scholarly journals The Cerný conjecture for automata respecting intervals of a directed graph

2013 ◽  
Vol Vol. 15 no. 3 (Automata, Logic and Semantics) ◽  
Author(s):  
Mariusz Grech ◽  
Andrzej Kisielewicz
Keyword(s):  
Directed Graph ◽  
International Audience ◽  
Synchronizing Automaton ◽  
Černý’S Conjecture

Automata, Logic and Semantics International audience The Cerný's conjecture states that for every synchronizing automaton with n states there exists a reset word of length not exceeding (n - 1)2. We prove this conjecture for a class of automata preserving certain properties of intervals of a directed graph. Our result unifies and generalizes some earlier results obtained by other authors.

scholarly journals The number of Euler tours of a random $d$-in/$d$-out graph

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Páidí Creed ◽  
Mary Cryan
Keyword(s):  
Asymptotic Distribution ◽  
Directed Graph ◽  
Polynomial Time ◽  
Simple Approach ◽  
Uniform Sampling ◽  
Concentration Result ◽  
International Audience ◽  
De Bruijn ◽  
Euler Tours

International audience In this paper we obtain the expectation and variance of the number of Euler tours of a random $d$-in/$d$-out directed graph, for $d \geq 2$. We use this to obtain the asymptotic distribution and prove a concentration result. We are then able to show that a very simple approach for uniform sampling or approximately counting Euler tours yields algorithms running in expected polynomial time for almost every $d$-in/$d$-out graph. We make use of the BEST theorem of de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte, which shows that the number of Euler tours of a $d$-in/$d$-out graph is the product of the number of arborescences and the term $[(d-1)!]^n/n$. Therefore most of our effort is towards estimating the asymptotic distribution of the number of arborescences of a random $d$-in/$d$-out graph.

scholarly journals An Algebraic Analogue of a Formula of Knuth

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Lionel Levine
Keyword(s):  
Directed Graph ◽  
Spanning Trees ◽  
Line Graph ◽  
Torsion Subgroup ◽  
Directed Line ◽  
De Bruijn Graphs ◽  
International Audience ◽  
De Bruijn ◽  
Kautz Graphs ◽  
Sandpile Group

International audience We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph $G$ and its directed line graph $\mathcal{L} G$. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when $G$ is regular of degree $k$, we show that the sandpile group of $G$ is isomorphic to the quotient of the sandpile group of $\mathcal{L} G$ by its $k$-torsion subgroup. As a corollary we compute the sandpile groups of two families of graphs widely studied in computer science, the de Bruijn graphs and Kautz graphs. Nous généralisons un théorème de Knuth qui relie les arbres couvrants dirigés d'un graphe orienté $G$ au graphe adjoint orienté $\mathcal{L} G$. On peut associer à tout graphe orienté un groupe abélien appelé groupe du tas de sable, et dont l'ordre est le nombre d'arbres couvrants dirigés enracinés en un sommet fixé. Lorsque $G$ est régulier de degré $k$, nous montrons que le groupe du tas de sable de $G$ est isomorphe au quotient du groupe du tas de sable de $\mathcal{L} G$ par son sous-groupe de $k$-torsion. Comme corollaire, nous déterminons les groupes de tas de sable de deux familles de graphes étudiées en informatique: les graphes de de Bruijn et les graphes de Kautz.

scholarly journals Coherent fans in the space of flows in framed graphs

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Vladimir I. Danilov ◽  
Alexander V. Karzanov ◽  
Gleb A. Koshevoy
Keyword(s):  
Simplicial Complex ◽  
Binary Relation ◽  
Directed Graph ◽  
The Other ◽  
Cluster Algebras ◽  
Linear Orders ◽  
International Audience ◽  
Acyclic Directed Graph ◽  
Extreme Rays ◽  
Space Of Flows

