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

numeric and mesh algorithms for the coxeter spectral study of positive edge bipartite graphs and their isotropy groups we develop algorithmic techniques for the coxeter spectral analysis of the class ubigr n of connected loop free positive edge bipartite graphs d with n 2 vertices i e signed graphs in particular we present numerical and graphical algorithms allowing us a computer search in the study of such graphs d by means of their gram matrix email protected d the complex spectrum specc email protected c of the coxeter matrix cox d email protected email protected email protected d t r and the geometry of weyl orbits in the set mor d d of matrix morsifications email protected m n z of a simply laced dynkin diagram email protected a n d n e 6 e 7 e 8 associated with d and mesh root systems of type email protected our algorithms construct the coxeter gram polynomials cox d t z t and mesh geometries of root orbits of small connected loop free positive edge bipartite graphs d we apply them to the study of the following coxeter spectral analysis problem does email protected email protected hold i e the email protected email protected d arez congruent for any pair of connected positive loop free edge bipartite graphs d d inubigr nsuch thatspecc d specc d the problem if any square integer matrix email protected m n z is z congruent with its transpose a t r is also discussed we present a solution for graphs in ubigr n with email protected 6

algebras with separating almost cyclic coherent auslander reiten components we prove that the class of artin algebras whose auslander reiten quiver admits a separating family of almost cyclic coherent components coincides with the class of generalized multicoil enlargements of concealed canonical algebras moreover the module category homological properties and the representation type of artin algebras with separating families of almost cyclic coherent auslander reiten components are described

periodicity in bilinear lattices and the coxeter formalism abstract we introduce and study in detail so called circulant coxeter periodic elements and circulant families in a bilinear lattice k as well as their dual versions called anti circulant we show that they form a natural environment for a systematic explanation of certain cyclotomic factors of the coxeter polynomial k of k and in consequence of coxeter polynomials of algebras of finite global dimension we discuss the properties of quadratic forms induced by circulant and anti circulant families moreover we interpret the results in the language of representation theory of algebras and point out applications facts concerning tubular families in auslander reiten quivers and quadratic forms of algebras abstract considerations in bilinear lattices are illustrated with a collection of non trivial examples arising from module and derived categories the results show that techniques of linear algebra and number theory provide efficient tools for explaining various representation theoretic facts

hereditary categories with tilting object let k be an algebraically closed field and h a connected abelian k category which is hereditary that is ext2 vanishes on h assume also that hom x y and ext1 x y are finite dimensional vector spaces over k for all x and y in h we consider such hereditary categories h which in addition have a tilting object t that is an object t such that x ext1 t x 0 fact the factors of finite direct sums of copies of t hereditary categories h with a tilting object t are of special interest in connection with the construction of the class of algebras called quasitilted algbras which was introduced in hrs1 they are by definition the algebras of the form endh t op an important property is that h and endh t op have equivalent bounded derived categories the main examples of such hereditary categories are the category modh of finitely generated modules over a finite dimensional hereditary k algebra h and the category cohx of coherent sheaves on a weighted projective line in the sense of gl1 there are also others derived equivalent to them because of the simple description of the corresponding bounded derived category db h it is possible to give a description of those in the same derived equivalence class see ls h2 it follows from hre that they automatically have a tilting object the quasitilted algebras provide common generalization for the tilted and the canonical finite dimensional algebras note that the tilted algebras are those coming from modh using an arbitrary

the spectrum of a coxeter transformation affine coxeter transformations and the defect map abstract the spectrum of the coxeter transformation r in a reflection group defined by a matrix a without odd cycles is determined in terms of spec a for affine a r determines the defect map which is given a simple formulation in terms of null roots the affine exponents determined by spec r are related to the coefficients of the fundamental co weight corresponding to the fork node of the coxeter dynkin diagram

butler groups of infinite rank ii a torsion free abelian group g is called a butler group if bext g t 0 for any torsion group t we show that every butler group g of cardinality ltl is a b2 group i e g is a union of a smooth ascending chain of pure subgroups g0 where g ga b0 b 0 a butler group of finite rank assuming the validity of the continuum hypothesis ch we show that every butler group of cardinality not exceeding n is a b2 group moreover we are able to prove that the derived functor bext2 a t 0 for any torsion group t and any torsion free a with jai this implies that under ch all balanced subgroups of completely decomposable groups of cardinality nt are b2 groups

categorification of a linear algebra identity and factorization of serre functors we provide a categorical interpretation of a well known identity from linear algebra as an isomorphism of certain functors between triangulated categories arising from finite dimensional algebras as a consequence we deduce that the serre functor of a finite dimensional triangular algebra a has always a lift up to shift to a product of suitably defined reflection functors in the category of perfect complexes over the trivial extension algebra of a

the strong global dimension of piecewise hereditary algebras let a be a finite dimensional piecewise hereditary algebra over an algebraically closed field this text investigates the strong global dimension of a this invariant is characterised in terms of the lengths of sequences of tilting mutations relating a to a hereditary abelian category in terms of the generating hereditary abelian subcategories of the derived category of a and in terms of the auslander reiten structure of that derived category

determining the optimal flexion extension axis of the elbow in vivo a study of interobserver and intraobserver reliability in the current study the interobserver and intraobserver reliability of a recently developed method to obtain the position and orientation vectors of the flexion extension axis of the elbow in vivo is determined the method uses the flock of birdstm six degrees of freedom electromagnetic tracking device ten subjects performed three trials comprising five flexion and extension cycles the movements of the forearm with respect to the upper arm were recorded observer a measured two trials and observer b measured one trial optimal instantaneous helical axes were calculated in a humeral coordinate system for each trial intraclass correlation coefficients and 99 confidence intervals were computed to compare the three measurements zero was in the range of all the narrow confidence intervals which is strong indication for resemblence interobserver intraclass correlation coefficients values for orientation vectors were good to excellent and intraobserver values were fair to good the intraclass correlation coefficients values for position vectors were lower probably due to the lack of variance between subjects it is concluded that the method is reliable and can be used in certain clinical settings

