pseudo hermiticity versus pt symmetry iii equivalence of pseudo hermiticity and the presence of antilinear symmetries we show that a diagonalizable non hermitian hamiltonian h is pseudo hermitian if and only if it has an antilinear symmetry i e a symmetry generated by an invertible antilinear operator this implies that the eigenvalues of h are real or come in complex conjugate pairs if and only if h possesses such a symmetry in particular the reality of the spectrum of h implies the presence of an antilinear symmetry we further show that the spectrum of h is real if and only if there is a positive definite inner product on the hilbert space with respect to which h is hermitian or alternatively there is a pseudo canonical transformation of the hilbert space that maps h into a hermitian operator

central limit theorems and uniform laws of large numbers for arrays of random fields over the last decades spatial interaction models have been increasingly used in economics however the development of a sufficiently general asymptotic theory for nonlinear spatial models has been hampered by a lack of relevant central limit theorems clts uniform laws of large numbers ullns and pointwise laws of large numbers llns these limit theorems form the essential building blocks towards developing the asymptotic theory of m estimators including maximum likelihood and generalized method of moments estimators the paper establishes a clt ulln and lln for spatial processes or random fields that should be applicable to a broad range of data processes

isospectrality of spherical mhd dynamo operators pseudo hermiticity and a no go theorem the isospectrality problem is studied for the operator of the spherical hydromagnetic 2 dynamo it is shown that this operator is formally pseudo hermitian j symmetric and lives in a krein space based on the j symmetry an operator intertwining ansatz with first order differential intertwining operators is tested for its compatibility with the structure of the 2 dynamo operator matrix an intrinsic structural inconsistency is obtained in the set of associated matrix riccati equations this inconsistency is interpreted as a no go theorem which forbids the construction of isospectral 2 dynamo operator classes with the help of first order differential intertwining operators

on the pseudo hermitian nondiagonalizable hamiltonians we consider a class of possibly nondiagonalizable pseudo hermitian operators with discrete spectrum and we establish for such a class the equivalence between the pseudo hermiticity property and the existence of an antilinear involutory symmetry moreover we prove that this class actually coincides with the one of possibly nondiagonalizable weak pseudo hermitian operators and that in no case unless they are diagonalizable and have a real spectrum they are hermitian with respect to a definite inner product finally we show that a typical degeneracy of the real eigenvalues which reduces to the well known kramers degeneracy in the hermitian case occurs whenever a fermionic possibly nondiagonalizable pseudo hermitian hamiltonian admits an antilinear symmetry like the time reversal operator t some consequences and applications are briefly discussed

pseudo hermiticity and generalized pt and cpt symmetries we study certain linear and antilinear symmetry generators and involution operators associated with pseudo hermitian hamiltonians and show that the theory of pseudo hermitian operators provides a simple explanation for the recent results of bender brody and jones quant ph 0208076 on the cpt symmetry of a class of pt symmetric non hermitian hamiltonians we present a natural extension of these results to the class of diagonalizable pseudo hermitian hamiltonians h with a discrete spectrum in particular we introduce generalized parity p time reversal t and charge conjugation c operators and establish the pt and cpt invariance of h

fragile pt symmetry in a solvable model one of the simplest pseudo hermitian models with real spectrum viz square well on a real interval i of coordinates is re examined a pt symmetric complex deformation c of i is introduced and shown tractable via an innovated approach to matching conditions the result is surprising an arbitrarily small deformation i c implies a sudden collapse i e the spontaneous pt symmetry breaking of virtually all the spectrum i e up to its low energy part

bayesian computation via the gibbs sampler and related markov chain monte carlo methods the use of the gibbs sampler for bayesian computation is reviewed and illustrated in the context of some canonical examples other markov chain monte carlo simulation methods are also briefly described and comments are made on the advantages of sample based approaches for bayesian inference summaries

