If $u+iv$ and $p+iq$ are holomorphic functions, with the image of the first inside the domain of the second, then $p(u,v)+iq(u,v)$ is holomorphic by the chain rule. Directly, their Cauchy-Riemann equations imply $[p(u,v){]}_x=[q(u,v){]}_y$ and $[p(u,v){]}_y=-[q(u,v){]}_x$. Differentiating these relations proves harmonicity of both parts. No nonvanishing derivative hypothesis is required.

For a harmonic function $u$, the function $g=u_x-i{u}_y$ is holomorphic. It is the derivative of a single-valued holomorphic function exactly when its integrals around all closed curves vanish. A primitive then has real part $u$ after adjusting a real constant. Simply connected domains satisfy this condition; on an annulus $u=\log |z|$ fails because the derivative has period $2\pi i$.

A zero-handle in an $n$-dimensional handle decomposition is $D^0\times D^n=D^n$. Its attaching region is empty, so it starts a connected component of the manifold. In dimension four it is a four-ball; adding one-handles and two-handles gives a four-dimensional handlebody with a two-dimensional CW spine.

A framed embedding of the left-hand region in an $m$-dimensional manifold specifies an index-$\lambda$ surgery. Remove its interior and glue the right-hand region along their common boundary $S^{\lambda -1}\times S^{m-\lambda }$. The cobordism realizing this change attaches an index-$\lambda$ handle of dimension $m+1$. On an oriented surface, index-two surgery cuts out an annular neighbourhood of a simple closed curve and caps its two boundary circles.

An index-$\lambda$ handle in dimension $d$ is this product of disks, attached along $S^{\lambda -1}\times D^{d-\lambda }$. Its core disk is $D^{\lambda }\times \{0\}$, and its belt sphere is $\{0\}\times S^{d-\lambda -1}$. The complementary boundary piece is $D^{\lambda }\times S^{d-\lambda -1}$. Crossing a critical point in the Morse handle-attachment theorem produces precisely this local change.

A four-handle is an index-four handle. In a four-dimensional handle decomposition it is a four-ball attached along its whole boundary, a three-sphere. It caps a spherical boundary component left by earlier handle attachments.

The binary linear code known as the Hamming code of length seven is the kernel of a $3\times 7$ parity-check matrix whose columns are all the nonzero vectors of ${\mathbb{F}}_2^3$. Its dimension is four. No one-column or two-column sum vanishes, but some three-column sum does, so its minimum Hamming distance is three. The syndrome of a one-bit error is its column, uniquely identifying the erroneous position.

Appending an overall parity bit to a binary Hamming code of redundancy $r\ge 2$ gives the extended Hamming code with parametersThe extension preserves dimension. Every resulting codeword has even Hamming weight, while every original nonzero codeword has weight at least three. A weight-three codeword extends to weight four, proving that the minimum Hamming distance is exactly four.

For $r=3$, extending the Hamming code of length seven gives the familiar $[8,4,4]$ linear code. A membership test detects every error of weight at most three, and nearest-word decoding corrects one error. Joint single-error correction and double-error detection uses the overall parity together with the original syndrome: a nonzero original syndrome with odd overall parity indicates a single error in an original coordinate, whereas a nonzero original syndrome with even overall parity detects a double error, under the assumption that at most two bits changed.

Here $N=2^l-1$, $m=2^{l-1}$ and the weight enumerator convention is $W_C(s,t)={\sum }_{c\in C}{s}^{\mathrm{wt}(c)}{t}^{N-\mathrm{wt}(c)}$. The dual code is the binary simplex code, with enumerator $t^N+(2^l-1)s^m{t}^{m-1}$. The MacWilliams identity gives the formula. For $l=3$, the ordinary enumerator $W_C(z,1)$ is $1+7z^3+7z^4+z^7$; for $l=1$, the degenerate one-word code has enumerator $t$.

Let $\omega$ generate ${\mathbb{F}}_{2^s}^{\times }$, $s\ge 2$, and set $n=2^s-1$. The binary cyclic code defined by $c(\omega )=0$ has a parity-check matrix consisting of every nonzero vector of ${\mathbb{F}}_2^s$ once, after choosing a basis and ordering its columns. It is therefore a Hamming code. Its generator polynomial is the minimal polynomial of an algebraic element $m_{\omega }(x)={\prod }_{j=0}^{s-1}(x-{\omega }^{2^j})$, of degree $s$.

For $s\ge 2$, use one nonzero representative from each one-dimensional subspace of ${\mathbb{F}}_3^s$ as a parity-check matrix column. The rank is $s$; no zero or proportional columns imply minimum Hamming distance at least three. Two independent columns and the representative of their sum give a weight-three relation, proving equality. Every nonzero syndrome uniquely identifies one column and one nonzero error value.

For an incompressible transport flow, every integrable function of the transported scalar has constant integral: the scalar is unchanged along characteristics and the flow's Jacobian determinant equals one. For a finite integral on infinite phase space, require integrability; $F(0)=0$ and compactly supported data are one common choice. For a whole-space conservation statement also require no escape or boundary flux at infinity.