inflation algorithm for loop free non negative edge bipartite graphs of corank at least two we continue the study of finite connected loop free edge bipartite graphs with m 3m 3 vertices a class of signed graphs we started in simson 2013 48 and m g siorek et al 2016 19 by means of the non symmetric gram matrix g mm z of its symmetric gram matrix g 12 g g tr mm 12z and the gram quadratic form q zm zq zm z in particular we study connected non negative edge bipartite graphs with n r 3n r 3 vertices of corank r 2r 2 in the sense that the symmetric gram matrix g mn r z g mn r z of is positive semi definite of rank n 1n 1 the edge bipartite graphs of corank r 2r 2 are studied up to the weak gram zz congruence z z where z z means that g btr g bg btr g b for some b mn r z b mn r z such that det b 1det b 1 our main result of the paper asserts that given a connected edge bipartite graph with n r 3n r 3 vertices of corank r 2r 2 there exists a suitably chosen sequence t of the inflation operators of the form tab such that the composite operator t reduces to a connected bigraph d n r such that zd n r and d n r is one of the canonical r vertex extensions a n r n 1n 1 d n r n 4n 4 e 6 r e 7 r and e 8 r with n rn r vertices of the simply laced dynkin diagrams an dn e6 e7 e8an dn e6 e7 e8 with n 1n 1 vertices the algorithm constructs also a matrix b mn r z b mn r z with det b 1det b 1 defining the weak gram zz congruence zd n r that is satisfying the equation gd n r btr g b

on the periodicity of coxeter transformations and the non negativity of their euler forms abstract we show that for piecewise hereditary algebras the periodicity of the coxeter transformation implies the non negativity of the euler form contrary to previous assumptions the condition of piecewise heredity cannot be omitted even for triangular algebras as demonstrated by incidence algebras of posets we also give a simple direct proof that certain products of reflections defined for any square matrix a with 2 on its main diagonal and in particular the coxeter transformation corresponding to a generalized cartan matrix can be expressed as a 1 a t where a a are closely associated with the upper and lower triangular parts of a

on the structure of loop free non negative edge bipartite graphs abstract we continue the study of a class of signed graphs called finite connected loop free edge bipartite graphs bigraphs for short started in simson 2013 33 and continued in g siorek et al 2016 14 and simson zaj c 2017 38 in this paper we present an algorithmic approach to the study of non negative bigraphs with n r 2 vertices of corank r 1 that is bigraphs with the symmetric gram matrix g 1 2 g g t r m n r 1 2 z positive semi definite of rank n 1 where g m n r z is the non symmetric gram matrix of one of the main results of this paper are theorem 2 1 theorem 2 17 asserting that every such a bigraph with n r 2 vertices can be obtained from a corank zero i e positive connected loop free bigraph with n 1 vertices by an r point extension procedure u 1 u r u 1 u r along the r 1 roots u 1 u r z n of the positive definite gram form q z n z v v g v t r it is also shown that the extension procedure yields a combinatorial algorithm allowing us to obtain inductively all connected loop free non negative corank r 1 bigraphs with n r 2 vertices from the corank zero bigraphs with n 1 vertices

lower extremity control during turns initiated with and without hip external rotation the pirouette turn is often initiated in neutral and externally rotated hip positions by dancers this provides an opportunity to investigate how dancers satisfy the same mechanical objectives at the whole body level when using different leg kinematics the purpose of this study was to compare lower extremity control strategies during the turn initiation phase of pirouettes performed with and without hip external rotation skilled dancers n 5 performed pirouette turns with and without hip external rotation joint kinetics during turn initiation were determined for both legs using ground reaction forces grfs and segment kinematics hip muscle activations were monitored using electromyography using probability based statistical methods variables were compared across turn conditions as a group and within dancer despite differences in grfs and impulse generation between turn conditions at least 90 of each grf was aligned with the respective leg plane a majority of the net joint moments at the ankle knee and hip acted about an axis perpendicular to the leg plane however differences in shank alignment relative to the leg plane affected the distribution of the knee net joint moment when represented with respect to the shank versus the thigh during the initiation of both turns most participants used ankle plantar flexor moments knee extensor moments flexor and abductor moments at the push leg s hip and extensor and abductor moments at the turn leg s hip representation of joint kinetics using multiple reference systems assisted in understanding control priorities

on the coefficients of the coxeter polynomial of an accessible algebra abstract let a be a finite dimensional algebra over an algebraically closed field k assume a is basic connected with n pairwise non isomorphic simple modules we consider the coxeter polynomial a t of a one point extension algebra a b m and the polynomial of the extension p t 1 t 1 t b t a t if m is exceptional then p t 1 p 1 t p n 3 t n 3 t n 2 in that case we call s a b p 1 the linear index of the extension a b m we give conditions for s a b 0 for a tower t k a 1 a 2 a n a of access to a that is a i is a one point co extension of a i 1 by an exceptional module the index s t i 2 n s a i a i 1 n 1 a 2 is an invariant depending on the derived equivalence class of a where a 2 is the quadratic coefficient of a t we show that in the case a is piecewise hereditary then a 2 1 if and only if a is derived equivalent to a quiver algebra of type a n

