polynomial time probabilistic controllability analysis of discrete time piecewise affine systems this paper proposes a probabilistic approach to the controllability analysis for discrete time piecewise affine pwa systems three kinds of randomized algorithms which are based on random sampling of the mode sequence and or the initial state for determining with a probabilistic accuracy if the system is controllable are presented a positive one sided error algorithm a negative one sided error algorithm and a two sided error algorithm it is proven that these are polynomial time algorithms with respect to several variables of the problem it is also shown with some examples for which it is hopeless to check the controllability in a deterministic way that these algorithms are efficient

some universal properties for restricted trace gaussian orthogonal unitary and symplectic ensembles consider fixed and bounded trace gaussian orthogonal unitary and symplectic ensembles closely related to gaussian ensembles without any constraint for three restricted trace gaussian ensembles we prove universal limits of correlation functions at zero and at the edge of the spectrum edge our argument also applies to restricted trace ensembles with monomial potentials in addition by using the universal result in the bulk for fixed trace gaussian unitary ensemble which has been obtained by gotze and gordin we also prove the universal limits of correlation functions everywhere in the bulk for bounded trace gaussian unitary ensemble

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

synthesis of optimal controllers for piecewise affine systems with sampled data switching this paper discusses the optimal control problem of the continuous time piecewise affine pwa systems with sampled data switching where the switching action is executed based upon a condition on the state at each sampling time first an algebraic characterization for the problem to be feasible is derived next an optimal continuous time controller is derived for a general class of pwa systems with sampled data switching for which the optimal control problem is feasible but whose subsystems in some modes may be uncontrollable in the usual sense finally as an application of the proposed approach the high speed and energy saving control problem of the cpu processing is formulated and the validity of the proposed methods is shown by numerical simulations

noise dressing of financial correlation matrices we show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate time series the central result of the present study which focuses on the case of financial price fluctuations is the remarkable agreement between the theoretical prediction based on the assumption that the correlation matrix is random and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the s 500 or other major markets in particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management

some consequences of exchangeability in random matrix theory properties of infinite sequences of exchangeable random variables result directly in explicit expressions for calculating asymptotic densities of eigenvalues of any ensemble of random matrices h whose distribution depends only on tr h h where h is the hermitian conjugate of h for real symmetric matrices and for hermitian matrices the densities are constructed by summing up wigner semicircles with varying radii and weights as confirmed by monte carlo simulations extensions to more general matrix ensembles are also considered

asymptotic short wave equivalence of one dimensional schrodinger equations by formal canonical transformations and its generalizations the authors define the asymptotic equivalence of systems of two ordinary first order differential equations with an arbitrary finite number and multiplicities of turning points the theory is exemplified by the short wave semiclassical equivalence of time independent one dimensional schrodinger equations they describe the set of all transformation matrices realizing the equivalence the use of their determinant properties simplifies the calculations needed for applications for the particular case of schrodinger equations the transformation matrix can be chosen to be canonical

on the efetov wegner terms by diagonalizing a hermitian supermatrix the diagonalization of hermitian supermatrices is studied such a change of coordinates is inevitable if one wishes to find certain structures in random matrix theory however it still poses serious problems since until now the calculation of all rothstein contributions known as efetov wegner terms in physics was quite cumbersome we derive the supermatrix bessel function with all efetov wegner terms for an arbitrary rotation invariant probability density function as applications we consider representations of generating functions for hermitian random matrices with and without an external field as integrals over eigenvalues of hermitian supermatrices all results are obtained with all efetov wegner terms which were previously unknown in such an explicit and compact representation

compact support probability distributions in random matrix theory we consider a generalization of the fixed and bounded trace ensembles introduced by bronk and rosenzweig up to an arbitrary polynomial potential in the large n limit we prove that the two are equivalent and that their eigenvalue distribution coincides with that of the canonical ensemble with measure exp ntr v m the mapping of the corresponding phase boundaries is illuminated in an explicit example in the case of a gaussian potential we are able to derive exact expressions for the one and two point correlator for finite n having finite support

recursive construction for a class of radial functions ii superspace we extend the recursion formula for matrix bessel functions which we obtained previously to superspace it is sufficient to do this for the unitary orthosymplectic supergroup by direct computations we show that fairly explicit results can be obtained at least up to dimension 8 8 for the supermatrices since we introduce a new technique we discuss various of its aspects in some detail

