analytical solutions to a class of nonlinear schrodinger equations with cal pt like potentials we present closed form solutions to a certain class of one and two dimensional nonlinear schr odinger equations involving potentials with broken and unbroken pt symmetry in the one dimensional case these solutions are given in terms of jacobi elliptic functions hyperbolic and trigonometric functions some of these solutions are possible even when the corresponding pt symmetric potentials have a zero threshold in two dimensions hyperbolic secant type solutions are obtained for a nonlinear schr odinger equation with a nonseparable complex potential

an object oriented and quadrilateral mesh based solution adaptive algorithm for compressible multi fluid flows in this paper an object oriented and quadrilateral mesh based solution adaptive algorithm for the simulation of compressible multi fluid flows is presented the hllc scheme harten lax and van leer approximate riemann solver with the contact wave restored is extended to adaptively solve the compressible multi fluid flows under complex geometry on unstructured mesh it is also extended to the second order of accuracy by using muscl extrapolation the node edge and cell are arranged in such an object oriented manner that each of them inherits from a basic object a home made double link list is designed to manage these objects so that the inserting of new objects and removing of the existing objects nodes edges and cells are independent of the number of objects and only of the complexity of o 1 in addition the cells with different levels are further stored in different lists this avoids the recursive calculation of solution of mother non leaf cells thus high efficiency is obtained due to these features besides as compared to other cell edge adaptive methods the separation of nodes would reduce the memory requirement of redundant nodes especially in the cases where the level number is large or the space dimension is three five two dimensional examples are used to examine its performance these examples include vortex evolution problem interface only problem under structured mesh and unstructured mesh bubble explosion under the water bubble shock interaction and shock interface interaction inside the cylindrical vessel numerical results indicate that there is no oscillation of pressure and velocity across the interface and it is feasible to apply it to solve compressible multi fluid flows with large density ratio 1000 and strong shock wave the pressure ratio is 10 000 interaction with the interface

finite volume treatment of dispersion relation preserving and optimized prefactored compact schemes for wave propagation in developing suitable numerical techniques for computational aero acoustics the dispersion relation preserving drp scheme by tam and co workers and the optimized prefactored compact opc scheme by ashcroft and zhang have shown desirable properties of reducing both dissipative and dispersive errors these schemes originally based on the finite difference attempt to optimize the coefficients for better resolution of short waves with respect to the computational grid while maintaining pre determined formal orders of accuracy in the present study finite volume formulations of both schemes are presented to better handle the nonlinearity and complex geometry encountered in many engineering applications linear and nonlinear wave equations with and without viscous dissipation have been adopted as the test problems highlighting the principal characteristics of the schemes and utilizing linear and nonlinear wave equations with different wavelengths as the test cases the performance of these approaches is documented for the linear wave equation there is no major difference between the drp and opc schemes for the nonlinear wave equations the finite volume version of both drp and opc schemes offers substantially better solutions in regions of high gradient or discontinuity

quantum graphs a simple model for chaotic scattering we connect quantum graphs with infinite leads and turn them into scattering systems we show that they display all the features which characterize quantum scattering systems with an underlying classical chaotic dynamics typical poles delay time and conductance distributions ericson fluctuations and when considered statistically the ensemble of scattering matrices reproduces quite well the predictions of the appropriately defined random matrix ensembles the underlying classical dynamics can be defined and it provides important parameters which are needed for the quantum theory in particular we derive exact expressions for the scattering matrix and an exact trace formula for the density of resonances in terms of classical orbits analogous to the semiclassical theory of chaotic scattering we use this in order to investigate the origin of the connection between random matrix theory and the underlying classical chaotic dynamics being an exact theory and due to its relative simplicity it offers new insights into this problem which is at the forefront of the research in chaotic scattering and related fields

an h r adaptive approximate riemann solver for the euler equations in two dimensions a new adaptive strategy for solving the euler equations of compressible flow is presented the method of roe j comput phys 43 1981 pp 357 372 is extended into two dimensions for an arbitr

