the binomial theorem and the extended pascal s triangle abstract with the binomial coefficients k n being defined for all integers n k several forms of the binomial theorem valid for all n are provided this is set up using algebraic means on infinitely long rows of numbers when diverging series result they are evaluated modulo an infinite number this modular arithmetic related to p adic arithmetic thus provides a new way to interpret diverging series

homology and closure properties of autostackable groups abstract autostackability for finitely presented groups is a topological property of the cayley graph combined with formal language theoretic restrictions that implies solvability of the word problem the class of autostackable groups is known to include all asynchronously automatic groups with respect to a prefix closed normal form set and all groups admitting finite complete rewriting systems although groups in the latter two classes all satisfy the homological finiteness condition f p we show that the class of autostackable groups includes a group that is not of type f p 3 we also show that the class of autostackable groups is closed under graph products and extensions

the action of the orthogonal group on planar vectors invariants covariants and syzygies the construction of invariant and covariant polynomials from the x y components of n planar vectors under the so 2 and o 2 orthogonal groups is addressed molien functions determined under the so 2 symmetry group are used as a guide to propose integrity bases for the algebra of invariants and the modules of covariants the molien functions that describe the structure of the algebra of invariants and the free modules of m covariants m n 1 are written as a ratio of a numerator in with positive coefficients over a 1 2 2n 1 denominator this form of single rational function is standard in invariant theory and has a clear symbolic interpretation however its usefulness is lost for the non free modules of m covariants m n due to negative coefficients in the numerator we propose for these non free modules a new representation of the molien function as a sum of n rational functions with positive coefficients in the numerators and different numbers of terms in the denominators this non standard form is symbolically interpreted in terms of a generalized integrity basis integrity bases are explicitly given for n 2 3 4 planar vectors and m ranging from 0 to 5 the integrity bases obtained under the so 2 symmetry group are subsequently extended to the o 2 group

on the knuth bendix completion for concurrent processes abstract we consider critical pairs for replacement systems over free partially commutative monoids this is done in order to apply the knuth bendix completion procedure to concurrent processes we will see that there are one rule systems which have no finite set of critical pairs therefore we develop a sufficient and computable condition such that finite trace replacement systems have a finite set of critical pairs this condition is always satisfied in the purely free case of semi thue systems or purely commutative case of vector replacement systems in fact it generalizes and unifies both cases we will give examples of how one can use this generalization

automorphisms of graph groups abstract given a graph v e the graph group on is the group generated by the vertex set v with the defining relations being that two adjacent vertices commute in the following the centralizer problem for graph groups is solved and it is shown that for some graph groups aut f is finitely generated by generators which are natural analogues to the nielsen automorphisms for a free group

the geometry of purely loxodromic subgroups of right angled artin groups we prove that finitely generated purely loxodromic subgroups of a right angled artin group a fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups mod s in particular such subgroups are quasiconvex in a in addition we identify a milder condition for a finitely generated subgroup of a that guarantees it is free undistorted and retains finite generation when intersected with a for subgraphs of these results have applications to both the study of convex cocompactness in mod s and the way in which certain groups can embed in right angled artin groups

nielsen realisation for untwisted automorphisms of right angled artin groups we prove nielsen realisation for finite subgroups of the groups of untwisted outer automorphisms of raags in the following sense given any graph gamma and any finite group g u0 a gamma out 0 a gamma we find a non positively curved cube complex with fundamental group a gamma on which g acts by isometries realising the action on a gamma

acylindrical hyperbolicity of groups acting on trees we provide new examples of acylindrically hyperbolic groups arising from actions on simplicial trees in particular we consider amalgamated products and hnn extensions one relator groups automorphism groups of polynomial algebras 3 manifold groups and graph products acylindrical hyperbolicity is then used to obtain some results about the algebraic structure analytic properties and measure equivalence rigidity of groups from these classes

model reference adaptive control for plants with unknown relative degree a parameterization scheme for model reference control which guarantees stable model plant transfer function matching is presented this scheme has a linear dynamic feedforward block whose numerator and denominator should be stable polynomials for closed loop stability an adaptive law is designed to update the controller parameters in the feedback loop and the coefficients of the numerator of the feedforward block as well as some additional parameters since for stability the inverse of the polynomial differential operator defined by some of the estimated parameters has to be stable the adaptive law is modified using a projection technique the stability analysis of the model reference adaptive control mrac scheme is given the robustness of the mrac scheme with respect to unmodeled dynamics is established

product systems over right angled artin semigroups we build upon mac lane s definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid g we show that the existing notions of product systems fit into our categorical framework as do the k graphs of kumjian and pask we then specialize to product systems over right angled artin semigroups these are semigroups that interpolate between free semigroups and free abelian semigroups for such a semigroup we characterize all product systems which take values in a given tensor groupoid g in particular we obtain necessary and sufficient conditions under which a collection of k 1 graphs form the coordinate graphs of a k graph

