pseudounitary symmetry and the gaussian pseudounitary ensemble of random matrices employing the currently discussed notion of pseudo hermiticity we define a pseudounitary group further we develop a random matrix theory that is invariant under such a group and call this ensemble of pseudo hermitian random matrices the pseudounitary ensemble we obtain exact results for the nearest neighbor level spacing distribution for 2 x 2 pt invariant hamiltonian matrices that have forms approximately sln 1 s near zero spacing for three independent elements and approximately s for four independent elements this shows a level repulsion in a marked distinction with an algebraic form s beta in the wigner surmise we believe that this paves the way for a description of varied phenomena in two dimensional statistical mechanics quantum chromodynamics and so on

asymptotic optimality of cl and generalized cross validation in ridge regression with application to spline smoothing the asymptotic optimality of mallows c l and generalized cross validation is demonstrated in the setting of ridge regression an application is made to spline smoothing in nonparametric regression a counterexample is given to help understand why sometimes gcv may not be asymptotically optimal the coefficient of variation for the eigenvalues of the information matrix must be large in order to guarantee the optimality of gcv the proff is based on the connection between gcv and stein s unbiased risk estimate

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

an efficient estimator for the generalized semilinear model abstract the generalized semilinear model gslim extends the generalized linear model by allowing the addition of an unknown smooth function to the predictor parameter producing a semiparametric model some previously proposed score based estimators for the parametric component in this model are not n consistent we propose efficient estimators for both the parametric and nonparametric components of the gslim the estimator for the parametric component is the solution to an estimated efficient score equation and is based on recent theoretical advances in semiparametric estimation the estimator for the nonparametric component is the maximizer of an estimated penalized likelihood of the form considered by cox and o sullivan a robust standard error estimate maintains valid inference under certain model violations an application and simulation results are given for data with binary outcomes using a modified generalized cross validation algorithm to select the smoothing parameter parameter and standard

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

the stratified canonical fuzzy topology on i i is uniformizable abstract in this paper we have constructed a hutton fuzzy uniformity on the fuzzy unit interval i i which in the sense of hutton generates the stratified canonical fuzzy topology on i i thus a positive answer to a question posed by s e rodabaugh j math anal appl 129 1988 37 70 in the case l i is obtained moreover we have also proved that has a countable base therefore the stratified canonical fuzzy topology in i i is also metrizable in the sense of hutton

transition between hermitian and non hermitian gaussian ensembles the transition between hermitian and non hermitian matrices of the gaussian unitary ensemble is revisited an expression for the kernel of the rescaled hermite polynomials is derived which expresses the sum in terms of the highest order polynomials from this christoffel darboux like formula some results are derived including an extension to the complex plane of the airy kernel

regularized reconstruction in electrical impedance tomography using a variance uniformization constraint this paper describes a new approach to reconstruction of the conductivity field in electrical impedance tomography our goal is to improve the tradeoff between the quality of the images and the numerical complexity of the reconstruction method in order to reduce the computational load we adopt a linearized approximation to the forward problem that describes the relationship between the unknown conductivity and the measurements in this framework we focus on finding a proper way to cope with the ill posed nature of the problem mainly caused by strong attenuation phenomena this is done by devising regularization techniques well suited to this particular problem first we propose a solution which is based on tikhonov regularization of the problem second we introduce an original regularized reconstruction method in which the regularization matrix is determined by space uniformization of the variance of the reconstructed conductivities both methods are nonsupervised i e all tuning parameters are automatically determined from the measured data tests performed on simulated and real data indicate that tikhonov regularization provides results similar to those obtained with iterative methods but with a much smaller amount of computations regularization using a variance uniformization constraint yields further improvements particularly in the central region of the unknown object where attenuation is most severe we anticipate that the variance uniformization approach could be adapted to iterative methods that preserve the nonlinearity of the forward problem more generally it appears as a useful tool for solving other severely ill posed reconstruction problems such as eddy current tomography

resolution and stability analysis of an inverse problem in electrical impedance tomography dependence on the input current patterns electrical impedance tomography is a procedure by which one finds the conductivity distribution inside a domain from measurements of voltages and currents at the boundary this work addresses the i

