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Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, abstract algebra and logic, and criticized Georg Cantor's work on set theory. Heinrich Weber quoted Kronecker as having said, "'" ("God made the integers, all else is the work of man").The English translation is from Gray. In a footnote, Gray attributes the German quote to "Weber 1891/92, 19, quoting from a lecture of Kronecker's of 1886". Weber, Heinrich L. 1891–1892Kronecker 2:5-23. (The quote is on p. 19.) Kronecker was a student and life-long friend of Ernst Kummer. Biography Leopold Kronecker was born ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liegnitz
Legnica (; , ; ; ) is a city in southwestern Poland, in the central part of Lower Silesia, on the Kaczawa River and the Czarna Woda. As well as being the seat of the county, since 1992 the city has been the seat of the Diocese of Legnica. Legnica had a population of 97,300 inhabitants. The city was first referenced in chronicles dating from the year 1004, although previous settlements could be traced back to the 7th century. The name "Legnica" was mentioned in 1149 under High Duke of Poland Bolesław IV the Curly. Legnica was most likely the seat of Bolesław and it became the residence of the dukes of Legnica from 1248 until 1675. Legnica is a city over which the Piast dynasty reigned the longest, for about 700 years, from the time of ruler Mieszko I of Poland after the creation of the Polish state in the 10th century, until 1675 and the death of the last Piast duke George William. Legnica is one of the historical burial sites of Polish monarchs and consorts. Legni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jules Molk
Jules Molk (8 December 1857 in Strasbourg, France – 7 May 1914 in Nancy) was a French mathematician who worked on elliptic functions. The French Academy of Sciences awarded him the Prix Binoux for 1913. He was appointed to the chair of applied mathematics at the University of Nancy upon the death of Émile Léonard Mathieu in 1890.H. Vogt''Jules Molk, 8. décembre 1857 - 7. mai 1914'' L´Enseignement mathématique, tome 16, 1914, 380–383 From 1902 until his death in 1914, Molk was the leader and editor-in-chief of the publication of a French encyclopedia of pure and applied mathematical sciences based upon Klein's encyclopedia. It was a translation of the volumes in German and required the collaboration of many mathematicians and theoretical physicists from France, Germany, and several other European countries. Among the noteworthy contributors are: Paul Appell, Felix Klein, Jacques Hadamard, David Hilbert, Émile Borel, Paul Montel, Maurice Fréchet, Édouard Goursat, E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker's Theorem
In mathematics, Kronecker's theorem is a theorem about diophantine approximation, introduced by . Kronecker's approximation theorem had been firstly proved by L. Kronecker in the end of the 19th century. It has been now revealed to relate to the idea of n-torus and Mahler measure since the later half of the 20th century. In terms of physical systems, it has the consequence that planets in circular orbits moving uniformly around a star will, over time, assume all alignments, unless there is an exact dependency between their orbital periods. Statement Kronecker's theorem is a result about Diophantine approximations that generalizes Dirichlet's approximation theorem to multiple variables. The Kronecker approximation theorem is classically formulated as follows. :''Given real ''n''-tuples \alpha_i=(\alpha_,\dots,\alpha_)\in\mathbb^n, i=1,\dots,m and \beta=(\beta_1,\dots,\beta_n)\in \mathbb^n , the condition: '' ::\forall \epsilon > 0 \, \exists q_i, p_j \in \mathbb Z : \biggl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker's Lemma
In mathematics, Kronecker's lemma (see, e.g., ) is a result about the relationship between convergence of infinite sums and convergence of sequences. The lemma is often used in the proofs of theorems concerning sums of independent random variables such as the strong Law of large numbers. The lemma is named after the Germany, German mathematician Leopold Kronecker. The lemma If (x_n)_^\infty is an infinite sequence of real numbers such that :\sum_^\infty x_m = s exists and is finite, then we have for all 0 0. Now choose ''N'' so that S_k is ''ε''-close to ''s'' for ''k'' > ''N''. This can be done as the sequence S_k converges to ''s''. Then the right hand side is: : S_n - \frac1\sum_^(b_ - b_k)S_k - \frac1\sum_^(b_ - b_k)S_k : = S_n - \frac1\sum_^(b_ - b_k)S_k - \frac1\sum_^(b_ - b_k)s - \frac1\sum_^(b_ - b_k)(S_k - s) : = S_n - \frac1\sum_^(b_ - b_k)S_k - \fracs - \frac1\sum_^(b_ - b_k)(S_k - s). Now, let ''n'' go to infinity. The first term goes to ''s'', which cancels with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Twelfth Problem
Hilbert's twelfth problem is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. It is one of the 23 mathematical Hilbert problems and asks for analogues of the roots of unity that generate a whole family of further number fields, analogously to the cyclotomic fields and their subfields. Leopold Kronecker described the complex multiplication issue as his , or "dearest dream of his youth", so the problem is also known as Kronecker's Jugendtraum. The classical theory of complex multiplication, now often known as the , does this for the case of any imaginary quadratic field, by using modular functions and elliptic functions chosen with a particular period lattice related to the field in question. Goro Shimura extended this to CM fields. In the special case of totally real fields, Samit Dasgupta and Mahesh Kakde provided a construction of the maximal abelian extension of totally real fields using the Brumer– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker's Congruence
