For finite alphabets, $H(X_1,\dots ,X_n)\ge {\max }_{i}H(X_i)$, soFor $n$ identical copies of one fair bit, the upper bound is attained and total correlation is $n-1$ bits. Adding another copy increases the result without introducing a new underlying random choice. The choice of coordinates and their grouping therefore matters when comparing systems of different sizes.

For more general random variables, the relative entropy against the product of the marginals supplies a definition of total correlation that may be infinite. Subtracting differential entropies is justified only when those quantities are finite and the entropy identity is valid; an undefined difference of infinities is not a definition. The finite-alphabet examples above avoid that problem entirely.

Let $P$ be a joint probability distribution on a finite product alphabet, with marginals $P_i$. Compare it with a product model $Q={⨂}_i{Q}_i$, assuming $Q_i(x_i)>0$ wherever $P_i(x_i)>0$. Adding and subtracting the logarithm of ${\prod }_i{P}_i(x_i)$ givesThe first term is the unavoidable error from discarding dependence; the other terms are errors in the individual marginal probability distributions. Averaging a term that depends only on $x_i$ under $P$ is the same as averaging it under $P_i$, which proves the identity directly.

Among product laws, the choice $Q_i=P_i$ minimizes this relative entropy, leaving exactly total correlation. The related cross-entropy equals entropy plus relative entropy identity interprets this as extra average logarithmic loss from using the independent model. It is a statement about a specified joint law and loss function, not a claim that fitting a finite sample automatically recovers that law.

For finite alphabets, expanding joint entropy with the chain rule for information entropy givesEach term is mutual information with an entire preceding block, rather than necessarily with one preceding variable. Changing the order changes these individual terms but leaves their sum unchanged.

For the noiseless parity constraint and total correlation, $I(Y;X)=0$ and $I(Z;X,Y)=1$ bit. The conditional block term supplies exactly the dependence that the pairwise comparisons missed. For three identical copies of a fair bit, the two block terms are both one bit. Thus total correlation counts total departure from the product law, not merely the presence of a single irreducible multiway relation.

Let $X,Y$ be independent fair bits and set $Z=X\oplus Y$, their exclusive or. The four possible triples are $000,011,101,110$, each with probability $1/4$. Every coordinate is fair, and every pair has all four outcomes with equal probability. Thus each pair's mutual information is zero, as in pairwise independence does not imply joint independence.

The full random vector has joint entropy ${\log }_{2}4=2$ bits, while the marginal information entropies sum to three bits. ConsequentlyThe product of the marginals predicts probability $1/8$ for every triple, including the four impossible ones. On each possible triple, the ratio of true to product probability is $(1/4)/(1/8)=2$; averaging its base-two logarithm gives the same one bit of relative entropy.

The example gives a concrete use for joint information measures. A lossless source coding scheme that treats the three coordinates separately needs three bits here. If the decoder knows the integer parity rule, the encoder can send just $X,Y$ and let the decoder reconstruct $Z$ exactly. Pairwise independence does not reveal that saving. This is a stipulated source law and an explicit two-bit code, rather than a promise that arbitrary data can be shortened.

Use $c=1$, $a_0=1$, positive pressureless matter and positive cosmological constant, with ${\Omega }_m+{\Omega }_{\Lambda }=1$ and no radiation. The Friedmann equation gives ${\chi }_{\max }=H_0^{-1}{\int }_0^{\infty }dy/({\Omega }_{\Lambda }+{\Omega }_m{y}^3{)}^{1/2}$. Set $y=({\Omega }_{\Lambda }/{\Omega }_m{)}^{1/3}u$, then $v=u^3$. The remaining integral is $\frac{1}{3}{\int }_0^{\infty }{v}^{-2/3}(1+v{)}^{-1/2}dv=\frac{1}{3}B(1/3,1/6)$, proving the formula by the beta function integral. Its two endpoints converge for positive densities, but the lifetime diverges in either pure-fluid limit.

A damped torsional oscillation is rotation about a fiber's fixed reference orientation with a restoring torque and dissipative torque. In the ideal viscous model, damping contributes $-b\dot{\theta }$ with constant $b\ge 0$. This is a rotational damped harmonic oscillator. For known constant moment of inertia $I$ and torsion constant $\kappa$, define $\gamma =b/(2I)$. In the oscillatory regime ${\gamma }^2<\kappa /I$,$A$ and $\phi$ are set by the initial state. Successive positive peaks measured from the same equilibrium have magnitudes $p_n$ and $p_{n+1}$; their ratio gives $\delta =\ln (p_n/p_{n+1})=\gamma T_d$. Consequently a measured damped oscillation period and decay together recoverUsing only $4{\pi }^{2}I/T_d^2$ underestimates the torsion constant in this model. The toy curve in torsion constant sets $I=2.0\times 10^{-6}\mathrm{kg}{\mathrm{m}}^2$, $\kappa =1.0\times 10^{-8}\mathrm{N}m/rad$, and $\gamma =0.020{\mathrm{s}}^{-1}$. It gives $T_0\simeq 88.86\mathrm{s}$, $T_d\simeq 92.64\mathrm{s}$, and $\delta \simeq 1.853$. Ignoring the decay recovers $9.2\times 10^{-9}\mathrm{N}m/rad$, an $8\%$ underestimate; including it recovers the stipulated torsion constant.

