Double-sided probing by map of Asplund's distances using Logarithmic Image Processing in the framework of Mathematical Morphology

TL;DR

This paper links Mathematical Morphology with Asplund's distances using Logarithmic Image Processing, showing how the distance map relates to dilations and erosions, and applies it to pattern matching.

Contribution

It introduces a novel connection between Asplund's distances and morphological operations within the Logarithmic Image Processing framework, simplifying computations with flat structuring elements.

Findings

The distance map is the logarithm of the ratio of dilation and erosion.

The approach simplifies with flat structuring elements.

Application demonstrated in pattern matching.

Abstract

We establish the link between Mathematical Morphology and the map of Asplund's distances between a probe and a grey scale function, using the Logarithmic Image Processing scalar multiplication. We demonstrate that the map is the logarithm of the ratio between a dilation and an erosion of the function by a structuring function: the probe. The dilations and erosions are mappings from the lattice of the images into the lattice of the positive functions. Using a flat structuring element, the expression of the map of Asplund's distances can be simplified with a dilation and an erosion of the image; these mappings stays in the lattice of the images. We illustrate our approach by an example of pattern matching with a non-flat structuring function.

Equations

$$f\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle+$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle+$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle+$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle+$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle+$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle+$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle+$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle+$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle+$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle+$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle+$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle+$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle+$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle+$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle+$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle+$}}$}}$}}}g=f+g-\frac{f.g}{M}$$

$$\lambda\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}f=M-M\left(1-\frac{f}{M}\right)^{\lambda}$$

$$d_{As}^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}(f,g)=\ln\left(\lambda/\mu\right)$$

$$\forall(f^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}},g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}})\in(I^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}})^{2},\quad d_{As}^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}(f^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}},g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}})=d_{As}^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}(f_{1},g_{1})$$

$$As_{t}^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}f:\left\{\begin{array}[]{ccc}\mathcal{I}^{*}\times(\mathcal{T}^{*})^{D_{t}}&\rightarrow&\left(\mathbb{R}^{+}\right)^{D}\\ (f,t)&\rightarrow&As_{t}^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}f(x)=d_{As}^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}(f_{\left|D_{t}(x)\right.},t)\\ \end{array}\right.$$

$$\begin{array}[]{ccc}\left(\wedge_{\overline{\mathcal{I}}}\mathscr{X}\right)(x)&=&\wedge_{[0,M]}\left\{f(x):f\in\mathscr{X},\>x\in D\right\}\\ \left(\vee_{\overline{\mathcal{I}}}\mathscr{X}\right)(x)&=&\vee_{[0,M]}\left\{f(x):f\in\mathscr{X},\>x\in D\right\}.\\ \end{array}$$

$$i_{x,t}(x)=t;i_{x,t}(y)=0\text{ if }x\ne y.$$

$${\delta }_W(f)=\vee \left\{W(i_{x,f(x)}),x\in D\right\}$$

$$\begin{array}[]{ccc}(\delta_{B}(f))(x)&=&\vee\left\{f(x-h)+B(h),h\in D_{B}\right\}=(f\oplus B)(x)\\ (\varepsilon_{B}(f))(x)&=&\wedge\left\{f(x+h)-B(h),h\in D_{B}\right\}=(f\ominus B)(x)\\ \end{array}$$

$$\lambda_{B}f:\left\{\begin{array}[]{ccc}\overline{\mathcal{I}}\times(\mathcal{T}^{*})^{D_{B}}&\rightarrow&\left(\overline{\mathbb{R}}^{+}\right)^{D}\\ (f,B)&\rightarrow&\lambda_{B}f(x)=\wedge\left\{\alpha(x),f(x+h)\leq\alpha(x)\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}B(h),h\in D_{B}\right\}.\\ \end{array}\right.$$

$$\mu_{B}f:\left\{\begin{array}[]{ccc}\overline{\mathcal{I}}\times(\mathcal{T}^{*})^{D_{B}}&\rightarrow&\left(\overline{\mathbb{R}}^{+}\right)^{D}\\ (f,B)&\rightarrow&\mu_{B}f(x)=\vee\left\{\beta(x),\beta(x)\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}B(h)\leq f(x+h),h\in D_{B}\right\}.\\ \end{array}\right.$$

$$As_{B}^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}f=\ln\left(\frac{\lambda_{B}f}{\mu_{B}f}\right),\text{ with }f>0.$$

$$\begin{array}[]{lcl}\alpha(x)\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}B(h)\geq f(x+h)&\Leftrightarrow&M-M\left(1-B(h)/M\right)^{\alpha(x)}\geq f(x+h),\>\text{ (from eq. \ref{eq:LIPtimes})}\\ &\Leftrightarrow&\alpha(x)\geq\frac{\ln{\left(1-\frac{f(x+h)}{M}\right)}}{\ln{\left(1-\frac{B(h)}{M}\right)}},\>\text{because }\left(1-\frac{B(h)}{M}\right)\in]0,1[.\end{array}$$

