distributed coverage games for energy aware mobile sensor networks inspired by current challenges in data intensive and energy limited sensor networks we formulate a coverage optimization problem for mobile sensors as a constrained repeated multiplayer game each sensor tries to optimize its own coverage while minimizing the processing energy cost the sensors are subject to the informational restriction that the environmental distribution function is unknown a priori we present two distributed learning algorithms where each sensor only remembers its own utility values and actions played during the last plays these algorithms are proven to be convergent in probability to the set of constrained nash equilibria and global optima of a certain coverage performance metric respectively numerical examples are provided to verify the performance of our proposed algorithms

analytic perturbation of generalized inverses abstract we investigate the analytic perturbation of generalized inverses firstly we analyze the analytic perturbation of the drazin generalized inverse also known as reduced resolvent in operator theory our approach is based on spectral theory of linear operators as well as on a new notion of group reduced resolvent it allows us to treat regular and singular perturbations in a unified framework we provide an algorithm for computing the coefficients of the laurent series of the perturbed drazin generalized inverse in particular the regular part coefficients can be efficiently calculated by recursive formulae finally we apply the obtained results to the perturbation analysis of the moore penrose generalized inverse in the real domain

evolutionary stability for large populations and backward induction it has been shown hart 2002 that the backward induction or subgame perfect equilibrium of a perfect information game is the unique stable outcome for dynamic models consisting of selection and mutation when the mutation rate is low and the populations are large under the assumption that the expected number of mutations per generation is bounded away from zero here it is shown that one can dispense with this last condition in particular it follows that the backward induction equilibrium is evolutionarily stable for large populations

rate distortion theory with generalized information measures via convex programming duality a new generalized average mutual information measure gamim is introduced in terms of csiszar phi divergence and the associated rate distortion function r phi is studied the main objective is to derive in a unified way a dual representation of r phi then to use it to generalize classical results corresponding to phi t t log t in rate distortion theory and extend results associated with other concepts of gamim our development uses the methodology of convex programming duality extensively

aggregation of the policy iteration method for nearly completely decomposable markov chains a steady state optimal control problem is considered for nearly completely decomposable markov chains in order to apply the policy iteration method of r a howard dynamic programming and markov processes cambridge ma mit press 1960 a high dimensional ill conditioned system of algebraic equations must be solved in the value determination step although algorithms exist for aggregation of the steady state probability distribution problem they only provide methods for computing the cost not the dual variables using a singular perturbation approach an aggregation method for the value determination equation is developed the aggregation method is developed in three steps first a class of similarity transformations that transform the system into a singularly perturbed form is developed second an aggregation method to compute the steady state probability distribution is derived third this aggregation method is applied to the value determination step of howard s method 1960

constrained discounted markov decision processes and hamiltonian cycles this paper establishes new links between stochastic and discrete optimization we consider the following three problems for discrete time markov decision processes with finite states and action sets i find an optimal deterministic policy for a discounted problem with constraints ii find an optimal stationary policy for a weighted discounted problem with constraints iii find an optimal deterministic policy for a weighted discounted problem with constraints we formulate mathematical programs for problems i iii and show that the hamiltonian cycle problem is a special case of each of these problems therefore problems i iii are np hard we also provide new mathematical programming formulations for the hamiltonian cycle and traveling salesman problems

resource constrained management of heterogeneous assets with stochastic deterioration we consider a collection of heterogeneous assets which exhibit independent stochastic behavior but are economically interdependent due to resource constraints on management decisions we represent the collection of assets as a network of nonhomogeneous markov decision processes linked by side constraints to facilitate a procedure for obtaining nonrandomized decision policies we express the resource limitations as integrated chance constraints and relax them into the objective function within a lagrangian penalty term we utilize subgradient optimization to establish upper bounds and a greedy randomized lagrangian repair heuristic to obtain feasible solutions we empirically validate the tightness of these a posteriori upper and lower bounds with computational experiments on a pair of applications for pavement maintenance we examine the effect that the reward structure of each constituent mdp has on the maintenance policy in the presence of budget constraints for equipment replacement we consider constraints on two resource types

computing cumulative measures of stiff markov chains using aggregation an aggregation method for computing transient cumulative measures of large stiff markov models is presented the method is based on classifying the states of the original problem into slow fast transient and fast current states the authors aggregate fast transient states and fast recurrent states so that an approximate value to the desired cumulative measure can be obtained by solving a nonstiff set of linear differential equations defined over a reduced subset of slow states only several examples are included to illustrate how stiffness arises naturally in actual queuing and reliability models and to show that cumulative measures provide a better characterization of the time dependent system behavior

