If $A$ is an integral domain with fraction field $K$ and $L/K$ is finite, a $K$-basis of $L$ can be rescaled to consist of integral elements over $A$. For a monic algebraic equation, choose $a\in A$ clearing the coefficient denominators; replacing $u$ by $au$ multiplies the coefficient of degree $r-j$ by $a^j$, putting it in $A$.

For a normal Noetherian ring $A$ that is an integral domain, the integral closure in a finite separable field extension of its fraction field is finite as an $A$-module. An integral field basis by denominator clearing gives a finite lattice. Integral traces place the closure in its finite trace-dual lattice; Noetherianity makes that submodule finitely generated.

Differentiating the Euclidean classical equation for an instanton gives a zero eigenfunction proportional to its time derivative. Its norm satisfies $\mu \int {\dot{q}}_I^{2}d\tau =S_I$, so replacing the Gaussian zero-mode integral by the center coordinate gives $\sqrt{S_I/(2\pi \hslash )}d{\tau }_0$. The nonzero modes belong in the primed functional determinant. This removes the false divergence without deleting the center's integration volume.

The canonical gradient is the influence-function representer in the statistical tangent space $\mathcal{T}$. It equals the orthogonal projection onto $\mathcal{T}$ of any representer. Every other representer differs from it by an element of ${\mathcal{T}}^{\perp }$, so the Pythagorean theorem in an inner-product space gives it minimum variance. For $\psi (P)=Pa$ in an unrestricted density model with bounded $a$, it is $a-Pa$.

For $a_j\ge 0$, the finite product satisfiesThe lower inequality follows by product expansion; the upper one uses $1+t\le e^t$. Since both the partial sums and partial products are increasing, one converges to a finite limit exactly when the other does. The monotone bounded sequence theorem proves both implications. The product cannot tend to zero in this nonnegative setting.

If a tail of holomorphic factors admits logarithms whose absolute values have a summable bound uniformly on every compact set, their product converges locally uniformly to the exponential of the logarithmic sum. The tail is holomorphic and nowhere zero. Finite initial factors may have zeros; those finite factors determine the locations and orders of all zeros near any given compact set. The scalar sufficient condition ${\sum }_j|u_j|<\infty$ for factors $1+u_j$ follows from $|\log (1+u_j)|\le 2|u_j|$ for $|u_j|\le 1/2$.

The Weierstrass elementary factor has a simple zero at $w=1$ and no other zeros. Its logarithmic series begins at degree $p+1$: $\log E_p(w)=-{\sum }_{\nu \ge p+1}{w}^{\nu }/\nu$ for $|w|<1$. Raising the factor order $p$ makes the tail arbitrarily small on a fixed compact set and enables infinite product convergence from logarithmic tails for unrestricted discrete zero data.

A nonempty path space $[T]$ is perfect if it has no isolated points: for every path and every finite prefix of it there is a different path sharing that prefix. This property of the closed subset of Cantor space does not imply that its particular finite-string presentation has no dead ends. Requiring every finite node to extend to incompatible nodes is the stronger pruned-tree convention for a perfect tree.

The hadronic R ratio normalizes the photon-channel hadronic relativistic cross-section to the leading massless muon-pair value. The free active-quark result is $R=N_c{\sum }_f{Q}_f^2$. It depends on flavor charges as well as the number of active flavors; it is not simply $N_c{N}_f$.

The measure $d\sigma /\sigma$ is invariant under multiplication by every positive constant. In the logarithmic coordinate it is Lebesgue measure. It cannot be normalized over $(0,\infty )$, although a bounded truncation is a log-uniform distribution.

If a linear functional $\ell$ vanishes on the image of a linear map $T$ but $\ell (b)\ne 0$, the equation $Tx=b$ has no solution. In Euclidean coordinates, a vector $n$ with $n^{T}A=0$ provides such an obstruction whenever $n^{T}b\ne 0$. In finite dimensions these annihilating functionals exactly characterize membership in the image: $b\in \mathrm{im}A$ precisely when it is orthogonal to every vector in $\ker A^T$.

For a morphism in an abelian category, its coimage $\mathrm{coker}(\ker f)$ is canonically isomorphic to its image $\ker (\mathrm{coker}f)$. This yields an epimorphism followed by a monomorphism, unique up to unique compatible isomorphism. Every epi-mono factorization has this same middle object.

An idele obtained by mapping the same nonzero element of a number field into every completion. Only finitely many local valuations are nonzero, so this diagonal tuple belongs to the idele group.

For $S=L/(c_p\Delta T)\gg 1$, $ϵ=\sqrt{D/\kappa }\ll S^{-1}$ and $R=m{C}_0/\Delta T>1$, the leading thickness is the displayed expression. The basal position is $2A\sqrt{Dt}$, with $A$ determined byOnly when $ϵSA\ll 1$ can the basal translation be omitted from the upper position. Then $A{e}^{A^2}\simeq \sqrt{\pi }RS/4$, exposing the logarithmic refinement to the scale ordering.

If forward-going surface-gravity-wave energy decays as $a^2(x)=a_0^2{e}^{-\kappa x}$ and local wave force is $F=C{\rho }_{w}g{R}^2{a}^2$, its spatial decrease is $-{\partial }_{x}F=C{\rho }_{w}g{R}^2\kappa a^2$. In a weak-reflection independent-layer approximation, $\kappa \simeq p{R}^2/d$. Standard deep-water wave radiation stress gives $C=1/2$, hence a compaction-force density ${\rho }_{w}gp{R}^4{a}^2/(2d)$. Different amplitude or force conventions change the coefficient; the force formula alone does not fix $\kappa$. This quantity is force per horizontal area, not automatically the three-dimensional stress in the ice skeleton.

For every $\eta >0$, some $\rho >0$ has this property: a three-uniform hypergraph on $v$ vertices with fewer than $\rho v^4$ copies of the three-uniform tetrahedron can be made tetrahedron-free by removing fewer than $\eta v^3$ hyperedges. The four-term progression hypergraph encoding converts this into the length-four case of the Szemerédi theorem.

Truncating the steady porous thermal plume profile below a head height and conserving supplied heat gives ${\dot{Z}}_h\simeq \int w^{2}dx/\int wdx=(2/3)W(Z_h)$. Thus $Z_h\propto t^{3/4}$. This is an integral bulk-head estimate, while fastest centreline parcels have characteristic speed $W$; the approximation does not resolve the transient nose.

Internal Horn models are finite-limit-preserving functors out of the Cartesian syntactic category. Since that category has finite limits, these are exactly flat functors; the presheaf Diaconescu equivalence for geometric morphisms proves the classifying property. Equivalently the classifier is the covariant functor category on finitely presented set-based models.

In the spherical approximation, equatorial centrifugal acceleration balances surface gravity when ${\Omega }_c^2{R}_c=GM/R_c^2$. For $I=\alpha M{R}^2$ and conserved angular momentum $J$, this gives the displayed critical radius. Real rotating gas can deform, lose matter and exchange torques before a spherical model remains accurate at this limit.

In a homogeneous Riemannian manifold, choose $r>0$ so that one closed metric ball $\overline{B}(o,r)$ is compact. Such a ball exists from local compactness and the metric topology. Every radius-$r$ ball is isometric to it. A Cauchy sequence eventually lies in one such compact ball, has a convergent subsequence, and therefore converges. This proves metric completeness; the Hopf-Rinow theorem gives geodesic completeness and minimizing geodesics between points.