a polynomial time approximation algorithm for the permanent of a matrix with nonnegative entries we present a polynomial time randomized algorithm for estimating the permanent of an arbitrary n n matrix with nonnegative entries this algorithm technically a fully polynomial randomized approximation scheme computes an approximation that is with high probability within arbitrarily small specified relative error of the true value of the permanent

symmetry and phaselocking in chains of weakly coupled oscillators weakly coupled chains of oscillators with nearest neighbor interactions are analyzed for phaselocked solutions it is shown that the symmetry properties of the coupling affect the qualitative form of the phaselocked solutions and the scaling behavior of the system as the number of oscillators grows without bound it is also shown that qualitative behavior of these solutions depends on whether the coupling is diffusive or i synaptic terms defined in the paper the methods include the demonstration that the equations for phaselocked solutions can be approximated by a singularly perturbed two point continuum boundary value problem that is easier to analyze the issue of convergence of the phaselocked solutions to solutions of the continuum equation is closely related to questions involving numerical entropy in computation schemes for a conservation law an application to the neurophysiology of motor behavior is discussed briefly

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

pseudo hermiticity versus pt symmetry iii equivalence of pseudo hermiticity and the presence of antilinear symmetries we show that a diagonalizable non hermitian hamiltonian h is pseudo hermitian if and only if it has an antilinear symmetry i e a symmetry generated by an invertible antilinear operator this implies that the eigenvalues of h are real or come in complex conjugate pairs if and only if h possesses such a symmetry in particular the reality of the spectrum of h implies the presence of an antilinear symmetry we further show that the spectrum of h is real if and only if there is a positive definite inner product on the hilbert space with respect to which h is hermitian or alternatively there is a pseudo canonical transformation of the hilbert space that maps h into a hermitian operator

synchronization in a system of globally coupled oscillators with time delay we study the synchronization phenomena in a system of globally coupled oscillators with time delay in the coupling the self consistency equations for the order parameter are derived which depend explicitly on the amount of delay analysis of these equations reveals that the system in general exhibits discontinuous transitions in addition to the usual continuous transition between the incoherent state and a multitude of coherent states with different synchronization frequencies in particular the phase diagram is obtained on the plane of the coupling strength and the delay time and ubiquity of multistability as well as suppression of the synchronization frequency is manifested numerical simulations are also performed to give consistent results

associative memory storing an extensive number of patterns based on a network of oscillators with distributed natural frequencies in the presence of external white noise we study associative memory based on temporal coding in which successful retrieval is realized as an entrainment in a network of simple phase oscillators with distributed natural frequencies under the influence of white noise the memory patterns are assumed to be given by uniformly distributed random numbers on 0 2 ensuremath pi so that the patterns encode the phase differences of the oscillators to derive the macroscopic order parameter equations for the network with an extensive number of stored patterns we introduce an effective transfer function by assuming a fixed point equation of the form of the thouless anderson palmer equation which describes the time averaged output as a function of the effective time averaged local field properties of the networks associated with synchronization phenomena for a discrete symmetric natural frequency distribution with three frequency components are studied based on the order parameter equations and are shown to be in good agreement with the results of numerical simulations two types of retrieval states are found to occur with respect to the degree of synchronization when the size of the width of the natural frequency distribution is changed

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 quantum cellular automata to quantum lattice gases a natural architecture for nanoscale quantum computation is that of a quantum cellular automaton motivated by this observation we begin an investigation of exactly unitary cellular automata after proving that there can be no nontrivial homogeneous local unitary scalar cellular automaton in one dimension we weaken the homogeneity condition and show that there are nontrivial exactly unitary partitioning cellular automata we find a one parameter family of evolution rules which are best interpreted as those for a one particle quantum automaton this model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the dirac equation we describe two generalizations of this automaton the second of which to multiple interacting particles is the correct definition of a quantum lattice gas

ito formula for one dimensional continuous time quantum random walk abstract in the paper we first extend the discrete time quantum random walk dqrw to continuous time quantum random walk cqrw then we establish an ito formula for the one dimensional cqrw as an application we present tanaka s formula with the ito formula and characterize the relation between quantum local time l t a and dirac delta function finally we discuss quantum stochastic calculus for the cqrw

a network of oscillators a model of neuronal behaviour capable of accounting for the oscillatory plateau and rebound properties of biological neurons is derived discussed and analysed the model is based on a piecewise linear form of the fitzhugh nagumo equations but reduces to a set of maps very similar to those of the hopfield model 1982 in particular the binary description of individual neurons and the well studied form of the synaptic current sigma jijsj are preserved although the model is capable of reproducing behaviours on the slow timescales characteristic of plateau and oscillation by coupling two model cells together a mutually inhibitory or half centred oscillator and an oscillator fixed phase follower pairs are constructed