congruences of a square matrix and its transpose abstract it is known that any square matrix a over any field is congruent to its transpose at stas for some nonsingular s moreover s can be chosen such that s2 i that is s can be chosen to be involutory we show that a and at are congruent over any field f of characteristic not two with involution a a the involution can be the identity a t s t as for some nonsingular s moreover s can be chosen such that s s i that is s can be chosen to be coninvolutory the short and simple proof is based on sergeichuk s canonical form for congruence math ussr izvestiya 31 3 1988 481 it follows that any matrix a over f can be represented as a eb in which e is coninvolutory and b is symmetric

coxeter energy of graphs abstract we study the concept of the coxeter energy of graphs and digraphs quivers as an analogue of gutman s adjacency energy which has applications in theoretical chemistry and is a recently widely investigated graph invariant coxeter energy ce g of a di graph g is defined to be the sum of the absolute values of all complex eigenvalues of the coxeter matrix associated with g our main inspiration for the study comes from the coxeter formalism appearing in group theory lie theory representation theory of algebras mathematical physics and other contexts we focus on the coxeter energy of trees and we prove that the path resp the maximal star has the smallest resp the greatest coxeter energy among all trees resp two large subclasses of trees with fixed number of vertices we provide several other related results as the characterization of trees with second smallest and second greatest coxeter energy bounds for coxeter energy and general facts on coxeter spectra of graphs extending known results e g for salem trees and for certain special real coxeter eigenvalues of trees additionally we discuss few other energy like quantities for the coxeter spectra of di graphs including coxeter energy normalized by the trace of the coxeter matrix and the quantities derived from variants of coulson integral formula

on spectral radii of coxeter transformations let t be a finite tree and dt the maximal branching degree of its vertices let t be the spectral radius of a coxeter transformation defined by t t hen t 4dt 6

wild canonical algebras canonical algebras p depending on a weight sequence p p1 pt of positive integers and a parameter sequence 3 t of pairwise distinct non zero elements from the base eld k were introduced and studied in 23 the nite dimensional representation theory mod is completely controlled by the category coh x of coherent sheaves on a non singular weighted projective line x x p since the derived categories d mod and d coh x are equivalent as triangulated categories 9 the curve x attached to has virtual genus gx 1 2 t 2 p t i 1 p pi where p l c m p1 pt the complexity of the classi cation problem for coh x and hence for mod is essentially determined by gx for gxi1 the algebra is concealed of extended dynkin type accordingly the classi cation problem for coh x and mod is equivalent to the classi cation of indecomposable modules over a tame hereditary algebra cf 18 9 or according to 10 to the classi cation problem for cohen macaulay modules over a simple surface singularity for gx 1 the algebra is of tubular type its representation theory is known from 23 while the classi cation problem for coh x 9 20 relates to atiyah s classi cation 1 of vector bundles over an elliptic curve very little was known on mod or coh x if gx 1 in this case 0 r is the one point extension of a wild hereditary algebra 0 accordingly the category mod 0 embeds into coh x as a full exact and extension closed subcategory implying that the classi cation problems for coh x and mod are wild from now on we assume gx 1 combining the sheaf theory 9 with the spectral investigation of wild hereditary algebras 6 we obtain a fairly complete picture on the structure of auslander reiten components for mod the asymptotic behaviour of the auslander reiten translations and

new least squares solutions for estimating the average centre of rotation and the axis of rotation a new method is proposed for estimating the parameters of ball joints also known as spherical or revolute joints and hinge joints with a fixed axis of rotation the method does not require manual adjustment of any optimisation parameters and produces closed form solutions it is a least squares solution using the whole 3d motion data set we do not assume strict rigidity but only that the markers maintain a constant distance from the centre or axis of rotation this method is compared with other methods that use similar assumptions in the cases of random measurement errors systematic skin movements and skin movements with random measurement noise simulation results indicate that the new method is superior in terms of the algorithm used the closure of the solution consistency and minimal manual parameter adjustment the method can also be adapted to joints with translational movements

symbolic computation of strong gram congruences for cox regular positive edge bipartite graphs with loops abstract the paper can be viewed as a second part of the author s paper simson 2018 43 our main aim is to construct symbolic algorithms for the coxeter spectral classification of a class of signed graphs called edge bipartite graphs with n 2 vertices we started in simson 2013 37 and bocian et al 2014 6 more precisely we construct algorithms for the classification up to the strong gram z congruence z of all finite connected cox regular edge bipartite graphs with at least one loop bigraphs for short that are positive in the sense that the associated symmetric gram matrix g 1 2 g g t r m n 1 2 z is positive definite where g m n z is the non symmetric gram matrix of defined in section 1 and z means that there is a z invertible matrix b m n z such that g b t r g b we recall from 43 that every such a bigraph with a loop is strongly gram z congruent with a bigraph d that is one of the positive cox regular bigraphs b n n 2 c n n 3 f 4 m 4 g 2 presented in section 1 here by applying the geometry of mesh root system technique we construct symbolic algorithms that compute the correspondence d and the set gl n z d of all z invertible matrices b m n z defining the strong gram z congruence z d for any connected positive bigraph with n 2 vertices and at least one loop

invariant theory of canonical algebras abstract based on the first fundamental theorem of classical invariant theory we present a reduction technique for computing relative invariants for quivers with relations this is applied to the invariant theory of canonical algebras and yields an explicit construction of the moduli spaces together with the quotient morphisms from the corresponding representation spaces for families of modules with a fixed dimension vector belonging to the central sincere separating subcategory by means of a tilting process we extend these results to the invariant theory of concealed canonical algebras thus covering the cases of tame hereditary tame concealed and tubular algebras respectively our approach yields in particular a uniform treatment to an essential part of the invariant theory of extended dynkin quivers a topic popular over the years but stretches far beyond since also concealed canonical algebras of tubular or wild representation type are covered

