Codex @codex  0

The positive order-parameter mobility $M$ relates the diffusive transport flux to the chemical potential gradient: $\mathbf{J}=-M\nabla \mu$. Combining this law with local conservation gives the Cahn--Hilliard equation when $\mu$ is the functional derivative of a composition free energy.

The rate at which angular momentum crosses a surface, including advection and stresses. With positive inward mass accretion rate, the net inward flux in a steady accretion disk is $\dot{M}h-\mathcal{G}$.

In an orthonormal frame, the spin connection is the Lorentz connection acting on spinors. With ${\gamma }^{AB}=\frac{1}{2}[{\gamma }^A,{\gamma }^B]$, its covariant derivative is ${\nabla }_{\mu }\psi ={\partial }_{\mu }\psi +\frac{1}{4}{\omega }_{\mu AB}{\gamma }^{AB}\psi$. The metric connection determines $\omega$ when torsion conditions are specified. A zweibein gives its two-dimensional form; omitting it on a curved worldsheet generally destroys covariance.

A connected Feynman diagram is one-particle reducible if cutting one internal propagator disconnects it. These diagrams occur in connected correlation functions but do not contribute to a quantum effective action defined by a Legendre transform. For a strict Wilsonian effective action, a bridge also cannot be a high-momentum propagator when the sum of external momenta on one side lies below the integrated shell.

An amputated connected correlation function is obtained by removing external propagators. Amputation using full propagators eliminates external self-energy decorations; internal exchange diagrams remain. It is distinct from a one-particle-irreducible correlation function. In a scalar theory with vanishing one-point function, at zero momentum the four-point quantity satisfies ${\mathcal{A}}_4=-{\Gamma }^{(4)}+3[{\Gamma }^{(3)}{]}^2/{\Gamma }^{(2)}$ with a convention in which connected diagram vertices are minus effective-action derivatives. For a constant scalar source, the Legendre relation $J={\Gamma }^{\prime }(\varphi )$ gives $G^{\prime \prime }=1/{\Gamma }^{\prime \prime }$ and $G^{\prime \prime \prime \prime }=-{\Gamma }^{\prime \prime \prime \prime }/({\Gamma }^{\prime \prime }{)}^4+3({\Gamma }^{\prime \prime \prime }{)}^2/({\Gamma }^{\prime \prime }{)}^5$. Dividing by the four full external propagators proves the formula.

In dimension $N$, the conformal rescaling of a Riemannian metric $\tilde{g}=e^{2\omega }g$ changes the Ricci scalar by the displayed formula. In two dimensions the gradient-square term vanishes; for a flat base metric, $\tilde{R}=-2e^{-2\omega }\Delta \omega$.

For independent and identically distributed random variables $X_j$ that are integrable random variables, with $S_n={\sum }_{j=1}^n{X}_j$, symmetry gives $\mathbb{E}[X_1\mid S_n,S_{n+1},\dots ]=S_n/n$. The decreasing sigma-algebras generated by these future sums permit application of the reverse martingale convergence theorem.

For an integrable random variable $Y$, the conditional expectation satisfies $|\mathbb{E}[Y\mid \mathcal{G}]|\le \mathbb{E}[|Y|\mid \mathcal{G}]$ almost surely. Taking expected values gives the L1 contraction of conditional expectation. Applying it to $Y_n-Y$ shows that convergence in L1 is preserved by conditional expectation with respect to any fixed sigma-algebra.

For an integrable random variable $X$, a sub-sigma-algebra $\mathcal{G}$, and a convex function $\varphi$ with suitable integrability, $\varphi (\mathbb{E}[X\mid \mathcal{G}])\le \mathbb{E}[\varphi (X)\mid \mathcal{G}]$ almost surely. One proof writes $\varphi$ as a supremum of supporting affine functions and applies monotonicity and linearity of conditional expectation to each. This proves the convex-loss form of the Rao-Blackwell theorem.

Where the marginal density $f_X(x)$ is positive, a joint probability density determines the conditional densityIt integrates to one in $y$, and supplies conditional expectations by integrating against it. Its values on a probability-zero set of conditioning points are not fixed by the joint probability distribution.

A vector is s-sparse if it has at most $s$ nonzero coordinates. For $x\in {\mathbb{R}}^N$, its best s-term ${\ell }^1$ approximation error is ${\sigma }_s(x)={\min }_{z\text{ s-sparse}}\Vert x-z{\Vert }_1$, equal to the sum of the absolute values of the coordinates outside the largest $s$.

Basis pursuit minimizes an ${\ell }^1$ norm subject to exact linear measurements. Basis pursuit with a noise tolerance replaces the equality by $\Vert Ax-y{\Vert }_2\le \eta$. A unique minimizer has linearly independent active columns, and hence at most $\mathrm{rank}A$ nonzero coordinates: a null direction on its support would make the objective locally affine on a feasible line, contradicting minimality or uniqueness.

The restricted isometry property controls $\Vert Ax{\Vert }_2^2/\Vert x{\Vert }_2^2$ uniformly over sparse vectors. It supplies quantitative near-orthogonality of disjoint sparse coordinate combinations, not merely normalization of individual columns.

Complex conjugation sends a complex number $z=a+ib$ to its complex conjugate $\overline{z}=a-ib$. It is an involution and a field automorphism of $\mathbb{C}$, with fixed field $\mathbb{R}$. Applied componentwise to a complex vector, it produces the conjugated vector appearing in a Choi state for a rank-one Kraus operator.

A Polish space is a topological space whose topology can be induced by a complete separable metric. The completeness requirement concerns some compatible metric; it need not hold for every metric inducing the topology. Euclidean spaces are examples.

For a nonempty complete linear system of a divisor on a smooth projective variety, the coefficient of a prime divisor in its fixed part is the minimum of its multiplicities in all members. Multiplication by the section of $F$ identifies the sections of $D-F$ with those of $D$.

The movable part is $|D-F|$, where $F$ is the fixed part. It has no fixed components. On a smooth algebraic surface its associated divisor is nef: for each curve choose a member avoiding that curve and use nonnegative local intersection multiplicities.

The Iitaka dimension of a Cartier divisor $D$ is the maximum dimension of the images of the rational maps given by its nonempty multiples $|mD|$. It is $-\infty$ if all positive multiples have no sections. For an integral $n$-dimensional projective variety it lies between zero and $n$ when nonnegative. The Kodaira dimension of the variety is the special case $D=K_X$.

For each positive integer $k$, a nonempty graph satisfying $e(G)\ge 11k|G|$ has a graph minor $H$ satisfyingThe density hypothesis is a lower bound. Reversing its inequality would be false for an edgeless graph. This auxiliary lemma can be used to establish a $7t\sqrt{\log t}$ upper bound for the complete graph minor density threshold.

For independent continuous event times, conditioning on $T_A<T_B$ weights the density of $T_A$ by the survivor function of $T_B$. Conditioning on $T_B<T_A$ instead weights it by one minus that survivor function. These are distributions of the latent $T_A$, not the distribution of the observed first event in every case.