For compact metric spaces and a continuous cost, define the marginal functional $\ell (f(x)+g(y))=\int fdP+\int gdQ$. Its majorant envelope $p(h)={\inf }_{s\ge h}\ell (s)$ is finite and sublinear. The dual value is $-p(-c)$. Assign that value to the cost direction and use the dominated Hahn-Banach theorem to obtain a positive linear functional on all continuous functions of two variables. The Riesz-Markov-Kakutani representation theorem turns it into an optimal transport plan with the prescribed marginals.

For a direction $e$, the Kakeya maximal function is the supremum over translations of normalized averages of $|f|$ over unit-length Kakeya tubes in direction $e$. It measures how large a function can be on at least one thin tube in each direction. Normalizing by tube volume is essential when comparing different thicknesses.

A complex two-dimensional linear operator whose two eigenvalues both equal $a$ is similar either to $aI$ or to $\left(\begin{array}{cc}a& 1\\ 0& a\end{array}\right)$. Choose an eigenvector and complete it to a basis. The matrix becomes $\left(\begin{array}{cc}a& c\\ 0& a\end{array}\right)$ because its characteristic polynomial is $(t-a{)}^2$. A nonzero $c$ can be normalized to one by rescaling the first basis vector. The two types are distinguished by eigenspace dimension.

Perturb each diagonal position of a Jordan normal form matrix by a different arbitrarily small scalar, avoiding the finitely many collisions between diagonal entries. The resulting upper triangular matrix has distinct eigenvalues, hence is diagonalizable. Conjugate back to approximate any complex matrix. Continuous identities proved for diagonalizable matrices can therefore extend to all matrices.

The outer boundary of a compact planar set $K$ is the boundary of the unbounded component $C_{\infty }(K)$ of its complement. Its filled hull is $\mathbb{C}\setminus C_{\infty }(K)$. When the outer boundary is a simple closed curve, the hull is its closed interior. For a simply connected open domain $U$, containment of a compact trace and of its filled hull are equivalent. This lets outer-boundary pushforwards retain simply connected domain-containment information.

Let a line collection have $M$ joints of a line collection and put $D=\lceil n{M}^{1/n}\rceil$. If $M>LD$, deleting each line with at most $D$ current joints of a line collection, together with those joints of a line collection, leaves a nonempty configuration in which every line has more than $D$ joints of a line collection. A nonzero polynomial of degree at most $D$ vanishes on the retained joints of a line collection by rank-nullity theorem. Choose one of minimum degree. Its restriction vanishes identically on every retained line, so its gradient is orthogonal to a spanning set of directions at every joint of a line collection and hence vanishes there. Each nonzero partial derivative would be a lower-degree polynomial with the same zeros, contradicting minimality. In characteristic zero, all derivatives vanishing forces a constant polynomial, another contradiction. Therefore $M\le LD$, giving the joints theorem. The argument's characteristic-zero assumption is essential to its derivative step.

Every zero in the smaller disk contributes at least $\log (R/r)$ in Jensen's formula. Bounding the outer circle average by its maximum proves the estimate. With $R=2r$, an exponential $O(r\log r)$ maximum bound gives an $O(r\log r)$ zero count.

For a finite-type integer algebra, a nonnilpotent element $r$ gives a nonzero localization of a ring $R[1/r]$. A maximal ideal there has finite residue field by the finite-field theorem for finitely generated integer algebras. The image of $R$ in that field is a finite integral domain, hence a field. Its kernel is therefore maximal in $R$ and avoids $r$. Nilpotents lie in every maximal ideal, proving the radical equality. Applying the same argument to every prime quotient shows that these algebras are Jacobson rings. Merely contracting a localized maximal ideal without the finite-image argument would not prove maximality.

For the Riemann curvature tensor convention $R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$, define ${\mathcal{R}}_X(v)=R(v,X)X$. Curvature pair interchange makes this operator self-adjoint. If $v\perp X$, then $⟨{\mathcal{R}}_{X}v,v⟩=K(v,X)|v{|}^2|X{|}^2$. The Jacobi field equation is $D_t^{2}J+{\mathcal{R}}_{\dot{\gamma }}J=0$. The operator $v\mapsto R(X,v)X$ is $-{\mathcal{R}}_X$ and is likewise self-adjoint; this sign distinction matters when testing curvature inequalities.

