An acoustic density perturbation is the deviation of fluid mass density from its quiescent reference value. In a homogeneous isentropic linear acoustic field, its corresponding pressure perturbation is $p^{\prime }=c_0^2{\rho }^{\prime }$. Nonlinear and entropy-dependent departures from this relation enter the Lighthill stress tensor.

For adiabatic perturbations of a static perfect gas, pressure obeysThe stationary balance is $\nabla (p_0+\chi )=0$. Density and entropy need not be spatially uniform. Expanding the divergence exposes the gradient terms that distinguish this equation from the homogeneous acoustic pressure wave equation. Its divergence form yields weighted acoustic Green-function reciprocity.

Apply Haar twirling conditional expectation to the complement of a neighbourhood around the initial operator support. A Lieb-Robinson bound on commutators with arbitrary complement-supported unitaries then bounds the approximation error without summing over sites. For a bound proportional to $e^{-\mu d}(e^{2kst}-1)$, choosing $v=4ks/\mu$ gives an error at most $\mu vt|X|\Vert A_X\Vert e^{-\mu l/2}$ outside radius $vt+l$.

The interaction distance between disjoint supports can count the minimum number of interacting hyperedges in a chain connecting them. Overlapping supports have distance zero in the usual support convention. Distances between interaction terms can instead assign shell zero only to the central term and positive shells to distinct terms. The convention must be specified when using a truncated neighborhood Hamiltonian: ordinary zero support distance includes distinct overlapping terms.

Iterating the integral commutator inequality gives ordered interaction chains with factors $(2|t|{)}^n/n!$. A per-site interaction weight at most $s{e}^{-\mu }$ bounds each successive chain extension by $ks{e}^{-\mu }$. No chain shorter than the interaction distance connects separated supports. Reversing chains supplies the smaller support prefactor, producing the Lieb-Robinson bound. Overlapping supports require retaining the equal-time commutator term.

A Duhamel comparison can bound truncation error using commutators evolved under the full Hamiltonian. This avoids assuming that a black-box Lieb-Robinson bound for $H$ also holds for every truncation. Polynomial shell weights and exponential distance suppression make the sum over omitted shells convergent. Comparing two truncations through the full evolution bounds their difference.

A Lagrangian is regular when its velocity Hessian matrix is invertible. The inverse function theorem then makes the fiber derivative $p_i=L_{v^i}$ locally invertible, allowing a local Legendre transform in mechanics to a Hamiltonian. If that fiber derivative is globally invertible, the Lagrangian is called a hyperregular Lagrangian. Regularity alone does not assert global invertibility, and singular Lagrangians may give constrained rather than ordinary unconstrained Hamilton's equations.

For symmetric $L$ and antisymmetric $A$ satisfying $\dot{L}=LA-AL$, solve $\dot{Q}=-AQ$, $Q(0)=I$. Then $Q$ is orthogonal and $Q^{T}LQ$ is constant. Hence every eigenvalue and every coefficient of the characteristic polynomial of $L$ is conserved.

A Puiseux series permits fractional exponents with a fixed common denominator: ${\sum }_{n\ge n_0}{a}_n{z}^{n/q}$. These are natural local expansions of branches of algebraic functions. A fractional-power root expansion uses such powers to resolve roots that coalesce as a parameter tends to zero.

Let $0<c<1$, $A>0$, $\sigma >0$ and $p\ge 2$. For design matrix columns of norm $\sqrt{n}$ and $\epsilon \sim N_n(0,{\sigma }^{2}I)$, the event $\Vert X^T\epsilon {\Vert }_{\infty }/n\le c\lambda$ has probability at least $1-p^{1-A^2{c}^2/2}$ when $\lambda =A\sigma \sqrt{\log (p)/n}$. Apply the sharp two-sided Gaussian tail bound to each standardized score and then the union bound; independence of the scores is not required.

The Lasso regularization path follows its coefficients as the penalty or constraint radius changes. Under the usual least-angle construction, the coefficient path is piecewise linear between knots, and variables can enter or leave the active set. Reading support at a selected cross-validation fraction requires checking which coefficients are actually zero at that fraction, rather than retaining only visibly large traces.

For the Lasso criterion $\Vert Y-Xb{\Vert }_2^2/(2n)+\lambda \Vert b{\Vert }_1$, the displayed subgradient optimality condition is necessary and sufficient. A nonzero coefficient has residual score $\lambda \mathrm{sign}(b_k)$, and a zero coefficient has residual score in $[-\lambda ,\lambda ]$. The subdifferential of the L1 norm supplies both cases.

For a minimization problem with $g_i(u)\le 0$ and $h_j(u)=0$, the optimization Lagrangian is $L=f+{\sum }_i{\lambda }_i{g}_i+{\sum }_j{\mu }_j{h}_j$ with ${\lambda }_i\ge 0$ and unrestricted equality Lagrange multipliers. Its infimum over the original variable domain gives a dual lower bound. For a maximization problem, use $L=f-{\sum }_i{\lambda }_i{g}_i-{\sum }_j{\mu }_j{h}_j$ to obtain upper bounds. The Lagrangian sufficiency theorem combines a global extremum of this function with feasibility and complementary slackness. This is distinct from a Lagrangian density in variational physics.

Transport a phase-space density along Hamilton's equations. The Hamiltonian vector field is $b=({\nabla }_{v}H,-{\nabla }_{x}H)$ and the equation is ${\partial }_{t}f+b\cdot {\nabla }_{x,v}f=h$. Its zero divergence gives volume-preserving transport. For $h=0$, the density is constant along characteristics; a source is integrated by the Duhamel formula for Hamiltonian transport. This is distinct from the elliptic Liouville equation.

The mass-normalized first velocity moment of a distribution on phase space. For a velocity-only distribution, omit the position integral. Nonzero finite mass and a finite first moment are required. For a probability distribution this is its expected value; its unnormalized counterpart is the momentum.

On the unit-period spatial circle, free streaming has solution $f_t(x,v)=f_0(x-tv,v)$. With Fourier phase $e^{-2\pi i(kx+\xi v)}$, its mixed Fourier transform is ${\hat{f}}_t(k,\xi )={\hat{f}}_0(k,\xi +kt)$. A nonzero spatial mode of the velocity-integrated density therefore samples increasingly large velocity frequencies. The conserved spatial zero mode remains, while extra velocity regularity supplies quantitative decay of the other modes.

This many-particle stochastic collision model rotates one uniformly selected velocity pair by a uniformly selected angle, conserving the sum of squared velocities. Its normalized angular averaging is the Kac collision operator, and its continuous-time evolution is the Kac master equation. In this version velocities range over ${\mathbb{R}}^N$ and initial energy is not restricted to one sphere; the evolution preserves the initial energy distribution.

For a dilute phase-mixed population, an identical-species collision rate counts unordered pairs: ${\Gamma }_{ii}=K_{ii}{N}_i(N_i-1)/2$. Different species have rate ${\Gamma }_{ij}=K_{ij}{N}_i{N}_j$. Here $K_{ij}=⟨{\sigma }_{ij}{v}_{ij}⟩/V$ uses the collision cross-section and relative speed. Destroying both identical particles gives ${\dot{N}}_i=-2{\Gamma }_{ii}$, not $-{\Gamma }_{ii}$. This distinction prevents inconsistent factors when using one kinetic kernel for several particle sizes.

Where a Killing vector field is ${\partial }_x$ and its squared norm $V=h_{xx}$ is nonzero, an invariant metric tensor decomposes as $h=V(dx+A{)}^2+\gamma$. The quotient one-form has $A_i=h_{xi}/V$ and the quotient metric has ${\gamma }_{ij}=h_{ij}-h_{xi}{h}_{xj}/V$. All coefficients are independent of $x$. Changing the fibre coordinate to $x^{\prime }=x+\lambda$ sends $A$ to $A-d\lambda$. A null Killing direction requires a different decomposition.

On three noncompact spatial dimensions times a circle of circumference $L$, the method of images gives ${\sum }_{j\in \mathbb{Z}}[r^2+j^2{L}^2{]}^{-1}=\pi \coth (\pi r/L)/(Lr)$ at equal compact coordinates. This interpolates from $1/r^2$ at short distances to $\pi /(Lr)$ at long distances. A gravitational circle-radius scalar can change the long-range coefficient if it remains unstabilized.