elliptic boundary value problems with λ dependent boundary conditions abstract in this paper second order elliptic boundary value problems on bounded domains r n with boundary conditions on depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework for a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure in the special case of rational nevanlinna or riesz herglotz functions on the boundary the solution operator is obtained in an explicit form in the product hilbert space l 2 l 2 m which is a natural generalization of known results on linear elliptic boundary value problems and rational boundary value problems for ordinary second order differential equations

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

slowly coupled oscillators phase dynamics and synchronization in this paper we extendthe results of frankel andkiemel siam j appl math 5 3 1993 pp 1436 1446 to a network of slowly coupledoscillators first we use malkin s theorem to derive a canonical phase model that describes synchronization properties of a slowly coupled network then we illustrate the result using slowly coupledoscillators 1 near and ronov hopf bifurcations 2 near saddle node on invariant circle bifurcations and 3 near relaxation oscillations we compare andcontrast synchronization properties of slowly andweakly coupledoscillators

on the negative squares of indefinite sturm liouville operators abstract the number of negative squares of all self adjoint extensions of a simple symmetric operator of defect one with finitely many negative squares in a krein space is characterized in terms of the behaviour of an abstract titchmarsh weyl function near 0 and these results are applied to a large class of symmetric and self adjoint indefinite sturm liouville operators with indefinite weight functions

instrumental variables regression with weak instruments this paper develops asymptotic distribution theory for instrumental variable regression when the partial correlation between the instruments and a single included endogenous variable is weak here modeled as local to zero asymptotic representations are provided for various instrumental variable statistics including the two stage least squares tsls and limited information maximum likelihood liml estimators and their t statistics the asymptotic distributions are found to provide good approximations to sampling distributions with just 20 observations per instrument even in large samples tsls can be badly biased but liml is in many cases approximately median unbiased the theory suggests concrete quantitative guidelines for applied work these guidelines help to interpret angrist and krueger s 1991 estimates of the returns to education whereas tsls estimates with many instruments approach the ols estimate of 6 the more reliable liml and tsls estimates with fewer instruments fall between 8 and 10 with a typical confidence interval of 6 14

the similarity problem for j nonnegative sturm liouville operators abstract sufficient conditions for the similarity of the operator a 1 r x d 2 d x 2 q x with an indefinite weight r x sgn x r x are obtained these conditions are formulated in terms of titchmarsh weyl m coefficients sufficient conditions for the regularity of the critical points 0 and of j nonnegative sturm liouville operators are also obtained this result is exploited to prove the regularity of 0 for various classes of sturm liouville operators this implies the similarity of the considered operators to self adjoint ones in particular in the case r x sgn x and q l 1 r 1 x d x we prove that a is similar to a self adjoint operator if and only if a is j nonnegative the latter condition on q is sharp i e we construct q 1 l 1 r 1 x d x such that a is j nonnegative with the singular critical point 0 hence a is not similar to a self adjoint operator for periodic and infinite zone potentials we show that j positivity is sufficient for the similarity of a to a self adjoint operator in the case q 0 we prove the regularity of the critical point 0 for a wide class of weights r this yields new results for forward backward diffusion equations

evolution equations with dynamic boundary conditions in this paper we study boundary problems with dynamic boundary conditions that is with boundary operators containing time derivatives the equations under consideration are transformed into abstract cauchy problems x cx f and x 0 x 0 abstract theoretical results concerning the operators c are obtained by the study of a naturally arising pseudodifferential operator for existence and uniqueness theorems concerning solutions of parabolic and hyperbolic equations we then apply the theory of semigroups in banach spaces some examples of semilinear and quasilinear problems to which our results apply are given

phase locking in networks of synaptically coupled mckean relaxation oscillators we use geometric dynamical systems methods to derive phase equations for networks of weakly connected mckean relaxation oscillators we derive an explicit formula for the connection function when the oscillators are coupled with chemical synapses modeled as the convolution of some input spike train with an appropriate synaptic kernel the theory allows the systematic investigation of the way in which a slow recovery variable can interact with synaptic time scales to produce phase locked solutions in networks of pulse coupled neural relaxation oscillators the theory is exact in the singular limit that the fast and slow time scales of the neural oscillator become effectively independent by focusing on a pair of mutually coupled mckean oscillators with alpha function synaptic kernels we clarify the role that fast and slow synapses of excitatory and inhibitory type can play in producing stable phase locked rhythms in particular we show that for fast excitatory synapses there is coexistence of a stable synchronous a stable anti synchronous and one stable asynchronous solution for slower synapses the anti synchronous solution can lose stability whilst for even slower synapses it can regain stability the case of inhibitory synapses is similar up to a reversal of the stability of solution branches using a return map analysis the case of strong pulsatile coupling is also considered in this case it is shown that the synchronous solution can co exist with a continuum of asynchronous states

