Sampling order-three Cardinal B-splines $N_j$ at $x_i=i+3/2$ gives $N_i(x_i)=3/4$, $N_{i-1}(x_i)=N_{i+1}(x_i)=1/8$, and all other entries zero. The B-spline collocation matrix therefore has row strict diagonal dominance margin at least $1/2$, giving inverse operator norm at most two. More precisely, with $q=3-2\sqrt{2}$, its absolute inverse row sums are $v_i=2[1-(q^i+q^{n+1-i})/(1+q^{n+1})]$. To see this, change signs by $D_{ii}=(-1{)}^i$: $DAD$ has negative off-diagonal entries and a nonnegative inverse given by a convergent Neumann series. Its inverse row sums solve $6v_i-v_{i-1}-v_{i+1}=8$, $v_0=v_{n+1}=0$. Thus $\Vert A^{-1}{\Vert }_{\infty }=v_{\lfloor (n+1)/2\rfloor }<2$, tending to two as the dimension grows.

If $X\stackrel{f}{\to }Y\stackrel{g}{\to }Z$ has proper composite and $g$ is separated, the graph $X\to X{\times }_{Z}Y$ is a closed immersion, being a base change of the diagonal of $g$. The projection $X{\times }_{Z}Y\to Y$ is a base change of the proper composite. Their composition is $f$, so $f$ is proper.

Suppose $gf$ is proper, $f$ is surjective, and $g$ is separated and of finite type. Surjectivity of scheme morphisms is preserved by base change: a nonempty fiber remains nonempty after a residue-field extension. After any base change to $Z$, for a closed subset $C$ of the new target of $f$, its inverse image is closed and has image $g(C)$ under the proper composite. Thus $g(C)$ is closed. This proves universal closedness of $g$, completing the three conditions for properness.

For a finite-type affine morphism of Noetherian schemes, apply its valuative lifting property to each domain quotient $B/\mathfrak{q}$ for a minimal prime and all valuation rings of its fraction field containing the image of $A$. The lift forces $B/\mathfrak{q}$ into every such valuation ring, hence makes it integral over that image. For each $b\in B$, multiply the finitely many resulting monic equations. Its value at $b$ lies in the nilradical. The nilradical of a Noetherian ring is nilpotent, so a power of this monic polynomial annihilates $b$. Thus $B$ is integral over $A$. Finitely many algebra generators, each satisfying a monic equation, have their powers reduced to finitely many monomials, proving that $B$ is a finite $A$-module. Nilpotents cannot be ignored in this proof.

A variety over a field $k$ is complete when its map to $\mathrm{Spec}k$ is universally closed: after any base change, projection sends closed subsets to closed subsets. For a separated finite-type variety, this is equivalent to being proper. In the classical algebraically closed setting one may test projections $X\times Y\to Y$ for arbitrary varieties. Images of rational points over a non-algebraically-closed field must not replace geometric images.

Over ${\mathbb{F}}_5$, putThese determinant-one matrices satisfy $i^2=j^2=-I$, $ij=-ji$, and $r^3=I$. Conjugation by $r$ cycles $i,ij,j$. Thus $⟨i,j⟩$ is a quaternion group of order eight, normalized by an element of order three. Their generated subgroup has order $24$ in $S{L}_2(5)$, hence order $12$ after quotienting by $\{\pm I\}$. Its index-five coset action, together with simplicity and perfectness, identifies $PS{L}_2(5)$ with the alternating group $A_5$.

For two projectively equivalent connections that are Levi-Civita connections, contraction of their difference gives $(n+1)V_c={\bar{\Gamma }}^a{}_{ac}-{\Gamma }^a{}_{ac}$. The Jacobi formula gives ${\Gamma }^a{}_{ac}={\partial }_c\log \sqrt{|\det g|}$. Subtraction proves the displayed expression. The two determinant densities transform by the same coordinate Jacobian factor, so their ratio is a scalar and its derivative a covector. The formula is a necessary consequence of projective equivalence, not a test asserting that arbitrary metric tensors have identical geodesics.

Let $\tau$ be an affine parameter for a geodesic of $\nabla$ and suppose ${\bar{\nabla }}_{X}Y-{\nabla }_{X}Y=V(X)Y+V(Y)X$. Then ${\bar{\nabla }}_{\dot{\gamma }}\dot{\gamma }=2V(\dot{\gamma })\dot{\gamma }$. For the new velocity $d\gamma /dt=\dot{\gamma }/t^{\prime }$, the acceleration is $(t^{\prime }{)}^{-2}[{\bar{\nabla }}_{\dot{\gamma }}\dot{\gamma }-(t^{\prime \prime }/t^{\prime })\dot{\gamma }]$, which vanishes under the displayed differential equation. Its nonzero solution $t^{\prime }=C\exp (2\int V(\dot{\gamma })d\tau )$ changes the parameter while preserving the curve.

The logarithm of the profile likelihood is the full log-likelihood evaluated at the nuisance fit constrained to the specified parameter of interest. It preserves likelihood ratios and maximizers, but is not generally a normalized sampling log-density. Conditioning and integrating out nuisance parameters are different operations.

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.