large sample properties of the matrix exponential spatial specification with an application to fdi this paper studies large sample properties of the matrix exponential spatial specification mess we find that the quasi maximum likelihood estimator qmle for the mess is consistent under heteroskedasticity a property not shared by the qmle of the sar model for the general model that has mess in both the dependent variable and disturbances labeled mess 1 1 the qmle can be consistent under unknown heteroskedasticity when the spatial weights matrices in the two mess processes are commutative we also consider the generalized method of moments estimator gmme in the homoskedastic case we derive a best gmme that is as efficient as the maximum likelihood estimator under normality and can be asymptotically more efficient than the qmle under non normality in the heteroskedastic case an optimal gmme can be more efficient than the qmle asymptotically the qml approach for the mess model has the computational advantage over that of a sar model the computational simplicity carries over to mess models with any finite order of spatial matrices no parameter range needs to be imposed in order for the model to be stable results of monte carlo experiments for finite sample properties of the estimators are reported finally the mess 1 1 is applied to belgium s outward fdi data and we observe that the dominant motivation of belgium s outward fdi lies in finding cheaper factor inputs

nonlinear generalized functions on manifolds in this work we adopt a new approach to the construction of a global theory of algebras of generalized functions on manifolds based on the concept of smoothing operators this produces a generaliz

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

microlocal properties of basic operations in colombeau algebras abstract the colombeau algebra of generalized functions allows us to unrestrictedly carry out products of distributions we analyze this operation from a microlocal point of view deriving a general inclusion relation for wave front sets of products in the algebra furthermore we give explicit examples showing that the given result is optimal i e its assumptions cannot be weakened finally we discuss the interrelation of these results with the concept of pullback under smooth maps

factorization little or great algorithm the progress of the factorization method since the 1935 work of dirac is briefly reviewed though linked with older mathematical theories the factorization seems an autonomous driving force offering substantial support to the present day darboux and backlund approaches

integration and microlocal analysis in colombeau algebras of generalized functions abstract we study integration and fourier transform in the colombeau algebra g of tempered generalized functions using a general damping factor this unifies different settings described earlier by j f colombeau m nedeljikov s pilipovic and m damsma for a simplified version further we prove characterizations of regularity for generalized functions in two situations compactly supported or in the image of s inside g finally we investigate the notion of wave front set in the colombeau algebra g an open subset of r n and show that it is in fact independent of the damping measure used for fourier transform

on multimodality of the likelihood in the spatial linear model summary a popular covariance scheme used for the spatial linear model in geostatistics has spherical form however the likelihood is not twice differentiable with respect to the range parameter and this raises some questions regarding the unimodality of the likelihood we compare the likelihoods of the spatial linear model for small samples under this scheme and the doubly geometric scheme also a power covariance with range parameter is proposed in view of potential multimodal likelihoods for small samples for this model a convenient profile likelihood is introduced and studied

the intrinsic bayes factor for model selection and prediction abstract in the bayesian approach to model selection or hypothesis testing with models or hypotheses of differing dimensions it is typically not possible to utilize standard noninformative or default prior distributions this has led bayesians to use conventional proper prior distributions or crude approximations to bayes factors in this article we introduce a new criterion called the intrinsic bayes factor which is fully automatic in the sense of requiring only standard noninformative priors for its computation and yet seems to correspond to very reasonable actual bayes factors the criterion can be used for nested or nonnested models and for multiple model comparison and prediction from another perspective the development suggests a general definition of a reference prior for model comparison

nonlinear pseudo supersymmetry in the framework of fold supersymmetry we recall the importance of recognizing the different mathematical nature of various concepts relating to symmetric quantum theories after clarifying the relation between supersymmetry and pseudo supersymmetry we prove generically that nonlinear pseudo supersymmetry recently proposed by sinha and roy is just a special case of fold supersymmetry in particular we show that all the models constructed by these authors have type a two fold supersymmetry furthermore we prove that an arbitrary one body quantum hamiltonian which admits two local solutions in closed form belongs to type a two fold supersymmetry irrespective of whether or not it is hermitian symmetric pseudo hermitian and so on

mathcal p mathcal t symmetry as a generalization of hermiticity the hilbert space in symmetric quantum mechanics is formulated as a linear vector space with a dynamic inner product the most general symmetric matrix hamiltonians are constructed for the 2 2 and 3 3 cases in the former case the symmetric hamiltonian represents the most general matrix hamiltonian with a real spectrum in both cases hermitian matrices are shown to be special cases of symmetric matrices this finding confirms and strengthens the early belief that the symmetric quantum mechanics is a generalization of the conventional hermitian quantum mechanics

