nonlinear wave interactions in a complex hamiltonian formalism a hamiltonian formalism is used to study the set of equations governing the changes in the amplitudes of some coherent waves due to their nonlinear interactions the main point of the present description is that it uses these complex wave amplitudes as canonical variables for a real hamiltonian thus greatly simplifying the algebra three examples are then dealt with first the hamiltonian for the four wave interactions is derived in accordance with previous studies such systems are then known to be integrable as a second example one looks at a very special case of three interacting waves where a fundamental interacts simultaneously with its second and third harmonic even though this system can be hamiltonian it seems to be nonintegrable finally attention is focused on the elliptically polarised third harmonic generation both in anisotropic and in isotropic media this set of equations can be hamiltonian under certain restrictions on the coupling coefficients but it also appears to be nonintegrable because not enough independent invariants are found

empirical process approach in a two sample location scale model with censored data to compare two samples of possibly right censored failure times a location scale model without assuming the distribution form is considered for the log transformed data this new accelerated failure time model is introduced to accommodate the possible heterogeneity between within treatment variations of the two groups considered one distinct feature of this model is that with the presence of heterogeneity statistical inferences on both location and scale parameters based on log rank tests or linear rank tests will become inappropriate in this paper we propose the empirical process approach to construct a regression setup derived from the theory of strong approximation of the kaplan meier product limit empirical quantile process a generalized least squares gls estimator is obtained and shown to be semiparametric efficient also it is shown that this estimation is adaptive for a special case at the end results on the two sample setting are applied to the k sample problem

generalized treatment effects for clinical trials practice in the analysis of clinical trials with continuously measured endpoints is to focus on the difference or percentage change in mean or median response however treatments may have effects on the distribution of responses other than on the average response we sought an approach to such generalized treatment effects that i targets a parameter that is easily understood by our clinical colleagues and ii employs confidence intervals as the basis for inference we consider one such approach based on work in reliability theory namely setting pr y x as the target parameter and compare this approach to an earlier one due to o brien the two approaches have similar properties when they both seek to reject the null hypothesis of no effect due to different variances but differ when the larger variance corresponds to the larger mean in that case our approach views that the larger variance attenuates the effect of the larger mean out suggested approach applies easily to positive control clinical equivalence trials copyright 2000 john wiley sons ltd

a two sample test sensitive to crossing hazards in uncensored and singly censored data savage score statistics are employed to develop a test for comparing survival distributions with right hand singly censored data the procedure is motivated by the interest in developing a powerful method for determining differences when true survival distributions cross examination of small sample characteristics under the null hypothesis indicate that asymptotic critical values yield a slightly conservative test power of the test compares favorably with other criteria including the modified smirnov procedure particularly if there is a single crossing of the survival curves

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

a new pt symmetric complex hamiltonian with a real spectrum we construct an isospectrum system in terms of a real and a complex potential to show that the underlying pt symmetric complex hamiltonian possesses a real spectrum which is shared by its real partner

maximally selected chi square statistics two samples can be compared by selecting a cut point and then forming a 2 x 2 table of the numbers of observations above and below the cut point in each sample when the cut point is selected so as to maximize the standard chi square statistic the chi square percentile points are inappropriate actual significance levels are computed for large samples and correct percentile poillts are tabulated

maximally selected chi square statistics for small samples two samples can be compared by selecting a cut point and forming a 2 x 2 table of the numbers of observations above and below the cut point in each sample different cut points correspond to the maximum chi square statistic the maximum squared log odds ratio and the generalized kolmogorov smirnov statistic we simulated the small sample null distribution of these statistics the distributions of the maximally selected chi square statistic and the squared log odds ratio statistic are compared to each other and to the large sample results of miller and siegmund 1982 biometrics 38 1011 1016 the distributions of and agreement among the three cut points are also considered

