the concept of duality for automata over a changing alphabet and generation of a free group by such automata in the paper we deal with the notion of an automaton over a changing alphabet which generalizes the concept of a mealy type automaton we modify the methods based on the idea of a dual automaton and its action used by b steinberg et al 2011 and m vorobets and ya vorobets 2007 2010 16 18 and adapt them to automata over a changing alphabet we show that this modification provides some naturally defined automaton representations of a free nonabelian group by a 2 state automaton over a changing alphabet

an iterative algorithm for g 2 multiwise merging of bezier curves this paper presents an iterative algorithm for g 2 multiwise merging of bezier curves by using the g 2 constraint the l 2 distance is represented after simplification as a quartic polynomial in two parameters relating to the magnitudes of end tangents of the merged curve these two parameters are restricted in a feasible region in order for the merged curve to preserve the specified directions of end tangents then g 2 multiwise merging is formulated as a constrained minimization problem and the classic projected newton method is applied to find the minimizer some extensions of multiwise merging using g 3 constraints other energy functionals and curve representations are also outlined several comparative examples are provided to demonstrate the effectiveness of the proposed method

automaton semigroups the concept of an automaton group generalizes easily to semigroups and the systematic study of this area is beginning this paper aims to contribute to that study the basic theory of automaton semigroups is briefly reviewed various natural semigroups are shown to arise as automaton semigroups the interaction of certain semigroup constructions with the class of automaton semigroups is studied semigroups arising from cayley automata are investigated various open problems and areas for further research are suggested

orbit automata as a new tool to attack the order problem in automaton groups abstract we introduce a new tool called the orbit automaton that describes the action of an automaton group g on the subtrees corresponding to the orbits of g on levels of the tree the connection between g and the groups generated by the orbit automata is used to find elements of infinite order in certain automaton groups for which other methods failed to work

on a 2 generated infinite 3 group the presentation problem the author and narain gupta constructed in 3 a family of residually finite 2generated infinite p groups for every odd prime p as groups of automorphisms of regular p trees this was inspired by grigorchuk s construction in z of a 3 generated infinite 2 group as a permutation group of subintervals of the unit interval in 6 merzlyakov relates grigorchuk s group to p groups constructed by aleshin in 1972 l by using automatas the regularity of our definitions has allowed us to generalize our earlier constructions see 4 and gave us ready access to some of their special properties see 51 the prototype of all our constructions has been the infinite 3 group

efficient merging of multiple segments of bezier curves this paper deals with the merging problem of segments of a composite bezier curve with the endpoints continuity constraints we present a novel method which is based on the idea of using constrained dual bernstein polynomial basis wo ny and lewanowicz 2009 12 to compute the control points of the merged curve thanks to using fast schemes of evaluation of certain connections involving bernstein and dual bernstein polynomials the complexity of our algorithm is significantly less than complexity of other merging methods

maximal subgroups of groups of intermediate growth abstract finding the number of maximal subgroups of infinite index of a finitely generated group is a natural problem that has been solved for several classes of geometric groups linear groups hyperbolic groups mapping class groups etc here we provide a solution for a family of groups with a different geometric origin groups of intermediate growth that act on rooted binary trees in particular we show that the non torsion iterated monodromy groups of the tent map a special case of some groups first introduced by sunic in 32 as siblings of the grigorchuk group have exactly countably many maximal subgroups of infinite index and describe them up to conjugacy this is in contrast to the torsion case e g grigorchuk group where there are no maximal subgroups of infinite index it is also in contrast to the above mentioned geometric groups where there are either none or uncountably many such subgroups along the way we show that all the groups defined by sunic have the congruence subgroup property and are just infinite

automaton semigroups new constructions results and examples of non automaton semigroups abstract this paper studies the class of automaton semigroups from two perspectives closure under constructions and examples of semigroups that are not automaton semigroups we prove that semigroup free products of finite semigroups always arise as automaton semigroups and that the class of automaton monoids is closed under forming wreath products with finite monoids we also consider closure under certain kinds of rees matrix constructions strong semilattices and small extensions finally we prove that no subsemigroup of n arises as an automaton semigroup previously n itself was the unique example of a semigroup having the general properties of automaton semigroups such as residual finiteness solvable word problem etc but that was known not to arise as an automaton semigroup