period doubling bifurcations chaos phase locking and devil s staircase in a bonhoeffer van der pol oscillator abstract the effect of a periodic input current a 1 cos t in the bonhoeffer van der pol oscillator along with a bias a 0 is investigated numerically as the parameter a 1 is varied in the absence of bias by holding the other parameters at constant values typical period doubling bifurcation sequence is found to occur leading to chaotic motion in agreement with the feigenbaum scenario when the bias is switched on at the transition to chaos frequency locking is observed in the system the frequency locked intervals exhibit complete devil s staircase similar to the one observed in belousov zhabotinsky reaction

absorption problems for quantum walks in one dimension this paper treats absorption problems for the one dimensional quantum walk determined by a 2 2 unitary matrix u on a state space 0 1 n where n is finite or infinite by using a new path integral approach based on an orthonormal basis p q r and s of the vector space of complex 2 2 matrices our method studied here is a natural extension of the approach in the classical random walk

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

the failure distribution in percolation models of breakdown the probability of finding a largest defect cluster of size n in a percolation network is calculated analytically using a new distribution function scaling equation from this result and the stress voltage enhancement at the tip of the most critical defect in the network the probability of failure in percolation models of breakdown is calculated for defect fractions less than the percolation point this distribution is found to be of the form exponential of an exponential numerical simulations on the two dimensional random fuse network confirm the new distribution function and convincingly distinguish between it and the weibull 1951 form most often used in the fitting of breakdown data

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

conjecture on the interlacing of zeros in complex sturm liouville problems the zeros of the eigenfunctions of self adjoint sturm liouville eigenvalue problems interlace for these problems interlacing is crucial for completeness for the complex sturm liouville problem associated with the schrodinger equation for a non hermitian pt symmetric hamiltonian completeness and interlacing of zeros have never been examined this paper reports a numerical study of the sturm liouville problems for three complex potentials the large n limit of a ix n potential a quasiexactly solvable x4 potential and an ix3 potential in all cases the complex zeros of the eigenfunctions exhibit a similar pattern of interlacing and it is conjectured that this pattern is universal understanding this pattern could provide insight into whether the eigenfunctions of complex sturm liouville problems form a complete set

the potential iz m generates real eigenvalues only under symmetric rapid decay boundary conditions we consider the non hermitian eigenvalue problems u z iz mu z u z m 3 under every rapid decay boundary condition that is symmetric with respect to the imaginary axis in the complex z plane we prove that the eigenvalues are all real and positive

bifurcations and chaos of the bonhoeffer van der pol model periodic and chaotic behaviour of the bonhoeffer van der pol model of a nerve membrane driven by a periodic stimulating current a1 cos omega t is investigated results show that there exist ordinary and reversed period doubling cascades and a mode locking state at low driving amplitudes a1 there are period doubling and chaotic states but no impulse solutions when a1 is larger than a0 0 749 there are chaotic reversed period doubling and mode locking states and there also exist impulse trains a mode locking state with period 4 over a very large range of amplitudes is also found at a1 1 7059 the system goes back to a one period state

finite dimensional pt symmetric hamiltonians this paper investigates finite dimensional pt symmetric hamiltonians it is shown here that there are two ways to extend real symmetric hamiltonians into the complex domain i the usual approach is to generalize such hamiltonians to include complex hermitian hamiltonians ii alternatively one can generalize real symmetric hamiltonians to include complex pt symmetric hamiltonians in the first approach the spectrum remains real while in the second approach the spectrum remains real if the pt symmetry is not broken both generalizations give a consistent theory of quantum mechanics but if d 2 a d dimensional hermitian matrix hamiltonian has more arbitrary parameters than a d dimensional pt symmetric matrix hamiltonian

solvability and pt symmetry in a double well model with point interactions the concept of point interactions offers one of the most suitable guides towards a quantitative analysis of properties of certain specific non hermitian so called symmetric quantum mechanical systems this is illustrated with a double well model the easy solvability of which is shown to lead to a clear picture of the mechanisms of the unavoided level crossing and of the spontaneous symmetry breaking at a certain strong non hermiticity boundary below this limit the model is shown to be suitable for an explicit illustration of technicalities related to the standard probabilistic physical interpretation of bound states in symmetric quantum mechanics