complexity of equivalence problems for concurrent systems of finite agents abstract a concurrent system of synchronous communicating agents is assembled from simpler sequential agents by parallel composition and hiding for example hide a1 alin p1 p2 pn describes the system of communicating agentsp1 pnin which the communication eventsa1 alare hidden consider descriptions of two systemspandqof synchronously communicating finite state agents assume that one wants to check whetherp qfor one of the commonly used equivalence we show that this question is pspace hard for all equivalences that lie between strong bisimulation and trace equivalences for some equivalences exponential lower and upper bounds are proven we also show that this problem is np hard and co np hard even for a class of very simple finite agents

the complexity of counting cuts and of computing the probability that a graph is connected several enumeration and reliability problems are shown to be p complete and hence at least as hard as np complete problems included are important problems in network reliability analysis namely computing the probability that a graph is connected and counting the number of minimum cardinality s t cuts or directed network cuts also shown to be p complete are counting vertex covers in a bipartite graph counting antichains in a partial order and approximating the probability that a graph is connected and the probability that a pair of vertices is connected

transition from localized to extended eigenstates in the ensemble of power law random banded matrices we study statistical properties of the ensemble of large n times n random matrices whose entries h ij decrease in a power law fashion h ij sim i j alpha mapping the problem onto a nonlinear sigma model with non local interaction we find a transition from localized to extended states at alpha 1 at this critical value of alpha the system exhibits multifractality and spectral statistics intermediate between the wigner dyson and poisson one these features are reminiscent of those typical for the mobility edge of disordered conductors we find a continuous set of critical theories at alpha 1 parametrized by the value of the coupling constant of the sigma model at alpha 1 all states are expected to be localized with integrable power law tails at the same time for 1 alpha 3 2 the wave packet spreading at short time scale is superdiffusive langle r rangle sim t frac 1 2 alpha 1 which leads to a modification of the altshuler shklovskii behavior of the spectral correlation function at 1 2 alpha 1 the statistical properties of eigenstates are similar to those in a metallic sample in d alpha 1 2 1 dimensions finally the region alpha 1 2 is equivalent to the corresponding gaussian ensemble of random matrices alpha 0 the theoretical predictions are compared with results of numerical simulations

optical lattices with pt symmetry are not transparent in diffraction of a plane wave by a non hermitian pt symmetric optical lattice the sum of the bragg beam intensities need not be conserved even though the gain and loss are equally distributed the evolution is not unitary instead different sums are conserved in which the intensities are weighted with real numbers positive or negative several such sum rules are derived two beam diffraction from a refractive index of the form constant a cos x ib sin x is studied in detail the sum rule depends on the balance between the real hermitian parameter a and the real anti hermitian parameter b

scattering reflection and impedance of waves in chaotic and disordered systems with absorption we review recent progress in analysing wave scattering in systems with both intrinsic chaos and or disorder and internal losses when the scattering matrix is no longer unitary by mapping the problem onto a nonlinear supersymmetric model we are able to derive closed form analytic expressions for the distribution of reflection probability in a generic disordered system one of the most important properties resulting from such an analysis is statistical independence between the phase and the modulus of the reflection amplitude in every perfectly open channel the developed theory has far reaching consequences for many quantities of interest including local green functions and time delays in particular we point out the role played by absorption as a sensitive indicator of mechanisms behind the anderson localization transition we also provide a random matrix based analysis of s matrix and impedance correlations for various symmetry classes as well as the distribution of transmitted power for systems with broken time reversal invariance completing previous works on the subject the results can be applied in particular to the experimentally accessible impedance and reflection in a microwave or an ultrasonic cavity attached to a system of antennas

complex mathcal pt symmetric extensions of the nonlinear ultra short light pulse model the short pulse equation is symmetric which arises in nonlinear optics for the ultra short pulse case we present a family of new complex symmetric extensions of the short pulse equation based on the complex symmetric extension principle some properties of these equations with some chosen parameters are studied including the hamiltonian structures and exact solutions such as solitary wave solutions doubly periodic wave solutions and compacton solutions our results may be useful to understand complex symmetric nonlinear physical models this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

statistics of resonances and delay times in random media beyond random matrix theory we review recent developments in quantum scattering from mesoscopic systems various spatial geometries whose closed analogues show diffusive localized or critical behaviour are considered these are the features that cannot be described by the universal random matrix theory results instead one has to go beyond this approximation and incorporate them in a non perturbative way here we pay particular attention to the traces of these non universal characteristics in the distribution of the wigner delay times and resonance widths the former quantity captures time dependent aspects of quantum scattering while the latter is associated with the poles of the scattering matrix

