properties of the solutions of the fourth order bessel type differential equation the structured bessel type functions of arbitrary even order were introduced by everitt and markett in 1994 these functions satisfy linear ordinary differential equations of the same even order the differential equations have analytic coefficients and are defined on the whole complex plane with a regular singularity at the origin and an irregular singularity at the point of infinity they are all natural extensions of the classical second order bessel differential equation further these differential equations have real valued coefficients on the positive real half line of the plane and can be written in lagrange symmetric formally self adjoint form in the fourth order case the lagrange symmetric differential expression generates self adjoint unbounded operators in certain hilbert function spaces these results are recorded in many of the papers here given as references it is shown in the original paper of 1994 that in this fourth order case one solution exists which can be represented in terms of the classical bessel functions of order 0 and 1 the existence of this solution further aided by computer programs in maple led to the existence of a linearly independent basis of solutions of the differential equation in this paper a new proof of the existence of this solution base is given on using the advanced theory of special functions in the complex plane the methods lead to the development of analytical properties of these solutions in particular the series expansions of all solutions at the regular singularity at the origin of the complex plane

spectra of sparse random matrices we compute the spectral density for ensembles of sparse symmetric random matrices using replica our formulation of the replica symmetric ansatz shares the symmetries of that suggested in a seminal paper by rodgers and bray symmetry with respect to permutation of replica and rotation symmetry in the space of replica but uses a different representation in terms of superpositions of gaussians it gives rise to a pair of integral equations which can be solved by a stochastic population dynamics algorithm remarkably our representation allows us to identify pure point contributions to the spectral density related to the existence of normalizable eigenstates our approach is not restricted to matrices defined on graphs with poissonian degree distribution matrices defined on regular random graphs or on scale free graphs are easily handled we also look at matrices with row constraints such as discrete graph laplacians our approach naturally allows us to unfold the total density of states into contributions coming from vertices of different local coordinations and an example of such an unfolding is presented our results are well corroborated by numerical diagonalization studies of large finite random matrices

synthesis of inference based decentralized control for discrete event systems in our past work we presented a framework for the decentralized control of discrete event systems involving inferencing over ambiguities about the system state of various local decision makers using the knowledge of the self ambiguity and those of the others each local control decision is tagged with a certain ambiguity level level zero being the minimum and representing no ambiguity a global control decision is taken to be a winning local control decision i e one with a minimum ambiguity level for the existence of a decentralized supervisor so that for each controllable event the ambiguity levels of all winning disablement or enablement decisions are bounded by some number n such a supervisor is termed n inferring the notion of n inference observability was introduced when the given specification fails to satisfy the iv inference observability property an iv inferring supervisor achieving the entire specification does not exist we first show that the class of iv inference observable sublanguages is not closed under union implying that the supremal n inference observable sublanguage need not exist we next provide a technique for synthesizing an n inferring decentralized supervisor that achieves an n inference observable sublanguage of the specification the sublanguage achieved equals the specification language when the specification itself is iv inference observable a formula for the synthesized sublanguage is also presented for the special cases of n 0 and n 1 the proposed supervisor achieves the same language as those reported in 25 31 for n 0 and 32 for n 1 the synthesized supervisor is parameterized by n the parameter bounding the ambiguity level and as n is increased the supervisor becomes strictly more permissive in general thus a user can choose n based on the degree of permissiveness and the degree of computational complexity desired

universality of level correlation function of sparse random matrices the statistical properties of sparse random matrices ensembles are investigated by means of a supersymmetric approach with the use of a functional generalization of the hubbard stratonovich transformation when used to calculate the density of states the method is shown to be absolutely equivalent to the replica trick the model turns out to bear a close resemblance to the anderson model on the bethe lattice it possesses a delocalization transition that occurs with an increase in the mean connectivity parameter in the delocalized phase the level level correlation function proves to have a universal dyson form with the full density of states replaced by the contribution from the infinite cluster

spectra of sparse non hermitian random matrices an analytical solution we present the exact analytical expression for the spectrum of a sparse non hermitian random matrix ensemble generalizing two standard results in random matrix theory this analytical expression constitutes a non hermitian version of the kesten mckay measure as well as a sparse realization of girko s elliptic law our exact result opens new perspectives in the study of several physical problems modelled on sparse random graphs which are locally treelike in this context we show analytically that the convergence rate of a transport process on a very sparse graph depends in a nonmonotonic way upon the degree of symmetry of the graph edges

level repulsion for band 3 3 random matrices in view of the great interest in band random matrices the authors investigate the level repulsion for the simplest case of 3 3 band matrices a non linear repulsion is found

eigenvalue separation in some random matrix models the eigenvalue density for members of the gaussian orthogonal and unitary ensembles follows the wigner semicircle law if the gaussian entries are all shifted by a constant amount s 2n 1 2 where n is the size of the matrix in the large n limit a single eigenvalue will separate from the support of the wigner semicircle provided s 1 in this study using an asymptotic analysis of the secular equation for the eigenvalue condition we compare this effect to analogous effects occurring in general variance wishart matrices and matrices from the shifted mean chiral ensemble we undertake an analogous comparative study of eigenvalue separation properties when the sizes of the matrices are fixed and s and higher rank analogs of this setting this is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions and using the interpretation of the latter as a green function in the dyson brownian motion model for the shifted mean gaussian unitary ensemble an

