semigroup characterization of besov type morrey spaces and well posedness of generalized navier stokes equations abstract the well posedness of generalized navier stokes equations with initial data in some critical homogeneous besov spaces and in some critical q spaces was known in this paper we establish a wavelet characterization of besov type morrey spaces under the action of semigroup as an application we obtain the well posedness of smooth solution for the generalized navier stokes equations with initial data in some critical homogeneous besov type morrey spaces b p p 1 2 n 1 2 1 1 2 1 2 1 p 2 and n p 2 2 2 n p or 2 p and max n p 2 2 1 2 n p with divergence free these critical homogeneous besov type morrey spaces are larger than corresponding classical besov spaces and cover q spaces

a general framework for codes involving redundancy minimization a framework with two scalar parameters is introduced for various problems of finding a prefix code minimizing a coding penalty function the framework encompasses problems previously proposed by huffman campbell nath and drmota and szpankowski shedding light on the relationships among these problems in particular nath s range of problems can be seen as bridging the minimum average redundancy problem of huffman with the minimum maximum pointwise redundancy problem of drmota and szpankowski using this framework two linear time huffman like algorithms are devised for the minimum maximum pointwise redundancy problem the only one in the framework not previously solved with a huffman like algorithm both algorithms provide solutions common to this problem and a subrange of nath s problems the second algorithm being distinguished by its ability to find the minimum variance solution among all solutions common to the minimum maximum pointwise redundancy and nath problems simple redundancy bounds are also presented

well posedness of quasi geostrophic equations with data in besov q spaces abstract by wavelets and multi resolution analysis we convert the bilinear estimate on tent spaces associated with besov q spaces b p p 1 2 r n to the estimates of the numerical quantities as an application we get the well posedness of the quasi geostrophic equations with initial data in b p p 1 2 r 2

an iterative algorithm to solve state perturbed stochastic algebraic riccati equations in lq zero sum games an iterative algorithm to solve a kind of state perturbed stochastic algebraic riccati equation sare in lq zero sum game problems is proposed in our algorithm we replace the problem of solving a sare with an indefinite quadratic term by the problem of solving a sequence of sares with a negative semidefinite quadratic term which can be solved by existing methods under some appropriate conditions we prove that our algorithm is globally convergent we give a numerical example to show the effectiveness of our algorithm our algorithm also has a natural game theoretic interpretation

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

conditions for optimality of the huffman algorithm a new general formulation of huffman tree construction is presented which has broad application recall that the huffman algorithm forms a tree in which every node has some associated weight by specifying at every step of the construction which nodes are to be combined to form a new node with a new combined weight we characterize a wide class of weight combination functions the quasilinear functions for which the huffman algorithm produces optimal trees under correspondingly wide classes of cost criteria in addition known results about huffman tree construction and related concepts from information theory and from the theory of convex functions are tied together suggestions for possible future applications are given

on the solution of the riccati differential equation arising from the lq optimal control problem in this paper we consider the matrix riccati differential equation rde that arises from linear quadratic lq optimal control problems in particular we establish explicit closed formulae for the solution of the rde with a terminal condition using particular solutions of the associated algebraic riccati equation we discuss how these formulae change as assumptions are progressively weakened an application to lq optimal control is briefly analysed

a short note on the redundancy of degreea it is known that there always exists an instantaneous uniquely decodable code such that the redundancy of the discrete source is bounded between 0 and 1 johnsen 5 provided sharper bounds on the redundancy by making assumptions on the most likely source letter in this paper we have obtained bounds on the redundancy of degree by applying johnsen s approach

q spaces and morrey spaces abstract we characterize functions in morrey space by p carleson measures we then reveal a simple relation between q space and morrey space that is q space can be viewed as a fractional integration of the morrey space therefore many results for morrey space can be translated onto q space for example we show that q space is a dual space by identifying its predual

algebraic riccati equation and j spectral factorization for h estimation abstract in this paper we investigate on the existence of the stabilizing solution of the algebraic riccati equation are related to the h filtering problem with a prescribed attenuation level it is well known that such a solution exists and is positive definite for larger than a certain f and it does not exist for smaller than a certain 0 we consider the intermediate case 0 f and show that in this interval the stabilizing solution does exist except for a finite number of values of we show how the solution of the are may be employed to obtain a minimum phase j spectral factor of the j spectrum associated with the h filtering problem