conservative smoothing on an adaptive quadrilateral grid the lax wendroff scheme can be freed of spurious oscillations by introducing conservative smoothing in this paper the approach is first tested in 1 d modeling equations and then extended to multidimensional flows by the finite volume method the scheme is discretized by a space splitting method on an adaptive quadrilateral grid the artificial viscosity coefficients in the conservative smoothing step are specially designed to capture slipstreams and vortices algorithms are programmed using a vectorizable data structure under which not only the flow solver but also the adaptation procedure is well vectorized the good resolution and high efficiency of the approach are demonstrated in calculating both unsteady and steady compressible flows with either weak or strong shock waves

a box scheme for transcritical flow the accurate computer simulation of river and pipe flow is of great importance in the design of urban drainage networks the use of implicit numerical schemes allows the time step to be chosen on the basis of accuracy rather than stability offering a potential computational saving over explicit methods the highly successful box scheme is an implicit method which can be used to model a wide range of subcritical and supercritical flows however care must be taken over the modelling of transcritical flows since unless the correct internal boundary conditions are imposed the scheme becomes unstable the necessity of accurately tracking all the critical interfaces and treating them accordingly can be algorithmically complex and in practice the underlying mathematical model is often modified to ensure that the flow remains essentially subcritical such a modification however inevitably leads to additional errors and incorrect qualitative behaviour can be observed in this paper we show how the technique of residual distribution can be successfully implemented in order to accurately model unsteady transcritical flow without the need to know a priori which regions of the computational domain correspond to subcritical and supercritical flow when used in conjunction with a form of artificial smoothing the resulting method generates very high resolution results even for transcritical problems involving shocks as can be seen in the numerical results

elements of a theory of simulation ii sequential dynamical systems we study a class of discrete dynamical systems that is motivated by the generic structure of simulations the systems consist of the following data a a finite graph y with vertex set 1 n where each vertex has a binary state b functions f i f 2 n f 2 n and c an update ordering p the functions f i update the binary state of vertex i as a function of the state of vertex i and its y neighbors and leave the states of all other vertices fixed the update ordering is a permutation of the y vertices by composing the functions f i in the order given by p one obtains the sequential dynamical system sds f y p email protected i 1nf p i f 2 n f 2 n we derive a decomposition result characterize invertible sds and study fixed points in particular we analyse how many different sds that can be obtained by reordering a given multiset of update functions and give a criterion for when one can derive concentration results on this number finally some specific sds are investigated

localization problem in one dimension mapping and escape a one dimensional schr odinger equation in a discontinuous quasiperiodic potential is reduced to a recursion relation for transfer matrices and then to one for traces of these matrices when the potential is periodic the bandwidth goes to zero as an algebraic function of the period with a critical index which depends upon the potential strength this critical index is also evaluated as the solution to an escape rate problem for the recursion relations

classical wave experiments on chaotic scattering we review recent research on the transport properties of classical waves through chaotic systems with special emphasis on microwaves and sound waves inasmuch as these experiments use antennas or transducers to couple waves into or out of the systems scattering theory has to be applied for a quantitative interpretation of the measurements most experiments concentrate on tests of predictions from random matrix theory and the random plane wave approximation in all studied examples a quantitative agreement between experiment and theory is achieved to this end it is necessary however to take absorption and imperfect coupling into account concepts that were ignored in most previous theoretical investigations classical phase space signatures of scattering are being examined in a small number of experiments

