replica treatment of non hermitian disordered hamiltonians we employ the fermionic and bosonic replicated nonlinear models to treat ginibre unitary symplectic and orthogonal ensembles of non hermitian random matrix hamiltonians using the saddle point approach combined with the borel resummation procedure we derive the exact large n results for microscopic density of states in all three ensembles we also obtain tails of the density of states as well as the two point function for the unitary ensemble

applications of vector bundles to factorization of rational matrices abstract we show that wiener hopf s and forney s factorization theorems are immediate consequences of grothendieck s theorem on vector bundles

from time series to linear system part i finite dimensional linear time invariant systems abstract in the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed in this context the notions of linearity time invariance and finite dimensionality will be introduced it will be shown that a given family of time series may be represented by a system of ar equations riw t l rl 1w t l 1 r0w t 0 or equivalently by a finite dimensional linear time invariant system x t 1 ax t bu t y t cx t du t w u y if and only if this family is linear shift invariant and complete or as is equivalent closed in the topology of pointwise convergence this yields a very high level and elegant set of axioms which characterize these familiar objects it is emphasized however that no a priori choice is made as to which components of w are inputs and which are outputs such a separation always exists in any specific linear time invariant model starting from these definitions the structural indices of such systems are introduced and it is shown how an ar representation of a system having a given behaviour can be constructed these results will be used in a modelling context in part ii of the paper

the klein gordon equation of generalized hulthen potential in complex quantum mechanics we have investigated the reality of exact bound states of complex and or pt symmetric non hermitian exponential type generalized hulthen potential the klein gordon equation has been solved by using the nikiforov uvarov method which is based on solving the second order linear differential equations by reduction to a generalized equation of hypergeometric type in many cases of interest negative and positive energy states have been discussed for different types of complex potentials

individual based model with global competition interaction fluctuation effects in pattern formation we present some numerical results obtained from a simple individual based model that describes clustering of organisms caused by competition our aim is to show that even when a deterministic description developed for continuum models predicts no pattern formation an individual based model displays well defined patterns as a consequence of fluctuation effects caused by the discrete nature of the interacting agents

generalised autoregressive models and their trajectories the set of classical ar models is not compact so that a sequence of ar models may fail to converge to anything reasonable a compactification leads to ar models with arbitrary singularity at infinity these limit ar models are represented by pairs of polynomial and proper rational matrices satisfying certain compatibility condition it is shown that they produce trajectories in a very natural way then the classical equivalence theorem is extended to them stating that two ar models having the same behaviour are equivalent

pattern formation and turing instability in an activator inhibitor system with power law coupling abstract we investigate activator inhibitor systems in two spatial dimensions with a non local coupling for which the interaction strength decreases with the lattice distance as a power law by varying a single parameter we can pass from a local laplacian to a global all to all coupling type we derived from a linear stability analysis of the fourier spatial modes a set of conditions for the occurrence of a turing instability by which a spatially homogeneous pattern can become unstable in nonlinear systems the growth of these modes is limited and pattern formation is possible we have studied some qualitative features of the patterns formed in non local coupled activator inhibitor systems described by the meinhardt gierer equations

fluctuations in the lattice gas for burgers equation the langevin equation for the biased random hopping of particles onto unoccupied sites of a lattice is obtained from the master equation for finite systems by invoking a limit theorem due to kurtz for a weakly asymmetric process with diffusive scaling the continuum limit of this equation yields a deterministic equation that is equivalent to burgers equation the stochastic equation for the leading lattice corrections is derived and the approach to the continuum limit is illustrated with simulations

the importance of being discrete life always wins on the surface many systems in chemistry biology finance and social sciences present emerging features that are not easy to guess from the elementary interactions of their microscopic individual components in the past the macroscopic behavior of such systems was modeled by assuming that the collective dynamics of microscopic components can be effectively described collectively by equations acting on spatially continuous density distributions it turns out that to the contrary taking into account the actual individual discrete character of the microscopic components of these systems is crucial for explaining their macroscopic behavior in fact we find that in conditions in which the continuum approach would predict the extinction of all of the population respectively the vanishing of the invested capital or the concentration of a chemical substance etc the microscopic granularity insures the emergence of macroscopic localized subpopulations with collective adaptive properties that allow their survival and development in particular it is found that in two dimensions life the localized proliferating phase always prevails

macroscopic description of particle systems with nonlocal density dependent diffusivity in this paper we study macroscopic density equations in which the diffusion coefficient depends on a weighted spatial average of the density itself we show that large differences not present in the local densitydependence case appear between the density equations that are derived from different representations of the langevin equation describing a system of interacting brownian particles linear stability analysis demonstrates that under some circumstances the density equation interpreted with the ito calculus has pattern solutions that never appear for the kinetic interpretation which is the other one typically appearing in the context of nonlinear diffusion processes we also introduce a discrete time microscopic model of particles that confirms the results obtained at the macroscopic density level

