98 research outputs found
Lower bound theorems for general polytopes
For a -dimensional polytope with vertices, , we
calculate precisely the minimum possible number of -dimensional faces, when
or . This confirms a conjecture of Gr\"unbaum, for these
values of . For , we solve the same problem when or ; the
solution was already known for . In all these cases, we give a
characterisation of the minimising polytopes. We also show that there are many
gaps in the possible number of -faces: for example, there is no polytope
with 80 edges in dimension 10, and a polytope with 407 edges can have dimension
at most 23.Comment: 26 pages, 3 figure
Proximal-point-like algorithms for abstract convex minimisation problems
In this paper we introduce two conceptual algorithms for minimising abstract
convex functions. Both algorithms rely on solving a proximal-type subproblem
with an abstract Bregman distance based proximal term. We prove their
convergence when the set of abstract linear functions forms a linear space.
This latter assumption can be relaxed to only require the set of abstract
linear functions to be closed under the sum, which is a classical assumption in
abstract convexity. We provide numerical examples on the minimisation of
nonconvex functions with the presented algorithms.Comment: 14 pages, 6 figure
Mathematics Yearbook 2021
The Deakin University Mathematics Yearbook publishes student reports and articles in all areas of mathematics with an aim of promoting interest and engagement in mathematics and celebrating student achievements. The 2021 edition includes 7 coursework articles, where students have extended upon submissions in their mathematics units, as well as 4 articles based on student research projects conducted throughout 2020 and 2021
Chebyshev multivariate polynomial approximation : alternance interpretation
In this paper, we derive optimality conditions for Chebyshev approximation of multivariate functions. The theory of Chebyshev (uniform) approximation for univariate functions was developed in the late nineteenth and twentieth century. The optimality conditions are based on the notion of alternance (maximal deviation points with alternating deviation signs). It is not clear, however, how to extend the notion of alternance to the case of multivariate functions. There have been several attempts to extend the theory of Chebyshev approximation to the case of multivariate functions. We propose an alternative approach, which is based on the notion of convexity and nonsmooth analysis
- …