eigenfunctions transfer matrices and absolutely continuous spectrum of one dimensional schrodinger operators on l r and its half line problem h on l2 0 with u 0 0 boundary conditions we will focus on a new approach to the absolutely continuous spectrum ac h and more generally ac h the essential support of the a c part of the spectral measures what is new in our approach is that it relies on estimates on the transfer matrix that is the 2 2 matrix te n m which takes u m 1 u m to u n 1 u n for solutions u of hu eu in the continuum case use u x u x instead of u x 1 u x we let te n te n 0 for example we will prove the following

on new relations between spectral properties of jacobi matrices and their coefficients we study the spectral properties of jacobi matrices by using higher order trace formulae we obtain a result relating the properties of the elements of jacobi matrices and the corresponding spectral measures complicated expressions for traces of some operators can be magically simplified allowing us to apply induction arguments our theorems are generalizations of a recent result of r killip and b simon 17

control of spiking regularity in a noisy complex neural network the effects of spatiotemporally correlated noise on the regularity of spiking oscillations are studied in a network composed of fitz hugh nagumo neurons the spiking regularity of the neural network becomes the best at a moderate noise intensity indicating the occurrence of coherence resonance cr the cr in a watts strogatz small world network is further improved by adding a small fraction of long range connections given a set of temporal correlation constant ensuremath tau and spatial correlation length ensuremath lambda of the noise there exists an optimal network topology randomness at which the spiking oscillations show the best regularity the optimal randomness of the network topology at different ensuremath tau and ensuremath lambda varies in a narrow range changing ensuremath lambda does not affect the optimal ensuremath tau for achieving the most regular spike train whereas varying ensuremath tau the best spiking regularity emerges at different optimal ensuremath lambda

asymptotics of the fourier transform of the spectral measure for schrodinger operators with bounded and unbounded sparse potentials we study the pointwise behaviour of the fourier transform of the spectral measure for discrete one dimensional schrodinger operators with unbounded sparse potentials particularly with the potentials of the special type which give rise to the spectra with hausdorff dimensionality between 1 2 and 1 operators with bounded sparse potentials are also considered

statistical properties of stochastic nonlinear dynamical models of single spiking neurons and neural networks dynamical stochastic models of single neurons and neural networks often take the form of a system of n 2 coupled stochastic differential equations we consider such systems under the assumption that third and higher order central moments are relatively small in the general case a system of 1 n n13 generally nonlinear coupled ordinary differential equations holds for the approximate means variances and covariances for the general linear system the solutions of these equations give exact results this is illustrated in a simple case generally the moment equations can be solved numerically results are given for a spiking fitzhugh nagumo model neuron driven by a current with additive white noise differential equations are obtained for the means variances and covariances of the dynamical variables in a network of n connected spiking neurons in the presence of noise s1063 651x 96 03511 8 pacs number s 87 10 1e 05 40 1j 02 50 2r

mean field approximation of two coupled populations of excitable units the analysis on stability and bifurcations in the macroscopic dynamics exhibited by the system of two coupled large populations composed of n stochastic excitable units each is performed by studying an approximate system obtained by replacing each population with the corresponding mean field model in the exact system one has the units within an ensemble communicating via the time delayed linear couplings whereas the interensemble terms involve the nonlinear time delayed interaction mediated by the appropriate global variables the aim is to demonstrate that the bifurcations affecting the stability of the stationary state of the original system governed by a set of 4n stochastic delay differential equations for the microscopic dynamics can accurately be reproduced by a flow containing just four deterministic delay differential equations which describe the evolution of the mean field based variables in particular the considered issues include determining the parameter domains where the stationary state is stable the scenarios for the onset and the time delay induced suppression of the collective mode as well as the parameter domains admitting bistability between the equilibrium and the oscillatory state we show how analytically tractable bifurcations occurring in the approximate model can be used to identify the characteristic mechanisms by which the stationary state is destabilized under different system configurations like those with symmetrical or asymmetrical interpopulation couplings

phase ordering on small world networks with nearest neighbor edges we investigate global phase coherence in a system of coupled oscillators on small world networks constructed from a ring with nearest neighbor edges the effects of both thermal noise and quenched randomness on phase ordering are examined and compared with the global coherence in the corresponding xy model without quenched randomness it is found that in the appropriate regime phase ordering emerges at finite temperatures even for a tiny fraction of shortcuts the nature of the phase transition is also discussed

the incorporation of the semi implicit linear equations into newton s method to solve radiation transfer equations abstract for large time steps the nonlinear equations of radiation transfer may not be solved adequately by the semi implicit linear approximation to yield physical solutions this deficiency is rectified in three steps the equations of the semi implicit linear method are modified the modified equations are incorporated into newton s method to solve nonlinear equations and the transfer equations are solved by the resulting method the new method also uses the photon free method to search for the solution in a lower dimensional space than the space of the underlying transfer equations two algorithms are developed from the new method they solve the modified semi implicit linear equations by different approaches the first is a physics approach it solves the linear equations approximately by the grey transport approximation the second is a mathematical approach it solves the linear equations exactly by the sherman morrison woodbury formula of linear algebra however both algorithms yield the solution to the nonlinear system derived by the photon free method therefore their solutions are equal to within the specified tolerance of the nonlinear solver moreover both methods can take advantage of the unconditional stability which comes with the implicit differencing of the time derivative the time step which both methods can take is much larger than the time step in which time discretization error is discernible we shall relate the mathematical approach to the photon free method numerical results for three test problems are presented

dynamics of encoding in neuron populations some general mathematical features the use of a population dynamics approach promises efficient simulation of large assemblages of neurons depending on the issues addressed and the degree of realism incorporated in the simulated neurons a wide range of different population dynamics formulations can be appropriate here we present a common mathematical structure that these various formulations share and that implies dynamical behaviors that they have in common this underlying structure serves as a guide toward efficient means of simulation as an example we derive the general population firing rate frequency response and show how it may be used effectively to address a broad range of interacting population response and stability problems a few specific cases will be worked out a summary of this work appears at the end before the appendix

singular continuous spectrum is generic in a variety of contexts we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators those with singular spectrum are a dense g delta

de localization in the prime schrodinger operator we report a combined numerical approach to study the localization properties of the one dimensional tight binding model with potential modulated along the prime numbers a localization delocalization transition was found as a function of the potential intensity it is also argued that there are delocalized states for any value of the potential intensity

