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For independent $X\sim \mathrm{Gamma}(m,\psi \lambda )$ and $Y\sim \mathrm{Gamma}(n,\lambda )$ in shape-rate notation, $T=X/Y$ has density proportional to ${\psi }^m{t}^{m-1}(1+\psi t{)}^{-(m+n)}$. Its distribution is nuisance-free in lambda. The nuisance maximum is $(m+n)/(\psi X+Y)$, so substituting into the full likelihood gives the same interest-parameter log-likelihood up to a parameter-free term. Equality here is special to the common-scale structure, not a general equality of profiling and marginalization.

For parameter of interest $\psi$, constrained nuisance fit ${\hat{\lambda }}_{\psi }$, and suitable ancillary sample coordinates $(\hat{\psi },\hat{\lambda },a)$, this adjustment includes both nuisance observed information and a sample-space Jacobian determinant. The derivative holds $\hat{\psi },a$ fixed. Implicit differentiation of the constrained nuisance score function gives the equivalent expression ${\ell }_p+\frac{1}{2}\log |j_{\lambda \lambda }|-\log |{\ell }_{\lambda ;\hat{\lambda }}|$. The semicolon distinguishes data-coordinate differentiation from parameter differentiation. Omitting the Jacobian requires an additional justified simplification; the information adjustment alone is not universal.

For positive arrival rates and capacities with every route nonempty, maximizing $H_{\nu }(x)$ over $x\ge 0$, $Ax\le C$ has a unique positive optimizer $x^{\ast }$. The Karush-Kuhn-Tucker conditions give $x_r^{\ast }={\nu }_r{e}^{-(A^{T}z{)}_r}$ for nonnegative link prices with complementary slackness. Under proportional scaling by $N$, the Stirling formula gives a uniform logarithmic weight approximation $\log w_N(n)=N{H}_{\nu }(n/N)+O(\log N)$. Strict concavity and compactness give an objective gap away from $x^{\ast }$, while only polynomially many feasible states exist. Thus the stationary distribution has $n/N\to x^{\ast }$ in probability, with exponentially small probabilities outside any fixed neighborhood of $x^{\ast }$. In particular $\mathbb{E}[n]=N{x}^{\ast }+o(N)$.

Let $Z(C)$ sum the unnormalized occupancy weights ${\prod }_r{\rho }_r^{n_r}/n_r!$ over feasible counts. The states admitting another type-$r$ call are exactly those feasible under capacities $C-A_r$. Poisson arrivals see time averages therefore gives the displayed blocking probability. Take $Z$ to be zero when the reduced capacities make its feasible set empty.

Use the determinant-normalized slash operator and $W_M=\left(\begin{array}{cc}0& -1\\ M& 0\end{array}\right)$. For units $u,v$ with $Nuv\equiv -1(\mathrm{mod}D)$, the matrix identity $U_u{W}_{N{D}^2}=D{\gamma }_u{W}_N{U}_v$ has ${\gamma }_u=\left(\begin{array}{cc}(1+Nuv)/D& u\\ Nv& D\end{array}\right)\in {\Gamma }_0(N)$. A positive scalar matrix acts trivially. The nebentypus supplies $\psi (D)$; reindexing gives $\overline{\chi }(u)=\chi (-N)\chi (v)$. Thus the factor is initially $\psi (D)\chi (-N)\tau (\chi )/\tau (\overline{\chi })$. The primitive Gauss sum of a Dirichlet character identity $\tau (\chi )\tau (\overline{\chi })=\chi (-1)D$ gives the displayed equivalent factor. No assumption that $f$ itself is a Fricke eigenform is needed.

For categorical presheaves $P,Q$ on a small category, restrictions in the displayed exponential object act by precomposition with $H_u\times 1_P$. Evaluation sends $(\alpha ,x)$ at $c$ to ${\alpha }_c(1_c,x)$. The currying of $t:R\times P\to Q$ sends $z\in R(c)$ to the natural transformation whose value on $(v:a\to c,x\in P(a))$ is $t_a(R(v)z,x)$. Evaluation and currying are inverse and natural, proving the Cartesian closed category structure.

