codimensions of algebras and growth functions abstract let a be an algebra over a field f of characteristic zero and let c n a n 1 2 be its sequence of codimensions we prove that if c n a is exponentially bounded its exponential growth can be any real number 1 this is achieved by constructing for any real number 1 an f algebra a such that lim n c n a n exists and equals the methods are based on the representation theory of the symmetric group and on properties of infinite sturmian and periodic words

varieties of special jordan algebras of almost polynomial growth abstract let j be a special jordan algebra and let c n j be its corresponding codimension sequence the aim of this paper is to prove that in case j is finite dimensional such a sequence is polynomially bounded if and only if the variety generated by j does not contain u j 2 the special jordan algebra of 2 2 upper triangular matrices as an immediate consequence we prove that u j 2 is the only finite dimensional special jordan algebra that generates a variety of almost polynomial growth

on some minimal supervarieties of exponential growth abstract in the present paper we investigate minimal supervarieties of given superexponent over fields of characteristic zero we show that any minimal supervariety of finite basic rank is generated by one of the minimal superalgebras introduced by giambruno and zaicev in 2003 furthermore it is proved that any minimal superalgebra whose graded simple components of the semisimple part are simple generates a minimal supervariety finally we state that the same conclusion holds when the semisimple part of a minimal superalgebra has exactly two arbitrary graded simple components

discrete monotonic optimization with application to a discrete location problem a general discrete optimization problem is investigated that includes integer polynomial programs as special cases to exploit the discrete monotonic structure of these problems a special class of cuts called monotonicity cuts are developed and then adjusted according to a suitable procedure to accommodate discrete requirements as illustration the method is applied to solve a discrete location problem which is also a variant of the well known engineering problem of design centering computational results are reported for instances of the latter problem with up to 100 variables and 500 constraints

exponential codimension growth of pi algebras an exact estimate abstract letabe an associative pi algebra over a fieldfof characteristic zero by studying the exponential behavior of the sequence of codimensions cn a ofa we prove thatinv a limn c n a always exists and is an integer we also give an explicit way for computing such integer letbbe a finite dimensionalz2 graded algebra whose grassmann envelopeg b satisfies the same identities ofa theninv a inv g b dim c 0 dim c 1 wherec 0 c 1 is a suitablez2 graded semisimple subalgebra ofb

association models and the bivariate normal for contingency tables with ordered categories summary association models considered by goodman 1979 for contingency tables with ordered categories are presented here in a somewhat different form to facilitate comparison with the bivariate normal association models can be applied when an underlying bivariate normal is assumed and also under more general conditions and they provide alternatives to tetrachoric and polychoric correlation these models agree more closely with the bivariate normal than does plackett s model 1965 when the row and column classifications arise from underlying univariate normal distributions the general utility of the association models is illustrated here

representability and specht problem for g graded algebras abstract let w be an associative pi algebra over a field f of characteristic zero graded by a finite group g let id g w denote the t ideal of g graded identities of w we prove 1 g graded pi equivalence there exists a field extension k of f and a finite dimensional z 2 z g graded algebra a over k such that id g w id g a where a is the grassmann envelope of a 2 g graded specht problem the t ideal id g w is finitely generated as a t ideal 3 g graded pi equivalence for affine algebras let w be a g graded affine algebra over f then there exists a field extension k of f and a finite dimensional algebra a over k such that id g w id g a

applications of hook young diagrams to p i algebras abstract a the multiplicities m in the cocharacter n a of any p i algebra a are polynomially bounded b a hook containing n a b is obtained from the hooks containing n a and n b these results are obtained by applying a theory of hook young diagrams to p i algebras and they generalize results known for algebras satisfying capelli identities