the concept of self similar automata over a changing alphabet and lamplighter groups generated by such automata abstract generalizing the idea of self similar groups defined by mealy automata we introduce the notion of a self similar automaton and a self similar group over a changing alphabet we show that every finitely generated residually finite group is self similar over an arbitrary unbounded changing alphabet we construct some naturally defined self similar automaton representations over an unbounded changing alphabet for any lamplighter group k z with an arbitrary finitely generated finite or infinite abelian group k

differential recurrence properties of dual bernstein polynomials abstract new differential recurrence properties of dual bernstein polynomials are given which follow from relations between dual bernstein and orthogonal hahn and jacobi polynomials using these results a fourth order differential equation satisfied by dual bernstein polynomials has been constructed also a fourth order recurrence relation for these polynomials has been obtained this result may be useful in the efficient solution of some computational problems

on centralizers in locally finite groups abstract the main result of the paper is the following theorem let g be a locally finite group containing a finite p subgroup a such that cg a is finite and a non cyclic subgroup b of order p2 such that cg b has finite exponent for all b b then g is almost locally solvable and has finite exponent

automata generating free products of groups of order 2 abstract we construct a family of automata with n states n 4 acting on a rooted binary tree that generate the free products of cyclic groups of order 2

finiteness properties of direct products of algebraic structures we consider the preservation of properties of being finitely generated being finitely presented and being residually finite under direct products in the context of different types of algebraic structures the structures considered include mal cev algebras including groups rings and other classical algebras as well as loops idempotent algebras including lattices semigroups and algebras in congruence modular varieties we aim to identify as broad classes as possible in which the expected preservation results a b a b satisfies property p p if and only if a and b satisfy p p hold and to exhibit ways in which they may fail outside those classes

piecewise automatic groups we introduce a notion of a piecewise automatic group among these groups we describe a new class of groups of intermediate growth we show that for any function f n n there exists a finitely generated torsion group of intermediate growth g for which the folner function satisfies folg s n f n for some generating set s and all sufficiently large n as a corollary we see that the asymptotic entropy of simple random walks on these groups could be arbitrarily close to being linear while the poisson boundary is trivial

fractal just infinite nil lie superalgebra of finite width abstract the grigorchuk and gupta sidki groups play fundamental role in modern group theory they are natural examples of self similar finitely generated periodic groups as their natural analogues there are constructions of nil lie p algebras over a field of characteristic 2 40 and arbitrary positive characteristic 52 in characteristic zero similar examples of lie algebras do not exist by a result of martinez and zelmanov 32 the second author constructed analogues of the grigorchuk and gupta sidki groups in the world of lie superalgebras of arbitrary characteristic the virtue of that construction is that the lie superalgebras have clear monomial bases 41 that lie superalgebras have slow polynomial growth and are graded by multidegree in the generators in particular a self similar lie superalgebra q is z 3 graded by multidegree in 3 generators its z 3 components lie inside an elliptic paraboloid in space the components are at most one dimensional thus the z 3 grading of q is fine an analogue of the periodicity is that homogeneous elements of the grading q q 0 q 1 are ad nilpotent in particular q is a nil finely graded lie superalgebra which shows that an extension of the mentioned result of martinez and zelmanov 32 to the lie superalgebras of characteristic zero is not valid but computations with q are rather technical in this paper we construct a similar but simpler and smaller example namely we construct a 2 generated fractal lie superalgebra r over arbitrary field we find a clear monomial basis of r and unlike many examples studied before we find also a clear monomial basis of its associative hull a the latter has a quadratic growth the algebras r and a are z 2 graded by multidegree in the generators positions of their z 2 components are bounded by pairs of logarithmic curves on plane the z 2 components of r are at most one dimensional thus the z 2 grading of r is fine as an analogue of periodicity we establish that homogeneous elements of the grading r r 0 r 1 are ad nilpotent in case of n graded algebras a close analogue to being simple is being just infinite unlike previous examples of lie superalgebras we are able to prove that r is just infinite but not hereditary just infinite our example is close to the smallest possible example because r has a linear growth with a growth function r m 3 m as m moreover r is of finite width 4 char k 2 in case char k 2 we obtain a lie algebra of width 2 that is not thin thus we have got a more handy analogue of the grigorchuk and gupta sidki groups the constructed lie superalgebra r is of linear growth of finite width 4 and just infinite it also shows that an extension of the result of martinez and zelmanov 32 to the lie superalgebras of characteristic zero is not valid

