The group Lasso partitions a coefficient vector into groups $G_1,\dots ,G_q$ and replaces the coordinatewise ${\ell }^1$ penalty with a weighted sum of group norms,It tends to retain or discard an entire group at once. Its dual norm isso the corresponding Holder inequality is $|u^{T}v|\le P(u)P^{\ast }(v)$.

A Komar integral constructs a conserved gravitational charge from the exterior derivative of the one-form dual to a Killing vector. In four dimensions, a stationary mass is conventionally normalized as

If $T^{ab}$ is symmetric and covariantly conserved and $K$ is a Killing vector field, then $J^a=T^a{}_b{K}^b$ is conserved. Its flux through a spacelike hypersurface is the energy or momentum charge generated by the isometry.

The Killing equation tests whether a candidate vector field generates a local isometry, without needing to solve for its flow of a vector field. For the Levi-Civita connection of a metric tensor $g$, it isParentheses mean averaging over the two index orders. The equation says the symmetric part of the covariant derivative vanishes, not that the entire covariant derivative vanishes. The Lie derivative of the metric tensor gives the coordinate formThe last two terms account for how tangent vectors move with the field; checking only $K^{\rho }{\partial }_{\rho }{g}_{\mu \nu }$ misses this. See Carroll, section 5, equations (5.33) and (5.41)-(5.42).

For the Euclidean Killing vector field $K=-y{\partial }_x+x{\partial }_y$, the Euclidean metric in Cartesian coordinates makes covariant derivatives ordinary partial derivatives. Its diagonal equations give ${\partial }_x{K}_x={\partial }_y{K}_y=0$, and its mixed equation gives ${\partial }_x{K}_y+{\partial }_y{K}_x=1-1=0$. Thus a varying vector field can preserve a constant metric tensor. In contrast, $K=x{\partial }_x+y{\partial }_y$ gives $({\mathcal{L}}_{K}g{)}_{xx}=2$ and fails the Killing equation, despite the constant metric components.

If $K={\partial }_z$ is a coordinate basis vector, its components are constant and the coordinate form reduces to ${\partial }_z{g}_{\mu \nu }=0$. Hence independence of the metric components from $z$ supplies a Killing vector field. This shortcut applies to that coordinate generator, not to an arbitrary varying vector field.

When kinematic viscosity and mass diffusivity are equal to $ϵ$, a monochromatic internal-wave beam with wavenumber $k$ and ray angle $\theta$ has leading stream-function amplitudeThe cubic dependence on $k$ makes slope-focused, short internal waves dissipate especially rapidly.

A fluid has stable density stratification when a small vertical displacement produces a restoring buoyancy force. In a gravitational field this normally means that mass density increases downward.

An atmospheric internal gravity wave is an internal gravity wave propagating through a stably stratified atmosphere. Flow across mountains can generate nearly stationary waves whose vertical propagation is controlled by the wind and buoyancy-frequency profiles.

The Radon transform integrates a function over affine hyperplanes:

The spherical mean of a function on ${\mathbb{R}}^n$ is its average over a sphere:

A Volterra integral equation has a variable integration limit, so its value at a point depends only on earlier values. Repeated integration produces factorial decay, which often makes its Neumann series converge without a smallness assumption.

Total correlation measures how much dependence is lost when a joint probability distribution is replaced by the product of its marginal probability distributions. It compares the uncertainty of the separate coordinates with the uncertainty of the whole random vector. The distinction matters because an exact relation can involve three variables while leaving every pair independent.

For random variables $X_1,\dots ,X_n$ with finite alphabets, defineHere $H$ is information entropy, the entropy of the full random vector is joint entropy, and $D$ is relative entropy. Base-two logarithms give bits; natural logarithms give nats. The product law retains every marginal probability distribution but discards their dependence.^[ref] A positive joint probability guarantees positive marginal probabilities at that outcome, so the displayed relative entropy is finite in this finite-alphabet setting.

For two variables, total correlation is exactly mutual information. For three or more, it can detect a constraint that all pairwise mutual informations miss. Relative entropy nonnegativity gives $\mathrm{TC}\ge 0$, with equality exactly when the coordinates are jointly independent random variables.

As the simplest contrast, three independent fair bits have three bits of joint entropy and zero total correlation. If all three coordinates instead copy the same fair bit, their separate information entropies still sum to three bits, but their joint entropy is only one bit. Their total correlation is therefore two bits. The separate uncertainties count the same underlying choice three times.

A longest increasing subsequence is an increasing subsequence of maximum length. Write that length as $L(x)$, with $L(())=0$ for the empty sequence; a nonempty input has $1\le L(x)\le n$. Several selections can attain the same maximum. Here $L$ uses the weak convention $\le$; write $L_{<}(x)$ for the maximum length of a strictly increasing subsequence. For $(2,2,2)$, these lengths are three and one respectively; for an input with distinct values they agree.

For $x=(4,1,3,2,2,5,0)$, the answer is $L(x)=4$, attained at indices $(2,4,5,6)$, while $L_{<}(x)=3$. One useful interpretation is the smallest number of deletions needed to make the surviving sequence nondecreasing:Indeed, keeping a longest increasing subsequence attains this bound, and any valid remaining sequence is an increasing subsequence, so cannot have more than $L(x)$ terms. In this example the answer is $7-4=3$. This makes $L(x)$ a measure of how much order can be recovered without moving any observation.

