scholarly journals Weighted growth functions of automatic groups

2017 ◽  
Author(s):  
Mikael Vejdemo-Johansson
Keyword(s):  
Normal Form ◽  
Growth Function ◽  
Braid Groups ◽  
Group Element ◽  
Systems Of Equations ◽  
Graphs Of Groups ◽  
Growth Functions ◽  
Automatic Groups ◽  
Novel Method ◽  
Context Free

The growth function is the generating function for sizes of spheres around the identity in Cayley graphs of groups. We present a novel method to calculate growth functions for automatic groups with normal form recognizing automata that recognize a single normal form for each group element, and are at most context free in complexity: context free grammars can be translated into algebraic systems of equations, whose solutions represent generating functions of their corresponding non-terminal symbols. This approach allows us to seamlessly introduce weightings on the growth function: assign different or even distinct weights to each of the generators in an underlying presentation, such that this weighting is reflected in the growth function. We recover known growth functions for small braid groups, and calculate growth functions that weight each generator in an automatic presentation of the braid groups according to their lengths in braid generators.

scholarly journals Weighted growth functions of automatic groups

2017 ◽  
Author(s):  
Mikael Vejdemo-Johansson
Keyword(s):  
Normal Form ◽  
Growth Function ◽  
Braid Groups ◽  
Group Element ◽  
Systems Of Equations ◽  
Graphs Of Groups ◽  
Growth Functions ◽  
Automatic Groups ◽  
Novel Method ◽  
Context Free

The growth function is the generating function for sizes of spheres around the identity in Cayley graphs of groups. We present a novel method to calculate growth functions for automatic groups with normal form recognizing automata that recognize a single normal form for each group element, and are at most context free in complexity: context free grammars can be translated into algebraic systems of equations, whose solutions represent generating functions of their corresponding non-terminal symbols. This approach allows us to seamlessly introduce weightings on the growth function: assign different or even distinct weights to each of the generators in an underlying presentation, such that this weighting is reflected in the growth function. We recover known growth functions for small braid groups, and calculate growth functions that weight each generator in an automatic presentation of the braid groups according to their lengths in braid generators.

MONOID GROWTH FUNCTIONS FOR BRAID GROUPS

1991 ◽  
Vol 01 (02) ◽  
pp. 201-205 ◽  
Author(s):  
MARCUS BRAZIL
Keyword(s):  
Braid Group ◽  
Growth Function ◽  
Braid Groups ◽  
Growth Functions

It is shown that for all n, the braid group on n strings, Bn, has rational growth with respect to a certain set of elements of the group which generate it as a monoid. In particular, the precise growth function for B4 is calculated.

A description of the growth of sheep

1998 ◽  
Vol 1998 ◽  
pp. 47-47
Author(s):  
R.M. Lewis ◽  
G.C. Emmans ◽  
G. Simm ◽  
W.S. Dingwall ◽  
J. FitzSimons
Keyword(s):  
Growth Curves ◽  
Growth Function ◽  
Single Equation ◽  
Rate Parameter ◽  
Growth Functions ◽  
Gompertz Equation ◽  
Lumped Parameter ◽  
Highly Correlated ◽  
Mature Size

The idea that an animal of a given kind has, and grows to, a final or mature size is a useful one and several equations have been proposed that describe such growth to maturity (Winsor, 1932; Parks, 1982; Taylor, 1982). The Gompertz is one of these growth functions and describes in a comparatively simple, single equation the sigmoidal pattern of growth. It has 3 parameters, only 2 of which are important - mature size A and the rate parameter B. Estimates of A and B, however, are highly correlated. Considering A and B as a lumped parameter (AB) may overcome this problem. A Gompertz, or any other, growth function is not expected to describe all growth curves. When the environment (e.g., feed, housing) is non-limiting, it may provide a useful and succinct description of growth. The objectives of this study were to examine: (i) if the Gompertz equation adequately describes the growth of two genotypes of sheep under conditions designed to be non-limiting; and, (ii) if the lumped parameter AB has more desirable properties for estimation than A and B separately.

scholarly journals One-dimensional game of life and its growth functions

1992 ◽  
Vol 15 (3) ◽  
pp. 499-508
Author(s):  
Mohammad H. Ahmadi
Keyword(s):  
Power Series ◽  
Rational Function ◽  
Formal Power Series ◽  
Infinite Product ◽  
Growth Function ◽  
The Other ◽  
Arbitrary Integer ◽  
One Dimensional ◽  
Growth Functions ◽  
Formal Power

We start with finitely many1's and possibly some0's in between. Then each entry in the other rows is obtained from the Base2sum of the two numbers diagonally above it in the preceding row. We may formulate the game as follows: Defined1,jrecursively for1, a non-negative integer, andjan arbitrary integer by the rules:d0,j={1     for   j=0,k         (I)0   or   1   for   0<j<kd0,j=0   for   j<0   or   j>k              (II)di+1,j=di,j+1(mod2)   for   i≥0.      (III)Now, if we interpret the number of1's in rowias the coefficientaiof a formal power series, then we obtain a growth function,f(x)=∑i=0∞aixi. It is interesting that there are cases for which this growth function factors into an infinite product of polynomials. Furthermore, we shall show that this power series never represents a rational function.

