A nuclear-encoded plastid sigma factor involved in activation of the psbD gene promoter in Arabidopsis thaliana.^[ref]

A perfect vector $a⟨110⟩/2$ can separate into two in-plane Shockley partials bounding a stacking-fault ribbon. The sum of squared Burgers magnitudes falls from $a^2/2$ to $a^2/3$, reducing the simplest elastic line-energy estimate, while stacking-fault area costs energy.

A shock wave with significant radiation losses in its downstream evolution. Often a narrow dissipative jump initially heats the gas according to adiabatic Rankine-Hugoniot conditions, followed by a much longer radiative cooling layer. The gas's specific entropy increases at the jump and may then fall as radiation carries entropy away. If both distant states have equal temperature, the combined mass and momentum balances give an isothermal shock; the gas's energy flux is not conserved across the whole radiating region.

Two identical cold clouds approach at speeds $\pm V$ in their center-of-momentum frame. Reflection symmetry makes the uniformly compressed gas stationary. A shock moving outward at $S$ sees upstream speed $V+S$ and downstream speed $S$; their ratio equals the strong adiabatic compression $(\gamma +1)/(\gamma -1)$. Solving gives the displayed speed. For a monatomic gas $S=V/3$. Finite-cloud breakout ends the steady upstream supply and requires a subsequent boundary-release calculation.

A horizontally compressive force per width $F$, carried uniformly through thickness $a$, gives additional horizontal stress $F/a$. Resolving its traction on a fault dipping at angle $\delta$ gives the displayed mean shear stress. An isotropic background pressure adds only normal traction. The integrated force alone does not determine the actual depth-dependent stress distribution or whether a fault fails.

A numerical semigroup is a subset of the nonnegative integers that contains zero, is closed under addition, and has finite complement within the nonnegative integers. For example,Its missing values are $1,2,4,7$. To verify that no larger value is missing, observe that $8,9,10$ are present and then repeatedly add three, using consecutive-base induction for additive semigroups.

A finite family of positive integer generators produces a numerical semigroup exactly when their greatest common divisor is one. The condition is necessary because all generated integers are multiples of that greatest common divisor. Sufficiency follows by finding a generated representative of each residue class modulo one generator and then adding multiples of that generator.

A description of earthquake source geometry, often strike, dip and rake for a double-couple shear fault. First-motion polarities constrain the radiation sign pattern, with a fundamental interchange ambiguity between fault and auxiliary planes.

An ideal compact volume source has $M_{ij}(t)=M(t){\delta }_{ij}$. In an unbounded homogeneous isotropic solid its far-field body radiation is P only. Boundaries and heterogeneity can create converted SV and surface-wave energy; a real explosion can have nonisotropic components.

A stress glut tensor describes a seismic source through an equivalent body-force density $\mathbf{f}=-\nabla \cdot \mathbf{\Gamma }$. For a symmetric stress glut tensor supported away from the boundary, integration by parts makes its coupling to a normal mode equal to $\int \mathbf{\Gamma }:{\mathbf{e}}^{\ast }dV$, where $\mathbf{e}$ is that mode's strain.

For mass-normalized normal modes, an instantaneous step in a point seismic moment tensor excites mode $\alpha$ with $C_{\alpha }=\mathbf{M}:{\mathbf{e}}_{\alpha }^{\ast }({\mathbf{x}}_s)$ and $\tau =t-t_s$. Convolving the causal oscillator Green's function with the step gives $a_{\alpha }=C_{\alpha }[1-\cos ({\omega }_{\alpha }\tau )]H(\tau )/{\omega }_{\alpha }^2$. The expression contains a static response and undamped oscillations; attenuation requires an extension of the conservative model.

A double-couple seismic source has a moment tensor proportional to $\mathbf{b}\otimes \mathbf{n}+\mathbf{n}\otimes \mathbf{b}$, with the slip direction $\mathbf{b}$ perpendicular to the fault plane's normal $\mathbf{n}$. Its eigenvalues are proportional to $(1,0,-1)$. A general symmetric seismic moment tensor need not have this restricted form.

For uniform slip on a shear fault, scalar seismic moment is $M_0=\mu Ad$, with shear modulus $\mu$, fault area $A$ and slip magnitude $d$. More generally it integrates rigidity-weighted slip. It measures source strength rather than radiated energy alone.

In a two-layer P wave model, the head-wave branch overtakes the direct upper-layer arrival at $X_c=2H\sqrt{({\alpha }_2+{\alpha }_1)/({\alpha }_2-{\alpha }_1)}$. It differs from the smaller minimum offset needed for the critical-ray construction.

For a P ray incident from a layer of speed ${\alpha }_1$ onto a faster layer ${\alpha }_2$, the transmitted ray is tangential when $\sin i_c={\alpha }_1/{\alpha }_2$. Angles are measured from the interface normal.

In an all-left-handed Weyl basis $({\nu }_L,{\nu }_R^c)$, the symmetric matrix has left and right Majorana entries $M_L,M_R$ and off-diagonal Dirac entry $m$. Real eigenvalues can have signs; physical Majorana masses are nonnegative after appropriate field rephasings.

In the type-I regime $M_L\simeq 0$, $M_R\gg m_D$, the light mass is $m_D^2/M_R$. Specifying only the right-handed scale does not fix a numerical light mass. A choice of Dirac mass is an extra assumption, not a result of matrix diagonalization.

For a one-to-one elementary reaction with concentrations x and h, the difference $\Delta =h_0-x_0$ remains constant. Integration gives $\ln (h{x}_0/(x{h}_0))=k\Delta t$ when $\Delta \ne 0$; the equal-initial-concentration limit is $1/x-1/x_0=kt$.

Motion of a screw segment from one allowed glide plane to another containing the same Burgers vector.

At $r=6r_g$, the local circular speed is $c/2$, so the receding direct-projection shift is $\sqrt{2}(1+\sin i/2)$. The upper envelope over inclinations is $3/\sqrt{2}$. It is not the shift for a face-on or arbitrary azimuthal emitter, and it excludes plunging-region emission and ray-tracing corrections.