path integration on hermitian hyperbolic space in this paper the path integral technique is applied to the quantum motion on the hermitian hyperbolic space hh 2 the schrodinger equation on this space separates in 12 coordinate systems which are closely related to the coordinate systems on the two dimensional hyperboloid for six coordinate systems out of the twelve it is possible to find a path integral solution

iterative algorithms for large stochastic matrices markov chains have always constituted an efficient tool to model discrete systems many performance criteria for discrete systems can be derived from the steady state probability vector of the associated markov chain however the large size of the state space of the markov chain often allows this vector to be determined by iterative methods only various iterative methods exist but none can be proved a priori to be the best in this paper we propose a practical measure which allows the convergence rate of the various existing methods to be compared this measure is an approximation of the modulus of the second largest eigenvalue of the iteration matrix and can be determined a priori the model of a queueing network is used as an example to compare the convergence of several iterative methods and to show the accuracy of the measure

cesaro limits of analytically perturbed stochastic matrices abstract let p 0 be a stochastic matrix with no eigenvalue other than 1 on the unit circle and let p e p 0 a e be an analytic perturbation which is still stochastic for small enough positive e we determine the limit lim e 0 1 n e l 1 n e p l e when n e at various rates

evolutionary games in economics evolutionary games are introduced as models for repeated anonymous strategic interaction actions or behaviors which are more fit given the current distribution of behaviors tend over time to displace less fit behaviors cone fields characterize the continuous time processes compatible with a given fitness or payoff function for large classes of dynamics it is shown that all stable steady states are nash equilibria and that all nash equilibria are steady states the biologists evolutionarily stable strategy condition is shown to be less closely related to the dynamic equilibria economic examples and a literature survey are also provided copyright 1991 by the econometric society

quantum motion on the three dimensional sphere the ellipso cylindrical bases we study the free quantum motion on the three dimensional sphere in ellipso cylindrical coordinates where we distinguish between prolate elliptic and oblate elliptic coordinates the oblate and prolate elliptic wavefunctions are constructed in terms of the hyper spherical and cylindrical wavefunctions a perturbation theory for the elliptic wavefunctions in powers of the elliptic parameter is established finally the interbasis expansions are used to set up path integral identities for the two ellipso cylindrical coordinate systems on

on nearly optimal controls of hybrid lqg problems the paper is concerned with hybrid control of a class of linear quadratic gaussian lqg systems modulated by a finite state markov chain it develops approximation schemes for systems involving singularly perturbed markov chains with weak and strong interactions which are useful for natural time scale separation and for large scale markovian systems the presentation encompasses three cases the recurrent markov chains inclusion of transient states and inclusion of absorbing states near optimality of the approximation schemes is obtained and a numerical example is presented computational results indicate our approximation schemes perform well

the first laurent series coefficients for singularly perturbed stochastic matrices there are a few procedures for computing the laurent series expansions for the mean passage time matrix and for the deviation matrix of a singularly perturbed markov chain we suggest here a method for computing the first terms in these expansions in a way which highlights the system dynamics in various time scales

online learning of feasible strategies in unknown environments define an environment as a set of convex constraint functions that vary arbitrarily over time and consider a cost function that is also convex and arbitrarily varying agents that operate in this environment intend to select actions that are feasible for all times while minimizing the cost s time average such action is said optimal and can be computed offline if the cost and the environment are known a priori an online policy is one that depends causally on the cost and the environment to compare online policies to the optimal offline action define the fit of a trajectory as a vector that integrates the constraint violations over time and its regret as the cost difference with the optimal action accumulated over time fit measures the extent to which an online policy succeeds in learning feasible actions while regret measures its success in learning optimal actions this paper proposes the use of online policies computed from a saddle point controller which are shown to have fit and regret that are either bounded or grow at a sublinear rate these properties provide an indication that the controller finds trajectories that are feasible and optimal in a relaxed sense concepts are illustrated throughout with the problem of a shepherd that wants to stay close to all sheep in a herd numerical experiments show that the saddle point controller allows the shepherd to do so