the dual algebraic riccati equations and the set of all solutions of the discrete time riccati equation abstractin this paper two new pairs of dual continuous time algebraic riccati equations cares and dual discrete time algebraic riccati equations dares are proposed the dual dares are first studied with some nonsingularity assumptions on the system matrix and the parameter matrix then in the case of singular matrices a generalised inverse is introduced to deal with the dual dare problem these dual ares can easily lead us to an iterative procedure for finding the anti stabilising solutions especially to dare by means of that for the stabilising solutions furthermore we provide the counterpart results on the set of all solutions to dare inspired by the results for care two examples are presented to illustrate the theoretical results

the generalised discrete algebraic riccati equation in linear quadratic optimal control this paper investigates the properties of the solutions of the generalised discrete algebraic riccati equation arising from the classic infinite horizon linear quadratic lq control problem in particular a geometric analysis is used to study the relationship existing between the solutions of the generalised riccati equation and the output nulling subspaces of the underlying system and the corresponding reachability subspaces this analysis reveals the presence of a subspace that plays an important role in the solution of the related optimal control problem which is reflected in the generalised eigenstructure of the corresponding extended symplectic pencil in establishing the main results of this paper several ancillary problems on the discrete lyapunov equation and spectral factorisation are also addressed and solved

state observability and observers of linear time invariant systems under irregular sampling and sensor limitations state observability and observer designs are investigated for linear time invariant systems in continuous time when the outputs are measured only at a set of irregular sampling time sequences the problem is primarily motivated by systems with limited sensor information in which sensor switching generates irregular sampling sequences state observability may be lost and the traditional observers may fail in general even if the system has a full rank observability matrix it demonstrates that if the original system is observable the irregularly sampled system will be observable if the sampling density is higher than some critical frequency independent of the actual time sequences this result extends shannon s sampling theorem for signal reconstruction under periodic sampling to system observability under arbitrary sampling sequences state observers and recursive algorithms are developed whose convergence properties are derived under potentially dependent measurement noises persistent excitation conditions are validated by designing sampling time sequences by generating suitable switching time sequences the designed state observers are shown to be convergent in mean square with probability one and with exponential convergence rates schemes for generating desired sampling sequences are summarized

a spectral factorization approach for h spl infin estimation problems in discrete time this note deals with the h filtering prediction and smoothing problems for discrete time linear systems the j spectral factorization approach is pursued and for each of the three problems a computationally efficient procedure to derive a j spectral factor is presented the construction of filter smoother and predictor hinges on the stabilizing solution of the same algebraic riccati equation this fact allows for the avoidance of any fictitious augmentation of the state space dimensions

well posedness for the navier stokes equations where u is the velocity and p is the pressure it is well known that the navierstokes equations are locally well posed for smooth enough initial data as long as one imposes appropriate boundary conditions on the pressure at for instance it is easy to see see 9 for much more general results that if s n 2 then for any h initial data there exists a unique c 0 t h r local solution with a pressure p c 0 t h r in the sequel we consider solutions for less regular initial data this has to be understood in the sense that the map from the initial data to the solution extends continuously to rougher function spaces the question we are interested in is the global well posedness for small data and local well posedness for large data with respect to a certain space of

precise minimax redundancy and regret recent years have seen a resurgence of interest in redundancy of lossless coding the redundancy regret of universal fixed to variable length coding for a class of sources determines by how much the actual code length exceeds the optimal ideal over the class code length in a minimax scenario one finds the best code for the worst source either in the worst case called also maximal minimax or on average we first study the worst case minimax redundancy over a class of stationary ergodic sources and replace shtarkov s bound by an exact formula among others we prove that a generalized shannon code minimizes the worst case redundancy derive asymptotically its redundancy and establish some general properties this allows us to obtain precise redundancy for memoryless markov and renewal sources for example we present the exact constant of the redundancy for memoryless and markov sources by showing that the integer nature of coding contributes log logm m 1 logm o 1 where m is the size of the alphabet then we deal with the average minimax redundancy and regret our approach here is orthogonal to most recent research in this area since we aspire to show that asymptotically the average minimax redundancy is equivalent to the worst case minimax redundancy for some classes of sources after formulating some general bounds relating these two redundancies we prove our assertion for memoryless and markov sources nevertheless we provide evidence that maximal redundancy of renewal processes does not have the same leading term as the average minimax redundancy however our general results show that maximal and average regrets are asymptotically equivalent

