A state with total spin angular momentum quantum number $S$ has $2S+1$ magnetic projections. A singlet state has $S=0$, a triplet state has $S=1$, and the $S=3$ chromium ground term is a septet.

A wavefunction intended for a specified total-spin state contains components of other total spins. For an unrestricted determinant with one alpha and one beta orbital, $⟨S^2⟩=1-|⟨{\varphi }_{\alpha }|{\varphi }_{\beta }⟩{|}^2$ in units of ${\hslash }^2$.

A spin operator describes intrinsic angular momentum. Its components obey angular-momentum commutators; for spin one half, ${\hat{S}}_j=\hslash {\sigma }_j/2$ in terms of the Pauli matrices.

A spin rotation acts on a quantum state through the unitary representation generated by its angular momentum operators. With $\hslash =1$, rotation by angle $\theta$ about axis $\hat{\mathbf{n}}$ is $e^{-i\theta \hat{\mathbf{n}}\cdot \mathbf{S}}$. Simultaneously rotating every site is a global symmetry of an isotropic Heisenberg model.

The total spin is the vector sum of subsystem spin operators and generates their simultaneous rotation. For a rotationally invariant Heisenberg model, both its squared magnitude and any chosen component commute with the Hamiltonian. An anisotropic exchange or field can preserve only a component.

For a sphere of radius $R>0$, the spherical cap above $z=R-h$, with $0\le h\le 2R$, has curved surface area $2\pi Rh$. On the surface use $r(z)=\sqrt{R^2-z^2}$. The surface of revolution area element is $2\pi r\sqrt{1+(dr/dz{)}^2}dz=2\pi Rdz$. Integrating from $R-h$ to $R$ proves the formula. It includes only the spherical surface; if the planar closing disk is required, add $\pi (2Rh-h^2)$. The coordinate expression is interpreted by a limit at a pole.

For a solid spherical cap $r^2+z^2\le R^2$, $z\ge c$, where $0\le c\le R$ and the mass density is $\rho =kz$ with $k\ge 0$, a volume integral in cylindrical coordinates givesHere the height is measured from the centre of the sphere, rather than from the bottom of the cap. Specifying that origin matters for a nonuniform mass density.

A rigid body rolling with effective radius $h$ has kinetic energy $(m+I/h^2)v^2/2$. If a transition changes the rolling constraint while preserving this total energy, the translational speed changes according to the title formula. Maintaining the same speed would generally require energy loss or gain. The energy-preserving condition is an assumption about the transition, not a generic property of sudden impacts.

The solid ball inside a radius-$a$ sphere has volume $4\pi a^3/3$. One derivation integrates concentric shell area: $V={\int }_0^{a}4\pi r^{2}dr$. The sphere itself is the boundary surface; its area is $4\pi a^2$, a different dimensional quantity.

A great circle passing through a chosen pair of antipodal poles, or the pole-to-pole half of that circle at fixed longitude. In spherical coordinates, a fixed azimuth selects such a half-circle. Meridians are special cases of a meridian of a surface of revolution and are geodesics of the sphere.

For a sphere with a chosen polar axis, its equator is the great circle in the plane through its center perpendicular to that axis. On the unit sphere in spherical coordinates it is $\theta =\pi /2$. Like every great circle, it is a geodesic.

Two distinct spheres are externally tangent when their center distance equals the sum of their radii. They are internally tangent when it equals the absolute difference of their radii. At the point of contact they have the same tangent plane.

A logarithmic comparison of specific power against specific energy for storage or conversion devices. With consistent mass and energy units, constant-duration lines satisfy $p_s=e_s/t$. The plotted capability depends on operating conditions and mass convention.

For coordinates $(t,x,y,z)$ and line element $d{s}^2=c^{2}d{t}^2-a^2(t)d{\mathbf{x}}^2$, these are the nonzero Christoffel symbols. The prime means differentiation with respect to $t$, and coordinate zero is $t$, not $ct$. They follow either from the metric-derivative formula or the Euler-Lagrange equations for the affine geodesic Lagrangian. Both mixed lower-index orders must be counted to obtain the factor of two in the spatial geodesic equation.

Spatial barcoding attaches a location-associated sequence label to a measured molecule or its derivative. In spatial transcriptomics, sequencing the barcode together with captured RNA connects expression measurements to tissue position. The barcode identifies the sampling location rather than necessarily the exact original molecular coordinate or a unique cell. Slide-seq implements the principle with individually barcoded, position-mapped beads. ^[ref]

Slide-seq measures gene expression with spatial information by transferring tissue-associated RNA onto a surface carrying position-mapped barcoded beads. Sequencing associates captured transcripts with bead barcodes and hence with locations. The original study used approximately ten-micrometre beads and demonstrated tissue expression patterns at high spatial resolution. A bead measurement is a capture region rather than an automatic identification of one intact cell; tissue geometry and capture efficiency still affect interpretation. ^[ref]

Write a Euclidean isometry as $\{R\mid t\}$, acting by $x\mapsto Rx+t$. Composition is $\{R\mid t\}\{R^{\prime }\mid t^{\prime }\}=\{R{R}^{\prime }\mid R{t}^{\prime }+t\}$. The translation subgroup is normal; the quotient is the crystallographic point group.

A space group for which an origin can be chosen so that representatives of all point-group operations have zero translation part. It is a semidirect product of its lattice translations and point group. Presence of centring translations in a conventional cell does not alone make a group nonsymmorphic.

A space group whose point-group operations cannot all be represented without essential fractional translations for any origin. screw axes and glide planes are characteristic examples. Its point group alone omits these translational data.

A constant space algorithm uses $O(1)$ working storage with respect to the stated input-size parameter. Under a machine-word model, an iterative binary search needs only a fixed number of indices, so it uses constant auxiliary words. Under a bit model, indices up to $n$ require $\Theta (\log n)$ bits, so constant words need not mean constant bits. Input and output storage may be excluded by an auxiliary-space convention, which must be stated. Keeping a fixed number of arbitrarily large integers is not a constant-bit-space guarantee.