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Chip-firing Game
The chip-firing game is a one-player game on a graph which was invented around 1983 and since has become an important part of the study of structural combinatorics. Each vertex has the number of "chips" indicated by its state variable. On each firing, a vertex is selected and one of its chips is transferred to each neighbour (vertex it shares an edge with). The number of chips on each vertex cannot be negative. The game ends when no firing is possible. Definition Let the finite graph ''G'' be connected and loopless, with vertices ''V'' = . Let deg(''v'') be the degree of a vertex, and e(''v,w'') the number of edges between vertices ''v'' and ''w''. A configuration or state of the game is defined by assigning each vertex a nonnegative integer ''s''(''v''), representing the number of chips on this vertex. A move starts with selecting a vertex ''w'' which has at least as many chips as its degree: ''s''(''w'') ≥ deg(''w''). The vertex ''w'' is fired, moving one chip from w along ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chip Firing Game Example Graph
Chromatin immunoprecipitation (ChIP) is a type of immunoprecipitation experimental technique used to investigate the interaction between proteins and DNA in the cell. It aims to determine whether specific proteins are associated with specific genomic regions, such as transcription factors on promoters or other DNA binding sites, and possibly define cistromes. ChIP also aims to determine the specific location in the genome that various histone modifications are associated with, indicating the target of the histone modifiers. ChIP is crucial for the advancements in the field of epigenomics and learning more about epigenetic phenomena. Briefly, the conventional method is as follows: # DNA and associated proteins on chromatin in living cells or tissues are crosslinked (this step is omitted in Native ChIP). # The DNA-protein complexes (chromatin-protein) are then sheared into ~500 bp DNA fragments by sonication or nuclease digestion. # Cross-linked DNA fragments associated with the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Mathematics Of Chip-Firing
''The Mathematics of Chip-Firing'' is a textbook in mathematics on chip-firing games and abelian sandpile models. It was written by Caroline Klivans, and published in 2018 by the CRC Press. Topics A chip-firing game, in its most basic form, is a process on an undirected graph, with each vertex of the graph containing some number of chips. At each step, a vertex with more chips than incident edges is selected, and one of its chips is sent to each of its neighbors. If a single vertex is designated as a "black hole", meaning that chips sent to it vanish, then the result of the process is the same no matter what order the other vertices are selected. The stable states of this process are the ones in which no vertex has enough chips to be selected; two stable states can be added by combining their chips and then stabilizing the result. A subset of these states, the so-called critical states, form an abelian group under this addition operation. The abelian sandpile model applies this m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Sandpile Model
The Abelian sandpile model (ASM) is the more popular name of the original Bak–Tang–Wiesenfeld model (BTW). BTW model was the first discovered example of a dynamical system displaying self-organized criticality. It was introduced by Per Bak, Chao Tang and Kurt Wiesenfeld in a 1987 paper. Three years later Deepak Dhar discovered that the BTW sandpile model indeed follows the abelian dynamics and therefore referred to this model as the Abelian sandpile model. The model is a cellular automaton. In its original formulation, each site on a finite grid has an associated value that corresponds to the slope of the pile. This slope builds up as "grains of sand" (or "chips") are randomly placed onto the pile, until the slope exceeds a specific threshold value at which time that site collapses transferring sand into the adjacent sites, increasing their slope. Bak, Tang, and Wiesenfeld considered process of successive random placement of sand grains on the grid; each such place ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by ''exactly one'' path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ''at most one'' path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A polytreeSee . (or directed tree or oriented treeSee .See . or singly connected networkSee .) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is call ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Sandpile Model
The Abelian sandpile model (ASM) is the more popular name of the original Bak–Tang–Wiesenfeld model (BTW). BTW model was the first discovered example of a dynamical system displaying self-organized criticality. It was introduced by Per Bak, Chao Tang and Kurt Wiesenfeld in a 1987 paper. Three years later Deepak Dhar discovered that the BTW sandpile model indeed follows the abelian dynamics and therefore referred to this model as the Abelian sandpile model. The model is a cellular automaton. In its original formulation, each site on a finite grid has an associated value that corresponds to the slope of the pile. This slope builds up as "grains of sand" (or "chips") are randomly placed onto the pile, until the slope exceeds a specific threshold value at which time that site collapses transferring sand into the adjacent sites, increasing their slope. Bak, Tang, and Wiesenfeld considered process of successive random placement of sand grains on the grid; each such place ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation \cdot that combines any two elements a and b of A to form another element of A, denoted a \cdot b. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Sandpile Model
The Abelian sandpile model (ASM) is the more popular name of the original Bak–Tang–Wiesenfeld model (BTW). BTW model was the first discovered example of a dynamical system displaying self-organized criticality. It was introduced by Per Bak, Chao Tang and Kurt Wiesenfeld in a 1987 paper. Three years later Deepak Dhar discovered that the BTW sandpile model indeed follows the abelian dynamics and therefore referred to this model as the Abelian sandpile model. The model is a cellular automaton. In its original formulation, each site on a finite grid has an associated value that corresponds to the slope of the pile. This slope builds up as "grains of sand" (or "chips") are randomly placed onto the pile, until the slope exceeds a specific threshold value at which time that site collapses transferring sand into the adjacent sites, increasing their slope. Bak, Tang, and Wiesenfeld considered process of successive random placement of sand grains on the grid; each such place ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–Roch Theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus ''g'', in a way that can be carried over into purely algebraic settings. Initially proved as Riemann's inequality by , the theorem reached its definitive form for Riemann surfaces after work of Riemann's short-lived student . It was later generalized to algebraic curves, to higher-dimensional varieties and beyond. Preliminary notions A Riemann surface X is a topological space that is locally homeomorphic to an open subset of \Complex, the set of complex numbers. In addition, the transition maps between these open subsets are required to be holomorphic. The latter condition allows one to transfer the notions and methods of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
European Journal Of Combinatorics
European, or Europeans, or Europeneans, may refer to: In general * ''European'', an adjective referring to something of, from, or related to Europe ** Ethnic groups in Europe ** Demographics of Europe ** European cuisine European cuisine comprises the cuisines of Europe "European Cuisine." [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chip Firing Game Example Sequence
Chromatin immunoprecipitation (ChIP) is a type of immunoprecipitation experimental technique used to investigate the interaction between proteins and DNA in the cell. It aims to determine whether specific proteins are associated with specific genomic regions, such as transcription factors on promoters or other DNA binding sites, and possibly define cistromes. ChIP also aims to determine the specific location in the genome that various histone modifications are associated with, indicating the target of the histone modifiers. ChIP is crucial for the advancements in the field of epigenomics and learning more about epigenetic phenomena. Briefly, the conventional method is as follows: # DNA and associated proteins on chromatin in living cells or tissues are crosslinked (this step is omitted in Native ChIP). # The DNA-protein complexes (chromatin-protein) are then sheared into ~500 bp DNA fragments by sonication or nuclease digestion. # Cross-linked DNA fragments associated with the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antimatroid
In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by including elements one at a time, and in which an element, once available for inclusion, remains available until it is included. Antimatroids are commonly axiomatized in two equivalent ways, either as a set system modeling the possible states of such a process, or as a formal language modeling the different sequences in which elements may be included. Dilworth (1940) was the first to study antimatroids, using yet another axiomatization based on lattice theory, and they have been frequently rediscovered in other contexts. The axioms defining antimatroids as set systems are very similar to those of matroids, but whereas matroids are defined by an '' exchange axiom'', antimatroids are defined instead by an ''anti-exchange axiom'', from which their name derives. Antimatroids can be viewed as a special case of greedoids and of semimodular lattices, and as a generalization of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
