level crossings and new exact solutions of the two photon rabi model an infinite family of exact solutions of the two photon rabi model was found by investigating the differential algebraic properties of the hamiltonian this family the new states corresponds to energy level crossings not covered by the juddian class which is given by elemetary functions in contrast the new states are expressible in terms of parabolic cylinder or bessel functions we discuss three approaches for discovering this hidden structure factorization of differential equations kimura transformation and a doubly infinite transcendental basis of the bargmann space

hypothesis testing in a mixture case control model summary we consider a problem of testing mixture proportions using two sample data one from group one and the other from a mixture of groups one and two with unknown proportion for being in group two various statistical applications including microarray study infectious epidemiological studies case control studies with contaminated controls clinical trials allowing nonresponders genetic studies for gene mutation and fishery applications can be formulated in this setup under the assumption that the log ratio of probability density functions from the two groups is linear in the observations we propose a generalized score test statistic to test the mixture proportion under some regularity conditions it is shown that this statistic converges to a weighted chi squared random variable under the null hypothesis of 0 where the weight depends only on the sampling fraction of both groups the permutation method is used to provide more reliable finite sample approximation simulation results and two real data applications are presented

quantal problems with partial algebraization of the spectrum we discuss a new class of spectral problems discovered recently which occupies an intermediate position between the exactly solvable problems e g harmonic oscillator and all others the problems belonging to this class are distinguished by the fact that a part of the eigenvalues and eigenfunctions can be found algebraically but not the whole spectrum the reason explaining the existence of the quasi exactly solvable problems is a hidden dynamical symmetry present in the hamiltonian for one dimensional motion this hidden symmetry issl 2 r it is shown that other groups lead to a partial algebraization in multidimensional quantal problems in particular sl 2 r sl 2 r so 3 andsl 3 r are relevant to two dimensional motion inducing a class of quasi exactly solvable two dimensional hamiltonians typically they correspond to systems in a curved space but sometimes the curvature turns out to be zero graded algebras open the possibility of constructing quasi exactlysolvable hamiltonians acting on multicomponent wave functions for example with a non minimal superextension ofsl 2 r we get a hamiltonian describing the motion of a spinor particle

pairwise fitting of mixed models for the joint modeling of multivariate longitudinal profiles a mixed model is a flexible tool for joint modeling purposes especially when the gathered data are unbalanced however computational problems due to the dimension of the joint covariance matrix of the random effects arise as soon as the number of outcomes and or the number of used random effects per outcome increases we propose a pairwise approach in which all possible bivariate models are fitted and where inference follows from pseudo likelihood arguments the approach is applicable for linear generalized linear and nonlinear mixed models or for combinations of these the methodology will be illustrated for linear mixed models in the analysis of 22 dimensional highly unbalanced longitudinal profiles of hearing thresholds

a new class of quasi exactly solvable potentials with a position dependent mass a new class of quasi exactly solvable potentials with a variable mass in the schrodinger equation is presented we have derived a general expression for the potentials including natanzon confluent potentials the general solution of the schrodinger equation is determined and the eigenstates are expressed in terms of the orthogonal polynomials

empirical likelihood ratio confidence intervals for a single functional summary the empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken in this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions it is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean for a class of m estimates that includes quantiles and for differentiable statistical functionals the results are nonparametric extensions of wilks s 1938 theorem for parametric likelihood ratios the intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on student s t statistic a hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced

on the solvability of the quantum rabi model and its 2 photon and two mode generalizations we study the solvability of the time independent matrix schrodinger differential equations of the quantum rabi model and its 2 photon and two mode generalizations in bargmann hilbert spaces of entire functions we show that the rabi model and its 2 photon and two mode analogs are quasi exactly solvable we derive the exact closed form expressions for the energies and the allowed model parameters for all the three cases in the solvable subspaces up to a normalization factor the eigenfunctions for these models are given by polynomials whose roots are determined by systems of algebraic equations

analytical energy spectrum for hybrid mechanical systems we investigate the energy spectrum for hybrid mechanical systems described by non parity symmetric quantum rabi models a set of analytical solutions in terms of the confluent heun functions and their analytical energy spectrum is obtained the analytical energy spectrum includes regular and exceptional parts which are both confirmed by direct numerical simulation the regular part is determined by the zeros of the wronskian for a pair of analytical solutions the exceptional part is relevant to the isolated exact solutions and its energy eigenvalues are obtained by analyzing the truncation conditions for the confluent heun functions by analyzing the energy eigenvalues for exceptional points we obtain the analytical conditions for the energy level crossings which correspond to two fold energy degeneracy

a fuzzy subgroupoid which is not a fuzzy group abstract we give an example to show that the proposition a fuzzy subgroupoid of a group must be a fuzzy group which appeared in fuzzy sets and systems 51 1992 105 109 structure and constructing of fuzzy subgroups of a group by l filep is wrong we also point out that the proof of theorem 6 in the same article is not true we give a valid proof of it

