selection of important variables and determination of functional form for continuous predictors in multivariable model building in developing regression models data analysts are often faced with many predictor variables that may influence an outcome variable after more than half a century of research the best way of selecting a multivariable model is still unresolved it is generally agreed that subject matter knowledge when available should guide model building however such knowledge is often limited and data dependent model building is required we limit the scope of the modelling exercise to selecting important predictors and choosing interpretable and transportable functions for continuous predictors assuming linear functions stepwise selection and all subset strategies are discussed the key tuning parameters are the nominal p value for testing a variable for inclusion and the penalty for model complexity respectively we argue that stepwise procedures perform better than a literature based assessment would suggest concerning selection of functional form for continuous predictors the principal competitors are fractional polynomial functions and various types of spline techniques we note that a rigorous selection strategy known as multivariable fractional polynomials mfp has been developed no spline based procedure for simultaneously selecting variables and functional forms has found wide acceptance results of fp and spline modelling are compared in two data sets it is shown that spline modelling while extremely flexible can generate fitted curves with uninterpretable wiggles particularly when automatic methods for choosing the smoothness are employed we give general recommendations to practitioners for carrying out variable and function selection while acknowledging that further research is needed we argue why mfp is our preferred approach for multivariable model building with continuous covariates

maximal abelian subalgebras of e p q algebras abstract maximal abclian subalgebras masas of one of the classical real inhomogencous lie algebras are constructed namely those of the pseudoeuclidean lie algebra e p q use is made of the semidireet sum structure of e p q with the translations t p q as an abclian ideal we first construct splitting masas that arc themselves direct sums of abelian subalgebras of o p q and of subalgebras of t p q the splitting subalgebras are used to construct the complementary nonsplitting ones here the results are less complete than in the splitting case we present general decomposition theorems and construct indecomposable masas for all algebras e p q p q 0 the case of q 0 and 1 were treated earlier in a physical context the case q 2 is analyzed here in detail as an illustration of the general results

on the digraphs of sign solvable linear systems abstract a linear system of equations ax b is sign solvable if it is solvable and both its solvability and the sign pattern of its solution vector x are uniquely determined by the sign patterns of a and b it was proved in l bassett j maybee j quirk economitrica 36 1968 544 563 r a bruali b l shader matrices of sign solvable linear systems cambridge university press cambridge 1995 that the sign solvability of a linear system ax b in standard form and with a square coefficient matrix a can be described in terms of the signed digraph s a of the matrix a with a specified vertex subset depending on the sign pattern of the vector b in this paper we define the notion of w sign solvable signed digraphs which represent precisely the sign solvable linear systems in standard form we give necessary and sufficient conditions for the underlying unsigned digraphs of w sign solvable signed digraphs in terms of forbidden subdigraphs we also give several necessary and sufficient conditions for strongly connected w sign solvable signed digraphs

an overall strategy based on regression models to estimate relative survival and model the effects of prognostic factors in cancer survival studies relative survival provides a measure of the proportion of patients dying from the disease under study without requiring the knowledge of the cause of death we propose an overall strategy based on regression models to estimate the relative survival and model the effects of potential prognostic factors the baseline hazard was modelled until 10 years follow up using parametric continuous functions six models including cubic regression splines were considered and the akaike information criterion was used to select the final model this approach yielded smooth and reliable estimates of mortality hazard and allowed us to deal with sparse data taking into account all the available information splines were also used to model simultaneously non linear effects of continuous covariates and time dependent hazard ratios this led to a graphical representation of the hazard ratio that can be useful for clinical interpretation estimates of these models were obtained by likelihood maximization we showed that these estimates could be also obtained using standard algorithms for poisson regression

