In a full regular minimal exponential family, a finite maximum-likelihood estimator exists precisely when the observed sufficient-statistic mean is in the interior of the convex support of an exponential family. It is unique because the log likelihood is strictly concave. For an interior mean, positive base-measure mass near finitely many support points surrounding the mean makes $\kappa (\theta )-{\theta }^T\bar{Y}$ grow at least linearly as $\Vert \theta \Vert \to \infty$. At a finite boundary of the full open natural domain, the normalizing integral diverges. Thus the likelihood attains its maximum in the domain. On a boundary face, no finite parameter can have the required mean. A regular restriction should instead use the exact mean-image criterion. These full-family conventions are also discussed in Geyer's exponential-family lecture notes.

For an adjunction $F⊣G$, applying $G$ to its standard counit presentation gives $T^{2}GD\rightrightarrows TGD\to GD$, split by the monad units. Reflection therefore makes every counit presentation a coequalizer and gives full faithfulness by descent. Conversely, monad algebra forgetful functor creates split coequalizers makes an existing G-split quotient a coequalizer in the Eilenberg-Moore category. Fullness and faithfulness of the comparison lift its universal factorization back to the original category.

A quasistatic approximation to the solute field neglects its perturbation time derivative while retaining the interface motion in the solute balance. For steady pulling, a lateral wavenumber $k$ then has decaying exponent $q=(V+\sqrt{V^2+4D^2{k}^2})/(2D)$. It is justified when the interface growth time is long compared with the relevant solute diffusion time. Fixing the thermal field does not by itself impose this solutal timescale ordering.

In an equal-density, no-slip lubrication approximation, repulsive disjoining pressure $A/(6\pi d^3)$ sustains a premelting film of thickness $d$ beneath a rigid disk of radius $a$. With imposed radial undercooling $\Delta T(1-r^2/a^2)$, local thermomolecular pressure is $P_T=\rho L\Delta T(1-r^2/a^2)/T_m$, and $d^3=A/(6\pi P_T)$. A constant vertical lifting rate $U$ requires inward volume flux $q_r=-Ur/2$. Integrating the resulting lubrication pressure and disk force balance gives the displayed rate, where $W$ is the apparent load, $\mu$ is liquid dynamic viscosity, $L$ is specific latent heat, and $T_m$ is the absolute bulk melting temperature. Positive lifting requires $W<\pi a^2\rho L\Delta T/(2T_m)$. For fixed positive load, the rate rises from zero at that threshold towards $A/(6\pi \mu a^2)$ as undercooling increases. That plateau is an asymptotic limit within this constitutive model, not a claim that continuum-film physics persists at arbitrarily low temperatures.

An odd-order finite group has only odd irreducible degrees, by irreducible character degree divides the group order. Its nontrivial characters occur in distinct conjugate pairs, by the Frobenius-Schur indicator. Therefore $|G|-k(G)={\sum }_{\chi }(\chi (1{)}^2-1)$ is a sum of pairs $2(d^2-1)$ with $d$ odd. Since $d^2\equiv 1(\mathrm{mod}8)$, each pair is divisible by sixteen. Here $k(G)$ is the number of conjugacy classes.

For a Frobenius complement $H$, a class function $\theta$ with $\theta (1)=0$ satisfies $({\theta }^G{)}_H=\theta$. For each nontrivial $\eta \in \mathrm{Irr}(H)$ of degree $d$, the virtual character $(\eta -d{1}_H{)}^G+d{1}_G$ has norm one and positive degree by Frobenius reciprocity. It is therefore irreducible, restricts to $\eta$ and is identically $d$ on the candidate kernel $N$. The intersection of the kernels of these characters is exactly $N$: outside $N$, separation follows from the character of the regular representation of $H$. This proves normality; $N\cap H=1$ and $|N|=[G:H]$ then give the semidirect product.

Let $N$ be a finite group and $\alpha$ an automorphism whose only fixed element is the identity. If $\alpha (a{)}^{-1}a=\alpha (b{)}^{-1}b$, then $\alpha (b{a}^{-1})=b{a}^{-1}$, so $a=b$. The displayed map is therefore a bijection. For conjugation by a nonidentity element of a Frobenius complement, it gives the commutator map $y\mapsto h^{-1}{y}^{-1}hy$.

For $1<N◃G$ proper, the displayed condition is equivalent to $N$ being a Frobenius kernel. For a normal subgroup the induction norm is $|I_G(\phi ):N|$. Trivial inertia for each nonprincipal character and Brauer's permutation lemma imply that every element outside $N$ has trivial centralizer in $N$. This forces $N$ to be a normal Hall subgroup; a complement is then fixed-point-free on $N$.

For a connected smooth algebraic group defined over a finite field, work over its algebraic closure and let $F$ be the finite-field Frobenius. Then the Lang map $g\mapsto g^{-1}F(g)$ is surjective. One proof uses the twisted action $x\cdot y=xyF(x{)}^{-1}$. Since $dF=0$, every orbit map has surjective differential, so each orbit is open; connectedness leaves only one orbit. The same differential calculation makes the Lang map étale, with finite fibres equal to left cosets of $G^F$.

Let a monic integer polynomial have distinct roots in its splitting field. Outside the finitely many primes dividing its polynomial discriminant or ramifying in that field, its roots reduce to distinct residue roots. The Frobenius automorphism permutes them by $a\mapsto a^p$; a reduced root belongs to ${\mathbb{F}}_p$ exactly when it is fixed. Distinctness of reduction identifies this with a fixed root of the Galois permutation itself. Combining this observation with a derangement in a transitive group action and the Chebotarev density theorem proves that an irreducible polynomial of degree at least two has rootless reduction at a positive Dirichlet density of primes. The degree bound is essential: every monic linear integer polynomial has a root modulo every prime.

