A rational representation is a group homomorphism whose matrix coefficients are regular functions on the affine algebraic group. For the general linear group, those functions are polynomials in matrix entries and the inverse determinant. The word rational does not permit poles at points of the group.

A rational structure over ${\mathbb{F}}_q$ is a model over that finite field together with its identification after extension to ${\overline{\mathbb{F}}}_q$. It determines a Frobenius endomorphism of an algebraic group, whose fixed points are the rational points. Conjugating a standard Frobenius by an algebraic group automorphism transports the rational structure, even if the automorphism is not itself rational for the original structure.

An affine algebraic group is linearly reductive when every finite-dimensional rational representation is completely reducible. Equivalently its invariants functor is exact. Over a field of characteristic zero, reductive algebraic groups are linearly reductive; in positive characteristic this stronger property fails for many reductive groups. An algebraic torus is linearly reductive in every characteristic because after extension to a separable closure its rational representations decompose into character spaces, and exactness of the invariants functor descends to the original field.

For every finite-dimensional rational representation $V$ and every nonzero $v\in V^G$, a geometrically reductive algebraic group admits a homogeneous invariant polynomial $F\in k[V{]}^G$ of positive degree with $F(v)\ne 0$. A linearly reductive group can take degree one. In positive characteristic, higher degree is essential: a power $p^r$ can separate a fixed vector even when an invariant linear functional cannot.

An algebraic one-parameter subgroup is a regular group homomorphism from the multiplicative algebraic group. In $S{L}_n$, a change of basis makes it $t\mapsto \mathrm{diag}(t^{a_1},\dots ,t^{a_n})$ for integers with sum zero. This is the algebraic multiplicative notion, distinct from an additive real-parameter subgroup of a Lie group.

Fix a Chevalley basis and its integral lattice. The adjoint Chevalley group over a field is generated by the root operators obtained from the integral divided-power polynomials $x_{\alpha }(t)={\sum }_j{t}^j(\mathrm{ad}{e}_{\alpha }{)}^j/j!$ on that lattice. The coefficients are first computed over the integers and then reduced to the field; factorials need not be invertible there. This is the adjoint construction; other weight lattices give other isogeny forms of groups of Lie type.

For a nonrotating self-gravitating star, let $p_1,{\rho }_1,{\varphi }_1$ be Eulerian perturbations and $\Delta p=-\gamma p\nabla \cdot \mathbf{\xi }$ the adiabatic Lagrangian perturbation. Integration of the linear momentum equation against ${\mathbf{\xi }}^{\ast }$, using hydrostatic balance and the perturbed Poisson equation, yields the displayed Rayleigh quotient. The gravitational integral is over all space and the other integrals over the star. The compression denominator is $\gamma p$, not $\gamma \rho$. Surface terms vanish for compactly supported trials or the physical free-surface and exterior potential conditions.

The method of images enforces impermeability at a plane wall by reflecting a line vortex across it and reversing its circulation. The two logarithmic stream functions cancel on the wall, which is consequently a streamline. For a vortex of circulation $\Gamma$ at $(0,b)$ above $y=0$, the image of circulation $-\Gamma$ at $(0,-b)$ induces velocity $(\Gamma /(4\pi b),0)$ at the real vortex. The signed drift parallel to the wall is therefore $\Gamma /(4\pi b)$.

Take vortex strength to mean circulation and $J(x,y)=(-y,x)$. Each point vortex moves with the velocity induced by the other, so for $\mathbf{d}={\mathbf{x}}_1-{\mathbf{x}}_2$ the displayed equation holds. Since $\mathbf{d}\cdot J\mathbf{d}=0$, separation is constant; weighting the two velocities also gives $d({\kappa }_1{\mathbf{x}}_1+{\kappa }_2{\mathbf{x}}_2)/dt=0$. Equal strengths at opposite points of a circle of radius $a$ rotate with angular speed $\kappa /(4\pi a^2)$.

Under an assumed algebraicity of $i\pi$, form the product ${\prod }_j(1+e^{{\alpha }_j})$ over its conjugates; its one known vanishing factor makes the whole product zero and regroup the subset sums as $k+\sum b_j{e}^{{\beta }_j}=0$, with nonzero exponents and Galois-invariant integral weights. After clearing their denominators, use $f_p(z)=(cz{)}^{p-1}{\prod }_j(cz-c{\beta }_j{)}^p/(p-1)!$. The sum of its derivatives at each ${\beta }_j$ is divisible by $p$ in the algebraic integers, whereas at zero its first nonzero derivative is a fixed nonzero residue for all but finitely many primes. Summing with the weights gives a nonzero rational integer. Integrating the derivative of $e^{-z}\sum f_p^{(j)}(z)$ expresses that integer as a quantity bounded by $C^p/(p-1)!$, which tends to zero. This is the contradiction.

Let $\bar{x}=m^{-1}{\sum }_i{x}_i$ and $S={\sum }_i(x_i-\bar{x}{)}^2>0$. With Lebesgue base measure $d\mu dv$ for $v={\sigma }^2$, the likelihood-power law has conditionalsThe joint law is proper for $T<m/3$, since its marginal $v$ is inverse-gamma distribution with shape $m/(2T)-3/2$ and scale $S/(2T)$. There is also a direct convergence proof for successive Gibbs sampling updates with deterministic $0<T_n\le m/10$ and $T_n\to 0$. Update ${\mu }_n$ from $v_{n-1}$, then $v_n$ from ${\mu }_n$. Inverse-gamma distribution moments and conditional normal moments giveThe coefficients of the preceding moments are bounded by $1/6$ and $1/8$, so these moments stay bounded from a finite deterministic initial variance. Letting $T_n\to 0$ gives $\mathbb{E}{v}_n\to S/m$ and $\mathbb{E}{v}_n^2\to (S/m{)}^2$. Also $\mathbb{E}({\mu }_n-\bar{x}{)}^2=T_n\mathbb{E}{v}_{n-1}/m\to 0$. Thus the iterates converge in mean square to the normal maximum-likelihood estimate.

For $m$ independent binomial distribution observations with fixed trial count $n$ and total successes $s$, the likelihood-power probability density function relative to $dp$ is $p^{s/T}(1-p{)}^{(mn-s)/T}$. Its normalized law is the displayed beta distribution. Its mean tends to $s/(mn)$ and its variance tends to zero as $T\downarrow 0$, including endpoint maxima. Energy for maximizing a log-likelihood is its negative; the opposite sign reverses the optimization.

Write a piecewise smooth Lighthill stress tensor as $T_{ij}=T_{ij}^{-}+D_{ij}H(S)$, where $D_{ij}=T_{ij}^{+}-T_{ij}^{-}$ and the regular level surface $S=0$ is a shock wave. The distributional derivative of the Heaviside step function givesHere repeated indices are summed, $S_i={\partial }_{i}S$ and $S_{ij}={\partial }_i{\partial }_{j}S$. Keep the coefficients as smooth extensions before multiplying distributions: prematurely replacing the coefficient of ${\delta }^{\prime }(S)$ by its surface trace can lose a $\delta (S)$ contribution. The extra surface delta distributions and their derivatives are supported entirely on the discontinuity.

For an isospin singlet quark $h$ of charge $q_h$, the antisymmetric light-flavour pair gives an $I=0,J=1/2$ state with charge $q_h+1/3$. The symmetric light-flavour pairs $uu,\{ud\},dd$ give $I=1$ triplets of spins $1/2$ and $3/2$, with charges $q_h+4/3,q_h+1/3,q_h-2/3$. These follow from light-diquark spin-isospin correlation and addition of constituent electric charges, with zero orbital angular momentum.

The future light cone is the future-directed part of the light cone of a spacetime event. Its boundary consists of the events reached by outgoing light rays. Events reached by slower causal signals lie inside the boundary; relativistic causality forbids a controllable signal from reaching the spacelike exterior.

A compact Lie group is a Lie group whose underlying manifold is compact. Averaging any positive inner product over its Haar measure supplies an invariant inner product on each finite-dimensional continuous representation of a topological group, including the Adjoint representation.

On a connected Lie group, a nondegenerate invariant symmetric bilinear form on a Lie algebra extends by translation to a metric preserved by both left and right translation. For a disconnected Lie group, the form must additionally be invariant under the full Adjoint representation of a Lie group. It may have indefinite signature. Its Levi-Civita connection is ${\nabla }_{X}Y=\frac{1}{2}[X,Y]$ for left-invariant vector fields, and its Ricci tensor is $-K/4$. Choosing the nondegenerate Killing form of a semisimple Lie algebra yields a canonical indefinite Einstein manifold; compact groups instead admit the positive metric $-K$.

The loop group of a Lie group is the group of smooth maps from the circle to that group, with pointwise multiplication and the smooth-function topology. Rotation of the source circle acts by group automorphisms. For $G=U(1)$, every loop has the form $z^w{e}^{ih(z)}$, where $w\in \mathbb{Z}$ is its winding number and $h$ is a smooth real periodic function determined up to addition of an integer multiple of $2\pi$.

Inside the truncated tensor algebra, let ${\mathfrak{g}}^N(V)$ be the span of iterated Lie brackets of elements of $V$ of lengths at most $N$. Its exponential is the connected simply connected free nilpotent Lie group of step $N$, with tensor multiplication. The bracket is the commutator in the tensor algebra. Its grading defines dilations that multiply degree $k$ by ${\lambda }^k$.

A real or complex Lie group is semisimple when its Lie algebra is semisimple. For a real group, the Killing form then defines a nondegenerate bi-invariant pseudo-Riemannian metric. Compact semisimple groups have negative-definite Killing form; noncompact examples such as $SL(2,\mathbb{R})$ have indefinite Killing form. A disconnected group can also have a semisimple Lie algebra, so connectedness is a separate hypothesis when integrating infinitesimal actions.