An integer multiple of an integer $d$ is a product $dk$ with $k$ an integer; equivalently, $d$ is a divisor of that product. The integer multiples of $12$ are $\dots ,-24,-12,0,12,24,\dots$. Zero is an integer multiple of every integer, and the only integer multiple of zero is zero. Shared positive integer multiples lead to the least common multiple.

A differential operator $P$ is hypoelliptic when $Pu$ smooth on an open set implies that $u$ is smooth there. Every elliptic differential operator is hypoelliptic, but the heat operator is hypoelliptic without being elliptic.

A distribution on an open set $U$ is a continuous linear functional on the space of test functions $\mathcal{D}(U)=C_c^{\infty }(U)$.

A first-order irreversible transfer from fluid concentration $C$ to stored concentration $S$, measured per matching fluid storage volume, obeys the displayed balances. A sustained inlet can produce a stationary decaying fluid profile, but the stored amount then grows rather than becoming stationary. A finite pulse instead has $C\to 0$ and a finite final deposit. Decay, desorption or finite adsorption capacity is additional physics, not a consequence of the linear capture term alone.

Time-reversal symmetry reverses the time coordinate and is represented antiunitarily in quantum theory.

For a map $F:X\to X$, its iterates are defined by $F^0$ equal to the identity map and $F^{n+1}=F\circ F^n$.

The displayed map has a symmetric periodic orbit $\{\pm \sqrt{\mu +1}\}$ for $\mu >-1$, with multiplier $(-2\mu -3{)}^2>1$. For $\mu >2$, put $s_{\pm }=(\mu \pm \sqrt{{\mu }^2-4})/2$. The two additional period-two orbits are $\{\sqrt{s_{+}},\sqrt{s_{-}}\}$ and their negatives. Their multiplier is $9-2{\mu }^2$, so they are stable for $2<\mu <\sqrt{5}$ and undergo a period-doubling bifurcation at $\sqrt{5}$. At $\mu =2$ these points coincide with fixed points, not genuine two-cycles.

Away from zero this discontinuous map is odd, and its branch derivatives are $2|x|$. With the convention $\mathrm{sgn}(0)=1$, its value at zero is $-\mu$, so it is not odd at zero unless $\mu =0$. A global transition at $\mu =0$ turns two small stable fixed points into one stable symmetric two-cycle. Its unsigned dynamics can be related to the even quadratic map $g_{\mu }(x)=\mu -x^2$, provided critical hits are treated separately.

// Target: dynamical-systems.bigb

If $g$ is even and an orbit does not hit zero before its final iterate, induction gives $f^n(x)=(-1{)}^n{g}^n(x){\prod }_{k=0}^{n-1}\mathrm{sgn}(g^k(x))$. The step uses oddness of $f$ on nonzero arguments. A one-sided assigned value of $f(0)$ can invalidate that step after a critical hit; for $g(x)=\mu -x^2$, $\mu >0$, $x=\sqrt{\mu }$, the claimed second-iterate formula would give $+\mu$ while the actual value is $-\mu$.

// Target: dynamical-systems.bigb

The Hénon map is a two-parameter quadratic planar map, conventionally $(X,Y)\mapsto (1-a{X}^2+Y,b_{H}X)$. The alternative family $(x,y)\mapsto (y,\mu -bx-y^2)$ has constant Jacobian determinant $b$. When $\mu \ne 0$, the change $X=y/\mu$, $Y=-bx/\mu$ gives the conventional parameters $a=\mu$, $b_H=-b$. Fixed points in the alternative normalization satisfy $x=y$ and $y^2+(1+b)y-\mu =0$. Its invertibility requires $b\ne 0$; at $b=1$ it preserves area.

For $F(x,y)=((a+1)x-x^3+y,(b-1)y-(b-1)x^3+y^3)$, the origin multipliers are $a+1$, $b-1$. A nonzero fixed point has $S=x^2>0$, $y=x(S-a)$, and $S[1-(S-a{)}^3]=a(2-b)$. Since the map is odd, a symmetric two-cycle has $F(x,y)=-(x,y)$; it satisfies $y=x(S-a-2)$ and $S[1+(S-a-2{)}^3]=b(a+2)$. Eliminating $y$ from the two map equations gives these identities directly. They locate the side of each pitchfork or flip without relying on a potentially misleading cubic sign in the original coordinates. At $(a,b)=(-1,0)$ the second identity becomes $3S^2-3S^3+S^4=b$, giving the quartic amplitude scaling of a generalized flip bifurcation.