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

pascal s triangle on a torus and on a binary tree abstract the extension of pascal s triangle by defining binomial coefficients rn for all integers n r has an interesting interpretation in terms of mod p arithmetic for p an infinite prime integer the different uses and interpretations of the classical binomial coefficients can be extended in varying degrees to the new ones two useful tools in this regard are a torus and a binary tree

morse theory and finiteness properties of groups we examine the finiteness properties of certain subgroups of right angled artin groups in particular we find an example of a group that is of type fp z but is not finitely presented

convex cocompactness in mapping class groups via quasiconvexity in right angled artin groups we characterize convex cocompact subgroups of mapping class groups that arise as subgroups of specially embedded right angled artin groups that is if the right angled artin group g mod s satisfies certain conditions that imply g is quasi isometrically embedded in mod s then a purely pseudo anosov h g is convex cocompact in mod s if and only if it is combinatorially quasiconvex in g we use this criterion to construct convex cocompact subgroups of mod s whose orbit maps into the curve complex have small lipschitz constants

applications of hysteresis switching in parameter adaptive control the hysteresis switching algorithm of r h middleton et al ibid vol 33 no 1 p 50 8 jan 1988 is reexamined in a broader context to demonstrate its utility the algorithm is applied to various families of identifier based parameterized controllers of both the direct and indirect control types application to the direct control type results in a model reference adaptive controller capable of stabilizing without excitation any siso process which can be modeled by a minimum phase linear system whose transfer function has relative and mcmillan degrees not exceeding prescribing integers m and n respectively it is shown that such processes can also be adaptively stabilized with indirect adaptive controllers and hysteresis switching a simple numerical example involving a non minimum phase process model is used to illustrate how hysteresis switching might be applied to implicitly tuned parameterized controllers to realize an adaptive controller with capabilities which might prove very difficult if not impossible to achieve without hysteresis switching or some other form of discontinuous control

separating quasiconvex subgroups of right angled artin groups a graph group or right angled artin group is a group given by a presentation where the only relators are commutators of the generators a graph group presentation corresponds in a natural way to a simplicial graph with each generator corresponding to a vertex and each commutator relator corresponding to an edge suppose that g is a graph group whose corresponding graph is a tree and h is a subgroup of g we show that if h is quasiconvex with respect to either the word metric on g or the cat 0 metric on the universal cover of the standard complex for g then h is separable that is h is the intersection of finite index subgroups of g we also discuss some consequences relating to certain 3 manifold groups

centralizers in graph products of groups abstract a graph product of groups is a construction which generalizes both free products and direct products of groups in this paper we study commuting elements in graph products of groups and find explicit formulas for the centralizer the subgroup consisting of elements which commute with a given element of elements of a graph product of groups in terms of the centralizers of elements in each vertex group and the structure of the graph

the mixed two qubit system and the structure of its ring of local invariants the local invariants of a mixed two qubit system are discussed these invariants are polynomials in the elements of the corresponding density matrix they are counted by means of group theoretic branching rules which relate this problem to one arising in spin isospin nuclear shell models the corresponding molien series and a refinement in the form of a four parameter generating function are determined a graphical approach is then used to construct explicitly a fundamental set of 21 invariants relations between them are found in the form of syzygies by using these the structure of the ring of local invariants is determined and complete sets of primary and secondary invariants are identified there are 10 of the former and 15 of the latter

artin groups of large type are shortlex automatic with regular geodesics we prove that any artin group of large type is shortlex automatic with respect to its standard generating set and that the set of all geodesic words over the same generating set satisfies the falsification by fellow traveller property fftp and hence is regular

