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Isospectral
In mathematics, two linear operators are called isospectral or cospectral if they have the same spectrum. Roughly speaking, they are supposed to have the same sets of eigenvalues, when those are counted with multiplicity. The theory of isospectral operators is markedly different depending on whether the space is finite or infinite dimensional. In finite-dimensions, one essentially deals with square matrices. In infinite dimensions, the spectrum need not consist solely of isolated eigenvalues. However, the case of a compact operator on a Hilbert space (or Banach space) is still tractable, since the eigenvalues are at most countable with at most a single limit point ''λ'' = 0. The most studied isospectral problem in infinite dimensions is that of the Laplace operator on a domain in R2. Two such domains are called isospectral if their Laplacians are isospectral. The problem of inferring the geometrical properties of a domain from the spectrum of its Laplacian is ofte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hearing The Shape Of A Drum
In theoretical mathematics, the conceptual problem of "hearing the shape of a drum" refers to the prospect of inferring information about the shape of a hypothetical idealized drumhead from the sound it makes when struck, i.e. from analysis of overtones. "Can One Hear the Shape of a Drum?" is the title of a 1966 article by Mark Kac in the ''American Mathematical Monthly'' which made the question famous, though this particular phrasing originates with Lipman Bers. Similar questions can be traced back all the way to physicist Arthur Schuster in 1882. For his paper, Kac was given the Lester R. Ford Award in 1967 and the Chauvenet Prize in 1968. The frequencies at which a drumhead can vibrate depend on its shape. The Helmholtz equation calculates the frequencies if the shape is known. These frequencies are the eigenvalues of the Laplacian in the space. A central question is whether the shape can be predicted if the frequencies are known; for example, whether a Reuleaux ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Milnor
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook University and the only mathematician to have won the Fields Medal, the Wolf Prize, the Abel Prize and all three Steele prizes. Early life and career Milnor was born on February 20, 1931, in Orange, New Jersey. His father was J. Willard Milnor, an engineer, and his mother was Emily Cox Milnor. As an undergraduate at Princeton University he was named a Putnam Fellow in 1949 and 1950 and also proved the Fáry–Milnor theorem when he was only 19 years old. Milnor graduated with an A.B. in mathematics in 1951 after completing a senior thesis, titled "Link groups", under the supervision of Ralph Fox. He remained at Princeton to pursue graduate studies and received his Ph.D. in mathematics in 1954 after completing a doctoral dissertation, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toshikazu Sunada
is a Japanese mathematician and author of many books and essays on mathematics and mathematical sciences. He is professor emeritus of both Meiji University and Tohoku University. He is also distinguished professor of emeritus at Meiji in recognition of achievement over the course of an academic career. Before he joined Meiji University in 2003, he was professor of mathematics at Nagoya University (1988–1991), at the University of Tokyo (1991–1993), and at Tohoku University (1993–2003). Sunada was involved in the creation of the School of Interdisciplinary Mathematical Sciences at Meiji University and is its first dean (2013–2017). Since 2019, he is President of Mathematics Education Society of Japan. Main work Sunada's work covers complex analytic geometry, spectral geometry, dynamical systems, probability, graph theory, discrete geometric analysis, and mathematical crystallography. Among his numerous contributions, the most famous one is a general construction of iso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinitesimal
In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the "infinity- th" item in a sequence. Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are the reciprocals of one another. Infinitesimal numbers were introduced in the development of calculus, in which the derivative was first conceived as a ratio of two infinitesimal quantities. This definition was not rigorously formalized. As calculus developed further, infinitesimals were replaced by limits, which can be calculated using the standard real numbers. In the 3rd century BC Archimedes used what ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century. One major area of study has been classification: the classification of finite simple groups (those with no nontrivial normal subgroup) was completed in 2004. History During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind Zeta Function