the poincare algebra in rank 3 simple lie algebras we classify embeddings of the poincare algebra p 3 1 into the rank 3 simple lie algebras up to inner automorphism we show that there are exactly two embeddings of p 3 1 into sl 4 c which are however related by an outer automorphism of sl 4 c next we show that there is a unique embedding of p 3 1 into so 7 c up to inner automorphism but no embeddings of p 3 1 into sp 6 c all embeddings are explicitly described as an application we show that each irreducible highest weight module of sl 4 c not necessarily finite dimensional remains indecomposable when restricted to p 3 1 with respect to any embedding of p 3 1 into sl 4 c

maximal abelian subgroups of the isometry and conformal groups of euclidean and minkowski spaces the maximal abelian subalgebras masas of the euclidean and pseudo euclidean lie algebras are classified into conjugacy classes under the action of the corresponding lie groups and and also under the conformal groups and respectively the results are presented in terms of decomposition theorems for orthogonally indecomposable masas exist only for p 1 and p 2 for on the other hand orthogonally indecomposable masas exist for all values of p the results are used to construct new coordinate systems in which wave equations and hamilton jacobi equations allow the separation of variables

causal diagrams for empirical research summary the primary aim of this paper is to show how graphical models can be used as a mathematical language for integrating statistical and subject matter information in particular the paper develops a principled nonparametric framework for causal inference in which diagrams are queried to determine if the assumptions available are sufficient for identifying causal effects from nonexperimental data if so the diagrams can be queried to produce mathematical expressions for causal effects in terms of observed distributions otherwise the diagrams can be queried to suggest additional observations or auxiliary experiments from which the desired inferences can be obtained

matrices with signed generalized inverses abstract a real matrix a is said to have a signed generalized inverse if the sign pattern of its generalized inverse a is uniquely determined by the sign pattern of a in this paper we give complete characterizations of the m n matrices a which have signed generalized inverses for both cases a n and a where a is the term rank of a and without loss of generality we assume n m and thus solve a problem proposed in b l shader siam j matrix anal appl 16 1995 1056 using these characterizations we are also able to show that the property of having a signed generalized inverse for a matrix a is inherited by all the submatrices b of a with b a and is also inherited by all those matrices a 1 with a 1 a which can be obtained from a by replacing some nonzero entries of a by zero we also consider several special cases of a problem proposed in r a brualdi b l shader matrices of sign solvable linear systems cambridge university press cambridge ma 1995 b l shader siam j matrix anal appl 16 1995 1056 about the characterization of the matrices in a special triangular block form to have signed generalized inverses

exact and quasiexact solvability of second order superintegrable quantum systems i euclidean space preliminaries we show that second order superintegrable systems in two dimensional and three dimensional euclidean space generate both exactly solvable es and quasiexactly solvable qes problems in quantum mechanics via separation of variables and demonstrate the increased insight into the structure of such problems provided by superintegrability a principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable most past separation of variables treatments of qes problems via partial differential equations have only incorporated separability not multiseparability also we propose another definition of es and qes the quantum mechanical problem is called es if the solution of schrodinger equation can be expressed in terms of hypergeometric functions fnm and is qes if the schrodinger equation admits polynomial solutions with coefficients necessarily satisfying a three term or higher order of recurrence relations in three dimensions we give an example of a system tha

on the nonsingularity of matrices with certain sign patterns abstract a new nonsingularity criterion for an n n real matrix a based on sign distribution and some conditions on the elements is given this work was motivated by the solution of poisson s equation for doubly connected regions we consider a matrix a arising in the above application to demonstrate the theory

on digraphs and forbidden configurations of strong sign nonsingular matrices a square real matrix a called a strong sign nonsingular matrix or s2ns matrix if all matrices with the same sign pattern as a nonsingular and the inverses of these matrices all have the same sign pattern a digraph which is the underlying digraph of the signed digraph of an s2ns matrix with a negative main diagonal is called an s2ns digraph in 9 thomassen gave a characterization of strongly connected s2ns digraphs in terms of the forbidden subdigraphs in 2 brualdi and shader constructed minimal forbidden configurations for s2ns digraphs for the general cases where the digraphs considered are not necessarily strongly connected they also proposed the problem about the existence of new minimal forbidden configurations other than those found in 2 9 in this paper we construct infinitely many new basic minimal forbidden configurations and thus obtain the answer this problem we also obtain several necessary conditions for minimal forbidden configurations and give a generalization of thomassen s theorem

