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Forwith the boundary conditions $y(0)=y^{\prime }(0)=0$, $y^{\prime \prime }(L)=0$, and $-A{y}^{\prime \prime \prime }(L)=F$, the higher-order Euler-Lagrange equation isIts solution splits as

If a time-dependent perturbation remains diagonal in a fixed eigenbasis, Hamiltonians at different times commute. Each component therefore acquires the exact phase $\exp [-i\int E_{\pm }(t)dt/\hslash ]$, with no time ordering required.

Two pure states can be distinguished with certainty in one measurement exactly when their inner product is zero.

For ${\rho }_j=|{\alpha }_j⟩⟨{\alpha }_j|$, projecting onto the positive and negative eigenspaces of ${\rho }_0-{\rho }_1$ attains

The maximal averaging operator has weak type $(1,1)$, controlling the measure of points where local averages are large.

Liouville integrability means having $n$ almost-everywhere independent first integrals in mutual Poisson involution.

For a Hamiltonian phase space of dimension $2n$, Liouville integrability means $n$ functionally independent conserved quantities with pairwise vanishing Poisson brackets. This is the Hamiltonian specialization of integrable systems.

A canonical transformation preserves the symplectic form and hence all Poisson brackets. In one degree of freedom, an invertible map $(q,p)\mapsto (Q,P)$ is canonical exactly when $\{Q,P{\}}_{q,p}=1$.

For a functional $H[u]$, its variational derivative is defined byafter integrating derivatives of $v$ by parts and discarding the boundary terms.

In canonical coordinates,

The Hamiltonian is a function of canonical position and momentum that generates time evolution through Hamilton's equations.

Phase space has one position coordinate $q_i$ and one conjugate momentum $p_i$ for every degree of freedom; a point $(q,p)$ specifies an instantaneous mechanical state.

If $M$ of $N$ items are marked and $\sin \theta =\sqrt{M/N}$, each Grover iteration rotates the state by $2\theta$ in the marked--unmarked plane, so after $r$ iterations the marked amplitude is $\sin ((2r+1)\theta )$.

For $|s⟩=H^{\otimes n}|0^n⟩$, the diffusion operator is $2|s⟩⟨s|-I=-H^{\otimes n}{I}_0{H}^{\otimes n}$, where $I_0=I-2|0^n⟩⟨0^n|$.

The order of an element is the least positive power equal to the identity and divides the order of a finite group.

This is the order of a group element in the finite setting, one use of order (group theory).

The exponent of a finite group is the least common multiple of the orders of all its elements.

Every finite group is a torsion group; its exponent is a common multiple of the order of a group element for every element.

A solvable group has a subnormal series whose factor groups are abelian.

A semidirect product combines a normal subgroup with a subgroup acting on it by automorphisms.

The projective linear group is $PG{L}_n(F)=G{L}_n(F)/F^{\times }$ and acts on projective space.

A Dedekind group is a group in which every subgroup is normal.

A subgroup $N\le G$ is a normal subgroup when $gN{g}^{-1}=N$ for every $g\in G$. In other words, conjugation by any group element preserves the subgroup as a set, although it may exchange its individual elements. Equivalently, $gN=Ng$ for every $g$: the left and right cosets of $N$ agree. Write $N⊴G$.

Every subgroup of an abelian group is a normal subgroup, because conjugation fixes each element. Normality can also hold in a nonabelian group: in $S_3$, the three even permutations form the alternating group $A_3=\{e,(123),(132)\}$, a normal subgroup as the group kernel of the sign of a permutation. Its two cosets separate the even and odd permutations.

In contrast, $H=\{e,(12)\}$ is a subgroup of $S_3$ but is not a normal subgroup. With function composition of permutations acting rightmost first, choose $g=(123)$. Thenwhich are different cosets. Also $g(12)g^{-1}=(23)\notin H$, directly contradicting invariance under conjugation. Thus a subgroup alone does not suffice for quotient group multiplication.

The practical value of normality is coset multiplication is well-defined exactly for normal subgroups. Each coset can act as one element of a quotient group, with results independent of representatives. Conversely every normal subgroup is the group kernel of its quotient map of groups, so normality also identifies which distinctions a group homomorphism can consistently forget.