character generators for elements of finite order in simple lie groups a1 a2 a3 b2 and g2 elements of finite order in the compact simple lie groups su 2 su 3 su 4 sp 4 or o 5 and g 2 are considered we provide the characters of the elements on irreducible representations of lie groups by assigning appro priate numerical values to the variables on which the characters of representations of the lie group depend in this way we specialize generating functions for the characters of the representations of the lie groups to the generating functions for characters of the elements of finite order particular attention is paid to rational elements all of whose characters are integers they are listed and the generating functions for their characters are obtained in a simplified form from which the characters can be read gaussian elements are also studied in detail their characters are complex valued with integer real and imaginary parts

discrete and continuous exponential transforms of simple lie groups of rank 2 we develop and describe continuous and discrete transforms of class functions on compact simple lie group g as their expansions into series of uncommon special functions called here e functions in recognition of the fact that the functions generalize common exponential functions the rank of g is the number of variables in the e functions a uniform discretization of the decomposition problem is described on lattices of any density and symmetry admissible for the lie group g

decay estimates of solutions to the compressible euler maxwell system in r 3 abstract we study the large time behavior of solutions near a constant equilibrium to the compressible euler maxwell system in r 3 we first refine a global existence theorem by assuming that the h 3 norm of the initial data is small but the higher order derivatives can be arbitrarily large if the initial data belongs to h s 0 s 3 2 or b 2 s 0 s 3 2 by a regularity interpolation trick we obtain the various decay rates of the solution and its higher order derivatives as an immediate byproduct the usual l p l 2 1 p 2 type of the decay rates follows without requiring that the l p norm of initial data is small

a simulation based algorithm for ergodic control of markov chains conditioned on rare events we study the problem of long run average cost control of markov chains conditioned on a rare event in a related recent work a simulation based algorithm for estimating performance measures associated with a markov chain conditioned on a rare event has been developed we extend ideas from this work and develop an adaptive algorithm for obtaining online optimal control policies conditioned on a rare event our algorithm uses three timescales or step size schedules on the slowest timescale a gradient search algorithm for policy updates that is based on one simulation simultaneous perturbation stochastic approximation spsa type estimates is used deterministic perturbation sequences obtained from appropriate normalized hadamard matrices are used here the fast timescale recursions compute the conditional transition probabilities of an associated chain by obtaining solutions to the multiplicative poisson equation for a given policy estimate further the risk parameter associated with the value function for a given policy estimate is updated on a timescale that lies in between the two scales above we briefly sketch the convergence analysis of our algorithm and present a numerical application in the setting of routing multiple flows in communication networks

on actor critic algorithms in this article we propose and analyze a class of actor critic algorithms these are two time scale algorithms in which the critic uses temporal difference learning with a linearly parameterized approximation architecture and the actor is updated in an approximate gradient direction based on information provided by the critic we show that the features for the critic should ideally span a subspace prescribed by the choice of parameterization of the actor we study actor critic algorithms for markov decision processes with polish state and action spaces we state and prove two results regarding their convergence

decay of the navier stokes poisson equations we establish the time decay rates of the solution to the cauchy problem for the compressible navier stokes poisson system via a refined pure energy method in particular the optimal decay rates of the higher order spatial derivatives of the solution are obtained the h s 0 s 3 2 negative sobolev norms are shown to be preserved along time evolution and enhance the decay rates as a corollary we also obtain the usual lp l2 1 p 2 type of the optimal decay rates compared to the compressible navier stokes system and the compressible irrotational euler poisson system our results imply that both the dispersion effect of the electric field and the viscous dissipation contribute to enhance the decay rate of the density our proof is based on a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis

distributive stochastic learning for delay optimal ofdma power and subband allocation in this paper we consider the distributive queue aware power and subband allocation design for a delay optimal ofdma uplink system with one base station k users and nf independent subbands each mobile has an uplink queue with heterogeneous packet arrivals and delay requirements we model the problem as an infinite horizon average reward markov decision problem mdp where the control actions are functions of the instantaneous channel state information csi as well as the joint queue state information qsi to address the distributive requirement and the issue of exponential memory requirement and computational complexity we approximate the subband allocation q factor by the sum of the per user subband allocation q factor and derive a distributive online stochastic learning algorithm to estimate the per user q factor and the lagrange multipliers lm simultaneously and determine the control actions using an auction mechanism we show that under the proposed auction mechanism the distributive online learning converges almost surely with probability 1 for illustration we apply the proposed distributive stochastic learning framework to an application example with exponential packet size distribution we show that the delay optimal power control has the multilevel water filling structure where the csi determines the instantaneous power allocation and the qsi determines the water level the proposed algorithm has linear signaling overhead and computational complexity o knf which is desirable from an implementation perspective

a sensitivity formula for risk sensitive cost and the actor critic algorithm abstract we propose for risk sensitive control of finite markov chains a counterpart of the popular actor critic algorithm for classical markov decision processes the algorithm is based on a sensitivity formula for the risk sensitive cost and is shown to converge with probability one to the desired solution the proof technique is an adaptation of the ordinary differential equations approach for the analysis of two time scale stochastic approximation algorithms

