community detection algorithms a comparative analysis uncovering the community structure exhibited by real networks is a crucial step toward an understanding of complex systems that goes beyond the local organization of their constituents many algorithms have been proposed so far but none of them has been subjected to strict tests to evaluate their performance most of the sporadic tests performed so far involved small networks with known community structure and or artificial graphs with a simplified structure which is very uncommon in real systems here we test several methods against a recently introduced class of benchmark graphs with heterogeneous distributions of degree and community size the methods are also tested against the benchmark by girvan and newman proc natl acad sci u s a 99 7821 2002 and on random graphs as a result of our analysis three recent algorithms introduced by rosvall and bergstrom proc natl acad sci u s a 104 7327 2007 proc natl acad sci u s a 105 1118 2008 blondel et al j stat mech theory exp 2008 p10008 and ronhovde and nussinov phys rev e 80 016109 2009 have an excellent performance with the additional advantage of low computational complexity which enables one to analyze large systems

constructive almost periodic factorization of some triangular matrix functions abstract a factorability criterion is obtained constructively and the respective factorization obtained explicitly for 2 2 triangular almost periodic matrix functions of the form e 0 f e here f c 1 e c 0 c 1 e e x e i x c j are non zero constants and 0 max with being irrational note that the factorization problem even for triangular matrix functions as above with an arbitrary trinomial f is open the result obtained is yet another step towards its solution

a new approach to factorization of a class of almost periodic triangular symbols and related riemann hilbert problems the factorization of almost periodic triangular symbols g associated to finite interval convolution operators is studied for two classes of operators whose fourier symbols are almost periodic polynomials with spectrum in the group z z z 0 1 1 q the factorization problem is solved by a method that is based on the calculation of one solution of the riemann hilbert problem g in l r and does not require solving the associated corona problems since a second linearly independent solution is obtained by means of an appropriate transformation on the space of solutions to the riemann hilbert problem some unexpected but interesting results are obtained concerning the fourier spectrum of the solutions of g in particular it is shown that a solution exists with fourier spectrum in the additive group z z whether this group contains z or not possible application of the method to more general classes of symbols is considered in the last section of the paper

almost periodic factorization of block triangular matrix functions revisited abstract let g be an n n almost periodic ap matrix function defined on the real line r by the ap factorization of g we understand its representation in the form g g g where g 1 g 1 is an ap matrix function with all fourier exponents of its entries being non negative respectively non positive and x diag ei 1x ei nx 1 n r this factorization plays an important role in the consideration of systems of convolution type equations on unions of intervals in particular systems of m equations on one interval of length lead to ap factorization of matrices 0 1 g x e i x i m 0 f x e i x i m we develop a factorization techniques for matrices of the form 0 1 under various additional conditions on the off diagonal block f the cases covered include f with the fourier spectrum f lying on a grid f h z and the trinomial f f with

caratheodory toeplitz and nehari problems for matrix valued almost periodic functions in this paper the positive and strictly contractive extension problems for almost periodic matrix functions are treated we present nece ssary and sufficient conditions for the existence of extensions in terms of toeplit z and hankel operators on besicovitch spaces and lebesgue spaces furthermore when a solution exists a special extension the band extension is constructed which enjoys a maximum entropy property a linear fractional parameterization of the set of all extensions is also provided the techniques used xn the proofs include factorizations of matrix valued almost periodic functions and a general algebraic scheme called the band method

algebras of almost periodic functions with bohr fourier spectrum in a semigroup hermite property and its applications abstract it is proved that the unital banach algebra of almost periodic functions of several variables with bohr fourier spectrum in a given additive semigroup is an hermite ring the same property holds for the wiener algebra of functions that in addition have absolutely convergent bohr fourier series as applications of the hermite property of these algebras we study factorizations of wiener hopf type of rectangular matrix functions and the toeplitz corona problem in the context of almost periodic functions of several variables

