generalized heisenberg algebras and k generalized fibonacci numbers it is shown how some of the recent results of de souza et al 2006 j phys a math gen 39 10415 can be generalized to describe hamiltonians whose eigenvalues are given as k generalized fibonacci numbers here k is an arbitrary integer and the cases considered by de souza et al correspond to k 2

simplicity and superrigidity of twin building lattices kac moody groups over finite fields are finitely generated groups most of them can naturally be viewed as irreducible lattices in products of two closed automorphism groups of non positively curved twinned buildings those are the most important but not the only examples of twin building lattices we prove that these lattices are simple if the corresponding buildings are irreducible and not of affine type i e they are not bruhat tits buildings many of them are finitely presented and enjoy property t our arguments explain geometrically why simplicity fails to hold only for affine kac moody groups moreover we prove that a nontrivial continuous homomorphism from a completed kac moody group is always proper we also show that kac moody lattices fulfill conditions implying strong superrigidity properties for isometric actions on non positively curved metric spaces most results apply to the general class of twin building lattices

an effective lie kolchin theorem for quasi unipotent matrices abstract we establish an effective version of the classical lie kolchin theorem namely let a b gl m c be quasi unipotent matrices such that the jordan canonical form of b consists of a single block and suppose that for all k 0 the matrix a b k is also quasi unipotent then a and b have a common eigenvector in particular a b gl m c is a solvable subgroup we give applications of this result to the representation theory of mapping class groups of orientable surfaces

homological and monodromy representations of framed braid groups in this paper we introduce two new classes of representations of the framed braid groups one is the homological representation constructed as the action of a mapping class group on a certain homology group the other is the monodromy representation of the confluent kz equation which is a generalization of the kz equation to have irregular singularities we also give a conjectural equivalence between these two classes of representations

machine calculations in weyl groups a geometrical description of the minimal automaton recognizing lexicographically minimal reduced expressions for elements of a weyl group is given this complements the construction of a related automaton in a recent preprint of brink and howlett by going on to present a practical way to construct it a transducer is described that will output in reverse the word corresponding to right multiplication by a generato r on such a word in a right to left scan the question is raised whether or not this transducer is finite

rigidity of coxeter groups let w be a coxeter group acting properly discontinuously and cocompactly on manifolds n and m 9m 0 such that the fixed point sets of finite subgroups are contractible let f n n mx dk m xsk1 be a w homotopy equivalence which restricts to a w homeomorphism on the boundary under an assumption on the three dimensional fixed point sets we show that then f is w homotopic to a w homeomorphism

fast block diagonalization of k tridiagonal matrices abstract in the present paper we give a fast algorithm for block diagonalization of k tridiagonalmatrices the block diagonalization provides us with some useful results e g another der ivation of a very recent result on generalized k fibonacci numbers in m e a el mikkawy t sogabe a new family of k fibonacci numbers appl math comput 215 2010 4456 4461 ef cient symbolic algorithm for computing the matrix determinant 2011 elsevier inc all rights reserved 1 introduction and objectiveslet t k n be the k tridiagonal matrix of order n n i e t k n d 1 0 0 a 1 0 00 d 2 0 0 a 2 0 0 00 d n k a n k b k 1 0 00 b k 2 0 0 0 0 d n 1 00 0 b n 0 0 d n 0bbbbbbbbbbbbbbbbbbb 1ccccccccccccccccccca 1 for example t 3 7 is given as follows t 3 7 d 1 00a 1 00 00 d 2 00a 2 0000d 3 00a 3 0b 4 00d 4 00a 4 0 b 5 00d 5 0000b 6 00d 6 0000b 7 00d 7 0bbbbbbbbbbb 1ccccccccccca 2 0096 3003 see front matter 2011 elsevier inc all rights reserved doi 10 1016 j amc 2011 08 014

coxeter systems for which the brink howlett automaton is minimal abstract in this note we classify the coxeter systems for which the brink howlett automaton is minimal we show that this automaton is minimal if and only if each elementary root is supported on a standard spherical subsystem thereby resolving a conjecture of hohlweg nadeau and williams

