Radio triangulation estimates a radio transmitter's position by combining directions measured from different known locations. Each radio direction finding observation supplies a bearing line, with the forward direction selecting a ray. Their intersection gives an ideal position estimate. This is one method of radio tracking; detecting a signal at one station does not determine its distance.^[ref]

For two stations at $(0,0)$ and $(L,0)$ in a horizontal Cartesian coordinate system, let their bearings make angles $\alpha$ and $\beta$ from the positive horizontal axis. Away from vertical bearings, their lines satisfy $y=x\tan \alpha$ and $y=(x-L)\tan \beta$. Solving givesFor $L=4\mathrm{km}$, $\alpha =45^{\circ }$ and $\beta =135^{\circ }$, the result is $(2,2)\mathrm{km}$. The intersection must also lie in each observed forward direction; intersecting the infinite lines alone is insufficient.

The denominator exposes a limitation: nearly parallel bearings can make the position estimate highly sensitive to angular error. Observations taken at different times can also refer to different target positions. More stations allow consistency checks or a fitted estimate when noisy lines do not meet at a point. Smithsonian bird-tracking researchers use this combination of bearings to locate animals without needing continuous visual contact.^[ref]

Spatially averaged radiant exposure is the received radiant energy divided by the specified receiving area. For a patch $A$ and local radiant exposure $H(\mathbf{x})$,Both local and averaged values have units $\mathrm{J}{\mathrm{m}}^{-2}$, but the average need not equal the local maximum. The radiant exposure definition uses incident energy and the receiving surface's area.^[ref]

For an original unequal-area example, let one $3\mathrm{m}{\mathrm{m}}^2$ region receive a uniform $2\mathrm{J}{\mathrm{m}}^{-2}$, and a neighbouring $1\mathrm{m}{\mathrm{m}}^2$ region receive $10\mathrm{J}{\mathrm{m}}^{-2}$. They receive $6$ and $10\mu \mathrm{J}$ respectively. The average over the combined patch is thereforeThe unweighted average of the two listed exposure values would be $6\mathrm{J}{\mathrm{m}}^{-2}$, which is wrong for these unequal areas. The area weighting of radiant exposure gives the larger region three times the influence. The actual peak remains $10\mathrm{J}{\mathrm{m}}^{-2}$ in the stipulated piecewise-uniform model.

A receiver integrating all incident energy over the complete patch can determine $Q_A$; division by its known collecting area then gives the average. That measurement alone cannot identify the peak or reconstruct the spatial distribution. Subdivision with a characterized spatial response supplies additional information. For a finite aperture intercepting a Gaussian beam, use energy actually entering the aperture, as calculated in radiant exposure of a Gaussian beam, rather than the total energy of the untruncated beam.

The radiant exposure of a Gaussian beam is its time-integrated transverse irradiance. At a fixed plane normal to a circular Gaussian beam, suppose the Gaussian beam radius $w$ is unchanged during a pulse and denotes the radius where irradiance is $e^{-2}$ of the on-axis value. With total laser pulse energy $U$,The primary beam account specifies this radius convention and the corresponding irradiance normalization; the radiant exposure parent derives its time integral.^[ref]

An original aperture example asks how much of that energy a finite receiver collects. Integrating over a centred disk of radius $R$ gives the encircled radiant energy of a Gaussian beam:For stipulated $U=2\mu \mathrm{J}$, $w=0.80\mathrm{mm}$ and $R=0.40\mathrm{mm}$, the receiver collects only $39.35$percent, or $0.7869\mu \mathrm{J}$. Its spatially averaged radiant exposure is about $1.566\mathrm{J}{\mathrm{m}}^{-2}$, compared with on-axis $H(0)\approx 1.989\mathrm{J}{\mathrm{m}}^{-2}$. Dividing the total $U$ by this receiver's area would count energy that missed it.

To collect a chosen fraction $0<f<1$ under this ideal model, the centred aperture needs $R=w\sqrt{-\log (1-f)/2}$. For95percent, $R\approx 1.224w$. A possible experimental check measures integrated received energy through several known apertures and compares their ratios. Aperture centring, detector linearity and any profile change during the pulse affect that comparison. A time-varying radius must remain inside the time integral rather than being replaced by an unspecified average radius.

A radiometric energy balance accounts for radiant energy crossing the specified measurement boundary during a specified interval. For a passive optic, suppose the relevant transient has finished, no electromagnetic energy remains stored, and no additional energy source is present. Incident radiation is partitioned into all outgoing radiation and energy transferred to matter. Resolving outgoing paths givesThe channels must be disjoint: a scattering term cannot include energy already counted by a reflection or transmission collector. The definition of radiant energy concerns radiation emitted, transferred or received, so the location and collection geometry belong in each term.^[ref]

For an original example, inject $10\mu \mathrm{J}$ and measure $6\mu \mathrm{J}$ in the transmitted path plus $2\mu \mathrm{J}$ in the specular reflection. Calling the missing $2\mu \mathrm{J}$ absorbed is premature. If a separate collector finds $1.5\mu \mathrm{J}$ of diffuse outgoing radiation outside those channels, the inferred absorbed energy is only $0.5\mu \mathrm{J}$. The uncollected scattering in a radiometric energy balance would otherwise be misidentified as optical absorption.