synchronization induced by periodic inputs in finite n unit bistable langevin models the augmented moment method we have studied the synchronization induced by periodic inputs applied to a finite n unit coupled bistable langevin model which is subjected to cross correlated additive and multiplicative noises the effects on the synchronization of the system size n the coupling strength and the cross correlation between additive and multiplicative noise have been investigated using the semi analytical augmented moment method amm which is a second order moment approximation for local and global variables h hasegawa phys rev e 67 2003 041903 a linear analysis of the stationary solution of the amm equations shows that the stability is improved degraded by positive negative couplings results of the nonlinear bistable langevin model are compared to those of the linear langevin model

augmented moment method for stochastic ensembles with delayed couplings i langevin model by employing a semianalytical dynamical mean field approximation theory previously proposed by the author h hasegawa phys rev e 67 041903 2003 we have developed an augmented moment method amm in order to discuss dynamics of an n unit ensemble described by langevin equations with delays in an amm original n dimensional stochastic delay differential equations sddes are transformed to infinite dimensional deterministic des for means and correlations of local as well as global variables infinite order des arising from the non markovian property of sdde are terminated at the finite level m in the level m amm mathrm amm m which yields 3 m dimensional deterministic des model calculations have been made for linear and nonlinear langevin models the stationary solution of amm for the linear langevin model with n 1 is nicely compared to the exact result in the nonlinear langevin ensemble the synchronization is shown to be enhanced near the transition point between the oscillating and nonoscillating states results calculated by amm6 are in good agreement with those obtained by direct simulations

on stable parametric finite element methods for the stefan problem and the mullins sekerka problem with applications to dendritic growth we introduce a parametric finite element approximation for the stefan problem with the gibbs thomson law and kinetic undercooling which mimics the underlying energy structure of the problem the proposed method is also applicable to certain quasi stationary variants such as the mullins sekerka problem in addition fully anisotropic energies are easily handled the approximation has good mesh properties leading to a well conditioned discretization even in three space dimensions several numerical computations including for dendritic growth and for snow crystal growth are presented

a homotopy perturbation method for solving a neutron transport equation the purpose of this paper consists in the finding of the solution for a stationary transport equation using the techniques of homotopy perturbation method hpm the results of a numerical example illustrate the accuracy and computational efficiency of the new proposed method

a linear discontinuous spatial differencing scheme forsnradiative transfer calculations various types of linear discontinuous spatial differencing schemes have been developed for thesn discrete ordinates equations approximating the linear boltzmann transport equation it has been shown through an asymptotic analysis that the 1d slab geometry lumped linear discontinuous scheme not only goes over to a convergent and robust differencing of the diffusion equation in the monoenergetic thick diffusion limit but it also yields the correct interior solution even when boundary layers are left unresolved by the spatial mesh in the present work we generalize this scheme to obtain a 1d slab geometry lumped linear discontinuous scheme for the nonlinear radiative transfer equation and the associated material temperature equation we present a full nonlinear energy dependent asymptotic analysis of the behavior of this scheme in the thick equilibrium diffusion limit we find that this scheme goes over to a convergent and robust differencing of the equilibrium diffusion equation on the interior of the mesh but it does not yield the exact interior solution when boundary layers are left unresolved by the spatial mesh nevertheless the interior solution obtained with spatially unresolved boundary layers is always well behaved and fairly accurate computational results are presented which test the predictions of our asymptotic analysis and demonstrate the efficiency of our solution technique

asymptotic equilibrium diffusion analysis of time dependent monte carlo methods for grey radiative transfer the equations of nonlinear time dependent radiative transfer are known to yield the equilibrium diffusion equation as the leading order solution of an asymptotic analysis when the mean free path and mean free time of a photon become small we apply this same analysis to the fleck cummings carter forest and n kaoua monte carlo approximations for grey frequency independent radiative transfer although monte carlo simulation usually does not require the discretizations found in deterministic transport techniques monte carlo methods for radiative transfer require a time discretization due to the nonlinearities of the problem if an asymptotic analysis of the equations used by a particular monte carlo method yields an accurate time discretized version of the equilibrium diffusion equation the method should generate accurate solutions if a time discretization is chosen that resolves temperature changes even if the time steps are much larger than the mean free time of a photon this analysis is of interest because in many radiative transfer problems it is a practical necessity to use time steps that are large compared to a mean free time our asymptotic analysis shows that i the n kaoua method has the equilibrium diffusion limit ii the carter forest method has the equilibrium diffusion limit if the material temperature change during a time step is small and iii the fleck cummings method does not have the equilibrium diffusion limit we include numerical results that verify our theoretical predictions

asymptotic diffusion limit of the symbolic monte carlo method for the transport equation we use asymptotic analysis to study the diffusion limit of the symbolic implicit monte carlo simc method for the transport equation for standard simc with piecewise constant basis functions we demonstrate mathematically that the solution converges to the solution of a wrong diffusion equation nevertheless a simple extension to piecewise linear basis functions enables to obtain the correct solution we present numerical examples which illustrate the analysis

dynamical mean field approximation to small world networks of spiking neurons from local to global and or from regular to random couplings by extending a dynamical mean field approximation previously proposed by the author h hasegawa phys rev e 67 041903 2003 we have developed a semianalytical theory which takes into account a wide range of couplings in a small world network our network consists of noisy n unit fitzhugh nagumo neurons with couplings whose average coordination number z may change from local z n to global couplings z n ensuremath 1 and or whose concentration of random couplings p is allowed to vary from regular p 0 to completely random p 1 we have taken into account three kinds of spatial correlations the on site correlation the correlation for a coupled pair and that for a pair without direct couplings the original 2n dimensional stochastic differential equations are transformed to 13 dimensional deterministic differential equations expressed in terms of means variances and covariances of state variables the synchronization ratio and the firing time precision for an applied single spike have been discussed as functions of z and p our calculations have shown that with increasing p the synchronization is worse because of increased heterogeneous couplings although the average network distance becomes shorter results calculated by our theory are in good agreement with those by direct simulations

modified prufer and efgp transforms and the spectral analysis of one dimensional schrodinger operators using control of the growth of the transfer matrices wediscuss the spectral analysis of continuum and discrete half line schrodinger operators with slowly decaying potentials among our results we show if where w has compact support and then h has purely a c resp purely s c spectrum on 0 if for n 1 2 a n potentials where a n are independent identically distributed random variables with e a n 0 e a 2 n 1 and 2 we find singular continuous spectrum with explicitly computable fractional hausdorff dimension

transient and recurrent spectrum abstract we deal primarily with spectral analysis of an abstract self adjoint operator h on a hilbert space h we propose a further refinement of the absolutely continuous subspace h ac into the transient absolutely continuous subspace h tac which is the closure of those with e ith o t n for all n and the recurrent absolutely continuous subspace h rac h ac h tac we discuss general features of this breakup in a subsequent paper we construct analytic almost periodic functions v on so that h d 2 dx 2 v x on l 2 has only recurrent absolutely continuous spectrum in the sense that h rac h