derivation of determinantal structures for random matrix ensembles in a new way there are several methods to treat ensembles of random matrices in symmetric spaces circular matrices chiral matrices and others orthogonal polynomials and the supersymmetry method are particular powerful techniques here we present a new approach to calculate averages over ratios of characteristic polynomials at first sight paradoxically one can call our approach supersymmetry without supersymmetry because we use structures from supersymmetry without actually mapping onto superspaces we address two kinds of integrals which cover a wide range of applications for random matrix ensembles for probability densities factorizing in the eigenvalues we find determinantal structures in a unifying way as a new application we derive an expression for the k point correlation function of an arbitrary rotation invariant probability density over the hermitian matrices in the presence of an external field

family of solvable generalized random matrix ensembles with unitary symmetry we construct a very general family of characteristic functions describing random matrix ensembles rme having a global unitary invariance and containing an arbitrary one variable probability measure which we characterize by a spread function various choices of the spread function lead to a variety of possible generalized rmes which show deviations from the well known gaussian rme originally proposed by wigner we obtain the correlation functions of such generalized ensembles exactly and show examples of how particular choices of the spread function can describe ensembles with arbitrary eigenvalue densities as well as critical ensembles with multifractality

random matrix ensembles from nonextensive entropy the classical gaussian ensembles of random matrices can be constructed by maximizing boltzmann gibbs shannon s entropy s text bgs ensuremath ensuremath int d mathbf h p mathbf h mathrm ln p mathbf h with suitable constraints here we construct and analyze random matrix ensembles arising from the generalized entropy s q 1 ensuremath ensuremath int d mathbf h p mathbf h q q ensuremath 1 thus s 1 s text bgs the resulting ensembles are characterized by a parameter q measuring the degree of nonextensivity of the entropic form making q ensuremath rightarrow 1 recovers the gaussian ensembles if q ensuremath ne 1 the joint probability distributions p mathbf h cannot be factorized i e the matrix elements of mathbf h are correlated in the limit of large matrices two different regimes are observed when ql1 p mathbf h has compact support and the fluctuations tend asymptotically to those of the gaussian ensembles anomalies appear for qg1 both p mathbf h and the marginal distributions p h ij show power law tails numerical analyses reveal that the nearest neighbor spacing distribution is also long tailed not wigner dyson and after proper scaling very close to the result for the 2 ifmmode times else texttimes fi 2 case char22 a generalization of wigner s surmise we discuss connections of these nonextensive ensembles with other non gaussian ones such as the so called l evy ensembles and those arising from soft confinement

family of generalized random matrix ensembles using the generalized maximum entropy principle based on the nonextensive q entropy a family of random matrix ensembles is generated this family unifies previous extensions of random matrix theory rmt and gives rise to an orthogonal invariant stable levy ensemble with new statistical properties some of them are analytically derived

observability and controllability of piecewise affine and hybrid systems we prove in a constructive way the equivalence between piecewise affine systems and a broad class of hybrid systems described by interacting linear dynamics automata and propositional logic by focusing our investigation on the former class we show through counterexamples that observability and controllability properties cannot be easily deduced from those of the component linear subsystems instead we propose practical numerical tests based on mixed integer linear programming

brief paper tracking and synchronisation for a class of pwa systems in this paper the tracking problem for a class of discontinuous piecewise affine pwa systems is addressed we propose an observer based output feedback control design consisting of a feedforward a piecewise affine feedback law and a model based observer solving the tracking problem these synthesis results can also be employed to tackle the master slave synchronisation problem for pwa systems it is shown that for certain classes of pwa systems the design is characterised in terms of linear matrix inequalities the results are illustrated by application to mechanical systems with discontinuous friction characteristics

on conditions for regularity of solutions for a piecewise linear system abstract we study a piecewise linear system approximating the behaviour of a switched dc dc converter we give conditions for a unique limit cycle in our different cases and give examples of complicated phenomena that might occur the stability of a state space trajectory controlled converter is addressed

uniform approximations for the natural modes and frequencies of spindle shaped acoustical resonators we wish to obtain uniform approxiations for the natural modes and frequencies of some elongated acoustic bodies or spindles these are bodies generated by plain curves of the type y h a a dxb also the end points are pointed finally we remark that when s o r ex r x is a body of infinite length with a nonzero cross section such a body is generated by a curve of the type y x i o kl x xx 1 2