A physical check compares calibrated incident and outgoing energy readouts while controlling apertures, backgrounds and the observation window. The scattered component may need a different collection geometry from the specular beam. Energy absorbed by matter can remain stored or later leave as heat or radiation; a more general control-volume balance must explicitly include storage, other heat transfer and work. It must not count absorbed energy twice by also treating the same material-energy increase as an independent loss.

The Rabin-Williams scheme is a family of modified Rabin cryptosystem constructions using a modulus $N=pq$ with distinct prime numbers $p\equiv 3(\mathrm{mod}8)$ and $q\equiv 7(\mathrm{mod}8)$. These congruences make sign and doubling adjustments useful for producing quadratic residues before taking modular square roots. In signature variants, the private integer factorization permits root extraction while the public verification uses squaring.

The arithmetic mechanism can be seen without fixing a full message encoding. For a unit $h$ modulo $N$, exactly one of $h,-h,2h,-2h$ is a quadratic residue modulo both primes: the two signs change both Legendre symbols, while multiplication by $2$ changes only one of them. For the toy modulus $77=11\cdot 7$ and $h=5$, the adjustment $-2h\equiv 67$ has square root $12$, since $12^2\equiv 67(\mathrm{mod}77)$.

A complete digital signature scheme must additionally specify message hashing or redundancy, the selected root, and how the adjustment is encoded and verified. The congruence calculation alone is not such a scheme. A specific modified-Rabin message-recovery construction is given in the primary handbook, rather than being identical to raw four-root Rabin encryption.^[ref]

For a single-server queue using first come first served, let $A_n$ be the arrival time of job $n$, $S_n\ge 0$ its service time, and $D_n$ its departure time. Number jobs in arrival order, with an explicit order for ties. Assume the server works whenever a job is waiting, service is not interrupted, and jobs are neither lost nor dropped. If the server is initially free at time 0 and arrivals are nonnegative, set $D_0=0$. ThenThe proof is a scheduling constraint: job $n$ cannot start before it arrives or before the preceding job has finished. Under the stated work-conserving rule it starts at the later of those times, then requires $S_n$ more units of time.

Its queueing delay and customer sojourn time are consequentlyFor an original four-job example, use arrivals $(0,1,1,6)$ and service times $(2,3,1,2)$. The third job follows the second in the tie:The fourth job arrives exactly when the third finishes and therefore has zero queueing delay.

The same recursion also gives a sample-path comparison without assuming a particular arrival or service probability distribution. If every arrival time and service time is increased, and the initial available time is not reduced, induction through the monotone max and addition operations shows that no corresponding departure time decreases. This does not imply that every queueing delay increases: a later arrival can spend less time waiting for the same departure opportunity.

An initially busy server requires the appropriate initial available time in place of 0. Multiple servers, priority changes, interruption or abandonment require additional state or a different recursion. These assumptions belong to the model, not to the algebraic max operation.

The Quantum Fisher information measures the local sensitivity of a parametrized density matrix ${\rho }_{\theta }$. The symmetric logarithmic derivative $L_{\theta }$ satisfies ${\partial }_{\theta }\rho =(L_{\theta }\rho +\rho L_{\theta })/2$, and $F_Q=\mathrm{Tr}(\rho L_{\theta }^2)$.

For $n$ independent preparations, a locally unbiased estimator obeys $\mathrm{Var}(\hat{\theta })\ge 1/(n{F}_Q)$ under the usual regularity assumptions. For a pure family $|{\psi }_{\theta }⟩=e^{-i\theta G}|{\psi }_0⟩$, $F_Q=4\mathrm{Var}(G)$. A state with definite $G$ acquires only a global phase and has zero information about $\theta$; a superposition can have observable relative-phase sensitivity.^[ref]

A quantum correlation is a statistical relation among observables of a composite quantum system. For local observables $A$ and $B$, a useful diagnostic is $⟨A\otimes B⟩-⟨A\otimes I⟩⟨I\otimes B⟩$. Product states make this expression vanish for every pair of local observables.

Not every correlation in a quantum state is quantum entanglement. The separable mixture $[|00⟩⟨00|+|11⟩⟨11|]/2$ gives perfectly correlated computational-basis outcomes but has no entanglement. The pure Bell state $|{\Phi }^{+}⟩$ has the same outcomes in that basis while retaining coherent relative-phase information detectable in complementary bases. The measurement choices therefore matter when characterizing the kind of correlation present.

