a new approach to inverse spectral theory i fundamental formalism we present a new approach distinct from gelfand levitan to the the orem of borg marchenko that the m function equivalently spectral measure for a finite interval or half line schrodinger operator determines the poten tial our approach is an analog of the continued fraction approach for the moment problem we prove there is a representation for the m function m 2 r b

the proper dissipative extensions of a dual pair let a and widetilde a be dissipative operators on a hilbert space mathcal h and let a widetilde a form a dual pair i e a subset widetilde a resp widetilde a subset a we present a method of determining the proper dissipative extensions widehat a of this dual pair i e a subset widehat a subset widetilde a provided that mathcal d a cap mathcal d widetilde a is dense in mathcal h applications to symmetric operators symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators are discussed finally we investigate the stability of the numerical range of the different dissipative extensions

the infinite dimensional evans function abstract we introduce generalized operator valued jost solutions of first order ill posed differential equations on hilbert spaces we then construct an infinite dimensional evans function for abstract differential equations as a 2 modified fredholm determinant of the operator obtained by adding the values at zero of the generalized operator valued jost solutions next we prove a formula that connects the 2 modified evans determinant and the 2 modified determinant related to the birman schwinger type operator associated to the ill posed equation using this formula we construct a holomorphic infinite dimensional evans function for second order differential operators on infinite cylinders whose zeros are the eigenvalues of the differential operators

meyer vietoris type formula for determinants of elliptic differential operators abstract for a closed codimension one submanifold of a compact manifold m let m be the manifold with boundary obtained by cutting m along let a be an elliptic differential operator on m and b and c be two complementary boundary conditions on if a b is an elliptic boundary valued problem on m then one defines an elliptic pseudodifferential operator r of neumann type on and prove the following factorization formula for the regularized determinants det a det a b k det r with k a local quantity depending only on the jets of the symbols of a b and c along the particular case when m has dimension 2 a is the laplace beltrami operator and b resp c is the dirichlet resp neumann boundary condition is considered

one dimensional characteristic boundary conditions using nonlinear invariants a new treatment of boundary conditions using the flow decomposition in characteristics is derived for inviscid one dimensional flows using nonlinear invariants the new set of equations is equivalent to the relations of thompson 1987 5 but it has the advantage to provide a physical interpretation of the characteristics for nonlinear perturbations this interpretation is a major advantage to deal with the nonlinear injection of waves in the domain in particular the limitation of the standard relations to the injection of waves without nonlinear interactions on the boundary is addressed associated errors are evaluated analytically and numerically and a clear improvement of the results is demonstrated with the new expressions to avoid a drift of the mean values relaxation terms are usually added in the relations and the boundary conditions become almost non reflecting the consequences of these relaxation terms on outgoing and ingoing waves are widely investigated in the present paper and a nonlinear correction is proposed to recover perfectly non reflecting conditions without drift to end simulations are performed on the generation of indirect combustion noise through a nozzle to illustrate the advantages of the new formulation

a remark on the dirichlet neumann decoupling and the integrated density of states abstract we prove an estimate on the difference of the number of eigenvalues for schrodinger operators with dirichlet and neumann boundary conditions in large boxes in the proof we use krein s theory of the spectral shift function as an application we show that the integrated density of states is independent of the choice of boundary conditions

scattering matrices and dirichlet to neumann maps abstract a general representation formula for the scattering matrix of a scattering system consisting of two self adjoint operators in terms of an abstract operator valued titchmarsh weyl m function is proved this result is applied to scattering problems for different self adjoint realizations of schrodinger operators on unbounded domains schrodinger operators with singular potentials supported on hypersurfaces and orthogonal couplings of schrodinger operators in these applications the scattering matrix is expressed in an explicit form with the help of dirichlet to neumann maps

stabilization of delay systems delay dependent impulsive control the stabilization problem of delay systems is studied under the delay dependent impulsive control the main contributions of this technical note are that for one thing it shows that time delays in impulse term may contribute to the stabilization of delay systems that is a control strategy which does not work without delay feedback in impulse term can be activated to stabilize some unstable delay systems if there exist some time delay feedbacks for another it shows the robustness of impulsive control that is the designed control strategy admits the existence of some time delays in impulse term which may do harm to the stabilization in this technical note from impulsive control point of view we firstly propose an impulsive delay inequality then we apply it to the delay systems which may be originally unstable and derive some delay dependent impulsive control criteria to ensure the stabilization of the addressed systems the effectiveness of the proposed strategy is evidenced by two illustrative examples