scaling properties of the eigenvalue spacing distribution for band random matrices the authors show that the spacing distribution for eigenvalues of band random matrices is described by a single parameter b2 n where b is the band half width and n is the size of the matrices it is also shown that the eigenvalue s density obeys the semicircle law the found scaling behaviour suggests that the fluctuation properties in the intermediate regime between wigner dyson and poisson are universal

random antagonistic matrices the ensemble of antagonistic matrices is introduced and studied in antagonistic matrices the entries and are real and have opposite signs or are both zero and the diagonal is zero this generalization of antisymmetric matrices is suggested by the linearized dynamics of competitive species in ecology

supervisory control of a class of discrete event processes this paper studies the control of a class of discrete event processes i e processes that are discrete asynchronous and possibly nondeter ministic the controlled process is described as the generator of a formal language while the controller or supervisor is constructed from the grammar of a specified target language that incorporates the desired closed loop system behavior the existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language two examples are provided

on the density of states of sparse random matrices the supersymmetric method of calculation of the density of states of a sparse random matrix is shown to be absolutely equivalent to the replica trick a functional generalization of the hubbard stratonovich transformation was used in the course of the solution

diluted banded random matrices scaling behavior of eigenfunction and spectral properties we demonstrate that the normalized localization length of the eigenfunctions of diluted sparse banded random matrices follows the scaling law the scaling parameter of the model is defined as where is the average number of non zero elements per matrix row n is the matrix size and additionally we show that also scales the spectral properties of the model up to certain sparsity characterized by the spacing distribution of eigenvalues

the statistics of eigenvector components of random band matrices analytical results by using a supersymmetric formalism the authors calculate analytically all higher moments pq sigma n mod psi n mod 2q generalizing the inverse participation ratio p2 where psi n 1 or n or n stands for the nth component of an eigenvector of a large random matrix with a band structure on this basis they reconstruct the whole probability distribution function of eigenvector components the relation with known numerical results is discussed

sparse sign consistent johnson lindenstrauss matrices compression with neuroscience based constraints johnson lindenstrauss jl matrices implemented by sparse random synaptic connections are thought to be a prime candidate for how convergent pathways in the brain compress information however to date there is no complete mathematical support for such implementations given the constraints of real neural tissue the fact that neurons are either excitatory or inhibitory implies that every so implementable jl matrix must be sign consistent i e all entries in a single column must be either all nonnegative or all nonpositive and the fact that any given neuron connects to a relatively small subset of other neurons implies that the jl matrix should be sparse we construct sparse jl matrices that are sign consistent and prove that our construction is essentially optimal our work answers a mathematical question that was triggered by earlier work and is necessary to justify the existence of jl compression in the brain and emphasizes that inhibition is crucial if neurons are to perform efficient correlation preserving compression

localization in random bipartite graphs numerical and empirical study we investigate adjacency matrices of bipartite graphs with a power law degree distribution motivation for this study is twofold first vibrational states in granular matter and jammed sphere packings second graphs encoding social interaction especially electronic commerce we establish the position of the mobility edge and show that it strongly depends on the power in the degree distribution and on the ratio of the sizes of the two parts of the bipartite graph at the jamming threshold where the two parts have the same size localization vanishes we found that the multifractal spectrum is nontrivial in the delocalized phase but still near the mobility edge we also study an empirical bipartite graph namely the amazon reviewer item network we found that in this specific graph the mobility edge disappears and we draw a conclusion from this fact regarding earlier empirical studies of the amazon network

centralized and decentralized supervisory control of nondeterministic systems under partial observation in this paper we extend our earlier work on supervisory control of nondeterministic systems using prioritized synchronization as the mechanism of control and trajectory model as the modeling formalism by considering design of supervisors under partial observation we introduce the notion of observation compatible systems and show that prioritized synchronous composition psc of observation compatible systems can be used as a mechanism of control of nondeterministic systems under partial observation in presence of driven events necessary and sufficient conditions that depend on the trajectory model as opposed to the language model of the plant are obtained for the existence of centralized as well as decentralized supervision our work on centralized control shows that the results of the traditional supervisory control can be extended to the above setting provided that the supervisor is deterministic and the observation mask is projection type on the other hand our work on decentralized control is based on a new relation between controllability observability co observability and psc that we derive in this paper

deciding co observability is pspace complete in this note we reduce the deterministic finite state automata intersection problem to the problem of deciding co observability for regular languages using a polynomial time many one mapping this demonstrates that the problem of deciding co observability for languages marked by deterministic finite state automata is pspace complete we use a similar reduction to reduce the deterministic finite state automata intersection problem to deciding other versions of co observability introduced in a previous paper these results imply that the co observability of regular languages most likely cannot be decided in polynomial time unless we make further restrictions on the languages these results also show that deciding decentralized supervisor existence is pspace complete and therefore probably intractable