positive solutions of operator equations on half line abstract this paper is concerned with an operator equation on half line some theorems for the existence of its positive solutions are obtained by using the krasnosel skii guo theorem on cone expansion and compression in a special function space

coxeter groups and hyperbolic manifolds the theory of coxeter groups is used to provide an algebraic construction of finite volume hyperbolic manifolds combinatorial properties of finite images of these groups can be used to compute the volumes of the resulting manifolds three examples in 4 5 and 6 dimensions are given each of very small volume and in one case of smallest possible volume

non linear residually finite groups abstract we give the first example of a non linear residually finite 1 related group a t a t 2 a 2

a homological representation formula of colored alexander invariants we give a formula of the colored alexander invariant in terms of the homological representation of the braid groups which we call truncated lawrence s representation this formula generalizes the famous burau representation formula of the alexander polynomial

parameters for which the lawrence krammer representation is reducible we show that the representation introduced by lawrence and krammer to show the linearity of the braid group is generically irreducible however for some values of its two parameters when these are specialized to complex numbers it becomes reducible we construct a representation of degree n n 1 2 of the bmw algebra of type a n 1 as a representation of the braid group on n strands it is equivalent to the lawrence krammer representation where the two parameters of the bmw algebra are related to those appearing in the lawrence krammer representation we give the values of the parameters for which the representation is reducible and give the proper invariant subspaces in some cases we use this representation to show that for these special values of the parameters the bmw algebra of type a n 1 is not semisimple

automata reduced words and garside shadows in coxeter groups abstract in this article we introduce and investigate a class of finite deterministic automata that all recognize the language of reduced words of a finitely generated coxeter system w s the definition of these automata is straightforward as it only requires the notion of weak order on w s and the related notion of garside shadows in w s an analog of the notion of a garside family then we discuss the relations between this class of automata and the canonical automaton built from brink and howlett s small roots we end this article by providing partial positive answers to two conjectures 1 the automata associated to the smallest garside shadow is minimal 2 the canonical automaton is minimal if and only if the support of all small roots is spherical i e the corresponding root system is finite

homological representations of the hecke algebra in this paper a topological construction of representations of thean 1 series of hecke algebras associated with 2 row young diagrams will be given this construction gives the representations in terms of the monodromy representation obtained from a vector bundle on which there is a natural flat connection the fibres of the vector bundle are homology spaces of configuration spaces of points in c with a suitable twisted local coefficient system it is also shown that there is a close correspondence between this construction and the work of tsuchiya and kanie who constructed hecke algebra representations from the monodromy ofn point functions in a conformal field theory onp1 this work has significance in relation to the one variable jones polynomial which can be expressed in terms of characters of the iwahori hecke algebras associated with 2 row young diagrams it gives rise to a topological description of the jones polynomial which will be discussed elsewhere l2

the burau representation of the braid group b_n is unfaithful for large n two fundamental theorems of classical knot theory by j alexander are that every knot is a closed braid and second that a certain procedure see 1 2 for both results assigns to each knot k a polynomial a k in t which only depends on the topological type of the knot once normalized by a power of t to take a positive value at t 0 with this normalization take t to be a transcendental number since 1926 2 it has been known that the function a knots c is not one to one however it has not become clear whether the composite

flat connections and brauer type algebras in this paper we introduce a brauer type algebra bg associated with every pseudo reflection group and every coxeter group g when g is a coxeter group of simply laced type we show that bg is isomorphic to the generalized brauer algebra of simply laced type introduced by cohen gijsbers and wales 2005 cgw1 we also prove that bg has a cellular structure and be semisimple for generic parameters when g is a dihedral group or the type h3 coxeter group moreover in the process of construction we introduce a further generalization of the lawrence krammer representation to complex braid groups associated with all pseudo reflection groups

a note on words in braid monoids abstract the purpose of this note is to give a self contained apart from simple facts about coxeter groups and we hope a bit shorter and more understandable account of some results of c1 c2 on normal forms of braids which are themselves based on the papers d1 t in particular a motivation was to give a proof of proposition 5 1 that we use in b m some proofs and results from section 2 onwards seem to be new i thank several people for improvements from earlier versions of the manuscript m geck for pointing out some errors f digne for pointing out that some results don t need the braid group to be of finite type and j y hee for suggesting and providing further improvements in that direction

on k fibonacci numbers of arithmetic indexes in this paper we study the sums of k fibonacci numbers with indexes in an arithmetic sequence say an r for fixed integers a and r this enables us to give in a straightforward way several formulas for the sums of such numbers

on twisted lawrence krammer representations abstract lawrence krammer representations are an important family of linear representations of artin tits groups of small type which are known under some assumptions on the parameters to be faithful when the type is spherical or more generally when they are restricted to the artin tits monoid and irreducible when the type is connected here we investigate an analogue of these representations introduced by digne in the spherical cases for every artin tits monoid that appears as the submonoid of fixed points of an artin tits monoid of small type under a group of graph automorphisms and for the corresponding artin tits group under the same assumptions on the parameters as in the small type cases we first show that these so called twisted lawrence krammer representations are faithful and we then prove by computing their formulas when the group of graph automorphisms is of order two or three their irreducibility in all the spherical and connected cases but one

braid groups and the co hopfian property abstract let b n be the braid group on n 4 strands we prove that b n modulo its center is co hopfian we then show that any injective endomorphism of b n is geometric in the sense that it is induced by a homeomorphism of a punctured disk we further prove that any injection from b n to b n 1 is geometric for n 7 additionally we obtain analogous results for mapping class groups of punctured spheres the methods use thurston s theory of surface homeomorphisms and build upon work of ivanov mccarthy

existence theory for nonlinear fredholm and volterra integral equations on half open intervals in this chapter we present existence theory for the nonlinear fredholm integral equation y t h t smallint 0 t k t s g s y s ds 4 1 1 and the nonlinear volterra integral equation y t h t smallint 0 t k t s g s y s ds 4 1 2 when both are defined on the compact interval 0 t naturally we first concern ourselves with existence principles for both equations

on the k generalized fibonacci numbers and high order linear recurrence relations in this paper using elementary symmetric function and complete homogeneous symmetric function we obtain a determinant formula for the k generalized fibonacci sequence the relationship between the k generalized fibonacci sequence and the order k linear homogeneous recurrence relation has been investigated and a general solution for the latter is also derived furthermore we obtain an explicit expression for the elements in the nth power of the companion matrix in terms of k generalized fibonacci numbers

some new biautomatic coxeter groups abstract a coxeter system is called two dimensional if every parabolic subgroup of rank at least 3 is infinite we apply results of niblo and reeves and of b t williams to show that coxeter groups with two dimensional systems are biautomatic if the corresponding presentation contains no visible euclidean triangle groups this provides a partial positive answer to a question posed by niblo and reeves in g niblo l reeves coxeter groups act on cat 0 cube complexes j group theory 6 3 2003 399 413

bmw algebras of simply laced type abstract it is known that the recently discovered representations of the artin groups of type a n the braid groups can be constructed via bmw algebras we introduce similar algebras of type d n and e n which also lead to the newly found faithful representations of the artin groups of the corresponding types we establish finite dimensionality of these algebras moreover they have ideals i 1 and i 2 with i 2 i 1 such that the quotient with respect to i 1 is the hecke algebra and i 1 i 2 is a module for the corresponding artin group generalizing the lawrence krammer representation finally we give conjectures on the structure the dimension and parabolic subalgebras of the bmw algebra as well as on a generalization of deformations to brauer algebras for simply laced spherical type other than a n

the automorphism group of a free group is not linear a linear group has ascending chain condition on centralizers malcev s p 511 finitely generated linear groups are residually finite malcev s p 453 a solvable linear group is nilpotent by abelian byfinite tits 5 pp 14551461 a linear group either contains a free group of rank two or is solvable by locally finite lubotzky 3 has characterized finitely generated linear groups over a field of characteristic zero by purely group theoretic conditions these conditions appear difficult to check although they have been used to show linearity for certain groups however we do not use his theorem the main result of this paper theorem 5 is that for n 2 3 the automorphism group of a free group of rank n is not a linear group the proof uses the representation theory of algebraic groups to show that a kind of diophantine equation between the irreducible representations of