These equations assume a single rotational mode, a stationary zero, linear elasticity, and viscous damping. Internal fiber losses need not behave as a constant viscous coefficient: Fleischer and colleagues model their precision apparatus using a loss angle. Their response is evidence for specifying a damping model, not for applying the viscous correction universally. ^[ref]

Given a dual pair of free lattices $N,M$ and a rational fan, glue $\mathrm{Spec}A[{\sigma }^{\vee }\cap M]$ over common-face charts. The monoid rings are semigroup algebras. A lattice map respecting cones gives a toric morphism. This construction works over a commutative ring $A$, including $\mathbb{Z}$, and also for an infinite fan in toric geometry.

// Target: toric-geometry.bigb

A toric lattice is a finite-rank free abelian group $N$, embedded as a discrete full-rank subgroup of $N_{\mathbb{R}}=N{\otimes }_{\mathbb{Z}}\mathbb{R}$. Its dual is $M=\mathrm{Hom}(N,\mathbb{Z})$. This lattice of integral vectors is distinct from an order-theoretic lattice. It supplies the integral structure for toric cones and their dual monomials.

Let $A$ be a ring, $I$ a right ideal, and $M$ a left module. Then $A/I$ and the resolving modules in this argument are right $A$-modules, while the second tensor factor is a left $A$-module. Start a free resolution of $A/I$ with $P_0=A$ and a free module $P_1$ surjecting onto $I$. The next differential presents the relations in $I$. Right exactness of the tensor product of modules identifies $\mathrm{coker}(P_2{\otimes }_{A}M\to P_1{\otimes }_{A}M)$ with $I{\otimes }_{A}M$. Taking degree-one homology then gives the displayed kernel formula. Hence vanishing for every ideal is equivalent to the flatness criterion using ideals.

For a commutative ring $A$ and a non-zero-divisor $a\in A$, resolve $A/(a)$ by $0\to A\stackrel{a}{\to }A\to A/(a)\to 0$. Tensoring with $M$ gives the kernel and cokernel in the displayed formula; all higher Tor functors vanish. Resolving either argument gives the same homology by the two filtrations of the double complex of the two free resolutions. For $M=A/(a)$, both degrees zero and one equal $A/(a)$.

A continuous linear operator is a linear map that is continuous for the topologies of its domain and codomain. Between normed vector spaces, continuity is equivalent to an operator bound $\Vert Tx\Vert \le C\Vert x\Vert$ for some finite $C$, so the terms bounded linear operator and continuous linear operator coincide there.

For general topological vector spaces, sending every bounded set in a topological vector space to a bounded set is a weaker property. For example, the identity from the weak topology on an infinite-dimensional Hilbert space to its norm topology maps bounded sets to bounded sets by the Uniform boundedness principle, but is not continuous. A basic weak zero neighbourhood only restricts finitely many linear functionals, leaving a nonzero common-kernel direction with arbitrarily large norm.

Thus the operator norm interpretation of boundedness used throughout numerical linear algebra belongs to the normed-space setting. In other vector topologies, specify continuity explicitly.

A dense subspace of a topological vector space $X$ is a linear subspace $D$ with $\overline{D}=X$. For a normed vector space, every vector of $X$ is a limit of a sequence from $D$. A continuous linear operator into a Hausdorff target is uniquely determined by its values on $D$.

Existence of an extension needs additional hypotheses. A bounded linear operator from $D$ into a Banach space extends uniquely to $X$ by taking limits, as in extension of a bounded linear operator from a dense subspace. An incomplete target can obstruct this construction, as the failure of dense-subspace extension into an incomplete codomain shows.

An untwisted band is the topological surface obtained from a rectangle by identifying $(0,v)$ with $(1,v)$. It is the Cartesian product of a circle and a closed interval, and is homeomorphic to a closed annulus. A finite portion of the lateral surface of a circular cylinder realizes it in three-dimensional space. Its two boundary circles are $S^1\times \{-1\}$ and $S^1\times \{1\}$, and the matching seam directions make it an orientable surface. The Möbius band changes just one identification sign and loses both this two-component boundary and its orientability.

The sphere with three points removed is an orientable noncompact surface of genus zero with three ends. It can be realized by pairing the two sides at each of two opposite ideal vertices of an ideal hyperbolic quadrilateral in the hyperbolic disk. The four ideal vertices fall into three equivalence classes; after adding these three points the cell structure has one face, two edges and three vertices, hence Euler characteristic two. Removing these added vertices recovers the original quotient.

A connected compact surface is determined by its orientability, number $b$ of boundary circles and genus. An orientable surface of genus $g$ has Euler characteristic $2-2g-b$; a surface which is not orientable with $r$ crosscaps has Euler characteristic $2-r-b$. In particular, a closed connected surface with vanishing first mod-two homology is a sphere, and a connected closed orientable surface of Euler characteristic zero is a torus.

The one-point compactification of a locally compact noncompact Hausdorff space adjoins one point whose neighbourhoods are complements of compact subsets. The result is compact Hausdorff.

A topological space is $T_0$ when every pair of distinct points is distinguished by an open set containing exactly one of them.

A net is a family indexed by a directed ordered set: every pair of indices has a common upper bound. It converges to $x$ when it is eventually in every neighbourhood of $x$. Unlike sequences, nets characterize closure in every topological space: a point is in the closure of a set precisely when some net in that set converges to it.

A topological space is sober when every nonempty irreducible closed subset is the closure of a unique point. Every Hausdorff space is sober. In contrast, an infinite set with the cofinite topology has the whole space irreducible and closed, while every singleton is closed, so it is not sober.

A space is sigma-compact if it is a countable union of compact subsets. A locally compact, sigma-compact metrizable space is second countable: each compact metrizable subset has a countable dense set, their union is countable and dense, and a separable metric space has a countable basis.