To deduce $D{u}_t=I+tDF+o(t)$ from $u_t(x)=x+tF(x)+o(t)$, the remainder must be small in a topology controlling its spatial derivatives, such as $C^1$ locally. Small displacement alone is insufficient: $u_t(x,y,z)=(x+t^2\sin (x/t^2),y,z)$ differs uniformly from the identity by $o(t)$, but has Jacobian determinant $2$ at $x=0$ for nonzero $t$. A smooth flow map generated by a smooth vector field has the required derivative control.

This convention for the Jacobi theta function is the usual theta constant evaluated at twice the argument. The Poisson summation formula gives $\theta (-1/(4\tau ))=\sqrt{-2i\tau }\theta (\tau )$, with the square root holomorphic on the half-plane and positive on the imaginary axis after evaluating its positive real argument.

In the $e^{\pi iz}$ convention, Poisson summation gives ${\Theta }_3(-1/z)=\sqrt{-iz}{\Theta }_3(z)$ and interchanges ${\Theta }_2,{\Theta }_4$ with the same factor. Translation by one sends ${\Theta }_2$ to $e^{\pi i/4}{\Theta }_2$ and interchanges ${\Theta }_3,{\Theta }_4$. Eighth powers remove every phase and square-root ambiguity when proving integral-weight modularity.

An object is j-separated when two maps into it which agree on a j-dense monomorphism are equal. Equivalently its diagonal is j-closed. Quotienting an arbitrary object by the j-closure of its diagonal gives its separated reflection, used in constructing the sheaf reflector for a local operator.

A Lagrangian boundary condition for a J-holomorphic curve requires specified boundary components of the source to map into specified Lagrangian submanifolds. For the strip, this is $u(s,0)\in L_0$ and $u(s,1)\in L_1$. For a compatible almost complex structure these target submanifolds are totally real. The condition accompanies the interior equation $u_s+J{u}_t=0$.

For a semimartingale with càdlàg paths $X$ and $f\in C^2$, the Itô formula uses the left-limit gradient in $\int f^{\prime }(X_{s-})d{X}_s$, the continuous quadratic variation in its second-order integral, and the jump correction ${\sum }_{s\le t}[f(X_s)-f(X_{s-})-f^{\prime }(X_{s-})\Delta X_s]$. In several dimensions use the corresponding gradient, Hessian and continuous bracket matrix. The correction is the part of each jump not captured by first-order linearization.

A second-rank Cartesian tensor invariant under every proper rotation matrix is a scalar multiple of the Kronecker delta. Half-turns about the coordinate axes kill its off-diagonal entries, and quarter-turns make its diagonal entries equal. A tensor of the form $v_i{w}_j$ is an outer product with matrix rank at most one, so it is isotropic in three dimensions only when it vanishes, equivalently when $v=0$ or $w=0$.

The regular spherical isothermal inflow from mass density ${\rho }_{\infty }$ at rest at infinity has sonic mass density $e^{3/2}{\rho }_{\infty }$. Combining it with the Bondi sonic point and mass conservation gives $\dot{M}=\pi e^{3/2}{G}^2{M}^2{\rho }_{\infty }/c_s^3$.

An isorotation is a global transformation generated by isospin. In the pion Skyrme model it acts as $U\mapsto AU{A}^{†}$, with constant $A\in SU(2)$, rotating the three pion components while preserving the scalar component. The matrices $A$ and $-A$ induce the same classical rotation, so this action factors through $SO(3)$; quantum states may carry half-integer isospin on its $SU(2)$ cover. This global internal rotation is distinct from a spatial rotation or a local gauge transformation.

A class of metrics is spectrally rigid if equality of their specified spectra forces the metrics to be isometric, or, in a deformation version, if continuous isospectral deformations are trivial. The spectrum of a flat torus determines every two-dimensional flat torus up to isometry. The Wolpert generic spectral rigidity theorem is a generic, rather than universal, uniqueness statement.