on l3 stability of strong solutions of the navier stokes equations with the navier type boundary conditions abstract we show that a strong solution u of the navier stokes initial boundary value problem on the time interval 0 t where 0 t is stable in the l 3 norm provided u l 2 0 t w 2 2 l 0 t w 1 2 the problem is considered with the so called navier type boundary conditions

adaptive importance sampling and control variates abstract we construct and investigate an adaptive variance reduction framework in which both importance sampling and control variates are employed the three lines monte carlo averaging and two variance reduction parameter search lines run in parallel on a common sequence of uniform random vectors on the unit hypercube given that these two variance reduction techniques are effective often in a complementary way their combined application is well expected to widen the applicability of adaptive variance reduction we derive convergence rates of the theoretical estimator variance towards its minimum as a fixed computing budget increases when stochastic approximation runs with optimal constant learning rates we derive sufficient conditions for the proposed algorithm to attain the minimal estimator variance in the limit by stochastic approximation with decreasing learning rates or by sample average approximation when computing budget is unlimitedly available numerical results support our theoretical findings and illustrate the effectiveness of the proposed framework

stability of 2d incompressible flows in r 3 abstract we investigate the global in time stability of regular solutions with large velocity vectors to the evolutionary navier stokes equation in r 3 the class of stable flows contains all two dimensional weak solutions the only assumption which is required is smallness of the l 2 norm of initial perturbation or its derivative with respect to the z coordinate in the same norm the magnitude of the rest of the norm of initial datum is not restricted

a weak lp prodi serrin type regularity criterion for the micropolar fluid equations we investigate regularity criteria for weak solutions of the micropolar fluid equations in a bounded three dimensional domain we show that the solution u w is strong on 0 t if either u ls 0 t lr or u ls 0 t lr is bounded from above by a specific constant where 3 r 2 s 1 and r 3

navier stokes equations stability and minimal perturbations of global solutions abstract in a recent paper we proved that assuming some initial data h 1 2 r 3 lead to a singularity for the 3d navier stokes equations there are also initial data with the minimal h 1 2 norm which produce a singularity and the set of such data is compact up to translations and the natural scaling of the equation the purpose of this paper is to analyze a more general case where the set of initial data potentially leading to a singularity is on a sphere centered at non zero initial data leading to a global solution

on the large time decay of asymmetric flows in homogeneous sobolev spaces abstract in this paper the large time decay of the micropolar fluid equations on h s r n where n 2 3 and s 0 are studied we show for each s 0 that t s 2 u w t h s r n 0 as t for leray global solutions with arbitrary initial data in l 2 r n when the vortex viscosity is present we obtain a faster decay for the micro rotational field w t h s r n o t s 1 2 some related results are also included

generalized mhd equations abstract solutions of the d dimensional generalized mhd gmhd equations t u u u p b b u t b u b b u b are studied in this paper we pay special attention to the impact of the parameters and on the regularity of solutions our investigation is divided into three major cases 1 0 and 0 2 0 and 0 and 3 0 and 0 when 0 and 0 the gmhd equations with any 0 and 0 possess a global weak solution corresponding to any l2 initial data furthermore weak solutions associated with 1 2 d 4 and 1 2 d 4 are actually global classical solutions when their initial data are sufficiently smooth as a special consequence smooth solutions of the 3d gmhd equations with 5 4 and 5 4 do not develop finite time singularities the study of the gmhd equations with 0 and 0 is motivated by their potential applications in magnetic reconnection a local existence result of classical solutions and several global regularity conditions are established for this case these conditions are imposed on either the vorticity u or the current density j b but not both and are weaker than some of current existing ones when 0 and 0 the gmhd equations reduce to the ideal mhd equations it is shown here that the ideal mhd equations admit a unique local solution when the prescribed initial data is in a holder space cr with r 1

new algorithms of the q learning type we propose two algorithms for q learning that use the two timescale stochastic approximation methodology the first of these updates q values of all feasible state action pairs at each instant while the second updates q values of states with actions chosen according to the current randomized policy updates a proof of convergence of the algorithms is shown finally numerical experiments using the proposed algorithms on an application of routing in communication networks are presented on a few different settings

on the supnorm form of leray s problem for the incompressible navier stokes equations we show that t3 4 u t l r3 0 as t for all leray hopf s global weak solutions u t of the incompressible navier stokes equations in 3 it is also shown that t u t e tu0 l r3 0 as t where e t is the heat semigroup as well as other fundamental new results in spite of the complexity of the questions our approach is elementary and is based on standard tools like conventional fourier and energy methods

discrete and continuous sine transform generalized to semisimple lie groups of rank two a new continuous group transform together with its discretization over a lattice of any density and admissible symmetry is defined and described for the four compact semisimple lie groups of rank 2 in the case of rank 1 the discrete version is the sine transform properties of the expansion functions of the transform called s functions are studied digital data processing is our motivating application

the boltzmann equation besov spaces and optimal time decay rates in rxn abstract we prove that k th order derivatives of perturbative classical solutions to the hard and soft potential boltzmann equation without the angular cut off assumption in the whole space r x n with n 3 converge in large time to the global maxwellian with the optimal decay rate of o t 1 2 k n 2 n r in the l x r l v 2 norm for any 2 r these results hold for any 0 n 2 as long as initially f 0 b 2 l v 2 in the hard potential case we prove faster decay results in the sense that if p f 0 b 2 l v 2 and i p f 0 b 2 1 l v 2 for n 2 n 2 2 then the solution decays the global maxwellian in l v 2 l x 2 with the optimal large time decay rate of o t 1 2