In 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 ar ..., Kronecker's congruence, introduced by Kronecker, states that : \Phi_p(x,y)\equiv (x-y^p)(x^p-y)\bmod p, where ''p'' is a prime and Φ''p''(''x'',''y'') is the modular polynomial of order ''p'', given by :\Phi_n(x,j) = \prod_\tau (x-j(\tau)) for ''j'' the elliptic modular function and τ running through classes of imaginary quadratic integers of discriminant ''n''. References * Modular arithmetic Theorems in number theory {{numtheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker Substitution
Kronecker substitution is a technique named after Leopold Kronecker for determining the coefficients of an unknown polynomial by evaluating it at a single value. If ''p''(''x'') is a polynomial with integer coefficients, and ''x'' is chosen to be both a power of two and larger in magnitude than any of the coefficients of ''p'', then the coefficients of each term of can be read directly out of the binary representation of ''p''(''x''). One application of this method is to reduce the computational problem of multiplying polynomials to the (potentially simpler) problem of multiplying integers. If ''p''(''x'') and ''q''(''x'') are polynomials with known coefficients, then one can use these coefficients to determine a value of ''x'' that is a large enough power of two for the coefficients of the product ''pq''(''x'') to be able to be read off from the binary representation of the number ''p''(''x'')''q''(''x''). Since ''p''(''x'') and ''q''(''x'') are themselves straightforward to deter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quiver (mathematics)
In mathematics, especially representation theory, a quiver is another name for a multidigraph; that is, a directed graph where Loop (graph theory), loops and multiple arrows between two vertex (graph theory), vertices are allowed. Quivers are commonly used in representation theory: a representation of a quiver assigns a vector space to each vertex of the quiver and a linear map to each arrow . In category theory, a quiver can be understood to be the underlying structure of a category (mathematics), category, but without composition or a designation of identity morphisms. That is, there is a forgetful functor from (the category of categories) to (the category of multidigraphs). Its left adjoint is a free functor which, from a quiver, makes the corresponding free category. Definition A quiver consists of: * The set of vertices of * The set of edges of * Two functions: giving the ''start'' or ''source'' of the edge, and another function, givin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker Product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map with respect to a standard choice of basis. The Kronecker product is to be distinguished from the usual matrix multiplication, which is an entirely different operation. The Kronecker product is also sometimes called matrix direct product. The Kronecker product is named after the German mathematician Leopold Kronecker (1823–1891), even though there is little evidence that he was the first to define and use it. The Kronecker product has also been called the ''Zehfuss matrix'', and the ''Zehfuss product'', after , who in 1858 described this matrix operation, but Kronecker product is currently the most widely used term. The misattribution to Kronecker rather than Zehfuss wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker Symbol
In number theory, the Kronecker symbol, written as \left(\frac an\right) or (a, n), is a generalization of the Jacobi symbol to all integers n. It was introduced by . Definition Let n be a non-zero integer, with prime factorization :n=u \cdot p_1^ \cdots p_k^, where u is a unit (i.e., u=\pm1), and the p_i are primes. Let a be an integer. The Kronecker symbol \left(\frac\right) is defined by : \left(\frac\right) := \left(\frac\right) \prod_^k \left(\frac\right)^. For odd p_i, the number \left(\frac\right) is simply the usual Legendre symbol. This leaves the case when p_i=2. We define \left(\frac\right) by : \left(\frac\right) := \begin 0 & \mboxa\mbox \\ 1 & \mbox a \equiv \pm1 \pmod, \\ -1 & \mbox a \equiv \pm3 \pmod. \end Since it extends the Jacobi symbol, the quantity \left(\frac\right) is simply 1 when u=1. When u=-1, we define it by : \left(\frac\right) := \begin -1 & \mboxa 0. Table of values The following is a table of values of Kronecker symbol \left(\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker Limit Formula
In mathematics, the classical Kronecker limit formula describes the constant term at ''s'' = 1 of a real analytic Eisenstein series (or Epstein zeta function) in terms of the Dedekind eta function. There are many generalizations of it to more complicated Eisenstein series. It is named for Leopold Kronecker. First Kronecker limit formula The (first) Kronecker limit formula states that :E(\tau,s) = + 2\pi(\gamma-\log(2)-\log(\sqrt, \eta(\tau), ^2)) +O(s-1), where *''E''(τ,''s'') is the real analytic Eisenstein series, given by :E(\tau,s) =\sum_ for Re(''s'') > 1, and by analytic continuation for other values of the complex number ''s''. *γ is Euler–Mascheroni constant *τ = ''x'' + ''iy'' with ''y'' > 0. * \eta(\tau) = q^\prod_(1-q^n), with ''q'' = e2π i τ is the Dedekind eta function. So the Eisenstein series has a pole at ''s'' = 1 of residue π, and the (first) Kronecker limit formula gives the constant term of the Laurent series at this pole. This formula has an inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an topological manifold, ''n''-manifold, the equivalence classes being connected, injective function, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the manifold decomposition, decomposition of the real coordinate space R''n'' into the cosets ''x'' + R''p'' of the standardly embedding, embedded subspace topology, subspace R''p''. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear manifold, piecewise-linear, differentiable manifold, differentiable (of class ''Cr''), or analytic manifold, analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class ''Cr'' it is usually understood that ''r'' ≥ 1 (otherwise, ''C''0 is a topological foliation). The number ''p'' (the dime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
