering and to the traditional bandit problem we consider two approaches to the problem the first is based on a myopic sequential design the second approach uses the best design out of a family of designs related to upper bounds for the predicted surface the family includes myopic and sequential versions of d optimal designs these approaches can be generalized to more broadly defined sequential problems

on the existence of low regularity solutions to semilinear generalized tricomi equations in mixed type domains abstract in 20 21 we have established the existence and singularity structures of low regularity solutions to the semilinear generalized tricomi equations in the degenerate hyperbolic regions and to the higher order degenerate hyperbolic equations respectively in the present paper we shall be concerned with the low regularity solution problem for the semilinear mixed type equation t 2 u t 2 l 1 u f t x u with an initial data u 0 x x h s r n 0 s n 2 where t x r r n n 2 l n f t x u is c 1 smooth in its arguments and has compact support with respect to the variable x under the assumption of the subcritical growth of f t x u on u we will show the existence and regularity of the considered solution in the mixed type domain t 0 t 0 r n for some fixed constant t 0 0

robust counterpart optimization for the redundancy allocation problem in series parallel systems with component mixing under uncertainty in this paper a robust optimization approach is used to solve the redundancy allocation problem rap in series parallel systems with component mixing where uncertainty exists in components reliabilities in real world the reliabilities of components are imprecisely estimated or the reliability of some components may vary due to some realistic factors therefore we may deal with a system where there are many components with uncertain values of reliabilities to deal with this problem for the first time a robust optimization approach is applied to rap with component mixing to produce a robust solution which is relatively insensitive with respect to uncertainty in reliability of components in addition the advantages of the proposed robust technique are illustrated by considering a series parallel system and finding the suitable redundancy levels and then monte carlo simulation is implemented to examine the quality of the robust solutions the results indicate that applying the proposed robust rap can be more reliable to determine system reliability in the designing phase of systems

a note on the fundamental solution for the tricomi type equation in the hyperbolic domain abstract barros neto and gelfand duke math j 98 3 1999 465 duke math j 117 2 2003 561 constructed for the tricomi operator y x 2 y 2 on the plane the fundamental solutions with the supports in the regions related to the geometry of the characteristics of the tricomi operator we give for the tricomi type operator t 2 t m x a fundamental solution relative to an arbitrary point of r n 1 with the support in the region t 0 where the operator is hyperbolic our key observation is that the fundamental solution for the tricomi type operator can be written like an integral of the distributions generated by the fundamental solution of the cauchy problem for the wave equation the application of that fundamental solution to the l p l q estimate for the forced tricomi type equation is given as well

nonlinear liouville theorems for grushin and tricomi operators the aim of this paper is to study necessary conditions for existence of weak solutions of the inequality l x y dx dy u u q x 1 y 2 x rd y rk where l is a quasi homogeneous differential operator q 1 1 2 r and k d 1 as special cases our results can be applied when l is tricomi or grushin type operators

fundamental solutions for the tricomi operator in this paper we explicitly calculate fundamental solutions for the tricomi operator relative to an arbitrary point in the plane and show that all such fundamental solutions originate from the hypergeometric function f 1 6 1 6 1 that is obtained when we look for homogeneous solutions to the reduced hyperbolic tricomi equation

the dirichlet problem for a 3d elliptic equation with two singular coefficients abstract in the present work we investigate the dirichlet problem for a three dimensional 3d elliptic equation with two singular coefficients we find four fundamental solutions of the equation containing hypergeometric functions of appell then using an a b c method the uniqueness for the solution of the dirichlet problem is proved applying a method of green s function we are able to find the solution of the problem in an explicit form moreover decomposition formulas formulas of differentiation and some adjacent relations for appell s hypergeometric functions were used in order to find the explicit solution for the formulated problem

bayesian sequential estimation of the reliability of a parallel series system we give a risk averse solution to the problem of estimating the reliability of a parallel series system we adopt a beta binomial model for components reliabilities assuming that the total sample size for the experiment is fixed the allocation at subsystems or components level may be random based on the sampling schemes for parallel and series systems separately we propose a hybrid sequential scheme for the parallel series system asymptotic optimality of the bayes risk associated with quadratic loss is proved with the help of martingale convergence properties a comparison is given with the best fixed design with numerical examples

bessel integrals and fundamental solutions for a generalized tricomi operator abstract the method of partial fourier transform is used to find explicit formulas for two remarkable fundamental solutions for a generalized tricomi operator these fundamental solutions reflect clearly the mixed type of the tricomi operator in proving these results we establish explicit formulas for fourier transforms of some functions involving bessel functions