$${\lambda }_{B}f=\wedge \left\{\alpha (x),\alpha (x)\ge \frac{\tilde{f}(x+h)}{\tilde{B}(h)},h\in D_B\right\}=\vee \left\{\frac{\tilde{f}(x+h)}{\tilde{B}(h)},h\in D_B\right\}.$$

$${\mu }_{B}f=\vee \left\{\beta (x),\beta (x)\le \frac{\tilde{f}(x+h)}{\tilde{B}(h)},h\in D_B\right\}=\wedge \left\{\frac{\tilde{f}(x+h)}{\tilde{B}(h)},h\in D_B\right\}.$$

$$\begin{array}[]{ccl}\lambda_{B_{0}}f&=&\frac{1}{\widetilde{B}_{0}}\wedge\left\{\widetilde{f}(x+h),h\in D_{B}\right\},\quad\text{ because }\widetilde{B}_{0}<0\\ &=&\frac{1}{\widetilde{B}_{0}}\ln{\left(1-\frac{\vee\{f(x-h),-h\in D_{B}\}}{M}\right)}\\ &=&\frac{1}{\widetilde{B}_{0}}\ln{\left(1-\frac{\delta_{\widecheck{D}_{B}}f}{M}\right)}\end{array}$$

$$\begin{array}[]{ccl}\mu_{B_{0}}f&=&\frac{1}{\widetilde{B}_{0}}\vee\left\{\widetilde{f}(x+h),h\in D_{B}\right\},\quad\text{ because }\widetilde{B}_{0}<0\\ &=&\frac{1}{\widetilde{B}_{0}}\ln{\left(1-\frac{\wedge\{f(x+h),h\in D_{B}\}}{M}\right)}\\ &=&\frac{1}{\widetilde{B}_{0}}\ln{\left(1-\frac{\varepsilon_{D_{B}}f}{M}\right)}.\end{array}$$

$$As_{B_{0}}^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}f=\ln{\left[\frac{\ln{\left(1-\frac{\delta_{\widecheck{D}_{B}}f}{M}\right)}}{\ln{\left(1-\frac{\varepsilon_{D_{B}}f}{M}\right)}}\right]},\>\text{ with }f>0.$$

$$As_{B}^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}f=\ln\left(\frac{\lambda_{B}f}{\mu_{B}f}\right)=\ln{\left(\frac{\vee\left\{\frac{\widetilde{f}(x+h)}{\widetilde{B}(h)},h\in D_{B}\right\}}{\wedge\left\{\frac{\widetilde{f}(x+h)}{\widetilde{B}(h)},h\in D_{B}\right\}}\right)},\>\text{ with }f>0.$$

$$\begin{array}[]{ccl}\lambda_{B}(f\vee g)&=&\vee\left\{\frac{\widetilde{f\vee g}(x+h)}{\widetilde{B}(h)},h\in D_{B}\right\}\\ &=&\vee\left\{\frac{\widetilde{f}(x+h)\wedge\widetilde{g}(x+h)}{\widetilde{B}(h)},h\in D_{B}\right\},\>\text{because $\widetilde{f}$ is decreasing}\\ &=&\vee\left\{\frac{\widetilde{f}(x+h)}{\widetilde{B}(h)}\vee\frac{\widetilde{g}(x+h)}{\widetilde{B}(h)},h\in D_{B}\right\},\>\text{because }\widetilde{B}(h)<0\\ &=&\left[\vee\left\{\frac{\widetilde{f}(x+h)}{\widetilde{B}(h)},h\in D_{B}\right\}\right]\vee\left[\vee\left\{\frac{\widetilde{g}(x+h)}{\widetilde{B}(h)},h\in D_{B}\right\}\right]\\ &=&\lambda_{B}(f)\vee\lambda_{B}(g).\end{array}$$

$${\lambda }_B(O)={\lambda }_B(f_0)=\wedge \left\{\alpha (x),\alpha (x)\ge \frac{\tilde{0}(x+h)}{\tilde{B}(h)},h\in D_B\right\}=0=O.$$

$$\begin{array}[]{ccl}\mu_{B}(f\wedge g)&=&\wedge\left\{\frac{\widetilde{f\wedge g}(x+h)}{\widetilde{B}(h)},h\in D_{B}\right\}\\ &=&\wedge\left\{\frac{\widetilde{f}(x+h)\vee\widetilde{g}(x+h)}{\widetilde{B}(h)},h\in D_{B}\right\},\>\text{because $\widetilde{f}$ is decreasing}\\ &=&\left[\wedge\left\{\frac{\widetilde{f}(x+h)}{\widetilde{B}(h)},h\in D_{B}\right\}\right]\wedge\left[\wedge\left\{\frac{\widetilde{g}(x+h)}{\widetilde{B}(h)},h\in D_{B}\right\}\right],\>\text{because }\widetilde{B}(h)<0\\ &=&\mu_{B}(f)\wedge\mu_{B}(g).\end{array}$$

