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Feynman rules translate each diagram into momentum-space factors for propagators, vertices, external states, momentum conservation and loop integration.

A ferromagnetic magnon is a quantized spin wave. Without a linear restoring term its long-wavelength frequency is proportional to $k^2$.

Easy-plane anisotropy penalizes spin components normal to a preferred plane, leaving a low-energy angular degree of freedom within that plane.

A Fermi gas consists of noninteracting fermions whose one-particle states are occupied according to the Fermi-Dirac distribution.

For a Fermat prime $p=2^{2^k}+1$, the multiplicative group modulo $p$ has order $2^{2^k}$. Its $\phi (p-1)=(p-1)/2$ primitive roots are all quadratic nonresidues; since there are also $(p-1)/2$ nonresidues, the two sets coincide.

Lenz's law expresses the minus sign in Faraday's law of induction: the induced electromotive force drives a passive resistive loop's electric current in the direction whose own magnetic field opposes the imposed change in magnetic flux. It opposes the change of flux, not invariably the external magnetic field. Choose an oriented surface and its positive boundary direction first, find the sign of $d{\Phi }_{\mathrm{ext}}/dt$, and then use the right-hand rule to translate the signed electric current into a physical direction. ^[ref].

In the fixed-loop example in Faraday's law of induction, out-of-page magnetic flux increases at $0.0060Wb/s$. With counterclockwise positive, $I=-0.0030\mathrm{A}$ is clockwise, producing an in-page magnetic field through the loop. Reverse the ramp to $B(t)=0.40\mathrm{T}-(0.30T/s)t$ on the same one-second interval: now $\mathcal{E}=+0.0060\mathrm{V}$ and $I=+0.0030\mathrm{A}$. The counterclockwise electric current produces an out-of-page magnetic field, supporting the decreasing external magnetic flux. A steady out-of-page magnetic field with fixed loop geometry gives no induced electromotive force in either direction.

For the increasing-area example in motional electromotive force, clockwise electric current runs down the right-hand rod. Its magnetic force on a current-carrying wire points left, opposing the imposed motion to the right. Writing $I_{\mathrm{mag}}=B\ell v/R$ for the current magnitude, the power required to maintain constant speed isThus the sign agrees with conservation of energy: the external drive supplies the Joule heating. The magnetic forces do no work on individual charges; mechanical work by the drive supplies the electrical energy. This also explains the usefulness of electromagnetic braking of a sliding rod and the terminal speed of a falling conducting loop. ^[ref].

The sign of induced electromotive force alone does not determine an arbitrary circuit's instantaneous electric current. Retaining constant self-inductance $L$ gives $LdI/dt+RI=-d{\Phi }_{\mathrm{ext}}/dt$, so a pre-existing electric current can persist while the induced drive opposes its change. The distinction matters in the lossless limit of magnetic loop braking. The resistive toy examples use $L=0$ as an approximation, not a general consequence of Lenz's law.

For a film of thickness $h$ flowing down a slope of angle $\alpha$, an upslope surface stress of magnitude $\tau$ givesThe surface velocity, total flux, and basal shear reverse atrespectively.

The extremal number $\mathrm{ex}(n,H)$ is the largest number of edges in an $n$-vertex graph containing no subgraph isomorphic to $H$.

A differential form $\alpha$ is closed when $d\alpha =0$.

For fixed positive integer $r$, the failures-before-the-$r$th-success distribution is a one-parameter exponential family withand carrier $h(x)=\left(\genfrac{}{}{0ex}{}{x+r-1}{x}\right)$.

The deviance from ${\theta }_1$ to ${\theta }_2$ is twice the Kullback-Leibler divergence:

For an expected value parameter $\mu >0$ and a shape parameter $\lambda >0$, the inverse Gaussian distribution has probability density functionIts mean is $\mu$ and its variance is ${\mu }^3/\lambda$.

The cumulant function, or log-partition function, is the normalizing termwith a sum in the discrete case.

In the canonical representation$\theta$ is the natural parameter and $T$ is the natural statistic.

