raising and lowering operators factorization and differential difference operators of hypergeometric type starting from the rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variables on a uniform lattice in order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight using the infeld hull factorization method we generate from the raising and lowering operators the second order self adjoint differential difference operator of hypergeometric type

a complete determination of the zeros of weight 1 6j coefficients it is shown how all zeros of weight 1 6j coefficients arise as particular cases of a four parameter family of such zeros the parametrization is given explicitly

a new statistic for influence in linear regression since the seminal article by cook the usual way to measure the influence of an observation in a statistical model is to delete the observation from the sample and compute a convenient norm of the change in the parameters or in the vector of forecasts in this article we define a new way to measure the influence of an observation based on how this observation is being influenced by the rest of the data more precisely the new statistic we propose is defined as the squared norm of the vector of changes of the forecast of one observation when each of the sample points are deleted one by one we prove that this new statistic has asymptotically a normal distribution and is able to detect a group of high leverage similar outliers that will be undetected by cook s statistic we show in several examples that the proposed statistic is useful for detecting heterogeneity in regression models in large high dimensional datasets

normal convexity and equations over groups a subgroups of a grouph is said to be normal convex inh if for any subsetr s the natural maps r s h r his injective

representations of the discrete inhomogeneous lorentz group and dirac wave equation on the lattice we propose the fundamental and two dimensional representation of the lorentz groups on a 3 1 dimensional hypercubic lattice from which representations of higher dimensions can be constructed for the unitary representation of the discrete translation group we use the kernel of the fourier transform from the dirac representation of the lorentz group including reflections we derive in a natural way the wave equation on the lattice for spin 1 2 particles finally the induced representation of the discrete inhomogeneous lorentz group is constructed by standard methods and its connection with the continuous case is discussed

non orientable surface plus one relation groups recently dicks linnell determined the l 2 betti numbers of the orientable surfaceplus one relation groups and their arguments involved some results that were obtained topologically by hempel and howie using algebraic arguments we now extend all these results of hempel and howie to a larger class of two relator groups and we then apply the extended results to determine the l 2 betti numbers of the non orientable surface plus one relation groups 2000 mathematics subject classification primary 20f05 secondary 16s34 20j05

the standardized influence matrix and its applications abstract in this article we introduce the standardized influence matrix sim of parameter estimators as a conjugate of the sample covariance matrix of standardized influence functions sifs evaluated at the data points we propose principal component analysis for the sim and its complement sim analysis as a diagnostic tool for regression or other statistical inference and provide theoretical insight to the local influence defined by cook we show that sim analysis reveals the multivariate structure of outlying and or influential points specifically sim analysis uncovers hidden structures of influence such as clustering that cannot be identified by the lengths of the standardized influences finally examples in linear regression show that a diagnostic method using sim is more effective if robust parameter estimates are used in calculating the sample sim

outlier detection using nonconvex penalized regression this article studies the outlier detection problem from the standpoint of penalized regression in the regression model we add one mean shift parameter for each of the n data points we then apply a regularization favoring a sparse vector of mean shift parameters the usual l1 penalty yields a convex criterion but fails to deliver a robust estimator the l1 penalty corresponds to soft thresholding we introduce a thresholding denoted by based iterative procedure for outlier detection ipod a version based on hard thresholding correctly identifies outliers on some hard test problems we describe the connection between ipod and m estimators our proposed method has one tuning parameter with which to both identify outliers and estimate regression coefficients a data dependent choice can be made based on the bayes information criterion the tuned ipod shows outstanding performance in identifying outliers in various situations compared with other existing approaches in addition ipod is much

raising and lowering operators and their factorization for generalized orthogonal polynomials of hypergeometric type on homogeneous and non homogeneous lattices we complete the construction of raising and lowering operators given in a previous work for the orthogonal polynomials of hypergeometric type on a non homogeneous lattice we extend these operators to the generalized orthogonal polynomials namely those difference orthogonal polynomials that satisfy a similar difference equation of hypergeometric type

a modification of a method for the detection of outliers in multivariate samples we modify a method for identifying outliers in multivariate samples proposed by hadi a simulation study shows that this modification controls the size of the test leads to substantially improved power and is more effective in dealing with the masking and swamping problems

on the zeros of 6 j coefficients an extensive computer search for non trivial zeros of the 6 j coefficients of su 2 has been performed to find their frequency as a function of the sum of their quantum numbers and of their polynomial degree for the zeros of degree 2 four of the seven cases already known taking into account regge symmetries of two relations between quantum numbers relevant to a pell equation have been generalized to deal with only one relation a new case of two relations and many new cases of three relations all with polynomial solutions have been found but all these formulae only account for one zero out of seven for degree 3 we found four new cases of two relations and some other cases of three relations with polynomial solutions with the already known case of a pell equation they account for almost half of the zeros the fast decrease of the number of zeros with the degree and the relative smallness of their quantum number is an indication that there should be no zeros of degree larger than 9