$${\mu }_B(I)={\mu }_B(f_M)=\vee \left\{\beta (x),\beta (x)\le \frac{\tilde{M}(x+h)}{\tilde{B}(h)},h\in D_B\right\}=+\infty =I.$$

$$\begin{array}[]{lll}\left.\begin{array}[]{l}d_{As}^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}(f^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}},g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}})=0\Rightarrow\lambda=\mu\\ (\text{def. \ref{def:asplund_metric}})\Rightarrow\lambda\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}\geq f^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}\geq\mu\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}\end{array}\right\}&\Rightarrow\lambda\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}=f^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}&\Rightarrow f^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}=g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}\text{ in }\mathcal{I}^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}\\ \end{array}$$

$$\begin{array}[]{ll}\left.\begin{array}[]{l}\forall f^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}},g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}\in\mathcal{I}^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}},f^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}=g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}\Rightarrow\lambda\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}=f^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}\\ (\text{def. \ref{def:asplund_metric}})\lambda\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}\geq f^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}\geq\mu\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}\end{array}\right\}&\Rightarrow\lambda\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}=f^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}=\mu\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}\\ \\ \Rightarrow\lambda=\mu\Rightarrow d_{As}^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}(f^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}},g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}})=0\end{array}$$

$$d_{As}^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}(f^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}},g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}})+d_{As}^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}(g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}},h^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}})=\ln\left(\frac{\lambda_{1}\lambda_{2}}{\mu_{1}\mu_{2}}\right)$$

$$\begin{array}[]{lcl}\text{Def. \ref{def:asplund_metric} }&\Rightarrow&\left\{\begin{array}[]{l}\lambda_{1}=\inf\left\{k_{1}:\forall x,k_{1}\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}(x)\geq f^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}(x)\right\}\\ \lambda_{2}=\inf\left\{k_{2}:\forall x,k_{2}\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}h^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}_{j}(x)\geq g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}(x)\right\}\\ \lambda_{3}=\inf\left\{k_{3}:\forall x,k_{3}\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}h^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}_{j}(x)\geq f^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}(x)\right\}\\ \end{array}\right.\\ \end{array}$$

$$\begin{array}[]{llll}\Rightarrow&\lambda_{1}\lambda_{2}&\leq&\inf_{k_{1}}\left\{\inf_{k_{2}}\left\{\forall x,k_{1}\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}(k_{2}\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}h^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}_{j}(x))\geq k_{1}\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}g^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}(x)\right\}\geq f^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}(x)\right\}\\ &&\leq&\inf\left\{k^{\prime}:\forall x,k^{\prime}\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 1.4pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}h^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}_{j}(x)\geq f^{\mathbin{\ooalign{$\bigtriangleup$\crcr\raise 0.98pt\hbox{$\mathchoice{\scalebox{1.0}[0.7]{$\displaystyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\textstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}{\scalebox{1.0}[0.7]{$\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{$\displaystyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\textstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptstyle\scriptscriptstyle\times$}}{\scalebox{0.7}[1.0]{$\scriptscriptstyle\scriptscriptstyle\times$}}$}}$}}}}(x)\right\}\text{, with }k^{\prime}=k_{1}\times k_{2}\\ \Rightarrow&\lambda_{1}\lambda_{2}&\leq&\lambda_{3}\text{, with }\lambda_{1},\lambda_{2},\lambda_{3}>0\\ \end{array}$$

$${\mu }_1{\mu }_2\ge {\mu }_3\text{, with }{\mu }_1,{\mu }_2,{\mu }_3>0$$

Full text

1 Introduction

Asplund’s metric is a useful method of pattern matching based on a double-sided probing, i.e. a probing by a greatest lower bound probe and a least upper bound probe. It was originally defined for binary shapes, or sets [1, 7], by using the smallest homothetic shape (probe) containing the shape to be analysed and the greatest homothetic probe contained by the shape. Jourlin et al. [13, 14] have extended this metric to functions and to grey-level images in the framework of the Logarithmic Image Processing (LIP) [15, 16] using a multiplicative or an additive LIP law [11]. Then, Asplund’s metric has been extended to colour and multivariate images by a marginal approach in [23, 11] or by a spatio-colour (i.e. vectorial [21, 22]) approach in [24].