eigenvector localization for random band matrices with power law band width it is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing fraction of standard basis vectors provided the band width w raised to a power remains smaller than the matrix size n for a gaussian band ensemble with matrix elements given by i i d centered gaussians within a band of width w the estimate 8 holds

finite length lyapunov exponents and conductance for quasi 1d disordered solids abstract the transfer matrix method is applied to finite quasi 1d disordered samples attached to perfect leads the model is described by structured band matrices with random and regular entries we investigate numerically the level spacing distribution for finite length lyapunov exponents as well as the conductance and its fluctuations for different channel numbers and sample sizes a comparison is made with theoretical predictions and with numerical results recently obtained with the scattering matrix approach the role of the coupling and finite size effects is also discussed

correlation between eigenvalue spectra and dynamics of neural networks this letter presents a study of the correlation between the eigenvalue spectra of synaptic matrices and the dynamical properties of asymmetric neural networks with associative memories for this type of neural network it was found that there are essentially two different dynamical phases the chaos phase with almost all trajectories converging to a single chaotic attractor and the memory phase with almost all trajectories being attracted toward fixed point attractors acting as memories we found that if a neural network is designed in the chaos phase the eigenvalue spectrum of its synaptic matrix behaves like that of a random matrix i e all eigenvalues lie uniformly distributed within a circle in the complex plan and if it is designed in the memory phase the eigenvalue spectrum will split into two parts one part corresponds to a random background the other part equal in number to the memory attractors the mechanism for these phenomena is discussed in this letter

random incidence matrices moments of the spectral density we study numerically and analytically the spectrum of incidence matrices of random labeled graphs on n vertices any pair of vertices is connected by an edge with probability p we give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in n and p for large n and fixed p the spectrum contains a large eigenvalue at np and a semicircle of small eigenvalues for large n and fixed average connectivity pn dilute or sparse random matrices limit we show that the spectrum always contains a discrete component an anomaly in the spectrum near eigenvalue 0 for connectivity close to e is observed we develop recursion relations to compute the moments as explicit polynomials in pn their growth is slow enough so that they determine the spectrum the extension of our methods to the laplacian matrix is given in appendix

density of states of sparse random matrices a supersymmetric formalism is used to derive a set of equations giving the density of states of any real symmetric sparse random matrix as a function of the distribution of non zero elements and the mean number of non zero elements per row p in the matrix where the non zero elements take the values or 1 with equal probability the equations are solved as p to infinity recovering results obtained previously with the replica method as e to 0 the density of states rho e behaves as 1 e ln e 2 the more general case where or 1 occur with unequal probabilities is also considered

brief paper formulae relating controllability observability and co observability we present formulae relating controllability observability and co observability arising in the context of supervisory control of discrete event systems given a discrete event plant g with event set e uncontrollable event set e u i and observation mask m i of the ith supervisor we first show that the infimal prefix closed l g i e u i controllable and l g e u i m i co observable superlanguage equals the intersection taken over all i s of the infimal prefix closed l g e u i controllable and l g m i observable superlanguage next we show that the infimal prefix closed and e m observable superlanguage computation preserves e e u controllability these results can be used to compute individual supervisors in the decentralized control setting

eigenfunction statistics of complex systems a common mathematical formulation we derive a common mathematical formulation for the eigenfunction statistics of hermitian operators represented by a multiparametric probability density the system information in the formulation enters through two parameters only namely system size and the complexity parameter a function of all system parameters including size the behavior is contrary to the eigenvalue statistics which is sensitive to the complexity parameter only and shows a single parametric scaling the existence of a mathematical formulation of both eigenfunctions and eigenvalues common to a wide range of complex systems indicates the possibility of a similar formulation for many physical properties this also suggests the possibility to classify them in various universality classes defined by the complexity parameter

eigenvalue spectra of random matrices for neural networks the dynamics of neural networks is influenced strongly by the spectrum of eigenvalues of the matrix describing their synaptic connectivity in large networks elements of the synaptic connectivity matrix can be chosen randomly from appropriate distributions making results from random matrix theory highly relevant unfortunately classic results on the eigenvalue spectra of random matrices do not apply to synaptic connectivity matrices because of the constraint that individual neurons are either excitatory or inhibitory therefore we compute eigenvalue spectra of large random matrices with excitatory and inhibitory columns drawn from distributions with different means and equal or different variances

technical communique a modified normality condition for decentralized supervisory control of discrete event systems in this paper we study nonblocking decentralized supervisory control of discrete event systems we introduce a modified normality condition defined in terms of a modified natural projection map the modified normality condition is weaker than the original one and stronger than the co observability condition moreover it is preserved under union given a marked language specification there exists a nonblocking decentralized supervisor for the supremal sublanguage which satisfies l m g closure controllability and modified normality such a decentralized supervisor is more permissive than the one which achieves the supremal l m g closed controllable and normal sublanguage