operators with singular continuous spectrum vii examples with borderline time decay we construct one dimensional potentials v x so that if h d 2 dx 2 v x on l 2 then h has purely singular spectrum but for a dense set d d implies that ith c t 1 2 in t for t 2 this implies the spectral measures have hausdorff dimension one and also following an idea of malozemov molchanov provides counterexamples to the direct extension of the theorem of simon spencer on one dimensional infinity high barriers

an extension of implicit monte carlo diffusion multigroup and the difference formulation implicit monte carlo imc and implicit monte carlo diffusion imd are approaches to the numerical solution of the equations of radiative transfer imd was previously derived and numerically tested on grey or frequency integrated problems 1 in this research we extend implicit monte carlo diffusion imd to account for frequency dependence and we implement the difference formulation 2 as a source manipulation variance reduction technique we derive the relevant probability distributions and present the frequency dependent imd algorithm with and without the difference formulation the imd code with and without the difference formulation was tested using both grey and frequency dependent benchmark problems the su and olson semi analytic marshak wave benchmark was used to demonstrate the validity of the code for grey problems 3 the su and olson semi analytic picket fence benchmark was used for the frequency dependent problems 4 the frequency dependent imd algorithm reproduces the results of both su and olson benchmark problems frequency group refinement studies indicate that the computational cost of refining the group structure is likely less than that of group refinement in deterministic solutions of the radiation diffusion methods our results show that applying the difference formulation to the imd algorithm can result in an overall increase in the figure of merit for frequency dependent problems however the creation of negatively weighted particles from the difference formulation can cause significant numerical instabilities in regions of the problem with sharp spatial gradients in the solution an adaptive implementation of the difference formulation may be necessary to focus its use in regions that are at or near thermal equilibrium

subdiffusive quantum transport for 3d hamiltonians with absolutely continuous spectra we exhibit a class of hamiltonians in dimension d 3 describing a quantum particle in an aperiodic medium with absolutely continuous spectrum and subdiffusive behavior the diffusion exponent which characterizes the time growth of the mean square displacement can be chosen slightly bigger than guarneri s lower bound these models are built out of 1d hamiltonians with well understood spectral and transport properties

dynamics of moments of fitzhugh nagumo neuronal models and stochastic bifurcations for the study of the behavior of noisy neuronal models rodriguez and tuckwell have introduced an elegant and systematic method which consists of replacing the system of stochastic differential equations with a system of deterministic equations representing the dynamics of the means variances and covariance of the state variables r rodriguez and h c tuckwell phys rev e 54 5585 1996 in this work we first report a modification of their method in the case of the fitzhugh nagumo model which enhances the accuracy of the approximation without including higher order moments this method is then combined with a self consistency argument in order to better characterize the behavior of the underlying stochastic processes through the computation of approximate auto and cross correlation functions of the state variables finally we argue that the moments equations can also reveal the existence of stochastic bifurcations i e qualitative changes in the dynamics of stochastic systems

the absolutely continuous spectrum of jacobi matrices i explore some consequences of a groundbreaking result of brei messer and pearson on the absolutely continuous spectrum of one dimensional schrodinger operators these include an oracle theorem that predicts the potential and rather general results on the approach to certain limit potentials in particular we prove a denisov rakhmanov type theorem for the general nite gap case the main theme is the following it is extremely dicult to produce abso lutely continuous spectrum in one space dimension and thus its existence has strong implications

dynamical analysis of schrodinger operators with growing sparse potentials we consider discrete half line schrodinger operators h with potentials of the form v n s n q n here q is any compactly supported real function open image in new window if n l n and s n 0 otherwise where 0 1 and l n is a very fast growing sequence we study in a rather detailed manner the time averaged dynamics exp ith for various initial states in particular for some we calculate explicitly the intermittency function p which turns out to be nonconstant the dynamical results obtained imply that the spectral measure of h has exact hausdorff dimension for all boundary conditions improving the result of jitomirskaya and last

network induced chaos in integrate and fire neuronal ensembles it has been shown that a single standard linear integrate and fire if neuron under a general time dependent stimulus cannot possess chaotic dynamics despite the firing reset discontinuity here we address the issue of whether conductance based pulsed coupled network interactions can induce chaos in an if neuronal ensemble using numerical methods we demonstrate that all to all homogeneously pulse coupled if neuronal networks can indeed give rise to chaotic dynamics under an external periodic current drive we also provide a precise characterization of the largest lyapunov exponent for these high dimensional nonsmooth dynamical systems in addition we present a stable and accurate numerical algorithm for evaluating the largest lyapunov exponent which can overcome difficulties encountered by traditional methods for these nonsmooth dynamical systems with degeneracy induced by e g refractoriness of neurons

a probabilistic approach to one dimensional schrodinger operators with sparse potentials we consider the one dimensional schrodinger equation with sparse potential v i e mainly v 0 it is shown that the asymptotics of the solutions corresponding to positive energies e can be approximately described by an infinite sum of independent random variables using results from probability theory we can then determine the spectral properties of the operators under consideration we prove absolute continuity for a general class of potentials and we also have examples with singular continuous spectrum

stationary and dynamical properties of finite n unit langevin models subjected to multiplicative noises abstract we have studied the finite n unit langevin model subjected to multiplicative noises by using the augmented moment method amm as a continuation of our previous paper h hasegawa j phys soc japan 75 2006 033001 effects of couplings on stationary and dynamical properties of the model have been investigated the difference and similarity between the results of diffusive and sigmoid couplings are studied in details time dependences of average and fluctuations in local and global variables calculated by the amm are in good agreement with those of direct simulations dss we also discuss stationary distributions of local and global variables with the use of the fokker planck equation fpe method and dss it is demonstrated that stationary distributions show much variety when multiplicative noise and external inputs are taken into account

spatial discretizations for self adjoint forms of the radiative transfer equations there are three commonly recognized second order self adjoint forms of the neutron transport equation the even parity equations the odd parity equations and the self adjoint angular flux equations because all of these equations contain second order spatial derivatives and are self adjoint for the mono energetic case standard continuous finite element discretization techniques have proved quite effective when applied to the spatial variables we first derive analogs of these equations for the case of time dependent radiative transfer the primary unknowns for these equations are functions of the angular intensity rather than the angular flux hence the analog of the self adjoint angular flux equation is referred to as the self adjoint angular intensity equation then we describe a general arbitrary order continuous spatial finite element approach that is applied to each of the three equations in conjunction with backward euler differencing in time we refer to it as the standard technique we also introduce an alternative spatial discretization scheme for the self adjoint angular intensity equation that requires far fewer unknowns than the standard method but appears to give comparable accuracy computational results are given that demonstrate the validity of both of these discretization schemes

