At nonzero along-field wavenumber, put $q=i{k}_y{\xi }_y$, $P=\gamma p$ and $C=B^2/{\mu }_0$. After pressure minimization the energy density completes to $CP/(P+C)|q-\rho g{\xi }_z/P{|}^2+[C{k}_y^2-g{\rho }^{\prime }-{\rho }^2{g}^2/P]|{\xi }_z{|}^2$. Choosing the square to vanish and then taking a sufficiently small nonzero $k_y$ gives negative energy exactly when the displayed strict criterion holds somewhere. A fixed prescribed wavenumber retains the stabilizing $C{k}_y^2$ tension term; the criterion asserts existence of an allowed nonzero wavenumber. The associated along-field displacement lets material drain while magnetic flux bends.

From a point light source $P$, a caster point $C(t)$ projects onto a receiving surface along the displayed ray. On a regular transverse branch, differentiating gives $[S_u,S_v,-(C-P)](u^{\prime },v^{\prime },{\alpha }^{\prime }{)}^T=\alpha C^{\prime }$. Newton method correction and certified patch/curve bounds support continuation and isolation of all visible branches. If the shadow's second derivative is bounded by $M$ on an interval of width $h$, its chord error is at most $M{h}^2/8$. A fixed sample grid can miss entire shadow components; surface boundaries, occlusion changes and grazing rays require explicit treatment.

Intersecting a parametric curve with a plane reduces to isolating zeros of a scalar function. A certified range for $F$ rejects intervals that do not contain zero. On an interval whose derivative range excludes zero, a sign-changing endpoint bracket gives exactly one root, which bisection method or a safeguarded Newton root-finding iteration refines. Other intervals are subdivided conservatively; tangencies and coplanar pieces need multiplicity or identity tests supported by the representation. An unresolved small interval is a possible contact, not a proof of an intersection. With finitely many transverse intersections and convergent enclosures, isolation terminates to any prescribed tolerance.

For $E_x=i{\Delta }_{\perp }E/(2k)+ik\mu WE$, a jointly Gaussian colored medium has covariance $C(s,\mathbf{r})$. Averaging the equation gives a term $⟨WE⟩$; the Furutsu–Novikov formula expresses it through the random propagator and the functional derivative of $E$. It does not generally close in the mean field alone. A phase-screen product followed by Gaussian averaging gives a formal exact mean propagator, with factor $\exp [-k^2{\mu }^2\Delta x^2{\sum }_{j,l}C(x_j-x_l,{\mathbf{r}}_j-{\mathbf{r}}_l)/2]$ for each transverse path. In the longitudinal white-noise limit the covariance collapses to equal slices, and the mean closes as $m_x=i{\Delta }_{\perp }m/(2k)-k^2{\mu }^{2}A(0)m/2$.

For $U_x=iL(x)U$ with a self-adjoint transverse differential operator $L(x)$, differentiating the squared L2 norm gives $i⟨U,LU⟩-i⟨LU,U⟩=0$. Reality of the refractive coefficient and vanishing transverse boundary flux are required. This is exact for the scalar envelope model. Its physical interpretation as leading paraxial power uses a fixed reference impedance; a varying local impedance or a longitudinal electromagnetic polarization requires a consistent Maxwell calculation and cannot be inferred from this invariant alone.

A parabolic wave equation driven by longitudinal Gaussian white noise with transverse covariance kernel $C$ has the displayed closed mutual coherence equation. The noise is the derivative of a longitudinal Brownian field with transverse covariance; the physical smooth-noise limit uses the Stratonovich integral. Conversion to the Itô integral supplies the attenuation drift and the cross variation supplies the $C({\mathbf{s}}_1-{\mathbf{s}}_2)$ term. For unit plane-wave incidence and transverse statistical homogeneity, $\Gamma =\exp [-k^2{\mu }^{2}x(C(0)-C({\mathbf{s}}_1-{\mathbf{s}}_2))]$. General Gaussian colored media do not have this local closure.

