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A right circular cone with vertex $O$ selects all directions within a half-opening angle $\theta$ of its axis. Here $\theta$ is measured from the axis to a boundary direction, so the full opening in a meridian plane is $2\theta$. For $0\le \theta \le \pi$, interpret the selected directions as a spherical cap; for $\theta >\pi /2$, this is a region larger than a hemisphere.

The spherical coordinate system gives the area element $dA=R^2\sin \alpha d\alpha d\varphi$ on the sphere. Dividing by $R^2$ and integrating over the cap yieldsThe factor $\sin \alpha$ matters: a thin angular strip near a pole contains fewer directions than an equally wide strip near the equator. Treating equal intervals of both spherical coordinates as equal solid angles would overcount the polar directions.

At $\theta =45^{\circ }$, the solid angle is $\pi (2-\sqrt{2})\approx 1.8403$ steradians, or about $14.64\%$ of all directions. At $\theta =90^{\circ }$ it is $2\pi$ steradians, and at $180^{\circ }$ it is $4\pi$ steradians. For a small $\theta$ measured in radians, the Taylor series $\cos \theta =1-{\theta }^2/2+O({\theta }^4)$ gives the small-angle approximation $\Omega \approx \pi {\theta }^2$. Doubling a narrow half-opening angle therefore approximately quadruples its coverage.

A crystal is a solid with long-range spatial order in its constituents. In a periodic crystal, translation by a lattice vector reproduces the structure; a unit cell and its constituent arrangement describe the repeating pattern. The atomic structure is distinct from the visible external shape: an irregularly cut specimen can remain a single ordered crystal.

Diffraction probes that order because radiation scattered from many regularly arranged sites can reinforce at selected directions. The resulting pattern reveals structural spacings and orientation, rather than merely showing the specimen's outer faces. A single crystal maintains one coherent orientation across the specimen, while a polycrystalline material contains many crystal grains. Crystal orientation can make properties anisotropic, so a test along one direction need not predict response along another. Surface shape additionally depends on equilibrium surface free energy and growth kinetics.

The dynamic modulus is the complex stress-amplitude to strain-amplitude ratio in steady oscillatory response. With the convention $e^{i\omega t}$, the real part is the storage modulus and the imaginary part is the loss modulus. For $ϵ={ϵ}_0\cos \omega t$, $\sigma ={ϵ}_0(G^{\prime }\cos \omega t-G^{\prime \prime }\sin \omega t)$. The average supplied power is $\omega {ϵ}_0^2{G}^{\prime \prime }/2$, fixing the loss-sign convention. The elastic coefficient is a shear or extension modulus according to the deformation measured; it is unrelated to the complex modulus meaning absolute value of a complex number.

For a real skew-symmetric matrix $M$, $M^2$ is symmetric and $x^T{M}^{2}x=-\Vert Mx{\Vert }^2$. On an eigenspace with eigenvalue $-{\mu }^2<0$, every unit vector $u$ has unit partner $v=Mu/\mu$, orthogonal to $u$, with $Mu=\mu v$ and $Mv=-\mu u$. Its plane and its orthogonal complement are invariant under $M$. Repeating gives real orthogonal two-dimensional blocks $\left(\begin{array}{cc}0& -\mu \\ \mu & 0\end{array}\right)$ and a zero block. Thus every nonzero eigenvalue of $M^2$ has even multiplicity, the zero multiplicity has the parity of $n$, and the number of distinct eigenvalues is at most $n/2$ for even $n$ and $(n+1)/2$ for odd $n$. Distinct positive block parameters attain the bound, with one zero direction added when $n$ is odd.

A real skew-symmetric matrix is an orthogonal matrix exactly when its square is minus the identity matrix. Indeed, $B^{\mathsf{T}}B=-B^2$. Such a matrix can exist only in even dimension: taking determinants of $B^2=-I$ would otherwise give $(\det B{)}^2=-1$. In two dimensions every example isThese are rotation matrices through opposite right angles. Their orthogonal similarity transforms preserve eigenvalues, but do not in general produce a diagonal matrix.

