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A Weierstrass equation presents an elliptic curve aswith nonzero discriminant, together with the point at infinity $O$.

Every complex number can be written uniquely as $x+y\omega$ with real $x,y$, where $\omega =(-1+i\sqrt{3})/2$. Round $x,y$ separately to integers $m,n$. The error is $s+t\omega$ with $|s|,|t|\le 1/2$, andFor Eisenstein integers $\alpha ,\beta$ with $\beta \ne 0$, apply this to $\alpha /\beta$ to choose $q=m+n\omega$. The remainder $r=\alpha -q\beta$ is an Eisenstein integer and $N(r)\le 3N(\beta )/4<N(\beta )$ if nonzero. Thus the Eisenstein norm is a Euclidean function and these integers form a Euclidean domain. Coordinate rounding suffices; no closest-point theorem is needed.

If $A(t)=A+tE$ and $\lambda (t)$ is a differentiable function defining a simple eigenvalue, thenThe Cauchy-Schwarz inequality and the operator norm give

A common-envelope phase occurs when unstable binary interaction engulfs both stellar cores in one envelope. Orbital energy and angular momentum may eject the envelope, leaving a compact binary, or the cores may merge.

For $S=({\mathbb{R}}^2\setminus \{0\})\times \{0\}\subset {\mathbb{R}}^3$ and $p=(1,0,0)$, geodesics are straight lines until they hit the missing origin. Thus $(-2,0,0)\notin {\mathcal{D}}_p$, and $(-1,0,0)$ is not in the image of ${\exp }_p$ because its unique straight segment from $p$ passes through the origin.

Dolbeault cohomology is

The value group of a valued field is the ordered abelian group formed by the values of its nonzero elements. A valuation is discrete when its value group is infinite cyclic with the order inherited from $\mathbb{Z}$.

For the $\{-1,1\}$-valued convention on the unbiased Boolean hypercube, the influence of coordinate $i$ is ${\mathrm{Inf}}_i(f)=\Vert D_{i}f{\Vert }_2^2$, where $D_{i}f$ is the discrete derivative of a Boolean function. It equals the probability that flipping coordinate $i$ changes $f$. For the $\{0,1\}$-valued convention, the same flip probability is instead $4\Vert D_{i}f{\Vert }_2^2$, since a nonzero discrete derivative has magnitude $1/2$.

If boundary fluxes decay sufficiently fast, two integrations by parts givewhile the diffusion term contributes zero to the first moment.

Near a stable deterministic equilibrium, replace the drift by its linearization and the diffusion coefficient by its equilibrium value. The resulting Ornstein-Uhlenbeck approximation has a stationary normal distribution.

The Debye temperature is $T_D=\hslash {\omega }_D/k_B$.

For a twice differentiable vector field $A$,

Exponential tilting by $\theta$ replaces a law $P$ byUnder the tilted law, the mean is ${\psi }^{\prime }(\theta )$ whenever the derivative exists.

The Coxeter graph has vertex set $I$, joins $i$ and $j$ when $m_{ij}\ge 3$, and labels the edge by $m_{ij}$ when $m_{ij}>3$. A Coxeter system is irreducible exactly when this graph is connected.

The Coxeter number is the order of a Coxeter element in a finite irreducible Coxeter group.

Every smooth closed differential form of positive degree is locally an exact differential form. On a coordinate ball centered at zero, put $F_t(x)=tx$ and let $R=x^i{\partial }_i$ be the radial vector field. Define the degree-lowering operatorFor a $p$-form with $p>0$, the integrand has a factor $t^{p-1}$ in its coordinate coefficients and is smooth at zero. Cartan's magic formula and differentiation of the pullback of a differential form give ${\partial }_t(F_t^{\ast }\alpha )=t^{-1}{F}_t^{\ast }(d{\iota }_R\alpha +{\iota }_{R}d\alpha )$. Integrating from zero to one gives $dK\alpha +Kd\alpha =\alpha$, since $F_0^{\ast }\alpha =0$ in positive degree. If $d\alpha =0$, take $\beta =K\alpha$. The local potential of a closed differential one-form gives the explicit degree-one construction. Global exactness can fail through nonzero periods, measured by de Rham cohomology.

For an invertible operator $\beta$ on $V$, the mapis a linear isomorphism. Conjugate choices of $\beta$ induce conjugate operators ${\varphi }_{\beta }$.

For a complex submanifold $Y\subseteq X$, the holomorphic normal bundle is the quotient holomorphic vector bundle

Convexity is essential. In $\mathbb{R}$, letThis set is closed, symmetric, and has gaps arbitrarily near zero. Given $x\ne 0$, choose $k$ so small-scale that the interval
$[|x|/(2\cdot 4^{-k}),|x|/4^{-k}]$ has length at least one; it contains an integer $n$, and then $x/n\in S$. Thus ${\bigcup }_{n}nS=\mathbb{R}$, although $S$ is not a neighbourhood of zero.

For a rank-$e+1$ vector bundle $E\to X$ and $p:\mathbb{P}(E)\to X$, powers of $\zeta =c_1(\mathcal{O}(1))$ give isomorphisms