robust reconstruction on trees is determined by the second eigenvalue consider a markov chain on an infinite tree t v e rooted at in such a chain once the initial root state is chosen each vertex iteratively chooses its state from the one of its parent by an application of a markov transition rule and all such applications are independent let j denote the resulting measure for j j is defined on configurations x x2v 2 a v where a is some finite set let n denote the restriction of to the sigma algebra generated by the variables x where x is at distance exactly n from letting n maxi j2a dtv n n where dtv denotes total variation distance we say that the reconstruction problem is solvable if liminfn 1 n 0 reconstruction solvability roughly means that the n th level of the tree contains a non vanishing amount of information on the root of the tree as n 1 in this paper we study the problem of robust reconstruction let be a non degenerate distribution on a and 0 let be chosen according to n and 0 be obtained from by letting for each node independently v 0 v with probability 1 and 0 v to be an independent sample from otherwise we denote by n the resulting measure on 0 the measure n is a perturbation of the measure n letting n maxi j2a dtv n n we say that the reconstruction problem is robust solvable if liminfn 1 n 0 for all 0 1 where 2 m is the second largest eigenvalue of m in absolute value then for all non degenerate the reconstruction problem is robust solvable we prove a converse and show that the reconstruction problem is not robust solvable if b 2 m 2 1 this proves a conjecture by the second author and peres we also consider other models of noise and general trees

non denseness of factorable matrix functions abstract it is proved that for certain algebras of continuous functions on compact abelian groups the set of factorable matrix functions with entries in the algebra is not dense in the group of invertible matrix functions with entries in the algebra assuming that the dual abelian group contains a subgroup isomorphic to z 3 these algebras include the algebra of all continuous functions and the wiener algebra more precisely it is shown that infinitely many connected components of the group of invertible matrix functions do not contain any factorable matrix functions again under the same assumption moreover these components actually are disjoint with the subgroup generated by the triangularizable matrix functions

lqg control with an h sup infinity performance bound a riccati equation approach an lqg linear quadratic gaussian control design problem involving a constraint on h sup infinity disturbance attenuation is considered the h sup infinity performance constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on l sub 2 performance in contrast to the pair of separated riccati equations of standard lqg theory the h sup infinity constrained gains are given by a coupled system of three modified riccati equations the coupling illustrates the breakdown of the separation principle for the h sup infinity constrained problem both full and reduced order design problems are considered with an h sup infinity attenuation constraint involving both state and control variables an algorithm is developed for the full order design problem and illustrative numerical results are given

h 2 optimal control with an h constraint the state feedback case in this paper we consider a mixed hz optimal control problem it is assumed that the plant as well as the feedback controller are finite dimensional and linear time invariant and that the plant state is available for feedback more specifically among all the state feedback controllers that minimize the he norm of a closed loop transfer matrix we give necessary and sufficient conditions for the existence of a controller that also satisfies a prescribed h norm bound on some other closed loop transfer matrix when these conditions are met the solution to the above problem is also a global solution to the contrained optimization problem of minimizing an he norm performance measure subject to an h norm constraint we also give state space formulae for computing the solutions some easily checkable sufficient conditions for the existence of solutions are given finally we give an example in which all solutions to the constrained optimization problem are necessarily dynamic i e there is no static gain solution even though plant state is available for feedback

polynomial almost periodic solutions for a class of riemann hilbert problems with triangular symbols abstract let g a e i b c e i with 0 1 such that 1 1 q and a b c c 0 in this paper the existence of almost periodic polynomial app solutions to the equation g h e l l with h h e h and l h is studied the natural space in which to seek a solution to the above problem is the space of almost periodic functions with spectrum in the group z z z due to the difficulty in dealing with the problem in that generality solutions are sought with spectrum in the group z z several interesting and totally new results are obtained it is shown that if 1 z z no polynomial solutions exist i e almost periodic polynomial solutions exist only if z z z z z keeping to this setting it is shown that app solutions exist if and only if the function g satisfies the simple spectral condition 1 2 the proof of this result is nontrivial and has a number theoretic flavour explicit formulas for the solution to the above problem are given in the final section of the paper the derivation of these formulas is to some extent a byproduct of the proof of the result on the existence of app solutions

