A complex group representation $\rho :G\to \mathrm{GL}(n,\mathbb{C})$ defines a real representation on ${\mathbb{R}}^{2n}$ by viewing each complex coordinate as its real and imaginary parts. In grouped coordinates its matrices are the realification of a complex matrix shown in the title. Direct block multiplication gives $\mathcal{R}(AB)=\mathcal{R}(A)\mathcal{R}(B)$ and $\mathcal{R}(I_n)=I_{2n}$, proving the representation property and preservation of invertibility.

Taking the trace gives its character ${\chi }_{\mathbb{R}}(g)=2\mathrm{Re}{\chi }_{\mathbb{C}}(g)$. After complexification of a real vector space, this realification splits as the original representation and its conjugate. Realification need not preserve irreducibility: the answer depends on whether the irreducible complex representation has real, complex or quaternionic representation type. The standard two-dimensional complex representation of the quaternion group gives a four-dimensional real example with character four at the identity, minus four at its central negative identity, and zero at the other six elements.

The fuel-temperature derivative of reactor reactivity associated with thermal broadening of nuclear resonances, with the other state variables specified. In ordinary uranium-fueled water reactors it is negative: heating the fuel increases effective resonance capture. Its magnitude depends on spectrum, fuel state and resonance self-shielding, so its sign does not determine how its magnitude changes under a lattice redesign.^[ref]

The change of reactor reactivity with a specified material temperature, with the other variables held as stated. Several component coefficients need not have the same sign.

A tetrahedral intermediate has a formerly trigonal carbonyl group carbon bonded to four groups after nucleophilic addition. Its subsequent elimination can restore a carbonyl group.

A hypersurface $s(q)=0$ separating selected reactant and product regions. A rate derived from signed flux and a suitable long-time commitment is independent of its location when the dynamics and normalization are treated consistently.

For isotropically oriented independent needles illuminated with $z$ polarization and viewed along $y$, the amplitudes through $x$ and $z$ analysers are proportional to $u_x{u}_z$ and $u_z^2$. The moments $⟨u_x^2{u}_z^2⟩=1/15$ and $⟨u_z^4⟩=1/5$ give intensity ratio $1:3$.

In a homogeneous medium driven by a plane wave, coherent scattering amplitudes carry phases $e^{i({\mathbf{k}}_i-{\mathbf{k}}_s)\cdot \mathbf{r}}$. An infinite uniform volume cancels arbitrary nonforward components. Finite shapes, boundaries and fluctuations determine any side signal; summing independent-particle intensities is not the continuous-medium limit.

An electrically small perfect conductor sphere has both an induced electric dipole moment and magnetic dipole moment. The latter contributes half the electric scattering amplitude in magnitude. The two amplitudes add in the backward direction and partially cancel in the forward direction. Thus the backward intensity is $9/4$ of the electric-only value and its radar cross section is $9\pi k^4{a}^6$. The correction persists at leading order as $ka\to 0$; discarding the magnetic term is a separate response approximation.^[ref]

In ideal single Rayleigh scattering of unpolarized incident light by an isotropic small scatterer, observed intensities perpendicular and parallel to the scattering plane are proportional to 1 and ${\cos }^2\chi$. Their normalized difference is $\mathcal{P}={\sin }^2\chi /(1+{\cos }^2\chi )$. At a right-angle scattering geometry the output is fully linear and perpendicular to that plane. Multiple scattering, large particles and anisotropic scatterers change this ideal sky-polarization law.

A random number generation technique that retains the unused equally likely outcomes after a rejected trial in rejection sampling. Instead of discarding those outcomes, it uses their conditional probability to obtain a new uniform distribution on a finite set for the input state of the next trial.

The Watts-Strogatz model starts with $n$ vertices arranged on a ring, each joined to a fixed even number $k$ of nearest neighbors. It then rewires one orientation of each original ring edge independently with probability $\beta$, choosing a new endpoint while avoiding loops and duplicate edges. The number of edges is preserved; individual degrees need not be.^[ref]

At $\beta =0$ the graph is a regular ring lattice. A small positive rewiring probability introduces shortcuts that can sharply reduce typical path lengths while retaining substantial local clustering. This offers a mechanism for small-world networks with both properties. It is a model of a structural effect, not a claim that every observed network was generated by literal rewiring.

The equivalent input noise temperature of the antenna-plus-receiver system is defined by available noise power $P_n=k_B{T}_{\mathrm{sys}}\Delta \nu$ in one matched polarization of waves mode. It includes receiver noise and the appropriate sky and instrumental contributions. It is not generally the physical temperature of a single component.

Radical polymerization in which growing chain ends reversibly exchange between active radicals and dormant species. Keeping the active-radical concentration small can reduce bimolecular termination and narrow the molar mass distribution. Termination and chain transfer may still occur, so reversible deactivation alone does not guarantee ideal living polymerization.

For $F(C)-F(C_0)=a\ln (C/C_0)$, a further annual concentration increment produces $a\ln [(C+\Delta C)/C]$, whereas the accumulated forcing from the original baseline is $a\ln (C/C_0)$. The former can decrease as $C$ rises while the latter increases. Maintaining a short-lived compensating aerosol burden also requires replacement, not only compensation of the next increment.

A radiation spectrum whose spectral flux density follows a power law over a stated frequency range. For $F_{\nu }\propto {\nu }^{-\alpha }$, the spectral energy distribution satisfies $\nu F_{\nu }\propto {\nu }^{1-\alpha }$. Thus $\alpha =1$ gives equal energy flux in equal logarithmic frequency intervals. A spectral break invalidates extrapolation with a single index.

The angular multipole of order two in a local radiation intensity distribution. A monopole is isotropic and a dipole has one directional axis; a quadrupole has a second-order angular pattern. Thomson scattering of an incident radiation quadrupole can generate linear polarization, which is central to Cosmic microwave background polarization.

At fixed specific angular momentum $j$, perturb the effective potential $\Phi (r)+j^2/(2r^2)$ about a circular orbit. Linear radial motion has $\ddot{x}+{\kappa }^{2}x=0$, where ${\kappa }^2={\Phi }^{\prime \prime }+3{\Phi }^{\prime }/r=r^{-3}d(j_c^2)/dr$. Positive ${\kappa }^2$ gives stable oscillations; axial symmetry alone does not force that sign.