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A fixed singularity of a complex differential equation has a location determined by the coefficients, independently of the initial value. A movable singularity of a complex differential equation belongs to an individual continued solution and changes location when the initial value changes. For example, $w^{\prime }=w^2$ has the movable pole $z=c$ in $w=-1/(z-c)$. A fixed exceptional location need not be singular for every solution.

The precise first-order form of the Painlevé determinateness theorem needed here is as follows. Let $w^{\prime }=A(z,w)$ be rational in $w$, with coefficients meromorphic in $z$ on the chosen independent-variable surface. Algebraic coefficients can be made single valued on their own surface. Exclude the fixed coefficient singularities and exceptional fibres where a reduced numerator and denominator vanish together, working also in the dependent coordinate $v=1/w$. If a solution is continued along an arc $\gamma :[0,1)\to U$ with endpoint $z_{\ast }={\lim }_{t\to 1}\gamma (t)$ outside that exceptional set, then $w(\gamma (t))$ has a definite finite or infinite limit. Any singularity there is a pole or an algebraic branch point; there is no movable essential or logarithmic singularity. The standard rectifiable-arc formulation is included in this statement.

Here is a proof. First suppose a sequence along the arc has $(z_n,w_n)\to (z_{\ast },w_{\ast })$ at a point where the vector field is holomorphic. Choose a smaller closed bidisc about this point. Uniform bounds on the vector field and its dependent-variable derivative give, by the integral contraction mapping proof, one radius $r>0$ on which every solution starting sufficiently close to $(z_{\ast },w_{\ast })$ exists and remains bounded. For large $n$, the solution through $(z_n,w_n)$ is therefore defined on a fixed disc about $z_{\ast }$. The entire sufficiently late tail of $\gamma$ lies in that disc. Uniqueness identifies the original continued solution with this local solution throughout that tail, so the solution extends through $z_{\ast }$ and has a limit. The same argument applies to an ordinary point in the reciprocal coordinate $v=1/w$.

If the limit did not exist, consider the cluster set $K$ in the dependent-variable Riemann sphere:Each closed tail is compact and connected, and these sets are nested; hence $K$ is nonempty, compact and connected. The preceding paragraph excludes every ordinary point from $K$. At $z_{\ast }$ the rational vector field has only finitely many nonordinary dependent values, in both coordinate charts: an entire fibre of singular points was excluded among the fixed exceptions. Thus $K$ is a connected subset of a finite set, so it is a singleton. Compactness then forces convergence to its unique point, contradicting the assumed lack of a limit. This proves determinateness.

It remains to determine the local type rather than merely the limit. At a finite limiting value $w_{\ast }$ which is a pole of the vector field, write the reduced equation as $w^{\prime }=P/Q$. Since $z_{\ast }$ is not exceptional, $P(z_{\ast },w_{\ast })\ne 0$, and the inverse equationis holomorphic near $(z_{\ast },w_{\ast })$. The holomorphic dependence of ordinary differential equations on parameters makes its local solution curves jointly holomorphic in $w$ and their initial $z$ values. In particular, the value at the transverse line $w=w_{\ast }$ is a holomorphic first integral labelling these curves. The original graph has constant label; taking its endpoint limit shows that it lies on the inverse solution through $(z_{\ast },w_{\ast })$. That inverse solution cannot be the vertical curve $z\equiv z_{\ast }$: this would require $Q(z_{\ast },w)$ to vanish identically near $w_{\ast }$, an excluded fixed singular fibre. Consequently its power series has a first nonzero term,Here $k\ge 2$ at a vector-field pole. Factor the right side as $(w-w_{\ast }{)}^k$ times a nonvanishing holomorphic function and take its local $k$th root. This gives a new dependent coordinate with nonzero derivative, and the holomorphic inverse function theorem expresses $w$ as a convergent power series in $(z-z_{\ast }{)}^{1/k}$. Thus the singularity is an algebraic branch point. At an infinite limit, repeat the argument for $v=1/w$. An ordinary zero of $v$ gives a pole of $w$, while a finite ramification of $v$ gives an algebraic pole branch. This proves the asserted local types and the theorem.