queue aware distributive resource control for delay sensitive two hop mimo cooperative systems in this paper we consider a queue aware distributive resource control algorithm for two hop mimo cooperative systems we shall illustrate that relay buffering is an effective way to reduce the intrinsic half duplex penalty in cooperative systems the complex interactions of the queues at the source node and the relays are modeled as an average cost infinite horizon markov decision process mdp the traditional approach solving this mdp problem involves centralized control with huge complexity to obtain a distributive and low complexity solution we introduce a linear structure which approximates the value function of the associated bellman equation by the sum of per node value functions we derive a distributive two stage two winner auction based control policy which is a function of the local csi and local qsi only furthermore to estimate the best fit approximation parameter we propose a distributive online stochastic learning algorithm using stochastic approximation theory finally we establish technical conditions for almost sure convergence and show that under heavy traffic the proposed low complexity distributive control is global optimal

stationary micropolar fluid with boundary data in l2 abstract we consider the dirichlet boundary value problem for the equations of a stationary micropolar fluid in a bounded three dimensional domain we show the existence and uniqueness of a distributional solution with boundary values in l 2

simulation based optimization of markov reward processes this paper proposes a simulation based algorithm for optimizing the average reward in a finite state markov reward process that depends on a set of parameters as a special case the method applies to markov decision processes where optimization takes place within a parametrized set of policies the algorithm relies on the regenerative structure of finite state markov processes involves the simulation of a single sample path and can be implemented online a convergence result with probability 1 is provided

the impact of stochastic noisy feedback on distributed network utility maximization the implementation of distributed network utility maximization num algorithms hinges heavily on information feedback through message passing among network elements in practical systems the feedback is often obtained using error prone measurement mechanisms and suffers from random errors in this paper we investigate the impact of noisy feedback on distributed num we first study the distributed num algorithms based on the lagrangian dual method and focus on the primal dual p d algorithm which is a single time scale algorithm in the sense that the primal and dual parameters are updated simultaneously assuming strong duality we study both cases when the stochastic gradients are unbiased or biased and develop a general theory on the stochastic stability of the p d algorithms in the presence of noisy feedback when the gradient estimators are unbiased we establish via a combination of tools in martingale theory and convex analysis that the iterates generated by distributed p d algorithms converge with probability one to the optimal point under standard technical conditions in contrast when the gradient estimators are biased we show that the iterates converge to a contraction region around the optimal point provided that the biased terms are asymptotically bounded by a scaled version of the true gradients we also investigate the rate of convergence for the unbiased case and find that in general the limit process of the interpolated process corresponding to the normalized iterate sequence is a stationary reflected linear diffusion process not necessarily a gaussian diffusion process we apply the above general theory to investigate stability of cross layer rate control for joint congestion control and random access next we study the impact of noisy feedback on distributed two time scale num algorithms based on primal decomposition we establish via the mean ode method the convergence of the stochastic two time scale algorithm under mild conditions for the cases where the gradient estimators in both time scales are unbiased numerical examples are used to illustrate the finding that compared to the single time scale counterpart the two time scale algorithm although having lower complexity is less robust to noisy feedback

on the regularity of weak solutions to the magnetohydrodynamic equations abstract in this paper we study the regularity of weak solution to the incompressible magnetohydrodynamic equations we obtain some sufficient conditions for regularity of weak solutions to the magnetohydrodynamic equations which is similar to that of incompressible navier stokes equations moreover our results demonstrate that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solution to the magneto hydrodynamic equations

stability of two dimensional navier stokes motions in the periodic case abstract we consider the motion described by the navier stokes equations in a box with periodic boundary conditions first we prove the existence of global strong two dimensional solutions next we show the existence of global strong three dimensional solutions under the assumption that the initial data and the external force are sufficiently close to the initial data and the external force of the two dimensional problem in appropriate spaces the second result can be treated as stability of strong two dimensional solutions in the set of suitably strong three dimensional motions

global existence of three dimensional incompressible magneto micropolar system with mixed partial dissipation magnetic diffusion and angular viscosity abstract the magneto micropolar fluid flows describe the motion of electrically conducting micropolar fluids in the presence of a magnetic field the issue of whether the strong solution of magneto micropolar equations in three dimensional can exist globally in time with large initial data is still unknown in this paper we deal with the cauchy problem of the three dimensional magneto micropolar system with mixed partial dissipation magnetic diffusion and angular viscosity more precisely the global existence of smooth solutions to the three dimensional incompressible magneto micropolar fluid equations with mixed partial dissipation magnetic diffusion and angular viscosity are obtained by energy method under the assumption that h 1 norm of the initial data u 0 b 0 w 0 sufficiently small namely u 0 b 0 0 h 1 r 3 2 e 2 where e is a sufficiently small positive number this work follows the techniques in the paper of cao and wu 2011

asymptotic stability of large solutions with large perturbation to the navier stokes equations abstract consider weak solutions w of the navier stokes equations in serrin s class w l 0 l q for 2 3 q 1 with 3 q where is a general unbounded domain in r 3 we shall show that although the initial and external disturbances from w are large every perturbed flow v with the energy inequality converges asymptotically to w as v t w t l 2 0 v t w t l 2 o t 1 2 as t