codimension growth and minimal superalgebras a celebrated theorem of kemer 1978 states that any algebra satisfying a polynomial identity over a field of characteristic zero is pi equivalent to the grassmann envelope g a of a finite dimensional superalgebra a in this paper by exploiting the basic properties of the exponent of a pi algebra proved by giambruno and zaicev 1999 we define and classify the minimal superalgebras of a given exponent over a field of characteristic zero in particular we prove that these algebras can be realized as block triangular matrix algebras over the base field the importance of such algebras is readily proved a is a minimal superalgebra if and only if the ideal of identities of g a is a product of verbally prime t ideals also such superalgebras allow us to classify all minimal varieties of a given exponent i e varieties v such that exp v d 2 and exp u d for all proper subvarieties u of v this proves in the positive a conjecture of drensky 1988 as a corollary we obtain that there is only a finite number of minimal varieties for any given exponent a classification of minimal varieties of finite basic rank was proved by the authors 2003 as an application we give an effective way for computing the exponent of a t ideal given by generators and we discuss the problem of what functions can appear as growth functions of varieties of algebras

monotonic optimization problems and solution approaches problems of maximizing or minimizing monotonic functions of n variables under monotonic constraints are discussed a general framework for monotonic optimization is presented in which a key role is given to a property analogous to the separation property of convex sets the approach is applicable to a wide class of optimization problems including optimization problems dealing with functions representable as differences of increasing functions d i functions

polynomial identities on superalgebras and exponential growth abstract let a be a finitely generated superalgebra over a field f of characteristic 0 to the graded polynomial identities of a one associates a numerical sequence cnsup a n 1 called the sequence of graded codimensions of a in case a satisfies an ordinary polynomial identity such sequence is exponentially bounded and we capture its exponential growth by proving that for any such algebra lim n c n sup a n exists and is a non negative integer we denote such integer by supexp a and we give an effective way for computing it as an application we construct eight superalgebras ai i 1 8 characterizing the identities of any finitely generated superalgebra a with supexp a 2 in the following way supexp a 2 if and only if idsup a idsup ai for some i 1 8 where idsup b is the ideal of graded identities of the algebra b we also compare the superexponent and the exponent see a giambruno m zaicev adv math 140 1998 145 155 of any finitely generated superalgebra

multivariate normal integrals and contingency tables with ordered categories ak dimensional multivariate normal distribution is made discrete by partitioning thek dimensional euclidean space with rectangular grids the collection of probability integrals over the partitioned cubes is ak dimensional contingency table with ordered categories it is shown that loglinear model with main effects plus two way interactions provides an accurate approximation for thek dimensional table the complete multivariate normal integral table is computed via the iterative proportional fitting algorithm from bivariate normal integral tables this approach imposes no restriction on the correlation matrix comparisons with other numerical integration algorithms are reported the approximation suggests association models for discretized multivariate normal distributions and contingency tables with ordered categories

local versus global association for bivariate ordered responses summary local and global parameters modelling association are examined for contingency tables with both margins ordered responses local association parameters are crossratios of 2 x 2 subtables of adjacent cell probabilities global association parameters are cross ratios of quadrant probabilities the interpretation and various properties of these measures of association are compared and illustrated

basis of the identities of the matrix algebra of order two over a field of characteristic p 2 abstract in this paper we prove that the polynomial identities of the matrix algebra of order 2 over an infinite field of characteristic p 2 admit a finite basis we exhibit a finite basis consisting of four identities and in almost all cases for p we describe a minimal basis consisting of two identities the only possibilities for p where we do not exhibit minimal bases of these identities are p 3 and p 5 we show that when p 3 one needs at least three identities and we conjecture a minimal basis in this case in the course of the proof we construct an explicit basis of the vector space of the central commutator polynomials modulo the ideal of the identities of the matrix algebra of order two

exponential growth for codimensions of some p i algebras abstract by the giambruno zaicev theorem for associative p i algebras the exponential rate of growth of the codimensions of such a p i algebra is always a positive integer here we calculate that integer for various generic p i algebras which are given by a single identity these include capelli type identities and the various powers of the standard polynomials

models and methods for the analysis of correlation coefficients abstract the normal distribution plays a very prominant role in statistics and in multivariate analysis the multinormal model is even more dominant than in the classical univariate or bivariate situations in this paper we show that the assumption of continuous multinormal observations can be relaxed in various ways for continuous multivariate analysis most of the existing large sample theory can be applied without assuming multivariate normality we also develop several discrete multinormal models the first one is the classical pearson model which results from discreticizing multinormal variables the second one is inspired by correspondence analysis and the third one by loglinear analysis the models are presented in terms of correlation coefficients and applied to the spearman model of factor analyses using an example from political science

finitely generated algebras with involution and multiplicities bounded by a constant abstract let a be an algebra with involution over a field f of characteristic zero and let n a n 1 2 be the sequence of cocharacters of a for every n 1 let l n a denote the nth colength of a which is the sum of the multiplicities in n a in this article we classify in two different ways the finitely generated algebras satisfying an ordinary polynomial identity whose multiplicities of the cocharacters n a are bounded by a constant as a consequence this also yields a characterization of the varieties whose colength l n a n 1 2 is bounded by a constant

algebras with involution and multiplicities bounded by a constant abstract let a be an algebra with involution over a field of characteristic zero in this paper we characterize in two different ways when the multiplicities of the cocharacter of a are bounded by a constant as a consequence we characterize the algebras with involution of bounded colength

graded polynomial identities and codimensions computing the exponential growth abstract let g be a finite abelian group and a a g graded algebra over a field of characteristic zero this paper is devoted to a quantitative study of the graded polynomial identities satisfied by a we study the asymptotic behavior of c n g a n 1 2 the sequence of graded codimensions of a and we prove that if a satisfies an ordinary polynomial identity lim n c n g a n exists and is an integer we give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple g z 2 graded algebra related to a

ordinal association in contingency tables some interpretive aspects abstract two families of models for ordered contingency tables goodman s association models and canonical correlation models are investigated and compared with respect to the interpretation of their parameters we show that the two families of models actually refer to different kinds of ordinal association stochastic order extremity for correlation models and stochastic order entropy for association models this difference is related to the way the two models scale interaction the scale difference is proven to be of substantial consequence especially under strong association

the design centering problem as a d c programming problem the following problem is studied given a compact sets inrn and a minkowski functionalp x find the largest positive numberr for which there existsx s such that the set of ally rn satisfyingp y x r is contained ins it is shown that whens is the intersection of a closed convex set and several complementary convex sets sets whose complements are open convex this design centering problem can be reformulated as the minimization of some d c function difference of two convex functions overrn in the case where moreover p x xtax 1 2 witha being a symmetric positive definite matrix a solution method is developed which is based on the reduction of the problem to the global minimization of a concave function over a compact convex set

pi algebras with slow codimension growth let cn a n 1 2 be the sequence of codimensions of an algebra a over a field f of characteristic zero we classify the algebras a up to pi equivalence in case this sequence is bounded by a linear function we also show that this property is closely related to the following if ln a n 1 2 denotes the sequence of colengths of a counting the number of sn irreducibles appearing in the nth cocharacter of a then limn ln a exists and is bounded by 2

group graded algebras and almost polynomial growth let f be a field of characteristic 0 g a finite abelian group and a a g graded algebra we prove that a generates a variety of g graded algebras of almost polynomial growth if and only if a has the same graded identities as one of the following algebras 1 fcp the group algebra of a cyclic group of order p where p is a prime number and p g 2 ut2g f the algebra of 2 2 upper triangular matrices over f endowed with an elementary g grading 3 e the infinite dimensional grassmann algebra with trivial g grading 4 in case 2 g ez2 the grassmann algebra with canonical z2 grading

some results concerning the multiplicities of cocharacters of superalgebras with graded involution abstract let a be a finitely generated superalgebra with graded involution over a field f of characteristic zero and let n 1 n 4 a n 1 n 4 n be the n 1 n 4 cocharacter of a in this paper we present some results concerning the multiplicities of cocharacters for this kind of algebras

a characterization of p i algebras with bounded multiplicities of the cocharacters abstract we characterize varieties of p i algebras with bounded multiplicities of the cocharacters a variety is such if and only if it does not contain the upper triangular 2 2 matrices this also yields a characterization of the varieties with bounded colength