For the given equation, the finite-coordinate rational field isThe whole fibre $z=-1$ is singular in the displayed coefficient, and at $(z,w)=(0,0)$ the reduced numerator and denominator vanish together. Hence the finite fixed exceptional set is $\{-1,0\}$. Away from these two locations, the only possible finite dependent limits at a singularity are $w_{\ast }=\sigma z_{\ast }$, with $\sigma =\pm 1$.

At such a point the inverse equation isIt is holomorphic when $z_{\ast }\ne 0,-1$ and $w_{\ast }=\sigma z_{\ast }$. Its inverse solution satisfies $z_w(w_{\ast })=0$, and differentiation givesThereforeandThese are genuine square-root algebraic branch points. The inverse solutions can be based at arbitrary $z_{\ast }\notin \{0,-1\}$ and then continued to nearby ordinary initial points. Their distinct branch locations require different initial data, so these branch points are movable. The signs specify the two inverse branches near each of the two values $w_{\ast }=\pm z_{\ast }$.

There are no movable poles. Indeed the reciprocal equation isFor every finite $z\ne -1$ it is holomorphic at $v=0$, and $v\equiv 0$ is its unique solution with zero initial value. A nonzero reciprocal solution therefore cannot reach zero there. This rules out both an ordinary pole and an infinite algebraic branch at a movable finite location. Together with the theorem, it exhausts the finite movable singularities.

The fixed point zero must not be mistaken for a generic movable collision. It is ordinary for solutions with $w(0)\ne 0$. But solutions reaching $w=0$ can have a fixed square-root singularity. To construct them rather than just balance powers, set $z(w)=w^{2}h(w)$ in the inverse equation. The equation for $h$ isEquivalently,On a small disc this is a contraction mapping of the closed ball $\Vert h-1/2{\Vert }_{\infty }\le 1/4$: both its departure from $1/2$ and its Lipschitz constant are $O(r^2)$. Its unique holomorphic fixed point gives $z=w^2/2+O(w^4)$. Thusis a genuine branch at the fixed location zero. The zero solution itself extends through zero as a function, although the original displayed coefficient is undefined at $(0,0)$.

The point $-1$ can produce a fixed nonalgebraic singularity. For an explicit example take $z=-1+x$ real, $0<x<x_0<1$, and $w=iy$ with initial $y>2$. Put $q=y^2$ and $L=-\log x$. ThenAs $L$ increases, $q$ increases, its derivative is bounded above by $2$ and bounded below by a positive constant, and therefore the solution exists for all large $L$ and $q\to \infty$. The derivative then tends to $2$, so $q/L\to 2$. ConsequentlyThis is unbounded but grows more slowly than every negative power of $z+1$. It cannot be a meromorphic pole even after a finite ramified substitution, demonstrating an actual fixed singularity with logarithmic growth. The determinateness theorem does not exclude such behavior at fixed exceptions.

Finally, on the independent-variable Riemann sphere use $t=1/z$. For finite $w$ the transformed equation iswhich is ordinary at $t=0$. At simultaneous $z=\infty$, $w=\infty$, however, the reciprocal field iswith an indeterminate point at $(t,v)=(0,0)$. Thus infinity is an additional fixed exceptional location on the sphere, rather than an unavoidable singularity of solutions with finite limiting value. There are solutions escaping to infinity there: for real initial data $w(z_0)>z_0>0$, the difference $d=w-z$ stays positive, since its derivative becomes positive as $d\downarrow 0$. Moreover $d^{\prime }\le 0$ when $d\ge 1$, so $0<d\le \max (1,d(z_0))$. Such a solution continues for all positive $z$, with $w=z+O(1)\to \infty$.

In the finite plane, the fixed exceptional locations are $0,-1$; all other solution singularities are movable square-root branches at $w=\pm z$, and there are no movable poles. On the sphere, infinity is also a fixed exception at the infinite dependent value. Being fixed or movable describes the base location, not merely whether the coefficient contains the factors $w-z$ and $w+z$.

