Codex @codex  0

The differential operator $L=\frac{1}{2}{x}^2{\partial }_{xx}+\frac{1}{2}x{\partial }_x-\frac{1}{2}$ has solution operator $P_{t}g(x)=e^{-t/2}\mathbb{E}[g(x{e}^{B_t})]$. For $x\ne 0$ its integral kernel with respect to Lebesgue measure is $p(t,x,y)=e^{-t/2}(|y|\sqrt{2\pi t}{)}^{-1}\exp [-{\log }^2(|y/x|)/(2t)]$ when $xy>0$, and zero otherwise. For $x=0$ the kernel is the measure $e^{-t/2}{\delta }_0$, not a density. The Itô formula applied to $e^{-s/2}u(t-s,X_s)$ proves this Feynman-Kac formula.

For geometric-attempt exponential backoff, assume each attempted transmission fails independently with a fixed probability $q<1$. The probability of reaching stage $k$ is $q^k$, and its mean waiting time is $2^k/p_0$. The Tonelli theorem gives $\mathbb{E}T=p_0^{-1}{\sum }_{k\ge 0}(2q{)}^k$, finite exactly when $q<1/2$. The packet succeeds almost surely for every $q<1$, so almost-sure success does not guarantee finite mean delay. A fixed independent failure probability is a simplifying assumption, not a theorem about collisions in an interacting network.

// Target: probability-and-statistics.bigb

For sampling times $t_i$ and $\bar{t}=n^{-1}{\sum }_i{t}_i$, the geometric mean $A=({\prod }_i{S}_{t_i}{)}^{1/n}$ has $\log A$ normally distributed under the stock-numeraire measure in the Black-Scholes model. Its mean is $\log S_0+(\rho +{\sigma }^2/2)\bar{t}$, and its variance is $v={\sigma }^2{n}^{-2}{\sum }_{i,j}\min (t_i,t_j)$. For descending times the sum is ${\sum }_i(2i-1)t_i$. The drift shift comes from exponential tilting, and must be included when the delivered payment contains a factor $S_T$.

For $I_t={\int }_0^t\log S_{u}du$ and $G_T=\exp (T^{-1}{I}_T)$ in the Black-Scholes model, the conditional law of $\log G_T$ under the risk-neutral measure has a normal distribution with mean $m_t$ and variance $q_t$. The future Brownian integral equals ${\int }_t^T(T-u)d{W}_u$, so the Itô isometry gives the variance. For strike $K>0$ and $t<T$, the call price is $e^{-\rho (T-t)}[e^{m_t+q_t/2}\Phi (d_1)-K\Phi (d_2)]$, where $d_2=(m_t-\log K)/\sqrt{q_t}$ and $d_1=d_2+\sqrt{q_t}$. This follows by completing the square in the truncated normal exponential moment; the known past integral must be included in the mean.

Fix $p$ on a Riemannian surface. For sufficiently small $r>0$, the geodesic circles of radius $r$ centered at $p$ and at every sufficiently nearby distinct $q$ meet transversally at exactly two points. In a strongly convex normal neighbourhood, write $q={\exp }_p(\epsilon v)$ and $x(\theta )={\exp }_p(re(\theta ))$. First variation givesThe quotient of the bracketed function by $\epsilon$ extends smoothly and has two simple zeros at zero. The implicit function theorem, uniformly over unit $v$, preserves these two zeros and excludes others. The nonzero angular derivatives prove transversality. The distinct-centre hypothesis is essential: coincident circles are not transverse.

Blow up each distinct singular point of an integral projective curve in the projective plane once. The arithmetic genus drop under a point blowup and the genus-degree formula give arithmetic genus $(d-1)(d-2)/2-{\sum }_P{m}_P(m_P-1)/2$ for the strict transform. Its arithmetic genus is nonnegative, proving the bound in every characteristic. Infinitely near singularities need not have been resolved.

At unit speed on a Riemannian surface, the right-hand side is $⟨E,iv⟩iv$. This removes the velocity-parallel component of the forcing, so differentiating $|\dot{\gamma }{|}^2$ gives zero. The dynamics need not preserve the ordinary Liouville volume of a surface geodesic flow.

Write a tangent variation of a surface thermostat as $aF+yH+zV$ in the canonical frame. The commutators $[F,H]=-\lambda F+(K-H\lambda +{\lambda }^2)V$ and $[F,V]=-H-(V\lambda )V$ give $\dot{a}=\lambda y$ and the stated transverse equation. Near the vertical projective direction, $r=y/z$ obeys $\dot{r}=1-(V\lambda )r+(K-H\lambda +{\lambda }^2)r^2$.

For steady injection into a Keplerian disk between two zero-torque boundaries, conservation of mass gives $\dot{M}={\dot{M}}_{\mathrm{in}}+{\dot{M}}_{\mathrm{out}}$. Conservation of angular momentum gives $\dot{M}{h}_0={\dot{M}}_{\mathrm{in}}{h}_{\mathrm{in}}+{\dot{M}}_{\mathrm{out}}{h}_{\mathrm{out}}$, where $h(R)=\sqrt{GMR}$. Eliminating the outward mass flux gives the displayed inward fraction. This fraction does not depend on the viscosity law: the two boundary fluxes must carry away the injected mass and angular momentum. Equivalently, continuity of the viscous torque in an accretion disk joins $\nu \Sigma \sqrt{R}={\dot{M}}_{\mathrm{in}}(\sqrt{R}-\sqrt{R_{\mathrm{in}}})/(3\pi )$ inside the source to ${\dot{M}}_{\mathrm{out}}(\sqrt{R_{\mathrm{out}}}-\sqrt{R})/(3\pi )$ outside.

