uniqueness of roots up to conjugacy for some affine and finite type artin groups let g be one of the artin groups of finite type b n c n and affine type tilde mathbf a n 1 tilde mathbf c n 1 in this paper we show that if and are elements of g such that k k for some nonzero integer k then and are conjugate in g for the artin group of type a n this was recently proved by gonzalez meneses in fact we prove a stronger theorem from which the above result follows easily by using descriptions of those artin groups as subgroups of the braid group on n 1 strands let p be a subset of 1 n an n braid is said to be p pure if its induced permutation fixes each i in p and p straight if it is p pure and it becomes trivial when we delete all the ith strands for i not in p exploiting the nielsen thurston classification of braids we show that if and are p pure n braids such that k k for some nonzero integer k then there exists a p straight n braid with 1 moreover if 1 in p the conjugating element can be chosen to have the first strand algebraically unlinked with the other strands especially in case of p 1 n our result implies the uniqueness of roots of pure braids which was known by bardakov and by kim and rolfsen

braid pictures for artin groups we define the braid groups of a two dimensional orbifold and introduce conventions for drawing braid pictures we use these to realize the artin groups associated to the spherical coxeter diagrams a n b n c n and d n and the affine diagrams a n b n c n and d n as subgroups of the braid groups of various simple orbifolds the cases d n b n c n and d n are new in each case the artin group is a normal subgroup with abelian quotient in all cases except a n the quotient is finite we also illustrate the value of our braid calculus by giving a picture proof of the basic properties of the garside element of an artin group of type d n

dual braid monoids mikado braids and positivity in hecke algebras we study the rational permutation braids that is the elements of an artin tits group of spherical type which can be written x 1 y where x and y are prefixes of the garside element of the braid monoid we give a geometric characterization of these braids in type a n and b n and then show that in spherical types different from d n the simple elements of the dual braid monoid for arbitrary choice of coxeter element embedded in the braid group are rational permutation braids we conjecture this to hold also in type d n this property implies positivity properties of the polynomials arising in the linear expansion of their images in the iwahori hecke algebra when expressed in the kazhdan lusztig basis in type a n it implies positivity properties of their images in the temperley lieb algebra when expressed in the diagram basis

cohomology of affine artin groups and applications the result of this paper is the determination of the cohomology of artin groups of type an b n and a n with non trivial local coefficients the main result is an explicit computation of the cohomology of the artin group of type b n with coefficients over the module q q 1 t 1 here the first n 1 standard generators of the group act by q multiplication while the last one acts by t multiplication the proof uses some technical results from previous papers plus computations over a suitable spectral sequence the remaining cases follow from an application of shapiro s lemma by considering some well known inclusions we obtain the rational cohomology of the artin group of affine type an as well as the cohomology of the classical braid group br n with coefficients in the n dimensional representation presented in tong yang and ma 1996 the topological counterpart is the explicit construction of finite cw complexes endowed with a free action of the artin groups which are known to be k 1 spaces in some cases including finite type groups particularly simple formulas for the euler characteristic of these orbit spaces are derived

k π 1 and word problems for infinite type artin tits groups and applications to virtual braid groups let be a coxeter graph let w s be its associated coxeter system and let a be its associated artin tits system we regard w as a reflection group acting on a real vector space v let i be the tits cone and let e be the complement in i iv of the reflecting hyperplanes recall that salvetti charney and davis have constructed a simplicial complex having the same homotopy type as e we observe that if t subset s then t naturally embeds into we prove that this embedding admits a retraction pi t omega gamma to omega gamma t and we deduce several topological and combinatorial results on parabolic subgroups of a from a family mathcal s of subsets of s having certain properties we construct a cube complex we show that has the same homotopy type as the universal cover of e and we prove that is cat 0 if and only if mathcal s is a flag complex we say that x subset s is free of infinity if x has no edge labeled by we show that if e gamma x is aspherical and a x has a solution to the word problem for all x subset s free of infinity then e is aspherical and a has a solution to the word problem we apply these results to the virtual braid group vb n in particular we give a solution to the word problem in vb n and we prove that the virtual cohomological dimension of vb n is n 1

on the connections between two classical notions of multidimensional system equivalence in this paper we study the connections between two classical notions of equivalence used in the literature of linear multidimensional nd systems on one hand we have the notion of zero coprime

braid groups and hodge theory this paper gives an account of the unitary representations of the braid group that arise via the hodge theory of cyclic branched coverings of mathbb p 1 highlighting their connections with ergodic theory complex reflection groups moduli spaces of 1 forms and open problems in surface topology

the asymptotic distribution of principal component roots under local alternatives to multiple roots the asymptotic distribution for the principal component roots under local alternatives to multiple population roots is derived the asymptotic theory assumes the estimate of the population covariance or scatter matrix to be asymptotically normal and to possess certain invariance properties these assumptions are satisfied for the affine invariant m estimates of scatter for an elliptical distribution the local alternative framework is used in deriving a local power function for the test for subsphericity

injectivity on the set of conjugacy classes of some monomorphisms between artin groups there are well known monomorphisms between the artin groups of finite type an bn cn and affine type a n 1 c n 1 the artin group a an is isomorphic to the n 1 strand braid group bn 1 and the other three artin groups are isomorphic to some subgroups of bn 1 the inclusions between these subgroups yield monomorphisms a bn a an a a n 1 a bn and a c n 1 a bn there are another type of monomorphisms a bd a amd 1 a bd a bmd and a bd a amd which are induced by isomorphisms between artin groups of type b and centralizers of periodic braids in this paper we show that the monomorphisms a bd a amd 1 a bd a bmd and a bd a amd induce injective functions on the set of conjugacy classes and that none of the monomorphisms a bn a an a a n 1 a bn and a c n 1 a bn does so

