integral bilinear forms coxeter transformations and coxeter polynomials of finite posets abstract linear algebra technique in the study of linear representations of finite posets is developed in the paper a concept of a quadratic wandering on a class of posets i is introduced and finite posets i are studied by means of the four integral bilinear forms b i b i b i b i z i z i z 1 1 the associated coxeter transformations and the coxeter polynomials in connection with bilinear forms of dynkin diagrams extended dynkin diagrams and irreducible root systems are also studied bilinear equivalences between some of the forms are established and equivalences with the bilinear forms of dynkin diagrams and extended dynkin diagrams are discussed a homological interpretation of the bilinear forms 1 1 is given and z bilinear equivalences between them are discussed by applying well known results of bongartz loupias and zavadskij shkabara we give several characterisations of posets i with the euler form q i x b i x x weakly positive resp with the reduced euler form q i x b i x x weakly positive and posets i with the tits form q i x b i x x weakly positive

reflection group presentations arising from cluster algebras we give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms of companion bases in the associated root system

prinjective modules reflection functors quadratic forms and auslander reiten sequences let a b be artinian rings and let a m b be an a b bimodule which is a finitely generated left a module and a finitely generated right b module a right a m b prinjective module is a finitely genrated module x r x a x b x a a m b x b over the triangular matrix ring r a o a m b b such that x a is a projective a module x b is an injective b module and is a b homomorphism we study the category prin r b a of right a m b prinjective modules it is an additive krull schmidt subcategory of mod r closed under extensions

cluster concealed algebras abstract the cluster tilted algebras have been introduced by buan marsh and reiten they are the endomorphism rings of cluster tilting objects t in cluster categories we call such an algebra cluster concealed in case t is obtained from a preprojective tilting module for example all representation finite cluster tilted algebras are cluster concealed if c is a representation finite cluster tilted algebra then the indecomposable c modules are shown to be determined by their dimension vectors for a general cluster tilted algebra c we are going to describe the dimension vectors of the indecomposable c modules in terms of the root system of a quadratic form the roots may have both positive and negative coordinates and we have to take absolute values

tame wild dichotomy of birkhoff type problems for nilpotent linear operators abstract inspired by recent results of ringel schmidmeier kussin lenzing meltzer xiao wu chen and pu zhang given a field k m 1 and a finite poset i i with a unique maximal element we study the category m o n i f m of mono representations of i over the frobenius k algebra f m k t t m of k dimension m viewed as k vector spaces u with an m nilpotent k linear operator t u u together with t invariant subspaces u i u j u for all i j in i the problem of when the krull schmidt k category m o n i f m is of wild resp tame representation type is called a wild resp tame birkhoff type problem for m nilpotent operators in case when k is algebraically closed we give a complete solution of the problem by describing all minimal pairs i m resp all pairs with m 1 such that category m o n i f m is of wild resp tame representation type we reduce the problem to a birkhoff type problem for the category fspr i f m m o n i f m of subprojective representations over f m of a larger poset i i the tame wild dichotomy for the category m o n i f m is also proved surprisingly in case when i i a b is the union of two incomparable chains i and i of length i a 1 1 and i b 1 1 with i i the problem is equivalent with the wildness resp tameness of the category coh x p of coherent sheaves over the weighted projective line x p for the weight triple p a b m with a b m 2 studied by kussin lenzing and meltzer 19 in relation with the hypersurface singularity f x 1 a x 2 b x 3 m

a new test for multivariate normality and homoscedasticity given data x ij j 1 to n i i 1 to g one may wish to test whether the x ij are normal with common covariance matrix letting x i denote the mean of the ith group and s the pooled covariance matrix the quantities v ij x ij x i s 1 x ij x i are used in the test the exact null distribution of v ij is noted and the anderson darling statistic is proposed for testing whether the data fit this null distribution the new test has acceptable power is flexible and is easily implemented