killing and the coxeter transformation of kac moody algebras on etudie le spectre de la transformation de coxeter associee a une algebre de kac moody arbitraire et plus particulierement a une algebre affine

the trace of the coxeter matrix and hochschild cohomology abstract for a basic finite dimensional k algebra of finite global dimension over an algebraically closed field k the coxeter matrix is defined the following formula is shown trm i 1 i dim k h i where tr is the trace and hi is the ith hochschild cohomology space

mckay correspondence for the poincare series of kleinian and fuchsian singularities we give a uniform and to a large extent geometrical proof that the poincare series of the coordinate algebra of a kleinian singularity and of a fuchsian singularity of genus 0 is the quotient of the characteristic polynomials of two coxeter elements these coxeter elements are interpreted geometrically using 2 calabi yau triangulated categories and spherical twist functors

triangle singularities ade chains and weighted projective lines abstract we study the singularity category of the triangle singularity f x a y b z c or s k x y z f attached to the weighted projective line x given by the weight triple a b c by investigating the stable category vect x of vector bundles on x obtained from the category of vector bundles by factoring out all the line bundles we show that vect x is fractional calabi yau whose cy dimension is a function of the euler characteristic of x and establish the existence of a tilting object having the shape of an a 1 b 1 c 1 cuboid the weight types 2 a b yield an explanation of happel seidel symmetry for a class of important nakayama algebras further the weight sequence 2 3 p corresponds to an ade chain the e n chain extrapolating the exceptional dynkin cases e 6 e 7 and e 8 to a whole sequence of triangulated categories

p critical integral quadratic forms and positive unit forms an algorithmic approach abstract we introduce the class of p critical integral unit forms q z n z containing the critical forms in the sense of ovsienko 13 several characterisations of p critical forms are given in particular it is proved that q is p critical if and only if there is a uniquely determined extended dynkin diagram a n d n e 6 e 7 e 8 and a special group isomorphism t z n z n such that q t is the quadratic form q z n z n 0 of the diagram a correspondence between positive forms p z n 1 z with a sincere root and p critical forms q z n z is described and efficient linear algebra algorithms for computing p critical unit forms and positive forms are constructed

root induced integral quadratic forms abstract given an integral quadratic unit form q z n z and a finite tuple of q roots r rj j j the induced q root form qr is considered as in p gabriel a v roiter representations of finite dimensional algebras in a i kostrikin i v shafarevich eds algebra viii encyclopaedia of the mathematical sciences vol 73 1992 springer chapter 6 we show that two non negative unit forms are of the same dynkin type precisely when they are root induced one from the other moreover there are only finitely many non negative unit forms without double edges of a given dynkin type root induction yields an interesting partial order on the dynkin types which is studied in the paper

coxeter spectral classification of almost tp critical one peak posets using symbolic and numeric computations following an idea of spectral graph theory we give several characterizations of finite one peak posets i with almost p critical quadratic tits form q i zi z by applying linear algebra methods combinatorial algorithms an old idea of s a ovsienko and recent results by v m bondarenko and m v stepochkina our study is mainly motivated by an important application of the quadratic tits form of a poset in constructing linear algebra invariants that measure a geometric complexity of nazarova roiter matrix problems over a field k in particular our study is inspired by a well known result of ju a drozd explained in introduction one of our main results asserts that given a finite poset i 1 n t such that is its unique maximal element and q i is almost p critical we have i the form q i is non negative and the subgroup kerq i v zn 1 q i v 0 of zi zn 1 is generated by a vector h h1 hn h with h1 0 hn 0 ii 5 i 9 and the coxeter polynomial coxi t z t coincides with the coxeter polynomial f i t of a uniquely determined euclidean diagram i d 4 e 6 e 7 e 8 iii the z bilinear tits form b i zn 1 zn 1 z of i is z equivalent to the gram z bilinear form b i zn 1 zn 1 z of the euclidean diagram i iv there is a z invertible matrix c such that c itr ctr c i c and c2 e where c i mn 1 z is the tits matrix of i and v i is one of the posets l1 l132 listed in the paper in particular we get in an alternative way the list of p critical posets t in the form t i presented by v m bondarenko and m v stepochkina 6 and we determine their coxeter euclidean types there are 115 one peak posets i l1 l115 with q i p critical and 17 exceptional posets i l116 l132 with q i almost p critical but not p critical our lists are obtained by using computer algebra tools mainly symbolic computation in maple and numeric computation in c

periodic coxeter matrices and their associated quadratic forms we study relations between the periodicity of a coxeter matrix and the positivity or the non negativity of a quadratic form with respect to a cartan matrix of an artinian ring we introduce the notion of the weak periodicity of a coxeter matrix induced by a non negative quadratic form and we characterize the weak periodicity condition and the periodicity condition in terms of eigenvalues of a coxeter matrix

a new method for estimating the axis of rotation and the center of rotation a new method is presented for estimating the parameters of two different models of a joint the two models are 1 a rotational joint with a fixed axis of rotation also referred to as a hinge joint and 2 a ball and socket model corresponding to a spherical joint given the motion of a set of markers it is shown how the parameters can be estimated utilizing the whole data set the parameters are estimated from motion data by minimizing two objective functions the method does not assume a rigid body motion but only that each marker rotates around the same fixed axis of rotation or center of rotation simulation results indicate that in situations where the rigid body assumption is valid and when measurement noise is present the proposed method is inferior to methods that utilize the rigid body assumption however when there are large skin movement artefacts simulation results show the proposed method to be more robust