Every finite subgroup of $K^{\times }$ is cyclic. If its exponent is $m$, all its elements are roots of $X^m-1$, so its size is at most $m$; finite abelian group theory supplies an element of order $m$, forcing equality.

For $q>0$ and a random variable $X$ with finite absolute $q$th moment, Tonelli theorem gives

Let $X_1,\dots ,X_n$ be an IID sample of real-valued random variables with finite expected value $\mu$ and finite variance ${\sigma }^2$, where $n\ge 1$. Its sample mean $\overline{X}=n^{-1}{\sum }_i{X}_i$ satisfiesLinearity of expectation gives the first identity. The variance of a sum and vanishing cross covariances give $\mathrm{Var}({\sum }_i{X}_i)=n{\sigma }^2$; scaling by $1/n$ scales the variance by $1/n^2$. Pairwise independent random variables with the same expected value and variance suffice for these identities, even without full independence. Grinstead and Snell state the corresponding finite-variance independent-trials result.^[ref]

The standard deviation of the sample mean is $\sigma /\sqrt{n}$. Four times as many observations halve this scale; they do not divide it by four. In the fair Bernoulli distribution example, $\sigma =1/2$, so sample sizes $4$ and $64$ give standard deviations $1/4$ and $1/16$, respectively. The common-coin averaging example shows why merely repeating observations with a common marginal law does not justify this calculation.

Chebyshev's inequality turns the variance identity into a probability bound: for $\epsilon >0$,This proves convergence in probability of the sample mean to $\mu$ in this finite-variance setting, a form of the weak law of large numbers. It is an upper bound, so values above one convey no improvement over the universal probability bound one. Cauchy sample mean stability lies outside these hypotheses.

For integrable random variables $X_1,\dots ,X_n$ on the same probability space and fixed real numbers $a_1,\dots ,a_n$, linearity of expectation statesThe random variables can be dependent random variables. The formula follows by the linearity of integration; a finite linear combination of integrable random variables remains integrable. For example, if $X$ takes $-1,2,8$ with probabilities $1/2,3/10,1/5$, then $X$ and $2X$ are dependent random variables, yet $\mathbb{E}[X+2X]=3\mathbb{E}[X]=5.1$.

An especially useful application counts how many events occur. Put $I_i=1_{A_i}$, an indicator random variable, and $N={\sum }_i{I}_i$. Since $\mathbb{E}[I_i]=\mathbb{P}(A_i)$, we get $\mathbb{E}[N]={\sum }_i\mathbb{P}(A_i)$. There is no need to compute the whole probability distribution of $N$, or to assume independence. For a uniform random permutation of $n\ge 1$ objects, each object stays fixed with probability $1/n$, so the fixed point count of a uniform random permutation has expected value one for every $n$. ^[ref].

Linearity of expectation concerns addition and multiplication by fixed scalars. Products instead satisfy $\mathbb{E}[XY]=\mathbb{E}[X]\mathbb{E}[Y]$ under the hypotheses of expectation of a product of independent random variables. For the example above, $\mathbb{E}[X^2]=14.5$, whereas $(\mathbb{E}[X]{)}^2=2.89$. Applying a nonlinear function before averaging can therefore change the answer substantially; use the law of the unconscious statistician to compute that average.

For nonnegative $Y$, $E[Y]={\int }_0^{\infty }P(Y\ge t)dt$; for nonnegative integer $N$, $E[N]={\sum }_{k\ge 1}P(N\ge k)$.

An indicator random variable is the indicator function of an event $A$ in a probability space: $1_A$ is one when $A$ occurs and zero otherwise. It has a Bernoulli distribution with parameter $p=\mathbb{P}(A)$, so $\mathbb{E}[1_A]=p$ and $\mathrm{Var}(1_A)=p(1-p)$. Even an event described without numbers becomes a numerical random variable in this way.

For a finite collection of events, their indicator random variables add up to the number that occur. Linearity of expectation therefore turns an average count into a sum of probabilities, without requiring independence. The fixed point count of a uniform random permutation provides an example where the indicators are dependent random variables.