Papers

We study online estimation of spiked covariance eigenvalues from observations distributed across $L$ nodes with heterogeneous and persistent effective sample sizes. In the proportional high-dimensional regime, local Rayleigh statistics are deterministically distorted by node-specific aspect ratios $c_{\ell,t}=p/N^{\mathrm{eff}}_{\ell,t}$, and direct aggregation of uncorrected statistics converges to the wrong limit. We propose a correct-then-aggregate framework in which each node removes its deterministic bias via an inverse Rayleigh transfer map, and the server fuses corrected estimates using adaptive soft-max weights based on predictable fluctuation metrics, transmitting only $O(k)$ scalars per active node per round. We establish consistency and asymptotic normality of the global estimator, enabling valid online inference, and derive non-asymptotic bounds quantifying how accuracy improves with the number of nodes and their effective sample sizes. The adaptive weights achieve variance reduction comparable to oracle inverse-variance weighting, confirming the data-driven construction is nearly efficient. Simulation studies validate these properties. An application to cross-venue monitoring of a dominant market factor shows the method tracks systemic risk in real time while substantially reducing communication cost relative to a centralized pooled approach.

We show that the ECH capacities of every closed symplectic 4-manifold are infinite. We also give the first examples of symplectic 4-manifolds whose alternative ECH capacities are infinite. We verify the ECH Weyl law holds with bounded subleading asymptotics for the alternative ECH capacities of any closed symplectic 4-manifold with $b_2^+=1$ or any smooth domain therewithin.

Let $(\mathcal{X},d,\mu)$ be a doubling metric measure space, $X$ a ball quasi-Banach function space on $\mathcal{X}$, and $L$ a non-negative self-adjoint operator on $L^2(\mathcal{X})$ whose heat kernel satisfies a Gaussian upper bound. In this article, we study the local Hardy space $h_{X,L}(\mathcal{X})$ associated with both $X$ and $L$. We first establish the atomic and molecular characterizations of $h_{X,L}(\mathcal{X})$. As applications of these characterizations, we obtain the relations between $h_{X,L}(\mathcal{X})$ and the global Hardy spaces $H_{X,L}(\mathcal{X})$ and $H_{X,L+mI}(\mathcal{X})$. We also establish the radial and non-tangential maximal function characterizations of $h_{X,L}(\mathcal{X})$. Using these maximal function characterizations, under the additional assumptions that the heat kernel satisfies the conservation property and a Hölder regularity estimate, we further show that $h_{X,L}(\mathcal{X})$ coincides with the local atomic Hardy space $h_{X,\mathrm{at}}^p(\mathcal{X})$ with equivalent quasi-norms in the corresponding range of indices. Finally, we apply the above results to Lebesgue spaces, Orlicz spaces, weighted Lebesgue spaces, and variable Lebesgue spaces.

We prove that the Ginzburg-Landau equation correctly predicts the dynamics of quasilinear or fully nonlinear pattern-forming systems close to the first instability. We present an approximation theory in uniform local Sobolev spaces which is applied to a quasilinear Swift-Hohenberg model, to a quasilinear version of Bénard's problem with velocity dependent viscosity, and to abstract quasilinear reaction-diffusion-advection systems.

Trace theorems are an indispensable tool for the analysis of kinetic equations. They have been established in wide generality by Cessenat and co-workers for radiative transfer and related applications. A variety of trace estimates have been established for the kinetic Fokker-Planck and Kolmogorov equation, typically requiring smoothness of the underlying domain; see the recent survey by Niebel & Valentini. In this work, we prove a new trace estimate for kinetic Sobolev spaces over the sphere for $\rho$-convex domains which, in general, may have a non-smooth boundary. Similar to the work of Cessenat, we use characteristics to obtain trace estimates in weighted trace spaces with explicit constants. For completeness, we also present a density result for the corresponding function spaces on Lipschitz domains.

