Lucas sequences satisfyThese identities follow from the root formulas for the Lucas U-sequence and Lucas V-sequence, or from their recurrence, so they are polynomial identities valid over a commutative ring. They compute terms with doubled indices from terms with about half the index.

A division-free algorithm carries $(U_n,U_{n+1})$. Put $a=U_n$, $b=U_{n+1}$, $c=a(2b-Pa)$ and $d=b^2-Q{a}^2$. For an even target return $(c,d)$; for an odd target return $(d,Pd-Qc)$. Starting from $(U_0,U_1)=(0,1)$ and following the binary digits of the target uses $O(\log n)$ ring operations. At $(P,Q)=(1,-1)$ this is the usual Fibonacci doubling method, and it also works in modular arithmetic.

The Lucas U-sequence has $U_0(P,Q)=0$, $U_1(P,Q)=1$ and $U_{n+2}=P{U}_{n+1}-Q{U}_n$. Its first terms are $0,1,P,P^2-Q,P^3-2PQ$. At $(P,Q)=(1,-1)$ it is the Fibonacci sequence.

If the roots $\alpha ,\beta$ of its characteristic polynomial of a linear recurrence are distinct, $U_n=({\alpha }^n-{\beta }^n)/(\alpha -\beta )$. This identity follows by checking the two initial conditions and the linear recurrence relation. At a repeated root $\alpha =\beta$, the corresponding formula is $U_n=n{\alpha }^{n-1}$ for $n\ge 1$.

The Lucas V-sequence has $V_0(P,Q)=2$, $V_1(P,Q)=P$ and $V_{n+2}=P{V}_{n+1}-Q{V}_n$. Thus its first terms are $2,P,P^2-2Q,P^3-3PQ$. For $(P,Q)=(1,-1)$ these are the Lucas numbers $2,1,3,4,7,\dots$.

For roots $\alpha ,\beta$ of $t^2-Pt+Q$, one has $V_n={\alpha }^n+{\beta }^n$, including the repeated-root case. This gives $V_{2n}=V_n^2-2Q^n$ immediately. The identity $V_n=2U_{n+1}-P{U}_n$ relates it to the Lucas U-sequence without requiring division.

A seismic shadow range is an interval of receiver distances missing a specified family of geometrical rays. A velocity decrease can separate shallow-turning and deep-turning arrivals; diffraction or other wave families can still deliver energy there.

For ideal oppositely charged parallel plates, the normal electric field is $\sigma /{ϵ}_0$. A boost along that normal leaves each transverse plate area unchanged. Invariance of total charge therefore preserves surface charge density and the normal electric field. The plate separation contracts, but the large-plate field does not depend on that separation. Including longitudinal contraction in a voltage argument while holding voltage fixed would impose an unsupported extra assumption.

An ideal long solenoid has $B={\mu }_{0}nI$, with axial turn density n. Under an axial boost, turn density becomes $n^{\prime }=\gamma n$. The transverse circulation path is unchanged while a turn's charge-circulation rate, measured relative to its moving material, becomes $I^{\prime }=I/\gamma$. Thus $n^{\prime }{I}^{\prime }=nI$ and the axial magnetic field remains unchanged. Equivalently, the ideal transverse current-sheet density is invariant under this boost. End effects are excluded.

The longitudinal rule of mixtures estimates the axial Young's modulus of continuous aligned fibres in a composite matrix:The model assumes perfect bonding, small elastic strain, no voids and loading parallel to the fibres. Equal extension over a common initial length gives equal axial strain $\epsilon$ in both phases. Adding their forces gives the average axial stress $\sigma =V_f{E}_f\epsilon +(1-V_f)E_m\epsilon$, which proves the formula. It is the axial Voigt elastic approximation.

For example, $V_f=0.6$, $E_f=70\mathrm{GPa}$ and $E_m=3\mathrm{GPa}$ give $E_L=43.2\mathrm{GPa}$. These are illustrative inputs, not universal constituent values. The formula makes reinforcement efficient because the stiffer constituent carries the larger stress at the common strain. Loading across the fibres changes the compatibility and traction conditions, so the same formula is not a general transverse-modulus law. Fibre waviness or poor fibre-matrix load transfer can also invalidate the ideal axial estimate.

A hybrid fibre composite contains two or more types of reinforcing fibre in a common composite matrix. Combining carbon fibres and glass fibres, for example, provides another design choice besides changing the amount or orientation of a single reinforcement. The word hybrid specifies the combination of reinforcements, not a guarantee that every mechanical property improves.

