scholarly journals Directed figure codes are decidable

2009 ◽  
Vol Vol. 11 no. 2 (Automata, Logic and Semantics) ◽  
Author(s):  
Michal Kolarz ◽  
Wlodzimierz Moczurad
Keyword(s):  
Two Dimensional ◽  
End Points ◽  
Constructive Algorithm ◽  
The Status ◽  
Finite Set ◽  
International Audience

Automata, Logic and Semantics International audience Two-dimensional structures of various kinds can be viewed as generalizations of words. Codicity verification and the defect effect, important properties related to word codes, are studied also in this context. Unfortunately, both are lost in the case of two common structures, polyominoes and figures. We consider directed figures defined as labelled polyominoes with designated start and end points, equipped with catenation operation that uses a merging function to resolve possible conflicts. We prove that in this setting verification whether a given finite set of directed figures is a code is decidable and we give a constructive algorithm. We also clarify the status of the defect effect for directed figures.

scholarly journals Asymptotics of lattice walks via analytic combinatorics in several variables

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Stephen Melczer ◽  
Mark C. Wilson
Keyword(s):  
Generating Functions ◽  
Kernel Method ◽  
Rational Functions ◽  
Two Dimensional ◽  
Combinatorial Properties ◽  
Analytic Combinatorics ◽  
Several Variables ◽  
Algebra Approach ◽  
Finite Set ◽  
International Audience

International audience We consider the enumeration of walks on the two-dimensional non-negative integer lattice with steps defined by a finite set S ⊆ {±1, 0}2 . Up to isomorphism there are 79 unique two-dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Pade ́-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.

scholarly journals Counting occurrences for a finite set of words: an inclusion-exclusion approach

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Frédérique Bassino ◽  
Julien Clément ◽  
J. Fayolle ◽  
P. Nicodème
Keyword(s):  
Generating Function ◽  
Exclusion Principle ◽  
Complete Proof ◽  
Detailed Proof ◽  
Finite Set ◽  
International Audience ◽  
The One ◽  
Maple Package

International audience In this paper, we give the multivariate generating function counting texts according to their length and to the number of occurrences of words from a finite set. The application of the inclusion-exclusion principle to word counting due to Goulden and Jackson (1979, 1983) is used to derive the result. Unlike some other techniques which suppose that the set of words is reduced (<i>i..e.</i>, where no two words are factor of one another), the finite set can be chosen arbitrarily. Noonan and Zeilberger (1999) already provided a MAPLE package treating the non-reduced case, without giving an expression of the generating function or a detailed proof. We give a complete proof validating the use of the inclusion-exclusion principle and compare the complexity of the method proposed here with the one using automata for solving the problem.

scholarly journals Graphs with many vertex-disjoint cycles

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Dieter Rautenbach ◽  
Friedrich Regen
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Linear Time ◽  
Simple Extension ◽  
Disjoint Cycles ◽  
Cyclomatic Number ◽  
Finite Set ◽  
Extension Rule ◽  
International Audience ◽  
Vertex Disjoint

Graph Theory International audience We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

scholarly journals Solutions de similitude d'un jeu différentiel stochastique

2006 ◽  
Vol Volume 5, Special Issue TAM... ◽  
Author(s):  
Mario Lefebvre
Keyword(s):  
Differential Equation ◽  
Stochastic Process ◽  
Partial Differential Equation ◽  
Differential Equations ◽  
Similarity Solutions ◽  
Explicit Solutions ◽  
Two Dimensional ◽  
Controlled Processes ◽  
International Audience ◽  
First Time

International audience A two-dimensional controlled stochastic process defined by a set of stochastic differential equations is considered. Contrary to the most frequent formulation, the control variables appear only in the infinitesimal variances of the process, rather than in the infinitesimal means. The differential game ends the first time the two controlled processes are equal or their difference is equal to a given constant. Explicit solutions to particular problems are obtained by making use of the method of similarity solutions to solve the appropriate partial differential equation. On considère un processus stochastique commandé bidimensionnel défini par un ensemble d'équations différentielles stochastiques. Contrairement à la formulation la plus fréquente, les variables de commande apparaissent dans les variances infinitésimales du processus, plutôt que dans les moyennes infinitésimales. Le jeu différentiel prend fin lorsque les deux processus sont égaux ou que leur différence est égale à une constante donnée. Des solutions explicites à des problèmes particuliers sont obtenues en utilisant la méthode des similitudes pour résoudre l'équation aux dérivées partielles appropriée.

A Reinforced Tabu Search Approach for 2D Strip Packing

2012 ◽  
pp. 171-188
Author(s):  
Giglia Gómez-Villouta ◽  
Jean-Philippe Hamiez ◽  
Jin-Kao Hao
Keyword(s):  
Tabu Search ◽  
Search Algorithm ◽  
Fitness Function ◽  
Packing Problem ◽  
Two Dimensional ◽  
Strip Packing ◽  
Benchmark Instances ◽  
Finite Set ◽  
Infinite Height ◽  
Search Approach

This paper discusses a particular “packing” problem, namely the two dimensional strip packing problem, where a finite set of objects have to be located in a strip of fixed width and infinite height. The variant studied considers regular items, rectangular to be precise, that must be packed without overlap, not allowing rotations. The objective is to minimize the height of the resulting packing. In this regard, the authors present a local search algorithm based on the well-known tabu search metaheuristic. Two important components of the presented tabu search strategy are reinforced in attempting to include problem knowledge. The fitness function incorporates a measure related to the empty spaces, while the diversification relies on a set of historically “frozen” objects. The resulting reinforced tabu search approach is evaluated on a set of well-known hard benchmark instances and compared with state-of-the-art algorithms.