A set S of vertices of a graph is odd independent if it is independent and every vertex outside S has either zero or an odd number of neighbors in S. The largest size of such a set is the odd independence number alpha_od. Caro, Petrusevski, Skrekovski and Tuza [2] conjectured that alpha_od = 1 for every finite Queen graph. They also asked whether the infinite Queen graph has alpha_od = 1 or alpha_od = infinity. We prove that alpha_od = 1 in both cases. In particular, in the case of an infinite board we prove that alpha_od = 1 holds on the quarter plane and on the whole plane.

This paper concerns the null control of quasi-linear parabolic systems where the diffusion coefficient depends on the gradient of the state variable. In our main theoretical result, with some assumptions on the regularity and growth of the diffusion coefficient and regular initial data, we prove that local null controllability holds. To this purpose, we consider the null controllability problem for the linearized system, we deduce new estimates on the control and the state and, then, we apply a Local Inversion Theorem. We also formulate an iterative algorithm of the quasi-Newton kind for the computation of a null control and an associated state. We apply this method to some numerical approximations of the problem and illustrate the results with several experiments.

Let $X$ be a vector field on an open set of $\mathbb{R}^n$ with $X(p)=0$ and Jacobian $J=DX(p)$ of rank $n-1$ having $0$ as an eigenvalue of algebraic multiplicity two. Let $q_0$ span $\ker J$ and let $e_k(A)$ denote the sum of the principal $k\times k$ minors of $A$. Under the usual hyperbolicity assumption on the transverse block, we prove that the quadratic coefficients $a,b$ of the Bogdanov–Takens normal form on the centre manifold are $a=-\frac{1}{2}\frac{D_{q_0}e_n}{e_{n-2}}$ and $b=\frac{D_{q_0}e_{n-1}}{e_{n-2}}-\frac{e_{n-3}D_{q_0}e_n}{e_{n-2}^2}$. The underlying spectral identity requires only invertibility of the transverse block and remains valid without hyperbolicity. Both coefficients arise as the lowest-order terms of a generating identity for the first-order spectral jet of $DX$ along $\ker J$; all higher coefficients depend on the transverse block. We prove that this dichotomy is sharp. The planar formulas $a=-\frac{1}{2}D_{q_0}\det$ and $b=D_{q_0}\operatorname{tr}$ are recovered when $n=2$. We also obtain a coordinate-free nondegeneracy test and a geometric interpretation in terms of the transversality of the kernel line to two invariant hypersurfaces. A self-contained Wolfram Language notebook accompanies the paper and verifies the results symbolically.

Occupancy-targeted curb pricing, including San Francisco's SFpark pilot, often relies on local threshold rules that do not account for spatial substitution or uncertainty in demand response. We estimate parking-price elasticities from disaggregated SFpark data using a hierarchical Bayesian demand model and use the resulting posterior distribution in a spatially coupled constrained pricing model. Partial pooling across districts gives a posterior mean elasticity of -0.241 with a 95% credible interval of [-0.325, -0.176]. The pricing problem is formulated as a constrained nonlinear program with spatial coupling, temporal price-change limits, neighbor price gaps, and a soft revenue floor, and is solved with Ipopt. Expected-loss and CVaR objectives are evaluated over joint posterior draws rather than a fixed elasticity. On held-out posterior scenarios, the posterior-based CVaR policy has a lower objective than the SFpark threshold rule and historical tariff with posterior probability 1.00, and than a literature-calibrated robust policy with probability 0.85. Epsilon-constraint frontiers show that posterior calibration changes the efficient pricing set. Results from 60 district-windows across Fillmore, Mission, and Marina also report the associated tradeoffs in occupancy-band performance, revenue, and spatial disparity.