analytic solutions of a nonlinear convective equation in population dynamics analytic solutions are presented for a simple nonlinear convective equation of use in population dynamics in spite of its simplicity the equation predicts rich behavior including a velocity inversion transition stability considerations are also presented

pt symmetric pseudo perturbation recipe an imaginary cubic oscillator with spikes the pseudo perturbative shifted l expansion technique pslet is shown to be applicable in the non hermitian symmetric context the construction of bound states for several symmetric potentials is presented with special attention paid to v r ir3 ir oscillators

from time series to linear system part iii approximate modelling abstract in the third part of this paper the problem of finding a linear time invariant complete system which models an observed time series will be continued however in this part it will be assumed that the model is required to explain the observations only approximately this raises the question of what is meant by an approximate model and how one should judge a model against data which in a strict sense falsify the model the initial sections of this paper introduce the methodology either the maximal admissible complexity is fixed and the best fitting model in the model class is sought or the maximally tolerated misfit is fixed and the least complex model in the model class is sought this is illustrated by means of finite dimensional models where natural complexity and misfit functions suggest themselves subsequently what is meant by the complexity of a linear time invariant complete system is discussed a concrete complexity function is proposed which is defined in terms of the richness of the behaviour on finite time intervals and is in one to one correspondence with the number of inputs the number of states and the observability indices of the underlying minimal i s o system the next definition is that of the misfit between an observed time series and a system a concrete misfit function is proposed which takes into consideration the lag structure of the system and in how far the observations fail to corroborate the ar equations which define the system s behaviour how the misfit may be computed by means of the correlation function of the data is then shown these concepts are subsequently used in order to set up two algorithms for approximate modelling of an observed q dimensional time series the first algorithm fixes the maximal complexity in a sense this comes down to fixing the number and the lags of the required ar equations the second algorithm fixes the tolerated misfit it is adaptive in nature in the sense that the number and the structure of the resulting ar equations will depend on the observations these ideas and algorithms are illustrated by means of two simulated examples

algebraic and scattering aspects of a symmetric solvable potential we study a particular solvable potential and analyse the effect of symmetry on its bound state as well as scattering solutions we determine the transmission and reflection coefficients for the symmetric case and also formulate the problem in terms of an su 1 1 potential group which allows unified treatment of the discrete and the continuous spectra in a natural way we find that bound and scattering states of the symmetric problem supply a basis for the unitary irreducible representations of the su 1 1 potential group and this gives a straightforward group theoretical interpretation of the fact that the complex invariant potential has a real energy spectrum

finite sections of random jacobi operators this article is about a problem in the numerical analysis of random operators we study a version of the finite section method for the approximate solution of equations ax b in infinitely many variables where a is a random jacobi i e tridiagonal operator in other words we approximately solve infinite second order difference equations with stochastic coefficients by reducing the infinite volume case to the large finite volume case via a particular truncation technique for most of the paper we consider non self adjoint operators a but we also comment on the self adjoint case when simplifications occur

random fibonacci sequences solutions to the random fibonacci recurrence xn 1 xn xn 1 decrease increase exponentially xn exp n for sufficiently small large in the limits 0 and we expand the lyapunov exponent in powers of and 1 respectively for the classical case of 1 we obtain exact non perturbative results in particular an invariant measure associated with ricatti variable rn xn 1 xn is shown to exhibit plateaux around all rational r

clustering determines who survives for competing brownian and levy walkers the competition between two ecologically similar species that use the same resources and differ from each other only in the type of spatial motion they undergo is studied the latter is assumed to be described either by brownian motion or levy flights competition is taken into account by assuming that individuals reproduce in a density dependent fashion it is observed that no influence of the type of motion occurs when the two species are in a well mixed unstructured state however as soon as the species develop spatial clustering the one forming more concentrated clusters gets a competitive advantage and eliminates the other a similar competitive advantage would occur between walkers of the same type but with different diffusivities if this leads also to different clustering the coexistence of both species is also possible under certain conditions

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

statistical properties of eigenvectors in non hermitian gaussian random matrix ensembles statistical properties of eigenvectors in non hermitian random matrix ensembles are discussed with an emphasis on correlations between left and right eigenvectors two approaches are described one is an exact calculation for ginibre s ensemble in which each matrix element is an independent identically distributed gaussian complex random variable the other is a simpler calculation using n 1 as an expansion parameter where n is the rank of the random matrix this is applied to girko s ensemble consequences of eigenvector correlations which may be of physical importance in applications are also discussed it is shown that eigenvalues are much more sensitive to perturbations than in the corresponding hermitian random matrix ensembles it is also shown that in problems with time evolution governed by a non hermitian random matrix transients are controlled by eigenvector correlations

