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For $L=mc|\dot{x}|-V(x)$,The momentum loses the speed, so the velocity cannot be recovered from $(x,p)$ and the ordinary inverse Legendre transform does not exist.

For $u_t=i{u}_{xx}$ on $x>0$, the half-line Fourier transform satisfiesSubtracting the relation at $-k$ cancels the unknown Neumann datum. Fourier inversion of the resulting odd combination gives the solution in terms of the initial transform and the prescribed Dirichlet datum, with boundary kernel $G(k,t)=kh(t)$.

For ${\Phi }_x+U\Phi =0$ and ${\Phi }_y+V\Phi =0$, compatibility is $V_x-U_y+[U,V]=0$.

For the auxiliary equations ${\psi }_x=U\psi$ and ${\psi }_t=V\psi$, equality of mixed derivatives gives $U_t-V_x+[U,V]=0$. This is the compatibility condition for a Lax pair.

For $\dot{L}=[L,A]$, the operator ${\partial }_t+A$ maps each eigenspace of $L$ into itself, so the spectrum is time-independent.

For complex fields $q,r$, the standard two-by-two AKNS pair has zero-curvature equationsThe reductions $q=\overline{r}$ and $q=-\overline{r}$ give the focusing and defocusing cubic nonlinear Schrödinger equations, respectively.

The Airy equation is the linear dispersive partial differential equation $q_t+q_{xxx}=0$. Under the spatial Fourier transform, each mode evolves by multiplication by $e^{i{k}^{3}t}$.

For positive time its fundamental solution is $(3t{)}^{-1/3}\mathrm{Ai}(x/(3t{)}^{1/3})$, connecting this evolution equation to the Airy function without identifying the two concepts.

An analytic function on an annulus $r<|z-a|<R$ has a unique locally uniformly convergent expansionwhere $C$ is any positively oriented circle around $a$ in the annulus.

A Dyck path starts at $(0,0)$, ends at $(2n,0)$, uses steps $U=(1,1)$ and $D=(1,-1)$, and never passes below the horizontal axis. Its semilength of a Dyck path is $n$: there are $n$ up steps and $n$ down steps. The nonnegative height records a constraint throughout the journey, not just the endpoint. The Catalan number $C_n$ counts the Dyck paths of this size.

Soft thresholding maps $u$ to $\mathrm{sgn}(u)(|u|-\lambda {)}_{+}$ and is the proximal operator of the absolute-value penalty.

The square-root Lasso uses the unsquared residual norm plus an ${\ell }^1$ penalty.

The Dantzig selector minimizes the coefficient ${\ell }^1$ norm subject to an ${\ell }^{\infty }$ bound on residual correlations.

A dark-matter halo is a gravitationally bound concentration of dark matter that hosts galaxies and larger structures.

Standard perturbation theory expands the density contrast and velocity divergence of a pressureless irrotational fluid in powers of the linear density field.

An Einstein-de Sitter universe is spatially flat and contains only pressureless matter. In conformal time, $a\propto {\tau }^2$ and $\mathcal{H}=a^{\prime }/a=2/\tau$.

Cosmological structure formation is the growth of primordial density perturbations into sheets, filaments, halos, galaxies, and clusters under gravity.

A gap equation determines a physical mass or excitation gap self-consistently from an interaction and a regulated fluctuation integral.

The Laplace transform of a matrix-valued function is obtained by transforming each entry. A constant matrix may be taken outside on the corresponding side of the integral; in particular, $\mathcal{L}\{AB\}=A\mathcal{L}\{B\}$ when $A$ is constant.

If $g$ has period $T$, then

For $a\ge 0$, $\mathcal{L}\{f(t-a)H(t-a)\}(s)=e^{-as}\mathcal{L}\{f\}(s)$.

Suppose $f$ is an absolutely continuous function on every finite interval $[0,T]$, both $e^{-st}f(t)$ and $e^{-st}{f}^{\prime }(t)$ are Lebesgue integrable functions, and $e^{-sT}f(T)\to 0$. Then integration by parts on $[0,T]$ followed by a limit givesHere $f(0+)$ is the right one-sided limit. Exponential order bounds on $f$ and $f^{\prime }$ suffice when $\mathrm{Re}s$ exceeds both bounds. For $f(t)=e^{-2t}$, the identity reads $-2/(s+2)=s/(s+2)-1$ on $\mathrm{Re}s>-2$. This is the ordinary derivative on nonnegative time; differentiating a causal extension by zero at its jump instead introduces a Dirac delta function. ^[ref].