International audience Let $G=(V,E)$ be a finite acyclic directed graph. Being motivated by a study of certain aspects of cluster algebras, we are interested in a class of triangulations of the cone of non-negative flows in $G, \mathcal F_+(G)$. To construct a triangulation, we fix a raming at each inner vertex $v$ of $G$, which consists of two linear orders: one on the set of incoming edges, and the other on the set of outgoing edges of $v$. A digraph $G$ endowed with a framing at each inner vertex is called $framed$. Given a framing on $G$, we define a reflexive and symmetric binary relation on the set of extreme rays of $\mathcal F_+ (G)$. We prove that that the complex of cliques formed by this binary relation is a pure simplicial complex, and that the cones spanned by cliques constitute a unimodular simplicial regular fan $Σ (G)$ covering the entire $\mathcal F_+(G)$. Soit $G=(V,E)$ un graphe orientè, fini et acyclique. Nous nous intéressons, en lien avec l’étude de certains aspects des algèbres amassées, à une classe de triangulations du cône des flots positifs de $G, \mathcal F_+(G)$. Pour construire une triangulation, nous ajoutons une structure en chaque sommet interne $v$ de $G$, constituée de deux ordres totaux : l'un sur l'ensemble des arcs entrants, l'autre sur l'ensemble des arcs sortants de $v$. On dit alors que $G$ est structurè. On définit ensuite une relation binaire réflexive et symétrique sur l'ensemble des rayons extrêmes de $\mathcal F_+ (G)$. Nous démontrons que le complexe des cliques formè par cette relation binaire est un complexe simplicial pur, et que le cône engendré par les cliques forme un éventail régulier simplicial unimodulaire $Σ (G)$ qui couvre complètement $\mathcal F_+(G)$.

scholarly journals Coloring Rings in Species

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Jacob White
Keyword(s):  
Directed Graph ◽  
Symmetric Function ◽  
Chromatic Polynomial ◽  
Connected Components ◽  
Expansion Formula ◽  
Strongly Connected Components ◽  
Set Partitions ◽  
Strongly Connected ◽  
International Audience ◽  
Chromatic Symmetric Function

International audience We present a generalization of the chromatic polynomial, and chromatic symmetric function, arising in the study of combinatorial species. These invariants are defined for modules over lattice rings in species. The primary examples are graphs and set partitions. For these new invariants, we present analogues of results regarding stable partitions, the bond lattice, the deletion-contraction recurrence, and the subset expansion formula. We also present two detailed examples, one related to enumerating subgraphs by their blocks, and a second example related to enumerating subgraphs of a directed graph by their strongly connected components.

scholarly journals Synchronizing random automata

2010 ◽  
Vol Vol. 12 no. 4 ◽  
Author(s):  
Evgeny Skvortsov ◽  
Yulia Zaks
Keyword(s):  
High Probability ◽  
Special Issue ◽  
International Audience ◽  
Synchronizing Automaton ◽  
Random Automaton ◽  
Random Automata

special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications International audience Conjecture that any synchronizing automaton with n states has a reset word of length (n - 1)(2) was made by. Cerny in 1964. Notwithstanding the numerous attempts made by various researchers this conjecture hasn't been definitively proven yet. In this paper we study a random automaton that is sampled uniformly at random from the set of all automata with n states and m(n) letters. We show that for m(n) > 18 ln n any random automaton is synchronizing with high probability. For m(n) > n(beta), beta > 1/2 we also show that any random automaton with high probability satisfies the. Cerny conjecture.

scholarly journals Cyclic inclusion-exclusion and the kernel of P -partitions

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Valentin Féray
Keyword(s):  
Directed Graph ◽  
Symmetric Function ◽  
Bipartite Graphs ◽  
Linear Map ◽  
Generating Series ◽  
International Audience ◽  
Decreasing Functions ◽  
Acyclic Directed Graph

International audience Following the lead of Stanley and Gessel, we consider a linear map which associates to an acyclic directed graph (or a poset) a quasi-symmetric function. The latter is naturally defined as multivariate generating series of non-decreasing functions on the graph (or of P -partitions of the poset).We describe the kernel of this linear map, using a simple combinatorial operation that we call cyclic inclusion- exclusion. Our result also holds for the natural non-commutative analog and for the commutative and non-commutative restrictions to bipartite graphs.

scholarly journals Semisimple Synchronizing Automata and the Wedderburn-Artin Theory

2016 ◽  
Vol 27 (02) ◽  
pp. 127-145 ◽  
Author(s):  
Jorge Almeida ◽  
Emanuele Rodaro
Keyword(s):  
Upper Bound ◽  
Broad Class ◽  
Regular Languages ◽  
Theoretic Approach ◽  
Synchronizing Automata ◽  
Natural Notion ◽  
Strongly Connected ◽  
Synchronizing Automaton ◽  
Synchronizing Word ◽  
Černý’S Conjecture