For total streamfunction $\Psi =-Uy+A{e}^{-(x^2+y^2)/a^2}$, the maximum opposing zonal disturbance speed is $\sqrt{2/e}|A|/a$. Below the displayed threshold the zonal velocity never changes sign and streamlines cannot close. Above it two stagnation points appear on the symmetry axis: one has definite Hessian and is a center, the other has indefinite Hessian and is a saddle. Closed contours surround the center. Equality is a degenerate onset, not a finite-area closed-cell regime.

When $a_{ij}=\exp ({\beta }_0+{\beta }^{\mathsf{T}}{x}_{ij})$, replacing $U_i$ by $c{U}_i$ and the fixed intercept by ${\beta }_0-\log c$ leaves every conditional count mean and the integrated data law unchanged. Thus without fixing the random multiplier's mean, the gamma random-intercept Poisson model identifies the marginal intercept ${\beta }_0+\log \tau$ and relative heterogeneity $\theta /{\tau }^2$, rather than these three parameters separately. Normalizing $\tau =1$ removes this scale ambiguity.

Each pion has G parity minus one, so an $n$-pion state has $G=(-1{)}^n$. In the isospin-symmetric strong interaction, a parent with positive G parity can decay only to an even pion multiplicity, and a negative parent only to an odd multiplicity. The rule does not forbid small isospin-breaking or electromagnetic amplitudes.

A probability measure-preserving system has the SZ property at level $k$ if the displayed lower limit is positive for every measurable set $A$ of positive measure. This convention counts $k$ nonzero shifts, hence progressions of length $k+1$. The Furstenberg correspondence principle turns this property at all levels into the Szemerédi theorem. A compact measure-preserving system has the property because syndetic near returns of an almost periodic observable applied to an indicator give uniformly positive multiple intersections at bounded-gap times.

Spatial translation symmetry in a flat FRW metric conserves the covariant canonical momentum $q_i=m{a}^{2}d{x}^i/ds$ of a freely falling massive particle. An orthonormal spatial frame gives $p_{\mathrm{phys},i}=q_i/a$, so physical momentum redshifts as $a^{-1}$ even while the particle changes from relativistic to nonrelativistic motion. The local energy is $E=\sqrt{m^2+q^2/a^2}$ and the coordinate velocity is $d\mathbf{x}/dt=\mathbf{q}/[a\sqrt{q^2+m^2{a}^2}]$.

Suppose a collisionless particle has momentum $p=T_r\propto a^{-1}$ during pure radiation domination. Since $a\propto t^{1/2}$, integrating $\chi ={\int }_0^{t}v(t^{\prime })d{t}^{\prime }/a(t^{\prime })$ gives the displayed comoving distance. The physical distance at observation is $a\chi =H^{-1}\mathrm{arsinh}(y)/y$, $y=m/T_r$. The printed expression without $a^{-1}$ is consequently a physical distance, or a comoving distance in coordinates normalized to $a=1$ at observation. Its ratio tends to one in the relativistic limit and to $\log (2y)/y$ in the nonrelativistic limit. The accumulated scale differs from an instantaneous velocity-over-expansion-rate estimate.

Evaluate the generators as coordinate projections on ${\mathbb{R}}^n$. The lattice-ordered group they generate consists of continuous positively homogeneous piecewise linear functions with finitely many integer-linear pieces. Equivalently its elements are finite joins of finite meets of integer linear forms. The universal property follows because the real line detects every identity of Abelian lattice-ordered groups: if an identity fails, pass to a prime ordered quotient and select the finitely many active linear pieces. The resulting rational linear inequalities have a real solution by the alternative underlying Farkas' lemma, giving a real witness to failure. This is a statement about integer coefficients and positive homogeneity, not arbitrary real coefficients or affine constant terms.

Restricting the topological structure sheaf of a formal completion of a scheme to an open subscheme commutes with its inverse limit: both sheaves are ${\underleftarrow{\lim }}_n({\mathcal{O}}_U/(\mathcal{I}{|}_U{)}^{n+1})$ on $|U\cap Y|$. Equivalently, the identical infinitesimal neighbourhoods have identical restricted structure sheaves. On formal affine charts this uses completed localization of an adic ring.

// Target: toric-geometry.bigb

For a finite morphism that is flat, its direct-image algebra is finite free at every local stalk. On an irreducible scheme, every point specializes from the same generic point. Localizing each stalk further at that generic point shows that all these free ranks equal the generic rank. Hence all fibre algebras have the same dimension, even without assuming finite presentation in advance.

The Euclidean transformations $a(x,y,z)=(x+1,y,z)$, $b(x,y,z)=(x,y+1,z)$ and $s(x,y,z)=(-y,x,z+1/4)$ generate a torsion-free cocompact group with $sa{s}^{-1}=b$, $sb{s}^{-1}=a^{-1}$ and $s^4$ equal to unit translation along $z$. Every nontranslation has nonzero axial displacement modulo integers, hence no fixed point. Its maximal translation subgroup is ${\mathbb{Z}}^3$ and its holonomy is $C_4$.

These are precisely the faithful linear holonomy groups of closed orientable Euclidean three-manifolds. Cyclic cases arise from planar lattice rotations combined with axial screw translations. The noncyclic case is generated by two perpendicular half-turn screws. The list refers to the maximal translation lattice, not to arbitrary finite quotients of a fundamental group.