symmetric hyperbolic systems in algebras of generalized functions and distributional limits we study existence uniqueness and distributional aspects of generalized solutions to the cauchy problem for first order symmetric or hermitian hyperbolic systems of partial differential equations with colombeau generalized functions as coefficients and data the proofs of solvability are based on refined energy estimates on lens shaped regions with spacelike boundaries we obtain several variants and also partial extensions of previous results in oberguggenberger 1989 lafon and oberguggenberger 1991 and hormann 2004 26 23 16 and provide aspects accompanying related recent work in oberguggenberger 2009 garetto and oberguggenberger 2011 28 10 9

bayesian model averaging for linear regression models abstract we consider the problem of accounting for model uncertainty in linear regression models conditioning on a single selected model ignores model uncertainty and thus leads to the underestimation of uncertainty when making inferences about quantities of interest a bayesian solution to this problem involves averaging over all possible models i e combinations of predictors when making inferences about quantities of interest this approach is often not practical in this article we offer two alternative approaches first we describe an ad hoc procedure occam s window which indicates a small set of models over which a model average can be computed second we describe a markov chain monte carlo approach that directly approximates the exact solution in the presence of model uncertainty both of these model averaging procedures provide better predictive performance than any single model that might reasonably have been selected in the extreme case where there are many candidate predictors but

a likelihood ratio test for spatial model selection abstract this paper develops a nondegenerate likelihood ratio test for model selection between two competitive spatial econometrics models it generalizes the test of vuong 1989 to models with spatial near epoch dependent ned data we do not make any structural assumption on the true model specification and allow for the cases where both or one of the two competing models are mis specified the test is valid whether two models are nested or non nested as a prerequisite of the test we first show that quasi maximum likelihood estimators qmle of spatial econometrics models are consistent estimators of their pseudo true values and are asymptotically normal under regularity conditions in particular we study spatial autoregressive models with spatial autoregressive errors sarar and matrix exponential spatial specification mess models we derive the limiting null distribution of the test statistic a spatial heteroskedastic and autoregressive consistent estimator of asymptotic variance of the test statistic under the null which is necessary to implement the test is constructed monte carlo experiments are designed to investigate finite sample performance of qmles for sarar and mess models as well as the size and power of the proposed test

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

p mathcal t symmetric non self adjoint operators diagonalizable and non diagonalizable with a real discrete spectrum consider in the operator family h0 a 1a1 a dad d 2 is the quantum harmonic oscillator with rational frequencies w is a symmetric bounded potential and g is a real coupling constant we show that if g being an explicitly determined constant the spectrum of h g is real and discrete moreover we show that the operator has a real discrete spectrum but is not diagonalizable

a matrix exponential spatial specification we introduce the matrix exponential as a way of modelling spatially dependent data the matrix exponential spatial specification mess simplifies the log likelihood allowing a closed form solution to the problem of maximum likelihood estimation and greatly simplifies the bayesian estimation of the model the mess can produce estimates and inferences similar to those from conventional spatial autoregressive models but has analytical computational and interpretive advantages we present maximum likelihood and bayesian approaches to the estimation of this spatial model specification along with methods of model comparisons over different explanatory variables and spatial specifications

exact pt symmetry is equivalent to hermiticity we show that a quantum system possessing an exact antilinear symmetry in particular pt symmetry is equivalent to a quantum system having a hermitian hamiltonian we construct the unitary operator relating an arbitrary non hermitian hamiltonian with exact pt symmetry to a hermitian hamiltonian we apply our general results to pt symmetry in finite dimensions and give the explicit form of the above mentioned unitary operator and hermitian hamiltonian in two dimensions our findings lead to the conjecture that non hermitian cpt symmetric field theories are equivalent to certain nonlocal hermitian field theories

spiked potentials and quantum toboggans even if the motion of a quantum quasi particle proceeds along a left right symmetric symmetric curved path in complex plane the spectrum of bound states may remain physical i e real and bounded below a generalization is outlined first we show how the topologically less trivial tobogganic contours may be allowed to live on several sheets of a riemann surface second the specification of a scattering regime is formulated for such a class of models

