sl 3 r as the group of symmetry transformations for all one dimensional linear systems ii realizations of the lie algebra the two dimensional space time realizations of the lie algebra of sl 3 r are obtained when the group acts as the maximal point symmetry group of any given one dimensional newtonian linear system it is shown that these realizations are isomorphic with the realization of the lie algebra of the projective group in the plane next an active point of view is introduced and sl 3 r is interpreted as a group of mappings that transform one admissible world line of the system into another thus a new mechanical realization of the sl 3 r algebra comes to the fore some miscellaneous examples are included

the em algorithm for kernel matrix completion with auxiliary data in biological data it is often the case that observed data are available only for a subset of samples when a kernel matrix is derived from such data we have to leave the entries for unavailable samples as missing in this paper the missing entries are completed by exploiting an auxiliary kernel matrix derived from another information source the parametric model of kernel matrices is created as a set of spectral variants of the auxiliary kernel matrix and the missing entries are estimated by fitting this model to the existing entries for model fitting we adopt the em algorithm distinguished from the em algorithm of dempster et al 1977 based on the information geometry of positive definite matrices we will report promising results on bacteria clustering experiments using two marker sequences 16s and gyrb

learning multiple tasks with kernel methods we study the problem of learning many related tasks simultaneously using kernel methods and regularization the standard single task kernel methods such as support vector machines and regularization networks are extended to the case of multi task learning our analysis shows that the problem of estimating many task functions with regularization can be cast as a single task learning problem if a family of multi task kernel functions we define is used these kernels model relations among the tasks and are derived from a novel form of regularizers specific kernels that can be used for multi task learning are provided and experimentally tested on two real data sets in agreement with past empirical work on multi task learning the experiments show that learning multiple related tasks simultaneously using the proposed approach can significantly outperform standard single task learning particularly when there are many related tasks but few data per task

manifold regularization a geometric framework for learning from labeled and unlabeled examples we propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution we focus on a semi supervised framework that incorporates labeled and unlabeled data in a general purpose learner some transductive graph learning algorithms and standard methods including support vector machines and regularized least squares can be obtained as special cases we use properties of reproducing kernel hilbert spaces to prove new representer theorems that provide theoretical basis for the algorithms as a result in contrast to purely graph based approaches we obtain a natural out of sample extension to novel examples and so are able to handle both transductive and truly semi supervised settings we present experimental evidence suggesting that our semi supervised algorithms are able to use unlabeled data effectively finally we have a brief discussion of unsupervised and fully supervised learning within our general framework

the ladder problem painleve integrability and explicit solution we consider the n dimensional ladder system that is the homogeneous quadratic system of first order differential equations of the form where aij i 1 j i j 1 n introduced by imai and hirata 2002 preprint nlin si 0212007 v1 3 the ladder system is found to be integrable for all n in terms of the painleve analysis and its solution is explicitly given

input output kernel regression supervised and semi supervised structured output prediction with operator valued kernels in this paper we introduce a novel approach called input output kernel regression iokr for learning mappings between structured inputs and structured outputs the approach belongs to the family of output kernel regression methods devoted to regression in feature space endowed with some output kernel in order to take into account structure in input data and benefit from kernels in the input space as well we use the reproducing kernel hilbert space theory for vector valued functions we first recall the ridge solution for supervised learning and then study the regularized hinge loss based solution used in maximum margin regression both models are also developed in the context of semi supervised setting in addition we derive an extension of generalized cross validation for model selection in the case of the least square model finally we show the versatility of the iokr framework on two different problems link prediction seen as a structured output problem and multi task regression seen as a multiple and interdependent output problem eventually we present a set of detailed numerical results that shows the relevance of the method on these two tasks