frequency locking in countable cellular systems localization of asymptotic quasi periodic solutions of autonomous differential systems we address the question of frequency locking in coupled differential systems and of the existence of some quasi periodic solutions of a certain kind of differential systems those systems are named cellular systems quite generally as they deal with countable numbers of coupled systems in some general banach spaces moreover the inner dynamics of each subsystem does not have to be specified we reach some general results about how the frequency locking phenomenon is related to the structure of the coupling map those results can be restated in terms of localization of a certain type of quasi periodic solution of differential systems that may be seen as cellular systems this paper gives some explanations about how and why synchronized behaviors naturally occur in a wide variety of complex systems

standing waves for the nls on the double bridge graph and a rational irrational dichotomy abstract we study standing waves of nls equation posed on the double bridge graph two semi infinite half lines attached at a circle at the two vertices kirchhoff boundary conditions are imposed the configuration of the graph is characterized by two lengths l 1 and l 2 we study the solutions with possibly nontrivial components on the half lines and a cnoidal component on the circle the problem is equivalent to a nonlinear boundary value problem in which the boundary condition depends on the spectral parameter after classifying the solutions with rational l 1 l 2 we turn to l 1 l 2 irrational showing that there exist standing waves only in correspondence to a countable set of negative frequencies n moreover we show that the frequency sequence admits cluster points and any negative real number can be a limit point of frequencies choosing a suitable irrational geometry l 1 l 2 these results depend on basic properties of diophantine approximation of real numbers

heat equation with dynamical boundary conditions of reactive diffusive type this paper deals with the heat equation posed in a bounded reg ular domain of r n n 2 coupled with a dynamical boundary condition of reactive diffusive type in particular we study the problem 8 utu 0 in 0 1 ut ku lu on 0 1 u 0 x u0 x on where u u t x t 0 x 2 x denotes the laplacian operator with respect to the space variable whiledenotes the laplace beltrami operator on is the outward normal to and k and l are given real constants l 0 well posedness is proved for data u0 2 h 1 such that u0 2 h 1 we also study higher regularity of the solution

spectral theory of schrodinger operators with infinitely many point interactions and radial positive definite functions abstract a number of results on radial positive definite functions on r n related to schoenberg s integral representation theorem are obtained they are applied to the study of spectral properties of self adjoint realizations of two and three dimensional schrodinger operators with countably many point interactions in particular we find conditions on the configuration of point interactions such that any self adjoint realization has purely absolutely continuous non negative spectrum we also apply some results on schrodinger operators to obtain new results on completely monotone functions

generalized empirical likelihood tests in time series models with potential identification failure abstract we introduce test statistics based on generalized empirical likelihood methods that can be used to test simple hypotheses involving the unknown parameter vector in moment condition time series models the test statistics generalize those in guggenberger and smith 2005 generalized empirical likelihood estimators and tests under partial weak and strong identification econometric theory 21 4 667 709 from the i i d to the time series context and are alternatives to those in kleibergen 2005a testing parameters in gmm without assuming that they are identified econometrica 73 4 1103 1123 and otsu 2006 generalized empirical likelihood inference for nonlinear and time series models under weak identification econometric theory 22 3 513 527 the main feature of these tests is that their empirical null rejection probabilities are not affected much by the strength or weakness of identification more precisely we show that the statistics are asymptotically distributed as chi square under both classical asymptotic theory and weak instrument asymptotics of stock and wright 2000 gmm with weak identification econometrica 68 5 1055 1096 we also introduce a modification to otsu s 2006 statistic that is computationally more attractive a monte carlo study reveals that the finite sample performance of the suggested tests is very competitive

type i membranes phase resetting curves and synchrony type i membrane oscillators such as the connor model connor et al 1977 and the morris lecar model morris and lecar 1981 admit very low frequency oscillations near the critical applied current hansel et al 1995 have numerically shown that synchrony is difficult to achieve with these models and that the phase resetting curve is strictly positive we use singular perturbation methods and averaging to show that this is a general property of type i membrane models we show in a limited sense that so called type ii resetting occurs with models that obtain rhythmicity via a hopf bifurcation we also show the differences between synapses that act rapidly and those that act slowly and derive a canonical form for the phase interactions