log multiplicative association models as item response models log multiplicative association lma models which are special cases of log linear models have interpretations in terms of latent continuous variables two theoretical derivations of lma models based on item response theory irt arguments are presented first we show that anderson and colleagues anderson vermunt 2000 anderson bockenholt 2000 anderson 2002 who derived lma models from statistical graphical models made the equivalent assumptions as holland 1990 when deriving models for the manifest probabilities of response patterns based on an irt approach we also present a second derivation of lma models where item response functions are specified as functions of rest scores these various connections provide insights into the behavior of lma models as item response models and point out philosophical issues with the use of lma models as item response models we show that even for short tests lma and standard irt models yield very similar to nearly identical results when data arise from standard irt models log multiplicative association models can be used as item response models and do not require numerical integration for estimation

graded algebras with polynomial growth of their codimensions abstract let a be an algebra over a field of characteristic 0 and assume a is graded by a finite group g we study combinatorial and asymptotic properties of the g graded polynomial identities of a provided a is of polynomial growth of the sequence of its graded codimensions roughly speaking this means that the ideal of graded identities is very large we relate the polynomial growth of the codimensions to the module structure of the multilinear elements in the relatively free g graded algebra in the variety generated by a we describe the irreducible modules that can appear in the decomposition we show that their multiplicities are eventually constant depending on the shape obtained by the corresponding multipartition after removing its first row we relate moreover the polynomial growth to the colengths finally we describe in detail the algebras whose graded codimensions are of linear growth

association models and canonical correlation in the analysis of cross classifications having ordered categories abstract the association models considered in goodman 1979a for the analysis of cross classifications having ordered categories are presented in a somewhat different form in the present article to facilitate comparison of the results obtained using these models with those obtained using the earlier canonical correlation approach both the association models and the canonical correlation approach can provide meaningful scores for the row and column categories and these scores can be used to partition into relevant components the usual chi squared statistic for testing the null hypothesis of statistical independence between the row classification and column classification however while the usual procedure for testing the statistical significance of chi squared components is invalid when these components are based on the canonical correlations a corresponding procedure is valid when these components are obtained with the association models the components of association obtained with the association mo

polynomial codimension growth of algebras with involutions and superinvolutions abstract let a be an associative algebra over a field f of characteristic zero endowed with a graded involution or a superinvolution and let c n a be its sequence of codimensions in 4 12 it was proved that if a is finite dimensional such sequence is polynomially bounded if and only if a generates a variety not containing a finite number of algebras the group algebra of z 2 and a 4 dimensional subalgebra of the 4 4 upper triangular matrices with suitable graded involutions or superinvolutions in this paper we focus our attention on such algebras since they are the only finite dimensional algebras up to t 2 equivalence generating varieties of almost polynomial growth i e varieties of exponential growth such that any proper subvariety has polynomial growth we classify the subvarieties of such varieties by giving a complete list of generating finite dimensional algebras along the way we classify all minimal varieties of polynomial growth and surprisingly we show that their number is finite for any given growth finally we describe the algebras whose codimensions are bounded by a linear function

a single general method for the analysis of cross classified data reconciliation and synthesis of some methods of pearson yule and fisher and also some methods of correspondence analysis and association analysis 1 2 3 4 5 6 7 8 9 10 contents introduction and summary a general measure of nonindependence for the cells in the table a general method of nonindependence analysis some simple examples some special cases of generalized nonindependence analysis standardized generalized nonindependence analysis overall indices and graphical displays of generalized nonindependence saturated and unsaturated models symmetric nonindependence and generalized quasi symmetry closing comments on 2 x 2 table measures of nonindependence and some brief historical notes

identities of graded algebras and codimension growth let a g g a g be a g graded associative algebra over a field of characteristic zero in this paper we develop a conjecture that relates the exponent of the growth of polynomial identities of the identity component a e to that of the whole of a in the case where the support of the grading is finite we prove the conjecture in several natural cases one of them being the case where a is finite dimensional and a e has polynomial growth

actions of taft s algebras on finite dimensional algebras abstract let f be a field containing a primitive m th root of the unit we characterize the actions of a taft s algebra h m of a certain order m on finite dimensional arbitrary algebras we describe the action in terms of gradings and actions by skew derivations moreover we prove the associative algebra u t 2 of 2 2 upper triangular matrices with entries from f does not generate a variety of h m module algebras of almost polynomial growth