If every slice category $\mathcal{C}/A$ is essentially small, the displayed exponential object and the set of sieves on a category at each $A$ are small even when $\mathcal{C}$ is large. Together with pointwise finite limits, they give the elementary-topos constructions in a universe admitting the large diagrams. This does not force the entire functor category to be locally small in the original universe: a large discrete category has terminal slices, but maps between its constant-one and constant-two presheaves are arbitrary class-indexed binary families.

For a nonsingular multivariate normal distribution with precision matrix $\Omega$, fixing all coordinates except $X_i$ leaves a density proportional to $\exp [-{\Omega }_{ii}(x_i-m_i{)}^2/2]$ after completing the square. Thus its conditional variance is the reciprocal diagonal precision entry. This is conditioning on every other coordinate, not the marginal variance $({\Omega }^{-1}{)}_{ii}$.

For a continuously differentiable curl-free vector field $F$ on a domain star-shaped about the origin, define $\Phi$ by the displayed integral. Differentiation and ${\partial }_i{F}_j={\partial }_j{F}_i$ give ${\partial }_i\Phi ={\int }_0^1[F_i(tx)+t{x}_j{\partial }_j{F}_i(tx)]dt={\int }_0^1{\partial }_t[t{F}_i(tx)]dt=F_i(x)$. Thus $F=\nabla \Phi$. To adopt the sign convention $F=-\nabla \varphi$, take $\varphi =-\Phi$. The star-shaped-domain hypothesis ensures every segment used in the construction lies in the domain; curl-free fields need not have global potentials on arbitrary domains.

Two positive bumps of an order automorphism are conjugate in the full order automorphism group of the real line exactly when their supports of a permutation have the same endpoint type: both endpoints finite, only the left endpoint finite, only the right endpoint finite, or neither endpoint finite. The actual finite endpoint locations impose no further restriction.

To construct the conjugacy on the supports, choose $x$ in the support of $f$ and $y$ in the support of $g$. Choose an increasing bijection $b_0:[x,xf)\to [y,yg)$ matching the endpoint limits, using the right-action convention $xf=f(x)$. Extend it byThe iterated fundamental intervals partition each support, so this defines an increasing bijection satisfying $fb=bg$. If the endpoint types agree, extend $b$ over the corresponding complementary rays, where both bumps are the identity. This gives a global order automorphism with $b^{-1}fb=g$.

Endpoint type is necessary because a global increasing bijection of the real line preserves finite endpoints and the two directions of infinity. For example, the translation $f(x)=x+1$ has no fixed points, while a positive bump supported on $(0,1)$ fixes every point outside that interval. They cannot be conjugate. Within the bounded-support case, the construction shows that interval length and endpoint positions do not distinguish conjugacy classes.

For $F(n)=n(n+2)$, every odd prime has two forbidden residue classes. Use Selberg sieve weights supported on odd squarefree integers $d\le R=x^{1/8}$. Their main quadratic form is $1/J$, where $J={\sum }_{d\le R}{\prod }_{p\mid d}2/(p-2)$. The truncated Euler-product lower bound applied to primes up to $R^{1/8}$ gives $J\gg {\log }^{2}R$: the mean logarithmic divisor size is $2{\sum }_{3\le p\le R^{1/8}}(\log p)/p<\frac{1}{2}\log R$, by the Mertens first theorem, and the full product is $\asymp {\log }^{2}R$, by the Mertens second theorem. The optimal Selberg weights have modulus at most one, so the remainder is at most $O({\sum }_{d,e\le R}[d,e])=O(R^4)$. The finitely many possible pairs with $p\le R$ contribute $O(R)$. Thus the count is $O(x/J+R^4+R)$, proving the bound.

For the polynomial $F(n)=n(n+2)$, take $\rho (2)=1$ and $\rho (p)=2$ at odd primes. On odd squarefree integers, the summand is at least $2^{\omega (d)}/d$. Put $y=\sqrt{z}$ and $L(y)={\sum }_{a\le y,a\text{ odd}}1/a=\frac{1}{2}\log y+O(1)$. The weight of odd pairs $a,b\le y$ is $L(y{)}^2$. The pairs for which $a$ is not squarefree, $b$ is not squarefree, or $(a,b)>1$ each have weight at most $L(y{)}^2{\sum }_{p\text{ odd}}{p}^{-2}$. Indeed, the weighted count of multiples of an odd $p^k$ is $p^{-k}L(y/p^k)\le p^{-k}L(y)$. Moreover,Hence the odd coprime squarefree pairs have weight at least $L(y{)}^2/6$. Their products $d=ab\le z$ are squarefree, and $2^{\omega (d)}$ counts all ordered coprime factorizations of $d$. This proves the displayed lower bound needed for the Selberg upper-bound sieve.

