scholarly journals A MEALY MACHINE WITH POLYNOMIAL GROWTH OF IRRATIONAL DEGREE

2008 ◽  
Vol 18 (01) ◽  
pp. 59-82 ◽  
Author(s):  
LAURENT BARTHOLDI ◽  
ILLYA I. REZNYKOV
Keyword(s):  
Polynomial Growth ◽  
Growth Function ◽  
Finitely Generated ◽  
Mealy Machine ◽  
A Chain

We consider a very simple Mealy machine (two nontrivial states over a two-symbol alphabet), and derive some properties of the semigroup it generates. It is an infinite, finitely generated semigroup, and we show that the growth function of its balls behaves asymptotically like ℓα, for [Formula: see text]; that the semigroup satisfies the identity g6 = g4; and that its lattice of two-sided ideals is a chain.

scholarly journals Growth of linear semigroups

1996 ◽  
Vol 60 (1) ◽  
pp. 18-30 ◽  
Author(s):  
Jan Okniński
Keyword(s):  
Finite Rank ◽  
Polynomial Growth ◽  
Growth Function ◽  
Unipotent Radical ◽  
Maximal Degree ◽  
Finitely Generated ◽  
Group Of Fractions

AbstractWe show that the growth function of a finitely generated linear semigroup S ⊆ Mn(K) is controlled by its behaviour on finitely many cancellative subsemigroups of S. If the growth of S is polynomially bounded, then every cancellative subsemigroup T of S has a group of fractions G ⊆ Mn (K) which is nilpotent-by-finite and of finite rank. We prove that the latter condition, strengthened by the hypothesis that every such G has a finite unipotent radical, is sufficient for S to have a polynomial growth. Moreover, the degree of growth of S is then bounded by a polynomial f(n, r) in n and the maximal degree r of growth of finitely generated cancellative T ⊆ S.

THE BURNSIDE PROBLEM FOR GROUPS OF LOW QUADRATIC GROWTH

1996 ◽  
Vol 06 (03) ◽  
pp. 369-377
Author(s):  
ROBERTO INCITTI
Keyword(s):  
Infinite Order ◽  
Polynomial Growth ◽  
Growth Function ◽  
Infinite Group ◽  
Quadratic Growth ◽  
Burnside Problem ◽  
Finitely Generated ◽  
Combinatorial Argument ◽  
Groups Of Polynomial Growth ◽  
Generating System

We show with a combinatorial argument that a finitely generated infinite group whose growth function relative to some finite generating system is less or equal to [Formula: see text], r<2, contains an element of infinite order. This result is aimed at investigating the combinatorial nature of M. Gromov’s theorem on groups of polynomial growth.

scholarly journals The primitivity index function for a free group, and untangling closed curves on hyperbolic surfaces.With the appendix by Khalid Bou–Rabee

2017 ◽  
Vol 166 (1) ◽  
pp. 83-121
Author(s):  
NEHA GUPTA ◽  
ILYA KAPOVICH
Keyword(s):  
Lower Bounds ◽  
Free Group ◽  
Growth Function ◽  
Residual Finiteness ◽  
Closed Geodesics ◽  
Hyperbolic Surfaces ◽  
Finitely Generated ◽  
Closed Curves ◽  
Index Function ◽  
Finite Covers

AbstractMotivated by the results of Scott and Patel about “untangling” closed geodesics in finite covers of hyperbolic surfaces, we introduce and study primitivity, simplicity and non-filling index functions for finitely generated free groups. We obtain lower bounds for these functions and relate these free group results back to the setting of hyperbolic surfaces. An appendix by Khalid Bou–Rabee connects the primitivity index functionfprim(n,FN) to the residual finiteness growth function forFN.

IDENTITIES AND A BOUNDED HEIGHT CONDITION FOR SEMIGROUPS

2003 ◽  
Vol 13 (05) ◽  
pp. 565-583 ◽  
Author(s):  
L. M. SHNEERSON
Keyword(s):  
Polynomial Growth ◽  
Nilpotent Groups ◽  
The Other ◽  
Finitely Generated ◽  
Different Types ◽  
Associative Rings ◽  
Nontrivial Identity

We consider two different types of bounded height condition for semigroups. The first one originates from the classical Shirshov's bounded height theorem for associative rings. The second which is weaker, in fact was introduced by Wolf and also used by Bass for calculating the growth of finitely generated (f.g.) nilpotent groups. Both conditions yield polynomial growth. We give the first two examples of f.g. semigroups which have bounded height and do not satisfy any nontrivial identity. One of these semigroups does not have bounded height in the sense of Shirshov and the other satisfies the classical bounded height condition. This develops further one of the main results of the author's paper (J. Algebra, 1993) where the first examples of f.g. semigroups of polynomial growth and without nontrivial identities were given.

scholarly journals Model of Profit Maximization of The Giant Gourami (Osphronemus goramy) Culture

