At zero temperature, linear-mode vacuum depletion scales as $\int dk{k}^{d-2}$; at nonzero temperature Bose occupation changes this to $\int dk{k}^{d-3}$. The two convergence criteria are different.

A horizontal neutron-reflecting mirror supplies the lower boundary of a quantum bouncer, with the Earth's approximately linear gravitational potential above it. An absorber above the mirror removes neutrons with sufficient vertical extent. Scanning the mirror-absorber gap and counting transmitted neutrons probes the lowest allowed gravitational states. The 2002 work led by Valery V. Nesvizhevsky reported this confinement; a subsequent detailed report describes suppression below the lowest-state height.^[ref][ref]

This is a count-versus-gap experiment, not a direct image of a wavefunction. Surface scattering, absorption and the incident vertical-motion distribution enter a quantitative transmission model.

Loss of a rotating neutron star's angular momentum and rotational energy through electromagnetic torque and a magnetized outflow. For constant moment of inertia $I$, differentiating $E_{\mathrm{rot}}=I(2\pi /P{)}^2/2$ gives the displayed positive loss rate when $\dot{P}>0$. It is not the same as the measured radiation luminosity in any one observing band.

Glutamine has a neutral side-chain amide. Its N–H groups can donate hydrogen bonds and its carbonyl oxygen can accept them; its amide nitrogen is not an ordinary amine base.

Tyrosine has a phenolic side chain. Its hydroxyl can participate in hydrogen bonding and, when the environment is suitable, acid/base chemistry.

A nonzero prime ideal $\mathfrak{p}$ of a number field $K$ is inert in a finite extension $L/K$ when $\mathfrak{p}{\mathcal{O}}_L$ remains a prime ideal. There is then one prime above it, with ramification index one and residue degree $[L:K]$.

For example, $3$ is inert in $\mathbb{Z}[i]$: its quotient is ${\mathbb{F}}_3[X]/(X^2+1)$, a field because $X^2+1$ has no root in ${\mathbb{F}}_3$. By contrast $5=(2+i)(2-i)$ splits. For quadratic extensions, irreducibility of a suitable defining polynomial modulo an unramified prime gives an elementary test for this behavior.

In a finite extension of number fields $L/K$, a nonzero prime ideal $\mathfrak{p}$ of ${\mathcal{O}}_K$ factors uniquely as $\mathfrak{p}{\mathcal{O}}_L={\prod }_i{\mathfrak{P}}_i^{e_i}$. The distinct prime ideals ${\mathfrak{P}}_i$ are the primes above $\mathfrak{p}$, the exponents $e_i$ are ramification indices, and $f_i=[{\mathcal{O}}_L/{\mathfrak{P}}_i:{\mathcal{O}}_K/\mathfrak{p}]$ are residue degrees. They satisfy ${\sum }_i{e}_i{f}_i=[L:K]$.

A prime splits completely if it has $[L:K]$ distinct factors, all with $e_i=f_i=1$. In the Gaussian integers, $(5)=(2+i)(2-i)$ splits, $(3)$ is an inert prime, and $(2)$ ramifies because $2=-i(1+i{)}^2$. The three behaviors distinguish factorization of ideals from factorization of ordinary integers.

For fixed cloud mass $M$ and isothermal sound speed $c_s$, approximate the binding energy by $-3G{M}^2/(5r)$. The boundary-pressure virial theorem gives $A=3M{c}_s^2/(4\pi )$ and $B=3G{M}^2/(20\pi )$. The required external pressure peaks at $r_c=4B/(3A)$. A homologous radial displacement is restoring on the branch $d{p}_{\mathrm{ext}}/dr<0$, or $r>r_c$; the smaller-radius positive-pressure branch is unstable. This is a one-radius virial model, not the exact hydrostatic isothermal-sphere sequence.

For a finite sequence $a_0,\dots ,a_{n-1}$, its suffix sums are $S_i={\sum }_{j=i}^{n-1}{a}_j$, with $S_n=0$. They satisfy $S_i=a_i+S_{i+1}$, so one backward pass computes all of them in $O(n)$ additions.

