There are 21 incident point-line pairs and 28 nonincident pairs in the Fano plane, with $G{L}_3(2)$ transitive on each type. Orbit-stabilizer theorem gives the two orders as $168/21=8$ and $168/28=6$.

The Fano plane duality group includes both incidence-preserving permutations of points and projective lines and the dualities that exchange those two types. Its type-preserving subgroup is $G{L}_3({\mathbb{F}}_2)$, of order $(8-1)(8-2)(8-4)=168$. A polarity supplies the inverse-transpose automorphism $A\mapsto (A^{-1}{)}^{\mathsf{T}}$, giving the semidirect productThere are $7\cdot 3=21$ unordered incident point-line pairs and $7\cdot 4=28$ unordered nonincident point-line pairs. The two transitive actions therefore have stabilizers of orders $336/21=16$ and $336/28=12$, respectively. These are dihedral groups. The normalizer of a Sylow subgroup of order seven has order42 and is $C_7⋊C_6$.

Every maximal proper subgroup is a conjugate subgroup of one of these three types or of the index-two collineation subgroup. The extension is also isomorphic to $PG{L}_2(7)$, identifying the geometric stabilizers with the standard finite-group classification.^[ref]

In multiple hypothesis testing, let $R$ count all rejections and $V$ count rejected true null hypotheses. The false discovery proportion is the random variable $V/(R\vee 1)$, defined to be zero when $R=0$. Its expectation is the false discovery rate. Distinguishing the realized proportion from its expectation matters: controlling the false discovery rate does not by itself give a high-probability bound on the realized proportion.

Expanding the ghost covariant derivative in Yang-Mills theory produces an interaction between the Faddeev-Popov ghost field, Faddeev-Popov antighost field, and gauge boson. For the displayed kinetic convention, Fourier factors $e^{-ipx}$ and all momenta incoming give vertex $-g{f}^{abc}{k}_{\mu }$, where $k$ is the antighost momentum. The structure constants encode the Adjoint representation. Propagator, vertex, Fourier, and ghost-order conventions must be chosen consistently.

With signature $(+---)$ and action $-\int {\partial }^{\mu }\bar{c}{\partial }_{\mu }c$, integration by parts gives the quadratic kernel $K_c=\square$, or $-p^2$ in Fourier space. The Grassmann Gaussian integral with odd sources gives $Z=\exp [-i\bar{\eta }{K}_{c,F}^{-1}\eta ]$. Left and right source derivatives, including their insertion factors, yield $⟨Tc\bar{c}⟩=i{K}_{c,F}^{-1}$ and hence the displayed sign. Replacing $\bar{c}$ by $-\bar{c}$ changes this propagator and the ghost-gluon vertex together; it is a convention change, not a physical discrepancy.

A ghost propagator is the two-point function $⟨Tc(x)\bar{c}(y)⟩$ of a Faddeev-Popov ghost field and its independent Faddeev-Popov antighost field. For quadratic action $\int \bar{c}Mc$ and source ordering consistent with that field ordering, the Grassmann integral gives $G_c=i{M}_F^{-1}$. The subscript specifies the causal pole prescription. For the negative derivative kinetic term $-\int \partial \bar{c}\cdot \partial c$, the Fourier kernel is $M=-p^2$. With signature $(-,+,+,+)$ this gives $G_c=-i/(p^2-i0)$. Field normalization multiplies the inverse kernel accordingly, while an antighost sign redefinition changes propagator and vertex signs together. Anticommuting statistics and loop signs do not turn these auxiliary fields into physical asymptotic particles.

Assume an invariant volume measure, an invariant integrand and one regular gauge slice representative per orbit. At that representative, let $M$ be the Jacobian matrix of $F$ along the group coordinates, whose Haar measure is normalized to ordinary coordinate measure at the identity. Changing coordinates from the group parameter to $F$ gives ${\int }_{G}d\mu (g)\delta (F(x_g))=1/|\det M(x_0)|$. Multiply by its reciprocal and insert this identity into the original integral. Invariance permits a change of variables $x\mapsto x_g$ for each $g$, factoring out the group volume. The absolute determinant is required for ordinary unoriented real integration; a consistently positive orientation permits writing $\det M$. Infinite group volume is divided out formally or treated with a regulator.