compatibility of characteristic boundary conditions with radial equilibrium in turbomachinery simulations setting up outlet boundary conditions in configurations that have a strong rotating motion is a crucial issue for turbomachinery simulations this is usually done using the so called radial equilibrium assumption which is used before the simulation and provides an approximate expression for the pressure profile to impose in the outlet plane this paper shows that recent methods developed for compressible flows based on characteristic methods including the effects of transverse terms can capture the radial equilibrium naturally without having to impose a precomputed pressure profile in addition these methods are also designed to control acoustic reflections on boundaries and the present work suggests that they could replace classical radial equilibrium assumption approximations when nonreflecting boundary conditions are required at the outlet of a turbomachine simulation for example in large eddy simulation this is demonstrated in two cases 1 a simple annulus flow with a swirl imposed at the inlet and 2 a transonic turbine vane

potential theory on lipschitz domains in riemannian manifolds sobolev besov space results and the poisson problem abstract we continue a program to develop layer potential techniques for pde on lipschitz domains in riemannian manifolds building on l p and hardy space estimates established in previous papers here we establish sobolev and besov space estimates on solutions to the dirichlet and neumann problems for the laplace operator plus a potential on a lipschitz domain in a riemannian manifold with a metric tensor smooth of class c 1 for some 0 we treat the inhomogeneous problem and extend it to the setting of manifolds results obtained for the constant coefficient laplace operator on a lipschitz domain in euclidean space with the dirichlet boundary condition by d jerison and c kenig

high order low dispersive and low dissipative explicit schemes for multiple scale and boundary problems explicit high order numerical schemes are proposed for the accurate computation of multiple scale problems and for the implementation of boundary conditions specific high order node centered finite differences and selective filters removing grid to grid oscillations are first designed for the discretization of the buffer region between a x grid domain and 2 x grid domain the coefficients of these matching schemes are chosen so that the maximum order of accuracy is reached non centered finite differences and selective filters are then developed with the aim of accurately computing boundary conditions they are constructed by minimizing the dispersion and the dissipation errors in the wave number space for waves down to four points per wavelength the dispersion and dissipation properties of the matching and the boundary schemes are described in detail and their accuracy limits are determined to show that these schemes calculate accurately waves with at least five points per wavelength test problems including linear convection wall reflection and acoustic scattering around a cylinder are finally solved to illustrate the accuracy of the schemes

short note on the application of explicit spatial filtering to the variables or fluxes of linear equations the need for filtering high frequency waves is a recurrent issue in numerical simulations these waves might indeed lead to instability and they are in general not calculated accurately by the discretization algorithms selective filters have therefore been designed in order to damp high frequency waves without affecting significantly low frequency disturbances 1 6 these filters are particularly used in computational aeroacoustics but they appear also suitable for large eddy simulations les in which only the scales larger than the grid size are computed and whose equations are derived formally by applying a filter operator to the navier stokes equations 7 moreover les based specially on explicit filtering have been also developed 8 10 in practice the flow variables are usually filtered explicitly after each time step consider for example the time integration of the following differential equation

on the determinant of elliptic differential and finite difference operators in vector bundles over s 1 for an elliptic differential operatora overs 1 a sum limits k 0 n a k x d k witha k x in end r and as a principal angle the regularized determinant det a is computed in terms of the monodromy mapp a associated toa and some invariant expressed in terms ofa n anda n 1 a similar formula holds for finite difference operators a number of applications and implications are given in particular we present a formula for the signature ofa whena is self adjoint and show that the determinant ofa is the limit of a sequence of computable expressions involving determinants of difference approximation ofa

analysis of sponge zones for computational fluid mechanics the use of sponge regions or sponge zones which add the forcing term email protected q q r e f to the right hand side of the governing equations in computational fluid mechanics as an ad hoc boundary treatment is widespread they are used to absorb and minimize reflections from computational boundaries and as forcing sponges to introduce prescribed disturbances into a calculation a less common usage is as a means of extending a calculation from a smaller domain into a larger one such as in computing the far field sound generated in a localized region by analogy to the penalty method of finite elements the method is placed on a solid foundation complete with estimates of convergence the analysis generalizes the work of israeli and orszag m israeli s a orszag approximation of radiation boundary conditions j comp phys 41 1981 115 135 and confirms their findings when applied as a special case to one dimensional wave propagation in an absorbing sponge it is found that the rate of convergence of the actual solution to the target solution with an appropriate norm is inversely proportional to the sponge strength a detailed analysis for acoustic wave propagation in one dimension verifies the convergence rate given by the general theory the exponential point wise convergence derived by israeli and orszag in the high frequency limit is recovered and found to hold over all frequencies a weakly nonlinear analysis of the method when applied to burgers equation shows similar convergence properties three numerical examples are given to confirm the analysis the acoustic extension of a two dimensional time harmonic point source the acoustic extension of a three dimensional initial value problem of a sound pulse and the introduction of unstable eigenmodes from linear stability theory into a two dimensional shear layer