Two-dimensional symmetric double contacts of elastically similar materials

2015 ◽  
Vol 230 (10) ◽  
pp. 1626-1633 ◽  
Author(s):  
P Ghanati ◽  
S Adibnazari
Keyword(s):  
Contact Problem ◽  
Singular Integral ◽  
Analytical Approach ◽  
Flat Surface ◽  
Consistency Condition ◽  
Singular Integral Equations ◽  
Two Dimensional ◽  
End Points ◽  
Side Condition ◽  
Similar Materials

The two-dimensional contact problem for an elastic body indenting an elastically similar half plane resulting in double contacts is important for various applications. In this paper, a generic quasi-static two-dimensional symmetric double contact problem with nonsingular end points between two elastically similar half planes, under the constant normal and oscillatory tangential loading, is analyzed. The classical singular integral equations approach is utilized to extract the pressure and shear functions in the contact zones; subsequently boundary conditions at end points are applied and a new side condition is derived and titled “the consistency condition” for symmetric double contacts. This condition is necessary for determining the extent of the contact and stick zones. Next, this analytical approach is applied to the symmetric indentation of a flat surface by two rigidly interconnected wedge-shaped punches.

On the right lower circular density of two-dimensional Jordan curves in the plane

1968 ◽  
Vol 64 (1) ◽  
pp. 67-70
Author(s):  
D. G. Larman
Keyword(s):  
Jordan Curve ◽  
Two Dimensional ◽  
End Points ◽  
Jordan Curves ◽  
End Point ◽  
Dimensional Measure ◽  
Image Position ◽  
The Right

We use J(a, b) to denote a Jordan curve of positive two-dimensional measure in the plane, with end-points a and b. If υ is a point of J(a, b), we define the right lower arc density at υ bywhere J( υ, υ′) is the largest arc, whose left-end point is υ, which is contained in the disc c(υ, r).

scholarly journals Visualisation of Collage Grammar to CellWorks: ET0L Mode and Part Sensitive Mode

2017 ◽  
Vol 16 (3) ◽  
pp. 47-56
Author(s):  
Ann Susa Thomas
Keyword(s):  
Computer Science ◽  
Human Life ◽  
Three Dimensional ◽  
Theoretical Computer Science ◽  
Two Dimensional ◽  
Theoretical Computer ◽  
Finite Set ◽  
A Cell ◽  
Hyperedge Replacement ◽  
Better Than

Images are an important aspect of human life as one remembers pictures better than words. Informally, a twodimensional string is called a picture. A two-dimensional language (or picture language) is a set of pictures. Picture generation and analysis has become a widely investigated field in Theoretical Computer Science and in Mathematics. Collage grammars are studied as devices that generate pictures by rewriting based on hyperedge replacement. A cell-work is a finite set of cells where each cell (being a three dimensional entity) is surrounded by one or more faces. This paper focuses on how cell work languages can be captured by collage grammar in ET0L and Part Sensitive modes.

scholarly journals A Formal Approach to the Description and Manipulation of Structured Mathematical Objects

2005 ◽  
Vol Volume 3, Special Issue... ◽  
Author(s):  
Bernard Fotsing Talla ◽  
Georges-Edouard Kouamou
Keyword(s):  
Square Root ◽  
Two Dimensional ◽  
Attribute Grammars ◽  
Formal Approach ◽  
Mathematical Symbols ◽  
Nous Montrons ◽  
The Matrix ◽  
Mathematical Formulas ◽  
International Audience ◽  
Structured Objects

International audience We present in this paper a formal approach of description, posting and handling of the mathematical structured objects; based on the formalism of attribute grammars. We are interested particularly in the problem of two-dimensional and bidirectional posting of certain expressions and mathematical formulas. Indeed, in more of the two-dimensional character that presents certain mathematical symbols like the square root or the matrix, we also note the problem of posting rightto-left of an Arab text in a context planned for a posting left-to-right of an Indo-European text, or a bidirectional posting mixing the two modes. After a study of some solutions suggested in the literature, we show how the method of attribute grammars adapts easily to these types of problem. Nous présentons dans ce papier une approche formelle de description, d'affichage et de manipulation des objets structurés mathématiques ; basée sur le formalisme des grammaires attribuées. Nous nous intéressons particulièrement au problème d'affichage bidimensionnel et bidirectionnel de certaines expressions et formules mathématiques. En effet, en plus du caractère bidimensionnel que présentent certains symboles comme la racine carrée ou la matrice, on note le problème d'affichage de droite à gauche d'un texte arabe dans un contexte prévu pour un affichage de gauche à droite d'un texte indo-européen, ou encore un affichage bidirectionnel mélangeant les deux modes. Après une étude de quelques méthodes proposées dans la littérature, nous montrons comment la méthode des grammaires attribuées s'adapte facilement à ces types de problèmes.

scholarly journals Partially persistent search trees with transcript operations

1999 ◽  
Vol Vol. 3 no. 3 ◽  
Author(s):  
Kim S. Larsen
Keyword(s):  
Efficient Implementation ◽  
Search Trees ◽  
Time And Space ◽  
The Status ◽  
International Audience ◽  
Over Time

International audience When dictionaries are persistent, it is natural to introduce a transcript operation which reports the status changes for a given key over time. We discuss when and how a time and space efficient implementation of this operation can be provided.

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