a studentized permutation test for three arm trials in the gold standard design the gold standard design for three arm trials refers to trials with an active control and a placebo control in addition to the experimental treatment group this trial design is recommended when being ethically justifiable and it allows the simultaneous comparison of experimental treatment active control and placebo parametric testing methods have been studied plentifully over the past years however these methods often tend to be liberal or conservative when distributional assumptions are not met particularly with small sample sizes in this article we introduce a studentized permutation test for testing non inferiority and superiority of the experimental treatment compared with the active control in three arm trials in the gold standard design the performance of the studentized permutation test for finite sample sizes is assessed in a monte carlo simulation study under various parameter constellations emphasis is put on whether the studentized permutation test meets the target significance level for comparison purposes commonly used wald type tests which do not make any distributional assumptions are included in the simulation study the simulation study shows that the presented studentized permutation test for assessing non inferiority in three arm trials in the gold standard design outperforms its competitors for instance the test based on a quasi poisson model for count data the methods discussed in this paper are implemented in the r package threearmedtrials which is available on the comprehensive r archive network cran copyright 2016 john wiley sons ltd

a general approach for sample size calculation for the three arm gold standard non inferiority design in the three arm gold standard non inferiority design an experimental treatment an active reference and a placebo are compared this design is becoming increasingly popular and it is whenever feasible recommended for use by regulatory guidelines we provide a general method to calculate the required sample size for clinical trials performed in this design as special cases the situations of continuous binary and poisson distributed outcomes are explored taking into account the correlation structure of the involved test statistics the proposed approach leads to considerable savings in sample size as compared with application of ad hoc methods for all three scale levels furthermore optimal sample size allocation ratios are determined that result in markedly smaller total sample sizes as compared with equal assignment as optimal allocation makes the active treatment groups larger than the placebo group implementation of the proposed approach is also desirable from an ethical viewpoint

asymptotic permutation tests in general factorial designs summary in general factorial designs where no homoscedasticity or a particular error distribution is assumed the well known wald type statistic is a simple asymptotically valid procedure however it is well known that it suffers from a poor finite sample approximation since the convergence to its 2 limit distribution is quite slow this becomes even worse with an increasing number of factor levels the aim of the paper is to improve the small sample behaviour of the wald type statistic maintaining its applicability to general settings as crossed or hierarchically nested designs by applying a modified permutation approach in particular it is shown that this approach approximates the null distribution of the wald type statistic not only under the null hypothesis but also under the alternative yielding an asymptotically valid permutation test which is even finitely exact under exchangeability finally its small sample behaviour is compared with competing procedures in an extensive simulation study

su 2 quantum kinematics rotation observable versus angular momentum generalized commutation relations the canonical commutation relations of quantum mechanics are generalized to the case where appropriate dynamical variables are angular momentum rotation angle and rotation axis observables to this end su 2 is quantized on the compact group manifold according to the standard procedure of non abelian quantum kinematics quantum kinematic invariant operators are introduced and their commutation relations with the rotation variables are found in an explicit manner the quantum kinematic invariants yield superselection rules in the form of eigenvalue equations of an isotopic structure which one should solve in the applications in order to get multiplets that carry the irreducible representations of the underlying quantum kinematic models a wide range of applicability of su 2 quantum kinematics is suggested

quantum kinematics and the lie group structure of non abelian quantum mechanics a formalism of quantum kinematics which rests on the regular representation of a lie group is proposed this representation leads to explicit canonical commutation relations for non abelian dynamical variables which together with the lie algebra define the kinematic algebra generalised equations of motion for the group parameter dependent operators as well as generalised wave equations are introduced over the group manifold considered as a background arena the formalism proposed affords a method of geometric quantisation stemming directly from the observed symmetries of a system

universal multi task kernels in this paper we are concerned with reproducing kernel hilbert spaces hk of functions from an input space into a hilbert space y an environment appropriate for multi task learning the reproducing kernel k associated to hk has its values as operators on y our primary goal here is to derive conditions which ensure that the kernel k is universal this means that on every compact subset of the input space every continuous function with values in y can be uniformly approximated by sections of the kernel we provide various characterizations of universal kernels and highlight them with several concrete examples of some practical importance our analysis uses basic principles of functional analysis and especially the useful notion of vector measures which we describe in sufficient detail to clarify our results

