Leopold Kronecker

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Kronecker, Leopold

 

Born Dec. 7, 1823, in Liegnitz, now Legnica, Poland; died Dec. 29, 1891, in Berlin. German mathematician. Member of the Berlin Academy of Sciences (1861). Professor at the University of Berlin from 1883.

Kronecker’s principal works were devoted to algebra and the theory of numbers, where he continued the studies of his teacher E. Kummer on the theory of quadratic forms and the theory of groups. His studies on the arithmetic theory of algebraic numbers are of extreme importance. Kronecker advocated an “arithmeticization” of mathematics, which he believed should be reduced to the arithmetic of integers. He asserted that only integers possess authentic reality. Defending these one-sided views, Kronecker strongly opposed the principles of the functional-theoretic school of K. Weierstrass and the set-theoretic school of G. Cantor.

WORKS

Werke, vols. 1–5. Leipzig, 1895–1930.
Vorlesungen über Mathematik, parts 1–2. Leipzig, 1894–1903.

REFERENCE

Frobenius, G. Gedächtnissrede auf Leopold Kronecker. Berlin, 1893.
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We remark that if A is the usual tangent bundle of N then (1.24) - (1.26) reduce to the formulae (1.7) - (1.9) since [rho] is the Kronecker endomorphism (1.17).
If [[PHI].sub.1], [[PHI].sub.2], ..., [[PHI].sub.D] are matrices with restricted isometry constants (RIP)), [[delta].sub.K] ([[PHI].sub.2]), ..., [[delta].sub.K]([[PHI].sub.D]), the structure of Kronecker product matrices yields simple bounds for their RIP that can be expressed as
via the Tan's (et al.) method [6], [7] or through the Kronecker summation method [10] and their combination with the sixteen plant theorem [3], [18].
Para esto, mostrara las posturas enfrentadas de autores como Kronecker y Dedekind, o en la actualidad, Connes y Gowers.
Where [M.sup.(2).sub.1] is the Kronecker power 2 of [M.sub.1] and the symbol [cross product] denotes the Kronecker product.
The topics include Kostka systems and exotic t-structures for reflection groups, quantum deformations of irreducible representations of GL(mn) toward the Kronecker problem, generic extensions and composition monoids of cyclic quivers, blocks of truncated q-Schur algebras of type A, a survey of equivariant map algebras with open problems, and forced gradings and the Humphrey-Verma conjecture.
The most common analytical approach used in forced vibration analysis is based on the assumed modes method in conjunction with the classical orthogonality conditions which may be expressed as [[integral].sup.L.sub.0] [Y.sub.i] [Y.sub.j] dx = [[delta].sub.ij], where L is the length of the beam, [Y.sub.i] and [Y.sub.j] are the fth and jth normal mode shapes, respectively, and [[delta].sub.ij] is the Kronecker delta function.
With [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we denote the generalized Kronecker delta: these isotropic tensors are symmetric with respect to all of their 2n subscripts.
where [C.sub.ijkl], [n.sub.i], [n.sub.j], [[delta].sub.ik], [rho], and v are the stiffness tensor, the unit vectors in the directions i and j, the Kronecker delta, wood density, and wave velocity, respectively.
Additionally, we set [[beta].sub.0,k] to be the Kronecker delta [[delta].sub.0,k], which is equal to 1 if k = 0 and 0 otherwise.
A comma in the subscript denotes the spatial derivative and [[delta].sub.ij] is the Kronecker delta.
is the expectation operator, and [delta](k, I) is Kronecker delta function, defined by