Size function

In size theory, the size function ℓ ( M , φ ) : Δ + = { ( x , y ) ∈ R 2 : x < y } → N {displaystyle ell _{(M,varphi )}:Delta ^{+}={(x,y)in mathbb {R} ^{2}:x<y} o mathbb {N} } associated with the size pair ( M , φ : M → R ) {displaystyle (M,varphi :M o mathbb {R} )} is defined in the following way. For every ( x , y ) ∈ Δ + {displaystyle (x,y)in Delta ^{+}} , ℓ ( M , φ ) ( x , y ) {displaystyle ell _{(M,varphi )}(x,y)} is equal to the number of connected components of the set { p ∈ M : φ ( p ) ≤ y } {displaystyle {pin M:varphi (p)leq y}} that contain at least one point at which the measuring function (a continuous function from a topological space M {displaystyle M} to R k {displaystyle mathbb {R} ^{k}} ) φ {displaystyle varphi } takes a value smaller than or equal to x {displaystyle x} .The concept of size function can be easily extended to the case of a measuring function φ : M → R k {displaystyle varphi :M o mathbb {R} ^{k}} , where R k {displaystyle mathbb {R} ^{k}} is endowed with the usual partial order. A survey about size functions (and size theory) can be found in. Size functions were introduced infor the particular case of M {displaystyle M} equal to the topological space of all piecewise C 1 {displaystyle C^{1}} closed paths in a C ∞ {displaystyle C^{infty }} closed manifold embedded in a Euclidean space. Here the topology on M {displaystyle M} is induced by the C 0 {displaystyle C^{0}} -norm, while the measuring function φ {displaystyle varphi } takes each path γ ∈ M {displaystyle gamma in M} to its length.Inthe case of M {displaystyle M} equal to the topological space of all ordered k {displaystyle k} -tuples of points in a submanifold of a Euclidean space is considered.Here the topology on M {displaystyle M} is induced by the metric d ( ( P 1 , … , P k ) , ( Q 1 … , Q k ) ) = max 1 ≤ i ≤ k ‖ P i − Q i ‖ {displaystyle d((P_{1},ldots ,P_{k}),(Q_{1}ldots ,Q_{k}))=max _{1leq ileq k}|P_{i}-Q_{i}|} . An extension of the concept of size function to algebraic topology was made inwhere the concept of size homotopy group was introduced. Here measuring functions taking values in R k {displaystyle mathbb {R} ^{k}} are allowed.An extension to homology theory (the size functor) was introduced in.The concepts of size homotopy group and size functor are strictly related to the concept of persistent homology groupstudied in persistent homology. It is worth to point out that the size function is the rank of the 0 {displaystyle 0} -th persistent homology group, while the relation between the persistent homology groupand the size homotopy group is analogous to the one existing between homology groups and homotopy groups. Size functions have been initially introduced as a mathematical tool for shape comparison in computer vision and pattern recognition, and have constituted the seed of size theory.The main point is that size functions are invariant for every transformation preserving the measuring function. Hence, they can be adapted to many different applications, by simply changing the measuring function in order to get the wanted invariance. Moreover, size functions show properties of relative resistance to noise, depending on the fact that they distribute the information all over the half-plane Δ + {displaystyle Delta ^{+}} . Assume that M {displaystyle M} is a compact locally connected Hausdorff space. The following statements hold: If we also assume that M {displaystyle M} is a smooth closed manifold and φ {displaystyle varphi } is a C 1 {displaystyle C^{1}} -function, the following useful property holds: . A strong link between the concept of size function and the concept of natural pseudodistance d ( ( M , φ ) , ( N , ψ ) ) {displaystyle d((M,varphi ),(N,psi ))} between the size pairs ( M , φ ) ,   ( N , ψ ) {displaystyle (M,varphi ), (N,psi )} exists