Reply to “Comments on ‘Analytical expression of explicit MPC solution via lattice piecewise-affine function’ [Automatica 45 (2009) 910–917]”

Automatica ◽  
2012 ◽  
Vol 48 (11) ◽  
pp. 2995-2996 ◽  
Author(s):  
Chengtao Wen ◽  
Xiaoyan Ma ◽  
B. Erik Ydstie
Keyword(s):  
Affine Function ◽  
Analytical Expression ◽  
Piecewise Affine

Analytical expression of explicit MPC solution via lattice piecewise-affine function

Automatica ◽  
2009 ◽  
Vol 45 (4) ◽  
pp. 910-917 ◽  
Author(s):  
Chengtao Wen ◽  
Xiaoyan Ma ◽  
B. Erik Ydstie
Keyword(s):  
Affine Function ◽  
Analytical Expression ◽  
Piecewise Affine

scholarly journals Prime numbers with a certain extremal type property

2018 ◽  
Vol 17 (1) ◽  
pp. 127-151
Author(s):  
Edward Tutaj
Keyword(s):  
Riemann Hypothesis ◽  
Convex Set ◽  
Counting Function ◽  
Numerical Data ◽  
Prime Numbers ◽  
Affine Function ◽  
Piecewise Affine ◽  
Order Of Magnitude ◽  
Type Property ◽  
Prime Counting Function

Abstract The convex hull of the subgraph of the prime counting function x → π(x) is a convex set, bounded from above by a graph of some piecewise affine function x → (x). The vertices of this function form an infinite sequence of points $({e_k},\pi ({e_k}))_1^\infty $ . The elements of the sequence (ek)1∞ shall be called the extremal prime numbers. In this paper we present some observations about the sequence (ek)1∞ and we formulate a number of questions inspired by the numerical data. We prove also two – it seems – interesting results. First states that if the Riemann Hypothesis is true, then ${{{e_k} + 1} \over {{e_k}}} = 1$ . The second, also depending on Riemann Hypothesis, describes the order of magnitude of the differences between consecutive extremal prime numbers.

scholarly journals Box-constrained optimization for minimax supervised learning

2021 ◽  
Vol 71 ◽  
pp. 101-113
Author(s):  
Cyprien Gilet ◽  
Susana Barbosa ◽  
Lionel Fillatre
Keyword(s):  
Constrained Optimization ◽  
Empirical Bayes ◽  
Optimization Procedure ◽  
Affine Function ◽  
Bayes Risk ◽  
Piecewise Affine ◽  
Empirical Risk ◽  
Polyhedral Domain ◽  
Subgradient Algorithm ◽  
Constrained Minimax

In this paper, we present the optimization procedure for computing the discrete boxconstrained minimax classifier introduced in [1, 2]. Our approach processes discrete or beforehand discretized features. A box-constrained region defines some bounds for each class proportion independently. The box-constrained minimax classifier is obtained from the computation of the least favorable prior which maximizes the minimum empirical risk of error over the box-constrained region. After studying the discrete empirical Bayes risk over the probabilistic simplex, we consider a projected subgradient algorithm which computes the prior maximizing this concave multivariate piecewise affine function over a polyhedral domain. The convergence of our algorithm is established.

scholarly journals The topological entropy of iterated piecewise affine maps is uncomputable

2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
Pascal Koiran
Keyword(s):  
Cellular Automata ◽  
Real Number ◽  
Topological Entropy ◽  
Linear Functions ◽  
Affine Function ◽  
Piecewise Affine ◽  
Affine Maps ◽  
International Audience

International audience We show that it is impossible to compute (or even to approximate) the topological entropy of a continuous piecewise affine function in dimension four. The same result holds for saturated linear functions in unbounded dimension. We ask whether the topological entropy of a piecewise affine function is always a computable real number, and conversely whether every non-negative computable real number can be obtained as the topological entropy of a piecewise affine function. It seems that these two questions are also open for cellular automata.

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