linear stochastic approximation driven by slowly varying markov chains we study a linear stochastic approximation algorithm that arises in the context of reinforcement learning the algorithm employs a decreasing step size and is driven by markov noise with time varying statistics we show that under suitable conditions the algorithm can track the changes in the statistics of the markov noise as long as these changes are slower than the rate at which the step size of the algorithm goes to zero

on immanant functions related to weyl groups of an in this work we recall the definition of matrix immanants a generalization of the determinant and permanent of a matrix we use them to generalize families of symmetric and antisymmetric orbit functions related to weyl groups of the simple lie algebras an the new functions and their properties are described in particular we give their continuous orthogonality relations several examples are shown

asymptotic stability for the navier stokes equations in the marginal class in this paper we consider the navier stokes equations in r n n 3 we prove the asymptotic stability for weak solutions in the marginal class u l 2 0 bmo where bmo denotes the bounded mean oscillation function with arbitrary initial and external perturbations

lp stability for the strong solutions of the navier stokes equations in the whole space abstract we consider the motion of a viscous fluid filling the whole space r3 governed by the classical navier stokes equations 1 existence of global in time regular solutions for that system of non linear partial differential equations is still an open problem from either the mathematical and the physical point of view an interesting property is the stability or not of the eventual global regular solutions here we assume that v1 t x is a solution with initial data a1 x for small perturbations of a1 we want the solution v1 t x being slightly perturbed too due to viscosity it is even expected that the perturbed solution v2 t x approaches the unperturbed one as time goes to infinity this is just the result proved in this paper to measure the distance between v1 t x and v2 t x at each time t suitable norms are introduced lp norms for fluids filling a bounded vessel exponential decay of the above distance is expected such a strong result is not reasonable for fluids filling the entire space

an actor critic algorithm for constrained markov decision processes abstract an actor critic type reinforcement learning algorithm is proposed and analyzed for constrained controlled markov decision processes the analysis uses multiscale stochastic approximation theory and the envelope theorem of mathematical economics

stability enhancement by boundary control in 2 d channel flow in this paper we stabilize the parabolic equilibrium profile in a two dimensional 2 d channel flow using actuators and sensors only at the wall the control of channel flow was previously considered by speyer and coworkers and bewley and coworkers who derived feedback laws based on linear optimal control and implemented by wall normal actuation with an objective to achieve global lyapunov stabilization we arrive at a feedback law using tangential actuation using teamed pairs of synthetic jets or rotating disks and only local measurements of wall shear stress allowing to embed the feedback in microelectromechanical systems mems hardware without need for wiring this feedback is shown to guarantee global stability in at least h sup 2 norm which by sobolev s embedding theorem implies continuity in space and time of both the flow field and the control as well as their convergence to the desired steady state the theoretical results are limited to low values of reynolds number however we present simulations that demonstrate the effectiveness of the proposed feedback for values five order of magnitude higher

dynamic cesaro wardrop equilibration in networks we analyze a routing scheme for a broad class of networks which converges in the cesaro sense with probability one to the set of approximate cesaro wardrop equilibria an extension of the notion of a wardrop equilibrium the network model allows for wireline networks where delays are caused by flows on links as well as wireless networks a primary motivation for us where delays are caused by other flows in the vicinity of nodes the routing algorithm is distributed using only the local information about observed delays by the nodes and is moreover impervious to clock offsets at nodes the scheme is also fully asynchronous since different iterates have their own counters and the orders of packets and their acknowledgments may be scrambled finally the scheme is adaptive to the traffic patterns in the network the demonstration of convergence in a fully dynamic context involves the treatment of two time scale distributed asynchronous stochastic iterations using an ordinary differential equation approach the invariant measures are identified due to a randomization feature in the algorithm a direct stochastic analysis shows that the algorithm avoids non wardrop equilibria finally some comments on the existence uniqueness stability and other properties of wardrop equilibria are made

exponential stability for magneto micropolar fluids abstract we prove existence uniqueness and exponential stability of mildly decaying global strong solutions for the magneto micropolar fluids system in three space dimensions our main objective is to study the convergence of nonstationary solutions to stationary solutions for the system fluids when t we show that under mild suitable conditions the convergence is indeed exponential

six types of e functions of the lie groups o 5 and g 2 new families of e functions are described in the context of the compact simple lie groups o 5 and g 2 these functions of two real variables generalize the common exponential functions and for each group only one family is currently found in the literature all the families are fully characterized their most important properties are described namely their continuous and discrete orthogonalities and decompositions of their products

global regularity for the two dimensional anisotropic boussinesq equations with vertical dissipation this paper establishes the global in time existence of classical solutions to the two dimensional anisotropic boussinesq equations with vertical dissipation when only vertical dissipation is present there is no direct control on the horizontal derivatives and the global regularity problem is very challenging to solve this problem we bound the derivatives in terms of the l infty norm of the vertical velocity v and prove that v l r with 2 leqq r infty does not grow faster than sqrt r log r at any time as r increases a delicate interpolation inequality connecting v l infty and v l r then yields the desired global regularity