For real polynomial $f(x)=c{x}^n+\cdots$, $c\ne 0$, the zero mode of $H=(p+if)(p-if)/(2m)$ is normalizable on $\mathbb{R}$ exactly when $n$ is odd and $c>0$. Its squared modulus has leading exponential $\exp [-2c{x}^{n+1}/((n+1)\hslash )]$. Odd $n$ gives the same tail sign at both ends; positivity of $c$ makes both tails decay. This kernel is one-dimensional because $(p-if)\psi =0$ is a first-order differential equation. On the operator domain where integration by parts is valid, the energy form is $\Vert (p-if)\psi {\Vert }^2/(2m)\ge 0$.

At fixed sphere rotation in an annular Stokes flow, add rigid particles with zero net force and torque. Extend their actual rigid velocities through their interiors. The extension is an admissible incompressible competitor in the particle-free annulus, with zero interior rate-of-strain tensor. The Minimum-dissipation theorem for Stokes flow gives $D\ge D_0$. The boundary-work identity gives $D=\mathbf{\Omega }\cdot \mathbf{G}$, because the stationary outer wall and the freely moving particles contribute no power. Equality would require the extension to equal the unique particle-free flow, whose strain cannot vanish on a finite particle volume when $\Omega \ne 0$. Thus the resistance component along the imposed rotation strictly increases.

On a compact oriented Riemannian manifold without boundary, the Generalized Stokes theorem applied to $f{\iota }_X{\omega }_g$ gives the displayed identity. The product expansion is $d(f{\iota }_X{\omega }_g)=[X(f)+f\mathrm{div}X]{\omega }_g$. With boundary, the omitted boundary integral is ${\int }_{\partial M}f{\iota }_X{\omega }_g$.

The linearization at a travelling wave $w_c$ is $A={\partial }_z^2+c{\partial }_z+k-2w_c$. Conjugation by $e^{\gamma z}$ changes this to ${\partial }_z^2+(c-2\gamma ){\partial }_z+{\gamma }^2-c\gamma +k-2w_c$. The choice $\gamma =c/2$ removes the first derivative and makes the operator self-adjoint. Its two limiting potentials are $-k-c^2/4$ and $k-c^2/4$, so essential-spectrum invariance under relatively compact perturbations gives ${\sigma }_{\mathrm{ess}}=(-\infty ,k-c^2/4]$. This lies strictly below zero for $c>2\sqrt{k}$ but touches zero at the critical speed. Stability of this essential spectrum alone is not a general nonlinear stability theorem.

For a small category, this formula defines the exponential object in its covariant set-valued functor category. An arrow $c\to d$ acts by precomposition with $\mathcal{C}(d,-)\to \mathcal{C}(c,-)$. Evaluation applies a natural transformation to $(1_c,z)$; currying a map $P\times G\to H$ sends $p\in P(c)$ to the transformation $(f,z)\mapsto \theta (P(f)p,z)$. The formulas prove the universal property directly, without an adjoint existence theorem.

For a real standard Gaussian random vector $Z$, a real symmetric matrix $Q$, and $I-2Q$ strictly positive,Diagonalize $Q$ and complete the square in each coordinate. Failure of strict positivity makes the positive exponential integral divergent. The same formula extends to a self-adjoint trace-class quadratic form of an isonormal Gaussian process, with square-summable linear coefficients and a Fredholm determinant. A linear-plus-quadratic Gaussian random field functional is reduced to this form by its covariance square root.

Let $S_n$ have integrable, mean-zero independent and identically distributed random variables as increments, taking integer values at most one, with $\mathbb{P}(X_1=1)>0$. For a positive integer $b$, put $T_b=\inf \{n:S_n=b\}$ and $M(\tau )=\mathbb{E}{e}^{\tau X_1}$, $\tau >0$. Then $T_b$ is finite almost surely. Indeed, $b-S_{n\wedge T_b}$ is a nonnegative martingale; finite convergence on $\{T_b=\infty \}$ would force its integer increments eventually to vanish, contrary to the Borel-Cantelli lemmas applied to $\{X_n=1\}$.

