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Homogenizing the potential vorticity over a finite constant bottom slope creates the odd anomaly displayed here while leaving the exterior unchanged. Inverting $({\partial }_y^2-L_R^{-2})\psi =\Delta Q$ with decay at both infinities and matching $\psi ,{\psi }_y$ gives a central return current and flanking jets. The deformation radius sets the smoothing length. For $L_R\ll a$, the central surface follows the bottom slope to leading order, but edge corrections and decaying exterior flow are essential.

Two stationary no-slip surfaces with local gap $H(x)=h+x^2/(2a)$ transmit pressure-driven volume flux $q=-H^3{p}_x/(12\mu )$ per unit transverse length. Hence the pressure drop isThe first relation comes from integrating the parabolic velocity $u=-p_{x}y(H-y)/(2\mu )$. Substitution $x=\sqrt{2ah}\tan \theta$ evaluates the resistance integral using ${\int }_{-\pi /2}^{\pi /2}{\cos }^4\theta d\theta =3\pi /8$. The formula gives the leading resistance when $h\ll a$.

Statistical homogeneity defines the primordial bispectrum by

A bispectrum is a Fourier-space third-order statistic; primordial perturbations supply the cosmological field in this instance.

The principal stretches are the eigenvalues of a stretch tensor, or equivalently the singular values of the deformation gradient. They measure local length ratios along the principal reference directions.

On a spacelike constant-target-time slice, integrate the induced spatial line element to obtain the geometric string length. It depends on the embedding and is distinct from the fundamental string length ${\ell }_s=\sqrt{{\alpha }^{\prime }}$. For transverse local motion, energy additionally includes the local Lorentz factor; the geometric length alone does not determine the total energy.

A deformation map twisting every cross-section of a cylinder through an angle proportional to its reference axial position, without radial or axial extension. In the local current and reference cylindrical coordinate systems, the deformation gradient is $F=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& \tau r\\ 0& 0& 1\end{array}\right)$. Its determinant is one. The twist per unit reference length is $\tau$; an end separation $L$ gives end rotation $\phi =\tau L$. The physical torque is $N=2\pi {\int }_0^A{T}_{\theta z}{r}^{2}dr$.

For a right triangle in Euclidean geometry with perpendicular side lengths $a,b>0$ and hypotenuse $c$, the Pythagorean theorem statesGeometrically, the area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides. Euclid's Elements, Book I, Proposition 47 states this area equality; Proposition 48 states its converse: a triangle satisfying the equality has a right angle opposite side $c$.^[ref][ref]

For $a=3$ and $b=4$, we compute $c=\sqrt{9+16}=5$. The triangle has area $ab/2=6$, and its side lengths form a Pythagorean triple. Conversely, a triangle measured to have sides $3,4,5$ must have a right angle between the sides $3$ and $4$. This makes the equality both a calculation tool and a test for perpendicularity.

The three squares in the next figure have areas $3^2=9$, $4^2=16$ and $5^2=25$ in squared length units. These are different from the central right triangle's area $3\cdot 4/2=6$. The Pythagorean theorem adds the areas of the attached squares, not their side lengths and not the area of the triangle. The square on the hypotenuse is tilted, but a rotation preserves its area. This makes the formula visible without suggesting that the hypotenuse has to align with an axis.

Historically, Euclid's Elements records an organized proof of the Pythagorean theorem as an equality of areas, rather than the modern equation between numbers representing lengths. Its placement in Book I makes it available before the treatment of similar triangles in Book VI. This is evidence of how the result was established within that text; the name Pythagorean theorem alone does not establish who first discovered or proved it. The precise historical anchor here is the surviving proposition, rather than an unsupported story of a single inventor.^[ref]

The dissection below makes the general identity tangible. Both outer squares have side $a+b$ and contain the same four congruent right triangles. In the first arrangement, their hypotenuses bound a square of side $c$: adjacent acute angles together make a right angle. In the second, the uncovered regions are squares of sides $a$ and $b$. Subtracting the same total triangle area from equal outer areas givesFor the pictured dimensions, the shared uncovered area is $49-24=25=9+16$.

The same mechanism survives in inner product spaces: for orthogonal vectors $u,v$, expanding the inner product gives $\Vert u+v{\Vert }^2=\Vert u{\Vert }^2+\Vert v{\Vert }^2$. This is why orthogonal projection separates a squared distance into independent contributions. The familiar right triangle is the simplest picture of a much more reusable principle.

Quantum confinement restricts a particle's wavefunction over a length comparable with its characteristic quantum length scales, altering the allowed energy levels relative to a bulk material. For a parabolic band confined over width $L$, a characteristic spacing is of order ${\hslash }^2/(m^{\ast }{L}^2)$, with boundary-dependent numerical factors. A quantum dot confines all three directions; confinement in only one direction can leave continuous motion in the other two. Finite barrier height and interactions modify the simplest infinite-wall estimates.

The quantum-critical regime is the finite-temperature region in which temperature exceeds the energy scale generated by detuning from a quantum critical point. Temperature then controls leading masses, relaxation rates, and correlation lengths.

Quantum modular exponentiation is the reversible computation that mapsIt can be implemented in polynomial time in the bit lengths of $x$ and $N$ by repeated squaring and controlled modular multiplication.