extending the binomial coefficients to preserve symmetry and pattern abstract we show how to extend the domain of thee binomial coefficients rn so that n and r may take any integer value we argue from two directions on the one hand we wish to preserve symmetry and pattern within pascal s triangle thus creating pascal s hexagon and on the other hand we wish the binomial coefficients to preserve their algebraic role in terms of the taylor series and laurent series expansions of 1 x n valid when x 1 respectively a geometric configuration within the pascal triangle called the pascal flower has some extraordinary properties these properties persist into the hexagon moreover the binomial coefficients may by the use of the function even be extended to all real or complex values of n and r with the conservation of their principal properties

the σ2 conjecture for metabelian groups the general case abstract the bieri neumann strebel invariant m g of a group g is a certain subset of a sphere that contains information about finiteness properties of subgroups of g in case of a metabelian group g the set 1 g completely characterizes finite presentability and it is conjectured that it also contains complete information about the higher finiteness properties fpm conjecture the m conjecture states how the higher invariants are obtained from 1 g in this paper we prove the 2 conjecture

conjugacy growth series and languages in groups in this paper we introduce the geodesic conjugacy language and geodesic conjugacy growth series for a finitely generated group we study the effects of various group constructions on rationality of both the geodesic conjugacy growth series and spherical conjugacy growth series as well as on regularity of the geodesic conjugacy language and spherical conjugacy language in particular we show that regularity of the geodesic conjugacy language is preserved by the graph product construction and rationality of the geodesic conjugacy growth series is preserved by both direct and free products 2010 mathematics subject classification 20f65 20e45

extending pascal s triangle an alternative is given to hilton and pedersen s method of defining binomial coefficients rn for any integers n r it involves using the usual factorial formula for rn via the defining of the factorial ratio n r for any integers n r in the process defining 00 1 turns out to have several useful applications an attempt to extend the binomial theorem to negative exponents suggests viewing the extended pascal s triangle as binomial coefficients modulo an infinite number

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

cohomological finiteness conditions for elementary amenable groups it is proved that every elementary amenable group of type fp admits a cocompact classifying space for proper actions

entropy determinants and l 2 torsion we show that for any amenable group and any z module m of type fl with vanishing euler characteristic the entropy of the natural action on the pontryagin dual of m is equal to the l 2 torsion of m as a particular case the entropy of the principal algebraic action associated with the module z z f is equal to the logarithm of the fuglede kadison determinant of f whenever f is a non zero divisor in z this conrms a conjecture of deninger as a key step in the proof we provide a general szeg o type approximation theorem for the fuglede kadison determinant on the group von neumann algebra of an amenable group as a consequence of the equality between l 2 torsion and entropy we show that the l 2 torsion of a non trivial amenable group with nite classifying space vanishes this was conjectured by luck finally we establish a milnor turaev formula for the l 2 torsion of a nite acyclic chain complex

subgroups of graph groups given a graph x we define a group gx as follows gx is generated by the vertices of x with a defining relation xy jt for each pair y 2 of ver tices joined by an edge of x a group is called a graph group if it is isomorphic to gx for some graph x we will not distinguish by notation in what follows between a vertex of x and the corresponding element of gx we remark that the extreme cases of graphs namely the complete and the completely disconnected graphs correspond respectively to free abelian and free groups it is well known that any subgroup of a group of either of these types is again of the same type and it is thus natural to ask to what extent and in what form this is true of the graph groups that lie in between these extreme cases in this article we will prove the following

the action of the special orthogonal group on planar vectors integrity bases via a generalization of the symbolic interpretation of molien functions the present article completes the mathematical description initiated in the paper by dhont and zhilinskii 2013 the action of the orthogonal group on planar vectors invariants covariants and syzygies j phys a math theor 46 455202 of the algebraic structures that emerge from the symmetry adapted polynomials in the coordinates of n planar vectors under the action of the so 2 group the set of covariant polynomials contains all the polynomials that transform according to the weight of so 2 and is a free module for but a non free module for the sum of the rational functions of the molien function for covariants describes the decomposition of the ring of invariants or the module of covariants as a direct sum of submodules a method for extracting the generating function for covariants from the comprehensive generating function for all polynomials is introduced the approach allows the direct construction of the integrity basis for the module of covariants decomposed as a direct sum of submodules and gives insight into the expressions for the molien functions found in our earlier paper in particular a generalized symbolic interpretation in terms of the integrity basis of a rational function is discussed where the requirement of associating the different terms in the numerator of one rational function with the same subring of invariants is relaxed