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

fuzzy e subgroups the aim of the paper is to investigate the known classical problem of a partition of a semigroup into a family of groups in the lattice valued fuzzy framework for that purpose a fuzzy subgroup of a fuzzy subsemigroup is introduced and characterized by cuts using the cutworthy approach as a main result it is proved that a fuzzy subsemigroup can be partitioned using special type of fuzzy partitions into a family of fuzzy e subgroups if and only if it is fuzzy completely regular

quasi exactly solvable lie algebras of differential operators in two complex variables the authors completely classify all quasi exactly solvable lie algebras of first order differential operators in two complex variables applications to quasi exactly solvable quantum problems are indicated

families of quasi exactly solvable extensions of the quantum oscillator in curved spaces we introduce two new families of quasi exactly solvable qes extensions of the oscillator in a d dimensional constant curvature space for the first three members of each family we obtain closed form expressions of the energies and wavefunctions for some allowed values of the potential parameters using the bethe ansatz method we prove that the first member of each family has a hidden sl 2 symmetry and is connected with a qes equation of the first or second type respectively one dimensional results are also derived from the d dimensional ones with d 2 thereby getting qes extensions of the mathews lakshmanan nonlinear oscillator

new quasi exactly solvable hamiltonians in two dimensions quasi exactly solvable schrodinger operators have the remarkable property that a part of their spectrum can be computed by algebraic methods such operators lie in the enveloping algebra of a finite dimensional lie algebra of first order differential operators the hidden symmetry algebra in this paper we develop some general techniques for constructing quasi exactly solvable operators our methods are applied to provide a wide variety of new explicit two dimensional examples on both flat and curved spaces of quasi exactly solvable hamiltonians corresponding to both semisimple and more general classes of lie algebras

type b 3 fold supersymmetry and non polynomial invariant subspaces we obtain the most general type b 3 fold supersymmetry by solving directly the intertwining relation we then show that it is a necessary and sufficient condition for a second order linear differential operator to have three linearly independent local analytic solutions we find that there are eight linearly independent non trivial linear differential operators of this kind as a by product we find new quasi solvable second order operators preserving a monomial or polynomial subspace one in type b two in type c and four in type x2 all of which have been missed in the existing literature in addition we show that type a type b and type c 3 fold supersymmetries are connected continuously via one parameter a few new quasi solvable models are also presented

perturbation results of critical elliptic equations of caffarelli kohn nirenberg type abstract we find for small e positive solutions to the equation div x 2a u x 2 1 a u 1 ek x u p 1 x bp in r n which branch off from the manifold of minimizers in the class of radial functions of the corresponding caffarelli kohn nirenberg type inequality moreover our analysis highlights the symmetry breaking phenomenon in these inequalities namely the existence of non radial minimizers

regression analysis under non standard situations a pairwise pseudolikelihood approach regression analysis is one of the most used statistical methods for data analysis there are however many situations in which one cannot base inference solely on f y x the conditional probability density function for the response variable y given x the covariates examples include missing data where the missingness is non ignorable sampling surveys in which subjects are selected on the basis of the y values and meta analysis where published studies are subject to selection bias the conventional approaches require the correct specification of the missingness mechanism sampling probability and probability for being published respectively in this paper we propose an alternative estimating procedure for based on an idea originated by kalbfleisch the novelty of this method is that no assumption on the missingness probability mechanisms etc mentioned above is required to be specified asymptotic efficiency calculations and simulation studies were conducted to compare the method proposed with the two existing methods the conditional likelihood and the weighted estimating function approaches

a note on an equivalence relation of fuzzy subgroups the purpose of this paper is to discuss the following three problems 1 the number of equivalent classes of fuzzy subgroups of a group g 2 the relationship between the number of equivalent classes of fuzzy subgroups and the order of a cyclic group g 3 the relationship between the order of fuzzy subgroups and the order of a cyclic group g 4 a classification of a group using membership values of fuzzy subgroups is given

