universal coding on infinite alphabets exponentially decreasing envelopes this paper deals with the problem of universal lossless coding on a countable infinite alphabet it focuses on some classes of sources defined by an envelope condition on the marginal distribution namely exponentially decreasing envelope classes with exponent the minimax redundancy of exponentially decreasing envelope classes is proved to be equivalent to then an adaptive algorithm is proposed whose maximum redundancy is equivalent to the minimax redundancy

overtaking optimal regulation and tracking of piecewise diffusion linear systems the infinite horizon optimal control of linear stochastic systems with quadratic cost integrand is studied and the tracking of a periodic signal on an infinite time interval is considered the system is exposed to three types of noises and is modeled by a nonhomogeneous linear stochastic control plant with modal and diffusion disturbances where the dynamics switch at random times within a finite number of descriptions according to a markov chain the overtaking optimality criterion is employed considering the expected cost the existence of a unique optimal control is established for the above noisy control systems with partial information this is realized as an affine feedback control of the best estimate of the state moreover in the case of partial information this feedback control is proved to be also almost surely overtaking optimal

submodular function minimization submodular functions often arise in various fields of operations research including discrete optimization game theory queueing theory and information theory in this survey paper we give overview on the fundamental properties of submodular functions and recent algorithmic devolopments of their minimization

the performance of universal encoding universal coding theory is surveyed from the viewpoint of the interplay between delay and redundancy the price for universality turns out to be acceptably small

global solutions to a game theoretic riccati equation of stochastic control abstract in this paper the existence of global solutions to game theoretic riccati equations associated to a controlled stochastic differential equation with state dependent white noise is investigated one proves that if a stabilizing and attenuating feed back exists then a matrix differential game theoretic riccati equation of stochastic control has a unique global bounded stabilizing positive semidefinite solution this solution is periodic if the coefficients are periodic functions and it solves an algebraic riccati equation if the coefficients are constant conversly if such a solution exists it allows an explicit construction of a stabilizing and disturbance attenuating state feedback

maximum entropy and conditional probability it is well known that maximum entropy distributions subject to appropriate moment constraints arise in physics and mathematics in an attempt to find a physical reason for the appearance of maximum entropy distributions the following theorem is offered the conditional distribution of x l given the empirical observation 1 n sum n i l h x i alpha where x 1 x 2 cdots are independent identically distributed random variables with common density g converges to f lambda x e lambda t h x g x suitably normalized where lambda is chosen to satisfy int f lambda x h x dx alpha thus the conditional distribution of a given random variable x is the normalized product of the maximum entropy distribution and the initial distribution this distribution is the maximum entropy distribution when g is uniform the proof of this and related results relies heavily on the work of zabell and lanford

a fast parametric maximum flow algorithm and applications the classical maximum flow problem sometimes occurs in settings in which the arc capacities are not fixed but are functions of a single parameter and the goal is to find the value of the parameter such that the corresponding maximum flow or minimum cut satisfies some side condition finding the desired parameter value requires solving a sequence of related maximum flow problems in this paper it is shown that the recent maximum flow algorithm of goldberg and tarjan can be extended to solve an important class of such parametric maximum flow problems at the cost of only a constant factor in its worst case time bound faster algorithms for a variety of combinatorial optimization problems follow from the result

speaking of infinity i i d strings we study the redundancy of three approaches to compression of independent and identically distributed i i d strings over large possibly infinite alphabets standard compression of the string itself and compression of the string s shape and pattern which describe its symbols relative magnitude and precedence respectively we determine the rate at which per symbol standard redundancy increases to infinity as the alphabet size increases show that the maximum per symbol shape redundancy is between 0 027 and 1 and compare these to results showing that per symbol pattern redundancy diminishes to zero for all alphabet sizes we relate these concepts to ordered and unordered partitions of integers and sets and use this framework to explore relations between several combinatorial quantities including the bell fubini and second type stirling numbers

coding on countably infinite alphabets this paper describes universal lossless coding strategies for compressing sources on countably infinite alphabets classes of memoryless sources defined by an envelope condition on the marginal distribution provide benchmarks for coding techniques originating from the theory of universal coding over finite alphabets we prove general upper bounds on minimax regret and lower bounds on minimax redundancy for such source classes the general upper bounds emphasize the role of the normalized maximum likelihood nml codes with respect to minimax regret in the infinite alphabet context lower bounds are derived by tailoring sharp bounds on the redundancy of krichevsky trofimov coders for sources over finite alphabets up to logarithmic resp constant factors the bounds are matching for source classes defined by algebraically declining resp exponentially vanishing envelopes effective and almost adaptive coding techniques are described for the collection of source classes defined by algebraically vanishing envelopes those results extend our knowledge concerning universal coding to contexts where the key tools from parametric inference are known to fail

