exploring estimator bias variance tradeoffs using the uniform cr bound we introduce a plane which we call the delta sigma plane that is indexed by the norm of the estimator bias gradient and the variance of the estimator the norm of the bias gradient is related to the maximum variation in the estimator bias function over a neighborhood of parameter space using a uniform cramer rao cr bound on estimator variance a delta sigma tradeoff curve is specified that defines an unachievable region of the delta sigma plane for a specified statistical model in order to place an estimator on this plane for comparison with the delta sigma tradeoff curve the estimator variance bias gradient and bias gradient norm must be evaluated we present a simple and accurate method for experimentally determining the bias gradient norm based on applying a bootstrap estimator to a sample mean constructed from the gradient of the log likelihood we demonstrate the methods developed in this paper for linear gaussian and nonlinear poisson inverse problems

mean and variance of implicitly defined biased estimators such as penalized maximum likelihood applications to tomography many estimators in signal processing problems are defined implicitly as the maximum of some objective function examples of implicitly defined estimators include maximum likelihood penalized likelihood maximum a posteriori and nonlinear least squares estimation for such estimators exact analytical expressions for the mean and variance are usually unavailable therefore investigators usually resort to numerical simulations to examine the properties of the mean and variance of such estimators this paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters we derive the approximations using the implicit function theorem the taylor expansion and the chain rule the expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation as illustrations we demonstrate that the approximations work well in two tomographic imaging applications with poisson statistics we also describe a plug in approximation that provides a remarkably accurate estimate of variability even from a single noisy poisson sinogram measurement the approximations should be useful in a wide range of estimation problems

on hamiltonian perturbations of hyperbolic systems of conservation laws ii universality of critical behaviour hamiltonian perturbations of the simplest hyperbolic equation u t a u u x 0 are studied we argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution moreover this behaviour is described by a special solution to an integrable fourth order ode

periodic solutions of the intermediate long wave equation a nonlinear superposition principle periodic stationary wave solutions of the intermediate long wave ilw equation are derived using the bilinear transformation method and a new expression for the dispersion relation is obtained the class of physically important real valued solutions is identified these solutions may be represented as an infinite superposition of solitary wave profiles a property shared by the related korteweg de vries kdv and benjamin ono bo equation this nonlinear superposition principle which has been the subject of various interpretations in the literature is discussed the ilw periodic solution approximates to a sinusoidal wave and a solitary wave in the limits of small and large amplitudes respectively for intermediate amplitudes the solution can be well approximated by either a sine wave or solitary wave in the shallow water kdv limit the ilw periodic solution leads to the familiar cnoidal wave whereas the deep water bo limit yields benjamin s periodic wave a previously unknown expression for the cnoidal wave dispersion relation in terms of theta functions is obtained

nonlinear evolution equations associated with a riemann hilbert scattering problem in an earlier paper nonlinear evolution equations associated with a riemann hilbert scattering problem which reduces in an appropriate limit to the zakharov shabat akns scattering problem were considered here we discuss certain necessary constraints associated with the scattering problem and their impact upon the associated evolution equations moreover the direct linearization of the nonlinear evolution equations and an algorithm to construct ann soliton solution are given

on the initial value problem for the modified benjamin ono equation the initial value problem for the modified benjamin ono equation is solved by exploiting the fokas ablowitz inverse scattering transform for a certain complex extension of the benjamin ono equation the asymptotics of the jost functions for small positive values of the spectral parameter are sharpened and lead to a previously overlooked nongeneric case

periodic ilw equation with discrete laplacian we study an integro differential equation which generalizes the periodic intermediate long wave ilw equation the kernel of the singular integral involved is an elliptic function written as a second order difference of the weierstrass function using sato s formulation we show the integrability and construct some special solutions an elliptic solution is also obtained we present a conjecture based on a poisson structure that gives an alternative description of this integrable hierarchy we note that this poisson algebra in turn is related to a quantum algebra related to the family of macdonald difference operators