varieties of superalgebras of almost polynomial growth abstract let v gr be a variety of superalgebras and let c n gr v gr n 1 2 be its sequence of graded codimensions such a sequence is polynomially bounded if and only if v gr does not contain a list of five superalgebras consisting of a commutative superalgebra the infinite dimensional grassmann algebra and the algebra of 2 2 upper triangular matrices with trivial and natural z 2 gradings in this paper we completely classify all subvarieties of the varieties generated by these five superalgebras by giving a complete list of finite dimensional generating superalgebras

on minimal varieties of quadratic growth abstract in this paper we investigate the polynomial identities of an important subalgebra of the pi algebra ut 3 of 3 3 upper triangular matrices in characteristic zero moreover we prove that the five algebras which were used in giambruno and la mattina a giambruno d la mattina pi algebras with slow codimension growth j algebra 284 2005 371 391 to classify up to pi equivalence the algebras whose sequence of codimensions is bounded by a linear function generate the only five minimal varieties of quadratic growth

correspondence analysis used complementary to loglinear analysis loglinear analysis and correspondence analysis provide us with two different methods for the decomposition of contingency tables in this paper we will show that there are cases in which these two techniques can be used complementary to each other more specifically we will show that often correspondence analysis can be viewed as providing a decomposition of the difference between two matrices each following a specific loglinear model therefore in these cases the correspondence analysis solution can be interpreted in terms of the difference between these loglinear models a generalization of correspondence analysis recently proposed by escofier will also be discussed with this decomposition which includes classical correspondence analysis as a special case it is possible to use correspondence analysis complementary to loglinear analysis in more instances than those described for classical correspondence analysis in this context correspondence analysis is used for the decomposition of the residuals of specific restricted loglinear models

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

simultaneously modeling joint and marginal distributions of multivariate categorical responses abstract we discuss model fitting methods for analyzing simultaneously the joint and marginal distributions of multivariate categorical responses the models are members of a broad class of generalized logit and loglinear models we fit them by improving a maximum likelihood algorithm that uses lagrange s method of undetermined multipliers and a newton raphson iterative scheme we also discuss goodness of fit tests and adjusted residuals and give asymptotic distributions of model parameter estimators for this class of models inferences are equivalent for poisson and multinomial sampling assumptions simultaneous models for joint and marginal distributions may be useful in a variety of applications including studies dealing with longitudinal data multiple indicators in opinion research cross over designs social mobility and inter rater agreement the models are illustrated for one such application using data from a recent general social survey regarding opinions about various types of government s

on weak equivalences of gradings abstract when one studies the structure e g graded ideals graded subspaces radicals or graded polynomial identities of graded algebras the grading group itself does not play an important role but can be replaced by any other group that realizes the same grading here we come to the notion of weak equivalence of gradings two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component the following question arises naturally when a group grading on a finite dimensional algebra is weakly equivalent to a grading by a finite group it turns out that this question can be reformulated purely group theoretically in terms of the universal group of the grading namely a grading is weakly equivalent to a grading by a finite group if and only if the universal group of the grading is residually finite with respect to a special subset of the grading group the same is true for all the coarsenings of the grading if and only if the universal group of the grading is hereditarily residually finite with respect to the same subset we show that if n 349 then on the full matrix algebra m n f there exists an elementary group grading that is not weakly equivalent to any grading by a finite semi group and if n 3 then any elementary grading on m n f is weakly equivalent to an elementary grading by a finite group

the probability integrals of bivariate normal distributions a contingency table approach summary after partitioning the plane into fine rectangular regions the bivariate normal probability integrals over all the rectangular regions can be considered as a population contingency table the row and column marginals of this contingency table are two discretized univariate normal distributions the logarithm of the local cross product ratios are approximately linear given the correlation coefficient of the bivariate normal a contingency table with discretized normal marginals and proper cross product ratios can be constructed which represents a table of probability integrals for the bivariate normal