the braid group b4 is linear we study a linear representation b n gl m z q 1 t 1 with m n n 1 2 we will show that for n 4 this representation is faithful we prove a relation with the new charney length function we formulate a conjecture implying that is faithful for all n

configurations braids and homotopy groups the main results of this article are certain connections between braid groups and the homotopy groups of the 2 sphere the connections are given in terms of brunnian braids over the disk and over the 2 sphere the techniques arise from the natural structure of simplicial and structures on fundamental groups of configuration spaces

cells in coxeter groups i abstract the purpose of this article is to shed new light on the combinatorial structure of kazhdan lusztig cells in infinite coxeter groups w our main focus is the set d of distinguished involutions in w which was introduced by lusztig in one of his first papers on cells in affine weyl groups we conjecture that the set d has a simple recursive structure and can be enumerated algorithmically starting from the distinguished involutions of finite coxeter groups moreover to each element of d we assign an explicitly defined set of equivalence relations on w that altogether conjecturally determine the partition of w into left right cells we are able to prove these conjectures only in a special case but even from these partial results we can deduce some interesting corollaries

algorithms and topology of cayley graphs for groups autostackability for finitely generated groups is defined via a topological property of the associated cayley graph which can be encoded in a finite state automaton autostackable groups have solvable word problem and an effective inductive procedure for constructing van kampen diagrams with respect to a canonical finite presentation a comparison with automatic groups is given another characterization of autostackability is given in terms of prefix rewriting systems every group which admits a finite complete rewriting system or an asynchronously automatic structure with respect to a prefix closed set of normal forms is also autostackable as a consequence the fundamental group of every closed 3 manifold with any of the eight possible uniform geometries is autostackable

on the representation of k generalized fibonacci and lucas numbers abstract in this paper we give some determinantal and permanental representations of k generalized fibonacci and lucas numbers we obtain the binet s formula for these sequences by using our representations

rationality and reciprocity for the greedy normal form of a coxeter group we show that the characteristic series for the greedy normal form of a coxeter group is always a rational series and prove a reciprocity formula for this series when the group is right angled and the nerve is eulerian as corollaries we obtain many of the known rationality and reciprocity results for the growth series of coxeter groups as well as some new ones

existence of positive solutions for sturm liouville boundary value problems on the half line in this paper we establish sufficient conditions to guarantee the existence of at least one positive solution a unique positive solution and multiple positive solutions for the sturm liouville boundary value problem on the half line by using an effective operator the fixed point theorems in cone especially krasnoselskii fixed point theorem can be applied to such systems and then existence criteria are established the interesting point of the results is that the nonlinear term f can be sign changing

braid groups are linear in a previous work 11 the author considered a representation of the braid group p bn glm z ql1 t l m n n 1 2 and proved it to be faithful for n 4 bigelow 3 then proved the same representation to be faithful for all n by a beautiful topological argument the present paper gives a different proof of the faithfulness for all n we establish a relation between the charney length in the braid group and exponents of t a certain bn invariant subset of the module is constructed whose properties resemble those of convex cones we relate line segments in this set with the thurston normal form of a braid

asymptotic entropy of random walks on fuchsian buildings and kac moody groups in this article we prove existence of the asymptotic entropy for isotropic random walks on regular fuchsian buildings moreover we give formulae for the asymptotic entropy and prove that it is equal to the rate of escape of the random walk with respect to the green distance when the building arises from a fuchsian kac moody group our results imply results for random walks induced by bi invariant measures on these groups however our results are proven in the general setting without the assumption of any group acting on the building the main idea is to consider the retraction of the isotropic random walk onto an apartment of the building to prove existence of the asymptotic entropy for this retracted walk and to lift this in order to deduce the existence of the entropy for the random walk on the building

the dominance hierarchy in root systems of coxeter groups abstract if x and y are roots in the root system with respect to the standard tits geometric realization of a coxeter group w we say that x dominates y if for all w w wy is a negative root whenever wx is a negative root we call a positive root elementary if it does not dominate any positive root other than itself the set of all elementary roots is denoted by e it has been proved by b brink and r b howlett b brink r b howlett a finiteness property and an automatic structure of coxeter groups math ann 296 1993 179 190 that e is finite if and only if w is a finite rank coxeter group amongst other things this finiteness property enabled brink and howlett to establish the automaticity of all finite rank coxeter groups later brink has also given a complete description of the set e for arbitrary finite rank coxeter groups b brink the set of dominance minimal roots j algebra 206 1998 371 412 however the set of non elementary positive roots has received little attention in the literature in this paper we answer a collection of questions concerning the dominance behavior between such non elementary positive roots in particular we show that for any finite rank coxeter group and for any non negative integer n the set of roots each dominating precisely n other positive roots is finite we give upper and lower bounds for the sizes of all such sets as well as an inductive algorithm for their computation

the lawrence krammer bigelow representations of the braid groups via uq sl2 abstract we construct representations of the braid groups b n on n strands on free z q 1 s 1 modules w n l using generic verma modules for an integral version of u q sl 2 we prove that the w n 2 are isomorphic to the faithful lawrence krammer bigelow representations of b n after appropriate identification of parameters of laurent polynomial rings by constructing explicit integral bases and isomorphism we also prove that the b n representations w n l are irreducible over the fractional field q q s

elements d algebre lineaire tropicale abstract we define a formal framework for the study of algebras of type max plus min plus tropical algebras and more generally algebras over a commutative idempotent semi field this work is motivated by the increasingly diversified use of these algebras which occur also in control theory automata theory as well as in algebraic geometry and in more specific ways in other parts of mathematics such as the theory of monoids in this first article we expecially re examine linear algebra over idempotent semi fields the most delicate but undoubtedly the most interesting point is the notion of a singular point seen as a generalization of the notion of zero we thus rediscover many notions of regularity already introduced for matrices and this allows us to define further notions new in this context such as that of the kernel of a linear form and to apply duality to obtain a good notion of tropical dimension of a submodule

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

embeddings of root systems ii permutation root systems abstract this paper studies a class of embeddings of permutation root systems with properties abstracting those of examples of embeddings of linear root systems over commutative rings as considered in the first paper in this series the root system of one coxeter group w regarded as a permutation representation is realized with its roots as sets of roots of another coxeter group w with action induced by an embedding of w in a completion of w

inversion of k tridiagonal matrices with toeplitz structure in this paper we consider an inverse problem with the k tridiagonal toeplitz matrices a theoretical result is obtained that under certain assumptions the explicit inverse of a k tridiagonal toeplitz matrix can be derived immediately two numerical examples are given to demonstrate the validity of our results

on injective homomorphisms for pure braid groups and associated lie algebras abstract the purpose of this article is to record the center of the lie algebra obtained from the descending central series of artin s pure braid group a lie algebra analyzed in work of kohno t kohno linear representations of braid groups and classical yang baxter equations in contemp math vol 78 1988 pp 339 363 t kohno vassiliev invariants and the de rham complex on the space of knots in symplectic geometry and quantization in contemp math vol 179 amer math soc providence ri 1994 pp 123 138 t kohno serie de poincare koszul associee aux groupes de tresses pures invent math 82 1985 57 75 and falk and randell m falk r randell the lower central series of a fiber type arrangement invent math 82 1985 77 88 the structure of this center gives a lie algebraic criterion for testing whether a homomorphism out of the classical pure braid group is faithful which is analogous to a criterion used to test whether certain morphisms out of free groups are faithful f r cohen j wu on braid groups free groups and the loop space of the 2 sphere in algebraic topology categorical decomposition techniques in progr math vol 215 birkhauser basel 2003 braid groups free groups and the loop space of the 2 sphere math at 0409307 however it is as unclear whether this criterion for faithfulness can be applied to any open cases concerning representations of p n such as the gassner representation

a new approach to generalized fibonacci and lucas numbers with binomial coefficients in this study fibonacci and lucas numbers have been obtained by using generalized fibonacci numbers in addition some new properties of generalized fibonacci numbers with binomial coefficients have been investigated to write generalized fibonacci sequences in a new direct way furthermore it has been given a new formula for some lucas numbers

a new approach to the word and conjugacy problems in the braid groups abstract a new presentation of the n string braid group b n is studied using it a new solution to the word problem in b n is obtained which retains most of the desirable features of the garside thurston solution and at the same time makes possible certain computational improvements we also give a related solution to the conjugacy problem but the improvements in its complexity are not clear at this writing