a piecewise linear finite element discretization of the diffusion equation for arbitrary polyhedral grids we develop a piecewise linear pwl galerkin finite element spatial discretization for the multi dimensional radiation diffusion equation it uses recently introduced piecewise linear weight and basis functions in the finite element approximation and it can be applied on arbitrary polygonal 2d or polyhedral 3d grids we first demonstrate some analytical properties of the pwl method and perform a simple mode analysis to compare the pwl method with palmer s vertex centered finite volume method and with a bilinear continuous finite element method we then show that this new pwl method gives solutions comparable to those from palmer s however since the pwl method produces a symmetric positive definite coefficient matrix it should be substantially more computationally efficient than palmer s method which produces an asymmetric matrix we conclude that the galerkin pwl method is an attractive option for solving diffusion equations on unstructured grids

symbolic implicit monte carlo radiation transport in the difference formulation a piecewise constant discretization the equations of radiation transport for thermal photons are notoriously difficult to solve in thick media without resorting to asymptotic approximations such as the diffusion limit one source of this difficulty is that in thick absorbing media thermal emission and absorption are almost completely balanced a new formulation for thermal radiation transport called the difference formulation was recently introduced in order to remove the stiff balance between emission and absorption in the new formulation thermal emission is replaced by derivative terms that become small in thick media it was proposed that the difficulties of solving the transport equation in thick media would be ameliorated by the difference formulation while preserving full rigor and accuracy of the transport solution in the streaming limit in this paper the transport equation is solved by the symbolic implicit monte carlo method and comparisons are made between the standard formulation and the difference formulation the method is easily adapted to the derivative source terms of the difference formulation and a remarkable reduction in noise is obtained when the difference formulation is applied to problems involving thick media

augmented moment method for stochastic ensembles with delayed couplings ii fitzhugh nagumo model dynamics of fitzhugh nagumo fn neuron ensembles with time delayed couplings subject to white noises has been studied by using both direct simulations and a semianalytical augmented moment method amm which has been proposed in a preceding paper h hasegawa phys rev e 70 021911 2004 for n unit fn neuron ensembles amm transforms original 2n dimensional stochastic delay differential equations sddes to infinite dimensional deterministic des for means and correlation functions of local and global variables infinite order recursive des are terminated at the finite level m in the level m amm mathrm amm m yielding 8 m 1 dimensional deterministic des when a single spike is applied the oscillation may be induced if parameters of coupling strength delay noise intensity and or ensemble size are appropriate effects of these parameters on the emergence of the oscillation and on the synchronization in fn neuron ensembles have been studied the synchronization shows the fluctuation induced enhancement at the transition between nonoscillating and oscillating states results calculated by amm5 are in fairly good agreement with those obtained by direct simulations

operators with singular continuous spectrum i general operators the baire category theorem implies that the family f of dense sets g in fixed metric space x is a candidate for generic sets since it is closed under countable intersections and if x is perfect has no isolated point then a f has uncountable intersections with any open ball in x

value distribution and spectral analysis of differential operators the theory of value distribution is applied through a study of boundary value distribution for the weyl titchmarsh m function to the spectral analysis of differential operators on the half line particular consequences are the theoretical underpinning of a numerical approach to spectral analysis of differential operators exhibiting absolutely continuous or singular spectra and new identities for the spectral measure in terms of solutions of the associated differential equation

sparse potentials with fractional hausdorff dimension abstract we construct non random bounded discrete half line schrodinger operators which have purely singular continuous spectral measures with fractional hausdorff dimension in some interval of energies to do this we use suitable sparse potentials our results also apply to whole line operators as well as to certain random operators in the latter case we prove and compute an exact dimension of the spectral measures

a residual monte carlo method for discrete thermal radiative diffusion residual monte carlo methods reduce statistical error at a rate of exp bn where b is a positive constant and n is the number of particle histories contrast this convergence rate with 1 n which is the rate of statistical error reduction for conventional monte carlo methods thus residual monte carlo methods hold great promise for increased efficiency relative to conventional monte carlo methods previous research has shown that the application of residual monte carlo methods to the solution of continuum equations such as the radiation transport equation is problematic for all but the simplest of cases however the residual method readily applies to discrete systems as long as those systems are monotone i e they produce positive solutions given positive sources we develop a residual monte carlo method for solving a discrete id non linear thermal radiative equilibrium diffusion equation and we compare its performance with that of the discrete conventional monte carlo method upon which it is based we find that the residual method provides efficiency gains of many orders of magnitude part of the residual gain is due to the fact that we begin each timestep with an initial guess equal to the solution from the previous timestep moreover fully consistent non linear solutions can be obtained in a reasonable amount of time because of the effective lack of statistical noise we conclude that the residual approach has great potential and that further research into such methods should be pursued for more general discrete and continuum systems

robust and accurate filtered spherical harmonics expansions for radiative transfer we present a novel application of filters to the spherical harmonics p n expansion for radiative transfer problems in the high energy density regime the filter we use is based on non oscillatory spherical splines and a filter strength chosen to i preserve the equilibrium diffusion limit and ii vanish as the expansion order tends to infinity our implementation is based on modified equations that are derived by applying the filter after every time step in a simple first order time integration scheme the method is readily applied to existing codes that solve the p n equations numerical results demonstrate that the solution to the filtered p n equations are i more robust and less oscillatory than standard p n solutions and ii more accurate than discrete ordinates solutions of comparable order in particular the filtered p 7 solution demonstrates comparable accuracy to an implicit monte carlo solution for a benchmark hohlraum problem in 2d cartesian geometry

the approach of a neuron population firing rate to a new equilibrium an exact theoretical result the response of a noninteracting population of identical neurons to a step change in steady synaptic input can be analytically calculated exactly from the dynamical equation that describes the population s evolution in time here for model integrate and fire neurons that undergo a fixed finite upward shift in voltage in response to each synaptic event we compare the theoretical prediction with the result of a direct simulation of 90 000 model neurons the degree of agreement supports the applicability of the population dynamics equation the theoretical prediction is in the form of a series convergence is rapid so that the full result is well approximated by a few terms

