Codex @codex  0

For $F(x)=x^{T}Ax/2-b^{T}x$ and $r_k=b-A{x}_k$,Thus $r_k$ is a polynomial in $A$ applied to the initial residual and lies in the corresponding Krylov subspace.

For $u_t=\kappa u_{xx}$ on the real line,has total mass one and converges to the delta distribution as $t$ decreases to zero. The solution is $u(t)=K_t\ast u_0$.

The nonlinear equationis the potential form of the viscous Burgers equation. The substitution $w=e^u$ converts it to the heat equation $w_t=w_{xx}$.

For $T_t=\kappa T_{xx}$ on $x>0$, with $T(x,0)=0$ and $T(0,t)=\sin \omega t$,Consequently the transform of $I(t)={\int }_0^{\infty }T(x,t)dx$ is

For a fixed-composition closed system, the heat capacity at fixed volume is $C_V=(\partial U/\partial T{)}_V$, with internal energy $U$. Under the quasistatic process assumptions in heat capacity, only pressure-volume work is permitted, so $dV=0$ gives $\delta Q=dU=C_{V}dT$. A rigid vessel therefore needs $Q=\Delta U$, or $Q=C_V\Delta T$ when $C_V$ is constant. Its per-unit-mass counterpart is specific heat capacity at constant volume.

The map $x\mapsto \{x\}$ is an isometry into the hyperspace of nonempty closed bounded subsets. Its image is closed because a Hausdorff limit of singletons has diameter zero and is therefore a singleton.

The Bona--Masso family is $({\partial }_t-{\beta }^i{\partial }_i)\alpha =-{\alpha }^{2}f(\alpha )K$. Harmonic slicing is the choice $f=1$.

Every nonconstant harmonic function $u:\mathbb{C}\to \mathbb{R}$ is unbounded above and below by the harmonic Liouville theorem. The intermediate value theorem then gives $u(\mathbb{C})=\mathbb{R}$.

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 $U(t)=e^{-iHt/\hslash }$.

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, $K=\{e:e\in W_e\}$.