For $\gamma (t)={\exp }_p(tV)$, every Jacobi field with $J(0)=0$ is uniquely of the formDifferentiate the geodesic variation ${\exp }_p(t(V+sB))$ in $s$ to obtain the formula. Uniqueness of the Jacobi equation with prescribed initial value and derivative shows that all such fields are obtained. A nontrivial field vanishing at $t=L>0$ therefore corresponds to a nonzero vector in the kernel of $(d{\exp }_p{)}_{LV}$, giving the conjugate point criterion.

A finitely generated module over $\Lambda ={\mathbb{Z}}_p[[T]]$ is pseudo-isomorphic towhere the $f_j$ are irreducible distinguished polynomials. Height-one localizations are discrete valuation rings, which give the elementary divisors; the remaining errors are supported only at the maximal ideal and are finite. This is a finite-error classification, not necessarily an actual direct-sum decomposition.

For a Zp-extension, let $L_{\infty }$ be the maximal unramified abelian pro-$p$ extension of $F_{\infty }$. Its Galois group is a compact Iwasawa module, identifiable by Artin reciprocity with the inverse limit of the $p$-primary ideal class groups of finite layers under norms.

Let $M_{\infty }$ be the maximal abelian pro-$p$ extension of $F_{\infty }$ unramified away from primes over $p$. Its Galois group is the p-ramified Iwasawa module. In the cyclotomic tower of $\mathbb{Q}({\mu }_p)$ for odd $p$, its Iwasawa-module rank is $(p-1)/2$. It is therefore quite different from the torsion unramified Iwasawa module.

The Mazur-Wiles theorem proves the classical Iwasawa main conjecture for abelian extensions of $\mathbb{Q}$. Its construction of class fields uses Galois representations associated with modular forms, congruences with Eisenstein series and the Eisenstein ideal.

Integrating free fermions in a constant auxiliary spin field gives $S_{\mathrm{eff}}=\beta N{M}^2/(2U)-{\sum }_{k\sigma }\ln [1+e^{-\beta ({ϵ}_k-\mu -\sigma M)}]$. Differentiation gives the self-consistent spin polarization. The extensive action is a mean-field restriction of the full auxiliary-field functional integral, not an exact omission of all spatial/time fluctuations.

In the homogeneous spin-vector mean-field scheme with field cost $M^2/(2U)$ and spin-summed density of states per site, the paramagnetic curvature is proportional to $U^{-1}-\int \nu (ϵ)[-n_F^{\prime }(ϵ)]dϵ$. It changes sign at $U{\chi }_0=1$. For a smooth low-temperature density of states, ${\chi }_0\simeq \nu (\mu )$, giving the familiar Fermi-level criterion. Interaction and spin-counting conventions must be specified before comparing coefficients.

An isotropic spin-polarized metal has a degenerate magnetization direction and a long-wavelength transverse magnon with quadratic dispersion. Nonzero spin density pairs the two transverse broken generators into one type-B Goldstone boson. Exchange-split Fermi surfaces also support fermionic low-energy excitations; the magnon is distinct from longitudinal amplitude motion and the spin-flip continuum.

For isotropic stellar mass loss the eccentricity vector obeys $\dot{\mathbf{e}}=-(\dot{\mu }/\mu )(\mathbf{e}+\hat{\mathbf{r}})$. With true anomaly $f$ and $e>0$, this gives $\dot{e}=-(\dot{\mu }/\mu )(e+\cos f)$ and $\dot{\varpi }=-(\dot{\mu }/\mu )\sin f/e$. The osculating pericentre satisfies $\dot{q}/q=-(\dot{\mu }/\mu )(1-\cos f)/(1+e)$, hence cannot decrease during mass loss and is instantaneously stationary at pericentre. The vector form is nonsingular at a circular orbit.

Time averaging over a Kepler orbit gives $⟨\cos f⟩=-e$, as follows from $dt=(1-e\cos E)dE/n$ and $\cos f=(\cos E-e)/(1-e\cos E)$. Consequently slow isotropic stellar mass loss preserves the secular orbital eccentricity. Conserved specific angular momentum then gives $\mu a=\mathrm{constant}$, so pericentre, apocentre and semimajor axis expand inversely with the remaining gravitational parameter.

In the ordered spin-one basis $(m=1,0,-1)$, introduce the dimensionless generator $j_2=J_2/\hslash$. The spin-one angular momentum matrices imply $j_2^3=j_2$, so the matrix exponential reduces toAt a half-turn,The rotation exchanges the two outer weights and negates the middle weight. Keeping $J_2/\hslash$ in the exponential distinguishes the dimensional angular momentum operator from its dimensionless representation matrix.

Applied to the pion isospin triplet, the same spin-one representation fixes the rotation factor entering G parity. That application uses the internal-symmetry generators rather than a mechanical rotation of the particles.