numerical analysis of time integration errors for nonequilibrium radiation diffusion numerical analysis of time integration errors for nonequilibrium radiation diffusion is considered two first order implicit time integration methods are studied asymptotic analysis and modified equation analysis are applied to both time integration methods numerical experiments are used to highlight the results of the analysis asymptotic analysis is used to highlight the source of temperature spiking when a hot radiation wave propagates into a cold material modified equation analysis is used to provide insight into the thermal wave speed coming from the two different first order methods

absolutely continuous convolutions of singular measures and an application to the square fibonacci hamiltonian author s damanik d gorodetski a solomyak b abstract 2015 we prove for the square fibonacci hamiltonian that the density of states measure is absolutely continuous for almost all pairs of small coupling constants this is obtained from a new result we establish about the absolute continuity of convolutions of measures arising in hyperbolic dynamics with exact dimensional measures

singular continuous spectrum for palindromic schrodinger operators we give new examples of discrete schrodinger operators with potentials taking finitely many values that have purely singular continuous spectrum if the hullx of the potential is strictly ergodic then the existence of just one potentialx inx for which the operator has no eigenvalues implies that there is a generic set inx for which the operator has purely singular continuous spectrum a sufficient condition for the existence of such anx is that there is az x that contains arbitrarily long palindromes thus we can define a large class of primitive substitutions for which the operators are purely singularly continuous for a generic subset inx the class includes well known substitutions like fibonacci thue morse period doubling binary non pisot and ternary non pisot we also show that the operator has no absolutely continuous spectrum for allx x ifx derives from a primitive substitution for potentials defined by circle maps x n 1 j 0 n we show that the operator has purely singular continuous spectrum for a generic subset inx for all irrational and every half open intervalj

schrodinger operators with sparse potentials asymptotics of the fourier transform of the spectral measure we study the pointwise behavior of the fourier transform of the spectral measure for discrete one dimensional schrodinger operators with sparse potentials we find a resonance structure which admits a physical interpretation in terms of a simple quasiclassical model we also present an improved version of known results on the spectrum of such operators

piecewise linear discretization of symbolic implicit monte carlo radiation transport in the difference formulation we describe a monte carlo solution for time dependent photon transport in the difference formulation with the material in local thermodynamic equilibrium that is piecewise linear in its treatment of the material state variable our method employs a galerkin solution for the material energy equation while using symbolic implicit monte carlo to solve the transport equation in constructing the scheme one has the freedom to choose between expanding the material temperature or the equivalent black body radiation energy density at the material temperature in terms of finite element basis functions the former provides a linear treatment of the material energy while the latter provides a linear treatment of the radiative coupling between zones subject to the conditional use of a lumped material energy in the vicinity of strong gradients possible with a linear treatment of the material energy our approach provides a robust solution for time dependent transport of thermally emitted radiation that can address a wide range of problems it produces accurate results in thick media

random long range connections induce activity of complex hindmarsh rose neural networks abstract in this paper we investigate how activity of complex neural networks depends on random long range connections network elements are described by hindmarsh rose hr neurons assumed to be inactive it is found that for a given coupling strength when the number of random connections or randomness is greater than a threshold the spiking neurons which are absent in the nearest neighbor neural network occur the spiking activity becomes stronger in intensity and higher in frequency as the randomness is further increased these phenomena imply that random long range connections can induce and enhance the activity of neural networks furthermore the possible mechanism behind the action of random long range connections is also addressed our results may provide a useful hint for understanding the properties of collective dynamics in coupled real neurons

a linear stability analysis for nonlinear grey thermal radiative transfer problems we present a new linear stability analysis of three time discretizations and monte carlo interpretations of the nonlinear grey thermal radiative transfer trt equations the widely used implicit monte carlo imc equations the carter forest cf equations and the ahrens larsen or semi analog monte carlo smc equations using a spatial fourier analysis of the 1 d implicit monte carlo imc equations that are linearized about an equilibrium solution we show that the imc equations are unconditionally stable undamped perturbations do not exist if the imc time discretization parameter satisfies 0 5 1 this is consistent with conventional wisdom however we also show that for sufficiently large time steps unphysical damped oscillations can exist that correspond to the lowest frequency fourier modes after numerically confirming this result we develop a method to assess the stability of any time discretization of the 0 d nonlinear grey thermal radiative transfer problem subsequent analyses of the cf and smc methods then demonstrate that the cf method is unconditionally stable and monotonic but the smc method is conditionally stable and permits unphysical oscillatory solutions that can prevent it from reaching equilibrium this stability theory provides new conditions on the time step to guarantee monotonicity of the imc solution although they are likely too conservative to be used in practice theoretical predictions are tested and confirmed with numerical experiments

sparse block jacobi matrices with arbitrarily accurate hausdorff dimension abstract we show that the hausdorff dimension of the spectral measure of a class of deterministic i e nonrandom block jacobi matrices may be determined with any degree of precision improving a result of zlatos andrej zlatos sparse potentials with fractional hausdorff dimension j funct anal 207 2004 216 252

dynamical mean field theory of noisy spiking neuron ensembles application to the hodgkin huxley model a dynamical mean field approximation dma previously proposed by the present author h hasegawa phys rev e 67 041903 2003 has been extended to ensembles described by a general noisy spiking neuron model ensembles of n unit neurons each of which is expressed by coupled k dimensional differential equations des are assumed to be subject to spatially correlated white noises the original mathrm kn dimensional stochastic des have been replaced by k k 2 dimensional deterministic des expressed in terms of means and the second order moments of local and global variables the fourth order contributions are taken into account by the gaussian decoupling approximation our dma has been applied to an ensemble of hodgkin huxley hh neurons k 4 for which effects of the noise the coupling strength and the ensemble size on the response to a single spike input have been investigated numerical results calculated by the dma theory are in good agreement with those obtained by direct simulations although the former computation is about a thousand times faster than the latter for a typical hh neuron ensemble with n 100

dynamical mean field theory of spiking neuron ensembles response to a single spike with independent noises a semianalytical dynamical mean field theory has been developed for a study of dynamics of an ensemble of n unit fitzhugh nagumo neurons subject to white noises assuming weak noises and the gaussian distribution of state variables we have driven equations of motions for moments of local and global variables dynamical mean field approximation dma has replaced original 2n dimensional stochastic differential equations des by eight dimensional deterministic des whereas the conventional moment method yields n 2n 3 dimensional deterministic des for local variables we have discussed the dependence of the spike firing precision and the synchronization on the noise intensity synaptic coupling and the size of the neuron ensemble the spike timing precision is shown to be improved by increasing the size of the neuron ensemble even when there are no couplings among neurons when the coupling is introduced neurons in ensembles respond to an input spike with a partial synchronization results calculated by our dma theory are in good agreement with those obtained by direct simulations dma theory is extended to a large cluster which can be divided into multiple subclusters according to their functions a model calculation has demonstrated that when the noise intensity is moderate the spike propagation with a fairly precise timing is possible among noisy subclusters with feed forward couplings as in the synfire chain we have compared dma theory with the conventional moment method showing that the former may be alternatively derived from the latter by a reduction in the number of moments with the mean field approximation