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

on the spectral theory of singular indefinite sturm liouville operators we consider a singular sturm liouville differential expression with an indefinite weight function and we show that the corresponding self adjoint differential operator in a krein space locally has the same spectral properties as a definitizable operator

elliptic eigenvalue problems with eigenparameter dependent boundary conditions abstract let a be a uniformly elliptic second order linear operator on a smooth bounded domain r n we study the eigenvalue problem au u subject to boundary conditions b0u b1u on where bj are linear boundary operators the problem is recast in the form a u u in a hilbert or krein space and results are given on the location and type of the spectrum full and half range completeness and regularity of critical points

regression rank scores and regression quantiles we show that regression quantiles which could be computed as solutions of a linear programming problem and the solutions of the corresponding dual problem which we call the regression rank scores generalize the duality of order statistics and of ranks from the location to the linear model noting this fact we study the regression quantile and regression rank score processes in the heteroscedastic linear regression model obtaining some new estimators and interesting comparisons with existing estimators

indefinite sturm liouville problems and half range completeness where the function h changes sign on i such problems occur in certain physical models particularly in transport theory and statistical physics see 2 9 11 131 and references therein as for the standard eigenvalue problem it is true under quite general conditions and not difficult to prove that there is a set of eigenfunctions un which is complete in an appropriate hilbert space of functions this is known as full range completeness for the physical problems the question of interest is generally that of half range completeness to formulate this we set

sturm liouville operators with indefinite weight functions and eigenvalue depending boundary conditions we consider a class of boundary value problems for sturm liouville operators with indenite weight functions the spectral parameter appears nonlinearly in the boundary condition in the form of a function which has the property that 7 is a generalized nevanlinna function we construct linearizations of these boundary value problems and study their spectral properties

a remark on discrete wave scattering we consider the spectral problem for the adjacency matrix of a discrete star graph composed of a compact cluster and a few semi infinite periodic leads attached based on the spectral properties of the adjacency matrix we develop the lax phillips scattering theory for the corresponding discrete wave equation

on symmetries in the theory of finite rank singular perturbations for a nonnegative self adjoint operator a0 acting on a hilbert space h singular perturbations of the form a0 v v 1nbij j i are studied under some additional requirements of symmetry imposed on the initial operator a0 and the singular elements j a concept of symmetry is defined by means of a one parameter family of unitary operators u that is motivated by results due to r s phillips the abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials v and the corresponding self adjoint realizations of a0 v the results are applied for the investigation of singular perturbations of the schrodinger operator in l2 r3 and for the study of a fractional p adic schrodinger type operator with point interactions

optimal inference for instrumental variables regression with non gaussian errors abstract this paper is concerned with inference on the coefficient on the endogenous regressor in a linear instrumental variables model with a single endogenous regressor nonrandom exogenous regressors and instruments and i i d errors whose distribution is unknown it is shown that under mild smoothness conditions on the error distribution it is possible to develop tests which are nearly efficient in the sense of andrews et al 2006 when identification is weak and consistent and asymptotically optimal when identification is strong in addition an estimator is presented which can be used in the usual way to construct valid indeed optimal confidence intervals when identification is strong the estimator is of the two stage least squares variety and is asymptotically efficient under strong identification whether or not the errors are normal

synchronization of weakly coupled canard oscillators synchronization has been studied extensively in the context of weakly coupled oscillators using the so called phase response curve prc which measures how a change of the phase of an oscillator is affected by a small perturbation this approach was based upon the work of malkin and it has been extended to relaxation oscillators namely synchronization conditions were established under the weak coupling assumption leading to a criterion for the existence of synchronous solutions of weakly coupled relaxation oscillators previous analysis relies on the fact that the slow nullcline does not intersect the fast nullcline near one of its fold points where canard solutions can arise in the present study we use numerical continuation techniques to solve the adjoint equations and we show that synchronization properties of canard cycles are different than those of classical relaxation cycles in particular we highlight a new special role of the maximal canard in separating two distinct synchronization regimes the hopf regime and the relaxation regime phase plane analysis of slow fast oscillators undergoing a canard explosion provides an explanation for this change of synchronization properties across the maximal canard