optimal prefix codes for infinite alphabets with nonlinear costs let p p i be a measure of strictly positive probabilities on the set of nonnegative integers although the countable number of inputs prevents usage of the huffman algorithm there are nontrivial p for which known methods find a source code that is optimal in the sense of minimizing expected codeword length for some applications however a source code should instead minimize one of a family of nonlinear objective functions beta exponential means those of the form loga sigmaip i an i where n i is the length of the ith codeword and a is a positive constant applications of such minimizations include a novel problem of maximizing the chance of message receipt in single shot communications a 1 this paper introduces methods for finding codes optimal for such exponential means one method applies to geometric distributions while another applies to distributions with lighter tails the latter algorithm is applied to poisson distributions and both are extended to alphabetic codes as well as to minimizing maximum pointwise redundancy the aforementioned application of minimizing the chance of buffer overflow is also considered

an application of informational divergence to huffman codes a classification of all probability distributions over the finite alphabet of an information source is given where the classes are the sets of distributions sharing the same binary huffman code such a classification can be used in noiseless coding when the distribution of the finite memoryless source varies in time or becomes gradually known instead of applying the huffman algorithm to each new estimate of the probability distribution if a simple test based on the above classification is passed then the huffman code used previously is optimal also for the new distribution

y spaces and global smooth solution of fractional navier stokes equations with initial value in the critical oscillation spaces abstract for fractional navier stokes equations and critical initial spaces x one used to establish the well posedness in the solution space which is contained in c r x in this paper for heat flow we apply parameter meyer wavelets to introduce y spaces y m where y m is not contained in c r b 1 2 consequently for 1 2 1 we establish the global well posedness of fractional navier stokes equations with small initial data in all the critical oscillation spaces the critical oscillation spaces may be any besov morrey spaces b p q 1 2 r n n or any triebel lizorkin morrey spaces f p q 1 2 r n n where 1 p q 0 2 n p 1 2 1 2 these critical spaces include many known spaces for example besov spaces sobolev spaces bloch spaces q spaces morrey spaces and triebel lizorkin spaces etc

a globally convergent matricial algorithm for multivariate spectral estimation in this paper we first describe a matricial newton type algorithm designed to solve the multivariable spectrum approximation problem we then prove its global convergence finally we apply this approximation procedure to multivariate spectral estimation and test its effectiveness through simulation simulation shows that in the case of short observation records this method may provide a valid alternative to standard multivariable identification techniques such as matlab s pem and matlab s n4sid

structure preserving numerical algorithm for solving discrete time lmi and dars abstract we present a numerical algorithm to solve a discrete time linear matrix inequality lmi and discrete time algebraic riccati system dars with a given system a b c d we associate a para hermitian matrix pencil then we transform it by an orthogonal transformation matrix into a block triangular para hermitian form under either of the two assumptions 1 matrix pair a b is controllable or 2 matrix pair a b is reachable and a b c d is a left invertible system we extract the solution of lmi and dars by the entries of the orthogonal transformation matrix

the renyi redundancy of generalized huffman codes huffman s algorithm gives optimal codes as measured by average codeword length and the redundancy can be measured as the difference between the average codeword length and shannon s entropy if the objective function is replaced by an exponentially weighted average then a simple modification of huffman s algorithm gives optimal codes the redundancy can now be measured as the difference between this new average and a renyi s 1961 generalization of shannon s entropy by decreasing some of the codeword lengths in a shannon code the upper bound on the redundancy given in the standard proof of the noiseless source coding theorem is improved the lower bound is improved by randomizing between codeword lengths allowing linear programming techniques to be used on an integer programming problem these bounds are shown to be asymptotically equal the results are generalized to the renyi case and are related to r g gallager s 1978 bound on the redundancy of huffman codes

on the factorization of rational discrete time spectral densities in this paper we consider an arbitrary matrix valued rational spectral density phi z we show with a constructive proof that phi z admits a factorization of the form phi z w top z 1 w z where w z is stochastically minimal moreover w z and its right inverse are analytic in regions that may be selected with the only constraint that they satisfy some symplectic type conditions by suitably selecting the analyticity regions this extremely general result particularizes into a corollary that may be viewed as the discrete time counterpart of the matrix factorization method devised by youla in his celebrated work 48

