Nice distance (as a perspective) images: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

Encyclopedia of Distances, CORRECTIONS, ADDITIONS AND UPDATES:

equivalence relation, called {\index{\bf metric identification}},

{\index{\bf premetric}} or {\index{\bf prametric}}.

Any compact metric space of topological dimension $n$ can be embedded isometrically in $\mathbb{E}^{2n+1}$.

The {\em aspect ratio} of a shape is the ratio of its longer and shorter dimensions, say, the length and diameter of a rod, major and minor axes of a torus or width and height of a rectangle (image, screen, etc.). In Physics, {\em aspect ratio} is the ratio of height-to-length scales characteristics.

The {\bf eccentricity graph} of $M$ (Akyiama-Ando-Avis, 1976) has, as vertices, all points $x \in M$ and, as edges, all pairs $(x,y)$ of points at distance $\min\{e(x),e(y)\}$.

\item{\index{\bf Inverse distance weighting}}

{\bf Inverse distance weighting} is a method for multi-varuate interpolation. Let $x_1, \dots, x_n$ be known points, $x$ be an interpolated point and $d(x,x_i)$ be a given distance. A general form of interpolated value $u(x)$ is

\begin{displaymath}\frac{\sum_{1\le i \le n}(d(x,x_i))^{-p}u_i}{\sum_{1\le i \le n}(d(x,x_i))^{-p}}\end{displaymath},

where $p\ge 1$ (usually, $p=2$) is a fixed number.

\item{\index{\bf Barbilian semi-metric}}

Given sets $X$ and $P$, the function $f: P\times X \rightarrow \mathbb{R}_{>0}$ is called an {\em influence} (of the set $P$ over $X$) if for any $x,y \in X$ the ratio $g_{xy}(p)=\frac{f(p,x)}{f(p,y)}$ has a maximum when $p\in P$.

The {\bf Barbilian semi-metric} is defined on the set $X$ by \begin{displaymath} \ln \frac {\max_{p\in P}g_{xy}(p)}{\min_{p\in P}g_{xy}(p)} \end{displaymath} for any $x, y \in X$. Barbilian, 1959, proved that above function is well-defined (moreover, $\min_{p\in P}g_{xy}(p)=\frac{1}{\max_{p\in P}g_{yx}(p)}$) and is a semi-metric. Also, it is a metric if the influence $f$ is {\em effective}, i.e., there is no pair $x,y \in X$ such that $g_{xy}(p)$ is constant for all $p\in P$. Cf. a special case {\bf Barbilian metric} in Chap. $6$.

{\em level set} (or {\em $t$-cut}) $A_{\mu}(t)=\{x\in \mathbb{R}:\mu(x)\ge t\}$ is convex

in general, for any given numbers $p\ge 1$ and $0\le q\le 1$, the distance

\begin{displaymath} (\sum_{i=1}^{n}(1-q)(\mu(x_i)-\mu'(x_i))^p+q(\nu(x_i)- \nu'(x_i)^p)^{\frac{1}{p}}. \end{displaymath}

{\em $\ge$-filter} of semi-metrics.

\le \int^b_a |f(x)|dx$ {\em continuous triangle

Liu, Sun and Yau, 2005, showed that all known complete metrics on the Teichm\"{u}ller space and moduli space (including {\bf Teichm\"{u}ller metric}, {\bf Bergman metric}, {\em Cheng-Yau-Mok} {\bf K\"{a}hler-Einstein metric}, {\bf Carath\'{e}odory metric}, {\em McMullen metric}) are equivalent since they are {\bf quasi-isometric} (cf. Chap. 1) to the {\em Ricci metric} and the {\em perturbed Ricci metric} introduced by them.

\item{\index{\bf Lagrange metric}}

Consider a real $n$-dimensional manifold $M^n$. A set of symmetric non-degenerated matrices $((g_{ij}(p,x)))$ define a {\index{\bf generalized Lagrange metric}} on $M^n$ if a change of coordinates $(p,x)\rightarrow (q,y)$, such that $q_i=q_i(p_1, \dots , p_n)$, $y_i=(\partial_jq_i)x_j$ and rank $(\partial_jq_i)=n$, implies $g_{ij}(p,x) = (\partial_iq_i) (\partial_jq_j)g_{ij}(q,y)$.

A generalized Lagrange metric is called a {\bf Lagrange metric} if there exists a {\em Lagrangian}, i.e., a smooth function $L(p,x)$ such that it holds \begin{displaymath} g_{ij}(p,x)=\frac{1}{2} \frac{\partial^2 L(p,x)}{\partial x_i \partial x_j}. \end{displaymath}

Every {\bf Finsler metric} is a Lagrange metric with $L = F^2$.

For toroidally confined plasma, as in Magnetic Confinement Fusion, the coordinates $u$, $v$ and $a$ correspond to the directions called , respectively, {\em toroidal} (long, as lines of latitude, way around the torus), {\em poloidal} (short way around the torus) and {\em radial}. The {\index{\bf poloidal distance}}, used in plasma context, is the distance in the poloidal direction.

In general, a chord $[a,b]$ of a convex body $C$ is called its {\em affine diameter} if there is a pair of different hyperplanes each containing one of the endpoints $a, b$ and supporting $C$.

the Albanese

In a large sense, a {\em statistical distance} is a measure of dissimilarity between distributions.

The {\index{\bf Lukaszyk-Karmovski metric}} is a metric on $\mathbf{Z}$ with $\mathcal{X} \subset \mathbb{R}$, defined by

\begin{displaymath} ((\sum_{(x,y)\in \mathcal{X}\times \mathcal{X}}|x-y|p(x)p(y). \end{displaymath}

{\em diameter} of $D$ is the maximal length of shortest directed $(u-v)$-path in it. The {\index{\bf oriented diameter}} of a graph $G$ is the smallest diameter among strong orientations of $G$. If it is equal to the diameter of $G$, then any orientation realizing this equality is called {\em tight}.

If all edge-weights are $1$, then $A$ is usual adjacency matrix of $G$. Let $J$ be the $n \times n$-matrix of all ones and let $-\mu$ be the smallest eigenvalue of $A$. Let $T=((t_{ij}))=\mu (J-I)-A$. Neumaier, 1980, considered distance $\sqrt{t_{ij}}$ between vertices $v_i$ and $v_j$ of $G$.

The {\em characteristic polynomial} of an $n \times n$-matrix $M$ is the determinant $det(M- \lambda I)$. The {\bf distance polynomial} of an $n$-vertex graph $G=(V,E)$ is the characteristic polynomial of the graph distance (usually, {\bf path metric}) matrix $D$.

The roots of the distance polynomial constitute the {\index{\bf distance spectrum}} of the graph. Sometimes, the distance polynomial is defined as $det(\lambda I-D)$ or $(-1)^ndet(D- \lambda I)$.

So, $\Omega_{u,v}=a_{uv}|((g_{ij}))|a_{uv}$, where the vectors $a_{uv}$ are with all elements $0$ except for $+1$ and $-1$ in the $u$th and $v$th positions.

{\index{\bf distance-balanced graph}} (i.e.,

\item{\index{\bf Distance-perfect graph}}

Cvetkovi/'{c} et al., 2007, observed that any graph of diameter $T$ has at most $k+T^k$ vertices, where $k$ is the size of its {\bf metric basis}, i.e., (cf. Chap. 1) a smallest set of vertices, the path distances from which uniquely determines any vertex. They called a graph {\bf distance-perfect} if it meets this upper bound and proved that such graph has $T \neq 2$.