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

non hermitian random matrix theory summation of planar diagrams the single ring theorem and the disc annulus phase transition i review aspects of work done in collaboration with a zee and r scalettar 1997 nucl phys b 504 579 1997 nucl phys b 501 643 2001 j math phys 42 5718 on complex non hermitian random matrices i open by explaining why the bag of tools used regularly in analysing hermitian random matrices cannot be applied directly to analyse non hermitian matrices and then introduce the method of hermitization which solves this problem then for rotationally invariant ensembles i derive a master equation for the average density of eigenvalues in the complex plane in the limit of infinitely large matrices this is achieved by resumming all the planar diagrams which appear in the perturbative expansion of the hermitized green function remarkably this resummation can be carried out explicitly for any rotationally invariant ensemble i prove that in the limit of infinitely large matrices the shape of the eigenvalue distribution is either a disc or an annulus this is the celebrated single ring theorem which of these shapes is realized is determined by the parameters coupling constants which determine the ensemble by varying these parameters a phase transition may occur between the two possible shapes i briefly discuss the universal features of this transition as the analysis of this problem relies heavily on summation of planar feynman diagrams i make a special effort at presenting a pedagogical exposition of the diagrammatic method which some readers may find useful

complexity of generalized satisfiability counting problems abstract the class of generalized satisfiability problems introduced in 1978 by schaefer presents a uniform way of studying the complexity of satisfiability problems with special conditions the complexity of each decision and counting problem in this class depends on the involved logical relations in 1979 valiant defined the class p the class of functions definable as the number of accepting computations of a polynomial time nondeterministic turing machine clearly all satisfiability counting problems belong to this class p we prove a dichotomy theorem for generalized satisfiability counting problems that is if all logical relations involved in a generalized satisfiability counting problem are affine then the number of satisfying assignments of this problem can be computed in polynomial time otherwise this function is p complete this gives us a comparison between decision and counting generalized satisfiability problems we can determine exactly the polynomial satisfiability decision problems whose number of solutions can be computed in polynomial time and also the polynomial satisfiability decision problems whose counting counterparts are already p complete moreover taking advantage of a similar dichotomy result proved in 1978 by schaefer for generalized satisfiability decision problems we get as a corollary the implication that the counting counterpart of each np complete generalized satisfiability decision problem is p complete

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

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 parameterized complexity of counting problems we develop a parameterized complexity theory for counting problems as the basis of this theory we introduce a hierarchy of parameterized counting complexity classes w t for t ge 1 that corresponds to downey and fellows s w hierarchy r g downey and m r fellows parameterized complexity springer verlag new york 1999 and we show that a few central w completeness results for decision problems translate to w completeness results for the corresponding counting problems counting complexity gets interesting with problems whose decision version is tractable but whose counting version is hard our main result states that counting cycles and paths of length k in both directed and undirected graphs parameterized by k is w 1 complete this makes it highly unlikely that these problems are fixed parameter tractable even though their decision versions are fixed parameter tractable more explicitly our result shows that most likely there is no f k cdot n c algorithm for counting cycles or paths of length k in a graph of size n for any computable function f mathbb n to mathbb n and constant c even though there is a 2 o k cdot n 2 376 algorithm for finding a cycle or path of length k n alon r yuster and u zwick j acm 42 1995 pp 844 856

the complexity of counting in sparse regular and planar graphs we show that a number of graph theoretic counting problems remain cal np hard indeed cal p complete in very restricted classes of graphs in particular we prove that the problems of counting matchings vertex covers independent sets and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree we obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2 cnf formulae to achieve these results a new interpolation based reduction technique which preserves properties such as constant degree is introduced

random matrices close to hermitian or unitary overview of methods and results the paper discusses recent progress in understanding statistical properties of eigenvalues of weakly non hermitian and non unitary random matrices the first type of ensembles is of the form i with being a large random n n hermitian matrix with independent entries deformed by a certain anti hermitian n n matrix i satisfying in the limit of large dimension n the condition tr 2 n tr 2 here can be either a random or just a fixed given hermitian matrix ensembles of such a type with 0 emerge naturally when describing quantum scattering in systems with chaotic dynamics and serve to describe resonance statistics related models are used to mimic complex spectra of the dirac operator with chemical potential in the context of quantum chromodynamics ensembles of the second type arising naturally in scattering theory of discrete time systems are formed by n n matrices with complex entries such that i for 0 this coincides with the circular unitary ensemble and 0 i describes deviation from unitarity our result amounts to answering statistically the following old question given the singular values of a matrix describe the locus of its eigenvalues we systematically show that the obtained expressions for the correlation functions of complex eigenvalues describe a non trivial crossover from wigner dyson statistics of real unimodular eigenvalues typical of hermitian unitary matrices to ginibre statistics in the complex plane typical of ensembles with strong non hermiticity tr 2 tr 2 when n finally we discuss scarce results available on eigenvector statistics for weakly non hermitian random matrices

