spectral statistics for unitary transfer matrices of binary graphs quantum graphs have recently been introduced as model systems to study the spectral statistics of linear wave problems with chaotic classical limits it is proposed here to generalize this approach by considering arbitrary directed graphs with unitary transfer matrices an exponentially increasing contribution to the form factor is identified when performing a diagonal summation over periodic orbit degeneracy classes a special class of graphs so called binary graphs is studied in more detail for these the conditions for periodic orbit pairs to be correlated including correlations due to the unitarity of the transfer matrix can be given explicitly using combinatorial techniques it is possible to perform the summation over correlated periodic orbit pair contributions to the form factor for some low dimensional cases gradual convergence towards random matrix results is observed when increasing the number of vertices of the binary graphs

hamiltonian mappings and circle packing phase spaces we introduce three area preserving maps with phase space structures which resemble circle packings each mapping is derived from a kicked hamiltonian system with one of the three different phase space geometries planar hyperbolic or spherical and exhibits an infinite number of coexisting stable periodic orbits which appear to pack the phase space with circular resonances

spectral behavior of contractive noise we study the behavior of the spectra corresponding to quantum systems subjected to a contractive noise i e the environment reduces the accessible phase space of the system but the total probability is conserved we find that the number of long lived resonances grows as a power law in h but surprisingly there is no relationship between the exponent of this power law and the fractal dimension of the corresponding classical attractor this is in disagreement with the predictions of the fractal weyl law which has been established for open systems where the probability is lost under the effect of a projective opening

finite time stability conditions for non autonomous continuous systems finite time stability is defined for continuous non autonomous systems starting with a result from haimo 1986 we then extend this result to n dimensional non autonomous systems through the use of smooth and non smooth lyapunov functions as in perruquetti and drakunov 2000 one obtains two different sufficient conditions and a necessary one for finite time stability of continuous non autonomous systems

parameter identification for semiconductor diodes by lbic imaging laser beam induced current lbic imaging is a nondestructive technique used for the characterization of the electrical structure within a semiconductor in this paper a model is formulated for this technique using the standard drift diffusion model and subsequently an approximate version and its dual are derived for the study of the inverse problem the formulation is then applied to a cross sectional model for n on p devices of finite depth to study in detail the relation between the lbic images and the device parameters numerical methods are developed for the simulation of the lbic image of a diode as well as for the identification of parameters from the lbic image by least squares formulation numerical examples are presented to illustrate the success of identifying parameters such as junction depth diffusion length and equilibrium potential of an abrupt p n junction diode from its lbic image the differentiability of the image with respect to the parameters also is established

fractal weyl laws in discrete models of chaotic scattering we analyse simple models of quantum chaotic scattering namely quantized open baker s maps we numerically compute the density of quantum resonances in the semiclassical regime this density satisfies a fractal weyl law where the exponent is governed by the fractal dimension of the set of trapped trajectories this type of behaviour is also expected in the physically more relevant case of hamiltonian chaotic scattering within a simplified model we are able to rigorously prove this weyl law and compute quantities related to the coherent transport through the system namely the conductance and shot noise the latter is close to the prediction of random matrix theory

quantum fractal eigenstates abstract we study quantum chaos in open dynamical systems and show that it is characterized by quantum fractal eigenstates located on the underlying classical strange repeller the states with longest life times typically reveal a scars structure on the classical fractal set

solving partial differential equations on irregular domains with moving interfaces with applications to superconformal electrodeposition in semiconductor manufacturing we present a numerical algorithm for solving partial differential equations on irregular domains with moving interfaces instead of the typical approach of solving in a larger rectangular domain our approach performs most calculations only in the desired domain to do so efficiently we have developed a one sided multigrid method to solve the corresponding large sparse linear systems our focus is on the simulation of the electrodeposition process in semiconductor manufacturing in both two and three dimensions our goal is to track the position of the interface between the metal and the electrolyte as the features are filled and to determine which initial configurations and physical parameters lead to superfilling we begin by motivating the set of equations which model the electrodeposition process building on existing models for superconformal electrodeposition we develop a model which naturally arises from a conservation law form of surface additive evolution we then introduce several numerical algorithms including a conservative material transport level set method and our multigrid method for one sided diffusion equations we then analyze the accuracy of our numerical methods finally we compare our result with experiment over a wide range of physical parameters