trace formulas for additive and non additive perturbations abstract trace formulas for pairs of self adjoint maximal dissipative and accumulative as well as other types of resolvent comparable operators are obtained in particular the existence of a complex valued spectral shift function for a pair h h of maximal accumulative operators has been proved we investigate also the existence of a real valued spectral shift function moreover we treat in detail the case of additive trace class perturbations assuming that h and h h v are maximal accumulative and v is trace class we prove the existence of a summable complex valued spectral shift function we also obtain trace formulas for pairs h h assuming only that h and h are resolvent comparable in this case the determinant of the characteristic function of h is involved in trace formulas the results rely on the technique of boundary triplets for densely and non densely defined closed symmetric operators the approach allows to express the spectral shift function of a pair of extensions in terms of the abstract weyl function and the boundary operator we improve and generalize certain classical results of m g kre n for pairs of self adjoint and dissipative operators results of a v rybkin for such pairs as well as of v m adamjan b s pavlov and m g kre n for pairs h h with a maximal dissipative operator h

a family of low dispersive and low dissipative explicit schemes for flow and noise computations explicit numerical methods for spatial derivation filtering and time integration are proposed they are developed with the aim of computing flow and noise with high accuracy and fidelity all the methods are constructed in the same way by minimizing the dispersion and the dissipation errors in the wavenumber space up to k x 2 corresponding to four points per wavelength they are shown to be more accurate and also more efficient numerically than most of the standard explicit high order methods for uniform and slowly non uniform grids two problems involving long range sound propagation are resolved to illustrate their respective precisions remarks about their practical applications are then made especially about the connection with the boundary conditions finally their relevance for the simulation of turbulent flows is emphasized

a computational study of three numerical methods for some advection diffusion problems three numerical methods have been used to solve two problems described by advection diffusion equations with specified initial and boundary conditions the methods used are the third order upwind scheme 5 fourth order upwind scheme 5 and non standard finite difference scheme nsfd 10 we considered two test problems the first test problem we considered has steep boundary layers near x 1 and this is challenging problem as many schemes are plagued by non physical oscillation near steep boundaries 16 many methods suffer from computational noise when modeling the second test problem we compute some errors namely l2 and l errors dissipation and dispersion errors total variation and the total mean square error for both problems we then use an optimization technique 1 to find the optimal value of the time step at a given value of the spatial step which minimizes the dispersion error and this is validated by numerical experiments

a uniqueness theorem for an inverse boundary value problem in electrical prospection we show that a near constant conductivity of a two dimensional body can be uniquely determined by steady state direct current measurements at the boundary mathematically we show that the coefficient in the operator is uniquely determined by its dirichlet integrals

generalized q functions and dirichlet to neumann maps for elliptic differential operators the classical concept of q functions associated to symmetric and selfadjoint operators due to m g krein and h langer is extended in such a way that the dirichletto neumann map in the theory of elliptic differential equations can be interpreted as a generalized q function for couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit krein type formulas for the difference of the resolvents and trace formulas in an h 2 framework are obtained

dirichlet to neumann maps on bounded lipschitz domains the dirichlet to neumann map associated to an elliptic partial differential equation becomes multivalued when the underlying dirichlet problem is not uniquely solvable the main objective of this paper is to present a systematic study of the dirichlet to neumann map and its inverse the neumann to dirichlet map in the framework of linear relations in hilbert spaces our treatment is inspired by abstract methods from extension theory of symmetric operators utilizes the general theory of linear relations and makes use of some deep results on the regularity of the solutions of boundary value problems on bounded lipschitz domains