on the use of the generalized t and generalized rank sum statistics in medical research we have used monte carlo methods to compare the type i error properties of the conditional and unconditional versions of the generalized t and the generalized rank sum tests to those of the independent samples t and wilcoxon rank sum tests results showed inflated type i errors for the conditional generalized tests but not for the unconditional tests we also compared the power of the unconditional generalized tests to that of the t and wilcoxon tests under a variety of conditions results showed the generalized tests to be much more efficient than their traditional counterparts in some circumstances but substantially less powerful in others based on these and other considerations we conclude that the application of these newer statistics in medical research needs further consideration

strong coupling expansions for the symmetric oscillators we study the traditional problem of convergence of perturbation expansions when the hermiticity of the hamiltonian is relaxed to a weaker symmetry an elementary and quite exceptional cubic anharmonic oscillator is chosen as an illustrative example of such models we describe its perturbative features paying particular attention to the strong coupling regime efficient numerical perturbation theory proves suitable for such a purpose

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

on non parametric and generalized tests for the two sample problem with location and scale change alternatives various tests have been proposed for the two sample problem when the alternative is more general than a simple shift in location non parametric tests o brien s generalized t and rank sum tests and other tests related to the t we show that the generalized tests are directly related to non parametric tests proposed by lepage as a result we obtain a wider more flexible class of o brien type procedures which inherit the level robustness property of non parametric tests we have also computed the tests empirical sizes and powers under several models the non parametric procedures and the related o brien type tests are valid and yield good power in the settings investigated they are preferable to the t test and related procedures whose type i errors differ noticeably from nominal size for skewed and long tailed distributions

smooth quantile ratio estimation in a study of health care expenditures attributable to smoking we seek to compare the distribution of medical costs for persons with lung cancer or chronic obstructive pulmonary disease cases to those without controls using a national survey which includes hundreds of cases and thousands of controls the distribution of costs is highly skewed toward larger values making estimates of the mean from the smaller sample dependent on a small fraction of the biggest values one approach to deal with the smaller sample is to rely on a simple parametric model such as the log normal but this makes the undesirable assumption that the distribution of the log expenditures is symmetric we propose a novel approach to estimate the mean difierence of two highly skewed distributions which we call smooth quantile ratio estimation square square is obtained by smoothing over percentiles the ratio of the cost quantiles of the cases and controls square deflnes a large class of estimators of including 1 the sample mean difference 2 the maximum likelihood estimate under log normal samples and 3 l estimates we detail asymptotic properties of square such as consistency and asymptotic normality and also provide a closed form expression for the asymptotic variance through a simulation study we show that square has lower mean squared error than several competitors including the sample mean difierence and log normal parametric estimates in several realistic situations we apply square to the 1987 national medicare expenditure survey to estimate the difierence in medical expenditures between persons sufiering from the smoking attributable diseases lung cancer and chronic obstructive pulmonary disease and persons without these diseases software in r ihaka and gentleman 1996 for the implementation of square and of all its special cases and the cost data used in this paper are available at http biostat jhsph edu fdominic square html

a class of rank test procedures for censored survival data summary a class of linear rank statistics is proposed for the k sample problem with rightcensored survival data the class contains as special cases the log rank test mantel 1966 cox 1972 and a test essentially equivalent to peto peto s 1972 generalization of the wilcoxon test martingale theory is used to establish asymptotic normality of test statistics under the null hypotheses considered and to derive expressions for asymptotic relative efficiencies under contiguous sequences of alternative hypotheses a class of distributions is presented which corresponds to the class of rank statistics in the sense that for each distribution there is a statistic with some optimal properties for detecting location alternatives from that distribution some monte carlo results are displayed which present small sample behaviour

on fuzzy multi valued functions part 1 introduction and general properties abstract in this paper a concept of fuzzy multi valued function is introduced and other related objects are defined next their properties that assume no topological structure are presented

quantum mechanics of complex hamiltonian systems in one dimension with a view to obtaining further insight into the nature of eigenvalues and eigenfunctions of a stationary state one dimensional schrodinger equation corresponding to a non hermitian hamiltonian h x p we investigate the ground state solutions for a variety of potentials within the framework of an extended complex phase space characterized by x x1 ip2 p p1 ix2 where x1 p1 and x2 p2 are real and considered as canonical pairs the analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunctions for different systems it is noted that the imaginary part of the eigenvalue ei turns out to be zero for all potentials v x with real couplings whereas it turns out to be nonzero for the case when the couplings are complex the prescription is also extended to study the excited states the problems related to the normalization of the eigenfunction and the boundary conditions to be used within this framework are also discussed