weyl asymptotics from closed to open systems we present microwave experiments on the symmetry reduced five disk billiard studying the transition from a closed to an open system the measured microwave reflection signal is analyzed by means of the harmonic inversion and the counting function of the resulting resonances is studied for the closed system this counting function shows the weyl asymptotic with a leading exponent equal to 2 by opening the system successively this exponent decreases smoothly to a noninteger value for the open systems the extraction of resonances by the harmonic inversion becomes more challenging and the arising difficulties are discussed the results can be interpreted as a first experimental indication for the fractal weyl conjecture for resonances

weyl law for fat fractals it has been conjectured that for a class of piecewise linear maps the closure of the set of images of the discontinuity has the structure of a fat fractal that is a fractal with positive measure an example of such maps is the sawtooth map in the elliptic regime in this work we analyze this problem quantum mechanically in the semiclassical regime we find that the fraction of states localized on the unstable set satisfies a modified fractal weyl law where the exponent is given by the exterior dimension of the fat fractal

nodal domains in open microwave systems nodal domains are studied both for real psir and imaginary part psii of the wave functions of an open microwave cavity and found to show the same behavior as wave functions in closed billiards in addition we investigate the variation of the number of nodal domains and the signed area correlation by changing the global phase phig according to psir ipsii eiphig psir ipsii this variation can be qualitatively and the correlation quantitatively explained in terms of the phase rigidity characterizing the openness of the billiard

role of short periodic orbits in quantum maps with continuous openings we apply a recently developed semiclassical theory of short periodic orbits to the continuously open quantum tribaker map in this paradigmatic system the trajectories are partially bounced back according to continuous reflectivity functions this is relevant in many situations that include optical microresonators and more complicated boundary conditions in a perturbative regime the shortest periodic orbits belonging to the classical repeller of the open map a cantor set given by a region of exactly zero reflectivity prove to be extremely robust in supporting a set of long lived resonances of the continuously open quantum maps moreover for steplike functions a significant reduction in the number needed is obtained similarly to the completely open situation this happens despite a strong change in the spectral properties when compared to the discontinuous reflectivity case in order to give a more realistic interpretation of these results we compare with a fresnel type reflectivity function

towards a semiclassical theory of the quantum baker s map abstract we develop semiclassical 0 treatments of the quantum baker s map paying special attention to a the discrete and finite nature of its hilbert space and b the asymptotic analysis of the true quantum objects themselves new techniques are described at the theoretical level exact symbolic decomposition of the propagator and at the numerical level color graphics display of operators in the kirkwood representation both of which are of general interest for other quantized chaotic maps periodic orbit theory is tested numerically and we find that log corrections need to be included before an accurate calculation of the spectrum from periodic orbits can be achieved however the asymptotic analysis of the traces of the propagator reveals unexpected departures from standard semiclassical theory those anomalies which we compute explicitly in a few cases are traced ultimately to the discontinuity of the map and to the compactness of the phase space the analysis is far from complete but the results point towards the necessity of careful assessment of semiclassical folklore when applied to chaotic maps

reconstruction of semiconductor doping profile from laser beam induced current image in this paper the authors study the reconstruction of a semiconductor doping profile or equivalently the equilibrium potential from its lbic laser beam induced current image for the one dimensional case the authors first characterize the attainable class of current measurements and from this they show the nonuniqueness of the inverse problem then the reconstruction of the equilibrium potential is reduced to finding two constants subject to some constraints a reconstruction algorithm is established based on a least squares formulation of the problem the case of noise collapsed data is also discussed for a special case of two dimensional domain the authors apply the one dimensional algorithm supplemented with a correction from the other spatial direction to establish an alternate direction iteration algorithm for reconstruction of the two dimensional equilibrium potential the authors also present some numerical examples to illustrate the reconstruction results by these algorithms