bogdanov takens singularity in the hindmarsh rose neuron with time delay abstract in this paper we study the bogdanov takens singularity in the hindmarsh rose neuron model with time delay we use the center manifold reduction and the normal form method by means of which the dynamics near this nonhyperbolic equilibrium can be reduced to the study of the dynamics of the corresponding normal form restricted to the associated two dimensional center manifold we show that changes in the time delay length can lead to the saddle node bifurcation to the hopf bifurcation and to the homoclinic bifurcation

on the effective accuracy of spectral like optimized finite difference schemes for computational aeroacoustics the present article aims at highlighting the strengths and weaknesses of the so called spectral like optimized explicit central finite difference schemes when the latter are used for numerically approximating spatial derivatives in aeroacoustics evolution problems with that view we first remind how differential operators can be approximated using explicit central finite difference schemes the possible spectral like optimization of the latter is then discussed the advantages and drawbacks of such an optimization being theoretically studied before they are numerically quantified for doing so two popular spectral like optimized schemes are assessed via a direct comparison against their standard counterparts such a comparative exercise being conducted for several academic test cases at the end general conclusions are drawn which allows us discussing the way spectral like optimized schemes shall be preferred or not to standard ones when it comes to simulate real life aeroacoustics problems

generalized conservative approximations of split convective derivative operators we propose a strategy to design locally conservative finite difference approximations of convective derivatives for shock free compressible flows with arbitrary order of accuracy that generalizes the approach of ducros et al 2000 1 and that can be applied as a building block of low dissipative hybrid shock capturing methods the approximations stem from application of standard central difference formulas to split forms of the convective terms in the compressible euler equations which guarantee strong numerical stability and near energy preservation in the inviscid limit a convenient implementation of the high order fluxes is suggested which guarantees improved computational efficiency over existing methods numerical tests performed for isotropic turbulence at zero viscosity show stability of schemes with order of accuracy up to ten and effectiveness of convective splitting of kennedy and gruber 2008 2 in providing extra stability in the presence of strong density variations numerical simulations of compressible turbulent boundary layer flow indicate suitability of the method for non uniform grids and overall support superior computational efficiency of high order schemes

spectral enclosures for non self adjoint extensions of symmetric operators abstract the spectral properties of non self adjoint extensions a b of a symmetric operator in a hilbert space are studied with the help of ordinary and quasi boundary triples and the corresponding weyl functions these extensions are given in terms of abstract boundary conditions involving an in general non symmetric boundary operator b in the abstract part of this paper sufficient conditions for sectoriality and m sectoriality as well as sufficient conditions for a b to have a non empty resolvent set are provided in terms of the parameter b and the weyl function special attention is paid to weyl functions that decay along the negative real line or inside some sector in the complex plane and spectral enclosures for a b are proved in this situation the abstract results are applied to elliptic differential operators with local and non local robin boundary conditions on unbounded domains to schrodinger operators with potentials of complex strengths supported on unbounded hypersurfaces or infinitely many points on the real line and to quantum graphs with non self adjoint vertex couplings

stabilization of a meshless method via weighted least squares filtering abstract stability of a meshless method proposed for strong form of a linear hyperbolic partial differential equation can be ensured via simple filtering procedure in the weighted least squares sense this approach overcomes the need for the upwind implementation of numerical fluxes and thus employs central type scheme that is more suitable for methods constructing the approximations on local clouds of points the behavior of the filter and the overall performance of the method is verified on a set of benchmark problems where the propagation of acoustic waves is modeled by linearized euler equations and the results are compared with analytical and or numerical reference solutions

spectral theory for perturbed krein laplacians in nonsmooth domains abstract we study spectral properties for h k the krein von neumann extension of the perturbed laplacian v defined on c 0 where v is measurable bounded and nonnegative in a bounded open set r n belonging to a class of nonsmooth domains which contains all convex domains along with all domains of class c 1 r r 1 2 in particular in the aforementioned context we establish the weyl asymptotic formula j n k j 2 n v n n 2 o n 1 2 2 as where v n n 2 n 2 1 denotes the volume of the unit ball in r n and k j j n are the non zero eigenvalues of h k listed in increasing order according to their multiplicities we prove this formula by showing that the perturbed krein laplacian i e the krein von neumann extension of v defined on c 0 is spectrally equivalent to the buckling of a clamped plate problem and using an abstract result of kozlov from the mid 1980s our work builds on that of grubb in the early 1980s who has considered similar issues for elliptic operators in smooth domains and shows that the question posed by alonso and simon in 1980 pertaining to the validity of the above weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting we also study certain exterior type domains r n k n 3 with k r n compact and vanishing bessel capacity b 2 2 k 0 to prove equality of friedrichs and krein laplacians in l 2 d n x that is c 0 has a unique nonnegative self adjoint extension in l 2 d n x