If a Freiman 2-isomorphism into ${\mathbb{F}}_2^m$ uses minimal $m$, every nonzero vector lies in the fourfold sumset. Otherwise quotienting by the line it spans would preserve all additive quadruple relations and would still be injective, contradicting minimality. Since zero also belongs to the fourfold sumset, that sumset is the full model group. Plünnecke-Ruzsa inequality then bounds its size by $K^4|A|$. A finite subset of the direct sum of countably many binary fields has a finite-dimensional model to start with.

For planar strong anchoring and a quarter-turn base profile, ${\varphi }^{\prime }=\pi /(2L)$ and $\theta =0$. Expanding the one-dimensional twisted nematic energy through quadratic order gives $\Delta F=\frac{1}{2}\int [K_1{\theta }^{\prime 2}+\{(K_3-2K_2)(\pi /(2L){)}^2-\Delta ϵE^2/(4\pi )\}{\theta }^2]dz$. The lowest zero-endpoint mode is $\sin (\pi z/L)$. Its coefficient vanishes at the displayed threshold, with $V=EL$, provided the bracket and anisotropy are positive. Halving the electric coupling doubles $V_c^2$.

For a Neumann screen on the negative horizontal axis, the Helmholtz equation has an exact solution in Fresnel integrals. With time dependence $e^{i\omega t}$, a wave incident from angle $0<{\theta }_0<\pi$ has total field $e^{ikr\cos (\theta -{\theta }_0)}\mathcal{F}(s_{-})+e^{ikr\cos (\theta +{\theta }_0)}\mathcal{F}(-s_{+})$, where $s_{\pm }=\sqrt{2kr}\cos [(\theta \pm {\theta }_0)/2]$ and $\mathcal{F}(s)=\frac{1}{2}\mathrm{erfc}(-e^{i\pi /4}s)$. The angular derivatives cancel on both faces of the screen. Away from the two shadow boundaries, the Fresnel tail expansion splits this into incident/reflected geometrical optics fields and an outgoing edge wave proportional to $r^{-1/2}{e}^{-ikr}$.

In a free group, a reduced word has no adjacent inverse letters. Successively cancelling such pairs gives the unique reduced representative in the chosen free generators. In a free product, a reduced word instead uses nonidentity factor elements as syllables in a free product, with consecutive syllables in a free product from different factors; the alternating syllables in a free product are unique.

For an amalgamated free product $A{\ast }_{C}B$, the usual reduced form relative to a common subgroup requires the maps $C\to A$ and $C\to B$ to be injective. Chosen representatives of cosets then distinguish the alternating factors and the remaining common-subgroup element. The exact form depends on the chosen left/right representative convention.

For general group pushouts defined by arbitrary homomorphisms, factors can collapse and this embedded-subgroup normal form is unavailable. For example, take $A=B=C=\mathbb{Z}$, map $C$ identically into $A$, and map it trivially into $B$. The pushout kills all of $A$ and is isomorphic to $B$, so words in nonidentity elements of $A$ cannot retain the usual nontrivial-factor normal form.

A subgroup $A$ is a free factor of $G$ if $G$ is isomorphic to a free product $A\ast B$ by an isomorphism restricting to the inclusion of $A$, for some subgroup $B$. The normal form theorem for a free product ensures that both factors embed. A free factor need not be a normal subgroup.

Suppose $P$ has a soluble word problem for a group and $z$ generates an infinite cyclic group. In $P\ast ⟨z⟩$, for $g\ne 1$ the words $gz$ and $[g,z]$ are cyclically reduced with two and four syllables respectively. Their nonzero powers have syllable lengths $2|k|$ and $4|k|$. Given an input's reduced free product form, its length bounds the possible exponents; compare the finitely many candidates using the base word algorithm. Membership and the exponent in each of these infinite cyclic subgroups are therefore computable, with no algorithm for the order of $g$ required.

A finite generating set and finitely many relators in a free pro-p group, divided out by their closed normal closure. Relators are elements of the free pro-p group; they need not be finite ordinary words. A pro-p group with such a presentation is finitely presented in the pro-$p$ category.

Embed a free pro-p group on $x_1,\dots ,x_d$ into the units of the noncommutative power-series algebra ${\mathbb{F}}_p⟨⟨X_1,\dots ,X_d⟩⟩$ by $x_i\mapsto 1+X_i$. Its Magnus degree is the lowest total degree occurring in $\iota (w)-1$, with $D(1)=\infty$. Since $(1+v{)}^{p^e}=1+v^{p^e}$ in characteristic $p$, one has $D(w^{p^e})\ge p^{e}D(w)$. High powers therefore impose relations with small weight at any fixed parameter $0<\tau <1$.

Writing the characteristic equation with reciprocal hyperbolic cosine avoids exponentially large products. For the positive roots indexed from one, $q_n=(n+1/2)\pi +2(-1{)}^{n+1}{e}^{-(n+1/2)\pi }+O(e^{-2(n+1/2)\pi })$. The apparent root at zero represents the separate affine kernel, not another positive bending mode. Evaluating hyperbolic pieces of high modes with decaying exponential forms also avoids cancellation.

For bounded operators with trace-class commutator, the multiplicative commutator differs from identity by a trace-class operator. Along $F(t)=e^{tA}{e}^B{e}^{-tA}{e}^{-B}$, the trace of its logarithmic derivative is the constant $\mathrm{Tr}[A,B]$. This follows by writing $e^{B}A{e}^{-B}-A$ as the integral of conjugates of $[B,A]$ and using trace invariance under bounded similarities. Integrating the Fredholm determinant derivative from $F(0)=I$ proves the formula. Individual traces of $A$ and $B$ need not exist.