a theory of fuzzy uniformities with applications to the fuzzy real lines abstract for each completely distributive lattice l with order reversing involution the fuzzy real line r l is uniformizable by a uniformity which both generates the canonical fuzzy topology and induces a pseudometric generating the canonical topology if l is also a chain the usual addition and multiplication defined on r r 0 1 extend jointly fuzzy continuously to and on r l three fundamental questions in fuzzy sets until now are question a if l 1 l 2 is r l 1 uniformly isomorphic to r l 2 in some sense question b for each chain l is jointly uniformly continuous in a sense which guarantees its joint continuity on r l question c is r l a complete pseudometric space in some sense we construct categories qu and u using the quasi uniformities of b hutton which enable us to answer these questions in the affirmative these results enhance the canonical standing of the fuzzy real lines and so give additional justification for answering in the affirmative question d does fuzzy topology have deep specific canonical examples

random matrix ensembles for pt symmetric systems recently much effort has been made towards the introduction of non hermitian random matrix models respecting pt symmetry here we show that there is a one to one correspondence between complex pt symmetric matrices and split complex and split quaternionic versions of hermitian matrices we introduce two new random matrix ensembles of a gaussian split complex hermitian and b gaussian split quaternionic hermitian matrices of arbitrary sizes we conjecture that these ensembles represent universality classes for pt symmetric matrices for the case of 2 2 matrices we derive analytic expressions for the joint probability distributions of the eigenvalues the one level densities and the level spacings in the case of real eigenvalues

regularized liml for many instruments the use of many moment conditions improves the asymptotic efficiency of the instrumental variables estimators however in finite samples the inclusion of an excessive number of moments increases the bias to solve this problem we propose regularized versions of the limited information maximum likelihood liml based on three different regularizations tikhonov landweber fridman and principal components our estimators are consistent and asymptotically normal under heteroskedastic error moreover they reach the semiparametric efficiency bound assuming homoskedastic error we show that the regularized liml estimators possess finite moments when the sample size is large enough the higher order expansion of the mean square error mse shows the dominance of regularized liml over regularized two staged least squares estimators we devise a data driven selection of the regularization parameter based on the approximate mse a monte carlo study and two empirical applications illustrate the relevance of our estimators

the stratified canonical fuzzy topology on i l is uniformizable abstract in this paper it is proved that for every completely distributive lattice l with an order reversing involution the stratified canonical fuzzy topology on i l is uniformizable in the sense of hutton

induced current electrical impedance tomography a 2 d theoretical simulation a reconstruction algorithm based on the modified newton raphson algorithm was developed for induced current electrical impedance tomography and studied in theoretical two dimensional geometry representing a human thorax the finite volume method was applied for the discretization of the physical domain resulting in a symbolic representation of the jacobian matrix which is accurate and fast to construct several system configurations differing in the number of excitation coils and electrodes were simulated and the performance in thoracic imaging was studied it was found that a six coil system shows a significant 40 improvement of conductivity values reconstruction over the three coil system an error of 2 06 spl omega sup 1 compared with 3 44 spl omega sup 1 a number of 32 electrodes was found to be sufficient being the smallest number of electrodes to still provide a reasonable performance only 4 2 degradation in average conductivity error compared with the maximum possible 106 electrode system

a categorical accommodation of various notions of fuzzy topology different definitions of fuzzy topology have been stated and developed in the literature but have not been satisfactorily related to each other and to ordinary topology to remedy this a new fuzzy topological category fuzz is defined it is a significant generalization of previous definitions it is the simplest setting yet constructed into which ordinary topology and these previous definitions may be placed by identifying each with a different subcategory of fuzz it therefore allows us to conveniently analyze the relative merits of previous definitions give coherence to known results and indicate appropriate directions for future development and it generates and may be viewed as both including and included in categories of fuzzy topological spaces which exhibit higher order fuzziness

