1,721,066 research outputs found
A note on the representability of compact sets by real sequences
No full textAbstractConsider the set K of all nonempty compact subsets of a compact metric space (M, d), endowed with the Hausdorff metric. In this paper, we prove that K is isometric to a subset of l∞(R). An approximation result is also proved
Size functions and formal series
No full textIn this paper we consider a mathematical tool for shape description called size function. We prove that every size function can be represented as a set of points and lines in the real plane, with multiplicities. This allows for an algebraic approach to size functions and the construction of new pseudo-distances between size functions for comparing shapes
An Introduction to the Notion of Natural Pseudo-distance in Topological Data Analysis
Full text linkThe natural pseudo-distance dG associated with a group G of self-homeomorphisms of a topological space X is a pseudo-metric developed to compare real-valued functions defined on X, when the equivalence between functions is expressed by the group G. In this paper, we illustrate dG, its role in topological data analysis, its main properties and its link with persistent homology
Connections between size functions and critical points
No full textA mathematical approach to the concept of shape of a submanifold.M of a Edclidean space had previously been given by means of 'measuring functions' (e.g. diameter or volume) and of the derived 'size functions'. This paper relates the study and the computation of any such size function to the structure of critical points of the associated measuring function
A distance for similarity classes of submanifolds of a Euclidean space
No full textA distance is defined on the quotient of the set of submanifolds of a Euclidean space, with respect to similarity. It is then related to a previously defined function which captures the metric behaviour of paths. © 1990, Australian Mathematical Society. All rights reserved
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