For a nonnegative classical solution $V$ of a finite-horizon Hamilton-Jacobi-Bellman equation with nonnegative running reward $r$, the process $V(X_t,t)+{\int }_0^{t}r(X_s,a_s)ds$ is a local supermartingale under every admissible control. The Itô formula gives nonpositive drift because the chosen control cannot exceed the supremum in the equation. Stop where the stochastic integral becomes a true martingale, take expectations and apply the Fatou lemma. Nonnegativity prevents loss of a lower bound when removing the stops, giving the verification upper bound. A maximizing control attains the bound when its stopped reward processes are uniformly integrable.

Two maximum-cardinality $\epsilon$-separated subsets $A,B$ of a compact metric space have a bijection that moves each point by less than $\epsilon$. Join $a$ to $b$ when their distance is below $\epsilon$. If $S\subseteq A$ had fewer than $|S|$ neighbours, replacing these neighbours in $B$ by $S$ would create a larger separated set. The Hall marriage theorem therefore supplies a perfect matching. This allows empirical averages on a compact isometry group to be made approximately invariant without assuming a prior Haar measure construction.

In the equal-part random bipartite graph with fixed edge probability $0<p<1$, rule out empty rectangles of side $\lceil \sqrt{n}\rceil$ and vertices with at least $n-\lceil \sqrt{n}\rceil$ nonneighbors. A union bound gives the displayed asymptotic estimate. Small vertex sets satisfy Hall's condition by the degree bound; intermediate sets do so because their omitted-neighbor rectangle would contain a forbidden square; large sets meet every vertex in the other part. Thus Hall's marriage theorem gives a perfect matching asymptotically almost surely. This does not mean probability one at any finite size.

For a uniform guide field, write the transverse perturbation in velocity units as $\mathbf{b}$. Linear momentum gives $\omega \mathbf{u}=-k_{\parallel }{v}_A\mathbf{b}$. Linear induction gives $\omega \mathbf{b}=-k_{\parallel }{v}_A\mathbf{u}+i{k}_{\parallel }{v}_A{d}_i\mathbf{k}\times \mathbf{b}$. The helicity decomposition of a transverse Fourier mode then yields the displayed dispersion relation. Its two positive frequency magnitudes are $|k_{\parallel }|v_A[\sqrt{1+(k{d}_i{)}^2/4}\pm k{d}_i/2]$, approaching the Alfvén wave frequency for $k{d}_i\ll 1$. For $k{d}_i\gg 1$, the fast branch is a whistler wave and the slow branch tends to $|k_{\parallel }|v_A/(k{d}_i)$.

For field amplitudes of order the guide field, the ion Lorentz-force response over time $\tau$ is $\Delta u/v_A=O(v_A\tau /l)\ll 1$. The electron-current velocity is of order $v_A{d}_i/l$, larger than a bulk velocity of order $v_A$. Thus the leading Hall magnetohydrodynamics induction equation becomes the closed electron magnetohydrodynamics equation, while ions remain fixed to leading order on this fast scale. The first nonzero ion response need not have zero acceleration. This limit retains the fast branch of incompressible Hall-MHD wave dispersion, not the slow branch; very oblique modes must separately satisfy the fast-time ordering.

Induct on the alphabet size $k$. For an $r$-colouring, choose an $(k-1,k)$-insensitive combinatorial subspace whose dimension forces a monochromatic combinatorial line over the first $k-1$ letters. The inductive Hales-Jewett theorem supplies that line. Changing all of its active parameters from $k-1$ to $k$, one at a time, preserves the colour by the alphabet insensitivity lemma. Thus the line extends to all $k$ letters. The one-letter alphabet is the base case.

For a nonempty radially open convex set $C$ disjoint from a vector subspace $U$, there is a nonzero linear functional $L$ whose kernel of a linear map contains $U$ and avoids $C$. Choose $a\in C$ and the Minkowski functional $p$ of $C-a$. On $U+\mathbb{R}a$, set $g(u+ta)=-t$. Disjointness implies $p(u-a)\ge 1$, which gives $g\le p$ for negative $t$; nonnegativity of $p$ gives it for nonnegative $t$. The algebraic Hahn-Banach theorem extends $g$ to $L\le p$. For $x\in C$, $L(x)+1=L(x-a)<1$, so $L(x)<0$. Also $L(a)=-1$ and $L{|}_U=0$.

If a convex set $E$ in a real normed vector space contains $B(0,ϵ)$, every $x\notin E$ admits a continuous linear functional $T$ with $Tx\ge 1\ge Te$ for all $e\in E$, even if $E$ is not closed. Its Minkowski functional $p(v)=\inf \{t>0:v\in tE\}$ is sublinear functional, satisfies $p(v)\le \Vert v\Vert /ϵ$, $p(e)\le 1$, and $p(x)\ge 1$. Extend $tx\mapsto tp(x)$ by the dominated Hahn-Banach theorem. The resulting domination $T\le p$ applied to $v$ and $-v$ makes $T$ continuous and supplies the separation inequalities.