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A gauge anomaly destroys a gauge redundancy needed to remove unphysical states and makes the quantum gauge theory inconsistent unless the anomaly cancels.

Polar coordinates give ${\mathbb{C}}^{\ast }\cong S^1\times \mathbb{R}$ after taking logarithmic radius.

The vector fields $(cx+d){\partial }_x-2cu{\partial }_u$ generateTheir second prolongations multiply the differential equation by $-4c$, so they are infinitesimal Lie symmetries.

On jet coordinates,It differentiates a differential function along prolonged solution graphs.

A smooth projective curve is a nonsingular projective curve. A smooth projective curve is a nonsingular projective variety of dimension one.

The product of projective varieties is projective via the Segre embedding. Dimensions add under products, and the product of smooth varieties is smooth.

For $X=V_{+}(I)\subseteq {\mathbb{P}}^n$, the homogeneous coordinate ring is the graded ring $k[x_0,\dots ,x_n]/I$.

A projective curve is a projective algebraic curve. A projective curve is a one-dimensional projective variety.

If projective varieties $X,Y\subseteq {\mathbb{P}}^n$ have dimensions $r,s$ and $r+s\ge n$, then $X\cap Y$ is nonempty and every component has dimension at least $r+s-n$.

The translations $x\mapsto x+b$ for $b\in {\mathbb{F}}_4$ form a Sylow $2$-subgroup of $PG{L}_2({\mathbb{F}}_4)$. Every nonidentity translation fixes infinity and exchanges the four finite points in two transpositions, so its action lies in $A_5$.

The projective line over a field $F$ is the set of one-dimensional subspaces of $F^2$ and may be identified with $F\cup \{\infty \}$.

At zero, $f(x)=x$ and $g(x)=1+|x|$ give a differentiable productwith derivative one, although $g$ is not differentiable.

The higher-order product rule is

Every invertible real matrix has the polar decompositionwhere $P$ is symmetric positive definite and $R$ is orthogonal.

This is the invertible real-matrix case of polar decomposition.

The conformal weights determine the response of a primary operator to holomorphic and antiholomorphic coordinate rescaling. Its scaling dimension is $\Delta =h+\tilde{h}$ and its spin is $h-\tilde{h}$.

A minimal primary decomposition of an ideal $I$ is an irredundant finite intersection $I={\bigcap }_i{Q}_i$ of primary ideals whose radicals are pairwise distinct.

The radical $\sqrt{I}=\{x:x^n\in I\text{ for some }n\ge 1\}$ of a primary ideal is prime. In a number ring this forces a nonzero primary ideal to have only one prime in its unique factorization.

The Press-Schechter differential abundance is

For positive-definite $D$ and no-flux boundaries,decreases as $\dot{\mathcal{F}}=-\int P\nabla (V+\log P{)}^{T}D\nabla (V+\log P)dx$. Its unique normalized minimizer is proportional to $e^{-V}$.

If $M⪰0$ has eigenpairs $({\mu }_i,v_i)$ and $\mathrm{tr}M>s$, then its Frobenius projection onto ${\mathcal{S}}_s$ iswhere $\rho >0$ is uniquely determined by ${\sum }_i({\mu }_i-\rho {)}_{+}=s$.