on balanced presentations of the trivial group to solve problems of magnus and stallings stated in the 1960s we construct a balanced presentation of the trivial group such that no defining relator of this presentation could be replaced by a free generator and the group given by the altered presentation would still be trivial some other related problems on balanced presentations of the trivial group are discussed

a multistage procedure for detecting several outliers in linear regression a new statistic fk is proposed for detecting multiple outliers in linear regression this statistic is incorporated into the following multistage procedure initially a subset of k observations is selected to be tested if fk is found to be significant the most extreme observation in the subset as determined by the largest studentized residual is deleted and the test repeated for the k 1 observations in the subset using the remaining sample the procedure is stopped when a test fails to reject the no outlier hypothesis a monte carlo study is used to evaluate the performance of this procedure

nontrivial zeros of the wigner 3j and racah 6j coefficients ii some nonlinear solutions additional formulas for nontrivial zeros in the 3j and 6j symbols have been found for some higher order cases i e where k 2 3 and 4 numerical examples only

on lie algebras built from so 3 tensor operators concerning the lie algebras generated by certain linear combinations of so 3 tensor operators two theorems are given which disprove some widely accepted results in the standard literature

the group g t e when g is torsion free abstract in this paper we discuss classes of nontrivial elements in the group with a relative presentation g t e where g is torsion free and e is solvable over g

adjoint boson realization of su n and a family of structural zeros of 6j coefficients for the fundamental boson realization of su n in its principal so 3 basis the generators of su n are so 3 tensor operators constructed out of one single type of boson namely a boson with so 3 multiplet label n 1 2 here the adjoint boson realization of su n is given this time the so 3 tensor operators are constructed by means of n 1 types of bosons those with so 3 labels 1 2 n 1 the validity of this realization depends upon a new identity between 6j coefficients from this realization or equivalently from this new identity one can deduce the existence of a family of structural zeros of 6j coefficients

continuum quantum systems as limits of discrete quantum systems iv affine canonical transforms affine canonical transforms complex order fourier transforms and their associated coherent states appear in two scenarios finite discrete and continuum we examine the relationship between the two scenarios making systematic use of inductive limits which were developed in the preceding articles in this series

procedures for the identification of multiple outliers in linear models abstract we consider the problem of identifying and testing multiple outliers in linear models the available outlier identification methods often do not succeed in detecting multiple outliers because they are affected by the observations they are supposed to identify we introduce two test procedures for the detection of multiple outliers that appear to be less sensitive to this problem both procedures attempt to separate the data into a set of clean data points and a set of points that contain the potential outliers the potential outliers are then tested to see how extreme they are relative to the clean subset using an appropriately scaled version of the prediction error the procedures are illustrated and compared to various existing methods using several data sets known to contain multiple outliers also the performances of both procedures are investigated by a monte carlo study the data sets and the monte carlo indicate that both procedures are effective in the detection of multiple outliers

the identification of influential subsets in regression using a fuzzy clustering strategy a fuzzy clustering strategy is used to identify subsets of influential observations in regression as part of the fuzzy clustering strategy the analyst can explore the uniqueness of selected subsets and the degree of belonging of observations to selected subsets this is accomplished by either varying a fuzzy parameter or the number of clusters once the observations or subsets have been identified the analyst can then compute regression diagnostics to confirm their degree of influence in regression bootstrapping and high breakdown procedures were used to confirm the influence of the previously identified subsets this fuzzy clustering strategy is applied to the modified data on wood specific gravity and an augmented production dataset both datasets have been previously presented in the literature

influence analysis based on the case sensitivity function the case sensitivity function approach to influence analysis is introduced as a natural smooth extension of influence curve methodology in which both the insights of geometry and the power of convex analysis are available in it perturbation is defined as movement between probability vectors defining weighted empirical distributions a euclidean geometry is proposed giving such perturbations both size and direction the notion of the salience of a perturbation is emphasized this approach has several benefits a general probability case weight analysis results answers to a number of outstanding questions follow directly rescaled versions of the three usual finite sample influence curve measures seen now to be required for comparability across different sized subsets of cases are readily available these new diagnostics directly measure the salience of the infinitesimal perturbations involved their essential unity both within and between subsets is evident geometrically finally it is shown how a relaxation strategy in which a high dimensional 0 ncm discrete problem is replaced by a low dimensional o n continuous problem can combine with convex optimization results to deliver better performance in challenging multiple case influence problems further developments are briefly indicated

weight 2 zeros of 3j coefficients and the pell equation all weight 2 zeros of the wigner 3j coefficients may be obtained from the quadratic diophantine equation known as pell s equation these zeros may then be classified by the orbits of a discrete infinite order subgroup of the lorentz group so 1 1 this is carried out by transforming the polynomial part of a weight 2 3j coefficient to pellian form and obtaining the fundamental zeros numerically the relation of this polynomial to a family of binary quadratic forms is also given together with a discussion of the invariance group

assessment of local influence summary statistical models usually involve some degree of approximation and therefore are nearly always wrong because of this inexactness an assessment of the influence of minor perturbations of the model is important we discuss a method for carrying out such an assessment the method is not restricted to linear regression models and it seems to provide a relatively simple unified approach for handling a variety of problems

a fast procedure for outlier diagnostics in large regression problems abstract we propose a procedure for computing a fast approximation to regression estimates based on the minimization of a robust scale the procedure can be applied with a large number of independent variables where the usual algorithms require an unfeasible or extremely costly computer time also it can be incorporated in any high breakdown estimation method and may improve it with just little additional computer time the procedure minimizes the robust scale over a set of tentative parameter vectors estimated by least squares after eliminating a set of possible outliers which are obtained as follows we represent each observation by the vector of changes of the least squares forecasts of the observation when each of the data points is deleted then we obtain the sets of possible outliers as the extreme points in the principal components of these vectors or as the set of points with large residuals the good performance of the procedure allows identification of multiple outliers avoiding masking effe

heegaard splittings of seifert fibered spaces with boundary we give the classification theorem for heegaard splittings of fiberwise orientable seifert fibered spaces with nonempty boundary a thin position argument yields a reducibility result which by induction shows that all heegaard splittings of such manifolds are vertical in the sense of lustig moriah algebraic arguments allow a classification of the vertical heegaard splittings

nontrivial zeros of the wigner 3 j and racah 6 j coefficients i linear solutions some formuals for nontrivial zeros in the 3 j and 6 j symbols have been found some formuals for nontrivial zeros in the 3 j and 6 j symbols have been found

unmasking multivariate outliers and leverage points abstract detecting outliers in a multivariate point cloud is not trivial especially when there are several outliers the classical identification method does not always find them because it is based on the sample mean and covariance matrix which are themselves affected by the outliers that is how the outliers get masked to avoid the masking effect we propose to compute distances based on very robust estimates of location and covariance these robust distances are better suited to expose the outliers in the case of regression data the classical least squares approach masks outliers in a similar way also here the outliers may be unmasked by using a highly robust regression method finally a new display is proposed in which the robust regression residuals are plotted versus the robust distances this plot classifies the data into regular observations vertical outliers good leverage points and bad leverage points several examples are discussed

a new test for asphericity and diagrammatic reducibility of group presentations we present a new test for studying asphericity and diagrammatic reducibility of group presentations our test can be applied to prove diagrammatic reducibility in cases where the classical weight test fails we use this criterion to generalize results of j howie and s m gersten on asphericity of lots and of adian presentations and derive new results on solvability of equations over groups we also use our methods to investigate a conjecture of s v ivanov related to kaplansky s problem on zero divisors we strengthen ivanov s result for locally indicable groups and prove a weak version of the conjecture

k clustering as a detection tool for influential subsets in regression this article describes a new methodology for the detection of influential subsets in regression the method is based on an adaptation of computational and graphical techniques used in cluster analysis and makes use of some general properties of influential subsets but it is independent of any specific measure of influence for small to moderate data sets the proposed method is computationally efficient compared to existing search methods and it identifies subset candidates that merit attention according to some or all measures of joint influence that have appeared in the literature to date examples are given illustrating the method applied to two data sets previously analyzed in published studies

a generalized extreme studentized residual multiple outlier detection procedure in linear regression this article is concerned with procedures for detecting multiple y outhers in linear regression a generalized extreme studentized residual gesr procedure which controls type i error rate is developed an approximate formula to calculate the percentiles is given for large samples and more accurate percentiles for n 25 are tabulated the performance of this procedure is compared with others by monte carlo techniques and found to be superior the procedure however fails in detecting y outliers that are on high leverage cases for this a two phase procedure is suggested in phase 1 a set of suspect observations is identified by gesr and one of the diagnostics applied sequentially in phase 2 a backward testing is conducted using the gesr procedure to see which of the suspect cases are outlicrs several examples are analyzed

