The main result of this article allows formulas of analytic geometry to be elegantly unified, by readily providing parametric as well as cartesian systems of equations. These systems characterize affine subspaces in Rp passing through specified affine subspaces of lower dimension. The problem solved is closely related to constrained principal component analysis. A few interesting applications a...

We consider the task of summing a non-negative function f over a discrete set Ω, e.g., to compute the partition function of a graphical model. Ermon et al. have shown that in a probabilistic approximate sense summation can be reduced to maximizing f over random subsets of Ω defined by parity (XOR) constraints. Unfortunately, XORs with many variables are computationally intractable, while XORs w...

In this paper, we first introduce the notion of $c$-affine functions for $c> 0$. Then we deal with some properties of strongly convex functions in real inner product spaces by using a quadratic support function at each point which is $c$-affine. Moreover, a Hyers–-Ulam stability result for strongly convex functions is shown.

The paper deals with the derivation of improved perturbation bounds for the matrix expression A1A −1 2 A3 and their application to the sensitivity analysis and the solution of fractional–affine matrix equations. An estimate of the overall error in the solution of matrix equations is also given.

We show several estimates on the probability distribution of some data at points in real complete intersection varieties: norms of real affine solutions, condition number of real solution of real systems of multivariate polynomial equations and convergence radius of Newton’s operator for under-determined system of multi-variate polynomial equations.

We investigate a class of nonlinear optimal control problems with pointwise affine control constraints. Necessary optimality conditions of first order and sufficient second order conditions are obtained. For the numerical solution of the optimal control problems a semismooth Newton method is proposed. Local superlinear convergence of the infinite dimensional method is proved. Finally, the prope...

Abstract Let $W$ be a finite Weyl group and $\widetilde W$ the corresponding affine group. A random element of can obtained as reduced walk on alcoves W$. By theorem Lam (Ann. Prob. 2015), such almost surely approaches one $|W|$ many directions. We compute these directions when is $B_n$, $C_n$, $D_n$ weighted by Kac dual labels. This settles Lam’s questions for types $B$ $C$ in affirmative type...