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A prime-power ideal has the form ${\mathfrak{p}}^m$ for a prime ideal $\mathfrak{p}$ and positive integer $m$.

Ideals $I$ and $J$ are comaximal when $I+J=R$.

The set-theoretic intersection of any family of ideals is again an ideal.

The product $IJ$ consists of finite sums of products $xy$ with $x\in I$ and $y\in J$.

A principal ideal is generated by one element: $(a)=\{ra:r\in R\}$.

A set $S\subseteq I$ generates an ideal $I$ when every element of $I$ is a finite sum of ring multiples of elements of $S$.

An ideal $I$ is primary when $ab\in I$ and $a\notin I$ imply $b^n\in I$ for some positive integer $n$.

For a geodesic hyperbolic $n$-gon of curvature $-1$ with interior angles ${\alpha }_1,\dots ,{\alpha }_n$, triangulation or the Gauss--Bonnet theorem gives

The identity ${\cosh }^{2}x-{\sinh }^{2}x=1$ implies

The identity follows directly from the exponential definitions of the hyperbolic functions.

The Taylor series at zero beginsso $\tanh x\sim x$ as $x\to 0$.

The gravitational scattering angle is the change between incoming and outgoing asymptotic velocity directions.

The Huber loss is quadratic near zero and linear beyond a fixed cutoff; its derivative is the clipped Huber score.

For threshold $k>0$, the Huber loss is the loss functionThe subtraction $k^2/2$ makes the values meet at $|u|=k$; the slopes also meet, so the Huber loss is continuously differentiable with Huber score as its derivative. It is a convex function, since that derivative is a monotone function. These quadratic and linear pieces define the original clipped location criterion.^[ref]

For $k=2$, ${\rho }_2(1)=1/2$, ${\rho }_2(2)=2$, and ${\rho }_2(5)=8$, whereas the squared criterion $u^2/2$ at $u=5$ is $12.5$. Increasing a standardized difference from $5$ to $6$ raises the Huber loss by $2$, but raises the squared criterion by $5.5$. The compromise preserves quadratic sensitivity to small differences while limiting the slope for large ones. This makes the Huber loss useful when ordinary observations should retain their detailed distances yet isolated outliers should exert limited pull.

For a fixed finite dataset and scale, taking $k$ large enough that every difference from the sample mean lies within $ks$ recovers that sample mean. Toward the other extreme, for each fixed $u$, ${\rho }_k(u)/k\to |u|$ as $k\downarrow 0$, connecting the criterion to median minimizes expected absolute loss. The normalization is essential: putting $k=0$ into the unscaled Huber loss gives zero everywhere, not a usable median criterion.

A weak homotopy equivalence between connected CW complexes is a homotopy equivalence. The analogous statement holds for Kan complexes.

If a path-connected space is $(n-1)$-connected for $n\ge 2$, then its first potentially nonzero homotopy group maps isomorphically to homology:

The rational homotopy group is obtained by tensoring a homotopy group with the rational numbers. It retains the free part and discards torsion.

For a based space $X$, adjunction gives

If $A\subseteq X$ is contractible and $(X,A)$ has the homotopy extension property, the quotient mapis a homotopy equivalence. Extend a contraction of $A$ to $X$; its endpoint is constant on $A$ and therefore factors through $q$ to supply a homotopy inverse.

The Sidorenko conjecture states that every bipartite graph $H$ satisfiesfor every bipartite host graph $G$. Thus a random map is at least as likely to preserve every edge as the heuristic that treats the edge constraints independently predicts.

If $g\in S{U}_2$ has eigenvalues $z,z^{-1}$, then

At nonzero charge density, a translationally invariant holographic state has infinite D.C. conductivity because an electric field continuously injects conserved momentum. Momentum relaxation broadens the zero-frequency delta function and makes the D.C. value finite.