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A tricritical point is where a line of continuous transitions meets a line of first-order transitions; in a ${\mathbb{Z}}_2$ Landau expansion both quadratic and quartic coefficients vanish and a positive sextic term stabilizes the free energy.

For an ordinary quartic Landau transition, the heat-capacity and order-parameter exponents are $\alpha =0$ and $\beta =1/2$. At a tricritical sextic point they are $\alpha =1/2$ and $\beta =1/4$.

Landau-Ginzburg theory promotes an order parameter to a spatial field and includes an energy cost for its variation. For a scalar field,The local terms select preferred phases, while the gradient term makes abrupt spatial changes costly. Minimizing this free-energy functional can describe a smooth domain wall between two ordered regions.

Expanding around a uniform stable minimum gives a quadratic fluctuation kernel and an Ornstein--Zernike correlation function. The framework therefore connects an equilibrium phase diagram with measurable spatial correlations and interfaces. Minimization alone is a mean-field treatment; integrating fluctuations can change the predictions and is central to the renormalization group. The derivative expansion assumes fields varying slowly compared with microscopic lengths.

Stokes drift is the difference between the mean velocity of moving fluid particles and the mean velocity measured at fixed spatial positions. An oscillating particle samples a slightly different part of the velocity field during each part of its cycle. Correlation between its displacement and the spatial variation of the field can therefore produce a net drift even when the first-order Eulerian coordinate velocity average is zero.

For a small-amplitude deep-water gravity wave with surface elevation $\eta =a\cos (kx-\omega t)$, no imposed mean current and mean particle height $z\le 0$, the leading drift isIt follows the propagation direction and decreases with depth. Its being quadratic in wave amplitude explains why the linear particle orbits nearly close but do not close exactly.^[ref]

A marked fluid particle and a fixed velocity probe measure different averages, so a wave-tank comparison must also account for any compensating return current. Surface float motion can include additional effects from wind or the float's own response. Stokes drift is consequently a precise difference of averages, rather than the name for every observed motion of floating debris.

The Euler-Lagrange field equation implements the principle of stationary action for a Lagrangian density depending on scalar fields and their first derivatives. For a single scalar field $\Phi$,

Vary $S=\int d^{4}x\mathcal{L}$. Integration by parts transfers the derivative from $\delta \Phi$ to $\partial \mathcal{L}/\partial ({\partial }_{\mu }\Phi )$. The boundary term vanishes for variations supported away from the boundary; arbitrary interior variations then force the displayed coefficient to vanish. For $\mathcal{L}=\frac{1}{2}{\partial }_{\mu }\Phi {\partial }^{\mu }\Phi -\frac{1}{2}{m}^2{\Phi }^2$, the result is the Klein-Gordon equation $({\partial }_{\mu }{\partial }^{\mu }+m^2)\Phi =0$, using natural units and Minkowski metric signature $(+---)$. ^[ref]

A field interaction term is part of the action coupling fluctuations beyond a chosen free reference theory. Cubic and quartic scalar terms provide standard examples; in a perturbative expansion their coefficients and derivatives determine the corresponding interaction vertex.

The split into free and interacting parts is a calculational choice. Quadratic mixing can be diagonalized into a new free basis or treated as an insertion relative to another basis, so “nonlinear” alone is not a universal definition of every interaction term. Gauge-covariant kinetic terms generate derivative interactions and two-gauge-field contact terms when expanded. Specifying the complete reference action, field normalization and interaction coefficients makes the resulting perturbation theory reproducible.

A mass term is a derivative-free quadratic field contribution whose coefficient fixes the linearized mass about a chosen background. For a real scalar with canonical kinetic normalization, $\mathcal{L}\supset -\frac{1}{2}{m}^2{\varphi }^2$ gives $(\square +m^2)\varphi =0$ and the relativistic dispersion $E^2={\mathbf{p}}^2+m^2$ in natural units.

Around an interacting vacuum, the mass squared comes from the potential curvature together with the kinetic normalization, not from every quadratic coefficient viewed in isolation. Several mixed fields require diagonalizing the corresponding quadratic problem. A negative curvature signals instability of that background toward a different configuration; it does not describe a stable particle simply traveling faster than light. The background and sign convention are therefore part of the statement.

A kinetic term contains derivatives of a field and determines its free propagation. Its quadratic differential operator becomes the inverse quantum field theory propagator.

For $x^3+ax+b$, the Lagrange-resolvent cubes are roots of $X^2+27bX-27a^3$.

For univariate normal distributions,

For a probability mass function $Q$ of full support on a finite alphabet and a real function $g$,where $D_e$ uses natural logarithms. The unique maximizing probability mass function is the exponential tilt $P^{\ast }(x)=Q(x)e^{g(x)}/{\mathbb{E}}_Q{e}^g$.

For nonnegative $a_i,b_i$, with $a={\sum }_i{a}_i$ and $b={\sum }_i{b}_i$, the log-sum inequality isIt proves both Gibbs inequality and the data processing inequality for relative entropy.

This is the data processing inequality for f-divergences specialized to relative entropy. Passing two probability distributions through the same Markov kernel cannot increase their relative entropy. For a kernel $K$,

For probability distributions $P$ and $Q$ on a product space, the Kullback-Leibler divergence decomposes into the divergence of a marginal and the expected divergence of the corresponding conditional distributions. Iteration gives

For probability distributions $p$ and $q$ on the same finite set,with equality exactly when $p=q$ on the support of $p$.

For a Noetherian local ring $(A,\mathfrak{m})$, the Krull dimension of $A$, the Krull dimension of the associated graded ring ${\mathrm{gr}}_{\mathfrak{m}}(A)$, and the order of the pole at $t=1$ of its Hilbert series are equal.

Height measures prime-chain length within Krull dimension theory.

The height of an ideal $I$ is the infimum of the heights of prime ideals containing it:The height of a prime is the supremum of the lengths of strict prime chains ending at that prime.

If one symmetric cell has reflection and transmission amplitudes $r,t$, thenAt a periodic or antiperiodic band edge, with $\mathrm{tr}M=2\sigma$ and $\sigma =\pm 1$, setting $z=e^{-ika}$ gives

Let $E={\hslash }^2{k}^2/(2m)>0$ for the Kronig-Penney model, and propagate the column $y=(\psi ,{\psi }^{\prime }{)}^T$ by one lattice cell. Free propagation through distance $a$ and passage across one delta potential respectively give the propagation and jump matricesFor a cell beginning just after a barrier, $M=DP$ takes it to just after the next barrier. Both matrices have determinant one, so their product has reciprocal eigenvalues. Multiplication gives the half-trace, or Floquet discriminant,A Bloch state has Floquet matrix eigenvalues $e^{\pm iqa}$, hence $\cos (qa)=\Delta (E)$. Real crystal wavenumber $q$ therefore requires $|\Delta (E)|\le 1$. The values $\Delta =\pm 1$ give periodic or antiperiodic band-edge conditions. When $|\Delta |>1$, the reciprocal eigenvalues have unequal magnitudes and one cell solution grows while the other decays, producing a band gap in the infinite periodic system. The displayed positive-energy formula continues to negative energy by replacing trigonometric functions with their hyperbolic counterparts; the limit at $k=0$ uses $\sin (ka)/k\to a$.

A knot is doubly slice when it is the equatorial cross-section of an unknotted smooth two-sphere in $S^4$.