Other approaches of double-sided probing have been previously defined in the framework of Mathematical Morphology [19, 27]. The well-known hit-or-miss transform [27] allows to extract all the pixels such that the first set of a structuring element fits the object while the second set misses it (i.e. fits its background). An extension based on two operations of dilation (for grey level images) has been proposed in [18]. It consists of a unique structuring element, which is used in the two dilations in order to match the signal from above and from below.

Banon et al. [2] use two structuring elements obtained by two translations of a unique template along the grey level axis. They use an erosion and an anti-dilation to count the pixels whose values are in between the two structuring elements.

Odone et al. [25] use an approach inspired by the computation of the Hausdorff distance. They consider a grey level image as a tridimensional (3D) graph. They dilate by a 3D ball a template in order to compute a 3D “interval”. Then, for any point of the image, they translate vertically the “interval” in order to contain the maximum number of points of the function and they count this number.

Barat et al. [4] present a unified framework for these last three methods. They show that they correspond to a neighbourhood of functions (i.e. a tolerance tube) with a different metric for each method. Their topological approach is named virtual double-sided image probing (VDIP) and they defined it as a difference between a grey-scale dilation and an erosion. For pattern matching, only the patterns which are in the tolerance tube are selected. It is a metric defined on the equivalence class of functions according to an additive grey level shift.

In [13], Jourlin et al. have introduced the logarithmic homothetic defined according to the LIP multiplication. This makes a compensation of the lighting variation due to a multiplicative effect, i.e. a thickening or a thinning of the object crossed by the light.

In the current paper, the important novelty introduced is the link between the map of Asplunds’ metrics defined in the LIP multiplicative framework and the operations of Mathematical Morphology. We will show that the map of Asplunds’ distances in the LIP multiplicative framework is the logarithm of the ratio between a dilation and an erosion.

This gives access to many other notions well defined in the corpus of Mathematical Morphology.

The paper is organised as follows: 1) a reminder of the main notions (LIP, Asplund’s metrics, fundamental operations and framework of Mathematical Morphology), 2) the demonstration of the link between the map of Asplund’s distances and Mathematical Morphology for flat structuring element (se) and for non-flat ones and 3) an illustration of pattern matching with Asplund’s metric.

2 Prerequisites

In the current section, we remind the different mathematical notions and frameworks to be used: LIP model, Asplund’s metric and the basis of Mathematical Morphology.

2.1 Logarithmic Image Processing (LIP)

The LIP model, created by Jourlin et al. [15, 17, 16, 10], is a mathematical framework for image processing based on the physical law of transmittance. It is perfectly suited to process images acquired with transmitted light (when the object is located between the source and the sensor) but also with reflected light, due to the consistency of the model with human vision [6]. The mathematical operations performed using the LIP model are consistent with the physical principles of image formation. Therefore the values of an image defined in $[0,M[$ stay in this bounded domain. For 8 bits images $M=256$ and the 256 grey levels are in the range of integers $[0,...,255]$.

A grey scale image $f$ is a function defined on a domain $D\subset\mathbb{R}^{N}$ with values in $\mathcal{T}=\mathcal{[}0,M[$, $M\in\mathbb{R}$. $f$ is a member of the space $\mathcal{I}=\mathcal{T}^{D}$.

Due to the link with the transmittance law: $T_{f}=1-f/M$, the grey-scale is inverted in the LIP framework, [math] corresponds to the white extremity of the grey scale, when no obstacle is located between the source and the sensor, while the other extremity $M$ corresponds to the black value, when the source cannot be transmitted through the obstacle.

In the LIP sense, the addition of two images corresponds to the superposition of two obstacles (objects) generating $f$ and $g$:

[TABLE]

From this law, we deduce the LIP multiplication of $f$ by a scalar $\lambda\in\mathbb{R}$:

[TABLE]

It corresponds to a thickness change of the observed object in the ratio $\lambda$. If $\lambda>1$, the thickness is increased and the image becomes darker than $f$, while if $\lambda\in[0,1[$, the thickness is decreased and the image becomes brighter than $f$.

The LIP laws satisfy strong mathematical properties. Let $\mathcal{F}(D,[-\infty,M[)$ be the set of functions defined on $D$ with values in $]-\infty,M[$. We equipped it with the two logarithmic laws and (\mathcal{F}(D,[-\infty,M[),\mathbin{\ooalign{\bigtriangleup\crcr\raise 1.4pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle+}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle+}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle+}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle+}}}}}}},\mathbin{\ooalign{\bigtriangleup\crcr\raise 1.4pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}) becomes a real vector space. (\mathcal{I},\mathbin{\ooalign{\bigtriangleup\crcr\raise 1.4pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle+}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle+}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle+}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle+}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle+}}}}}}},\mathbin{\ooalign{\bigtriangleup\crcr\raise 1.4pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}) is the positive cone of this vector space [16].

There exists a colour version of the LIP model [12].