A table that ends at each position. Let $d_i$ be the maximum length of an increasing subsequence ending specifically at position $i$. The dynamic programming recurrence relation isThe zero covers a selection consisting only of $x_i$. Otherwise its penultimate index is an eligible $j$; append $x_i$ to an optimal selection ending there. Fill entries in increasing index order. For strictly increasing subsequences, change the comparison to $x_j<x_i$. The strict version and reconstruction are developed in University of Toronto, CSC373 tutorial 4 solutions (2018), exercise 2; the weak version follows by allowing equality at the append step.

In the worked input, the weak table is $(1,1,2,2,3,4,1)$ and the strict table is $(1,1,2,2,2,3,1)$. At the second $2$, equality extends the first $2$ in the weak table. The final entry is only one, although the optimum is four: the best increasing subsequence need not finish at the last input position. Record a maximizing predecessor $p_i=j$ whenever $d_i>1$. Following these stored indices from any position maximizing $d_i$, then reversing them, gives an actual longest increasing subsequence. For the weak example the backward chain is $6,5,4,2$.

Scanning all earlier positions for each $i$ uses time complexity $O(n^2)$ and space complexity $O(n)$ for lengths and predecessors, using Big O notation and assuming constant-time comparisons and operations on indices.

Keep the easiest ending to extend. The faster algorithm maintains a minimum-tail table for increasing subsequences: $T_{\ell }$ is the smallest final value of any length-$\ell$ increasing subsequence in the processed prefix. For equal lengths, a smaller ending permits every future extension allowed by a larger ending. Thus only one best ending per length is needed. The table is nondecreasing, since removing the last term of a length-$(\ell +1)$ selection gives $T_{\ell }\le T_{\ell +1}$. Start with an empty table. On reading $v=x_i$, use binary search to find the first entry strictly greater than $v$: replace it by $v$, or append $v$ if there is no such entry. Its position is one beyond the longest existing length whose ending is at most $v$. See Liben-Nowell, Vee and Zhu, section 3.1 and Figure 1.

Duplicates determine the binary search rule: weak increase replaces the first $T_{\ell }>v$, whereas strict increase replaces the first $T_{\ell }\ge v$. In Python these positions are returned by bisect_right(T, v) and bisect_left(T, v) respectively, counting positions from zero. Replace an entry or append; do not insert and shift the remaining entries. The documented equality rules are in Python documentation, bisect_left and bisect_right. Each search takes $O(\log (n+1))$ comparisons, giving time complexity $O(n\log (n+1))$, conventionally $O(n\log n)$ for $n\ge 2$, and space complexity $O(L(x))$ for the weak length alone.

A length certificate needs separate reconstruction. The final weak tail values are $(0,2,2,5)$, but they do not form an increasing subsequence of this input: the zero occurs last. Each table entry certifies a selection of its own length; the entries need not come from one compatible selection. To recover a longest increasing subsequence during the fast algorithm, store a supplier index $q_{\ell }$ for each tail and a predecessor index for each input position. Before replacing slot $r$, save $p_i=q_{r-1}$, or no predecessor if $r=1$, then set $q_r=i$. Later replacements leave saved $p_i$ unchanged. Follow predecessors from the final $q_{L(x)}$ and reverse. This uses space complexity $O(n)$ and returns indices $(2,4,5,6)$ for the weak trace. The same construction works with the strict comparator; compare the two auxiliary arrays in CP Algorithms contributors, restoring the subsequence after binary search.

Deleting or changing one term. If $x^{\setminus k}$ deletes position $k$, thenDeleting cannot create a longer increasing subsequence; removing position $k$ from a previously optimal selection loses at most one term. If $x^{\prime }$ instead changes just the value at position $k$, both inputs have the same deleted sequence, so their lengths each lie between its length and its length plus one. Consequently $|L(x^{\prime })-L(x)|\le 1$. These bounds also hold for strictly increasing subsequences. In the weak example, deleting either $2$ lowers the length to three, while deleting the final zero leaves it four. This is a Lipschitz bound for single-term edits, regardless of the numerical size of a replacement. Insertion has the same bound by reversing deletion, so $m$ insertions, deletions or replacements can change $L$ by at most $m$.

For historical context, ^[ref] studies the comparison count for distinct elements of a totally ordered set. Its distinctness assumption makes the weak and strict conventions coincide. No distinctness assumption is needed for the comparator-specific algorithms above.

For $m$ lines and $n$ points in the real plane, the number of point-line incidences isConnecting consecutive incidence points on each line and applying the Crossing lemma proves the bound.

An isothermal atmosphere has constant temperature. Under constant gravity and ideal-gas hydrostatic equilibrium, its pressure and density decay exponentially with altitude.

A hyperbolic metric is a complete Riemannian metric of constant sectional curvature $-1$. On a surface it supplies local charts modeled on the hyperbolic plane.

For inviscid axisymmetric disturbances to a rotating flow, the Rayleigh discriminant measures the radial gradient of squared specific angular momentum.

Dissipation-induced instability occurs when damping one component of a coupled system moves an eigenvalue into the growing half-plane. Selective damping can alter the phase relation that previously stabilized two interacting waves or oscillators.

The Eady model describes quasi-geostrophic baroclinic instability in a uniformly stratified layer with constant vertical shear, zero interior potential-vorticity gradient, and two rigid horizontal boundaries.

Symmetric instability occurs in rotating stratified flow when perturbations aligned with the basic current can extract energy from vertical shear. It is excluded when the Ertel potential vorticity has the stable sign; for constant gradients this gives a condition relating stratification, rotation, and horizontal buoyancy gradient.

The energy-stability threshold is the parameter value below which a positive perturbation energy decreases monotonically for every disturbance. It can be lower than the linear instability threshold when non-normal transient growth is possible.