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

the t 2 law for quantum random walks on the line starting in the classical state based on the theory of unitary matrices our treatment of the theory of quantum random walks simplifies and clarifies certain prior derivations based on fourier transform methods given a quantum random walk on the line determined by a 2 2 unitary matrix u we show how the first two moments of the position probability distribution are determined by the eigenvalues of u by varying the coin operator a we show that the leading term of the standard deviation of the position probability distribution is ct where t denotes time and 0 c 1 however it turns out that the maximum value of c namely c 1 is achievable when and only when the coin operator a is diagonal and the initial state is unbiased starting in the classical state 0 1 our approach confirms that the maximum value of the leading term of the standard deviation of the position probability distribution is which by way of known examples is verified to be achievable

analysis of oscillator neural network without lyapunov function a new oscillator neural network is proposed on the basis of a system of coupled complex ginzburg landau equations uchiyama s and fujisaka h 1997 phys rev e 56 99 it is shown that this network can retrieve fixed point attractors and has an excellent mathematical tractability just as the hopfield model although the system has no lyapunov function we use self consistent signal to noise analysis instead of the replica method to investigate the storage capacity and demonstrate the usefulness of the present approach to the neural network exhibiting complex dynamical behaviours

on the reality of the eigenvalues for a class of symmetric oscillators we study the eigenvalue problem with the boundary conditions that decays to zero as z tends to infinity along the rays where is a real polynomial and we prove that if for some we have for all then the eigenvalues are all positive real we then sharpen this to a larger class of polynomial potentials

coupled channel version of the pt symmetric square well a coupled pair of pt symmetric square wells is studied as a prototype of a quantum system characterized by two manifestly non hermitian commuting observables via the diagonalization of our hamiltonian h h and spin like observable we demonstrate that there exists a domain of couplings where both the respective sets of eigenvalues en n 0 1 energies and 1 spin projections remain real in such a measurable regime the model acquires a consistent probabilistic interpretation mediated by our selection of one of many available interaction dependent scalar products

a random polynomial time algorithm for approximating the volume of convex bodies a randomized polynomial time algorithm for approximating the volume of a convex body k in n dimensional euclidean space is presented the proof of correctness of the algorithm relies on recent theory of rapidly mixing markov chains and isoperimetric inequalities to show that a certain random walk can be used to sample nearly uniformly from within k

distributional borel summability of odd anharmonic oscillators it is proved that the divergent rayleigh schrodinger perturbation expansions for the eigenvalues of any odd anharmonic oscillator are borel summable in the distributional sense to the resonances naturally associated with the system

eigenvalues of pt symmetric oscillators with polynomial potentials we study the eigenvalue problem u z iz m pm 1 iz u z u z with the boundary condition that u z decays to zero as z tends to infinity along the rays in the complex plane where pm 1 z a1zm 1 a2zm 2 am 1z is a polynomial and integers m 3 we provide an asymptotic expansion of the eigenvalues n as n and prove that for each real polynomial pm 1 the eigenvalues are all real and positive with only finitely many exceptions

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

learning algorithms with optimal stability in neural networks to ensure large basins of attraction in spin glass like neural networks of two state elements xi imu or 1 the authors propose to study learning rules with optimal stability delta where delta is the largest number satisfying delta or sigma j jij xi jmu xi imu mu 1 p i 1 n where n is the number of neurons and p is the number of patterns they motivate this proposal and provide optimal stability learning rules for two different choices of normalisation for the synaptic matrix jij in addition numerical work is presented which gives the value of the optimal stability for random uncorrelated patterns

three routes to the exact asymptotics for the one dimensional quantum walk we demonstrate an alternative method for calculating the asymptotic behaviour of the discrete one coin quantum walk on the infinite line via the jacobi polynomials that arise in the path integral representation we calculate the asymptotics using a method that is significantly easier to use than the darboux method it also provides a single integral representation for the wavefunction that works over the full range of positions n including throughout the transitional range where the behaviour changes from oscillatory to exponential previous analyses of this system have run into difficulties in the transitional range because the approximations on which they were based break down here the fact that there are two different kinds of approach to this problem path integral versus schrodinger wave mechanics is ultimately a manifestation of the equivalence between the path integral formulation of quantum mechanics and the original formulation developed in the 1920s we also discuss how and why our approach is related to the two methods that have already been used to analyse these systems

space of state vectors in symmetric quantum mechanics the space of states of symmetric quantum mechanics is examined the requirement that eigenstates with different eigenvalues must be orthogonal leads to the conclusion that eigenfunctions belong to a space with an indefinite metric self consistent expressions for the probability amplitude and the average value of operator are suggested further specification of the space of state vectors yields a superselection rule redefining the notion of the superposition principle an expression for the probability current density satisfying the equation of continuity and vanishing for the bound state is proposed

some properties of eigenvalues and eigenfunctions of the cubic oscillator with imaginary coupling constant comparison between the exact value of the spectral zeta function zh 1 5 6 5 3 2cos 5 2 1 5 3 5 and the results of numeric and wkb calculations supports the conjecture by daniel bessis 1995 private communication that all the eigenvalues of this pt invariant hamiltonian are real for one dimensional schrodinger operators with complex potentials having a monotonic imaginary part the eigenfunctions and the imaginary parts of their logarithmic derivatives have no real zeros

