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Every toric ray $\rho$ of a normal toric variety determines a torus-invariant prime Weil divisor $D_{\rho }$. An invariant toric toric divisor is an integer combination $D={\sum }_{\rho }{a}_{\rho }{D}_{\rho }$.

Such a toric divisor is Cartier exactly when each toric cone $\sigma$ admits a algebraic torus character $m_{\sigma }\in M$ satisfyingThen the corresponding local algebraic torus character sections trivialize its divisor line bundle. Without this integral local condition, the toric divisor gives a divisorial rank-one reflexive sheaf and need not define a line bundle. The positive class generator of $\mathbb{P}(1,1,2)$, for example, has Cartier index two. Thus an arbitrary combination of invariant prime toric divisors must not be silently identified with invertible-sheaf data.^[ref]

A lattice homomorphism that maps each cone of one fan into a cone of another induces an equivariant morphism of the corresponding toric varieties.

For an integer $k\ge 0$, the Hirzebruch surface is the ruled surfaceIts smooth complete toric toric fan has toric rays $(1,0),(0,1),(-1,k),(0,-1)$. The ruling has projective lines as fibers. Its distinguished negative section of a Hirzebruch surface has self-intersection $-k$; together with a fiber class of a Hirzebruch surface $F$ it gives the relations $F^2=0$ and $S\cdot F=1$.

For $k=0$ the bundle is trivial and ${\mathbb{F}}_0\cong {\mathbb{P}}^1\times {\mathbb{P}}^1$. Inserting the sum of two adjacent toric rays in the toric fan of the projective plane gives ${\mathbb{F}}_1$, the blowup at a point; its exceptional curve is the section of self-intersection minus one. The nonnegative indexing convention avoids calling a positive-self-intersection section negative when changing the sign of the bundle parameter.^[ref]

This is a two-dimensional weighted projective space. The weighted projective plane $\mathbb{P}(a,b,c)$ is the quotient of ${\mathbb{A}}^3\setminus \{0\}$ by $\lambda \cdot (x,y,z)=({\lambda }^{a}x,{\lambda }^{b}y,{\lambda }^{c}z)$. It is generally singular but is a proper toric surface.

A normal affine toric variety $U_{\sigma }$ is smooth exactly when the primitive toric ray generators of $\sigma$ extend to a lattice basis of the ambient toric lattice $N$. In that case a choice of basis givesFor a two-dimensional toric cone in a rank-two toric lattice with primitive generators $u,v$, the criterion becomes $|\det (u,v)|=1$. In higher rank, a rectangular coordinate matrix has no such single determinant test; the generators must span a saturated subgroup of the correct rank. For example $(1,0),(1,2)$ span a real basis but have determinant two, giving a singular chart. This distinguishes smoothness from simpliciality.^[ref]

A toric variety is proper exactly when its fan is complete.

For a normal split toric variety $X_{\Sigma }$ of dimension $n$, the orbit-cone correspondence assigns an orbit $O(\sigma )$ of the algebraic torus to each toric cone $\sigma \in \Sigma$, withThe zero toric cone gives the dense algebraic torus; a ray of a fan gives a codimension-one orbit. The torus-fixed points correspond exactly to full-dimensional toric cones. A maximal toric cone need not be full-dimensional: the toric fan containing only the zero toric cone gives the algebraic torus itself and has no fixed point when $n>0$.

For affine space ${\mathbb{A}}^2$, the first-quadrant toric fan has a two-dimensional zero-cone orbit $(k^{\ast }{)}^2$, two one-dimensional toric ray orbits on the punctured axes, and one zero-dimensional orbit at the origin. The face-to-orbit reversal also explains the formula for an orbit closure in a toric variety. ^[ref]

For baryon-loading ratio $R=3{\rho }_b/(4{\rho }_{\gamma })$, the photon-baryon sound speed is $c_s^2=1/[3(1+R)]$.

Tidal stripping removes satellite material outside the tidal radius. The resulting mass loss weakens dynamical friction and can greatly delay or prevent a merger.

If a real algebraic number $\alpha$ has degree $d\ge 3$, then for every $\epsilon >0$ there are only finitely many reduced fractions $p/q$ such thatThe Thue-Siegel auxiliary polynomial proof chooses two such approximations, constructs an auxiliary polynomial nonzero at their pair, and compares its denominator lower bound with the upper bound supplied by its high-order zero at $(\alpha ,\alpha )$.

Let $F(X,Y)=P(X)+YQ(X)$ with $P,Q\in \mathbb{Z}[X]$ linearly independent, ${\mathrm{deg}}_{X}F\le n$, and $H(F)\le C^n$. For every $\delta >0$, if a reduced rational $p/q$ has sufficiently large denominator in terms of $C$ and $\delta$, then $F(X,y)$ has multiplicity at most $\delta n+1$ at $p/q$, for every fixed real $y$.