a gram classification of non negative corank two loop free edge bipartite graphs we continue the coxeter spectral study of finite connected loop free edge bipartite graphs with m 2 3 vertices a class of signed graphs started in simson 2013 49 by means of the non symmetric gram matrix g mm 2 z of its symmetric gram matrix g 12 g g tr mm 2 12z the gram quadratic form q zm 2 z and the coxeter spectrum specc c i e the complex spectrum of the coxeter matrix cox g g tr gl m 2 z in the present paper we study non negative edge bipartite graphs of corank two in the sense that the symmetric gram matrix g mm 2 z of is positive semi definite of rank m 1 one of our aims is to get a complete classification of all connected corank two loop free edge bipartite graphs with m 2 3 vertices up to the weak gram z congruence z where z means that g btr g b for some b mm 2 z such that det b 1 by one vertex extensions of the simply laced euclidean diagrams a m m 1 d m m 4 e 6 e 7 e 8 we construct a family of connected loop free corank two diagrams a m 2 d m 2 e 6 2 e 7 2 e 8 2 called simply extended euclidean diagrams such that they classify all connected corank two loop free edge bipartite graphs with m 2 3 vertices up to the weak gram z congruence z a structure of connected corank two loop free edge bipartite graphs is described it is shown that every such contains a connected positive edge bipartite subgraph that is z congruent with a simply laced dynkin diagram dyn called the dynkin type of such that is a two point extension u w of along two roots u w of the positive definite gram form q zm z this yields a combinatorial algorithm u w u w allowing us to construct all connected corank two loop free edge bipartite graphs with m 2 3 vertices and d dyn from the triples u w where is positive of the dynkin type d and u w are roots of the positive definite gram form q zm z

hereditary noetherian categories of positive euler characteristic we develop the fundamentals of hereditary noetherian categories with serre duality over an arbitrary field k where the category of coherent sheaves over a smooth projective curve over k serves as the prime example and others are coming from the representation theory of finite dimensional algebras the proper way to view such a category is to think of coherent sheaves on a possibly non commutative smooth projective curve we define for each such category notions like function field and euler characteristic determine its auslander reiten components and study stable and semistable bundles for an appropriate notion of degree we provide a complete classification of hereditary noetherian categories for the case of positive euler characteristic by relating these to finite dimensional representations of locally bounded hereditary k algebras whose underlying valued quiver admits a positive additive function

a new method for estimating joint parameters from motion data joint centers and axes of rotation joint parameters are central to all branches of movement analysis in gait analysis the standard protocol used to determine hip and knee joint parameters is prone to errors arising from palpation anthropometric regression equations and misplaced alignment devices several alternative methods have been proposed but to date none have been shown to be accurate and reliable enough for use in the clinical setting this article describes a new method for joint parameter estimation the new method can be summarized as follows i the motions of two adjacent segments spanning a single joint are tracked ii the axis of rotation between every pair of observed segment configurations is computed iii the most likely intersection of all axes effective joint center and most likely orientation of the axes effective joint axis is found initial validation of the method was conducted on a hinged mechanical analog and a single healthy adult subject for the analog the center was found to be within 3 8 mm of the geometric center and 2 0 degrees of the geometric axis standard deviation for the adult subject hip centers varied on the order of 1 3 mm knee centers by 3 9 mm and knee axes by 2 0 degrees the results suggest that the new method is an objective precise and practical alternative to the standard clinical approach

a chebysheff recursion formula for coxeter polynomials abstract a polynomial f t z t is represented by q t z t if f t 2 q t t deg q q t t 1 f t is graphically represented if f t 2 m t for m t the characteristic polynomial of a symmetric matrix m many instances of coxeter polynomials f a t for a a finite dimensional algebra are graphically representable we study the case of extended canonical algebras a see h lenzing j a de la pena extended canonical algebras and fuchsian singularities in press show that the corresponding polynomials f a t are representable and satisfy a chebysheff type recursion formula we get consequences for the eigenvalues of the coxeter transformation of a showing for instance that at most four eigenvalues may lie outside the unit circle

canonical matrices for linear matrix problems abstract we consider a large class of matrix problems which includes the problem of classifying arbitrary systems of linear mappings for every matrix problem from this class we construct belitskii s algorithm for reducing a matrix to a canonical form which is the generalization of the jordan normal form and study the set c mn of indecomposable canonical m n matrices considering c mn as a subset in the affine space of m n matrices we prove that either c mn consists of a finite number of points and straight lines for every m n or c mn contains a 2 dimensional plane for a certain m n

bounds related to coxeter spectral measures of graphs abstract let m be a real square matrix we give upper bounds for the sum of the absolute values of the real part of the eigenvalues of m quantity in some particular cases known as real energy of m from these results we obtain a combinatorial bound for the real energy of the coxeter matrix q of a tree digraph q with n vertices e r e q min n 2 a b c d 2 where a is the number of edges b and c are respectively the number of bifurcation and congregation paths of q as defined below d i 1 n i 1 i 2 with i the degree of a vertex i and where the minimum is taken over all possible orientations of edges in q as particular case we consider dynkin euclidean and star graphs obtaining practical bounds for the real coxeter energy of these classes of digraphs

