The Wigner-Seitz cell of a reciprocal lattice centered at the origin. It contains wavevectors at least as close to zero as to any other reciprocal lattice point. Boundary sharing conventions do not change its area or volume.

The extended-zone region reached from the first zone by crossing one Bragg plane from the origin, excluding boundaries. For a one-dimensional lattice with spacing $a$, its two components are $\pi /a<|k|<2\pi /a$. It is not simply a translated first-zone cell.

A reduced Brillouin zone is the reciprocal primitive cell for an enlarged real-space unit cell. Doubling a one-dimensional unit cell halves the momentum interval but adds a second internal sublattice degree of freedom. Band folding therefore preserves the number of normal modes.

The complex logarithm has infinite-sheeted branching: continuing once around its zero changes its value by $2\pi i$. Multiplication by a function vanishing at that point need not remove the branching. For example $z^m\log z$ with positive integer $m$ tends to zero but changes by $2\pi i{z}^m$ after continuation around zero. This remains a branch point, rather than a removable singularity, because no single-valued holomorphic function exists on a full punctured neighbourhood. The coordinate $w=1/z$ tests logarithmic branching at infinity in the same way.

Consider a frictionless track from rest between equal-height endpoints separated horizontally by $l>0$. Let depth $y$ be positive downward and let the maximum depth be $h>0$. Conservation of energy gives speed $\sqrt{2gy}$. The Cauchy-Schwarz inequality, applied to the tangent displacement, givesThe squared norm of the coefficient vector is $1/h+(h-y)/(hy)=1/y$. The track goes from depth zero to $h$ and returns to zero, so the second term integrates to at least $2{\int }_0^h\sqrt{(h-y)/(hy)}dy=\pi \sqrt{h}$. Integrating the first term gives $l/\sqrt{h}$, even if the track backtracks horizontally. Division by $\sqrt{2g}$ proves the first bound. The arithmetic-geometric mean inequality proves the second.

Equality requires $h=l/\pi$ and tangent slope $|dy/dx|=\sqrt{(h-y)/y}$, on the descending and ascending halves respectively. Substitution $y=h{\sin }^2\theta$ integrates to the cycloid $x=h(\theta -\frac{1}{2}\sin 2\theta )$, $0\le \theta \le \pi$. It attains the bound, proving global minimality of the single-arch brachistochrone problem between these endpoints. Endpoint singularities are handled as limits of the integrals; the cycloid has finite travel time.

Measure polar angle $\theta$ from the downward vertical on the lower hemisphere of radius $R$. A particle released from rest at the rim and sliding without energy loss has speed $\sqrt{2gR\cos \theta }$. The spherical arc length along $\varphi =\varphi (\theta )$ is $R\sqrt{1+{\sin }^2\theta {\varphi }^{\prime 2}}|d\theta |$. Thus its travel time has integrand proportional to $\sqrt{(1+{\sin }^2\theta {\varphi }^{\prime 2})/\cos \theta }$. Since the azimuth is a cyclic coordinate, the Euler-Lagrange equation givesSolving for ${\varphi }^{\prime 2}$ gives the displayed first integral on a segment where $\theta$ is a valid path parameter. For finite $C$, approaching the rim $\theta \uparrow \pi /2$ gives ${\varphi }^{\prime }\to 0$. Thus a regular stationary path begins along the meridian toward the bottom. This is an initial direction condition; a path joining other rim points can turn before reaching the bottom. Monotone segments use positive $|d\theta |$ in the time integral, and a turning point requires another path parameter or a split into segments.

A correction for the difference between real surrounding relief and the assumed simple topographic slab. Its computation needs terrain geometry and an assumed density, and can matter in rough surface topography.

Two weakly coupled condensates in a double-well potential realize one bosonic Josephson junction. The 2005 experiment measured the density distribution in situ and inferred the evolving relative phase from interference fringes. It distinguished tunneling oscillations from nonlinear suppression of large-amplitude transfer, called macroscopic quantum self-trapping.^[ref]

An ordinary normal-ordered coherent-state Gaussian for energy $\epsilon a^{†}a$ gives $(1-e^{-\beta \epsilon }{)}^{-1}$. The physical oscillator with zero-point energy $\epsilon /2$ requires the factor $e^{-\beta \epsilon /2}$. A symmetric determinant prescription may encode that factor implicitly; it must not be counted twice.

With exact one-slice weight $q=e^{-\beta \epsilon /M}$ and periodic shift matrix $P$, $\det (I-qP)=1-q^M$. This finite-dimensional identity fixes the normal-ordered thermal determinant without an ambiguous unregulated frequency product.

Physical imaginary time runs over $[0,\beta \hslash ]$ with an action divided by $\hslash$. Rescaling it by $\hslash$ gives a variable in $[0,\beta ]$ and the operator ${\partial }_{\tau }+\epsilon$. Matsubara eigenvalues in the latter variable have energy units.

Bosonic traces impose periodic complex fields and inverse determinants. Fermionic traces impose antiperiodic Grassmann fields and determinants. The distinction encodes quantum statistics, not a freely chosen boundary condition.

In a bosonic coherent-state path integral, normal ordering determines which fields occur on the two neighboring time slices. For unnormalized coherent states, the short-step kernel obeyswhere $K$ is the normal-ordered grand Hamiltonian and $ϵ$ has inverse-energy units. Creation operators supply the later conjugate field and annihilation operators the earlier field.

The overlaps yield the temporal derivative term in the continuum action, but replacing the two times by one too early can lose equal-time ordering information. Retaining the discrete prescription fixes contact terms and the meaning of occupation insertions. A continuum expression is therefore used together with its time-slice convention, especially when evaluating products at coincident times.

A gas whose indistinguishable constituents are bosons. The ideal model follows Bose-Einstein statistics, while interactions modify its excitations and thermodynamics. Macroscopic ground-state occupation can occur under suitable dimension and density conditions, but it does not alone determine a superfluid response.

The single proximal element of the upper limb and the corresponding pectoral appendage in relevant fossil comparisons.

One of the two principal forearm bones; movement relative to the ulna contributes to forearm rotation.

One of the principal forearm bones, paired with the radius bone.