asymptotic semiclassical equivalence for schrodinger equations with singular potentials and for related systems of two first order equations the asymptotic equivalence of systems of two ordinary first order linear differential equations with complex independent variable and a small parameter at the derivatives is analyzed in the case of arbitrary numbers and multiplicities of turning points and singular points the set of all the transformation matrices realizing the equivalence is described and a recursive procedure for constructing these matrices is developed by persistently using the determinant properties of the transformation matrices the number of integration operations at each step of this procedure is halved compared with the algorithms known before the theory is specialized to the case of time independent one dimensional schrodinger equations with singular potentials some generalizations to multichannel schrodinger equations are also presented

convex dynamic programming for hybrid systems a classical linear programming approach to optimization of flow or transportation in a discrete graph is extended to hybrid systems the problem is finite dimensional if the state space is discrete and finite but becomes infinite dimensional for a continuous or hybrid state space it is shown how strict lower bounds on the optimal loss function can be computed by gridding the continuous state space and restricting the linear program to a finite dimensional subspace upper bounds can be obtained by evaluation of the corresponding control laws

observability of the discrete state for dynamical piecewise hybrid systems abstract in this paper we deal with the observability of piecewise affine hybrid systems our aim is to give sufficient conditions to observe the discrete and continuous states in terms of algebraic and geometrical conditions firstly we will give the algebraic conditions to observe the discrete state based on the switch function reconstruction for linear hybrid systems secondly we will give a geometrical condition based on the transversality concept for nonlinear hybrid systems throughout this paper we illustrate our propositions with examples and simulations

piecewise linear quadratic optimal control the use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control lower bounds on the optimal control cost are obtained by semidefinite programming based on the bellman inequality this also gives an approximation to the optimal control law an upper bound to the optimal cost is obtained by another convex optimization problem using the given control law a compact matrix notation is introduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyze smooth nonlinear dynamics with arbitrary accuracy

switching relay and complementarity systems a tutorial on their well posedness and relationships this paper reviews the relationships between switching systems defined from a partition of the state space into convex cells and relay or complementarity dynamical systems which are other classes of discontinuous systems well posedness results i e results on the existence and the uniqueness of solutions for different classes of relay and complementarity systems which are also switching systems are presented the reverse issue when can a switching system be rewritten equivalently as a relay or a complementarity system is also tackled many examples from mechanics circuits and biology illustrate the developments throughout the paper the paper focuses on systems with continuous solutions i e with no state jumps convexity is the central property

control of systems integrating logic dynamics and constraints this paper proposes a framework for modeling and controlling systems described by interdependent physical laws logic rules and operating constraints denoted as mixed logical dynamical mld systems these are described by linear dynamic equations subject to linear inequalities involving real and integer variables mld systems include linear hybrid systems finite state machines some classes of discrete event systems constrained linear systems and nonlinear systems which can be approximated by piecewise linear functions a predictive control scheme is proposed which is able to stabilize mld systems on desired reference trajectories while fulfilling operating constraints and possibly take into account previous qualitative knowledge in the form of heuristic rules due to the presence of integer variables the resulting on line optimization procedures are solved through mixed integer quadratic programming miqp for which efficient solvers have been recently developed some examples and a simulation case study on a complex gas supply system are reported

existence and uniqueness of solutions of piecewise nonlinear systems abstract in this paper we consider the existence and the uniqueness of solutions of piecewise nonlinear systems we will present some necessary and sufficient conditions for the existence and the uniqueness of solutions of this class of systems

asymptotic stability of piecewise affine systems with filippov solutions via discontinuous piecewise lyapunov functions asymptotic stability of continuous time piecewise affine systems defined over a polyhedral partition of the state space with possible discontinuous vector field on the boundaries is considered in the first part of the paper the feasible filippov solution concept is introduced by characterizing single mode caratheodory sliding mode and forward zeno behaviors then a global asymptotic stability result through a possibly discontinuous piecewise lyapunov function is presented the sufficient conditions are based on pointwise classifications of the trajectories which allow the identification of crossing unreachable and caratheodory boundaries it is shown that the sign and jump conditions of the stability theorem can be expressed in terms of linear matrix inequalities by particularizing to piecewise quadratic lyapunov functions and using the cone copositivity approach several examples illustrate the theoretical arguments and the effectiveness of the stability result

a result on the hyperstability of a class of hybrid dynamic systems this paper presents a hyperstability theorem for a class of hybrid dynamic systems composed of coupled differential and difference equations subject to possibly time varying nonlinearities satisfying a popov type inequality the nonlinear controller generates the plant input at all times from its sampled values by defining an extended discrete system the hyperstability results are obtained from this discrete system of special type whose state consists of the sampled continuous substate and the digital substate of the given hybrid system some corollaries and related physical interpretations are also given

