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A linear polyhedron is an intersection of finitely many closed affine half-spaces in finite-dimensional real space. It may be empty, unbounded, or lower-dimensional. This usage differs from a three-dimensional geometric polyhedron. The strict separation of disjoint linear polyhedra follows from linear programming duality.

A vector cosmological perturbation is the transverse-vector sector of linear metric and matter perturbations. For a Fourier mode with nonzero wavevector $\mathbf{k}$, transversality removes the component parallel to $\mathbf{k}$, leaving two helicity components. In a gauge with $d{s}^2=a^2[-d{\eta }^2+2B_{i}d\eta d{x}^i+{\delta }_{ij}d{x}^{i}d{x}^j]$, ${\partial }_i{B}_i=0$ describes the vector metric perturbation.

A cosmological gauge transformation changes the identification of a perturbed spacetime with its homogeneous background. Individual field perturbations can change without changing physical observables. At first order the metric change is the linear metric gauge-transformation law; matter scalars obey the corresponding background-shift rule.

In a flat matter-dominated universe with $a\propto t^{2/3}$ and $P\propto {\rho }^{4/3}$, the squared sound speed scales as $t^{-2/3}$. Each comoving Fourier mode then satisfies an Euler-Cauchy equation, with gravitational growth at long wavelengths and logarithmic-time acoustic oscillations at short wavelengths.

The Lighthill acoustic analogy rewrites fluid mass and momentum conservation as a constant-speed wave equation driven by a nonlinear Lighthill stress tensor. It is an exact rearrangement before making source approximations. The outgoing field can be constructed with a retarded acoustic Green function.

Acoustic sources may be organized by the number of spatial derivatives in their effective wave equation forcing. A direct scalar source is an acoustic monopole, a divergence of a force density an acoustic dipole, and a double divergence of a stress tensor an acoustic quadrupole.

An acoustic source is compact when its size is much smaller than the acoustic wavelength. The propagation delay across the source can then be neglected in leading source integrals. This condition is distinct from the radiation-region requirement $k_{0}r\gg 1$.

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.