REPETITIONS, FULLNESS, AND UNIFORMITY IN TWO-DIMENSIONAL WORDS

2004 ◽  
Vol 15 (02) ◽  
pp. 355-383 ◽  
Author(s):  
ARTURO CARPI ◽  
ALDO de LUCA
Keyword(s):  
Essential Role ◽  
Finite Alphabet ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Fixed Size ◽  
Open Problems ◽  
One Dimensional ◽  
Combinatorial Properties ◽  
The Difference ◽  
The One

We consider some combinatorial properties of two-dimensional words (or pictures) over a given finite alphabet, which are related to the number of occurrences in them of words of a fixed size (m,n). In particular a two-dimensional word (briefly, 2D-word) is called (m,n)-full if it contains as factors (or subwords) all words of size (m,n). An (m,n)-full word such that any word of size (m,n) occurs in it exactly once is called a de Bruijn word of order (m,n). A 2D-word w is called (m,n)-uniform if the difference in the number of occurrences in w of any two words of size (m,n) is at most 1. A 2D-word is called uniform if it is (m,n)-uniform for all m,n>0. In this paper we extend to the two-dimensional case some results relating the notions above which were proved in the one-dimensional case in a preceding article. In this analysis the study of repeated factors in a 2D-word plays an essential role. Finally, some open problems and conjectures are discussed.

Regions-based Two-dimensional Continua: The Euclidean Case

2018 ◽  
Author(s):  
Geoffrey Hellman ◽  
Stewart Shapiro
Keyword(s):  
Mathematical Induction ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Entire Space ◽  
Euclidean Case ◽  
Free Basis ◽  
One Dimensional ◽  
Archimedean Property ◽  
The One

This chapter develops a Euclidean, two-dimensional, regions-based theory. As with the semi-Aristotelian account in Chapter 2, the goal here is to recover the now orthodox Dedekind–Cantor continuum on a point-free basis. The chapter derives the Archimedean property for a class of readily postulated orientations of certain special regions, what are called “generalized quadrilaterals” (intended as parallelograms), by which the entire space is covered. Then the chapter generalizes this to arbitrary orientations, and then establishes an isomorphism between the space and the usual point-based one. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit “extremal clause”, and we have no axiom of induction other than ordinary numerical (mathematical) induction.

A Graphical Exposition of the Ising Problem

1971 ◽  
Vol 12 (3) ◽  
pp. 365-377 ◽  
Author(s):  
Frank Harary
Keyword(s):  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Higher Dimensional ◽  
The One

Ising [1] proposed the problem which now bears his name and solved it for the one-dimensional case only, leaving the higher dimensional cases as unsolved problems. The first solution to the two dimensional Ising problem was obtained by Onsager [6]. Onsager's method was subsequently explained more clearly by Kaufman [3]. More recently, Kac and Ward [2] discovered a simpler procedure involving determinants which is not logically complete.

scholarly journals Revisiting the 1D and 2D Laplace Transforms

2020 ◽  
Author(s):  
Manuel Duarte Ortigueira ◽  
José Tenreiro Machado
Keyword(s):  
Transfer Function ◽  
Laplace Transform ◽  
Linear Systems ◽  
Laplace Transforms ◽  
Dimensional Case ◽  
Initial Condition ◽  
Two Dimensional ◽  
One Dimensional ◽  
The One

This paper reviews the unilateral and bilateral, one- and two-dimensional Laplace transforms. The unilateral and bilateral Laplace transforms are compared in the one-dimensional case, leading to the formulation of the initial-condition theorem. This problem is solved with all generality in the one- and two-dimensional cases with the bilateral Laplace transform. General two-dimensional linear systems are introduced and the corresponding transfer function defined.

A Two-Dimensional Model of the Fluid Dynamics of an Air Reverser

1998 ◽  
Vol 65 (1) ◽  
pp. 171-177 ◽  
Author(s):  
S. Mu¨ftu¨ ◽  
T. S. Lewis ◽  
K. A. Cole ◽  
R. C. Benson
Keyword(s):  
Design Guidelines ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Web Handling ◽  
One Dimensional ◽  
Air Supply ◽  
Two Dimensional Model ◽  
The One ◽  
The Web

A theoretical analysis of the fluid mechanics of the air cushion of the air reversers used in web-handling systems is presented. A two-dimensional model of the air flow is derived by averaging the equations of conservation of mass and momentum over the clearance between the web and the reverser. The resulting equations are Euler’s equations with nonlinear source terms representing the air supply holes in the surface of the reverser. The equations are solved analytically for the one-dimensional case and numerically for the two-dimensional case. Results are compared with an empirical formula and the one-dimensional airjet theory developed for hovercraft. Conditions that maximize the air pressure supporting the web are analyzed and design guidelines are deduced.

scholarly journals On the notion of the dimension with respect to a dynamical system

1984 ◽  
Vol 4 (3) ◽  
pp. 405-420 ◽  
Author(s):  
Ya. B. Pesin
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Invariant Sets ◽  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Common Features ◽  
Dynamical Properties ◽  
The One ◽  
Hyperbolic Sets

AbstractFor the invariant sets of dynamical systems a new notion of dimension-the so-called dimension with respect to a dynamical system-is introduced. It has some common features with the general topological notion of the dimension, but it also reflects the dynamical properties of the system. In the one-dimensional case it coincides with the Hausdorff dimension. For multi-dimensional hyperbolic sets formulae for the calculation of our dimension are obtained. These results are generalizations of Manning's results obtained by him for the Hausdorff dimension in the two-dimensional case.