confidence intervals for p y1 y2 with normal outcomes in linear models recently there is an emerging interest in the inference of p y1 y2 where y1 and y2 stand for two independent continuous random variables so far most of the research in this field focuses on simply comparing two outcomes without adjusting for covariates this paper mainly presents a large sample approach based on a noncentral t distribution for the confidence interval estimation of p y1 y2 with normal outcomes in linear models furthermore the performance of the proposed large sample approach is compared with that of a generalized variable approach and a bootstrap approach simulation studies demonstrate that for small to medium sample sizes both the large sample approach and the generalized variable approach provide confidence intervals with satisfying coverage probabilities whereas the bootstrap approach can be slightly liberal for certain scenarios the proposed approaches are applied to three real life data sets copyright 2008 john wiley sons ltd

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

comparing two samples extensions of the t rank sum and log rank tests abstract i consider the problem of testing that two populations are identical with respect to the distribution of a continuous variable against the alternative that values tend to be larger in one population i observe that the t rank sum and log rank tests are insensitive for a large class of alternatives that may be expected to occur commonly in practice propose a criterion for identifying this problem in a particular data set and propose corresponding extensions of the conventional methods the proposed methods should be useful for both identifying and interpreting group differences

on the choice of variance for the log rank test summary the log rank test is widely used for comparison of survival curves this paper examines various estimators for the variance of the log rank test statistic these include the mantel haenszel variance the permutation variance of peto peto and several newly proposed estimators the results generalize to a class of test statistics proposed by tarone ware the log rank test is widely used for comparison of survival curves there are two versions of the test which differ in the choice of variance for the same test statistic t these are the mantel haenszel variance an estimate of var t which is unbiased independently of differences in censoring and the permutation variance of peto peto 1972 for which the variance is computed under the assumption of equal censoring this paper examines the choice of variance it is observed that for equal sample sizes the mantel haenszel variance tends to underestimate the true variance when t is large in absolute value as this is the situation where rejection of the null hypothesis is to be seriously considered it appears that the true type i error will tend to be higher than that assumed and the reported p value will tend to be too low this phenomena is discussed also by morton 1978 in 4 it is shown that under equal sample sizes the permutation variance tends to overestimate the true variance when censoring is unequal it thus appears prudent to use the permutation variance rather than the mantel haenszel variance in assessing significance in 5 alternative variance estimators are proposed the results immediately generalize to a class of test statistics proposed by tarone ware 1977 this point is discussed in 6

the solution of the schrodinger equation for complex hamiltonian systems in two dimensions we investigate the ground state solutions of the schrodinger equation for complex non hermitian hamiltonian systems in two dimensions within the framework of an extended complex phase space approach the eigenvalues and eigenfunctions of some two dimensional complex potentials are found

additive expectancy regression regression is a useful tool for studying association between an outcome variable and its covariates in a classical linear regression model the outcome s expectation is usually a function of the covariates and some regression parameters and the underlying distribution is often normal in this article we propose a new class of additive expectancy regression models in which the outcomes tend to be positively skewed in the new regression models regression parameters are practically meaningful and also useful with inferences on the means of skewed outcomes parametric and semiparametric methods are developed for model estimation and inferences model based prediction and model adequacy assessment are discussed the proposed methodologies are demonstrated by two real data analyses

inference based on simple rank step score statistics for the location model abstract a class of estimation procedures that are robust relatively efficient and yet computationally simple is proposed for the one and two sample location problems particular emphasis is placed on the problem of determining confidence intervals a topic that has seen limited exposure in the literature as a by product some interesting point estimates are obtained a class of score functions that are ordinary step functions is considered for the location model point estimates and confidence intervals are obtained by inverting the corresponding rank statistics efficiency and robustness properties of the procedures are investigated simple computational schemes for the estimates and intervals are provided

