For a simple continued fraction, the complete quotients satisfy ${\theta }_0=x$, $a_n=\lfloor {\theta }_n\rfloor$ and ${\theta }_{n+1}=1/({\theta }_n-a_n)$ whenever the denominator is nonzero. Thus ${\theta }_n=[a_n;a_{n+1},\dots ]$.

The map $T(x)=1/x-\lfloor 1/x\rfloor$ on $(0,1]$, with $T(0)=0$. Its digit $N(x)=\lfloor 1/x\rfloor$ gives the successive partial quotients $N(T^{j}x)$ of a continued fraction. This is distinct from the geometric Gauss map assigning surface normals.

A hyperbola is a conic section with two unbounded branches. In suitable coordinates its equation is $x^2/a^2-y^2/b^2=1$ with $a,b>0$. The unit hyperbola is parametrized by $x=\cosh t$, $y=\sinh t$ on its right branch, relating it to the hyperbolic functions.

For example, $x^2/9-y^2/16=1$ has foci of a conic at $(\pm 5,0)$, because $5^2=3^2+4^2$. Its focal-distance difference has absolute value $6$, while its eccentricity of a conic is $5/3>1$. At the point $(3,0)$, the focal distances are $8$ and $2$, verifying that difference. The asymptotes are $y=\pm 4x/3$; a distant point approaches those directions while remaining on one of the two branches. The focal-distance difference, rather than a constant sum, distinguishes this hyperbola from an ellipse.

A parabola is a plane curve whose points are equally distant from a focus and a directrix. In suitable coordinates it has equation $y=a{x}^2$, with $a\ne 0$. Its quadratic geometry can also describe a contour deformation through a saddle point.

For example, take a focus of a conic at $(1,0)$ and a directrix of a conic $x=-1$. Equality of squared Euclidean distances gives $(x-1{)}^2+y^2=(x+1{)}^2$, hence $y^2=4x$. The point $(1,2)$ has distance $2$ to both the focus and the directrix. Its eccentricity of a conic is one. Unlike an ellipse, a parabola is unbounded and has only one matched focus-directrix description.

Before the Loewner swallowing time of $z$, set $Z_t=g_t(z)-U_t=X_t+i{Y}_t$ and ${{\rm Y}}_t=Y_t/|g_t^{\prime }(z)|$. Since the conformal radius of the complex upper half-plane at $Z_t$ is $2Y_t$, the transformation rule gives $2{{\rm Y}}_t={\mathrm{crad}}_{D_t}(z)$. The Koebe quarter theorem and the interior-disc comparison give $\mathrm{dist}(z,\partial D_t)/2\le {{\rm Y}}_t\le 2\mathrm{dist}(z,\partial D_t)$. Differentiating the Chordal Loewner equation gives $\boxed{{\partial }_t\log {{\rm Y}}_t=-4Y_t^2/|Z_t{|}^4}$. Thus this quantity decreases as the remaining domain shrinks.

An ellipsoidal confidence region defined by a positive-definite matrix $A$ and a calibrated threshold $c$. For a known-covariance multivariate normal distribution sample, $A=n{\Sigma }^{-1}$ and the $1-\alpha$ quantile of the chi-squared distribution with $d$ degrees of freedom give exact coverage for the $d$-dimensional mean.

For two independent observations uniform on $[\theta -a,\theta +a]$, the displayed interval of possible parameter values contains the true parameter almost surely. Intersecting any confidence interval with it preserves that interval's coverage, while excluding values inconsistent with the observations. The order statistics interval $[U_{(1)},U_{(2)}]$ has coverage $1/2$, since the observations straddle $\theta$ with probability $1/2$. If their separation exceeds $a$, the entire interval of possible parameters lies between the observations; this explains why the nominal one-half procedure can be unnecessarily wide on that event.

For $\mathcal{L}={\partial }_{\mu }{\psi }^{\ast }{\partial }^{\mu }\psi -V(|\psi {|}^2)$, constant $\alpha$ preserves both terms. This is an internal symmetry of a classical field theory with circle group $U(1)$. A spacetime-dependent phase would introduce derivative terms, so the ordinary-derivative density has only the global symmetry. The associated current is the Noether charge of a complex scalar field construction; the orientation $e^{-i\alpha }$ fixes its overall sign.

The principal cube root is $z^{1/3}=\exp (\mathrm{Log}z/3)$ using the principal complex logarithm. In the right half-plane, its complex argument lies between $-\pi /6$ and $\pi /6$. Consequently the roots of $r^3=-z$ consist of one root with negative real part, $-z^{1/3}$, and two with positive real part, $z^{1/3}{e}^{\pm i\pi /3}$.