on the geometry of normal forms in discrete groups we study normal forms in finitely generated groups from the geometric viewpoint of combings we introduce notions of combability considerably weaker than those commonly in use we prove that groups which satisfy these conditions are finitely presented and satisfy isoperimetric and isodiametric inequalities of a controlled nature

metric systolicity and two dimensional artin groups we introduce the notion of metrically systolic simplicial complexes we study geometric and large scale properties of such complexes and of groups acting on them geometrically we show that all two dimensional artin groups act geometrically on metrically systolic complexes as direct corollaries we obtain new results on two dimensional artin groups and all their finitely presented subgroups we prove that the conjugacy problem is solvable and that the dehn function is quadratic we also show several large scale features of finitely presented subgroups of two dimensional artin groups lying background for further studies concerning their quasi isometric rigidity

graph groups coherence and three manifolds 1 introou t n we study groups given by presentations each of whose defining relations is of the form ql yx for some generators y and y to such a presentation we associate a graph x whose vertices are the generators two vertices y and 1 being adjacent in x if and only if xy yx is a defining relation given a graph x we denote by gx the group defined by the presentation associated to x in this way we call gx a graph group these groups have been studied by kim and roush s and by dicks 3 in this paper we prove the following

tits alternatives for graph products we discuss various types of tits alternative for subgroups of graph products of groups and prove that under some natural conditions a graph product of groups satisfies a given form of tits alternative if and only if each vertex group satisfies this alternative as a corollary we show that every finitely generated subgroup of a graph product of virtually solvable groups is either virtually solvable or large as another corollary we prove that every non abelian subgroup of a right angled artin group has an epimorphism onto the free group of rank 2 in the course of the paper we develop the theory of parabolic subgroups which allows to describe the structure of subgroups of graph products that contain no non abelian free subgroups we also obtain a number of results regarding the stability of some group properties under taking graph products

homological and homotopical dehn functions are different the homological and homotopical dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space the homological dehn function measures fillings of cycles by chains whereas the homotopical dehn function measures fillings of curves by disks because the two definitions involve different sorts of boundaries and fillings there is no a priori relationship between the two functions however before this work there were no known examples of finitely presented groups for which the two functions differ this paper gives such examples constructed by amalgamating a free by cyclic group with several bestvina brady groups

improving the robustness of nussbaum type regulators by the use of σ modification local results abstract the bursting phenomenon in a nussbaum type regulator is discussed its cause is associated with some properties of the equilibrium manifold a modification factor is introduced and the resulting equilibrium set is completely characterized using linearization techniques local robustness against bounded disturbances is proved by malkin s theorem and verified by simulation

algorithms detecting stability and morseness for finitely generated groups abstract the notions of stable and morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word hyperbolic group for a word hyperbolic group g kapovich 31 provided a partial algorithm which on input a finite set s of g halts if s generates a quasiconvex subgroup of g and runs forever otherwise in this paper we give various detection and decidability algorithms for stability and morseness of a finitely generated subgroup of mapping class groups right angled artin groups toral relatively hyperbolic groups and finitely generated groups discriminated by a locally quasiconvex torsion free hyperbolic group for example ordinary limit groups

generalising some results about right angled artin groups to graph products of groups we prove three results about the graph product g g gv v v of groups gv over a graph the first result generalises a result of servatius droms and servatius proved by them for right angled artin groups we prove a necessary and sufficient condition on a finite graph for the kernel of the map from g to the associated direct product to be free one part of this result already follows from a result in s kim s phd thesis the second result generalises a result of hermiller and sunic again from right angled artin groups we prove that for a graph with finite chromatic number g has a series in which every factor is a free product of vertex groups the third result provides an alternative proof of a theorem due to meier which provides necessary and sufficient conditions on a finite graph for g to be hyperbolic

a 4 n 1 dimensional model reference adaptive stabilizer for any relative degree one or two minimum phase system of dimension n or less a 4 n 1 dimensional model reference controller is developed which can adaptively stabilize any single input single output process while forcing the process output to follow the output of a prespecified reference model provided the process can be modelled by a finite dimensional strictly proper stabilizable minimum phase linear system with relative degree not exceeding two and dimension not exceeding n