on properties not inherited by monoids from their schutzenberger groups we give an example of a monoid with finitely many left and right ideals all of whose schutzenberger groups are presentable by finite complete rewriting systems and so each have finite derivation type but such that the monoid itself does not have finite derivation type and therefore does not admit a presentation by a finite complete rewriting system the example also serves as a counterexample to several other natural questions regarding complete rewriting systems and finite derivation type specifically it allows us to construct two finitely generated monoids m and n with isometric cayley graphs where n has finite derivation type respectively admits a presentation by a finite complete rewriting system but m does not this contrasts with the case of finitely generated groups for which finite derivation type is known to be a quasi isometry invariant the same example is also used to show that neither of these two properties is preserved under finite green index extensions

the conjugacy problem in automaton groups is not solvable abstract free abelian by free self similar groups generated by finite self similar sets of tree automorphisms and having unsolvable conjugacy problem are constructed along the way finitely generated orbit undecidable free subgroups of gl d z for d 6 and aut f d for d 5 are constructed as well

automata groups limit spaces and tilings abstract we explore the connections between automata groups limit spaces of self similar actions and tilings in particular we show how a group acting nicely on a tree gives rise to a self covering of a topological groupoid and how the group can be reconstructed from the groupoid and its covering the connection is via finite state automata these define decomposition rules or self similar tilings on leaves of the solenoid associated with the covering

a just nonsolvable torsion free group defined on the binary tree abstract a two generator torsion free subgroup of the group of finite state automorphisms of the binary tree is constructed having the properties of being just nonsolvable and residually torsion free solvable a presentation is produced for this subgroup in two generators and two relations together with their images under the iterated application of a certain simple substitution

the automorphism tower of groups acting on rooted trees the group of isometries aut t n of a rooted n ary tree and many of its subgroups with branching structure have groups of automorphisms induced by conjugation in aut t n this fact has stimulated the computation of the group of automorphisms of such well known examples as the group studied by r grigorchuk and the group f studied by n gupta and the second author in this paper we pursue the larger theme of towers of automorphisms of groups of tree isometries such as and we describe this tower for all subgroups of aut t 2 which decompose as infinitely iterated wreath products furthermore we fully describe the towers of and r more precisely the tower of is infinite countable and the terms of the tower are 2 groups quotients of successive terms are infinite elementary abelian 2 groups in contrast the tower of f has length 2 and its terms are 2 3 groups we show that aut 2 aut is an elementary abelian 3 group of countably infinite rank while aut 3 aut 2

syntactic and rees indices of subsemigroups abstract we define two different notions of index for subsemigroups of semigroups the right syntactic index and the rees index we investigate the relationships between them and with the group index in particular we show that the syntactic index is a generalisation of both the group index and the rees index we use this fact to prove further similarities between the group index and the rees index it turns out however that very few of the nice properties of the group index are inherited by the syntactic index

on subgroups of finite index in branch groups we give a structural description of the normal subgroups of subgroups of finite index in branch groups in terms of rigid stabilizers this gives further insight into the structure lattices of branch groups introduced by the second author we derive a condition concerning abstract commensurability of branch groups acting on the p ary tree for any prime p

profinite completions of some groups acting on trees abstract we investigate the profinite completions of a certain family of groups acting on trees it turns out that for some of the groups considered the completions coincide with the closures of the groups in the full group of tree automorphisms however we introduce an infinite series of groups for which that is not so and describe the kernels of natural homomorphisms of the profinite completions onto the aforementioned closures of respective groups

examples of self iterating lie algebras abstract we construct examples of self iterating lie algebras similar to the grigorchuk group in case of characteristic 2 we construct a two generated restricted lie algebra of polynomial growth that has a nil p mapping

orbit expandability of automaton semigroups and groups abstract we introduce the notion of expandability in the context of automaton semigroups and groups a word is k expandable if one can append a suffix to it such that the size of the orbit under the action of the automaton increases by at least k this definition is motivated by the question which words admit infinite orbits for such a word every prefix is expandable in this paper we show that given a word u an automaton t and a number k as input it is decidable to check whether u is k expandable with respect to the action of t in fact this can be done in exponential nondeterministic space from this nondeterministic algorithm we obtain a bound on the length of a potential orbit increasing suffix x moreover we investigate the situation if the automaton is invertible and generates a group in this case we give an algebraic characterization for the expandability of a word based on its shifted stabilizer we also give a more efficient algorithm to decide the expandability of a word in the case of automaton groups which allows us to improve the upper bound on the maximal orbit increasing suffix length then we investigate the situation for reversible and complete automata and obtain that every word is expandable with respect to such automata finally we give a lower bound example for the length of an orbit increasing suffix