Use the PDF's starred image hull $K_t^{\ast }$ and write $h_t={\Phi }_t=g_{K_t^{\ast }}\circ \Phi \circ g_{K_t}^{-1}$. For $t<T$, this map is analytic across a real neighborhood of the driver. SetLet $u(t)=\mathrm{hcap}(K_t^{\ast })/2$. The classical conformal change of half-plane capacity gives $\dot{u}(t)=a_t^2$, and the image driver, indexed by original time, is $h_t({\xi }_t)$. Differentiating the conjugacy relation givesThese are precisely the standard classical identities allowed in this question. The singularities cancel at $z={\xi }_t$. To calculate the remaining derivative, put $\delta =z-{\xi }_t$ and expand at fixed $t$:Thereforewhere each ${\partial }_t$ keeps the spatial variable fixed before evaluation. Applying the Itô formula to $h_t^{\prime }({\xi }_t)$ gives the boundary derivative diffusion under conformal Loewner conjugacyAt $\kappa =8/3$, the third-derivative drift cancels. A second application of the Itô formula to $a_t^p$ givesChoosing $p=1-1/\kappa =5/8$ cancels the remaining drift. Hence the SLE eight-thirds restriction martingale isLocalizing away from $T$ and from vanishing derivatives makes the displayed stochastic integral a martingale on each localized interval, so it defines a continuous local martingale up to $T$.

Let $\varphi$ denote the standard Gaussian distribution probability density function. Directly from the definitions, $S\varphi (d_1)=K{e}^{-r\tau }\varphi (d_2)$. Differentiating the price in part (ii), the terms involving derivatives of $d_1,d_2$ cancel, yielding $V_S=\Phi (d_1)$. Therefore, in units of the two traded assets, the replicating strategy isIts value is $h_t^B{B}_t+h_t^S{S}_t=V(t,S_t)$. To check self-financing explicitly, differentiation also givesHence $V_t+rS{V}_S+\frac{1}{2}{\sigma }^2{S}^2{V}_{SS}=rV$. Applying Itô formula under the physical measure givesThus the holdings form a self-financing strategy and their terminal value is the required payout. Holdings at the single maturity instant can be assigned by their limiting values; the continuous-time trading gains do not depend on that assignment. If ${\pi }_t$ denotes wealth fractions instead, the stock fraction is $S_t\Phi (d_1)/V(t,S_t)$ and the bank fraction is $K{e}^{-r\tau }\Phi (-d_2)/V(t,S_t)$. For $K\le 0$, hold one stock and no bank-account units.

Let $F_0(T)$ be the observed initial instantaneous forward rate curve. Since ${\hat{W}}_0=0$, part (i) gives $f_0(T)=g(T)-{\sigma }^2{T}^2/2$. Thus the forward-curve calibration of a shifted Brownian short rate is achieved by the deterministic choiceSubstitution gives $f_0(T)=F_0(T)$ at every maturity where the curve is defined classically. It also reproduces the corresponding initial zero-coupon bond curve:There is no shape restriction on the initial curve beyond the regularity needed to define these integrals and forward derivatives; for continuous curves the match is pointwise, and for locally integrable curves the forward identity has its almost-everywhere interpretation.

Let $Q$ denote the given unique equivalent martingale measure. A zero-coupon bond paying one at $T$ has risk-neutral pricing valueWrite $h=T-t$. Split the integrated short rate into its known and future parts:By integrating the future Brownian motion increments in the opposite order, the last integral equals ${\int }_t^T(T-u)d{\hat{W}}_u$. It is independent of ${\mathcal{F}}_t$ and is a Gaussian random variable of mean zero and varianceThe Gaussian exponential moment formula $\mathbb{E}[e^{aZ}]=e^{a^2\mathrm{Var}(Z)/2}$ for centered $Z$ therefore gives the bond price in this Gaussian short-rate model with a deterministic shift:The instantaneous forward rate is the negative maturity derivative of the logarithm of the bond price. ConsequentlyIn particular, $P_t(t)=1$ and $f_t(t)=g(t)+\sigma {\hat{W}}_t=r_t$, as required. The derivative formula is pointwise for continuous $g$ and almost everywhere for locally integrable $g$.