Omni-Akuatika ◽  
2017 ◽  
Vol 13 (1) ◽  
Author(s):  
Dian Wijayanto ◽  
Faik Kurohman ◽  
Ristiawan Nugroho
Keyword(s):  
Polynomial Growth ◽  
Profit Maximization ◽  
Growth Function ◽  
Fish Growth ◽  
Research Model ◽  
First Derivative ◽  
Culture Time ◽  
Maximum Profit ◽  
Culture Cycle ◽  
Size Target

The research purpose was to develop a model of profit maximization that can be applied to the giant gourami culture. The development of fish growth model used polynomial growth function. Profit maximization process used the first derivative of profit equation to culture time equal to zero. This research also developed the equations to estimate the culture time to reach the size target of fish. The research has been proven that this research model could be applied in the giant gouramy culture. In the case of this study, the giant gouramy culture can achieve the maximum profit at 324 days and the profit of IDR. 7 847 700 per culture cycle. If we used a size target 500 g, the culture of the giant gouramy need 135 days of culture time.

scholarly journals Complexity and cohomology for cut-and-projection tilings

2009 ◽  
Vol 30 (2) ◽  
pp. 489-523 ◽  
Author(s):  
ANTOINE JULIEN
Keyword(s):  
Polynomial Growth ◽  
Finitely Generated ◽  
Complexity Function ◽  
Tiling Space ◽  
Dimension One ◽  
Natural Complexity

AbstractWe consider a subclass of tilings: the tilings obtained by cut-and-projection. Under somewhat standard assumptions, we show that the natural complexity function has polynomial growth. We compute its exponentαin terms of the ranks of certain groups which appear in the construction. We give bounds forα. These computations apply to some well-known tilings, such as the octagonal tilings, or tilings associated with billiard sequences. A link is made between the exponent of the complexity, and the fact that the cohomology of the associated tiling space is finitely generated over ℚ. We show that such a link cannot be established for more general tilings, and we present a counterexample in dimension one.

scholarly journals GROUPS AND SEMIGROUPS WITH A ONE-COUNTER WORD PROBLEM

2008 ◽  
Vol 85 (2) ◽  
pp. 197-209 ◽  
Author(s):  
DEREK F. HOLT ◽  
MATTHEW D. OWENS ◽  
RICHARD M. THOMAS
Keyword(s):  
Word Problem ◽  
Linear Growth ◽  
Elementary Proof ◽  
Growth Function ◽  
Finitely Generated ◽  
Strong Restriction

AbstractWe prove that a finitely generated semigroup whose word problem is a one-counter language has a linear growth function. This provides us with a very strong restriction on the structure of such a semigroup, which, in particular, yields an elementary proof of a result of Herbst, that a group with a one-counter word problem is virtually cyclic. We prove also that the word problem of a group is an intersection of finitely many one-counter languages if and only if the group is virtually abelian.

Some questions concerning the cofinality of Sym(κ)

1995 ◽  
Vol 60 (3) ◽  
pp. 892-897 ◽  
Author(s):  
James D. Sharp ◽  
Simon Thomas
Keyword(s):  
Analogous Result ◽  
Infinite Cardinal ◽  
Finitely Generated ◽  
Regular Cardinals ◽  
Uncountable Cardinals ◽  
A Chain

Suppose that G is a group that is not finitely generated. Then the cofinality of G, written c(G), is denned to be the least cardinal λ such that G can be expressed as the union of a chain of λ proper subgroups. If κ is an infinite cardinal, then Sym(κ) denotes the group of all permutations of the set κ = {α∣α < κ}. In [1], Macpherson and Neumann proved that c(Sym(κ)) > κ for all infinite cardinals κ. In [4], we proved that it is consistent that c(Sym(ω)) and 2ω can be any two prescribed regular cardinals, subject only to the obvious requirement that c(Sym(ω)) ≤ 2ω. Our first result in this paper is the analogous result for regular uncountable cardinals κ.Theorem 1.1. Let V ⊨ GCH. Let κ, θ, λ ∈ V be cardinals such that(i) κ and θ are regular uncountable, and(ii) κ < θ ≤ cf(λ).Then there exists a notion of forcing ℙ, which preserves cofinalities and cardinalities, such that if G is ℙ-generic then V[G] ⊨ c(Sym(κ)) = θ ≤ λ = 2κ.Theorem 1.1 will be proved in §2. Our proof is based on a very powerful uniformization principle, which was shown to be consistent for regular uncountable cardinals in [2].

scholarly journals What makes a complex a virtual resolution?

2021 ◽  
Vol 8 (28) ◽  
pp. 885-898
Author(s):  
Michael Loper
Keyword(s):  
Classical Theory ◽  
Toric Variety ◽  
Chain Complex ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Content Type ◽  
Graded Modules ◽  
Cox Ring ◽  
A Chain ◽  
Free Modules

Virtual resolutions are homological representations of finitely generated Pic ( X ) \text {Pic}(X) -graded modules over the Cox ring of a smooth projective toric variety. In this paper, we identify two algebraic conditions that characterize when a chain complex of graded free modules over the Cox ring is a virtual resolution. We then turn our attention to the saturation of Fitting ideals by the irrelevant ideal of the Cox ring and prove some results that mirror the classical theory of Fitting ideals for Noetherian rings.

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