Nice distances (as a perspective) images: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

Encyclopedia of Distances (4th edition) - CORRECTIONS, ADDITIONS AND UPDATES:

The space $(H,G)$ is called a {\index{\bf Kre\u{i}n space}} (or {\em $J$-space}). It is a generalization of a Hilbert space as a {\bf pseudo-Euclidean space} (Sect. 7.1) is a generalization of a Euclidean space. A Kre\u{i}n space with finite rank of indefiniteness is called a {\em Pontryagin space}.

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The space $(H,G)$ is called a {\em $J$-space}. A $J$-space with finite rank of indefiniteness is called a {\em Pontryagin space}.

\item{\index{\bf Minkowski length of lattice polytope}}

Let $P$ be a convex $d$-dimensional {\em lattice polytope} in $\mathbb {R}^d$, i.e., a convex hull of finitely many points in the integer lattice $\mathbb{Z}^d \subset \mathbb{R}^d$. The {\em lattice diameter} $l((P)$ is defined as one less than the largest number of collinear lattice points in $P$.

For any $1\le n\le d$, the {\em $n$-th Minkowski length} $L_n(P)$ is defined (Soprunov--Soprunova, 2009) as the largest number of lattice polytopes of positive dimension whose Minkowski sum is contained in $P$, i.e., the largest number of lattice segments whose Minkowski sum is at most $n$-dimensional and is contained in $P$; so, $L_1(P)=l(P)$. The {\bf Minkowski length} of $P$ is $L_d(P)$.

A $D$-distance graph is called a {\index {\bf graph of diameters}} if the {\bf diameter} $Diam$ of $(X,d)$ is finite and $D=\{Diam \}$.

\item{\index{\bf Radio labeling}}

A {\em radio labeling} of a graph $G=(V,E)$ is (Chartrand {\em et al.}, 2001) an assignment of labels $f(v)$ from the set $\{0, 1, \dots , \lambda\}$ of integers to the vertices $v \in V$, such that

$$d(u,v)+|f(u)-f(v)|\ge diam(G)+1$$

holds for any $u,v\in V$. The {\em radio number} of $G$ is defined as $rn(G)= \min_f span(f)$, where $span(f)$ is defined as $\max_{u,v\in V} |f(u)-f(v)|$. A radio labeling $f$ of $G$ is called {\em optimal} if $span(f)= rn(G)$.

The {\index{\bf leaf distance}} of a tree is the maximum $d$ such that the distance between any pair of its leaves is at least $d$.

\item{\index{\bf Correlation matrix distance}}

The {\bf correlation matrix distance} between two correlation matrices $R$ and $Q$ is defined (Herdin {\em et al.}, 2005) by

\begin{displaymath} 1-\frac{Tr(RQ)}{||R|| _{Fr}\cdot ||Q||_{Fr}}= 1- \frac{\langle \vec{R}, \vec{Q}\rangle}{||\vec{R}|| _2\cdot ||\vec{Q}||_2}, \end{displaymath} where $||.||_{Fr}$ denotes the Frobenius norm and $\vec{A}$ denotes vectorized matrix $A$.

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\item{\index{\bf Dynamic time warping distance}}

The largest synchronized movement of biomass on Earth is diel vertical migration of zooplankton in the ocean and lakes.

\item{\index{\bf Isoline}}

An {\bf isoline} (or {\em contour line}, {\em isopleth}) of a function of two variables is a curve along which the function has a constant value. For example, an {\bf isotherm}, {\em isobar}, {\em isopycnal} and {\em isotach} are the lines that connects points on a map that have the same temperature, pressure, density and wind direction, respectively.

``Contour line'' is the most common usage in cartography, but {\bf isobath} for underwater depths on bathymetric maps and {\bf isohypse} for elevations are also used. An {\em isocline} is a line joining points with equal slope.

The {\em gradient} of the function is always perpendicular to the isolines.

$50-300$ km: unexplored slice of atmosphere, where the air is too thin to support research balloons, but it is too thick for satellites to survive the drag forces for more than a few months.

At $100,000$ km $\approx 15.5$ Earth radii: the {\em geocorona}, a layer of UV-luminescent hydrogen atoms, the end of the luminous part of the exosphere.

$$B[(\vec{u},v),r,t] =B[\vec{u},r]\times [v-t,v+t]= \{(\vec{a},b)\in \mathbb{R}^n \times \mathbb{R}: ||\vec{u}-\vec{a}||_2 \le r, |v-b|\le t\} ,$$

where $B[\vec{u},r]=\{\vec{a}\in \mathbb{R}^n: |\vec{u}-\vec{a}||_2 \le r\}$ is the Euclidean {\bf metric ball} (Sect. 1.2).

Guterstam {\em et al.}, 2016, showed that our {\em proprioception} (the sense of the relative position of neighboring parts of the body and strength of effort being employed in movement) extends to a part, from touching to $30$ cm, of the intimate space.

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