large hereditarily just infinite groups abstract it is shown that there is a hereditarily just infinite group in which all countably based profinite groups can be embedded and also that there exist hereditarily just infinite prosoluble groups that are not topologically finitely generated

fractal nil graded lie superalgebras abstract the grigorchuk and gupta sidki groups play fundamental role in modern group theory they are natural examples of self similar finitely generated periodic groups the author constructed their analogue in case of restricted lie algebras of characteristic 2 31 shestakov and zelmanov extended this construction to an arbitrary positive characteristic 41 thus we have examples of self similar finitely generated restricted lie algebras with a nil p mapping in characteristic zero similar examples of lie algebras do not exist by a result of martinez and zelmanov 27 the goal of the present paper is to construct analogues of the grigorchuk and gupta sidki groups in the world of lie superalgebras of an arbitrary characteristic we construct two examples r q of finitely generated self similar lie superalgebras over a field k of an arbitrary characteristic and study their properties and properties of their associative hulls in case char k 2 these examples turn into restricted lie algebras the virtue of the present construction is that the lie superalgebras have clear monomial bases these lie superalgebras have slow polynomial growth and are multigraded by multidegree in the generators the lie superalgebra r is z 2 graded while q has a multidegree z 3 gradation and a z 2 gradation both algebras r and q have similar constructions computations for r are simpler but q enjoys some more specific interesting properties the z 3 components of q lie inside an elliptic paraboloid in space they are at most one dimensional thus the z 3 grading of q is fine in the z 2 gradation of q all components q n m n m z are infinite dimensional except for q 00 0 the z 2 gradation also yields a continuum of different decompositions into a direct sum of two locally nilpotent subalgebras q q q in both examples ad a is nilpotent a being even or odd with respect to the z 2 grading as lie superalgebras this property is an analogue of the periodicity of the grigorchuk and gupta sidki groups in particular q is a nil finely graded lie superalgebra which shows that an extension of a theorem due to martinez and zelmanov 27 for the lie superalgebras of characteristic zero is not valid both lie superalgebras are self similar contain infinitely many copies of themselves let us also call them fractal due to this property

large scale properties for bounded automata groups abstract in this paper we study some large scale properties of the mother groups of bounded automata groups first we give two methods to prove every mother group has infinite asymptotic dimension then we study the decomposition complexity of certain subgroup in the mother group we prove the subgroup belongs to d

generators and relations of direct products of semigroups the purpose of this paper is to give necessary and sufficient conditions for the direct product of two semigroups to be finitely generated and also for the direct product to be finitely presented as a consequence we construct a semigroup s of order 11 such that s x t is finitely generated but not finitely presented for every finitely generated infinite semigroup t by way of contrast we show that if s and t belong to a wide class of semigroups then s x t is finitely presented if and only if both s and t are finitely presented exactly as in the case of groups and monoids

on finite complete rewriting systems and large subsemigroups abstract let s be a semigroup and t be a subsemigroup of finite index in s that is the set s t is finite the subsemigroup t is also called a large subsemigroup of s it is well known that if t has a finite complete rewriting system then so does s in this paper we will prove the converse that is if s has a finite complete rewriting system then so does t our proof is purely combinatorial and also constructive

the smallest mealy automaton of intermediate growth abstract in this paper we study the automaton i 2 the smallest mealy automaton of intermediate growth first considered by the last two authors i i reznykov v i sushchansky the two state mealy automata over the two symbol alphabet of the intermediate growth mat zametki 72 2002 102 117 we describe the automatic transformation monoid defined by i 2 give a formula for the generating series for the ball volume growth function of i 2 and give sharp asymptotics for the growth function of i 2 namely i 2 n 2 5 2 3 3 4 2 n 1 4 exp n 6 with the ratios of left to right hand side tending to 1 as n

