On the complexity function of leading digits of powers of one-digit primes

2021 ◽  
Author(s):  
Mehdi Golafshan ◽  
Alexei Kanel-Belov ◽  
Alex Heinis ◽  
Georgy Potapov
Keyword(s):  
Complexity Function

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

scholarly journals Subshifts on Infinite Alphabets and Their Entropy

Entropy ◽  
2020 ◽  
Vol 22 (11) ◽  
pp. 1293
Author(s):  
Sharwin Rezagholi
Keyword(s):  
Growth Rate ◽  
Markov Chains ◽  
Topological Entropy ◽  
Spectral Radius ◽  
Symbolic Dynamics ◽  
Infinite Graphs ◽  
Complexity Function ◽  
Countably Infinite

We analyze symbolic dynamics to infinite alphabets by endowing the alphabet with the cofinite topology. The topological entropy is shown to be equal to the supremum of the growth rate of the complexity function with respect to finite subalphabets. For the case of topological Markov chains induced by countably infinite graphs, our approach yields the same entropy as the approach of Gurevich We give formulae for the entropy of countable topological Markov chains in terms of the spectral radius in l2.

scholarly journals A New Asymptotic Notation: Weak Theta

2015 ◽  
Vol 2015 ◽  
pp. 1-15
Author(s):  
Andrei-Horia Mogoş ◽  
Bianca Mogoş ◽  
Adina Magda Florea
Keyword(s):  
Artificial Intelligence ◽  
Computer Science ◽  
Main Tool ◽  
Complexity Of Algorithms ◽  
Complexity Function ◽  
Theoretical Results

Algorithms represent one of the fundamental issues in computer science, while asymptotic notations are widely accepted as the main tool for estimating the complexity of algorithms. Over the years a certain number of asymptotic notations have been proposed. Each of these notations is based on the comparison of various complexity functions with a given complexity function. In this paper, we define a new asymptotic notation, called “Weak Theta,” that uses the comparison of various complexity functions with two given complexity functions. Weak Theta notation is especially useful in characterizing complexity functions whose behaviour is hard to be approximated using a single complexity function. In addition, in order to highlight the main particularities of Weak Theta, we propose and prove several theoretical results: properties of Weak Theta, criteria for comparing two complexity functions, and properties of a new set of complexity functions (also defined in the paper) based on Weak Theta. Furthermore, to illustrate the usefulness of our notation, we discuss an application of Weak Theta in artificial intelligence.

scholarly journals THE AUTOMORPHISM GROUP OF A SHIFT OF LINEAR GROWTH: BEYOND TRANSITIVITY

2015 ◽  
Vol 3 ◽  
Author(s):  
VAN CYR ◽  
BRYNA KRA
Keyword(s):  
Automorphism Group ◽  
Finite Groups ◽  
Linear Growth ◽  
Finite Alphabet ◽  
Finitely Generated ◽  
Complexity Function ◽  
Algebraic Properties

For a finite alphabet ${\mathcal{A}}$ and shift $X\subseteq {\mathcal{A}}^{\mathbb{Z}}$ whose factor complexity function grows at most linearly, we study the algebraic properties of the automorphism group $\text{Aut}(X)$. For such systems, we show that every finitely generated subgroup of $\text{Aut}(X)$ is virtually $\mathbb{Z}^{d}$, in contrast to the behavior when the complexity function grows more quickly. With additional dynamical assumptions we show more: if $X$ is transitive, then $\text{Aut}(X)$ is virtually $\mathbb{Z}$; if $X$ has dense aperiodic points, then $\text{Aut}(X)$ is virtually $\mathbb{Z}^{d}$. We also classify all finite groups that arise as the automorphism group of a shift.

Reaction time (RT) and IQ: Shape of the task complexity function

1995 ◽  
Vol 18 (3) ◽  
pp. 339-345 ◽  
Author(s):  
Richard H. Lindley ◽  
Steven M. Wilson ◽  
William Ray Smith ◽  
Kay Bathurst
Keyword(s):  
Reaction Time ◽  
Task Complexity ◽  
Complexity Function

Design and Realization of the Intelligent Scheduling and Management System of Transmission Lines

2011 ◽  
Vol 381 ◽  
pp. 109-113
Author(s):  
Xing Wang ◽  
Ling Xu
Keyword(s):  
Information Processing ◽  
Resource Management ◽  
Transmission Line ◽  
Transmission Lines ◽  
Management System ◽  
Optical Network ◽  
Resource Management System ◽  
Complexity Function ◽  
Intelligent Scheduling ◽  
Intelligent Distribution

Transmission line intelligent distribution and management system is the latest systematic product for optical network intelligent distribution, with the function of detecting, warning and information processing, whose purpose is to realize the delicacy management of transmission line resources and solve the problems appearing in the process of using resource management system., such as input complexity, function singleness and so on. The thesis introduces different functions of each module in this system, puts forward the solutions to this systematic function and provides some realization codes.

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