equivalence classes of second order ordinary differential equations with only a three dimensional lie algebra of point symmetries and linearisation there are seven equivalence classes of second order ordinary differential equations possessing only three lie point symmetries and hence not linearisible by means of a point transformation we examine the representatives of these classes for linearisibility by means of other types of transformation in particular we compare the potential for linearisibility and the possession of the painleve property the complete symmetry group is realised in the standard algebra for each of the equivalence classes

blinded sample size re estimation in three arm trials with gold standard design in this article we study blinded sample size re estimation in the gold standard design with internal pilot study for normally distributed outcomes the gold standard design is a three arm clinical trial design that includes an active and a placebo control in addition to an experimental treatment we focus on the absolute margin approach to hypothesis testing in three arm trials at which the non inferiority of the experimental treatment and the assay sensitivity are assessed by pairwise comparisons we compare several blinded sample size re estimation procedures in a simulation study assessing operating characteristics including power and type i error we find that sample size re estimation based on the popular one sample variance estimator results in overpowered trials moreover sample size re estimation based on unbiased variance estimators such as the xing ganju variance estimator results in underpowered trials as it is expected because an overestimation of the variance and thus the sample size is in general required for the re estimation procedure to eventually meet the target power to overcome this problem we propose an inflation factor for the sample size re estimation with the xing ganju variance estimator and show that this approach results in adequately powered trials because of favorable features of the xing ganju variance estimator such as unbiasedness and a distribution independent of the group means the inflation factor does not depend on the nuisance parameter and therefore can be calculated prior to a trial moreover we prove that the sample size re estimation based on the xing ganju variance estimator does not bias the effect estimate copyright 2017 john wiley sons ltd

complete symmetry group and nonlocal symmetries for some two dimensional evolution equations abstract the complete symmetry group of an 1 1 evolution equation of maximal symmetry has been demonstrated to be represented by the six dimensional lie algebra of point symmetries sl 2 r s w where w is the three dimensional heisenberg weyl algebra we construct a complete symmetry group of a 1 2 evolution equation u t f u u x y for some functions f using the point symmetries admitted by the equation the 1 2 equation is not completely specifiable by point symmetries alone for some specific functions f we make use of ansatze already reported by myeni and leach s m myeni p g l leach nonlocal symmetries and complete symmetry groups of evolution equations j nonlinear math phys 13 2006 377 392 which provide a route to the determination of the required generic nonlocal symmetries necessary to supplement the point symmetries for the complete specification of these 1 2 evolution equations further we find that taking some suitable linear combination of lie point symmetries helps to optimise the procedure of specifying the equation a general result concerning the number of symmetries required to form a complete symmetry group of evolution is presented in the conclusion

complete symmetry groups of ordinary differential equations and their integrals some basic considerations abstract the concept of the complete symmetry group of a differential equation introduced by j krause 1994 j math phys 35 5734 5748 is extended to integrals of such equations this paper is devoted to some aspects characterising complete symmetry groups the algebras of the symmetries of both differential equations and integrals are studied in the context of equations for which the elements are represented by point or contact symmetries so that there is no ambiguity about the group both algebras and groups are found to be nonunique

the complete symmetry group of the generalised hyperladder problem abstract we further consider the n dimensional ladder system that is the homogeneous quadratic system of first order differential equations of the form x i x i j 1 n a ij x j i 1 n where aij i 1 j i j 1 n introduced by imai and hirata nlin si 0212007 we establish the most general system of first order ordinary differential equations invariant under the algebra which characterises the ladder system of imai and hirata and the algebra of minimal dimension required to specify completely this most general system we provide the complete symmetry group of the generalised hyperladder system and discuss its integrability