efficient variational constitutive updates for allen cahn type phase field theory coupled to continuum mechanics abstract this paper deals with efficient variational constitutive updates for allen cahn type phase field theory coupled to a geometrically exact description of continuum mechanics the starting point of the implementation is a unified variational principle a time continuous potential is introduced the minimizers of which describe naturally every aspect of the aforementioned coupled model including the homogenization assumptions defining the mechanical response of the bulk material in the diffuse interface region with regard to these assumptions classic models such as the one by voigt taylor or the one by reuss sachs are included additionally more sound homogenization approaches falling into the range of rank 1 convexification are also included in the unified framework based on a direct discretization of this time continuous potential in time and space an efficient numerical finite element implementation is proposed in order to guarantee admissible order parameters of the phase field the unconstrained optimization problem is supplemented by respective constraints they are implemented by means of lagrange parameters combined with the fischer burmeister ncp functions this results in an exact fulfillment of the aforementioned constraints without considering any inequality several numerical examples show the predictive capabilities as well as the robustness and efficiency of the final algorithmic formulation furthermore the influence of the homogenization assumption is analyzed in detail it is shown that the choice of the homogenization assumption does influence the predicted microstructure in general however all models converge to the same solution in the limiting case

merging of bezier curves with box constraints in this paper we present a novel approach to the problem of merging of bezier curves with respect to the l 2 norm we give illustrative examples to show that the solution of the conventional merging problem may not be suitable for further modification and applications as in the case of the degree reduction problem we apply the so called restricted area approach proposed recently in gospodarczyk 2015 to avoid certain defects and make the resulting curve more useful a method of solving the new problem is based on box constrained quadratic programming approach we present a novel approach to the problem of merging of bezier curves we impose constraints of a new type we give illustrative examples to explain the purpose of the new restrictions the merged curve is suitable for further modification and applications

on the implementation of rate independent standard dissipative solids at finite strain variational constitutive updates this paper is concerned with an efficient variationally consistent implementation for rate independent dissipative solids at finite strain more precisely focus is on finite strain plasticity theory based on a multiplicative decomposition of the deformation gradient adopting the formalism of standard dissipative solids which allows to describe constitutive models by means of only two potentials being the helmholtz energy and the yield function or equivalently a dissipation functional finite strain plasticity is recast into an equivalent minimization problem in contrast to previous models the presented framework covers isotropic and kinematic hardening as well as isotropic and anisotropic elasticity and yield functions based on this approach a novel numerical implementation representing the main contribution of the paper is given in sharp contrast to by now classical approaches such as the return mapping algorithm and analogously to the theoretical part the numerical formulation is variationally consistent i e all unknown variables follow naturally from minimizing the energy of the considered system consequently several different numerically efficient and robust optimization schemes can be directly employed for solving the resulting minimization problem extending previously published works on variational constitutive updates the advocated model does not rely on any material symmetry and therefore it can be applied to a broad range of different plasticity theories as two examples an anisotropic hill type and a barlat type model are implemented numerical examples demonstrate the applicability and the performance of the proposed implementation

on self similarity of p adic analytic pro p groups of small dimension abstract given a torsion free p adic analytic pro p group g with dim g p we show that the self similar actions of g on regular rooted trees can be studied through the virtual endomorphisms of the associated z p lie lattice we explicitly classify 3 dimensional unsolvable z p lie lattices for p odd and study their virtual endomorphisms together with lazard s correspondence this allows us to classify 3 dimensional unsolvable torsion free p adic analytic pro p groups for p 5 and to determine which of them admit a faithful self similar action on a p ary tree in particular we show that no open subgroup of s l 1 1 p admits such an action on the other hand we prove that all the open subgroups of s l 2 z p admit faithful self similar actions on regular rooted trees

on amenability of automata groups we show that the group of bounded automatic automorphisms of a rooted tree is amenable which implies amenability of numerous classes of groups generated by finite automata the proof is based on reducing the problem to showing amenability just of a certain explicit family of groups mother groups which is done by analyzing the asymptotic properties of random walks on these groups

on a 2 generated infinite 3 group subgroups and automorphisms this is the second paper dedicated to a detailed study of the infinite 3 group j 74 r 1 3 rr 3 1 y y ct 2 l a recursive definition to be explained later which was introduced by the author and narain gupta in 2 as a group of automorphisms of an infinite regular tree f as was shown in 4 y has the remarkable properties of containing a copy of every finite 3 group and having all its proper quotients finite the first paper s in this detailed study 99 dealt with the presentation problem here we study further aspects of its subgroup structure our prime objective though is to determine the group of automorphisms of 9 this task represents a large portion of our present paper the main results are as follows