the o d e method for convergence of stochastic approximation and reinforcement learning it is shown here that stability of the stochastic approximation algorithm is implied by the asymptotic stability of the origin for an associated ode this in turn implies convergence of the algorithm several specific classes of algorithms are considered as applications it is found that the results provide i a simpler derivation of known results for reinforcement learning algorithms ii a proof for the first time that a class of asynchronous stochastic approximation algorithms are convergent without using any a priori assumption of stability iii a proof for the first time that asynchronous adaptive critic and q learning algorithms are convergent for the average cost optimal control problem

remarks on singularities dimension and energy dissipation for ideal hydrodynamics and mhd for weak solutions of the incompressible euler equations there is energy conservation if the velocity is in the besov space b 3 s with s greater than 1 3 b 3 s consists of functions that are lip s i e holder continuous with exponent s measured in the l p norm here this result is applied to a velocity field that is lip 0 except on a set of co dimension on which it is lip agr 1 with uniformity that will be made precise below we show that the frisch parisi multifractal formalism is valid at least in one direction for such a function and that there is energy conservation if analogous conservation results are derived for the equations of incompressible ideal mhd i e zero viscosity and resistivity for both energy and helicity in addition a necessary condition is derived for singularity development in ideal mhd generalizing the beale kato majda condition for ideal hydrodynamics

asymptotic stability of non newtonian flows with large perturbation in r2 consider asymptotic stability of weak solutions for the incompressible non newtonian fluid motion in whole spaces r 2 we shall show that although the initial data disturbance from u are large every perturbed flow v with the energy inequality converges asymptotically to u as v t u t 0 as t

learning algorithms for markov decision processes with average cost this paper gives the first rigorous convergence analysis of analogues of watkins s q learning algorithm applied to average cost control of finite state markov chains we discuss two algorithms which may be viewed as stochastic approximation counterparts of two existing algorithms for recursively computing the value function of the average cost problem the traditional relative value iteration rvi algorithm and a recent algorithm of bertsekas based on the stochastic shortest path ssp formulation of the problem both synchronous and asynchronous implementations are considered and analyzed using the ode method this involves establishing asymptotic stability of associated ode limits the ssp algorithm also uses ideas from two time scale stochastic approximation

recursive algorithms for estimation of hidden markov models and autoregressive models with markov regime this paper is concerned with recursive algorithms for the estimation of hidden markov models hmms and autoregressive ar models under the markov regime convergence and rate of convergence results are derived acceleration of convergence by averaging of the iterates and the observations are treated finally constant step size tracking algorithms are presented and examined

global well posedness for magneto micropolar system in 2 1 2 dimensions whether or not the classical solution of the 3d magneto micropolar equations exists globally or develops the finite time singularity is still a difficult problem in this paper we investigated the global well posedness for 2 1 2 d incompressible magneto micropolar system by a standard energy method we obtain the global existence and uniqueness for 2 1 2 d incompressible magneto micropolar system

natural actor critic algorithms we present four new reinforcement learning algorithms based on actor critic natural gradient and function approximation ideas and we provide their convergence proofs actor critic reinforcement learning methods are online approximations to policy iteration in which the value function parameters are estimated using temporal difference learning and the policy parameters are updated by stochastic gradient descent methods based on policy gradients in this way are of special interest because of their compatibility with function approximation methods which are needed to handle large or infinite state spaces the use of temporal difference learning in this way is of special interest because in many applications it dramatically reduces the variance of the gradient estimates the use of the natural gradient is of interest because it can produce better conditioned parameterizations and has been shown to further reduce variance in some cases our results extend prior two timescale convergence results for actor critic methods by konda and tsitsiklis by using temporal difference learning in the actor and by incorporating natural gradients our results extend prior empirical studies of natural actor critic methods by peters vijayakumar and schaal by providing the first convergence proofs and the first fully incremental algorithms

on discretization of tori of compact simple lie groups the discrete orthogonality of special function families called c and s functions which are derived from the characters of compact simple lie groups is described in hrivnak and patera 2009 j phys a math theor 42 385208 here the results of hrivnak and patera are extended to two additional recently discovered families of special functions called ss and sl functions the main result is an explicit description of their pairwise discrete orthogonality within each family when the functions are sampled on finite fragments fsm and flm of a lattice in any dimension n 2 and of any density controlled by m and of the symmetry of the weight lattice of any compact simple lie group with two different lengths of roots

discrete and continuous cosine transform generalized to lie groups su 3 and g 2 in the paper we complete the development and description of the four variants of two dimensional generalization of the cosine transform started in patera and zaratsyan j math phys 46 053514 2005 each variant is based on a compact semisimple lie group g of rank 2 here the groups are su 3 and g 2 the cosines are generalized as the corresponding c functions of the lie group a c function is the contribution to an irreducible character from one orbit of the appropriate weyl group an explicit description is provided for expansions of functions given on the fundamental region f of the two compact simple lie groups into series of c functions the fundamental region f is an equilateral triangle for su 3 and half of such a triangle for g 2 expansion coefficients are calculated using orthogonality of c functions on f discrete expansions are set up on a grid fm f the grid is defined group theoretically for all positive integers m it consists of points in f that represent conjugacy classes of ele