For a metric space $(X, d)$ and a positive number $t$, {\em signed distance graph} is (Fiedler, 1969) a signed graph with the vertex-set $X$ in which vertices $x,y$ are joined by a positive edge if $t>d(x,y)$, by a negative edge if $d(x,y)>t$, and not joined if $d(x,y)=t$.

f(L_G(u),L_G(v))

\item{\index{\bf Wirelength problem}}

Let $G=(V_1,E_1)$ and $H=(V_2,E_2)$ be finite graphs with the same number $|V_1|=|V_2|$ of vertices. An {\em embedding} of $G$ into $H$ is a bijective map $f:V_1 \rightarrow V_2$ which is a one-to-one map from $\{(u,v)\in E_1\}$ to the set of all paths in $H$ between $f(u)$ and $f(v)$. The {\em wirelength} of an embedding $f$ is given by

\begin{displaymath} \sum_{(u,v)\in E_1}d_H(f(u),f(v)). \end{displaymath}

The {\bf wirelength problem} of a graph $G$ into $H$ is to find an embedding of $G$ into $H$ that induces the minimum wirelength.

\item{\index{\bf Joint angle metric}}

For a given frame (or pose) $i$ in an animation, let us define $p_i \in \mathbb{R}^3$ as the global (root) position and $q_{i,k}\in S^3$ as the unit quaternion describing the orientation of a joint $k$ from the joint set $J$. Cf. {\bf unit quaternions metric} and {\bf 3D point cloud distance}. The {\bf joint angle metric} between a frame $x$ and a frame $y$ is defined as follows:

\begin{displaymath} |p_x-p_y|^2+\sum_{k\in J}w_k|log(q_{y,k}^{-1}q_{x,k})|^2. \end{displaymath}

The second term describes the weighted sum of the orientation differences; cf. {\bf weighted Euclidean distance}. Sometimes, the terms expressing differences in derivatives, such as joint velocity and acceleration, are added.

\item{\index{\bf Lanzon-Papageorgiou quasi-distance}}

Given a plant $P$, a {\em perturbed plant} $\hat{P}$ and an uncertainty structure expressed via a {\em generalized plant} $H$, let $\Delta$ be the set of all possible perturbations that explain the disprepancy between $P$ and $\hat{P}$. Then {\bf Lanzon-Papageorgiou quasi-distance} (Lanzon and Papageorgiou, 2009) between $P$ and $\hat{P}$ is defined as $\infty$ if $\Delta=\emptyset$ and $\inf_{\delta \in \Delta}||\delta||_{\infty}$, otherwise.

This quasi-distance corresponds to the worst-case degradation of the stability margin due to a plant perturbation. For standard uncertainity structures $H$, it is a metric, but it is only a quasi-metric for multiplicative uncertainity.

the largest cardinality of an {\em isosceles set} in $\mathbb{R}^{2}$, i.e., a set of points, any three of which form an isosceles triangle;

The visible spectrum is about $380-740$ nm. It matches the range of wavelengths sustaining photosynthesis; also, at those wavelenths opacity often coincides with impenetrability.

{\bf Poincar\'{e} metric}

The {\em reach} of $M$ is the {\bf set-set distance} (cf. Chap.1) between $M$ and $MA(X)$.

\item{\index {\bf WikiDistance}}

The {\bf WikiDistance} is the directed path quasi-metric of the {\em Wikipedia digraph}, having about $3.1$ billion (as in November 2009) vertices (English Wikipedia articles) with $xy$ being an arc if the article $x$ contains an hyperlink to the article $y$; cf. \url{http://software.tanos.co.uk/wikidistance} and the {\bf Web hyperlink quasi-metric} on the vertices of the {\em Web digraph}. Dolan (\url{http://www.netsoc.tcd.ie/~mu/wiki}) observed that in March 2008, this digraph has $\approx 2.3$ billion vertices with $\approx 2.1$ articles forming the largest strongly connected component with an average of $4.573$ clicks to get from one to another.

Gabrilovich and Markovich, 2007, proposed to measure semantic relatedness of two texts by the {\bf cosine distance} (cf. {\bf Web similarity metrics}) between weighted vectors, interpreting those texts in terms of affinity with a host of Wikipedia concepts.

Berman, 1996, introduced {\em scheduled network}: a directed network (of, say, airports), in which each edge (say, flight) is labeled by departure and arrival times. Kempe-Kleinberg-Kumar, 2002, introduced (in fact, more general) {\em temporal network}: an edge-weighted graph, in which the weight of an edge is the time at which its endpoints communicated. A path is {\em time-respecting} if the weights of its edges are non-decreasing. Besides Scheduling and Epidemiology, such networks occur in Distributed Systems (say, dissemination of information using node-to-node ommunication).

In order to handle large temporal data on human behavior, Kostakos, 2009, introduced {\em temporal graph}: an arc-weighted directed graph, where the vertices are instances $a_it_k$ (person $a_i$ in point $t_k$ of time), and the arcs are $(t_{k+1}-t_k)$-weighted ones $(a_it_k,a_it_{k+1})$ linking time-consecutive pairs and unweighted ones $(a_it_k,a_jt_k)$ representing a communication (say, E-mail) from $a_i$ to $a_j$ at time $t_k$. In order to handle also {\em temporally disconnected} (i.e., not connected by a time-respecting path) nodes, Tang-Musolesi-Mascolo-Latora, 2009, introduced {\em time-varying network}: an ordered set $\{D_t\}_{t=1,2,...,T}$ of directed (or not) graphs $D_t=(X,A_t)$, where the arc-sets $A_t$ may change in time and the arcs have temporal duration. As real-world examples, they considered brain cortical and social interaction networks.

Cf. also {\bf time series video distances} in Chap. 21 and {\bf time-distance relation, in Psychology} in Chap. 28.

The primary metric tool in MMOG and Virtual Worlds is the proximity sensor recording when an avatar is within its specified range.

\item{\index {\bf Normalized Google distance}}

The {\bf normalized Google distance} between two search terms $x$ and $y$ is defined (Cilibrasi and Vitanyi, 2005) by

\begin{displaymath}\frac{\max\{\log f(x),\log f(y)\}-\log f(x,y)}{\log m- \min\{\log f(x),\log f(y)\}}, \end{displaymath}

where $m$ is the total number of web pages searched by Google search engine; $f(x)$ and $f(y)$ are the number of hits for terms $x$ and $y$, respectively; and $f(x,y)$ is the number of web pages on which both $x$ and $y$ occur. Cf. {\bf normalized information distance} in Chap. 11.

\item{\index {\bf SimRank similarity}}

Consider a directed multi-graph $D$ representing a cross-referred document corpus, say, a set of citation-related scientific papers, hyperlink-related web pages, etc. {\bf SimRank similarity} $s(x,y)$ between vertices $x$ and $y$ of $D$ is defined (Jeh and Widom, 2002) recursively as $1$ if $x=y$, $0$ if $|I(x)||I(y)|=0$ and

\begin{displaymath} \frac{C}{|I(x)||I(y)|}\sum_{a\in I(x),b\in I(y)}s(a,b),

\end{displaymath} otherwise, where $I(v)$ is the set of in-neighbors of a vertex $v$ and $C$ is a constant, $0$$<$$C$$<1$ (usually, $C=0.8$ or $0.6$ is used).