engel elements in weakly branch groups abstract we study properties of engel elements in weakly branch groups lying in the group of automorphisms of a spherically homogeneous rooted tree more precisely we prove that the set of bounded left engel elements is always trivial in weakly branch groups in the case of branch groups the existence of non trivial left engel elements implies that these are all p elements and that the group is virtually a p group and so periodic for some prime p we also show that the set of right engel elements of a weakly branch group is trivial under a relatively mild condition also we apply these results to well known families of weakly branch groups like the multi ggs groups

green index and finiteness conditions for semigroups let s be a semigroup and let t be a subsemigroup of s then t acts on s by left and by right multiplication if the complement s t has finitely many strong orbits by both these actions we say that t has finite green index in s this notion of finite index encompasses subgroups of finite index in groups and also subsemigroups of finite rees index complement therefore the question of s and t inheriting various finiteness conditions from each other arises in this paper we consider and resolve this question for the following finiteness conditions finiteness residual finiteness local finiteness periodicity having finitely many right ideals and having finitely many idempotents

gram matrix of bernstein basis this note presents explicit expressions for the inverses of the gram matrix of the bernstein basis and its principal submatrices by taking the advantages of the transformations between the bernstein basis and the constrained dual bernstein basis using the symmetry property fast calculation of these matrices and their inverses is achieved significant improvements are obtained for applications including polynomial approximation of functions and degree reduction of bezier curves

self similar associative algebras abstract famous self similar groups were constructed by grigorchuk gupta and sidki these examples lead to interesting examples of associative algebras the authors suggested examples of self similar lie algebras in terms of differential operators recently sidki introduced an example of an associative algebra of self similar matrices we construct families of self similar associative algebras c d generalizing the example of sidki we prove that our algebras are z z graded and have polynomial growth our approach is the weight strategy developed by the authors for self similar lie algebras and their envelopes in particular we obtain similar triangular decompositions into direct sums of three subalgebras c c c 0 c d d d 0 d we prove that some of our algebras are direct sums of two locally nilpotent subalgebras c c c d d d we show that in some cases the zero components c 0 d 0 are nontrivial and not nil algebras we show that our construction includes the example of sidki and the examples of self similar lie algebras and their associative hulls constructed by the authors before

presentations for subgroups of monoids abstract the main result of this paper gives a presentation for an arbitrary subgroup of a monoid defined by a presentation it is a modification of the well known reidemeister schreier theorem for groups some consequences of this result are explored it is proved that a regular monoid with finitely many left and right ideals is finitely presented if and only if all its maximal subgroups are finitely presented an inverse monoid with finitely many left and right ideals is finitely presented as an inverse monoid if and only if it is finitely presented as a monoid an example of a finitely presented monoid with a finitely generated but not finitely presented group of units is exhibited

weakly maximal subgroups in regular branch groups abstract let g be a finitely generated regular branch group acting by automorphisms on a regular rooted tree t it is well known that stabilizers of infinite rays in t aka parabolic subgroups are weakly maximal subgroups in g that is maximal among subgroups of infinite index we show that given a finite subgroup q g g possesses uncountably many automorphism equivalence classes of weakly maximal subgroups containing q in particular for grigorchuk gupta sidki type groups this implies that they have uncountably many automorphism equivalence classes of weakly maximal subgroups that are not parabolic

explicit algorithms for multiwise merging of bezier curves this paper presents a novel scheme called c r s multiwise merging for merging multiple segments of bezier curves using a single bezier curve it is considered as an extension of the existing pairwise merging to avoid the limitations caused by recursively applying pairwise merging to the multiple case an explicit algorithm is developed to obtain the merged curve which preserves c r and c s continuity at the endpoints and is optimal in the sense that the l 2 or l 2 distance is minimized as an application we develop explicit algorithms for g 1 multiwise merging always producing better results than c 1 multiwise merging

bezier representation of the constrained dual bernstein polynomials abstract explicit formulae for the bezier coefficients of the constrained dual bernstein basis polynomials are derived in terms of the hahn orthogonal polynomials using difference properties of the latter polynomials efficient recursive scheme is obtained to compute these coefficients applications of this result to some problems of cagd is discussed

twisted burnside frobenius theory for discrete groups for a wide class of groups including polycyclic and finitely generated poly nomial growth groups it is proved that the reidemeister number of an automorphism is equal to the number of finite dimensional fixed points of the induced map b on the unitary dual if one of these numbers is finite this theorem is a natural generalization of the classical burnside frobenius theorem to infinite groups this theorem also has important consequences in topological dynamics and in some sense is a reply to a remark of j p serre the main technical results proved in the paper yield a tool for a further progress