For $(3,-1,4)$, the suffix sums are $(6,3,4,0)$ including the empty suffix. A range sum on $[i,j)$ is $S_i-S_j$. Thus suffix sums provide the same constant-time range queries as prefix sums, with the accumulation direction reversed.

For finite-radius regular polytropes of one fixed index $n$ and composition, let ${\xi }_1$ be the first zero of the Lane-Emden function and $Y_n=-{\xi }_1^2{\theta }^{\prime }({\xi }_1)$. Hydrostatic scaling gives $GM/R=(n+1)(Y_n/{\xi }_1)\mathcal{R}{T}_c$. Fixed central temperature therefore implies $M\propto R$. This is not the fixed-entropy-constant sequence; if the index or composition changes, their prefactors cannot be discarded.

A reciprocal optical circular polarizer can be assembled from a linear polarizer followed by a quarter-wave plate whose axes make $45^{\circ }$ to the transmitted linear state. Forward passage converts unpolarized light into circular light of half the ideal incident intensity. Reverse passage encounters the retarder first; it leaves unpolarized light unpolarized until the linear polarizer transmits half as linear light. Thus the one-pass intensity can be equal in both directions even though the outgoing polarization differs. Use a coherency matrix, not a single Jones vector, for unpolarized input.

A slender high-permittivity particle responds mainly along its axis $\mathbf{u}$, giving $\mathbf{\alpha }\simeq {\alpha }_{\parallel }\mathbf{u}{\mathbf{u}}^{\mathsf{T}}$. Far-field projection into an observation plane leaves one real polarization direction. Every nonzero output is therefore linearly polarized in this leading approximation, even for circular or unpolarized input.

On a disk of radius $a$, prescribe $V_1$ on the upper open semicircle and $V_2$ on the lower one. The bounded harmonic extension has the formula in the title for $r<a$. The boundary data have only a constant term and odd sine Fourier coefficients, giving $u=(V_1+V_2)/2+[2(V_1-V_2)/\pi ]{\sum }_{m\ge 0}(r/a{)}^{2m+1}\sin ((2m+1)\theta )/(2m+1)$. The complex series is half of $\log [(1+w)/(1-w)]$, where $w=(r/a)e^{i\theta }$. Its quotient has positive real part, so taking the imaginary part yields the displayed principal arctangent without an argument ambiguity. At the two boundary jumps, the radial limit is the mean; the harmonic function has no continuous extension matching both adjacent constant arcs there.

The derivative $D_t={\partial }_t+\{\cdot ,H\}$ is a derivation of the Poisson bracket: $D_t\{f,g\}=\{D_{t}f,g\}+\{f,D_{t}g\}$. The time derivative part obeys the product rule, and the Hamiltonian part obeys it by the Jacobi identity. Therefore explicitly time-dependent conserved quantities also have a conserved Poisson bracket.

A sum of independent Bernoulli random variables whose success probabilities may differ. Its distribution is the Poisson binomial distribution.

The polhode is the contact curve traced on the body's energy ellipsoid during Poinsot's torque-free rolling construction.

The herpolhode is the contact curve traced on the fixed invariable plane in Poinsot's torque-free construction. It is distinct from the polhode, which is described in body coordinates.

A sphere intercepts a parallel stellar beam over projected area $\pi R_p^2$ and radiates over surface area $4\pi R_p^2$. Dividing total absorbed power by total surface area gives the factor one-quarter, independent of planet radius. Under uniform-temperature reradiation this implies $T=T_{\ast }\sqrt{R_{\ast }/(2a)}(1-A{)}^{1/4}$. Without heat redistribution it constrains the area mean of $T^4$, not the arithmetic mean of $T$.

The ratio of top-of-atmosphere outgoing thermal flux to the blackbody flux at a chosen representative surface temperature. In global equilibrium with a uniform surface, ${ϵ}_{\mathrm{eff}}=(T_{\mathrm{eq}}/T_s{)}^4$. It parametrizes the relation between surface and radiating temperatures and need not equal the material emissivity of the surface or an atmospheric layer. An effective value below one is compatible with a black surface when colder atmospheric layers contribute to radiation escaping to space.