local enrichment and its nonlocal consequences for victim exploiter metapopulations abstract the stabilizing effects of local enrichment are revisited diffusively coupled host parasitoid and predator prey metapopulations are shown to admit a stable fixed point limit cycle or stable torus with a rich bifurcation structure a linear toy model that yields many of the basic qualitative features of this system is presented the further nonlinear complications are analyzed in the framework of the marginally stable lotka volterra model and the continuous time analog of the unstable host parasitoid nicholson bailey model the dependence of the results on the migration rate and level of spatial variations is examined and the possibility of nonlocal effect of enrichment where local enrichment induces stable oscillations at a distance is studied a simple method for basic estimation of the relative importance of this effect in experimental systems is presented and exemplified

ribbed elastic structures under a mean flow the problem of a ribbed membrane or plate submerged in a fluid with mean flow is studied we first derive a method which can be used to reduce this and similar problems to a band matrix inversion we then find the pass and stop band structure found in the case of static fluid persists when a mean flow is introduced and we give an explanation in terms of the eigenvalues of the transfer matrix of the system we then study disordered structures and observe the phenomenon of anderson localization in some parameter regimes the addition of disorder causes significant delocalization

disturbance propagation in chaotic extended systems with long range coupling propagation of initially localized perturbations is investigated in chaotic coupled map lattices with long range couplings decaying as a power of the distance the initial perturbation propagates exponentially fast along the lattice with a rate given by the ratio of the maximal lyapunov exponent and the power of the coupling a complementary description in terms of a suitable comoving lyapunov exponent is also given

linear dynamical systems compactness theorem abstract we define a natural compact topology on the set of all behaviours with a fixed input number and macmillan degree contained in a given behaviour and obtain a continuous bejective parametrization of this set by a behaviour we mean a certain generalization of what willems terms by the same word

dynamics of a globally coupled oscillator array abstract we study a set of n globally coupled ordinary differential equations of the form encountered in circuit analysis of superconducting josephson junction arrays particular attention is paid to two kinds of simple time periodic behavior known as in phase and splay phase states some results valid for general n as well as further results for n 2 and n are presented a recurring theme is the appearance of very weak dynamics near the periodic states the implications for josephson junction arrays are discussed

the spectra of large toeplitz band matrices with a randomly perturbed entry this paper is concerned with the union sp j k tn a of all possible spectra that may emerge when perturbing a large n n toeplitz band matrix tn a in the j k site by a number randomly chosen from some set the main results give descriptive bounds and in several interesting situations even provide complete identifications of the limit of sp j k tn a as n also discussed are the cases of small and large sets as well as the discontinuity of the infinite volume case which means that in general sp j k tn a does not converge to something close to sp j k tn a as n where t a is the corresponding infinite toeplitz matrix illustrations are provided for tridiagonal toeplitz matrices a notable special case

large order perturbation theory for a non hermitian pt symmetric hamiltonian a precise calculation of the ground state energy of the complex pt symmetric hamiltonian h p2 14x2 i x3 is performed using high order rayleigh schrodinger perturbation theory the energy spectrum of this hamiltonian has recently been shown to be real using numerical methods here we present convincing numerical evidence that the rayleigh schrodinger perturbation series is borel summable and show that pade summation provides excellent agreement with the real energy spectrum pade analysis provides strong numerical evidence that the once subtracted ground state energy considered as a function of 2 is a stieltjes function the analyticity properties of this stieltjes function lead to a dispersion relation that can be used to compute the imaginary part of the energy for the related real but unstable hamiltonian h p2 14 x2 ex3

striated populations in disordered environments with advection abstract growth in static and controlled environments such as a petri dish can be used to study the spatial population dynamics of microorganisms however natural populations such as marine microbes experience fluid advection and often grow up in heterogeneous environments we investigate a generalized fisher kolmogorov petrovsky piscounov fkpp equation describing single species population subject to a constant flow field and quenched random spatially inhomogeneous growth rates with a fertile overall growth condition we analytically and numerically demonstrate that the non equilibrium steady state population density develops a flow driven striation pattern the striations are highly asymmetric with a longitudinal correlation length that diverges linearly with the flow speed and a transverse correlation length that approaches a finite velocity independent value linear response theory is developed to study the statistics of the steady states theoretical predictions show excellent agreement with the numerical steady states of the generalized fkpp equation obtained from lattice boltzmann simulations these findings suggest that although the growth disorder can be spatially uncorrelated correlated population structures with striations emerge naturally at sufficiently strong advection

on large toeplitz band matrices with an uncertain block abstract this paper investigates the possible spectra of large finite dimensional toeplitz band matrices with perturbations impurities uncertainties in the upper left m m block the main result shows that the asymptotic spectrum of such a matrix is not affected by these perturbations provided they have sufficiently small norm this follows from analysis of structured pseudospectra structured spectral value sets in contrast for typical non hermitian toeplitz matrices there exist certain rank one perturbations of arbitrarily small norm that move an eigenvalue away from the asymptotic spectrum in the large dimensional limit