(concatenation of their directed paths to a common ancestor).

the stewardship of the Internet, and in 2009, US Department of Commerce accepted privatization/internationalization of ICANN, the body responsible for domain names in the Internet.

only. It is expected to exhaust in 2010-2011 but new Internet Protocol IPv6 has address space $2^{128} \approx 4.4 \times 10^{}38$. IPv6 is not interoperable with existing protocol IPv4, and in 2008, it was deployed by less than $1\%$ of Internet-enabled hosts.

social sites (as Facebook, Twitter, MySpace) and Wikipedia (the collaborative encyclopedia). Original Web-as-nformation-source is often referred as {\em Web $1.0$}, while {\em Web $2.0$} means present Web-as-paticipation-platform as, for example, web-based communities, blogs, social-networking (and video-sharing) sites, wikis, hosted services and web applications.

In September 2009, nearly $27$ billion hours were spent on the Internet globally by a online population of $1.2$ billion users age $15$ and older. Top global properties were Microsoft, Google, Yahoo, Facebook with $14.5\%$, $9.3\%$, $6.3\%$, $5.1\%$ of total time online.

IM (instant messaging) is real-time text-based communication over the Internet or some internal network/intranet. Leskovec and Horvitz, 2008, studied $30$ billion anonymized conversations among $240$ million people, which occured within the Microsoft Messenger IM system in June 2006. From the data, they constructed a connected graph ($uv$ being edge if user $v$ sent to, or received from, user $u$ an instant message) with $180$ million vertices and $1.3$ billion edges. The average path length among Messenger users was $6.6$ with $78\%$ of the pairs being separated by at most $7$ steps. It validated Milgram's theory of {\em six degrees of separation} on a planetary scale.

It is one of {\em distance-vector routing protocols}.

SMM corresponds to high-rate evolution, while IAM corresponds to low-rate and short-term evolution.

Besides {\em vertical gene transfer} (reproduction within species), the evolution is affected by {\em HGT} (horizontal gene transfer when an organism incorporates genetic material from another one without being its offspring) and {\em hybridization} (extra-species sexual reproduction). HGT is common among unicellular and viruses, even across large {\bf taxonomic distances}. HGT happen also in plants and animals, usually, by viruses. $40-50\%$ of the human genome consists of DNA imported horizontally by viruses. The most taxonomically distant fertile hybrids are (very rare) interfamilial ones, for instance, blue-winged parrot $\times$ cocktail, chicken $\times$ guineafowl in birds and (under UV irradiation) carrot with tobacco, rice or barley.

In an edge-weighted phylogenetic tree, the {\index{\bf additive distance}} between two taxa is the minimal sum of edge-weights in a path connecting them.

In Genetic Genealogy, {\em genetic distance between individuals} (of the same species) is the percentage of genome inherited from common ancestors.

\item{\index{\bf Shared allele distance}}

The {\bf shared allele distance} $D_{SA}$ (Stephens et al., 1992, corrected by Chakraborty-Jin, 1993) between individuals $a,b$ is $1-SA(a,b)$, while between populations $x,y$ it is defined by

\item{\index{\bf MHC genetic dissimilarity}}

MHC is the most gene-dense and fast evolving region of the mammalian genome. In humans, it is $3.6$ Mb region containing $140$ genes on chromosome 6 and called HLA ({\em human leucocyte antigen system}). MHC has largest {\em polymorphism} (allelic diversity) found in the population; for example, $1,178$ variant alleles of the locus HLA-B were known in 2009. This diversity is essential for immune function since it broadens the range of {\em antigens} (proteins bound by MHC and presented to T-cells for destruction); cf. {\bf immunological distance}. MHC-diversity allows to mark each individual of a species with a unique body odor permitting kin recognition and mate selection. {\em MHC-negative assortative mating} (tendency to select MHC-dissimilar mates) increases MHC variation and so, provide progeny with an enhanced immunological surveillance and reduced disease levels.

\item{\index{\bf $Dps$ distance}}

The {\index{\bf Thorpe similarity}} (proportion of shared alleles)

{\bf Prevosti-Ocana-Alonso distance} (1975)

{\bf Roger distance} $D_R$ (1972)

{\bf Cavalli-Sforza--Edvards chord distance} $D_{CH}$ (1967)

{\bf Bhattacharya and Nei distance} (1987) is $(\arccos\left(\sum\sqrt{x_{ij} y_{ij}}\right))^2$.

{\bf Nei-Tajima-Tateno distance} $D_A$ (1983)

The {\index{\bf Tomiuk-Loeschcke distance}} (1998) is $-\ln{\frac{1}{n}}\sqrt{\sum{{x_{ij}}\sum{y_{ij}}}}$.

{\bf Nei minimum genetic distance} $D_m$ (1973)

{\bf Nei standard genetic distance} $D_s$ (1972)

The {\index{\bf kinship distance}} is defined by $-\ln{\langle x,y \rangle}$. Caballero and Toro, 2002, defined {\em molecular kinship coefficient} between $x$ and $y$ as the probability that two randomly sampled alleles from the same locus in them are identical by state. Computing it as $\langle x,y \rangle$ and using analogy with {\bf kinship coeficient} defined via identity by descent, they proposed several distances adapted to molecular markers (polymorphisms of microsatellites).

{\bf Sangvi $\chi^{2}$ distance} (1953)

Cf. {\bf Tanimoto distance} in Chap. 17.

\item{\index{\bf Goldstein et al. distance}}

{\bf Goldstein et al. distance} $(\delta\mu)^2$ (Goldstein, Ruiz Linares, Cavalli-Sforza, and Feldman, 1995)

It is the loci-averaged value $(\delta\mu)^2=(\mu(x)_j-\mu(y)_j)^2$, where $\mu(z)_j=\sum_iiz_{ij}$ is the mean number of repeats of allele at the $j$-th (microsatellite) locus in population $z$. This and next distance assume high-rate SMM (stepwise mutation model of evolution) with mutations-drift balance.

\item{\index{\bf Shriver et al. stepwise distance}}

The {\bf Shriver et al. stepwise distance} $D_{SW}$ (Shriver, Jin, Boerwinkle, Deka, Ferrel and Chakraborty, 1995)

{\bf Latter F-statistics distance} (1972)

The {\bf Reynolds-Weir-Cockerham distance} (1983)

$K2P$ (Kimura, 1980) between DNA

\frac{1}{4}\ln\sqrt{1-2Q}.

No. of DNA differences

Two genes at map distance $\approx 50$ cM are unlinked.

Recombination frequency is dependent on physical distance between {\em markers}, i.e., microsatellite polymorphic loci used to detect exchanges without influencing them. Weissenbach et al. 1992 estimated, for the human genome in the average, that a map distance of $1$ cM corresponds to a physical {\bf genome distance} of $\approx 1$ Mb (million base pairs). In human female this recombination rate (and so map distances) are twice that of the male. In {\em Drosophila} male there is no recombination.

The {\em intermarker meiotic recombination distance} (Dib et al. 1992) counts only meiotic crossovers. {\em Mitotic crossover}, i.e., a crossover resulting from the pairing of homologous chromosomes in a diploid during mitosis, is rare, but some asexually reproducing organisms use it as a source of variation. Also, in mammalian cells it allows recessive cancer-causing mutations to become expressed.