non perturbative response chaos versus disorder quantized chaotic systems are generically characterized by two energy scales the mean level spacing and the bandwidth b this implies that with respect to driving such systems have an adiabatic a perturbative and a non perturbative regimes a strong nonlinearity in the response due to a quantal non perturbative effect is found for disordered systems that are described by random matrix theory models is there a similar effect for quantized chaotic systems theoretical arguments cannot exclude the existence of an analogous weak version of the above mentioned nonlinear response effect but our numerics demonstrates an unexpected degree of semiclassical correspondence

sparse random matrices the eigenvalue spectrum revisited we revisit the derivation of the density of states of sparse random matrices we derive a recursion relation that allows one to compute the spectrum of the matrix of incidence for finite trees that determines completely the low concentration limit using the iterative scheme introduced by biroli and monasson 1999 j phys a math gen 32 l255 we find an approximate expression for the density of states expected to hold exactly in the opposite limit of large but finite concentration the combination of the two methods yields a very simple geometric interpretation of the tails of the spectrum we test the analytic results with numerical simulations and we suggest an indirect numerical method to explore the tails of the spectrum

decentralized control of discrete event systems with bounded or unbounded delay communication we introduce problems of decentralized control with delayed communication where delays are either unbounded or bounded by a given constant k in the k bounded delay model between the transmission of a message and its reception the plant can execute at most k events in the unbounded delay model the plant can execute any number of events between transmission and reception we show that our framework yields an infinite hierarchy of control problems where the containments are strict cc is the set of control problems solvable with a single controller centralized case and dcc sub k respectively dcuc dc is the set of problems solvable with two controllers in a k bounded delay network respectively in an unbounded delay network without communication the hierarchy is a result of the following property controllers which work in a given network will also work in a less nondeterministic network this property does not hold when nonblockingness is introduced checking the existence of controllers in the unbounded delay case or in the case without communication are undecidable problems however a related decentralized observation problem with bounded delay communication is decidable

energy level statistics in the transition region between integrability and chaos a generic one parameter family of billiards discovered and introduced by robnik 1983 is used to study the spectral properties of corresponding quantum systems when the parameter is varied a smooth transition from an integrable system over a typical kam system to an almost ergodic system can be observed the authors calculate up to 7600 lowest reliable energy levels a detailed analysis of the numerical data shows significant deviation from the semiclassical berry robnik formulae for the nearest neighbour level spacing distribution p s except for large level spacings s 1 which can only be explained by a very slow convergence towards the semiclassical regime where these formulae are predicted to be correct at small s the power law level repulsion is clearly observed and a fit by the phenomenological formula by izrailev 1988 1989 is statistically significant

spectral fluctuation properties of hamiltonian systems the transition region between order and chaos the authors study numerically the classical dynamical behaviour and the spectral fluctuation properties of a class of hamiltonian systems with two degrees of freedom the quantum mechanical properties of these systems are monotonic but non universal functions of the fraction of classical phase space filled by chaotic trajectories it is found that the observed spectral fluctuation measures can be reproduced by a random matrix model which depends on one parameter only

dynamical localization in chaotic systems spectral statistics and localization measure in the kicked rotator as a paradigm for time dependent and time independent systems we study the kicked rotator in the classically fully chaotic regime using izrailev s n dimensional model for various n 4000 which in the limit n tends to the quantized kicked rotator we do treat not only the case k 5 as studied previously but also many different values of the classical kick parameter 5 k 35 and many different values of the quantum parameter ke 5 60 we describe the features of dynamical localization of chaotic eigenstates as a paradigm for other both time periodic and time independent autonomous fully chaotic or and mixed type hamilton systems we generalize the scaling variable l n to the case of anomalous diffusion in the classical phase space by deriving the localization length l for the case of generalized classical diffusion we greatly improve the accuracy and statistical significance of the numerical calculations giving rise to the following conclusions 1 the level spacing distribution of the eigenphases or quasienergies is very well described by the brody distribution systematically better than by other proposed models for various brody exponents br 2 we study the eigenfunctions of the floquet operator and characterize their localization properties using the information entropy measure which after normalization is given by loc in the interval 0 1 the level repulsion parameters br and loc are almost linearly related close to the identity line 3 we show the existence of a scaling law between loc and the relative localization length now including the regimes of anomalous diffusion the above findings are important also for chaotic eigenstates in time independent systems batistic and robnik j phys a math gen 43 215101 2010 arxiv 1302 7174 2013 where the brody distribution is confirmed to a very high degree of precision for dynamically localized chaotic eigenstates even in the mixed type systems after separation of regular and chaotic eigenstates