In this magnetic circuit approximation, each circular path has air-gap length $d$, core length $2\pi r-d$, and the same normal magnetic flux density in air and core. Ampère's law then gives $B(r)\simeq {\mu }_0{\mu }_{r}NI/[2\pi r+({\mu }_r-1)d]$; integrating across the section gives the displayed mutual inductance. Even $d\ll a$ can strongly reduce inductance when ${\mu }_{r}d/a$ is appreciable. This is a path estimate, not an exact local field solution in the slit geometry. It neglects magnetic fringing and redistribution between paths. In particular, a literal field proportional to $\hat{\mathbf{\varphi }}/(r+q)$ everywhere in a current-free core is not curl-free; transverse corrections are required. Narrow-gap circuit approximations and their limitations are discussed in ^[ref].

A steady two-dimensional purely radial incompressible Newtonian fluid flow has $u_r=g(\theta )/r$ by the continuity equation. The polar Navier-Stokes equation givesEquality of the mixed pressure derivatives yields the displayed third-order equation. On fixed walls $\theta =\pm \alpha$, no-slip boundary conditions require $g(\pm \alpha )=0$. A source with volume flux $Q$ per unit out-of-plane length requires ${\int }_{-\alpha }^{\alpha }gd\theta =Q$. Consequently $g=QG$ leaves the single inertial parameter $ϵ=Q/\nu$ in $G^{(3)}+4G^{\prime }+2ϵG{G}^{\prime }=0$. The origin is a source singularity; the radial ansatz describes the wedge away from that point, not the detailed source apparatus.

An angle in radians is the ratio of its subtended arc length to the radius of a circle. A full turn is $2\pi$ radians and a right angle is $\pi /2$ radians.

A ratio compares two quantities $a,b$, written $a:b$ and corresponding to division $a/b$ when $b\ne 0$. Equal ratios of corresponding lengths characterize the scaling of similar triangles.

For example, $6:4$ and $3:2$ specify the same ratio because $6/4=3/2$. Multiplying both entries by the same nonzero scale preserves the quotient: $(ca)/(cb)=a/b$. This is useful when comparing mixtures or scaling a recipe; doubling both ingredient amounts preserves their ratio, while adding the same amount to both generally does not. For quantities with the same physical units, their ratio is dimensionless, connecting this elementary calculation with dimensional analysis.

The real case of the finite-dimensional spectral theorem says that a self-adjoint operator on a finite-dimensional real inner product space has a real orthonormal eigenbasis. In orthonormal coordinates this is the spectral theorem for real symmetric matrices, $A=Q\Lambda Q^T$. The worked ellipse under finite-dimensional spectral theorem shows why both orthogonal diagonalization and orthogonality matter: the new coordinates separate the quadratic form into squares without distorting lengths or angles.

A real normal matrix need not have real eigenvalues. For example, the quarter-turn $\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ is an orthogonal matrix, but its characteristic polynomial is $z^2+1$ and its eigenvalues are $i,-i$. Over the complex numbers it has an orthonormal eigenbasis; over the real numbers it has no eigenvector. Self-adjointness supplies the reality conclusion in the real spectral theorem.

Let a plane incident wavefront first meet a flat interface at $A$, and let another point $B$ on that front reach $C$ on the interface after $\Delta t$. Its path has length $BC=v_1\Delta t$. The secondary wavelet from $A$ has radius $AD=v_2\Delta t$ in the second medium; the new wavefront through $C$ is tangent to it at $D$. Right triangles with common hypotenuse $AC$ give $\sin {\theta }_1=BC/AC$ and $\sin {\theta }_2=AD/AC$, where both ray angles are measured from the interface normal. Consequently $\sin {\theta }_1/\sin {\theta }_2=v_1/v_2=n_2/n_1$, which is Snell's law. The construction uses phase velocities in homogeneous isotropic media.

For two lines measured with the same interplanar spacing at the same temperature, division of Bragg's law givesThe common $d$ cancels. Consequently a precise wavelength ratio can coexist with an uncertain conversion to SI length. In contrast, angular calibration errors generally survive: a shared small zero offset $ϵ$ changes the inferred ratio fractionally by approximately $(\cot {\theta }_1-\cot {\theta }_2)ϵ$. Nearby lines partially cancel this effect, but widely separated angles need not. Different crystals, temperatures or refraction corrections also prevent unqualified cancellation.

For a near-axis beam of equal incoming speed $v$ through a uniform solenoid of length $l$, define $\omega =|qB|/(2\gamma m)$. Neglecting the small axial-speed changes, the internal radius is $\rho ={\rho }_0\cos (\omega z/v)$. The exit fringe removes the azimuthal velocity for zero incident canonical angular momentum. The field-free radial trajectory then reaches the axis a distancebeyond the exit, when $0<\omega l/v<\pi /2$. In the weak-lens limit, $f\simeq 4(\gamma mv{)}^2/(q^2{B}^{2}l)$. Without the paraxial approximation, the radius-dependent axial speed produces aberration; the ideal common focus is then only approximate.

For a linear-radius taper from $R$ to $r$ over length $s$, constant absolute permeability $\mu$, and a common approximately axial flux, integrate $dx/[\mu \pi \{R-(R-r)x/s{\}}^2]$ to obtain the displayed reluctance. Leakage, fringing, nonuniform transverse fields and saturation limit this one-dimensional magnetic-circuit approximation.

The remnant profile is the time integral of a decaying heat kernel. Its length scale is $\sqrt{D/\mu }$ and its total mass is $N\lambda /\mu$, the initial population times the mean deposition over an exponential lifetime.