"$17$ $CI$" and "$18$ $NI$"

\item{\index{\bf Distance sampling}}

{\bf Distance sampling} is a widely-used group of methods for estimating the density and abundance of biological populations. It is an extension of quadrat-based sampling.

A standardized survey along a series of lines or points is performed, searching for objects of interest (animals, plants or their clusters). {\em Detection distances} $r$ are measured to each detected object. The {\em detection function} $g(r)$ (the probability that an object at distance $r$ is detected) is fit then to the observed distances, and this fitted function is used to estimate the proportion of objects missed by the survey. It gives point and interval estimates for the density and abundance of objects in the survey area.

Microbes are carried by wind thousands of kilometers on dust particles protecting them. It limits the nummber of endemic species of free-living micro-organisms: they occupy every ecological niche but their biodiversity is low.

\item{\index{\bf Human migration distances}}

{\bf Human migration distances} are the distances between birthplaces of paired persons. If the pairs are spouses (gametes) or siblings, we have {\index{\bf marital distance}} or {\index{\bf sib distance}}, respectively. Also, the {\index{\bf parent-offspring distance}} is used to describe gene migration per generation.

Those distances are measured either in km, or, say, as the number of municipalities crossed by a straight line between municipality midpoints of pair's birthplaces.

The {\em marital migration distance} is the distance between the birthplace of a person and its marriage place.

{\bf Spatial fidelity zones} specific to individuals (say, at a given distance from the colony centre, or within a particular zone of the total foraging area) were observed in some social insect species, molluscan communities, birds, etc.

An animal is {\bf territorial} if it consistently occupies, marks and defends a {\em territory}, i.e., an area containing, say, a nest, den, mating site or sufficient food resources. Related {\em dear enemy recognition} is a situation in which a territorial animal responds more strongly to strangers than to its neighbors from adjacent territories.

Matioli's {\index{\bf mitotic length}} is the distance, in number of intervening mitoses, from the normal (i.e., neither immortal nor malignant) cells in the immature precursor stage to their progeny in a state of {\em mitotic death} (terminal differentiation) and phenotypic maturity.

\item {\index{\bf Read length}}

In Gene Sequencing, automated sequencers transform electropherograms (obtained by {\em electrophoresis} using fluorescent dyes) into a $4$-color chromatogram where peaks represent each of the DNA bases A, T, C, G.

The {\bf read length} is the length, in the number of bases, of resulting sequence obtained from an individual chosen clone. Computers assemble then those short blocks into long continuous stretches which are analyzed for errors, gene-coding regions, etc.

On the other hand, the variant E4 of gene ApoE is the only known genetic risk factor for developing Alzheimer's and heart disease.

Another example: eye, brain and body sizes are closely correlated in vertebrates.

A time period temperature

Last two forces can be both push and pull, depending on the charges of involved bodies. The nucleon-nucleon interaction (or {\em residual strong force}) is attractive but becomes repulsive at very small distances keeping the nucleons apart. Dark matter is attractive while dark energy is repulsive.

A (particle) {\bf displacement} is a vector version of Euclidean distance

opened 10 September 2008

\item {\index{\bf Turbulence length scales}}

The turbulent field consists of the superposition of interacting eddies of different length scales. The turbulence kinetic energy cascades from the eddies of largest scales down to the smallest ones generated from the larger ones through the nonlinear process of vortex stretching.

The {\bf turbulence length scales} are measures of the eddy scale sizes in turbulent flow. Such standard length scales for largest, smallest and intermediate eddy sizes are called {\bf integral length scale}, {\bf Kolmogorov microscale} and {\bf Taylor microscale}, respectively. The corresponding ranges are called {\em energy-containing}, {\em dissipation} and {\em inertial range}.

Integral length scale measures the largest separation distance over which components of the eddy velocities at two distinct points are correlated; it depends usually of the flow geometry. For example, it is $\int{R(r)dr}$ over the (perpendicular) autocorrelation function, where $R(r)$ is autocorrelation coefficient and $r$ is the distance between the fixed and the moving probe volumes.

At the smallest scale, the dynamics of the small eddies become independent of the large-scale eddies, and the rate at which energy is supplied is equal to the rate at which it is dissipated into heat by viscosity. The Kolmogorov length microscale is given by $\tau= (\frac{v^3}{\epsilon})^{\frac{1}{4}}$, where $\epsilon$ is the average rate of energy dissipation per unit mass and $v$ is the kinematic viscosity of the fluid.

On intermediate Taylor microscale, turbulence kinetic energy is neither generated nor destroyed but is transferred from larger to smaller scales.

(around of {\em Planck energy} $\approx 1.22\times 10^{19}$ GeV corresponding to

Salart et al., 2008, estimated that such hidden signal should be $10,000$ times faster than light.

Wrachtrup et al., 2008, entangled three diamond nuclei. Grimm et al., 2006, detected {\em Efimov states} (stable $3$-particle quantum mechanical systems, in which any $2$-particle subsystem is unstable) in scattering using caesium atoms at very low temperature. $2$-particle entanglement occurs in any temperature.

\item{\index{\bf Large scale entanglement and superposition}}

{\em Quantum superposition} is the addition of the amplitudes of wavefunctions, occuring when an object simultaneously "possesses" two or more values for an observable quantity, say, the position or energy of a particle. If the system interacts with its environment in a {\em thermodynamically irreversible way} (say, the quantity is measured), then {\em quantum decoherence} occurs: the state randomly collapses onto one of those values. Reynaud et al., 2003, conjectured that {\em gravitational waves} (ripples of space-time generated by Univers rapid expansion, colliding black holes, etc.) induce this decoherence but effect is significant only for systems with large mass. Schrödinger coined the term {\em entanglement} in 1936 developing his thought experiment about a cat that might be alive or dead, depending on an earlier random event. Superposition and {\em entanglement} (non-local correlation which cannot be described by classical communication or common causes) were observed at atomic scale. The maximum size for objects demonstrating these and other quantum effects (and even its existence) is a hot research topic.

Entanglement was obtained first by coordinating the spin of electrons and the polarisation of photons. Jost and al., 2009, entangled the vibrational motions of two (separated by $240$ micrometers) mechanical oscillators, each consisting of two {\em ions} (electrically charged atoms) and behaving like a spring $4$ micrometers long. First macroscale entanglement: Martinis et al., 2009, entangled (at very low temperature) the electric current directions of two (separated by a few millimeters) superconductors, each $\approx 1$ mm across with billions of flowing electrons.

Superposition results in many observable effects and it was tested with electrons, photons, atoms, ions and molecules. Much larger object is hard to put in superposition because air molecules and photons bounce off it. Aspelmeyer et al., 2009, test superposition on cooled micro-mirrors, $0.5$ mm across and $0.01$ mm wide, containing billions of atoms. Romero-Isart et al., 2009, proposed to create superpositions of the sub-wavelength motion with dielectric objects trapped inside a high--finesse cavity at a very low pressure and temperature. It can be a flu virus or even a tardigrade (an water animal $\le 1$ mm in size) that can survive vacuum for several days.

Specifically, DX can mean {\index{\bf distance unknown}}, short for DXing and a far-away station that is hard to hear.

The {\index{\bf official distance}}: officially recognized (by, say, an employer or an unsurance company) driving distance between two locations that will be used for travel or mileage reimbursement. Distance data (shortest paths between locations) are taken from a large web map service (say, MapQuest, Google, Yahoo or Bing) which uses a variation of {\em Dijkstra algoritm}; cf. {\bf Steiner ratio} in Chap. 1.