systems of matrix rational differential equations arising in connection with linear stochastic systems with markovian jumping abstract in this paper a class of systems of matrix nonlinear differential equations containing as particular cases the systems of coupled riccati differential equations arising in connection with control of some linear stochastic systems is considered the system of differential equations considered in this paper are converted in a suitable nonlinear differential equation on a finite dimensional hilbert space adequately choosen this allows us to use the positivity properties of the linear evolution operator defined by the linear differential equations of lyapunov type our aim is to investigate properties of stabilizing and bounded solutions of the considered differential equations and to obtain some conditions ensuring the existence of such solutions conditions providing the existence of a maximal solution minimal solution respectively with respect to some classes of global solutions are presented it is shown that if the coefficients of the equations are periodic functions all these special solutions stabilizing maximal minimal are periodic functions too whenever possible the probabilistic arguments were avoided and so the results proved in the paper appear as results in the field of differential equations with interest in themselves

a new code for encoding all monotone sources with a fixed large alphabet size the problem of designing a fixed code for encoding all monotone sources with a large alphabet size n is studied it is proved that if the codelengths of a code sequence are of order o log n then the redundancy for almost all monotone sources with n symbols is very close to that for a specific distribution so called average distribution noting this point for any large alphabet size n a new code with a very simple structure is proposed whose redundancy is close to that of the huffman code for almost all monotone sources with alphabet size n

optimal attack and reinforcement of a network in a nonnegative edge weighted network the weight of an edge represents the effort required by an attacker to destroy the edge and the attacker derives a benefit for each new component created by destroying edges the attacker may want to minimize over subsets of edges the difference between or the ratio of the effort incurred and the benefit received this idea leads to the definition of the strength of the network a measure of the resistance of the network to such attacks efficient algorithms for the optimal attack problem the problem of computing the strength and the problem of finding a minimum cost reinforcement to achieve a desired strength are given these problems are also solved for a different model in which the attacker wants to separate vertices from a fixed central vertex

computing the strength of a graph the strength of a graph is a measure of its vulnerability strictly generalizing the edge connectivity of a weighted graph cunningham j assoc comput mach 32 1985 pp 549 562 showed how to compute the strength of a graph in o v 4 e time using ideas from polymatroids and network flow in this paper his polymatroid approach is modified a modified version of the goldberg tarjan network flow algorithm j assoc comput mach 4 1988 pp 136 146 is used then using ideas developed by gallo grigoriadis and tarjan siam j comput 18 1989 pp 30 55 and by gusfield martel and fernandez baca in siam j comput 16 1987 pp 237 251 it is shown that the solution runs in o v 3 e time analogous speedups for sparse case bounds are also obtainable

eigensets and power products of a bimatroid abstract we consider a bimatroid linking system which has a natural one to one corre spondence between the row set and the column set such bimatroid can be regarded as a combinatorial counterpart of a matrix which is subject to similarity transformations for such a bimatroid we can consider the power product introduce the notion of an eigenset which is a combinatorial analogue of an eigenvector and define the jordan type in particular the maximum size of an eigenset is characterized by the rank of the power products

generalized polymatroids and submodular flows polyhedra related to matroids and sub or supermodular functions play a central role in combinatorial optimization the purpose of this paper is to present a unified treatment of the subject the structure of generalized polymatroids and submodular flow systems is discussed in detail along with their close interrelation in addition to providing several applications we summarize many known results within this general framework

a fast parametric submodular intersection algorithm for strong map sequences this paper presents a fast algorithm to solve the intersection problem for a pair of nondecreasing and nonincreasing strong map sequences of submodular systems the worst case time bound is the same as that of the push relabel algorithm for a single intersection problem this extends the gallo grigoriadis tarjan ggt method for parametric maximum flow problems and reveals an algorithmic significance of the concept of strong maps for submodular systems

stochastic h infty we consider stochastic linear plants which are controlled by dynamic output feedback and subjected to both deterministic and stochastic perturbations our objective is to develop an h infty type theory for such systems we prove a bounded real lemma for stochastic systems with deterministic and stochastic perturbations this enables us to obtain necessary and sufficient conditions for the existence of a stabilizing compensator which keeps the effect of the perturbations on the to be controlled output below a given threshhold gamma 0 in the deterministic case the analogous conditions involve two uncoupled linear matrix inequalities but in the stochastic setting we obtain coupled nonlinear matrix inequalities instead the connection between h infty theory and stability radii is discussed and leads to a lower bound for the radii which is shown to be tight in some special cases