geometric bounds on the density of resonances for semiclassical problems in this paper we shall give upper bounds on the number of resonances in certain regions in the complex plane close to the real axis for semiclassical operators like h 2 v x when h is small

truncations of random unitary matrices we analyse properties of non hermitian matrices of size m constructed as square submatrices of unitary orthogonal random matrices of size n m distributed according to the haar measure in this way we define ensembles of random matrices and study the statistical properties of the spectrum located inside the unit circle in the limit of large matrices this ensemble is characterized by the ratio m n for the truncated cue we analytically derive the joint density of eigenvalues and all correlation functions in the strongly non unitary case universal ginibre behaviour is found for n m fixed and n to the universal resonance width distribution with n m open channels is recovered

quantum decay rates in chaotic scattering we study quantum scattering on manifolds equivalent to the euclidean space near infinity in the semiclassical regime we assume that the corresponding classical flow admits a non trivial trapped set and that the dynamics on this set is of axiom a type uniformly hyperbolic we are interested in the distribution of quantum resonances near the real axis in two dimensions we prove that if the trapped set is sufficiently thin then there exists a gap between the resonances and the real axis that is quantum decay rates are bounded from below in higher dimension the condition for this gap is given in terms of a certain topological pressure associated with the classical flow under the same assumption we also prove a resolvent estimate with a logarithmic loss compared to non trapping situations

google matrix dynamical attractors and ulam networks we study the properties of the google matrix generated by a coarse grained perron frobenius operator of the chirikov typical map with dissipation the finite size matrix approximant of this operator is constructed by the ulam method this method applied to the simple dynamical model generates directed ulam networks with approximate scale free scaling and characteristics being in certain features similar to those of the world wide web with approximate scale free degree distributions as well as two characteristics similar to the web a power law decay in pagerank that mirrors the decay of pagerank on the world wide web and a sensitivity to the value ensuremath alpha in pagerank the simple dynamical attractors play here the role of popular websites with a strong concentration of pagerank a variation in the google parameter ensuremath alpha or other parameters of the dynamical map can drive the pagerank of the google matrix to a delocalized phase with a strange attractor where the google search becomes inefficient

fractal weyl law for quantum fractal eigenstates the properties of the resonant gamow states are studied numerically in the semiclassical limit for the quantum chirikov standard map with absorption it is shown that the number of such states is described by the fractal weyl law and their husimi distributions closely follow the strange repeller set formed by classical orbits nonescaping in future times for large matrices the distribution of escape rates converges to a fixed shape profile characterized by a spectral gap related to the classical escape rate

evaluation of the coulomb force via the fredholm integral equation the electrostatic force between two conducting spheres is solved exactly from coulomb s law the force is evaluated in two steps i the charge distribution on each of the spheres is calculated as a function of separation distance by way of a solution to the fredholm integral equation ii the effect of the known charge distribution is integrated to obtain the electrostatic force evaluation of the electrostatic force is fast because the series expression for the charge distribution is super converging a characteristic trait for some fredholm type equations

chaotic sets of continuous and discontinuous maps abstract this paper discusses li yorke chaotic sets of continuous and discontinuous maps with particular emphasis to shift and subshift maps scrambled sets and maximal scrambled sets are introduced to characterize li yorke chaotic sets the orbit invariant for a scrambled set is discussed some properties about maximality equivalence and uniqueness of maximal scrambled sets are also discussed it is shown that for shift maps the set of all scrambled pairs has full measure and chaotic sets of some discontinuous maps such as the gauss map interval exchange transformations and a class of planar piecewise isometries are studied finally some open problems on scrambled sets are listed and remarked

fractal weyl law for open quantum chaotic maps we study a semiclassical quantization of poincar e maps arising in scattering problems with fractal hyperbolic trapped sets the main application is the proof of a fractal weyl upper bound for the number of resonances scattering poles in small domains near the real axis this result includes the case of several convex hard obstacles satisfying a no eclipse condition

