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String tension is energy per unit length. The Regge-slope parameter ${\alpha }^{\prime }$ is conventionally defined by $T=(2\pi {\alpha }^{\prime }{)}^{-1}$.

For a product state, probabilities of local measurement outcomes multiply.

For two labelled orthonormal bases, the event of equal labels is represented by ${\sum }_j|x_j{y}_j⟩⟨x_j{y}_j|$.

Starting the answer qubit in $|0⟩$, the sequencemaps $|x⟩|0⟩$ to $(-1{)}^{f(x)}|x⟩|0⟩$. It realizes a phase oracle with two standard-oracle queries while returning the answer register to its initial state.

The Bondi news is the retarded-time derivative of the leading asymptotic shear and measures outgoing gravitational radiation at null infinity.

Expanding an angular distribution in multipoles converts the Boltzmann equation into a hierarchy in which free streaming couples neighboring multipoles and collisions damp or source selected moments.

The first Brillouin zone is a fundamental cell in reciprocal space. For a one-dimensional lattice of spacing $a$, one conventional choice is

The delta-comb Kronig-Penney potential is

For a potential of period $a$, the Floquet matrix $M(E)$ maps Cauchy data through one cell:For a real Schrodinger equation, $\det M=1$.

The nearly-free electron model treats a weak periodic potential by degenerate perturbation theory. At a Bragg crossing, a Fourier coefficient $V_n$ couples the two degenerate plane waves and opens an energy gap of width $2|V_n|$.

An energy band is a continuous branch $E_j(k)$ of the dispersion relation of a periodic quantum system.

A periodic potential satisfies $V(x+a)=V(x)$ for some lattice period $a$. Its Fourier coefficients couple plane waves whose wavevectors differ by reciprocal-lattice vectors.

The Kerr black hole is the asymptotically flat stationary axisymmetric vacuum black hole with mass $M$ and angular momentum $J=Ma$.

A Kerr black hole is the Kerr metric with mass $M>0$ and spin parameter satisfying $|a|\le M$ in geometrized units; its outer horizon is $r_{+}=M+\sqrt{M^2-a^2}$.

Black-hole thermodynamics relates horizon geometry to thermodynamic laws, assigning temperature and entropy to a stationary black hole.

A Penrose diagram conformally compactifies a spacetime while preserving causal directions, placing null infinity, timelike infinity, spacelike infinity, horizons, and singularities at finite coordinate locations.

An event horizon is the null boundary of the causal past of future null infinity. Its global definition makes it respond teleologically to matter that falls in later.

If coaxial loops of radii $R$ and $2R$, separated by $D$, carry currents $I$ and $2I$ in opposite senses, their on-axis fields cancel between the loops at distance $D/3$ from the smaller loop.

A circular loop of radius $R$ carrying electric current $I$ has, on its axis,where the sign of $\hat{z}$ is fixed by the right-hand rule. At its centre, $B(0)={\mu }_{0}I/(2R)$.

Put the loop in the $xy$ plane, centered at the origin, with positive electric current counterclockwise when viewed from $+z$. The right-hand rule then gives the central magnetic field along $+z$. Parameterize source positions by the angle $\varphi$ and observe at $(0,0,z)$:Here $R$ without bold type is the loop radius, whereas $\mathbf{R}$ is the source-to-observer displacement. Their cross product isThe denominator $|\mathbf{R}{|}^3=(R^2+z^2{)}^{3/2}$ is constant around the loop. The integrals of sine and cosine over a full turn vanish, while the axial contribution adds to $2\pi R^2$. Substitution into the Biot-Savart law proves the on-axis formula. Symmetry cancels transverse components; it does not cancel the axial components.

For an original one-turn example in the same vacuum approximation, choose $R=0.04\mathrm{m}$ and $I=2\mathrm{A}$. With ${\mu }_0\approx 4\pi \times 10^{-7}\mathrm{T}m/A$,At $z=0.03\mathrm{m}$, the source-to-observer distance is $0.05\mathrm{m}$, so $B(z)/B(0)=64/125=0.512$ and $B(z)\approx 16.08\mu \mathrm{T}$. At $z=-0.03\mathrm{m}$, the magnetic field has the same magnitude and still points along $+z$. The point has crossed the loop plane, but the loop's electric current has not reversed. Reversing the electric current reverses the magnetic field on both sides.

For $|z|\ll R$, a Taylor expansion gives $B(z)/B(0)=1-3z^2/(2R^2)+O(z^4/R^4)$: the center is a smooth maximum, with zero first derivative. For $|z|\gg R$,This is the axial magnetic dipole field of magnetic dipole moment $\mathbf{m}=I\pi R^2\hat{\mathbf{z}}$.^[ref] A compact closed loop has an inverse-cube axial far field, in contrast to the inverse-distance field of the infinite straight wire. The SI units of the exact loop formula are ${\mu }_{0}I{R}^2/(R^2+z^2{)}^{3/2}\sim {\mu }_{0}I/\text{length}$, hence tesla.

For $N$ coincident turns carrying the same electric current, the superposition principle multiplies the ideal result by $N$. A real coil has radial width and axial length: if those are not negligible, integrate or sum over its individual turns. This is why a single measured coil diameter and a turn count may not determine the full spatial magnetic field precisely. A collection of loops becomes a solenoid; integrating their axial contributions gives the on-axis magnetic field of a finite solenoid.

A pinned filament of length $L$ and bending modulus $A$ first buckles under compressive force $F_c={\pi }^{2}A/L^2$.

The worm-like chain models an inextensible filament with bending energy $A\int {\kappa }^{2}ds/2$ and thermal tangent fluctuations.