on finite rank perturbations of definitizable operators it was shown by p jonas and h langer that a selfadjoint definitizable operator a in a krein space remains definitizable after a finite rank perturbation in resolvent sense if the perturbed operator b is selfadjoint and the resolvent set b is nonempty it is the aim of this note to prove a more general variant of this perturbation result where the assumption on b is dropped as an application a class of singular ordinary differential operators with indefinite weight functions is studied

exactly distribution free inference in instrumental variables regression with possibly weak instruments abstract this paper introduces a rank based test for the instrumental variables regression model that dominates the anderson rubin test in terms of finite sample size and asymptotic power in certain circumstances the test has correct size for any distribution of the errors with weak or strong instruments the test has noticeably higher power than the anderson rubin test when the error distribution has thick tails and comparable power otherwise like the anderson rubin test the rank tests considered here perform best relative to other available tests in exactly identified models

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

on kreĭn s formula abstract kre n s formula provides a parametrization of the generalized resolvents and straus extensions of a closed symmetric operator with equal possibly infinite defect numbers in a hilbert space in terms of nevanlinna families in a parameter space the aim of this note is to give a simple complete analytical proof of kre n s formula

relative phase behavior of two slowly coupled oscillators two oscillators coupled via slowly varying scalars are investigated it is shown that the dynamics of the slow variables is approximated by a two dimensional averaged system if the oscillators have similar frequencies at a fixed point of the averaged system then the relative phase behavior of the oscillators is approximated by a three dimensional flow under the stronger assumption that the oscillators themselves are similar and that the fixed point of the averaged system is symmetric hyperbolic and stable a further reduction to a flow on a cylinder is possible this cylinder flow is computed for an example in which phase entrainment relative phase oscillations and bidirectional wrapping of relative phase are demonstrated

changes of firing rate induced by changes of phase response curve in bifurcation transitions we study dynamical mechanisms responsible for changes of the firing rate during four different bifurcation transitions in the two dimensional hindmarsh rose 2dhr neuron model the saddle node on an invariant circle snic bifurcation to the supercritical andronov hopf ah one the snic bifurcation to the saddle separatrix loop ssl one the ah bifurcation to the subcritical ah sah one and the ssl bifurcation to the ah one for this purpose we study slopes of the firing rate curve with respect to not only an external input current but also temperature that can be interpreted as a timescale in the 2dhr neuron model these slopes are mathematically formulated with phase response curves prcs expanding the firing rate with perturbations of the temperature and external input current on the one dimensional space of the phase in the 2dhr oscillator by analyzing the two different slopes of the firing rate curve with respect to the temperature and external input current we find that during changes of the firing rate in all of the bifurcation transitions the calculated slope with respect to the temperature also changes this is largely dependent on changes in the prc size that is also related to the slope with respect to the external input current furthermore we find phase transition like switches of the firing rate with a possible increase of the temperature during the ssl to ah bifurcation transition

a krein space approach to symmetric ordinary differential operators with an indefinite weight function on considere les proprietes spectrales du probleme differentiel l f 1 n p 0 f n n 1 n 1 p 1 f n 1 n 1 p n f rf sur un intervalle fini ou infini a b avec des coefficients reels localement sommables 1 p 0 p 1 p n r sous l hypothese que p 0 0 et que la fonction poids r change de signe sur a b

gel statistics under weak identification abstract the central concern of this paper is the provision in a time series moment condition framework of practical recommendations of confidence regions for parameters whose coverage probabilities are robust to the strength or weakness of identification to this end we develop pearson type test statistics based on gel implied probabilities formed from general kernel smoothed versions of the moment indicators we also modify the statistics suggested in guggenberger and smith 2008 for a general kernel smoothing function importantly for our conclusions we provide gel time series counterparts to gmm and gel conditional likelihood ratio statistics given in kleibergen 2005 and smith 2007 our analysis not only demonstrates that these statistics are asymptotically conditionally pivotal under both classical asymptotic theory and weak instrument asymptotics of stock and wright 2000 but also provides asymptotic power results in the weakly identified time series context consequently the empirical null rejection probabilities of the associated tests and thereby the coverage probabilities of the corresponding confidence regions should not be affected greatly by the strength or otherwise of identification a comprehensive monte carlo study indicates that a number of the tests proposed here represent very competitive choices in comparison with those suggested elsewhere in the literature

sharp eigenvalue estimates for rank one perturbations of nonnegative operators in krein spaces abstract let a and b be selfadjoint operators in a krein space and assume that the resolvent difference of a and b is of rank one in the case that a is nonnegative and i is an open interval such that a i consists of isolated eigenvalues we prove sharp estimates on the number and multiplicities of eigenvalues of b in i the general result is illustrated with eigenvalue estimates for singular indefinite sturm liouville problems

partial synchronization in populations of pulse coupled oscillators i study the long term behavior of populations of nonlinear oscillators with all to all noninstantaneous pulse coupling with fast enough excitatory coupling both the fully synchronized and the asynchronous state are unstable in this case individual units fire quasiperiodically even though the network as a whole shows a periodic firing pattern the behavior of networks with three or more units is different in this regard from that of two unit networks with inhibitory coupling the network can break up into a variable number of fully synchronized clusters for fast inhibition the number of clusters tends to be large while the number of clusters is smaller for slow inhibition textcopyright 1996 the american physical society

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

misspecification tests in econometrics based on ranks abstract this paper introduces a class of rank statistics for use in testing econometric models for misspecification these statistics are ideally suited to survey data since endogenous and exogenous variables in econometric models may therefore be considered exchangeable thus residuals from a wide class of models e g linear regression non linear regression simultaneous equations etc are also exchangeable since rank statistics are distribution free when based on exchangeable data it follows the rank statistics based on these residuals are also distribution free simulation studies in the linear regression case suggest that they have good power characteristics and are distributionally robust in comparison to the usual classical procedures

optimal two sided invariant similar tests for instrumental variables regression this paper considers tests of the parameter on an endogenous variable in an instrumental variables regression model the focus is on determining tests that have some optimal power properties we start by considering a model with normally distributed errors and known error covariance matrix we consider tests that are similar and satisfy a natural rotational invariance condition we determine a two sided power envelope for invariant similar tests this allows us to assess and compare the power properties of tests such as the conditional likelihood ratio clr the lagrange multiplier and the anderson rubin tests we find that the clr test is quite close to being uniformly most powerful invariant among a class of two sided tests the finite sample results of the paper are extended to the case of unknown error covariance matrix and possibly nonnormal errors via weak instrument asymptotics strong instrument asymptotic results also are provided because we seek tests that perform well under both weak and strong instruments

piecewise linear differential equations and integrate and fire neurons insights from two dimensional membrane models we derive and study two dimensional generalizations of integrate and fire models which can be found from a piecewise linear idealization of the fitzhugh nagumo or morris lecar model these models give rise to new properties not present in one dimensional integrate and fire models a detailed analytical study of the models is presented in particular i for the piecewise linear fitzhugh nagumo model we determine analytically the bistability regime between stationary solutions and oscillations that is typical for class ii models ii in the piecewise morris lecar model we find a noncanonical class i transition from a stationary state to oscillations with logarithmic dependence similar to that found for leaky integrate and fire models iii furthermore we establish a relation to the recently proposed resonate and fire model and show that a short input current pulse can trigger several spikes

a conditional likelihood ratio test for structural models this paper develops a general method for constructing exactly similar tests based on the conditional distribution of nonpivotal statistics in a simultaneous equations model with normal errors and known reduced form covariance matrix these tests are shown to be similar under weak instrument asymptotics when the reduced form covariance matrix is estimated and the errors are non normal the conditional test based on the likelihood ratio statistic is particularly simple and has good power properties like the score test it is optimal under the usual local to null asymptotics but it has better power when identification is weak

synchronization in two neurons results for a two component dynamical model with time delayed inhibition abstract we study synchronization in two neurons using two dimensional neuronal models coupled through threshold delayed inhibition acting for a fixed duration t we explain the mechanism of synchronization between the two coupled units by means of phase plane analysis using a piecewise linear approximation we compute the firing times of two mutually coupled neurons which allows a quantitative study of synchronization the results are in good agreement both quantitatively and qualitatively with the numerical simulations of the full nonlinear coupled system describing the two neurons we first show that with simple inhibition only phase lagged synchronization can be achieved except for a particular set of t values we then show that coincidence synchronization can rapidly be achieved with an improved coupling model which takes into account the increase of membrane conductance during inhibition

