The Gordon identity rewrites a vector-current matrix element between equal-mass on-shell Dirac spinors using a momentum term and an antisymmetric spin term. In the mostly-minus convention $\{{\Gamma }^a,{\Gamma }^b\}=2{\eta }^{ab}$, $(/p-m)u=0$, and ${\Gamma }^{ab}=[{\Gamma }^a,{\Gamma }^b]/2$, it reads $2m{\bar{u}}^{\prime }{\Gamma }^{a}u={\bar{u}}^{\prime }[(p^{\prime }+p{)}^a-{\Gamma }^{ab}(p_b^{\prime }-p_b)]u$. Derive it by adding the two external Dirac equation identities. The equivalent formula in the mostly-plus Dirac convention is ${\bar{u}}^{\prime }[{\gamma }^{ab}(p_b^{\prime }-p_b)+2m{\gamma }^a+p^a+p^{\prime a}]u=0$. The identity before division by $m$ remains valid for massless equal-mass states.

A Dirac bracket on a graded phase space eliminates second-class constraints while respecting the graded Poisson bracket. The fermionic kinetic term $i{\psi }^m{\dot{\psi }}_m/2$ gives $\{{\psi }^m,{\psi }^n{\}}_D=-i{\eta }^{mn}$. Quantization then gives $\{{\hat{\psi }}^m,{\hat{\psi }}^n{\}}_{+}={\eta }^{mn}$.

The Dirac scalar bilinear is invariant under the simultaneous spinor and adjoint transformations because ${\bar{\Psi }}^{\prime }{\Psi }^{\prime }=\bar{\Psi }{S}^{-1}S\Psi$. It is the mass term of the Dirac action. At fixed coordinates it transforms as a scalar field, with an orbital transport term and no spin term.

The dimensionless spectrum is $\mathcal{P}(k)=k^{3}P(k)/(2{\pi }^2)$. If it rises as $k^4$ below equality and as ${\ln }^{2}k$ above equality, then the dimensional spectrum rises as $k$ and falls as $k^{-3}{\ln }^{2}k$. Thus the dimensional spectrum has a turnover while the dimensionless spectrum only bends to a much slower rise. A schematic transition must smoothly join the two asymptotic regimes.

Evaluation of bounded-degree multivariate polynomials on a finite set defines a linear map $E_P:f\mapsto (f(p){)}_{p\in P}$. Its rank is at most $|P|$ and its kernel is the space of polynomials vanishing on $P$. The rank-nullity theorem gives $\dim \ker E_P\ge \left(\genfrac{}{}{0ex}{}{d+n}{n}\right)-|P|$ for degree at most $d$ in $n$ variables. Dependent point constraints only enlarge the kernel.

For the dicyclic group ${\mathrm{Dic}}_n=⟨a,b\mid a^{2n}=1,b^2=a^n,ba{b}^{-1}=a^{-1}⟩$, with $n\ge 2$, there are four linear characters and $n-1$ irreducible complex characters of degree two.

The linear characters are determined by $a\mapsto ϵ\in \{1,-1\}$ and $b\mapsto \beta$, where ${\beta }^2={ϵ}^n$. Their values are ${ϵ}^r$ on $a^r$ and ${ϵ}^r\beta$ on $a^{r}b$. When $n$ is even both choices of $ϵ$ allow $\beta =\pm 1$; when $n$ is odd, $ϵ=-1$ requires $\beta =\pm i$.

Put $\zeta =e^{\pi i/n}$. For $1\le j<n$, take the group representationIts character is $2\cos (\pi jr/n)$ at $a^r$ and zero at every $a^{r}b$. The distinct eigenlines of $a$ are exchanged by $b$, proving that this group representation is an irreducible representation. The degree-square sum $4+4(n-1)=4n$ proves completeness. These formulas specify every row of the character table; the elements outside $⟨a⟩$ form two conjugacy classes, rather than two distinct cosets of that index-two cyclic subgroup.

For a differentiable invertible matrix $A(t)$, $\frac{d}{dt}\log \det A=\mathrm{tr}(A^{-1}\dot{A})$ wherever a consistent logarithm is defined. In particular a positive spatial metric tensor satisfies $h^{ij}{\dot{h}}_{ij}=\dot{h}/h$. This separates its local volume change from shape change.

The block matrix $M=\left(\begin{array}{cc}A& -B\\ B& A\end{array}\right)$ is similar over $\mathbb{C}$ to $\mathrm{diag}(A+iB,A-iB)$ via $T=\left(\begin{array}{cc}I& I\\ -iI& iI\end{array}\right)$. Therefore its determinant is $\det (A+iB)\det (A-iB)$ for arbitrary square matrices $A,B$. The argument uses a constant change-of-basis matrix and multiplicativity of the determinant.