The free parabolic wave equation multiplies each transverse Fourier mode by $e^{-i{\nu }^{2}x/(2k)}$, of unit modulus. A statistically stationary input therefore keeps its uncentered spectral measure and mean intensity. A pure phase screen of unit incident amplitude has total mean intensity one even when its coherent intensity is smaller. This statement does not require a Gaussian joint phase distribution; second-order stationarity and free paraxial propagation suffice.

For $P(z)=z-z^2$, the disc $\Delta =\{|z-1/2|<1/2\}$ corresponds under $w=1/z$ to $\Re w>1$. The transformed map is $T(w)=w+1+1/(w-1)$, so $\Re T(w)>\Re w+1$. Thus $P(\Delta )\subseteq \Delta$ and $|P^n(z)|<1/(n+1)$ there. The Fatou component $U$ containing the disc has $P^n\to 0$ locally uniformly by normality and the identity theorem. It is forward invariant. The point $1/4$ in the disc has the entire fiber $\{1/2\}$, so the Fatou component containing an entire fiber is completely invariant criterion gives $P^{-1}(U)=U$. Zero itself is a Julia point: a normal subsequential limit near zero would be zero by the disc limit, but every iterate has derivative one at zero, contradicting derivative convergence.

For positive integer severities with probability mass function $f_j$ and independent count in the Panjer claim-count class, the aggregate probability mass function starts at $g_0=p_0$ and satisfies the displayed recurrence. If $P,F,G$ are the count, severity and aggregate probability generating functions, respectively, then $G=P\circ F$ and $(1-aF)G^{\prime }=(a+b)F^{\prime }G$. Equating the coefficient of $z^{r-1}$ gives $r{g}_r={\sum }_{j=1}^r[a(r-j)+(a+b)j]f_j{g}_{r-j}$. The positivity assumption excludes zero severities; otherwise the initial value and recursion denominator change.

Let $X,Y$ be independent fair bits, and let $Z=X\oplus Y$ be their exclusive or: $Z=1$ precisely when $X\ne Y$. The four possible triples are $000,011,101,110$, each with probability $1/4$. Each coordinate is a fair bit. Writing out all eight patterns givesEvery pair has all four cells $00,01,10,11$ with probability $1/4$:For $(X,Z)$, the four allowed triples project to $00,01,11,10$; for $(Y,Z)$ they project to $00,11,01,10$. Each projection lists each pair exactly once, proving pairwise independence. But $\mathbb{P}(X=0,Y=0,Z=0)=1/4$ differs from the product $1/8$, so the three are not independent random variables. Indeed, any two determine the third. A single coordinate reveals nothing about another, while two coordinates together reveal it completely.

The exclusive or example is the complementary-bit version of the two-toss exercise in Grinstead and Snell, Introduction to Probability, section 4.1, exercise 7, printed page 151. It is a compact reason to check the full product rule whenever a calculation uses three-way or higher events.

A Y-piece is a compact curvature-$-1$ hyperbolic surface homeomorphic to a pair of pants, with three geodesic boundary curves. Its labelled boundary lengths ${\ell }_1,{\ell }_2,{\ell }_3>0$ determine it up to isometry. Cutting along the three mutually perpendicular seams produces two congruent right-angled hyperbolic hexagons with alternating sides ${\ell }_i/2$. Conversely, doubling the hexagon along its other three sides constructs the Y-piece for arbitrary positive lengths.

For a steady accretion disk with arbitrary rotation law and a zero-torque inner boundary condition at $R=3R_S$, the outward viscous torque in an accretion disk is $\mathcal{G}=\dot{M}(j-j_{\mathrm{in}})$. The dissipation per unit planar area summed over both disk faces is $D=-\mathcal{G}{\Omega }^{\prime }/(2\pi R)=\nu \Sigma (R{\Omega }^{\prime }{)}^2$. Integrating $2\pi RD$ gives $L=\dot{M}GM/(8R_S)=\dot{M}{c}^2/16$ in the Paczyński-Wiita potential. The radiative flux from one face is $D/2$, preventing a factor-of-two error in the luminosity.