fluctuations impact on a pattern forming model of population dynamics with non local interactions a model of interacting random walkers is presented and shown to give rise to patterns consisting in periodic arrangements of fluctuating particle clusters the model represents biological individuals that die or reproduce at rates depending on the number of neighbors within a given distance we evaluate the importance of the discrete and fluctuating character of this particle model on the pattern forming process to this end a deterministic mean field description including a linear stability and a weakly nonlinear analysis is given and compared with the particle model the deterministic approach is shown to reproduce some of the features of the discrete description in particular the existence of a finite wavelength instability stochasticity in the particle dynamics however has strong effects in other important aspects such as the parameter values at which pattern formation occurs or the nature of the homogeneous phase

pattern formation and nonlocal logistic growth logistic growth process with nonlocal interactions is considered in one dimension spontaneous breakdown of translational invariance is shown to take place at some parameter region and the bifurcation regime is identified for short and long range interactions domain walls between regions of different order parameter are expressed as soliton solutions of the reduced dynamics for nearest neighbor interactions the analytic results are confirmed by numerical simulations

novel computation of the growth rate of generalized random fibonacci sequences a new formulation is proposed for the computation of average growth rates of generalized random fibonacci sequences based on the new formula a novel numerical scheme is designed and successfully implemented and interesting analytic asymptotic expansions are obtained for several examples

poles zeros and sheaf cohomology abstract the fundamental pole zero exact sequence for a transfer function matrix first enunciated in 1989 supplies a structural meaning to the hope that the number of zeros of a transfer function is the same as the number of poles this principle requires the inclusion of generic zeros counted using wedderburn forney spaces which measure the kernel and cokernel of the transfer function in this brief note we identify the wedderburn forney space as a cohomology group and give a new approach to the fundamental pole zero exact sequence in terms of exact sequences of sheaf cohomology groups

on the remarkable spectrum of a non hermitian random matrix model a non hermitian random matrix model proposed a few years ago has a remarkably intricate spectrum various attempts have been made to understand the spectrum but even its dimension is not known using the dyson schmidt equation we show that the spectrum consists of a non denumerable set of lines in the complex plane each line is the support of the spectrum of a periodic hamiltonian obtained by the infinite repetition of any finite sequence of the disorder variables our approach is based on the theory of words we make a complete study of all four letter words the spectrum is complicated because our matrix contains everything that will ever be written in the history of the universe including this particular paper

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

spectra pseudospectra and localization for random bidiagonal matrices there has been much recent interest initiated by work of the physicists hatano and nelson in the eigenvalues of certain random non hermitian periodic tridiagonal matrices and their bidiagonal limits these eigenvalues cluster along a bubble with wings in the complex plane and the corresponding eigenvectors are localized in the wings delocalized in the bubble here in addition to eigenvalues pseudospectra are analyzed making it possible to treat the nonperiodic analogues of these random matrix problems inside the bubble the resolvent norm grows exponentially with the dimension outside it grows subexponentially in a bounded region that is the spectrum of the infinite dimensional operator localization and delocalization correspond to resolvent matrices whose entries exponentially decrease or increase respectively with distance from the diagonal this article presents theorems that characterize the spectra pseudospectra and numerical range for the four cases of finite bidiagonal matrices infinite bidiagonal matrices stochastic toeplitz operators finite periodic matrices and doubly infinite bidiagonal matrices stochastic laurent operators c 2001 john wiley sons inc

smooth impulsive linear systems interconnections smooth impulsive sm imp linear systems are the smallest class containing all classical linear systems and stable with respect to limits in this article we show that they behave well under the operation of interconnection namely it turns out that the mcmillan degree of a regular interconnection is always equal to the sum of the mcmillan degrees of the components in other words all regular interconnections of sm imp linear systems automatically are regular feedback interconnections for this more general class of linear systems we also study the regular interconnection problem

fisher equation with turbulence in one dimension as an example of life at high reynolds number we investigate the dynamics of the fisher equation for the spreading of micro organisms in one dimension subject to both turbulent convection and diffusion we show that for strong enough turbulence bacteria for example track in a quasilocalized fashion with remarkably long persistence times sinks in the turbulent field an important consequence is a large reduction in the carrying capacity of the fluid medium we analytically determine the regimes where this quasi localized behavior occurs and test our predictions by numerical simulations

predictability in spatially extended systems the predictability problem is discussed in turbulent fluids and in spatially extended systems we perform a numerical analysis of the spatial propagation of a small perturbation initially localized at a point of a one dimensional lattice of coupled maps where the interactions can be either local or non local in space in both cases the nonlinear terms in the dynamics of the perturbation cannot be neglected and the predictability problem cannot be reduced to the linearized evolution equations only for non local interactions is the predictability time proportional to the inverse lyapunov exponent lambda as in dynamical systems with a few degrees of freedom for local interactions it is proportional to the size of the system and is practically independent of the chaoticity degree lambda

