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Magnetocentrifugal acceleration occurs when matter tied to a rotating inclined magnetic field moves outward under the effective centrifugal force. For a Keplerian disc, cold launching is possible when the field tilts by more than $30^{\circ }$ from the vertical.

An Alfvén surface is where the poloidal flow speed equals the poloidal Alfvén speed. Smooth passage through it imposes a regularity condition relating field-line angular velocity and angular momentum.

The total specific angular momentum transported by matter and magnetic stress isand is constant on each magnetic surface.

Ideal induction in a steady axisymmetric flow makes each magnetic surface rotate with a constant field-line angular velocity $\omega (\psi )$.

When $\rho {\mathbf{u}}_p=k{\mathbf{B}}_p$, the flux-surface function $k$ is the mass flux per unit poloidal magnetic flux.

For ${\mathbf{B}}_p=\nabla \psi \times \nabla \varphi$, surfaces $\psi =\text{constant}$ are poloidal magnetic surfaces and $2\pi \psi$ is magnetic flux up to the choice of axial reference value.

The average treatment effect on the treated is $\mathbb{E}[Y(1)-Y(0)\mid A=1]$.

A first-order structure is rigid when its identity map is its only automorphism. Every finite linear order is rigid, whereas $(\mathbb{Z},<)$ is nonrigid because every translation is an order automorphism.

Rigidity here means absence of nonidentity automorphisms of a first-order structure; it does not assert the mechanical rigidity of a rod-and-hinge framework.

A family $(f_g{)}_{g\in G}$ of automorphisms satisfying $f_g{f}_h=f_{gh}$ defines a group action. It is faithful exactly when distinct indices define distinct automorphisms, equivalently when $G$ embeds in the automorphism group.

After adjoining a unary function symbol $f$, injectivity, surjectivity, preservation of each operation, and preservation in both directions of each relation form an axiom scheme whose models interpret $f$ as an automorphism.

For an $\mathfrak{m}$-primary ideal $I$ in a Noetherian local ring, $\mathrm{length}(R/I^n)$ agrees for all sufficiently large $n$ with the Hilbert-Samuel polynomial of $I$.

For a Noetherian local ring $(A,\mathfrak{m})$, the pole dimension $d({\mathrm{gr}}_{\mathfrak{m}}(A))$ is the order of the pole at $t=1$ ofThe Hilbert-Serre theorem makes this series rational, so the order is defined.

A constant added to a function changes its reference level without changing its derivative:Geometrically, it is a vertical translation of the graph. For $F(x)=x^3/3$, changing $C$ moves every plotted value by the same amount while preserving every tangent slope. This is the geometric reason that antiderivatives form a family.

When $F$ represents a physical quantity, the physical dimension of $C$ must match that of $F$. When position is obtained from velocity, $C$ is a position offset. An unknown starting position remains unknown even when the rate of change is known exactly.

When describing the family of antiderivatives on an interval, the arbitrary additive constant is the constant of integration. On a disconnected open domain, the corresponding freedom is a locally constant function, as in antiderivatives on disconnected domains, rather than necessarily one constant on the whole domain.

Forthe Fourier convention $\hat{u}(\theta )={\sum }_m{e}^{-im\theta }{u}_m$ givesprovided the denominator does not vanish.

The velocity of a surface curve is tangent. If its acceleration is normal, thenso every affinely parametrized geodesic has constant speed.

The free product $A\ast B$ is the amalgamated free product over the trivial group. Its nonidentity elements have unique reduced alternating normal forms.

The alternating harmonic series ${\sum }_{n\ge 1}(-1{)}^{n+1}/n$ converges. Its even partial sums increase, its odd partial sums decrease, and the two limits agree because their difference is the next term. In particular, every even partial sum is a lower bound and every odd partial sum is an upper bound for the limit.

This alternates the signs of the terms of the harmonic series, which itself diverges.

An Alfvén number is the ratio of a flow speed to the corresponding Alfvén speed. For a poloidal flow parallel to a poloidal magnetic field, $A^2={\mu }_0\rho u_p^2/B_p^2$.

This group has seven conjugacy classes: the identity; two classes of five nonidentity translations, distinguished by square class of the translation parameter; and, for each of the four $r\in Q\setminus \{1\}$, one class $\{(r,b):b\in {\mathbb{F}}_{11}\}$ of size eleven.

An operator is normal when $T{T}^{\ast }=T^{\ast }T$. Equivalently,for every $x$; the converse follows by polarizing the quadratic form of the self-adjoint commutator $T^{\ast }T-T{T}^{\ast }$.