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

well posedness and regularity of generalized navier stokes equations in some critical q spaces we study the well posedness and regularity of the generalized navier stokes equations with initial data in a new critical space q 1 rn q rn n 12 1 which is larger than some known critical homogeneous besov spaces here q rn is a space defined as the set of all measurable functions with sup l i 2 1 n i i f x f y 2 x y n 2 1 dxdy where the supremum is taken over all cubes i with edge length l i and edges parallel to the coordinate axes in rn in order to study the well posedness and regularity we give a carleson measure characterization of q rn by investigating a new type of tent spaces and an atomic decomposition of the predual for q rn in addition our regularity results apply to the incompressible navier stokes equations with initial data in q 1 1 rn

on the convergence of an efficient algorithm for kullback leibler approximation of spectral densities this paper deals with a method for the approximation of a spectral density function among the solutions of a generalized moment problem a la byrnes georgiou lindquist the approximation is pursued with respect to the kullback leibler pseudo distance which gives rise to a convex optimization problem after developing the variational analysis we discuss the properties of an efficient algorithm for the solution of the corresponding dual problem based on the iteration of a nonlinear map in a bounded subset of the dual space our main result is the proof of local convergence of the latter established as a consequence of the central manifold theorem supported by numerical evidence we conjecture that in the mentioned bounded set the convergence is actually global

characterization of multiplier spaces by wavelets and logarithmic morrey spaces in this paper we consider multipliers from sobolev spaces to lebesgue spaces we establish some wavelet characterization of multiplier spaces without using capacity further we give a sharp logarithmic morrey space condition for multipliers which lessens fefferman s morrey space condition to the logarithm level and generalizes lemarie s counter example to non integer cases and expresses his results in a more precise way

on the discrete time bounded real lemma with application in the characterization of static state feedback h controllers abstract this paper deals with the discrete time bounded real lemma and its application in the characterization of all static state feedback h controllers for discrete time systems we first present two versions of the discrete time bounded real lemma the results apply to non minimal stable state space realizations and are given in terms of either the positive semi definite strong solution or the positive semi definite stabilizing solution of a discrete algebraic riccati equation based on the discrete time bounded real lemma we obtain a characterization of all static state feedback h controllers for discrete time systems as an application of the parameterization of all static state feedback h controllers we develop a solution to the discrete time h control problem via static output feedback also a numerical procedure is proposed to determine an output feedback gain

some new tent spaces and duality theorems for fractional carleson measures and qα rn several duality questions for fractional carleson measures and the spaces q rn are resolved using a new type of tent spaces these tent spaces are defined in terms of choquet integrals with respect to hausdorff capacity a predual for q rn is then defined as a space of distributions containing the hardy space h1 and an atomic decomposition is proved

source coding for quasiarithmetic penalties whereas huffman coding finds a prefix code minimizing mean codeword length for a given finite item probability distribution quasiarithmetic or quasilinear coding problems have the goal of minimizing a generalized mean of the form rho 1 sigmaipirho li where li denotes the length of the ith codeword p i denotes the corresponding probability and rho is a monotonically increasing cost function such problems proposed by campbell have a number of diverse applications several cost functions are shown here to yield quasiarithmetic problems with simple redundancy bounds in terms of a generalized entropy a related property also shown here involves the existence of optimal codes for well behaved cost functions optimal codes always exist for possibly infinite alphabet sources having finite generalized entropy an algorithm is introduced for finding binary codes optimal for convex cost functions this algorithm which can be extended to other minimization utilities can be performed using quadratic time and linear space this reduces the computational complexity of a problem involving minimum delay in a queue allows combinations of previously considered problems to be optimized and greatly expands the set of problems solvable in quadratic time and linear space

redundancy related bounds for generalized huffman codes this paper presents new lower and upper bounds for the compression rate of binary prefix codes optimized over memoryless sources according to various nonlinear codeword length objectives like the most well known redundancy bounds for minimum average redundancy coding huffman coding these are in terms of a form of entropy and or the probability of an input symbol often the most probable one the bounds here some of which are tight improve on known bounds of the form l h h 1 where h is some form of entropy in bits or in the case of redundancy objectives 0 and l is the length objective also in bits the objectives explored here include exponential average length maximum pointwise redundancy and exponential average pointwise redundancy also called dth exponential redundancy the first of these relates to various problems involving queueing uncertainty and lossless communications the second relates to problems involving shannon coding and universal modeling also explored here for these two objectives is the related matter of individual codeword length

discrete time h algebraic riccati equation and parametrization of all h filters this paper is concerned with the algebraic riccati equations ares related to the h filtering problem a necessary and sufficient condition for the h problem to be solvable is that the h are has a positive semidefinite stabilizing solution with an additional condition that a certain matrix is positive definite it is shown that such a stabilizing solution is a monotonically non increasing convex function of the prescribed h norm bound this property of the h are is very important for the analysis of the performance of the h filter in this paper the size of the set of all h filters is considered on the basis of the monotonicity of the above riccati solution it turns out that under a certain condition the degree of freedom of the h filter reduces a1 the optimal h norm bound these results provide a guideline for selecting the value of some numerical examples are included