pseudounitary operators and pseudounitary quantum dynamics we consider pseudounitary quantum systems and discuss various properties of pseudounitary operators in particular we prove a characterization theorem for block diagonalizable pseudounitary operators with finite dimensional diagonal blocks furthermore we show that every pseudounitary matrix is the exponential of i 1 times a pseudo hermitian matrix and determine the structure of the lie groups consisting of pseudounitary matrices in particular we present a thorough treatment of 2 2 pseudounitary matrices and discuss an example of a quantum system with a 2 2 pseudounitary dynamical group as other applications of our general results we give a proof of the spectral theorem for symplectic transformations of classical mechanics demonstrate the coincidence of the symplectic group sp 2n with the real subgroup of a matrix group that is isomorphic to the pseudounitary group u n n and elaborate on an approach to second quantization that makes use of the underlying pseudounitary dynamical groups

the matrix logarithmic covariance model abstract a flexible method is introduced to model the structure of a covariance matrix c and study the dependence of the covariances on explanatory variables by observing that for any real symmetric matrix a the matrix exponential transformation c exp a is a positive definite matrix because there is no constraint on the possible values of the upper triangular elements on a any possible structure of interest can be imposed on them the method presented here is not intended to replace the existing special models available for a covariance matrix but rather to provide a broad range of further structures that supplements existing methodology maximum likelihood estimation procedures are used to estimate the parameters and the large sample asymptotic properties are obtained a simulation study and two real life examples are given to illustrate the method introduced

problems with likelihood estimation of covariance functions of spatial gaussian processes summary maximum likelihood has frequently been suggested as a way to estimate covariance parameters in spatial gaussian processes the results of some simple examples based on simulated data are discussed it is shown by an example with real data that the profile likelihood can be multimodal and that the global maximum may not correspond to a sensible value of the parameters mardia marshall 1984 worked in the following framework we have a gaussian process z x i x e x with mean given by a regression equation e z x f x t for a p dimensional vector the covariance function cov z x z y c x y 0 is also specified by a q dimensional vector 0 this process is observed at n points xi xn and from these observations we wish to estimate 3 and 0 let zn z xi z xn t be the vector of observations and k c xi xj 0 be its covariance matrix finally let f be the matrix with rows f xi t then zn has a multivariate normal distribution with mean f18 dispersion matrix k and the log likelihood is up to an additive constant

the signed root deviance profile and confidence intervals in maximum likelihood analysis abstract we investigate a natural extension of the profile t plot of bates and watts to a general parametric function g of the parameters in a general maximum likelihood analysis although the basic purpose of the extension called the signed root deviance profile srdp is to construct likelihood ratio lr confidence intervals for g it has various other applications that significantly extend its usefulness the tangent to the plot of the srdp at the maximum likelihood estimate of g gives the linear approximation la interval based on the observed information matrix the plot may be used as a diagnostic tool to compare la and lr intervals and to suggest transformations of g whose la intervals when inverted are close to the lr intervals for g the standard way to construct any profile is through repeated optimizations but problems associated with nonlinear constraints can make this difficult an alternative method based on integration is presented that avoids these problems an exam

benchmark priors for bayesian model averaging in contrast to a posterior analysis given a particular sampling model posterior model probabilities in the context of model uncertainty are typically rather sensitive to the speciflcation of the prior in particular difiuse priors on model speciflc parameters can lead to quite unexpected consequences here we focus on the practically relevant situation where we need to entertain a large number of sampling models and we have or wish to use little or no subjective prior information we aim at providing an automatic or benchmark prior structure that can be used in such cases we focus on the normal linear regression model with uncertainty in the choice of re gressors we propose a partly noninformative prior structure related to a natural conjugate g prior speciflcation where the amount of subjective information requested from the user is limited to the choice of a single scalar hyperparameter g0j the consequences of difierent choices for g0j are examined we investigate theoretical properties such as consistency of the implied bayesian procedure links with classical information criteria are provided in addition we examine the flnite sample implications of several choices of g0j in a simulation study the use of the mc 3 algorithm of madigan and york 1995 combined with e cient coding in fortran makes it feasible to conduct large simulations in addition to posterior criteria we shall also compare the predictive performance of difierent priors a classic exam ple concerning the economics of crime will also be provided and contrasted with results in the literature the main flndings of the paper will lead us to propose a benchmark prior speciflcation in a linear regression context with model uncertainty