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

finite time stability of discrete autonomous systems abstract finite time stability involves dynamical systems whose trajectories converge to a lyapunov stable equilibrium state in finite time in this paper we address finite time stability of discrete time dynamical systems specifically we show that finite time stability leads to uniqueness of solutions in forward time furthermore we provide lyapunov and converse lyapunov theorems for finite time stability of discrete autonomous systems involving scalar difference fractional inequalities and minimum operators in addition lower semicontinuity of the settling time function capturing the finite settling time behavior of the dynamical system is studied and illustrated through several examples in particular it is shown that the regularity properties of the lyapunov function and those of the settling time function are related consequently converse lyapunov theorems for finite time stability of discrete time systems can only assure the existence of lower semicontinuous lyapunov functions

identification of contact regions in semiconductor transistors by level set methods in this paper we present the formulation of level set methods for the inverse problem of identifying an interface in the coefficient of an elliptic equation from a boundary measurement this problem arises from the modeling of the identification of contact regions by boundary measurements for semiconductor transistors we propose the gauss newton direction as the interface velocity and implement the scheme for a parameterized class of interfaces

the immersed interface multigrid methods for interface problems new multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic pdes that involve interfaces and discontinuities for elliptic interface problems the multigrid solver developed in this paper works while some other multigrid solvers do not for linear parabolic equations we have developed the second order maximum principle preserving finite difference scheme in this paper we use the crank nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives numerical examples are also presented

a multigrid ghost point level set method for incompressible navier stokes equations on moving domains with curved boundaries abstract in this paper we present a numerical approach to solve the navier stokes equations on moving domains with second order accuracy the space discretization is based on the ghost point method which falls under the category of unfitted boundary methods since the mesh does not adapt to the moving boundary the equations are advanced in time by using crank nicholson the momentum and continuity equations are solved simultaneously for the velocity and the pressure by adopting a proper multigrid approach to avoid the checkerboard instability for the pressure a staggered grid is adopted where velocities are defined at the sides of the cell and the pressure is defined at the centre the lack of uniqueness for the pressure is circumvented by the inclusion of an additional scalar unknown representing the average divergence of the velocity and an additional equation to set the average pressure to zero several tests are performed to simulate the motion of an incompressible fluid around a moving object as well as the lid driven cavity tests around steady objects the object is implicitly defined by a level set approach that allows a natural computation of geometrical properties such as distance from the boundary normal directions and curvature different shapes are tested circle ellipse and flower numerical results show the second order accuracy for the velocity and the divergence that decays to zero with second order and the efficiency of the multigrid that is comparable with the tests available in literature for rectangular domains without objects showing that the presence of a complex shaped object does not degrade the performance

brief paper finite time stability and instability of stochastic nonlinear systems this paper presents a new definition of finite time stability for stochastic nonlinear systems this definition involves stability in probability and finite time attractiveness in probability an important lyapunov theorem on finite time stability for stochastic nonlinear systems is established a theorem extending the stochastic lyapunov theorem is also proved moreover an example and a lemma are presented to illustrate the scope of extension a useful inequality extended from bihari s inequality is derived which plays an important role in showing the lyapunov theorem finally a lyapunov theorem on finite time instability is proved which states that almost surely globally asymptotical stability is not equivalent to finite time stability for some stochastic systems two simulation examples are given to illustrate the theoretical analysis

maximum principle preserving schemes for interface problems with discontinuous coefficients new finite difference methods using cartesian grids are developed for elliptic interface problems with variable discontinuous coefficients singular sources and nonsmooth or even discontinuous solutions the new finite difference schemes are constructed to satisfy the sign property of the discrete maximum principle using quadratic optimization techniques the methods are shown to converge under certain conditions using comparison functions the coefficient matrix of the resulting linear system of equations is an m matrix and is coupled with a multigrid solver numerical examples are also provided to show the efficiency of the proposed methods

distribution of resonances for open quantum maps we analyze a simple model of quantum chaotic scattering system namely the quantized open baker s map this model provides a numerical confirmation of the fractal weyl law for the semiclassical density of quantum resonances the fractal exponent is related to the dimension of the classical repeller we also consider a variant of this model for which the full resonance spectrum can be rigorously computed and satisfies the fractal weyl law for that model we also compute the shot noise of the conductance through the system and obtain a value close to the prediction of random matrix theory