comparison between splines and fractional polynomials for multivariable model building with continuous covariates a simulation study with continuous response in observational studies many continuous or categorical covariates may be related to an outcome various spline based procedures or the multivariable fractional polynomial mfp procedure can be used to identify important variables and functional forms for continuous covariates this is the main aim of an explanatory model as opposed to a model only for prediction the type of analysis often guides the complexity of the final model spline based procedures and mfp have tuning parameters for choosing the required complexity to compare model selection approaches we perform a simulation study in the linear regression context based on a data structure intended to reflect realistic biomedical data we vary the sample size variance explained and complexity parameters for model selection we consider 15 variables a sample size of 200 1000 and r 2 0 2 0 8 is the scenario with the smallest largest amount of information for assessing performance we consider prediction error correct and incorrect inclusion of covariates qualitative measures for judging selected functional forms and further novel criteria from limited information a suitable explanatory model cannot be obtained prediction performance from all types of models is similar with a medium amount of information mfp performs better than splines on several criteria mfp better recovers simpler functions whereas splines better recover more complex functions for a large amount of information and no local structure mfp and the spline procedures often select similar explanatory models

separation of variables and subgroup bases on n dimensional hyperboloids a graphical formalism is introduced for describing subgroup type coordinates on n dimensional lorentzian hyperboloids imbedded into n 1 dimensional minkowski spaces the o n 1 group element is parametrized according to different subgroup chains involving lorentz rotation and euclidean subgroups the coordinates are then induced by the corresponding group action eigenfunctions of the laplace beltrami operator are obtained as products of jacobi functions associated legendre functions and modified bessel functions

contractions of lie algebras applications to special functions and separation of variables we investigate the consequences of contraction of the lie algebras of the orthogonal groups to the lie algebras of the euclidean groups in terms of separation of variables for laplace beltrami eigenvalue equations and the solutions of these equations that arise through separation of variables techniques on the n sphere and in n dimensional euclidean space general ellipsoidal and paraboloidal coordinates are included not just the subgroup type coordinates that have been the concern of most investigations of contractions as applied to special functions we pay special attention to the case n 2 where we show in detail for example how lame polynomials contract to periodic mathieu functions our point of view emphasizes the characterization of separable polynomial eigenfunctions in terms of the zeros of these eigenfunctions we also consider all possible separable coordinate systems on the complex two sphere which includes real hyperboloids as special cases and their contraction to flat space coordinates

when the sign pattern of a square matrix determines uniquely the sign pattern of its inverse abstract we give a precise characterization in terms of parity digraphs of those square matrices a such that for every matrix b with the same sign pattern as a b 1 exists and has the same sign pattern as a 1

strengthening analytical thinking for observational studies the stratos initiative the validity and practical utility of observational medical research depends critically on good study design excellent data quality appropriate statistical methods and accurate interpretation of results statistical methodology has seen substantial development in recent times unfortunately many of these methodological developments are ignored in practice consequently design and analysis of observational studies often exhibit serious weaknesses the lack of guidance on vital practical issues discourages many applied researchers from using more sophisticated and possibly more appropriate methods when analyzing observational studies furthermore many analyses are conducted by researchers with a relatively weak statistical background and limited experience in using statistical methodology and software consequently even standard analyses reported in the medical literature are often flawed casting doubt on their results and conclusions an efficient way to help researchers to keep up with recent methodological developments is to develop guidance documents that are spread to the research community at large these observations led to the initiation of the strengthening analytical thinking for observational studies stratos initiative a large collaboration of experts in many different areas of biostatistical research the objective of stratos is to provide accessible and accurate guidance in the design and analysis of observational studies the guidance is intended for applied statisticians and other data analysts with varying levels of statistical education experience and interests in this article we introduce the stratos initiative and its main aims present the need for guidance documents and outline the planned approach and progress so far we encourage other biostatisticians to become involved

