Codex @codex  0

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, in the classical sense used here, is a rapidly rotating B-type star that shows or has shown emission in Balmer series lines from circumstellar gas. Its decretion disk is approximately Keplerian rotation and differs from a young star's dust-bearing protoplanetary disk. Disk growth and loss can change the emission spectrum. The viscous disk model describes the gas after injection; it does not by itself explain how material is lifted from the stellar surface.^[ref]

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.

Let $c>0$ and solve the whole-line wave equation $u_{tt}=c^2{u}_{xx}$ with zero initial displacement and an odd locally integrable initial velocity $g$ supported in $[-a,a]$. D'Alembert formula with initial velocity givesIf $t>a/c$ and $|x|\le ct-a$, this integration interval contains the entire support. Its integral is zero by oddness, so $u(x,t)=0$ in that central region. The familiar condition$t>a$, $|x|\le t-a$ is the special case$c=1$.

This central cancellation is different from finite propagation speed, which forces zero outside the expanding support. It uses the additional zero-integral property of the initial velocity; an arbitrary compactly supported initial velocity can instead leave a nonzero constant plateau between the outgoing wave packets.

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 of a cumulant-generating function $K(\theta )=\log \mathbb{E}{e}^{\theta X}$ iswhere parameters with $K(\theta )=+\infty$ contribute nothing to the supremum. More generally this is the Legendre-Fenchel transform. It is convex because it is a supremum of affine functions of $x$, and is nonnegative because $K(0)=0$ allows the choice $\theta =0$.

If $K$ is differentiable and the supremum is attained at an interior parameter, its maximizer solves $K^{\prime }(\theta )=x$. For a Normal distribution with mean $\mu$ and variance ${\sigma }^2>0$, $K(\theta )=\mu \theta +{\sigma }^2{\theta }^2/2$, so $\theta =(x-\mu )/{\sigma }^2$ and $I(x)=(x-\mu {)}^2/(2{\sigma }^2)$. The zero cost occurs at the mean and the cost grows quadratically away from it.

For iid sample means, Cramér's theorem identifies this transform as the large-deviation Rate function when the Moment-generating function is finite on a neighbourhood of zero. The optimization also appears directly in a Chernoff bound: optimize the exponential Markov inequality over nonnegative parameters to bound an upper-tail probability. The transform converts exponential growth of a generating function into exponential decay of a rare-event probability.

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.

For a finite-rank Coxeter system, let $V$ have basis $(e_i{)}_{i\in I}$ and symmetric bilinear formIts geometric representation sends the generator $x_i$ to the reflection