For two assemblies of congruent Euclidean tiles, let $M_s,N_s$ encode the gluing or boundary reflection at each labelled face. An invertible constant matrix $C$ satisfying $C{M}_s=N_{s}C$ carries tile restrictions of Laplacian eigenfunctions bijectively to those on the second assembly. Boundary values satisfy $M_{s}u=u$ and outward normal derivatives satisfy $M_{s}v=-v$, so the intertwining identities preserve matching and boundary conditions. Use diagonal $-1$ for a Dirichlet boundary condition and $+1$ for a Neumann boundary condition.

Suppose a finite group $G$ acts by Riemannian isometries on a compact Riemannian manifold $M$, and Gassmann equivalent subgroups $H_1,H_2$ act freely. The quotient manifolds $H_i\backslash M$ have equal Laplace-Beltrami operator spectra. For an eigenfunction space $E$, the quotient multiplicity is $|H_i{|}^{-1}{\sum }_{h\in H_i}\mathrm{tr}(h|E)$, and the equality follows by summing its character of a representation over conjugacy classes. Nonconjugacy inside $G$ alone does not guarantee that the quotients are nonisometric.

An isometric extension of a connected metric manifold is a proper embedding onto an open subset of a larger connected manifold of the same dimension, preserving the metric tensor. Smooth complete connected Riemannian manifolds admit no such proper extension; Lorentzian geodesic completeness has different subtleties.

A Markov chain with transition kernel $P$ is $\phi$-irreducible if $\phi$ is a nonzero sigma-finite measure and every measurable set $A$ with $\phi (A)>0$ satisfies ${\sum }_{n\ge 1}{P}^n(x,A)>0$ from every starting state $x$. This is positive reachability, not certain eventual hitting.

For left state zero and right state $U$, negative-flux Inviscid Burgers equation gives a shock at $\theta =-UZ/2$ when $U>0$, and a rarefaction wave $f=-\theta /Z$ on $0<\theta <-UZ$ when $U<0$. This is the standard positive-flux equation after setting $u=-f$; entropy solution selection reverses which step direction is compressive.

For the negative-flux equation $f_Z-f{f}_{\theta }=0$, a period-two increasing initial ramp develops rarefaction fans at odd points and steepens at even points. Before $Z=1$, its central ramp is $f=(\theta -2m)/(1-Z)$ on $|\theta -2m|<1-Z$, and its odd-point fans are $f=-(\theta -(2m+1))/Z$. After $Z=1$, stationary compressive shocks occupy the even points and $f=(2m+1-\theta )/Z$ between consecutive shocks. The amplitude decays as $1/Z$. The Rankine-Hugoniot condition and incoming characteristic speeds certify the post-breaking entropy solution.

For a constant-coefficient single-asset investment-consumption problem with constant relative risk aversion utility, put $\kappa =(\mu -r)/\sigma$ and $\delta =[\rho -(1-R)(r+{\kappa }^2/(2R))]/R$. When $\delta >0$, the infinite-horizon value is ${\delta }^{-R}{w}^{1-R}/(1-R)$ and the optimal controls are $c^{\ast }=\delta w$ and ${\theta }^{\ast }=(\mu -r)w/(R{\sigma }^2)$. The case $R=1$ uses logarithmic utility.

A high-water mark investment tax charges increments of historical maximum wealth ${\bar{w}}_t=\max ({\bar{w}}_0,{\sup }_{s\le t}{w}_s)$ rather than every positive instantaneous return. For the convention charging $\tau d\bar{w}$, the high-water mark tax boundary condition is $V_{\bar{w}}=\tau V_w$ when the optimal policy raises the maximum. At sufficiently high tax, a wealth-cap investment boundary can make further maximum increases suboptimal.

Intertemporal hedging demand adjusts the myopic risky position when investment opportunities depend on a stochastic state. Shared noise between wealth and that state contributes a cross derivative to the Hamilton-Jacobi-Bellman equation. In a complete diffusion market the optimal exposures combine the myopic market price of risk term and the value-gradient hedge term.

When asset correlation varies with a traded market index, the index is both an asset and a state variable. Writing $m=\log M$ and optimizing Brownian portfolio exposures accounts for the cross derivative between wealth and $m$. Power homogeneity reduces the Hamilton-Jacobi-Bellman equation to an ordinary differential equation in $m$.