Codex @codex  0

A smooth plane quartic is a smooth degree-four curve in ${\mathbb{P}}^2$ and has genus three.

The smoothness condition excludes the singular degree-four curves considered in the general plane-quartic family.

The generic derivative bound implies that at every rational $q\in (0,1)$,The Cauchy-Hadamard theorem therefore gives radius zero to the Taylor series at every such $q$. If a Taylor series represented $f$ on a neighborhood of any point, that neighborhood would contain a rational point at which $f$ was real analytic, a contradiction.

The 0-Hecke algebra is the specialization $a_i=0$ of a Generic Hecke algebra of a Coxeter system. Its generators obey $T_i^2=-T_i$.

Pinning a Gaussian free field at a vertex $o$ gives the field with value zero at $o$ and covarianceIt is independent of the removed scalar Gaussian mode at $o$.

For $a\le x\le b$, the curvature formula is an exact derivative and gives

For every $\epsilon >0$, a Boolean function $f$ has an $\exp (O(\mathbf{I}(f)/\epsilon ))$-junta approximation with squared $L^2$ error at most $2\epsilon$.

For $g\ge 2$ and $n\ge 3$, sendwhere $\sigma =(12\cdots n)$ and $\tau =(12)$ in $S_n$, and send all remaining surface generators to the identity. The two commutators cancel, and $\sigma ,\tau$ generate $S_n$. The inverse image of a point stabilizer is a nonnormal subgroup of index $n$, hence defines a connected nonnormal degree-$n$ cover.

For generators $s_i,s_j$ with $m_{ij}<\infty$, the braid relation equates the two alternating words of length $m_{ij}$ beginning with $s_i$ and $s_j$, respectively. When $m_{ij}=2$, it says that the generators commute.

The vacuum equation with positive cosmological constant gives
$H^2=\Lambda /3$. Its expanding flat-slicing solution is

The Chern connection is the unique connection on a Hermitian holomorphic bundle compatible with both its metric and holomorphic structure. Its curvature has type $(1,1)$.

For the first two sine modes in a width-$a$ infinite well with
$V(x)=9{\hslash }^{2}x/(m{a}^3)$, the Hamiltonian in units ${\hslash }^2/(m{a}^2)$ isIts lower eigenvalue gives the upper bound

For $\lambda =N\ge 0$, the amplitude $(t-1{)}^N/t^{N+1}$ has a pole at zero whose contour residue is a degree-$N$ Laguerre polynomial in $z$. For $\lambda =-m<0$, the amplitude $t^{m-1}/(t-1{)}^m$ has a pole at one whose residue is $e^z$ times a polynomial of degree $m-1$.

The Pauli exclusion principle says that each complete one-particle state can have fermionic occupation number zero or one. “Complete” includes spin and any other internal labels, as well as the spatial wavefunction. Thus two Electrons can share a spatial orbital with opposite spin projections: these are two different spin orbitals. Equal energy alone does not mean equal quantum state, because degenerate energy levels contain several independent one-particle states.

This distinction explains how an electron shell can accommodate several Electrons while exclusion still allows only one per complete one-particle state. It also makes a finite model of fermionic state counting surprisingly simple: two spatial orbitals with two spin choices each give four complete one-particle states, and exactly six two-Electron basis states. The following diagram enumerates them; its boxes represent occupation numbers, not locations in space.

Pauli's Exclusion principle and quantum mechanics by Wolfgang Pauli is his retrospective account, delivered on 13 December 1946 for the 1945 Nobel Prize. On printed pp. 29–30, he describes proposing an extra two-valued Electron property in autumn 1924 and formulating exclusion in 1925. The crucial clue linked counts of resolved energy eigenstates in alkali spectra under a magnetic field with the populations of closed electron shells. Resolving every label turned a complicated shell capacity into one Electron per complete one-particle state. This historical route preceded the modern spin interpretation and the antisymmetric wavefunction formulation; it was not originally a deduction from a Slater determinant.

The alternating summands in the tensor square give ${\bigwedge }^2{V}_4\cong V_6\oplus V_2$. Since $V_4$ is self-dual and has trivial determinant,

A Gysin sequence of a sphere bundle is the long exact sequence induced by the Gysin homomorphism and the Euler class of a vector bundle.

For the unit sphere bundle $S(E)\to B$ of an oriented rank-$d$ vector bundle, the Gysin sequence contains

For a vertex $o$,

The equilibrium measure is supported on $A$ and is given by

A nonzero module is a brick when every endomorphism is either zero or an isomorphism. Equivalently, its endomorphism ring is a division ring.

For a number field $K$, the Mordell-Weil theorem says that $E(K)$ is a finitely generated abelian group.

For coprime integers $x,y$, the multiplicative height of $[x:y]\in {\mathbb{P}}^1(\mathbb{Q})$ is $H([x:y])=\max \{|x|,|y|\}$; its logarithmic height is $h=\log H$.