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A hybrid infrared detector bonds a photosensitive semiconductor array to a separate silicon readout circuit. Each pixel has its own electrical connection; charge need not be transferred across the array as in a charge-coupled device.

Correlated double sampling differences a signal reading and its associated reset or baseline reading. It suppresses correlated offsets and reset noise, while the noise of the two measurements must still be propagated.

Dark current is charge produced in a detector without illumination, often by thermally excited carriers. Cooling reduces it, although cooling requirements also depend on the wavelength range and thermal background.

A photodiode is a semiconductor diode producing photocurrent from photon absorption. Exciting an electron across the band gap leaves an electron hole. The depletion region's electric field separates the charge carriers; others can arrive by diffusion before electron-hole recombination. Contacts complete the electrical circuit.^[ref]

The useful bridge is from light to a measurable electric current: spectral responsivity converts incident radiant flux into photocurrent, while quantum efficiency counts collected signal charge carriers per incident photon. These answer different questions: how much electric current one watt produces, and how many incident photons contribute to the signal. The worked calculation below makes their relationship explicit.

The worldline einbein imposes the relativistic mass-shell condition. With the Minkowski metric of signature $(-,+,\dots ,+)$, its first-class constraint is $\phi =(p^2+M^2)/2$.

For a $C^4$ real map $f(u)=-u+c_2{u}^2+c_3{u}^3+O(u^4)$, the second iterate isIf $c_3+c_2^2>0$, sufficiently small nonzero $u$ retains its sign under $f^2$ and strictly decreases in absolute value. Its iterates therefore converge to zero, proving local asymptotic stability even though $f^{\prime }(0)=-1$. If $c_3+c_2^2<0$, the second iterate moves small positive points away from zero, proving instability. The zero-coefficient case requires higher-order terms. With strictly stable transverse directions, the centre manifold theorem for a discrete dynamical system applies this scalar test to a higher-dimensional discrete dynamical system.

For $F_{\mu }(x)=\mu x+b{x}^2+a{x}^3$ near $\mu =-1$, write $\mu =-1+ϵ$. Then $F_{\mu }^2(x)-x=-2ϵx-2(a+b^2)x^3+O({ϵ}^{2}x,ϵx^2,x^4)$. The small two-cycle is attracting on $\mu <-1$ if $a+b^2>0$ and repelling on $\mu >-1$ if $a+b^2<0$. Vanishing of $a+b^2$ gives a degenerate flip requiring higher terms.

A path graph $P_n$ has $n$ vertices in a single chain and $n-1$ edges. Its endpoint graph distance is $n-1$ and, with unit resistances, its endpoint effective resistance is also $n-1$.

For a partition regular matrix $A$ and any property $P$ of its positive solution vectors, either every finite coloring admits a monochromatic solution with $P$, or every finite coloring admits one without $P$. If the first alternative fails, take a witnessing coloring and pair it with any other coloring. Partition regularity of the product coloring forces a solution avoiding $P$ in the other coloring. The alternatives need not be exclusive.

If a rational matrix is a partition regular matrix, then for each fixed number of colors $k$ some finite integer interval $[T]$ already forces a monochromatic positive solution. Otherwise the solution-free finite colorings of successive intervals form a finitely branching tree with every level nonempty. The König infinity lemma gives an infinite branch, contradicting partition regularity. More generally, this argument applies to any family of configurations each using finitely many positive integers.

If a rational matrix $A$ is a partition regular matrix, every finite coloring of the positive integers admits monochromatic $y_1,\dots ,y_n$ such thatChoose a compactness bound for partition regularity $T$ for the number of colors, put $S=\mathrm{lcm}(1,\dots ,T)$, and pull back the coloring by $t\mapsto S/t$ on $[T]$. A monochromatic positive solution $x$ of $Ax=0$ gives $y_i=S/x_i$; then $(1/y_i)=x_i/S$. The least common multiple guarantees that every $y_i$ is a positive integer.

For every partial computable function $f(\mathbf{x})$, there are total primitive recursive functions $C,U$ such thatHere $C$ is zero exactly on codes of valid halting computation histories for the chosen program and input, and $U$ extracts the final output. Coding finite configurations and finite histories makes checking every local step a bounded primitive recursive operation. If a computation halts, some history code passes and every passing code has the same output; if it does not halt, none passes. This is the usual normal-form version with the program index fixed; a universal predicate can also retain that index as an input.

The bracket enumerates the positive integers not divisible by $10$. Every positive integer has exactly one decomposition into a power of $10$ times such an integer, so this formula is a bijection from ${\mathbb{N}}^2$ to $\mathbb{N}$. Subtracting one in the codomain includes zero without requiring a trailing-zero convention for its decimal representation. For the inverse, write $N+1=10^{i}q$ with $10\nmid q$, then $j=q-1-\lfloor q/10\rfloor$.

In the limit where scalar field fluctuations dominate metric fluctuations, put $\zeta =-H\delta \varphi /\dot{\bar{\varphi }}$ and neglect derivatives of slowly varying background coefficients. Splitting $\delta X=\delta X_1+\delta X_2$ givesThe cubic terms $P_{XX}\delta X_1\delta X_2+P_{XXX}\delta X_1^3/6$ therefore yieldThe expression before factoring remains well-defined when $P_{XX}=0$. These interactions can generate substantial primordial non-Gaussianity when the Sound speed of a P(X, phi) scalar perturbation is small.

An oscillatory integral amplitude multiplies $e^{i\Phi }$ in an oscillatory integral with phase function $\Phi$. A finite symbol class order controls its frequency growth and the order of a distribution obtained by integration by parts.

An oscillatory integral with a degree-one homogeneous phase function, $k$ frequency variables, and oscillatory integral amplitude in fixed symbol class order $N$ has order of a distribution at most any nonnegative integer $r>N+k$. The integer works on every compact set; the continuity constant can vary with the compact set.

A zweibein is a two-dimensional orthonormal coframe in spacetime: its components obey $h_{\mu \nu }={\eta }_{AB}{e}^A{}_{\mu }{e}^B{}_{\nu }$. It converts worldsheet coordinate indices into local Lorentz-frame indices and defines curved gamma matrices ${\gamma }^{\mu }=e_A{}^{\mu }{\gamma }^A$. The associated spin connection supplies the covariant derivative of worldsheet spinors. In flat conformal gauge the coframe can be constant.

An Orr-Sommerfeld mode has nonzero wall-normal velocity satisfying the Orr-Sommerfeld equation. Its wall-normal vorticity satisfies the accompanying forced Squire equation. The triangular coupling separates the velocity eigenproblem from the vorticity forcing; the vorticity need not vanish in three dimensions.

The Squire equation governs wall-normal vorticity $\eta ={\partial }_{z}u-{\partial }_{x}w$ in the same normal mode convention as the Orr-Sommerfeld equation. It follows by taking ${\partial }_z$ of the streamwise momentum equation minus ${\partial }_x$ of the spanwise momentum equation. At a rigid no-slip wall, $\hat{\eta }=0$.

For nondimensional constant shear $U=y$ and streamwise wavenumber one, put $W=(D^2-1)\hat{v}$. The Orr-Sommerfeld equation gives $W^{\prime \prime }=[1+iRe(y-c)]W$. The displayed substitution transforms it into the Airy ordinary differential equation, because $(e^{i\pi /6}{)}^3=i$. The scale $R{e}^{-1/3}$ balances shear advection against viscous diffusion.