Such a quantum state remains an eigenstate of $H_0+f(t)H_1$ for every control input. Its projector commutes with all available generators, so its population is conserved. This obstructs full-space pure-state controllability even when the orthogonal bright subspace can be controlled. An instantaneous dark state of a slowly varying Quantum Hamiltonian instead supports adiabatic quantum control if a spectral gap and slowness condition are maintained.

For $\alpha >0$, let $L$ be the upper branch $\mathrm{Im}k=(\alpha +\sqrt{{\alpha }^2+4(\mathrm{Re}k{)}^2})/2$, oriented left to right. Above $L$, $i\alpha -k$ lies in the lower half-plane, where a decaying half-line Fourier transform is analytic. For $x>0$, closing upward proves that the displayed integral vanishes. An additional multiplier analytic above $L$ with at most polynomial growth is allowed, using exponential decay and, if needed, regularized contours. This supplies a null term in a Fokas method inversion formula without changing the function in the interior.

For $Q(k)={\int }_0^{\infty }{e}^{-ikx}q(x)dx$, the Half-range Fourier transform is holomorphic in the lower half-plane. Let $\alpha =e^{2\pi i/3}$, $E=\{0<\arg k<\pi /3\}$ and $D=\{2\pi /3<\arg k<\pi \}$, with their boundaries oriented counterclockwise. The rotated arguments ${\alpha }^{2}k$ on $E$ and $\alpha k$ on $D$ remain in the lower half-plane. For $x>0$ and transform-admissible smooth decay, the Cauchy integral theorem and Jordan lemma give the displayed null integrals. Arbitrary multiples can therefore be added to Fourier inversion without changing the recovered function on the positive half-line.

// Target: integrable-systems.bigb

Evaluating the Lax pair connection formula at $(0,T)$ gives $S^{-1}s=e^{2i{k}^{2}T{\hat{\sigma }}_3}{\mu }_3(0,T,k)$. The upper off-diagonal entry is the displayed relation, where $c_T=[{\mu }_3(0,T){]}_{12}$ is analytic in the upper half-plane and is $O(1/k)$. A finite-time relation must retain this final-state term.

In a real normed vector space with a specified countable dense subset, a bounded linear functional on any vector subspace admits a norm-preserving extension without the axiom of choice. Add the dense vectors one at a time using the supremum endpoint in the one-dimensional dominated extension of a real linear functional. This makes the recursion deterministic. Extend from the resulting dense subspace by continuity, using the least index in the dense list which approximates a given vector to within $2^{-n}$.

If $p$ is a sublinear functional, $q$ a superlinear functional and $q\le p$ on a real vector space, there exists a linear functional $U$ between them. More generally, a linear functional $S$ on a subspace extends with this domination whenever $S(y)\le p(x+y)-q(x)$ for all ambient $x$ and subspace $y$. In a one-dimensional extension the admissible values for a sandwich extension form a nonempty real interval. Positive homogeneity checks the extension on both signs of the new coordinate. Chain unions preserve linearity and the mixed domination, so Zorn's lemma gives a maximal extension, whose domain must be the whole space. Finally $x=0$ in the mixed inequality gives the upper bound; using the pair $(x,-x)$ gives the lower bound. Conversely, any linear $U$ between $q,p$ obeys the mixed inequality by $U(y)=U(x+y)-U(x)$.

A linear functional $f$ dominated by a finite real convex function $\varphi$ on a vector subspace $W$ extends with the same domination to the whole real vector space. To add $v\notin W$, choose $g(v)$ between the supremum of $(f(w)-\varphi (w-tv))/t$ and the infimum of $(\varphi (w+sv)-f(w))/s$, over $w\in W$ and $s,t>0$. Convexity, applied to $(s(w_1-tv)+t(w_2+sv))/(s+t)$, makes every lower candidate at most every upper candidate. The candidates at $w=0,s=t=1$ bound the interval by finite numbers. Chain unions preserve domination, so Zorn's lemma gives a full extension. No homogeneity of $\varphi$ is assumed.

For the Hadamard test with a density operator $\rho$ in the auxiliary register, the output control qubit has Pauli Z gate expectation $\mathrm{Re}m$ and Pauli Y gate expectation $-\mathrm{Im}m$, where $m=\mathrm{Tr}(\rho U)$. The minus sign comes from the final Hadamard gate, since $HYH=-Y$. Measuring in $(|0⟩\pm i|1⟩)/\sqrt{2}$ therefore estimates the imaginary part using the minus-outcome frequency minus the plus-outcome frequency.

A flat reference over $k$ adjacent Fock states has overlap $r_k$ with a one-photon shift and $s_k$ with a two-photon shift. These overlaps, obtained by counting shared number states, determine the reduced qubit channel of a number-conserving Hadamard interaction. For $k\ge 2$, its squared fidelity with the desired pure output is $1-(1+xz)/(2k)$ for input Bloch vector $(x,y,z)$. Its worst-case infidelity is $3/(4k)$ because $|xz|\le 1/2$. At $k=1$ the two-shift overlap is zero, not $-1$, and the fidelity is $1/2+(y^2-xz)/4$. Broad coherent number support, rather than merely a large fixed photon number, suppresses distinguishability of the outgoing field states.

