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A stellar core in which radiative diffusion in a star carries most of the energy, rather than convection. Its formation in a pre-main-sequence star breaks the fully convective assumption behind an idealized Hayashi track.

For a steady point-like astronomical source with isotropy of emission, in transparent space with negligible relativistic or cosmological effects, conservation of energy spreads its luminosity over a sphere of surface area $4\pi d^2$. Its bolometric observed radiative flux isThis inverse-square law reflects geometric dilution: the same emitted power crosses increasingly large spheres. It does not require light to lose energy on the journey. For a fixed luminosity, doubling Euclidean distance quarters the observed radiative flux. ^[ref].

With $L_0=10^{26}\mathrm{W}$ and $d_0=20$ parsecs, the result is $F_0\simeq 2.09\times 10^{-11}\mathrm{W}{\mathrm{m}}^{-2}$. At $40$ parsecs, the same source gives $F_0/4\simeq 5.22\times 10^{-12}\mathrm{W}{\mathrm{m}}^{-2}$. An ideal collecting area of $1{\mathrm{m}}^2$ at $20$ parsecs intercepts approximately $2.09\times 10^{-11}\mathrm{W}$, before detector losses.

Multipole radiation expands radiation from localized time-dependent sources into electric and magnetic multipoles. Electric dipole radiation is the leading electric term, while magnetic dipole radiation and higher electric multipoles contribute when the leading term vanishes or higher precision is needed.

A 2013 experiment by Paul A. Williams and collaborators mounted a 25-mm dielectric mirror on a commercial force-restoration balance. Removing its pan-weight compensation spring allowed horizontal force sensing with a horizontal laser path. They measured mirror optical reflectance and varied incidence from about $7^{\circ }$ to $45^{\circ }$. A 1071-nm laser, capable of 530 W, was switched at 0.1 Hz. Digitized mass-equivalent readings were converted to force using $g$; quadrature demodulation isolated the switched signal from drift. ^[ref]

Measurement calibration used a 20-g mass, with checks down to 1 mg. The optical forces lay below that checked range, limiting the accuracy claim. An angle scan tested the predicted cosine dependence at fixed intercepted power. Replacing the mirror with a carbon nanotube coated absorber increased absorbed power while reducing the predicted optical force; no modulation peak was detected. This bounded gas-heating artifacts under those conditions, rather than proving their universal absence. The fitted slope was about 7% below $2/c$. Inferring laser power requires calibrated force, optical reflectance, radiometric absorptance and angle of incidence, with measurement uncertainty for each. ^[ref]

For radiation in vacuum with isotropy, energy is distributed equally per solid angle: write $u_{\Omega }=u/(4\pi )$ for energy density per unit solid angle. Let $\theta$ be the angle between a ray's path and the wall normal. Its normal velocity $c\cos \theta$ determines the arrival rate, while normal momentum per energy is $\cos \theta /c$. Their product is ${\cos }^2\theta$. For a perfectly reflecting cavity wall,The leading two is the normal momentum reversal; the incoming hemisphere contains half of the field's directional energy density. This is also the normal component of the bulk pressure tensor, $P_{nn}={\int }_{4\pi }{u}_{\Omega }{\cos }^2\theta d\Omega =u/3$. Isotropy makes all three diagonal components equal, while their sum is $u$ because the squared direction components sum to one. The factor one third therefore depends on three-dimensional angular geometry, not on the shape of the blackbody radiation spectrum.

A perfectly absorbing wall exposed to this incoming hemisphere receives only $u/6$ from absorption. If it is in thermal equilibrium with the isotropic radiation, its outgoing emission supplies another $u/6$ of recoil, recovering $u/3$. A cold absorber before equilibrium is a different boundary condition. Similarly, the one-way incident energy flux is $cu/4$, whereas the net energy flux across a plane in the undisturbed isotropic field is zero. Dividing $cu/4$ by $c$ misses the extra normal-momentum projection and does not give the cavity pressure.

