inversion of the pieri formula for macdonald polynomials we give the explicit analytic development of macdonald polynomials in terms of modified complete and elementary symmetric functions these expansions are obtained by inverting the pieri formula specialization yields similar developments for monomial jack and hall littlewood symmetric functions

further results on the asymptotic agreement problem the problem of reaching a concensus between two decision makers provided with different information is considered the problem in which the decision makers may have different underlying probability models is studied results are developed to characterize the likelihood of an agreement being reached eventually in terms of the nature of the inter decision maker communications the problem in which the decision makers are aware of the possibility that they may have different models is treated it is found that in this case a deadlock can be reached where neither decision maker can learn additional information from the concensus process and they cannot reach a concensus decision this result indicates that incorporating human uncertainty in probability assessment into the asymptotic agreement problem can lead to outcomes not anticipated in the general theory previously developed

convergence results for the linear consensus problem under markovian random graphs this paper discusses the linear asymptotic consensus problem for a network of dynamic agents whose communication network is modeled by a randomly switching graph the switching is determined by a finite state markov process each topology corresponding to a state of the process we address the cases where the dynamics of the agents is expressed both in continuous time and in discrete time we show that if the consensus matrices are doubly stochastic average consensus is achieved in the mean square sense and the almost sure sense if and only if the graph resulting from the union of graphs corresponding to the states of the markov process is strongly connected the aim of this paper is to show how techniques from the theory of markovian jump linear systems in conjunction with results inspired by matrix and graph theory can be used to prove convergence results for stochastic consensus problems

asynchronous distributed algorithms for solving linear algebraic equations two asynchronous distributed algorithms are presented for solving a linear equation of the form ax b with at least one solution the equation is simultaneously and asynchronously solved by m agents assuming that each agent knows only a subset of the rows of the partitioned matrix a b the estimates of the equation s solution generated by its neighbors and nothing more neighbor relationships are characterized by a time dependent directed graph whose vertices correspond to agents and whose arcs depict neighbor relationships each agent recursively updates its estimate of a solution at its own event times by utilizing estimates generated by its neighbors which are transmitted with delays the event time sequences of different agents are not assumed to be synchronized it is shown that for any matrix vector pair a b for which the equation has a solution and any repeatedly jointly strongly connected sequence of neighbor graphs defined on the merged sequence of all agents event times the algorithms cause all agents estimates to converge exponentially fast to the same solution to ax b the first algorithm requires a specific initialization step at each agent and the second algorithm works for arbitrary initializations explicit expressions for convergence rates are provided and a relation between local initializations and limiting consensus solutions is established which is used to solve the least 2 norm solution

consensus based linear distributed filtering we address the consensus based distributed linear filtering problem where a discrete time linear stochastic process is observed by a network of sensors we assume that the consensus weights are known and we first provide sufficient conditions under which the stochastic process is detectable i e for a specific choice of consensus weights there exists a set of filtering gains such that the dynamics of the estimation errors without noise is asymptotically stable next we develop a distributed sub optimal filtering scheme based on minimizing an upper bound on a quadratic filtering cost in the stationary case we provide sufficient conditions under which this scheme converges conditions expressed in terms of the convergence properties of a set of coupled riccati equations

singular solutions for some semilinear elliptic equations let b r x n x r with n 2 consider a function u which satisfies cd egin array 20 c u n c 2 b r ackslash 0 u eqslant 0 on b r ackslash 0 elta u u p 0 on b r ackslash 0 nd array 1

unitary representations of affine hecke algebras related to macdonald spherical functions abstract for any reduced crystallographic root system we introduce a unitary representation of the extended affine hecke algebra given by discrete difference reflection operators acting in a hilbert space of complex functions on the weight lattice it is shown that the action of the center under this representation is diagonal on the basis of macdonald spherical functions as an application we compute an explicit pieri formula for these spherical functions