qualitative changes in phase response curve and synchronization at the saddle node loop bifurcation prominent changes in neuronal dynamics have previously been attributed to a specific switch in onset bifurcation the bogdanov takens bt point this study unveils another relevant and so far underestimated transition point the saddle node loop bifurcation which can be reached by several parameters including capacitance leak conductance and temperature this bifurcation turns out to induce even more drastic changes in synchronization than the bt transition this result arises from a direct effect of the saddle node loop bifurcation on the limit cycle and hence spike dynamics in contrast the bt bifurcation exerts its immediate influence upon the subthreshold dynamics and hence only indirectly relates to spiking we specifically demonstrate that the saddle node loop bifurcation i ubiquitously occurs in planar neuron models with a saddle node on invariant cycle onset bifurcation and ii results in a symmetry breaking of the system s phase response curve the latter entails an increase in synchronization range in pulse coupled oscillators such as neurons the derived bifurcation structure is of interest in any system for which a relaxation limit is admissible such as josephson junctions and chemical oscillators

bounding the difference between true and nominal rejection probabilities in tests of hypotheses about instrumental variables models abstract this paper presents a simple method for carrying out inference in a wide variety of possibly nonlinear iv models under weak assumptions the method provides a finite sample bound on the difference between the true and nominal probabilities of rejecting a correct null hypothesis the method is a non studentized version of the s test of stock and wright 2000 but is implemented and analyzed differently it does not require restrictive distributional assumptions linearity of the estimated model simultaneous equations or information about whether the instruments are strong or weak it can be applied to quantile iv models that may be nonlinear and can be used to test a parametric iv model against a nonparametric alternative it provides information about the relation between the degree of weakness of the instruments and the power of the test the bound presented here holds in finite samples regardless of the strength of the instruments

boundary relations and their weyl families the concepts of boundary relations and the corresponding weyl families are introduced let s be a closed symmetric linear operator or more generally a closed symmetric relation in a hilbert space h let h be an auxiliary hilbert space let graphics and let jh be defined analogously a unitary relation g from the krein space h 2 j h to the kre in space h 2 j h is called a boundary relation for the adjoint s if ker gamma s the corresponding weyl family m lambda is de ned as the family of images of the defect subspaces n over cap lambda lambda is an element of c r under gamma here gamma need not be surjective and is even allowed to be multi valued while this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex hilbert space h and the class of unitary relations gamma h 2 j h h 2 j h it also generalizes the notion of so called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems moreover these new notions yield for instance the following realization theorem every h valued maximal dissipative for lambda is an element of c holomorphic family of linear relations is the weyl family of a boundary relation which is unique up to unitary equivalence if certain minimality conditions are satisfied further connections between analytic and spectral theoretical properties of weyl families and geometric properties of boundary relations are investigated and some applications are given

clustering in globally coupled inhibitory neurons abstract a model of a large population of identical excitable neurons with a global slowly decaying inhibitory coupling is studied and its patterns of synchrony are examined in addition to converging to a homogeneous fixed point and a homogeneous limit cycle the system exhibits cluster states in which it breaks spontaneously into a few macroscopically big clusters each of which is fully synchronized a method for calculating the stability of cluster states is described and used for investigating the dynamical behavior of the network versus the parameters that describe the neurons and synapses effects of stochastic noise on the network dynamics are discussed at large enough noise the system goes to a globally stationary state low levels of noise preserve the cluster like neuron trajectories in the regime where a noiseless system converges to a fully synchronized periodic state relatively low noise levels cause neurons to burst only every two or more consecutive time periods

generalization of coupled spiking models and effects of the width of an action potential on synchronization phenomena the integrate and fire if neuron model and the piecewise linear version of fitzhugh nagumo pl neuron model with a time scale parameter mu are frequently being used in the study of synchronization phenomena although the two models are regarded to be different type we show a certain equivalence between them by deriving the coupled if model of an improved version with a firing duration from the recovery variable coupled system of the pl model under taking the limit of mu 0 without a loss of any coupling properties in the coupled if model with the duration time the synchronization behavior of a pair of neurons with excitatory or inhibitory synaptic coupling can be systematically explored in terms of three parameters in the model synaptic strength decaying relaxation rate of the synaptic coupling and the parameter exhibiting the firing duration we find some irregularly synchronous behavior with or without constant firing order alternations we show that the duration of an impulse plays an important role in synchronization phenomena