Complex Order from Disorder and from Simple Order in Coarse-Graining Invariant Orbits of Certain Two-Dimensional Linear Cellular Automata

1997 ◽  
Vol 07 (07) ◽  
pp. 1451-1496 ◽  
Author(s):  
André Barbé
Keyword(s):  
Cellular Automata ◽  
Three Dimensional ◽  
Coarse Graining ◽  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Complex Order ◽  
Simple Order ◽  
Problems And Solutions ◽  
The One

This paper considers three-dimensional coarse-graining invariant orbits for two-dimensional linear cellular automata over a finite field, as a nontrivial extension of the two-dimensional coarse-graining invariant orbits for one-dimensional CA that were studied in an earlier paper. These orbits can be found by solving a particular kind of recursive equations (renormalizing equations with rescaling term). The solution starts from some seed that has to be determined first. In contrast with the one-dimensional case, the seed has infinite support in most cases. The way for solving these equations is discussed by means of some examples. Three categories of problems (and solutions) can be distinguished (as opposed to only one in the one-dimensional case). Finally, the morphology of a few coarse-graining invariant orbits is discussed: Complex order (of quasiperiodic type) seems to emerge from random seeds as well as from seeds of simple order (for example, constant or periodic seeds).

scholarly journals Alternating and variable controls for the wave equation

2020 ◽  
Vol 26 ◽  
pp. 38 ◽  
Author(s):  
Antonio Agresti ◽  
Daniele Andreucci ◽  
Paola Loreti
Keyword(s):  
Wave Equation ◽  
Sufficient Conditions ◽  
Dimensional Case ◽  
Equivalent Condition ◽  
Open Problems ◽  
One Dimensional ◽  
Constant Rate ◽  
Exact Observability ◽  
Single Exchange ◽  
The One

The present article discusses the exact observability of the wave equation when the observation subset of the boundary is variable in time. In the one-dimensional case, we prove an equivalent condition for the exact observability, which takes into account only the location in time of the observation. To this end we use Fourier series. Then we investigate the two specific cases of single exchange of the control position, and of exchange at a constant rate. In the multi-dimensional case, we analyse sufficient conditions for the exact observability relying on the multiplier method. In the last section, the multi-dimensional results are applied to specific settings and some connections between the one and multi-dimensional case are discussed; furthermore some open problems are presented.

scholarly journals COMBINATORIAL RECONSTRUCTION OF TWO-DIMENSIONAL WORDS IN THE HYPOTHESIS SHIFT 1

2021 ◽  
Vol 1 (73) ◽  
pp. 59-61
Author(s):  
M. Ulyanov
Keyword(s):  
Linear Size ◽  
Reconstruction Method ◽  
Two Dimensional ◽  
Reconstruction Problem ◽  
Fixed Size ◽  
One Dimensional ◽  
The One

The article considers the formulation of the problem of reconstruction of two-dimensional words by a given multiset of subwords, under the hypothesis that this subset is generated by the displacement of a two-dimensional window of fixed size by an unknown two-dimensional word with a shift 1. A variant of the combinatorial solution of this reconstruction problem is proposed, based on a two-fold application of the one-dimensional word reconstruction method using the search for Eulerian paths or cycles in the de Bruyne multiorgraph. The efficiency of the method is discussed under the conditions of a square two-dimensional shift window one having a large linear size.

scholarly journals The Statistical Properties of theq-Deformed Dirac Oscillator in One and Two Dimensions

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Abdelmalek Boumali ◽  
Hassan Hassanabadi
Keyword(s):  
Quantum Dynamics ◽  
Relativistic Quantum ◽  
Small Deformation ◽  
Two Dimensions ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Dirac Oscillator ◽  
One Dimensional ◽  
Pure Phase ◽  
The One

We study the behavior of the eigenvalues of the one and two dimensions ofq-deformed Dirac oscillator. The eigensolutions have been obtained by using a method based on theq-deformed creation and annihilation operators in both dimensions. For a two-dimensional case, we have used the complex formalism which reduced the problem to a problem of one-dimensional case. The influence of theq-numbers on the eigenvalues has been well analyzed. Also, the connection between theq-oscillator and a quantum optics is well established. Finally, for very small deformationη, we (i) showed the existence of well-knownq-deformed version of Zitterbewegung in relativistic quantum dynamics and (ii) calculated the partition function and all thermal quantities such as the free energy, total energy, entropy, and specific heat. The extension to the case of Graphene has been discussed only in the case of a pure phase (q=eiη).

UNCONVENTIONAL INVERTIBLE BEHAVIORS IN REVERSIBLE ONE-DIMENSIONAL CELLULAR AUTOMATA

2008 ◽  
Vol 18 (12) ◽  
pp. 3625-3632
Author(s):  
JUAN CARLOS SECK TUOH MORA ◽  
MANUEL GONZÁLEZ HERNÁNDEZ ◽  
GENARO JUÁREZ MARTÍNEZ ◽  
SERGIO V. CHAPA VERGARA ◽  
HAROLD V. McINTOSH
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Temporal Evolution ◽  
Complete Characterization ◽  
Dimensional Case ◽  
Neighborhood Size ◽  
One Dimensional ◽  
Combinatorial Properties ◽  
Reversible Cellular Automata ◽  
The One

Reversible cellular automata are discrete invertible dynamical systems determined by local interactions among their components. For the one-dimensional case, there are classical references providing a complete characterization based on combinatorial properties. Using these results and the simulation of every automaton by another with neighborhood size 2, this paper describes other types of invertible behaviors embedded in these systems different from the classical one observed in the temporal evolution. In particular, spatial reversibility and diagonal surjectivity are studied, and the generation of macrocells in the evolution space is analyzed.

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