443 research outputs found
Random polytopes in smooth convex bodies
There is an error in the proof of Theorem 2 of my paper [1]. It appears on; age 91, lines 10 and 11: the application of the affine transformation T changes he measure on the Grassmannian G(d, d-i) which is not taken into account. As a result, in the statement of Theorem 2 the coefficient is not correct
Central limit theorems for Gaussian polytopes
Choose random, independent points in according to the standard
normal distribution. Their convex hull is the {\sl Gaussian random
polytope}. We prove that the volume and the number of faces of satisfy
the central limit theorem, settling a well known conjecture in the field.Comment: to appear in Annals of Probabilit
Magic Mirrors, Dense Diameters, Baire Category
Abstract. An old result of Zamfirescu says that for most convex
curves C in the plane most points in R
2
lie on infinitely many
normals to C, where most is meant in Baire category sense. We
strengthen this result by showing that ‘infinitely many’ can be
replaced by ‘continuum many’ in the statement. We present further
theorems in the same spirit
CONVEX-BODIES, ECONOMIC CAP COVERINGS, RANDOM POLYTOPES
Let K be a convex compact body with nonempty interior in the d-dimensional Euclidean space Rd and let x1, …, xn be random points in K, independently and uniformly distributed. Define Kn = conv {x1, …, xn}. Our main concern in this paper will be the behaviour of the deviation of vol Kn from vol K as a function of n, more precisely, the expectation of the random variable vol (K\Kn). We denote this expectation by E (K, n)
The cocked hat: formal statements and proofs of the theorems
Navigators have been taught for centuries to estimate the location of their craft on a map from three lines of position, for redundancy. The three lines typically form a triangle, called a cocked hat. How is the location of the craft related to the triangle? For more than 80 years navigators have also been taught that, if each line of position is equally likely to pass to the right and to the left of the true location, then the likelihood that the craft is in the triangle is exactly 1/4. This is stated in numerous reputable sources, but was never stated or proved in a mathematically formal and rigorous fashion. In this paper we prove that the likelihood is indeed 1/4 if we assume that the lines of position always intersect pairwise. We also show that the result does not hold under weaker (and more reasonable) assumptions, and we prove a generalisation to lines
'Fair' Partitions of Polygons - an Introduction
We address the question: Given a positive integer , can any 2D convex
polygonal region be partitioned into convex pieces such that all pieces
have the same area and same perimeter? The answer to this question is easily
`yes' for =2. We prove the answer to be `yes' for =4 and also discuss
higher powers of 2.Comment: 7 pages. 1 figure. This version (v6) is mostly a formal reworking of
the main proof in v2 which was uploaded in December 200
On the equivalence of strong formulations for capacitated multi-level lot sizing problems with setup times
Several mixed integer programming formulations have been proposed for modeling capacitated multi-level lot sizing problems with setup times. These formulations include the so-called facility location formulation, the shortest route formulation, and the inventory and lot sizing formulation with (l,S) inequalities. In this paper, we demonstrate the equivalence of these formulations when the integrality requirement is relaxed for any subset of binary setup decision variables. This equivalence has significant implications for decomposition-based methods since same optimal solution values are obtained no matter which formulation is used. In particular, we discuss the relax-and-fix method, a decomposition-based heuristic used for the efficient solution of hard lot sizing problems. Computational tests allow us to compare the effectiveness of different formulations using benchmark problems. The choice of formulation directly affects the required computational effort, and our results therefore provide guidelines on choosing an effective formulation during the development of heuristic-based solution procedures
A computational analysis of lower bounds for big bucket production planning problems
In this paper, we analyze a variety of approaches to obtain lower bounds for multi-level production planning problems with big bucket capacities, i.e., problems in which multiple items compete for the same resources. We give an extensive survey of both known and new methods, and also establish relationships between some of these methods that, to our knowledge, have not been presented before. As will be highlighted, understanding the substructures of difficult problems provide crucial insights on why these problems are hard to solve, and this is addressed by a thorough analysis in the paper. We conclude with computational results on a variety of widely used test sets, and a discussion of future research
Why 'scaffolding' is the wrong metaphor : the cognitive usefulness of mathematical representations.
The metaphor of scaffolding has become current in discussions of the cognitive help we get from artefacts, environmental affordances and each other. Consideration of mathematical tools and representations indicates that in these cases at least (and plausibly for others), scaffolding is the wrong picture, because scaffolding in good order is immobile, temporary and crude. Mathematical representations can be manipulated, are not temporary structures to aid development, and are refined. Reflection on examples from elementary algebra indicates that Menary is on the right track with his ‘enculturation’ view of mathematical cognition. Moreover, these examples allow us to elaborate his remarks on the uniqueness of mathematical representations and their role in the emergence of new thoughts.Peer reviewe
Mixed integer programming in production planning with backlogging and setup carryover : modeling and algorithms
This paper proposes a mixed integer programming formulation for modeling the capacitated multi-level lot sizing problem with both backlogging and setup carryover. Based on the model formulation, a progressive time-oriented decomposition heuristic framework is then proposed, where improvement and construction heuristics are effectively combined, therefore efficiently avoiding the weaknesses associated with the one-time decisions made by other classical time-oriented decomposition algorithms. Computational results show that the proposed optimization framework provides competitive solutions within a reasonable time
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