The balanced complete multipartite blow-up $K_r(t)$ replaces each vertex of the complete graph $K_r$ by an independent set of $t$ vertices, placing every possible edge between different sets and none inside a set. Every fixed graph of chromatic number at most $r$ embeds into $K_r(t)$ for sufficiently large $t$.

A compact exhaustion of an open subset $U$ of a finite-dimensional Euclidean space is an increasing sequence of compact subsets $K_n\subset U$ with $K_n\subset \mathrm{int}{K}_{n+1}$ and ${\bigcup }_n{K}_n=U$. For example, choose smoothed bounds or suitable nested levels of distance from the complement and distance from the origin. Exit times from these sets express the lifetime of a maximal local solution of a stochastic differential equation; every finite-time continuous path lying in $U$ has compact image and is contained in some $K_n$.

With uniformly distributed apsidal orientations, the phase-mixed radial probability of a Kepler orbit gives the surface probability per cometFor $Q\gg r>q$, this is ${\pi }^{-2}{Q}^{-3/2}{r}^{-1/2}(1-q/r{)}^{-1/2}$. Multiply by the population size for a number surface density of a disk.

For a fixed alphabet size $m>1$, a family $\mathcal{F}\subseteq \mathcal{P}([n])$ is adequate if every two-color finite coloring of $[m{]}^n$ has a monochromatic combinatorial line whose nonempty active coordinate set belongs to $\mathcal{F}$.

An intersecting family is never adequate. Color a word red when the coordinates carrying letter $1$ contain a member of $\mathcal{F}$, and blue otherwise. For a line with active set $A\in \mathcal{F}$, its letter-$1$ word is red and its letter-$2$ word is blue, because no member of an intersecting family can lie wholly outside $A$.

Repeat every coordinate of a word $q$ times. A combinatorial line maps to a combinatorial line whose active-set size is multiplied by $q$. Pulling back a finite coloring along this map and using the Hales-Jewett theorem forces monochromatic lines with active-set size divisible by any prescribed positive integer $q$.

For any $N\ge 1$, color $w\in [3{]}^N$ by its first letter together with its number of runs of a word modulo $3$. This finite coloring uses at most nine colors and has no monochromatic combinatorial line whose active coordinates form an interval. An active interval starting at the first coordinate changes the first letter. Otherwise, its left boundary and optional right boundary contribute $0$ versus $1$, $0$ versus $2$, or $1$ versus $2$ to the run count as the active letter varies. These differences remain nonzero modulo $3$. The first-letter component is essential for intervals that occupy the entire word.

An inelastic collision changes total kinetic energy. In the dissipative case it decreases, transferring energy to internal energy, including deformation and microscopic motion. Isolated colliding bodies still obey momentum conservation; conservation of energy requires accounting for the transferred energy. Bodies need not stick in every inelastic collision. The decrease in translational kinetic energy drives dissipative spreading of a planetary ring.

For the incoming masses and velocities in the elastic collision example, stipulate that the bodies stick. This perfectly inelastic collision has a shared outgoing velocityThe outgoing momenta are $4$ and $2\mathrm{kg}m/s$, totaling $6\mathrm{kg}m/s$. The outgoing kinetic energies are $4$ and $2\mathrm{J}$, totaling $6\mathrm{J}$. Thus $9-6=3\mathrm{J}$ of kinetic energy becomes internal energy within the isolated pair. Nothing in momentum conservation requires this $3\mathrm{J}$ to remain as kinetic energy. ^[ref]

Substitute a static soliton family with slowly time-dependent collective coordinates into the action. The kinetic energy induces a Riemannian metric $g$ on the family, after imposing any Gauss law constraint in gauge theory constraint and projecting out gauge directions. If its static energy is constant, the Euler-Lagrange equations are the geodesic equations of $g$. An approximately flat family instead carries a potential $V(q)$. This approximation neglects excitations of the other field modes and is justified only when their effects are small on the time and energy scales studied.

Relative cohomology is the cohomology of cochains on $X$ vanishing on chains in $A$. A pair has a natural long exact sequence in cohomology, and excision identifies relative groups for suitable neighborhood pairs.

Integral cohomology is cohomology with integer coefficients, $H^{\ast }(X;\mathbb{Z})$. Its cup product makes it a graded cohomology ring, retaining torsion and integral information that can disappear after changing coefficients to a field.

A sequentially closed set is one that contains every limit of its convergent sequences. This is weaker than being a closed set in a general topological space, but the two conditions agree in metric spaces.