The limiting inward departure from the marginal Paczyński-Wiita circular orbit at $r=3a$ conserves the displayed specific angular momentum and specific orbital energy. In the Paczyński-Wiita potential, its inward speed satisfies $v_r^2=GM(3a-r{)}^3/[4a{r}^2(r-a)]$. The exactly marginal circular orbit does not depart without a perturbation; the formula describes the separatrix limit of inward releases.

For a stable isotropic solid in $y<0$, use $u_x={\varphi }_x+{\psi }_y$, $u_y={\varphi }_y-{\psi }_x$. An incident P wave with frequency $\omega$, tangential wavenumber $\kappa$ and upward normal wavenumber $a$ reflects into a P potential $A{e}^{i(\kappa x-ay-\omega t)}$ and an SV potential $B{e}^{i(\kappa x-by-\omega t)}$, where $b^2={\omega }^2/c_s^2-{\kappa }^2$. Vanishing of both displacements at the rigid plane gives $\kappa (1+A)-bB=0$, $a(1-A)-\kappa B=0$. Hence $A=(ab-{\kappa }^2)/(ab+{\kappa }^2)$ and $B=2a\kappa /(ab+{\kappa }^2)$. Because $c_p>c_s$ and ${\kappa }^2\le {\omega }^2/c_p^2$, $b^2>0$ and the converted S wave propagates rather than being evanescent.

Let $t=T(\mathbf{x})$ be a smooth wavefront and suppose the first jump in time derivatives of displacement field is $[{\partial }_t^k\mathbf{u}]=a^{[k]}\mathbf{n}$, with $k\ge 2$. Substitution of $\mathbf{u}=a^{[k]}\mathbf{n}(t-T{)}_{+}^k/k!+\cdots$ into the elastic equation first gives the eikonal equation $|\nabla T|=1/\alpha$ and longitudinal polarization $\mathbf{n}=\alpha \nabla T$. Projecting the next-order compatibility condition onto $\mathbf{n}$ gives $2\rho \alpha \mathbf{n}\cdot \nabla a^{[k]}+a^{[k]}\nabla \cdot (\rho \alpha \mathbf{n})=0$, hence the displayed conservation law. Coefficients and wavefront must be sufficiently differentiable for this classical transport equation; its local form fails at a caustic.

A p-subgroup is p-radical if it equals the p-core of its normalizer. Every defect group of a block is p-radical by defect groups are centralizer-conjugate Sylow intersections. Being an intersection of arbitrary Sylow subgroups alone is weaker than the centralizer-conjugate statement used in that argument.

The integer $h=|G:G^{\prime }{|}_p$ divides $s={\sum }_{\chi \in {\mathrm{Irr}}_{p^{\prime }}(G)}\chi (1{)}^2$. The group is p-nilpotent exactly when $p\nmid s/h$. To see this, let $N=O^p(G)$. Invariant p-prime-degree characters of $N$ extend, and each has exactly $h$ extensions of p-prime degree. Conjugation by the p-group $G/N$ then gives $s/h\equiv |N|(\mathrm{mod}p)$. Thus the quotient is prime to $p$ precisely when $N$ is a normal p-complement.

If a prime $p$ divides the degree of every nonlinear irreducible character of a finite group, the group has a normal p-complement. Indeed, the p-prime-degree characters are exactly the linear characters. Their squared-degree sum is $|G:G^{\prime }|$, whose p-part is $|G:G^{\prime }{|}_p$; Isaacs' character-degree criterion for p-nilpotency applies.

Put $\overline{G}=G/O_{p^{\prime }}(G)$ using the p-prime core of a finite group. A finite group is p-constrained if $C_{\overline{G}}(O_p(\overline{G}))\le O_p(\overline{G})$. If $O_{p^{\prime }}(G)=1$, the p-core of a finite group has its centralizer contained in itself. Then block idempotents centralize a normal p-subgroup places every block idempotent in $k{O}_p(G)$, a local ring, so $kG$ has only the principal block.

On a profinite group, convolution is first defined for functions on a finite quotient by summing products of the two measures' coset coefficients, then extended by uniform approximation. Finite-quotient associativity proves associativity. For an abelian group it is commutative; the Dirac measure at the identity is the identity element. Under the measure realization of an Iwasawa algebra, it becomes group-algebra multiplication.