fractal weyl law behavior in an open hamiltonian system we numerically show fractal weyl law behavior in an open hamiltonian system that is described by a smooth potential and which supports numerous above barrier resonances this behavior holds even relatively far away from the classical limit the complex resonance wave functions are found to be localized on the fractal classical repeller

a second order cartesian method for the simulation of electropermeabilization cell models in this paper we present a new finite differences method to simulate electropermeabilization models like the model of neu and krassowska or the recent model of kavian et al these models are based on the evolution of the electric potential in a cell embedded in a conducting medium the main feature lies in the transmission of the voltage potential across the cell membrane the jump of the potential is proportional to the normal flux thanks to the well known kirchoff law an adapted scheme is thus necessary to accurately simulate the voltage potential in the whole cell notably at the membrane separating the cell from the outer medium we present a second order finite differences scheme in the spirit of the method introduced by cisternino and weynans for elliptic problems with immersed interfaces this is a cartesian grid method based on the accurate discretization of the fluxes at the interface through the use of additional interface unknowns the main novelty of our present work lies in the fact that the jump of the potential is proportional to the flux and therefore is not explicitly known the original use of interface unknowns makes it possible to discretize the transmission conditions with enough accuracy to obtain a second order spatial convergence we prove the second order spatial convergence in the stationary linear one dimensional case and the first order temporal convergence for the dynamical non linear model in one dimension we then perform numerical experiments in two dimensions that corroborate these results

immersed interface methods for neumann and related problems in two and three dimensions we develop and apply a finite difference method to discretize the laplacian operator with neumann boundary conditions on an irregular domain using a regular cartesian grid the method is an extension of the immersed interface method developed by leveque and li siam j numer anal 31 1994 1019 1044 with careful selection of stencils the method is second order accurate and produces a matrix that is stable diagonally semidominant the method is illustrated on several two dimensional problems and one three dimensional problem

high order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources this paper introduces a novel high order interface scheme the matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and singular sources on cartesian grids by appropriate use of auxiliary line and or fictitious points physical jump conditions are enforced at the interface unlike other existing interface schemes the proposed method disassociates the enforcement of physical jump conditions from the discretization of the differential equation under study to construct higher order interface schemes the proposed mib method bypasses the major challenge of implementing high order jump conditions by repeatedly enforcing the lowest order jump conditions the proposed mib method is of arbitrarily high order in principle in treating straight regular interfaces we construct mib schemes up to 16th order for more general elliptic problems with curved irregular interfaces and boundary up to 6th order mib schemes have been demonstrated by employing the standard high order finite difference schemes to discretize the laplacian the present mib method automatically reduces to the standard central difference scheme when the interface is absent the immersed interface method iim is regenerated for a comparison study of the proposed method the robustness of the mib method is verified against the large magnitude of the jump discontinuity across the interface the nature of high efficiency and low memory requirement of the mib method is extensively validated via solving various elliptic immersed interface problems in two and three dimensions ee dimensions

on the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values given any fixed n n positive semi definite diagonal matrix g 0 we derive the explicit formula for the density of complex eigenvalues for random matrices a of the form where the random unitary matrices u are distributed on the group u n according to the haar measure

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

polygonal invariant curves for a planar piecewise isometry we investigate a remarkable new planar piecewise isometry whose generating map is a permutation of four cones for this system we prove the coexistence of an infinite number of periodic components and an uncountable number of transitive components the union of all periodic components is an invariant pentagon with unequal sides transitive components are invariant curves on which the dynamics are conjugate to a transitive interval exchange the restriction of the map to the invariant pentagonal region is the first known piecewise isometric system for which there exist an infinite number of periodic components but the only aperiodic points are on the boundary of the region the proofs are based on exact calculations in a rational cyclotomic field we use the system to shed some light on a conjecture that pwis can possess transitive invariant curves that are not smooth

