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The jump process with total collision rate $N$ and normalized pair-angle operator $Q$ evolves by ${\partial }_t{F}_N=N(Q-I)F_N$. It preserves mass and the total squared velocity. Its one-coordinate distribution satisfies the Kac marginal evolution equation, and its Gaussian relative entropy in Kac's model decreases at the rate given by Kac entropy production.

A permutation-symmetric sequence of probability laws is chaotic with one-particle law $f$ when, for every fixed $k$, its $k$-coordinate marginal distributions converge weakly to the product law $f^{\otimes k}$ as $N\to \infty$. This is an asymptotic independence condition, not a consequence of symmetry alone. A rigorous kinetic limit also requires propagation of this property under the dynamics and sufficient convergence to pass collision integrals.

Use ${\gamma }_N=(2\pi {)}^{-N/2}{e}^{-|v{|}^2/2}$ as the reference probability density. The relative entropy is nonnegative by writing $F={\gamma }_{N}u$ and integrating $u\log u-u+1$. Its expansion is $\int F\log F+(N/2)\log (2\pi )+(1/2)\int |v{|}^{2}F$. Equality holds only at $F={\gamma }_N$, whereas other radial densities may still be stationary for the Kac master equation.

For real $s$ and vectors $\xi ,\eta$, the displayed bound follows from $⟨a+b⟩\le \sqrt{2}⟨a⟩⟨b⟩$, applied in the opposite direction when $s<0$. One can take $C_s=2^{|s|/2}$. Combined with Young's convolution inequality, it proves Sobolev multiplication by a smooth cutoff for all real orders: the rapidly decreasing transform of the cutoff absorbs the extra weight. The negative-order case is particularly useful when a distribution is not initially a function.

If $I$ is a nilpotent ideal and $J(A/I)=0$, then $I=J(A)$. Every $1-ax$, $x\in I$, has a finite geometric-series inverse, giving $I\subseteq J(A)$. The radical's image lies in $J(A/I)=0$, giving the reverse inclusion. A semisimple algebra as quotient supplies the required zero radical.

The normalized even kernelis nonnegative, has integral one, and is a trigonometric polynomial of degree $2(n-1)$. Its mass is concentrated on a scale $1/n$: $J_n(t)\le C\min (n,n^{-3}|t{|}^{-4})$ for $|t|\le \pi$. This controls a weighted second moment and produces the Jackson operator estimate.

The specific enthalpy is equal on both sides of an isothermal perfect-gas shock. The energy removed per unit area per unit time is $j(u_1^2-u_2^2)/2$, with $j>0$ the normal mass flux. Net cooling therefore selects compression shocks; an expansion jump requires heating.

The mean-field approximation neglects products of Ising spin deviations, replacing each Ising spin's neighbours by the common spin magnetization. With coordination number of a lattice $q$, the effective field is $h+qJm$. The self-consistency equation equates the assumed mean Ising spin with its independent-spin expectation in that effective field. It predicts $k_B{T}_c=qJ$ and the usual quartic mean-field critical exponents, while neglecting spatially correlated fluctuations.

An Ising Ising spin variable is a classical two-state degree of freedom. Its thermal mean can lie anywhere in $[-1,1]$ even though each microscopic value is $-1$ or $1$. It should not be confused with an arbitrary quantum-spin operator. Products of neighbouring Ising spins supply the pair interactions of the Ising model, and their average defines spin magnetization.

This is the supremum of lengths of strict chains of nonempty irreducible closed subsets. Closedness is relative to the given space. This is the relevant dimension for locally closed subsets of algebraic varieties, rather than a general Hausdorff-space notion of dimension.

If $f$ is an irreducible polynomial over a field $F$, then a nonzero residue $[g]$ modulo $(f)$ has $\gcd (f,g)=1$. The Euclidean algorithm supplies $uf+vg=1$, giving inverse $[v]$ for $[g]$. Thus the quotient ring $F[X]/(f)$ is a field. If $\mathrm{deg}f=d$ and $F$ has $q$ elements, polynomial division supplies exactly $q^d$ residue classes.

In a one-period normal-return model with common beliefs and unrestricted exponential utility investors, their risky holdings are proportional to $V^{-1}b$, where $b$ is the expected excess-gain vector and $V$ its positive definite covariance matrix. The aggregate risky holding defines the common market portfolio.

The expected excess return of a portfolio divided by its return standard deviation. Under the capital asset pricing model, the ratio for one asset equals its correlation with the market times the market's ratio.

Brownian portfolio exposures are the coefficients $y_j$ of independent Brownian motions in the wealth equation. An invertible asset volatility matrix lets one optimize directly over these exposures. The drift risk premium is then the dot product of exposures with the market price of risk vector.

The inverse of the tangent restricted to $(-\pi /2,\pi /2)$, taking real values in that interval. Its derivative is $(1+x^2{)}^{-1}$. A formula involving an inverse tangent of a ratio may need a continuous branch of a multivalued function to describe a smooth field through a pole of that ratio.

A dust tail is an extended distribution of grains released from a localized moving source. Radiation pressure changes the grains' orbits; their release times, grain properties and survival probabilities determine the spatial profile.

For two incompressible inviscid layers at rest between rigid horizontal boundaries, with the heavier fluid below and no surface tension, the displayed dispersion relation follows from Laplace equation for the velocity potentials, no normal flow at the walls, equal normal velocities at the interface, and pressure continuity. The positive density contrast supplies the restoring force. In the long-wave limit the squared wave speed is $g({\rho }_2-{\rho }_1)/({\rho }_1/h_1+{\rho }_2/h_2)$.

A monochromatic internal gravity wave in uniform stratification has a spatially linear phase. In the nonrotating inviscid Boussinesq approximation, its dispersion relation is ${\omega }^2=N^2|{\mathbf{k}}_h{|}^2/(|{\mathbf{k}}_h{|}^2+m^2)$. The wave vector is normal to its phase planes, while the group velocity lies along them.

Nonlinear disruption of an internal gravity wave, for example when large displacement gradients overturn the stable density arrangement or strong shear becomes unstable. The resulting turbulence dissipates wave energy and mixes the stratification, limiting ideal geometric focusing.

Specific internal energy is internal energy per unit mass. For a perfect gas with constant specific-heat ratio, $e=p/[(\gamma -1)\rho ]$. Its material balance is $\rho De/Dt=-p\nabla \cdot \mathbf{u}+Q$, where $Q$ is heat supplied per unit volume per unit time.