an analytic characterization of the eigenvalues of self adjoint extensions let a be a self adjoint extension in k of a fixed symmetric operator a in k k an analytic characterization of the eigenvalues of a is given in terms of the q function and the parameter function in the krein naimark formula here k and k are krein spaces and it is assumed that a locally has the same spectral properties as a self adjoint operator in a pontryagin space the general results are applied to a class of boundary value problems with dependent boundary conditions

threshold effects on synchronization of pulse coupled oscillators the threshold effects of synchronous firing are studied in a class of models the main result is that for almost all initial conditions the oscillation system approaches synchrony when the threshold condition is below a critical value

spectral analysis of singular ordinary differential operators with indefinite weights abstract in this paper we develop a perturbation approach to investigate spectral problems for singular ordinary differential operators with indefinite weight functions we prove a general perturbation result on the local spectral properties of selfadjoint operators in krein spaces which differ only by finitely many dimensions from the orthogonal sum of a fundamentally reducible operator and an operator with finitely many negative squares this result is applied to singular indefinite sturm liouville operators and higher order singular ordinary differential operators with indefinite weight functions

symmetry based inference in an instrumental variable setting we describe exact inference based on group invariance assumptions that specify various forms of symmetry in the distribution of a disturbance vector in a general nonlinear model it is shown that such mild assumptions can be equivalently formulated in terms of exact confidence sets for the parameters of the functional form when applied to the linear model this exact inference provides a unified approach to a variety of parametric and distribution free tests in particular we consider exact instrumental variable inference based on symmetry assumptions the unboundedness of exact confidence sets is related to the power to reject a hypothesis of underidentification in a multivariate instrumental variables context generalizations of anderson rubin confidence sets are considered

completion extension factorization and lifting of operators the well known results of m g kre n concerning the description of selfadjoint contractive extensions of a hermitian contraction t 1 and the characterization of all nonnegative selfadjoint extensions widetilde a of a nonnegative operator a via the inequalities a k le widetilde a le a f where a k and a f are the kre n von neumann extension and the friedrichs extension of a are generalized to the situation where widetilde a is allowed to have a fixed number of negative eigenvalues these generalizations are shown to be possible under a certain minimality condition on the negative index of the operators i t 1 t 1 and a respectively these conditions are automatically satisfied if t 1 is contractive or a is nonnegative respectively the approach developed in this paper starts by establishing first a generalization of an old result due to yu l shmul yan on completions of nonnegative block operators the extension of this fundamental result allows us to prove analogs of the above mentioned results of m g kre n and in addition to solve some related lifting problems for j contractive operators in hilbert pontryagin and kre n spaces in a simple manner also some new factorization results are derived for instance a generalization of the well known douglas factorization of hilbert space operators in the final steps of the treatment some very recent results concerning inequalities between semibounded selfadjoint relations and their inverses turn out to be central in order to treat the ordering of non contractive selfadjoint operators under cayley transforms properly

tests with correct size when instruments can be arbitrarily weak abstract this paper applies classical exponential family statistical theory to develop a unifying framework for testing structural parameters in the simultaneous equations model under the assumption of normal errors with known reduced form variance matrix the results can be divided into the limited information and full information categories in the limited information model it is possible to characterize the entire class of similar tests in a model with only one endogenous explanatory variable in the full information framework this paper proposes a family of similar tests for subsets of endogenous variables coefficients for both limited and full information models there exist power upper bounds for unbiased tests when the model is just identified the anderson rubin score and pseudo conditional likelihood ratio tests are optimal when the model is over identified the pseudo conditional likelihood ratio test has power close to the power envelope when identification is strong

global competition and local cooperation in a network of neural oscillators abstract an architecture of locally excitatory globally inhibitory oscillator networks is proposed and investigated both analytically and by computer simulation the model for each oscillator corresponds to a standard relaxation oscillator with two time scales oscillators are locally coupled by a scheme that resembles excitatory synaptic coupling and each oscillator also inhibits other oscillators through a common inhibitor oscillators are driven to be oscillatory by external stimulation the network exhibits a mechanism of selective gating whereby an oscillator jumping up to its active phase rapidly recruits the oscillators stimulated by the same pattern while preventing the other oscillators from jumping up we show analytically that with the selective gating mechanism the network rapidly achieves both synchronization within blocks of oscillators that are stimulated by connected regions and desynchronization between different blocks computer simulations demonstrate the model s promising ability for segmenting multiple input patterns in real time this model lays a physical foundation for the oscillatory correlation theory of feature binding and may provide an effective computational framework for scene segmentation and figure ground segregation

