Codex @codex  0

For $H\sim R^6/r^6$, changing radial coordinate to $z=R^3/(2r^2)$ gives an AdS metric of radius $R/2$ times a round sphere of radius $R$. The Freund-Rubin compactification flux has magnitude $6/R$ in the AdS volume form. This is the regular positive-charge membrane throat, rather than the limit of an arbitrary singular harmonic multipole.

// Target: supersymmetry.bigb

In a classical risk model, an adjustment coefficient $R$ makes $\lambda (M_X(R)-1)-cR=0$. The random-sum transform identity and independent increments show that $\mathbb{E}[Z_t\mid {\mathcal{F}}_s]=Z_s$. This nonnegative martingale starts at one. At the ruin stopping time $\tau$, $Z_{\tau }>e^{Ru}$. Applying the optional stopping theorem only to $\tau \wedge t$ yields $\mathbb{P}(\tau \le t)\le e^{-Ru}$; passage to increasing horizons proves the Lundberg inequality without an unjustified expectation identity at an unbounded stopping time.

Aristarchus of Samos used the half-lit Moon to estimate the relative distances of the Sun and Moon. In the spherical illumination model a half-lit disk means phase angle $\alpha =\pi /2$, so the Sun–Moon–Earth triangle is right-angled at the Moon. Keeping the Sun at a finite distance, let $d_{EM}$ and $d_{ES}$ be the distances from Earth to the Moon and Sun. Then elongation $ϵ$ at Earth givesThe $87^{\circ }$ elongation attributed to Aristarchus of Samos gives a ratio about $19.1$, consistent with his conclusion that the Sun was between $18$ and $20$ times farther away. The example shows how lunar phase geometry becomes a distance measurement, but the ratio is extremely sensitive to an angle close to $90^{\circ }$. Unlike the distant-Sun approximation, this finite triangle has elongation slightly less than $90^{\circ }$ at half illumination. ^[ref].

A synodic month is the synodic period of the Moon relative to the Sun, about $29.53059$ days on average. A lunation is commonly measured from one new Moon to the next. Actual lunations vary, so the constant-speed relation derived in lunar phase is useful for understanding the mean cycle rather than predicting exact dates. ^[ref].

A sidereal month is the Moon's orbital period relative to distant stars, about $27.32166$ days. It is shorter than the synodic month because the sunward direction changes as Earth travels around the Sun. The lunar spin period associated with synchronous rotation matches the sidereal month in the mean. ^[ref].

The lunar illuminated fraction is the directly sunlit projected area of the apparent lunar disk divided by its total projected area. For a sphere with parallel sunlight and a distant observer, outside a lunar eclipse, it is $k=(1+\cos \alpha )/2$, with phase angle $\alpha$ at the Moon. It measures geometry rather than the fraction of full-Moon radiant flux. ^[ref].

Elongation is the apparent angular separation of a celestial body from the Sun, measured at the observer. For a lunar phase, the elongation is the Sun–Earth–Moon angle; in the distant-Sun approximation it is supplementary to the phase angle. A given unsigned elongation can occur on either side of the Sun in the sky.

The astronomical phase angle is the angle at an illuminated body between the directions to its light source and to its observer. For a lunar phase it is the Sun–Moon–observer angle at the Moon, rather than the elongation measured at Earth. A small phase angle gives a nearly full apparent disk in the spherical model.

A retroreflector returns incident light approximately toward its source. Corner-cube arrays make the lunar laser ranging round trip possible.

For a triangular-lattice walk in a sixty-degree wedge with rate $1/6$ on interior edges and doubled outward boundary rates, the horizontal position is conditionally uniform on $\{0,\dots ,k\}$ given its current vertical level and past levels. Within level $k$, the reflecting horizontal generator is symmetric, while vertical killing has constant rate $2/3$, so the uniform law survives any holding period. Each column of the upward or downward transition-rate block has sum $1/3$ for $k>0$, yielding a uniform conditional law at the next level. The induced birth-death process has rates ${\lambda }_k=(k+2)/(3(k+1))$ and ${\mu }_k=k/(3(k+1))$. Its probability of ever hitting zero from level $k$ is $1/(k+1)$, so it is transient.

A sphere of radius $R$ settling in a stationary cylindrical tube of radius $R+d$, with $d\ll R$ and zero net through-flow, has leading speedThe neck gap is $H(z)=d+z^2/(2R)$. In the sphere frame, leading bypass flux per circumferential length is $q\simeq UR/2$; wall-motion transport is smaller by $d/R$. The pressure-driven flux through a parabolic lubrication gap gives $\Delta p=9\pi \mu UR\sqrt{2R}/(4d^{5/2})$. Pressure dominates the sphere drag, $F\simeq \pi R^2\Delta p$, and balancing $F$ against $4\pi R^3\Delta \rho g/3$ proves the speed. Tube end conditions allowing appreciable net flow change the bypass balance and can give other drag regimes.