For a stipulated isotropic radiation energy density $u=6\mu \mathrm{J}{\mathrm{m}}^{-3}$, the bulk pressure is $2\mu \mathrm{Pa}$ regardless of its spectral shape. In thermal equilibrium, blackbody radiation supplies $u=a_{\mathrm{rad}}{T}^4$ with radiation constant $a_{\mathrm{rad}}=4\sigma /c$, so $p_{\mathrm{rad}}=a_{\mathrm{rad}}{T}^4/3$; Stefan–Boltzmann law defines $\sigma$. The equilibrium photon gas derives this same equation of state from its free energy.^[ref]

For normal incidence in vacuum, a uniform beam delivers energy $IA\Delta t$ to an illuminated surface area $A$ during time $\Delta t$. A perfect absorber receives momentum $IA\Delta t/c$, so force divided by surface area gives $p_{\mathrm{abs}}=I/c$. A perfect stationary mirror sends the photons back with reversed momentum, so momentum conservation gives twice the normal transfer, $p_{\mathrm{refl}}=2I/c$. These are time-averaged pressures; the microscopic arrivals need not be uniform.

For a stipulated model light intensity $I=1000\mathrm{W}{\mathrm{m}}^{-2}$ and the exact SI speed of light $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$,These are computed predictions, not measured data or a universal sunlight value. If the beam fully covers a normally oriented surface area $A=0.010{\mathrm{m}}^2$, the intercepted radiant flux is $\Phi =IA=10\mathrm{W}$ and the absorber's force is $\Phi /c\simeq 33.356\mathrm{nN}$. Using the total source power would be wrong if only part of its beam reaches the receiver. The absorption prediction assumes complete photon absorption, while the reflection prediction assumes complete specular reflection; transmission, scattering of light and directed thermal photon emission require their own outgoing momentum accounting.

Measure the angle of incidence $\theta$ between the incoming path and the surface's normal line, with $0\le \theta <\pi /2$. There are two separate cosine factors. A fixed actual surface area $A$ intercepts only its projected area $A\cos \theta$, so $\Phi =IA\cos \theta$. Each incoming photon supplies normal momentum $(E_{\gamma }/c)\cos \theta$ on absorption, or twice that on specular reflection. Thus, for radiometric absorptance $a$ and reflected power fraction $R$,This model assumes transmitted light continues straight in the same medium and carries away its original momentum. It omits thermal recoil and assumes a stationary receiver. At $\theta =60^{\circ }$, fixed beam light intensity therefore gives one quarter of the normal-incidence pressure. For fixed intercepted radiant flux $\Phi$, however, the normal force has only one cosine factor. Equivalently, if the given quantity is receiving-surface irradiance $E_{\mathrm{inc}}=I\cos \theta$, then $p_{\perp }=(a+2R)E_{\mathrm{inc}}\cos \theta /c$. The distinction matters when interpreting an experiment's angle scan. ^[ref]

Absorption also transfers a tangential momentum component at oblique incidence. It contributes a tangential force, rather than normal pressure. Ideal specular reflection preserves the tangential momentum component, so it transfers only normal momentum.

Simultaneous trilinear baryon number and lepton number violation permits a squark to connect the two interactions and induce a dimension-six proton-decay operator. Its coefficient is of order ${\lambda }^{\prime }{\lambda }^{\prime \prime }/m_{\tilde{q}}^2$. A dimensional decay-width estimate then gives ${\Gamma }_p\sim |{\lambda }^{\prime }{\lambda }^{\prime \prime }{|}^2{m}_p^5/m_{\tilde{q}}^4$, up to phase space and hadronic matrix elements. The flavor antisymmetry of $U^c{D}^c{D}^c$ matters: the two down-type flavor indices must differ. For example, ${\lambda }_{112}^{\prime \prime }$ and ${\lambda }_{112}^{\prime }$ allow strange-squark exchange in $p\to e^{+}{\pi }^0$.

For fixed $u\ne 0$, the parallel affine lines $l_x=x+\mathrm{span}(u)$ are precisely the cosets of the indicated quotient vector space. Changing representatives adds a multiple of $u$, so $l_x+l_y=l_{x+y}$ and $a{l}_x=l_{ax}$ are well-defined. The zero is the whole line $l_0$, not a single point. If $(u,b_1,b_2)$ is a basis, the two line classes $l_{b_1},l_{b_2}$ form a basis of the quotient by spanning and independence modulo $\mathrm{span}(u)$.

For independent Poisson process arrivals and exponential distribution claim sizes of mean $\mu$, minimizing the logarithm of the negative expected exponential utility gives $H^{\prime }(\alpha )=\lambda \beta \mu [(1-\beta \mu \alpha {)}^{-2}-(1+{\theta }_R)]$. Its strictly positive second derivative makes the displayed stationary point unique. On $0\le \alpha \le 1$, the optimum is $\min (1,{\alpha }_{\ast })$. On the strictly open interval $0<\alpha <1$, there is no maximizer when ${\alpha }_{\ast }\ge 1$, only a supremum at full retention. The stationary retention increases with reinsurer loading and is independent of the direct insurer's loading.