convergence theory of some classes of iterative aggregation disaggregation methods for computing stationary probability vectors of stochastic matrices this contribution is a natural follow up of the paper of the same authors entitled convergence theory of an aggregation disaggregation methods for computing stationary probability vectors of stochastic matrices published in numer linear algebra appl 5 1998 253 in contrast to that paper in which the algorithm studied was based on the splitting whose iteration matrix was identical with the matrix whose stationary probability vectors are computed the present paper presents a convergence analysis of algorithms based on fully general splittings of nonnegative type together with this generalization another feature of the older paper and namely the independence of the convergence results on the size of the elements of the examined stochastic matrix is shown to remain valid for the new algorithms as well this concerns in particular the possibility of computing stationary probability vectors of markov chains containing rare events i e events whose stationary probabilities are substantially smaller than some of elements of the transition matrix of the chain

iterative aggregation disaggregation techniques for nearly uncoupled markov chains iterative aggregation disaggregation methods provide an efficient approach for computing the stationary probability vector of nearly uncoupled also known as nearly completely decomposable markov chains three such methods that have appeared in the literature recently are considered and their similarities and differences are outlined specifically it is shown that the method of takahashi corresponds to a modified block gauss seidel step and aggregation whereas that of vantilborgh corresponds to a modified block jacobi step and aggregation the third method that of koury et al is equivalent to a standard block gauss seidel step and iteration for each of these methods a lemma is established which shows that the unique fixed point of the iterative scheme is the left eigenvector corresponding to the dominant unit eigenvalue of the stochastic transition probability matrix in addition conditions are established for the convergence of the first two of these methods convergence conditions for the third having already been established by stewart et al all three methods are shown to have the same asymptotic rate of convergence

integration of large deviation kernels and applications to large deviations for evolutionary games we investigate large deviation properties for systems of probability kernels and these kernels when integrated with respect to a system of measures that also satisfy the large deviation principle the results of our analysis are then used to demonstrate the large deviation property and derive rate functions for systems of measures derived from the distributions of evolutionary game processes

the evolution of conventions the author shows how a group of individuals can learn to play a coordination game without any common knowledge and with only a small amount of rationality the game is repeated many times by different players each player chooses an optimal reply based on incomplete information about what other players have done in the past occasionally they make mistakes when the likelihood of mistakes is very small typically one coordination equilibrium will be played almost all of the time over the long run this stochastically stable equilibrium can be computed analytically using a general theorem the author proves on perturbed markov processes copyright 1993 by the econometric society

support recovery for the drift coefficient of high dimensional diffusions consider the problem of learning the drift coefficient of a p dimensional stochastic differential equation from a sample path of length t we assume that the drift is parametrized by a high dimensional vector and study the support recovery problem when both p and t can tend to infinity in particular we prove a general lower bound on the sample complexity t by using a characterization of mutual information as a time integral of conditional variance due to kadota zakai and ziv for linear stochastic differential equations the drift coefficient is parametrized by a p times p matrix that describes which degrees of freedom interact under the dynamics in this case we analyze a ell 1 regularized least squares estimator and prove an upper bound on t that nearly matches the lower bound on specific classes of sparse matrices

perturbation results for nearly uncoupled markov chains with applications to iterative methods the standard perturbation theory for linear equations states that nearly uncoupled markov chains numcs are very sensitive to small changes in the elements indeed some algorithms such as standard gaussian elimination will obtain poor results for such problems a structured perturbation theory is given that shows that numcs usually lead to well conditioned problems it is shown that with appropriate stopping criteria iterative aggregation disaggregation algorithms will achieve these structured error bounds a variant of gaussian elimination due to grassman taksar and heyman was recently shown by o cinneide to achieve such bounds

learning mutation and long run equilibria in games an evolutionary model with a finite number of players and with stochastic mutations is analyzed the expansion and contraction of strategies is linked to their current relative success but mutuation perturbing the system from its deterministic evolution are present as well the focus is on the long run implications of ongoing mutations which drastically reduce the set of equilibria for 2 by 2 symmetric games with two symmetric strict nash equilibria the risk dominant equilibrium is selected in particular if both strategies have equal security levels the pareto dominant nash equilibrium is selected in particular if both strategies have equal security levels the pareto dominant nash equilibrium is selected even though there is another strict nash equilibrium copyright 1993 by the econometric society

on a rayleigh ritz refinement technique for nearly uncoupled stochastic matrices abstract an analysis is presented for the convergence of an iterative technique for computing the dominant left eigenvector of a nearly uncoupled nearly completely decomposable stochastic matrix the technique combines the power method and rayleigh ritz refinement to force rapid convergence to the stationary probability vector of the associated markov chain