nonlocal competition and logistic growth patterns defects and fronts logistic growth of diffusing reactants on spatial domains with long range competition is studied the bifurcations cascade involved in the transition from the homogeneous state to a spatially modulated stable solution is presented and a distinction is made between a modulated phase dominated by single or few wave numbers and the spiky phase where localized colonies are separated by depleted region the characteristic defects in the periodic structure are presented for each phase together with the invasion dynamics in the case of local initiation it is shown that the basic length scale that controls the bifurcation is the width of the fisher front and that the total population grows as this width decreases a mix of analytic results and extensive numerical simulations yields a comprehensive examination of the possible phases for logistic growth in the presence of nonlocal competition

regular spacings of complex eigenvalues in the one dimensional non hermitian anderson model we prove that in dimension one the non real eigenvalues of the non hermitian anderson nha model with a selfaveraging potential are regularly spaced the class of selfaveraging potentials which we introduce in this paper is very wide and in particular includes stationary potentials with probability one as well as all quasi periodic potentials it should be emphasized that our approach here is much simpler than the one we used before it allows us a to investigate the above mentioned spacings b to establish certain properties of the integrated density of states of the hermitian anderson models with selfaveraging potentials and c to obtain as a by product much simpler proofs of our previous results concerned with non real eigenvalues of the nha model

on the infinite dimensional qr algorithm spectral computations of infinite dimensional operators are notoriously difficult yet ubiquitous in the sciences indeed despite more than half a century of research it is still unknown which classes of operators allow for the computation of spectra and eigenvectors with convergence rates and error control recent progress in classifying the difficulty of spectral problems into complexity hierarchies has revealed that the most difficult spectral problems are so hard that one needs three limits in the computation and no convergence rates nor error control is possible this begs the question which classes of operators allow for computations with convergence rates and error control in this paper we address this basic question and the algorithm used is an infinite dimensional version of the qr algorithm indeed we generalise the qr algorithm to infinite dimensional operators we prove that not only is the algorithm executable on a finite machine but one can also recover the extremal parts of the spectrum and corresponding eigenvectors with convergence rates and error control this allows for new classification results in the hierarchy of computational problems that existing algorithms have not been able to capture the algorithm and convergence theorems are demonstrated on a wealth of examples with comparisons to standard approaches that are notorious for providing false solutions we also find that in some cases the iqr algorithm performs better than predicted by theory and make conjectures for future study

quantum mechanics of complex hamiltonian systems in one dimension with a view to obtaining further insight into the nature of eigenvalues and eigenfunctions of a stationary state one dimensional schrodinger equation corresponding to a non hermitian hamiltonian h x p we investigate the ground state solutions for a variety of potentials within the framework of an extended complex phase space characterized by x x1 ip2 p p1 ix2 where x1 p1 and x2 p2 are real and considered as canonical pairs the analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunctions for different systems it is noted that the imaginary part of the eigenvalue ei turns out to be zero for all potentials v x with real couplings whereas it turns out to be nonzero for the case when the couplings are complex the prescription is also extended to study the excited states the problems related to the normalization of the eigenfunction and the boundary conditions to be used within this framework are also discussed

self organized memories in coupled map lattices abstract spatially extended dynamical systems subjected to a periodic external input are able to store short term memories as a result of self organization this phenomenon has been used to interpret sliding charge density wave experiments and has been found in lattices of coupled maps and oscillators in this paper we aim to describe the influence of a stochastic perturbation in the self organized memories in a lattice of coupled linear and weakly nonlinear maps we consider the case in which the coupling between maps depends on the lattice distance as a power law

the contact problem in thermoviscoelastic materials abstract we stablish an existence result for the thermoviscoelastic degenerated contact problem the nonlinear stress strain relation we will consider here is given by m l 0 u x 2 dx u x u xt where m is a function satisfying m c 1 0 c 0 m s c s p moreover we show that the solution decays uniformly to zero that is denoting by e t the first order energy associated to the equation we show that there exist positive constants c satisfying e t c e 0 1 t p 2 p where p is a positive number which depends on nonlinear terms of the system

turing instability in oscillator chains with nonlocal coupling we investigate analytically and numerically the conditions for the turing instability to occur in a one dimensional chain of nonlinear oscillators coupled nonlocally in such a way that the coupling strength decreases with the spatial distance as a power law a range parameter makes it possible to cover the two limiting cases of local nearest neighbor and global all to all couplings we consider an example from a nonlinear autocatalytic reaction diffusion model