Every real $2\times 2$ antisymmetric matrix has the form $A=sJ$, where $J=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$. Thus $A^2=-s^{2}I$, $\mathrm{tr}A=0$, and $\det A=s^2$. In particular,A convergent scalar power series applied to $A$ can therefore be reduced into an even scalar multiple of $I$ plus an odd scalar multiple of $A$. For real coefficients and $s\ne 0$, this is $f(A)=\mathrm{Re}f(is)I+[\mathrm{Im}f(is)/s]A$, provided the series converges at $\pm is$. Absolute convergence justifies separating even and odd terms. This converts many small matrix series problems into ordinary scalar series, without computing eigenvectors.

Let $B$ be an invertible matrix and $C$ a real square matrix of the same odd order. If the displayed identity holds, multiply it on the left by $B$ to obtain $B{B}^{\mathsf{T}}{C}^{\mathsf{T}}=-C$. Taking determinants givesThus $((\det B{)}^2+1)\det C=0$. Since $\det B$ is real, the first factor is positive, forcing $C$ to be a singular matrix. Both odd dimension and reality matter: an invertible real antisymmetric matrix exists in even dimension, while over the complex numbers a squared determinant can equal $-1$.

In the product basis $|m_1,m_2⟩$ for two spin-one particles, total magnetic quantum number zero allows only $|1,-1⟩$, $|0,0⟩$ and $|-1,1⟩$. Applying the total ladder operators to $\alpha |1,-1⟩+\beta |0,0⟩+\gamma |-1,1⟩$ givesA singlet state is annihilated by $S_{+}$ and $S_{-}$, so $\alpha =\gamma =-\beta$. Wavefunction normalization gives the displayed state, up to an overall phase. Here the ket on the left of the title formula has total-spin labels, while its right-hand kets have individual magnetic labels. The state is symmetric under exchanging the particles, although no exchange constraint is needed for distinguishable particles.

A barycentric coordinate system assigns to a point in the affine span of the vertices of a nondegenerate simplex its unique vertex weights ${\lambda }_0,\dots ,{\lambda }_d$, constrained to sum to one. Uniqueness follows because $v_1-v_0,\dots ,v_d-v_0$ form a basis of the translation space: expand $x-v_0$ in that basis and set ${\lambda }_0=1-{\sum }_{i=1}^d{\lambda }_i$. A point lies in the closed simplex exactly when every weight is nonnegative. An affine function satisfies $f(x)={\sum }_i{\lambda }_{i}f(v_i)$, which makes these coordinates useful for interpolation and the integral of an affine function over a simplex.

A point mass constrained to a straight guide through the axis of a uniformly rotating reference frame, with signed central displacement $x$ and Hooke's law force $-kx$, satisfies $m\ddot{x}=-kx+m{\Omega }^{2}x$. The transverse Coriolis force is balanced by the guide. The central equilibrium is stable for ${\Omega }^2<k/m$ and unstable above it. At equality the restoring coefficient vanishes and $x(t)=x_0+v_{0}t$: a small initial velocity causes unbounded drift in the ideal guide. The threshold therefore has no positive oscillation frequency. The angular frequency is the positive square root in the title.

For $x^{\prime \prime }+{\omega }_0^{2}x=F$ during $0<t<T$ and zero force afterwards, with zero initial displacement and velocity, the causal Green function gives $x(t)=F[1-\cos ({\omega }_{0}t)]/{\omega }_0^2$ during the pulse. For $t\ge T$ it gives $x(t)=F[\cos ({\omega }_0(t-T))-\cos ({\omega }_{0}t)]/{\omega }_0^2$. For $F\ne 0$, both outgoing displacement and velocity vanish precisely when $T$ is an integer number of full natural periods. Cancellation requires both initial data for the subsequent homogeneous equation, not only zero displacement at switch-off.