a survey of strategies for modeling cross classifications having ordinal variables abstract a survey is given of the main strategies for modeling cross classifications that contain ordinal variables models are described for two way tables in which one or both classifications are ordered and for multidimensional tables in which at least one classification is ordered primary emphasis is given to construction interpretation and implications of loglinear and logit models

using association models to analyse agreement data two examples two examples demonstrate how one can use association models to analyse agreement data the first example concerns intra rater variability in the classification of sputum cytology slides and the second deals with variability associated with the reporting of passive smoking histories the paper emphasizes models in which one estimates category scores from the data that is models that are not in the log linear family of models such models have use in assessment of category distinguishability and provide insights not easily obtained with log linear models

on codimension growth of finitely generated associative algebras abstract letabe a pi algebra over a fieldf we study the asymptotic behavior of the sequence of codimensionscn a ofa we show that ifais finitely generated overftheninv a limn c n a always exists and is an integer we also obtain the following characterization of simple algebras ais finite dimensional central simple overfif and only ifinv a dim a

some models for the analysis of association in multiway cross classifications having ordered categories abstract goodman recently presented a class of models for the analysis of association between two discrete ordinal variables the association was measured in terms of the odds ratios in 2 2 subtables formed from adjacent rows and adjacent columns of the cross classification and models were devised that allowed the odds ratios to depend on an overall effect on row effects on column effects and on other effects this article presents some generalizations of this approach appropriate for multiway cross classifications including a models for the analysis of conditional association b models for the analysis of partial association and c models for the analysis of symmetric association three cross classifications are analyzed with these models and methods and rather simple interpretations of the association in each are provided

gee for multinomial responses using a local odds ratios parameterization in this article we propose a generalized estimating equations gee approach for correlated ordinal or nominal multinomial responses using a local odds ratios parameterization our motivation lies upon observing that i modeling the dependence between correlated multinomial responses via the local odds ratios is meaningful both for ordinal and nominal response scales and ii ordinary gee methods might not ensure the joint existence of the estimates of the marginal regression parameters and of the dependence structure to avoid ii we treat the so called working association vector as a nuisance parameter vector that defines the local odds ratios structure at the marginalized contingency tables after tabulating the responses without a covariate adjustment at each time pair to estimate and simultaneously approximate adequately possible underlying dependence structures we employ the family of association models proposed by goodman in simulations the parameter estimators with the proposed gee method for a marginal cumulative probit model appear to be less biased and more efficient than those with the independence working model especially for studies having time varying covariates and strong correlation

noncommutative invariant theory of symplectic and orthogonal groups abstract we present a method for computing the hilbert series of the algebra of invariants of the complex symplectic and orthogonal groups acting on graded noncommutative algebras with homogeneous components which are polynomial modules of the general linear group we apply our method to compute the hilbert series for different actions of the symplectic and orthogonal groups on the relatively free algebras of the varieties of associative algebras generated respectively by the grassmann algebra and the algebra of 2 2 upper triangular matrices these two varieties are remarkable with the property that they are the only minimal varieties of exponent 2

graded polynomial identities of matrices we consider g graded polynomial identities of the p p matrix algebra mp k over a field k of characteristic 0 graded by an arbitrary group g we find relations between the g graded identities of the g graded algebra mp k and the g h graded identities of the tensor product of mp k and the h graded algebra mq k with a fine h grading we also find a basis of the g graded identities of mp k with an elementary grading such that the identity component coincides with the diagonal of mp k

quasisymmetric models for the analysis of square contingency tables summary a parametric model is presented for the analysis of square contingency tables where there is a one to one correspondence between the categories of the row variable and the categories of the column variable the model is applicable to cross classifications where the categories are unordered partially ordered or completely ordered but the interpretation of the model and associated parameters is most straightforward when the categories are at least partially ordered connections with the mover stayer model and the usual model of quasisymmetry are described two data sets are used to illustrate the utility of the model

hook young diagrams with applications to combinatorics and to representations of lie superalgebras definition et etude des tableaux k l semi standards applications a la combinatoire et aux representations des superalgebres de lie des diagrammes de young a crochets