A quadrupole stability diagram plots the parameter combinations for which an ideal quadrupole mass analyzer gives bounded transverse ion motion. The useful region is the overlap of stability in both transverse directions: confinement along only one direction still allows loss along the other. The physical selection mechanism is stability in combined static and high-frequency quadrupole electric fields.^[ref]

Using the potential convention stated in quadrupole mass analyzer, introduce dimensionless time $\tau =\Omega t/2$ and parametersHere $Q$ is electric charge, $m$ is mass, $r_0$ describes geometry, $\Omega$ is drive angular frequency, and $U,V$ specify the static and alternating potentials. The transverse equations becomeThese are Mathieu equations. In the convention $u^{\prime \prime }+(a-2q\cos 2\tau )u=0$, the first equation has $q=-b$; quoting a diagram requires checking this sign and the potential's numerical convention.

A useful limiting test is $b=0$, $a>0$. The $x$ equation oscillates, but the $y$ equation has exponentially growing solutions. Static focusing alone therefore fails to provide a stable overlap. Periodic focusing changes this conclusion; its stability regions cannot be read off from the force at one instant.

At fixed positive $U/V$, changing mass-to-charge ratio moves a point along a ray with $a/b=2U/V$. Doubling $m/Q$ halves both coordinates. Doubling both applied potentials moves the same ion twice as far along that ray. These scaling relations explain how scanning the potentials selects successive mass-to-charge ratios. A stable region is a property of the ideal differential equations; finite rod length, entrance conditions and field imperfections still determine actual transmission and mass resolving power.

The Witt index of a nondegenerate quadratic form $q$ on a finite-dimensional vector space $V$ is the largest dimension of a totally isotropic subspace: a linear subspace $U$ for which $q(u)=0$ for every $u\in U$. In characteristic different from two, the Witt decomposition writeswhere $H$ is a hyperbolic plane of a quadratic form, $q_{\mathrm{an}}$ is an anisotropic quadratic form, and $r$ is the Witt index. The anisotropic part has no nonzero vector on which the form vanishes.^[ref]

For the real form $x_1^2+x_2^2-x_3^2-x_4^2$, the subspace spanned by $(1,0,1,0)$ and $(0,1,0,1)$ is totally isotropic. Projection onto the two negative coordinates is injective on any totally isotropic subspace, giving the matching upper bound two. More generally, a real form of signature $(a,b)$ has Witt index $\min (a,b)$.

The base field matters: $x^2+y^2$ has index zero over $\mathbb{R}$ but index one over $\mathbb{C}$, using the line spanned by $(1,i)$. In characteristic two, vanishing of the associated polar form on $U$ alone is insufficient; the condition remains $q{|}_U=0$. Over a finite field, the two nondegenerate types in dimension $2m$ have Witt indices $m$ and $m-1$.^[ref]

For a finite-dimensional nonsingular quadratic form $q$ over a field of characteristic different from two, a Witt decomposition is an orthogonal splittingwhere $H(u,v)=uv$ is a hyperbolic plane of a quadratic form and $q_{\mathrm{an}}$ is an anisotropic quadratic form. The integer $r$ is the Witt index. Both $r$ and the isometry class of the anisotropic part are determined by $q$, although the particular subspaces realizing the splitting need not be unique.^[ref]

Over $\mathbb{Q}$, take $q(x,y,z)=x^2+y^2-z^2$. Set $x=(u+v)/2$, $z=(u-v)/2$ and $y=w$. Thenso $q\cong H\perp ⟨1⟩$. The one-dimensional remainder is anisotropic and the Witt index is one. This is an isometry of quadratic spaces, stronger than merely expressing the underlying vector space as a direct sum.

The construction starts from an isotropic vector, finds a partner with nonzero polar pairing, and splits off the hyperbolic plane they generate. Repeating the process ends when no nonzero isotropic vector remains. The decomposition makes the isotropic directions and the residual arithmetic obstruction visible separately.

A quadratic form $q$ over a field $k$ is an anisotropic quadratic form if $q(v)=0$ forces $v=0$. An isotropic quadratic form instead has a nonzero zero vector. This is a property of the form together with its coefficient field, rather than just the displayed polynomial.^[ref]

For example, $q(x,y)=x^2-2y^2$ is anisotropic over $\mathbb{Q}$. If a nonzero rational solution had $y=0$, then $x=0$ too; otherwise $x/y$ would be a rational square root of two. Over $\mathbb{R}$, however, $(\sqrt{2},1)$ is a nonzero zero vector. The coordinate change $u=x+\sqrt{2}y$, $v=x-\sqrt{2}y$ gives $q=uv$, a hyperbolic plane of a quadratic form.

This example also separates anisotropy from positive definiteness: the rational form takes the positive value $1$ and the negative value $-2$, despite having no nonzero rational zero. In a Witt decomposition, an anisotropic remainder is what remains after all hyperbolic planes have been split off. Its Witt index is zero. In characteristic two, one must test vanishing of the quadratic form itself; properties of the associated polar bilinear form alone do not supply the same test.