a language hierarchy and kitchens type theorem for self similar groups abstract we generalize the notion of self similar groups of infinite tree automorphisms to allow for groups which are defined on a tree but may not act faithfully on it the elements of such a group correspond to labeled trees which may be recognized by a tree automaton e g rabin buchi etc or considered as elements of a tree shift e g of finite type sofic as in symbolic dynamics we give examples to show how self similar groups defined in this way can be separated into different tree language hierarchies as the main result extending the classical result of kitchens on one dimensional group shifts we provide a sufficient condition for a self similar group whose elements form a sofic tree shift to be a tree shift of finite type as an application we show that the closures of certain self similar groups of rooted k ary tree automorphisms that satisfy an algebraic law are not rabin recognizable that is they can not be described within the second order theory of k successors in both the main result and in the application a crucial role is played by a distinguished branched subgroup structure of the groups under consideration

finite complete rewriting systems and finite derivation type for small extensions of monoids abstract letsbe a monoid and lettbe a submonoid of finite index ins the main results in this article state thatscan be presented by a finite complete rewriting system iftcan andshas finite derivation type ifthas

finite derivation type for rees matrix semigroups this paper introduces the topological finiteness condition finite derivation type fdt on the class of semigroups this notion is naturally extended from the monoid case with this new concept we are able to prove that if a rees matrix semigroup m s i j p has fdt then the semigroup s also has fdt given a monoid s and a finitely presented rees matrix semigroup m s i j p we prove that if the ideal of s generated by the entries of p has fdt then so does m s i j p in particular we show that for a finitely presented completely simple semigroup m the rees matrix semigroup m m s i j p has fdt if and only if the group s has fdt

relatively free semigroups of intermediate growth abstract the problem of calculating the growth of a finitely generated f g semigroup satisfying the given system of identities is considered examples of relatively free semigroups having intermediate growth new growth criteria and constructions are given the main ideas employed in the proofs are based on some new points of view on growth our method is also useful for constructing various examples of f g nilpotent and nil semigroups having intermediate growth and arbitrarily small gelfand kirillov superdimension

groups acting on rooted trees and their representations on the boundary abstract we consider groups that act on spherically symmetric rooted trees and study the associated representation of the group on the space of locally constant functions on the boundary of the tree we introduce and discuss the new notion of locally 2 transitive actions assuming local 2 transitivity our main theorem yields a precise decomposition of the boundary representation into irreducible constituents the method can be used to study gelfand pairs and enables us to answer a question of grigorchuk to provide examples we analyse in detail the local 2 transitivity of ggs groups moreover our results can be used to determine explicit formulae for zeta functions of induced representations defined by klopsch and the author

an explicit method for g 3 merging of two bezier curves this paper presents an explicit method for the g 3 merging problem of two bezier curves the main idea is to express the l 2 distance as a quadratic function of some parameters provided by g 3 continuity conditions an efficient non iterative algorithm is proposed to obtain the optimal merged curve when the l 2 distance is minimized the uniqueness of the global minimum is also proven this method can be applied to two adjacent curves with different degrees and has the ability to obtain satisfactory merging results by using curves of lower degree the efficiency and accuracy of the proposed explicit method are illustrated through several comparative examples

freeness of automaton groups vs boundary dynamics abstract we prove that the boundary dynamics of the semi group generated by the enriched dual transducer characterizes the algebraic property of being free for an automaton group we specialize this result to the class of bireversible transducers and we show that the property of being not free is equivalent to the existence of a finite schreier graph in the boundary of the enriched dual pointed at some essentially non trivial point from these results we derive some consequences from the algebraic algorithmic and dynamical points of view

multi degree reduction of tensor product bezier surfaces with general boundary constraints abstract we propose an efficient approach to the problem of multi degree reduction of rectangular bezier patches with prescribed boundary control points we observe that the solution can be given in terms of constrained bivariate dual bernstein polynomials the complexity of the method is o mn 1 n 2 with m min m1 m2 where n1 n2 and m1 m2 is the degree of the input and output bezier surface respectively if the approximation with appropriate boundary constraints is performed for each patch of several smoothly joined rectangular bezier surfaces the result is a composite surface of global cr continuity with a prescribed r 0 in the detailed discussion we restrict ourselves to r 0 1 which is the most important case in practical application some illustrative examples are given