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

a note on performance limitations in bandit problems with side information we consider a sequential adaptive allocation problem which is formulated as a traditional two armed bandit problem but with one important modification at each time step t before selecting which arm to pull the decision maker has access to a random variable xt which provides information on the reward in each arm performance is measured as the fraction of time an inferior arm generating lower mean reward is pulled we derive a minimax lower bound that proves that in the absence of sufficient statistical diversity in the distribution of the covariate x a property that we shall refer to as lack of persistent excitation no policy can improve on the best achievable performance in the traditional bandit problem without side information

on fundamental solutions for multidimensional helmholtz equation with three singular coefficients abstract the main result of the present paper is the construction of fundamental solutions for a class of multidimensional elliptic equations with three singular coefficients which could be expressed in terms of a confluent hypergeometric function of four variables in addition the order of the singularity is determined and the properties of the found fundamental solutions that are necessary for solving boundary value problems for degenerate elliptic equations of second order are found

bandit problems with side observations an extension of the traditional two armed bandit problem is considered in which the decision maker has access to some side information before deciding which arm to pull at each time t before making a selection the decision maker is able to observe a random variable x sub t that provides some information on the rewards to be obtained the focus is on finding uniformly good rules that minimize the growth rate of the inferior sampling time and on quantifying how much the additional information helps various settings are considered and for each setting lower bounds on the achievable inferior sampling time are developed and asymptotically optimal adaptive schemes achieving these lower bounds are constructed

reliability equivalence with a basic series parallel system this paper extends the concept of reliability equivalence from simple series and parallel systems to some complex systems the reliability equivalence factors of a complex system are obtained we consider a radar system in an aircraft which consists of three independent and non identical components both survival function and mean time to failure are used as references of reliability equivalence the case of independent and identical components is studied as a special case theoretical comparisons among the used methods of improving the system reliability are introduced numerical examples are presented to interpret how one can utilize the obtained results

sequential allocation for an estimation problem with ethical costs the problem of designing an experiment to estimate the difference between the means of two normal populations with unit variances is considered when the cost of drawing a sample from either population may depend on unknown parameters a quasi bayesian approach is adopted in which the mean difference is estimated by its maximum likelihood estimator but the design allocation rule is evaluated in bayesian decision theoretic terms a three stage procedure is proposed and its risk evaluated up to terms which are small compared to the cost of a single observation

model of pair aggregation on the bethe lattice departamento de ciencias de la tierra universidad de zaragoza 50009 zaragoza spain received 16 july 1996 revised manuscript received 18 november 1996 we extend a recent model of aggregation of pairs of particles analyzing the case in which the supportingframework is a bethe lattice the model exhibits a critical behavior of the percolation theory type s1063 651x 97 03403 x pacs number s 05 50 1qi introduction

randomized allocation with nonparametric estimation for a multi armed bandit problem with covariates we study a multi armed bandit problem in a setting where covariates are available we take a nonparametric approach to estimate the functional relationship between the response reward and the covariates the estimated relationships and appropriate randomization are used to select a good arm to play for a greater expected reward randomization helps balance the tendency to trust the currently most promising arm with further exploration of other arms it is shown that with some familiar nonparametric methods e g histogram the proposed strategy is strongly consistent in the sense that the accumulated reward is asymptotically equivalent to that based on the best arm which depends on the covariates almost surely 1 introduction multi armed bandit problems have been extensively studied in probability and statistics in the past few decades and still receive considerable interest in these and related fields readers are referred to berry and fristedt 1985 and gittins 1989 and the references cited therein for the history many elegant results and applications in clinical trials scheduling and other industrial problems some recent developments have been reported in for example lai and yakowitz 1995 who considered the situation in which observations are dependent berry chen zame heath and shepp 1997 who considered infinitely many arms and auer cesa bianchi freund and schapire 1995 who considered worst case performances in classic bandit problems each arm sequentially generates rewards based on a distribution with some unknown parameters and one needs to sequentially select one arm to play for the maximum expected reward in a majority of the earlier work no auxiliary information beyond the observed rewards was considered when selecting an arm to play exceptions include woodroofe 1979 sarkar 1991 and clayton 1989 who considered one armed bandit problems with covariates the first two papers studied bayesian sequential allocation in non bernoulli bandit models with parametric frameworks and showed that the myopic rule is asymptotically optimal the third studied bernoulli bandit problems

