Events turn Boolean operations into operations on sets. For a die with sample space $U=\{1,2,3,4,5,6\}$, let $A=\{2,4,6\}$ be the event “even outcome” and $B=\{4,5,6\}$ the event “outcome at least four”. At outcome $u$, use the indicator functions $P=1_A(u)$ and $Q=1_B(u)$. ThenThus logical conjunction becomes set intersection, logical disjunction becomes set union, exclusive or becomes symmetric difference, and logical negation becomes the complement of a set within $U$. Outcome four satisfies both events, so inclusive logical disjunction includes it and exclusive or excludes it. If the die is fair, these four events have probabilities $1/3$, $2/3$, $1/3$, and $1/2$, respectively. No assumption of independent events is needed for the set identities; indeed $A$ and $B$ here are not independent, since $1/3\ne (1/2)(1/2)$.

Section of Biology 2e by Mary Ann Clark and others, whose whole-book authors are Mary Ann Clark, Matthew Douglas and Jung Choi. No separately credited section author is asserted.

Online text: openstax.org/books/biology-2e/pages/30-6-plant-sensory-systems-and-responses

Cited passages: Biology 2e, §30.6, The Phytochrome System and the Red/Far-Red Response, Figure 30.38: interconversion of Pr and Pfr and cellular signaling.

Section of Biology 2e by Mary Ann Clark and others, whose whole-book authors are Mary Ann Clark, Matthew Douglas and Jung Choi. No separately credited section author is asserted.

Online text: openstax.org/books/biology-2e/pages/32-2-pollination-and-fertilization

Cited passages: Biology 2e, §32.2, Development of the Seed, Figure 32.20; endospermic and non-endospermic seeds, pea cotyledons and early reserve dependence.

Online text: openstax.org/books/biology-2e/pages/10-3-control-of-the-cell-cycle

The linked §10.3 discusses internal cell-cycle checkpoints. The year 2018 identifies the existing book publication rather than a chapter revision.

Cited passages: Biology 2e, §10.3, Regulation at Internal Checkpoints, Figure 10.11 and The M Checkpoint.

Online text: openstax.org/books/biology-2e/pages/30-5-transport-of-water-and-solutes-in-plants

Cited passages: Biology 2e, §30.5, Solute Potential: negative concentration contribution; Biology 2e, §30.5, Water Potential, Solute Potential, Pressure Potential and Figure 30.32: component balance and direction of transfer.

A nonempty binary tree in which every level, including the deepest, contains all its possible vertices. Equivalently, it is a full binary tree whose leaves are all at the same graph distance from the root of a tree.^[ref]

Every perfect binary tree is both a full binary tree and a complete binary tree; the five-vertex example above shows that the converse fails even when both properties hold. These definitions fix the terminology used here.

For height of a binary tree $h\ge 0$, the level sizes are $1,2,4,\dots ,2^h$, giving $n=2^{h+1}-1$ by a geometric series. There are $2^h$ leaves. The seven-vertex example has level sizes $1,2,4$ and height of a binary tree $2$; the single-vertex binary tree is already perfect, with $h=0$. More generally, joining two perfect binary trees of equal height of a binary tree under a fresh root of a tree gives the next such shape. Equal heights are essential: fullness alone allows unequal leaf distances. A perfect binary tree realizes the maximum number of vertices for its height of a binary tree, making the exponential capacity behind short lookup graph paths visible.

// Target: probability-and-statistics.bigb

One of the two distinguished positions, left or right, at a vertex of a binary tree. Each binary tree child slot is occupied by a child and its subtree, or is empty. A reference representation can use null for an empty binary tree child slot. Two slots are capacity for at most two children in a rooted tree, rather than a requirement for two actual vertices.

Online text: iupac.qmul.ac.uk/stereo/A.html

Cited passages: Basic Terminology of Stereochemistry, entry Axial Chirality: nonplanar pairs about an axis, including allenes.