transition from poissonian to gaussian orthogonal ensemble level statistics in a modified artin s billiard one wall of an artin s billiard on the poincar e half plane is replaced by a one parameter mathit c mathit p family of nongeodetic walls a brief description of the classical phase space of this system is given in the quantum domain the continuous and gradual transition from the poisson like to gaussian orthogonal ensemble goe level statistics due to the small perturbations breaking the symmetry responsible for the arithmetic chaos at mathit c mathit p 1 is studied another goe ensuremath rightarrow poisson transition due to the mixed phase space for large perturbations is also investigated a satisfactory description of the intermediate level statistics by the brody distribution was found in both cases the study supports the existence of a scaling region around mathit c mathit p 1 a finite size scaling relation for the brody parameter as a function of 1 mathit c mathit p and the number of levels considered can be established

on the factorizations of ordinary linear differential operators etude des modes de factorisation de certains operateurs differentiels lineaires d ordre n a l aide des wronskiens

scattering and transport properties of tight binding random networks we study numerically scattering and transport statistical properties of tight binding random networks characterized by the number of nodes n and the average connectivity we use a scattering approach to electronic transport and concentrate on the case of a small number of single channel attached leads we observe a smooth crossover from insulating to metallic behavior in the average scattering matrix elements the conductance probability distribution w t the average conductance the shot noise power p and the elastic enhancement factor f by varying from small 0 to large 1 values we also show that all these quantities are invariant for fixed n moreover we proposes a heuristic and universal relation between and p and the disorder parameter

recent directions in matrix stability abstract matrix stability has been intensively investigated in the past two centuries we review work that has been done in this topic focusing on the great progress that has been achieved in the last decade or two we start with classical stability criteria of lyapunov routh and hurwitz and lienard and chipart we then study recently proven sufficient conditions for stability with particular emphasis on p matrices we investigate conditions for the existence of a stable scaling for a given matrix we review results on other types of matrix stability such as d stability additive d stability and lyapunov diagonal stability we discuss the weak principal submatrix rank property shared by lyapunov diagonally semistable matrices we also discuss the uniqueness of lyapunov scaling factors maximal lyapunov scaling factors cones of real positive semidefinite matrices and their applications to matrix stability and inertia preserving matrices

the integrated density of states of the random graph laplacian we analyse the density of states of the random graph laplacian in the percolating regime a symmetry argument and knowledge of the density of states in the nonpercolating regime allows us to isolate the density of states of the percolating cluster dspc alone thereby eliminating trivially localised states due to finite subgraphs we derive a nonlinear integral equation for the integrated dspc and solve it with a population dynamics algorithm we discuss the possible existence of a mobility edge and give strong evidence for the existence of discrete eigenvalues in the whole range of the spectrum

physiological gain leads to high isi variability in a simple model of a cortical regular spiking cell to understand the interspike interval isi variability displayed by visual cortical neurons softky koch 1993 it is critical to examine the dynamics of their neuronal integration as well as

density of states for band random matrices with electric field we present a long range hopping tight binding model in which the ensemble averaged density of states rho e undergoes a gradual transition from a homogeneous form with almost no variation to strongly peaked behaviour as an external electric field grows the model consists of n n band random matrices of bandwidth b the corresponding matrix elements hij have all vanishing average except on the diagonal where hii alpha i here the parameter alpha plays the role of the electric field we approximate the behaviour of the width sigma e of the local density of states rho l e and use it to predict the value of alpha around which the transition is centred

critique of the replica trick it is shown that the replica trick fails to give the correct non perturbative result for the two point function s2 of the gaussian unitary ensemble of n n random matrices the failure arises from an incorrect description of the symmetries of the random matrix system in the limit n to infinity the correct description which involves integration over both non compact and compact degrees of freedom is obtained by using the method of superfields some implications for the localisation transition in disordered electronic systems and the theory of the quantised hall effect are suggested

decentralized control of discrete event systems using prioritized composition with exclusion a necessary and sufficient condition for decentralized control of discrete event systems via prioritized composition with exclusion pcx is provided pcx was extended from prioritized synchronous composition psc in order to model various boolean modes of decision fusion by allowing the exclusivity of participation in system interactions a notion of pcx coobservability is introduced that captures the nature of pcx based decision fusion and a testing algorithm with polynomial complexity in the states of plant and specification is presented the result is extended for multiple supervisors with multiple exclusivity sets based on an extended pcx definition

random matrix theory by the supersymmetry method beyond the sigma model abstract the leading correction to the smoothed connected energy density density correlation function is obtained for the large energy difference within the context of the gaussian random matrix theory in order to achieve this result the supersymmetry method is extended beyond the sigma model to include small quadratic fluctuations around the saddle point special care is taken to avoid the potential divergence arising from the unbounded nature of the saddle point also in the small energy difference regime the leading correction to a two point correlation function is obtained

maximal solutions in decentralized supervisory control the decentralized supervisory control problem is to construct for a discrete event system a set of supervisors each observing only part of the system and each controlling only part of the events such that the interconnection of the system and the supervisors meets control objectives of safety and liveness definitions are provided of the concepts of a maximal solution of a nash equilibrium and of a strong nash equilibrium for a set of supervisors with as order relation the inclusion relation on the set of closed loop languages the main result is that a set of supervisors is a maximal solution if and only if it is a strong nash equilibrium a procedure to determine a nash equilibrium is described and illustrated by an example there is no guarantee that the procedure halts in finite time however in the case that it halts in finite time then it is proven that a nash equilibrium is obtained

