Action-Graph Games (AGGs) (Bhat & Leyton-Brown 2004) are a fully expressive game representation which can compactly express strict and context-specific independence and anonymity structure in players’ utility functions. We present an efficient algorithm for computing expected payoffs under mixed strategy profiles. This algorithm runs in time polynomial in the size of the AGG representation (whi...

A procedure is presented in this paper for developing a representation of lognormal stochastic processes via the polynomial chaos expansion. These are processes obtained by applying the exponential operator to a gaussian process. The Polynomial Chaos expansion results in a representation of a stochastic process in terms of mul-tidimensional polynomials orthogonal with respect to the Gaussian me...

Sparse grid interpolation is widely used to provide good approximations to smooth functions in high dimensions based on relatively few function evaluations. By using an efficient conversion from the interpolating polynomial provided by evaluations on a sparse grid to a representation in terms of orthogonal polynomials (gPC representation), we show how to use these relatively few function evalua...

Let S = {x ∈ R | g1(x) ≥ 0, . . . , gm(x) ≥ 0} be a basic closed semialgebraic set defined by real polynomials gi. Putinar’s Positivstellensatz says that, under a certain condition stronger than compactness of S, every real polynomial f positive on S posesses a representation f = P m i=0 σigi where g0 := 1 and each σi is a sum of squares of polynomials. Such a representation is a certificate fo...

We use the Krammer representation of the braid group in Libgober’s invariant and construct a new multivariate polynomial invariant for curve complements: Krammer polynomial. We show that the Krammer polynomial of an essential braid is equal to zero. We also compute the Krammer polynomials of some certain n-gonal curves.

A given explicit piecewise affine representation of an MPC feedback law is approximated by a single polynomial, computed using linear programming. This polynomial state feedback control law guarantees closed-loop stability and constraint satisfaction. The polynomial feedback can be implemented in real time even on very simple devices with severe limitations on memory storage.

8 Matrices 16 8.1 Representation of vector spaces and linear transformations . . . . . . . . . . . . . . . 16 8.2 Similarity transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 8.3 Direct sums of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 8.4 The companion matrix of a polynomial . . . . . . . . . . . . . . . . . . . . . . . ....

The representation polynomial, developed by K. Johnson, is a mathematical equation corresponding to a directed acyclic graph. Johnson’s algorithm is based on decomposing the graph into a set of chain graphs which have a known equation for the representation polynomial. However, the analysis time and computer memory requirements for calculating the representation polynomial for an arbitrary grap...

Given a polynomial x ∈ Rn 7→ p(x) in n = 2 variables, a symbolic-numerical algorithm is first described for detecting whether the connected component of the plane sublevel set P = {x : p(x) ≥ 0} containing the origin is rigidly convex, or equivalently, whether it has a linear matrix inequality (LMI) representation, or equivalently, if polynomial p(x) is hyperbolic with respect to the origin. Th...