The divisible group $D=\mathbb{Q}/\mathbb{Z}$ is injective: every homomorphism $n\mathbb{Z}\to D$ extends to $\mathbb{Z}$ by choosing an $n$th division of its value at $n$, and the Baer criterion applies. It is a cogenerator because any nonzero element generates a cyclic subgroup admitting a character nonzero on that element, and injectivity extends that character to the whole group. Consequently a map $u:T\to U$ of abelian groups is injective exactly when restriction $U^{\ast }\to T^{\ast }$ is surjective: injectivity gives extension of characters, while a nonzero kernel element is detected by a character that cannot extend along $u$.

A prescribed PDE source term changes the heat equation to $u_t=D{u}_{xx}+f(x,t)$, or the transport equation to $u_t+c{u}_x=f(x,t)$. If $u$ is a temperature, $f$ has units of temperature per time. The formulas below assume $f=0$; a nonzero PDE source term can change the spatial average even with periodic boundary conditions.

Let $E$ be the forward shift on sequences and $P$ the polynomial defining a constant-coefficient linear recurrence relation. For a nonzero base $r$, $P(E)r^n=P(r)r^n$, so $r^n/P(r)$ is a particular solution when $P(r)\ne 0$. At a simple root of the characteristic equation of a linear recurrence, the correct trial acquires a factor $n$:The identity follows by substituting $(n+j)r^{n+j}$ into each shift term. A root of multiplicity $m$ analogously requires a degree-$m$ polynomial factor; the lower-degree homogeneous terms are supplied by the repeated characteristic root of a linear recurrence.

The unit-response sequence $a_1=a_2=1$, $a_{n+2}-a_{n+1}+a_n=0$ is six-periodic. In the displayed convolution, all older terms cancel under $P_{k+2}-P_{k+1}+P_k$ by this recurrence; the two new boundary terms reduce to $f_{k+2}$. Thus it gives a particular solution of the inhomogeneous linear recurrence with forcing $f_{k+2}$. Add the two-dimensional homogeneous span of $\cos (k\pi /3)$ and $\sin (k\pi /3)$ for the general solution. This is a discrete Green function construction.

A singularity of a massless scattering or radiation integral when momenta of two particles become parallel. In quantum chromodynamics, unresolved quark-gluon splittings cause such divergences. A mass regulator or dimensional regularization exposes them; they cancel between suitable inclusive real and virtual contributions or are absorbed into factorized distributions where appropriate.

An observable unchanged by adding a particle with vanishing energy or replacing one massless particle by exactly collinear particles with the same total momentum. This property permits cancellation of real and virtual infrared divergences and collinear divergences in its perturbative cross-section. Fully inclusive electron-positron hadronic rates and suitably defined jet observables are examples.

For a vertex law $P$ on a graph, choose $X\sim P$ and an independent set valued random variable $Y$ containing $X$ almost surely. Minimize their mutual information over all such joint laws. This defines graph entropy. It differs from the entropy of a proper coloring, because the random independent set need not be a deterministic color class.

The self-information of a positive-probability outcome is the negative logarithm of its probability. Independent outcomes have additive self-information. Its expectation is Shannon entropy.

// Target: probability-and-statistics.bigb

This is the entropy in base $q$ of the distribution with one symbol of probability $1-x$ and each of the other $q-1$ symbols of probability $x/(q-1)$. It increases on $[0,1-1/q]$ from zero to one, and governs the exponential volume of a Hamming ball. For $q=2$ it is the binary entropy function.

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If $U,V$ are independent random variables taking values in a countable additive group, conditioning on $V$ makes $U+V$ a translate of $U$. Consequently $H(U+V\mid V)=H(U)$, and conditional entropy gives $H(U+V)\ge H(U)$. If the two information entropies are finite, equality holds exactly when $U+V$ and $V$ are independent.

// Target: probability-and-statistics.bigb

Cross-entropy measures the average logarithmic loss incurred when outcomes follow one probability distribution $P$, but a model assigns probabilities using another probability distribution $Q$. For a finite alphabet, write their probability mass functions as $p_i$ and $q_i$. With base-two logarithms,The expectation is under the true or stipulated source law $P$, rather than the model $Q$. Terms with $p_i=0$ contribute nothing. If $p_i>0$ but $q_i=0$ for any outcome, the cross-entropy is $+\infty$: the model rules out an event that can occur. The result is measured in bits; natural logarithms give nats.