continuum of weakly coupled oscillatory mckean neurons the mckean model of a neuron possesses a one dimensional fast voltage like variable and a slow recovery variable a recent geometric analysis of the singularly perturbed system has allowed an explicit construction of its phase response curve s coombes 2001 phase locking in networks of synaptically coupled mckean relaxation oscillators physica d vol 160 173 188 here we use tools from coupled oscillator theory to study weakly coupled networks of mckean neurons using numerical techniques we show that the mckean system has traveling wave phase locked solutions consistent with that of a network of more biophysically detailed hodgkin huxley neurons

a kind of structural frequency locking in generalized spatial automata abstract the classical concept of synchronization is usually related to the locking of the basic frequencies and instantaneous phases of regular oscillations and this question is addressed by studying specific kinds of coupled systems this work presents a different point of view we do not study the convergence of coupled systems to a synchronized behaviour but try to answer the following question in a population of coupled differential systems when each cell subsystem exhibits a periodic behaviour is the whole trajectory periodic we define generalized spatial automata with reference to continuous spatial automata by means of coupling maps and associated measures on the set of cells the main idea is the fact that a cell interprets its own environment via the states of the whole population and according to its own state a natural partition of periods is such that cells belong to the same class if their trajectories share a common period we demonstrate that in a general case where cells belong to an a priori unstructured set and their trajectories evolve in possibly distinct banach spaces the set of classes of periods is generally countable in particular when the set of cells is endowed with a borelian structure all the cells necessarily share a common period

dirichlet to neumann maps abstract weyl titchmarsh m functions and a generalized index of unbounded meromorphic operator valued functions abstract we introduce a generalized index for certain meromorphic unbounded operator valued functions the class of functions is chosen such that energy parameter dependent dirichlet to neumann maps associated to uniformly elliptic partial differential operators particularly non self adjoint schrodinger operators on bounded lipschitz domains and abstract operator valued weyl titchmarsh m functions and donoghue type m functions corresponding to closed extensions of symmetric operators belong to it the principal purpose of this paper is to prove index formulas that relate the difference of the algebraic multiplicities of the discrete eigenvalues of robin realizations of non self adjoint schrodinger operators and more abstract pairs of closed operators in hilbert spaces with the generalized index of the corresponding energy dependent dirichlet to neumann maps and abstract weyl titchmarsh m functions respectively

phase equations for relaxation oscillators we use the malkin theorem to derive phase equations for networks of weakly connected relaxation oscillators we find an explicit formula for the connection functions when the oscillators have one dimensional slow variables the functions are discontinuous in the relaxation limit mu rightarrow 0 which provides a simple alternative illustration to the major conclusion of the fast threshold modulation ftm theory by somers and kopell biological cybernetics 68 1993 pp 393 407 that synchronization of relaxation oscillators has properties that are quite different from those of smooth nonrelaxation oscillators we use bonhoeffer van der pol relaxation oscillators to illustrate the theory numerically

sturm liouville problems with boundary conditions rationally dependent on the eigenparameter ii necessary and sufficient conditions are given for two sequences n and n to be the eigenvalues and norming constants of a sturm liouville boundary value problem y qy y y 0 cos y 0 sin and y 1 f y 1 where f is a rational function of herglotz nevanlinna type it is also proved that q and f are uniquely determined by the sequences n and n

synchronization rates in classes of relaxation oscillators relaxation oscillators arise frequently in physics electronics mathematics and biology their mathematical definitions possess a high degree of flexibility in the sense that through appropriate parameter choices relaxation oscillators can be made to exhibit qualitatively different kinds of oscillations we study numerically four different classes of relaxation oscillators through their synchronization rates in one dimensional chains with a heaviside step function interaction and obtain the following results relaxation oscillators in the sinusoidal and relaxation regime both exhibit an average time to synchrony spl sim n where n is the chain length relaxation oscillators in the singular limit exhibit spl sim n sup p where p is a numerically obtained value less than 0 5 relaxation oscillators in the singular limit with parameters modified so that they resemble spike oscillations exhibit spl sim log n in chains and spl sim log l in two dimensional square networks of length l finally using a sigmoid interaction results in spl sim n sup 2 for relaxation oscillators in the sinusoidal and relaxation regimes indicating that the form of the coupling is a controlling factor in the synchronization rate