on the complete symmetry group of the classical kepler system a rather strong concept of symmetry is introduced in classical mechanics in the sense that some mechanical systems can be completely characterized by the symmetry laws they obey accordingly a complete symmetry group realization in mechanics must be endowed with the following two features 1 the group acts freely and transitively on the manifold of all allowed motions of the system 2 the given equations of motion are the only ordinary differential equations that remain invariant under the specified action of the group this program is applied successfully to the classical kepler problem since the complete symmetry group for this particular system is here obtained the importance of this result for the quantum kinematic theory of the kepler system is emphasized

kepler orbits and the harmonic oscillator it is shown that the kustaanheimo stiefel transformation which transforms the three dimensional kepler problem into that for a four dimensional harmonic oscillator may be expressed in terms of quaternions in this form the transformation represents a continuous mapping of a triad of orthonormal vectors fixed in space into a rotating triad of orthogonal vectors where one of the unit vectors is mapped into the position vector of a moving particle the reduction of the kepler problem to that of a harmonic oscillator is straightforward and direct in the quaternion formalism the complex stereographic transformation used recently by the author in the corresponding reduction of schrodinger s equation for the hydrogen atom is shown to be closely related to the quaternion representation

continuum versus discrete networks graph laplacians and reproducing kernel hilbert spaces abstract motivated by applications to machine learning we construct a reversible and irreducible markov chain whose state space is a certain collection of measurable sets of a chosen l c h space x we study the resulting network connected undirected graph including transience royden and riesz decompositions and kernel factorization we describe a construction for hilbert spaces of signed measures which comes equipped with a new notion of reproducing kernels and there is a unique solution to a regularized optimization problem involving the approximation of l 2 functions by functions of finite energy the latter has applications to machine learning for markov random fields for example

the ermanno bernoulli constants and representations of the complete symmetry group of the kepler problem the expression of the components of the equation of motion of the classical kepler problem in terms of the natural variables associated with the ermanno bernoulli constants leads naturally to the same equations as are obtained by the technique of reduction of order developed by nucci j math phys 37 1772 1996 reported by nucci and leach j math phys 42 746 2001 three representations of the complete symmetry group of the kepler problem are obtained from the three standard representations of the complete symmetry group of the simple harmonic oscillator the algebra of the complete symmetry group of the two dimensional kepler problem is identified to be a1 a3 3 the applicability of the results to other classes of problem such as the kepler problem with drag which possess a conserved vector of laplace runge lenz type is indicated the three dimensional kepler problem is shown to be completely specified by six symmetries rather than the eight previously reported by krause j math phys

sl 3 r as the group of symmetry transformations for all one dimensional linear systems the converse problem of similarity analysis is solved in general for the finite symmetry transformations of any inhomogeneous ordinary linear differential equation of the second order x f2 t x f1 t x f0 t the eight parameter realizations of the symmetry group are obtained in the form f 1p2 f where f stands for transformations of t x that depend exclusively on the fundamental solutions of the equation and where p2 is an arbitrary projective transformation in the plane thus it is shown that the full point symmetry group corresponds to sl 3 r indeed without recourse to the lie algebra also a technique is obtained for calculating the finite point symmetry realization of sl 3 r for any given one dimensional linear system some miscellaneous examples are given

the complete kepler group can be derived by lie group analysis it is shown that the complete symmetry group for the kepler problem as introduced by krause can be derived by lie group analysis the same result is true for any autonomous system

the determination of nonlocal symmetries by the technique of reduction of order autonomous systems of ordinary differential equations can be rewritten as systems of first order ordinary differential equations and one of the dependent variables chosen as a new independent variable some of the variables are eliminated to give a mixed system of first and second order equations for which the determination of point symmetries can be automated without having to make an ansatz on the detailed structure of the symmetry because the coefficient function for the original independent variable appears only as its derivative in the reduced system symmetries which are nonlocal in this variable become local symmetries of the reduced system and can be computed algorithmically