The LIP framework has been successfully applied to numerous problems for industry, medical applications, digital photography, etc. It gives access to new notions of contrast and metrics which take into account the variation of light, for example the Asplund’s metric for functions.

2.2 Asplund’s metric for functions using the LIP multiplicative law

Let us remind the novel notion of Asplund’s metric defined in [13, 14] for functions in place of sets. It consists of using the logarithmic homothetic \lambda\mathbin{\ooalign{\bigtriangleup\crcr\raise 1.4pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}f.

Let $\mathcal{T}^{*}=]0,M[$ and the space of positive images be $\mathcal{I}^{*}={\mathcal{T}^{*}}^{D}$.

Definition 1

Asplund’s metric Given two images $f$, $g\in\mathcal{I}^{*}$, $g$ is chosen as the probing function for example, and we define the two numbers: \lambda=\inf\left\{\alpha,f\leq\alpha\mathbin{\ooalign{\bigtriangleup\crcr\raise 1.4pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}g\right\} and \mu=\sup\left\{\beta,\beta\mathbin{\ooalign{\bigtriangleup\crcr\raise 1.4pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}g\leq f\right\}. The corresponding “functional Asplund’s metric” d_{As}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}} (with the LIP multiplication) is:

[TABLE]

From a mathematical point of view [11], d_{As}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}} is a metric if the images $f,g\in\mathcal{I}^{*}$ are replaced by their equivalence classes f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}=\{g/\exists k>0,k\mathbin{\ooalign{\bigtriangleup\crcr\raise 1.4pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}g=f\} and g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}.

The relation (\exists k>0,k\mathbin{\ooalign{\bigtriangleup\crcr\raise 1.4pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}g=f) is clearly an equivalence relation written $fRg$, because it satisfies the three properties: i) reflexivity $\forall f\in\mathcal{I},fRf$, ii) symmetry $\forall(f,g)\in\mathcal{I}^{2}$, $fRg\Leftrightarrow gRf$ and iii) transitivity $\forall(f,g,h)\in\mathcal{I}^{3}$, $(fRg$ and $gRh)\Rightarrow fRh$. Let us now give a rigorous definition of the multiplicative Asplund’s metric using the space of equivalence classes \mathcal{I}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}

[TABLE]

d_{As}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}(f_{1},g_{1}) is defined by eq. 3 between two elements $f_{1}$ and $g_{1}$ of the equivalence classes f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}} and g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}.

The demonstration of the metric properties are in the appendix (section 6).

Several examples have shown the interest of using Asplund’s metric for pattern matching between a template function $t:D_{t}\rightarrow\mathcal{T}$ and the function $f$. For each point $x$ of $D$, the distance d_{As}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}(f_{\left|D_{t}(x)\right.},t) is computed in the neighbourhood $D_{t}(x)$ centred in $x$, with $f_{\left|D_{t}(x)\right.}$ being the restriction of $f$ to $D_{t}(x)$. Therefore, one can define a map of Asplund’s distances [23].

Definition 2

Map of Asplund’s distances Given a grey-level image $f\in\mathcal{I}^{*}$ and a probe $t\in(\mathcal{T}^{*})^{D_{t}}$, $t>0$, their map of Asplund’s distances is:

[TABLE]

$D_{t}(x)$ is the neighbourhood associated to $D_{t}$ centred in $x\in D$.

One can notice that the template $t$ is acting like a structuring element.

2.3 Short reminder on Mathematical Morphology

In this subsection we give a reminder of the basis notions used in Mathematical Morphology (MM) [19]. MM is defined in complete lattices [27, 9, 3, 5].

Definition 3

Complete lattice Given a set $\mathscr{L}$ and a binary relation $\leq$ defining a partial order on $\mathscr{L}$, we say that $(\mathscr{L},\leq)$ is a partially ordered set or poset. $\mathscr{L}$ is a complete lattice if any non empty subset $\mathscr{X}$ of $\mathscr{L}$ has a supremum (a least upper bound) and an infimum (a greatest lower bound). The infimum and the supremum will be denoted, respectively, by $\wedge\mathscr{X}$ and $\vee\mathscr{X}$. Two elements of the complete lattice $\mathscr{L}$ are important: the least element $O$ and the greatest element $I$.

The set of images from $D$ to $[0,M]$, $\overline{\mathcal{I}}=[0,M]^{D}$, is a complete lattice with the partial order relation $\leq$, by inheritance of the complete lattice structure of $[0,M]$. The least and greatest elements are the constant functions $f_{0}$ and $f_{M}$ whose values are equal respectively to [math] and $M$ for all elements of $D$. The supremum and infimum are respectively, for any $\mathscr{X}\subset\overline{\mathcal{I}}$

[TABLE]

The set of functions $\overline{\mathbb{R}}^{D}$ is also a complete lattice with $\overline{\mathbb{R}}=\mathbb{R}\cup\{-\infty,+\infty\}$ with the usual order $\leq$, like the set of all subsets of $D$, written $\mathcal{P}(D)$, with the set inclusion $\subset$.

Definition 4

Erosion, dilation, anti-erosion, anti-dilation [3] Given $\mathscr{L}_{1}$ and $\mathscr{L}_{2}$ two complete lattices, a mapping $\psi\in\mathscr{L}_{2}^{\mathscr{L}_{1}}$ is

an erosion iff         $\forall\mathscr{X}\subset\mathscr{L}_{1}$, $\psi(\wedge\mathscr{X})=\wedge\psi(\mathscr{X})$, then we write $\varepsilon=\psi$; 2. 2.

a dilation iff          $\forall\mathscr{X}\subset\mathscr{L}_{1}$, $\psi(\vee\mathscr{X})=\vee\psi(\mathscr{X})$, then we write $\delta=\psi$; 3. 3.

an anti-erosion iff   $\forall\mathscr{X}\subset\mathscr{L}_{1}$, $\psi(\wedge\mathscr{X})=\vee\psi(\mathscr{X})$, then we write $\varepsilon^{a}=\psi$; 4. 4.

an anti-dilation iff $\forall\mathscr{X}\subset\mathscr{L}_{1}$, $\psi(\vee\mathscr{X})=\wedge\psi(\mathscr{X})$, then we write $\delta^{a}=\psi$.

As the definitions of these mappings apply even to the empty subset of $\mathscr{L}_{1}$, we have: $\varepsilon(I)=I$, $\delta(O)=O$, $\varepsilon^{a}(I)=O$ and $\delta^{a}(O)=I$.

Erosions and dilations are increasing mappings: $\forall\mathscr{X},\mathscr{Y}\subset\mathscr{L}_{1}$, $\mathscr{X}\leq\mathscr{Y}\Rightarrow\psi(\mathscr{X})\leq\psi(\mathscr{Y})$ while anti-erosions and anti-dilations are decreasing mappings: $\forall\mathscr{X},\mathscr{Y}\subset\mathscr{L}_{1}$, $\mathscr{X}\leq\mathscr{Y}\Rightarrow\psi(\mathscr{Y})\leq\psi(\mathscr{X})$.

Definition 5

Structuring element [26, 9, 29] Let us define a pulse function $i_{x,t}\in\overline{\mathcal{I}}$ of level $t$ at the point $x$:

[TABLE]

The function $f$ can be decomposed into the supremum of its pulses $f=\vee\left\{i_{x,f(x)},x\in D\right\}$. It is easy to define dilations and erosions which are not translation-invariant (in the domain $D$). Let $W$ be a map $\overline{\mathcal{I}}\rightarrow\overline{\mathcal{I}}$ associating to each pulse function $i_{x,t}\in\overline{\mathcal{I}}$ a (functional) “window” $W(i_{x,t})$. Then the operator $\delta_{W}:\overline{\mathcal{I}}\rightarrow\overline{\mathcal{I}}$ defined by:

[TABLE]

is a dilation. When all ‘windows” $W(i_{x,f(x)})$ are translation invariant (in $D$), they take the form $W(i_{x,f(x)})=B(x)$ with $B(x)=B_{x}$ being a structuring element (or structuring function).

In this case the previously defined dilation $\delta$ and erosion $\varepsilon$, in the same lattice $(\overline{\mathcal{I}},\leq)$, can be simplified:

[TABLE]

$D_{B}\subset D$ is the definition domain of the structuring function $B:D_{B}\rightarrow\overline{\mathcal{T}}$. The symbols $\oplus$ and $\ominus$ represent the extension to functions [27] of Minkowski operations between sets [20, 8]. Notice: in the case of a flat structuring element with its values equal to zero (i.e. $\forall x\in D_{B}$, $B(x)=0$), we have $\delta_{B}(f)(x)=\vee\left\{f(x-h),h\in D_{B}\right\}=\delta_{D_{B}}(f)(x)$ and $\varepsilon_{B}(f)(x)=\wedge\left\{f(x+h),h\in D_{B}\right\}=\varepsilon_{D_{B}}(f)(x)$.

3 Map of Aplund’s distances and mathematical morphology

We now link the map of Asplund’s distances with Mathematical Morphology.

