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If $\dot{\Phi }=A(t)\Phi$ and $\Phi (0)=I$, differentiation of the determinant gives $\det \Phi (t)=\exp ({\int }_0^t\mathrm{tr}A(s)ds)$. For a planar periodic orbit this gives the nontrivial Floquet multiplier, since the tangent multiplier is one.

The constant diagram functor sends $C$ to the functor on $J$ with constant object value $C$ and identity arrow values; a morphism $f$ becomes the natural transformation with every component $f$. When all $J$-shaped categorical limits exist, it is a left adjoint to the limit functor.

In a full-sky cubic bispectrum estimator, pairwise Gaussian covariance within one triple gives a sum of Gaunt integrals with opposite $m$ indices. The spherical harmonic addition theorem turns their pair into the constant $(2\ell +1)/(4\pi )$; its integral against $Y_{LM}$ vanishes for $L>0$. A zero temperature monopole removes $L=0$. Thus only the six pairings connecting the two triples survive in the Gaussian variance; isotropic weights are essential for the cancellation.

A nowhere-zero differential one-form $n$ is hypersurface orthogonal if locally $n=Fdt$ with $F\ne 0$. Its kernel consists of tangent vectors to the level hypersurfaces of $t$. Frobenius theorem makes this equivalent to $n\wedge dn=0$, where $d$ is the exterior derivative. With a torsion-free Levi-Civita connection, this is $n_{[\alpha }{\nabla }_{\beta }{n}_{\gamma ]}=0$. For a timelike unit normal, projecting the derivative on both indices gives zero antisymmetric part, which explains why hypersurface orthogonality implies symmetric extrinsic curvature.

A terminating Frobenius series is a Frobenius method solution whose analytic power series factor has only finitely many nonzero coefficients. Its coefficient linear recurrence relation must allow the first omitted coefficient and all following coefficients to be zero. At an integer separation of characteristic exponents at a regular singular point, a resonant coefficient equation can become $0=0$, permitting such a choice rather than forcing a logarithmic term. With exponent zero, termination produces a polynomial solution.

In a finite Galois extension $L/K$, choose a prime ideal $\mathfrak{P}$ above an unramified $\mathfrak{p}$. The Frobenius automorphism at $\mathfrak{P}$ is characterized on its residue field by $x\mapsto x^{N\mathfrak{p}}$. Changing $\mathfrak{P}$ conjugates this element, so its conjugacy class is well-defined. It is the identity class exactly at a completely split prime.

In a boundary-adapted temporal gauge, a free open string endpoint has $X^{\prime m}=0$. The Nambu–Goto phase-space constraints then imply $P^2=0$, while $\dot{X}=eP$ implies ${\dot{X}}^2=0$. For $X^0=t$, its spatial speed is one. This null boundary is compatible with a timelike interior worldsheet.

If fragments are injected with number spectrum $S_N(D)\propto D^{-{\alpha }_r}$ and escape faster than they collide, steady balance gives $n(D)=S_N(D)t_y(D)$. Thus $t_y\propto D^{\gamma }$ changes the differential number index to ${\alpha }_r-\gamma$. A removed population with size-independent subsequent lifetime retains the source index ${\alpha }_r$.

For fractional Brownian motion, $H\in (0,1)$ controls the variance $|t-s{|}^{2H}$ of increments and the scaling of its finite-dimensional distributions. Larger $H$ means smaller increment variance on intervals shorter than one and greater sample-path Hölder continuity.

For independent coordinate flips at rate $1/2$, a Walsh character $w_A$ is an eigenfunction with eigenvalue $-|A|$. The generator is self-adjoint and negative semidefinite. Its Dirichlet form of a Markov chain is $-\mathbb{E}[fLf]=\frac{1}{4}{\sum }_i\mathbb{E}(f({\omega }^{(i)})-f(\omega ){)}^2$.

The Fourier transform on a finite group convention $\hat{f}(\rho )={\mathbb{E}}_{x}f(x)\rho (x)$ has the displayed inversion formula. The sum is over one representative from each equivalence class of unitary irreducible representations. It follows from the Schur orthogonality relations and the regular representation decomposition, and holds at every group element without a limiting argument.