The SRP receptor receives the SRP–ribosome–nascent chain complex for handoff to the translocon.^[ref]

The dimensionless parameter adjusting the forward slope in the Shockley diode equation. Its ideal junction law sets $n=1$.

The positive electric current scale in the Shockley diode equation.

The electric potential difference corresponding to thermal energy $k_{B}T$ per elementary charge $q$.

For a long, nondegenerate, ideal p-n junction under low injection, the diode saturation current is set by minority carriers supplied from the neutral regions. Their equilibrium densities are $n_{p0}=n_i^2/N_A$ and $p_{n0}=n_i^2/N_D$. Solving the steady diffusion and recombination equations gives $I_S=e\mathcal{A}(D_n{n}_{p0}/L_n+D_p{p}_{n0}/L_p)$, where $\mathcal{A}$ is junction area, $D_n,D_p$ are diffusion coefficients, and $L_n,L_p$ are diffusion lengths. Reverse bias suppresses opposite majority-carrier injection exponentially while collecting this minority supply. Thus $I_S$ is approximately bias independent before breakdown, but depends strongly on temperature through $n_i^2$ and on transport, doping, and geometry.

The infinitesimal strain tensor $\mathrm{diag}(-e,e)$ compresses one principal direction and stretches the perpendicular one by the same amount. Rotating the coordinate axes through $45^{\circ }$ turns it into a tensor with zero normal strains and off-diagonal entries of magnitude $e$. The engineering shear strain is twice the tensor shear component, so its shear angle has magnitude $\gamma =2e$. Likewise principal stresses $(-S,S)$ give zero normal stress and shear stress of magnitude $S$ on those rotated faces. The shear modulus law $S=G\gamma$ therefore gives $S=2Ge$.

For inviscid nonrotating shallow water equations, columnar horizontal flow has material vorticity equation $D\zeta /Dt=-\zeta {\nabla }_h\cdot {\mathbf{u}}_h$ and depth equation $Dh/Dt=-h{\nabla }_h\cdot {\mathbf{u}}_h$. Thus $\zeta /h$ is conserved, not $\zeta$ itself. Squashing a column to much smaller depth decreases its vertical vorticity. This permits a late draining flow to approximate an irrotational $u_{\theta }=C/r$ profile outside the drain, despite nonzero initial solid-body vorticity. At finite nonzero depth the residual vorticity need not be exactly zero. In a rotating frame, replace $\zeta$ by absolute vorticity $\zeta +f$ in the conserved ratio. Kelvin's circulation theorem remains compatible with these changes because it follows a material loop, not a fixed spatial circle.

Write ${\omega }_0^2=1/(LC)$ and $\gamma =R/L$ for a series RLC circuit. The squared charge response has denominator $D=({\omega }_0^2-{\omega }^2{)}^2+{\gamma }^2{\omega }^2$. Its nonzero resonance peak occurs at ${\omega }_r^2={\omega }_0^2-{\gamma }^2/2$. The two squared half-height frequencies, when both exist, areTheir difference in angular frequencies is only approximately $\gamma$ at large quality factor. The electric current instead has $|I{|}^2=|V{|}^2/[R^2+(\omega L-1/(\omega C){)}^2]$. Its exact half-power frequencies satisfy $|\omega L-1/(\omega C)|=R$, whose positive roots differ by $R/L$, independently of capacitance. Thus an exact capacitance-independent bandwidth applies to current or resistor-voltage output; capacitor-voltage output is a low-pass response with a possible resonant peak.

A double series is a series whose terms carry two indices. For nonnegative terms, define its sum as the supremum of all finite subsums; iterated sums in either order give this same value, possibly infinity. For absolutely summable terms, both orders also agree and give a finite sum. Without these hypotheses, an order of summation must be specified and changing it can change the result. A triangular index set, such as $0\le n\le m$, can be represented by setting the other terms to zero.