stochastic nestedness and the belief sharing information pattern solutions to decentralized stochastic optimization problems lead to recursions in which the state space enlarges with the time horizon thus leading to non tractability of classical dynamic programming a common joint information state supplied to each of the agents leads to a tractable recursion as is evident in the one step delayed information sharing structure case or when deterministic nestedness in information holds when there is a causality relationship as in the case of partially nested information structure however communication requirements for such conditions require exchange of very large data noiselessly hence these assumptions are generally impractical in this paper we present a weaker notion of nestedness which we term as stochastic nestedness which is characterized by a sequence of markov chain conditions it is shown that if the information structure is stochastically nested then an optimization problem is tractable and in particular for lqg problems the team optimal solution is linear despite the lack of deterministic nestedness or partial nestedness one other contribution of this paper is that by regarding the multiple decision makers as a single decision maker and using witsenhausen s equivalent model for discrete stochastic control it is shown that the common state required need not consist of observations and it suffices to share beliefs on the state and control actions a pattern we refer to as k stage belief sharing pattern we discuss the minimum amount of information exchange required to achieve such an information pattern for k 1 the information exchange needed is generally strictly less than what is needed for deterministic nestedness and is zero whenever stochastic nestedness applies in view of nestedness we present a discussion on the monotone values of information channels

identification and adaptive control of markov chains consider a countable state controlled markov chain whose transition probability is specified up to an unknown parameter alpha taking values in a compact metric space a to each alpha is associated a prespecified stationary control law zeta alpha the adaptive control law selects at each time t the control action zeta alpha t x t where x t is the state and alpha t is the maximum likelihood estimate of alpha the asymptotic behavior of this control scheme is investigated for the cases when the true parameter value alpha 0 does or does not belong to a and for the case when zeta is chosen to minimize an average cost criterion the analysis uses an appropriate extension of the notions of recurrence to nonstationary markov chains

consensus of second order discrete time multi agent systems with fixed topology abstract this paper studies the consensus of second order discrete time multi agent systems with fixed topology first we formulate the problem and give some preliminaries then by algebraic graph theory and matrix theory the convergence of system matrix is analyzed our main results indicate that the consensus of second order system can be achieved if and only if the topology graph has a directed spanning tree and the values of the scaling parameters satisfy a range the eigenvalues of the corresponding laplacian matrix play a key role in reaching consensus finally numerical simulations are given to illustrate the results

technical communique a linear distributed filter inspired by the markovian jump linear system filtering problem in this paper we introduce a consensus based distributed filter executed by a sensor network inspired by the markovian jump linear system filtering theory we show that the optimal filtering gains of the markovian jump linear system can be used as an approximate solution of the optimal distributed filtering problem this parallel allows us to interpret each filtering gain corresponding to a mode of operation of the markovian jump linear system as a filtering gain corresponding to a sensor in the network the approximate solution can be implemented distributively and guarantees a quantifiable level of performance

buildings and hecke algebras abstract in this paper we establish a strong connection between buildings and hecke algebras by studying two algebras of averaging operators on buildings to each locally finite regular building we associate a natural algebra b of chamber set averaging operators and when the building is affine we also define an algebra a of vertex set averaging operators we show that for appropriately parametrised hecke algebras h and h the algebra b is isomorphic to h and the algebra a is isomorphic to the centre of h on the one hand these results give a thorough understanding of the algebras a and b on the other hand they give a nice geometric and combinatorial understanding of hecke algebras and in particular of the macdonald spherical functions and the centre of affine hecke algebras our results also produce interesting examples of association schemes and polynomial hypergroups in later work we use the results here to study random walks on affine buildings

the asymptotic consensus problem on convex metric spaces a consensus problem consists of a group of dynamic agents who seek to agree upon certain quantities of interest the agents exchange information according to a communication network modeled as a directed time varying graph and evolve in a convex metric space a metric space endowed with a convex structure in this paper we generalize the asymptotic consensus problem to convex metric spaces under weak connectivity assumptions we show that if at each iteration an agent updates its state by choosing a point from a particular subset of the generalized convex hull generated by the agent s current state and the states of its neighbors then agreement is achieved asymptotically in addition we present several examples of convex metric spaces and their corresponding agreement algorithms

