Consider the problem of minimizing a polynomial $f$ over compact semialgebraic set $\mathbf{X} \subseteq \mathbb{R}^n$. Lasserre introduces hierarchies semidefinite programs to approximate this hard optimization problem, based on classical sum-of-squares certificates positivity polynomials due Putinar and Schmüdgen. When $\mathbf{X}$ is unit ball or standard simplex, we show that Schmüdgen-type...

Indonesia adalah negara dengan sistem demokrasi dalam pemerintahannya. Adanya pemilihan presiden yang dilakukan selama 5 tahun sekali dari masa kemerdekaan sampai sekarang. Pemilihan atau sering disebut pemilu (pemilihan umum) ini berguna untuk memilih calon dan wakil sebuah negara. Mengingat adanya pergantian setelah 2 periode, para remaja jaman sekarang cenderung mengikuti millennial. Sehingg...

Two unitary integral transforms with a very-well poised 7F6-function as a kernel are given. For both integral transforms the inverse is the same as the original transform after an involution on the parameters. The 7F6-function involved can be considered as a non-polynomial extension of the Wilson polynomial, and is therefore called a Wilson function. The two integral transforms are called a Wil...

We address the question: Does there exist a non-trivial knot with a trivial Jones polynomial? To find such a knot, it is almost certainly sufficient to find a non-trivial braid on four strands in the kernel of the Burau representation. I will describe a computer algorithm to search for such a braid.

Recall some basic facts about eigenspaces — we say that Eα is an eigenspace of a matrixM of dimension d if the dimension of the nullspace of αI −M has dimension d. Moreover, by the theory covered in class, we must have that the polynomial (λ− α) divides the characteristic polynomial of M , which is defined to be det(λI −M). We first calculate the eigenspaces of Bn. Observe that if α = 1, then w...

In this paper we study strictly positive definite functions on the unit sphere of the m-dimensional Euclidean space. Such functions can be used for solving a scattered data interpolation problem on spheres. Since positive definite functions on the sphere were already characterized by Schoenberg some fifty years ago, the issue here is to determine what kind of positive definite functions are act...

We present a first theoretical analysis of the power of polynomial-time preprocessing for important combinatorial problems from various areas in AI. We consider problems from Constraint Satisfaction, Global Constraints, Satisfiability, Nonmonotonic and Bayesian Reasoning. We show that, subject to a complexity theoretic assumption, none of the considered problems can be reduced by polynomial-tim...