optimal interpolatory subdivision schemes in multidimensional spaces we analyze the approximation and smoothness properties of fundamental and refinable functions that arise from interpolatory subdivision schemes in multidimensional spaces in particular we provide a general construction of bivariate interpolatory refinement masks such that the corresponding fundamental and refinable functions attain the optimal approximation order and smoothness order in addition these interpolatory refinement masks are minimally supported and enjoy full symmetry several examples are explicitly computed

how to solve path integrals in quantum mechanics a systematic classification of feynman path integrals in quantum mechanics is presented and a table of solvable path integrals is given which reflects the progress made during the last 15 years including of course the main contributions since the invention of the path integral by feynman in 1942 an outline of the general theory is given which will serve as a quick reference for solving path integrals explicit formulas for the so called basic path integrals are presented on which the general scheme to classify and calculate path integrals in quantum mechanics is based

decision accuracy and the role of spatial interaction in opinion dynamics the opinions and actions of individuals within interacting groups are frequently determined by both social and personal information when sociality or the pressure to con form is strong and individual preferences are weak groups will remain cohesive until a consensus decision is reached when group decisions are subject to a bias representing for example private information known by some members of the population or imperfect infor mation known by all then the accuracy achieved for a fixed level of bias will increase with population size in this work we determine how the scaling between accuracy and group size can be related to the microscopic properties of the decision making process by simulating a spatial model of opinion dynamics we show that the relationship between the instantaneous fraction of leaders in the population l system size n and accuracy depends on the fre quency of individual opinion switches and the level of population viscosity when social mixing is slow and individual opinion changes are frequent accuracy is determined by the absolute number of informed individuals as mixing rates increase or the rate of opinion updates decrease a transition occurs to a regime where accuracy is determined by the value of l n we investigate the transition between different scaling regimes analytically by

constrained undiscounted stochastic dynamic programming in this paper we investigate the computation of optimal policies in constrained discrete stochastic dynamic programming with the average reward as utility function the state space and action sets are assumed to be finite constraints which are linear functions of the state action frequencies are allowed in the general multichain case an optimal policy will be a randomized nonstationary policy an algorithm to compute such an optimal policy is presented furthermore sufficient conditions for optimal policies to be stationary are derived there are many applications for constrained undiscounted stochastic dynamic programming e g in multiple objective markovian decision models

quincunx fundamental refinable functions and quincunx biorthogonal wavelets we analyze the approximation and smoothness properties of quincunx fundamental refinable functions in particular we provide a general way for the construction of quincunx interpolatory refinement masks associated with the quincunx lattice in r2 their corresponding quincunx fundamental refinable functions attain the optimal approximation order and smoothness order in addition these examples are minimally supported with symmetry for two special families of such quincunx interpolatory masks we prove that their symbols are nonnegative finally a general way of constructing quincunx biorthogonal wavelets is presented several examples of quincunx interpolatory masks and quincunx biorthogonal wavelets are explicitly computed

multidimensional interpolatory subdivision schemes this paper presents a general construction of multidimensional interpolatory subdivision schemes in particular we provide a concrete method for the construction of bivariate interpolatory subdivision schemes of increasing smoothness by finding an appropriate mask to convolve with the mask of a three direction box spline br r r of equal multiplicities the resulting mask for the interpolatory subdivision exhibits all the symmetries of the three direction box spline and with this increased symmetry comes increased smoothness several examples are computed for r 2 8 regularity criteria in terms of the refinement mask are established and applied to the examples to estimate their smoothness

achieving target state action frequencies in multichain average reward markov decision processes in this paper we address a basic problem that arises naturally in average reward markov decision processes with constraints and or nonstandard payoff criteria given a feasible state action frequency vector the target construct a policy whose state action frequencies match those of the target vector while it is well known that the solution to this problem cannot in general be found in the space of stationary randomized policies we construct a solution that has ultimately stationary structure it consists of two stationary policies where the first one is used initially and then the switch to the second one is made at a certain random switching time the computational effort required to construct this solution is minimal we also show that our problem can always be solved by a stationary policy if the original mdp is extended by adding certain states and actions the solution in the original mdp is obtained by mapping the solution in the extended mdp back to the original process

a unified perturbation analysis framework for countable markov chains abstract in this paper we are devoted to singular perturbation analysis for discrete time or continuous time markov chains we modify and extend the drift condition method well known for regular perturbation to develop a new framework for singular perturbation analysis our results extend and improve the corresponding ones in 2 for singularly perturbed markov chains by allowing a general perturbation form less restrictive conditions and more computable bounds our analysis covers the regular perturbation analysis and hence unifies singular and regular perturbation analysis furthermore our results are illustrated by two two dimensional markov chains including a discrete time queue and a continuous time level dependent quasi birth death process