dispersive shock waves in the kadomtsev petviashvili and two dimensional benjamin ono equations abstract dispersive shock waves dsws in the kadomtsev petviashvili kp equation and two dimensional benjamin ono 2dbo equation are considered using step like initial data along a parabolic front employing a parabolic similarity reduction exactly reduces the study of such dsws in two space one time 2 1 dimensions to finding dsw solutions of 1 1 dimensional equations with this ansatz the kp and 2dbo equations can be exactly reduced to the cylindrical korteweg de vries ckdv and cylindrical benjamin ono cbo equations respectively whitham modulation equations which describe dsw evolution in the ckdv and cbo equations are derived and riemann type variables are introduced dsws obtained from the numerical solutions of the corresponding whitham systems and direct numerical simulations of the ckdv and cbo equations are compared with very good agreement obtained in turn dsws obtained from direct numerical simulations of the kp and 2dbo equations are compared with the ckdv and cbo equations again with good agreement it is concluded that the 2 1 dsw behavior along self similar parabolic fronts can be effectively described by the dsw solutions of the reduced 1 1 dimensional equations

exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method the authors present a systematic and formal approach toward finding solitary wave solutions of nonlinear evolution and wave equations from the real exponential solutions of the underlying linear equations the physical concept is one of the mixing of these elementary solutions through the nonlinearities in the system the emphasis is however on the mathematical aspects i e the formal procedure necessary to find single solitary wave solutions by means of examples it is shown how various cases of pulse type and kink type solutions are to be obtained by this method an exhaustive list of equations so treated is presented in tabular form together with the particular intermediate relations necessary for deriving their solutions the extension of the technique to construct n soliton solutions and indicate connections with other existing methods is outlined

final state problem for class of nonlinear nonlocal dispersive equation abstract we study the large time asymptotics of the solutions to nonlinear dispersive equations of the form 0 1 t u p i x x u x u 3 t 0 x r where p i x is a pseudo differential operator defined by the symbol p in this paper we consider the case that p is nonlocal and nonhomogeneous and show the existence of the solution to 0 1 around a given modified free solution

the interacton equation given the lie algebra lfp of vector fields of a free particle then in a joint field of interacting particles an interacton is defined as a direct summand isomorphic to lfp for several soliton equations it is shown how interactons can be obtained from various methods of symmetry analysis and how their dynamics can be expressed in terms of self interaction alone this procedure leads to new integrable systems furthermore in case of nonlinear schrodinger equations it is shown how the dynamics of interactons can be formulated in terms of self consistent potentials with antlinear parts

eigenfunctions and eigenvalues for a scalar riemann hilbert problem associated to inverse scattering a complete set of eigenfunctions is introduced within the riemann hilbert formalism for spectral problems associated to some solvable nonlinear evolution equations in particular we consider the time independent and time dependent schrodinger problems which are related to the kdv and kpi equations possessing solitons and lumps respectively non standard scalar products orthogonality and completeness relations are derived for these problems the complete set of eigenfunctions is used for perturbation theory and bifurcation analysis of eigenvalues supported by the potentials under perturbations we classify two different types of bifurcations of new eigenvalues and analyze their characteristic features one type corresponds to thresholdless generation of solitons in the kdv equation while the other predicts a threshold for generation of lumps in the kpi equation

singularity confinement analysis of integrodifferential equations of benjamin ono type we analyse the integrability of various types of intregrodifferential equations belonging to the benjamin ono family the method used is a combination of two well known integrability detectors the painleve method for continuous systems and the singularity confinement for discrete systems we confirm the results of hietarinta based on the study of multisoliton solutions in particular we show that the third order extension to benjamin ono that he proposed does pass the test and is thus an excellent candidate for integrability