In Transportation Engineering, the {\em normal visual acuity} is the ability of a person to recognize a letter (or an object) of size $25$ mm from a distance $12$ m. The {\bf visibility distance} of a traffic control device is the maximum distance at which one can see it, while its {\bf legibility distance} is the distance from which the driver can clearly discern the intended message in order to have time to take the nesessary action.

\item{\index{\bf Acceleration-deceleration distance}}

The {\bf acceleration-deceleration distance} of a vehicle, say, a car or aircraft, is (Drezner-Drezner-Vesolowsky, 2009) the cruising speed $v$ times the travel time. For the large origin-destination distances $d$, it is $d+\frac{v^2}{2}(\frac{1}{a}+\frac{1}{b})$, where $a$ is acceleration in the beginning and $-b$ is deceleration at the end. Cf. {\bf vehicle distances} in Chap. 29.

$100$ km: ``edge of space'', the point where the relatively gentle winds of Earth's atmosphere give way to more violent flows of charged particles in outer space.

The {\bf aerodynamic roughness length} (or {\em roughness length}) $z_0$ is the height at which a wind profile assumes a zero velocity.

A possible origin of the rings of Saturn ($70,000-80,000$ km above its equator) is a moon ($\approx 300$ km in diameter) which came ($\approx 4$ billion years ago) closer to Saturn than its Roche limit.

The {\em Roche lobe} of a star in a binary system is the region of space around the star within which orbiting material is gravitationally bound to it.

Above notion of surface habitability is modeled from temperature/humidity and quatifies the odds of a technological civilization. But Mendez, 2009, proposed an index of subsurface microbial habitability. It is $0.5$ and $0.3$ for icy moons Saturn's Enceladus and Jupiter's Europa; for Earth and Mars, it is $0.4$ and $0.3$. The best compromise between such habitability and accessibility is given by Mars ($\approx 6$ km underground) and Europa (in its subsurface ocean).

The stars in the solar neighbourhood are typically $M$-dwarfs ($\approx 10,000$ within $35$ parsec) with habitable zone radius $\approx 0.1$ of the solar one. So, their super-earths are tidally-locked (no day-night cycle) and under heavy ultraviolet radiation.

(in particular, for dark matter and dark energy measurements). Main other techniques to estimate the {\bf angular diameter distance} to galaxies are {\em gravitational lensing} (cf. {\bf Einstein radius}) and using {\em baryon acoustic oscillations} matter clustering (due to acoustic waves which propagated in the early universe), $\approx 150$ Mpc now, as a standard ruler.

in the presence of gravity the geometry of space-time is curved. (But Bean, 2009, found evidence that over extragalactic distances gravity exerts a greater pull on time than on space.)

A {\em metric ton} (or {\em metric tonne}, {\em tonne}) is a unit of mass equal to $1,000$ kg; this non-SI unit is used instead of SI term {\em megagram} ($10^6$ grams). For capacity, the {\em litre}

{\em Multimetric crystallography}: to consider (Janner, 1991), in addition to the Euclidean metric tensor, other {\em pseudo-Euclidean tensors} (hyperbolic rotations) attached to the same basis; cf. {\bf pseudo-Euclidean distance} in Chap. 7.

Some models challenge Special Relativity on Planck scale supposing a variation of the speed of light with photon energy. But gamma-ray observations (Abdo et al., 204 authors, 2009) gave an upper bound $\frac{5}{6}l_P$ for the length scale of such effect making it implausible.

The {\em Maldacena duality} is the conjectured equivalence between a string theory defined on a ("large, relativistic") space, and a (quantum, without gravity) conformal field theory defined on its (lower dimensional) conformal boundary.

$5.8 \times 10^{-2}$: length of a sperm of the fruit fly {\em Drosophilla bifurca} of the total length $0.15 \times 10^{-2}$" (it is $10,000$ times longer than a human sperm);

$4.83 \times 10^{5}$: diameter of Wilkes Land (Antarctica), the largest crater. This impact is suspected to cause the worst, Permian, mass extinction of life 250 millions years ago;

rotational center of our Milky Way galaxy (in Sagittarius $A^{*}$, putative supermassive black hole). The {\em galactic anticenter} is the point that lies directly opposite the galactic center;

$10^{5}-3\times 10^{5}$ light-years: hypothetical dark matter halo around Milky Way;

The {\bf whiffing distance} (or {\bf spitting distance}) is a very close distance.

Humans and monkeys with amygdala lesions have much smaller than average preferred interpersonal distance.

See \url{http://www.surveymonkey.com/s.aspx?sm=Nd8c_2fazsxMZfK9ryhvzPlw_3d_3d} (related online questionare ).

\item{\index{\bf Psychological distance}}

CLT ({\em construal level theory}) in Liberman-Trope, 2003, defines {\bf psychological distance} from an event or object as a common meaning of spatial (``where''), temporal (``when''), social (``who'') and hypotheticality (``whether'') distance from it.

Expanding spatial, temporal, social and hypotheticality horizons in human evolution, history and child development is enabled by our capacity for {\em mental construals}, i.e., abstract mental representation. Any event or object can be represented at {\em lower-level} (concrete, contextualized, secondary) or {\em higher-level} (abstract, more schematic, primary) construal.

More abstract construals lead to think of more distant (spatially, temporally, socially and hypothetically) objects and vice versa. People construe events at greater, say, temporal distance in terms of their abstract, central, goal-related features and pro-arguments, while nearer events are treated situation-specifically at lower level of counter-arguments. Examples are: greater moral concern over a distant future event, more likely victim's forgiveness of the earlier transgression, more intense affective consumer's reaction when a positive outcome is just missed.

CLT implied that judgments along the four dimensions are conceptually related, i.e., the dimensions are functionaly similar. For example, increase of distance in only one dimension leads to greater moral concern.

But Zhang and Wang, 2008, observed that stimulating people to consider spatial distance influences their judgments along three other dimensions, but the reverse is not true. It is consistent with a claim by Boroditsky, 2000, that the human cognitive system is structured around only concepts emerging directly out of experience, and that other concepts are then built in a metaphorical way. Williams and Bargh, 2008, also claim that psychological distance is a derivative of spatial distance. Spatial concepts such as ``near/far'' are present at $3$ to $4$ months of age since the relevant information is readily available to the senses, whereas abstract concepts related to internal states are more difficult to understand. Also, spatial relations between oneself, one's caretakers and potential predators have primary adaptive sinificance.

\item{\index{\bf Time-distance relation, in Psychology}}

People often talk about time using spatial linguistic metaphors (a long vacation, a short concert) but much less talk about space in terms of time. This bidirectional but asymmetric relation suggests that spatial representations are primary, and are later co-opted for other uses such as time. Casasanto and Boroditsky, 2008, showed that people, in tasks not involving any linguistic stimuli or responses, are unable to ignore irrelevant spatial information when making judgments about duration, but not the converse. So, the metaphorical space-time relationship observed in language also exists in our more basic representations of distance and duration. Mentally representing time as a linear spatial path may enable us to conceptualize abstract (as moving a meeting forward, pushing a deadline back) and impossible (as time-travel) temporal events.