For the Fourier transform on a finite group, the displayed identity uses uniform expectation in the original function space and a weighted Hilbert-Schmidt inner product in the matrix components. In particular $\Vert f{\Vert }_2^2={\sum }_{\rho }{d}_{\rho }\Vert \hat{f}(\rho ){\Vert }_{\mathrm{HS}}^2$. For an abelian group, all $d_{\rho }=1$, and the identity becomes a sum of squared scalar coefficients.

For an orthogonal matrix $R$, a unit-Jacobian change of variables gives $\widehat{\phi \circ R^t}=\hat{\phi }\circ R^t$. The dual definition of the Fourier transform of a tempered distribution consequently givesSince the Fourier transform is invertible on the Schwartz space and its dual, a tempered distribution is invariant under a rotation group exactly when its transform is invariant. This includes radial tempered distributions.

For a test function $\phi$ supported in a fixed ball of radius $R$ and a fixed height bound $H$, repeated integration by parts givesApply $(1-\Delta {)}^L$ to the compactly supported smooth function $e^{x\cdot \eta }\phi (x)$ inside the real-frequency Fourier integral. The bounded $\eta$ absorbs its derivative and exponential factors into the constant. This estimate makes the numerator of a Hörmander staircase integral absolutely integrable and makes truncated contour endpoints vanish.

Let a distribution have support in a compact convex set $K$. Fixed neighborhood cutoffs initially give an exponential bound larger than the support function $H_K$. Cutoffs of width $\epsilon$ cost powers of ${\epsilon }^{-1}$ in derivative estimates but enlarge the exponential by only $e^{C\epsilon |\mathrm{Im}z|}$. Choosing $\epsilon =(1+|z|{)}^{-1}$ absorbs the cutoff cost into a polynomial and retains the exact exponential $e^{H_K(\mathrm{Im}z)}$.

On an $N$-pixel line with last forward difference zero, $D^{\ast }p$ has first entry $-p_1$, interior entries $p_{i-1}-p_i$, and last entry $p_{N-1}$, with $D^{\ast }=0$ for $N=1$. On a square grid add these expressions along rows and columns. Unused last-edge components contribute nothing. The identity $⟨Du,p⟩=⟨u,D^{\ast }p⟩$ fixes every boundary sign. On the unscaled square grid, $\Vert D{\Vert }^2=8{\cos }^2(\pi /(2N))\le 8$ for $N>1$.

A formal power series ring is a reduced ring if and only if its coefficient ring is reduced. A nonzero constant nilpotent proves one direction. In the other direction, a nonzero series with first nonzero coefficient $a_r$ has first coefficient $a_r^m$ in its $m$th power; this is nonzero over a reduced ring.

Opposite forces $\pm F\mathbf{e}$ applied to the fluid at $\pm (a/2)\mathbf{e}$ have no net force or torque. Their leading Stokeslet difference is $-Fa(\mathbf{e}\cdot \nabla )[J(\mathbf{r})\mathbf{e}]$, equal to the axial stresslet with strength $S=Fa$. Positive $S$ drives outward flow on the axis and inward flow in the equatorial plane, corresponding to a pusher microswimmer; negative $S$ gives a puller microswimmer.

The angular factor of an axial stresslet is linear in the symmetric traceless rank-two tensor $\mathbf{ee}-I/3$. Hence averaging a collection of directors amounts to averaging this tensor. For uniformly distributed in-plane directions, $⟨\mathbf{ee}⟩=(I-\mathbf{nn})/2$, so $⟨S(\mathbf{ee}-I/3)⟩=-(S/2)(\mathbf{nn}-I/3)$. The averaged director is normal to the plane and its signed strength is reversed and halved.

If $P(k)$ is analytic in the upper half-plane with suitable growth, paired integrals along the positive real and imaginary rays cancel when their exponential is $e^{ik(x+iy)}$ with $x,y>0$. For $0<y<\ell$, the analogous pair along the imaginary and negative real rays cancels with $e^{ik(x+i(y-\ell ))}$. These two applications of the Cauchy integral theorem eliminate reflected unknown transforms from a strip global relation; an individual ray integral need not vanish.