multidimensional nonlinear evolution equations and inverse scattering abstract in this paper we will review some recent work done in the field of integrable nonlinear evolution equations and inverse scattering we will concentrate on the basic underlying areas and refer interested readers to suitable references for complete details specifically background material can be found in various texts on this subject e g 1 by ablowitz and segur more recent references will be given as necessary the outline of the paper is as follows 1 1 introductory remarks 2 2 a discussion of two separate but related issues namely a solving certain nonlinear evolutions equations in infinite space and b inverse scattering these are important problems having many physical applications moreover they are related to each other by what we refer to as the inverse scattering transform ist 3 3 at the end of the paper we will make some remarks on the possibility of solving nonlinear evolution equations in high dimensions i e equations with more than two spatial and one time variable by using the ist method as we now understand it

an application of the inverse scattering transform to the modified intermediate long wave equation the modified intermediate long wave milw equation is a 1 1 dimensional nonlinear singular integro differential equation that possesses soliton solutions in an appropriate limit the milw equation reduces to the well known modified korteweg de vries equation in this paper we solve the initial value problem for the milw equation through a suitable implementation of the inverse scattering transform and use of the miura type transformation that maps solutions of the milw equation into solutions of a complexified version of the standard intermediate long wave ilw equation the initial value used for the milw equation is assumed to be real valued sufficiently smooth and decaying to zero as the absolute value of the spatial variable approaches large values an interesting feature of the procedure we develop is that soliton solutions for the ilw and milw equations can be derived by appropriate specializations of a master set of equations

modulation solutions for the benjamin ono equation abstract in this work modulation solutions for three initial value problems for the benjamin ono equation are studied the first problem studied is the dispersive resolution of a step initial condition an explicit solution of gurevich pitaevskii type is derived which explains the dispersive resolution of the step in terms of modulations the second problem is the dispersive resolution of a breaking initial condition while the third problem is the generation of a second phase as the result of the evolution of a modulated single phase wave again explicit modulation solutions for these problems are derived in the case of the third problem the modulation solution explicitly exhibits the formation of the new phase in contrast to situation for the korteweg de vries equation for which the formation of a second phase has only been solved for phase loss in the dispersive analogue of shock merging these explicit solutions are possible since the modulation theory of dobrokhotov and krichever for the bo equation gives uncoupled modulation equations for the different phases which is not the case for the kdv equation this de coupling means that the matching of known explicit solutions enables the full description of two phase problems this extends results which have recently been obtained from analytic solutions of the whitham equations for the kdv equation for shock formation and shock merging

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

on the initial value problem for a class of nonlinear integral evolution equations including the sine hilbert equation a method for solving a class of nonlinear singular integral evolution equations for decaying initial values on the line is presented the underlying scattering problem is a matrix riemann hilbert problem scattering analysis shows that the spectrum is purely discrete an application is to the so called sine hilbert equation h t c sin where c is a constant and h denotes the hilbert transform

the scattering transform for the benjamin ono equation in the small dispersion limit abstract using exact formulae for the scattering data of the benjamin ono equation valid for general rational potentials recently obtained in miller and wetzel 17 we rigorously analyze the scattering data in the small dispersion limit in particular we deduce precise asymptotic formulae for the reflection coefficient the location of the eigenvalues and their density and the asymptotic dependence of the phase constant associated with each eigenvalue on the eigenvalue itself our results give direct confirmation of conjectures in the literature that have been partly justified by means of inverse scattering and they also provide new details not previously reported in the literature

whitham modulation theory for 2 1 dimensional equations of kadomtsev petviashvili type whitham modulation theory for certain two dimensional evolution equations of kadomtsev petviashvili kp type is presented three specific examples are considered in detail the kp equation the two dimensional benjamin ono 2dbo equation and a modified kp m2kp equation a unified derivation is also provided in the case of the m2kp equation the corresponding whitham modulation system exhibits features different from the other two the approach presented here does not require integrability of the original evolution equation indeed while the kp equation is known to be a completely integrable equation the 2dbo equation and the m2kp equation are not known to be integrable in each of the cases considered the whitham modulation system obtained consists of five first order quasilinear partial differential equations the riemann problem i e the analogue of the gurevich pitaevskii problem for the one dimensional reduction of the m2kp equation is studied for the m2kp equation the system of modulation equations is used to analyze the linear stability of traveling wave solutions

