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For a radiation-era transition at $T_i\sim \eta$, use $H_i=1.66\sqrt{g_{\ast }}{\eta }^2/m_{\mathrm{pl}}$ and initial number density $n_i=C_n{H}_i^3$. Stable monopoles of mass $C_M\eta$ redshift as nonrelativistic matter. With comoving entropy conservation,The fourth-power scale dependence makes even a small causal initial abundance incompatible with conventional high-scale grand unification unless inflation, later entropy production or efficient destruction reduces the relic population.

// Target: cosmology.bigb

A relativistic relic that decouples between pion and muon annihilation keeps $a{T}_H$ constant. The coupled bath initially contains photons, electron-positron pairs, muon-antimuon pairs and three neutrino-antineutrino species, with entropy degrees of freedom $57/4$. After muon annihilation this is $43/4$, giving the displayed temperature ratio. After neutrino decoupling, electron-positron annihilation heats photons by $(11/4{)}^{1/3}$, so the final relic-to-photon ratio becomes $(172/627{)}^{1/3}$. This counting assumes the species are relativistic when counted and treats annihilations and decoupling as sharp equilibrium transitions.

Geometrical distances to nearby calibrators determine the luminosities of indicators that can be observed farther away. Those indicators calibrate still more luminous indicators in turn. A ladder may connect parallax or maser distances to Cepheid variables and then to Type Ia supernovae. Each step propagates its calibration uncertainty to the final distance scale and the inferred Hubble constant.

The cosmic microwave background temperature is about $2.725\mathrm{K}$, corresponding to the displayed thermal energy. The conversion uses $k_B\simeq 8.61733\times 10^{-5}eV/K$. Thus $2.35\mathrm{meV}$ would correspond to about $27.3\mathrm{K}$, rather than the present CMB. Primary references are NASA's CMB temperature measurements and NIST's Boltzmann constant in electronvolts per kelvin.

Hot electrons in galaxy clusters scatter photons of the cosmic microwave background, producing a characteristic spectral distortion. The thermal contribution measures a line-of-sight integral of electron pressure, whereas X-ray emission depends approximately on the integral of electron density squared times a temperature-dependent emissivity. Combining the two with a geometrical model can constrain a cluster's physical size and hence its angular diameter distance. Temperature structure and departures from the assumed geometry introduce systematic uncertainty.

The bilinear hybrid superpotential couples an inflaton $S$ to two waterfall fields $u,v$. In global supersymmetry, $V=|g{|}^2(|uv-\zeta {|}^2+|S{|}^2(|u{|}^2+|v{|}^2))$. The branch $u=v=0$ breaks supersymmetry and is transversely stable for $|S{|}^2>|\zeta |$, while the zero-energy branch $S=0$, $uv=\zeta$ is supersymmetric. A canonical Kähler potential in supergravity slightly lifts the inflationary valley. It must not be described as two isolated stationary minima of the full potential.

Power-law inflation is cosmic inflation with a scale factor proportional to $t^p$ for constant $p>1$. Its Hubble parameter is $H=p/t$ and its first Hubble slow-roll parameter is ${ϵ}_H=1/p$. The comoving Hubble radius decreases as $t^{1-p}$. A canonical inflaton with an exponential inflaton potential admits an exact example. Because ${ϵ}_H$ is constant, an indefinitely maintained power law has no graceful exit from inflation.

A graceful exit from inflation ends sustained accelerating expansion and connects it to a hot radiation-dominated universe. Crossing ${ϵ}_H=1$ ends acceleration in a homogeneous expanding model; reheating requires an additional mechanism to transfer inflaton energy into other species. Pure exponential-potential power-law inflation keeps ${ϵ}_H$ constant and its inflaton potential has no finite minimum, so neither an automatic endpoint nor oscillatory reheating follows from that potential alone.

The coronagraph inner working angle describes how close to the blocked source a specified coronagraph can usefully transmit or detect a faint neighbor. It requires a criterion, such as a chosen fraction of the far off-axis coronagraph throughput, and is often expressed in units of $\lambda /D$. It is not automatically the coronagraphic focal plane mask radius or a guarantee of detection. Small separations make useful source transmission and accurate control of wavefront errors particularly demanding. ^[ref].

For a specified photodetector measurement, coronagraph throughput $T(\theta )$ is the fraction of a source's light transmitted into that measurement at angular separation $\theta$ from the blocked source, relative to a stated reference configuration. Total transmitted power and power inside a small image opening can give different fractions; one must specify which is used. Throughput usually depends on wavelength and separation. A lower stellar residual is useful only in relation to the retained companion signal and noise. Guyon's worked designs plot transmitted off-axis light against separation to expose that tradeoff. ^[ref].

A classical Lyot coronagraph first forms an image of a bright source on the optical axis. A coronagraphic focal plane mask hides its core. Relay optics then form an image of the entrance pupil, where a Lyot stop blocks the outer rim and, when needed, borders around internal obstructions. Finally an optical camera forms the science image. The focal plane sorts incoming directions into image positions; the optical pupil sorts contributions across the aperture. This change of plane makes it possible to reject additional stellar light that survives the first mask. ^[ref].