asymptotic diffusion limit of cell temperature discretisation schemes for thermal radiation transport abstract this paper attempts to unify the asymptotic diffusion limit analysis of thermal radiation transport schemes for a linear discontinuous representation of the material temperature reconstructed from cell centred temperature unknowns in a process known as source tilting the asymptotic limits of both monte carlo continuous in space and deterministic approaches based on linear discontinuous finite elements for solving the transport equation are investigated in slab geometry the resulting discrete diffusion equations are found to have nonphysical terms that are proportional to any cell edge discontinuity in the temperature representation based on this analysis it is possible to design accurate schemes for representing the material temperature for coupling thermal radiation transport codes to a cell centred representation of internal energy favoured by ale arbitrary lagrange eulerian hydrodynamics schemes

operators with singular continuous spectrum ii rank one operators for an operator a with cyclic vector we studya p wherep is the rank one projection onto multiples of if spec a anda has no a c spectrum we prove thata p has purely singular continuous spectrum on for a denseg of s

a monte carlo synthetic acceleration method for solving the thermal radiation diffusion equation we present a novel synthetic acceleration based monte carlo method for solving the equilibrium thermal radiation diffusion equation in three spatial dimensions the algorithm performance is compared against traditional solution techniques using a marshak benchmark problem and a more complex multiple material problem our results show that our monte carlo method is an effective solver for sparse matrix systems for solutions converged to the same tolerance it performs competitively with deterministic methods including preconditioned conjugate gradient and gmres we also discuss various aspects of preconditioning the method and its general applicability to broader classes of problems

trace class perturbations and the absence of absolutely continuous spectra we show that various hamiltonians and jacobi matrices have no absolutely continuous spectrum by showing that under a trace class perturbation they become a direct sum of finite matrices

a deterministic photon free method to solve radiation transfer equations a new method to solve radiation transfer equations is presented in the absence of scattering material motion and heat conduction the photon variables can be eliminated from the fully implicit multi group discrete ordinate finite difference finite element equations of continuum radiation transfer to yield a smaller set of equations which depends only on temperature the solution to this smaller set of equations is used to generate the solution to the original set of equations from which the reduced set is derived the reduced system simplifies to a nonlinear heat equation in the regime of strong absorption and strong emission we solve the reduced set of equations by the newton gmres method in which the jacobian update is preconditioned by a linearization of this nonlinear heat equation the performances of this new method and of the semi implicit linear method which is preconditioned by grey transport acceleration combined with diffusion synthetic acceleration are compared on two test problems the test results indicate that the new method can take larger time steps requires less memory is more accurate and is competitive in speed with the semi implicit linear method

operators with singular continuous spectrum iii almost periodic schrodinger operators we prove that one dimensional schrodinger operators with even almost periodic potential have no point spectrum for a dense g in the hull this implies purely singular continuous spectrum for the almost mathieu equation for coupling larger than 2 and a dense g in even if the frequency is an irrational with good diophantine properties

a hybrid transport diffusion method for monte carlo radiative transfer simulations discrete diffusion monte carlo ddmc is a technique for increasing the efficiency of monte carlo particle transport simulations in diffusive media if standard monte carlo is used in such media particle histories will consist of many small steps resulting in a computationally expensive calculation in ddmc particles take discrete steps between spatial cells according to a discretized diffusion equation each discrete step replaces many small monte carlo steps thus increasing the efficiency of the simulation in addition given that ddmc is based on a diffusion equation it should produce accurate solutions if used judiciously in practice ddmc is combined with standard monte carlo to form a hybrid transport diffusion method that can accurately simulate problems with both diffusive and non diffusive regions in this paper we extend previously developed ddmc techniques in several ways that improve the accuracy and utility of ddmc for nonlinear time dependent radiative transfer calculations the use of ddmc in these types of problems is advantageous since due to the underlying linearizations optically thick regions appear to be diffusive first we employ a diffusion equation that is discretized in space but is continuous in time not only is this methodology theoretically more accurate than temporally discretized ddmc techniques but it also has the benefit that a particle s time is always known thus there is no ambiguity regarding what time to assign a particle that leaves an optically thick region where ddmc is used and begins transporting by standard monte carlo in an optically thin region also we treat the interface between optically thick and optically thin regions with an improved method based on the asymptotic diffusion limit boundary condition that can produce accurate results regardless of the angular distribution of the incident monte carlo particles finally we develop a technique for estimating radiation momentum deposition during the ddmc simulation a quantity that is required to calculate correct fluid motion in coupled radiation hydrodynamics problems with a set of numerical examples we demonstrate that our improved ddmc method is accurate and can provide efficiency gains of several orders of magnitude over standard monte carlo

power law bounds on transfer matrices and quantum dynamics in one dimension ii abstract we establish quantum dynamical lower bounds for a number of discrete one dimensional schrodinger operators these dynamical bounds are derived from power law upper bounds on the norms of transfer matrices we develop further the approach from part i and study many examples particular focus is put on models with finitely or at most countably many exceptional energies for which one can prove power law bounds on transfer matrices the models discussed in this paper include substitution models sturmian models a hierarchical model the prime model and a class of moderately sparse potentials

population dynamics under the laplace assumption in this paper we describe a generic approach to modelling dynamics in neuronal populations this approach models a full density on the states of neuronal populations but finesses this high dimensional problem by re formulating density dynamics in terms of ordinary differential equations on the sufficient statistics of the densities considered c f the method of moments the particular form for the population density we adopt is a gaussian density c f the laplace assumption this means population dynamics are described by equations governing the evolution of the population s mean and covariance we derive these equations from the fokker planck formalism and illustrate their application to a conductance based model of neuronal exchanges one interesting aspect of this formulation is that we can uncouple the mean and covariance to furnish a neural mass model which rests only on the populations mean this enables us to compare equivalent mean field and neural mass models of the same populations and evaluate quantitatively the contribution of population variance to the expected dynamics the mean field model presented here will form the basis of a dynamic causal model of observed electromagnetic signals in future work

