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Černý Conjecture
In computer science, more precisely, in the theory of deterministic finite automata (DFA), a synchronizing word or reset sequence is a word in the input alphabet of the DFA that sends any state of the DFA to one and the same state.Avraham Trakhtman''Synchronizing automata, algorithms, Cerny Conjecture'' Accessed May 15, 2010. That is, if an ensemble of copies of the DFA are each started in different states, and all of the copies process the synchronizing word, they will all end up in the same state. Not every DFA has a synchronizing word; for instance, a DFA with two states, one for words of even length and one for words of odd length, can never be synchronized. Existence Given a DFA, the problem of determining if it has a synchronizing word can be solved in polynomial time using a theorem due to Ján Černý. A simple approach considers the power set of states of the DFA, and builds a directed graph where nodes belong to the power set, and a directed edge describes the action of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Road Coloring Conjecture
A road is a thoroughfare used primarily for movement of traffic. Roads differ from streets, whose primary use is local access. They also differ from stroads, which combine the features of streets and roads. Most modern roads are paved. The words "road" and "street" are commonly considered to be interchangeable, but the distinction is important in urban design. There are many types of roads, including parkways, avenues, controlled-access highways (freeways, motorways, and expressways), tollways, interstates, highways, and local roads. The primary features of roads include lanes, sidewalks (pavement), roadways (carriageways), medians, shoulders, verges, bike paths (cycle paths), and shared-use paths. Definitions Historically, many roads were simply recognizable routes without any formal construction or some maintenance. The Organization for Economic Co-operation and Development (OECD) defines a road as "a line of communication (travelled way) using a stabilized base other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Journal On Computing
The ''SIAM Journal on Computing'' is a scientific journal focusing on the mathematical and formal aspects of computer science. It is published by the Society for Industrial and Applied Mathematics (SIAM). Although its official ISO abbreviation is ''SIAM J. Comput.'', its publisher and contributors frequently use the shorter abbreviation ''SICOMP''. SICOMP typically hosts the special issues of the IEEE Annual Symposium on Foundations of Computer Science (FOCS) and the Annual ACM Symposium on Theory of Computing (STOC), where about 15% of papers published in FOCS and STOC each year are invited to these special issues. For example, Volume 48 contains 11 out of 85 papers published in FOCS 2016. References External linksSIAM Journal on Computing on |
Transformation Semigroup
In algebra, a transformation semigroup (or composition semigroup) is a collection of transformations ( functions from a set to itself) that is closed under function composition. If it includes the identity function, it is a monoid, called a transformation (or composition) monoid. This is the semigroup analogue of a permutation group. A transformation semigroup of a set has a tautological semigroup action on that set. Such actions are characterized by being faithful, i.e., if two elements of the semigroup have the same action, then they are equal. An analogue of Cayley's theorem shows that any semigroup can be realized as a transformation semigroup of some set. In automata theory, some authors use the term ''transformation semigroup'' to refer to a semigroup acting faithfully on a set of "states" different from the semigroup's base set. There is a correspondence between the two notions. Transformation semigroups and monoids A transformation semigroup is a pair (''X'',' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Israel Journal Of Mathematics
'' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem ( Magnes Press). History Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the journal publishes articles on all areas of mathematics. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.70, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... was 0.754. External links * Mathematics journals Academic journals established in 1963 Academic journals of Israel English-language journals Bimonthly journals Hebrew University of Jerusalem {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Avraham Trahtman
Avraham Naumovich Trahtman (Trakhtman) (; 10 February 1944 – 17 July 2024) was a Soviet-born Israeli mathematician and academic at Bar-Ilan University (Israel). In 2007, Trahtman solved a problem in combinatorics that had been open for 37 years, the Road Coloring Conjecture posed in 1970. Trahtman died in Jerusalem on 17 July 2024, at the age of 80. Road coloring problem posed and solved Trahtman's solution to the road coloring problem was accepted in 2007 and published in 2009 by the ''Israel Journal of Mathematics''. The problem arose in the subfield of symbolic dynamics, an abstract part of the field of dynamical systems. The road coloring problem was raised by R. L. Adler and L. W. Goodwyn from the United States, and the Israeli mathematician B. Weiss. The proof used results from earlier work. Černý conjecture The problem of estimating the length of synchronizing word has a long history and was posed independently by several authors, but it is commonly known as the Č ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aperiodic Graph