Given $\overline{\mathbb{R}}^{+}=[0,+\infty]$ a complete lattice with the natural order $\leq$, the map of the least upper bounds $\lambda_{B}$ between the probe $B\in(\mathcal{T}^{*})^{D_{B}}$ and the function $f\in\overline{\mathcal{I}}$ is defined as:

[TABLE]

The map of the greatest lower bounds $\mu_{B}$ between the probe $B\in(\mathcal{T}^{*})^{D_{B}}$ and the function $f\in\overline{\mathcal{I}}$ is defined as:

[TABLE]

The two mappings $\lambda_{B}$ and $\mu_{B}$ are defined between two complete lattices $\mathscr{L}_{1}=(\overline{\mathcal{I}},\leq)$ and $\mathscr{L}_{2}=(\left(\overline{\mathbb{R}}^{+}\right)^{D},\leq)$ with the natural order $\leq$. Therefore, the least element of $(\mathscr{L}_{1},\leq)$ corresponds to the constant function equal to zero, $O=f_{0}$ and the greatest element is the constant function equal to $M$, $I=f_{M}$.

Using the equations 10 and 11 the map of Asplund’s distances (eq. 5) can be simplified:

[TABLE]

In addition, $\forall x\in D$, $\forall h\in D_{B}$, $\forall\alpha\in\mathbb{R}^{+}$, we have:

[TABLE]

We assume that $\widetilde{f}=\ln{\left(1-f/M\right)}$. Using equation 13, equation 10 becomes:

[TABLE]

In a similar way:

[TABLE]

3.1 Case of a flat structuring element

In the case of a flat structuring element $B=B_{0}\in\mathcal{T}^{*}$ ($\forall x\in D_{B}$, $B(x)=B_{0}$), the equations 14 and 15 can be simplified.

[TABLE]

Notice: the infimum $\wedge$ is changed into a supremum $\vee$ because the function $\widetilde{f}:x\rightarrow\ln(1-x/M)$ is a continuous decreasing mapping. The reflected (or transposed) domain $\widecheck{D}_{B}$ is $\widecheck{D}_{B}=\{-h,h\in D_{B}\}$ and the reflected structuring function $\widecheck{B}$ is defined by the reflection of its definition domain $\forall x\in\widecheck{D}_{B}$, $\widecheck{B}(x)=B(-x)$ [28]. Similarly:

[TABLE]

With the equations 16 and 17, the map of Asplund’s distances (eq. 12) becomes:

[TABLE]

This important result shows that, with a flat probe, the map of Aplünd’s distances can be computed using logarithms and operations of morphological erosion and dilation of an image. From an implementation point of view, the programming of the map of Aplünd’s distances becomes easier, because the majority of image processing libraries contains morphological operations.

Notice: by replacing the dilation and erosion by rank-filters [26] one can compute the map of Asplund’s distances with a tolerance [14, 23].

3.2 General case: a structuring function

Using a general structuring function in the equations 14 and 15, the map of Asplund’s distances is expressed as:

[TABLE]

Let us study the properties of mappings $\lambda_{B}$, $\mu_{B}\in\mathscr{L}_{2}^{\mathscr{L}_{1}}$, $\forall f,g\in\overline{\mathcal{I}}$

[TABLE]

According to definition 4, 2 (p. 4), $\lambda_{B}$ is a dilation. In addition,

[TABLE]

Similarly, we have:

[TABLE]

According to definition 4, 1 (p. 4), $\mu_{B}$ is an erosion. In addition,

[TABLE]

Therefore, the map of Asplund’s distances is the logarithm of the ratio between a dilation and an erosion of the function $f$ by the structuring function $B$. The map of the least upper bounds $\lambda_{B}$ is a dilation and the map of the greatest lower bounds $\mu_{B}$ is an erosion. The two maps are defined from the lattice $(\mathscr{L}_{1}=\overline{\mathcal{I}},\leq)$ and the lattice $(\mathscr{L}_{2}=(\overline{\mathbb{R}}^{+})^{D},\leq)$ with their respective natural orders.

4 Illustration

In figure 1 (a), we extract a tile (i.e. the probe or the structuring function) in an image $f$ and we look for the similar ones in a darken image, $f^{d}$, by means of a LIP multiplication of 0.3. Physically, it corresponds to an object with a stronger light absorption. Importantly, the probe has a non convex domain shape and is not flat. We compute the map of Asplund’s distances between the probe $B$ and the image $f^{d}$ with a tolerance, As_{B,p}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}f^{d}, as introduced in [14, 23]. This metric, robust to noise, is computed by discarding $p=30\%$ of the points which are the closest to the least upper bounds and to the greatest lower bounds. The tiles are located at the local minima of the distance map which are extracted by a threshold of 0.7 (fig. 1 (b)). The tiles similar to the probe, according to the Asplund’s distance, have been correctly detected (fig. 1 (c)). Notice that the domain of the probe is slightly smaller than the domain of the tiles.

5 Conclusion

In the current paper, we have shown that the map of Asplund’s distances between a probe and a function using the LIP multiplication is linked with morphological operations. The probe corresponds to a structuring function and the map of Asplund’s distances is the logarithm of the ratio between a dilation and an erosion of the function by the structuring function into the lattice of positive functions $((\overline{\mathbb{R}}^{+})^{D},\leq)$. The dilation is the map of the least upper bounds $\lambda_{B}f$, between the function $f$ and the probe $B$, while the erosion is the map of the greatest lower bounds $\mu_{B}f$.