on codings and dynamics of planar piecewise rotations abstract in this paper we investigate codings and dynamics of the piecewise rotations belonging to a sub class of piecewise isometries we show that the irrational set is empty for some piecewise rational rotations under some assumptions while a piecewise irrational rotation has at least one irrational coding by extending the definition of the coding map onto the entire phase space we further prove that the cell corresponding to irrational coding is a single point set for a piecewise irrational rotation

multifractal eigenstates of quantum chaos and the thue morse sequence we analyze certain eigenstates of the quantum baker s map and demonstrate using the walsh hadamard transform the emergence of the ubiquitous thue morse sequence a simple sequence that is at the border between quasiperiodicity and chaos and hence is a good paradigm for quantum chaotic states we show a family of states that are also simply related to the thue morse sequence and are strongly scarred by short periodic orbits and their homoclinic excursions we give approximate expressions for these states and provide evidence that these and other generic states are multifractal

a fast iterative algorithm for elliptic interface problems a fast second order accurate iterative method is proposed for the elliptic equation grad cdot beta x y grad u f x y in a rectangular region omega in two space dimensions we assume that there is an irregular interface across which the coefficient beta the solution u and its derivatives and or the source term f may have jumps we are especially interested in the cases where the coefficients beta are piecewise constant and the jump in beta is large the interface may or may not align with an underlying cartesian grid the idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by leveque and li siam j numer anal 4 1994 pp 1019 1044 in order to take advantage of fast poisson solvers on a rectangular region an intermediate unknown function the jump in the normal derivative across the interface is introduced our discretization is equivalent to using a second order difference scheme for a corresponding poisson equation in the region and a second order discretization for a neumann like interface condition thus second order accuracy is guaranteed a gmres iteration is employed to solve the schur complement system derived from the discretization a new weighted least squares method is also proposed to approximate interface quantities from a grid function numerical experiments are provided and analyzed the number of iterations in solving the schur complement system appears to be independent of both the jump in the coefficient and the mesh size

global stability of a class of discontinuous dual billiards an infinite parameter family of discontinuous area preserving maps is studied using geometrical methods necessary and sufficient conditions are determined for the existence of some bounding invariant sets which guarantee global stability it is shown that under some additional constraints all orbits become periodic most of them lyapounov stable and with a maximal period in any bounded domain of phase space this yields a class of maps acting on a discrete phase space

a new multiscale finite element method for high contrast elliptic interface problems we introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes and without resolving the interfaces a typical application would be the modelling of flow in a porous medium containing a number of inclusions of low or high permeability embedded in a matrix of high respectively low permeability our method is h 1 conforming with degrees of freedom at the nodes of a triangular mesh and requiring the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface but which use standard linear approximation otherwise a key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems under moderate assumptions we prove that our methods have optimal convergence rate of o h in the energy norm and o h 2 in the l 2 norm where h is the coarse mesh diameter and the hidden constants in these estimates are independent of the contrast i e ratio of largest to smallest value of the pde coefficient for standard elements the best estimate in the energy norm would be o h 1 2 e with a hidden constant which in general depends on the contrast the new interior boundary conditions depend not only on the contrast of the coefficients but also on the angles of intersection of the interface with the element edges

technical communique finite time stability theorem of stochastic nonlinear systems a new concept of finite time stability called stochastically finite time attractiveness is defined for a class of stochastic nonlinear systems described by the ito differential equation the settling time function is a stochastic variable and its expectation is finite a theorem and a corollary are given to verify the finite time attractiveness of stochastic systems based on lyapunov functions two simulation examples are provided to illustrate the applications of the theorem and the corollary established in this paper