weak limits for quantum random walks we formulate and prove a general weak limit theorem for quantum random walks in one and more dimensions with x n denoting position at time n we show that x n n converges weakly as stackrel ensuremath rightarrow n ensuremath infty to a certain distribution which is absolutely continuous and of bounded support the proof is rigorous and makes use of fourier transform methods this approach simplifies and extends certain preceding derivations valid in one dimension that make use of combinatorial and path integral methods

a moment based approach to the dynamical solution of the kuramoto model we examine the dynamics of the kuramoto model with a new analytical approach by defining an appropriate set of moments the dynamical equations can be exactly closed we discuss some applications of the formalism such as the existence of an effective hamiltonian for the dynamics we also show how this approach can be used to numerically investigate the dynamical behaviour of the model without finite size effects

a path integral formula with applications to quantum random walks in z d we consider general quantum random walks in a d dimensional half space we first obtain a path integral formula for general quantum random walks in a d dimensional space our path integral formula is valid for general quantum random walks on cayley graphs as well then the path integral formula is applied to obtain the scaling limit of the exit distribution the expectation of exit time and the asymptotic behaviour of the exit probabilities for general quantum random walks in a half space under some conditions on amplitude functions the conditions are shown to be satisfied by both the hadamard and grover quantum random walks in two dimensional half spaces for the two dimensional case we show that the critical exponent for the scaling limit of the hitting distribution is 1 as the lattice spacing tends to zero i e the natural magnitude of the hitting position is of order o 1 if the lattice spacing is set to be 1 n we also show that the rate of convergence of the total hitting probability has lower bound n 2 and upper bound n 2 for any 0 for a quantum random walk with a fixed starting point we show that the probability of hitting times at the hyperplane decays faster than that of the classical random walk in both one and two dimensions given the event of a hit the conditional expectation of hitting times is finite in contrast to being infinite for the classical case in the one dimensional case we also obtain an exact order of the probability distribution of the hitting time at 0

temporal association in a network of neuronal oscillators we investigate temporal association in a set of globally coupled neuronal oscillators which retrieves time sequences of stored patterns the asymmetry to symmetry ratio in the synaptic coupling as well as the total strength turns out to play a major role in these phenomena by means of numerical simulations we identify four different states the memory retrieval state the temporal association state the mixed memory state and the no memory state and obtain the corresponding phase boundaries on the plane of the asymmetry ratio and the coupling strength

the operator p2 ix ν on l2 r reply to comment by bender and wang the pt invariant operator h p2 ix defined on l2 r has a hilbert schmidt resolvent for real 1 which is trace class if 2 is not an integer divisible by 4 for 2 4 the spectrum of h coincides with the eigenvalues of the problem posed by bender and boettcher on a contour in the lower complex half plane we give the value of the spectral zeta function z 1 h and investigate the limiting case

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

calculation of the hidden symmetry operator in symmetric quantum mechanics in a recent paper it was shown that if a hamiltonian h has an unbroken symmetry then it also possesses a hidden symmetry represented by the linear operator the operator commutes with both h and the inner product with respect to is associated with a positive norm and the quantum theory built on the associated hilbert space is unitary in this paper it is shown how to construct the operator for the non hermitian symmetric hamiltonian h 1 2p2 1 2x2 ix3 using perturbative techniques it is also shown how to construct the operator for h 1 2p2 1 2x2 x4 using nonperturbative methods

extension of a spectral bounding method to the pt invariant states of the ix n non hermitian potential the eigenvalue moment method emm developed by handy j phys a math gen 2001 34 l271 j phys a math gen 2001 34 5065 handy et al j phys a math gen 2001 34 5593 and handy and xiao qian wang j phys a math gen 2001 34 8297 which generates converging lower and upper bounds to the complex discrete state energies is extended to the case of discrete states with non real support in particular bender and boettcher phys rev lett 1998 80 5243 have argued on the reality of the discrete state spectrum for the ix n potential for n integer 4 such pt invariant solutions can only exist on appropriate complex contours we develop and apply the necessary emm formalism to such cases in particular the restriction of emm to the anti stokes angles significantly increases the convergence rate of the bounds

inertia effects on periodic synchronization in a system of coupled oscillators we study analytically the synchronization phenomena in a set of globally coupled oscillators under external periodic driving with emphasis on the effects of small inertia we examine in detail both the integer and fractional mode locking present in the system with inertia and derive the self consistency equation for the order parameter which reveals variation of the magnitude of the order parameter according to the external periodic driving in particular it is found that the inertia induces discontinuous transitions between the coherent and incoherent states