In Psychology, the {\em Kappa effect} is that among two journeys of the same duration, one covering more distance appears to take longer, and the {\em Tau effect} is that among two equidistant journeys, one taking more time to complete appears to have covered more distance. Jones and Huang, 1982, consider them as effects of {\em imputed velocity} (subjects impute uniform motion to discontinuous displays) on judgements of both time and space, rather than direct effect of time (distance) on distance (time) jugement. In Physics, {\em velocity} is the rate of change of position; it is a vector (speed and direction) measuring change in distance over an interval of time.

Fleet, Hallet and Jepson, 1985, found spatio-temporal inseparability in early visual processing by retinal cells. Also, Maruya and Sato, 2002, reported new illusion (time difference of two motion stimuli is converted in the illusory spatial offset) indicating interchangeability of space and time in early visual processing. The differences appear at the level of higher processing because of different representations: space is represented in retinotopic maps within the visual system, while time is processed in cerebellum, basal ganglia and cortical structures. Evidence from lesion and human functional brain imaging/interference studies point towards the posterior parietal cortex as the main site where spatial and temporal information converge and interact with each other. Cf. also {\bf spatial-temporal reasoning}.

\item{\index{\bf Distance as a metaphor}}

Lakoff and N\'{u}\~{n}ez, 2000, claim that mathematics emerged via conceptual metaphors grounded in the human body, its motion through space and time, and in human sense perceptions. In particular, the mathematical idea of distance comes from the activity of measuring, and corresponding mathematical technique consists of rational number and metric space. The idea of proximity/connection comes from connecting and corresponds to topological space. The idea of symmetry comes from looking at objects and corresponds to invariance and isometries.

\item{\index{\bf Metaphoric distance}}

A {\bf metaphoric distance} is any notion in which a degree of similarity between two difficult-to-compare things is expressed using spatial notion of distance as an implicite bidirectional and understandable metaphor. Some practical examples are:

{\em Internet and Web bring people closer}: proximity in subjective space is at-handness;

{\em Healthy professional distance} teacher-student, terapist-patient, manager-employee;

{\em Competitive distance} (uncomparability) between two airline product offerings;

{\em Metaphoric distance} that a creative thinker take the thinking away from the problem, i.e., degree of intuitivity, required to evolve/reshape concepts into new ideas.

The {\index{\bf distance-similarity metaphor}} (Montello-Fabrikant-Ruocco-Middleton, 2003) is a design principle, where relatedness (say, similarity) in non-spatial data content is projected onto distance, so that semantically similar documents are placed closer to one another in an information space. It is the invers of the {\em first law of geography} (Tobler, 1970); cf. {\bf nearness principle}. This metaphor is used in Data Mining, Pattern Recognition and Spatialization (information display) of non-spatial data.

Possible mental lines, explaining such confusion, are {\em linear-scalar} (the psychologic distance $d(a,a+1)$ between adjacent values is constant but the amount of noise increases as $ka$) or {\em logarithmic} (amount of noise is constant but $d(a,a+1)$ decreases logarithmically).

{\bf Distancing} (from the verb {\em to distance}, i.e., to move away from) is any behavior or attitude causing to be or appearing to be at a distance.

{\index{\bf Distantness}} is the state or quality of being distant or remote. A similar notion is {\index{\bf distancy}}, which is also rare/obsolete word for {\bf distance}.

{\bf Distancing from role identities} is the first step of 7th (individualistic) of 9 stages of ego development proposed by Loevinger, 1976.

In books by Kantor, {\bf distancing} refers to APD (Avoidant Personality Disorder): fear of intimacy and commitment (confirmed bachelors, ``femmes fatales'', etc.) {\bf Associational distancing} refers to individual's dissociation with those in the group inconsistent with his desired social identity.

The {\index{\bf distancing language}} is phrasing used by a person to avoid thinking about the subject or content of his own statement (for example, referring to death).

{\bf Distancing by scare quotes} is a placing quotation marks around an item (single word or phrase) to indicate that the item does not signify its literal or conventional meaning. The purpose could be to distance the writer from the quoted content, to alert the reader that the item is used in an unusual way, or to represent the writer's concise paraphrasing. {\bf Neutral distancing} convey a neutral writer's attitude, while distancing him from item's terminology, in order to call attention to a neologism, jargon, a slang usage, etc; sometimes italics are used for it. Cf. {\bf technology-related distancing}, {\bf antinomy of distance}, {\bf distanciation}, {\bf role distance}.

Vygotsky's {\em zone of proximal development} is the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance, or in collaboration with more capable peers.

For example (Mulgan, 1991), the centres of global cities are socially closer to each other than to their own peripheries.

But Shenkar, Luo and Yeheskel, 2008, claim that above cultural distance is merely a measure how much a country strayed from the core culture of the multinational enterprise. They propose instead (especially, as a regional construct) the {\em cultural friction} linking goal incongruity and the nature of cultural interaction.

Going deeper in the past, Parker, 2003, proposed that the sudden diversification in animal fossil forms at the start of the Cambrian Era, was due to the development of the vision faculty and the consequent intensification of predation.

Goldenberg-Levy, 2009, show that IT (Information Technology) revolution which occured in the 1990s, actually increased effect of distance on the volume of such social interactions as email, Facebook communications and baby names diffusion. They argue that IT increased local communications to a greater degree than long-distance ones.

{\index{\bf Virtual distance}} is the perceived distance between individuals when their primary way of communication is not face-to-face. The main markers of virtual distance are physical, operational and affinity distances.

{\bf map's distance}.

In Ballistics (cf. {\bf ballistics distances} in Chap. 24), {\em drop distance} is the height the bullet loses between the moment it leaves the rifle and the moment it reaches the target.

Human mental maps, used to find out distance and direction, rely mainly, instead of geometric realities, on real landscape understanding, via webs of landmarks. Ellard, 2009, suggests that this loss of natural navigation skills, coupled with unique ability to imagine themselves in another location, may have given modern humans the freedom to create a reality of their own.

They relate mainly to the perception and cognition of the {\index{\bf environmental distances}}, i.e.,

(turns are often memorized as straight lines or right angles);

Thorndyke and Hayes-Roth, 1982, compared distance judgements of people with navigation- and map-derived spatial knowledge. The subjects were asked route and Euclidean distances between the centers of rooms on the 1st floor of a Rand Corporation building. Navigation-derived route distance estimates were more accurate than Euclidean judgements, and this difference diminished with increased exploration. The reverse was true for map subjects, and no improvement was observed in the map learning. Turner and Turner, 1997, made similar experiment in the plane virtual building. Exploration-derived Euclidean jugements were good but route distances were much underestimated; repeated exposure not helped. Authors suggest that exploration of virtual environments is simular to navigation in real world but with restricted field of view, as in tunnels, caves or wearing a helmet, watching TV.

Krishna, Zhou and Zhang, 2008, compared spatial judgements of {\em self-focused} (say, ``Western'') and {\em relationship-focused} (say, ``Eastern'') people. The former ones were more likely to misjudge distance when multiple features should be considered; they were more likely to pay attention to only focal aspects of stimuli and ignore the context and background information.

\item {\index{\bf Spatial cognition}}

{\bf Spatial cognition} concerns the knowledge about spatial properties of objects and events: location, size, distance, direction, separation/connection, shape, pattern, and movement. For instance, it consider navigation (locomotion and wayfinding) and orientation during it: recognition of landmarks and {\em path integration} (an internal measuring/computing process of integrating information about movement). Spatial cognition addresses also our (spatial) understanding of the World Wide Web and computer-simulated virtual reality.

