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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]

An infinite descent rules out a nonempty set of positive integer solutions by constructing from any solution another with a strictly smaller positive integer measure. Choosing a solution of minimal measure gives a contradiction. Common measures are a hypotenuse, a denominator, or the size of a primitive solution.

For a number field $K$ and fixed nonzero coefficients $\alpha ,\beta \in K$, the equationhas finitely many solutions in the unit group of its ring of integers. The coefficients remain fixed; varying them gives a different finiteness problem.

Here is the logarithmic-form mechanism. The Dirichlet unit theorem writes each unit as a root of unity times a product of fixed fundamental units with integer exponents. If the unit rank is zero there are already only finitely many possibilities. Otherwise set $B$ to the maximum of $3$ and the absolute values of their exponents. The logarithmic embedding and the product formula imply that a unit with a large exponent vector is exponentially small at some archimedean place, after possibly interchanging $u$ and $v$.

At that place, $\alpha u+\beta v=1$ makes the other term exponentially close to one. A nonzero logarithm of that term is a linear form in logarithms of algebraic numbers, including a suitable multiple of $2\pi i$ when complex logarithm branches are adjusted. Its integer coefficients are $O(B)$. The Baker lower bound for a homogeneous linear form in logarithms bounds its nonzero size below by $B^{-C}$ for fixed data, contradicting an exponentially small upper bound when $B$ grows. The logarithm cannot vanish, since then the other term would be exactly one and the remaining nonzero term zero. Thus the exponent vectors are bounded, leaving finitely many solutions. A finite torsion subgroup and finitely many archimedean places allow uniform constants.

An irreducible integral binary form of degree at least three has only finitely many integral solutions at a fixed nonzero value. Factor the form in a splitting field. Each linear factor generates an ideal dividing a fixed ideal, so it is one of finitely many chosen factors times a unit. The linear relation among any three distinct factors gives a unit equation over a number field, hence finitely many ratios and finitely many rational directions. Homogeneity and the fixed nonzero value then bound the scale on each direction. The same argument works for a binary form with at least three distinct projective roots, even without irreducibility.

For every $ϵ>0$, the abc conjecture asserts that there is a constant $K_{ϵ}>0$ such that all coprime positive integers $A+B=C$ satisfyThe radical of an integer is the product of its distinct prime factors. The condition $\gcd (A,B)=1$ makes the three entries pairwise coprime. Repeated prime powers enlarge a number without enlarging its radical in the same way, so the conjecture constrains how often addition can relate unusually concentrated prime factorizations.

The example $(A,B,C)=(1,8,9)$ has $\mathrm{rad}(ABC)=6<9$ and is an abc-hit. Such examples are compatible with the conjecture: its bound includes both the constant $K_{ϵ}$ and the exponent $1+ϵ$. Conditional applications include the primitive Fermat bound from abc and the Catalan power bound from abc.

More generally, fix positive integers $a,b$, a nonzero integer $c$, and positive integers $x,y,n$ satisfying $a{x}^n-b{y}^n=c$. Put $g=\gcd (a{x}^n,b{y}^n)$, which divides $|c|$. Applying the conjecture to the normalized positive triple formed from $a{x}^n$, $b{y}^n$ and $|c|$ gives, for $M=\max (x,y)$,Indeed, the largest normalized entry is at least a fixed multiple of $M^n$, while the radical is at most $\mathrm{rad}(ab|c|)xy\le \mathrm{rad}(ab|c|)M^2$. Taking $ϵ=1/4$ makes the exponent at least $1/2$ for $n\ge 3$, bounding $M$ uniformly; if $M\ge 2$, it also bounds $n$. The separate case $x=y=1$ can satisfy the equation for every $n$ when $a-b=c$.

There are no positive integer solutions to the displayed equation when the exponent is an integer greater than two. Homogeneity means that any hypothetical solution would give infinitely many common multiples; statements bounding solutions by arithmetic height must therefore first distinguish primitive solutions from their multiples.

A Diophantine equation is soluble at a place of a number field if it has a solution over the corresponding completion of a valued field. It is locally soluble everywhere if this holds at all finite and infinite places. Over the rational numbers, these are the real field and all p-adic fields. Local solubility is necessary for a rational point, but the local-to-global gap in isogeny descent shows it need not suffice.

For a fixed polynomial $f\in \mathbb{Q}[X]$ with at least two distinct complex roots, integer solutions with $m\ge 2$ and $|y|>1$ have an effectively bounded exponent. Factorization over a splitting field and the fixed differences between distinct roots control common ideal divisors. Outside a fixed set of primes, each factor's valuation is divisible by $m/\gcd (m,e_i)$, where $e_i$ is its root multiplicity. The generalized power-equation estimates from linear forms in logarithms of algebraic numbers, including unit and exceptional-prime factors, give the exponent bound. A single-root polynomial and the values $y=0,\pm 1$ cannot be included: $4^m=(2^m{)}^2$, and $1^m=f(0)$ for $f(X)=X(X-1)+1$, give infinite exponent families.

Let $K$ be a number field of degree $D$, and let $L_1,\dots ,L_M$ be linear forms in $N>DM$ variables with coefficients in $K$ and projective height at most $\mathcal{H}\ge 1$. There is a nonzero $\mathbf{x}\in {\mathbb{Z}}^N$ satisfying every $L_j(\mathbf{x})=0$ andFor $K=\mathbb{Q}$, this follows by applying the pigeonhole principle to the images of the integer box $\{0,\dots ,Y{\}}^N$ under $(L_1,\dots ,L_M)$. Expanding coefficients in a rational basis of $K$ gives the factor $D$ in the general count.

For every irrational number $\alpha$ and positive integer $N$, there are integers $p,q$ with $1\le q\le N$ such thatThe proof applies the pigeonhole principle to the $N+1$ fractional parts of $0,\alpha ,\dots ,N\alpha$.

Transcendence theory proves that particular numbers are not algebraic numbers, or that several numbers satisfy no nonzero polynomial relation with algebraic coefficients. A recurring method constructs an auxiliary expression which is simultaneously too small analytically and too large arithmetically if the proposed values were algebraic.

The zero-mode dimensional reduction on a flat $(D-4)$-torus splits a massless p-form gauge field by the number $r$ of spacetime indices. Each sector has $\left(\genfrac{}{}{0ex}{}{D-4}{p-r}\right)$ copies and $\left(\genfrac{}{}{0ex}{}{2}{r}\right)$ local polarizations per copy. Comparing coefficients in $(1+t{)}^{D-4}(1+t{)}^2$ gives ${\sum }_r\left(\genfrac{}{}{0ex}{}{D-4}{p-r}\right)\left(\genfrac{}{}{0ex}{}{2}{r}\right)=\left(\genfrac{}{}{0ex}{}{D-2}{p}\right)$, proving polarization matching. Four-dimensional three- and four-form potentials contribute no local propagating states.

// Target: supersymmetry.bigb