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The strictly positive case states that a real square matrix $W$ with $W_{ij}>0$ has a positive simple eigenvalue $\lambda$ and positive left and right eigenvectors, with all other eigenvalues strictly smaller in modulus. The Brouwer fixed-point theorem applied to $v\mapsto Wv/(1^{T}Wv)$ on the nonnegative unit simplex gives $v>0$ and $Wv=\lambda v$. For another eigenvector $z$, maximize $|z_i|/v_i$; the triangle inequality gives $|\mu |\le \lambda$. Equality forces every modulus ratio and complex argument to agree because every entry is positive, so $z$ is proportional to $v$ and $\mu =\lambda$. Apply the same argument to $W^T$ for a positive left eigenvector; its positive pairing with $v$ rules out a generalized eigenvector at $\lambda$, establishing algebraic simplicity. For merely nonnegative matrices a nonnegative leading eigenvector exists. An irreducible nonnegative matrix has a positive leading eigenvector and simple Perron eigenvalue; a primitive nonnegative matrix has the strict modulus gap.

A square nonnegative matrix $A$ is irreducible if for each pair $(i,j)$ there is an integer $k\ge 0$ with $(A^k{)}_{ij}>0$. This says that every index can reach every other through positive entries. The Perron–Frobenius theorem then gives a positive leading eigenvector and an algebraically simple eigenvalue. Irreducibility alone permits other eigenvalues of the same modulus: the cyclic permutation matrix $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ has eigenvalues $1,-1$.

For every commutative ring $R$,Every nilpotent element lies in every prime ideal. If $a$ is not nilpotent, an ideal maximal among those disjoint from $\{1,a,a^2,\dots \}$ is a prime ideal avoiding $a$, by the Zorn lemma and the maximality argument.

With $X^2=0$ and invertible $F_X$, the additional chiral superfield $Y=y+\sqrt{2}\theta \chi +{\theta }^2{F}_Y$ satisfying $XY=0$ has $y=(G\chi )/F_X-(GG)F_Y/(2F_X^2)$. Its Weyl spinor $\chi$ and auxiliary field $F_Y$ remain independent. This eliminates the independent scalar. It does not force $Y^2=0$.

The Landau-de Gennes free energy is a rotational symmetry preserving Landau free energy of the nematic order parameter. One three-dimensional normalization through fourth order isThe cubic invariant generically produces a first-order phase transition. Its absence in two dimensions is an algebraic identity, not a guarantee that a mean-field approximation captures all fluctuations.

Under rotation by a rotation matrix, a symmetric second-rank tensor transforms by conjugation, so its traces of powers are invariant. For a three-dimensional traceless second-rank tensor, the Cayley-Hamilton theorem implies $\mathrm{Tr}{Q}^4=(\mathrm{Tr}{Q}^2{)}^2/2$. In two dimensions its eigenvalues are $s,-s$, so $\mathrm{Tr}{Q}^3=0$. These identities organize the Landau-de Gennes free energy.

A uniaxial nematic order parameter has one distinguished nematic director and equal eigenvalues in its perpendicular plane. In the displayed normalization the eigenvalues are $\lambda ,-\lambda /2,-\lambda /2$, giving $\mathrm{Tr}{Q}^2=3{\lambda }^2/2$ and $\mathrm{Tr}{Q}^3=3{\lambda }^3/4$. Positive and negative $\lambda$ describe prolate and oblate orientational anisotropy, respectively; reversing the nematic director leaves the tensor unchanged.

For $g=N\mathbf{v}$ and any $\tilde{g}\in C^2[a,b]$ taking the same values at at least two distinct knots, $J(\tilde{g})=J(g)+J(\tilde{g}-g)$. To prove it, set $r=\tilde{g}-g$, integrate $g^{\prime \prime }{r}^{\prime \prime }$ by parts on each polynomial interval and use $g^{(4)}=0$, $r(x_i)=0$, continuous $g^{\prime \prime }$ and linear exterior pieces. Boundary terms cancel. Expanding the square proves the identity. Equality forces $r^{\prime \prime }=0$, so $r$ is an affine function and its two prescribed zeros make it identically zero.

