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To establish whether a form h(x) is SOS amounts to solving a convex optimization problem. Indeed, any h(x) can be written as = {} ′ (+ ()) {} where {} is a vector containing a base for the forms of degree m in x (such as all monomials of degree m in x), the prime ′ denotes the transpose, H is any symmetric matrix satisfying = {} ′ {} and () is a linear parameterization of the linear ...
For each natural number the corresponding convex relaxation is known as the th level or -th round of the SOS hierarchy. The 1 {\textstyle 1} st round, when d = 1 {\textstyle d=1} , corresponds to a basic semidefinite program , or to sum-of-squares optimization over polynomials of degree at most 2 {\displaystyle 2} .
Polynomial SOS, polynomials that are sums of squares of other polynomials; The Brahmagupta–Fibonacci identity, representing the product of sums of two squares of polynomials as another sum of squares; Hilbert's seventeenth problem on characterizing the polynomials with non-negative values as sums of squares
This following list features abbreviated names of mathematical functions, function-like operators and other mathematical terminology. This list is limited to abbreviations of two or more letters (excluding number sets).
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of statements within those systems.It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations such as addition and multiplication.
Hilbert's basis theorem (commutative algebra,invariant theory) Hilbert's Nullstellensatz (theorem of zeroes) (commutative algebra, algebraic geometry) Hilbert–Schmidt theorem (functional analysis) Hilbert–Speiser theorem (cyclotomic fields) Hilbert–Waring theorem (number theory) Hilbert's irreducibility theorem (number theory)
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Proceedings of Symposia in Pure Mathematics. Vol. XXVIII.2. American Mathematical Society. pp. 483–489. ISBN 0-8218-1428-1. Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. Zbl 1068.11023. Lorenz, Falko (2008). Algebra.
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