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

on the symmetry lines of the standard mapping abstract the taylor and chirikov standard mapping is used as a model to represent the dynamics of non integrable systems by means of the poincare section of the trajectories intersecting a fixed plane of the phase space this study is based on a systematic use of the decomposition of the standard mapping in a product of two involutions as was birkhoff s in the restricted three body problem an infinite family of involutions is generated by these two involutions forming a group including the iterates of the mapping and their inverses the symmetry lines of the family of involutions determine the position of the periodic points and explain the onset of new chains of periodic points produced by the winding of the symmetry lines around the stable periodic points

finite element methods and their convergence for elliptic and parabolic interface problems in this paper we consider the finite element methods for solving second order elliptic and parabolic interface problems in two dimensional convex polygonal domains nearly the same optimal l 2 norm and energy norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth though the regularities of the solutions are low on the whole domain the assumptions on the finite element triangulation are reasonable and practical

identifying invariant ergodic subsets and barriers to mixing by cutting and shuffling study in a birotated hemisphere mixing by cutting and shuffling can be mathematically described by the dynamics of piecewise isometries pwis higher dimensional analogs of one dimensional interval exchange transformations in a two dimensional domain under a pwi the exceptional set e over which is created by the accumulation of cutting lines the union of all iterates of cutting lines and all points that pass arbitrarily close to a cutting line defines where mixing is possible but not guaranteed there is structure within e over that directly influences the mixing potential of the pwi here we provide computational and analytical formalisms for examining this structure by way of measuring the density and connectivity of fattened cutting lines that form an approximation of e over for the example of a pwi on a hemispherical shell studied here this approach reveals the subtle mixing behaviors and barriers to mixing formed by invariant ergodic subsets confined orbits within the fractal structure of the exceptional set some pwis on the shell have provably nonergodic exceptional sets which prevent mixing while others have potentially ergodic exceptional sets where mixing is possible since ergodic exceptional sets have uniform cutting line density for these latter exceptional sets we show the connectivity of orbits in the pwi map through direct examination of orbit position and shape and through a two dimensional return plot to explain the necessity of orbit connectivity for mixing

a novel concept for the design of immersed interface methods the objective of this paper is to present a novel robust high order accurate immersed interface method iim for advection diffusion type equations in contrast to other immersed methods that were designed for consistency and accuracy with a posteriori check of the numerical stability we combine local taylor series expansion at irregular grid points with a local stability constraint as part of the design process stability investigations of the iim are employed to demonstrate that the local stability constraint is sufficient for obtaining a globally stable method as long as the neumann number is less than its limiting value one of the key aspects of this iim is that the irregular finite difference stencils can be isolated from the rest of the computational domain to validate our novel immersed interface approach two dimensional and three dimensional test cases for model equations are presented in addition this method is applied to the incompressible navier stokes equations to conduct stability investigations of a boundary layer flow over a rough surface and for investigations of pulsatile stenotic flows stability investigations of wall bounded flows are challenging for immersed methods because the near wall accuracy is important for correctly capturing the characteristics of the hydrodynamic instability mechanisms in particular regarding the wave relation between the wave velocity components close to the wall

resonance expansions of propagators in the presence of potential barriers abstract we show that the schrodinger propagator can be expanded in terms of resonances at energy levels at which a barrier separates the interaction region from infinity the expansions hold for all times with errors small in the semi classical parameter as a byproduct we obtain a result on the approximation of clusters of resonant states by clusters of eigenfunctions of a self adjoint reference operator

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

the selberg zeta function for convex co compact schottky groups we give a new upper bound on the selberg zeta function for a convex co compact schottky group acting on the hyperbolic space n 1 in strips parallel to the imaginary axis the zeta function is bounded by exp c s where is the dimension of the limit set of the group this bound is more precise than the optimal global bound exp c s n 1 and it gives new bounds on the number of resonances scattering poles of n 1 the proof of this result is based on the application of holomorphic l2 techniques to the study of the determinants of the ruelle transfer operators and on the quasi self similarity of limit sets we also study this problem numerically and provide evidence that the bound may be optimal our motivation comes from molecular dynamics and we consider n 1 as the simplest model of quantum chaotic scattering

on global attractors for a class of nonhyperbolic piecewise affine maps dynamical behavior of a class of nonhyperbolic discrete systems are considered these systems are generated by iterating planar maps that are piecewise isometries and they arise as mathematical models for signal processing digital filters and modulator dynamics planar piecewise isometries may be discontinuous and or non invertible first the authors consider attraction caused by discontinuity in planar piecewise isometries namely they have shown that the maximal invariant set can induce an invariant measure and all the lyapunov exponents are zero under this invariant measure second they discuss various definitions of global attractors and their existence and uniqueness for discontinuous maps and introduce a few examples in which the attractors are created due to discontinuity third they study the relation between invariance and invertibility for various nonhyperbolic maps and finally they investigate decomposability of global attractors for certain nonhyperbolic systems