hamiltonian cycles random walks and discounted occupational measures we develop a new random walk based algorithm for the hamiltonian cycle problem the random walk is on pairs of extreme points of two suitably constructed polytopes the latter are derived from geometric properties of the space of discounted occupational measures corresponding to a given graph the algorithm searches for a measure that corresponds to a common extreme point in these two specially constructed polyhedral subsets of that space we prove that if a given undirected graph is hamiltonian then with probability one this random walk algorithm detects its hamiltonicity in a finite number of iterations we support these theoretical results by numerical experiments that demonstrate a surprisingly slow growth in the required number of iterations with the size of the given graph

the effect of increasing returns to scale in public goods investment on threshold values of cooperation under social exclusion mechanism abstract this paper establishes a stochastic evolutionary public goods game pgg model with exclusion type strategies in a well mixed and finite size population the effect of increasing returns to scale in public goods investment is considered we focus on the stochastic stable equilibrium sse of the evolutionary system by numerical experiments we observe that there are two types of sse in the system namely all individuals choosing defection all d state and the coexistence of cooperation and exclusion c e states and three types of phases namely choosing the all d state with probability one d phase choosing the c e states with probability one c e phase and choosing the all d and c e states with non zero probabilities c e d phase we study the combined effects of four parameters probability of exclusion success unit exclusion cost increasing returns to scale coefficient investment amplification factor on the phase selection of the system we get the boundary curves or surfaces for each pair or set of combined parameters thus conditions for the evolution of cooperative strategies depending on these parameters can be given notably we observe that in a well mixed population with exclusion type strategies the increasing returns to scale coefficient cannot change the parameter boundary between the c e phase and the c e d phase within our parameters corresponding simulation experiments are also carried out in a structured population and it is found that the increasing returns to scale effect can greatly reduce critical values of the amplification factor under which cooperation can emerge and the increase in the increasing returns to scale coefficient can induce the system to transfer directly from the d phase to the c e phase which is different from the well mixed situation this research can help us better understand the emergence of cooperation in the pgg with increasing returns to scale effect under social exclusion mechanism

hamiltonian cycles and markov chains in this paper we derive new characterizations of the hamiltonian cycles of a directed graph and a new lp relaxation of the traveling salesman problem our results are obtained via an embedding of these combinatorial optimization problems in suitably perturbed controlled markov chains this embedding lends probabilistic interpretation to many of the quantities of interest which in turn lead naturally to the introduction of a quadratic entropy like function

aggregation disaggregation methods for computing the stationary distribution of a markov chain we implement and analyse aggregation disaggregation procedures constructed to accelerate the convergence of successive approximation methods suitable for computing the stationary distribution of a finite markov chain we define six of these methods and analyse them in detail in particular we show that some existing procedures lie in the aggregation disaggregation framework we set and hence can be considered as special cases also for all described methods we identify cases where they are promising numerical examples for the applications of some of the methods for nearly completely decomposable stochastic matrices are given as well

optimal c 2 two dimensional interpolatory ternary subdivision schemes with two ring stencils for any interpolatory ternary subdivision scheme with two ring stencils for a regular triangular or quadrilateral mesh we show that the critical holder smoothness exponent of its basis function cannot exceed log 3 11 2 18266 where the critical holder smoothness exponent of a function f r 2 r is defined to be v f sup v f lipv on the other hand for both regular triangular and quadrilateral meshes we present several examples of interpolatory ternary subdivision schemes with two ring stencils such that the critical holder smoothness exponents of their basis functions do achieve the optimal smoothness upper bound log 3 11 consequently we obtain optimal smoothest c 2 interpolatory ternary subdivision schemes with two ring stencils for the regular triangular and quadrilateral meshes our computation and analysis of optimal multidimensional subdivision schemes are based on the projection method and the l p norm joint spectral radius

laurent expansion of the inverse of perturbed singular matrices in this paper we describe a numerical algorithm to compute the laurent expansion of the inverse of singularly perturbed matrices the algorithm is based on the resolvent formalism used in complex analysis to study the spectrum of matrices the input of the algorithm are the matrix coefficients of the power series expansion of the perturbed matrix the matrix coefficients of the laurent expansion of the inverse are computed using recursive analytical formulae we show that the computational complexity of the proposed algorithm grows algebraically with the size of the matrix but exponentially with the order of the singularity we apply this algorithm to several matrices that arise in applications we make special emphasis to interpolation problems with radial basis functions