testing ignorable missingness in estimating equation approaches for longitudinal data summary we address the matter of determining whether or not missing data in longitudinal studies are ignorable with regard to quasilikelihood or estimating equations approaches this involves testing for whether or not the zero mean property of estimating equations holds true chen little 1999 proposed testing for significant differences among parameter estimators calculated from sample subsets with different patterns of missing data whereas we propose a more unified generalised score type test this avoids exhaustive estimation of parameters for each missing data pattern testing instead with a single quadratic score test statistic whether or not there is a common parameter under which the means of all the pattern specific estimating equations are zero comparisons are made for the two approaches with both simulations and real data examples

degenerations in a class of matrix varieties and prinjective modules abstract let k be an algebraically closed field and let i be a finite partially ordered set to any vector v n i we associate the irreducible variety mat v i with an action of the algebraic group g v i see section 6 the main aim of this paper is to give a description of degenerations of orbits of this action in terms of homomorphisms and extensions of prinjective ki modules

a test of missing completely at random for longitudinal data with missing observations liang and zeger proposed a generalized estimating equations approach to the analysis of longitudinal data their models assume that missing observations are missing completely at random in the sense of rubin however when this assumption does not hold their analysis may yield biased results in this paper we develop a simple and practical procedure for testing this assumption the proposed procedure is related to that of park and davis

brief paper some properties of the output error method the output error identification method is studied in various respects the stationary points of the associated loss function are investigated sufficient conditions for a unique local minimum are given the loss functions can be minimized using a quasilinearization algorithm such an algorithm will give good local convergence it is however shown that global convergence does not always occur the output error method is also compared with some other estimation methods from the accuracy point of view it is proved that a prediction error method will give better accuracy an instrumental variable technique may give better or worse accuracy depending on the actual noise correlation

error surfaces of recursive adaptive filters for an adaptive filter with n adjustable coefficients or weights the error surface is a plot in n 1 dimensions of the mean squared error versus the n coefficient values if the adaptive filter is nonrecursive the error surface is a quadratic function of the coefficients with recursive adaptive filters the error surface is not quadratic and may even have local minima in this correspondence we discuss the nature of the recursive error surface and give examples of conditions under which local minima may exist we conclude with a discussion of the effects of the nonquadratic error surface on gradient search algorithms for recursive adaptive filters

a test of missing completely at random for generalised estimating equations with missing data we consider inference from generalised estimating equations when data are incomplete a test for missing completely at random is proposed to help decide whether or not we should adjust estimating equations to correct the possible bias introduced by a missing data mechanism that is not missing completely at random likelihood ratio tests have been introduced to test the missing completely at random hypothesis fuchs 1982 little 1988 for the estimating equation setting following the basic idea of little 1988 we propose a wald type test based on an information decomposition and recombination procedure which also provides an alternative method for estimating parameters one application of the test is to assess the adequacy of the marginal generalised estimating equation for longitudinal data with missing values simulations are done to evaluate its performance

almost split sequences in subcategories let r be a commutative artin ring and let i be an r algebra which is a finitely generated r module in 5 auslander and reiten introduced the notion of an almost split sequence in mod a the category of finitely generated modules over a and the existence and uniqueness of such short exact sequences were established there the theory of almost split sequences developed further and similar results for certain subcategories of mod i were proved by bautista and martinez 8 and roggenkamp 9 the main purpose of this paper is to develop a more general theory for subcategories of mod i having almost split sequences which has the previous examples as special cases notions closely related to that of almost split sequences are those of minimal left and right almost split morphisms these notions were introduced in 61 for mod i and further developed in 7 for subcategories of mod ii this paper is based upon 7 and we therefore use the same notations and conventions as used there we now give some definitions and then proceed to describe the content of the paper section by section let c be a subcategory of mod ii then a morphism g b c in c is said to be a right almost split morphism in c if i g is not a splittable epimorphism and ii whenever there is a nonsplittable epimorphism h c c in c there exists an h c b such that gh h dually a morphismfi a f b in c is said to be left almost split if i f is not a splittable monomorphism and ii whenever there is a nonsplittable monomorphism h a a there exists an h b a such that h f h we say that c has right almost split morphisms if for all indecomposable objects