real symmetric random matrices and replicas various ensembles of random matrices with independent entries are analyzed by the replica formalism in the large n limit a result on the laplacian random matrix with wigner rescaling is generalized to arbitrary probability distribution

eigenspectrum bounds for semirandom matrices with modular and spatial structure for neural networks the eigenvalue spectrum of the matrix of directed weights defining a neural network model is informative of several stability and dynamical properties of network activity existing results for eigenspectra of sparse asymmetric random matrices neglect spatial or other constraints in determining entries in these matrices and so are of partial applicability to cortical like architectures here we examine a parameterized class of networks that are defined by sparse connectivity with connection weighting modulated by physical proximity i e asymmetric euclidean random matrices modular network partitioning and functional specificity within the excitatory population we present a set of analytical constraints that apply to the eigenvalue spectra of associated weight matrices highlighting the relationship between connectivity rules and classes of network dynamics

spectral properties of random reactance networks and random matrix pencils our goal is to study the statistical properies of dielectric resonances which are poles of conductance of a large random lc network such poles are a particular example of eigenvalues n of matrix pencils h w with w being a positive definite matrix and h a random real symmetric one we first consider spectra of the matrix pencils with independent identically distributed entries of h then we concentrate on an infinite range full connectivity version of a random lc network in all cases we calculate the mean eigenvalue density and the two point correlation function in the framework of efetov s supersymmetry approach fluctuations in spectra turn out to be the same as those provided by the wigner dyson theory of usual random matrices

handbook of differential equations chapter headings definitions and concepts transformations exact analytical methods exact methods for odes exact methods for pdes approximate analytical methods numerical methods concepts numerical methods for odes numerical methods for pdes list of tables list of programs list of figures subject index

think globally act locally decentralized supervisory control decentralized supervisory control is investigated by considering problem formulations that model systems whose specifications are given as global constraints but whose solution is described by local controllers a necessary and sufficient condition is given for the existence of a solution to the problem of finding decentralized supervisors that ensure that the behavior of the closed loop system lies in a given range where the range of behavior can be described by regular languages it can be effectively tested whether the decentralized control problem is solvable in this case a procedure is given to compute the associated supervisors

inference based ambiguity management in decentralized decision making decentralized control of discrete event systems decentralized decision making requires the interaction of various local decision makers in order to arrive at a global decision limited sensing capabilities at each local site can create ambiguities in a decision making process at each local site we argue that such ambiguities are of differing gradations we propose a framework for decentralized decision making applied to decentralized control in particular that allows computation of such ambiguity gradations and utilizes their knowledge in arriving at a global decision each local decision is tagged with a certain grade or level of ambiguity with zero being the minimum ambiguity level a global decision is taken to be the same as a ldquowinningrdquo local decision i e one having the minimum level of ambiguity the computation of an ambiguity level for a local decision requires an assessment of the self ambiguities as well as the ambiguities of the others and an inference based upon such knowledge for the existence of a decentralized supervisor so that for each controllable event the ambiguity levels of all winning disablement or enablement decisions are bounded by some number n such a supervisor is termed n inferring the notion of n inference observability is introduced we show that the conjunctive and permissive c p v disjunctive and antipermissive d a co observability is the same as the zero inference observability whereas the conditional c p v d a co observability is the same as the unity inference observability we also present examples of higher order inference observable languages our framework does not require the existence of any a priori partition of the controllable events into permissive antipermissive sets nor does it require a global control computation based on conjunction disjunction of local decisions exhibiting that our ambiguity based approach is more efficient

semiempirical theory of level spacing distribution beyond the berry robnik regime modeling the localization and the tunneling effects in this work we study the level spacing distribution in the classically mixed type quantum systems which are generic exhibiting regular motion on invariant tori for some initial conditions and chaotic motion for the complementary initial conditions in the asymptotic regime of the sufficiently deep semiclassical limit sufficiently small effective planck constant the berry and robnik 1984 j phys a math gen 17 2413 picture applies which is very well established we present a new quasi universal semiempirical theory of the level spacing distribution in a regime away from the berry robnik regime the near semiclassical limit by describing both the dynamical localization effects of chaotic eigenstates and the tunneling effects which couple regular and chaotic eigenstates the theory works extremely well in the 2d mixed type billiard system introduced by robnik 1983 j phys a math gen 16 3971 and is also tested in other systems mushroom billiard and prosen billiard

variation of parameters and solutions of composite products of linear differential equations given a basis of solutions to k ordinary linear differential equations lj y 0 j 1 2 k we show how green s functions can be used to construct a basis of solutions to the homogeneous differential equation l y 0 where l is the composite product l l1l2 lk the construction of these solutions is elementary and classical in particular we consider the special case when l l1k remarkably in this case if y1 y2 yn is a basis of l1 y 0 then our method produces a basis of l1k y 0 for any k n we illustrate our results with several classical differential equations and their special function solutions

