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Bass's Theorem
In geometric group theory, Gromov's theorem on groups of polynomial growth, first proved by Mikhail Gromov, characterizes finitely generated groups of ''polynomial'' growth, as those groups which have nilpotent subgroups of finite index. Statement The growth rate of a group is a well-defined notion from asymptotic analysis. To say that a finitely generated group has polynomial growth means the number of elements of length (relative to a symmetric generating set) at most ''n'' is bounded above by a polynomial function ''p''(''n''). The ''order of growth'' is then the least degree of any such polynomial function ''p''. A nilpotent group ''G'' is a group with a lower central series terminating in the identity subgroup. Gromov's theorem states that a finitely generated group has polynomial growth if and only if it has a nilpotent subgroup that is of finite index. Growth rates of nilpotent groups There is a vast literature on growth rates, leading up to Gromov's theorem. An earlier ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Group Theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such group (mathematics), groups and topology, topological and geometry, geometric properties of spaces on which these groups Group action (mathematics), act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces). Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects. This is usually done by studying the Cayley graphs of groups, which, in addition to the graph (discrete mathematics), graph structure, are endowed with the structure of a metric space, given by the so-called word metric. Geometric group theory, as a distinct area, is relatively new, and became a clearly identifiable branch of mathematics in the late 1980s and early 1990s. Geometric group theory closely interacts with low-dimens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank Of An Abelian Group
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group ''A'' is the cardinality of a maximal linearly independent subset. The rank of ''A'' determines the size of the largest free abelian group contained in ''A''. If ''A'' is torsion-free then it embeds into a vector space over the rational numbers of dimension rank ''A''. For finitely generated abelian groups, rank is a strong invariant and every such group is determined up to isomorphism by its rank and torsion subgroup. Torsion-free abelian groups of rank 1 have been completely classified. However, the theory of abelian groups of higher rank is more involved. The term rank has a different meaning in the context of elementary abelian groups. Definition A subset of an abelian group ''A'' is linearly independent (over Z) if the only linear combination of these elements that is equal to zero is trivial: if : \sum_\alpha n_\alpha a_\alpha = 0, \quad n_\alpha\in\mathbb, where all but finitely many coef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grigorchuk's Group
In the mathematical area of group theory, the Grigorchuk group or the first Grigorchuk group is a finitely generated group constructed by Rostislav Grigorchuk that provided the first example of a finitely generated group of intermediate (that is, faster than polynomial but slower than exponential) growth. The group was originally constructed by Grigorchuk in a 1980 paper and he then proved in a 1984 paper that this group has intermediate growth, thus providing an answer to an important open problem posed by John Milnor in 1968. The Grigorchuk group remains a key object of study in geometric group theory, particularly in the study of the so-called branch groups and automata groups, and it has important connections with the theory of iterated monodromy groups.Volodymyr Nekrashevych''Self-similar groups.''Mathematical Surveys and Monographs, 117. American Mathematical Society, Providence, RI, 2005. . History and significance The growth of a finitely generated group measures the asym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annales Scientifiques De L'École Normale Supérieure
''Annales Scientifiques de l'École Normale Supérieure'' is a French scientific journal of mathematics published by the Société Mathématique de France. It was established in 1864 by the French chemist Louis Pasteur and published articles in mathematics, physics, chemistry, biology, and geology. In 1900, it became a purely mathematical journal. It is published with help of the Centre national de la recherche scientifique. Its web site is hosted by the mathematics department of the École Normale Supérieure École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, Savoi .... External links * Archive(1864–2013) Mathematics journals Publications established in 1864 Multilingual journals Multidisciplinary scientific journals Société Mathématique de France academic journals {{mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Narutaka Ozawa
(born 1974) is a Japanese mathematician, known for his work in operator algebras and discrete groups. He has been a professor at Kyoto University since 2013. He earned a bachelor's degree in mathematics in 1997 from the University of Tokyo and a Ph.D. in mathematics in 2000 from the same institution. One year later he received a Ph.D. in mathematics from Texas A&M University. He was selected for one of the prestigious Sloan Research Fellowships in 2005 and was an invited speaker at the 2006 ICM in Madrid where he gave a talk on "Amenable actions and Applications". He has won numerous prizes including the Mathematical Society of Japan (MSJ) Spring Prize and the Japan Society for the Promotion of Science (JSPS) Prize. Before becoming a full professor at Kyoto University in 2013, he was an associate professor at the University of Tokyo and at University of California, Los Angeles The University of California, Los Angeles (UCLA) is a public land-grant research university in L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous function, continuous, unitary operator, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximate Group