extended canonical algebras and fuchsian singularities the authors introduce a new class of finite dimensional algebras called extended canonical and determine the shape of their derived categories extended canonical algebras arise from a canonical algebra by onepoint extension or coextension by an indecomposable projective module our main results concern the case of negative euler characteristic of the corresponding weighted projective line mathbb x more specifically we establish for a base field of arbitrary characteristic a link to the fuchsian singularity r of mathbb x which for the base field of complex numbers is isomorphic to an algebra of automorphic forms by means of a recent result of orlov we show that the triangulated category of the graded singularities of r in the sense of buchweitz and orlov admits a tilting object whose endomorphism ring is the corresponding extended canonical algebra of particular interest are those cases where the attached coxeter transformation has spectral radius one a k theoretic analysis then shows that this happens exactly for 38 cases including arnold s 14 exceptional unimodal singularities the paper is related to recent independent work by kajiura saito and takahashi

methods to determine coxeter polynomials abstract a reduction formula for the characteristic polynomials of the coxeter matrices of certain split finite dimensional algebras is proved in the hereditary case this yields explicit expressions for the coxeter polynomials of large families of quivers moreover a combinatorial interpretation of the entries of the coxeter matrices of path algebras gives formulas for coxeter polynomials of some quivers which cannot be treated by the above mentioned reduction process

generalizations of the stacked bases theorem let h be a subgroup of the free abelian group g in order for there to exist a basis xi iei of g for which h e3iei nixi for suitable nonnegative integers ni it is obviously necessary for g h to be a direct sum of cyclic groups in the 1950 s kaplansky raised the question of whether this condition on g h is sufficient for the existence of such a basis j cohen and h gluck demonstrated in 1970 that the answer is yes their result is known as the stacked bases theorem and it extends the classical and well known invariant factor theorem for finitely generated abelian groups in this paper we develop a theory that contains and in fact generalizes in several directions the stacked bases theorem our work includes a complete classification using numerical invariants of the various free resolutions of any abelian group

on the coxeter polynomials of wild stars abstract the spectral radius of a coxeter transformation which plays an important role in the representation theory of hereditary algebras see v dlab c m ringel eigenvalues of coxeter transformations and the gelfand kirillov dimension of the preprojective algebras proc ams 83 1990 228 232 is its important invariant this paper provides both upper and lower bounds for the spectral radii of the coxeter transformations of wild stars i e trees that have a single branching point and are neither of dynkin nor of euclidean type in addition the paper determines limit of the spectral radii of particular infinite sequences of wild stars and shows different classes of graphs with the same limit the basic idea is to reduce the study of spectral radii of trees to the spectral radii of particular valued graphs with indefinite type of associated generalized cartan matrix

perpendicular categories with applications to representations and sheaves this paper is concerned with the omnipresence of the formation of the subcategories right left perpendicular to a subcategory of objects in an abelian category we encounter these subcategories in various contexts the formation of quotient categories with respect to localizing subcategories cf section 2 the deletion of vertices and shrinking of arrows see 37 in the representation theory of finite dimensional algebras cf section 5 the comparison of the representation theories of different extended dynkin quivers cf section 10 the theory of tilting cf sections 4 and 6 the study of homological epimorphisms of rings cf section 4 the passage from graded modules to coherent sheaves on a possibly weighted projective variety or scheme cf section 7 and 21 the study of maximal cohen macaulay modules over surface singularities cf sections the comparison of weighted projective lines for different weight sequences cf section 9 the formation of affine and local algebras attached to path algebras of extended dynkin quivers canonical algebras and weighted projective lines cf section 11 and 21 and the concept of universal localization in 40 formation of the perpendicular category has many aspects in common with localization and allows one to dispose of localization techniques in situations not accessible to any of the classical concepts of localization this applies in particular to applications in the domain of finite dimensional algebras and their representations several applications of the methods presented in this paper are already in existence partly published or appearing in print in the near future see for instance 40 39 4 45 26 49 46 and have show the versatility of the notion of a perpendicular category it seems that right perpendicular categories first appeared as the subcategories of so called closed objects in the process of the formation of the quotient cateogory of an abelian category with respect to a localizing serre subcategory see 18 47 34 another natural occurrence is encountered in commutative algebra forming the possibly infinitely generated modules of depth 2 cf section 7 the concept and some of the central applications were first presented in a talk given by the first author at the honnef meeting in january 1985 we also note that the perfectly matching nomination perpendicular category was coined by a schofield who discovered independently the usefulness of this concept in dealing with hereditary algebras see 39 cf also section 7 the authors further acknowledge the support of the deutsche forschungsgemeinschaft spp darstellungstheorie von endlichen gruppen und endlichdimensionalen algebren throughout this paper rings are associative with unit and modules are unitary right modules mod r respectively mod r denotes the category of all respectively all finitely presented right r modules

validation of a functional method for the estimation of hip joint centre location the present study assesses the accuracy with which the subject specific coordinates of the hip joint centre hjc in a pelvic anatomical frame can be estimated using different methods the functional method was applied by calculating the centre of the best sphere described by the trajectory of markers placed on the thigh during several trials of hip rotations different prediction methods proposed in the literature and in the present investigation which estimate the hjc of adult subjects using regression equations and anthropometric measurements were also assessed the accuracy of each of the above mentioned methods was investigated by comparing their predictions with measurements obtained on a sample of 11 male adult able bodied volunteers using roentgen stereophotogrammetric analysis rsa assumed to provide the true hjc locations prediction methods estimated the hjc location at an average rms distance of 25 30 mm the functional method performed significantly better and estimated hjcs within a rms distance of 13 mm on average this result may be confidently generalised if the photogrammetric experiment is carefully conducted and an optimal analytical approach used the method is therefore suggested for use in motion analysis when the subject s hip range of motion is not limited in addition the facts that it is not an invasive technique and that it has relatively small and un biased errors make it suitable for regression equations identification with no limit to sample size and population typology