on the fastest finite markov processes consider a finite irreducible markov chain with invariant probability define its inverse communication speed as the expectation to go from x to y when x y are sampled independently according to in the discrete time setting and when is the uniform distribution litvak and ejov have shown that the permutation matrices associated to hamiltonian cycles are the fastest markov chains here we prove a that the above optimality is with respect to all processes compatible with a fixed graph of permitted transitions assuming that it does contain a hamiltonian cycle not only the markov chains and b that this result admits a natural extension in both discrete and continuous time when is close to the fastest markov chains processes are those moving successively on the points of a hamiltonian cycle with transition probabilities jump rates dictated by nevertheless the claim is no longer true when is significantly different from

symmetry property and construction of wavelets with a general dilation matrix in this note we are interested in the symmetry property of a refinable function with a general dilation matrix we investigate the symmetry group of a mask so that its associated refinable function with a general dilation matrix has a certain kind of symmetry given two dilation matrices which produce the same lattice we demonstrate that if a mask has a certain kind of symmetry then its associated refinable functions with respect to the two dilation matrices are the same therefore the two corresponding derived wavelet systems are essentially the same finally we illustrate that for any dilation matrix orthogonal masks as well as interpolatory masks having nonnegative symbols can be easily constructed with any preassigned order of sum rules by employing a linear transform without solving any equations the method in this note on constructing masks with certain desirable properties is simple painless and general examples of quincunx wavelets and wavelets with respect to the checkerboard lattice are presented to illustrate the general theory

perturbation and stability theory for markov control problems a unified approach to the asymptotic analysis of a markov decision process disturbed by an epsilon additive perturbation is proposed irrespective of whether the perturbation is regular or singular the underlying control problem that needs to be understood is the limit markov control problem the properties of this problem are studied

path integration over compact and noncompact rotation groups applications of group theoretical methods in the path integral formalism of nonrelativistic quantum theory are considered analysis of the symmetry of the lagrangian leads to the expansion of the short time propagator in matrix elements of unitary irreducible representations of the symmetry group identification of the coordinates with the group parameters transforms the path integral to integrals over the group manifold the integration is performed using the orthogonality of the representations compact and noncompact rotation groups are considered where the corresponding path integral is embedded in euclidean and pseudo euclidean spaces respectively the unit sphere and unit hyperboloid may either be viewed as the group manifold itself or at least as a group quotient in the first case fourier analysis leads to an expansion in group characters in the second case an expansion in zonal spherical functions is obtained as examples the groups so n su 2 so n 1 1 and su 1 1 are explicitly discussed the path integral on so n m and so n m in bispherical coordinates is also treated

an interlacing theorem for reversible markov chains reversible markov chains are an indispensable tool in the modeling of a vast class of physical chemical biological and statistical problems examples include the master equation descriptions of relaxing physical systems stochastic optimization algorithms such as simulated annealing chemical dynamics of protein folding and markov chain monte carlo statistical estimation very often the large size of the state spaces requires the coarse graining or lumping of microstates into fewer mesoscopic states and a question of utmost importance for the validity of the physical model is how the eigenvalues of the corresponding stochastic matrix change under this operation in this paper we prove an interlacing theorem which gives explicit bounds on the eigenvalues of the lumped stochastic matrix

spherical geometry zernike s separability and interbasis expansion coefficients free motion on a 3 sphere properly projected on the 2 dimensional manifold of a disk yields the zernike system which exhibits the fundamental properties of superintegrability these include separability in a variety of coordinate systems polynomial solutions and a particular subset of clebsch gordan coefficients as interbasis expansion coefficients that are higher orthogonal polynomials from the askey scheme deriving these results from the initial formulation in spherical geometry provides the zernike system with interest beyond its optical applications

blockwise perturbation theory for nearly uncoupled markov chains and its application abstract let p be the transition matrix of a nearly uncoupled markov chain the states can be grouped into aggregates such that p has the block form p p ij i j 1 k where p ii is square and p ij is small for i j let t be the stationary distribution partitioned conformally as t 1 t k t in this paper we bound the relative error in each aggregate distribution i t caused by small relative perturbations in p ij the error bounds demonstrate that nearly uncoupled markov chains usually lead to well conditioned problems in the sense of blockwise relative error as an application we show that with appropriate stopping criteria iterative aggregation disaggregation algorithms will achieve such structured backward errors and compute each aggregate distribution with high relative accuracy

