For a finite group $G$ represented unitarily by $U(g)$, the central projector onto the isotypic subspace of an irreducible representation $\lambda$ is $P_{\lambda }=(d_{\lambda }/|G|){\sum }_g{\chi }_{\lambda }(g{)}^{\ast }U(g)$, where $d_{\lambda }$ is its dimension. Character orthogonality implies that the operator is identity on copies of $\lambda$ and zero on other irreducibles. For a one-dimensional irreducible, applying it to an atomic orbital gives either zero or a symmetry-adapted linear combination of that symmetry, to be normalized afterward. A multidimensional central projector selects an entire isotypic component rather than one uniquely chosen component function.

A matrix group acts on row vectors on the right by $v\cdot g=vg$, with $(v\cdot g)\cdot h=v\cdot (gh)$. To turn this into a left group action, use $g\cdot v=v{g}^{-1}$. The inverse is necessary to preserve the order of composition.

In Young symmetrizers, a row group instead consists of the permutations preserving each row of a tableau as a set. Its action on entries supports row symmetrization. These two usages concern different objects, so the row representation or tableau convention should be specified.

A column group action often means the action of a matrix group on column vectors by left multiplication: $g\cdot v=gv$. The rule $(gh)v=g(hv)$ makes this a left group action, and the identity matrix fixes every column.

The same term is used in Young symmetrizers for the group permuting entries within each column of a tableau. In that setting the action permutes labels rather than multiplying a coordinate column. The surrounding representation must specify which meaning is intended. For a concrete coordinate example, a permutation matrix exchanges two entries of a column vector.

Addition of an organomagnesium reagent to carbon dioxide followed by protonation gives a carboxylic acid with one additional carbon. A free acid cannot remain present during a subsequent Grignard addition without consuming reagent.

Two organomagnesium additions to a suitable ester give a tertiary alcohol after work-up. The first addition–elimination produces a ketone, which accepts the second carbon substituent.

The whole-space form of Green's second identity requires the displayed boundary limit, together with the existence of the relevant volume integrals. Pointwise decay of the fields does not by itself control their normal derivatives or the growing sphere area. In three dimensions, sufficient power bounds are $u=O(r^{-a})$, $\nabla u=O(r^{-a-1})$, $v=O(r^{-b})$ and $\nabla v=O(r^{-b-1})$, with $a+b>1$: the surface integral is then $O(R^{1-a-b})$.

For a smooth counterexample to pointwise decay alone, take $u=(1+r^2{)}^{-1/2}$ and $v=\cos r(1+r^2{)}^{-1/2}$. Both are smooth radial functions on all of ${\mathbb{R}}^3$ and tend to zero. Nevertheless $u{v}^{\prime }-v{u}^{\prime }=-\sin r/(1+r^2)$, giving surface integral $-4\pi R^2\sin R/(1+R^2)$, which has no zero limit. Screened Green functions with exponential decay avoid this particular problem.

For a surface load above a uniform-density crust over a denser substrate, a simple lithospheric flexure model gives the indicated gravity-topography admittance, with ${\ell }_f^4=D/[g({\rho }_m-{\rho }_c)]$. The surface mass contributes positive gravity; downward displacement of the density interface contributes a compensating negative signal attenuated by $e^{-kt}$. Hence $Z\to 0$ at long wavelength and $Z\to 2\pi G{\rho }_c$ at short wavelength. Here $t$ is the interface depth, not automatically the effective elastic thickness.

For cross-spectrum $S_{gh}$ and autospectra $S_{gg},S_{hh}$, squared coherence is $|S_{gh}{|}^2/(S_{gg}{S}_{hh})$. It measures the fraction of spectral association between gravity and topography. Low coherence can signal uncorrelated internal loading, noise or shortcomings of a simple surface-load model; interpreting it as elastic thickness requires a loading model.

Setting the gravitational Péclet number to one gives the displayed particle-scale estimate. A macroscopic sample instead requires its height to be small compared with the sedimentation scale height. These different observational criteria give different size thresholds; “no sedimentation” is not an exact universal cutoff.

A gas disk near $Q_T\sim 1$ need not form bound fragments: gravitational compression and shocks can heat it into a self-regulated spiral state. Fast enough cooling, often parameterized by $\beta =\Omega t_{\mathrm{cool}}$, permits collapse to continue. A numerical critical beta depends on thermodynamics, dimensionality and the cooling model; it is not universal.^[ref]

A layered material with guest species inserted between graphite sheets. Alkali-metal guests can donate electron density to graphitic bands, altering carrier concentration, spacing and scattering.

Two coupled graphene layers. Their electronic bands depend on relative stacking and external electrostatic conditions, rather than on layer count alone.

graphene oxide that has undergone partial removal of oxygen-containing groups; residual functional groups and lattice defects remain.

Separating a layered material into dispersed thin flakes by mechanical processing in a liquid. For graphite, liquid choice, shear or sonication, classification and recovery jointly control the graphene flake distribution. Preserving the lattice does not guarantee a high monolayer yield or a clean final electrode.^[ref]

The local energy difference between the conduction and valence bands along the boundary of the first Brillouin zone. In a nearest-neighbour model it vanishes at the corner valleys and reaches $2t$ at edge midpoints; this angular variation is compatible with a globally gapless material.

A graph coloring with two available colors in which adjacent vertices receive different colors. A graph admits a two-coloring exactly when it is a bipartite graph. Every nonempty connected bipartite graph has two such colorings when the two color names are distinguished.

Differentiating the grand partition function at fixed temperature and volume gives $⟨N⟩=-{\partial }_{\mu }\Phi$ and $\mathrm{Var}N=-k_{B}T{\partial }_{\mu }^2\Phi =k_{B}T{\partial }_{\mu }⟨N⟩$. It follows because derivatives of the logarithm of the state sum generate cumulants of particle number. Nonnegative variance makes the grand potential concave in chemical potential in a stable grand-canonical description.

For a single one-particle state with $x_k\ll 1$, the empty and singly occupied contributions dominate. A fermion state gives ${\Xi }_k=1+x_k$, a boson state gives $(1-x_k{)}^{-1}$, and Maxwell–Boltzmann statistics give $e^{x_k}$. Thus all have the same leading dilute response, $\ln {\Xi }_k=x_k+O(x_k^2)$. This approximation does not impose an exclusion principle on classical particles.

For one independent fermion level, occupations 0 and 1 form a Bernoulli distribution and give $\mathrm{Var}n=\bar{n}(1-\bar{n})$. For one ideal boson level in contact with a reservoir, its geometric distribution gives $\mathrm{Var}n=\bar{n}(1+\bar{n})$. Both approach variance equal to mean at small occupation. Pauli exclusion suppresses fluctuations of nearly full fermion states, whereas large boson occupation gives variance of order mean squared. Fixed-total-number constraints change condensate fluctuation statements.