on decidability of distributed diagnosis under unbounded delay communication in this note we show that the problem of distributed diagnosis under unbounded communication delay is decidable when there is no inferencing involved among the diagnosers the notion of jointinfin diagnosability is introduced to capture the diagnosability property in this setting we show the equivalence of jointinfin diagnosability and codiagnosability which captures the diagnosability property in the decentralized setting i e one involving no communication thus the decidability result follows from the decidability of codiagnosability established in a previous paper we also show that the property of jointinfin diagnosability is stronger than decentralized diagnosability introduced in a previous paper

dynamical localization of chaotic eigenstates in the mixed type systems spectral statistics in a billiard system after separation of regular and chaotic eigenstates we study the quantum mechanics of a billiard robnik 1983 j phys a math gen 16 3971 in the regime of mixed type classical phase space the shape parameter 0 15 at very high lying eigenstates starting at about 1 000 000th eigenstate and including the consecutive 587654 eigenstates by calculating the normalized poincare husimi functions of the eigenstates and comparing them with the classical phase space structure we introduce the overlap criterion which enables us to separate with great accuracy and reliability the regular and chaotic eigenstates and the corresponding energies the chaotic eigenstates appear all to be dynamically localized meaning that they do not uniformly occupy the entire available chaotic classical phase space component but are localized on a proper subset we find with unprecedented precision and statistical significance that the level spacing distributions of the regular levels obey the poisson statistics and the chaotic ones obey the brody statistics as anticipated in a recent paper by batistic and robnik 2010 j phys a math theor 43 215101 where the entire spectrum was found to obey the berry robnik brody statistics there are no effects of dynamical tunneling in this regime due to the high energies as they decay exponentially with the inverse effective planck constant which is proportional to the square root of the energy

distribution of the ratio of two consecutive level spacings in orthogonal to unitary crossover ensembles the ratio of two consecutive level spacings has emerged as a very useful metric in investigating universal features exhibited by complex spectra it does not require the knowledge of density of states and is therefore quite convenient to compute in analyzing the spectrum of a general system the wigner surmise like results for the ratio distribution are known for the invariant classes of gaussian random matrices however for the crossover ensembles which are useful in modeling systems with partially broken symmetries corresponding results have remained unavailable so far in this work we derive exact results for the distribution and average of the ratio of two consecutive level spacings in the gaussian orthogonal to unitary crossover ensemble using a 3 3 random matrix model this crossover is useful in modeling time reversal symmetry breaking in quantum chaotic systems although based on a 3 3 matrix model our results can also be applied in the study of large spectra provided the symmetry breaking parameter facilitating the crossover is suitably scaled we substantiate this claim by considering gaussian and laguerre crossover ensembles comprising large matrices moreover we apply our result to investigate the violation of time reversal invariance in the quantum kicked rotor system

robust diagnosis of discrete event systems against intermittent loss of observations in the usual approaches to fault diagnosis of discrete event systems it is assumed that not only all sensors work properly but also all information reported by sensors always reaches the diagnoser any bad sensor operation or communication failure between sensors and the diagnoser can be regarded as loss of observations of events initially assumed as observable in such situations it may be possible that either the diagnoser stands still or report some wrong information regarding the fault occurrence in this paper we assume that intermittent loss of observations may occur and we propose an automaton model based on a new language operation language dilation that takes it into account we refer to this problem as robust diagnosability against intermittent loss of observations or simply robust diagnosability where the context allows we present a necessary and sufficient condition for robust diagnosability in terms of the language generated by the original automaton and propose two tests for robust language diagnosability one that deploys diagnosers and another one that uses verifiers we also extend the results to robust codiagnosability against intermittent loss of observations

a generalized framework for inference based diagnosis of discrete event systems capturing both disjunctive and conjunctive decision making we have previously introduced an inference based framework for decentralized decision making comprising of multiple observers each with its own partial observation where inferencing over the ambiguities of the self and the others is used to issue local decisions and a global decision is taken to be the one possessing the minimum ambiguity level in this setting we previously introduced the notion of n inference vee diagnosability to characterize the existence of a disjunctive decentralized diagnosis scheme so that any fault can be detected within a bounded delay using at most n levels of inferencing by one of the diagnosers while the disjunctive scheme relies on one of the diagnosers making the failure decision the dual conjunctive scheme relies on none of the diagnosers making the nonfailure decision it is known that the two schemes are incomparable and in this paper we extend our earlier work to provide a more general framework introducing the notion of n inference diagnosability capturing both disjunctive and conjunctive schemes we also develop a method for verifying n inference diagnosability as well as discuss several of its useful properties

scaling properties of multilayer random networks multilayer networks are widespread in natural and manmade systems key properties of these networks are their spectral and eigenfunction characteristics as they determine the critical properties of many dynamics occurring on top of them here we numerically demonstrate that the normalized localization length ensuremath beta of the eigenfunctions of multilayer random networks follows a simple scaling law given by ensuremath beta x 1 x with x ensuremath gamma b mathrm eff 2 l ensuremath delta ensuremath delta ensuremath sim 1 and b mathrm eff being the effective bandwidth of the adjacency matrix of the network whose size is l the scaling law for ensuremath beta that we validate on real world networks might help to better understand criticality in multilayer networks and to predict the eigenfunction localization properties of them