Online text: www.jirka.org/ra/html/sec_liminter.html

Cited passages: Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl, §6.2.1, Theorem 6.2.2; Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl, §6.2.2, Theorem 6.2.4; Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl, §6.2.3, Theorem 6.2.10 and the unbounded-interval observation following its proof.

Online text: www.jirka.org/ra/html/sec_puconv.html

Cited passages: Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl, §6.1.1, Definition 6.1.1 and Proposition 6.1.5; Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl, §6.1.1, Example 6.1.2, even powers on the interval from minus one to one; Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl, §6.1.2, Definition 6.1.6 and Proposition 6.1.7.

Online text: www.jirka.org/ra/html/sec_bw.html

Cited passages: Basic Analysis, Volume I by Jiří Lebl, opening footnote to §2.3: the theorem's two namesakes; Basic Analysis, Volume I by Jiří Lebl, §2.3.3, Theorem 2.3.8: every bounded real sequence has a convergent subsequence; Lebl, Basic Analysis I, §2.3: equality of finite bounds and extended limits; Lebl, Basic Analysis I, §2.3: extreme subsequential limits.

Laser cooling uses laser light to reduce the random velocities of chemical atoms, and therefore their translational temperature. The striking ingredient is selective photon absorption: an atom moving towards a beam sees a Doppler shift that can make it absorb more strongly. Each absorbed photon transfers momentum along the beam; repeated absorption and spontaneous emission can then remove kinetic energy. Doppler cooling makes this selection with the atom's motion, while sub-Doppler cooling also exploits its internal quantum states.

Cooling makes chemical atoms move slowly enough for longer interrogation in atomic clocks and atom interferometry, and easier loading into an optical trap. The same radiation pressure that pushes an atom can cool it: with suitably tuned opposing beams, whichever direction the atom moves, it receives the stronger push against that motion.

A Raman transition changes an atom's internal state through a two-photon interaction involving an intermediate level. In a stimulated Raman transition, absorption from one applied beam and stimulated emission into another supply a net energy $\hslash ({\omega }_1-{\omega }_2)$ and momentum $\hslash ({\mathbf{k}}_1-{\mathbf{k}}_2)$.

The beams can be detuned from the intermediate state so that it is only weakly populated while their difference frequency is resonant with the desired transition. This couples long-lived states without requiring a directly allowed optical transition between them. Co-propagating and counter-propagating beams give different momentum transfer, which matters for velocity sensitivity and atom interferometry.

Atom interferometry measures relative wave phase by splitting, evolving and recombining coherent atomic matter-wave amplitudes. The paths may differ in position, momentum or internal quantum state; an observable population depends on their interference when they are recombined.

In a light-pulse implementation, stimulated Raman transitions couple internal states while transferring a controlled momentum difference. A split–redirect–recombine pulse sequence makes the output sensitive to acceleration and to the phases of the applied beams. Varying one pulse phase and observing oscillating state populations tests coherent recombination rather than ordinary loss of chemical atoms.^[ref]

Optical pumping redistributes populations among internal quantum states by selective photon absorption followed by spontaneous emission. It can prepare atoms for measurement and participate in sub-Doppler cooling.

The practical aim is to make otherwise almost equally populated magnetic sublevels behave differently. Repeated absorption and emission can store a population imbalance in the long-lived ground state; a change in photon absorption or fluorescence then reveals a perturbation of that prepared distribution. This combines preparation and measurement: a weak radio wave can reorganize the populations, while abundant optical photons supply the readout.

An imbalance among ground state magnetic sublevels need not be a population inversion between optical energy levels. Optical pumping also supplies excitation in some lasers, where an appropriate population inversion matters for gain. The preparation problem here concerns the distribution within the ground state. A two-level system with only one lower quantum state cannot describe this redistribution.

Online text: www.math.toronto.edu/ivrii/APM-textbook/Chapter9/L9.1.html

Cited passages: §9.1, Case epsilon=0 as a limit, identity following (9.1.10); §9.1, Problem set-up, (9.1.1)–(9.1.2); Cole-Hopf transform, (9.1.11)–(9.1.13).