parameter estimation in stochastic systems some recent results and applications some recent work on the characterization of almost sure limit sets for maximum likelihood estimates for stochastic systems is reviewed applications to allied topics such as input selection for identification model selection self tuning etc are briefly discussed

singular solutions of some nonlinear elliptic equations abstract we give a classification of isolated singularities of any solution of the equation u u u q 1 0 if 0 is the singular point we prove that there exist two types of singularities when 1 n n 2 and that we have essentially the following in 0 i either u x l q n x 2 q 1 with l q n 2 q 1 2q q 1 n 1 q 1 ii or u x c x where c is any constant and the fundamental solution of the laplacian in r n

asymptotic stability of ltv systems with applications to distributed dynamic fusion we investigate the behavior of linear time varying ltv systems with randomly appearing sub stochastic system matrices motivated by dynamic fusion over mobile networks we develop conditions on the system matrices that lead to asymptotic stability of the underlying ltv system by partitioning the sequence of system matrices into slices we obtain the stability conditions in terms of slice lengths and introduce the notion of unbounded connectivity i e the time intervals over which the multi agent network is connected do not have to be bounded as long as they do not grow faster than a certain exponential rate we further apply the above analysis to derive the asymptotic behavior of a dynamic leader follower algorithm

solutions with time dependent singular sets for the heat equation with absorption abstract we consider the heat equation with a superlinear absorption term t u u u p in r n and study the existence of nonnegative solutions with an m dimensional time dependent singular set where n m 3 we prove that if 1 p n m n m 2 then there are two types of singular solutions moreover we show the uniqueness of the solutions and specify the exact behavior of the solutions near the singular set

coordination and consensus of networked agents with noisy measurements stochastic algorithms and asymptotic behavior this paper considers the coordination and consensus of networked agents where each agent has noisy measurements of its neighbors states for consensus seeking we propose stochastic approximation type algorithms with a decreasing step size and introduce the notions of mean square and strong consensus although the decreasing step size reduces the detrimental effect of the noise it also reduces the ability of the algorithm to drive the individual states towards each other the key technique is to ensure a trade off for the decreasing rate of the step size by following this strategy we first develop a stochastic double array analysis in a two agent model which leads to both mean square and strong consensus and extend the analysis to a class of well studied symmetric models subsequently we consider a general network topology and introduce stochastic lyapunov functions together with the so called direction of invariance to establish mean square consensus finally we apply the stochastic lyapunov analysis to a leader following scenario

resolution of a semilinear equation in l1 let n or be a bounded regular open set of n and let x s be a continuous nondecreasing function in s measurable in x such that x 0 0 almost everywhere we solve for f l 1 the problem p u u f in u 0 on in fact for this result instead of assuming that is nondecreasing in s we need only that x s s 0 we deduce an almost necessary and sufficient condition on in order that p has a solution roughly speaking this condition is f v g with g l 1 and v l 1

nonexistence theorems for singular solutions of quasilinear partial differential equations on etudie les etats fondamentaux singuliers de l equation de poisson semilineaire u f u 0 et de son analogue quasi lineaire div a du du f u 0 ou est le laplacien a n dimensions et a p et f u sont des fonctions donnees

isolated singularities for weighted quasilinear elliptic equations abstract we classify all the possible asymptotic behavior at the origin for positive solutions of quasilinear elliptic equations of the form div u p 2 u b x h u in 0 where 1 p n and is an open subset of r n with 0 our main result provides a sharp extension of a well known theorem of friedman and veron for h u u q and b x 1 and a recent result of the authors for p 2 and b x 1 we assume that the function h is regularly varying at with index q that is lim t h t h t q for every 0 and the weight function b x behaves near the origin as a function b 0 x varying regularly at zero with index greater than p this condition includes b x x and some of its perturbations for instance b x x log x m for any m r our approach makes use of the theory of regular variation and a new perturbation method for constructing sub and super solutions