Let $\mathcal V\subseteq M_6(\mathbb C)$ be a $17$-dimensional linear subspace such that $ \operatorname{rank}[S,T]\leq2 \quad(S,T\in\mathcal V). $ We prove that $\mathcal V$, or its transpose, is conjugate to the algebra $ \left\{ \begin{pmatrix} A&B&C\\ 0&\lambda I_2&D\\ 0&0&\lambda I_2 \end{pmatrix}: A,B,C,D\in M_2(\mathbb C),\ \lambda\in\mathbb C \right\}. $ Consequently, the corresponding closed algebraic locus in $\operatorname{Gr}(17,M_6(\mathbb C))$ is the disjoint union of two nonsingular irreducible components, each isomorphic to $\operatorname{Fl}(2,4;6)$. We also prove that the Zariski tangent space at $\mathcal A$ of the corresponding closed algebraic locus is equal to the tangent space to the conjugacy orbit of $\mathcal A$.

This paper studies the problem of robust stabilization of linear input-state systems in discrete time. We prove that for any compact set of stabilizable systems, there exists a dynamic state-feedback controller that globally asymptotically stabilizes all systems in the set. In addition, we show that for some compact sets of stabilizable systems, no nonlinear static or linear dynamic state-feedback law can achieve this task. This proves that, in general, robust stabilization requires a feedback law that is both nonlinear and dynamic. We extend our study to robust exponential stabilization with a given rate of decay. Finally, for polytopic sets of systems, we introduce an algorithm for the design of robust feedback laws.

We identify a previously unnoticed gradient-flow structure of the Fokker–Planck equation arising in the spontaneous particle aggregation model. For an exponential response function, the equation is a generalized Wasserstein gradient flow of the attractive McKean–Vlasov free energy with a nonlocal mobility. We show that, within a natural class of local entropies and symmetric interaction energies, this structure essentially singles out the exponential response. We discuss consequences for stationary states, global minimizers, the single-cluster structure in one dimension, and formally outline convergence to equilibrium using a Wasserstein–\mboxŁojasiewicz inequality.

Radial clipping does two things at once: it cuts off the part of a vector outside a ball and leaves a bounded vector inside the ball. The part cut off defines the residual. For a random vector, we study the magnitude of the bias in the mean and the energy, defined as the centered second moment of the clipped vector. Controlling these effects is important in stochastic optimization with heavy tails, where the choice of the clipping threshold affects convergence bounds for algorithms. In the deterministic problem, for a fixed threshold $\tau>0$, $1\le p\le2$, and nonnegative coefficients $\alpha,\beta$, we find the smallest constant that simultaneously bounds the residual and energy in terms of $\lVert x\rVert^p$. This same constant remains sharp in the stochastic counterpart for random vectors in a Hilbert space. We also find the exact supremum of the corresponding linear combination of normalized bias and energy for arbitrary $\alpha,\beta\in\mathbb R$. We then give a geometric picture in the energy-bias plane. Varying the weights, we describe the smallest closed convex region containing all normalized pairs and the attainable set itself, identifying unattainable boundary values. Every attainable pair at a fixed threshold can be realized by a real-valued random variable taking at most two values. We also consider all thresholds simultaneously. For any integrable random vector in a Hilbert space, we construct a single $\mathbb R^3$-valued random vector preserving its norm distribution, bias magnitude, and energy simultaneously at every threshold $\tau>0$. Finally, we apply the joint bound to risk-adjusted reinsurance costs for a prescribed retention threshold and to the logarithmic shortfall of a clipped portfolio strategy relative to any constant-fraction strategy. AI was used in preparing this paper (see Section 8).

This paper relates the minimal Harder–Narasimhan slope to the surjectivity of real-linear Cauchy–Riemann operators. We establish a conformally invariant $L^2$ criterion and an asymptotic slope criterion, which yield higher-rank automatic transversality criteria for pseudoholomorphic curves beyond the classical rank-one framework. Applications to pseudoholomorphic spheres in $S^6$ provide quantitative $L^2$ obstructions to the integrability of almost complex structures.