multiple imputation for an incomplete covariate that is a ratio we are concerned with multiple imputation of the ratio of two variables which is to be used as a covariate in a regression analysis if the numerator and denominator are not missing simultaneously it seems sensible to make use of the observed variable in the imputation model one such strategy is to impute missing values for the numerator and denominator or the log transformed numerator and denominator and then calculate the ratio of interest we call this passive imputation alternatively missing ratio values might be imputed directly with or without the numerator and or the denominator in the imputation model we call this active imputation in two motivating datasets one involving body mass index as a covariate and the other involving the ratio of total to high density lipoprotein cholesterol we assess the sensitivity of results to the choice of imputation model and as an alternative explore fully bayesian joint models for the outcome and incomplete ratio fully bayesian approaches using winbugs were unusable in both datasets because of computational problems in our first dataset multiple imputation results are similar regardless of the imputation model in the second results are sensitive to the choice of imputation model sensitivity depends strongly on the coefficient of variation of the ratio s denominator a simulation study demonstrates that passive imputation without transformation is risky because it can lead to downward bias when the coefficient of variation of the ratio s denominator is larger than about 0 1 active imputation or passive imputation after log transformation is preferable 2013 the authors statistics in medicine published by john wiley sons ltd

contractions of lie algebras and the separation of variables interbase expansions lie algebra contractions from o n 1 to e n are used to obtain asymptotic limits of interbase expansions between bases corresponding to different subgroup chains for the group o n 1 the contractions lead to interbase expansions for different subgroup chains of the euclidean group e n they provide asymptotic formulae for quantities such as wigner rotation matrices clebsch gordan coefficients and racah coefficients

the central role of the propensity score in observational studies for causal effects abstract the results of observational studies are often disputed because of nonrandom treatment assignment for example patients at greater risk may be overrepresented in some treatment group this paper discusses the central role of propensity scores and balancing scores in the analysis of observational studies the propensity score is the estimated conditional probability of assignment to a particular treatment given a vector of observed covariates both large and small sample theory show that adjustment for the scalar propensity score is sufficient to remove bias due to all observed covariates applications include matched sampling on the univariate propensity score which is equal percent bias reducing under more general conditions than required for discriminant matching multivariate adjustment by subclassification on balancing scores where the same subclasses are used to estimate treatment effects for all outcome variables and in all subpopulations and visual representation of multivariate adjustment by a two dimensional plot author

sign properties of metzler matrices with applications abstract several results about sign properties of metzler matrices are obtained it is first established that checking the sign stability of a metzler sign matrix can be either characterized in terms of the hurwitz stability of the unit sign matrix in the corresponding qualitative class or in terms the negativity of the diagonal elements of the metzler sign matrix and the acyclicity of the associated directed graph similar results are obtained for the case of metzler block matrices and metzler mixed matrices the latter being a class of metzler matrices containing both sign and real type entries the problem of assessing the sign stability of the convex hull of a finite and summable family of metzler matrices is also solved and a necessary and sufficient condition for the existence of common lyapunov functions for all the matrices in the convex hull is obtained the concept of sign stability is then generalized to the concept of ker b sign stability a problem that arises in the analysis of certain jump markov processes a sufficient condition for the ker b sign stability of metzler sign matrices is obtained and formulated using inverses of sign matrices and the concept of l matrices several applications of the results are discussed in the last section

the moore penrose inverse of matrices with an acyclic bipartite graph abstract the moore penrose inverse of a real matrix having no square submatrix with two or more diagonals is described in terms of bipartite graphs for such a matrix the sign of every entry of the moore penrose inverse is shown to be determined uniquely by the signs of the matrix entries i e the matrix has a signed generalized inverse necessary and sufficient conditions on an acyclic bipartite graph are given so that each nonnegative matrix with this graph has a nonnegative moore penrose inverse nearly reducible matrices are proved to contain no submatrix having two or more diagonals implying that a nearly reducible matrix has a signed generalized inverse furthermore it is proved that the term rank and rank are equal for each submatrix of a nearly reducible matrix