semi implicit runge kutta schemes development and application to compressible channel flow time stepconstraintdueto thefine meshnear thewall the explicitsubschemeis asix stage fourth order low storage runge kutta scheme based on analysis in fourier space and results obtained for propagation test cases the semiimplicitschemesareshowntobeoforder3and forwavesdiscretizedbyanumberofpointsperperiodbetween4and 16 tobeasaccurateas ormoreaccuratethan thestandardexplicitfourth orderrunge kuttaalgorithmintermsof dissipationanddispersion thelarge eddysimulationofacompressibleturbulentchannelflowatafrictionreynolds numberof360andamachnumberof0 1isthencarriedoutwithoneoftheproposedalgorithms thecomputational time is reduced by a factor 1 33 with respect to a large eddy simulation using the explicit subscheme in all directions wall pressure and velocity spectra from the large eddy simulation are presented to give insights into the flow turbulent structures in particular wave number frequency spectra are calculated acoustic components appear to be identified in these spectra

inverse spectral theory for symmetric operators with several gaps scalar type weyl functions let s be the orthogonal sum of infinitely many pairwise unitarily equivalent symmetric operators with non zero deficiency indices let j be an open subset of r if there exists a self adjoint extension s0 of s such that j is contained in the resolvent set of s0 and the associated weyl function of the pair s s0 is monotone with respect to j then for any self adjoint operator r there exists a self adjoint extension s such that the spectral parts s j and rj are unitarily equivalent it is shown that for any extension s of s the absolutely continuous spectrum of s0 is contained in that one of s moreover for a wide class of extensions the absolutely continuous parts of s and s are even unitarily equivalent

on the convergence and optimization of the baker campbell hausdorff formula abstract in this paper the problem of the convergence of the baker campbell hausdorff series for z log e x e y is revisited we collect some previous results about the convergence domain and present a new estimate which improves all of them we also provide a new expression of the truncated lie presentation of the series up to sixth degree in x and y requiring the minimum number of commutators numerical experiments suggest that a similar accuracy is reached with this approximation at a considerably reduced computational cost

group velocity in finite difference schemes the relevance of group velocity to the behavior of finite difference models of time dependent partial differential equations is surveyed and illustrated applications involve the propagation of wave packets in one and two dimensions numerical dispersion the behavior of parasitic waves and the stability analysis of initial boundary value problems

inverse spectral theory for one dimensional schrodinger operators the a function we link recently developed approaches to the inverse spectral problem due to simon and myself respectively we obtain a description of the set of simon s a functions in terms of a positivity condition this condition also characterizes the solubility of simon s fundamental equation

eigenvalue problems on exterior domains and dirichlet to neumann maps we consider a schroedinger equation on an exterior domain in the case where the potential which may be complex valued has a limit at infinity associated with the problem is a dirichlet to neumann map on the inner boundary we examine approximations to the problem obtained by a truncating the domain b replacing the potential by its value at infinity outside some set c a combination of a and b these approximations give rise to approximate dirichlet to neumann maps on the inner boundary we analyze the convergence of these maps and deduce results on the approximation of the spectrum of the original problem by the spectra of the approximating problems we show that in theory spurious eigenvalues cannot be generated by these procedures numerical experiments show however that the inherent ill conditioning of a problem may cause spurious eigenvalues to appear due to the discretization of the truncated problems even when the potential decays exponentially fast

boundary value problems for elliptic partial differential operators on bounded domains for a symmetric operator or relation a with infinite deficiency indices in a hilbert space we develop an abstract framework for the description of symmetric and self adjoint extensions a of a as restrictions of an operator or relation t which is a core of the adjoint a this concept is applied to second order elliptic partial differential operators on smooth bounded domains and a class of elliptic problems with eigenvalue dependent boundary conditions is investigated

generalized resolvents and the boundary value problems for hermitian operators with gaps a hermitian operator a with gaps j j 1 j m is studied the self adjoint extensions which put exactly kj eigenvalues into each gap j j in particular for kj 0 1 j m the extensions preserving the gaps are described in terms of boundary conditions the generalized resolvents of the extensions with the indicated properties are described also a solvability criterion and description of all the solutions of the hamburger moment problem with supports in r j 1m j j are obtained in terms of the nevanlinna matrix

