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The waist circle $z=0$ is a geodesic. Its Gaussian curvature is $-1$, whereasoff the waist. Since isometries preserve Gaussian curvature, every isometry preserves this circle setwise.

For a unit-speed geodesic on $x^2+y^2=z^2+1$, Clairaut's constant $c$ givesIf $c>1$, the geodesic has a turning point at $z=\sqrt{c^2-1}$ and remains entirely in one of the regions $z>0$ or $z<0$.

On $x^2+y^2=z^2+1$, an axial plane such as $y=0$ cuts out two disjoint meridian geodesics. The tangent plane $x=1$ cuts out the two straight ruling lines $y=\pm z$, which are geodesics meeting orthogonally at $(1,0,0)$.

A scale-free linear matter spectrum has $P(k)=A{k}^n$. Dimensional analysis then makes its one-loop $P_{22}$ contribution proportional to $A^2{k}^{2n+3}$ whenever the loop integral converges.

A boost of rapidity $\phi$ acts on null coordinates by $u\mapsto e^{\phi }u$ and $v\mapsto e^{-\phi }v$, preserving $uv=x^2-y^2$.

The FENE-P model replaces infinitely extensible Hookean polymers by finitely extensible nonlinear elastic springs with a mean-field closure. Finite extensibility regularizes the Oldroyd-B extensional catastrophe and produces shear thinning.

If $N|\nu ⟩=\nu |\nu ⟩$, then $A$ lowers the eigenvalue by one and $\Vert A|\nu ⟩{\Vert }^2=\nu \Vert \nu ⟩{\Vert }^2$. Positivity forces the lowering ladder to terminate at eigenvalue zero, so every eigenvalue is a nonnegative integer.

For a hypersurface-orthogonal null congruence under the null energy condition in $D$ spacetime dimensions, an initially negative expansion ${\theta }_0$ diverges to $-\infty$ within affine distance at most $(D-2)/|{\theta }_0|$.

A map $f:S^{n-1}\to X$ extends to $D^n$ exactly when it is null-homotopic. An extension contracts radially through the disc; conversely, a null-homotopy descends through the cone quotient $C{S}^{n-1}\cong D^n$.

A null geodesic is both a null curve and a geodesic. In geometric optics, light rays follow null geodesics.

For the affine curve $C:y^2=x(x-1{)}^2$, the rational functiongives $x=t^2$ and $y=t(t^2-1)$. Thus its coordinate ring embeds as $k[t^2,t(t^2-1)]\subset k[t]$, and the function field is $k(t)$. Since $t$ satisfies the monic equation $t^2-x=0$, $k[t]$ is integral over the coordinate ring. It is an integrally closed domain with that same fraction field, so the affine normalization is ${\mathbb{A}}_k^1$.

If $\mathrm{char}k\ne 2$, the singular point $(1,0)$ has two preimages $t=1,-1$, which the normalization separates. In local coordinates $u=x-1$, its tangent cone is $y^2-u^2$, giving two distinct branches. The smooth projective model of $k(t)$ is ${\mathbb{P}}_k^1$, so the geometric genus is zero. The affine and projective normalizations must be distinguished.

In characteristic two the same normalization formula still works, but $1=-1$: there is only one geometric preimage and the tangent cone is a repeated line. The two-branch nodal description therefore requires the stated characteristic restriction.

Orthogonality separates total quadratic wave energy into a sum of independent modal energies.

Every finite-dimensional complex normal matrix has an orthonormal eigenbasis, equivalentlyfor some unitary $U$ and diagonal $D$.

If ${\hat{\beta }}_{(i)}$ is the least-squares estimate after deleting observation $i$, Cook's distance isIt measures the deleted estimate in the same quadratic metric as the coefficient confidence ellipsoid.

The standard normal distribution is $N(0,1)$: the normal distribution with mean zero, variance one and standard deviation one. Its standard normal density $\varphi$ integrates to one; its standard normal distribution function $\Phi$ records the area accumulated from $-\infty$ to a threshold. The distinction matters: $\varphi (0)=1/\sqrt{2\pi }\approx 0.398942$, $\Phi (0)=1/2$, and $\mathbb{P}(Z=0)=0$ for a standard normal random variable $Z$.

