In mathematics the symmetrization methods are algorithms of transforming a set to a ball with equal volume and centered at the origin. B is called the symmetrized version of A, usually denoted . These algorithms show up in solving the classical isoperimetric inequality problem, which asks: Given all two-dimensional shapes of a given area, which of them has the minimal perimeter (for details see Isoperimetric inequality). The conjectured answer was the disk and Steiner in 1838 showed this to be true using the Steiner symmetrization method (described below). From this many other isoperimetric problems sprung and other symmetrization algorithms. For example, Rayleigh's conjecture is that the first eigenvalue of the Dirichlet problem is minimized for the ball (see Rayleigh–Faber–Krahn inequality for details). Another problem is that the Newtonian capacity of a set A is minimized by and this was proved by Polya and G. Szego (1951) using circular symmetrization (described below).
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In mathematics the symmetrizationmethods are algorithms of transforming a set A ⊂ R n {\displaystyle A\subset \mathbb {R} ^{n}} to a ball B ⊂ R n {\displaystyle...
U_{ijk\dots }=U_{(ij)k\dots }+U_{[ij]k\dots }.} A shorthand notation for anti-symmetrization is denoted by a pair of square brackets. For example, in arbitrary dimensions...
Contraction of indices is represented by joining the index lines together. Symmetrization of indices is represented by a thick zig-zag or wavy bar crossing the...
{\textstyle s} , this would only be true in the absence of symmetrization postulate. Due to the symmetrization postulate, the choice of s {\textstyle s} will influence...
solution was made by Swiss geometer Jakob Steiner in 1838, using a geometric method later named Steiner symmetrisation. Steiner showed that if a solution existed...
using a symmetrized/antisymmetrized wavefunction and that independent treatment of these wavefunctions gives the same result. Hence the symmetrization postulate...
redundant names of the same quantum many-body state. So the symmetrization (or anti-symmetrization) must be introduced to eliminate this redundancy in the...
{\displaystyle \tau } . If it is not symmetrical, it can be replaced by a symmetrized kernel, which is the average over the n! permutations of these n variables...
and can't be described with Schrödinger mechanics). When symmetrization or anti-symmetrization is unnecessary, N-particle spaces of states can be obtained...
inconsistent. When applying a rule to a number of indices (differentiation, symmetrization etc., shown next), the bracket or punctuation symbols denoting the rules...
are the main constituents of ordinary matter. Anti-symmetrization of wave functions Symmetrization of wave functions Pauli exclusion principle Bose–Einstein...
brackets and parentheses on the indices denote the antisymmetrization and symmetrization operators, respectively. If there is nonzero torsion, the Bianchi identities...
Nagel, Federalism Beyond Federations: Asymmetry and Processes of Re-symmetrization in Europe (Aldershot: Ashgate, 2011), p. 39. R. Quinault, "Scots on...
semicolon represents a covariant derivative, and the brackets denote anti-symmetrization. The first equation asserts that the 4-divergence of the 2-form F is...
formula Mixed volume Power of a point theorem Steiner curve Steiner symmetrization Steiner system Steiner surface Steiner conic Steiner's conic problem...
Killing tensor – symmetric (0,2)-tensor field T such that the total symmetrization of its covariant derivative vanishesPages displaying wikidata descriptions...
2022-10-09. Crane, K.; Weischedel, C.; Wardetzky, M. (2017). "The Heat Method for Distance Computation". Communications of the ACM. 60 (11): 90–99. doi:10...
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