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

gaussian ensemble of 2 2 pseudo hermitian random matrices we present a random matrix theory for systems invariant under the joint action of parity and time reversal and more generally for pseudo hermitian systems this brings out the appearance of the metric in a systematic way so that consistency with the postulates of quantum mechanics is maintained here we specialize only to 2 2 matrices and we construct a pseudo unitary group with explicit examples nearest neighbour level spacing distributions for various classes of ensembles are found to exhibit a degree of level repulsion different from those hitherto known this work is not only relevant to quantum chaos but also to two dimensional statistical mechanics and consistent non local relativistic theories

spline smoothing and optimal rates of convergence in nonparametric regression models linear estimation is considered in nonparametric regression models of the form yi f xi ei xi e a b where the zero mean errors are uncorrelated with common variance a2 and the response function f is assumed only to have a bounded square integrable qth derivative the linear estimator which minimizes the maximum mean squared error summed over the observation points is derived and the exact minimax rate of convergence is obtained for practical problems where bounds on 11 f2q 11 and a2 may be unknown generalized cross validation is shown to give an adaptive estimator which achieves the minimax optimal rate under the additional assumption of normality 1 the model consider the nonparametric regression model

methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation the application of regularization to ill conditioned problems necessitates the choice of a regularization parameter which trades fidelity to the data with smoothness of the solution the value of the regularization parameter depends on the variance of the noise in the data the problem of choosing the regularization parameter and estimating the noise variance in image restoration is examined an error analysis based on an objective mean square error mse criterion is used to motivate regularization two approaches for choosing the regularization parameter and estimating the noise variance are proposed the proposed and existing methods are compared and their relationship to linear minimum mean square error filtering is examined experiments are presented that verify the theoretical results

weighted least squares criteria for electrical impedance tomography methods are developed for the design of electrical impedance tomographic reconstruction algorithms with specified properties assuming a starting model with constant conductivity or some other specified background distribution an algorithm with the following properties is found 1 the optimum constant for the starting model is determined automatically 2 the weighted least squares error between the predicted and measured power dissipation data is as small as possible 3 the variance of the reconstructed conductivity from the starting model is minimized 4 potential distributions with the largest volume integral of gradient squared have the least influence on the reconstructed conductivity and therefore distributions most likely to be corrupted by contact impedance effects are deemphasized 5 cells that dissipate the most power during the current injection tests tend to deviate least from the background value for a starting model with nonconstant conductivity the reconstruction algorithm has analogous properties

matrix models for beta ensembles this paper constructs tridiagonal random matrix models for general 0 hermite gaussian and laguerre wishart ensembles these generalize the well known gaussian and wishart models for 1 2 4 furthermore in the cases of the laguerre ensembles we eliminate the exponent quantization present in the previously known models we further discuss applications for the new matrix models and present some open problems

from stein s unbiased risk estimates to the method of generalized cross validation this paper concerns the method of generalized cross validation gcv a promising way of choosing between linear estimates based on stein estimates and the associated unbiased risk estimates stein 1981 a new approach to gcv is developed many consistency results are obtained for the cross validated steinized estimates in the contexts of nearest neighbor nonparametric regression model selection ridge regression and smoothing splines moreover the associated stein s unbiased risk estimate is shown to be uniformly consistent in assessing the true loss not the risk consistency properties are examined as well when the sampling error is unknown finally we propose a variant of gcv to handle the case that the dimension of the raw data is known to be greater than that of their expected values

image reconstruction and restoration overview of common estimation structures and problems developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined particular attention is paid to common estimation structures and to practical problems not properly solved yet the problem of image reconstruction and restoration is first formulated some of the current regularization approaches used to solve the problem are then described the concepts of a priori information and compound criterion are introduced a bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated the practical aspects of computing the solution first when the hyperparameters are known and second when they must be estimated are then considered conclusions are drawn and points that still need to be investigated are outlined

non hermitian localization and population biology the time evolution of spatial fluctuations in inhomogeneous d dimensional biological systems is analyzed a single species continuous growth model in which the population disperses via diffusion and convection is considered time independent environmental heterogeneities such as a random distribution of nutrients or sunlight are modeled by quenched disorder in the growth rate linearization of this model of population dynamics shows that the fastest growing localized state dominates in a time proportional to a power of the logarithm of the system size using an analogy with a schr odinger equation subject to a constant imaginary vector potential we propose a delocalization transition for the steady state of the nonlinear problem at a critical convection threshold separating localized and extended states in the limit of high convection velocity the linearized growth problem in d dimensions exhibits singular scaling behavior described by a d ensuremath 1 dimensional generalization of the noisy burgers equation with universal singularities in the density of states associated with disorder averaged eigenvalues near the band edge in the complex plane the burgers mapping leads to unusual transverse spreading of convecting delocalized populations