on transient error surfaces of output error iir adaptive filtering addresses the convergence of iir output error adaptive algorithms we argue that convergence to global or local optimum is simultaneously determined by the shape of the transient error surface and the convergence speed of the algorithm this argument is confirmed by simulation examples comparing the convergence of lms based and ls based algorithms

schurian sp representation finite right peak pi rings and their indecomposable socle projective modules such that soc r is an essential right ideal in r and it is isomorphic to a direct sum of finitely many copies of p called a right peak of r here f f are local rings f f is a division ring imj is an fi fj bimodule and the multiplication in r is given by fi f bimodule maps cii imj im jm satisfying the natural associativity conditions we denote by p p p the right indecomposable row ideals of r

a peak reduction functor for socle projective representations throughout this paper k is a fixed algebraically closed field given a finite poset i e partially ordered set j with the partial order u we denote by max j the set of all maximal elements of j with respect to the order u and we set jys j max j the poset j is said to be an r peak poset provided max j s r throughout this paper we suppose that j s 4 4 1 n p p and max j s p p without loss of generality 1 r 1 r we can assume that the partial order u in j is such that i u j implies 4 i f j in the natural order for all i j g 1 n w x we recall from 18 that the category j spr of peak j spaces or socle projective representations of j over the field k is defined as follows the objects of j spr are systems m s m of finite dimensional k vector j jg j spaces m such that j

a reduced tits quadratic form and tameness of three partite subamalgams of tiled orders let d be a complete discrete valuation domain with the unique maximal ideal p we suppose that d is an algebra over an algebraically closed field k and d p k subamalgam d suborders of a tiled d order are studied in the paper by means of the integral tits quadratic form q zn1 2n3 2 z a criterion for a subamalgam d order to be of tame lattice type is given in terms of the tits quadratic form q and a forbidden list 1 17 of minor d suborders of presented in the tables

tests of homogeneity of means and covariance matrices for multivariate incomplete data existing test statistics for assessing whether incomplete data represent a missing completely at random sample from a single population are based on a normal likelihood rationale and effectively test for homogeneity of means and covariances across missing data patterns the likelihood approach cannot be implemented adequately if a pattern of missing data contains very few subjects a generalized least squares rationale is used to develop parallel tests that are expected to be more stable in small samples three factors were varied for a simulation number of variables percent missing completely at random and sample size one thousand data sets were simulated for each condition the generalized least squares test of homogeneity of means performed close to an ideal type i error rate for most of the conditions the generalized least squares test of homogeneity of covariance matrices and a combined test performed quite well also

a narrow over ring adjustment functor such that xi is finitely generated xl is finitely cogenerated 9 is surjective and the a homomorphism cp xl hom an xg adjoint to rp is injective the category of adjusted right r modules will be denoted by adj r note that if r is an artin algebra then x is in adj r if and only if x is finitely generated sot x is a b module and top x is an a module via the natural epimorphisms a t r b in this case adj i r is one of the classes of modules considered by auslander and smal l and therefore adj has the auslander reiten sequences 1 proposition 6 51 the category adjg r was introduced in 25 26 in order to give a module theoretical interpretation of arbitrary bimodule matrix problems in the sense of drozd and roiter it was shown in 26 that if n kop x k mod k is an additive k functor k is a field k and l are additive krull schmidt k categories and mat jv is the category of matrices then there are a faithfully right b traced ring r and a full dense functor 0 mat wl adj r preserving the representation type and such that 0 vanishes only on a small set of trivial idecomposable matrices see also 14 this allows to show in 14 261 that the category mat n has auslander reiten sequences see also 30 moreover a coxeter map is defined in terms of 0 which provides us a useful computa