asymptotic agreement in distributed estimation each of several agents updates his estimate of the same random variable whenever he makes a new observation or receives the estimate made by another agent in turn each agent transmits his estimate to a randomly chosen subset of the other agents a subset of agents forms a communicating ring if for every pair underline m underline p of ring members there is a sequence of ring members underline m underline m 1 underline m 2 underline m n 1 underline p such that underline m i sends his estimate to underline m i 1 infinitely often if each ring member knows that he is a ring member then the estimates of all the ring members asymptotically agree however this common limit can depend upon the order in which estimates are transmitted

asymptotic behavior of solutions of nonlinear elliptic equations we study and obtain formulas for the asymptotic behavior as x of c2 solutions of the semilinear equation u f x u xe where is the complement of some ball in n and f is continuous and nonlinear in u if for large x f is nearly radially symmetric in x we give conditions under which each positive solution of is asymptotic as x to some radially symmetric function our results can also be useful when f is only bounded above or below by a function which is radially symmetric in x or when the solution oscillates in sign examples when f has power like growth or exponential growth in the variables x and u usefully illustrate our results

pieri formulas for macdonald s spherical functions and polynomials we present explicit pieri formulas for macdonald s spherical functions or generalized hall littlewood polynomials associated with root systems and their q deformation the macdonald polynomials for the root systems of type a our pieri formulas recover the well known pieri formulas for the hall littlewood and macdonald symmetric functions due to morris and macdonald as special cases

adaptive control and parameter estimation in markov chains a linear case we consider a controlled markov chain with finite state space whose transition probabilities are assumed to depend linearly upon an unknown real parameter in particular we study the asymptotic behavior of the maximum likelihood estimate of under an arbitrary realizable control when is known to belong to a given bounded interval on the line we show that the estimate converges with probability one and characterize those realizations for which convergence does not lead to the true value we also suggest corrections to the control policy which guarantee almost sure convergence to the true value for the adaptive situation where the control depends only on the current estimate of and the present state we show that the maximum likelihood estimate converges to a value indistinguishable from the true one under the feedback law induced by

the partial order of dominant weights abstract the weight lattice of a crystallographic root system is partially ordered by the rule that if is a nonnegative integer linear combination of positive roots in this paper we study the subposet formed by the dominant weights in particular we prove that covers in this partial order only if belongs to a distinguished subset of the positive roots also if the root system is irreducible we prove that the mobius function of the partial order takes on only the values 0 1 2

adaptive control with a compact parameter set we consider the problem of the adaptive control of a markov chain with unknown transition probabilities we suppose the transition probabilities p i j u alpha to be dependent on an unknown parameter alpha at each time instant t a maximum likelihood estimate hat alpha t of the unknown parameter is made and a control input u t phi x t hat alpha t is applied where for each alpha phi cdot alpha is a good feedback control law it is shown that if a ranges over a compact set s then hat alpha t may diverge with probability one in the event however that the parameter estimates converge or more generally just the control laws phi cdot hat alpha t converge to some psi then under psi the closed loop transition probabilities for the true model are indistinguishable from those of any limit point of the parameter estimates

exact singular behavior of positive solutions to nonlinear elliptic equations with a hardy potential abstract in this paper we study the singular behavior at x 0 of positive solutions to the equation u x 2 u x u p x 0 where r n n 3 is a bounded domain with 0 and p 1 2 are given constants for the case n 2 2 4 the singular behavior of all the positive solutions is completely classified in the recent paper 5 here we determine the exact singular behavior of all the positive solutions for the remaining case n 2 2 4 in sharp contrast to the case n 2 2 4 where several converging blow up rates of u x are possible as x 0 we show that when n 2 2 4 every positive solution u x blows up in the same fashion lim x 0 x 2 p 1 u x 2 p 1 2 p 1 2 n 1 p 1

