LLVM is a collection of reusable compiler infrastructure built around a typed intermediate representation. Its static single-assignment form gives each register-like value one definition, making data dependencies explicit for compiler optimization. The same intermediate representation has an in-memory form, a compact bitcode form and a readable textual form.^[ref]

A compiler front end translates a source programming language into the intermediate representation; optimization passes transform it while preserving the required semantics; a compiler back end produces code for a target computer architecture. Sharing the middle stages lets several source programming languages reuse the same analyses and target support. For a small textual example, %y = add i32 %x, 1 defines a new 32-bit integer value from an earlier one. The value %y is defined once even if the source program repeatedly assigns to a source variable.

A program representation used between input and final output, convenient for analysis, transformation or execution. An abstract syntax tree and bytecode can each serve this role.

The rule of sum counts alternatives in disjoint finite sets by adding their sizes. For a disjoint union $A\bigsqcup B$, each element belongs to exactly one alternative, so $|A\bigsqcup B|=|A|+|B|$. Overlapping alternatives require accounting for repeated elements, as in the inclusion-exclusion principle. The groups indexed by first-return position in the Catalan recurrence are disjoint alternatives.

An inclined plane is a flat surface tilted through an angle $\theta$ above horizontal. A ramp exchanges a smaller lifting force for a longer travel distance; for a released body, it reduces the downhill component of gravitational weight and makes the resulting acceleration easier to measure. Its simplest dynamical model is a nonrotating block on a stationary, rigid surface, with constant mass $m>0$, uniform gravitational acceleration of magnitude $g>0$, and $0\le \theta <\pi /2$. Assume continued contact, negligible drag force, no tipping, and no applied pull. The only forces are gravitational weight, the normal force, and any tangential friction.

The central distinction is between a block resting on the inclined plane and one already sliding. Static friction supplies the force needed to maintain rest, up to its bound; kinetic friction has a prescribed magnitude in the ideal constant-coefficient model. These laws are empirical approximations whose range depends on the contacting surfaces.^[ref]

A trajectory describes a body's position as it evolves with time, such as $\mathbf{r}(t)$ for a point mass. Its first time derivative is velocity and its second is acceleration. A geometrical path records which positions are visited; the time-parametrized trajectory additionally records how fast and in which direction the body moves along that path.

For a projectile in a uniform gravitational field without air resistance, horizontal uniform motion and vertical constant acceleration combine into a parabolic path. Changing the initial velocity selects a different trajectory even under the same force law. A phase-space trajectory retains momentum as well as position, so a particle passing through the same spatial point with different velocities occupies different phase-space states. Returning to one position alone therefore does not establish that the full motion is periodic.

Online text: www.damtp.cam.ac.uk/user/tong/dynamics/three.pdf

Cited passages: Classical Dynamics by David Tong, §3.4.3, The Asymmetric Top: Stability, printed pp. 57–58, equations (3.40)–(3.45); Classical Dynamics by David Tong, §3.4.3, equations (3.41)–(3.45), and §3.4.4, invariants (3.46)–(3.47); the invariant-bound argument here is an original short derivation.

An actogram displays activity across successive time intervals, commonly in one row per calendar day.

In masking of circadian rhythm, an external condition changes the observed output without necessarily changing the underlying circadian clock.

A circadian phase specifies position within a circadian rhythm's cycle relative to a chosen reference event.

A circadian clock is an endogenous timekeeping system that generates circadian rhythms.

Entrainment of circadian rhythm synchronizes a circadian clock to a repeating cue, a zeitgeber. One-to-one entrainment of circadian rhythm gives one cycle per environmental cycle with a stable relative circadian phase.^[ref] The stable relation need not put an activity onset exactly at lights-off: a constant offset also expresses phase locking.

For an original local model, let $x_n$ be the activity-onset offset from lights-off on environmental cycle $n$, with a negative value meaning an earlier onset. Choose an intrinsic period $\tau =23.5\mathrm{h}$ and environmental period $T=24\mathrm{h}$. Let $D(x)$ be a stipulated daily delay, measured in hours; positive $D$ moves the next onset later. The recurrence relation isA fixed phase relationship $x_{\ast }$ requires $D(x_{\ast })=T-\tau$: here a half-hour delay balances the half-hour intrinsic advance each cycle. This equality alone describes balance; restoration after a small disturbance also requires fixed point stability for an iteration.