comparing efficiencies of genetic crossover operators for one machine total weighted tardiness problem in this study the well known one machine problem with the performance criterion of minimizing total weighted tardiness is considered this problem is known to be np hard and consists of one machine and n independent jobs each of these jobs has a distinct integer processing time a distinct integer weighting factor and a distinct integer due date the purpose of this problem is to find a sequence of these jobs minimizing the sum of the weighted tardiness using benchmarking problems this study compares performances of eleven genetic crossover operators which have been widely used to solve other types of hard scheduling problems

the role of crossover operator in the genetic optimization of magnetic models the ising model introduced almost 100 years ago by wilhelm lenz and ernst ising is the formalism still popular as a tool to describe magnetic properties of a wide class of materials among many issues which arise when using this model there exist problems related to the process of finding minimum energy of the system since these problems are np hard optimizations can either be performed for some approximated cases or be the subject of global optimization techniques in this paper we present an analysis of the effect of different crossover operators on the efficiency of genetic algorithm used to minimize energy in the ising model although it is not a benchmark tool we hope it may be interesting as a testing tool

performance of modularity maximization in practical contexts although widely used in practice the behavior and accuracy of the popular module identification technique called modularity maximization is not well understood in practical contexts here we present a broad characterization of its performance in such situations first we revisit and clarify the resolution limit phenomenon for modularity maximization second we show that the modularity function q exhibits extreme degeneracies it typically admits an exponential number of distinct high scoring solutions and typically lacks a clear global maximum third we derive the limiting behavior of the maximum modularity q max for one model of infinitely modular networks showing that it depends strongly both on the size of the network and on the number of modules it contains finally using three real world metabolic networks as examples we show that the degenerate solutions can fundamentally disagree on many but not all partition properties such as the composition of the largest modules and the distribution of module sizes these results imply that the output of any modularity maximization procedure should be interpreted cautiously in scientific contexts they also explain why many heuristics are often successful at finding high scoring partitions in practice and why different heuristics can disagree on the modular structure of the same network we conclude by discussing avenues for mitigating some of these behaviors such as combining information from many degenerate solutions or using generative models

stochastic blockmodels and community structure in networks stochastic blockmodels have been proposed as a tool for detecting community structure in networks as well as for generating synthetic networks for use as benchmarks most blockmodels however ignore variation in vertex degree making them unsuitable for applications to real world networks which typically display broad degree distributions that can significantly affect the results here we demonstrate how the generalization of blockmodels to incorporate this missing element leads to an improved objective function for community detection in complex networks we also propose a heuristic algorithm for community detection using this objective function or its non degree corrected counterpart and show that the degree corrected version dramatically outperforms the uncorrected one in both real world and synthetic networks

oscillatory riemann hilbert problems and the corona theorem the paper is devoted to the riemann hilbert problem with matrix coefficient g l r 2 2 having detg 1 in hardy spaces hp 2 1 p on half planes c under the assumption of existence of a non trivial solution of corresponding homogeneous riemann hilbert problem in h 2 we study the solvability of the non homogeneous riemann hilbert problem in hp 2 1 p and get criteria for the existence of a generalized canonical factorization and bounded canonical factorization for g as well as explicit formulas for its factors in terms of solutions of two associated corona problems in c and c a separation principle for constructing corona solutions from simpler ones is developed and corona solutions for a number of corona problems in h are obtained making use of these results we construct explicitly canonical factorizations for triangular bounded measurable or almost periodic 2 2 matrix functions whose diagonal entries do not possess factorizations such matrices arise e g in the theory of convolution type equations on finite intervals

community detection in complex networks using extremal optimization we propose a method to find the community structure in complex networks based on an extremal optimization of the value of modularity the method outperforms the optimal modularity found by the existing algorithms in the literature giving a better understanding of the community structure we present the results of the algorithm for computer simulated and real networks and compare them with other approaches the efficiency and accuracy of the method make it feasible to be used for the accurate identification of community structure in large complex networks

asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications in this paper we extend our previous work on the stochastic block model a commonly used generative model for social and biological networks and the problem of inferring functional groups or communities from the topology of the network we use the cavity method of statistical physics to obtain an asymptotically exact analysis of the phase diagram we describe in detail properties of the detectability undetectability phase transition and the easy hard phase transition for the community detection problem our analysis translates naturally into a belief propagation algorithm for inferring the group memberships of the nodes in an optimal way i e that maximizes the overlap with the underlying group memberships and learning the underlying parameters of the block model finally we apply the algorithm to two examples of real world networks and discuss its performance

a branch and bound algorithm for the total weighted tardiness problem this paper presents a new branch and bound algorithm for the single machine total weighted tardiness problem it obtains lower bounds using a lagrangian relaxation approach with subproblems that are total weighted completion time problems the well known subgradient optimization technique is replaced by a multiplier adjustment method that leads to an extremely fast bound calculation the method incorporates various devices for checking dynamic programming dominance in the search tree extensive computational results for problems with up to 50 jobs show the superiority of the algorithm over existing methods

a maximum entropy principle for contractive interpolants abstract in this paper a maximum entropy principle is established for a class of continuous contractive m n matrix valued functions g with preassigned fourier coefficients gj j for j 0 1 the function g in this class which achieves the maximum entropy is characterized in a number of different ways and its entropy is computed in terms of a hankel operator based on the prescribed matrices 0 1 a pair of seemingly new szego type formulas are obtained enroute and some applications to nevanlinna pick interpolation are indicated

benchmark graphs for testing community detection algorithms community structure is one of the most important features of real networks and reveals the internal organization of the nodes many algorithms have been proposed but the crucial issue of testing i e the question of how good an algorithm is with respect to others is still open standard tests include the analysis of simple artificial graphs with a built in community structure that the algorithm has to recover however the special graphs adopted in actual tests have a structure that does not reflect the real properties of nodes and communities found in real networks here we introduce a class of benchmark graphs that account for the heterogeneity in the distributions of node degrees and of community sizes we use this benchmark to test two popular methods of community detection modularity optimization and potts model clustering the results show that the benchmark poses a much more severe test to algorithms than standard benchmarks revealing limits that may not be apparent at a first analysis

the effective energy transformation scheme as a special continuation approach to global optimization with application to molecular conformation this paper discusses a generalization of the function transformation scheme used in coleman shalloway and wu comput optim appl 2 1993 pp 145 170 j global optim 4 1994 pp 171 185 and shalloway global optimization c floudas and p pardalos eds princeton university press 1992 pp 433 477 global optim 2 1992 pp 281 311 for global energy minimization applied to the molecular conformation problem a mathematical theory for the method as a special continuation approach to global optimization is established we show that the method can transform a nonlinear objective function into a class of gradually deformed but smoother or easier functions an optimization procedure can then be successively applied to the new functions to trace their solutions back to the original function two types of transformation are defined isotropic and anisotropic we show that both transformations can be applied to a large class of nonlinear partially separable functions including energy functions for molecular conformation methods to compute the transformation for these functions are given

constructive factorization of some almost periodic triangular matrix functions with a quadrinomial off diagonal entry using an abbreviation e to denote the function ei x on the real line r let g e 0fe where f is a linear combination of the functions e e e e with some 0 the criterion for g to admit a canonical factorization was established recently by avdonin bulanova and moran 2007 1 we give an alternative approach to the matter proving the existence when it does take place via deriving explicit factorization formulas the non existence of the canonical factorization in the remaining cases then follows from the continuity property of the geometric mean

estimation and prediction for stochastic blockstructures a statistical approach to a posteriori blockmodeling for digraphs and valued digraphs is proposed the probability model assumes that the vertices of the digraph are partitioned into several unobserved latent classes and that the probability distribution of the relation between two vertices depends only on the classes to which they belong a bayesian estimator based on gibbs sampling is proposed the basic model is not identified because class labels are arbitrary the resulting identifiability problems are solved by restricting inference to the posterior distributions of invariant functions of the parameters and the vertex class membership in addition models are considered where class labels are identified by prior distributions for the class membership of some of the vertices the model is illustrated by an example from the social networks literature kapferer s tailor shop

