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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.

Four one-dimensional characters and $n-1$ two-dimensional characters exhaust a dicyclic group of order $4n$. In the two-dimensional rows, the values at $a^r$ are $2\cos (\pi jr/n)$ and those on the two noncyclic cosets are zero. The one-dimensional rows depend on whether $n$ is odd or even.

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 of a computational graph using the chain rule. Reverse mode is backpropagation and is useful for 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.

In a Grothendieck topos, an object is a quotient of a decidable object when it has an epic cover by one. These objects are closed under subobjects, small coproducts, quotients and finite limits. Their union inside any fixed object gives the largest subobject of this kind, defining a coreflective subcategory. The induced idempotent comonad is left exact, so the full subcategory is a topos.

A dbar equation in a complex spectral parameter, rather than the original spatial coordinate. In spectral reconstruction, its right-hand side is transformed data. A Cauchy-Pompeiu formula in the spectral plane reconstructs the auxiliary function; an asymptotic coefficient or the original differential equation then recovers the spatial field.

With permeability $K$, depth $d$, kinematic viscosity $\nu$ and the relevant effective thermal diffusivity ${\kappa }_T$, the Darcy Rayleigh number compares buoyancy-driven Darcy advection to thermal diffusion. It differs from the clear-fluid Rayleigh number by replacing $d^3$ with $Kd$. Effective heat capacity and porosity factors may be absorbed into the definition of ${\kappa }_T$ and the thermal-time scale, and must be kept consistent with the chosen Darcy law convention.