non nested testing of spatial correlation we develop non nested tests in a general spatial spatio temporal or panel data context the spatial aspect can be interpreted quite generally in either a geographical sense or employing notions of economic distance or when parametric modelling arises in part from a common factor or other structure in the former case observations may be regularly spaced across one or more dimensions as is typical with much spatio temporal data or irregularly spaced across all dimensions both isotropic models and non isotropic models can be considered and a wide variety of correlation structures in the second case models involving spatial weight matrices are covered such as spatial autoregressive models the setting is sufficiently general to potentially cover other parametric structures such as certain factor models and vector valued observations and here our preliminary asymptotic theory for parameter estimates is of some independent value the test statistic is based on a gaussian pseudo likelihood ratio and is shown to have an asymptotic standard normal distribution under the null hypothesis that one of the two models is correct this limit theory rests strongly on a central limit theorem for the gaussian pseudo maximum likelihood parameter estimates a small monte carlo study of finite sample performance is included

pseudo hermiticity versus pt symmetry ii a complete characterization of non hermitian hamiltonians with a real spectrum we give a necessary and sufficient condition for the reality of the spectrum of a non hermitian hamiltonian admitting a complete set of biorthonormal eigenvectors

prediction diagnostics for spatial linear models case deletion diagnostics are developed for detecting observations that are influential for prediction in linear models with a general covariance matrix a primary application of such results is in universal kriging and the related methodologies of ordinary kriging and intrinsic random function kriging in these applications the linear model and the covariance matrix of the data are determined by the locations at which observations are taken computational formulae are given that make the procedures feasible the diagnostics are illustrated using an example

microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients we investigate microlocal properties of partial differential operators with generalized functions as coefficients the main result is an extension of a corresponding microlocalized distribution theoretic result on operators with smooth hypoelliptic symbols methodological novelties and technical refinements appear embedded into classical strategies of proof in order to cope with most delicate interferences by non smooth lower order terms we include simplified conditions which are applicable in special cases of interest

pseudo hermitian hamiltonians indefinite inner product spaces and their symmetries we extend the definition of generalized parity p charge conjugation c and time reversal t operators to nondiagonalizable pseudo hermitian hamiltonians and use these generalized operators to describe the full set of symmetries of a pseudo hermitian hamiltonian according to a fourfold classification in particular we show that t p and ct p are the generators of the p antiunitary symmetries moreover a necessary and sufficient condition is provided for a pseudo hermitian hamiltonian h to admit a p reflecting symmetry which generates the p pseudounitary and the p pseudoantiunitary symmetries finally a physical example is considered and some hints on the p unitary evolution of a physical system are also given

irregular n2sls and lasso estimation of the matrix exponential spatial specification model abstract in this paper we consider estimation of the matrix exponential spatial specification model with the durbin and endogenous regressors we find that the nonlinear two stage least squares n2sls estimator is in general consistent and asymptotically normal however when the durbin and endogenous regressors are irrelevant the gradient vector of the n2sls criterion function has a singular covariance matrix with probability approaching one w p a 1 some components of the n2sls estimator have slower rates of convergence and their asymptotic distributions are nonstandard the distance difference and gradient test statistics which have irregular asymptotic distributions are derived to test for the irrelevance of the durbin and endogenous regressors as an alternative estimation and model selection approach we propose the adaptive group lasso which penalizes the coefficients of the durbin and endogenous explanatory variables we show that the estimator has the oracle properties so the true model can be selected w p a 1 and the estimator always has the n rate of convergence and asymptotic normal distribution we propose to select the tuning parameter for the adaptive group lasso by minimizing an information criterion