the integral theorem for supersymmetric invariants a supersymmetric integral theorem that extends results of parisi sourlas efetov wegner and others is rigorously proved in particular arbitrary generators are allowed in the integrand instead of canonical ones and the invariance condition is very much relaxed the connection with cauchy s integral formula is made transparent in passing the unitary lie supergroup is studied by using elementary methods applications in the theory of disordered systems are discussed

trajectory tracking control of bimodal piecewise affine systems this paper deals with a trajectory tracking problem for a class of bimodal piecewise affine systems which is inherently difficult because of the discontinuous changes of their vector fields first we introduce an error variable and an error system as a generalization of the tracking error and its system as an error variable a function switched by the mode of a piecewise affine system is adopted to overcome the inherent difficulty in trajectory tracking control of piecewise affine systems next we design a tracking controller which stabilizes the error system using a lyapunov like function which can be applied to systems including state jumps furthermore the feasibility condition of tracking for siso piecewise linear systems is simplified finally a numerical example is given to illustrate the effectiveness of the proposed method

well posedness analysis of switch driven piecewise affine systems the well posedness problem existence and uniqueness of solutions of a class of multi modal piecewise affine systems is addressed where binary switches individually act under autonomous switching first a new transition rule on the discrete state called the switch based transition rule is introduced and some relations with the mode based transition rule are discussed next a sufficient condition for such a multi modal system to be well posed for all external inputs is derived in terms of well posedness of its subsystems of lower complexity bimodal systems finally an easily checkable condition for the bimodal system to be well posed for all external inputs is given which consequently allows us to algebraically determine well posedness of the multi modal systems in question

regulator synthesis for bimodal linear systems this technical note presents a regulator synthesis approach for switched bimodal linear systems where it is desired to achieve regulation against known disturbance or reference signals first a set of observer based q parameterized stabilizing controllers for the switched system is constructed then regulation conditions for the switched system are obtained and a regulator synthesis method is derived based on iteratively solving a set of linear matrix inequalities finally a numerical example is presented to illustrate the effectiveness of the proposed method

global analysis of piecewise linear systems using impact maps and surface lyapunov functions this paper presents an entirely new constructive global analysis methodology for a class of hybrid systems known as piecewise linear systems pls this methodology infers global properties of pls solely by studying the behavior at switching surfaces associated with pls the main idea is to analyze impact maps i e maps from one switching surface to the next switching surface such maps are known to be unfriendly maps in the sense that they are highly nonlinear multivalued and not continuous we found however that an impact map induced by an linear time invariant flow between two switching surfaces can be represented as a linear transformation analytically parametrized by a scalar function of the state this representation of impact maps allows the search for surface lyapunov functions sulf to be done by simply solving a semidefinite program allowing global asymptotic stability robustness and performance of limit cycles and equilibrium points of pls to be efficiently checked this new analysis methodology has been applied to relay feedback on off and saturation systems where it has shown to be very successful in globally analyzing a large number of examples in fact it is still an open problem whether there exists an example with a globally stable limit cycle or equilibrium point that cannot be successfully analyzed with this new methodology examples analyzed include systems of relative degree larger than one and of high dimension for which no other analysis methodology could be applied this success in globally analyzing certain classes of pls has shown the power of this new methodology and suggests its potential toward the analysis of larger and more complex pls

arbitrary unitarily invariant random matrix ensembles and supersymmetry we generalize the supersymmetry method in random matrix theory to ensembles which are unitarily invariant but otherwise arbitrary our exact approach extends a previous contribution in which we constructed a supersymmetric representation for the class of norm dependent random matrix ensembles here we derive a supersymmetric formulation under very general circumstances a reduced probability density and a projector are identified that map the probability density from ordinary to superspace furthermore it is demonstrated that setting up the theory in fourier superspace has considerable advantages general and exact expressions for the correlation functions are given we also show how the use of hyperbolic symmetry can be circumvented in the present context in which the nonlinear model is not used we construct exact supersymmetric integral representations of the correlation functions for arbitrary positions of the imaginary increments in the green s functions