a test of missing completely at random for multivariate data with missing values abstract a common concern when faced with multivariate data with missing values is whether the missing data are missing completely at random mcar that is whether missingness depends on the variables in the data set one way of assessing this is to compare the means of recorded values of each variable between groups defined by whether other variables in the data set are missing or not although informative this procedure yields potentially many correlated statistics for testing mcar resulting in multiple comparison problems this article proposes a single global test statistic for mcar that uses all of the available data the asymptotic null distribution is given and the small sample null distribution is derived for multivariate normal data with a monotone pattern of missing data the test reduces to a standard t test when the data are bivariate with missing data confined to a single variable a limited simulation study of empirical sizes for the test applied to normal and nonnormal data suggests th

on lie algebras associated with representation finite algebras abstract let be a representation finite c algebra which has hall polynomials with the universal cover which is a locally bounded directed c algebra in this paper we prove that the z lie algebra l associated with which is defined by riedtmann in 17 and the z lie algebra k associated with which is defined by ringel in 19 are isomorphic as an application we show that if is a representation finite generalized cluster tilted algebra or representation finite trivial extension algebra then k l

quadratic bimodules and quadratic orders in some branches of mathematics especially the representation theory of artin algebras and orders it has been an important problem to study the geometric quotient gln mn m for a module m g m gmg 1 for g gln and m mn m an effective formulation of such problems is a bimodule problem 4 9 which is a study of the matrix category mat c m associated to an additive category c and a c c module m section 2 1 the bimodule problems have nice module theoretical interpretation by means of prinjective modules 22 29 30 they have already many useful applications in the representation theory of finite dimensional algebras artinian rings orders vector space categories and bocses see 6 7 9 16 19 25 28 29 in this paper we introduce the concept of quadratic extensions of rings section 1 2 which are closely related to many rings known in representation theory for example bass orders 10 17 backstrom orders 28 green orders 27 special biserial algebras 31 and clannish algebras 5 see sections 1 3 3 4 and 3 5 using quadratic extensions we introduce the concept of quadratic bimodules section 3 1 where we can easily formulate self reproducing systems 21 and clans 5 in terms of quadratic bimodules then we study their matrix category mat c m associated to a quadratic c c module m especially we give a description of all indecomposable objects in terms of walks sections 3 3 6 1 which can be regarded as a generalization of green walks 13 section 3 4 strings and bands 5 31 section 3 5 as an application we give a description of finitely generated indecomposable modules over quadratic orders section 1 3 2 in section 7 3 in forthcoming papers our results will be applied in construction of large new classes of orders of tame representation type some of the results of this paper were presented at the conference and workshop constanta 2000 and were announced in 19 without proof

on borcherds type lie algebras for each even lattice mathcal l there is a canonical way to construct an infinite dimensional lie algebra via lattice vertex operator algebra theory we call this lie algebra and its subalgebras the borcherds type lie algebras associated to mathcal l in this paper we apply this construction to even lattices arising from representation theory of finite dimensional associative algebras this is motivated by the different realizations of kac moody algebras by borcherds using lattice vertex operators and by peng xiao using ringel hall algebras respectively for any finite dimensional algebra a of finite global dimension we associate a borcherds type lie algebra mathfrak bl a to a in contrast to the ringel hall lie algebra approach mathfrak bl a only depends on the symmetric euler form or tits form but not the full representation theory of a however our results show that for certain classes of finite dimensional algebras whose representation theory is controlled by the euler bilinear forms or tits forms their borcherds type lie algebras do have close relations with the representation theory of these algebras beyond the class of hereditary algebras these algebras include canonical algebras representation directed algebras and incidence algebras of finite prinjective types

the existence of hall polynomials for posets of finite prinjective type abstract we prove the existence of hall polynomials for prinjective representations of finite partially ordered sets of finite prinjective type in section 4 we shortly discuss consequences of the existence of hall polynomials in particular we are able to define a generic ringel hall algebra for prinjective representations of posets of finite prinjective type