In mathematics, an approximate group is a subset of a group which behaves like a subgroup "up to a constant error", in a precise quantitative sense (so the term approximate subgroup may be more correct). For example, it is required that the set of products of elements in the subset be not much bigger than the subset itself (while for a subgroup it is required that they be equal). The notion was introduced in the 2010s but can be traced to older sources in additive combinatorics. Formal definition Let G be a group and K \ge 1; for two subsets X, Y \subset G we denote by X\cdot Y the set of all products xy, x\in X, y\in Y. A non-empty subset A \subset G is a ''K-approximate subgroup'' of G if: # It is symmetric, that is if g \in A then g^ \in A; # There exists a subset X \subset G of cardinality , X, \le K such that A \cdot A \subset X\cdot A. It is immediately verified that a 1-approximate subgroup is the same thing as a genuine subgroup. Of course this definition is only inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geom
GEOM is the main storage framework for the FreeBSD operating system. It is available in FreeBSD 5.0 and later releases, and provides a standardized way to access storage layers. GEOM is modular and allows for ''geom modules'' to connect to the framework. For example, the geom_mirror module provides RAID1 or mirroring functionality to the system. A number of modules are provided as part of FreeBSD and others have been developed independently and are distributed via (e.g.) GitHub. GEOM was developed for the FreeBSD Project by Poul-Henning Kamp and NAI Labs, the Security Research Division of Network Associates, Inc. under DARPA/ SPAWAR contract N66001-01-C-8035 ("CBOSS"), as part of the DARPA CHATS research program. The name symbolizes its impact on disk geometry. Stacked design Because of geom's modular design, modules can be 'stacked' together to form a chain of geom layers. For example, on top of the geom_mirror module an encryption module can be added, such as geom_eli to pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yehuda Shalom
Judah or Yehuda is the name of a biblical patriarch, Judah (son of Jacob). It may also refer to: Historical ethnic, political and geographic terms * Tribe of Judah, one of the twelve Tribes of Israel; their allotment corresponds to Judah or Judaea * Judea, the name of part of the Land of Israel ** Kingdom of Judah, an Iron Age kingdom of the Southern Levant *** History of ancient Israel and Judah ** Yehud (Persian province), a name introduced in the Babylonian period ** Judaea (Roman province) People * Judah (given name), or Yehudah, including a list of people with the name * Judah (surname) Other uses * Judah, Indiana, a small town in the United States * N Judah, a light trail line in San Francisco, U.S. * Yehuda Matzos, an Israeli matzo company See also * Juda (other) * Judas (other) * Jude (other) * Jews, an ethnoreligious group and nation originating from the Israelites and Hebrews of historical Israel and Judah * Judas Iscariot Judas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Terence Tao
Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes topics in harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing and analytic number theory. Tao was born to ethnic Chinese immigrant parents and raised in Adelaide. Tao won the Fields Medal in 2006 and won the Royal Medal and Breakthrough Prize in Mathematics in 2014. He is also a 2006 MacArthur Fellow. Tao has been the author or co-author of over three hundred research papers. He is widely regarded as one of the greatest living mathematicians and has been referred to as the "Mozart of mathematics". Life and career Family Tao's parents are first-generation immigrants from Hong Kong to Australia.''Wen Wei Po'', Page A4, 24 Au ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The American Mathematical Society
The ''Journal of the American Mathematical Society'' (''JAMS''), is a quarterly peer-reviewed mathematical journal published by the American Mathematical Society. It was established in January 1988. Abstracting and indexing This journal is abstracted and indexed in: 2011. American Mathematical Society. * * * * ISI Ale ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bruce Kleiner
Bruce Alan Kleiner is an American mathematician, working in differential geometry and topology and geometric group theory. He received his Ph.D. in 1990 from the University of California, Berkeley. His advisor was Wu-Yi Hsiang. Kleiner is a professor of mathematics at New York University. Kleiner has written expository papers on the Ricci flow. Together with John Lott of the University of Michigan, he filled in details of Grigori Perelman's proof of the Geometrization conjecture (from which the Poincaré conjecture follows) in the years 2003–2006. Theirs was the first publication acknowledging Perelman's accomplishment (in May, 2006), which was shortly followed by similar papers by Huai-Dong Cao and Xi-Ping Zhu (in June) and John Morgan and Gang Tian (in July). Kleiner found a relatively simple proof of Gromov's theorem on groups of polynomial growth In geometric group theory, Gromov's theorem on groups of polynomial growth, first proved by Mikhail Gromov, characteriz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
