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For a random field $(Y_x)$, level-set percolation studies the random vertex set $\{x:Y_x\ge a\}$ as the threshold $a$ varies.

For an affine variety over a perfect field, the tangent-space codimension at a point is the rank of the Jacobian matrix of defining equations. The smooth locus of a pure $d$-dimensional variety in ${\mathbb{A}}^N$ is where this rank is $N-d$.

If an elliptic function $g$ has degree $d$ and $r$ distinct poles in a fundamental parallelogram, then each pole order increases by one under differentiation, soBecause $1\le r\le d$, this gives $d+1\le \mathrm{deg}{g}^{\prime }\le 2d$.

Let a nonconstant rational function $f$ have degree $d$, let $r$ be its number of distinct finite poles, and let $m_{\infty }$ be its pole order at infinity, taken as zero when infinity is not a pole. Differentiation raises each finite pole order by one and changes a pole of order $m_{\infty }>0$ at infinity into one of order $m_{\infty }-1$. HenceIn particular, $d-1\le \mathrm{deg}{f}^{\prime }\le 2d$.

If the base of a covering is path-connected, a path from $x_0$ to $x_1$ gives a bijectionby sending each initial lift point to the endpoint of its unique lifted path. Lifting the reversed path gives the inverse bijection.

Let three equal masses move on a circle of radius $r$, and let the potential be $U(\alpha )+U(\beta )+U(\gamma )$ with $\alpha +\beta +\gamma =2\pi$. At equal spacing $s_0=2\pi /3$, angular displacements have a zero-frequency rigid-rotation mode $(1,1,1)$ and a two-dimensional sum-zero eigenspace withThe two vibrational modes are degenerate by the permutation symmetry.

A Be star is a rapidly rotating B-type star showing Balmer emission from a circumstellar gaseous decretion disk.

For two polarizations, $n$ atoms in area $A$, and $\omega =C|k{|}^{\alpha }$, mode counting givesAt low temperature its energy scales as $T^{1+2/\alpha }$ and its constant-volume heat capacity as $T^{2/\alpha }$.

The Debye frequency is the upper frequency cutoff fixed by equating the continuum mode count to the crystal's number of vibrational degrees of freedom.

For $k>0$, the trajectory beginning at $(x,y)=(1,y_0)$ with $y_0>0$ remains in $x>1$, $y>0$ and obeysIt enters every strip $0<y<\epsilon$: otherwise $\dot{E}=-k{y}^2\le -k{\epsilon }^2$ would eventually make the nonnegative energy negative.

If $u(x,0)=0$, $u_t(x,0)=g(x)$, and the odd function $g$ is supported on $[-a,a]$, thenThe interval $[x-t,x+t]$ then contains the entire support, whose integral vanishes by oddness.

For the parametrizationthe Gaussian curvature and area element are

For an arc-length profile with radius $Y(u)>0$, the equation $K=c$ isThus $Y=A\cos u$ gives $K=1$ locally, while $Y=A\cosh u$ gives $K=-1$ wherever $|Y^{\prime }|<1$. At $u=0$ their mean curvatures are respectivelyso different choices of $A$ give locally noncongruent surfaces with the same constant Gaussian curvature.

The curvature of a connection satisfies $d_{A}F(A)=0$. In a local frame this follows directly by expanding $d(dA+A\wedge A)+A\wedge F-F\wedge A$.

The Legendre transform is the convex function

For $S=\{{\sum }_i{l}_i{a}_i:-L_i/2\le l_i\le L_i/2\}$ and ${\alpha }_i=Q\cdot a_i$,For large $L_i$, each factor is sharply peaked when ${\alpha }_i\in 2\pi \mathbb{Z}$, exactly the condition $Q\in {\Lambda }^{\ast }$.

The critical line is the symmetry line $\Re s=1/2$ inside the critical strip.

A Coxeter element is a product of all simple generators in some order. For a finite irreducible Coxeter group, all Coxeter elements are conjugate.

For involutory generators, the folding condition requires that simple multiplication never preserve length and that simultaneous left and right ascents either combine to a two-step ascent or fold: if both $s_{i}w$ and $w{s}_j$ have length $\ell (w)+1$, then either $\ell (s_{i}w{s}_j)=\ell (w)+2$ or $w=s_{i}w{s}_j$.

For $J\subseteq I$, the standard parabolic subgroup is $W_J=⟨s_j:j\in J⟩$. Its cosets index faces of a fixed type in the Coxeter complex.