The identities $F_{2k}=F_k(2F_{k+1}-F_k)$ and $F_{2k+1}=F_k^2+F_{k+1}^2$ compute $(F_n,F_{n+1})$ recursively from the pair at $\lfloor n/2\rfloor$. Recursion depth is $O(\log n)$, with a constant number of large integer operations per level; the bit cost still depends on operand length.

For the Fibonacci numbers $F_0=0$, $F_1=1$, the identity $F_{n+1}{F}_{n-1}-F_n^2=(-1{)}^n$ holds for $n\ge 1$. It follows by induction: replacing each term using the Fibonacci recurrence negates the left-hand side when $n$ increases by one, and the value at $n=1$ is $-1$.

Every positive integer is a unique sum of nonconsecutive Fibonacci numbers, using the sequence $1,2,3,5,8,\dots$ with one copy of $1$. The greedy procedure selects the largest allowable Fibonacci number and repeats on the remainder.

The Zeckendorf representation expresses a nonnegative integer as a sum of nonconsecutive Fibonacci numbers, using the sequence that starts $1,2,3,5,\dots$. Every nonnegative integer has exactly one such representation; repeatedly taking the largest available Fibonacci number constructs it.

For an integer $m\ge 2$, the Fibonacci entry point $z(m)$ is the smallest positive index with $m\mid F_{z(m)}$. It records the first zero residue, whereas the Pisano period records the first return of the whole consecutive pair $(F_n,F_{n+1})$ to $(0,1)$.

For $m=3$, the first positive divisible term is $F_4=3$, so $z(3)=4$; the pair period is eight. Existence follows because the pair update is an invertible permutation of a finite set of residue pairs, so its orbit returns to $(0,1)$. By Fibonacci gcd reduction, $m\mid F_n$ holds precisely when $z(m)\mid n$.

The Pisano period $\pi (m)$ is the period of the Fibonacci number sequence in modular arithmetic modulo $m$, with $F_0=0$ and $F_1=1$. Consecutive residues evolve by the invertible update $(a,b)\mapsto (b,a+b)$ on a finite set, so the pair sequence returns to $(0,1)$ and is periodic from the start.

With $F_0=0$, $F_1=1$, the Fibonacci matrix advances consecutive Fibonacci numbers:Multiplying by $Q$ proves the formula by induction. Since matrix powers obey $Q^{m+n}=Q^m{Q}^n$, their entries give Fibonacci addition formulas. Exponentiation by squaring computes $Q^n$ with $O(\log n)$ matrix multiplications, including directly in modular arithmetic. The determinant identity $\det Q=-1$ also makes the update invertible modulo every positive integer.

A receptor important in early mouse lineage responses and primitive-streak morphogenesis. Its effects on fate and movement require separate measurements.

A soft magnetic material changes its magnetization relatively easily and has a low coercive field. High useful permeability and small hysteresis losses favour such materials in electromagnets and alternating-field cores. “Soft” describes magnetic response rather than mechanical softness; electrically resistive ferrimagnets can also be magnetically soft.

A hard magnetic material strongly resists reversal of its magnetization, giving high coercivity and useful remanence for permanent magnets. The relevant hardness is magnetic, not a mechanical strength rating.

A cluster of sporangia on a fern sporophyte.

The Fermi transition operator is an isospin ladder operator. Acting between normalized members of one isospin multiplet gives the displayed squared matrix element. An edge transition in a $T=3/2$ multiplet has Fermi strength three, whereas the transition between the two $T=1/2$ mirror members has strength one. This fixes the vector contribution; nonzero-spin analogue transitions can also have a Gamow-Teller transition contribution that is not fixed by this counting.

For $s=Q/(2R)$, each face cap has area $C=R^2[\arccos s-s\sqrt{1-s^2}]$. The total second-band electron area is $4C$; the first-band hole area is $Q^2-\pi R^2+4C$. Multiplying either area by the spin degeneracy two and dividing by $(2\pi {)}^2$ gives the carrier densities. This is an infinitesimal-potential count, not an exact finite-gap formula.

For a tree-level left-handed three-body decay with one massive final fermion and two massless ones, the integrated width relative to the all-massless result includes $f(r)=1-8r+8r^3-r^4-12r^2\ln r$, where $r$ is the squared final-to-parent mass ratio. It satisfies $f(0)=1$ and $f(1)=0$. It is not a universal factor for channels with several massive daughters.

A model with separate filled proton and neutron momentum seas in a nuclear mean field. Each spin-half species has two spin states, so counting momentum-space cells gives $p_{F,i}=\hslash (3{\pi }^2{n}_i{)}^{1/3}$. It describes typical nucleon motion and a kinetic contribution to nuclear symmetry energy, but binding requires additional interactions.

For a three-dimensional ideal gas with quadratic dispersion and fixed particle number, the total density of states at the Fermi energy is $g(E_F)=3N/(2E_F)$. The Sommerfeld coefficient then gives $C_V=({\pi }^2/2)N{k}_B(T/T_F)$ to leading order, where $T_F=E_F/k_B$. The fixed-number energy correction is $\Delta U=({\pi }^2/4)N(k_{B}T{)}^2/E_F$. Only states near the Fermi surface change occupation, explaining the small active fraction.

The circular zero-temperature occupation boundary of an isotropic two-dimensional electron gas. The occupied disk has area $\pi k_F^2$; converting its area to particle number requires the reciprocal-space state density and a separately specified spin degeneracy.

The fault strike is the azimuth of the intersection of a fault plane with a horizontal plane. A right-hand convention chooses the along-strike direction for which the fault dips to the observer's right.

The fault dip is the angle between a fault plane and a horizontal plane, measured perpendicular to fault strike towards the descending side. It lies between $0$ and $90^{\circ }$.

The fault rake specifies the slip direction within a fault plane. With an along-strike unit vector $\mathbf{s}$ and a down-dip unit vector $\mathbf{d}$, one explicit convention is $\mathbf{b}/b=\cos \lambda \mathbf{s}-\sin \lambda \mathbf{d}$. Positive $90^{\circ }$ then denotes up-dip slip. A convention must accompany a signed fault rake.