parametrization of the linear zeros of 6j coefficients the linear zeros of 6j coefficients are fully parametrized apart from a multiplicative factor in terms of four integers

location of several outliers in multiple regression data using elemental sets the outlying tendency of any case in a multiple regression of p predictors may be estimated by drawing all subsets of size p from the remaining cases and fitting the model each such subset yields an elemental residual for the case in question and a suitable summary statistic of them can be used as an estimate of the case s outlying tendency we propose two such summary statistics an unweighted median which is of bounded influence and a weighted median which is more efficient but less robust the computational load of the procedure is reduced by using random samples in place of the full set of subsets of size p as a byproduct the method yields useful information on the influence or leverage of cases and the mutual masking of high leverage points

on the complexity of the heegaard structure of hyperbolic 3 manifolds a well known result of waldhausen 22 states that all heegaard splittings of s 3 and of s 2 s 1 of a given genus g are homeomorphic and even isotopic recent work of various authors see 20 19 2 shows that there are more simple 3 manifolds where such a uniqueness holds however this is not true for most 3 manifolds see 12 15 pitts rubinstein 18 have announced a proof of the waldhausen conjecture 23 that for any 3 manifold m the maximal number hm g of non homeomorphic heegaard splittings is finite for any given genus g n this was shown by hass 10 for genus 2 manifolds and g 2 and by johannson 11 for haken manifolds it is expected that the number hm g grows for generic m i e not the above mentioned simple ones with the complexity of m in this note we consider heegaard splittings of minimal genus gmin m and show that for a large class of hyperbolic 3 manifolds hm gmin m grows exponentially theorem a let k s 3 be the montesinos knot m e 1 1 g g such that all i are mutually distinct odd primes and let g c d 1 g 22g 1 1 g let m be the closed 3 manifold obtained from k by m n surgery where m is even then for each m tupel e1 eg 1 1 g 1 1 there exists a minimal genus heegaard splitting y e1 eg and these are pairwise non homeomorphic

on locally indicable groups a group is said to be locally indicable if each of its non trivial finitely generated subgroups has the infinite cyclic group as a homomorphic image such groups were studied by higman 9 in connection with the zero divisor and unit problems for group rings more recently they have arisen i 2 12 in the study of equations over groups in 12 i proved a freiheitsatz for locally indicable groups this has been proved independently by brodskii 2 and short 22 the present paper arises from an investigation of a question put to me by s j pride whether torsionfree 1 relator groups are locally indicable the question was raised originally by baumslag i 1 problem 19 and an affirmative solution has recently been announced by brodski 2 as a consequence the group algebra rg of a torsion free 1 relator group g over an integral domain r has no non trivial zero divisors and no non trivial units using higman s results 9 the first fact was also proved by lewin and lewin 16 by embedding rg in a division ring the second appears to be new as was pointed out to me by k a brown a second consequence is that no 1 relator group has a non trivial finitely generated perfect subgroup which answers 1 problem 7 and 10 question 1 in fact using the freiheitsatz and the tower method described in 12 it is possible to prove the following general version of brodskii s theorem

on diophantine equations and nontrivial racah coefficients some families of zeros of weight 1 6j coefficients are given each in terms of four parameters they arise from a geometrical investigation of certain diophantine equations some general remarks on the solutions of diophantine equations are also made

the detection of influential subsets in linear regression by using an influence matrix this paper presents a new method to identify influential subsets in linear regression problems the procedure uses the eigenstructure of an influence matrix which is defined as the matrix of uncentred covariances of the effect on the whole data set of deleting each observation normalized to include the univariate cook statistics on the diagonal it is shown that the eigenstructure of the influence matrix is useful to identify influential subsets and a procedure for detecting influential sets is proposed the method is illustrated with two examples

nontrivial zeros of the racah quadrupole invariant it is shown that a class of nontrivial zeros of the racah quadrupole invariant operator is given by two orbits of the group action of an infinite discrete subgroup of the proper two dimensional lorentz group so 1 1 on the hyperbola 4x2 3y2 11 4

a new parametrization and all integral realizations of the lorentz group a simple linear transformation of the biquaternionic parameters and of the group sl 2 c which is two to one homomorphic to the restricted lorentz group l is used to express each element of sl 2 c and l in terms of the first column of an element of l and the quaternionic parameters this parametrization is shown to bring the determination of the integral realizations of the restricted lorentz group under the purview of classical diophantine analysis involving the expression of a given pair of integers respectively as a sum of three squares and a sum of four squares these solutions are further constrained by the condition that a set of four integers which is linear in the solution of the three squares problem and in the solution of the four squares problem have a common factor these are all classical problems addressed and solved by euclid euler fermat gauss jacobi and others the corresponding realizations of sl 2 c fall into three distinct classes those in which the elements of