the reduction of perturbed markov generators an algorithm exposing the role of transient states a new algorithm for the hierarchical aggregation of singularly perturbed finite state markov processes is derived the approach taken bridges the gap between conceptually simple results for a relatively restricted class of processes and the significantly more complex results for the general case the critical role played by almost transient states is exposed resulting in a straightforward algorithm for the construction of a sequence of aggregate generators associated with various time scales these generators together provide a uniform asymptotic approximation of the original probability transition function

applications of singular perturbation techniques to control problems this paper discusses typical applications of singular perturbation techniques to control problems in the last fifteen years the first three sections are devoted to the standard model and its time scale stability and controllability properties the next two sections deal with linear quadratic optimal control and one with cheap near singular control then the composite control and trajectory optimization are considered in two sections and stochastic control in one section the last section returns to the problem of modeling this time in the context of large scale systems the bibliography contains more than 250 titles

markov chains and optimality of the hamiltonian cycle we consider the hamiltonian cycle problem hcp embedded in a controlled markov decision process in this setting hcp reduces to an optimization problem on a set of markov chains corresponding to a given graph we prove that hamiltonian cycles are minimizers for the trace of the fundamental matrix on a set of all stochastic transition matrices in case of doubly stochastic matrices with symmetric linear perturbation we show that hamiltonian cycles minimize a diagonal element of a fundamental matrix for all admissible values of the perturbation parameter in contrast to the previous work on this topic our arguments are primarily based on probabilistic rather than algebraic methods

evolutionary dynamics of populations with conflicting interactions classification and analytical treatment considering asymmetry and power evolutionary game theory has been successfully used to investigate the dynamics of systems in which many entities have competitive interactions from a physics point of view it is interesting to study conditions under which a coordination or cooperation of interacting entities will occur be it spins particles bacteria animals or humans here we analyze the case where the entities are heterogeneous particularly the case of two populations with conflicting interactions and two possible states for such systems explicit mathematical formulas will be determined for the stationary solutions and the associated eigenvalues which determine their stability in this way four different types of system dynamics can be classified and the various kinds of phase transitions between them will be discussed while these results are interesting from a physics point of view they are also relevant for social economic and biological systems as they allow one to understand conditions for 1 the breakdown of cooperation 2 the coexistence of different behaviors subcultures 3 the evolution of commonly shared behaviors norms and 4 the occurrence of polarization or conflict we point out that norms have a similar function in social systems that forces have in physics

perturbation of null spaces with application to the eigenvalue problem and generalized inverses we consider properties of a null space of an analytically perturbed matrix in particular we obtain taylor expansions for the eigenvectors which constitute a basis for the perturbed null space furthermore we apply these results to the calculation of puiseux expansion of the perturbed eigenvectors in the case of general eigenvalue problem as well as to the calculation of laurent series expansions for the perturbed group inverse and pseudoinverse matrices

algorithms for singularly perturbed limiting average markov control problems a singularly perturbed markov decision process with the limiting average reward criterion is considered it is assumed that the underlying process is composed of n separate irreducible processes and that the small perturbation is such that it unites these processes into a single irreducible process two algorithms for the solution of the underlying limit markov control problem are presented the first of these is a linear program possessing the wolfe dantzig structure inherited from the ergodic nearly decomposable assumption in the model the second is an aggregation disaggregation policy improvement algorithm

a reduction process for perturbed markov chains given a perturbed markov chain with generator a varepsilon a 0 varepsilon a 1 varepsilon 2 a 2 cdots it is shown that to each time scale t t varepsilon t varepsilon 2 cdots is associated a reduced markov chain this process of reduction has only finitely many steps it is given in a recursive way and each reduced chain admits as state space the recurrent classes of the reduced chain defined at the previous stage it is shown that if lambda is an eigenvalue different from zero of the generator of the kth reduced chain with multiplicity p then a varepsilon admits p eigenvalues lambda varepsilon such that lambda varepsilon varepsilon k lambda o varepsilon k a basis of ker a 0 resp ker a 0 which is robust to the perturbation is constructed

stochastic learning dynamics and speed of convergence in population games we study how long it takes for large populations of interacting agents to come close to nash equilibrium when they adapt their behavior using a stochastic better reply dynamic prior work considers this question mainly for 2 2 games and potential games here we characterize convergence times for general weakly acyclic games including coordination games dominance solvable games games with strategic complementarities potential games and many others with applications in economics biology and distributed control if players better replies are governed by idiosyncratic shocks the convergence time can grow exponentially in the population size moreover this is true even in games with very simple payoff structures however if their responses are sufficiently correlated due to aggregate shocks the convergence time is greatly accelerated in fact it is bounded for all sufficiently large populations we provide explicit bounds on the speed of convergence as a function of key structural parameters including the number of strategies the length of the better reply paths the extent to which players can influence the payoffs of others and the desired degree of approximation to nash equilibrium

