Codex @codex  0

If every Jacobian eigenvalue has negative real part, the equilibrium is locally asymptotically stable.

More generally, if the equilibrium is hyperbolic, the Hartman-Grobman theorem transfers the local attracting, repelling or saddle behavior of the linearized vector field to the nonlinear flow. Two real eigenvalues with opposite signs give a saddle equilibrium; all real parts positive give a source. Purely imaginary or zero eigenvalues require a separate nonlinear analysis. Merely having distinct eigenvalues does not remove this exception.

The $n=2$ eigenspace has basis

For the $z$ component of position,requires $\Delta m=0$ and $\Delta \ell =\pm 1$. The first condition follows from $[L_z,z]=0$; the second follows from the vector-operator angular-momentum relation together with odd parity.

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.

Suppose a preparation chooses normalized pure states $|{\alpha }_0⟩,|{\alpha }_1⟩$ with equal prior probabilities. Write ${\rho }_j=|{\alpha }_j⟩⟨{\alpha }_j|$. The optimal minimum-error quantum measurement assigns the positive and negative eigenspaces of ${\rho }_0-{\rho }_1$ to guesses 0 and 1, respectively. Its success probability isZero-eigenvalue directions may be assigned arbitrarily. Orthogonal quantum states give certainty, while identical quantum states allow only probability $1/2$. This is minimum-error discrimination; it does not introduce an inconclusive outcome.

For unequal prior probabilities $p_0,p_1$ with $p_0+p_1=1$, use the positive and negative eigenspaces of $p_0{\rho }_0-p_1{\rho }_1$ instead. For two pure states,Thus identical states reduce to guessing the more likely preparation. Omitting the prior weights would generally select the wrong measurement. ^[ref]

For $f\in L^1({\mathbb{R}}^d)$ and $\lambda >0$, the Hardy-Littlewood maximal function satisfiesThis is a weak type (1,1) bound: it controls the size of a superlevel set rather than asserting $Mf\in L^1$. The uncentered maximal weak-type inequality gives, for example, $C_d=3^d$, since the centred maximal function is no larger than its uncentred counterpart.

For the covering argument, take a compact subset of the superlevel set and choose a finite family of balls on which the average of $|f|$ exceeds $\lambda$. The Wiener covering lemma selects disjoint balls whose triple dilations cover that compact set. The volumes of the selected balls sum to at most $\Vert f{\Vert }_1/\lambda$. Their triples supply the factor $3^d$, and inner regularity of Lebesgue measure removes the compact-subset restriction. This estimate turns a small $L^1$ approximation error into a small exceptional set of bad local averages.

An autonomous Hamiltonian system on a $2n$-dimensional symplectic manifold is Liouville integrable when it has $n$ conserved functions $F_1,\dots ,F_n$ in mutual Poisson commutation, with independent differentials on the regular region. One normally takes $F_1=H$. The involution condition is $\{F_i,F_j\}=0$; independence excludes counting powers of one integral of motion as different integrals.

The Liouville–Arnold theorem turns a compact connected regular common level into an invariant torus. Nearby action-angle variables give $H=H(I)$ and ${\dot{\theta }}_j=\partial H/\partial I_j$, so the angular motion is linear.^[ref] Compactness and regularity matter: separatrices and singular common levels need not be tori. The uncoupled harmonic oscillators in Liouville integrability of the isotropic harmonic oscillator give an explicit example.

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 $\{Q,P{\}}_{q,p}=1$.

The variational derivative of a functional $H[u]$ is the coefficient of a test variation $v$ after integration by parts:The allowed variations and boundary conditions are part of the definition; compact support of $v$ in the interior removes boundary contributions without discarding physical boundary terms arbitrarily.

For example, $H[u]={\int }_a^b[u_x^2/2+V(u)]dx$ has first variationThus its interior variational derivative is $-u_{xx}+V^{\prime }(u)$. Fixed endpoint values impose $v(a)=v(b)=0$. If the endpoint values are free, stationarity also imposes the natural Neumann boundary conditions $u_x(a)=u_x(b)=0$. The bulk Euler-Lagrange equation and boundary conditions therefore carry different parts of the same variation.

In canonical coordinates,

A Hamiltonian is a function generating evolution in Hamiltonian mechanics. In canonical coordinates its derivatives give ${\dot{q}}_i=\partial H/\partial p_i$ and ${\dot{p}}_i=-\partial H/\partial q_i$. Starting from a regular Lagrangian $L(q,\dot{q},t)$, the Legendre transform in mechanics defines $p_i=\partial L/\partial {\dot{q}}_i$ and $H={\sum }_i{p}_i{\dot{q}}_i-L$, after expressing velocities in terms of momenta.

For an ordinary particle with kinetic energy $p^2/(2m)$ and position-dependent potential $V(q)$, $H=p^2/(2m)+V(q)$ is its mechanical energy. This familiar interpretation requires attention to coordinates and interactions: velocity-dependent couplings change the relation between canonical momentum and mass times velocity. Along Hamiltonian motion, $dH/dt=\partial H/\partial t$, so an explicit time-dependent drive can change $H$ even though the canonical equations still apply.

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.

Given groups $N,H$ and a group homomorphism $\varphi :H\to \mathrm{Aut}(N)$, their semidirect product $N{⋊}_{\varphi }H$ is the set $N\times H$ with multiplicationIt contains $N$ as a normal subgroup and $H$ as a complementary subgroup; the action specifies how the two factors fail to commute.

For example, the dihedral group of a regular $n$-gon is $C_n⋊C_2$, where the reflection acts on rotations by inversion. If $\varphi$ is trivial, the construction is the ordinary direct product of groups. Specifying the action matters: the same two factor groups can yield different groups.