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Ihara Zeta Function
In mathematics, the Ihara zeta function is a zeta function associated with a finite graph. It closely resembles the Selberg zeta function, and is used to relate closed walks to the spectrum of the adjacency matrix. The Ihara zeta function was first defined by Yasutaka Ihara in the 1960s in the context of discrete subgroups of the two-by-two p-adic special linear group. Jean-Pierre Serre suggested in his book ''Trees'' that Ihara's original definition can be reinterpreted graph-theoretically. It was Toshikazu Sunada who put this suggestion into practice in 1985. As observed by Sunada, a regular graph is a Ramanujan graph if and only if its Ihara zeta function satisfies an analogue of the Riemann hypothesis. Definition The Ihara zeta function is defined as the analytic continuation of the infinite product \zeta_\left(u\right)=\prod_\frac The product in the definition is taken over all prime closed geodesics p of the graph G = (V, E), where geodesics which differ by a cyclic rotati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yasutaka Ihara
Yasutaka Ihara (伊原 康隆, ''Ihara Yasutaka''; born 1938, Tokyo Prefecture) is a Japanese mathematician and professor emeritus at the Research Institute for Mathematical Sciences. His work in number theory includes Ihara's lemma and the Ihara zeta function. Career Ihara received his PhD at the University of Tokyo in 1967 with thesis ''Hecke polynomials as congruence zeta functions in elliptic modular case''. From 1965 to 1966, Ihara worked at the Institute for Advanced Study. He was a professor at the University of Tokyo and then at the Research Institute for Mathematical Science (RIMS) of the University of Kyōto. In 2002 he retired from RIMS as professor emeritus and then became a professor at Chūō University. In 1970, he was an invited speaker (with lecture ''Non abelian class fields over function fields in special cases'') at the International Congress of Mathematicians (ICM) in Nice. In 1990, Ihara gave a plenary lecture ''Braids, Galois groups and some arithmetic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle (graph Theory)
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an ''acyclic graph''. A directed graph without directed cycles is called a ''directed acyclic graph''. A connected graph without cycles is called a ''tree''. Definitions Circuit and cycle * A circuit is a non-empty trail in which the first and last vertices are equal (''closed trail''). : Let be a graph. A circuit is a non-empty trail with a vertex sequence . * A cycle or simple circuit is a circuit in which only the first and last vertices are equal. Directed circuit and directed cycle * A directed circuit is a non-empty directed trail in which the first and last vertices are equal (''closed directed trail''). : Let be a directed graph. A directed circuit is a non-empty directed trail with a vertex sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Science+Business Media
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Journal Of Mathematics
The ''International Journal of Mathematics'' was founded in 1990 and is published monthly (with the exception of June and December) by World Scientific. The journal covers mathematics in general. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 0.688. Abstracting and indexing The journal is abstracted and indexed in: * Science Citation Index * ISI Alerting Services * CompuMath Citation Index * Current Contents/Physical, Chemical & Earth Sciences * Zentralblatt MATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastruct ... * Mathematical Reviews * CSA Aerospace Sciences Abstracts Mathematics journals Academic journals established in 1990 World Scientific academic journals English-language journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lecture Notes In Mathematics
''Lecture Notes in Mathematics'' is a book series in the field of mathematics, including articles related to both research and teaching. It was established in 1964 and was edited by A. Dold, Heidelberg and B. Eckmann, Zürich. Its publisher is Springer Science+Business Media (formerly Springer-Verlag). The intent of the series is to publish not only lecture notes, but results from seminars and conferences, more quickly than the several-years-long process of publishing polished journal papers in mathematics. In order to speed the publication process, early volumes of the series (before electronic publishing) were reproduced photographically from typewritten manuscripts. According to Earl Taft it has been "enormously successful" and "is considered a very valuable service to the mathematical community". there have been 2232 volumes in this series. See also * ''Lecture Notes in Physics'' * ''Lecture Notes in Computer Science ''Lecture Notes in Computer Science'' is a series of com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ruelle Zeta Function
In mathematics, the Ruelle zeta function is a zeta function associated with a dynamical system. It is named after mathematical physicist David Ruelle. Formal definition Let ''f'' be a function defined on a manifold ''M'', such that the set of fixed points Fix(''f'' ''n'') is finite for all ''n'' > 1. Further let ''φ'' be a function on ''M'' with values in ''d'' × ''d'' complex matrices. The zeta function of the first kind isTerras (2010) p. 28 : \zeta(z) = \exp\left( \sum_ \frac \sum_ \operatorname \left( \prod_^ \varphi(f^k(x)) \right) \right) Examples In the special case ''d'' = 1, ''φ'' = 1, we have : \zeta(z) = \exp\left( \sum_ \frac m \left, \operatorname(f^m)\ \right) which is the Artin–Mazur zeta function. The Ihara zeta function In mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symbolic Dynamics
In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (evolution) given by the shift operator. Formally, a Markov partition is used to provide a finite cover for the smooth system; each set of the cover is associated with a single symbol, and the sequences of symbols result as a trajectory of the system moves from one covering set to another. History The idea goes back to Jacques Hadamard's 1898 paper on the geodesics on surfaces of negative curvature. It was applied by Marston Morse in 1921 to the construction of a nonperiodic recurrent geodesic. Related work was done by Emil Artin in 1924 (for the system now called Artin billiard), Pekka Myrberg, Paul Koebe, Jakob Nielsen, G. A. Hedlund. The first formal treatment was developed by Morse and Hedlund in their 1938 paper. George Birkhoff, No ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Graph Theory
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. The adjacency matrix of a simple undirected graph is a real symmetric matrix and is therefore orthogonally diagonalizable; its eigenvalues are real algebraic integers. While the adjacency matrix depends on the vertex labeling, its spectrum is a graph invariant, although not a complete one. Spectral graph theory is also concerned with graph parameters that are defined via multiplicities of eigenvalues of matrices associated to the graph, such as the Colin de Verdière number. Cospectral graphs Two graphs are called cospectral or isospectral if the adjacency matrices of the graphs are isospectral, that is, if the adjacency matrices have equal multisets of eigenvalues. Cospectral graphs need not be isomorphic, but isomorphic graphs a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Group
In mathematics, the free group ''F''''S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1''t'', but ''s'' ≠ ''t''−1 for ''s'',''t'',''u'' ∈ ''S''). The members of ''S'' are called generators of ''F''''S'', and the number of generators is the rank of the free group. An arbitrary group ''G'' is called free if it is isomorphic to ''F''''S'' for some subset ''S'' of ''G'', that is, if there is a subset ''S'' of ''G'' such that every element of ''G'' can be written in exactly one way as a product of finitely many elements of ''S'' and their inverses (disregarding trivial variations such as ''st'' = ''suu''−1''t''). A related but different notion is a free abelian group; both notions are particular instances of a free object from universal algebra. As such, free groups are defined by their universal property. History Free ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
