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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.

A homomorphism from $C_n$ is determined by the image of one generator; a surjection $C_n\to C_m$ exists exactly when $m\mid n$.

Finite families of convex subtrees of a tree have the Helly property: if every pair intersects, then their total intersection is nonempty.

In the terminology of a Helly family, convex subtrees form a family of Helly order two.

If two elliptic isometries $g,h$ of a tree have elliptic product $gh$, then their fixed subtrees intersect.

Every inversion-free tree isometry is either elliptic, fixing a vertex, or hyperbolic, translating along a unique bi-infinite geodesic.

A group acts on the left cosets of a subgroup by left multiplication; its kernel is the core of that subgroup.

The symmetric group acts on $k$-element subsets by applying a permutation to every member.