a new zk ilw equation for algebraic gravity solitary waves in finite depth stratified atmosphere and the research of squall lines formation mechanism abstract the research of algebraic gravity solitary waves is an advanced field which has important practical and theoretical value in physical oceanography aerology and etc by calculation condition and theoretical method limit previous researches mainly focused on the 1 1 dimensional models and 2 1 dimensional models were few considered in this paper from the non static equilibrium equation a new zk ilw equation is derived by using multi scale analysis and perturbation method in finite depth stratified atmosphere which is the first time obtained the model can reduce to zk bo model h and zk model h 0 and is the generalization of the above two models in order to further understand the nature of algebraic gravity solitary waves we get the analytical solution of zk ilw equation by using the trial function method and discuss the conservation laws furthermore the fission process of algebraic gravity solitary waves is studied and we can judge that one of the possible formation mechanism of squall lines is the fission of algebraic gravity solitary waves

a spectral transform for the intermediate nonlinear schrodinger equation a new spectral transform system is introduced to solve the initial value problem for the intermediate nonlinear schrodinger inls equation describing envelope waves in a deep stratified fluid the spectral system is a combination of the zakharov shabat linear system and a local riemann hilbert problem in a strip of the complex plane the inverse scattering transform technique is developed and the backlund darboux transformation soliton solutions and an infinite number of conservation laws are constructed it is shown that all these properties of the inls equation are closely related to those of the intermediate long wave equation

direct and inverse scattering problems of the nonlinear intermediate long wave equation the inverse scattering transformation method associated with a nonlinear singular integrodifferential equation is discussed the equation describes long internal gravity waves in a stratified fluid of finite depth and reduces to the korteweg de vries equation as shallow water limit and the benjamin ono equation as deep water limit both limits of the method and novel aspects of the theory are also discussed

numerical stability analysis and computation of hopf bifurcation points for delay differential equations we present a numerical technique for the stability analysis and the computation of branches of hopf bifurcation points in nonlinear systems of delay differential equations with several constant delays the stability analysis of a steady state solution is done by a numerical implementation of the argument principle which allows to compute the number of eigenvalues with positive real part of the characteristic matrix the technique is also used to detect bifurcations of higher singularity hopf and fold bifurcations during the continuation of a branch of hopf points this allows to trace new branches of hopf points and fold points

uniqueness properties of solutions to the benjamin ono equation and related models abstract we prove that if u 1 u 2 are real solutions of the benjamin ono equation defined in x t r 0 t which agree in an open set r 0 t then u 1 u 2 we extend this uniqueness result to a general class of equations of benjamin ono type in both the initial value problem and the initial periodic boundary value problem this class of 1 dimensional non local models includes the intermediate long wave equation we relate our uniqueness results with those for a water wave problem finally we present a slightly stronger version of our uniqueness results for the benjamin ono equation

backlund transformation conservation laws and inverse scattering transform of a model integrodifferential equation for water waves the backlund transformation bt an infinite number of conservation laws and the inverse scattering transform ist of a model integrodifferential equation for water waves in fluids of finite depth y matsuno j math phys 29 49 1989 are constructed by employing the bilinear transformation method the model equation is also shown to pass the painleve test these facts prove the complete integrability of the equation both the deep and shallow water limits of various results thus obtained are then investigated in detail in addition a new method to evaluate conserved quantities for pure n soliton is developed by utilizing actively the time part of the bt it is found that the structure of conservation laws exhibits peculiar characteristics in comparison with those of usual water wave equations such as the benjamin ono and the korteweg de vries equations the most important problem left open in this paper is to solve various ist equations