nonlocal robin laplacians and some remarks on a paper by filonov on eigenvalue inequalities abstract the aim of this paper is twofold first we characterize an essentially optimal class of boundary operators which give rise to self adjoint laplacians in l 2 d n x with nonlocal and local robin type boundary conditions on bounded lipschitz domains r n n n n 2 second we extend friedlander s inequalities between neumann and dirichlet laplacian eigenvalues to those between nonlocal robin and dirichlet laplacian eigenvalues associated with bounded lipschitz domains following an approach introduced by filonov for this type of problems

generalized characteristic relaxation boundary conditions for unsteady compressible flow simulations we develop numerical boundary conditions for the compressible navier stokes equations based on a generalized relaxation approach grcbc which hinges on locally one dimensional characteristic projection at the computational boundaries supplemented with available information from the flow exterior the basic idea is to estimate the amplitude of incoming characteristic waves through first order one sided finite difference approximations which involve the value of the reference flow state at the first exterior ghost point unlike other characteristic based relaxation methods the present one requires minimal user supplied input including the reference flow state which may be totally or partially known and in general may vary both in space and time furthermore it can be applied to any type of computational boundary either inflow or outflow either subsonic or supersonic the method is theoretically predicted to convey reduced reflection of waves at computational boundaries compared to other ones and to have better properties of frequency response to injected disturbances numerical tests confirm the improvement of the nonreflecting performance and demonstrate high degree of flexibility also for problems with non trivial far field boundary conditions e g flows in rotating reference frames and for the artificial stimulation of subsonic turbulent boundary layers

the baker campbell hausdorff formula and nested commutator identities the coefficients of the baker campbell hausdorff expansion are calculated by using various methods comparison of the results yields several remarkable identities satisfied by multiple commutators which in turn allow us to greatly simplify the form of the expansion up to order eight

a new approach to inverse spectral theory ii general real potentials and the connection to the spectral measure we continue the study of the a amplitude associated to a half line schrodinger operator d 2 dx 2 q in l 2 0 b b a is related to the weyl titchmarsh m function via m k 2 k a 0 a e 2 k d o e 2 k for all 0 we discuss five issues here first we extend the theory to general q in l 1 0 for all a including q s which are limit circle at infinity second we prove the following relation between the a amplitude and the spectral measure p a 2 1 2 sin 2 dp since the integral is divergent this formula has to be properly interpreted third we provide a laplace transform representation for m without error term in the case b fourth we discuss m functions associated to other boundary conditions than the dirichlet boundary conditions associated to the principal weyl titchmarsh m function finally we discuss some examples where one can compute a exactly

studying the performance of critical slowing down indicators in a biological system with a period doubling route to chaos abstract this paper aims to investigate critical slowing down indicators in different situations where the system s parameters change variation of the bifurcation parameter is important since it allows finding bifurcation points a system s parameters can vary through different functions in this paper five cases of bifurcation parameter variation are considered in a biological model with a period doubling route to chaos the first case is a slow and small stepwise variation of the bifurcation parameter the second case is a cyclic state dependent variation of the bifurcation parameter in the third case a small cyclic variation is combined with a sizeable stochastic resonance the fourth case involves variations by a large noise and finally in the fifth case significant stepwise changes in the parameter are studied to identify the conditions under which critical slowing down occurs an improved version of four well known critical slowing down indicators autocorrelation at lag 1 variance kurtosis and skewness are used the results show that when bifurcations are caused by a sudden change in a parameter or state critical slowing down cannot be observed before the bifurcation points however in cases with slowly varying parameters critical slowing down can be detected before the bifurcation points thus critical slowing down indicators can predict these bifurcation points in other words in three cases the system approaches bifurcation points slowly in other cases the bifurcations occur suddenly because of a significant shift in the parameter or state thus critical slowing down indicators cannot predict those bifurcation points however critical slowing down indicators can predict the bifurcation points in other cases

spectral analysis of selfadjoint elliptic differential operators dirichlet to neumann maps and abstract weyl functions abstract the spectrum of a selfadjoint second order elliptic differential operator in l 2 r n is described in terms of the limiting behavior of dirichlet to neumann maps which arise in a multi dimensional glazman decomposition and correspond to an interior and an exterior boundary value problem this leads to pde analogs of renowned facts in spectral theory of odes the main results in this paper are first derived in the more abstract context of extension theory of symmetric operators and corresponding weyl functions and are applied to the pde setting afterwards