minimax redundancy for the class of memoryless sources let x sup n x sub 1 x sub n be a memoryless source with unknown distribution on a finite alphabet of size k we identify the asymptotic minimax coding redundancy for this class of sources and provide a sequence of asymptotically minimax codes equivalently we determine the limiting behavior of the minimax relative entropy min sub qxn max sub pxn d p sub xn spl par q sub xn where the maximum is over all independent and identically distributed i i d source distributions and the minimum is over all joint distributions we show in this paper that the minimax redundancy minus k 1 2 log n 2 spl pi e converges to log spl int spl radic det i spl theta d spl theta log spl gamma 1 2 sup k spl gamma k 2 where i spl theta is the fisher information and the integral is over the whole probability simplex the bayes strategy using jeffreys prior is shown to be asymptotically maximin but not asymptotically minimax in our setting the boundary risk using jeffreys prior is higher than that of interior points we provide a sequence of modifications of jeffreys prior that put some prior mass near the boundaries of the probability simplex to pull down that risk to the asymptotic minimax level in the limit

a new approach to the maximum flow problem all previously known efficient maximum flow algorithms work by finding augmenting paths either one path at a time as in the original ford and fulkerson algorithm or all shortest length augmenting paths at once using the layered network approach of dinic an alternative method based on the preflow concept of karzanov is introduced a preflow is like a flow except that the total amount flowing into a vertex is allowed to exceed the total amount flowing out the method maintains a preflow in the original network and pushes local flow excess toward the sink along what are estimated to be shortest paths the algorithm and its analysis are simple and intuitive yet the algorithm runs as fast as any other known method on dense graphs achieving an o n 3 time bound on an n vertex graph by incorporating the dynamic tree data structure of sleator and tarjan we obtain a version of the algorithm running in o nm log n 2 m time on an n vertex m edge graph this is as fast as any known method for any graph density and faster on graphs of moderate density the algorithm also admits efficient distributed and parallel implementations a parallel implementation running in o n 2 log n time using n processors and o m space is obtained this time bound matches that of the shiloach vishkin algorithm which also uses n processors but requires o n 2 space

layered augmenting path algorithms augmenting path algorithms first introduced by ford and fulkerson ford l r d r fulkerson 1962 flows in networks princeton university press princeton n j are widely used in optimization examples include schonsleben s polymatroid intersection algorithm schonsleben p 1980 ganzzahlige polymatroid intersection algorithmen ph d thesis eidgenossischen technischen hochschule zurich the maximum polymatroidal network flow algorithm of lawler and martel lawler e l c u martel 1982 computing maximal polymatroidal network flows math oper res 7 334 347 frank s algorithm for the edmonds giles polyhedron frank a 1984 finding feasible vectors of edmonds giles polyhedra j combin theory ser b36 221 239 and cunningham s algorithm for testing membership in matroid polyhedra cunningham h w 1984 testing membership in matroid polyhedra j combin theory ser b36 161 188 here we give an order of magnitude improvement for the above algorithms by using an approach analogous to that of dinits maximum flow algorithm dinits e a 1970 algorithm for solution of a problem of maximum flow in a network with power estimation soviet math dokl 11 1277 1280

comparison theorems for the matrix riccati equation abstract we derive comparison theorems for the matrix valued riccati equations of the form r i z b i z a i z r i z r i z d i z r i z c i z r i z r i 0 r 0 i i 1 2 such equations arise in transport problems and other applications sufficient conditions under which r 1 z r 2 z 0 and r 1 z r 2 z 0 in the componentwise sense for all z are given respectively the comparison theorems in which the positivity is defined in terms of positive semidefiniteness were obtained by royden because of the intrinsic disparity in the nature of positivity structure the techniques developed there cannot be applied to our problem

minimax compression and large alphabet approximation through poissonization and tilting this paper introduces a convenient strategy for coding and predicting sequences of independent identically distributed random variables generated from a large alphabet of size m in particular the size of the sample is allowed to be variable the employment of a poisson model and tilting method simplifies the implementation and analysis through independence the resulting strategy is optimal within the class of distributions satisfying a moment condition and it is close to optimal for the class of all i i d distributions on strings of a given length the method also can be used to code and predict strings with a condition on the tail of the ordered counts and it can be applied to distributions in an envelope class moreover we show that our model permits exact computation of the minimax optimal code for all alphabet sizes when conditioning on the size of the sample

a structure theory for the parametric submodular intersection problem a linearly parameterized polymatroid intersection problem appears in the context of principal partitions we consider a submodular intersection problem on a pair of strong map sequences of submodular functions which is an extension of the linearly parameterized polymatroid intersection problem to a nonlinearly parameterized one we introduce the concept of a basis frame on a finite nonempty set of cardinality n that gives a mapping from the set of all base polyhedra in n dimensional space into n dimensional vectors such that each base polyhedron is mapped to one of its bases we show the existence of a simple universal representation of all optimal solutions of the parameterized submodular intersection problem by means of basis frames