sonic supersonic solutions for the two dimensional steady full euler equations this paper focuses on the structure of classical sonic supersonic solutions near sonic curves for the two dimensional full euler equations in gas dynamics in order to deal with the parabolic degeneracy near the sonic curve a novel set of dependent and independent variables are introduced to transform the euler equations into a new system of governing equations which displays a clear regularity singularity structure with the help of technical characteristic decompositions the existence of a local smooth solution for the new system is first established in a weighted metric space by using the iteration method and then expressed in terms of the original physical variables this is the first construction of a classical sonic supersonic solution near a sonic curve for the full euler equations

mean field theory for a simple model of evolution a simple dynamical model for darwinian evolution on its slowest time scale is analyzed its mean field theory is formulated and solved a random neighbor version of the model is simulated as is a one dimensional version in one dimension the dynamics can be described in terms of a repetitious random walker and anomalous diffusion with exponent 0 4 in all cases the model self organizes to a robust critical attractor

analyticity of solutions of semi linear equations with double characteristics we prove the analyticity and gevrey regularity of solutions of elliptic degenerate semi linear differential equations principle part of which is a linear operator with double characteristics considered first by gilioli and treves a new elementary proof for hypoellipticity in the weak sense is given

on closed boundary value problems for equations of mixed elliptic hyperbolic type for partial differential equations of mixed elliptic hyperbolic type we prove results on existence and existence with uniqueness of weak solutions for closed boundary value problems of dirichlet and mixed dirichlet conormal types such problems are of interest for applications to transonicflow and are overdetermined for solutions with classical regularity the method employed consists in variants of the a b c integral method of friedrichs in sobolev spaces with suitable weights particular attention is paid to the problem of attaining results with a minimum of restrictions on the boundary geometry and the form of the type change function in addition interior regularity results are also given in the important special case of the tricomi equation c

integral transform approach to generalized tricomi equations abstract in this article we introduce the integral transform that can be used to obtain solutions of the generalized tricomi equation via solutions of a simpler equation we employed a particular version of this transform in 13 14 41 42 43 44 45 46 to investigate in a unified way several equations such as the linear and semilinear tricomi equations gellerstedt equation the wave equation in the einstein de sitter spacetime the wave and the klein gordon equations in the de sitter and anti de sitter spacetimes

estimations of parameters in a three state reliability semi markov model maximum likelihood and bayes estimates of the parameters included in a three state semi markov reliability model are presented the renewal kernel which depends upon a vector of unknown parameters is used to define a semi markov process which can be used to describe some reliability model moreover some of the obtained results are compared with those obtained in other literature special case of the obtained results is presented

the weighted w2 p estimate on the solution of the gellerstedt equation in the upper half space abstract in this note we are concerned with the weighted w 2 p estimate for the gellerstedt equation which is degenerate on the boundary r 2 based on the estimates of the generalized airy functions and mihlin s multiplier theorem we establish a weighted w 2 p a priori estimate

a two dimensional model of pair aggregation abstract we modify the usual percolation theory pt by subtituting single particles which occupy sites on a two dimensional square lattice by pairs of particles linked by strings whose maximum length is a multiple of the basic lattice spacing the crossing of random length strings of the pairs leads to entanglements and clusters are formed we analyze the critical behaviour of this model as a function of the pair occupation fraction p within the accuracy provided by the monte carlo simulations all the critical exponents obtained are compatible with their standard pt counterparts thus the ordinary scaling relations are fulfilled