For a $C^2$ solution $u$ of the minimal surface equation for a graph on the closure of a bounded domain,Write $a^{ij}(Du)u_{ij}=0$ with $a^{ij}(p)={\delta }_{ij}-p_i{p}_j/(1+|p{|}^2)$. Its differentiated operator $\mathcal{L}=a^{ij}{D}_{ij}+b^{\ell }{D}_{\ell }$, where $b^{\ell }=(\partial a^{ij}/\partial p_{\ell })u_{ij}$, satisfiesThe weak maximum principle for elliptic operators gives the result. Interior Schauder estimates justify differentiating a solution initially assumed only $C^2$.

A component in a minimal primary decomposition is embedded if its radical of an ideal strictly contains another radical occurring in that decomposition. Such components need not be unique. For example,in $k[x,y]$ gives different $(x,y)$-primary ideals as embedded components, with the same isolated component $(x)$.

A factor map $\pi$ is a measurable map satisfying $\pi \circ T=S\circ \pi$ and ${\pi }_{\ast }\mu =\nu$, from one measure-preserving system to another. The target system is the factor; its observable functions pull back to observable functions of the source.

The image of the linear map $A+B$ is contained in the sum of the images of $A$ and $B$. Since the dimension of a vector space of this sum is at most the sum of their dimensions, matrix rank is subadditive. Iteration shows that a sum of $r$ matrices of matrix rank at most one has matrix rank at most $r$, over any field.

For positive-definite Hermitian matrices $A,B$, Klein's inequality giveswith equality exactly when $A=B$. To prove it, let $a_i,b_j$ be their eigenvalues and $u_i,v_j$ their orthonormal eigenvectors. The weights $w_{ij}=|⟨u_i,v_j⟩{|}^2$ have row and column sums one. The difference between the two sides isby the scalar logarithm inequality $\ln t\le t-1$. Equality requires $a_i=b_j$ whenever $w_{ij}>0$, implying $A=B$. Limits extend the result to positive semidefinite matrices with the appropriate support condition. Applying it to trace-one matrices proves nonnegativity of quantum relative entropy.

In a flattened power-law gravitational potential $\Phi ={\Phi }_0{R}_0^{\beta }(R^2+{\lambda }^2{z}^2{)}^{-\beta /2}$, a magnetic surface corotating with its disk footpoint has an effective potential with meridional Hessian matrix ${\Omega }_f^2\mathrm{diag}[-(\beta +2),{\lambda }^2]$ there. A cold outward displacement along a field inclined by $\alpha$ to the vertical is downhill when ${\tan }^2\alpha >{\lambda }^2/(\beta +2)$. Equality is marginal and can depend on higher derivatives and field-line curvature; this is a local launching criterion, not a guarantee of a global escaping wind.

For a primitive Lubin–Tate torsion point ${\lambda }_n$, every Galois automorphism has $\sigma ({\lambda }_n)=[a_{\sigma }]({\lambda }_n)$ for a unique unit class $a_{\sigma }(\mathrm{mod}{\pi }^n)$. All torsion points lie in $K({\lambda }_n)$ because the integral endomorphism series converge there; hence the torsion field is Galois. The displayed map is an injective homomorphism. Its domain has order $(q-1)q^{n-1}$ by the primitive Eisenstein polynomial, equal to the number of unit classes, so it is an isomorphism.

Choose compatible primitive Lubin–Tate torsion points with $[\pi ]{\lambda }_{n+1}={\lambda }_n$. Restriction of the Lubin–Tate Galois action on primitive torsion is reduction of unit classes modulo ${\pi }^n$. The inverse limit givesas topological groups. This describes the Galois group without invoking local reciprocity.

For $K={\mathbb{Q}}_p$ and $\pi =p$, choose the Lubin–Tate series $f(T)=(1+T{)}^p-1$. The associated formal group law is the formal multiplicative group. Its torsion points are $\zeta -1$ for ${\zeta }^{p^n}=1$, so its Lubin–Tate extension is the cyclotomic extension of a p-adic field ${\mathbb{Q}}_p({\mu }_{p^n})$. The Galois action $\zeta \mapsto {\zeta }^a$ gives $(\mathbb{Z}/p^n\mathbb{Z}{)}^{\times }$, including the trivial first layer when $p=2$.