hyperbolic systems with discontinuous coefficients generalized solutions and a transmission problem in acoustics where v v v g g g a and f are n x n matrices a is real valued and diagonal in the case of discontinuous coeffkients more precisely we wish to assume solely that a is a bounded measurable function and f is the distributional derivative of such a function the example of hurd and sattinger 3 p 1661 shows that in general such a system will not have solutions in the sense of distributions even if it is in the form of a conservation law observe that a priori 8 v will not be a function and so multiplication of distributions is involved in 0 1 our approach will be to solve the problem in the colombeau algebra y a differential algebra which contains the distributions and in addition has the algebra of infinitely differentiable functions as a sub algebra in section 1 we shall show that if a f g are generalized functions belonging to 9 lr2 and a belongs to 9 r then system 0 1 has a unique solution ve a r provided the generalized functions a and f satisfy some assumptions corresponding to a e l lr fe w l m lr distributional solutions are recovered from the generalized solutions via a partial projection 452 0022 247x 89 3 00

on existence of a biorthonormal basis composed of eigenvectors of non hermitian operators we present a set of necessary conditions for the existence of a biorthonormal basis composed of eigenvectors of non hermitian operators as an illustration we examine these conditions in the case of normal operators we also provide a generalization of the conditions which is applicable to non diagonalizable operators by considering not only eigenvectors but also all root vectors

automatic bayesian curve fitting summary a method of estimating a variety of curves by a sequence of piecewise polynomials is proposed motivated by a bayesian model and an appropriate summary of the resulting posterior distribution a joint distribution is set up over both the number and the position of the knots defining the piecewise polynomials throughout we use reversible jump markov chain monte carlo methods to compute the posteriors the methodology has been successful in giving good estimates for smooth functions i e continuous and differentiable as well as functions which are not differentiable and perhaps not even continuous at a finite number of points the methodology is extended to deal with generalized additive models

the wave equation on singular space times we prove local unique solvability of the wave equation for a large class of weakly singular locally bounded space time metrics in a suitable space of generalised functions

two component formulation of the wheeler dewitt equation the wheeler dewitt equation for the minimally coupled friedman robertson walker massive scalar field minisuperspace is written as a two component schrodinger equation with an explicitly time dependent hamiltonian this reduces the solution of the wheeler dewitt equation to the eigenvalue problem for a nonrelativistic one dimensional harmonic oscillator and an infinite series of trivial algebraic equations whose iterative solution is easily found the solution of these equations yields a mode expansion of the solution of the original wheeler dewitt equation further analysis of the mode expansion shows that in general the solutions of the wheeler dewitt equation for this model are doubly graded i e every solution is a superposition of two definite parity solutions moreover it is shown that the mode expansion of both even and odd parity solutions is always infinite it may be terminated artificially to construct approximate solutions this is demonstrated by working out an explicit example which turns out to satisfy dewitt s boundary condition at initial singularity

projective hilbert space structures at exceptional points a non hermitian complex symmetric 2 2 matrix toy model is used to study projective hilbert space structures in the vicinity of exceptional points eps the bi orthogonal eigenvectors of a diagonalizable matrix are puiseuxexpanded in terms of the root vectors at the ep it is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behaviour in the ep limit can be resolved by projectively extending the original hilbert space the complementary normalization conditions correspond then to two different affine charts of this enlarged projective hilbert space geometric phase and phase jump behaviour are analysed and the usefulness of the phase rigidity as measure for the distance to ep configurations is demonstrated finally ep related aspects of pt symmetrically extended quantum mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated

geometric understanding of likelihood ratio statistics abstract it is well known that twice a log likelihood ratio statistic follows asymptotically a chi square distribution the result is usually understood and proved via taylor s expansions of likelihood functions and by assuming asymptotic normality of maximum likelihood estimators mles we obtain more general results by using a different approach the wilks type of results hold as long as likelihood contour sets are fan shaped the classical wilks theorem corresponds to the situations in which the likelihood contour sets are ellipsoidal this provides a geometric understanding and a useful extension of the likelihood ratio theory as a result even if the mles are not asymptotically normal the likelihood ratio statistics can still be asymptotically chi square distributed our technical arguments are simple and easily understood