a studentized permutation test for the comparison of spatial point patterns in this study a new test is proposed for the hypothesis that two or more observed point patterns are realizations of the same spatial point process model to this end the point patterns are divided into disjoint quadrats on each of which an estimate of ripley s k function is calculated the two groups of empirical k functions are compared by a permutation test using a studentized test statistic the proposed test performs convincingly in terms of empirical level and power in a simulation study even for point patterns where the k function estimates on neighboring subsamples are not strictly exchangeable it also shows improved behavior compared with a test suggested by diggle et al for the comparison of groups of independently replicated point patterns in an application to two point patterns from pathology that represent capillary positions in sections of healthy and cancerous tissue our studentized permutation test indicates statistical significance although the patterns cannot be clearly distin

the assessment of non inferiority in a gold standard design with censored exponentially distributed endpoints the objective of this paper is to develop statistical methodology for non inferiority hypotheses to censored exponentially distributed time to event endpoints motivated by a recent clinical trial in depression we consider a gold standard design where a test group is compared to an active reference and to a placebo group the test problem is formulated in terms of a retention of effect hypothesis thus the proposed wald type test procedure assures that the effect of the test group is better than a pre specified proportionof the treatment effect of the reference group compared to the placebo group a sample size allocation rule to achieve optimal power is presented which only depends on the pre specifiedand the probabilities for the occurrence of censoring in addition a pretest is presented for either the reference or the test group to ensure assay sensitivity in the complete test procedure the actual type i error and the sample size formula of the proposed tests is explored asymptotically and by means of a simulation study showing good small sample characteristics to illustrate the procedure a randomized double blind clinical trial in depression is evaluated an r package for implementation of the proposed tests and for sample size determination accompanies this paper on the author s web page

nonparametric testing for multiple survival functions with non inferiority margins new nonparametric tests for the ordering of multiple survival functions are developed with the possibility of right censorship taken into account the motivation comes from non inferiority trials with multiple treatments the proposed tests are based on nonparametric likelihood ratio statistics which are known to provide more powerful tests than wald type procedures but in this setting have only been studied for pairs of survival functions or in the absence of censoring we introduce a novel type of pool adjacent violator algorithm that leads to a complete solution of the problem the limit distributions can be expressed as weighted sums of squares involving projections of certain gaussian processes onto the given ordered alternative a simulation study shows that the new procedures have superior power to a competing combined pairwise cox model approach we illustrate the proposed methods using data from a three arm non inferiority trial

assessing non inferiority of a new treatment in a three arm clinical trial including a placebo in non inferiority trials where non inferiority of a new experimental drug compared to an active control has to be shown it may be advisable to use an additional placebo group for internal validation if ethically justifiable the focus of this paper is on such designs assuming normality and homogeneity of variances we will derive a statistical test procedure which turns out to be equivalent to the assessment based on fieller s confidence interval based on the power function of this test sample size calculations are carried out to achieve a given power additionally the optimal allocation of the total sample size is derived as an alternative to this parametric procedure the bootstrap percentile interval is discussed and finally compared with fieller s confidence interval in a study on mildly asthmatic patients

planning and analysis of three arm non inferiority trials with binary endpoints three arm trials including an experimental treatment an active control and a placebo group are frequently preferred for the assessment of non inferiority in contrast to two arm non inferiority studies these designs allow a direct proof of efficacy of a new treatment by comparison with placebo as a further advantage the test problem for establishing non inferiority can be formulated in such a way that rejection of the null hypothesis assures that a pre defined portion of the unknown effect the reference shows versus placebo is preserved by the treatment under investigation we present statistical methods for this study design and the situation of a binary outcome variable asymptotic test procedures are given and their actual type i error rates are calculated approximate sample size formulae are derived and their accuracy is discussed furthermore the question of optimal allocation of the total sample size is considered power properties of the testing strategy including a pre test for assay sensitivity are presented the derived methods are illustrated by application to a clinical trial in depression

symmetries of first integrals and their associated differential equations the relationship between the reduction of order through point symmetries and integration is explored with particular emphasis on the loss and gain of point contact for third order symmetries to and from nonlocal symmetries it is seen that reduction of order can even lead to the loss of all point symmetries at the third order level and their replacement at the second order level from nonlocal symmetries it is evident that nonlocal symmetries should be given more attention in applications