a littlewood richardson rule for macdonald polynomials macdonald polynomials are orthogonal polynomials associated to root systems and in the type a case the symmetric macdonald polynomials are a common generalization of schur functions macdonald spherical functions and jack polynomials we use the combinatorics of alcove walks to calculate products of monomials and intertwining operators of the double affine hecke algebra from this we obtain a product formula for macdonald polynomials of general lie type

classification of singular solutions of a nonlinear heat equation on considere toutes les solutions non negatives de l equation de la chaleur non lineaire avec absorption u t u m u p dans q r n 0 t pour n 1 m p 1 avec des conditions initiales u x 0 0 pour x 0

a generalized gossip algorithm on convex metric spaces a consensus problem consists of a group of dynamic agents who seek to agree upon certain quantities of interest this problem can be generalized in the context of convex metric spaces that extend the standard notion of convexity in this paper we introduce and analyze a randomized gossip algorithm for solving the generalized consensus problem on convex metric spaces where the communication between agents is controlled by a set of poisson counters we study the convergence properties of the algorithm using stochastic calculus in particular we show that the distances between the states of the agents converge to zero with probability one and in the r th mean sense in the special case of complete connectivity and uniform poisson counters we give upper bounds on the dynamics of the first and second moments of the distances between the states of the agents in addition we introduce instances of the generalized consensus algorithm for several examples of convex metric spaces together with numerical simulations

the dimension of the boundary of super brownian motion we show that the hausdorff dimension of the boundary of d dimensional super brownian motion is 0 if d 1 4 2 sqrt 2 if d 2 and 9 sqrt 17 2 if d 3

consensus in distributed estimation with inconsistent beliefs two people sequentially revise and exchange their estimates of the same random variable they may have different models of the underlying probability structure the two sets of estimates then will converge to the same value or the people will realize that their beliefs are inconsistent

scattering theory of discrete pseudo laplacians on a weyl chamber to a crystallographic root system we associate a system of multivariate orthogonal polynomials diagonalizing an integrable system of discrete pseudo laplacians on the weyl chamber we develop the time dependent scattering theory for these discrete pseudo laplacians and determine the corresponding wave operators and scattering operators in closed form as an application we describe the scattering behavior of certain hyperbolic ruijsenaars schneider type lattice calogero moser models associated with the macdonald polynomials

decentralized parameter estimation by consensus based stochastic approximation in this paper an algorithm for decentralized multi agent estimation of parameters in linear discrete time regression models is proposed in the form of a combination of local stochastic approximation algorithms and a global consensus strategy an analysis of the asymptotic properties of the proposed algorithm is presented taking into account both the multi agent network structure and the probabilities of getting local measurements and implementing exchange of inter agent messages in the case of non vanishing gains in the stochastic approximation algorithms an asymptotic estimation error covariance matrix bound is defined as the solution of a lyapunov like matrix equation in the case of asymptotically vanishing gains the mean square convergence is proved and the rate of convergence estimated in the discussion the problem of additive communication noise is treated in a methodologically consistent way it is also demonstrated how the consensus scheme in the algorithm can contribute to the overall reduction of measurement noise influence some simulation results illustrate the obtained theoretical results

asymptotic behavior of solutions of semilinear elliptic equations near an isolated singularity we consider the semilinear elliptic equationu h u in 0 w here is an open subset of r n n 2 containing the origin and h is locally lipschitz continuous on 0 positive in 0 w e give a complete classification of isolated singularities of positive solutions when h varies regularly at infinity of in dex q 1 c n that is limu h u h u q f or every 0 where cn denotes either n n 2 if n 3o r if n 2 our result extends a well known theorem of veron for the case h u u q

asymptotic behavior of the least energy solution of a problem with competing powers we consider the problem e2 u uq up 0 in u 0 in u 0 on here is a smooth bounded domain in rn 1 q p n 2n 2 if n 3 and e is a small positive parameter we study the asymptotic behavior of the least energy solution as e goes to zero in the case q nn 2 we show that the limiting behavior is dominated by the singular solution g gq 0 in p g 0 on the reduced energy is of nonlocal type