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If an entire harmonic function on ${\mathbb{R}}^n$ grows like $O(1+|x{|}^{k+\alpha })$ with $k\in \mathbb{N}$ and $0<\alpha <1$, then it is a harmonic polynomial of degree at most $k$. Apply the interior derivative estimate of order $k+1$ on expanding balls.

If $u$ is harmonic on $B_r(x_0)$, then for every integer $m\ge 1$,The first-order estimate follows by differentiating the ball mean-value formula and applying the divergence theorem; iteration gives higher orders.

The Weyl lemma says that every locally integrable distributional solution of $\Delta u=0$ agrees almost everywhere with a smooth harmonic function. Convolution with a mollifier and the harmonic mean value property provide a standard proof.

For every closed ball contained in the domain of a harmonic function, the value at its centre equals both its average over the ball and its average over the boundary sphere.

In harmonic gauge, the vacuum Einstein equations have principal part $-g^{\mu \nu }{\partial }_{\mu }{\partial }_{\nu }{g}_{\alpha \beta }/2$, forming a quasilinear wave system supplemented by constraints.

Harmonic slicing imposes the harmonic-coordinate condition only on time. In 3+1 form it obeys $({\partial }_t-{\beta }^i{\partial }_i)\alpha =-{\alpha }^{2}K$.

For canonical coordinates $(q_i,p_i)$ and Hamiltonian $H$,

A point outside a nonempty closed convex subset of a normed space can be strictly separated from it by the real part of a bounded linear functional.

Point evaluation at zero is bounded on $C([0,1])$ with the essential-supremum norm, and Hahn--Banach extends it to a bounded functional on $L^{\infty }([0,1])$. No $g\in L^1$ represents this extension: continuous functions supported in shrinking neighbourhoods of zero have value one at zero while their integrals against $g$ tend to zero. Hence $(L^{\infty }{)}^{\prime }$ is strictly larger than $L^1$.

For a normed space $X$, the map $J:X\to X^{\prime \prime }$ defined byis a linear isometry. The easy inequality is $\Vert Jx\Vert \le \Vert x\Vert$; Hahn--Banach extends the norm-one functional on $\mathrm{span}\{x\}$ that takes $x$ to $\Vert x\Vert$, proving the reverse inequality.

Adding a phase gate to the Hadamard test rotates the interference signal and measures the imaginary part of $⟨\psi |U|\psi ⟩$.

If $U|\psi ⟩=e^{2\pi i\theta }|\psi ⟩$, the ordinary Hadamard test returns ancilla zero with probability

A bias-free network with $p$ inputs, $m$ sign-activated hidden nodes, and a thresholded linear output satisfiesThe hidden units contribute $s(\mathcal{G},n{)}^m\le (n+1{)}^{pm}$ sample patterns, and the output weights contribute $(n+1{)}^m$.

For $N=4$ and one marked item, $\theta =\pi /6$. One Grover iteration gives marked amplitude $\sin (3\theta )=1$, so measurement returns the target with certainty.

The diffusion operator sends every computational-basis amplitude $a_x$ to $2\overline{a}-a_x$, where $\overline{a}$ is their arithmetic mean.

Coarsening continuous observations changes the delta-method variance; for floored exponential data the rate estimator has limiting variance $(e^{\theta }-1{)}^2/e^{\theta }$.

A representation is semisimple when it is a direct sum of irreducible representations. Equivalently, every invariant subspace has an invariant complement.

For a word $w\in F(S)$ representing the identity in $⟨S\mid R⟩$, its area is the least $N$ for whichin $F(S)$, where $r_i\in R$ and ${\epsilon }_i\in \{-1,1\}$.

The image of a group homomorphism $\varphi :G\to H$ is the subgroup $\{\varphi (g):g\in G\}$ of $H$.

The kernel of a group homomorphism is the group version of a kernel. For $\phi :G\to H$, it isIt is a normal subgroup of the domain: for $n$ in the group kernel, $\phi (gn{g}^{-1})=\phi (g)e_H\phi (g{)}^{-1}=e_H$. The group kernel describes inputs whose effect disappears after mapping. For reduction modulo three from $\mathbb{Z}/6\mathbb{Z}$, it is $\{0,3\}$; for the sign of a permutation on $S_3$, it is $A_3$. A group homomorphism is injective exactly when its group kernel contains only the identity element.