tensor product of group representations and structural zeros of racah coefficients it is shown that structural or nontrivial zeros of racah coefficients 6 j symbols can be related to decompositions of tensor products of group representations reduced to so 3 this extends a previous relation between structural zeros of 6 j symbols and boson realizations of exceptional lie groups in the present case both classical and exceptional groups may appear and examples are given for su 3 so 7 so 9 and f4

nontrivial zeros of weight 1 3j and 6j coefficients relation to diophantine equations of equal sums of like powers the nontrivial zeros of weight 1 3j and 6j coefficients given previously are shown to be the set of all such zeros the relation of these zeros to the solutions of well known diophantine equations is also discussed

a general method for local sensitivity analysis with application to regression models and other optimization problems this article introduces a method for sensitivity analysis of general applicability the method is based on the well known duality property of mathematical programming which states that the partial derivatives of the primal objective function with respect to the constraints on the right side parameters are the negative of the optimal values of the dual problem variables for the parameters or data for which sensitivities are sought to appear on the right side they are converted into artificial variables and set to their actual values thus obtaining the desired constraints the method is applicable to linear and nonlinear models to normal and nonnormal models and to least squares and other methods of estimation in addition to its general applicability the method is also computationally inexpensive because the necessary information becomes available without extra calculations the theoretical basis for the method is given and illustrated by its application to least squares least absolute value an

the double torus as a 2d cosmos groups geometry and closed geodesics the double torus provides a relativistic model for a closed 2d cosmos with topology of genus 2 and constant negative curvature its unfolding into an octagon extends to an octagonal tessellation of its universal covering the hyperbolic space h2 the tessellation is analysed with tools from hyperbolic crystallography actions on h2 of groups subgroups are identified for su 1 1 for a hyperbolic coxeter group acting also on su 1 1 and for the homotopy group 2 whose extension is normal in the coxeter group closed geodesics arise from links on h2 between octagon centres the direction and length of the shortest closed geodesics is computed dedicated to marcos moshinsky on the occasion of his 80th birthday

branched coverings of 2 complexes and diagrammatic reducibility the condition that all spherical diagrams in a 2 complex be reducible is shown to be equivalent to the condition that all finite branched covers be aspherical this result is related to the study of equations over groups furthermore large classes of 2 complexes are shown to be diagrammatically reducible in the above sense in particular every 2 complex has a subdivision which admits a finite branched cover which is diagrammatically reducible a consequence of the classical riemann hurwitz formula for riemann surfaces 6 p 301 is the fact that a finite branched cover of an aspherical riemann surface is also aspherical however the property of asphericity of a 2 complex implying asphericity of finite branched covers fails for general 2 complexes in fact we shall prove theorem 4 5 below that this property holds precisely for those 2 complexes which are diagrammatically reducible a 2 complex x is said to be diagrammatically reducible abbreviated dr x if for every combinatorial map f s2 x of the 2 sphere into x a spherical diagram in x there is a pair of faces in the domain with an edge in common which are mapped mirrorwise by f across that edge a closely related notion was introduced by lyndon and schupp 13 but this notion appears in its present form in a paper by sieradski 14 in that same paper sieradski asked a question which would have implied that all classical knot complements in the 3 sphere are homotopy equivalent to diagrammatically reducible 2 complexes we shall prove this latter assertion theorem 6 5 making use of our amalgamation theorem for diagrammatic reducibility theorem 5 4 the characterization of diagrammatic reducibility in terms of branched covers llows from a lifting theorem which may be of general interest if f x y is a combinatorial map of combinatorial 2 complexes then f induces a map of the link complexes of vertices lf lx ly the star graphs or coinitial graph in group theoretical language the map f is said to be reduced if lf is an immersion we prove theorem 4 2 th t if f x y is a reduced map of 2 complexes with x finite then there is a commutative diagram see below where qt z y is a finite branched cover and where g is injective off the zero skeleton of x this result is received by the editors july 9 1986 1980 muthenluticssulviectclussificwtios 1985 retvisios primary20f05 57m12 secondary57m20 1987 american mathematical scoclety 0002 9947 87 1 0 25 pel page

quadratic zeros of racah 6j coefficients a geometrical approach it is shown that the projective symmetries of the polynomial of quadratic 6j coefficients form the symmetrical group s6 nonlinear rational symmetries of are found partial parametrizations of the zeros of are presented