We present a ring theoretic approach to Černý's conjecture via the Wedderburn-Artin theory. We first introduce the radical ideal of a synchronizing automaton, and then the natural notion of semisimple synchronizing automata. This is a rather broad class since it contains simple synchronizing automata like those in Černý's series. Semisimplicity gives also the advantage of “factorizing” the problem of finding a synchronizing word into the sub-problems of finding “short” words that are zeros into the projection of the simple components in the Wedderburn-Artin decomposition. In the general case this last problem is related to the search of radical words of length at most [Formula: see text] where n is the number of states of the automaton. We show that the solution of this “Radical Conjecture” would give an upper bound [Formula: see text] for the shortest reset word in a strongly connected synchronizing automaton. Finally, we use this approach to prove the Radical Conjecture in some particular cases and Černý's conjecture for the class of strongly semisimple synchronizing automata. These are automata whose sets of synchronizing words are cyclic ideals, or equivalently, ideal regular languages that are closed under taking roots.

Somethin’ about Scandinavia: Danish Shorts on the Post-war International Scene

2018 ◽  
Author(s):  
C. Claire Thomson
Keyword(s):  
United Nations ◽  
International Collaboration ◽  
Marshall Plan ◽  
The Third ◽  
Animated Film ◽  
International Audience ◽  
Post War ◽  
Nordic Cooperation

Building on the picture of post-war Anglo-Danish documentary collaboration established in the previous chapter, this chapter examines three cases of international collaboration in which Dansk Kulturfilm and Ministeriernes Filmudvalg were involved in the late 1940s and 1950s. They Guide You Across (Ingolf Boisen, 1949) was commissioned to showcase Scandinavian cooperation in the realm of aviation (SAS) and was adopted by the newly-established United Nations Film Board. The complexities of this film’s production, funding and distribution are illustrative of the activities of the UN Film Board in its first years of operation. The second case study considers Alle mine Skibe (All My Ships, Theodor Christensen, 1951) as an example of a film commissioned and funded under the auspices of the Marshall Plan. This US initiative sponsored informational films across Europe, emphasising national solutions to post-war reconstruction. The third case study, Bent Barfod’s animated film Noget om Norden (Somethin’ about Scandinavia, 1956) explains Nordic cooperation for an international audience, but ironically exposed some gaps in inter-Nordic collaboration in the realm of film.

Conclusion

2018 ◽  
Author(s):  
Alistair Fox
Keyword(s):  
National Identity ◽  
New Zealand ◽  
Identity Formation ◽  
Essential Role ◽  
Coming Of Age ◽  
Financial Returns ◽  
Self Recognition ◽  
International Audience ◽  
National Identity Formation

The conclusion reaffirms the essential role played by cinema generally, and the coming-of-age genre in particular, in the process of national identity formation, because of its effectiveness in facilitating self-recognition and self-experience through a process of triangulation made possible, for the most part, by a dialogue with some of the nation’s most iconic works of literature. This section concludes by point out the danger posed, however, by an observable trend toward generic standardization in New Zealand films motivated by a desire to appeal to an international audience out of consideration for the financial returns expected by funding bodies under current regimes.

Early Cinema in Scotland

2018 ◽  
Keyword(s):  
Social History ◽  
Small Towns ◽  
Early Cinema ◽  
Popular Song ◽  
Feature Film ◽  
Local Newspapers ◽  
The Social ◽  
Production History ◽  
History Of ◽  
International Audience

This collection of essays, drawn from a three-year AHRC research project, provides a detailed context for the history of early cinema in Scotland from its inception in 1896 till the arrival of sound in the early 1930s. It details the movement from travelling fairground shows to the establishment of permanent cinemas, and from variety and live entertainment to the dominance of the feature film. It addresses the promotion of cinema as a socially ‘useful’ entertainment, and, distinctively, it considers the early development of cinema in small towns as well as in larger cities. Using local newspapers and other archive sources, it details the evolution and the diversity of the social experience of cinema, both for picture goers and for cinema staff. In production, it examines the early attempts to establish a feature film production sector, with a detailed production history of Rob Roy (United Films, 1911), and it records the importance, both for exhibition and for social history, of ‘local topicals’. It considers the popularity of Scotland as an imaginary location for European and American films, drawing their popularity from the international audience for writers such as Walter Scott and J.M. Barrie and the ubiquity of Scottish popular song. The book concludes with a consideration of the arrival of sound in Scittish cinemas. As an afterpiece, it offers an annotated filmography of Scottish-themed feature films from 1896 to 1927, drawing evidence from synopses and reviews in contemporary trade journals.

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