For continuous, parallel fibres, perfect bonding and elastic axial loading, equal strain gives the extension of the longitudinal rule of mixturesHere the $V_i$ are constituent fractions by volume and the $E_i$ are axial Young's moduli. This estimate describes the initial elastic response. It does not by itself predict tensile strength: when one fibre population breaks, its load is redistributed through the composite matrix, changing the stress carried by the surviving population.

The fibre volume fraction is the fraction of a composite material's total volume occupied by reinforcing fibres:For a specimen with only fibres and a composite matrix, their fractions sum to one. Voids require a third fraction; they must not be silently counted as composite matrix. The fibre volume fraction is different from the fibre mass fraction because the constituents can have different mass densities.

For a void-free material, a measured fibre mass fraction $w_f$ giveswhere ${\rho }_f$ and ${\rho }_m$ are the constituent mass densities. For example, masses in the ratio $3:2$ and mass densities in the ratio $3:1$ give $V_f=1/3$, not $3/5$. Inserting $V_f$ into the longitudinal rule of mixtures estimates axial Young's modulus; orientation and fibre-matrix load transfer remain separate requirements.

A stereochemically active lone pair is a nonbonding electron pair associated with an asymmetric local arrangement of bonds or neighboring atoms. In ions such as lead(II), the occupied valence states can interact with ligand and empty metal states in a way that favors off-center coordination.

An s-electron pair is therefore not adequately pictured as a rigid lump that mechanically pushes ligands aside. The energetic preference depends on orbital mixing and the surrounding lattice or coordination environment, explaining why related compounds can show different distortions.

Combining the static London equations with the magnetostatic Ampère law and $\nabla \cdot \mathbf{B}=0$ gives the displayed bulk equation for a uniform superconducting carrier density. A planar boundary gives exponentially decaying field on the London penetration depth scale; boundary geometry and vortices require their corresponding field solutions.

The Smith-Waterman algorithm finds an optimal local sequence alignment for a specified additive scoring scheme. For a linear gap penalty $e>0$, a dynamic programming score satisfiesZero permits an alignment to restart; tracing back from the largest entry to zero recovers a highest-scoring segment pair. Every optimal ending must have one of the listed final steps, which proves the recurrence by induction over the table. Optimality concerns this score, not guaranteed biological ancestry. ^[ref]

BLAST, the Basic Local Alignment Search Tool, searches a sequence database for high-scoring local similarities. It uses promising short matches to guide extension rather than exhaustively evaluating every possible alignment. Reported scores and expected counts help evaluate matches under a background model; they do not directly establish common function. A short conserved domain can produce a strong local match even when the complete proteins have different overall architectures. ^[ref]

The FASTA sequence search algorithm locates promising matching words and diagonals before refining a limited set of alignments. The FASTA format is a separate sequence-file convention.

An auxiliary entangled pure state can enable an otherwise forbidden deterministic bipartite LOCC transformation while being returned exactly and uncorrelated with the output. Nielsen's pure-state conversion theorem reduces this to the displayed majorization condition on sorted squared Schmidt coefficients. It does not follow from a scalar comparison of entanglement entropies: tensoring changes which products occupy the largest cumulative-weight sums. An exact returned catalyst can be reused; merely restoring its marginal while allowing correlations is a different resource task.^[ref]

Lithospheric flexure is bending of a mechanically strong outer shell under loads. A flat elastic plate approximation balances bending, buoyancy and applied loading. In the Fourier domain, bending contributes a term proportional to the flexural rigidity times the fourth power of horizontal wavenumber; it therefore suppresses short-wavelength deflection particularly strongly.

The half-space cooling model solves the one-dimensional heat equation for initially hot material cooled at a fixed-temperature surface. For constant thermal diffusivity $\kappa$, a fixed interior-temperature isotherm deepens proportional to $\sqrt{\tau }$. Applied to an oceanic lithosphere, it explains the leading age dependence of cooling; finite plate thickness and basal heat supply limit its extrapolation to old ages.

An oceanic plate cooling model solves the heat equation in a layer of fixed thickness $a$, with fixed cold surface and hot basal temperatures. A hot initial interior cools towards a linear steady profile. Unlike a half-space cooling model, it has a finite limiting thermal thickness and a nonzero old-age surface heat flux. Its transient sine coefficients must reproduce the chosen initial profile as well as the boundary values.