purely imaginary eigenvalues of zakharov shabat systems zakharov shabat systems with single hump and real but not necessarily symmetric potentials are shown to have purely imaginary eigenvalues only coupled with examples of double hump potentials with nonimaginary eigenvalues this establishes that confinement of zakharov shabat eigenvalues to the imaginary axis is a characteristic of potentials whose energy is concentrated in a single region of the time axis

on the limit from the intermediate long wave equation to the benjamin ono equation the intermediate long wave equation is a physically important singular integrodifferential equation containing a parameter referred to here as for it reduces to the benjamin ono equation it has been recently shown that the inverse scattering transform schemes of the above equations have certain significant differences here it is shown that for the inverse scattering transform scheme of the intermediate long wave equation reduces to that of the benjamin ono equation

single and multipinhole collimator design evaluation method for small animal spect high resolution functional imaging of small animals is often obtained by single pinhole spect with circular orbit acquisition multipinhole spect adds information due to its improved sampling and can improve the trade off between resolution and sensitivity to evaluate different pinhole collimator designs an efficient method is needed that quantifies the reconstruction image quality in this paper we propose a fast approximate method that examines the quality of individual voxels of a postsmoothed maximum likelihood expectation maximization mlem reconstruction by studying their linearized local impulse response llir and co variance for a predefined target resolution for validation the contrast to noise ratios cnrs in some voxels of a homogeneous sphere and of a realistic rat brain software phantom were calculated for many single and multipinhole designs a good agreement was observed between the cnrs obtained with the approximate method and those obtained with postsmoothed mlem reconstructions of simulated noisy projections this good agreement was quantified by a least squares fit through these results which yielded a line with slope 1 02 1 00 expected and a y intercept close to zero 0 expected 95 4 of the validation points lie within three standard deviations from that line using the approximate method the influence on the cnr of varying a parameter in realistic single and multipinhole designs was examined the investigated parameters were the aperture diameter the distance between the apertures and the axis of rotation the focal distance the acceptance angle the position of the apertures the focusing distance and the number of pinholes the results can generally be explained by the change in sensitivity the amount of postsmoothing and the amount of overlap in the projections the method was applied to multipinhole designs with apertures focusing at a single point but is also applicable to other designs

kinks of the sine hilbert equation and their dynamical motions an exact method for constructing exact solutions of the sine hilbert equation is developed it is shown that the sh equation can be transformed into a system of non linear ordinary differential equations through a dependent variable transformation furthermore this system of equations is transformed into a system of decoupled linear ordinary differential equations exact solutions of the sh equation are then constructed by means of a simple integration the dynamical properties of the solutions are also studied in detail the solution is interpreted as a superposition of n pulses which may be termed kinks where n is a positive integer asymptotically for large time it is composed of a pulse with a constant velocity and n 1 static pulses this is a novel characteristic of the solution unlike the well known n soliton solution which behaves like n moving pulses for large time

stability of nonlinear periodic internal waves in a deep stratified fluid stability of the nonlinear periodic solutions of the benjamin ono equation with respect to one dimensional perturbations and taking into account transverse modulations is analytically studied the full set of eigenfrequencies and of small perturbations is found in the one dimensional case the dispersion equation for long wave oblique perturbations is obtained these perturbations are shown to be unstable under either sign of the dispersion of the linear dispersion relation the instability limits and the maximum growth rates are determined in the extreme cases of strongly and weakly nonlinear waves

on the generation of dispersive shock waves abstract we review various methods for the analysis of initial value problems for integrable dispersive equations in the weak dispersion or semiclassical regime some methods are sufficiently powerful to rigorously explain the generation of modulated wavetrains so called dispersive shock waves as the result of shock formation in a limiting dispersionless system they also provide a detailed description of the solution near caustic curves that delimit dispersive shock waves revealing fascinating universal wave patterns