salem numbers defined by coxeter transformation abstract a real algebraic integer 1 is called a salem number if all its remaining conjugates have modulus at most 1 with at least one having modulus exactly 1 it is known j a de la pena coxeter transformations and the representation theory of algebras in v dlab et al eds finite dimensional algebras and related topics proceedings of the nato advanced research workshop on representations of algebras and related topics ottawa canada kluwer august 10 18 1992 pp 223 253 j f mckee p rowlinson c j smyth salem numbers and pisot numbers from stars number theory in progress in k gy ry et al eds proc int conf banach int math center diophantine problems and polynomials vol 1 de gruyter berlin 1999 pp 309 319 p lakatos on coxeter polynomials of wild stars linear algebra appl 293 1999 159 170 that the spectral radii of coxeter transformation defined by stars which are neither of dynkin nor of extended dynkin type are salem numbers we prove that the spectral radii of the coxeter transformation of generalized stars are also salem numbers a generalized star is a connected graph without multiple edges and loops that has exactly one vertex of degree at least 3

structure and a coxeter dynkin type classification of corank two non negative posets inspired by famous results of drozd 1974 13 nazarova ro ter 1975 42 and bondarenko stepochkina 2005 8 on the importance of finite posets i with the non negative tits quadratic form q i zi z in the representation theory we study the finite non negative posets i of corank two that is such posets i that the symmetric gram matrix gi 12 ci citr mi q of i is positive semi definite of corank two where ci mi z is the incidence matrix of i a structure of such posets is described in the paper it is shown that every such a poset i can be obtained in a natural way from a corank zero poset j and a pair of roots of j following main ideas of the coxeter spectral analysis of posets we associate with any connected corank two poset i the incidence quadratic form qi zi z the coxeter matrix coxi ci ci tr the coxeter polynomial coxi t det t e coxi z t its complex coxeter spectrum specci the coxeter number ci n the reduced coxeter number c i n the incidence defect homomorphism i zi kerqi zi and a simply laced dynkin diagram dyni in particular we study the question when the coxeter spectrum specci of i together with dyni determines the incidence matrix ci of i hence the poset i uniquely up to a z congruence by applying linear algebra techniques and a combinatorial computer aided analysis of posets we prove that the question has an affirmative answer for connected corank two posets i with at most 15 elements a complete list of such posets is presented and their coxeter type invariants are computed it follows that i 6 there is only one such a poset i of cardinality 6 and 8 such posets of cardinality 7 up to the poset duality i iop the existence of a i mesh translation quiver ri kerqi i on the set ri kerqi is discussed where ri v zi qi v 1 is the set of roots of qi

tubes in derived categories and cyclotomic factors of the coxeter polynomial of an algebra let be a k algebra of finite global dimension we study tubular families in the auslander reiten quiver of the bounded derived category db satisfying certain natural axioms in particular we precisely describe their influence on the cyclotomic factors of the coxeter polynomial of and discuss several numerical limitations for their possible shapes moreover we show that our results provide an alternative relatively simple proof of non trivial classical facts concerning tubular families in module categories and also extend them slightly

one peak posets with positive quadratic tits form their mesh translation quivers of roots and programming in maple and python by applying linear algebra and computer algebra tools we study finite posets with positive quadratic tits form our study is motivated by applications of matrix representations of posets in representation theory where a matrix representation of a partially ordered set t p1 pn with a partial order means a block matrix m m1 m2 mn over a field k of size d d1 dn determined up to all elementary row transformations elementary column transformations within each of the substrips m1 m2 mn and additions of linear combinations of columns of mi to columns of mj if pi pj drozd 10 proves that t has only a finite number of direct sum indecomposable representations if and only if its quadratic tits form q x1 xn x x12 xn2 x 2 pi pjxixj x x1 xn is weakly positive i e q a1 an a 0 for all non zero vectors a1 an a with integral non negative coefficients in this case there exists an indecomposable representation m of size d d1 dn if and only if d1 dn d is a root of q i e q d1 dn d 1 bondarenko and stepochkina 8 give a list of posets t with positive tits form consisting of four infinite series and 108 posets up to duality in the paper we construct this list in an alternative way by applying computational algorithms implemented in maple and python moreover given any poset t of the list we show that a the coxeter polynomial coxi t of the poset i t obtained from t by adding a unique maximal element called a peak is a coxeter polynomial f i t of a simply laced dynkin diagram i an dn e6 e7 e8 and b the set of orbits of the set rq of integral roots of q admits a mesh translation quiver structure rq of a cylinder shape where is the coxeter transformation of i t in the sense of drozd 10

systems of subspaces of a unitary space abstract for a finite poset p p 1 p t we study systems u 1 u t u of subspaces of a unitary space u such that u i u j if p i p j two systems u 1 u t u and v 1 v t v are said to be isometric if there exists an isometry u v such that u i v i we classify such systems up to isometry if p is a semichain we prove that the problem of their classification is unitarily wild if p is not a semichain a classification problem is called unitarily wild if it contains the problem of classifying linear operators on a unitary space which is hopeless in a certain sense

periodic coxeter matrices abstract let a kq i be a finite dimensional triangular k algebra consider the cartan matrix c a and the coxeter matrix a c a t c a let t det t id a be the coxeter polynomial of a we study conditions on spec a in order that a is a periodic matrix we show that in case a is periodic then the euler quadratic form q a x xc a t x t is non negative and q a 0 if and only if 1 spec a

a least squares algorithm for the equiform transformation from spatial marker co ordinates the present paper describes an algorithm for estimating the translation vector and the rotation matrix of a moving body from noisy measurements on the spatial co ordinates of at least three non collinear markers a sensitivity analysis of the estimated parameters and of the helical axis is presented in terms of characteristics of the marker distribution the implementation of the proposed algorithm in a fortran subroutine is appended