the evolutionary game of pressure or interference resistance and collaboration in this paper we extend the framework of the evolutionary inspection game put forward recently by the author and coworkers to a large class of conflict interactions to address the pressure executed by the major player or principal on the large group of small players who can resist this pressure or collaborate with the major player we prove rigorous results on the convergence of various markov decision models of interacting small agents including evolutionary growth i e pairwise in groups and by coalition formation to a deterministic evolution on the distributions of the state spaces of small players paying main attention to situations with an infinite state space of small players we supply precise rates of convergence the theoretical results of the paper are applied to the analysis of the processes of inspection corruption cyber security counter terrorism banks and firms merging strategically enhanced preferential attachment and many 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

hamiltonian cycles and singularly perturbed markov chains we consider the hamiltonian cycle problem embedded in a singularly perturbed markov decision process we also consider a functional on the space of deterministic policies of the process that consists of the 1 1 entry of the fundamental matrices of the markov chains induced by the same policies we show that when the perturbation parameter e is less than or equal to 1 n2 the hamiltonian cycles of the directed graph are precisely the minimizers of our functional over the space of deterministic policies in the process we derive analytical expressions for the possiblen distinct values of the functional over the typically much larger space of deterministic policies

payoff based dynamics for multiplayer weakly acyclic games we consider repeated multiplayer games in which players repeatedly and simultaneously choose strategies from a finite set of available strategies according to some strategy adjustment process we focus on the specific class of weakly acyclic games which is particularly relevant for multiagent cooperative control problems a strategy adjustment process determines how players select their strategies at any stage as a function of the information gathered over previous stages of particular interest are payoff based processes in which at any stage players know only their own actions and noise corrupted payoffs from previous stages in particular players do not know the actions taken by other players and do not know the structural form of payoff functions we introduce three different payoff based processes for increasingly general scenarios and prove that after a sufficiently large number of stages player actions constitute a nash equilibrium at any stage with arbitrarily high probability we also show how to modify player utility functions through tolls and incentives in so called congestion games a special class of weakly acyclic games to guarantee that a centralized objective can be realized as a nash equilibrium we illustrate the methods with a simulation of distributed routing over a network

consistent behavior of certain perturbed determinants induced by graphs abstract we show that the determinant objective function introduced in ejov et al v ejov j a filar w murray g t nguyen determinants and longest cycles of graph siam j discrete math 22 33 2008 1215 1225 performs well under a certain symmetric linear perturbation that means sub graphs corresponding to hamiltonian cycles of a given graph are maximizers over the hull of all sub graphs with perturbation parameter e 0 1 note that in other optimization formulations see for example v s borkar v ejov j a filar directed graphs hamiltonicity and doubly stochastic matrices random structures algorithms 25 2004 376 395 v ejov j a filar m nguyen hamiltonian cycles and singularly perturbed markov chains math oper res 29 1 2004 114 131 j a filar k liu hamiltonian cycle problem and singularly perturbed markov decision process in statistics probability and game theory papers in honor of david blackwell ims lecture notes monograph series usa 1996 e in the corresponding perturbation was required to be significantly small

markov chains with rare transitions and simulated annealing we consider approximate stationary markov chains in which the entries of the one step transition probability matrix are known to be of different orders of magnitude and whose structure that is the orders of magnitude of the transition probabilities does not change with time for such markov chains we present a method for generating order of magnitude estimates for the t step transition probabilities for any t we then notice that algorithms of the simulated annealing type may be represented by markov chains which are approximately stationary over fairly long time intervals using our results we obtain a characterization of the convergent cooling schedules for the most general class of algorithms of the simulated annealing type

interpolation through an iterative scheme abstract a new method of interpolation is generated through an iterative scheme on dyadic rationals the process is linear local and produces almost twice differentiable functions a fundamental interpolating function is analyzed error formulas are derived the method is applied for curve fitting

approximation properties of multivariate wavelets wavelets are generated from refinable functions by using multiresolution analysis in this paper we investigate the approximation properties of multivariate refinable functions we give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask we connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators finally we demonstrate that a refinable function in w k 1 1 r s provides approximation order k

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods