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.

For one species of particles of mass $m$, let $f_N(\mathbf{x},\mathbf{v})$ be a nonnegative number-normalized phase-space distribution function. If its total particle number $N$ is nonzero and its first velocity moment is finite, defineThis global mean velocity is an expected value under the normalized particle distribution. Its numerator is $N\mathbf{u}$; the total momentum is instead $\mathbf{P}=mN\mathbf{u}$. If the distribution is mass-normalized, $f_M=m{f}_N$, its zeroth moment is total mass $M$ and its first moment is directly $\mathbf{P}$. Stating the convention prevents a missing particle-mass factor.

A local mean velocity at $\mathbf{x}$ uses only the velocity integral, dividing by the local number density $n(\mathbf{x})=\int f_N(\mathbf{x},\mathbf{v})d^{3}v$ where $n>0$. A multispecies barycentric mean velocity weights species by their mass densities, not just their particle numbers.

The mean velocity does not measure random motion. Two equally populated narrow beams at $+{\mathbf{v}}_0$ and $-{\mathbf{v}}_0$ have zero mean velocity while retaining nonzero kinetic energy. Central velocity moments use $\mathbf{v}-\mathbf{u}$ to describe this motion relative to the flow.

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.

For an integer $a$ and positive integer $b$, Euclidean division writes $a=bq+r$ with unique integer quotient $q$ and remainder $0\le r<b$. For $a\ge 0$, repeated quotient-and-remainder steps yield the digits of a base-p expansion when $b=p$. The uniqueness follows because the difference of two permitted remainders cannot be a nonzero multiple of $b$.

There are injections $e_{\alpha }:\alpha \to \omega$ for $\alpha <{\omega }_1$, each with coinfinite range, with $e_{\alpha }\upharpoonright \beta$ equal to $e_{\beta }$ outside a finite set. At countable limits, extend partial functions along a cofinal sequence, repair finitely many collisions, and reserve a fresh omitted natural number at each step. The reserved sequence makes the union coinfinite and permits all later successor steps.

In a well $-a<x<a$, the normalized state proportional to $x$ on $0<x<a$ and zero elsewhere has stationary-state coefficients in the normalized energy eigenfunction basis ${\varphi }_n(x)=a^{-1/2}\sin [n\pi (x+a)/(2a)]$, with $n=1,2,\dots$,They follow from the orthonormal basis inner products. Their squared magnitudes give energy-measurement probabilities by the Born rule; splitting even and odd indices in the Parseval identity gives convergent integer-coefficient series identities. The expansion is interpreted in the square-integrable state norm.

Here the wavefunction is $\sqrt{3}x/a^{3/2}$ for $0<x<a$ and zero for $-a<x<0$. Its nonzero trace at the right wall means it is a valid square-integrable state with divergent energy expectation, as in the linear ramp state in an infinite square well.

Self-inductance is inductance associated with a coil's own electric current, rather than magnetic flux supplied by another coil. In the linear model, fixed geometry and magnetic response give constant $L$ in $\Lambda =LI$. With the turn orientations defined in magnetic flux linkage, Faraday's law of induction gives the self-induced electromotive forceThe final equality requires constant $L$; if the geometry changes, use $-d\Lambda /dt$. The minus sign expresses Lenz's law: the induced electromotive force opposes the change in electric current. It does not always oppose the electric current itself. When positive electric current decreases, the induced electromotive force acts in its direction. ^[ref].

For terminal sign bookkeeping, label the ideal inductor's terminals $a,b$, choose positive electric current entering $a$ and leaving $b$, and define $v_L$ as the terminal electric potential difference drop from $a$ to $b$ in the lumped electrical circuit model. This passive convention givesThus a coil with self-inductance $0.5\mathrm{H}$ and increasing electric current $dI/dt=12A/s$ has terminal drop $v_L=6\mathrm{V}$ and induced electromotive force ${\mathcal{E}}_{\mathrm{ind}}=-6\mathrm{V}$. These signs describe the same process with different reference conventions. The RL circuit uses the terminal-drop convention throughout.

A finite terminal electric potential difference cannot make the electric current of an ideal inductor with constant $L>0$ jump. Integrating its differential equation givesIf $v_L$ stays bounded, the integral tends to zero as the interval shrinks, ensuring continuity of electric current. A step in terminal electric potential difference changes $dI/dt$ immediately, while electric current remains continuous. At constant electric current, the self-induced electromotive force is zero, although the inductor's magnetic energy can remain nonzero.

Let $f$ take values in the additive cyclic group $\mathbb{Z}/n\mathbb{Z}$ on a cycle with $n$ vertices. A constraint on edge $j$ prescribes $f(j)-f(j-1)=c_j$. Any proper selection of $k<n$ edges leaves disjoint paths, hence $n-k$ free initial values and $n^{n-k}$ solutions. The full cycle has $n$ solutions if ${\sum }_j{c}_j=0$ and none otherwise. Combining these intersection counts with the inclusion-exclusion principle giveswhere $F$ is $n$ or zero according to the full-cycle consistency condition. This is a useful example in which every proper collection of local constraints is independent, but the full collection has one global obstruction.