For a toy measurement $X\sim N(100,15^2)$, leave the measurement unit unspecified. The standardization of a normal random variable $Z=(X-100)/15$ converts $x=85,100,115$ into standard scores $z=-1,0,1$. Using the standard normal distribution function,The second area is half the first because the normal distribution is a symmetric probability distribution about its mean. The numerical heights differ: $f_X(115)=\varphi (1)/15$, yet the corresponding shaded areas are equal because $dx=15dz$ in the change of variables.

A positive Random variable is log-normal if its logarithm has a Normal distribution. For parameters $\mu \in \mathbb{R}$ and $\sigma >0$, write $X=e^{\mu +\sigma Z}$ with $Z$ standard normal. The change of variables $z=(\log x-\mu )/\sigma$ gives the densityThe parameters describe the logarithm, not the mean and standard deviation of $X$ itself.

The normal Moment-generating function gives every real power moment:Consequently the mean is $e^{\mu +{\sigma }^2/2}$, the variance is $(e^{{\sigma }^2}-1)e^{2\mu +{\sigma }^2}$, and the median is $e^{\mu }$. For $\mu =0,\sigma =1$, the median is one but the mean is $e^{1/2}$, illustrating the right-skewed shape.

All power moments are finite, yet $\mathbb{E}{e}^{tX}=\infty$ for every $t>0$. Indeed the normal-coordinate integral contains $\exp (t{e}^{\mu +\sigma z}-z^2/2)$, whose exponential-of-exponential growth defeats the Gaussian tail as $z\to \infty$. Thus log-normal laws are heavy-tailed in the positive-exponential-moment convention despite having finite variance.

independent log-normal factors multiply to a log-normal variable because their logarithms add as independent normal variables. This is the natural setting for positive quantities with multiplicative variability. The case $\sigma =0$ is a deterministic positive limit, rather than a law with the displayed density.

The half-normal distribution describes the absolute value of a centered Gaussian random variable. Its unit-scale version is $H=|Z|$ for a standard normal random variable $Z$. It folds the negative half of the standard normal distribution onto the positive half, retaining the size of a deviation while discarding its sign. If a centered Gaussian random variable models measurement error, its magnitude therefore has a half-normal distribution.Here $\varphi$ is the standard normal density and $\Phi$ is the standard normal distribution function. For $x\ge 0$, the event $H\le x$ is $-x\le Z\le x$, so the symmetric probability distribution of $Z$ gives $F_H(x)=\Phi (x)-\Phi (-x)=2\Phi (x)-1$. The upper-tail probability is therefore $\mathbb{P}(H>x)=2(1-\Phi (x))$. For example, $\mathbb{P}(H\le 1)\approx 0.6826894921$. The doubled probability density function integrates to one on $[0,\infty )$, and $\mathbb{P}(H=0)=0$.

The mean and variance of the unit-scale half-normal distribution areFor a short derivation, $\mathbb{E}[H]=2{\int }_0^{\infty }x\varphi (x)dx=2\varphi (0)$ because ${\varphi }^{\prime }(x)=-x\varphi (x)$, while $H^2=Z^2$ gives $\mathbb{E}[H^2]=1$. More generally, $\sigma |Z|$ for $\sigma >0$ has a half-normal distribution with probability density function $2\varphi (x/\sigma )/\sigma$ for $x\ge 0$. Its mean is $\sigma \sqrt{2/\pi }$ and its variance is ${\sigma }^2(1-2/\pi )$; here $\sigma$ is the scale of the underlying Gaussian random variable, not the standard deviation of its absolute value.

For a regular closed embedding $i:X\hookrightarrow Y$ of codimension $d$, the self-intersection formula is $i^{\ast }{i}_{\ast }(\alpha )=c_d(N_{X/Y})\cap \alpha$.

The classical zero-free region asserts that for some $c>0$,Together with bounds for the logarithmic derivative, it permits contour arguments with exponentially small errors in $\sqrt{\log x}$.

The Riemann hypothesis asserts that every Nontrivial zero of the Riemann zeta function lies on the critical line $\Re s=1/2$.