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For $\ddot{x}=-U^{\prime }(x)-\mu \dot{x}$ with $\mu >0$,If $U$ is radially unbounded, every trajectory is bounded, and LaSalle's principle reduces its omega-limit set to invariant points with zero velocity.

The Heaviside function is zero before its switching time and one after it, allowing delayed signals to be written algebraically.

The delayed unit step has

After $t=zs$, the phase in the integral for ${\Gamma }^{\prime }(z)$ becomes $\log s-s$, with maximum $-1$ and second derivative $-1$ at $s=1$. Hence

A Laplacian eigenfunction satisfies $-\Delta u=\lambda u$ together with specified boundary conditions.

The spherical-coordinate Laplacian splits into a radial operator and the sphere’s angular Laplacian divided by $r^2$.

For radial $u(r)$ in $n$ dimensions, $\Delta u=u\prime \prime +(n-1)u\prime /r$.

Laplace's equation isIts solutions are harmonic functions.

If a twice differentiable real function has a local maximum at an interior point of a Riemannian manifold, its Hessian matrix is negative semidefinite there and hence its Laplacian is nonpositive.

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 supplements the local Landau potential with gradient terms and treats the order parameter as a spatial field.

Stokes drift is the difference between the mean velocity of fluid particles and the Eulerian mean velocity at fixed positions. It appears at second order because particles sample the oscillatory velocity at displaced positions.

For a field $\Phi$,

A field interaction term is nonlinear in the fields and produces the interaction vertices of perturbation theory.

A mass term is a quadratic term without derivatives whose coefficient sets the mass scale of a free field.

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 $P^{\ast }(x)=Q(x)e^{g(x)}/{\mathbb{E}}_Q{e}^g$.