In the mathematical area of graph theory, a directed graph is said to be aperiodic if there is no integer ''k'' > 1 that divides the length of every cycle of the graph. Equivalently, a graph is aperiodic if the greatest common divisor of the lengths of its cycles is one; this greatest common divisor for a graph ''G'' is called the ''period'' of ''G''. Graphs that cannot be aperiodic In any directed bipartite graph, all cycle lengths are even. Therefore, no directed bipartite graph can be aperiodic. In any directed acyclic graph, it is a vacuous truth that every ''k'' divides all cycles (because there are no directed cycles to divide) so no directed acyclic graph can be aperiodic. And in any directed cycle graph, there is only one cycle, so every cycle's length is divisible by ''n'', the length of that cycle. Testing for aperiodicity Suppose that ''G'' is strongly connected and that ''k'' divides the lengths of all cycles in ''G''. Consider the results of per ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly Connected Graph
In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of a directed graph form a partition into subgraphs that are themselves strongly connected. It is possible to test the strong connectivity of a graph, or to find its strongly connected components, in linear time (that is, Θ(''V'' + ''E'')). Definitions A directed graph is called strongly connected if there is a path in each direction between each pair of vertices of the graph. That is, a path exists from the first vertex in the pair to the second, and another path exists from the second vertex to the first. In a directed graph ''G'' that may not itself be strongly connected, a pair of vertices ''u'' and ''v'' are said to be strongly connected to each other if there is a path in each direction between them. The binary relation of being strongly connected is an equivalence relation, and the i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roy Adler
Roy Lee Adler (February 22, 1931 – July 26, 2016) was an American mathematician. Adler earned his Ph.D. in 1961 from Yale University under the supervision of Shizuo Kakutani (''On some algebraic aspects of measure preserving transformations''). He then worked as a mathematician for IBM at the Thomas J. Watson Research Center. Adler studies dynamical systems, ergodic theory, symbolic and topological dynamics and coding theory. The road coloring problem that was solved by Avraham Trakhtman in 2007 came from him, along with L. W. Goodwyn and Benjamin Weiss. He was a fellow of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, .... A paper was written on his work and the impact of his work by Bruce Kitchens and others. Writings *With Brian M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree (graph Theory)
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v is denoted \deg(v) or \deg v. The maximum degree of a graph G is denoted by \Delta(G), and is the maximum of G's vertices' degrees. The minimum degree of a graph is denoted by \delta(G), and is the minimum of G's vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A complete graph (denoted K_n, where n is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, n-1. In a signed graph, the number of positive edges connected to the vertex v is called positive deg(v) and the number of connected negative edges is enti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Graph
In graph theory, a regular graph is a Graph (discrete mathematics), graph where each Vertex (graph theory), vertex has the same number of neighbors; i.e. every vertex has the same Degree (graph theory), degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each internal vertex are equal to each other. A regular graph with vertices of degree is called a graph or regular graph of degree . Special cases Regular graphs of degree at most 2 are easy to classify: a graph consists of disconnected vertices, a graph consists of disconnected edges, and a graph consists of a disjoint union of graphs, disjoint union of cycle (graph theory), cycles and infinite chains. A graph is known as a cubic graph. A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number of neighbors in common, and every non-adjacent pair of vertices has the same number of neighbors in common. The smal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Road Coloring Problem
In graph theory the road coloring theorem, known previously as the road coloring conjecture, deals with synchronized instructions. The issue involves whether by using such instructions, one can reach or locate an object or destination from any other point within a network (which might be a representation of city streets or a maze). In the real world, this phenomenon would be as if you called a friend to ask for directions to his house, and he gave you a set of directions that worked no matter where you started from. This theorem also has implications in symbolic dynamics. The theorem was first conjectured by Roy Adler and Benjamin Weiss. It was proved by Avraham Trahtman. Example and intuition The image to the right shows a directed graph on eight vertices in which each vertex has out-degree 2. (Each vertex in this case also has in-degree 2, but that is not necessary for a synchronizing coloring to exist.) The edges of this graph have been colored red and blue to cre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
