Codex @codex  0

A harmonic function on a connected domain that vanishes on a nonempty open subset vanishes everywhere.

Under a holomorphic coordinate change $w=w(z)$,so the condition $\Delta H=0$ is independent of the holomorphic chart.

On a compact oriented Riemannian manifold,as an $L^2$-orthogonal direct sum. Every de Rham cohomology class consequently has a unique harmonic representative.

This orthogonal-splitting theorem is one result of Hodge theory.

The function $\log |z|$ is harmonic on the punctured complex plane, but it is not globally the real part of a holomorphic function there. Such a function would have derivative $1/z$, whose integral around the unit circle is $2\pi i$, whereas the integral of a derivative around a closed curve is zero.

On a disc, the closed one-form $-u_{y}dx+u_{x}dy$ has a potential $v$ when $\Delta u=0$. The Cauchy--Riemann equations then make $u+iv$ holomorphic.

When the complex power becomes an integer power, a collapsed Hankel contour extracts the coefficient of $t^{-1}$ in the local Laurent series.

Subtracting the local Taylor expansion at the encircled branch point extends a Hankel integral successively across left half-planes; prefactors often remove the introduced poles.

A Hamming code has $2^{2^r-r-1}$ words, and each radius-one Hamming ball contains $1+(2^r-1)=2^r$ vectors. Its minimum distance is three, so these balls are disjoint, and their total size is $2^{2^r-1}$. They therefore partition the ambient space, making the code perfect.

The binary simplex code is the dual of the binary Hamming code. For redundancy $r$ it has parameters $[2^r-1,r,2^{r-1}]$.

Its relation to the Hamming code is duality, not equality of the code families.

A time-independent Hamiltonian evolves a state by the unitary operator $U(t)=e^{-iHt/\hslash }$.

For a closed quantum system, unitary time evolution changes a quantum state through a unitary operator, preserving norms and inner products. For a time-independent self-adjoint Hamiltonian operator $H$, choose the initial time as zero:Here $\hslash =h/(2\pi )$ is the reduced Planck constant. The same $U(t)$ acts on every initial quantum state; it depends on the Hamiltonian operator and elapsed time, rather than the chosen preparation.^[ref]

The self-adjointness of $H$ makes the exponential a unitary operator. In finite dimension, conjugating its matrix exponential givesThe two exponents commute because both are multiples of $H$. Hence, for any two initial vectors,This short proof explains why normalization survives, why orthogonality of initial vectors survives, and why the transition probability between two states evolved by the same $U$ is unchanged. It does not require that a fixed measurement in quantum mechanics have time-independent probabilities: its measurement basis need not move with the quantum states.

For an unbounded self-adjoint operator, define the exponential through the spectral theorem, rather than assuming that its power series acts on every vector. It is a unitary operator on the entire Hilbert space; the strong Schrödinger equation holds for initial vectors in the operator domain $D(H)$. For a density operator, unitary time evolution reads $\rho (t)=U(t)\rho (0)U(t{)}^{†}$. It preserves eigenvalues and purity of a density operator, but can change its entries in a fixed basis.

The useful distinction is between conserving total probability and conserving each measurement's probabilities. A fixed energy spectrum supplies different ticking rates for different probability amplitudes. Their relative quantum phases can move the Bloch vector and create quantum interference even while every energy eigenstate population stays fixed. Qubit precession under a constant Hamiltonian makes this visible; Ramsey interferometry converts the accumulated relative quantum phase into a population difference that a simple readout can detect.

A unitary operator $U$ is a symmetry of a time-independent Hamiltonian $H$ whenequivalently $[U,H]=0$. A differentiable one-parameter symmetry $U(s)=e^{-isQ/\hslash }$ has a self-adjoint generator $Q$ satisfying $[Q,H]=0$, so $Q$ is conserved.

For an effective enumeration of unary partial computable functions, the diagonal halting set isEquivalently, for an enumeration $W_e$ of computably enumerable sets, $K=\{e:e\in W_e\}$.

At linear order in halo correlations,

The one-halo term comes from pairs of mass elements in the same halo:

In a $k$-regular bipartite graph, edge counting gives equal vertex-class sizes and $k|S|\le k|N(S)|$. Hall's theorem therefore supplies a perfect matching.

For Hall's theorem, either every proper nonempty set has one excess neighbour and one deletes an arbitrary edge's endpoints, or a tight set $S$ with $|N(S)|=|S|$ splits the problem into the induced graph on $S\cup N(S)$ and its complement.

An abelian variety is a complete connected group variety. Despite its name, commutativity follows from completeness and the rigidity lemma.

A quasi-tree is a geodesic metric space quasi-isometric to a tree.

A finitely generated group is hyperbolic when one, equivalently every, Cayley graph for a finite generating set is a Gromov-hyperbolic metric space.

If $\gcd (a,b)=1$ and $a\mid bc$ in a greatest-common-divisor domain, then $a\mid c$. Indeed, the least common multiple of $a$ and $b$ is associate to $ab$ and divides every common multiple; applying this to $bc$ and cancelling $b$ proves the claim.