characterisation of the algebraic properties of first integrals of scalar ordinary differential equations of maximal symmetry abstract we undertake a study of the first integrals of linear n th order scalar ordinary differential equations with maximal symmetry we establish patterns for the first integrals associated with these equations it is shown that second and third order equations are the pathological cases in the study of higher order differential equations the equivalence of contact symmetries for third order equations to non cartan symmetries of second order equations is highlighted

euclidean embedding of co occurrence data embedding algorithms search for a low dimensional continuous representation of data but most algorithms only handle objects of a single type for which pairwise distances are specified this paper describes a method for embedding objects of different types such as images and text into a single common euclidean space based on their co occurrence statistics the joint distributions are modeled as exponentials of euclidean distances in the low dimensional embedding space which links the problem to convex optimization over positive semidefinite matrices the local structure of the embedding corresponds to the statistical correlations via random walks in the euclidean space we quantify the performance of our method on two text data sets and show that it consistently and significantly outperforms standard methods of statistical correspondence modeling such as multidimensional scaling isomap and correspondence analysis

on learning vector valued functions in this letter we provide a study of learning in a hilbert space of vectorvalued functions we motivate the need for extending learning theory of scalar valued functions by practical considerations and establish some basic results for learning vector valued functions that should prove useful in applications specifically we allow an output space y to be a hilbert space and we consider a reproducing kernel hilbert space of functions whose values lie in y in this setting we derive the form of the minimal norm interpolant to a finite set of data and apply it to study some regularization functionals that are important in learning theory we consider specific examples of such functionals corresponding to multiple output regularization networks and support vector machines for both regression and classification finally we provide classes of operator valued kernels of the dot product and translation invariant type

reflections on the symmetry conservation law duality and the runge lenz vector following recent interest in an earlier work on the kepler problem prince and eliezer 1981 a number of remarks are made concerning the many faceted symmetry conservation law duality in classical mechanics in particular the advantages of associating a number of symmetries with the runge lenz vector are discussed

on the systematic approach to the classification of differential equations by group theoretical methods complete symmetry groups enable one to characterise fully a given differential equation by considering the reversal of an approach based upon complete symmetry groups we construct new classes of differential equations which have the equations of bateman monge ampere and born infeld as special cases we develop a symbolic algorithm to decrease the complexity of the calculations involved

quantum kinematics of the harmonic oscillator the formalism of non abelian quantum kinematics is applied to the newtonian symmetry group of the harmonic oscillator within the regular ray representation of the group the schrodinger operator as well as two other new invariant operators are obtained as casimir operators of the extended kinematic algebra superselection rules are then introduced which permit the identification and the explicit calculation of the physical states of the system next a complementary ray representation attached to the space time realization of the group casts the schrodinger operator into the familiar time dependent space time differential operator of the harmonic oscillator and thus by means of the superselection rules one obtains the time dependent schrodinger equation of the sytem finally the evaluation of a hurwitz invariant integral over the group manifold affords the well known feynman space time propagator t x t x of the simple harmonic oscillator everything comes out from the assumed symmetries of the system the whole approach is group theoretic and relativistic no classical analog is used in this quantization scheme

randomization tests for weak null hypotheses in randomized experiments the fisher randomization test frt is appropriate for any test statistic under a sharp null hypothesis that can recover all missing potential outcomes however it is often sought after to test a

non abelian group quantization and quantum kinematic invariants of some noncompact lie groups the formalism of non abelian group quantization is briefly revisited within the regular representation of noncompact lie groups it is shown that some of such r dimensional groups have a set of r basic quantum kinematic invariants which substantially differ from the traditional invariants the relation of the traditional invariants of the lie algebra with the new quantum kinematic invariants is also briefly examined this paper contains two miscellaneous examples of quantum kinematic invariant operators