A planar diffeomorphism is area-preserving if the phase-space area of every measurable set is unchanged. The change-of-variables formula proves this property when $|\det DF|=1$ everywhere, and conversely gives that determinant condition for a smooth map preserving all sufficiently small areas. Such a map cannot have an asymptotically attracting fixed point with a positive-area local basin: iterates of a small invariant neighborhood would have to enter arbitrarily small disks while retaining its phase-space area. For a map with constant positive determinant $b$, invariance of a bounded region of positive phase-space area instead forces $b=1$.

The Perron method constructs a harmonic solution with prescribed boundary data as the pointwise supremum of all subharmonic functions lying below those data. Harmonic lifting proves interior harmonicity; barriers determine whether the supremum attains the boundary values.

Unbounded-domain Dirichlet problems can need a condition at infinity. On all of ${\mathbb{R}}^n$, constant functions have identical empty boundary data and are distinct finite-energy stationary points of $\frac{1}{2}\int |Du{|}^2$. With nonempty boundary, on the exterior of the unit ball in dimension at least three, $1-|x{|}^{2-n}$ is a nonzero harmonic function vanishing on the sphere, with finite Dirichlet energy and a nonzero limit at infinity. It is stationary for compactly supported variations. The permitted class of variations and the behaviour at infinity must be specified for a global variational uniqueness claim.

For a bounded regular region and a sufficiently smooth real harmonic function, apply the divergence theorem to $u\nabla u$. Its divergence is $|\nabla u{|}^2+u\Delta u$. The boundary term vanishes, hence the integral of the nonnegative gradient square is zero. The gradient vanishes and the boundary values fix each component's constant as zero. Taking the difference of two solutions gives uniqueness for the Poisson equation with Dirichlet boundary data. Unbounded regions require an additional condition at infinity.

For $a>0$, a harmonic function on $x\ge 0$, $0\le y\le a$ with zero data on the horizontal sides, sine data $\sin (n\pi y/a)$ at the end, and decay at infinity is given by the displayed expression for every integer $n$. The second derivatives cancel since the decay exponent squared equals the transverse wavenumber squared. The absolute value is necessary for negative integers; $n=0$ gives the zero solution. The solution is bounded and decays uniformly in $y$.

The bounded harmonic function on the open unit disk with boundary values one on its upper semicircle and zero on its lower semicircle is the displayed function. The Möbius transformation $w=i(1-z)/(1+z)$ has real part $2y/|1+z{|}^2$ and imaginary part $(1-|z{|}^2)/|1+z{|}^2$. Pulling back the upper-half-plane step-data solution proves the formula by conformal invariance of harmonicity. The discontinuity points $z=\pm 1$ have no path-independent boundary limit.

A topological sorting is the task of constructing a topological order of a finite directed graph, or reporting that a directed cycle makes one impossible. Kahn's algorithm repeatedly takes a vertex with no unfinished prerequisites. This is useful whenever an operation consumes results produced by earlier operations: once a vertex is reached in a topological order, every prerequisite along a directed path has already been processed. ^[ref]

The reason a choice is always available in a nonempty finite directed acyclic graph is simple. If every vertex had an incoming directed edge, repeatedly following predecessors would eventually revisit a vertex, giving a directed cycle. Removing a vertex and its incident directed edges cannot create a directed cycle, so the argument repeats until every vertex is listed. This proves existence of a topological order and explains why finiteness matters to this procedure.

// Target: combinatorics.bigb

The apparatus and dipole inference are described under dipole evidence for bond polarization. The rigid rotor supplies a general starting model for rotational quantum states; rotational spectroscopy measures their transition frequencies, while molecular beam electric resonance can select and analyze states through their Stark effects.

An actual implementation gives substance to the apparatus description above. In Johannes Jacobus van Vaals's experiments, heated samples expanded with argon through a nozzle, then molecular beam skimmers formed the molecular beam in vacuum. Nonuniform electric fields selected and analyzed quantum states; parallel plates housed the microwave interaction. A surface ionization detector supplied electric current to an electronic amplifier and a lock-in amplifier. Resonant transfer into a rejected quantum state reduced the detected flux. Phase locking stabilized the microwave frequencies; an absorber suppressed reflections that could distort spectral lines. The thesis reports preliminary electric dipole moment components for gaseous potassium cyanide from measurements in nonzero electric fields. Such inference uses the Stark effect, rather than measuring electronegativity. The source notes restrictions from electric-dipole selection rules and molecular beam transit time; its preliminary dipole report is not a full calibration record. These are documented features of this apparatus, not proposed controls for the earlier hydrogen chloride experiment.^[ref]

The debye is a unit of electric dipole moment: $1\mathrm{D}\approx 3.33564\times 10^{-30}\mathrm{C}\mathrm{m}$.^[ref]