separation of variables on n dimensional riemannian manifolds i the n sphere sn and euclidean n space rn the following problem is solved what are all the different separable coordinate systems for the laplace beltrami eigenvalue equation on the n sphere sn and euclidean n space rn and how are they constructed this is achieved through a combination of differential geometric and group theoretic methods a graphical procedure for construction of these systems is developed that generalizes vilenkin s construction of polyspherical coordinates the significance of these results for exactly soluble dynamical systems on these manifolds is pointed out the results are also of importance for the analysis of the special functions appearing in the separable solutions of the laplace beltrami eigenvalue equation on these manifolds

maximal abelian subalgebras of pseudoorthogonal lie algebras abstract the classification of maximal abelian subalgebras masas of o p q is reduced to the construction of orthogonally indecomposable oid masas five types of oid masas exist and their classification amounts to an analysis of maximal abelian nilpotent subalgebras manss of o p q su p q sl n r or sl n c theorems are provided making possible the classification of all manss of o p q and low dimensional examples are analyzed in detail

on maximal sign nonsingular matrices abstract we derive several new results on maximal sign nonsingular sns matrices these include two methods for constructing new sns matrices from smaller sns matrices the results are used to produce a matrix definition for the c 4 cockades and to introduce new families of sns matrices not previously defined all of our results were deduced with the help of tables of maximal irreducible sns matrices of orders three through nine which we previously computed the interesting matrices in these tables for orders three through eight are reproduced at the end of an appendix in which we describe the computations

generalized survival models for correlated time to event data our aim is to develop a rich and coherent framework for modeling correlated time to event data including 1 survival regression models with different links and 2 flexible modeling for time dependent and nonlinear effects with rich postestimation we extend the class of generalized survival models which expresses a transformed survival in terms of a linear predictor by incorporating a shared frailty or random effects for correlated survival data the proposed approach can include parametric or penalized smooth functions for time time dependent effects nonlinear effects and their interactions the maximum penalized marginal likelihood method is used to estimate the regression coefficients and the variance for the frailty or random effects the optimal smoothing parameters for the penalized marginal likelihood estimation can be automatically selected by a likelihood based cross validation criterion for models with normal random effects gauss hermite quadrature can be used to obtain the cluster level marginal likelihoods the akaike information criterion can be used to compare models and select the link function we have implemented these methods in the r package rstpm2 simulating for both small and larger clusters we find that this approach performs well through 2 applications we demonstrate 1 a comparison of proportional hazards and proportional odds models with random effects for clustered survival data and 2 the estimation of time varying effects on the log time scale age varying effects for a specific treatment and two dimensional splines for time and age

methods for dealing with time dependent confounding longitudinal studies where data are repeatedly collected on subjects over a period are common in medical research when estimating the effect of a time varying treatment or exposure on an outcome of interest measured at a later time standard methods fail to give consistent estimators in the presence of time varying confounders if those confounders are themselves affected by the treatment robins and colleagues have proposed several alternative methods that provided certain assumptions hold avoid the problems associated with standard approaches they include the g computation formula inverse probability weighted estimation of marginal structural models and g estimation of structural nested models in this tutorial we give a description of each of these methods exploring the links and differences between them and the reasons for choosing one over the others in different settings

joint estimation of time dependent and non linear effects of continuous covariates on survival in order to yield more flexible models the cox regression model lambda t x lambda 0 t exp betax has been generalized using different non parametric model estimation techniques one generalization is the relaxation of log linearity in x lambda t x lambda 0 t exp r x another is the relaxation of the proportional hazards assumption lambda t x lambda 0 t exp beta t x these generalizations are typically considered independently of each other we propose the product model lambda t x lambda 0 t exp beta t r x which allows for joint estimation of both effects and investigate its properties the functions describing the time dependent beta t and non linear r x effects are modelled simultaneously using regression splines and estimated by maximum partial likelihood likelihood ratio tests are proposed to compare alternative models simulations indicate that both the recovery of the shapes of the two functions and the size of the tests are reasonably accurate provided they are based on the correct model by contrast type i error rates may be highly inflated and the estimates considerably biased if the model is misspecified applications in cancer epidemiology illustrate how the product model may yield new insights about the role of prognostic factors