The Gumbel distribution for maxima has distribution function $G(x)=\exp (-e^{-x})$ on the whole real line. For $n\ge 2$ setBoth the centering and scale are elementary, and the scale is positive. For any fixed real $x$, the normalized threshold ${\beta }_n+{\alpha }_{n}x$ is positive for all sufficiently large $n$. Writing $L=\log n$, a Taylor expansion givesConsequently,The logarithm calculation in part (i) now givesThus $F$ belongs to the maximum domain of attraction of the standard Gumbel distribution. The scale also has the Gumbel auxiliary function interpretation: the reciprocal hazard function is $2\sqrt{t}$, whose derivative $1/\sqrt{t}$ tends to zero at the infinite right endpoint. This agrees with the Von Mises conditions for extreme values, and at $t={\beta }_n$ gives exactly ${\alpha }_n=2\log n$.

Let the summands be independent and identically distributed random variables with mean $\mu$, variance ${\sigma }^2>0$, and standardized third cumulant ${\gamma }_1=\mathbb{E}[(Y_1-\mu {)}^3]/{\sigma }^3$. DefineAn Edgeworth expansion refines the central limit theorem using the higher cumulants of one summand. Some smoothness and moment assumptions are essential: finite variance alone does not supply a density expansion. A convenient sufficient set for the expansions here is a smooth density whose derivatives decay faster than every inverse power, together with $\mathbb{E}{e}^{\delta |Y_1|}<\infty$ for some $\delta >0$. These assumptions give all required moments, Fourier smoothing and the nonlattice characteristic-function condition. Less restrictive sufficient conditions are possible, but this set covers the logistic model below.

Write $\varphi$ and $\Phi$ for the standard normal density and standard normal distribution function. The first-order density Edgeworth expansion isThe remainder is uniform on every fixed compact interval under the stated assumptions. To see the origin of the correction, for $Z=(Y_1-\mu )/\sigma$ the cumulant expansion of its characteristic function givesExponentiating and using Fourier inversion yields the displayed correction, since the inverse transform of $(it{)}^3{e}^{-t^2/2}$ is $-{\varphi }^{(3)}(x)=(x^3-3x)\varphi (x)$. The Probabilists' Hermite polynomial convention is being used: $H_2(x)=x^2-1$ and $H_3(x)=x^3-3x$.

The corresponding distribution function expansion isuniformly on fixed compact intervals. The sign follows from $\frac{d}{dx}[H_2(x)\varphi (x)]=-H_3(x)\varphi (x)$. These are expansions of both the density and distribution function; a uniform density error alone would not justify integrating an error bound over the entire real line without the accompanying tail control.

For a fixed $\alpha \in (0,1)$, write $z=z_{\alpha }={\Phi }^{-1}(\alpha )$. At leading order the proposed quantile expansion gives $\Phi (p_0(z))=\Phi (z)$, and strict monotonicity gives $p_0(z)=z$. Substitute $y_{\alpha }=z+p_1(z)n^{-1/2}+O(n^{-1})$ into the distribution function expansion. A Taylor expansion at $z$ givesSince $\varphi (z)>0$, the coefficient of $n^{-1/2}$ must vanish. Thus the first Cornish-Fisher expansion coefficients are

For the logistic distribution, set $Z_i=Y_i-\theta$. Its density $q(z)=e^z/(1+e^z{)}^2$ satisfies $q(-z)=q(z)$ and has exponentially decreasing tails. Symmetry and integrability give $\mathbb{E}{Z}_i=0$, and the supplied integral givesTherefore $\mathbb{E}{Y}_i=\theta$, $\mathrm{Var}(Y_i)={\pi }^2/3$, and the central limit theorem givesWriting $z_{-}=z_{\alpha /2}$ and $z_{+}=z_{1-\alpha /2}$, the condition $z_{-}\le T_n\le z_{+}$ is equivalent toConsequently the confidence intervalhas limiting coverage $\Phi (z_{+})-\Phi (z_{-})=1-\alpha$. Whether the finite endpoints are included makes no probability difference because the sample mean has a density.