the vlasov poisson landau system in a periodic box the classical vlasov poisson landau system describes dynamics of a collisional plasma interacting with its own electrostatic eld as well as its grazing collisions such grazing collisions are modeled by the famous landau fokker planck collision kernel proposed by landau in 1936 we construct global unique solutions to such a system for initial data which have small weighted h norms but can have large h k 3 norms with high velocity moments our construction is based on accumulative study on the landau kernel in the past decade g1 sg1 3 with four extra ingredients to overcome the speci c mathematical di culties present in the vlasov poisson landau system a new exponential weight of electric potential to cancel the growth of the velocity a new velocity weight to capture the weak velocity di usion in the landau kernel a decay of the electric eld to close the energy estimate and a new bootstrap argument to control the propagation of the high moments and regularity with large amplitude

single sample fictitious play this paper is concerned with distributed learning and optimization in large scale settings the well known fictitious play fp algorithm has been shown to achieve nash equilibrium learning in certain classes of multiagent games however fp can be computationally difficult to implement when the number of players is large sampled fp sfp is a variant of fp that mitigates the computational difficulties arising in fp by using a monte carlo i e sampling based approach despite its computational advantages a shortcoming of sfp is that the number of samples that must be drawn at each iteration grows without bound as the algorithm progresses in this paper we propose single sample fp ssfp a variant of sfp in which only one sample needs to be drawn in each round of the algorithm convergence of ssfp to the set of nash equilibria is proven simulation results show the performance of ssfp is comparable to that of sfp despite drawing far fewer samples

lower bounds for an integral involving fractional laplacians and the generalized navier stokes equations in besov spaces when estimating solutions of dissipative partial differential equations in l p related spaces we often need lower bounds for an integral involving the dissipative term if the dissipative term is given by the usual laplacian lower bounds can be derived through integration by parts and embedding inequalities however when the laplacian is replaced by the fractional laplacian the approach of integration by parts no longer applies in this paper we obtain lower bounds for the integral involving by combining pointwise inequalities for with bernstein s inequalities for fractional derivatives as an application of these lower bounds we establish the existence and uniqueness of solutions to the generalized navier stokes equations in besov spaces the generalized navier stokes equations are the equations resulting from replacing in the navier stokes equations by

global well posedness and large time decay for the 2d micropolar equations abstract this paper studies the global in time regularity and large time behavior of solutions to the 2d micropolar equations with only angular viscosity dissipation micropolar equations model a class of fluids with nonsymmetric stress tensor such as fluids consisting of particles suspended in a viscous medium when there is no kinematic viscosity in the momentum equation the global regularity problem is not easy due to the lack of suitable bounds on the derivatives the idea here is to fully exploit the structure of the system and control the vorticity via the evolution equation of a combined quantity of the vorticity and the micro rotation angular velocity to understand the large time behavior we overcome two main difficulties the lack of kinematic viscosity and the presence of linear terms classical tools such as the fourier splitting method of schonbek and kato s approach for the decay of small solutions do not apply here we introduce a diagonalization process to eliminate the linear terms and rely on the uniform bounds for the first derivatives of the solutions to generate suitable decay rates

on the regularity criteria for weak solutions to the magnetohydrodynamic equations abstract we study the regularity criteria for weak solutions to the incompressible magnetohydrodynamic equations some regularity criteria are obtained for weak solutions to the magnetohydrodynamic equations which generalize the results in c he z xin on the regularity of solutions to the magneto hydrodynamic equations j differential equations 213 2 2005 235 254 our results reveal that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solutions to the magnetohydrodynamic equations

global existence and optimal convergence rates of solutions for 3d compressible magneto micropolar fluid equations abstract the cauchy problem for the three dimensional compressible magneto micropolar fluid equations is considered existence of global in time smooth solutions is established under the condition that the initial data are small perturbations of some given constant state moreover we obtain the time decay rates of the higher order spatial derivatives of the solution by combining the l p l q estimates for the linearized equations and the fourier splitting method if the initial perturbation is small in h 3 norm and bounded in l 1 norm

regularity criteria of weak solutions to the three dimensional micropolar flows regularity criteria of weak solutions to the three dimensional micropolar fluid motion equations are discussed sufficient conditions for the regularity of weak solutions are presented by imposing serrin s type growth conditions on the velocity field in lorentz spaces multiplier spaces bounded mean oscillation spaces and besov spaces respectively the findings demonstrate that the velocity field plays a dominant role in the regularity problem of micropolar fluid motion equations

two timescale analysis of the alopex algorithm for optimization alopex is a correlation based gradient free optimization technique useful in many learning problems however there are no analytical results on the asymptotic behavior of this algorithm this article presents a new version of alopex that can be analyzed using techniques of two timescale stochastic approximation method it is shown that the algorithm asymptotically behaves like a gradient descent method though it does not need or estimate any gradient information it is also shown through simulations that the algorithm is quite effective

a regularity criterion for the 3d magneto micropolar fluid equations in triebel lizorkin spaces abstract we consider the regularity criterion for the 3d magneto micropolar fluid equations in triebel lizorkin spaces it is proved that if u l p 0 t f q 2 q 3 0 with 2 p 3 q 2 3 2 q then the solution remains smooth in 0 t as a corollary we obtain the classical beal kato majda criterion that is the condition u l 1 0 t b 0 ensures the smoothness of the solution