The exponential martingale of a random walk $e^{\tau S_n}/M(\tau {)}^n$ stopped at $T_b$ is bounded by $e^{\tau b}$ because $S_{n\wedge T_b}\le b$ and $M(\tau )\ge 1$. The dominated convergence theorem therefore preserves its mean one at the limit and proves the displayed first-passage time transform. This argument accommodates arbitrarily large negative jumps; the restriction concerns upward jumps. For a simple symmetric random walk, $M(\tau )=\cosh \tau$, giving $\mathbb{E}{e}^{-\alpha T_b}=(e^{\alpha }-\sqrt{e^{2\alpha }-1}{)}^b$.

For a random walk with independent and identically distributed random variables as increments, let $M_n=e^{\lambda S_n}/\varphi (\lambda {)}^n$ define its finite-time exponential tilting. Under the tilted law, $L_n=M_n^{-1}$ is a martingale: the conditional mean of $\varphi (\lambda )e^{-\lambda X_{n+1}}$ is one. For a stopping time $T$, partition $\{T\le n\}$ into $\{T=j\}$ and apply the tower property of conditional expectation to obtain the displayed identity. Only finite-time absolute continuity of measures is needed; there need not be an infinite-horizon likelihood-ratio density.

The strong law for Brownian motion gives $B_s/s\to 0$. For every $0<\epsilon <\mu$, eventually $e^{(2\mu -2\epsilon )s}\le e^{2B_s+2\mu s}\le e^{(2\mu +2\epsilon )s}$. Integrating these bounds, taking logarithms and dividing by $t$ squeezes the growth rate between $2\mu -2\epsilon$ and $2\mu +2\epsilon$. Let $\epsilon \downarrow 0$. The corresponding inverse clock grows like $(\log t)/(2\mu )$.

A packet with $k$ previous collisions attempts independently in each slot with probability $p_k=p_0{2}^{-k}$. On failure it advances to stage $k+1$, and on success it leaves. The Markov chain state records the number of packets at every stage, rather than only the total backlog. This memoryless-window model is related to binary exponential backoff, but its geometric waiting times differ from uniform counters in finite retry windows.

// Target: probability-and-statistics.bigb

For $d\sigma =-\sigma dt/\tau +\sqrt{\kappa }dW$ with ${\sigma }_0=0$, the explicit stochastic integral is ${\sigma }_t=\sqrt{\kappa }{\int }_0^t{e}^{-(t-u)/\tau }d{W}_u$. The Itô isometry on the overlap of the two integration intervals gives the displayed covariance. The second exponential records the zero initial condition; removing it gives the stationary Ornstein-Uhlenbeck process covariance.

The logarithmic coordinate requires the displayed chain-rule factors for option delta and option gamma. If only a value error bound $\Vert e_h{\Vert }_{\infty }=O(h^2)$ is known, central differentiation gives $D_x{e}_h=O(h)$ and $D_{xx}{e}_h=O(1)$. Thus first and second Greek convergence do not follow with the same order from a value bound alone. With a smooth spatial error expansion $e_h=h^{2}E+o(h^2)$ and corresponding derivative control, both Greeks can instead retain second-order convergence away from a nonsmooth payoff or boundary. Refinement and Greek-specific regularity are essential to distinguish these situations.

Constant extra light makes an exoplanet transit appear shallower. If the host star supplies out-of-transit observed radiative flux $F_{\star }$ and an unresolved source supplies $F_c$, let $D=F_c/F_{\star }$. Then ${\delta }_{\mathrm{obs}}={\delta }_{\mathrm{host}}/(1+D)$. For $D=1$, a true exoplanet transit depth of $0.010$ appears as $0.005$; ignoring transit dilution gives a radius ratio $\sqrt{0.005}\simeq 0.07071$ instead of $0.10$, under the same uniform-disc assumptions.