numerical bifurcation analysis of delay differential equations numerical methods for the bifurcation analysis of delay differential equations ddes have only recently received much attention partially because the theory of ddes smoothness boundedness stability of solutions is more complicated and less established than the corresponding theory of ordinary differential equations as a consequence no established software packages exist at present for the bifurcation analysis of ddes we outline existing numerical methods for the computation and stability analysis of steady state solutions and periodic solutions of systems of ddes with several constant delays

generalized intermediate long wave hierarchy in zero curvature representation with noncommutative spectral parameter the simplest generalization of the intermediate long wave hierarchy ilw is considered to show how to extend the zakharov shabat dressing method to nonlocal i e integro partial differential equations the purpose is to give a procedure of constructing the zero curvature representation of this class of equations this result obtains by combining the drinfeld sokolov formalism together with the introduction of an operator valued spectral parameter namely a spectral parameter that does not commute with the space variable x this extension provides a connection between the ilwk hierarchy and the saveliev vershik continuum graded lie algebras in the case of ilw2 the fairlie zachos sinh algebra was found

universality of the break up profile for the kdv equation in the small dispersion limit using the riemann hilbert approach we obtain an asymptotic expansion for the solution of the cauchy problem for the korteweg de vries kdv equation u t 6uu x epsilon 2 u xxx 0 u x t 0 epsilon u 0 x for epsilon small near the point of gradient catastrophe x c t c for the solution of the dispersionless equation u t 6uu x 0 the sub leading term in this expansion is described by the smooth solution of a fourth order ode which is a higher order analogue to the painleve i equation this is in accordance with a conjecture of dubrovin suggesting that this is a universal phenomenon for any hamiltonian perturbation of a hyperbolic equation using the deift zhou steepest descent method applied on the riemann hilbert problem for the kdv equation we are able to prove the asymptotic expansion rigorously in a double scaling limit

nonlocal models for nonlinear dispersive waves abstract considered herein are model equations for the unidirectional propagation of small amplitude nonlinear dispersive long waves such as those governed by the classical korteweg de vries equation of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial so that the operator modelling dispersive effect is nonlocal particular cases in view here are the benjamin ono equation and the intermediate long wave equation which arise in internal wave theory and smith s equation which governs certain types of continental shelf waves the initial value problem for these equations is shown to be globally well posed in the classical sense including continuous dependence upon the initial data and in certain cases upon the modelling of nonlinear and dispersive effects whilst the results are stated for the specific equations listed above the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear dispersive evolution equations particularly worthy of note is our theorem implying that solutions of the intermediate long wave equation converge strongly to solutions of the korteweg de vries equation or to solutions of the benjamin ono equation in appropriate asymptotic limits

periodic solutions in a model of recurrent neural feedback an equation which plant used to model recurrent neural feedback with delay is analyzed to determine its periodic solution structure as a two parameter hopf bifurcation problem the model exhibits

bifurcations of new eigenvalues for the benjamin ono equation a criterion for the emergence of new eigenvalues is found for the linear scattering problem associated with the benjamin ono bo equation this bifurcation occurs due to perturbations of nongeneric potentials which include the soliton solutions of the bo equation the asymptotic approximation of an exponentially small new eigenvalue is derived the method is based on the expansion of a localized function through a complete set of unperturbed eigenfunctions explicit expressions are obtained for the soliton potentials

recursive algorithms for computing the cramer rao bound computation of the cramer rao bound crb on estimator variance requires the inverse or the pseudo inverse fisher information matrix fim direct matrix inversion can be computationally intractable when the number of unknown parameters is large in this correspondence we compare several iterative methods for approximating the crb using matrix splitting and preconditioned conjugate gradient algorithms for a large class of inverse problems we show that nonmonotone gauss seidel and preconditioned conjugate gradient algorithms require significantly fewer flops for convergence than monotone bound preserving algorithms