testing for improvement in prediction model performance authors have proposed new methodology in recent years for evaluating the improvement in prediction performance gained by adding a new predictor y to a risk model containing a set of baseline predictors x for a binary outcome d we prove theoretically that null hypotheses concerning no improvement in performance are equivalent to the simple null hypothesis that y is not a risk factor when controlling for x h0 p d 1 x y p d 1 x therefore testing for improvement in prediction performance is redundant if y has already been shown to be a risk factor we also investigate properties of tests through simulation studies focusing on the change in the area under the roc curve auc an unexpected finding is that standard testing procedures that do not adjust for variability in estimated regression coefficients are extremely conservative this may explain why the auc is widely considered insensitive to improvements in prediction performance and suggests that the problem of insensitivity has to do with use of invalid procedures for inference rather than with the measure itself to avoid redundant testing and use of potentially problematic methods for inference we recommend that hypothesis testing for no improvement be limited to evaluation of y as a risk factor for which methods are well developed and widely available analyses of measures of prediction performance should focus on estimation rather than on testing for no improvement in performance copyright 2013 john wiley sons ltd

full sign invertibility and symplectic matrices abstract an n n sign pattern h is said to be sign invertible if there exists a sign pattern h 1 called the sign inverse of h such that for all matrices a q h a 1 exists and a 1 q h 1 if in addition h 1 is sign invertible implying h 1 1 h h is said to be fully sign invertible and h h 1 is called a sign invertible pair given an n n sign pattern h a symplectic pair in q h is a pair of matrices a d such that a q h d q h and a t d i symplectic pairs are a pattern generalization of orthogonal matrices which arise from a special symplectic matrix found in n body problems in celestial mechanics 1 we discuss the digraphical relationship between a sign invertible pattern h and its sign inverse h 1 and use this to cast a necessary condition for full sign invertibility of h we proceed to develop sufficient conditions for h s full sign invertibility in terms of allowed paths and cycles in the digraph of h and conclude with a complete characterization of those sign patterns that require symplectic pairs

integrable and superintegrable quantum systems in a magnetic field integrable quantum mechanical systems with magnetic fields are constructed in two dimensional euclidean space the integral of motion is assumed to be a first or second order hermitian operator contrary to the case of purely scalar potentials quadratic integrability does not imply the separation of variables in the schrodinger equation moreover quantum and classical integrable systems do not necessarily coincide the hamiltonian can depend on the planck constant in a nontrivial manner

quantum superintegrable systems with quadratic integrals on a two dimensional manifold there are two classes of quantum integrable systems on a manifold with quadratic integrals the liouville and the lie integrable systems as it happens in the classical case the quantum liouville quadratic integrable systems are defined on a liouville manifold and the schrodinger equation can be solved by separation of variables in one coordinate system the lie integrable systems are defined on a lie manifold and are not generally separable ones but can be solved therefore there are superintegrable systems with two quadratic integrals of motion not necessarily separable in two coordinate systems the quantum analogs of the two dimensional superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion there are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones analytic formulas for the involved integrals are calculated in all the cases all the known quan

subgroups of lie groups and separation of variables separable systems of coordinates for the helmholtz equation d e in pseudo riemannian spaces of dimension d have previously been characterized algebraically in terms of sets of commuting second order symmetry operators for the operator d they have also been characterized geometrically by the form that the metric ds2 gik x dxidxk can take we complement these characterizations by a group theoretical one in which the second order operators are related to continuous and discrete subgroups of g the symmetry group of d for d 3 we study all separable coordinates that can be characterized in terms of the lie algebra l of g and show that they are of eight types seven of which are related to the subgroup structure of g our method clearly generalizes to the case d 3 although each separable system corresponds to a pair of commuting symmetry operators there do exist pairs of commuting symmetries s1 s2 that are not associated with separable coordinates for subgroup related operators we show in detail just