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

a comparison of gcv and gml for choosing the smoothing parameter in the generalized spline smoothing problem the partially improper prior behind the smoothing spline model is used to obtain a generalization of the maximum likelihood gml estimate for the smoothing parameter then this estimate is compared with the generalized cross validation gcv estimate both analytically and by monte carlo methods the comparison is based on a predictive mean square error criteria it is shown that if the true unknown function being estimated is smooth in a sense to be defined then the gml estimate undersmooths relative to the gcv estimate and the predictive mean square error using the gml estimate goes to zero at a slower rate than the mean square error using the gcv estimate if the true function is rough then the gcv and gml estimates have asymptotically similar behavior a monte carlo experiment was designed to see if the asymptotic results in the smooth case were evident in small sample sizes mixed results were obtained for n 32 gcv was somewhat better than gml for n 64 and gcv was decidedly superior for n 128 in the n 32 case gcv was better for smaller sigma 2 and the comparison close for larger sigma 2 the theoretical results are shown to extend to the generalized spline smoothing model which includes the estimate of functions given noisy values of various integrals of them

a backprojection algorithm for electrical impedance imaging the authors study a two dimensional reconstruction algorithm due to d c barber and b h brown applied to a linearized electrostatic inverse problem first the authors demonstrate how this algorithm fits within the framework of inverses of generalized radon transforms studied by g beylkin second an iterative improvement of the barber brown algorithm is constructed based on the conjugate residual method several numerical results obtained with this iterative algorithm are presented

oracle inequalities for inverse problems we consider a sequence space model of statistical linear inverse problems where we need to estimate a function f from indirect noisy observations let a finite set a of linear estimators be given our aim is to mimic the estimator in a that has the smallest risk on the true f under general conditions we show that this can be achieved by simple minimization of an unbiased risk estimator provided the singular values of the operator of the inverse problem decrease as a power law the main result is a nonasymptotic oracle inequality that is shown to be asymptotically exact this inequality can also be used to obtain sharp minimax adaptive results in particular we apply it to show that minimax adaptation on ellipsoids in the multivariate anisotropic case is realized by minimization of unbiased risk estimator without any loss of efficiency with respect to optimal nonadaptive procedures

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

monte carlo sure a black box optimization of regularization parameters for general denoising algorithms we consider the problem of optimizing the parameters of a given denoising algorithm for restoration of a signal corrupted by white gaussian noise to achieve this we propose to minimize stein s unbiased risk estimate sure which provides a means of assessing the true mean squared error mse purely from the measured data without need for any knowledge about the noise free signal specifically we present a novel monte carlo technique which enables the user to calculate sure for an arbitrary denoising algorithm characterized by some specific parameter setting our method is a black box approach which solely uses the response of the denoising operator to additional input noise and does not ask for any information about its functional form this therefore permits the use of sure for optimization of a wide variety of denoising algorithms we justify our claims by presenting experimental results for sure based optimization of a series of popular image denoising algorithms such as total variation denoising wavelet soft thresholding and wiener filtering smoothing splines in the process we also compare the performance of these methods we demonstrate numerically that sure computed using the new approach accurately predicts the true mse for all the considered algorithms we also show that sure uncovers the optimal values of the parameters in all cases

spectral analysis of the complex cubic oscillator using the exact semiclassical analysis we study the spectrum of a one parameter family of complex cubic oscillators the invariance property of the complex hamiltonians and the reality property of the spectrum are discussed analytic continuations of the spectrum in the complex parameter and their connections with the resonance problem for the real cubic oscillator are investigated the global analytic structure of the spectrum yields a branch point structure similar to the multivalued analytic structure discovered by bender and wu for the quartic oscillator