Assume exactly two operating-system processes, $N$ initially resident private memory pages of $P$ bytes each, initially distinct page frames shared by A and B. Neither unmaps, exits, forks again, or encounters page replacement; no other owner exists, allocation succeeds, and a first write copies one whole shared memory page. Once a page frame is exclusively owned it is reused on writes, without another copy; writable mappings remain writable. Count data storage only, excluding page tables and other bookkeeping. With B as the only writer, writing $k$ distinct memory pages givesThis depends on distinct written memory pages, not the number of store instructions or reads. For the diagrams, take $N=3$ and an illustrative $P=4096$ bytes:

For both writers, let $S_A,S_B$ be the sets of initial memory page numbers written by A and B. Under these assumptions, new allocations and copies both equal $|S_A\cup S_B|$, the cardinality of their set union, while resident data storage is $(N+|S_A\cup S_B|)P$. Each initially shared page frame needs at most one split: after the first writer copies it, the other owns the original exclusively. Both writing the same memory page therefore produces two versions with one copy, even if the final values happen to match. The last row can incur two protection page faults, one for each writer, while making only one copy. A handler that always allocates on a protected write can make a second copy and free the now-unreferenced original: allocation counts and final resident storage are different observables.

These are model counts, not measured Linux costs or a guaranteed bound on total computer memory. More descendants, other owners, demand paging, page replacement, different memory page sizes, and different exclusive-ownership rules change the accounting. The useful claim is conditional: deferred data copying can cost in proportion to the distinct shared memory pages that are written, while creating separate page tables has its own cost.

// Target: computer-science.bigb

The operating-system kernel must distinguish a temporarily protected copy-on-write mapping from one whose writes are forbidden. The teaching procedure in MIT's xv6 lab protects both page tables, copies the shared page frame on a valid write page fault, and installs a writable page table entry. Originally read-only code remains read-only. Its reference counting increments on sharing and decrements on removal; storage is freed only after the last reference disappears. The lab also requires kernel writes into user storage to respect copy-on-write, and specifies failure if no free storage remains. ^[ref]

Every write must respect the sharing policy, but every write need not copy. Repeated writes to B's already private writable F3 need no further copy-on-write page fault. If A later writes F0, an implementation can reuse F0 when it establishes exclusive, writable ownership; A's remaining read-only page table entry may still need a page fault to enable the write. This exclusive-reuse rule is an assumption of copy-on-write page accounting below, not a promise that every operating system uses that path. A count of one alone is not a universal permission to alter storage belonging to a memory-mapped file or another owner.

Changing mappings or permissions must also make cached address translations consistent. ^[ref] Concurrent software threads require synchronization around ownership checks, copying, and installation to prevent a race condition; neither diagrams nor counts specify a production operating-system kernel's locking or hardware invalidation protocol.

Projections onto convex sets is an iterative feasibility method that successively applies a Euclidean projection onto a convex set to each closed convex constraint set. With two sets it uses $x_{n+1}=P_D(P_C(x_n))$. It seeks a point in the intersection; a single projection onto one set is a different operation, and the method does not generally compute the closest intersection point to the initial iterate.

A subset $C$ of a real vector space is radially open when, for each $x\in C$ and each vector $v$, the points $x+tv$ remain in $C$ for all sufficiently small real $t$. No ambient topology is required. If $C$ is also convex and contains zero, it is absorbing and its Minkowski functional $p_C(x)=\inf \{s>0:x\in sC\}$ is finite and sublinear, with $C=\{p_C<1\}$. Convexity gives subadditivity; radial openness permits a slight dilation of any point of $C$, giving the strict inequality.

The relative interior of a convex set is its interior in its affine span. A polyhedral cone has nonempty relative interior in its linear span, even when its ordinary interior in the ambient vector space is empty.

// Target: geometry-and-topology.bigb

A convex set in a finite-dimensional space has a nonzero supporting linear functional at each boundary point. If zero lies outside a nonempty convex set $C$, weak separation gives $p\ne 0$ with $p\cdot c\ge 0$ for all $c\in C$, even when zero belongs to its closure. Strict separation needs stronger hypotheses.

Two nonempty disjoint convex sets in a finite-dimensional real vector space admit a separating nonzero linear functional: for an appropriate $h\ne 0$ and $a$, $h\cdot x\le a\le h\cdot y$ on the respective sets. Strict separation requires additional assumptions, such as one set being compact and the other closed. For an open convex cone $C$ excluding zero, separation from $\{0\}$ gives $h\cdot c\ge 0$ for all $c\in C$; scaling cone elements forces the separating level to be zero. This finite-dimensional result is related to the supporting hyperplane theorem.

A convex set that is also a closed subset in the specified topology. In a Hilbert space with its norm topology, every nonempty closed convex set has a unique nearest point to every ambient vector, by the Hilbert projection theorem.

Checking only midpoints is not enough for an arbitrary subset of a real vector space. The rational numbers contain the midpoint of every pair of rational numbers, but the line segment from $0$ to $1$ also contains irrational numbers. For instance $\sqrt{2}/2$ lies on that line segment and is not rational. This shows why the definition quantifies over every real $t\in [0,1]$.

There is a useful additional hypothesis: a closed subset $C\subseteq {\mathbb{R}}^d$ containing the midpoint of every pair of its points is a convex set. Repeated bisection includes all points $(1-t)x+ty$ with dyadic rational $t\in [0,1]$. These coefficients approach every real $t\in [0,1]$; being a closed subset then includes each limit. This is a short proof outline, and the rational numbers example identifies precisely why a closed subset was required.