population rate codes carried by mean fluctuation and synchrony of neuronal firings abstract a population of firing neurons is expected to carry information not only by the mean firing rate but also by fluctuation and synchrony among neurons in order to examine this possibility we have studied responses of neuronal ensembles to three kinds of inputs mean fluctuation and synchrony driven inputs the generalized rate code model including additive and multiplicative noise h hasegawa phys rev e 75 2007 051904 has been studied by direct simulations dss and the augmented moment method amm in which equations of motion for mean firing rate fluctuation and synchrony are derived results calculated by the amm are in good agreement with those by dss the independent component analysis ica of our results has shown that mean firing rate fluctuation or variability and synchrony may carry independent information in the population rate code model the input output relation of mean firing rates is shown to have higher sensitivity for larger multiplicative noise as recently observed in prefrontal cortex a comparison is made between results obtained by the integrate and fire if model and our rate code model

analysis of a monte carlo method for nonlinear radiative transfer it has recently been proved that solutions of nonlinear radiative transfer problems satisfy a maximum principle andreev kozmanov and rachilov u s s r comput math math phys 23 104 1983 mercier siam j math anal in press in this article it is shown that monte carlo solutions of such problems obtained using the method of fleck and cummings j comput phys 8 313 1971 must satisfy this maximum principle for sufficiently small time steps but can violate it for sufficiently large time steps analyses of the frequency dependent and grey cases are given and a numerical solution violating the maximum principle is discussed

second order discretization in space and time for radiation hydrodynamics second order accurate discretizations for radiation hydrodynamics are currently an area of interest in the high energy density laboratory physics and astrophysics communities second order methods used to solve the hydrodynamics equations and second order methods used to solve the radiation transport equation often differ fundamentally making it difficult to combine them in a second order manner here we present an implicitexplicit imex method for solving the 1d equations of radiation hydrodynamics that is second order accurate in space and time our radiation hydrodynamics model consists of the 1d euler equations coupled with a gray radiation s2 approximation our rh method combines the muscl hancock method for solving the euler equations with the tr bdf2 time integration scheme and the linear discontinuous galerkin finite element spatial discretization scheme for the s2 radiation equations the muscl hancock method is commonly used for hydrodynamic calculations and the linear discontinuous galerkin scheme is the standard for the sn equations of radiative transfer while somewhat similar these schemes vary fundamentally with respect to the treatment of spatial slopes we address the challenges inherent to coupling these different numerical methods and demonstrate how these challenges can be overcome using the method of manufactured solutions we show that the method is second order accurate in space and time for both the equilibrium diffusion and streaming limit and we show that the method is capable of computing radiative shock solutions accurately by comparing our results with semi analytic solutions

stochastic bifurcation in fitzhugh nagumo ensembles subjected to additive and or multiplicative noises abstract we have studied the dynamical properties of finite n unit fitzhugh nagumo fn ensembles subjected to additive and or multiplicative noises reformulating the augmented moment method amm with the fokker planck equation fpe method h hasegawa j phys soc japan 75 2006 033001 in the amm original 2 n dimensional stochastic equations are transformed to eight dimensional deterministic ones and the dynamics is described in terms of averages and fluctuations of local and global variables the stochastic bifurcation is discussed by a linear stability analysis of the deterministic amm equations the bifurcation transition diagram of multiplicative noise is rather different from that of additive noise the former has the wider oscillating region than the latter the synchronization in globally coupled fn ensembles is also investigated results of the amm are in good agreement with those of direct simulations dss

linear multifrequency grey acceleration recast for preconditioned krylov iterations the linear multifrequency grey acceleration lmfga technique is used to accelerate the iterative convergence of multigroup thermal radiation diffusion calculations in high energy density simulations although it is effective and efficient in one dimensional calculations the lmfga method has recently been observed to significantly degrade under certain conditions in multidimensional calculations with large discontinuities in material properties to address this deficiency we recast the lmfga method in terms of a preconditioned system that is solved with a krylov method lmfgk results are presented demonstrating that the new lmfgk method always requires fewer iterations than the original lmfga method the reduction in iteration count increases with both the size of the time step and the inhomogeneity of the problem however for reasons later explained the lmfgk method can cost more per iteration than the lmfga method resulting in lower but comparable efficiency in problems with small time steps and weak inhomogeneities in problems with large time steps and strong inhomogeneities the lmfgk method is significantly more efficient than the lmfga method

asymptotic analysis of a computational method for time and frequency dependent radiative transfer we consider a time dependent energy dependent nonlinear radiative transfer problem in which opacities are large o 1 and interior sources are small o an asymptotic analysis of this problem as 0 leads to the equilibrium diffusion equation in the interior of the system along with boundary conditions and initial conditions for this equation we apply the same asymptotic analysis to a discrete version of the problem in which the frequency variable is discretized by the multigroup method the direction variable by the discrete ordinates method the time variable by the fully implicit method and the spatial variable by a subcell balance method we find that as 0 the discrete solution satisfies a robust discretized version of the correct equilibrium diffusion equation with boundary conditions and initial conditions that are remarkably accurate the analysis thus predicts that if a spatial grid is chosen that resolves interior temperature gradients then the numerical method obtains an accurate solution in the interior of the system even though the optical thickness of the spatial cells tends to and boundary layers in the transport solution are not resolved we go a step further to analyze problems that are optically thin at some photon frequencies but thick at others and show that once again the discrete solution is remarkably accurate we present numerical results that verify these and other predictions of the analyses

semi implicit time integration for pn thermal radiative transfer implicit time integration involving the solution of large systems of equations is the current paradigm for time dependent radiative transfer in this paper we present a semi implicit linear discontinuous galerkin method for the spherical harmonics p n equations for thermal radiative transfer in planar geometry our method is novel in that the material coupling terms are treated implicitly via linearizing the emission source and the streaming operator is treated explicitly using a second order accurate runge kutta method the benefit of this approach is that each time step only involves the solution of equations that are local to each cell this benefit comes at the cost of having the time step limited by a cfl condition based on the speed of light to guarantee positivity and avoid artificial oscillations we use a slope limiting technique we present analysis and numerical results that show the method is robust in the diffusion limit when the photon mean free path is not resolved by the spatial mesh also in the diffusion limit the time step restriction relaxes to a less restrictive explicit diffusion cfl condition we demonstrate with numerical results that away from the diffusion limit our method demonstrates second order error convergence as the spatial mesh is refined with a fixed cfl number