inferences for case control and semiparametric two sample density ratio models summary we consider inference in general binary response regression models under retrospective sampling plans prentice pyke 1979 discovered that inference for the odds ratio parameter in a logistic model can be based on a prospective likelihood even though the sampling scheme is retrospective we show that the estimating function obtained from the prospective likelihood is optimal in a class of unbiased estimating functions also we link casecontrol sampling with a two sample biased sampling problem where the ratio of two densities is assumed to take a known parametric form connections between this model and the cox proportional hazards model are pointed out large and small sample size behaviour of the proposed estimators is studied

normalizability of one dimensional quasi exactly solvable schrodinger operators we completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi exactly solvable schrodinger operator on the line methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasi exactly solvable hamiltonians and explicit normalizability conditions in general coordinate systems

semiparametric likelihood ratio inference likelihood ratio tests and related confidence intervals for a real parameter in the presence of an infinite dimensional nuisance parameter are considered in all cases the estimator of the real parameter has an asymptotic normal distribution however the estimator of the nuisance parameter may not be asymptotically gaussian or may converge to the true parameter value at a slower rate than the square root of the sample size nevertheless the likelihood ratio statistic is shown to possess an asymptotic chi squared distribution the examples considered are tests concerning survival probabilities based on doubly censored data a test for presence of heterogeneity in the gamma frailty model a test for significance of the regression coefficient in cox s regression model for current status data and a test for a ratio of hazards rates in an exponential mixture model in both of the last examples the rate of convergence of the estimator of the nuisance parameter is less than the square root of the sample size

a proportional likelihood ratio model we propose a semiparametric proportional likelihood ratio model which is particularly suitable for modelling a nonlinear monotonic relationship between the outcome variable and a covariate this model extends the generalized linear model by leaving the distribution unspecified and has a strong connection with semiparametric models such as the selection bias model gilbert et al 1999 the density ratio model qin 1998 fokianos kaimi 2006 the single index model ichimura 1993 and the exponential tilt regression model rathouz gao 2009 a maximum likelihood estimator is obtained for the new model and its asymptotic properties are derived an example and simulation study illustrate the use of the model copyright 2012 oxford university press

fuzzy groups and collections of subgroups abstract the aim of the paper is to investigate fuzzy subgroups of a group from a general point of view using collections of ordinary subgroups fuzzy subgroups are taken to be special mappings from a group to a partially ordered set which can also be a lattice and particularly the unit interval 0 1 as it is known every fuzzy subgroup uniquely determines a poset of level subgroups of the same group properties of that poset are investigated on the other hand it is proved that every collection of subgroups can be used for the construction of a suitable fuzzy subgroup of a group for that purpose particular fuzzy completions of collections of subsets are introduced it turns out that fuzzy subgroup is an intrinsic notion expressible in terms of collections of subgroups under set theoretic relations and operations

differential equations for singular values of products of ginibre random matrices it was proved by akemann et al 2013 phys rev e 88 052118 that squared singular values of products of m complex ginibre random matrices form a determinantal point process whose correlation kernel is expressible in terms of meijer s g functions kuijlaars and zhang arxiv 1308 1003 recently showed that at the edge of the spectrum this correlation kernel has a remarkable scaling limit which can be understood as a generalization of the classical bessel kernel of random matrix theory in this paper we investigate the fredholm determinant of the operator with the kernel where j is a disjoint union of intervals and is the characteristic function of the set j this fredholm determinant is equal to the probability that j contains no particles of the limiting determinantal point process defined by the gap probability we derive the hamiltonian differential associated with the corresponding fredholm determinant and relate them with the monodromy preserving deformation equations of the jimbo miwa mori ueno and sato theory in the special case we give a formula for the gap probability in terms of a solution of a system of nonlinear ordinary differential equations

full spectrum of the two photon and the two mode quantum rabi models this paper is concerned with the rigorous analytical determination of the spectrum of the two photon and the two mode quantum rabi models to reach this goal we exploit the hidden symmetries in these models by means of the unitary and similarity transformations in addition to the bargmann fock space description in each case the purely quantum mechanical problem of the rabi model studied is reduced to solutions for differential equations this eventually gives a third order differential equation for each of these models which is reduced to a second order differential equation by additional transformations the analytical expressions of the wave functions describing the energy levels are obtained in terms of the confluent hypergeometric functions