An antichain in the Boolean lattice has Lubell mass one if and only if it is a full rank level. Equality means every maximal chain in a Boolean lattice meets the family exactly once. Swapping two adjacent entries in the defining permutation then forces the family to contain every set obtained from one member by exchanging an included and an excluded element, hence every set of that rank.

In the localized wave-drag response in a stratified channel, the full vertical integral of zonal acceleration equals the deposited wave momentum, $-G_0\sin (\pi y/L)$. The fraction inside the directly forced layer is $1-(1-e^{-2s})/(2s)$, where $s=\pi ND/(|f_0|L)$. A small $s$ spreads most acceleration outside the dissipation layer; a large $s$ confines most of it inside.

For every $0<a<1$, choose $1<p<1/a$. On the Bernoulli shift over independent uniform coordinates in $(0,1)$, the observable $f(x)=x_0^{-1/p}$ is integrable, but $\mu (f(T^{n}x)>n^a)=n^{-ap}$. The events are independent and their probabilities have divergent sum, so the Borel-Cantelli lemma implies that $f(T^{n}x)/n^a>1$ infinitely often almost surely. No exponent below one gives a universal bound for all integrable observables.

On an integral scheme with generic point $\eta$ and function field $K$, a rational section of a line bundle $\mathcal{L}$ is an element of the one-dimensional $K$-space ${\mathcal{L}}_{\eta }$. A nonzero rational section writes as $a_i{e}_i$ in local trivializations, with $a_i\in K^{\ast }$. Ratios of these coefficients are regular units, and therefore define a Cartier divisor.

A line bundle is globally generated if at each point some global section has a germ generating its stalk as a module over the local ring. Equivalently its evaluation map from global sections tensored with the structure sheaf is surjective.

The trivial representation sends every element of the Lie algebra to the zero linear map. For a complex semisimple Lie algebra, the irreducible representation of highest weight zero is the one-dimensional trivial representation.

The dual of a Lie algebra representation on $V$ acts on its dual space by $(x\varphi )(v)=-\varphi (xv)$. The minus sign makes this a Lie algebra homomorphism: taking transposes reverses the order in a commutator. It is the infinitesimal version of the dual representation of a group.

A bilinear form on a Lie algebra representation is invariant when it satisfies the displayed identity. Equivalently, $T_B(v)=B(v,-)$ is an intertwining operator from the representation to its dual Lie algebra representation. For a finite-dimensional irreducible representation over an algebraically closed field, any nonzero such form is nondegenerate, and the Schur lemma makes all invariant forms proportional. In characteristic different from two, transposing twice then proves that the form is a symmetric bilinear form or an alternating bilinear form.

For a finite-dimensional real Lie algebra, a finite-dimensional Lie algebra representation integrates uniquely to a smooth representation of the connected simply connected Lie group with that Lie algebra. For another connected group with the same Lie algebra, integration is possible exactly when the representation of its universal cover is trivial on the covering kernel. This separates local Lie-algebra classification from global representation restrictions.

With a consistent surgery orientation convention, $L(p,q)$ and $L(p,q^{\prime })$ are orientation-preservingly related by a homeomorphism exactly when $q^{\prime }\equiv q^{\pm 1}(\mathrm{mod}p)$. An orientation-reversing homeomorphism instead requires $q^{\prime }\equiv -q^{\pm 1}(\mathrm{mod}p)$.

Diverging $R_{ab}+2{\nabla }_a{\nabla }_b\phi =0$ and using the contracted Bianchi identity and Ricci identity gives ${\nabla }_b(R+4{\square }_g\phi -4|\nabla \phi {|}^2)=0$. The trace equation $R+2{\square }_g\phi =0$ then gives ${\square }_g\phi -2|\nabla \phi {|}^2=C$, with $C$ constant on each connected component. Zero does not follow from the tensor equation alone. The additional dilaton anomaly coefficient fixes $C=(D-26)/(3{\alpha }^{\prime })$ at leading order for the bosonic string without an antisymmetric background. In critical dimension it is zero; noncritical backgrounds have a central-charge-deficit term.