on the burnside problem for periodic groups narain gupta 1 and said sidki 2 1 university of manitoba department of mathematics winnipeg manitoba r3t 2n2 canada 2 university of brasilia department of mathematics brasilia d f brazil introduction the generalized burnside problem refers to the question are finitely generated periodic groups finite this was answered in the negative by golod 1 who proved that for each prime p there exists a finitely generated infinite p group golod s construction in not however direct and is based on his celebrated work with safarevi recently grigor uk 2 has given a direct and elegant construction of an infinite 2 group which is generated by three elements of order 2 in this paper we give for each odd prime p a direct construction of an infinite p group on two generators each of order p our group is a subgroup of the automorphism group of a regular tree of degree p and as might be expected it is residually finite and has infinite exponent preliminaries let p be an odd prime and let t 0 be the infinite regular tree of degree p with vertex 0 so that through each vertex u of t 0 there are p regular subtrees

self similar graphs a unified treatment of katsura and nekrashevych c algebras abstract given a graph e an action of a group g on e and a g valued cocycle on the edges of e we define a c algebra denoted o g e which is shown to be isomorphic to the tight c algebra associated to a certain inverse semigroup s g e built naturally from the triple g e as a tight c algebra o g e is also isomorphic to the full c algebra of a naturally occurring groupoid g tight s g e we then study the relationship between properties of the action of the groupoid and of the c algebra with an emphasis on situations in which o g e is a kirchberg algebra our main applications are to katsura algebras and to certain algebras constructed by nekrashevych from self similar groups these two classes of c algebras are shown to be special cases of our o g e and many of their known properties are shown to follow from our general theory

on maximal subgroups of infinite index in branch and weakly branch groups abstract we generalise a technical tool originally developed by pervova for the study of maximal subgroups in grigorchuk and ggs groups to all weakly branch groups satisfying a natural condition and in particular to all branch groups we then use this tool to prove that every maximal subgroup of infinite index of a branch group is also a branch group as a further application of this result we show that every maximal subgroup of the basilica group is of finite index

the variational formulation of viscoplastic constitutive updates we present a class of constitutive updates for general viscoplastic solids including such aspects of material behavior as finite elastic and plastic deformations non newtonian viscosity rate sensitivity and arbitrary flow and hardening rules the distinguishing characteristic of the proposed constitutive updates is that by construction the corresponding incremental stress strain relations derive from a pseudo elastic strain energy density this in turn confers the incremental boundary value problem a variational structure in particular the incremental deformation mapping follows from a minimum principle this minimum principle may conveniently be taken as a basis for error estimation and mesh adaption the accuracy and robustness of the variational constitutive updates is demonstrated with the aid of convergence tests involving the finitely deforming mises solid and ductile single crystals the ability of the updates to resolve the complex patterns of slip activity which arise in the latter application is particularly noteworthy

on the isolated points in the space of groups we investigate the isolated points in the space of finitely generated groups we give a workable characterization of isolated groups and study their hereditary properties various examples of groups are shown to yield isolated groups we also discuss a connection between isolated groups and solvability of the word problem

ellipticity of words abstract we prove that an arbitrary word v of the free pro p group is elliptic on a pro p completion of a discrete finitely generated torsion residually p group

on the automorphism group of the one rooted binary tree abstract let a be the automorphism group of the one rooted regular binary tree t 2 and g the subgroup of a consisting of those automorphisms admitting a finite description in their action on t 2 let n a g be the normaliser of g in a let aut g be the group of automorphisms of g and let end a g be the semigroup of endomorphisms of g induced by conjugation by elements of a then g is the infinite iterated wreath product c 2 c 2 and a is the topological limit of g we study in some detail the structure of g aut g and end a g in particular we prove n a g is isomorphic to aut g contains a copy of a itself and is a proper subgroup of end a g furthermore we discuss connections with automata and introduce the notion of functionally recursive automorphisms of t 2

the complexity of grigorchuk groups with application to cryptography abstract the turing complexity of the word problems of a class of groups introduced by grigorchuk 1985 is examined in particular it is shown that such problems of permutation groups of the infinite complete binary tree yield natural complete sets that separate time and space complexity classes if they are distinct a refinement of savitch s translation theorem as well as a similar result restricted for time complexity follow new families of nonfinitely presented groups are shown to have word problems uniformly solvable in simultaneous logspace and quadratic time a new family of public key cryptosystems based on these word problems is constructed