level spacing for band random matrices the authors examine the spacing distribution of the eigenvalues of tridiagonal real symmetric random matrices the elements of which are distributed according to a gaussian law they show explicitly that for 4 4 matrices the distribution at small spacings behaves as s log2 s they surmise that for n n matrices the behaviour is s logn 2s and they present numerical results which support this conjecture

energy level statistics and localization in sparsed banded random matrix ensemble the matrix elements of the quantum hamiltonian which corresponds to the classical kam system h h0 epsilon v where h0 is integrable in an energy ordered eigenbasis of h0 are considered their statistical properties are comprised in the definition of the sparsed banded random matrix ensemble sbrme in analysing the spectral statistics of this ensemble the authors find the power law level repulsion and the level spacing distribution which is well described by the brody distribution the numerical study of sbrme shows two interesting scaling laws i the connection between the scaling variable x alpha b3 2 alpha mean diagonal increment b bandwidth and the level repulsion parameter beta as deduced from the brody distribution and ii the connection between the same scaling variable x and an expression which contains two types of localization length entropy and geometric the universal aspects and the importance of sbrme for the hamiltonian systems in the transition region between integrability and chaos are discussed

the computational complexity of decentralized discrete event control problems computational complexity results are obtained for decentralized discrete event control problems these results generalize the earlier work of tsitsiklis 1989 who showed that for a special class of centralized supervisory control problems under partial observation there is an algorithm for determining in polynomial time whether or not a solution exists the negative complexity results associated with tsitsiklis work also carry over to the decentralized case so that solution existence for the more general class is not decidable in polynomial time nor does there exist a polynomial time algorithm for producing supervisor solutions when such solutions exist

decentralized supervisory control with intersection based architecture we investigate a new decentralized control architecture called intersection based architecture we propose a decentralized control protocol under this architecture called state estimator intersection based protocol sei protocol where each local supervisor sends its state estimate to the fusion sites and these sites take the intersection of these estimates in order to make a control decision the necessary and sufficient conditions for the achievability of the specification under this protocol is provided this condition is termed as state estimator intersection based coobservability sei coobservability a polynomial time algorithm for the verification of sei coobservability is provided we show that the languages that can be achieved under the sei protocol are incomparable with languages that can be achieved under existing architectures

extreme value statistics of brain networks importance of balanced condition despite the key role played by inhibitory excitatory couplings in the functioning of brain networks the impact of a balanced condition on the stability properties of underlying networks remains largely unknown we investigate properties of the largest eigenvalues of networks having such couplings and find that they follow completely different statistics when in the balanced situation based on numerical simulations we demonstrate that the transition from weibull to frechet via the gumbel distribution can be controlled by the variance of the column sum of the adjacency matrix which depends monotonically on the denseness of the underlying network as a balanced condition is imposed the largest real part of the eigenvalue emulates a transition to the generalized extreme value statistics independent of the inhibitory connection probability furthermore the transition to the weibull statistics and the small world transition occur at the same rewiring probability reflecting a more stable system

localization and spectral statistics in a banded random matrix ensemble we investigate the localization properties of the eigenvectors of a banded random matrix ensemble in which the diagonal matrix elements increase along the diagonal we relate the results to a transition in the spectral statistics which is observed as a parameter is vaned and discuss the relevance of this model to the quantum mechanics of chaotic hamiltonian systems

on codiagnosability and coobservability with dynamic observations codiagnosability and coobservability in discrete event systems where observations are dynamic are considered instead of having a fixed set of observable events the observation of an event is dynamic trace dependent in this paper a procedure is developed to transform the problem of coobservability to the problem of codiagnosability in the context of dynamic observations this proves that problems of coobservability are transformable to problems of codiagnosability and enables us to leverage the large literature available for codiagnosability to solve problems of coobservability furthermore in the case of dynamic observations the known polynomial complexity tests for the property of codiagnosability based on verifier automata with fixed observable event set s are no longer directly applicable a new testing procedure is developed that can handle transition based dynamic observations and remains of polynomial complexity in the state space of the system this new testing procedure employs a covering of the state space of the system based on cluster automata which enhances its computational efficiency based on cluster automata a new type of verifier automaton is built called the c verifier for verification of codiagnosability as an application of the above mentioned transformation the c verifier becomes a unified method for verifying both codiagnosability and coobservability

intermediate statistics of the quasi energy spectrum and quantum localisation of classical chaos the statistical properties of the quasi energy spectrum in a simple quantum model are investigated for the case when the corresponding classical system is fully chaotic while quantum chaos is restricted by the localisation effects it is shown that the level spacing distribution depends effectively on some parameter which is the ratio of the dimension of the eigenfunctions mean localisation length to the total number of the quasi energy levels numerical data for a wide range of parameters of the system are given