dichotomy results for fixed point counting in boolean dynamical systems we present dichotomy theorems regarding the computational complexity of counting fixed points in boolean discrete dynamical systems i e finite discrete dynamical systems over the domain 0 1 for a class f of boolean functions and a class g of graphs an f g system is a boolean dynamical system with local transitions functions lying in f and graphs in g we show that if local transition functions are given by lookup tables then the following complexity classification holds let f be a class of boolean functions closed under superposition and let g be a graph class closed under taking minors if f contains all min functions all max functions or all self dual and monotone functions and g contains all planar graphs then it is p complete to compute the number of fixed points in an f g system otherwise it is computable in polynomial time we also prove a dichotomy theorem for the case that local transition functions are given by formulas over logical bases this theorem has a significantly more complicated structure than the theorem for lookup tables a corresponding theorem for boolean circuits coincides with the theorem for formulas

elements of a theory of simulation iii equivalence of sds in the development of mathematical foundations for a theory of simulation a certain class of discrete sequential dynamical systems sds is of particular importance these systems which we refer to as sds consist of a a graph y with vertex set 1 2 n where each vertex has associated a binary state b a vertex labeled set of functions f i y f 2 n f 2 n and c a permutation p s n the function f i y update the state of vertex i as a function of the states of vertex i and its y neighbors and leaves all other states invariant by composing these functions in the order given by p we obtain the sds f y p i 1 nf p i y f 2 n f 2 n in this paper we give a combinatorial upper bound for the number of non equivalent sds for a given graph and we compute this bound explicitly for certain classes of graphs we give a full characterization of invertible sds and analyze the set of fixed points of sequential and parallel cellular automata further we introduce the concept of maj type sds and show that systems of this class only have fixed points as attractors finally we analyze sds that are fixed point free for arbitrary base graphs

runge kutta smoother for suppression of computational mode instability of leapfrog scheme abstract the runge kutta smoother is applied to suppress nonlinear numerical instabilities in the leapfrog scheme for time integration of the korteweg de vries equation the accuracy of integration is compared with that by the use of the second order smoother the runge kutta smoother enables us to make long time integration of the korteweg de vries equation for large amplitudes

elements of a theory of computer simulation i sequential ca over random graphs this paper is a first step in the development of mathematical foundations for a theory of simulation we employ sequentially updated cellular automata sca over arbitrary graphs as a paradigmatic framework in the development of the theory we focus on the properties of causal dependencies among local mappings in a simulation let y be a graph in which a each vertex i has a state x i0 1 and b there exists a local map i defined on the states of the y neighbors and x i that returns the state of y i0 1 suppose i 1 i n is a permutation of the y vertices then the order of application of the local maps i k induces a sequential or asynchronous cellular automaton sca over y in this paper we introduce a mapping between the base graph y over which the sca is defined and which represents the mutual dependencies of the local maps and the update graph u y whose vertices are permutations of all y vertices two permutations p 1 p 2 are adjacent in u y if they differ in exactly two consecutive coordinates i k i k 1 and i k i k 1 ey u y represents a conceptual framework which allows to determine equivalence classes of sca for a given graph x independent of the family of local maps i 1 i n we consider a u as a random variable over g n p and analyze the induced random graph u g n p the update graph and show b that u exhibits properties of a covariant functor

on the suppression of numerical oscillations using a non linear filter the idea of using a non linear filtering algorithm to eliminate numerically generated oscillations is investigated a detailed study is conducted to follow the development of numerical oscillations and their interaction with the filter a relaxation procedure is also proposed to enhance the effectiveness of the filter three model problems a 2d steady state scalar convection diffusion equation a 1d unsteady gas dynamics flow with shock and a 1d linear wave equation have been designed to test the performance of the filtering algorithm the effectiveness of the filter is assessed for convection schemes of different dispersive and diffusive characteristics demonstrating that it is effective in eliminating oscillations with short wavelength but oscillations of longer wavelengths are virtually unaffected it is concluded that a proper combination of non linear filter and dispersive numerical scheme is a viable alternative to dissipative schemes in resolving flows with sharp gradients and discontinuities