comparison of two tests useful in situations where treatment is expected to increase variability relative to controls the type i error and power characteristics of the modified t test were compared with those of the generalized t test results suggested that in contrast to the generalized t test the modified t test can be seriously non robust to departures from population normality with such departures often producing anti conservative results neither test held an absolute power advantage over the other when responses were from normal distributions but the modified t test was generally more powerful for these conditions in comparison with the pooled samples t test both tests were usually much more efficient when treatment caused increases both in mean response and between subject variance and suffered only small disadvantages when between subject variance was unchanged by treatment given these results and other considerations recommendations for use of these recently devised tests are given

harmonic oscillator well with a screened coulombic core is quasi exactly solvable in the quantization scheme which weakens the hermiticity of a hamiltonian to its mere t invariance the superposition v x x2 ze2 x of the harmonic and coulomb potentials is defined at the purely imaginary effective charges ze2 if and regularized by a purely imaginary shift of x this model is quasi exactly solvable we show that at each excited n 1 th harmonic oscillator energy e 2n 3 there exists not only the well known harmonic oscillator bound state at the vanishing charge f 0 but also a normalizable n 1 plet of the further elementary sturmian eigenstates n x at eigencharges f f n 0 n 0 1 n beyond the smallest multiplicities n we recommend perturbative methods for their construction

pseudo reality and pseudo adjointness of hamiltonians we define pseudo reality and pseudo adjointness of a hamiltonian h as h 1 h and h 1 h respectively we prove that the former yields the necessary condition for a spectrum to be real whereas the latter helps in fixing a definition for the inner product of the eigenstates here we separate out the adjointness of an operator from its hermitian adjointness it turns out that a hamiltonian possessing a real spectrum is first pseudo real further it could be hermitian pt symmetric or pseudo hermitian

non hermitian susy hydrogen like hamiltonians with real spectra it is shown that the radial part of the hydrogen hamiltonian factorizes as the product of two not mutually adjoint first order differential operators plus a complex constant t he 1 susy approach is used to construct non hermitian operators with hydrogen spectra other non hermitian hamiltonians are shown to admit an extra complex energy at n ew self adjoint hydrogen like hamiltonians are also derived by using a 2 susy transformation with complex conjugate pairs

the effects of transformations and preliminary tests for non linearity in regression non linear relationships between two variables are often detected as a result of a preliminary statistical test for linearity common approaches to dealing with non linearity are to a make a linearizing transformation in the independent variable or b fit a relationship that is non linear in the independent variable such as including a quadratic term with either approach the resulting test for association between the two variables can have an inflated type i error we consider testing the significance of the quadratic term in a quadratic model as a preliminary test for non linearity using simulation experiments and asymptotic arguments we quantify the type i error inflation and suggest simple modifications of standard practice to protect the size of the type i error in the case of quadratic regression the type i error will be increased by roughly 50 per cent the simple strategy of appropriately correcting the alpha level is shown to have minimal loss of power if the relationship is truly linear in the case of a linearizing transformation the impact on the type i error will depend on the values of the independent variable and on the set of potential linearizing transformations considered simulation results suggest that a procedure which adjusts the test statistic according to the results of the preliminary test may offer adequate protection

the diffusion reaction d r hamiltonian and the solutions of certain types of linear and nonlinear d r equations in one dimension with a view to having further insight into the mathematical content of the non hermitian hamiltonian associated with the diffusion reaction d r equation in one dimension we investigate a the solitary wave solutions of certain types of its nonlinear versions and b the problem of real eigenvalue spectrum associated with its linear version or with this class of non hermitian hamiltonians for case a we use the standard techniques to handle the quadratic and cubic nonlinearities in the d r equation whereas for case b a newly proposed method based on an extended complex phase space is employed for a particular class of solutions an ermakov system of equations is also found for the linear case further corresponding to the classical version of the above one dimensional complex hamiltonian an equivalent integrable system of two two dimensional real hamiltonians is suggested