a transformation for 2 d linear systems and a generalization of a theorem of rosenbrock the transformation of zero coprime system equivalence z c s e with its various characterizations is shown to have at least one important role for two dimensional linear systems theory this paper shows that it is z c s e which forms the basis of the generalization of rosenbrock s characterization of all least order polynomial realizations of a transfer function matrix for the case of 2 d systems the definition of what consistutes the least order is shown to be crucial

a note on the validity of statistical bootstrapping for estimating the uncertainty of tensor parameters in diffusion tensor images diffusion tensors are estimated from magnetic resonance images mris that are diffusion weighted and those images inherently contain noise therefore noise in the diffusion weighted images produces uncertainty in estimation of the tensors and their derived parameters which include eigenvalues eigenvectors and the trajectories of fiber pathways that are reconstructed from those eigenvalues and eigenvectors although repetition and wild bootstrap methods have been widely used to quantify the uncertainty of diffusion tensors and their derived parameters we currently lack theoretical derivations that would validate the use of these two bootstrap methods for the estimation of statistical parameters of tensors in the presence of noise the aim of this paper is to examine theoretically and numerically the repetition and wild bootstrap methods for approximating uncertainty in estimation of diffusion tensor parameters under two different schemes for acquiring diffusion weighted images whether these bootstrap methods can be used to quantify uncertainty in some diffusion tensor parameters such as fractional anisotropy fa depends critically on the morphology of the diffusion tensor that is being estimated the wild and repetition bootstrap methods in particular cannot quantify uncertainty in the principal direction pd of isotropic or oblate tensor we also examine the use of bootstrap methods in estimating tensors in a voxel containing multiple tensors demonstrating their limitations when quantifying the uncertainty of tensor parameters in those locations simulation studies are also used to understand more thoroughly our theoretical results our findings raise serious concerns about the use of bootstrap methods to quantify the uncertainty of fiber pathways when those pathways pass through voxels that contain either isotropic tensors oblate tensors or multiple tensors

artin groups of euclidean type this article resolves several long standing conjectures about artin groups of euclidean type specifically we prove that every irreducible euclidean artin group is a torsion free centerless group with a decidable word problem and a finite dimensional classifying space we do this by showing that each of these groups is isomorphic to a subgroup of a group with an infinite type garside structure the garside groups involved are introduced here for the first time they are constructed by applying semi standard procedures to crystallographic groups that contain euclidean coxeter groups but which need not be generated by the reflections they contain

pca consistency in high dimension low sample size context principal component analysis pca is an important tool of dimension reduction especially when the dimension or the number of variables is very high asymptotic studies where the sample size is fixed and the dimension grows i e high dimension low sample size hdlss are becoming increasingly relevant we investigate the asymptotic behavior of the principal component pc directions hdlss asymptotics are used to study consistency strong inconsistency and subspace consistency we show that if the first few eigenvalues of a population covariance matrix are large enough compared to the others then the corresponding estimated pc directions are consistent or converge to the appropriate subspace subspace consistency and most other pc directions are strongly inconsistent broad sets of sucient conditions for each of these cases are specified and the main theorem gives a catalogue of possible combinations in preparation for these results we show that the geometric representation of hdlss data holds under general conditions which includes a mixing condition and a broad range of sphericity measures of the covariance matrix

nonparametric estimation of non exchangeable latent variable models we propose a two step method to nonparametrically estimate multivariate models in which the observed outcomes are independent conditional on a discrete latent variable applications include microeconometric models with unobserved types of agents regime switching models and models with misclassification error in the first step we estimate weights that transform moments of the marginal distribution of the data into moments of the conditional distribution of the data for given values of the latent variable in the second step these conditional moments are estimated as weighted sample averages we illustrate the method by estimating a model of wages with unobserved heterogeneity on psid data

on strict system equivalence for multidimensional systems strict system equivalence describes the connection between least order realizations of a transfer function in terms of a general linear constant system rosenbrock h h 1970 state space and multivariable theory london nelson the paper generalizes this and related results to the case of multidimensional systems i e systems of linear partial difference or differential equations with constant coefficients

on wielandt s inequality and its application to the asymptotic distribution of the eigenvalues of a random symmetric matrix a relatively obscure eigenvalue inequality due to wielandt is used to give a simple derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix the asymptotic distributions are obtained under a fairly general setting an application of the general theory to the bootstrap distribution of the eigenvalues of the sample covariance matrix is given 1 introduction and summary the derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix arises in many papers in multivariate analysis although the main idea behind most of the derivations is quite basic i e the expansion of the sample roots about the population roots the derivations themselves are often quite involved these complications are primarily due to the mathematical rather than statistical nature of the eigenvalue problem one of the main objectives of this paper is to introduce a simple method for obtaining the asymptotic distribution of the eigenvalue of random symmetric matrices the method is based upon a relatively obscure eigenvalue inequality

braid groups of imprimitive complex reflection groups abstract we obtain new presentations for the imprimitive complex reflection groups of type d e e r and their braid groups b d e e r for d r 2 diagrams for these presentations are proposed the presentations have much in common with coxeter presentations of real reflection groups they are positive and homogeneous and give rise to quasi garside structures diagram automorphisms correspond to group automorphisms the new presentation shows how the braid group b d e e r is a semidirect product of the braid group of affine type a r 1 and an infinite cyclic group elements of b d e e r are visualised as geometric braids on r 1 strings whose first string is pure and whose winding number is a multiple of e we classify periodic elements and show that the roots are unique up to conjugacy and that the braid group b d e e r is strongly translation discrete