graph theoretical approach to qualitative solvability of linear systems the structure of sign solvable and strongly sign solvable systems is studied here by a refinement of the graph theoretic approach first suggested by maybee both sign solvability and strong sign solvability are characterized in terms of an associated digraph the problem of recognizing sign solvability is reduced to two subproblems recognizing strong sign solvability and recognizing sign nonsingularity under fairly general conditions on the sign patterns of a it is possible to determine all sign patterns for b which render the system ax b sign solvable

algebraic constructions of the minimal forbidden digraphs of strong sign nonsingular matrices a square real matrix a is called a strong sign nonsingular matrix s2ns matrix if all the matrices with the same sign pattern as a are nonsingular and all the inverses of these matrices have the same sign pattern s2ns digraphs are digraphs associated with those s2ns matrices with negative main diagonals in this paper we define the associated linear system of equations l d over the finite field f2 for each digraph d and then define an undirected graph g l d representing certain relations between the equations of l d we obtain algebraic criteria to recognize the minimal forbidden configurations of s2ns digraphs in terms of the solvability of the linear system l d and some of its subsystems and the connectedness of the undirected graph g l d these algebraic criteria together with a conjunction operation of digraphs can be used to construct infinitely many new minimal forbidden configurations

sign nonsingular matrices and even cycles in directed graphs a sign nonsingular matrix or l mahix a is a real m x n matrix such that the columns of any real m x n matrix with the same sign pattern as a are linearly independent the problem of recognizing square l matrices is equivalent to that of finding an even cycle in a directed graph in this paper we use graph theoretic methods to investigate l matrices in particular we determine the maximum number of nonzero elements in square l matrices and we characterize completely the semicomplete l matrices i e the square l matrices a j such that at least one of a and ajl is nonzero for any i j and those square l matrices which are combinatorially symmetric i e the main diagonal has only nonzero entries and a j 0 iff a jr 0 we also show that for any n x n l matrix there is an i such that the total number of nonzero entries in the i th row and i th column is less than n unless the matrix has a completely specified structure finally we discuss the algorithmic aspects

second order superintegrable systems in conformally flat spaces iv the classical 3d stackel transform and 3d classification theory this article is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems both classical and quantum in conformally flat spaces in the first part of the article we study the stackel transform or coupling constant metamorphosis as an invertible mapping between classical superintegrable systems on different three dimensional spaces we show first that all superintegrable systems with nondegenerate potentials are multiseparable and then that each such system on any conformally flat space is stackel equivalent to a system on a constant curvature space in the second part of the article we classify all the superintegrable systems that admit separation in generic coordinates we find that there are eight families of these systems

quantum superintegrability and exact solvability in n dimensions a family of maximally superintegrable systems containing the coulomb atom as a special case is constructed in n dimensional euclidean space two different sets of n commuting second order operators are found overlapping in the hamiltonian alone the system is separable in several coordinate systems and is shown to be exactly solvable it is solved in terms of classical orthogonal polynomials the hamiltonian and n further operators are shown to lie in the enveloping algebra of a hidden affine lie algebra

contractions of lie algebras and separation of variables the n dimensional sphere inonu wigner contractions from the rotation group o n 1 to the euclidean group e n are used to relate the separation of variables in laplace beltrami operators on n dimensional spheres and euclidean spaces respectively in this article we consider all subgroup type coordinates corresponding to different chains of subgroups of o n 1 and e n respectively in particular the contractions relate the graphical formalism of trees on spheres to the clusters on euclidean spaces introduced in this article the contractions are considered analytically on several levels the vector fields realizing the lie algebras the complete sets of commuting operators characterizing separable coordinate systems the coordinate systems themselves and the separated eigenfunctions