reinforcement learning for long run average cost abstract a large class of sequential decision making problems under uncertainty can be modeled as markov and semi markov decision problems smdps when their underlying probability structure has a markov chain they may be solved by using classical dynamic programming dp methods however dp methods suffer from the curse of dimensionality and break down rapidly in face of large state spaces in addition dp methods require the exact computation of the so called transition probabilities which are often hard to obtain and are hence said to suffer from the curse of modeling as well in recent years a simulation based method called reinforcement learning rl has emerged in the literature it can to a great extent alleviate stochastic dp of its curses by generating near optimal solutions to problems having large state spaces and complex transition mechanisms in this paper a simulation based algorithm that solves markov and smdps is presented along with its convergence analysis the algorithm involves a step size based transformation on two time scales its convergence analysis is based on a recent result on asynchronous convergence of iterates on two time scales we present numerical results from the new algorithm on a classical preventive maintenance case study of a reasonable size where results on the optimal policy are also available in addition we present a tutorial that explains the framework of rl in the context of long run average cost smdps

two time scale stochastic approximation with controlled markov noise and off policy temporal difference learning we present for the first time an asymptotic convergence analysis of two time scale stochastic approximation driven by controlled markov noise in particular the faster and slower recursions have nonadditive controlled markov noise components in addition to martingale difference noise we analyze the asymptotic behavior of our framework by relating it to limiting differential inclusions in both time scales that are defined in terms of the ergodic occupation measures associated with the controlled markov processes finally we present a solution to the off policy convergence problem for temporal difference learning with linear function approximation using our results

stochastic compositional gradient descent algorithms for minimizing compositions of expected value functions classical stochastic gradient methods are well suited for minimizing expected value objective functions however they do not apply to the minimization of a nonlinear function involving expected values or a composition of two expected value functions i e the problem min x mathbf e v left f v big mathbf e w g w x big right minxevfv ew gw x in order to solve this stochastic composition problem we propose a class of stochastic compositional gradient descent scgd algorithms that can be viewed as stochastic versions of quasi gradient method scgd update the solutions based on noisy sample gradients of f v g w fv gw and use an auxiliary variable to track the unknown quantity mathbf e w left g w x right ewgw x we prove that the scgd converge almost surely to an optimal solution for convex optimization problems as long as such a solution exists the convergence involves the interplay of two iterations with different time scales for nonsmooth convex problems the scgd achieves a convergence rate of mathcal o k 1 4 o k 1 4 in the general case and mathcal o k 2 3 o k 2 3 in the strongly convex case after taking k samples for smooth convex problems the scgd can be accelerated to converge at a rate of mathcal o k 2 7 o k 2 7 in the general case and mathcal o k 4 5 o k 4 5 in the strongly convex case for nonconvex problems we prove that any limit point generated by scgd is a stationary point for which we also provide the convergence rate analysis indeed the stochastic setting where one wants to optimize compositions of expected value functions is very common in practice the proposed scgd methods find wide applications in learning estimation dynamic programming etc

queue aware dynamic clustering and power allocation for network mimo systems via distributed stochastic learning in this paper we propose a two timescale delay optimal dynamic clustering and power allocation design for downlink network mimo systems the dynamic clustering control is adaptive to the global queue state information gqsi only and computed at the base station controller bsc over a longer time scale on the other hand the power allocations of all the bss in each cluster are adaptive to both intracluster channel state information ccsi and intracluster queue state information cqsi and computed at each cluster manager cm over a shorter time scale we show that the two timescale delay optimal control can be formulated as an infinite horizon average cost constrained partially observed markov decision process cpomdp by exploiting the special problem structure we derive an equivalent bellman equation in terms of pattern selection q factor to solve the cpomdp to address the distributed requirement and computational complexity we approximate the pattern selection q factor by the sum of per cluster potential functions and propose a novel distributed online learning algorithm to estimate them distributedly we show that the proposed distributed online learning algorithm converges almost surely by exploiting the birth death structure of the queue dynamics we further decompose the per cluster potential function into the sum of per cluster per user potential functions and formulate the instantaneous power allocation as a per stage qsi aware interference game played among all the cms the proposed qsi aware simultaneous iterative water filling algorithm qsiwfa is shown to achieve the nash equilibrium ne

actor critic type learning algorithms for markov decision processes algorithms for learning the optimal policy of a markov decision process mdp based on simulated transitions are formulated and analyzed these are variants of the well known actor critic or adaptive critic algorithm in the artificial intelligence literature distributed asynchronous implementations are considered the analysis involves two time scale stochastic approximations

global existence and decay estimate of solutions to magneto micropolar fluid equations abstract we are concerned with magneto micropolar fluid equations 1 3 1 4 the global existence of solutions to the cauchy problem is investigated under certain conditions precisely for the magneto micropolar navier stokes mmns system we obtain global existence and large time behavior of solutions near a constant states in r 3 appealing to a refined pure energy method we first obtain a global existence theorem by assuming that the h 3 norm of the initial data is small but the higher order derivatives can be arbitrary large if the initial data belongs to homogeneous sobolev norms h s 0 s 3 2 or homogeneous besov norms b 2 s 0 s 3 2 we obtain the optimal decay rates of the solutions and its higher order derivatives as an immediate byproduct we also obtain the usual l p l 2 1 p 2 type of the decay rates without requiring that the l p norm of initial data is small at last we derive a weak solution to 1 3 1 4 in r 2 with large initial data