lie algebras of differential operators and lie algebraic potentials abstract an explicit characterisation of all second order differential operators on the line which can be written as bilinear combinations of the generators of a finite dimensional lie algebra of first order differential operators is found solving a problem arising in the lie algebraic approach to scattering theory and molecular dynamics one dimensional potentials corresponding to these lie algebras are explicitly classified which include the harmonic oscillator morse one soliton poschl teller mathieu lame confluent hypergeometric and bessel potentials

algebraic equations for the exceptional eigenspectrum of the generalized rabi model we obtain the exceptional part of the eigenspectrum of the generalized rabi model also known as the driven rabi model in terms of the roots of a set of algebraic equations this approach provides a product form for the wavefunction components and allows an explicit connection with recent results obtained for the wavefunction in terms of truncated confluent heun functions other approaches are also compared for particular parameter values the exceptional part of the eigenspectrum consists of doubly degenerate crossing points we give a proof for the number of roots of the constraint polynomials and discuss the number of crossing points

lie algebras and polynomials in one variable a classification of linear differential and difference equations in one variable having polynomial solutions the generalized bochner problem is presented the idea of the approach is based on consideration of the eigenvalue problem for a polynomial in generators of the algebra sl2 r for differential equations or sl2 r q for finite difference equation given in the finite dimensional projectivized representation

algebraic approach to the scattering matrix a purely algebraic procedure is presented for calculating recursion relations for s matrices belonging to problems associated with the group su 1 1 the procedure involves supplementing a recently introduced group theoretic approach to scattering by an algebraic framework that characterizes asymptotic behavior our formulation makes use of the euclidean group which contains the symmetry transformations for a free particle the coulomb and poeschl teller potentials are discussed to illustrate our methods

galois theory for fuzzy fields abstract the aim of this paper is to present a generalized form of the galois connection between the intermediate fields of a field extension and the subgroups of the corresponding galois group we prove that the fundamental theorem of galois theory accepts a generalization in the fuzzy framework

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

a semiparametric model for vqtl mapping summary quantitative trait locus analysis has been used as an important tool to identify markers where the phenotype or quantitative trait is linked with the genotype most existing tests for single locus association with quantitative traits aim at the detection of the mean differences across genotypic groups however recent research has revealed functional genetic loci that affect the variance of traits known as variability controlling quantitative trait locus in addition it has been suggested that many genotypes have both mean and variance effects while the mean effects or variance effects alone may not be strong enough to be detected the existing methods accounting for unequal variances include the levene s test the lepage test and the d test but suffer from their limitations of lack of robustness or lack of power we propose a semiparametric model and a novel pairwise conditional likelihood ratio test specifically the semiparametric model is designed to identify the combined differences in higher moments among genotypic groups the pairwise likelihood is constructed based on conditioning procedure where the unknown reference distribution is eliminated we show that the proposed pairwise likelihood ratio test has a simple asymptotic chi square distribution which does not require permutation or bootstrap procedures simulation studies show that the proposed test performs well in controlling type i errors and having competitive power in identifying the differences across genotypic groups in addition the proposed test has certain robustness to model mis specifications the proposed test is illustrated by an example of identifying both mean and variances effects in body mass index using the framingham heart study data

exact solution of non adiabatic model hamiltonians in solid state physics and optics simple non adiabatic model hamiltonians are treated using the realisation of the vibrational modes on bargmann s hilbert space of analytic functions in this formulation the schrodinger equation is a system of linear first order differential equations the energy eigenvalues are selected by the requirement that the solutions belong to the space of entire functions the solutions are given in terms of neumann series the recurrence relations for the expansion coefficients have a simple structure under particular conditions for the interaction constant they allow for terminating neumann series isolated exact solutions in the general case the conditions for the eigenvalues are transcendental equations involving a continued fraction the continued fraction can be approximated to any desired degree of accuracy in a rapidly convergent process based on worpitzky s theorem and its relation to conformal mapping the eigenvalues are calculated a physical interpretation of the solutions which makes use of intuitive arguments is also given

structure and construction of fuzzy subgroups of a group abstract we define the order of a fuzzy set and give conditions for different order fuzzy sets to be fuzzy subgroups through their level subgroups finally we give a list of fuzzy subgroups of order 1 6