Related papers: Optimal Presentations of Mathematical Objects
Optimal geometrical arrangements, such as the stacking of atoms, are of relevance in diverse disciplines. A classic problem is the determination of the optimal arrangement of spheres in three dimensions in order to achieve the highest…
In the past few decades there has been a good deal of papers which are concerned with optimization problems in different areas of mathematics (along 0-1 words, finite or infinite) and which yield - sometimes quite unexpectedly - balanced…
We formulate explicit predictions concerning the symmetry of optimal codes in compact metric spaces. This motivates the study of optimal codes in various spaces where these predictions can be tested.
For any finite dimensional basic associative algebra, we study the presentation spaces and their relation with the representation spaces. We prove two propositions about a general presentation, one on its subrepresentations and the other on…
This introductory survey deals with mathematical and physical properties of discrete structures such as point sets and tilings. The emphasis is on proper generalizations of concepts and ideas from classical crystallography. In particular,…
Systems of equations with sets of integers as unknowns are considered. It is shown that the class of sets representable by unique solutions of equations using the operations of union and addition $S+T=\makeset{m+n}{m \in S, \: n \in T}$ and…
This article proposes a new discrete framework for approximating solutions to shape optimization problems under convexity constraints. The numerical method, based on the support function or the gauge function, is guaranteed to generate…
In this paper we propose to represent a scene as an abstraction of 'things'. We start from 'things' as generated by modern object proposals, and we investigate their immediately observable properties: position, size, aspect ratio and color,…
In information visualization, the position of symbols often encodes associated data values. When visualizing data elements with both a numerical and a categorical dimension, positioning in the categorical axis admits some flexibility. This…
Recent advances in visual representation learning allowed to build an abundance of powerful off-the-shelf features that are ready-to-use for numerous downstream tasks. This work aims to assess how well these features preserve information…
We explore the idea of using automatic and similar kind of presentations of structures to deal with the conceptual problem of natural proof-theoretic ordinal notations. We conclude that this approach still does not meet the goals.
We discuss an extension of classical combinatorics theory to the case of spatially distributed objects.
Being mathematics a natural language to Mankind and to physics, it must be constantly adapted to our necessities and our natural perception. Then, mathematical concepts are not absolute to reality. Although mathematical theories are…
This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…
Experimental evidence indicates that simple models outperform complex deep networks on many unsupervised similarity tasks. We provide a simple yet rigorous explanation for this behaviour by introducing the concept of an optimal…
The concept of representing a polytope that is associated with some combinatorial optimization problem as a linear projection of a higher-dimensional polyhedron has recently received increasing attention. In this paper (written for the…
The information technology explosion has dramatically increased the application of new mathematical ideas and has led to an increasing use of mathematics across a wide range of fields that have been traditionally labeled "pure" or…
This note being devoted to some aspects of the inverse problem of representation theory contains a new insight into it illustrated by two topics. The attention is concentrated on the manner of representation of abstract objects by the…
We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an…
The aim of this short lecture series is to expose the students to the beautiful theory of lattices by, on one hand, demonstrating various basic ideas that appear in this theory and, on the other hand, formulating some of the celebrated…