Choose the illustrative local law $D(x)=0.5\mathrm{h}-0.4x$. Thenfor the chosen starting offset $x_0=-3\mathrm{h}$. The error shrinks by a factor of $0.6$ each cycle, proving fixed point stability for an iteration in this model. After five cycles $x_5=-0.23328\mathrm{h}$, about fourteen minutes early. At the fixed point, the imposed delay is $0.5\mathrm{h}$ and the corrected onset interval is $\tau +D(x_{\ast })=24\mathrm{h}$, although the uncoupled period remains $23.5\mathrm{h}$.

The uncorrected comparison has $D=0$ and $x_n=-3\mathrm{h}+n(\tau -T)$, so after five cycles it is $5.5$ hours early. The two traces separate ongoing timekeeping from adjustment to a zeitgeber. The chosen linear delay law is only a local teaching construction: it is not a measured response to light, does not account for wrapping of circadian phase, and does not predict an organism's entrainment range. Nor does it include masking of circadian rhythm. In a real study, tracking several cycles after removal of the cue or after a timed perturbation supplies information that a single activity value under illumination cannot provide.

A free-running circadian rhythm persists without an entraining cycle. Its period $\tau$ separates corresponding circadian phases, such as successive activity onsets, under specified constant conditions.^[ref] A circadian phase tells where the system is within its cycle; a period tells how long one cycle takes. Choosing onset rather than peak as the reference changes the phase label, not the period of an ideal repeating signal.

Consider an original, deliberately idealized model with $\tau =23.5\mathrm{h}$. For seven days prescribe a 12:12 LD cycle with activity starting at hour 12 of each 24-hour day. Then omit the next scheduled dawn and keep the environment dark. Stipulate that the first DD onset retains hour 12, and that every later onset is exactly $23.5\mathrm{h}$ after the previous one. Activity lasts eight hours after each onset. These are chosen model values, not recorded mouse or human data, and the fixed activity window represents an output of the model rather than the molecular mechanism of a circadian clock.

An actogram displays activity against time in successive rows. Here each row spans $T=24\mathrm{h}$, regardless of the circadian rhythm's period. If $t_0$ is the first DD onset and $n$ counts completed free-running cycles, thenThe unwrapped onset displacement $x_n$ is therefore $-0.5n\mathrm{h}$. After eight cycles the onset is four hours earlier than a 24-hour continuation would predict; after sixteen cycles it is eight hours earlier. Because these onsets remain within successive calendar rows in the displayed interval, their row positions are $12,11.5,11,\dots ,4$ hours. The visual slant comes from comparing two different periods, not from cessation of the oscillation.

An alternative phase oscillator description makes the same distinction explicit. Set the circadian phase to zero at $t_0$ and defineThe angular frequency $\omega$ is constant in this model. Onset occurs whenever the unwrapped circadian phase completes another turn. A period longer than $T$ would instead produce later onsets; equality would produce aligned rows even without entrainment of circadian rhythm. Thus aligned rows by themselves do not identify the cause of a daily pattern. Real activity can be noisy, and a free-running circadian rhythm's period can vary with conditions or with time after release; the stipulated straight line assumes a stationary period.

A centromere is the chromosome region where a kinetochore assembles.^[ref] Following DNA replication, both sister chromatids have their own copied centromere regions; the counting convention groups the attached sisters as one replicated chromosome. Cohesin maintains sister association: a centromere is not an extra DNA bridge joining two separate DNA molecules.^[ref]

Chromosome segregation allocates chromosomes to the products of division. Mitosis separates sister chromatids, preserving ploidy in completed daughter cell nuclei. Meiosis I separates homologous chromosomes with their sister chromatids still associated, reducing diploid to haploid ploidy; meiosis II separates the sister chromatids. Only the first of the two divisions reduces the number of complete sets.^[ref]

For example, correct chromosome segregation of the stipulated three homologous chromosome pairs gives each meiosis I product one $A$, one $B$ and one $D$, each replicated. Giving one product both $A$ members and the other none would preserve the total DNA across products while failing to give each a complete set. Conservation of material alone does not prove correct chromosome segregation.