time varying performance surfaces for adaptive iir filters geometric properties and implications for filter stability we present a new framework for understanding the performance of adaptive iir filters which enhances the understanding of filter stability during on line operation this new understanding arises from examining geometric properties of time varying performance surfaces which are defined by the data rather than the standard steady state error surfaces as defined by statistics of the data data dependent descent directions used in adaptive algorithms to update filter coefficients typically are viewed as functions of gradients defined on fixed performance surfaces and data is used to approximate these gradients at each iteration in contrast we view data dependent descent directions at each iteration as functions of exact gradients on time varying performance surfaces by examining the shape of these time varying performance surfaces near filter stability boundaries we are able to identify the origin of on line stability problems associated with existing adaptive iir filtering formulations and suggest corrective measures specifically by using exact z domain methods we define time varying performance surfaces which geometrically enforce filter stability and maintain the geometric and physical properties of the true error surface at each iteration development of adaptive algorithms based on this measure is expected to result in adaptive filters having improved stability performance during on line operation

a nonparametric test of missing completely at random for incomplete multivariate data missing data occur in many real world studies knowing the type of missing mechanisms is important for adopting appropriate statistical analysis procedure many statistical methods assume missing completely at random mcar due to its simplicity therefore it is necessary to test whether this assumption is satisfied before applying those procedures in the literature most of the procedures for testing mcar were developed under normality assumption which is sometimes difficult to justify in practice in this paper we propose a nonparametric test of mcar for incomplete multivariate data which does not require distributional assumptions the proposed test is carried out by comparing the distributions of the observed data across different missing pattern groups we prove that the proposed test is consistent against any distributional differences in the observed data simulation shows that the proposed procedure has the type i error well controlled at the nominal level for testing mcar and also has good power against a variety of non mcar alternatives

tame prinjective type and tits form of two peak posets ii throughout this paper we use the terminology and notation introduced w x in 5 7 11 13 in particular k is an algebraically closed field and we denote by i u a finite poset i e partially ordered set with respect to the partial order u we shall write i j if i u j and i j for the sake of simplicity we write i instead of i u we denote by max i the set of all maximal elements of i and i will be called an n peak poset if max i s n let ki be the incidence k algebra of the poset i that is ki is the k subalgebra of the full i i matrix algebra m k consisting of all i w x matrices l in m k such that l s 0 if i uu j in i u we recall from i j i i j w x 13 that the socle soc ki of the right ki module ki is projective and the global dimension of the algebra ki is finite we denote by mod ki the category of finitely generated right ki mod ules and by prin ki the full subcategory of mod ki consisting of all prinjective modules that is modules x in mod ki having a projective resolution of the form

a test of the missing data mechanism for repeated categorical data due to the occurrence of missing observations longitudinal data are rarely balanced and complete weighted least squares analyses described by grizzle starmer and koch 1969 biometrics 25 489 504 have been developed for the analysis of incomplete longitudinal categorical data stanish gillings and koch 1978 biometrics 34 305 317 woolson and clarke 1984 journal of the royal statistical society series a 147 87 99 however all these analyses have assumed that missing observations are missing completely at random in the sense of rubin 1976 biometrika 63 581 592 when the occurrence of missing observations is related to the unobserved response values these analyses may result in biased results in this paper we develop a simple and practical test of the missing mechanism in incomplete repeated categorical data the proposed test is an extension of the test of little 1988 journal of the american statistical association 83 1198 1202 and uses a test criterion given in general form by wald the test is illustrated using data from a longitudinal investigation of obesity in school age children

missing data mechanisms and homogeneity of means and variances covariances unless data are missing completely at random mcar proper methodology is crucial for the analysis of incomplete data consequently methods for effectively testing the mcar mechanism become important and procedures were developed via testing the homogeneity of means and variances covariances across the observed patterns e g kim bentler in psychometrika 67 609 624 2002 little in j am stat assoc 83 1198 1202 1988 the current article shows that the population counterparts of the sample means and covariances of a given pattern of the observed data depend on the underlying structure that generates the data and the normal distribution based maximum likelihood estimates for different patterns of the observed sample can converge to the same values even when data are missing at random or missing not at random although the values may not equal those of the underlying population distribution the results imply that statistics developed for testing the homogeneity of means and covariances cannot be safely used for testing the mcar mechanism even when the population distribution is multivariate normal