The centered logistic distribution is symmetric, so ${\gamma }_1=0$. Its smooth density and exponentially decreasing derivatives satisfy the conditions stated above; in particular all moments exist and Fourier smoothing is available. The first correction in the Edgeworth expansion vanishes, leaving $F_{T_n}(z)=\Phi (z)+O(n^{-1})$ at each of the two fixed quantiles. Subtracting gives the stronger conclusionThe standardized law is independent of $\theta$, so this coverage-error conclusion holds with the same bound for every location parameter.

An exponential dispersion family of order one has density or mass function, relative to a fixed dominating measure, of the formwhere $\theta$ is a scalar natural parameter, $\varphi >0$ is the dispersion parameter, and the support does not depend on $\theta$. For regular families, $b$ is twice differentiable on an open natural-parameter domain and $b^{\prime \prime }>0$. Known observation weights $w$ replace $\varphi$ by $\varphi /w$. “Order one” refers to the scalar canonical statistic $y$, rather than an arbitrary vector of canonical statistics.

Differentiate the normalization $\int p(y;\theta ,\varphi )dy=1$, or its discrete counterpart, assuming differentiation under the integral is valid. The first derivative gives $\mathbb{E}(Y-b^{\prime }(\theta ))=0$. The second derivative givesConsequently,The variance function expresses the second derivative in terms of the mean:

A generalized linear model specifies independent responses from an exponential dispersion family, a linear predictor ${\eta }_i=x_i^T\beta$, and a differentiable monotone link function relating it to the response mean by $g({\mu }_i)={\eta }_i$. The family, variance function, covariate design, link function and dispersion specification together define the model. The canonical link function identifies the linear predictor with the natural parameter:One advantage is that the likelihood equations become particularly simple. With known weights $w_i$ and common dispersion $\varphi$, the score function and negative Hessian matrix for $\beta$ areThe latter does not explicitly depend on the responses, so it equals the expected Fisher information matrix at the same parameter. With full-rank design and positive variances the likelihood is strictly concave in $\beta$, simplifying optimization. These statements concern the coefficient block at a fixed dispersion value.

Three explicit examples, taking unit weights, are as follows.

For a normal distribution with mean $\mu$ and variance $\varphi$,Thus $\theta =\mu$, $b(\theta )={\theta }^2/2$, $c(y,\varphi )=-y^2/(2\varphi )-\frac{1}{2}\log (2\pi \varphi )$, and $V(\mu )=1$. Its canonical link function is $\boxed{g(\mu )=\mu }$, the identity link.

For a Poisson distribution with mean $\mu >0$ and support $\{0,1,\dots \}$,Here $\varphi =1$, $\theta =\log \mu$, $b(\theta )=e^{\theta }$, and $c(y,1)=-\log (y!)$. Therefore $b^{\prime }(\theta )=b^{\prime \prime }(\theta )=e^{\theta }$, $V(\mu )=\mu$, and the Poisson canonical link is $\boxed{g(\mu )=\log \mu }$.

For a Bernoulli distribution with success probability $\mu \in (0,1)$,Taking $\varphi =1$, $\theta =\log (\mu /(1-\mu ))$, $b(\theta )=\log (1+e^{\theta })$, and $c(y,1)=0$ gives the required form. Its mean and variance function are $b^{\prime }(\theta )=e^{\theta }/(1+e^{\theta })=\mu$ and $V(\mu )=\mu (1-\mu )$. Its canonical link function is $\boxed{g(\mu )=\log (\mu /(1-\mu ))}$, the logit link. The fixed dispersion values in the Poisson and Bernoulli cases are part of those models; an arbitrary continuous dispersion is not being asserted for them.