Men surpass women on test of spatial relations and mental rotation, while women have better memory for objects and their location. It should come from a division of labor in Pleistocene groups: hunting for men and gathering for women. One of {\em cultural universals} (traits common to all human cultures) is that men on average travel greater distances over lifetime.

spatial-temporal reasoning

It usually expresses unprecise and context-dependent information about space.

Computer Vision.

Translations of the English noun {\em distance}, for example, into French, Italian, German, Swedish, Spanish, Interlingua, Esperanto are: distance, distanza, distanz, distans, distancia, distantia, distanco.

In Language, {\bf long-distance dependence} (or {\em syntactic binding}) is a construction, including wh-questions (as ``Who do you think he like''), topicalizations (as "Mary, he like''), {\em easy}-adjectives (as ``Mary is easy to talk to''), relative clauses (as ``I saw the woman who I think he likes'') - which permits an element in one position ({\em filler}) to fill the grammatical role associated with another not adjacent position ({\em gap}). The {\em filler-gap distance}, in terms of the number of intervening clauses or words between them in a sentence, can be arbitrarly large.

"Pens\'{e}es"

Cf. earlier Montaigne's {\em nothing-smallest-largest} distances in {\em Essais, III:11 On the lame}: ``Yet the distance is greater from nothing to the minutest thing in the world than it is from the minutest thing to the biggest.''

Calvin's {\em Eucharistic theology} (doctrine on the meaning of bread and vine that Christ offered to his disciples during the last supper before his arrest) also relies on spatial distance as a metaphor that best conveys the separation of the world from Christ and of the earthly, human from the heavenly, divine.

``Our main business is not to see what lies dimly at a distance, but to do what lies clearly at hand'' (Thomas Carlyle)

``The closer the look one takes at a world, the greater distance from which it looks back.'' (Karl Kraus)

Erubin 53b,

Hadith 2687,

``Do we need distance to get close?'' (Sarah Jessica Parker)

``Once the realization is accepted that even between the closest human beings infinite distances continue, a wonderful living side by side can grow, if they succeed in loving the distance between them which makes it possible for each to see the other whole against the sky.'' (Rainer Maria Rilke)

``The human voice can never reach the distance that is covered by the still small voice of conscience.'' (Mohandas Gandhi)

``Truth is always the shortest distance between two points.'' (Sun Myung Moon)

``The shortest distance between two points is under construction.'' (Leo Aikman)

``Time is the longest distance between two places.'' (Tennessee Williams)

``Everywhere is within walking distance if you have the time.'' (Steven Wright)

``Fill the unforgiving minute with sixty seconds worth of distance run.'' (Rudyard Kipling)

``Distance not only gives nostalgia, but perspective, and maybe objectivity.'' (Robert Morgan)

``Age, like distance lends a double charm.'' (Oliver Wendell Holmes)

``Yet the distance is greater from nothing to the minutest thing in the world than it is from the minutest thing to the biggest.'' (Michel de Montaigne)

Antinomy between inspiration and technique (embracement and estrangement) in performance theory is called {\em Ion hook} since Ion of Ephesus (a reciter of rhapsodic poetry, in a Platon's dialogue) employed a double-consciousness, being ecstatic and rational. The acting model of Stanislavsky and Brecht are, respectively, incarnating the role truthfully and standing artfully distanced from it. Cf. {\bf role distance}.

gift of ecstasy." Interplay of proximity and distance to the Other is central also in Levinas ethics.

\item{\index{\bf Role distance}}

In Sociology, Goffman, 1961, using a dramaturgical metaphor, defined {\bf role distance} (or {\em role distancing}) as actions which effectively convey some disdainful detachment of the (real life) performer from a role he is performing. For example, the actor may only play the role in a tongue and cheek fashion. An example of social role distancing is when a teacher explains to students that his disciplinary actions are due only to his role as a teacher. So, the occupant of a role try to de-emphasize the importance of that role and communicate that his actions should be attributed rather to the role.

Goffman observed that children are capable to merge doing and being, i.e., {\em embracement of the performer's role}, only from 3-4 years. Starting from about 5, their role distance (distinguishing being from doing) appear and expands, especially, at age 8, 11 and adult years.

Beside role embracement and role distance, one can play role cynically in order to manage the outcomes of the situation (impression management). The most likely cause of role distancing is {\em role conflict}, i.e., the pressure exerted from another role to act inconsistently from the expectations of the first role.

``Distance to Fault'' (DTF) is a metal/indie rock band based in Hampshire UK.

The {\index{\bf intercanine distance}}: the distance between the distal surfaces of the maxillary canines on the curve, i.e., the cicumferential distance of the six maxillary anterior teeth.

The {\index{\bf interalar distance}} (or {\em nose width}, {\em nostril-to-nostril distance}, cf. {\bf head and face measurement distances}): the straight line distance between the outer points of the ala of the nose. Gomez et al., 2009, found that it, when increased by $31\%$, is usually equal to the {\bf intercanine distance}.

between the eyes (the average is about $6.5$ cm for men and $5.5$ cm for women).

The {\index{\bf intercornual distance}}: the distance between uterine horns (normally, $2-4$ cm). The {\index{\bf C-V distance}}: the distance between clitoris and vagina (usually, $2.5-3$ cm, greater than in other primates).

The {\index{\bf distance factor}} is a crude measure of arterial tortuosity defined as $\frac{l}{d}-1$, where $l$ is the vessel length and $d$ is Euclidean distance between the vessel end points.

Examples of neuropsychological spatial disorders are: {\em Balint's syndrome} (unability to localize objects in space), {\em unilateral spatial neglect} (bias of awareness and attention to the side of the hemispheric lesion) and left-right disorientation.

The ideal {\em TV viewing distance} is $1.9$ times the screen width, since then this width occupies a $30$-degree angle from the viewing position. For multiple-row seating in the home theater, a viewing angle $26-36$ degrees is recommended. The {\index{\bf Lechner distance}} is the optimal viewing distance at which the human eye can best process the details given by High Definition TV resolution. For example, it is $5.5$ or $9$ feet (i.e., about $1.68$ or $2.74$ m) for a $1080$ HD TV with a screen size of $42$ or $69$ inches, respectively.

In general, a face with larger $B$ is perceived as a baby-like and less dominant one.

females.

According to Patriquin-Steyn-Loth 2005, the main pelvic linear dimensions are: width (and depth) of sciatic notch, iliac breadth, total height, ischial length, acetabulum diameter and pubic length, height, width.

The main predictor for developmental instability, increasing with age, is FA ({\em fluctuating asymmetry}), i.e., the degree to which the size of bilateral body parts deviates from the population mean, aggregated across several traits. Penke et al., 2009, found that at age $79-83$ men (but not women) with lower facial FA have better cognitive ability and reaction time.

\item{\index{\bf Running distances}}

In Running, usually, {\em sprinting} is divided into 100, 200, 400 m, {\em middle distance} into 800, 1500, 3000 m and {\em long distance} into 5, 10 km.

{\em LSD} (long slow distance) is a slang term for a training method for runners that involves running distances longer, and at a slower pace, than those of races.

{\em Racewalking} is divided into 10, 20, 50 km, and {\em relay races} into $4\times 100, 4\times 200, 4\times 300, 4\times 400$ m.

\item{\index{\bf Distance swimming}}

{\bf Distance swimming} is any swimming race longer than $1.5$ km; usually, within $24-59$ km.

Also, {\em the distance} is boxing slang for a match that lasts the maximum number ($12$ or $10$) of scheduled rounds.

\item{\index{\bf Distance casting}}

{\bf Distance casting} is a sport of throwing fishing line with an atached sinker (usually, on land) as far as possible.

\item{\index{\bf Bat-and-ball game distances}} <\p>

Best known bat-and-ball games are cricket and baseball. In cricket, the field position of a player is named roughly according to its polar coordinates: one word ({\em leg, cover, mid-wicket}) specifies the angle from the batsman, and this word is preceded by an adjective describing the distance from the batsman. This distance is called {\em deep} (or {\em long}), {\em short} and {\em silly distance} if it is, respectively, farther away, closer and very close to the batsman. Distance further or closer to an extension of an imaginary line along the middle of the pitch bisecting the stumps, is called {\em wide} or {\em fine distance}, respectively. <\p>

In baseball, standard professional {\em pitching distance}, i.e., the distance between the front (near) side of the pitching rubber, where a pitcher must start his delivery, and home plate is $60$ feet, $6$ inches. The distance between bases is $90$ feet. <\p>

\item{\index{\bf Amazing greatest distances}}

Typical examples of {\bf amazing greatest distances} among Guinness world records are the greatest distances: run on a static cycle in one minute ($2.04$ km), moonwalked (as Michael Jackson) in one hour ($5.125$ km), covered three-legged (the left leg of one runner strapped to the right leg of another runner) in $24$ hours ($33$ km), jumped with pogo stick ($37.18$ km), walked with a milk bottle balanced on the head ($130.3$ km), covered by a car driven on its side on two wheels ($371.06$ km).

{\bf Mounting distance}: the distance from the crossing point of the axes to a locating surface of a gear, which may be at either back or front.

\item{\index{\bf Distance-to-fault}}

In Cabling, DTF ({\bf distance-to-fault}) is a test using time or frequency domain reflectometers to locate a fault, i.e., discontinuity caused by, say, damaged cable, water ingress or improperly installed/mated connectors. The amount of time pulse (output by tester into the cable) takes for the signal (reflected by a discontinuity) to return can be converted to distance along the line and provides an approximate location of the reflection point. Modern frequency domain testers measure signal attenuation (total return loss at the fault site).

It can be either a {\em tolerance} (the limit of an acceptable unplanned deviation from the nominal or theoretical dimension), or an {\em allowance} (planned deviation).

\item{\index{\bf Shooting distance}}

The {\bf shooting distance} (or {\em shot distance}) is the distance achieved by, say, a bullet or a golf ball after shot. The range of a Taser projectile, delivering an uncapacitating shock, is called the {\em shocking distance}.

In Photography, {\em shooting distance} is the distance from the camera to the subject.

\item{\index{\bf Distance spacer}}

A {\bf distance spacer} is a device for holding two objects at a given distance from each other.

Examples of related material components are: male-female {\em distance bolt}, {\em distance bush}, {\em distance ring}, {\em distance plate}, {\em distance sleeve}, {\em distance finger}, {\em distance gage}.

In Mathematics, {\em range} is the set of values of a function (or variable); more specifically, it means the difference (or interval, area) between maximum and minimum.

\item{\index{\bf Scale height}}

A {\bf scale height} is a distance over which a quantity decreases by a factor of $e$.

In Geophysics, the {\em scale height} (or {\em $e$-folding height}) is the height interval over which the pressure changes by a factor of $\frac{1}{e}$.

A general {\em L\'{e}vy flight} is a random walk in which the increments have power law probability distribution.

\item{\index{\bf L\'{e}vy walks in human mobility}}

A {\em jump} is a longest straight line trip from one location to another done without a directional change or pause. Consider a 2D {\em random walk} (taking successive jumps, each in a random direction) model that involves two distributions: a uniform one for the {\em turning angle} $\theta_i$ and a power law $P(l_i)\sim l_i^{-\alpha}$ for the {\em jump length} $l_i$. Brownian motion has $\alpha\ge 3$ and {\em normal diffusion}, i.e., MSD (mean squared displacement) grows linearly with time $t$: $MSD \sim t^{\gamma}, \gamma=1$. A {\em L\'{e}vy walk} has $1<\alpha<3$. Its jump length is {\em scale-free}, i.e., lacks an average scale $\overline{l_i}$, and it is {\em superdiffusive}: $MSD\sim t^{\gamma}, \gamma > 1$. Intuitively, L\'{e}vy walks consist of many short jumps and, exceptionally, long jumps eliminating the effect of short ones in average jump lengths.

L\'{e}vy walk dispersal was observed in foraging animals (spider monkeys, jackals, deer and many marine predators) since it might be optimal for finding patches of randomly dispersed food sources. L\'{e}vy walk might be a result of sequential visit pattern of the locations of meaningful contexts, i.e., walkers save time and effort by clustering closely located activities. So, they make many short jumps within the clustered areas and a few long jumps among areas.

Human mobility occurs on many length scales, ranging from walking to air travel. To predict it is important for planning (cities, pathways, marketing locations) and study (user distribution, virus spread, social networks). Brockmann, Hafnagel and Geisel, 2006, used the geographic circulation of money as a proxy for long-range human traffic. To track a bill, a user stamps it and enters data (serial number, series and local ZIP code) in a computer. Then the site \url{www.wheresgeorge.com} reports the time and distance between bill's consequtive sightings. $57\%$ of all $\approx 465,000$ considered bills traveled $50-800$ km over $9$ months in US. Obtained probability of a bill traversing a distance $r$ (an estimate of the probability of humans moving such distance) followed, over $10-3,500$ km, a power law $P(r)=r^{-1.6}$. Bank note dispersal was fractal, and the bill trajectories resembled L\'{e}vy walks.

Gonz\'{a}lez, Hidalgo and Barab\'{a}si, 2008, studied the trajectory of $100,000$ anonymized mobile phone users (random sample of $6$ million) over $6$ months. Obtained probability of finding a user at a location of {\em rank} $k$ (by the number of times an user was recorded in vicinity) was $P(k)\sim \frac{1}{k}$. $40\%$ of the time users were found at their first two preferred locations (home, work), while spending remaining time in $5-50$ places. Authors suggest that the L\'{e}vy statistics in bank note dispersal capture a convolution of the population heterogeneity, shown in their truncated power law, and the regularity of motion of individuals.

Jiang, Yin and Zhao, 2009, analyzed people's moving trajectories, obtained from GPS data of $50$ taxicabs over $6$ months in a large street network. They found a L\'{e}vy behavior in walks (both, origin-destination and between streets) and attributed it to the fractal property of underlying street network, not to the goal-directed nature of human movement. Rhee, Shin, Hong, Lee and Chong, 2009, analyzed $\approx 1000$ hours of GPS traces of walks of $44$ participants. They also got L\'{e}vy walks but explain that by human intentions in deciding travel destinations (and distance and sojourn time thereof) suggesting that geographical constraints (roads, buildings, etc.) only cause truncations of flight lengths.

A Scottish company Distance Labs has announced the ``intimate communication over a distance'', an interactive installation {\em Mutsugoto} which draws, using a custom computer vision and projection system, lines of light on a body of a person. {\em Sports over distance} is another example of implemented computer-supported movement-based collaborative interaction between remote players. Besides light, haptic technology provides a degree of touch communication between remote users.