species survival and scaling laws in hostile and disordered environments in this work we study the likelihood of survival of single species in the context of hostile and disordered environments population dynamics in this environment as modeled by the fisher equation is characterized by negative average growth rate except in some random spatially distributed patches that may support life in particular we are interested in the phase diagram of the survival probability and in the critical size problem i e the minimum patch size required for surviving in the long time dynamics we propose a measure for the critical patch size as being proportional to the participation ratio of the eigenvector corresponding to the largest eigenvalue of the linearized fisher dynamics we obtain the extinction survival phase diagram and the probability distribution function pdf of the critical patch sizes for two topologies namely the one dimensional system and the fractal peano basin we show that both topologies share the same qualitative features but the fractal topology requires higher spatial fluctuations to guarantee species survival we perform a finite size scaling and we obtain the associated scaling exponents in addition we show that the pdf of the critical patch sizes has an universal shape for the 1d case in terms of the model parameters diffusion growth rate etc in contrast the diffusion coefficient has a drastic effect on the pdf of the critical patch sizes of the fractal peano basin and it does not obey the same scaling law of the 1d case

non hermitian localization and population biology the time evolution of spatial fluctuations in inhomogeneous d dimensional biological systems is analyzed a single species continuous growth model in which the population disperses via diffusion and convection is considered time independent environmental heterogeneities such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size using an analogy with a schr odinger equation subject to a constant imaginary vector potential we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states in the limit of high convection velocity the linearized growth problem in d dimensions exhibits singular scaling behavior described by a d ensuremath 1 dimensional generalization of the noisy burgers equation with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane the burgers mapping leads to unusual transverse spreading of convecting delocalized populations

smooth impulsive linear systems axiomatic description abstract paper concerns with linear systems having arbitrary singularity at infinity these systems deserve to study since they are limits of familiar classical linear systems an axiomatic description of such systems is obtained

spectral properties of random non self adjoint matrices and operators i describe some numerical experiments which determine the degree of spectral instability of medium sized randomly generated matrices which are far from self adjoint the conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size i also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the eigenvectors generate a dense linear subspace but the eigenvalues do not determine the spectrum my results imply that the spectrum of the non self adjoint anderson model changes suddenly on passing to the infinite volume limit

optical structures with local pt symmetry we propose a new class of optical synthetic materials that are described by non hermitian hamiltonians the building blocks of such systems are coupled symmetric elements dimers with coupling t despite the lack of global symmetry these systems have a robust parameter region of real spectra exact phase even in cases where the complex refractive index n i of each dimer is random the validity of our proposition is confirmed for representative cases where we calculate the borders of the exact phase in terms of and t

nonlocal competition and the speciation transition on random networks a simple model for competition induced speciation is presented and analyzed logistic growth with nonlocal interaction is studied on regular and random networks and the large scale structure of the emerging genomic frequencies is examined the neutrality assumption is violated if the network is random and the competition is nonlocal instead hubs in the sequence space are suppressed by the competition more than nodes of a lower degree thus speciation is unavoidable for large scale free networks the emerging genetic mixture depends strongly on the initial conditions the frequency of hubs is much greater in a population that evolved from a single nucleation event than in a population that recovered from a catastrophe

smooth impulsive linear systems controllability in this article a fairly comprehensive study of the concept of controllability for smooth impulsive linear systems is undertaken

on the solution and impulsive behaviour of polynomial matrix descriptions of free linear multivariable systems in this note we examine the solution and the impulsive behaviour of autonomous linear multivariable systems whose pseudo state beta t obeys a linear matrix differential equation a rho beta t 0 where a rho is a polynomial matrix in the differential operator rho d dt we thus generalize to the general polynomial matrix case some results obtained by verghese and colleagues which regard the impulsive behaviour of the generalized state vector x t of input free generalized state space systems

generic unfolding of the thermoelastic contact instability the classic model of a one dimensional thermoelastic rod suspended between a hot and cold wall is revisited in this model the rod is held in place at the cold end while at the hot end it is allowed to separate from or make contact with the wall when the model includes the contact and gap dependent thermal boundary condition known as the barber condition it serves to illustrate the well known thermoelastic contact instability all previous studies of this instability have focused upon the symmetric case where as a control parameter is varied the system undergoes a pitchfork bifurcation that is a new pair of linearly stable steady state solutions bifurcate symmetrically from a previously unique solution which has changed from stable to unstable here it is shown that this behavior is not generic rather for typical contact resistance functions a fold bifurcation is encountered this represents a generic unfolding of the classic pitchfork bifurcation and contains the pitchfork as a special case

anderson localization non linearity and stable genetic diversity in many models of genotypic evolution the vector of genotype populations satisfies a system of linear ordinary differential equations this system of equations models a competition between differential replication rates fitness and mutation mutation operates as a generalized diffusion process on genotype space in the large time asymptotics the replication term tends to produce a single dominant quasi species unless the mutation rate is too high in which case the asymptotic population becomes de localized we introduce a more macroscopic picture of genotypic evolution wherein a random fitness term in the linear model produces features analogous to anderson localization when coupled with density dependent non linearities which limit the population of any given genotype we obtain a model whose large time asymptotics display stable genotypic diversity