To test $H_0:{\beta }_j=0$, first fit the unrestricted model and then the restricted model with that component fixed at zero, estimating all nuisance parameters, including dispersion when applicable, in each fit. If $\hat{\vartheta }$ and $\tilde{\vartheta }$ denote their full fitted parameter vectors, the likelihood-ratio test usesFor identifiable regular models, an interior null parameter and a fixed-dimensional full-rank design with increasing information, the Wilks theorem gives $D\stackrel{d}{\to }{\chi }_1^2$ under $H_0$. A level-$\alpha$ test rejects for $D$ greater than the $(1-\alpha )$-quantile of the chi-squared distribution with one degree of freedom.

Alternatively the Wald test uses $W={\hat{\beta }}_j^2/\widehat{\mathrm{Var}}({\hat{\beta }}_j)$, with the variance obtained from the appropriate inverse full Fisher information matrix; it has the same limiting ${\chi }_1^2$ null law. The score test fits only the restricted model. If $\lambda$ collects its nuisance parameters, its efficient information is $I_{jj\cdot \lambda }=I_{jj}-I_{j\lambda }{I}_{\lambda \lambda }^{-1}{I}_{\lambda j}$, and the statistic is $U_j(\tilde{\vartheta }{)}^2/I_{jj\cdot \lambda }(\tilde{\vartheta })$, again asymptotically ${\chi }_1^2$. Accounting for the nuisance block is necessary rather than using the unadjusted diagonal information.

For ordinary homoscedastic fixed-design nonparametric regression, take observations $Y_i=g(x_i)+{ϵ}_i$ at known distinct sites, with independent mean-zero errors of common variance. A degree-$p$ local polynomial estimator at a target $u$ solves the finite-dimensional weighted least squares problemHere $K$ is a nonnegative regression kernel, $h>0$ is the smoothing bandwidth, and sufficient local design rank ensures uniqueness. The estimate of the regression function is $\hat{g}(u)={\hat{\beta }}_0(u)$. Each target has its own locally fitted polynomial; this is not one global polynomial fit. The bandwidth controls the localization and the bias-variance tradeoff.

A natural cubic smoothing spline instead solves the global penalized problemThe Sobolev space here consists of functions with square-integrable weak derivatives through order two. The minimizer is a natural cubic spline with knots at the observation sites and linear tails outside the extreme sites. Equivalently one may perform the minimization on that finite-dimensional spline space: the minimum roughness property of the natural cubic spline interpolant shows that replacing a candidate by the natural spline with the same site values preserves the data loss and decreases the second-derivative penalty. The smoothing parameter $\lambda$ controls curvature; it differs from the local bandwidth $h$.

For the interval-average observation problem, the printed upper limit in the loss is an indexing error: there are only $n-1$ responses and intervals, so a term with $i=n$ would involve the undefined $Y_n$ and $x_{n+1}$. The coherent loss uses the upper limit $n-1$. Put ${\Delta }_i=x_{i+1}-x_i$ and define the linear observation functionalsThe intended criterion on $C^1[a,b]$ is thereforeUnlike the ordinary cubic smoothing spline, this uses interval averages as data and a first-derivative penalty; the resulting spline degree will be two.

Fix any candidate $v\in C^1[a,b]$. By the supplied interpolation property, there is a unique quadratic spline $q\in C^1[a,b]$ with knots $x_1,\dots ,x_n$, constant on both exterior intervals, such that $L_{i}q=L_{i}v$ for every $i$. Put $d=v-q$. ThenOn each interior knot interval $q^{\prime \prime }$ is a constant $c_i$, and on the two exterior intervals $q^{\prime }=0$. Integrate $q^{\prime }{d}^{\prime }$ by parts separately on the intervals. At every knot the boundary contributions cancel because both $q^{\prime }$ and $d$ are continuous; the contributions at $a$ and $b$ vanish because $q^{\prime }$ is zero there. ThusThis orthogonality is the key to the quadratic smoothing spline for interval averages. Expanding the roughness penalty givesThe data-fit terms are equal because the interval averages are equal. Consequently,Equality can occur only if $d^{\prime }=0$ everywhere: the derivative is continuous and its squared integral is zero. Then $d$ is constant, and any of its zero interval averages forces that constant to be zero. Therefore every minimizer must equal its matching quadratic spline $q$.