the caratheodory toeplitz problem for almost periodic functions abstract in this paper we consider the caratheodory toeplitz positive extension problem in the wiener algebra of almost periodic functions we derive necessary and sufficient conditions for the existence of a positive band extension which is also characterized by a maximum entropy principle in addition a linear fractional parametrization for the set of all positive extensions of a given function is obtained

invertibility of toeplitz operators and corona conditions in a strip abstract a toeplitz operator with symbol g such that det g 1 is invertible if there is a non trivial solution to a riemann hilbert problem g with and satisfying the corona conditions in c and c respectively however determining such a solution and verifying that the corona conditions are satisfied are in general difficult problems in this paper on one hand we establish conditions on which are equivalent to the corona conditions but easier to verify if g 1 are analytic and bounded in a strip this happens in particular with almost periodic symbols on the other hand we identify new classes of symbols g for which a non trivial solution to g can be explicitly determined and the corona conditions can be verified by the above mentioned approach thus obtaining invertibility criteria for the associated toeplitz operators

the corona theorem and the existence of canonical factorization of triangular ap matrix functions abstract using the corona theorem canonical factorizability criteria for triangular almost periodic matrix valued functions of the form formula are obtained this allows the authors to obtain effective criteria for the existence of canonical factorization as well as to construct explicit factorizations for new classes of almost periodic matrix functions in a companion paper

gibbs states and the set of solutions of random constraint satisfaction problems abstract an instance of a random constraint satisfaction problem defines a random subset the set of solutions of a large product space x n the set of assignments we consider two prototypical problem ensembles random k satisfiability and q coloring of random regular graphs and study the uniform measure with support on s as the number of constraints per variable increases this measure first decomposes into an exponential number of pure states clusters and subsequently condensates over the largest such states above the condensation point the mass carried by the n largest states follows a poisson dirichlet process for typical large instances the two transitions are sharp we determine their precise location further we provide a formal definition of each phase transition in terms of different notions of correlation between distinct variables in the problem the degree of correlation naturally affects the performances of many search sampling algorithms empirical evidence suggests that local monte carlo markov chain strategies are effective up to the clustering phase transition and belief propagation up to the condensation point finally refined message passing techniques such as survey propagation may also beat this threshold message passing algorithms phase transitions random graphs

conditional expectations and the corona problem of ergodic hardy spaces abstract using the regular corona theorem for the upper half plane we solve the coronatype problem for generalized analytic functions associated with flows

statistical mechanical approach to image processing the basic frameworks and techniques of the bayesian approach to image restoration are reviewed from the statistical mechanical point of view first a few basic notions in digital image processing are explained to convince the reader that statistical mechanics has a close formal similarity to this problem second the basic formulation of the statistical estimation from the observed degraded image by using the bayes formula is demonstrated the relationship between bayesian statistics and statistical mechanics is also listed particularly it is explained that some correlation inequalities on the nishimori line of the random spin model also play an important role in bayesian image restoration third the framework of bayesian image restoration for binary images by means of the ising model is reviewed some practical algorithms for binary image restoration are given by employing the mean field and the bethe approximations finally bayesian image restoration for a grey level image using the gaussian model is reviewed and the gaussian model is extended to a more practical probabilistic model by introducing the line state to treat the effects of edges the line state is also extended to quantized values

applications of the schur algorithm to mixed h2 and h nehari problems abstract in this paper we will use the schur algorithm to compute a solution to the kaftal larson weiss mixed h 2 h problem in the nehari setting first we will use the schur algorithm to solve a mixed h 2 h caratheodory interpolation problem by passing limits we will then obtain an explicit formula in terms of hankel matrices to solve the rational h 2 h nehari problem explicit solutions in terms of minimal state space realization will be given