The useful identity is cross-entropy equals entropy plus relative entropy:The source's information entropy $H(P)$ is the baseline uncertainty. The Kullback-Leibler divergence is the extra average cost of a mismatched model, with the true law first and model second. Thus optimizing $Q$ for a fixed $P$ reduces the mismatch, rather than changing the source's own uncertainty. For finite alphabets, the minimum is $H(P)$, attained exactly at $Q=P$.

Take a three-symbol source with $P=(1/2,1/4,1/4)$. A matched binary prefix code assigns $A\mapsto 0$, $B\mapsto 10$, $C\mapsto 11$, with lengths $(1,2,2)$. A mismatched prefix code instead assigns $A\mapsto 10$, $B\mapsto 11$, $C\mapsto 0$, with lengths $(2,2,1)$, corresponding to $Q=(1/4,1/4,1/2)$. Both prefix codes decode exactly, but the most common symbol is shorter in the matched code. Under the same source $P$,The mismatched code costs an extra (1/4) bit per source symbol on average. No information is lost; the penalty is inefficient representation of this source. The mismatch penalty in lossless coding explains when this coding interpretation is exact.

Cross-entropy is also useful when evaluating a predictive distribution. A report that places tiny probability on the realized outcome receives a large logarithmic loss, even when it usually predicts the most common outcome correctly. The loss convention is the negative of the reward convention in the logarithmic scoring rule. Its expected loss is uniquely minimized by the correct probability distribution, making it a strictly proper scoring rule. Strictly Proper Scoring Rules, Prediction, and Estimation by Tilmann Gneiting and others treats this connection in §3.1, Example 3; its discussion uses rewards, so the signs are reversed for the losses here.^[ref]

I. J. Good's Rational Decisions by I. J. Good appeared in 1952. Its published summary describes a method for encouraging accurate probability estimates and a connection with information theory. Strictly Proper Scoring Rules, Prediction, and Estimation by Tilmann Gneiting and others traces the logarithmic scoring rule at least as far back as this 1952 paper.^[ref]

Joint entropy is the information entropy of an entire random vector, treating its coordinates together as one outcome. For random variables with finite alphabets and joint probability mass function $p$,It is measured in bits. The distinction from separate information entropies is whether shared uncertainty is counted once or repeatedly. Two copies of the same fair bit have joint entropy one bit, since only $00$ and $11$ can occur. Two independent fair bits have joint entropy two bits, since all four pairs are possible and equiprobable.

For a less symmetric example, assign probabilities $1/2,1/4,1/4$ to $(X,Y)=(0,0),(1,0),(1,1)$ respectively. ThenHere $X$ is fair, while $\mathbb{P}(Y=1)=1/4$. Thus $H(X)=1$ and $H(Y)=h_2(1/4)\approx 0.811278$ bits, where $h_2$ is binary entropy. Adding the separate information entropies would give approximately $1.811278$ bits, overcounting by the mutual information $I(X;Y)\approx 0.311278$ bits.

The chain rule for information entropy makes the shared information explicit:In this example $Y$ is certain given $X=0$ and fair given $X=1$, so its average conditional entropy is $H(Y\mid X)=\frac{1}{2}\cdot 0+\frac{1}{2}\cdot 1=1/2$ bit. Sending the first coordinate changes how much uncertainty remains in the second. A prefix code with codewords 0, 10, 11 for the three possible pairs has average length $3/2$ bits, exactly the joint entropy in this stipulated law.

For finite alphabets,The lower bound follows because knowing the full random vector determines every coordinate; the upper bound follows from conditioning reduces entropy in the chain rule for information entropy. Equality in the upper bound means joint independence. The difference from that upper bound is total correlation, whose noisy parity total correlation example shows why separate pairwise comparisons need not find all the dependence.

A homogeneous canonical scalar field with positive scalar potential has energy density ${\rho }_{\varphi }={\dot{\varphi }}^2/2+V$ and pressure $P_{\varphi }={\dot{\varphi }}^2/2-V$. When its potential energy dominates its kinetic energy, the equation-of-state parameter approaches $-1$. If other components and spatial curvature are negligible, the Friedmann acceleration equation gives accelerating expansion. Small initial kinetic energy is not enough to ensure long cosmic inflation; the field must also remain on a sufficiently flat part of the scalar potential, as measured by the potential slow-roll parameters.

The inflaton potential is the potential energy density of an inflaton as a function of its field value. Its derivative enters the inflaton equation of motion, while its slope and curvature enter the potential slow-roll parameters.

Choose a primitive counterexample with minimal $z$ and $y$ even. Its Pythagorean triple $(x^2,y^2,z)$ has $x^2=m^2-n^2$, $y^2=2mn$. Modulo four forces $m$ odd and $n$ even, so $m=u^2$ and $n=2v^2$. The primitive triangle $(x,2v^2,u^2)$ then has a parametrization with $v^2=rs$ and $u^2=r^2+s^2$. Coprimality makes $r=a^2,s=b^2$, giving $a^4+b^4=u^2$ with $u<z$. This contradicts infinite descent.

The essential infimum and essential supremum describe the lower and upper bounds of a measurable function after disregarding null sets for a specified measure. A pointwise infimum or supremum sees every input; an essential bound sees whether values below or above a proposed threshold occur on a set of positive measure. Thus "essential" means insensitive to changes on a null set, rather than insensitive to unusual values. This distinction makes essential bounds compatible with Lebesgue integrals and the equivalence classes used in Lp spaces. ^[ref].

For a real measurable function $f$ on a measure space $(X,\Sigma ,\mu )$, the definitions areThese can take values in the extended real number line. On a space of positive total measure, finite essential bounds obey ${\mathrm{essinf}}_{\mu }f\le f\le {\mathrm{esssup}}_{\mu }f$ almost everywhere. If $s={\mathrm{esssup}}_{\mu }f$ is finite, every threshold $b<s$ has $\mu (\{f>b\})>0$; the corresponding lower-edge statement is $\mu (\{f<a\})>0$ for every $a>{\mathrm{essinf}}_{\mu }f$. These are the measure-based counterparts of the approximation property of a supremum and approximation property of an infimum.

Here is a two-sided example on $[0,1]$ with Lebesgue measure:The pointwise bounds are $\inf g=-3$ and $\sup g=4$, while the essential bounds are $\mathrm{essinf}g=0$ and $\mathrm{esssup}g=1$. Both exceptional singletons are null sets. But for $0<a<1$, the inputs where $g<a$ still have measure $a$, and those where $g>a$ have measure $1-a$: deleting or adding two individual inputs cannot change those lengths. Hence neither essential bound can be moved inward.

The choice of measure is part of the answer. The same $g$ has essential bounds $-3$ and $4$ under counting measure, because every nonempty set has positive measure. Under the Dirac measure at $1/4$, both essential bounds equal $-3$. A null set for one measure need not be a null set for another.

Nor must the exceptional inputs be isolated. The Dirichlet function, equal to one at rational numbers and zero at irrational numbers, has pointwise supremum one on $[0,1]$, but essential supremum zero for Lebesgue measure: the rational numbers form a countable set that is a null set, even though they are dense. Conversely, a value ten on an interval of length $0.001$ affects the essential supremum just as decisively as it would on half the domain. Small positive measure is not zero measure.

For a character $\psi$ of $H\le G$, the induced character isThis is the character of a representation of $\mathbb{C}[G]{\otimes }_{\mathbb{C}[H]}{V}_{\psi }$. Substituting this expression into the character inner product and putting $g=xh{x}^{-1}$ proves Frobenius reciprocity.

Every complex irreducible representation of a finite p-group is induced from a one-dimensional representation of a subgroup. Induct on the group order. Nonfaithful representations reduce to a smaller quotient. For a faithful representation of a nonabelian group, choose an abelian normal subgroup not contained in the centre. Restriction cannot be isotypic, since it would make that subgroup scalar and faithfulness would then force it central. The Clifford correspondence induces the representation from a proper subgroup; induction there and transitivity of induction complete the proof. Abelian groups have only one-dimensional complex irreducible representations.

For a smooth linear character $\chi$ of a parabolic subgroup $B$, unnormalized induction consists of smooth functions satisfying $f(bg)=\chi (b)f(g)$, with right-translation action. The quotient $B\backslash G$ is compact for the local general linear groups in question. Normalized induction inserts a square root of the parabolic modulus in the covariance, changing formulas for the inducing linear character.

Let $H$ be an open normal subgroup of a compact topological group $G$, and let $\tau :H\to {\mathrm{GL}}_m({\mathbb{Z}}_p)$ be a continuous faithful representation. Choose representatives $t_j$ for the finite left cosets $G/H$. Write $g{t}_j=t_i{h}_{ij}$ and act on the $j$-th summand of ${⨁}_j{\mathbb{Z}}_p^m$ by $\tau (h_{ij})$ into the $i$-th summand. The cocycle identity proves this is a continuous group representation. An element outside $H$ has a nontrivial coset permutation, while an element of $H$ acting trivially has $\tau (g)=1$ on the identity-coset summand. Hence the representation of $G$ is faithful and integral.