For independent units with lower and upper energy levels separated by $\Delta >0$, let their degeneracies be $g_0$ and $g_1$. Their shifted partition function is $q=g_0+g_1{e}^{-x}$, with $x=\Delta /(k_{\mathrm{B}}T)$. Differentiation givesThe heat capacity vanishes at both low and high temperature. Degeneracy changes both the maximum and its temperature; the equal-degeneracy result is not universal. Independent lower-level degeneracy also leaves residual entropy $N{k}_{\mathrm{B}}\ln g_0$ in the ideal model.

An orthonormal subsystem basis appearing in a Schmidt decomposition. Its vectors diagonalize the corresponding reduced density matrix. Degenerate Schmidt coefficients permit a freedom to rotate the paired bases together; zero-weight basis vectors complete the subsystem space without contributing to the state.

A conducting tip probes the energy-dependent local density of states by recording tunnel current versus applied bias, often measuring $dI/dV$ with a small bias modulation and lock-in detection. At low temperature and for slowly varying tip density and matrix element, $dI/dV$ is proportional to sample local density of states at the corresponding bias energy. Thermal broadening, surface structure and energy-dependent tunnelling selectivity limit a direct interpretation as bulk band structure.

Controlled positioning or modification of individual surface atoms or molecules, including manipulation with a scanning probe. A weakly coupled imaging pass locates the adsorbate; stronger tip interaction or suitable excitation changes its motion or binding. Subsequent imaging checks the result. Temperature, substrate, tip state and atomic species determine the useful mechanism.

Detection of transmitted electrons within an angular annulus excluding the direct beam. Contrast can include diffraction, atomic-number and thickness contributions depending on the collection angles.

An electron microscope that scans a focused probe across an electron-transparent specimen and records transmitted-electron signals at each scan position. Its image magnification is set by the displayed-to-scanned length ratio, separately from angular detector collection.

At constant radiative efficiency of black-hole accretion $\eta$ and the Eddington luminosity, a black hole grows exponentially with the Salpeter time $t_S=\eta t_E/(1-\eta )$, where $t_E={\sigma }_{T}c/(4\pi G{m}_p)$. Its doubling time is $t_S\log 2$. The factor $1-\eta$ accounts for the supplied rest energy that escapes as radiation instead of increasing the hole's mass.

For fixed efficiency, growth from $M_i$ to $M_f$ within $\Delta t$ at no more than the Eddington luminosity requires $[\eta /(1-\eta )]t_E\log (M_f/M_i)\le \Delta t$. Solving gives an upper bound, because greater radiative efficiency reduces retained mass growth at a fixed luminosity. A duty cycle below unity tightens the requirement; mergers and super-Eddington episodes invalidate its stated premises.

For an isotropic layered medium, SH displacement is perpendicular to the vertical propagation plane and SV displacement lies within it. P-SV coupling occurs at interfaces, while SH remains a separate polarization sector.

The epoch when the gaseous envelope mass becomes comparable with the solid core mass. Envelope self-gravity then changes structure and can accelerate gas capture through cooling and contraction. Crossover is a useful stage in classical models, rather than a universal core mass independent of opacity and accretion history.

For a nonzero complex number $z=r{e}^{i\theta }$ with $r>0$, its two square roots are $\pm \sqrt{r}{e}^{i\theta /2}$. They are distinct and differ by a sign. At $z=0$ the only square root is zero. A principal value convention chooses $\theta =\mathrm{Arg}z\in (-\pi ,\pi ]$ and the plus sign; continuity across the negative real axis then fails. A holomorphic square root is a different, domain-dependent notion: it requires a holomorphic function whose square is $z$ on the specified domain. It cannot exist on every punctured neighborhood surrounding zero.

A Newtonian rocket collecting stationary matter at rate $\mu$ and combining it with the same rate of carried fuel ejects mass at rate $2\mu$. If relative exhaust speed is $u_0$, momentum conservation gives the title equation and mass conservation gives $\dot{m}=-\mu$. With constant $\mu$ and initial rest, $m=m_i-\mu t$ and $u=2u_0\mu t/m_i$. The limiting speed $2u_0$ would require zero remaining mass in this ideal solution; positive dry mass terminates it sooner. A constant collection rate per time is distinct from a constant collection cross-section through a uniform medium.

The Mediator complex is a multiprotein coactivator involved in coupling regulatory proteins to RNA-polymerase-II transcription. A requirement for mediator activity can be tested by removal and restoration in a reconstituted transcription system.

An RNA-dependent RNA polymerase contributing to double-stranded precursors for canonical RdDM small RNAs.

Its RNA-copying activity helps convert a single-stranded RNA substrate into a double-stranded precursor for later processing. This step differs from copying DNA into RNA, which is performed by a DNA-dependent polymerase. ^[ref]

An RNA sequence element enriched in adenine and uracil that can recruit regulatory proteins affecting RNA stability.

The ridge-push force in a plate model is the depth integral of the horizontal pressure difference between hot and cold compensated columns. To leading order, the old plate's linear excess density gives $F=g{\int }_0^{a}s\delta \rho (s)ds={\rho }_{m}g\alpha \Delta T{a}^2/6$. Here $s$ is depth below the old seafloor. Retaining the geometrical contribution from its lowered surface adds $g({\rho }_m-{\rho }_w)h^2/2$ within the linear-density model. Force per ridge length has units $\mathrm{N}{\mathrm{m}}^{-1}$, distinct from stress.

For a spatially uniform line current $I\Theta (t)$, the causal potential outside the line is $A_z=({\mu }_{0}I/2\pi )\mathrm{arcosh}(ct/r)$ for $ct>r$, and zero before arrival. The finite causal integration length makes it finite at finite time, unlike an absolute magnetostatic potential for an infinite wire.

The generally complex, frequency- and wavevector-dependent relative permittivity describing linear polarization response. Its imaginary part represents dissipative response with the sign fixed by the time convention; optical prefactors connect it to an absorption coefficient.

A P-recursive sequence satisfies a linear recurrence with polynomial coefficients:where the $p_j$ are polynomials and not all zero. The recurrence is required for all sufficiently large relevant integers, allowing exceptional initial indices. Constant-coefficient linear recurrences are a special case.

For $a_n=n!$, the recurrence $a_{n+1}-(n+1)a_n=0$ proves P-recursiveness. By contrast, a simple-looking nonlinear recurrence is not automatically P-recursive. Over a field of characteristic zero, the ordinary generating function of a P-recursive sequence is D-finite, linking discrete recurrences with linear differential equations having polynomial coefficients.