bayesian confidence intervals for smoothing splines abstract the frequency properties of wahba s bayesian confidence intervals for smoothing splines are investigated by a large sample approximation and by a simulation study when the coverage probabilities for these pointwise confidence intervals are averaged across the observation points the average coverage probability acp should be close to the nominal level from a frequency point of view this agreement occurs because the average posterior variance for the spline is similar to a consistent estimate of the average squared error and because the average squared bias is a modest fraction of the total average squared error these properties are independent of the bayesian assumptions used to derive this confidence procedure and they explain why the acp is accurate for functions that are much smoother than the sample paths prescribed by the prior this analysis accounts for the choice of the smoothing parameter bandwidth using cross validation in the case of natural splines an adaptive method for avo

on estimating the condition of eigenvalues and eigenvectors abstract a method is developed for estimating the accuracy of computed eigenvalues and eigenvectors that are obtained via certain eispack subroutines it does this at a cost of o n 2 flops per eigenpair assuming that the eigenpair is known and assuming that the original matrix has been reduced to hessenberg form the heart of the technique involves estimating the smallest singular value of a certain nearly triangular submatrix this is accomplished by some standard zero chasing with givens transformations and with a 2 norm version of the linpack condition estimator an eispack compatible code has been developed and its performance is discussed

iterative reconstruction methods using regularization and optimal current patterns in electrical impedance tomography an iterative reconstruction method which minimizes the effects of ill conditioning is discussed based on the modified newton raphson algorithm a regularization method which integrates prior information into the image reconstruction was developed this improves the conditioning of the information matrix in the modified newton raphson algorithm optimal current patterns were used to obtain voltages with maximal signal to noise ratio snr a complete finite element model fem was used for both the internal and the boundary electric fields reconstructed images from phantom data show that the use of regularization optimal current patterns and a complete fem model improves image accuracy the authors also investigated factors affecting the image quality of the iterative algorithm such as the initial guess image iteration and optimal current updating

from finite sample to asymptotics a geometric bridge for selection criteria in spline regression this paper studies under the setting of spline regression the connection between finite sample properties of selection criteria and their asymptotic counterparts focusing on bridging the gap between the two we introduce a bias variance decomposition of the prediction error using which it is shown that in the asymptotics the bias term dominates the variability term providing an explanation of the gap a geometric exposition is provided for intuitive understanding the theoretical and geometric results are illustrated through a numerical example

asymptotic optimality for c_p c_l cross validation and generalized cross validation discrete index set c p c l cross validation and generalized cross validation are useful data driven techniques for selecting a good estimate from a proposed class of linear estimates the asymptotic behaviors of these procedures are studied some easily interpretable conditions are derived to demonstrate the asymptotic optimality it is argued that cross validation and generalized cross validation can be viewed as some special ways of applying c l applications in nearest neighbor nonparametric regression and in model selection are discussed in detail

a pseudo hermitian β hermite family of matrices abstract a family of pseudo hermitian random matrices is constructed whose eigenvalues are the same of the hermite ensemble of random matrix theory it is shown that matrices of this family are unstable in such a way that small perturbations cause the eigenvalues to move into the complex plane in conjugate pairs

2 2 random matrix ensembles with reduced symmetry from hermitian to pt symmetric matrices a possibly fruitful extension of conventional random matrix ensembles is proposed by imposing symmetry constraints on conventional hermitian matrices or parity time symmetric matrices to illustrate the main idea we first study 2 2 complex hermitian matrix ensembles with o 2 invariant constraints yielding novel level spacing statistics such as singular distributions the half gaussian distribution distributions interpolating between the goe gaussian orthogonal ensemble distribution and half gaussian distributions as well as the gapped goe distribution such a symmetry reduction strategy is then used to explore 2 2 symmetric matrix ensembles with real eigenvalues in particular symmetric random matrix ensembles with u 2 invariance can be constructed with the conventional complex hermitian random matrix ensemble being a special case in two examples of symmetric random matrix ensembles the level spacing distributions are found to be the standard gue gaussian unitary ensemble statistics or the truncated gue statistics this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators