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.

A forward problem computes observable data $f=Ku$ from a known state or parameter $u$ and a specified model $K$. Its associated inverse problem reconstructs $u$ from $f$ and the model. Inverse problems can fail existence, uniqueness or continuous dependence on the data.

Multiparticle quantum scattering is factorized when it decomposes into two-body scatterings without particle production or a new irreducible many-body amplitude. The independence of the answer from the order of exchanges imposes the Yang-Baxter equation. Classical pairwise additivity of soliton shifts is a corresponding property of integrable soliton collisions.

A cutting-plane method repeatedly adds valid inequalities that exclude a current relaxed solution while preserving all feasible integer solutions. Gomory fractional cuts provide such inequalities from a fractional simplex dictionary.

The knapsack problem selects items of prescribed weights and profits under a total-weight capacity. The 0-1 knapsack problem permits each item at most once; bounded and unbounded multiplicity variants use different integer restrictions. Nonnegative weights and capacity are assumed here.

A branch-and-bound search partitions a feasible set, maintains a feasible incumbent, and prunes subproblems using bounds that cannot improve that incumbent. For a quadratic knapsack problem, fixing a selected set $S$ adds its exact quadratic value as a constant and replaces remaining profits by $v_i+2{\sum }_{j\in S}{p}_{ij}$ under the ordered-pair convention. Residual fractional row bounds for quadratic knapsack and a Lagrangian knapsack bound can then be recomputed. Worst-case search remains exponential.