distribution of eigenvalues in non hermitian anderson models we develop a theory which describes the behavior of eigenvalues of a class of one dimensional random non hermitian operators introduced recently by hatano and nelson we prove that the eigenvalues are distributed along a curve in the complex plane an equation for the curve is derived and the density of complex eigenvalues is found in terms of spectral characteristics of a reference hermitian disordered system the generic properties of the eigenvalue distribution are discussed

free random variables and molecular spectra abstract we briefly review the method of free random variables and its relation to random matrices to demonstrate use of the approach we apply the formalism to study the effect of static disorder on the density of states in a model hamiltonian system

can spectral value sets of toeplitz band matrices jump abstract the spectral value set speb ca of a bounded linear operator a on l2 is the union of the spectra of all operators of the form a bkc where k

isolation effects in a system of two mutually communicating identical patches abstract starting from the fisher kolmogorov petrovskii piskunov equation fkpp we model the dynamic of a diffusive system with two mutually communicating identical patches and isolated of the remaining matrix for this system we find the minimal size of each fragment in the explicit form and compare with the explicit results for similar problems found in the literature from this comparison emerges an unexpected result that for a same set of the parameters the isolated system studied in this work with size l can be better or worst than the non isolated systems with the same size l uniquely depending on the parameter a 0 internal conditions of the patches due to the fact that this result is unexpected we propose an experimental verification

generalized swanson models and their solutions we analyse a class of non hermitian quadratic hamiltonians which are of the form where are real constants with and and are generalized creation and annihilation operators respectively thus these hamiltonians may be classified as generalized swanson models it is shown that the eigenenergies are real for a certain range of values of the parameters a similarity transformation mapping the non hermitian hamiltonian h to a hermitian one h is also obtained it is shown that h and h share identical energies as explicit examples the solutions of a couple of models based on the trigonometric rosen morse i and the hyperbolic rosen morse ii type potentials are obtained we also study the case where the non hermitian hamiltonian is symmetric

arbitrary nonlinearities in convective population dynamics with small diffusion convective counterparts of variants of the nonlinear fisher equation which describes reaction diffusion systems in population dynamics are studied with the help of an analytic prescription and shown to lead to interesting consequences for the evolution of population densities the initial value problem is solved explicitly for some cases and for others it is shown how to find traveling wave solutions analytically the effect of adding diffusion to the convective equations is first studied through exact analysis through a piecewise linear representation of the nonlinearity using an appropriate small parameter suggested by that analysis a perturbative treatment is developed to treat the case in which the convective evolution is augmented by a small amount of diffusion

global existence and decay for the semilinear thermoelastic contact problem in this paper a class of semilinear thermoelastic contact problems is considered and the existence and exponential decay of the weak solutions are obtained

when are linear differentiation invariant spaces differential it is shown that a linear differentiation invariant subspace of a c trajectory space is differential i e can be represented as the kernel of a linear constant coefficient differential operator if and only if its mcmillan degree is finite

existence and decay to contact problems for thermoviscoelastic plates we consider the dynamic contact problem for thermoviscoelastic plates we show the existence of weak solutions as well as exponential decay of the solution

impulsive smooth behavior in multimode systems part ii minimality and equivalence this is the second part of a two part study of linear multimode systems in the first part it was argued that the behavior of such a system on an interval between switches should be described in a framework that allows for impulses at the switching instant and both first order and polynomial representations were introduced that satisfy this requirement here we determine the conditions under which first order representations are minimal we also show how two minimal representations of the same behavior are related this leads in particular to an appropriate state space isomorphism theorem the minimality conditions are given a dynamic interpretation

singular state models and singular ltid systems in this paper we offer a systematic development of state representation theory for singular linear systems in the behavioural setting of willems

impulsive smooth behavior in multimode systems part i state space and polynomial representations abstract a switched or multimode system is one that can switch between various modes of operation we consider here switched systems in which the modes of operation are characterized as linear finite dimensional systems not necessarily all of the same mcmillan degree when a switch occurs from one of the modes to another of lower mcmillan degree the state space collapses and an impulse may result followed by a smooth evolution under the new regime this paper is concerned with the description of such impulsive smooth behavior on a typical interval we propose an algebraic framework modeled on the class of impulsive smooth distributions as defined by hautus both state space and polynomial representations are considered and we discuss transformations between the two forms

bounding biomass in the fisher equation the fisher kolmogorov petrovskii piskunov equation with a variable growth rate and advection by an incompressible velocity field is considered as a model for plankton dispersed by ocean currents if the average growth rate is negative then the model has a survival extinction transition the location of this transition in the parameter space is constrained using variational arguments and delimited by simulations the statistical steady state reached when the system is in the survival region of parameter space is characterized by integral constraints and upper and lower bounds on the biomass and productivity that follow from variational arguments and direct inequalities in the limit of zero decorrelation time the velocity field is shown to act as fickian diffusion with an eddy diffusivity much larger than the molecular diffusivity this allows a one dimensional model to predict the biomass productivity and extinction transitions all results are illustrated with a simple growth and stirring model

coburn s lemma and the finite section method for random jacobi operators abstract we study the spectra and pseudospectra of semi infinite and bi infinite tridiagonal random matrices and their finite principal submatrices in the case where each of the three diagonals varies over a separate compact set say u v w c such matrices are sometimes termed stochastic toeplitz matrices a in the semi infinite case and stochastic laurent matrices a in the bi infinite case their spectra spec a and spec a are independent of a and a as long as a and a are pseudoergodic in the sense of davies 2001 20 which holds almost surely in the random case this was shown in davies 2001 20 for a that the same holds for a is one main result of this paper although the computation of and in terms of u v and w is intrinsically difficult we give upper and lower spectral bounds and we explicitly compute a set g that fills the gap between and in the sense that g we also show that the invertibility of one and hence all operators a implies the invertibility and uniform boundedness of the inverses of all finite tridiagonal square matrices with diagonals varying over u v and w this implies that the so called finite section method for the approximate solution of a system a x b is applicable as soon as a is invertible and that the finite section method for estimating the spectrum of a does not suffer from spectral pollution both results illustrate that tridiagonal stochastic toeplitz operators share important properties of classical toeplitz operators indeed one of our main tools is a new stochastic version of the coburn lemma for classical toeplitz operators saying that a stochastic tridiagonal toeplitz operator if fredholm is always injective or surjective in the final part of the paper we bound and compare the norms and the norms of inverses of bi infinite semi infinite and finite tridiagonal matrices over u v and w this in particular allows the study of the resolvent norms and hence the pseudospectra of these operators and matrices

applicability of the fisher equation to bacterial population dynamics the applicability of the fisher equation which combines diffusion with logistic nonlinearity to population dynamics of bacterial colonies is studied with the help of explicit analytic solutions for the spatial distribution of a stationary bacterial population under a static mask the mask protects bacteria from ultraviolet light the solution which is in terms of jacobian elliptic functions is used to provide a practical prescription to extract fisher equation parameters from observations and to decide on the validity of the fisher equation

on the eigenproblems of pt symmetric oscillators we consider the non hermitian hamiltonian h d2 dx2 p x2 ix 2n 1 on the real line where p x is a polynomial of degree at most n 1 with all non negative real coefficients possibly p 0 it is proved that the eigenvalues must be in the sector arg 2n 3 also for the cubic case h d2 dx2 ix 3 we establish a zero free region of the eigenfunction u and its derivative u and we find some other interesting properties of eigenfunctions

the diffusion reaction d r hamiltonian and the solutions of certain types of linear and nonlinear d r equations in one dimension with a view to having further insight into the mathematical content of the non hermitian hamiltonian associated with the diffusion reaction d r equation in one dimension we investigate a the solitary wave solutions of certain types of its nonlinear versions and b the problem of real eigenvalue spectrum associated with its linear version or with this class of non hermitian hamiltonians for case a we use the standard techniques to handle the quadratic and cubic nonlinearities in the d r equation whereas for case b a newly proposed method based on an extended complex phase space is employed for a particular class of solutions an ermakov system of equations is also found for the linear case further corresponding to the classical version of the above one dimensional complex hamiltonian an equivalent integrable system of two two dimensional real hamiltonians is suggested

disordered contact process with asymmetric spreading an asymmetric variant of the contact process where the activity spreads with different and independent random rates to the left and to the right is introduced a real space renormalization scheme is formulated for the model by means of which it is shown that the local asymmetry of spreading is irrelevant on large scales if the model is globally statistically symmetric otherwise in the presence of a global bias in either direction the renormalization method predicts two distinct phase transitions which are related to the spreading of activity in and against the direction of the bias the latter is found to be described by an infinite randomness fixed point while the former is not

a pseudo hermitian β hermite family of matrices abstract a family of pseudo hermitian random matrices is constructed whose eigenvalues are the same of the hermite ensemble of random matrix theory it is shown that matrices of this family are unstable in such a way that small perturbations cause the eigenvalues to move into the complex plane in conjugate pairs

a quasistatic contact problem in thermoelasticity with a radiation condition for the temperature abstract an abstract existence theorem is proved for variational inequalities of monotone operators with a nonlinear perturbation it is used to prove the existence of a strong solution to the problem of quasistatic one dimensional contact in linearized thermoelasticity the problem after reformulation consists of the heat equation with a nonlocal and nonlinear term the uniqueness and stability of the solution are also established