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A magnetohydrodynamic shock is a discontinuity satisfying conservation of mass, momentum, energy, normal magnetic field, and tangential electric field.

The Alfvén speed associated with magnetic-field magnitude $B$ and density $\rho$ is $v_A=B/\sqrt{{\mu }_0\rho }$.

The radical of an ideal $I$ isAn ideal equal to its radical is a radical ideal.

An ideal of a graded ring is homogeneous when it is generated by homogeneous elements of a graded ring, meaning elements that belong to individual graded components. Equivalently, whenever an element of the ideal is written as a finite sum of homogeneous components, every one of those components belongs to the ideal.

For the usual total-degree grading on $k[X,Y]$, the ideal $(X+Y)$ is homogeneous because its generator has degree one. In $k[X]$, the ideal $(X+1)$ is not homogeneous: it contains $X+1$ but does not contain its constant component one. The definition concerns arbitrary graded rings; it does not require that homogeneous elements be polynomials.

Reduction modulo an ideal $I$ is the quotient homomorphism $R\to R/I$.

Let $R$ be a Dedekind domain, let $P_1,\dots ,P_r$ be distinct nonzero prime ideals, and choose nonnegative integers $n_1,\dots ,n_r$. There is a nonzero $a\in R$ withChoose $c_i\in P_i^{n_i}\setminus P_i^{n_i+1}$. The prime ideals are maximal, so their powers are pairwise comaximal. The Chinese remainder theorem gives $a\equiv c_i(\mathrm{mod}{P}_i^{n_i+1})$ for every $i$, which gives exactly the requested prime-ideal valuations in a Dedekind domain. No restrictions are imposed on primes outside the list.

For example, $12$ has valuation two at the prime two and valuation one at the prime three in $\mathbb{Z}$. Prescribed negative valuations can likewise be realized in the fraction field: first clear them by a common nonzero denominator, apply the integral version, and divide back. The Dedekind-domain hypothesis supplies the valuations and the comaximality used in this argument.

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 $y\in J$.

In a commutative ring, a principal ideal is generated by one element:For example, every ideal of $\mathbb{Z}$ has the form $(n)$, whereas $(X,Y)$ in a two-variable polynomial ring over a field is not principal.

In a noncommutative ring, the two-sided ideal generated by $a$ consists of finite sums ${\sum }_i{r}_{i}a{s}_i$; it need not equal the principal left ideal $Ra$. In $M_2(k)$, the matrix unit $E_{11}$ generates the entire matrix ring as a two-sided ideal, since $E_{i1}{E}_{11}{E}_{1j}=E_{ij}$. The left ideal $M_2(k)E_{11}$ only contains matrices with zero second column. The side is therefore part of a principal-ideal statement outside the commutative setting.

In a commutative ring $R$, a subset $S$ generates an ideal $I$ when every element of $I$ is a finite sum ${\sum }_i{r}_i{a}_i$ with $r_i\in R$ and $a_i\in S$. This is the smallest ideal containing $S$. Even when $S$ is infinite, each individual expression uses only finitely many generators.

For a general noncommutative ring, the two-sided ideal generated by $S$ instead consists of finite sums ${\sum }_i{r}_i{a}_i{t}_i$ with $r_i,t_i\in R$. Left and right ideals use only the corresponding one-sided coefficients. For example, $(2,X)$ in $\mathbb{Z}[X]$ has the two-element generating set $\{2,X\}$ but has no one-element generating set, by nonprincipal ideal generated by a prime and an indeterminate.

A proper ideal $I$ of a commutative ring is primary when $ab\in I$ and $a\notin I$ imply $b^n\in I$ for some positive integer $n$. Equivalently, every zero divisor of the nonzero quotient ring $R/I$ is nilpotent. Its radical of an ideal is a prime ideal.

For example, $(p^r)$ is primary in $\mathbb{Z}$ when $p$ is prime and $r\ge 1$. In contrast, $(6)$ is not primary: $2\cdot 3\in (6)$, but $2\notin (6)$ and no power of three lies in $(6)$. Requiring the ideal to be proper excludes the unit ideal, for which the elementwise implication would otherwise hold vacuously.

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: