On polynomial integrals over the orthogonal group
نویسندگان
چکیده
منابع مشابه
On polynomial integrals over the orthogonal group
We consider integrals of type ∫ On u1 11 . . . u an 1nu b1 21 . . . u bn 2n du, with respect to the Haar measure on the orthogonal group. We establish several remarkable invariance properties satisfied by such integrals, by using combinatorial methods. We present as well a general formula for such integrals, as a sum of products of factorials. Introduction The computation of polynomial integral...
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A recursion formula is derived which allows one to evaluate invariant integrals over the orthogonal group O(N), where the integrand is an arbitrary finite monomial in the matrix elements of the group. The value of such an integral is expressible as a finite sum of partial fractions in N . The recursion formula largely extends presently available integration formulas for the orthogonal group. © ...
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The large N limit of some matrix integrals over the orthogonal group O(N) and its relation with those pertaining to the unitary group U(N) are reexamined. It is proved that limN→∞ N −2 ∫ DO expN Tr JO is half the corresponding function in U(N), with a similar relation for limN→∞ ∫ DO expN Tr (AOBO), for A and B both symmetric or both skew symmetric. Large N limit of integrals over the orthogona...
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We study the integrals of type I(a) = ∫ On ∏ u aij ij du, depending on a matrix a ∈Mp×q(N), whose exact computation is an open problem. Our results are as follows: (1) an extension of the “elementary expansion” formula from the case a ∈M2×q(2N) to the general case a ∈Mp×q(N), (2) the construction of the “best algebraic normalization” of I(a), in the case a ∈M2×q(N), (3) an explicit formula for ...
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The little Grothendieck problem (a special case of Boolean quadratic optimization) consists of maximizing ∑ ij Cijxixj over binary variables xi ∈ {±1}, where C is a positive semidefinite matrix. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C ∈ Rdn×dn a positive semidefinite matrix, the objective is to maximi...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series A
سال: 2011
ISSN: 0097-3165
DOI: 10.1016/j.jcta.2010.11.015