single machine scheduling to minimize total weighted tardiness abstract the problem of minimizing the total weighted tardiness in single machine scheduling is a well known strongly np hard problem in this paper we present an o n 2 time approximation algorithm for the problem where n is the number of jobs we further investigate different sub models of the problem and obtain interesting properties and results we begin with the model where the job due dates are affine linear functions of their processing times and the job tardiness weights are proportional to their processing times for this model we classify the np hardness of the problem with respect to different affine linear functions and provide an o n log n algorithm for each of the polynomially solvable problems the second model we investigate is one where all job due dates have equal slacks namely the slk due date model results for this model include the problem is np hard in the ordinary sense a pseudo polynomial algorithm with time complexity o n 2 p is derived where p is the sum of all of the processing times and a fully polynomial time approximation scheme is constructed

a maximum entropy principle in the general framework of the band method abstract a maximum entropy principle is developed in the general framework of the band method known maximum entropy principles appear as corollaries and new ones are derived

hiding quiet solutions in random constraint satisfaction problems we study constraint satisfaction problems on the so called planted random ensemble we show that for a certain class of problems e g graph coloring many of the properties of the usual random ensemble are quantitatively identical in the planted random ensemble we study the structural phase transitions and the easy hard easy pattern in the average computational complexity we also discuss the finite temperature phase diagram finding a close connection with the liquid glass solid phenomenology

efficient and principled method for detecting communities in networks a fundamental problem in the analysis of network data is the detection of network communities groups of densely interconnected nodes which may be overlapping or disjoint here we describe a method for finding overlapping communities based on a principled statistical approach using generative network models we show how the method can be implemented using a fast closed form expectation maximization algorithm that allows us to analyze networks of millions of nodes in reasonable running times we test the method both on real world networks and on synthetic benchmarks and find that it gives results competitive with previous methods we also show that the same approach can be used to extract nonoverlapping community divisions via a relaxation method and demonstrate that the algorithm is competitively fast and accurate for the nonoverlapping problem

quasitriangular subalgebras of semifinite von neumann algebras are closed arveson has shown in 1 that if a is a nest algebra of operators acting on a separable hilbert space h with nest of order type the extended natural numbers and if k h is the ideal of compact operators on h then the quasitriangular algebra a k h is norm closed in 6 fall arveson and muhly extended arveson s result proving that in b h general quasitriangular algebras i e compact perturbations of nest algebras are always norm closed a key step in their proof and a result of independent interest first obtained by erdos in s is the a weak density of a n k h in a from this fact they deduced that quasitriangular algebras were norm closed by using an argument depending on the duality k h b h every semifinite von neumann algebra m has an ideal of compact operators that behaves like k h namely the norm closed two sided ideal k generated by the finite projections of the algebra it is a natural question

the nishimori line and bayesian statistics the nishimori line is a line or hypersurface in the parameter space of systems with quenched disorder where simple expressions of the averages of physical quantities over the quenched random variables are obtained it has been playing an important role in the theoretical studies of the random frustrated systems since its discovery in around 1980 in this paper an interpretation of the nishimori line from the viewpoint of statistical information processing is developed our main aim is the reconstruction of the whole theory of the nishimori line from the viewpoint of bayesian statistics or almost equivalently from the viewpoint of the theory of error correcting codes as a byproduct of the interpretation counterparts of the nishimori line in models without gauge invariance are given we also discussed the issues on the finite temperature decoding of error correcting codes and clarify the role of gauge invariance in this topic

community detection as an inference problem we express community detection as an inference problem of determining the most likely arrangement of communities we then apply belief propagation and mean field theory to this problem and show that this leads to fast accurate algorithms for community detection

a new crossover approach for solving the multiple travelling salesmen problem using genetic algorithms abstract this paper proposes a new crossover operator called two part chromosome crossover tcx for solving the multiple travelling salesmen problem mtsp using a genetic algorithm ga for near optimal solutions we adopt the two part chromosome representation technique which has been proven to minimise the size of the problem search space nevertheless the existing crossover method for the two part chromosome representation has two limitations firstly it has extremely limited diversity in the second part of the chromosome which greatly restricts the search ability of the ga secondly the existing crossover approach tends to break useful building blocks in the first part of the chromosome which reduces the ga s effectiveness and solution quality therefore in order to improve the ga search performance with the two part chromosome representation we propose tcx to overcome these two limitations and improve solution quality moreover we evaluate and compare the proposed tcx with three different crossover methods for two mtsp objective functions namely minimising total travel distance and minimising longest tour the experimental results show that tcx can improve the solution quality of the ga compared to three existing crossover approaches