residual finiteness growths of virtually special groups let g be a virtually special group then the residual finiteness growth of g is at most linear this result cannot be found by embedding g into a special linear group indeed the special linear group mathrm sl k mathbb z for k 2 has residual finiteness growth n k 1

the infimum supremum and geodesic length of a braid conjugacy class abstract algorithmic solutions to the conjugacy problem in the braid groups bn n 2 3 4 were given in earlier work this note concerns the computation of two integer class invariants known as inf and sup a key issue in both algorithms is the number m of times one must cycle resp decycle in order to either increase inf resp decrease sup or to be sure that it is already maximal resp minimal for the class our main result is to prove that m is bounded above by n2 n 2 1 in the situation stated by e a elrifai and h r morton 1994 quart j math oxford45 479 497 and by n 2 in the situation stated by authors 1998 adv math 139 322 353 it follows immediately that the computation of inf and sup is polynomial in both word length and braid index in both algorithms the integers inf and sup determine but are not determined by the shortest geodesic length for elements in a conjugacy class and so we also obtain a polynomial time algorithm for computing this length

on lawrence krammer representations lawrence krammer representations or simply lk representations for short are linear representations of artin tits groups of small type over the ring z x 1 y 1 which are of particular interest since they are known to be faithful when the type is spherical and irreducible when the type is spherical and connected here we define and study lk representations over an arbitrary commutative ring r under some conditions on r and on the parameters of an lk representation which generalize the classical settings we recover the faithfulness property and give a new proof of irreducibility that works for every non necessarily spherical connected type we then study the equivalence of distinct lk representations of a given artin tits group for distinct choices of parameters finally we generalize to our setting the known classification of lk representations in the spherical cases and proceed with their classification in the affine cases

positive solutions of singular boundary value problems on the half line in this paper by using the fixed point theory in the cone with a special norm and space the existence of positive solutions for a class of singular boundary value problems on the half line is established our results improve many known results including singular and non singular cases

a class of generalized tribonacci sequences applied to counting problems abstract generalized tribonacci numbers with the third order linear recurrence with constant coefficients t k n t k n 1 t k n 2 k t k n 3 for n 2 are investigated for some sets of the initial triples t0 t1 t2 in particular generating functions the binet formula and the limit of ratio of consecutive terms t k n 1 t k n are discussed these numbers are related to numbers of path graphs colorings with k 2 colors or equivalently to counting of q ary sequences of length n for q k 2 satisfying requirements which follow the problem of degeneration in the ising model with the second neighbor interactions it is shown that the results obtained can be considered as the base for considerations of cycle graph colorings cyclic q ary sequences these are counting problems so t0 t1 t2 and k should be natural numbers but these sequences can be considered for any real numbers the special cases k 0 1 lead to the fibonacci and the usual tribonacci numbers respectively so the results can be applied to binary and ternary sequences

reading the dual garside length of braids from homological and quantum representations we show that lawrence s representation and linear representations of the braid groups from generic highest weight representations of u q mathfrak sl 2 detect the dual garside length of braids in a simple and natural way that is by expressing a representation as a matrix over a laurent polynomial ring using a certain natural basis the span of the variable is equal to a constant multiple of the dual garside length

algorithms and geometry for graph products of groups recent work of gromov epstein cannon thurston and many others has generated strong interest in the geometric and algorithmic structure of finitely generated infinite groups see 16 17 and 14 many of these structures are preserved by taking graph products the graph product of groups not to be confused with the fundamental group of a graph of groups is a product mixing direct and free products whether the product between two groups in the graph product is free or direct is determined by a simplicial graph given a simplicial graph we say that two vertices are adjacent if they are joined by a single edge

krammer representations for complex braid groups abstract let b be the generalized braid group associated to some finite complex reflection group w we define a representation of b of dimension the number of reflections of the corresponding reflection group which generalizes the krammer representation of the classical braid groups and is thus a good candidate in view of proving the linearity of b we decompose this representation in irreducible components and compute its zariski closure as well as its restriction to parabolic subgroups we prove that it is faithful when w is a coxeter group of type ade and odd dihedral types and conjecture its faithfulness when w has a single class of reflections if true this conjecture would imply various group theoretic properties for these groups that we prove separately to be true for the other groups

on positive solutions of boundary value problems on the half line we study the existence of positive solutions of boundary value problems on the half line for differential equations of second order the krasnoselskii fixed point theorem on cone compression and expansion is used

bipolar coxeter groups we consider the class of those coxeter groups for which removing from the cayley graph any tubular neighbourhood of any wall leaves exactly two connected components we call these coxeter groups bipolar they include the virtually poincare duality coxeter groups the pseudo manifold coxeter groups and the infinite irreducible 2 spherical ones we show in a geometric way that a bipolar coxeter group admits a unique conjugacy class of coxeter generating sets moreover we provide a characterisation of bipolar coxeter groups in terms of the associated coxeter diagram

a note on a fast breakdown free algorithm for computing the determinants and the permanents of k tridiagonal matrices k tridiagonal matrices have attracted much attention in recent years which are a generalization of tridiagonal matrices in this note a breakdown free numerical algorithm of o n is presented for computing the determinants and the permanents of k tridiagonal matrices even though the algorithm is not a symbolic algorithm it never suffers from breakdown furthermore it produces exact values when all entries of the k tridiagonal matrices are given in integer in addition the algorithm can be simplified for a general symmetric toeplitz case and it generates the kth powers of fibonacci pell and jacobsthal numbers for a certain symmetric toeplitz case

on krammer s representation of the braid group a connection is made between the krammer representation and the birman murakami wenzl algebra inspired by a dimension argument a basis is found for a certain irrep of the algebra and relations which generate the matrices are found following a rescaling and change of parameters the matrices are found to be identical to those of the krammer representation the two representations are thus the same proving the irreducibility of one and the faithfulness of the other

positive solutions of a singular boundary value problem for systems of second order differential equations abstract in this paper we establish the conditions for the existence of positive solutions of a singular boundary value problem with two second order differential equations the development is based on a new maximum principle for the operator l 2 u u 2 au a 2 b 2 u under periodic boundary conditions and a fixed point theorem in cones when the eigenvalue lies in certain range the boundary value problem in question has at least one positive solution our results include extend and improve some previous results

symbolic algorithms for the inverses of general k tridiagonal matrices two symbolic algorithms for inverting k tridiagonal matrices have been recently found by el mikkawy and atlan 2014 2015 these two algorithms are mainly based on the doolittle lu factorization of the k tridiagonal matrix in the current paper we present a new explicit analytic expression for the inverses of general tridiagonal matrices at first by using a block diagonalization technique we then relate k tridiagonal matrix inversion to tridiagonal matrix inversion meanwhile an efficient algorithm is derived for computing the inverses of nonsingular k tridiagonal matrices with the help of any algorithm for computing the inverses of tridiagonal matrices three examples are given in order to illustrate the performance and efficiency of the proposed algorithms

on determinants of matrices with general fibonacci numbers entries in this paper we investigate the properties of the determinants of matrices obtained by generalized order k k step fibonacci numbers

a note on generalized k horadam sequence in this paper we define generalized k horadam sequence h k n n n after that we study the properties of the generalized k horadam sequence and prove some of these properties by means of determinant also we obtain a generating function for the generalized k horadam sequence

on a homogeneous recurrence relation for the determinants of general pentadiagonal toeplitz matrices pentadiagonal toeplitz matrices frequently arise in many application areas and have been attracted much attention in recent years in this paper we present a numerical algorithm of o log n for computing the determinants of general pentadiagonal toeplitz matrices without imposing any restrictive conditions in addition we investigate some special pentadiagonal toeplitz determinants and their relations to well known number sequences and give a few identity formulas for the ordinary fibonacci sequences and the generalized k fibonacci sequences

a new family of k fibonacci numbers in the present paper we give a new family of k fibonacci numbers and establish some properties of the relation to the ordinary fibonacci numbers furthermore we describe the recurrence relations and the generating functions of the new family for k 2 and k 3 and presents a few identity formulas for the family and the ordinary fibonacci numbers