tests of homoscedasticity normality and missing completely at random for incomplete multivariate data test of homogeneity of covariances or homoscedasticity among several groups has many applications in statistical analysis in the context of incomplete data analysis tests of homoscedasticity among groups of cases with identical missing data patterns have been proposed to test whether data are missing completely at random mcar these tests of mcar require large sample sizes n and or large group sample sizes n i and they usually fail when applied to non normal data hawkins 1981 proposed a test of multivariate normality and homoscedasticity that is an exact test for complete data when n i are small this paper proposes a modification of this test for complete data to improve its performance and extends its application to test of homoscedasticity and mcar when data are multivariate normal and incomplete moreover it is shown that the statistic used in the hawkins test in conjunction with a nonparametric k sample test can be used to obtain a nonparametric test of homoscedasticity that works well for both normal and non normal data it is explained how a combination of the proposed normal theory hawkins test and the nonparametric test can be employed to test for homoscedasticity mcar and multivariate normality simulation studies show that the newly proposed tests generally outperform their existing competitors in terms of type i error rejection rates also a power study of the proposed tests indicates good power the proposed methods use appropriate missing data imputations to impute missing data methods of multiple imputation are described and one of the methods is employed to confirm the result of our single imputation methods examples are provided where multiple imputation enables one to identify a group or groups whose covariance matrices differ from the majority of other groups

bipartite posets of finite prinjective type one of the main results of this paper is theorem 1 2 which contains a characterization of finite bipartite posetsi i i for which the category prin ki of prinjective modules over the incidencek algebrakiofiis of finite representation type wherekis a field in particular it is shown that for any such poseti the auslander reiten quiver of the category prin ki has no oriented cycle and there is a bijection between the isomorphism classes of indecomposable objects in prin ki and the positive roots of the quadratic formqiassociated withi an existence of a preprojective component in prin ki is proved for faithful posetsiwhich area free

on weakly non negative unit forms and tame algebras in this paper we consider integral quadratic forms q ip z of the form q x 2i x i jqijx xj and call them unit forms such a unit form q is said to be weakly non negative if q x q whenever jt2 0 that is whenever x o for i 1 n our main purpose is to formulate a criterion for weak non negativity of unit forms this question arises in various areas of representation theory in connection with the characterization of tame representation type in the representation theory of quivers 17 6 partially ordered sets 18 19 certain classes of bimodules 8 23 1 3 and certain classes of algebras 20 14 24 16 for each of these structures q there is a unit form qn naturally attached to q such that the weak non negativity of qn is an equivalent or at least a necessary condition for q to be of tame representation type a class of algebras for which tame representation type is equivalent to the weak non negativity of the associated unit form is described in the last section of this paper a general criterion for weak non negativity using determinants was given by zeldich 28 see also 21 another criterion was obtained by de la pena 21 our approach is similar to that in 11 where we gave a characterization of weakly positive unit forms based on the classification of the critical forms let us recall some notions given a unit form q ip tl and a non empty subset ix im of 1 n we denote by qt the quadratic form qd where d z it maps the cth natural basis vector ek onto eik such a unit form q is called the restriction of q to we call a unit form q on ip a hypercritical form if q is weakly non negative for every proper subset 1 n though q itself is not in these terms a unit form q is weakly non negative if and only if q admits no hypercritical restriction so we have to classify the hypercritical forms in 1 we show that they can be obtained by certain elementary transformations from the unit forms naturally attached to the hyperbolic graphs of fig 1 on p 50 in particular there are only finitely many hypercritical forms however their number is large by a simple process we reduce the problem to the classification of the stretched hypercritical forms in 2 we show that except for a few of them these are of the shape q x q x xn xn ixn where q q n d is critical in a similar way the unit forms attached to the hyperbolic graphs am dm em correspond to those