asymptotic analysis of the spatial discretization of radiation absorption and re emission in implicit monte carlo we perform an asymptotic analysis of the spatial discretization of radiation absorption and re emission in implicit monte carlo imc a monte carlo technique for simulating nonlinear radiative transfer specifically we examine the approximation of absorption and re emission by a spatially continuous artificial scattering process and either a piecewise constant or piecewise linear emission source within each spatial cell we consider three asymptotic scalings representing i a time step that resolves the mean free time ii a courant limit on the time step size and iii a fixed time step that does not depend on any asymptotic scaling for the piecewise constant approximation we show that only the third scaling results in a valid discretization of the proper diffusion equation which implies that imc may generate inaccurate solutions with optically large spatial cells if time steps are refined however we also demonstrate that for a certain class of problems the piecewise linear approximation yields an appropriate discretized diffusion equation under all three scalings we therefore expect imc to produce accurate solutions for a wider range of time step sizes when the piecewise linear instead of piecewise constant discretization is employed we demonstrate the validity of our analysis with a set of numerical examples

universal lower bounds for quantum diffusion abstract we study the connections between dynamical properties of schrodinger operators h on separable hilbert space h and the properties of corresponding spectral measures our main result establishes a relation for the moment of order p of the form x p t t t 1 t 0 x p 2 e ith 2 h dt l p d t 1 here l p d t is a function connected to the behavior of the fourier transform of measures in the subclass of measures absolutely continuous with respect to the spectral measure beyond the intrinsic interest of the general formulation 1 this result allows us to derive necessary conditions for dynamical localization in the presence of a pure point spectrum on the other hand if we focus on subsequences of time t k we can exhibit lower bounds which are in certain cases strictly larger than the well known power law lower bound for all t expressed in terms of the hausdorff dimension of spectral measures

relations between average distance heterogeneity and network synchronizability abstract by using the random interchanging algorithm we investigate the relations between average distance standard deviation of degree distribution and synchronizability of complex networks we find that both increasing the average distance and magnifying the degree deviation will make the network synchronize harder only the combination of short average distance and small standard deviation of degree distribution ensures strong synchronizability some previous studies assert that the maximal betweenness is the right quantity to estimate network synchronizability the larger the maximal betweenness the poorer the network synchronizability here we address an interesting case which strongly suggests that the single quantity maximal betweenness may not give a comprehensive description of network synchronizability

bounded solutions and absolute continuity of sturm liouville operators this work is a contribution to the theory of the absolutely continuous spectrum of one dimensional schriidinger operators i e sturm liouville operators generated by differential expressions of the form ru u qu in proofs of absolute continuity for such operators and also for the closely related dirac systems always asymptotic properties of solutions of zu lu for real or complex values of 1 have played an important role see for example 18 19 8 4 9 lo in these works and a number of others particular classes of potentials q were treated using properties of solutions in the course of the proof of absolute continuity somewhat more recently the connection between asymptotic properties of solutions and absolute continuity was given a theory i e results of the following form were proved let the solutions of zu lu satisfy some specific asymptotic property for a varying in some subset of r no or almost no a priori assumptions on q then operators generated by z are absolutely continuous in this set a first result of this type was given by carmona 3 who proved that local uniform boundedness of solutions together with their derivatives in some open a set implies pure absolute continuity in this set here we will mostly rely on the concept of non or racy of solutions which was shown by gilbert and pearson 7 12 61 to be a powerful tool in the investigation of the continuous spectrum of sturm liouville operators in section 2 we give some consequences of their results whi ch are useful in proofs of absolute continuity these consequences range from non existence of imbedded eigenvalues and the existence of absolutely continuous parts to pure absolute continuity and the existence of transient 210 0022 247x 92 5 00

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

value distribution and spectral theory of schrodinger operators with l 2 sparse potentials we apply the methods of value distribution theory to the spectral asymptotics of schrodinger operators with l2 sparse potentials

one dimensional schrodinger operators with random or deterministic potentials new spectral types abstract we prove sufficient conditions involving only potential asymptotic near one of the infinities in order to have purely absolutely continuous components in the spectrum these deterministic results are then applied to random cases and exhibit classes of models for which with probability one one component of the spectrum is purely absolutely continuous and the rest is dense pure point with exponentially decaying eigenfunctions

generalized rate code model for neuron ensembles with finite populations we have proposed a generalized langevin type rate code model subjected to multiplicative noise in order to study stationary and dynamical properties of an ensemble containing a finite number n of neurons calculations using the fokker planck equation have shown that owing to the multiplicative noise our rate model yields various kinds of stationary non gaussian distributions such as ensuremath gamma inverse gaussian like and log normal like distributions which have been experimentally observed the dynamical properties of the rate model have been studied with the use of the augmented moment method amm which was previously proposed by the author from a macroscopic point of view for finite unit stochastic systems in the amm the original n dimensional stochastic differential equations des are transformed into three dimensional deterministic des for the means and fluctuations of local and global variables the dynamical responses of the neuron ensemble to pulse and sinusoidal inputs calculated by the amm are in good agreement with those obtained by direct simulation the synchronization in the neuronal ensemble is discussed the variabilities of the firing rate and of the interspike interval are shown to increase with increasing magnitude of multiplicative noise which may be a conceivable origin of the observed large variability in cortical neurons

rank one perturbations with infinitesimal coupling we consider a positive self adjoint operator a and formal rank one perturbations b a where h 2 a but h 1 a with h s a the usual scale of spaces we show that b can be defined for such and what are essentially negative infinitesimal values of in a sense we will make precise every rank one perturbation is one of three forms i h 1 a r ii h 1 or iii the new type we consider here

the effects of slope limiting on asymptotic preserving numerical methods for hyperbolic conservation laws many hyperbolic systems of equations with stiff relaxation terms reduce to a parabolic description when relaxation dominates an asymptotic preserving numerical method is a discretization of the hyperbolic system that becomes a valid discretization of the parabolic system in the asymptotic limit we explore the consequences of applying a slope limiter to the discontinuous galerkin dg method with linear elements for hyperbolic systems with stiff relaxation terms without a limiter the dg method gives an accurate discretization of the chapman enskog approximation of the system when the relaxation length scale is not resolved it is well known that the first order upwind or step method fails to obtain the proper asymptotic limit we show that using the minmod slope limiter also fails but that using double minmod gives the proper asymptotic limit despite its effectiveness in the asymptotic limit the double minmod limiter allows artificial extrema at cell interfaces referred to as sawteeth we present a limiter that eliminates the sawteeth but maintains the proper asymptotic limit the systems that we analyze are the hyperbolic heat equation and the p n thermal radiation equations numerical examples are used to verify our analysis

some schrodinger operators with power decaying potentials and pure point spectrum we construct deterministic potentials such that the schrodinger equation on has dense pure point spectrum in for almost all boundary conditions at as a by product we also obtain power decaying potentials for which the spectrum is purely singular continuous on for all boundary conditions