uniform asymptotic solutions of second order linear ordinary differential equations with singular points i formal theory abstract a formal uniform asymptotic solution of the differential equation d 2 u dz 2 2 r z u 0 z d for large when d contains any number and order of poles and turning points is constructed the development is a generalization of langer s method an asymptotic relation between the wronskian of any pair of linearly independent solutions of the differential equation and the wronskian of the pair of corresponding solutions of the comparison equation is derived a formal uniform asymptotic solution of a system of two first order ordinary differential equation is also obtained

sufficient conditions for lyapunov s global stability of time varying hybrid linear systems this paper deals with the derivation of sufficient conditions for global stability and global exponential stability for any bounded initial conditions in the whole state space of time varying hybrid systems by using lyapunov s second method in general those systems consist of coupled analogue and digital substates the proofs of global stability are based on guaranteeing firstly that the uncoupled analogue and digital substates are stable or exponentially stable and then the couplings between both substates are sufficiently weak to maintain the global stability properties

a unified framework for hybrid control model and optimal control theory we propose a very general framework that systematizes the notion of a hybrid system combining differential equations and automata governed by a hybrid controller that issues continuous variable commands and makes logical decisions we first identify the phenomena that arise in real world hybrid systems then we introduce a mathematical model of hybrid systems as interacting collections of dynamical systems evolving on continuous variable state spaces and subject to continuous controls and discrete transitions the model captures the identified phenomena subsumes previous models yet retains enough structure to pose and solve meaningful control problems we develop a theory for synthesizing hybrid controllers for hybrid plants in all optimal control framework in particular we demonstrate the existence of optimal relaxed and near optimal precise controls and derive generalized quasi variational inequalities that the associated value function satisfies we summarize algorithms for solving these inequalities based on a generalized bellman equation impulse control and linear programming

the supersymmetry method for chiral random matrix theory with arbitrary rotation invariant weights in the past few years the supersymmetry method has been generalized to real symmetric hermitian and hermitian self dual random matrices drawn from ensembles invariant under the orthogonal unitary and unitary symplectic groups respectively we extend this supersymmetry approach to chiral random matrix theory invariant under the three chiral unitary groups in a unifying way thereby we generalize a projection formula providing a direct link and hence a short cut between the probability density in ordinary space and that in superspace we emphasize that this point was one of the main problems and shortcomings of the supersymmetry method since only implicit dualities between ordinary space and superspace were known before to provide examples we apply this approach to the calculation of the supersymmetric analogue of a lorentzian cauchy ensemble and an ensemble with a quartic potential moreover we consider the partially quenched partition function of the three chiral gaussian ensembles corresponding to four dimensional continuum quantum chromodynamics we identify a natural splitting of the chiral lagrangian in its lowest order into a part for the physical mesons and a part associated with source terms generating the observables e g the level density of the dirac operator

dyson s correlation functions and graded symmetry a new derivation of dyson s k level correlation functions of the gaussian unitary ensemble gue is given the method uses matrices with graded symmetry the number of integrations needed for the ensemble average becomes independent of the level number n for arbitrary level number n the k level correlation function is expressed as an integral involving the eigenvalues of a 2k 2k graded matrix the limit of infinitely many levels n is calculated by a simple saddle point approximation of this integral avoiding the introduction of hermite polynomials and oscillator wave functions

norm dependent random matrix ensembles in external field and supersymmetry the class of norm dependent random matrix ensembles is studied in the presence of an external field the probability density of the ensemble depends on the trace of the squared random matrices but is otherwise arbitrary an exact mapping to superspace is performed a transformation formula is derived which gives the probability density in superspace as a single integral over the probability density in ordinary space this is done for orthogonal unitary and symplectic symmetries in the case of unitary symmetry some explicit results for the correlation functions are derived

analysis of zeno behaviors in a class of hybrid systems this note investigates conditions for existence of zeno behaviors where a system undergoes an unbounded number of discrete transitions in a finite length of time in a class of hybrid systems zeno behavior occurs for example when a controller unsuccessfully attempts to satisfy an invariance specification by switching the system among different configurations faster and faster two types of zeno systems are investigated 1 strongly zeno systems where all runs of the system are zeno and 2 weakly zeno systems where only some runs of the system are zeno for constant rate and bounded rate hybrid systems and some nonlinear generalizations necessary and sufficient conditions for both zenoness and strong zenoness are derived the analysis is based on studying the trajectory set of a certain equivalent continuous time system that is associated with the dynamic equations of the hybrid system the relation between the possibility of existence of zeno behaviors in a system and the problem of existence of non zeno safety controllers that keep the system in a specified region of its operating space is also examined it is shown that in certain zeno systems a minimally interventive safety controller may not exist even if a safety controller exists disproving a conjecture made earlier in the literature