The dilation and the erosion are mappings between the complete lattices of the images $(\overline{\mathcal{I}},\leq)$ and the lattice $((\overline{\mathbb{R}}^{+})^{D},\leq)$ with the natural order. When using a flat structuring element, the expression of the map of Asplund’s distances can be simplified with a dilation and an erosion of the image into the same lattice of the images $(\overline{\mathcal{I}},\leq)$. An example of pattern matching has been presented with a non-flat structuring function.

The obtained results set the pattern matching approach by Asplund’s distances in the well established framework of Mathematical Morphology. The current reasoning can be extended to the double-sided probing by Asplund’s distances for colour and multivariate images using the LIP multiplicative or the LIP additive framework [23, 11, 24]. This will be presented in a coming paper.

6 Appendix

Le us demonstrate that the Aplünd’s metric d_{As}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}} is a metric in the space of equivalence classes \mathcal{I}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}. In order to be a metric on (\mathcal{I}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}\times\mathcal{I}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}})\rightarrow\mathbb{R}^{+}, d_{As}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}} must satisfy the four following properties:

(positivity): \forall f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}\neq g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}\in\mathcal{I}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}, $\forall x\in D$, \lambda\mathbin{\ooalign{\bigtriangleup\crcr\raise 1.4pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}(x)>\mu\mathbin{\ooalign{\bigtriangleup\crcr\raise 1.4pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}(x) (def. 3, p. 3), because g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}>0

$\Rightarrow\lambda>\mu$ because \mathcal{I}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}} is an ordered set with the order $\leq$

\Rightarrow\forall f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}\neq g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}\in\mathcal{I}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}, d_{As}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}(f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}},g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}})>0. 2. 2.

(Axiom of separation):

[TABLE]

Reciprocally:

[TABLE]

Eq. 24 and 25 $\Rightarrow$ \left\{\forall f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}},g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}\in\mathcal{I}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}\right., \left.d_{As}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}(f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}},g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}})=0\Leftrightarrow f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}=g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}\right\}. 3. 3.

(Triangle inequality): Let us define: d_{As}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}(f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}},g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}})=\ln(\lambda_{1}/\mu_{1}), d_{As}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}(g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}},h^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}})=\ln(\lambda_{2}/\mu_{2}) and d_{As}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}(f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}},h^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}})=\ln(\lambda_{3}/\mu_{3}). We have

[TABLE]

[TABLE]

[TABLE]

In the same way:

[TABLE]

Eq. 26, 27 28 $\Rightarrow$ $\frac{\lambda_{1}\lambda_{2}}{\mu_{1}\mu_{2}}\geq\frac{\lambda_{3}}{\mu_{3}}$

\Rightarrow\forall f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}},g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}},h^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}\in\mathcal{I}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}},d_{As}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}(f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}},h^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}})\leq d_{As}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}(f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}},g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}})+d_{As}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}(g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}},h^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}). 4. 4.

(Axiom of symmetry): Let us define: d_{As}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}(f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}},g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}})=\ln(\lambda_{1}/\mu_{1}), d_{As}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}(g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}},f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}})=\ln(\lambda_{2}/\mu_{2}).

Def. 3 $\Rightarrow$ \lambda_{1}=\inf\left\{k:\forall x,g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}\geq\frac{1}{k}\mathbin{\ooalign{\bigtriangleup\crcr\raise 1.4pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}\right\}, because $k>0$

\Rightarrow\frac{1}{\lambda_{1}}=\sup\left\{k^{\prime}:\forall x,g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}\geq k^{\prime}\mathbin{\ooalign{\bigtriangleup\crcr\raise 1.4pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}\right\} $\Rightarrow\frac{1}{\lambda_{1}}=\mu_{2}$.

In the same way, we have $\frac{1}{\mu_{1}}=\lambda_{2}$.

Therefore, \forall f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}},g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}\in\mathcal{I}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}, d_{As}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}(f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}},g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}})=\ln(\lambda_{1}/\mu_{1})=\ln(\lambda_{2}/\mu_{2})=d_{As}^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}(g^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}},f^{\mathbin{\ooalign{\bigtriangleup\crcr\raise 0.98pt\hbox{\mathchoice{\scalebox{1.0}[0.7]{\displaystyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\textstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}{\scalebox{1.0}[0.7]{\scriptscriptstyle\mathchoice{\scalebox{0.7}[1.0]{\displaystyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\textstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptstyle\scriptscriptstyle\times}}{\scalebox{0.7}[1.0]{\scriptscriptstyle\scriptscriptstyle\times}}}}}}}}).