Optical activity is the ability to rotate the direction of linear polarization of transmitted light. In an achiral solvent, the two members of an enantiomer pair have equal and opposite optical rotations at the same wavelength, temperature and concentration. The measured angle concerns a bulk sample; it is not a geometric rotation of the molecules.^[ref]

The rotating object in this description is the direction along which the electric field oscillates. At each position the transmitted light can still have linear polarization: its electric field moves back and forth along a line, while the orientation of that line changes along the sample. Nothing requires the molecules to turn bodily through the observed angle. This makes polarimetry a bridge from molecular chirality to a visible bulk measurement, and, with a suitable calibration, a way to follow concentration or the balance of enantiomers.

In a transparent, isotropic chemical solution, natural optical rotation can be understood as circular birefringence: the two opposite circular polarizations accumulate different wave phases, and their equal-amplitude sum remains linearly polarized along a rotated direction. Circular dichroism instead means unequal absorption of those components and can produce elliptical polarization. These are related responses, not interchangeable measurements. General birefringence need not produce this simple rotation of linear polarization.^[ref]

An allene has a $\mathrm{C}=\mathrm{C}=\mathrm{C}$ unit whose terminal substituents lie in perpendicular planes. In $\mathrm{abC}=\mathrm{C}=\mathrm{Ccd}$, with $a\ne b$ and $c\ne d$, the two terminal pairs can define enantiomers without any tetrahedral carbon stereocenter. The groups need be different within each end pair, not all four mutually different. This is axial chirality.^[ref]

A meso compound is an achiral molecule despite having stereocenters. Tartaric acid, $\mathrm{HOOC}-\mathrm{CH}(\mathrm{OH})-\mathrm{CH}(\mathrm{OH})-\mathrm{COOH}$, has two tetrahedral carbon stereocenters. Its $(2R,3R)$ and $(2S,3S)$ forms are enantiomers, but $(2R,3S)$ and $(2S,3R)$ name the same meso form after equivalent ends are identified. A symmetric spatial arrangement has a mirror plane exchanging its two halves. Thus there are three stereoisomers, not four.^[ref]

Opposite R and S configurations alone do not guarantee a meso compound: the rest of the molecule must permit mirror superposition. Likewise, a meso compound is one achiral chemical compound, whereas a racemate is a mixture of two chiral molecule populations. Both can give zero optical rotation for different reasons.

Take a central chemical atom at the origin and four substituents at the vertices of an ideal regular tetrahedron:These dimensionless coordinates describe shape, not measured bond lengths. For distinct labels A, B, C and D, define a signed determinantIt is six times the oriented volume of the labelled tetrahedron. A translation cancels from the differences. A rotation $Q$ has $\det Q=+1$ and preserves $\Delta$, whereas the mirror matrix $S=\mathrm{diag}(-1,1,1)$ has $\det S=-1$ and changes it to $-16$. Consequently no rotation can place every distinct label on its mirror partner. This argument uses three-dimensional coordinates, not clockwise order on a page.

Now let C and D both be the same substituent H. Swapping these identical groups is allowed. For this model the half-turnabout the axis $(0,1,1)$ maps the reflected A and B to their original positions and exchanges the two H positions. The models superpose. With distinct C and D, the same half-turn fails precisely at those two labels; the determinant argument rules out every other rotation as well. This is the local reason two identical hydrogen chemical atoms prevent glycine's amino acid alpha carbon from being a tetrahedral stereocenter.

For a flexible molecule, also distinguish conformational isomerism from a persistent enantiomer pair. A particular frozen shape may be chiral while ordinary internal motions interconvert mirror shapes. Turning the entire model is always allowed in a spatial comparison; an internal motion must be considered with its actual accessibility and timescale.

Achirality means that a molecule is superposable on its mirror image. A reflection plane that leaves the complete labelled spatial structure unchanged establishes achirality. Absence of such a plane alone is not a complete test: other orientation-reversing spatial symmetries can also make a rigid structure achiral.