well posedness of piecewise linear systems with multiple modes and multiple criteria the results of a previous paper imura and van der schaft 2000 are generalized to obtain necessary and sufficient conditions for the well posedness of piecewise linear systems with multiple modes and multiple criteria to check the necessary and sufficient conditions we present algorithmic procedures by making use of the fourier motzkin elimination technique

optimal control of sampled data piecewise affine systems this paper discusses the optimal continuous time control problem of a class of piecewise affine pwa systems where the switching action of the discrete state is determined at each sampling time according to a condition on the continuous state such a system is called here the sampled data pwa sd pwa system first important remarks on the control design of this system via the continuous or discrete time pwa model are pointed out which motivate us to use the sd pwa model next based on the good properties of the proposed model an optimal continuous time controller of the sd pwa systems is proposed

characterization of well posedness of piecewise linear systems one of the basic issues in the study of hybrid systems is the well posedness existence and uniqueness of solutions problem of discontinuous dynamical systems the paper addresses this problem for a class of piecewise linear discontinuous systems under the definition of solutions of caratheodory the concepts of jump solutions or of sliding modes are not considered here in this sense the problem to be discussed is one of the most basic problems in the study of well posedness for discontinuous dynamical systems first we derive necessary and sufficient conditions for bimodal systems to be well posed in terms of an analysis based on lexicographic inequalities and the smooth continuation property of solutions next its extensions to the multimodal case are discussed as an application to switching control in the case that two state feedback gains are switched according to a criterion depending on the state we give a characterization of all admissible state feedback gains for which the closed loop system remains well posed

hybrid model reference adaptive control of piecewise affine systems this paper is concerned with the derivation of a model reference adaptive control mrac scheme for multimodal piecewise affine pwa and piecewise linear systems the control allows the plant to track asymptotically the states of a multimodal piecewise affine or smooth reference model the reference model can be characterized by a number and geometry of phase space regions that can be entirely different from those of the plant numerical simulations on a set of representative examples confirm the theoretical derivation and proof of stability

superstatistical generalizations of wishart laguerre ensembles of random matrices using beck and cohen s superstatistics we introduce in a systematic way a family of generalised wishart laguerre ensembles of random matrices with dyson index 1 2 and 4 the entries of the data matrix are gaussian random variables whose variancesfluctuate from one sample to another according to a certain probability density f and a single deformation parameter three superstatistical classes for f are usually considered 2 inverse 2 and log normal distributions while the first class already considered by two of the authors leads to a power law decay of the spectral density we here introduce and solve exactly a superposition of wishart laguerre ensembles with inverse 2 distribution the corresponding macroscopic spectral density is given by a deformation of the semi circle and marcenko pastur laws on a non compact support with exponential tails after discussing in detail the validity of wigner s surmise in the wishart laguerre class we introduce a generalised dependent surmise with stretched exponential tails which well approximates the individual level spacing distribution in the bulk the analytical results are in excellent agreement with numerical simulations to illustrate our findings we compare the 2 and inverse 2 class to empirical data from financial covariance matrices

a lie backlund approach to equivalence and flatness of nonlinear systems a new system equivalence relation using the framework of differential geometry of jets and prolongations of infinite order is studied in this setting two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives this is a lie backlund isomorphism the authors prove that although the state dimension is not preserved the number of input channels is kept fixed they also prove that a lie backlund isomorphism can be realized by an endogenous feedback the differentially flat nonlinear systems introduced by the authors 1992 via differential algebraic techniques are generalized and the new notion of orbitally flat systems is defined they correspond to systems which are equivalent to a trivial one with time preservation or not the endogenous linearizing feedback is explicitly computed in the case of the vtol aircraft to track given reference trajectories with stability

spectral correlations in the crossover between gue and poisson regularity on the identification of scales motivated by questions of present interest in nuclear and condensed matter physics we consider the superposition of a diagonal matrix with independent random entries and a gue the relative strength of the two contributions is determined by a parameter suitably defined on the unfolded scale using results for the spectral two point correlator of this model obtained in the framework of the supersymmetry method we focus attention on two different regimes for 1 the correlations are given by dawson s integral while for 1 we derive a novel analytical formula for the two point function in both cases the energy scales in units of the mean level spacing at which deviations from pure gue behavior become noticeable can be identified we also derive an exact expansion of the local level density for a finite level number