When the tube radius is $a$ and the sphere radius is $a(1-ϵ)$, the minimum gap is $aϵ$ and its leading neck profile is $h=aϵ+z^2/(2a)$. Substitution $z=a\sqrt{2ϵ}t$ in $I_n^{\ast }=a^{n-1}\int h^{-n}dz$ gives the displayed coefficients. Constants change if the gap parameter is instead twice the radius deficit; the divergent powers do not.

For $h=H_0(1-x^2/a^2{)}^{1/6}$, elliptic coordinates make $h^3/\sin \eta$ constant on the upper crack face. Pressure continuity and harmonicity reduce the boundary condition to $F^{\prime }(0)=\alpha F(0)$ for the cosine mode, while uniform far flow gives $F\sim -e^{\xi }/2$. Solving $F^{\prime \prime }=F$ yields the displayed pressure. The midpoint crack flux is $2aU\alpha /(1+\alpha )$; small $\alpha$ leaves the background pressure gradient nearly unchanged, while large $\alpha$ makes the crack nearly equipotential.

For a bounded measurable function with positive essential supremum on a finite measure space, put $M_n=\int |f{|}^n$. The Cauchy-Schwarz inequality makes these moments log-convex, so successive ratios are nondecreasing. Their product satisfies $M_n/M_0\le (M_{n+1}/M_n{)}^n$, while each ratio is at most the essential supremum. Positive-measure level sets arbitrarily close to that supremum make $\mu (E{)}^{-1/n}\Vert f{\Vert }_n$ converge to it. Squeezing then proves the ratio limit.

Suppose $1\le p<q<r<\infty$, $\Vert f{\Vert }_p\le C_p$, $\Vert f{\Vert }_q\ge C_q>0$ and $\Vert f{\Vert }_r\le C_r$. Set $\epsilon =(C_q^q/(2C_p^p){)}^{1/(q-p)}$. The part of $\int |f{|}^q$ where $|f|\le \epsilon$ is at most ${\epsilon }^{q-p}{C}_p^p=C_q^q/2$. On the complementary level set, the Holder inequality bounds the integral by $C_r^q\mu \{|f|>\epsilon {\}}^{1-q/r}$, proving the bound. For $r=\infty$ the bound becomes $C_q^q/(2C_r^q)$. The lower-exponent bound prevents spreading into low amplitudes, while the higher-exponent bound prevents concentration into arbitrarily narrow spikes.

For $n\ge 2$ risk-neutral bidders with independent uniform distributions on $[0,1]$, the lowest-price auction has a symmetric Bayesian Nash equilibrium $b(v)=(n-1)v$. A deviation bidding $(n-1)z$, with $0\le z\le 1$, wins with probability $z^{n-1}$ and has expected payment $(n-1)z^n/n$. Its expected profit is $v{z}^{n-1}-(n-1)z^n/n$, whose derivative is $(n-1)z^{n-2}(v-z)$, proving optimality at $z=v$. Bids above $n-1$ give the same outcome as $z=1$. The seller receives $(n-1)$ times the minimum of $n$ independent uniform valuations, with expected revenue $(n-1)/(n+1)$.

// Target: mathematical-optimization.bigb

The concave biconjugate is an upper envelope of $F$, and equals $F$ for a proper closed concave function under the usual duality hypotheses. It equals $-(-F{)}^{\ast \ast }$ in terms of the ordinary biconjugate. If $e=D{F}_{\ast }(\tau )$, differentiability and concavity of $F_{\ast }$ imply $F_{\ast }(\tau )+F_{\ast \ast }(e)=\tau :e$. For a finite differentiable dual the equality follows because $\tau$ minimizes $s:e-F_{\ast }(s)$.

A lower bound in a partially ordered set which is greater than or equal to every other lower bound. For a pair it is written $a\wedge b$; an empty meet, when it exists, is the greatest element. In a complete lattice, every subset has a meet. In the open-set frame, this is the interior of the intersection, with finite meets given by the intersections themselves.

For a slow elastic layer above a faster half-space, put $q=\sqrt{c^2/{\bar{c}}_s^2-1}$ and $q_{\max }=\sqrt{c_s^2/{\bar{c}}_s^2-1}$. The positive branches of $\tan (khq)$ intersect the positive strictly decreasing traction ratio once whenever their left endpoint $m\pi$ is strictly below $kh{q}_{\max }$. The fundamental exists for every positive $kh$; the second requires the displayed strict inequality. At equality $c=c_s$ and the half-space decay exponent vanishes, so that endpoint is a cutoff, not a trapped mode satisfying decay at minus infinity.

For a slower surface layer supporting propagating SH-waves with vertical wavenumber $q>0$ above an evanescent half-space with decay exponent $p>0$, continuity of displacement field and traction gives the displayed reflection coefficient for a downward incident displacement field wave. Its modulus is one and its wave phase is $-2\arctan (\mu p/({\mu }_{0}q))$. A traction-free top reflects displacement field with coefficient $+1$. The round-trip condition $R_h{e}^{2iqh}=1$ then gives $\tan (qh)=\mu p/({\mu }_{0}q)$, equivalent to direct Love wave mode matching.