A binary form is a homogeneous polynomial in two variables. Degree-$d$ binary forms give the symmetric-power representation of the standard two-dimensional representation of $S{L}_2$, after using its invariant alternating form to identify the standard representation with its dual.

For the standard normal density, $f^{\prime \prime }(x)=(x^2-1)f(x)$ and $R(f^{\prime \prime })=3/(8\sqrt{\pi })$. The fifth power of the bandwidth ratio is $3\sqrt{2}{e}^{x^2/2}/[8(x^2-1{)}^2]$. Its global minima occur at $x=\pm \sqrt{5}$; $x=0$ is only the minimum within the interval between the two inflection points. The ratio diverges as $x$ approaches either inflection point.

The Scott continuous maps from $D$ to $E$, ordered pointwise. A nonempty directed family has pointwise supremum, and the constant-bottom function is least. Joint evaluation is Scott continuous: for a directed family of pairs $(f_i,x_i)$, directedness makes diagonal values cofinal among all $f_i(x_j)$. Evaluation and currying make the category of pointed complete partial orders and non-strict Scott continuous maps a Cartesian closed category.

For Scott continuous maps $e:D\to E$, $p:E\to D$, these two identities say that $D$ is embedded in $E$ and that projection followed by embedding approximates from below. Such pairs lift to endomorphism spaces by $f\mapsto efp$ and $g\mapsto pge$. Their composite on the smaller space is the identity, and the other composite is $g\mapsto (ep)g(ep)\le g$.

Zero Euler characteristic and $H_3=0$ give free rank $b_1-1$. The universal coefficient theorem for cohomology identifies $H^3(M;\mathbb{Z})=\mathbb{Z}/2$ with $\mathrm{Ext}(H_2(M;\mathbb{Z}),\mathbb{Z})$, so the only torsion in $H_2$ is one copy of $\mathbb{Z}/2$. In particular $b_1\ge 1$.

For a closed oriented manifold, the rational middle-dimensional cup-product pairing is nondegenerate by Poincare duality. Its degree is odd, so graded commutativity makes it skew-symmetric. A nondegenerate skew-symmetric form has even dimension. In particular a closed oriented six-manifold cannot have a rank-one third cohomology group.

// Target: algebra.bigb

A thin disk with alpha disk viscosity $\nu \sim \alpha H^2\Omega$ has a viscous timescale much longer than its dynamical infall time in the plunging region of a black-hole accretion disk. At fixed nonzero accretion rate, rapid inflow reduces its surface density of a disk. The resulting viscous torque in an accretion disk is smaller by the displayed factor than in the slowly drifting circular disk. This estimate concerns radii outside the narrow matching layers.

For a high-sound-speed, low-density atmosphere over a denser stellar envelope, set ${\omega }^2=gk(1-\epsilon )$ in the plane-parallel Cowling displacement-pressure equations. Define $I_a={\int }_{-\infty }^{z_0}{e}^{2k(z-z_0)}{\rho }^{-1}dz$ and $I_b={\int }_{z_0}^{\infty }{e}^{-2k(z-z_0)}\rho dz$. The lower leading solution has $\Pi /\xi =-2\epsilon gk{I}_b$ at the match. Above, write $\rho =\epsilon \hat{\rho }$, $\Pi =\epsilon \psi$, $\xi =\mu \psi$. The upward-decaying solution satisfies ${\mu }^{\prime }+2k\mu =-k/(g\hat{\rho })$, so $\mu (z_0)=-(k/g)\epsilon I_a$. Writing $f=\Pi /(\epsilon \xi )$ below, continuity of displacement and Lagrangian pressure requires $\mu (z_0)f(z_0)=1$, giving the displayed condition. The integrals must converge and the upper displacement must actually decay. For two constant-density incompressible layers it gives $\epsilon \simeq 2{\rho }_a/{\rho }_b$, the expansion of the exact interfacial gravity-wave dispersion relation.