lp exponential stability for the equilibrium solutions of the navier stokes equations the exponential stability of the solution of the navier stokes equation in lp p 2 p 3 in bounded domain is considered in this paper under some assumptions on the external force it can be shown that the bounded solution of the navier stokes equation with initial and boundary conditions approaches the stationary solution of the system exponentially when time t goes to infinite

a convergent online single time scale actor critic algorithm actor critic based approaches were among the first to address reinforcement learning in a general setting recently these algorithms have gained renewed interest due to their generality good convergence properties and possible biological relevance in this paper we introduce an online temporal difference based actor critic algorithm which is proved to converge to a neighborhood of a local maximum of the average reward linear function approximation is used by the critic in order estimate the value function and the temporal difference signal which is passed from the critic to the actor the main distinguishing feature of the present convergence proof is that both the actor and the critic operate on a similar time scale while in most current convergence proofs they are required to have very different time scales in order to converge moreover the same temporal difference signal is used to update the parameters of both the actor and the critic a limitation of the proposed approach compared to results available for two time scale convergence is that convergence is guaranteed only to a neighborhood of an optimal value rather to an optimal value itself the single time scale and identical temporal difference signal used by the actor and the critic may provide a step towards constructing more biologically realistic models of reinforcement learning in the brain

a stability criterion for two timescale stochastic approximation schemes we present the first sufficient conditions that guarantee stability of two timescale stochastic approximation schemes our analysis is based on the ordinary differential equation ode method and is an extension of the results in borkar and meyn 2000 for single timescale schemes as an application of our result we show the stability of iterates in a two timescale stochastic approximation scheme arising in reinforcement learning

new results of general n dimensional incompressible navier stokes equations let u u x t u0 represent the global strong weak solutions of the cauchy problems for the general n dimensional incompressible navier stokes equations ut e 2nut u u u p u 0 u 0 u x 0 u0 x u0 0 where the spatial dimension n 2 0 e 1 is a constant and 1 2 n t rn is a constant vector note that if e 0 and 0 then the problem reduces to the traditional navier stokes equations let the scalar functions ij c2 rn l1 rn ij xj l1 rn h2n rn i j 1 2 n define the real vector valued functions i i1 i2 in t let the initial data u0 x j 1n 1j xj x j 1n 2j xj x j 1n nj xj x t satisfy i 1n j 1n 2 ij xi xj x 0 then limt 1 t 1 n 2 rn u x t 2 e nu x t 2 dx 1 2 n 2 n 214 k 1n rn k x dx 2 for any integer m 1 we will establish the following limit limt 1 t 2m 1 n 2 rn mu x t 2 e m nu x t 2 dx 1 2 n 2 n 2 14 2m 1 l 12m 2l n k 1n rn k x dx 2 this kind of exact limit will have great influence on the hausdorff dimension of the global attractor of the model equations

multivariate stochastic approximation using a simultaneous perturbation gradient approximation the problem of finding a root of the multivariate gradient equation that arises in function minimization is considered when only noisy measurements of the function are available a stochastic approximation sa algorithm for the general kiefer wolfowitz type is appropriate for estimating the root the paper presents an sa algorithm that is based on a simultaneous perturbation gradient approximation instead of the standard finite difference approximation of keifer wolfowitz type procedures theory and numerical experience indicate that the algorithm can be significantly more efficient than the standard algorithms in large dimensional problems

q learning for continuous time linear systems a model free infinite horizon optimal control approach abstract in this paper we propose an online q learning algorithm to solve the infinite horizon optimal control problem of a linear time invariant system with completely uncertain unknown dynamics we first formulate the q function by using the hamiltonian and the optimal cost an integral reinforcement learning approach is used to develop an actor critic approximator structure to estimate the parameters of the q function online while also guaranteeing closed loop asymptotic stability and convergence to the optimal solution

global well posedness for the micropolar fluid system in critical besov spaces abstract we prove the global well posedness for the 3 d micropolar fluid system in the critical besov spaces by making a suitable transformation of the solutions and using the fourier localization method especially combined with a new l p estimate for the green matrix to the linear system of the transformed equation this result allows to construct global solutions for a class of highly oscillating initial data of cannone s type meanwhile we analyze the long time behavior of the solutions and get some decay estimates

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

four families of weyl group orbit functions of b 3 and c 3 the properties of the four families of special functions of three real variables called here c s ss and sl functions are studied the ss and sl functions are considered in all details required for their exploitation in fourier expansions of digital data sampled on finite fragment of lattices of any density and of the 3d symmetry imposed by the weight lattices of b3 and c3 simple lie algebras groups the continuous interpolations which are induced by the discrete expansions are exemplified and compared for some model functions

stability of two dimensional solutions to the navier stokes equations in cylindrical domains under navier boundary conditions abstract the navier stokes motions in a cylindrical domain with navier boundary conditions are considered first the existence of global regular two dimensional solutions is proved the solutions are such that norms bounded with respect to time are controlled by the same constant for all t r assuming that the initial velocity and the external force are sufficiently close to the initial velocity and the external force of a two dimensional solution we prove existence of global three dimensional solutions which remain close to the two dimensional solution for all time in this sense we have stability of two dimensional solutions thanks to the navier boundary conditions the nonlinear term in the two dimensional navier stokes equations does not influence the energy estimate this implies that the global two dimensional solution is proved without any structural restrictions on the external force initial data or viscosity