Indeed, the nonzero Wronskianhas degree below $2n$ and height at most $C_0^n$. A zero of multiplicity $r$ of $F(X,y)$ forces a zero of multiplicity at least $r-1$ of $W$. By Gauss lemma for polynomials, $(qX-p{)}^{r-1}$ then divides $W$ in $\mathbb{Z}[X]$, so $q^{r-1}$ divides the leading coefficient of $W$ and is at most $C_0^n$.

An isopycnal displacement is the vertical displacement $\xi$ of a constant-density surface. In linear stratified quasi-geostrophic flow,

A quasi-geostrophic vertical mode separates horizontal and vertical dependence of the streamfunction. With rigid horizontal boundaries and constant stratification, the vertical eigenfunctions are $\cos (n\pi z/H)$.

Canonical energy variance equals ${\partial }_{\beta }^2\log Z=k_B{T}^2{C}_V$.

Fourier's law predicts the local conductive heat flux from the local temperature gradient. In a continuum approximation with isotropic thermal conductivity $k>0$, its vector form and one-dimensional form areThe heat flux $q$ has units $\mathrm{W}{\mathrm{m}}^{-2}$, the temperature gradient has units $\mathrm{K}{\mathrm{m}}^{-1}$ and the thermal conductivity has units $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$. The rate of heat transfer $\dot{Q}$ has units $\mathrm{W}$. These units distinguish a local material response from the total power crossing a surface.

The minus sign makes heat flow toward lower temperature: for $k>0$, $\mathbf{q}\cdot \nabla T=-k|\nabla T{|}^2\le 0$. In the worked bar in thermal conduction, $dT/dx=-200\mathrm{K}{\mathrm{m}}^{-1}$ therefore gives $q=+400\mathrm{W}{\mathrm{m}}^{-2}$. Reversing the end temperatures reverses the heat flux. A zero temperature gradient gives zero conductive heat flux in this model.

Fourier's law is a constitutive relation expressing measured material behavior; it is not a consequence of conservation of energy alone. It applies locally and does not require a steady state. The straight-line bar solution additionally assumes constant thermal conductivity, constant cross-sectional area, no sources or side losses, and fixed end temperatures. Anisotropic materials require the directional response described by the thermal conductivity tensor. The scalar $k$ used in the examples expresses the stated isotropy assumption. ^[ref]

Historically, Joseph Fourier's Théorie analytique de la chaleur was published in 1822. In Chapter II, article 112, he writes the heat crossing a spherical surface as the product of thermal conductivity, elapsed time, surface area and the negative radial temperature gradient; article 113 balances the incoming and outgoing heat to obtain the heat equation for a sphere. The enduring idea is this connection between a local flow rule and a global prediction of changing temperature. ^{[ref] [ref]}

The temperature gradient is the vector $\nabla T$ describing the first-order spatial variation of a temperature field. It points toward fastest increase, and its magnitude is the increase per unit distance in that direction, measured in $\mathrm{K}{\mathrm{m}}^{-1}$. Along a small displacement $d\mathbf{x}$, $dT\simeq \nabla T\cdot d\mathbf{x}$.

For a uniform plate with $T(x)=T_0+ax$, the temperature gradient is $a{\mathbf{e}}_x$. In isotropic Fourier's law, heat flux is $\mathbf{q}=-k\nabla T$, so a larger gradient produces larger conductive transport at fixed thermal conductivity. In an anisotropic material, the thermal conductivity tensor can turn the flux away from the negative-gradient direction. A finite thermometer spacing estimates an average gradient over that spacing; resolving a thin thermal boundary layer requires a correspondingly fine spatial measurement.

Tensor product need not commute with an infinite direct product. For example,The element $(1\mathrm{mod}n{)}_n$ has infinite order in the product, so it survives rational localization.

For modules over a commutative ring, there is a natural isomorphismgiven by $f\mapsto (m\mapsto (n\mapsto f(m\otimes n)))$.

For a right module $M_R$, a left module ${}_{R}N$, and an abelian group $P$, every balanced map $b:M\times N\to P$ factors uniquely through an additive homomorphismExistence follows because additivity and the balancing equation are exactly the relations imposed on the generating symbols $m\otimes n$. Uniqueness follows because those symbols generate the tensor product as an abelian group.

For modules over a commutative ring, if $b$ is $R$-bilinear and the target is an $R$-module, the induced map is an R-module homomorphism. This is the linear version of the universal property; the additive version alone must not silently impose an action on the target.

For a right module $M_R$, a left module ${}_{R}N$, and an abelian group $P$, a balanced map is a map $b:M\times N\to P$ additive in each variable and satisfyingIt is exactly the sort of map represented by the tensor product of modules. The balancing equation transfers a scalar between the two inputs without assuming that $R$ is commutative.

When $R$ is a commutative ring and $P$ is an $R$-module, an $R$-bilinear map is balanced and induces an $R$-linear map on the tensor product. Mere additive balancedness gives an additive induced map; it does not by itself require linearity in the target. For example, $(z,w)\mapsto \overline{zw}$ is additive and balanced over $\mathbb{C}$, but its induced map is conjugate-linear.