effect of delayed impulses on input to state stability of nonlinear systems abstract this paper studies the input to state stability iss and integral input to state stability iiss of nonlinear systems with delayed impulses by using lyapunov method and the analysis technique proposed by hespanha et al 2005 some sufficient conditions ensuring iss iiss of the addressed systems are obtained those conditions establish the relationship between impulsive frequency and the time delay existing in impulses and reveal the effect of delayed impulses on iss iiss an example is provided to illustrate the efficiency of the obtained results

proximity to periodic windows in bifurcation diagrams as a gateway to coherence resonance in chaotic systems we show that chaotic states situated in the proximity of periodic windows in bifurcation diagrams are eligible for the observation of coherence resonance in particular additive gaussian noise of appropriate intensity can enhance the temporal order in such chaotic states in a resonant manner results obtained for the logistic map and the lorenz equations suggest that the presented mechanism of coherence resonance is valid beyond particularities of individual systems we attribute the findings to the increasing attraction of imminent periodic orbits and the ability of noise to anticipate their existence and use a modified wavelet analysis to support our arguments

high accuracy algorithms for computational aeroacoustics this paper presents an analysis of high bandwidth operators developed for use with an essentially nonoscillatory eno method the spatial operators of a sixth order eno code are modified to resolve waves with as few as 7 points per wavelength ppw by decreasing the formal order of the algorithm numerical and analytical solutions are compared for the model problems of plane wave propagation and sound generation by an oscillating sphere these problems involve linear propagation wave steepening and shock formation an analysis of the ppw required for sufficient accuracy shows that low order algorithms need an excessive number of grid points to produce acceptable solutions in contrast high order codes provide good predictions on relatively coarse grids the high bandwidth operators produce only modest improvements over the original sixth order operators for nonlinear problems in which wave steepening is significant however they clearly outperform the original operators for long distance linear propagation because the high bandwidth operators have the same stencil as the original sixth order operators these gains are achieved with no additional computational work

two dimensional scattering the number of bound states from scattering data relations are found between scattering data and the spectrum for the two dimensional schrodinger operator v x where v is a local noncentral potential in particular a two dimensional version of the levinson theorem is obtained this theorem gives the number of bound states in terms of the change in phase of the determinant of the scattering operator

dynkin s method of computing the terms of the baker campbell hausdorff series the infinite series for log exp x exp y for noncommuting x and y is expressible in terms of iterated commutators of x and y except for the linear term x y dynkin derived an explicit expression for the terms as a sum of iterated commutators over a certain set of sequences this paper presents a practical algorithm for applying dynkin s formula and gives several illustrative examples

functional determinants and geometry in this paper we study the zeta function determinant in the context of elliptic boundary value problems our main technique is to relate the determinant of an operator or a ratio of determinants to the boundary values of the solutions of the operator this has the advantage of restricting attention to the solutions of the operator which do not depend on the boundary conditions and can often be written down explicitly rather than the eigenvalues which are usually difficult to work with in addition the problem is reduced to a calculation over the boundary of the manifold which is a closed manifold of dimension one less than the original manifold this has special significance in the case that the manifold is a finite interval in this case the boundary is a pair of points and the determinant of an ordinary differential operator is expressed in terms of the determinant of a finite matrix

boundary value problems and hardy spaces associated to the helmholtz equation in lipschitz domains abstract in this paper we introduce and discuss in the clifford algebra framework certain hardy like spaces which are well suited for the study of the helmholtz equation u k 2 u 0 in lipschitz domains of r n 1 in particular in the second part of the paper these results are used in connection with the classical boundary value problems for the helmholtz equation in lipschitz domains in arbitrary space dimensions in this setting existence uniqueness and optimal estimates are obtained by inverting the corresponding layer potential operators on l p for sharp ranges of p s also a detailed discussion of the helmholtz eigenvalues of lipschitz domains is presented

relative zeta functions relative determinants and scattering theory we use the method of zeta function regularization to regularize the ratio det a det a0 of the determinants of two elliptic self adjoint operators a a0 satisfying certain natural assumptions this is of interest especially if the regularized determinants of the individual operators don t exist as for example in the case of elliptic operators on a noncompact manifold