A compact smooth h-cobordism of dimension at least six between connected simply connected closed manifolds is diffeomorphic to a product, relative to its incoming boundary. The boundary manifolds consequently are diffeomorphic. The dimension and simple-connectivity hypotheses are sufficient conditions, not necessary properties of a product cylinder. The two boundary inclusions being homotopy equivalences are necessary for a trivial cobordism.

For a spherical cell with its centre of mass displaced by $-{\ell }_g\mathbf{p}$ from its geometric centre, gravity produces torque $mg{\ell }_g\mathbf{p}\times \mathbf{k}$. Rotational resistance ${\zeta }_r$ gives angular velocity $\mathbf{\omega }/2+(mg{\ell }_g/{\zeta }_r)\mathbf{p}\times \mathbf{k}$. Taking its cross product with $\mathbf{p}$ gives the displayed orientation equation with $B={\zeta }_r/(mg{\ell }_g)$. Steady orientation therefore requires the two terms on the right to sum to zero; placing the vorticity term alone on the other side changes its sign.

For orientation probability density $f$ on the unit sphere, deterministic drift $\dot{\mathbf{p}}$ and isotropic rotational diffusion coefficient $D_R$ give ${\partial }_{t}f+{\nabla }_p\cdot (f\dot{\mathbf{p}}-D_R{\nabla }_{p}f)=0$. The derivatives are tangential to the sphere, and normalization is $\int fd\Omega =1$. The spherical divergence of $\mathbf{k}-p_3\mathbf{p}$ is $-2p_3$.

For fluid velocity $\Omega (-x_3,0,x_1)$ and $B\Omega <1$, the stable deterministic orientation is $\mathbf{p}=(-B\Omega ,0,\sqrt{1-B^2{\Omega }^2})$. Once this orientation has relaxed, swimming at speed $V_s$ shifts the centre of the circular material trajectories to $(-V_s\sqrt{1-B^2{\Omega }^2}/\Omega ,-V_{s}B)$ in the $(x_1,x_3)$ plane. Its distance from the cylinder axis is $V_s/\Omega$. General initial orientations produce an orientation transient before these circular paths.

For a bottom-heavy spherical swimmer in axial flow $u_z=W_0(-1+r^2+w)$, the bottom-heavy spherical-cell orientation dynamics with aligning rate $1/B$ gives $\sin \theta =-\lambda (2r+w^{\prime })$, $\lambda =B{W}_0/(2R)$. On the upward-stable branch $\cos \theta >0$, the cells tilt inward in a downward Poiseuille flow. Zero radial cell flux with scalar diffusivity gives $n^{\prime }=\chi n\sin \theta$, $\chi =V_{c}R/D$. For $w=0$, integration and the average normalization $2{\int }_0^{1}nrdr=1$ give the displayed concentration. A steady deterministic orientation requires $|\lambda (2r+w^{\prime })|<1$; beyond this shear threshold the stationary orientation closure fails.

For one marked element among $N\ge 2$ candidates, set $\theta =\arcsin (N^{-1/2})$. The first nonnegative Grover search algorithm iteration count attaining marked probability at least ${\cos }^2\theta$ is the displayed integer. Before the first success interval, the angle $(2k+1)\theta$ is below $\pi /2-\theta$; entering that interval requires $k\ge \pi /(4\theta )-1$. Rounding up gives an angle less than $\pi /2+\theta$, so it cannot skip the interval. Moreover $k_{\min }<\pi /(4\theta )<\pi \sqrt{N}/4$. For $N=1$, the known candidate is already the answer and no iteration is needed; the two-dimensional marked--unmarked plane degenerates.

If the marked fraction is $p$ and $\theta =\arcsin \sqrt{p}$, the success probability after $n$ Grover search algorithm iterations is ${\sin }^2((2n+1)\theta )$. For $0<p\le 1/2$, its first value at least $1/2$ occurs atBefore this integer the angle is below $\pi /4$, so success is below $1/2$. At this integer, the angle lies in $[\pi /4,\pi /4+2\theta )$, contained in $[\pi /4,3\pi /4)$, where squared sine is at least $1/2$. If $p\ge 1/2$, no iteration is needed. For small $p$, $n_{\min }$ is asymptotic to $\pi /(8\sqrt{p})$.

A complex group representation $\pi :G\to \mathrm{GL}(V)$ is smooth when every vector is fixed by some compact-open subgroup. Equivalently each vector's stabilizer is open. Smoothness is local constancy of orbit maps with the vector space regarded discretely, not ordinary continuity into a finite-dimensional complex topology.

A two-dimensional representation of an abelian group fitting into an extension of two distinct linear characters splits. Choose an element on which the linear characters differ. Its matrix has two distinct eigenvalues, and all commuting group operators preserve its two eigenlines. Those eigenlines supply the two invariant linear character summands. The argument does not apply when the two linear characters coincide.

If a representation has one-dimensional endomorphism algebra, it cannot be a direct sum of two nonzero invariant subspaces: the two coordinate projections would be independent endomorphisms. This obstruction does not require either summand to be irreducible, and does not assert that scalar endomorphisms alone imply irreducibility.

A representation over the rational numbers is a group representation on a finite-dimensional rational vector space. Choosing a basis gives rational matrices. It differs from a rational representation of an algebraic group, where rational refers to regular matrix-coefficient functions rather than the ground field. Scalar extension yields representations over every characteristic-zero field.