Let $b_i=N{e}_i$ be the interpolating cardinal basis and set ${\Gamma }_{ij}={\int }_a^b{b}_i^{\prime \prime }{b}_j^{\prime \prime }dx$. Linearity gives $J(N\mathbf{v})={\mathbf{v}}^T\Gamma \mathbf{v}$, and this Gram matrix is a positive semidefinite matrix. Its null space consists precisely of vectors $(\alpha +\beta x_i{)}_i$, by the null-space statement for the second derivative roughness penalty. Thus its rank is $n-2$, including the zero matrix when $n=2$.

A straight rigidly rotating classical string has speed of light at a free endpoint. Its energy and angular momentum satisfy a Regge trajectory. A segment with one anchored and one free end has $J=E^2/(\pi T)$, twice the slope of the string with two free ends.

A real Cartier class on a projective scheme is ample if and only if its top self-intersection on every positive-dimensional integral subvariety is positive. For the converse curve tests give nefness. Small ample perturbations and rational approximation give ample rational $B,C$ with $D-A=B-C$ satisfying the algebraic Morse inequality for ample divisors, hence bigness. Induction gives ample restrictions on codimension-one subvarieties; uniform ample subtraction from a big divisor with ample exceptional restrictions makes $D-ϵA$ nef. The nef-plus-ample ampleness lemma concludes. For reducible schemes perform the finite component tests simultaneously.

For $n\ge 3$, a nonnegative entire weak subsolution of a bounded measurable uniformly elliptic divergence-form equation that belongs to $L^p({\mathbb{R}}^n)$ for some $p>1$ is zero almost everywhere. Rescaling the local Moser iteration estimate gives $\Vert u{\Vert }_{L^{\infty }(B_{R/2})}\le C{R}^{-n/p}\Vert u{\Vert }_{L^p(B_R)}$. Let $R\to \infty$.

A proper morphism $f:X\to Y$ factors as $X\stackrel{g}{\to }Z\stackrel{h}{\to }Y$, with $h$ finite and $g_{\ast }{\mathcal{O}}_X={\mathcal{O}}_Z$; its fibers are connected. When $X$ is normal, $Z$ is normal. For a morphism from a smooth surface onto a curve, $Z$ is therefore a smooth projective curve over an algebraically closed field.

A monoid automorphism is a bijective operation- and identity-preserving self-map of a monoid. Its inverse is also a monoid homomorphism. Permuting prime numbers and extending by prime factorization gives all automorphisms of the multiplicative monoid of positive natural numbers; extending by $0\mapsto 0$ gives automorphisms of the multiplicative monoid including zero.

The Ramanujan tau function gives the Fourier coefficients at positive indices of the modular discriminant: $\Delta (z)={\sum }_{n\ge 1}\tau (n)q^n$, with $\tau (1)=1$. The Serre derivative and vanishing of weight-fourteen level-one cusp forms give $qd\Delta /dq=E_2\Delta$ and hence the convolution recurrence

For a random classifier $C$ determined by training information $\mathcal{T}$, and an independent test pair $(X,Y)$, its conditional misclassification risk is $L(C)=\mathbb{P}(C(X)\ne Y\mid \mathcal{T})$. It is itself a training-dependent random variable. The tower property of conditional expectation gives $\mathbb{E}L(C)=\mathbb{P}(C(X)\ne Y)$.

On a complete connected Riemannian manifold, let $c(v)$ for a unit tangent vector $v$ be the supremum of times for which $t\mapsto {\exp }_p(tv)$ minimizes Riemannian distance from $p$. The cut locus consists of the endpoints ${\exp }_p(c(v)v)$ with $c(v)<\infty$. Radial minimization fails beyond this first cut time; distance spheres can cease to be smooth there.

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.