resonances in one dimension and fredholm determinants we discuss resonances for schrodinger operators in whole and half line problems one of our goals is to connect the fredholm determinant approach of froese to the fourier transform approach of zworski another is to prove a result on the number of antibound states namely in a half line problem there are an odd number of antibound states between any two bound states

the morse and maslov indices for multidimensional schrodinger operators with matrix valued potentials we study the schrodinger operator l v on a star shaped domain in rd with lipschitz boundary the operator is equipped with quite general dirichletor robin type boundary conditions induced by operators between h1 2 and h 1 2 and the potential takes values in the set of symmetric n n matrices by shrinking the domain and rescaling the operator we obtain a path in the fredholm lagrangian grassmannian of the subspace of h1 2 h 1 2 corresponding to the given boundary condition the path is formed by computing the dirichlet and neumann traces of weak solutions to the rescaled eigenvalue equation we prove a formula relating the number of negative eigenvalues of l the morse index the signed crossings of the path the maslov index the number of negative eigenvalues of the potential matrix evaluated at the center of the domain and the number of negative eigenvalues of a bilinear form related to the boundary operator

effective computation of traces determinants and ζ functions for sturm liouville operators abstract the principal aim in this paper is to develop an effective and unified approach to the computation of traces of resolvents and resolvent differences fredholm determinants functions and function regularized determinants associated with linear operators in a hilbert space in particular we detail the connection between fredholm and function regularized determinants concrete applications of our formalism to general i e three coefficient regular sturm liouville operators on bounded intervals with various separated and coupled boundary conditions and schrodinger operators on a half line are provided and further illustrated with an array of examples

structural properties of statistically validated empirical information networks abstract we construct the empirical information network ein of traders using the order flow data of the constituent stocks of szse 100 index in 2013 a statistical validation method is applied to the edges of the network to filter out noises and uncover the intrinsic interaction behaviors of traders we investigate the correlation between topological structures and statistical properties for their largest connected components we find that the statistical validated network shows an assortative mixing pattern while the original network exhibits a disassortative mixing pattern we consider two definitions of edge weight for comparison but there is no significant difference in a same network we also analyze the mutual relationships among node degree edge weight and node strength

a high resolution code for turbulent boundary layers a new high resolution code for the direct simulation of incompressible boundary layers over a flat plate is described it can accommodate a wide range of pressure gradients and general time dependent boundary conditions such as incoming wakes or wall forcing the consistency orders of the advective and pressure correction steps are different but it is shown that the overall resolution is controlled by the higher order advection step the formulation of boundary conditions to ensure global mass conservation in the presence of arbitrary forcing is carefully analyzed two validation boundary layers with and without a strong adverse pressure gradient are presented with maximum reynolds numbers re q 2000 they agree well with the available experiments turbulent inflow conditions for the zero pressure case are implemented by a recycling method and it is shown that at least the initial 300 momentum thicknesses have to be discarded before the effect of the artificial inflow is forgotten it is argued that this is not a defect of the method used to generate the inflow but a property of the boundary layer

m operators a generalisation of weyl titchmarsh theory the theory of the weyl titchmarsh m function for second order ordinary differential operators is generalized and applied to partial differential operators of the form q x acting in three space dimensions weyl operators m z are defined as maps from l2 s 1 to l2 s 1 s 1 unit sphere in r3 for exterior and interior boundary value problems and for the partial differential operator acting in l2 r 3 with the standard weyl titchmarsh m function recovered in the special case that q is spherically symmetric the analysis is carried out rather explicitly allowing for the determination of precise norm bounds for m operators and for the proof of higher dimensional analogues of a number of the fundamental results of standard weyl titchmarsh theory

variations on a theme of jost and pais abstract we explore the extent to which a variant of a celebrated formula due to jost and pais which reduces the fredholm perturbation determinant associated with the schrodinger operator on a half line to a simple wronski determinant of appropriate distributional solutions of the underlying schrodinger equation generalizes to higher dimensions in this multi dimensional extension the half line is replaced by an open set r n n n n 2 where has a compact nonempty boundary satisfying certain regularity conditions our variant involves ratios of perturbation determinants corresponding to dirichlet and neumann boundary conditions on and invokes the corresponding dirichlet to neumann map as a result we succeed in reducing a certain ratio of modified fredholm perturbation determinants associated with operators in l 2 d n x n n to modified fredholm determinants associated with operators in l 2 d n 1 n 2 applications involving the birman schwinger principle and eigenvalue counting functions are discussed