Comparing Polynomial and von Bertalanffy Growth Functions for Fitting Tag–Recapture Data

1994 ◽  
Vol 51 (8) ◽  
pp. 1689-1691 ◽  
Author(s):  
William S. Hearn ◽  
George M. Leigh
Keyword(s):  
Growth Function ◽  
Growth Increment ◽  
Bias Reduction ◽  
Release Time ◽  
Von Bertalanffy ◽  
Growth Functions ◽  
Straight Line ◽  
Von Bertalanffy Growth Function ◽  
Multiple Measurements ◽  
The One

The properties of polynomial and von Bertalanffy growth functions are compared for analysing data from tag–recapture experiments in which fish are recaptured once. For the quadratic and von Bertalanffy growth functions, explicit formulae are obtained for the expected growth increment in terms of length-at-release, time-at-liberty, and the function parameters. If the least-squares fitting technique is used the von Bertalanffy function fits tag–recapture data with no more bias (probably less) than any other growth function, including polynomial growth functions. A bias-reduction technique for fitting the von Bertalanffy growth function to tag–recapture data is not applicable to other growth functions. We conclude that, apart from the straight line, the von Bertalanffy growth function is the one with the most desirable mathematical and statistical properties for fitting to tag–recapture data. The matter of the function that best characterises the way a specific fish species grows can be adequately addressed only by analyses of multiple measurements of individual fish.

scholarly journals MARKOV AND ARTIN NORMAL FORM THEOREM FOR BRAID GROUPS

2007 ◽  
Vol 17 (05n06) ◽  
pp. 951-961 ◽  
Author(s):  
L. A. BOKUT ◽  
V. V. CHAYNIKOV ◽  
K. P. SHUM
Keyword(s):  
Normal Form ◽  
Braid Groups ◽  
Form Theorem ◽  
Normal Form Theorem

In this paper we will present the results of Artin–Markov on braid groups by using the Gröbner–Shirshov basis. As a consequence we can reobtain the normal form of Artin–Markov–Ivanovsky as an easy corollary.

A NORMAL FORM FOR BRAIDS

2008 ◽  
Vol 17 (06) ◽  
pp. 697-732 ◽  
Author(s):  
XAVIER BRESSAUD
Keyword(s):  
Normal Form ◽  
Regular Language ◽  
Braid Groups ◽  
New Normal ◽  
Geometric Action

We present a seemingly new normal form for braids, where every braid is expressed using a word in a regular language on some simple alphabet of elementary braids. This normal form stems from analysing the geometric action of braid groups on curves in a punctured disk.

MULTILINEAR EQUATIONS IN AMALGAMS OF FINITE INVERSE SEMIGROUPS

2011 ◽  
Vol 21 (01n02) ◽  
pp. 35-59 ◽  
Author(s):  
A. CHERUBINI ◽  
C. NUCCIO ◽  
E. RODARO
Keyword(s):  
Normal Form ◽  
Inverse Semigroups ◽  
Satisfiability Problem ◽  
Finite Set ◽  
Context Free ◽  
Schützenberger Automaton ◽  
Amalgamated Product

Let S = S1 *U S2 = Inv〈X; R〉 be the free amalgamated product of the finite inverse semigroups S1, S2 and let Ξ be a finite set of unknowns. We consider the satisfiability problem for multilinear equations over S, i.e. equations wL ≡ wR with wL, wR ∈ (X ∪ X-1 ∪ Ξ ∪ Ξ-1)+ such that each x ∈ Ξ labels at most one edge in the Schützenberger automaton of either wL or wR relative to the presentation 〈X ∪ Ξ|R〉. We prove that the satisfiability problem for such equations is decidable using a normal form of the words wL, wR and the fact that the language recognized by the Schützenberger automaton of any word in (X ∪ X-1)+) relative to the presentation 〈X|R〉 is context-free.

Study on Optimum Multi-Parameter Growth Functions of Deformation Prediction for Loess Tunnel Initial Structure

2012 ◽  
Vol 170-173 ◽  
pp. 1769-1772 ◽  
Author(s):  
Fang Biao Liu ◽  
Shun Chuan Wu ◽  
Jian Li ◽  
Jie Hu
Keyword(s):  
Regression Analysis ◽  
Growth Function ◽  
Hyperbolic Function ◽  
Initial Structure ◽  
Deformation Prediction ◽  
Structure Deformation ◽  
North West ◽  
Growth Functions ◽  
Comparison Results ◽  
The Stability

Initial structure deformation prediction based on the regression analysis is an important means to evaluate the stability of tunnel, but the commonly used regression functions, such as exponential, logarithmic and hyperbolic function, have disadvantages of low accuracy. Aiming at the drawbacks, 3 kinds of multi-parameter growth functions (Weibull growth function, Richard function and Gompertz growth function) and 3 kinds of commonly used functions (exponential, logarithmic and hyperbolic function) are used to conduct the regression analysis for a loess tunnel’s initial structure in north-west China. Comparison results show that, (1) regressions based on multi-parameter growth functions are far more precise than the commonly used functions, reflecting its superiority in initial structure deformation prediction of the loess tunnel, (2) the more coefficients the function have, the higher the regression correlation is, and (3) the similarity of iterative and error analysis algorithm in different multi-parameter growth functions lays foundations for computer programming realization of regression analysis and model selection automation.

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