moduli spaces for representations of concealed canonical algebras abstract continuing m domokos and h lenzing j algebra 228 2000 738 762 we investigate the relative invariants and moduli spaces introduced in a d king quart j math oxford ser 2 45 1994 515 530 which are related to a separating family of stable tubes in the module category of a concealed canonical algebra the class of concealed canonical algebras consists of representation infinite algebras of tame or wild representation type and contains in particular the representation theory of extended dynkin quivers we present a uniform characteristic free approach avoiding case by case analysis we introduce the notion of admissible weight for such algebras these are the weights i e elements of the dual of the grothendieck group of the module category attached to the infinite moduli spaces for families of modules from the separating subcategory by using an explicit description of the semigroup of admissible weights the corresponding notion of semistability is analyzed and is related to the formation of perpendicular categories we construct relative invariants belonging to admissible weights and prove that the constructed system is complete in the case of canonical algebras it turns out that for any concealed canonical algebra and for any admissible weight all the corresponding moduli spaces are isomorphic to some projective space moreover any such moduli space can be naturally identified with the moduli space belonging to a special weight called the rank as a consequence we show that in the case of a tame concealed algebra any infinite moduli space for families of modules is a projective space and all fields of rational invariants on irreducible components of representation spaces are purely transcendental

the coxeter polynomial for a one point extension algebra let be a finite dimensional algebra over an algebraically closed field k of finite global dimension let m be a finitely generated module and let m be the one point extension algebra we show how to compute the coxeter polynomial for from the coxeter polynomial of and homological invariants of m

bias compensated least squares estimate of the center of rotation abstract it is shown that a recently published least squares method for the estimation of the average center of rotation is biased consequently a correction term is proposed and an iterative algorithm is derived for finding a bias compensated solution to the least squares problem the accuracy of the proposed bias compensated least squares method is compared to the previously proposed least squares method by monte carlo simulations the tests show that the new method gives a substantial improvement in accuracy

on the perpendicular category of a partial tilting module abstract the perpendicular category of a partial tilting module is studied in particular that of a regular module over a hereditary algebra preferably of wild representation type the results are used to establish the existence of a unique preprojective component for a large class of tilted algebras

comparison of anatomical functional and regression methods for estimating the rotation axes of the forearm numerous methods exist to estimate the pose of the axes of rotation of the forearm these include anatomical definitions such as the conventions proposed by the isb and functional methods based on instantaneous helical axes which are commonly accepted as the modelling gold standard for noninvasive in vivo studies we investigated the validity of a third method based on regression equations to estimate the rotation axes of the forearm we also assessed the accuracy of both isb methods axes obtained from a functional method were considered as the reference results indicate a large inter subject variability in the axes positions in accordance with previous studies both isb methods gave the same level of accuracy in axes position estimations regression equations seem to improve estimation of the flexion extension axis but not the pronation supination axis overall given the large inter subject variability the use of regression equations cannot be recommended

additive functions on quivers abstract an integral function on the set of vertices of a graph is additive if twice is value at any vertex v equals the sum of its values at all adjacent vertices counting multiple edges it is well known that among finite connected graphs exactly the extended dynkin graphs admit a positive additive function whereas the dynkin diagrams themselves only allow almost additive functions violating additivity in a single vertex in the present paper we study usually non positive additive or non additive functions on finite quivers and relate the concept of additivity to the radical of the homological euler form our main results concern the existence and construction of such functions for wild quivers our results are most specific in case the underlying graph is a tree possibly with multiple edges

on the mahler measure of the coxeter polynomial of an algebra abstract let a be a finite dimensional algebra over an algebraically closed field k assume a is a basic connected and triangular algebra with n pairwise non isomorphic simple modules we consider the coxeter transformation a t as the automorphism of the grothendieck group k 0 a induced by the auslander reiten translation in the derived category der b mod a of the module category mod a of finite dimensional left a modules we say that a is of cyclotomic type if the characteristic polynomial a of a is a product of cyclotomic polynomials equivalently if the mahler measure m a 1 in 6 we have considered many examples of algebras of cyclotomic type in the representation theory literature in this paper we study the mahler measure of the coxeter polynomial of accessible algebras in 1933 d h lehmer found that the polynomial t 10 t 9 t 7 t 6 t 5 t 4 t 3 t 1 has mahler measure 0 1 176280 and he asked whether there exist any smaller values exceeding 1 in this paper we prove that for any accessible algebra a either m a 1 or m b 0 for some convex subcategory b of a we introduce interlaced tower of algebras a m a n with m n 2 satisfying a s 1 t 1 a s t a s 1 for m 1 s n 1 we prove that if spec a n s 1 r and a n is not of cyclotomic type then m a m m a n we construct several examples

on tensor products of path algebras of type a abstract we derive a formula for the coxeter polynomial of the s fold tensor product i 1 s f a n i 1 of path algebras of linearly oriented quivers of dynkin type a n i 1 in terms of the weights n 1 n s 2 and show that conversely the weights can be recovered from the coxeter polynomial of the tensor product our results have relevance in singularity theory in particular these algebras occur as endomorphism algebras of tilting objects in certain stable categories of vector bundles

the structure and homological properties of generalized standard auslander reiten components abstract we describe the structure and homological properties of arbitrary generalized standard auslander reiten components of artin algebras in particular we prove that for all but finitely many indecomposable modules in such components the euler characteristic is defined and nonnegative further we provide a handy criterion for an infinite auslander reiten component of an artin algebra to be generalized standard we solve also the long standing open problem concerning the structure of artin algebras admitting a separating family of auslander reiten components