5,431 research outputs found
How to Integrate a Polynomial over a Simplex
This paper settles the computational complexity of the problem of integrating
a polynomial function f over a rational simplex. We prove that the problem is
NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin
and Straus. On the other hand, if the polynomial depends only on a fixed number
of variables, while its degree and the dimension of the simplex are allowed to
vary, we prove that integration can be done in polynomial time. As a
consequence, for polynomials of fixed total degree, there is a polynomial time
algorithm as well. We conclude the article with extensions to other polytopes,
discussion of other available methods and experimental results.Comment: Tables added with new experimental results. References adde
Software for Exact Integration of Polynomials over Polyhedra
We are interested in the fast computation of the exact value of integrals of
polynomial functions over convex polyhedra. We present speed ups and extensions
of the algorithms presented in previous work. We present the new software
implementation and provide benchmark computations. The computation of integrals
of polynomials over polyhedral regions has many applications; here we
demonstrate our algorithmic tools solving a challenge from combinatorial voting
theory.Comment: Major updat
Genuinely sharp heat kernel estimates on compact rank-one symmetric spaces, for Jacobi expansions, on a ball and on a simplex
We prove genuinely sharp two-sided global estimates for heat kernels on all
compact rank-one symmetric spaces. This generalizes the authors' recent result
obtained for a Euclidean sphere of arbitrary dimension. Furthermore, similar
heat kernel bounds are shown in the context of classical Jacobi expansions, on
a ball and on a simplex. These results are more precise than the qualitatively
sharp Gaussian estimates proved recently by several authors.Comment: 16 page
Sobolev orthogonal polynomials on a simplex
The Jacobi polynomials on the simplex are orthogonal polynomials with respect
to the weight function W_\bg(x) = x_1^{\g_1} ... x_d^{\g_d} (1-
|x|)^{\g_{d+1}} when all \g_i > -1 and they are eigenfunctions of a second
order partial differential operator L_\bg. The singular cases that some, or
all, \g_1,...,\g_{d+1} are -1 are studied in this paper. Firstly a complete
basis of polynomials that are eigenfunctions of L_\bg in each singular case
is found. Secondly, these polynomials are shown to be orthogonal with respect
to an inner product which is explicitly determined. This inner product involves
derivatives of the functions, hence the name Sobolev orthogonal polynomials.Comment: 32 page
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