toeplitz operators of finite interval type and the table method abstract we solve a riemann hilbert problem with almost periodic coefficient g associated to a toeplitz operator t g in a class which is closely connected to finite interval convolution equations based on a generalization of the so called table method the explicit determination of solutions to that problem allows one to establish necessary and sufficient conditions for the invertibility of the corresponding toeplitz operator and to determine an appropriate factorization of g providing explicit formulas for the inverse of t g some unexpected properties of the fourier spectrum of the solutions are revealed which are not apparent through other approaches to the same problem

factorization of matrix functions with subgroup supported fourier coefficients abstract it is proved that if a wiener class matrix function on a connected compact abelian group admits a factorization with respect to a connected compact abelian supergroup then it admits a factorization with respect to the original group in particular all factorization indices are characters of the original group an illustrative example is considered

the noncommutative wiener lemma linear independence and spectral properties of the algebra of time frequency shift operators in this paper we analyze the banach algebra of time frequency shifts with absolutely summable coecients we prove a noncommutative version of the wiener lemma we also construct a faithful tracial state on this algebra which proves the algebra contains no compact operators as a corollary we obtain a special case of the heil ramanathan topiwala conjecture regarding linear independence of nitely many time frequency shifts of one l 2 function we also estimate the coecient decay of the inverse of nite linear combinations of time frequency shifts

phase transitions in the coloring of random graphs we consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color using the cavity method we present a detailed and systematic analytical study of the space of proper colorings solutions we show that for a fixed number of colors and as the average vertex degree number of constraints increases the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses first at the clustering transition the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard afterward the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state eventually above the coloring threshold no more solutions are available

one sided invertibility of matrices over commutative rings corona problems and toeplitz operators with matrix symbols abstract conditions are established under which fredholmness coburn s property and one or two sided invertibility are shared by a toeplitz operator with matrix symbol g and the toeplitz operator with scalar symbol det g these results are based on one sided invertibility criteria for rectangular matrices over appropriate commutative rings and related scalar corona type problems

spectral factorization of bi infinite multi index block toeplitz matrices in this paper we formulate a theory of lu and cholesky factorization of bi infinite block toeplitz matrices a ai j i j zd indexed by i j z d and develop two numerical methods to compute such factorizations 2002 elsevier science inc all rights reserved

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

the corona theorem and the canonical factorization of triangular ap matrix functions effective criteria and explicit formulas abstract using results of the companion paper the authors obtain effective criteria for the existence of canonical factorization for new classes of triangular almost periodic matrix functions of the form formula the important role in the proof of necessity in these criteria is played by a result on the stability of canonical ap factorizability obtained in the present paper by finding corona solutions for a pair of binomial almost periodic functions it is possible to calculate explicitly the factors of a canonical factorization for a class of considered matrix functions with trinomial f

almost periodic factorization of certain block triangular matrix functions let formula math where c j e c m m 0 and a for rational a v such matrices g are periodic and their wiener hopf factorization with respect to the real line r always exists and can be constructed explicitly for irrational a v a certain modification called an almost periodic factorization can be considered instead the case of invertible co and commuting c 1 c 0 1 c 1 c 0 1 was disposed of earlier it was discovered that an almost periodic factorization of such matrices g does not always exist and a necessary and sufficient condition for its existence was found this paper is devoted mostly to the situation when co is not invertible but the c j commute pairwise j 0 1 the complete description is obtained when m 3 for an arbitrary m certain conditions are imposed on the jordan structure of c j difficulties arising for m 4 are explained and a classification of both solved and unsolved cases is given the main result of the paper existence criterion is theoretical however a significant part of its proof is a constructive factorization of g in numerous particular cases these factorizations were obtained using maple the code is available from the authors upon request