three dimensional matched interface and boundary mib method for treating geometric singularities this paper reports the three dimensional 3d generalization of our previous 2d higher order matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and non smooth interfaces new mib algorithms that make use of two sets of interface jump conditions are proposed to remove the critical acute angle constraint of our earlier mib scheme for treating interfaces with sharp geometric singularities such as sharp edges sharp wedges and sharp tips the resulting 3d mib schemes are of second order accuracy for arbitrarily complex interfaces with sharp geometric singularities of fourth order accuracy for complex interfaces with moderate geometric singularities and of sixth order accuracy for curved smooth interfaces a systematical procedure is introduced to make the mib matrix optimally symmetric and banded by appropriately choosing auxiliary grid points consequently the new mib linear algebraic equations can be solved with fewer number of iterations the proposed mib method makes use of cartesian grids standard finite difference schemes lowest order interface jump conditions and fictitious values the interface jump conditions are enforced at each intersecting point of the interface and mesh lines to overcome the staircase phenomena in finite difference approximation while a pair of fictitious values are determined along a mesh at a time an iterative procedure is proposed to determine all the required fictitious values for higher order schemes by repeatedly using the lowest order jump conditions a variety of mib techniques are developed to overcome geometric constraints the essential strategy of the mib method is to locally reduce a 2d or a 3d interface problem into 1d like ones the proposed mib method is extensively validated in terms of the order of accuracy the speed of convergence the number of iterations and cpu time numerical experiments are carried out to complex interfaces including the molecular surfaces of a protein a missile interface and van der waals surfaces of intersecting spheres

quantum iterated function systems an iterated function system ifs is defined by specifying a set of functions in a classical phase space which act randomly on an initial point in an analogous way we define a quantum ifs qifs where functions act randomly with prescribed probabilities in the hilbert space in a more general setting a qifs consists of completely positive maps acting in the space of density operators this formalism is designed to describe certain problems of nonunitary quantum dynamics we present exemplary classical ifss the invariant measure of which exhibits fractal structure and study properties of the corresponding qifss and their invariant states

a homogenization model for laser beam induced current imaging and detection of non uniformities in semiconductor arrays we present an approximate model for laser beam induced current lbic imaging of arrays of semiconductor devices based on homogenization lbic is a non destructive technique useful for the characterization and quality control of semiconductor focal plane arrays a key component in modern imaging systems the model provides not only an efficient alternative for lbic image simulation of large uniform arrays but also an effective method for detection of non uniformities among arrays numerical examples are presented to illustrate the effectiveness of this method in detecting array non uniformities due to variations in dislocation size or doping level among p n junction diodes

using the hadamard and related transforms for simplifying the spectrum of the quantum baker s map we rationalize the somewhat surprising efficacy of the hadamard transform in simplifying the eigenstates of the quantum baker s map a paradigmatic model of quantum chaos this allows us to construct closely related but new transforms that do significantly better thus nearly solving many states of the quantum baker s map these transforms which combine the standard fourier and hadamard transforms in an interesting manner are constructed from eigenvectors of the shift permutation operator that are also simultaneous eigenvectors of bit flip parity and possess bit reversal time reversal symmetry

classical 1d maps quantum graphs and ensembles of unitary matrices we study a certain class of classical one dimensional piecewise linear maps for these systems we introduce an infinite family of markov partitions in equal cells the symbolic dynamics generated by these systems is described by bi stochastic doubly stochastic matrices we analyse the structure of graphs generated from the corresponding symbolic dynamics we demonstrate that the spectra of quantized graphs corresponding to the regular classical systems have locally poissonian statistics while quantized graphs derived from classically chaotic systems display statistical properties characteristic of the circular unitary ensemble even though the corresponding unitary matrices are sparse