monte carlo estimates of the log determinant of large sparse matrices maximum likelihood estimates of parameters of some spatial models require the computation of the log determinant of positive definite matrices of the formi d whered is a large sparse matrix with eigenvalues in 1 1 and where 0 1 with extremely large matrices the usual direct methods of obtaining the log determinant require too much time and memory we propose a monte carlo estimate of the log determinant this estimate is simple to program very sparing in its use of memory easily computed in parallel and can estimate log det i d for many values of simultaneously using this estimator we estimate the log determinant for a 1 000 000 1 000 000 matrixd for 100 values of in 23 1 min on 133 mhz pentium with 64 mb of memory using matlab

pseudo hermiticity for a class of nondiagonalizable hamiltonians we give two characterization theorems for pseudo hermitian possibly nondiagonalizable hamiltonians with a discrete spectrum that admit a block diagonalization with finite dimensional diagonal blocks in particular we prove that for such an operator h the following statements are equivalent 1 h is pseudo hermitian 2 the spectrum of h consists of real and or complex conjugate pairs of eigenvalues and the geometric multiplicity and the dimension of the diagonal blocks for the complex conjugate eigenvalues are identical 3 h is hermitian with respect to a positive semidefinite inner product we further discuss the relevance of our findings for the merging of a complex conjugate pair of eigenvalues of diagonalizable pseudo hermitian hamiltonians in general and the pt symmetric hamiltonians and the effective hamiltonian for a certain closed frw minisuperspace quantum cosmological model in particular

first order hyperbolic pseudodifferential equations with generalized symbols abstract we consider the cauchy problem for a hyperbolic pseudodifferential operator whose symbol is generalized resembling a representative of a colombeau generalized function such equations arise for example after a reduction decoupling of second order model systems of differential equations in seismology we prove existence of a unique generalized solution under log type growth conditions on the symbol thereby extending known results for the case of differential operators j math anal appl 160 1991 93 106 j math anal appl 142 1989 452 467

estimating a product of means bayesian analysis with reference priors abstract suppose that we observe x n 1 and independently y n 1 and are concerned with inference mainly estimation and confidence statements about the product of means this problem arises most obviously in situations of determining area based on measurements of length and width it also arises in other practical contexts however for instance in gypsy moth studies the hatching rate of larvae per unit area can be estimated as the product of the mean of egg masses per unit area times the mean number of larvae hatching per egg mass approximately independent samples can be obtained for each mean see southwood 1978 noninformative prior bayesian approaches to the problem are considered in particular the reference prior approach of bernardo 1979 an appropriate reference prior for the problem is developed and relatively easily implementable formulas for posterior moments e g the posterior mean and variance and credible sets are derived comparisons with alternative noninf

weak pseudo hermiticity and antilinear commutant we inquire into some properties of diagonalizable pseudo hermitian operators showing that their definition can be relaxed and that the pseudo hermiticity property is strictly connected with the existence of an antilinear symmetry this result is then illustrated by considering the particular case of the complex morse potential

a krein space related perturbation theory for mhd α2 dynamos and resonant unfolding of diabolical points the spectrum of the spherically symmetric 2 dynamo is studied in the case of idealized boundary conditions starting from the exact analytical solutions of models with constant profiles a perturbation theory and a galerkin technique are developed in a krein space approach with the help of these tools a very pronounced resonance pattern is found in the deformations of the spectral mesh as well as in the unfolding of the diabolical points located at the nodes of this mesh non oscillatory as well as oscillatory dynamo regimes are obtained an estimation technique is developed for obtaining the critical profiles at which the eigenvalues enter the right spectral half plane with non vanishing imaginary components at which overcritical oscillatory dynamo regimes form finally frechet derivative gradient based methods are developed suitable for further numerical investigations of krein space related setups such as mhd 2 dynamos or models of symmetric quantum mechanics

adaptive inference on pure spatial models abstract in a general class of semiparametric pure spatial models having no explanatory variables allowing nonlinearity in the parameter and the weight matrix we propose adaptive tests and estimates which are asymptotically efficient in the presence of unknown nonparametric distributional form feasibility of adaptive estimation is verified and its efficiency improvement over gaussian pseudo maximum likelihood is shown to be either less than or more than for models with explanatory variables depending on properties of the spatial weight matrix an adaptive lagrange multiplier testing procedure for lack of spatial dependence is proposed and this and our adaptive parameter estimate are extended to cover regression with spatially correlated errors