In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function (which is obtained in the case where ''K'' is the field of rational numbers Q). It can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at ''s'' = 1, and its values encode arithmetic data of ''K''. The extended Riemann hypothesis states that if ''ζ''''K''(''s'') = 0 and 0 1. In the case ''K'' = Q, this definition reduces to that of the Riemann zeta function. Euler product The Dedekind zeta function of K has an Euler product which is a product over all the non-zero prime ideals \mathfrak of \mathcal_K :\zeta_K (s) = \prod_ \frac,\text(s)>1. This is the expression in analytic terms of the uniqueness of prime factorizati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Selberg Zeta Function
The Selberg zeta-function was introduced by . It is analogous to the famous Riemann zeta function : \zeta(s) = \prod_ \frac where \mathbb is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics instead of the prime numbers. If \Gamma is a subgroup of SL(2,R), the associated Selberg zeta function is defined as follows, :\zeta_\Gamma(s)=\prod_p(1-N(p)^)^, or :Z_\Gamma(s)=\prod_p\prod^\infty_(1-N(p)^), where ''p'' runs over conjugacy classes of prime geodesics (equivalently, conjugacy classes of primitive hyperbolic elements of \Gamma), and ''N''(''p'') denotes \exp(\textp) (equivalently, the square of the bigger eigenvalue of ''p''). For any hyperbolic surface of finite area there is an associated Selberg zeta-function; this function is a meromorphic function defined in the complex plane. The zeta function is defined in terms of the closed geodesics of the surface. The zeros and poles of the Selberg zeta-function, ''Z''(''s''), can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covering Space
In topology, a covering or covering projection is a continuous function, map between topological spaces that, intuitively, Local property, locally acts like a Projection (mathematics), projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphisms. If p : \tilde X \to X is a covering, (\tilde X, p) is said to be a covering space or cover of X, and X is said to be the base of the covering, or simply the base. By abuse of terminology, \tilde X and p may sometimes be called covering spaces as well. Since coverings are local homeomorphisms, a covering space is a special kind of étalé space. Covering spaces first arose in the context of complex analysis (specifically, the technique of analytic continuation), where they were introduced by Bernhard Riemann, Riemann as domains on which naturally multivalued function, multivalued complex functions become single-valued. These spaces are now called Riemann surfaces. Covering spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Class Field Theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credited as one of pioneers of the notion of a class field. However, this notion was already familiar to Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out. The relevant ideas were developed in the period of several decades, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin (with the help of Chebotarev's theorem). One of the major results is: given a number field ''F'', and writing ''K'' for the maximal abelian unramified extension of ''F'', the Galois group of ''K'' over ''F'' is canonically isomorphic to the ideal class group of ''F''. This statement was generalized to the so called Artin reciprocity law; in the idelic language, writing '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Selberg Trace Formula
In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of a Lie group on the space of square-integrable functions, where is a cofinite discrete group. The character is given by the trace of certain functions on . The simplest case is when is cocompact, when the representation breaks up into discrete summands. Here the trace formula is an extension of the Frobenius formula for the character of an induced representation of finite groups. When is the cocompact subgroup of the real numbers , the Selberg trace formula is essentially the Poisson summation formula. The case when is not compact is harder, because there is a continuous spectrum, described using Eisenstein series. Selberg worked out the non-compact case when is the group ; the extension to higher rank groups is the Arthur–Selberg trace formula. When is the fundamental group of a Riemann surface, the Selberg trace formula describes the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Witt
Ernst Witt (26 June 1911 – 3 July 1991) was a German mathematician, one of the leading algebraists of his time. Biography Witt was born on the island of Alsen, then a part of the German Empire. Shortly after his birth, his parents moved the family to China to work as missionaries, and he did not return to Europe until he was nine. After his schooling, Witt went to the University of Freiburg and the University of Göttingen. He joined the NSDAP (Nazi Party) and was an active party member. Witt was awarded a Ph.D. at the University of Göttingen in 1933 with a thesis titled: "Riemann-Roch theorem and zeta-Function in hypercomplexes" (Riemann-Rochscher Satz und Zeta-Funktion im Hyperkomplexen) that was supervised by Gustav Herglotz, with Emmy Noether suggesting the topic for the doctorate. He qualified to become a lecturer and gave guest lectures in Göttingen and Hamburg. He became associated with the team led by Helmut Hasse who led his habilitation. In June 1936, he gave his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
