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Under the hypothesis of Paris models are well-founded when their complete theory has a well-founded model, a Paris model $M$ is well-founded. Every automorphism of $M$ is then the identity by external epsilon induction: once all members of $x$ are fixed, extensionality gives $\pi (x)=\{\pi (y):y{\in }^{M}x\}=x$. Hence $M$ is a rigid first-order structure.

Let $N$ be a well-founded model of set theory of a complete theory $T\supseteq \mathsf{ZFC}$, and let $M$ be a Paris model of $T$. If the internal ordinals of $M$ admitted an external descending sequence ${\alpha }_0{\ni }^M{\alpha }_1{\ni }^M\cdots$, choose formulas ${\phi }_n$ defining them. Completeness makes $M$ and $N$ satisfy the same sentences, so the unique ordinals defined by the ${\phi }_n$ in $N$ would form an externally descending membership sequence. This contradicts well-foundedness of $N$.

A simple closed curve in ${\mathbb{R}}^3$ preserved by a nonconstant smooth family of proper rigid motions is a circle. Differentiating the normalized family at the identity gives a nonzero infinitesimal proper Euclidean motion tangent to the curve. Compactness rules out its line and nonclosed helical orbits, leaving a circular rotation orbit.

For every positive integer $n$,Under Local Artin reciprocity, the cyclotomic character identifies the quotient with $(\mathbb{Z}/2^n\mathbb{Z}{)}^{\times }$: a uniformizer acts trivially on the totally ramified extension, while a 2-adic unit acts through its residue class modulo $2^n$.

If $F:{\mathbb{P}}^1\to {\mathbb{P}}^1$ is a morphism of degree $d$ over $\mathbb{Q}$, there are positive constants $c_1,c_2$ such thatfor every $P\in {\mathbb{P}}^1(\mathbb{Q})$.

There is no nonconstant four-term arithmetic progression of rational square numbers. If $r^2,s^2,1,t^2$ formed one, then $(-2s^2,2rst)$ would lie on $y^2=x(x+1)(x+4)$; its explicitly known Mordell-Weil group forces $s^2=1$ and the common difference to vanish.

Put $\rho =\lambda /\mu$. The unique stationary distribution of an $M/M/s/K$ queue iswhere $Z$ is the sum of the displayed weights. Adjacent terms satisfy detailed balance for a birth-death process, so the queue is reversible in equilibrium.

Equality in the three-dimensional Loomis--Whitney inequality forces a measurable body to agree up to a null set with a Cartesian product $E_1\times E_2\times E_3$. This follows from the equality conditions in the two Cauchy-Schwarz inequalities used in its proof. If the body is connected and is a finite union of positive-volume axis-parallel boxes, each $E_i$ is an interval and equality up to a null set upgrades to exact equality with one box.

An additive class-effect logistic network model gives an edge joining classes $k$ and $\ell$ the log odds ${\beta }_k+{\beta }_{\ell }$. A further coefficient multiplying $1_{\{k=\ell \}}$ represents a common within-class log-odds effect.

Repeatedly applying the Local LYM inequality to upper shadows givesfor $\mathcal{A}\subseteq [n{]}^{(r)}$ and $0\le s\le n-r$. The analogous statement for iterated lower shadows follows directly from the stated lower-shadow form.

If $(g_1,\dots ,g_d)$ is an ordered basis of a finite-rank p-saturated group, thenwhere a primitive element $x$ satisfies $\Delta (x)=x\otimes 1+1\otimes x$ for the completed Hopf algebra coproduct.

The initial form of $g-1$ depends only on the initial form of $g$ in $\mathrm{gr}G$. The resulting Lie map extends to a surjective graded algebra homomorphismFor a complete finite-rank p-valued group, ordered-basis expansions show that it is an isomorphism.

The Koszul formula uniquely determines $g({\nabla }_{X}Y,Z)$ for all vector fields $X,Y,Z$, so nondegeneracy of the Riemannian metric determines ${\nabla }_{X}Y$. The resulting operation is $C^{\infty }$-linear in $X$, obeys the Leibniz rule in $Y$, is compatible with the metric, and is torsion-free. It is therefore the unique Levi-Civita connection.

The Pinsker sigma-algebra consists of the measurable sets $A$ whose binary partition $\{A,X\setminus A\}$ has entropy rate zero. It is a sigma-algebra and is the largest invariant factor with zero Kolmogorov-Sinai entropy.

The degree-one Hochschild cocycles are exactly the maps that are a derivation into a bimodule, while degree-one coboundaries are exactly the inner derivations. Therefore

For every simple finite-dimensional $kG$-module $S$,

Forwith the four displayed generators, $R_j$ has basis $Y_1^a{Y}_2^b$ with $|a|+|b|\le j$. The number of integer lattice points in this diamond isIts quadratic growth agrees with $\dim R=2$.

The l-p sequence space ${\ell }^1$ has the Schur property. If a weakly null sequence stayed bounded below in norm, coordinatewise convergence and summability would select disjoint blocks containing almost all of successive terms. A sequence in ${\ell }^{\infty }$ matching their signs on those blocks would pair uniformly positively with a subsequence, contradicting weak convergence through the duality of l1 and l infinity.

Let $S\subset {\mathbb{R}}^3$ be a smooth surface. Suppose its Gaussian curvature tends to infinity in absolute value along every maximal geodesic with a finite affine-parameter endpoint. Then $S$ is an inextendible embedded surface: a smooth extension would let some geodesic reach the extension boundary in finite time while its curvature remained locally bounded.

For $G_{\lambda }=\{g:\omega (g)\ge \lambda \}$ and $G_{\lambda +}=\{g:\omega (g)>\lambda \}$, the associated graded groupis a graded Lie algebra with bracket induced by the group commutator. For a p-valuation, $t{\mathrm{gr}}_{\lambda }(g)={\mathrm{gr}}_{\lambda +1}(g^p)$ makes it a graded ${\mathbb{F}}_p[t]$-Lie algebra.