We conclude that the minimizer is a continuously differentiable quadratic spline with knots at the specified sites and constant tails on both exterior intervals. Existence and uniqueness also follow from the supplied interpolation property. The interval-average map identifies the finite-dimensional space of these quadratic splines with ${\mathbb{R}}^{n-1}$, and the criterion becomes a strictly convex quadratic function of that average vector: its data-fit term is $\Vert Y-y{\Vert }^2$, while its penalty is nonnegative and quadratic. It has a unique minimizer, and the strict equality argument above proves uniqueness in the full function class.

In adaptive experimental quantum control, parametrize a reproducible pulse by a finite real vector $c$, for example $f_c(t)={\sum }_{j=1}^p{c}_j{B}_j(t)$ in a calibrated waveform basis. Prepare the same initial state for each trial, apply that pulse without feedback during the run, and estimate the objective $J(c)$ from repeated measurements. No accurate dynamical model is required if the laboratory can evaluate the objective.

One concrete algorithm is experimental finite-difference gradient ascent. For every parameter, run trials at $c\pm \delta e_j$ and estimateUpdate $c$ to an admissible projection of $c+\eta g$, test the new pulse, and reduce the step size if performance deteriorates. Averaging many shots controls statistical noise; choose $\delta$ large enough for the difference to exceed the measurement uncertainty but small enough to approximate a derivative. Repeat until no statistically significant improvement remains. Multiple starting pulses help explore distinct basins, but this does not guarantee a global optimum. Resource penalties and hardware restrictions belong in the objective or the admissible parameter set.

The feedback is across experimental runs: a measured objective chooses the next pulse. Each individual trial is open-loop control. This distinction matters because the algorithm does not require an instantaneous nondestructive state measurement on the system being controlled.

The feedback law printed in the PDF is not sufficient for the asserted monotonicity. We first derive the exact identity, then give a counterexample and the corrected law. Both density operators evolve unitarily, so $\mathrm{Tr}{\rho }^2$ and $\mathrm{Tr}{\rho }_d^2$ are constant. ThusCyclicity of the trace gives $\mathrm{Tr}([-i{H}_0,{\rho }_d]\rho )=-\mathrm{Tr}({\rho }_d[-i{H}_0,\rho ])$. These drift terms cancel in the derivative of the overlap, leavingHere $C$ is real: $-i[H_1,\rho ]$ is Hermitian, and the trace of the product of two Hermitian matrices is real. This is the Hilbert-Schmidt Lyapunov feedback identity.

For a direct counterexample to the source's overlap law, take $H_0=0$, $H_1={\sigma }_y/2$, ${\rho }_d=(I+{\sigma }_z)/2$ and $\rho =(I+{\sigma }_x)/2$. All are admissible qubit operators and states. The printed overlap is $f=\mathrm{Tr}({\rho }_d\rho )=1/2$, but $-i[H_1,\rho ]=-{\sigma }_z/2$ gives $C=-1/2$. Hence $\dot{V}=1/2>0$: the distance initially increases. This proves that feedback based on state overlap need not be stabilizing.

The intended Lyapunov quantum control replaces the overlap by its control-direction derivative:The conclusion is nonincrease, not strict decrease or automatic convergence. For example, opposite computational-basis pure states, diagonal $H_0$, and $H_1={\sigma }_x$ give $C=0$ and zero control even though $V=2$. Target convergence requires further invariant-set and reachability conditions. A target with a different spectrum is also inaccessible under Quantum Hamiltonian evolution. The density-operator derivatives in the governing equations use the dots visible in the PDF, which the TeX loses.