If insurer and reinsurer use the same positive relative safety loading $\theta$, retained claim sizes and net premiums both scale by the retention $\alpha$: $X_{\alpha }=\alpha X$ and $c_{\alpha }=\alpha c$. Substitution into the adjustment coefficient equation gives $R_{\alpha }=R_1/\alpha$. Thus there is no maximizing positive retention on $(0,1]$; the supremum is approached as $\alpha \downarrow 0$. At full cession, if allowed, both retained claims and net premiums vanish, and the insurer's nonnegative capital remains constant. The coefficient is then not uniquely defined by the degenerate equation, although ultimate ruin probability is zero in this idealized model.

A flat universe dominated by a canonical homogeneous scalar field with constant equation-of-state parameter $w>-1$ has $\rho ={\rho }_0(a/a_0{)}^{-3(1+w)}$ and ${\dot{\varphi }}^2=(1+w)\rho$. The positive rolling branch gives the displayed relation and $V(\varphi )=(1-w){\rho }_0\exp [-\sqrt{24\pi G(1+w)}(\varphi -{\varphi }_0)]/2$. The opposite rolling branch reverses the exponential slope. For $w=-1$ the field is constant; $w<-1$ cannot be realized by a real scalar with this positive kinetic term. An exponential slope $\lambda \sqrt{8\pi G}$ yields $w={\lambda }^2/3-1$, with accelerating nonphantom expansion when $0<{\lambda }^2<2$.

A homogeneous scalar with $V=m^2{\varphi }^2/2$ obeys $\ddot{\varphi }+3H\dot{\varphi }+m^2\varphi =0$. If it is to provide potential-dominated dark energy today rather than rapidly oscillating matter, $m\lesssim H_0\sim 10^{-33}\mathrm{eV}$ is required. Its present density gives ${\varphi }_0\simeq \sqrt{6{\Omega }_X}{M}_{\mathrm{Pl}}{H}_0/m$. Taking $m\sim H_0$ gives a Planck-scale field; a much smaller mass requires a larger field value, so density alone does not fix a unique mass.

// Target: cosmology.bigb

The queue workload of a queue is the total remaining service work of all present customers, including residual work in service. With a work-conserving server of rate $C$ and stationary input $A$, the stationary queue workload is represented byThis is the amount by which past offered work exceeds available service, maximized over possible starts of a busy period. It is the reflection of net input at zero. Under first come first served, an arriving infinitesimal customer waits $W/C$; a priority customer instead waits for the remaining work ahead of it under its service discipline.

The queueing delay is the time between a customer's arrival and the start of its service, excluding its own service time. Under first come first served with service rate $C$, its queueing delay is the queue workload ahead of it divided by $C$. The customer sojourn time also includes the customer's own service. In a preemptive priority queue with a shared buffer, a voice customer's initial waiting time depends only on the voice queue workload already present.

The amount of work a queue can serve per unit of time. In a slotted model the same units can describe work served per slot.

The maximum stored work or number of waiting jobs, according to the model's convention. A bufferless queue has no room to retain excess input for a later slot.

An arrival exceeds the available service and storage capacity, causing work to be lost or blocked. For a bufferless queue with capacity $c$ per slot, overflow means input work strictly greater than $c$. Stationary queue workload tails in an infinite-buffer comparison model are often used to estimate finite-buffer overflow probabilities.

A queue which retains no unfinished work between slots. Given nonnegative input $A$ and capacity $c$, its bufferless queue output is $\min (A,c)$ and its lost work is $(A-c{)}^{+}$.

The size and frequency of unusually large arrivals or sustained overload episodes. It depends on the time and threshold scales being examined, rather than just on the one-slot variance. A cap can leave smaller rare deviations unchanged while removing larger positive bursts.

The population $N(t)$ of a queue counts every admitted customer still inside the selected system at time $t$, including customers receiving service when the server is inside that boundary. Its long-run time average is $L$. The waiting population $N_q(t)$ excludes customers currently receiving service, and its long-run time average is $L_q$. For a single server that serves whenever work is present,where $\rho$ is the server utilization, assuming at most one customer receives service at a time. In a stationary process with ergodicity and finite expected values, these time averages equal $\mathbb{E}N(t)$ and $\mathbb{E}{N}_q(t)$. The workload of a queue instead counts unfinished work, so two queues with the same population can have very different remaining workloads.