new integrable nonlinear integrodifferential equations and related solvable finite dimensional dynamical systems a new integrable nonlinear integrodifferential equation nide is proposed this equation may be interpreted as a model equation for deep water waves the n periodic and n soliton solutions for the equation are constructed by means of the bilinear transformation method these solutions have the same structure as that for the benjamin ono equation which describes internal waves in stratified fluids of great depth furthermore it is shown that the motion of the positions of the poles of solutions is related to certain solvable finite dimensional dynamical systems described by first order nonlinear ordinary differential equations the discussion is also made on a more general nide that may be interpreted as a model equation describing nonlinear waves in fluids of finite depth

a modified uniform cramer rao bound for multiple pinhole aperture design this paper presents a modified uniform cramer rao bound ucrb for studying estimator spatial resolution and variance tradeoffs we proposed to use a resolution constraint that is imposed on mean gradient vectors of achieved estimators and derived the minimum achievable variance for any estimator satisfies this resolution constraint this approach partially overcomes the limitations of the former ucrb approach based on a bias gradient norm constraint we applied this method in a feasibility study of using multiple pinhole apertures for small animal spect imaging applications the spect system studied was based on an existing gamma camera the achievable spatial resolution and variance tradeoffs for systems with different design parameters such as number of pinholes and pinhole size were studied

period doublings and possible chaos in neural models a formal perturbation method is derived for the study of bifurcation near a degenerate zero eigenvalue this results in a third order differential equation with a single quadratic nonlinearity numerical solutions of this show successive period doubling bifurcations and eventual chaos the problem arises in the study of neural equations when an additional excitatory channel or cell is added to standard two component models which oscillate it also applies to a van der pol oscillator coupled to a simple rc circuit numerical simulations of a modified fitzhugh nagumo system agree qualitatively with the behavior of the bifurcation equations

on genus two solutions for the ilw equation the existence of theta function solutions of genus two for the intermediate long wave equation is established a numerical example is also presented the method basically goes along with krichever s construction of theta function solutions for soliton equations such as the kronecker product equation this idea leads us to a question whether a riemann surface exists which allows a peculiar abelian integral of the third kind the answer is affirmative at least for genus two curves

darboux transformations infinitesimal symmetries and conservation laws for the nonlocal two dimensional toda lattice the technique of darboux transformation is applied to the nonlocal partner of the two dimensional periodic an 1 toda lattice this system is shown to admit a representation as the compatibility conditions of direct and dual overdetermined linear systems with a quantized spectral parameter we give the generalization of the darboux transformation technique on linear equations of this type we present the connections between the solutions of overdetermined linear systems and their expansions in series in the neighbourhood of singular points the solutions of the nonlocal toda lattice and infinite hierarchies of the infinitesimal symmetries and conservation laws are obtained

on the zero dispersion limit of the benjamin ono cauchy problem for positive initial data we study the cauchy initial value problem for the benjamin ono equation in the zero dispersion limit and we establish the existence of this limit in a certain weak sense by developing an appropriate analogue of the method invented by lax and levermore to analyze the corresponding limit for the korteweg de vries equation 2010 wiley periodicals inc

critical behavior for scalar nonlinear waves abstract in the long wave regime nonlinear waves may undergo a phase transition from a smooth behavior to a fast oscillatory behavior in this study we consider this phenomenon which is commonly known as dispersive shock in the light of dubrovin s universality conjecture dubrovin 2006 dubrovin and elaeva 2012 and we argue that the transition can be described by a special solution of a model universal partial differential equation this universal solution is constructed using the string equation we provide a classification of universality classes and an explicit description of the transition with special functions thereby extending dubrovin s universality conjecture to a wider class of equations in particular we show that the benjamin ono equation belongs to a novel universality class with respect to those known previously and we compute its string equation exactly we describe our results using the language of statistical mechanics where we show that dispersive shocks share many of the features of the tricritical point in statistical systems and we also build a dictionary of the relations between nonlinear waves and statistical mechanics