Let independent exposure sums $A,B$ have mean $N$ ADU at each detector pixel, with pure Poisson distribution noise and fixed gain $g$. The delta method gives $A/B\simeq 1+(A-N)/N-(B-N)/N$, so $\mathrm{Var}(A/B)\simeq 2/(gN)$ and $g\simeq 2/(N{E}^2)$. For nonuniform detector pixel means $N_i$, an unweighted spatial ratio variance is instead $(2/g)⟨1/N_i⟩$. This high-count approximation requires linear unsaturated data, bias subtraction and negligible read noise.

Differentiation computes the derivative of a function. A difference quotient approximates a derivative using finite increments; its limit is the derivative when the function is differentiable.

Automatic differentiation evaluates derivatives by applying the chain rule to the elementary operations executed by a numerical program. It propagates local sensitivities through the calculation; it does not estimate them by choosing a finite increment as in numerical differentiation.

For $f(x)=\sin (x^2)$, record $v_1=x^2$, $v_2=\sin v_1$. A forward calculation carries both values and derivatives:It returns $f^{\prime }(x)=2x\cos (x^2)$. More generally, seeding an input direction $r$ computes the product $J_{f}r$ of a Jacobian matrix and a vector. A reverse calculation starts from an output sensitivity and accumulates contributions toward the inputs; for a scalar objective it computes the full gradient efficiently. This is the mechanism of backpropagation. ^[ref]

The chain-rule result is exact in exact arithmetic for the differentiable operations used; a floating-point implementation still has roundoff error. Branches and nondifferentiable operations require care: a derivative propagated through the executed branch need not be an ordinary derivative of the whole program at a branch boundary. The same chain-rule machinery can differentiate reparameterization gradients through simulated trajectories.

The tangent line to a differentiable function at $a$ is its affine first-order approximation $f(a)+f^{\prime }(a)(x-a)$. For a differentiable concave function, it is an upper bound throughout a convex interval: the secant-slope inequality and passage to the derivative give $f(x)\le f(a)+f^{\prime }(a)(x-a)$. This upper bound constructs valid envelopes in adaptive rejection sampling.

A complete dense linear order without endpoints with the countable chain condition for a linear order, but with no countable order-dense subset.

For a map of oriented disk pairs, the multiplier on top relative homology is its relative mapping degree. Naturality of the connecting homomorphism identifies it with the mapping degree of the boundary map.

For $u_{xx}+y^2{u}_{yy}=0$ with zero vertical-side values on $0<x<\pi$, the Fourier coefficient $b_n(y)$ obeys the Cauchy-Euler differential equation $y^2{b}_n^{\prime \prime }=n^2{b}_n$. Boundedness at $y=0$ removes the negative indicial root, leaving $b_n(y)=A_n{y}^{(1+\sqrt{1+4n^2})/2}$. Bounded top coefficients give absolute convergence of the Fourier sine series below the top boundary.

For actuator pitch $D/N$, full-cycle spatial frequency components are limited to $N/2$ by sampling. Integer lattice counting in a circular frequency disk gives approximately $\pi N^2/4$ signed points; conjugate pairing halves the number of pairs, and two real wave phase quadratures per pair restore approximately $\pi N^2/4$ real coefficients. Counting only positive-quadrant frequency representatives gives $\pi N^2/16$, but it omits other orientations or quadratures unless a restricted convention is specified. Area approximations have boundary corrections and do not give exact integer ranks. A square sampling cutoff and a circular illuminated actuator footprint are distinct geometric restrictions.

For an independent exponential distribution default time of rate $\lambda$ and a continuous nonnegative martingale $S$, this process is a martingale. The compensating pre-default factor $e^{\lambda t}$ balances the survival probability $e^{-\lambda t}$. The strict survival inequality gives a càdlàg price process.

In deep inelastic scattering, the scaling variable is $x=Q^2/(2P\cdot q)$ with $Q^2=-q^2$. In the massless collinear parton model, the outgoing on-shell condition $(xP+q{)}^2=0$ identifies $x$ with the struck parton's incoming longitudinal momentum fraction.

The inelasticity in deep inelastic scattering is $y=P\cdot q/(P\cdot p)$, where $p$ is the incoming lepton momentum, $P$ the target momentum and $q$ the transfer. In the target rest frame it is the fraction of the incoming lepton energy transferred to the target. It is unchanged by replacing $P$ by a collinear parton momentum $\xi P$, and $Q^2=xyS$ in massless kinematics.

For a fixed source distribution and code alphabet, an optimal uniquely decodable code minimizes expected codeword length. Huffman coding gives an optimal binary prefix code, and the equivalence of attainable uniquely-decodable and prefix-code lengths gives the same minimum for both classes.