The strong asymptotic freeness established in [28,53] allows for the study of norms of Gaussian matrix polynomials via corresponding free operator counterparts. For spectral edges of semicircular free counterparts to symmetric Kronecker-Gaussian matrices, Lehner in [39] determined closed-form analytical characterizations. As an alternative to classical spectral methods, in [62], we created a Random Duality Theory (RDT) based framework for studying these problems and reproved Lehner's formula for definite matrix coefficients. In this work, we develop the RDT machinery further and achieve strong progress in several key directions: (i) Symmetric Square Case: We consider indefinite matrix coefficients and provide spectral edges lower bounds that match Lehner's formula. (ii) Asymmetric Non-Square Variants: We establish RDT asymmetric analogues to Lehner formulas and prove that they lower-bound the spectral edges. (iii) Interlaced Decoupling Property: We uncover that the obtained analogues exhibit a remarkable interlaced decoupling property. Additionally, we consider semi-definite programming (SDP) formulations that allow for the practical solution of the obtained analytical characterizations. Theoretical predictions obtained through solving SDPs are compared to numerical simulations, showing a striking agreement even for problem dimensions on the order of a few hundreds.

The projective spectrum of a tuple $A=(A_0,A_1,\dots,A_n)$ with elements in a unital Banach algebra $\mathcal{A}$ is the collection of $[z_0:z_1:\cdots:z_n]\in \mathbb{P}^n$ such that $z_0A_0+z_1A_1+\dots+z_nA_n$ is not invertible in $\mathcal{A}$. For the tuple $A_{\rho}=(\rho({\bf e}),\rho({\bf a}),\rho({\bf b}))$, where ${\bf e},{\bf a},{\bf b}$ are the three states of the automaton generating the Basilica group $\mathcal{B}$ and $\rho$ is the Koopman representation, a dynamical map $F$ preserving the projective spectrum of $A_{\rho}$ is established. Further, a substantial common subset of the Julia set of $F$ and the projective spectrum of $A_{\rho}$, as well as a large common subset of the Fatou set of $F$ and the projective resolvent set of $A_{\rho}$ is identified. The result provides partial support for the conjecture of equality of the Julia set of $F$ and the projective spectrum of $A_{\rho}$.

We say that a matrix over a finite field $\mathbb{F}_q$ is positive definite if it is symmetric and each of its leading principal minors is a nonzero square in $\mathbb{F}_q$. In previous work of the authors [J. Algebra, 2025], the entrywise positivity preservers on $M_n(\mathbb{F}_q)$ were classified in every case except when $n=2$, $q\equiv1\pmod4$, and $q$ is not a square. We settle this remaining case, thereby completing the classification of entrywise positivity preservers over every finite field and in every dimension $n\ge2$. Our proof is based on a novel idempotent reduction that not only resolves the remaining case but also yields a self-contained proof of the complete classification, while avoiding several technical results used in the earlier arguments. As a further application of the same reduction, we classify the entrywise preservers of strongly nonsingular matrices, i.e., matrices whose leading principal minors are all nonzero. We also prove a more general theorem in odd characteristic: for every prescribed sign pattern of nonzero leading principal minors of matrices of a fixed dimension $n\ge2$, the entrywise preservers are precisely the positive scalar multiples of field automorphisms. Thus, in odd characteristic, preserving any nonzero leading-principal-minor sign pattern surprisingly forces the preservation of every such sign pattern.

In this paper, we study an optimal control problem related to an ODE model of glioblastoma growth influenced by the oxygen. The model considers a couple of controls describing the chemotherapy and antiangiogenic therapies. The cost functional aims to reduce the tumor growth, bring the oxygen concentration close to a desired value, and penalize the use of therapies. We solve the optimal control problem, proving the existence of a global optimal control and deriving first-order necessary optimality conditions through the Pontryagin Minimum Principle. For the numerical approximation, we use Physics-Informed Neural Networks (PINNs) to solve the state and adjoint systems, together with a gradient descent method with Armijo line search for the controls. To address the strategy of PINNs we consider the methodology proposed in [15], making a decomposition of the time domain into several subintervals, using different neural networks in each subinterval and enforcing continuity conditions between successive time subintervals. This strategy, called sequential PINN formulations in time, including soft and hard-constrained versions, is compared with the corresponding approximation results of classical solvers and traditional PINNs counterparts.

Norin and Turcotte's asymptotically sharp bound for graph burning [J. Combin. Theory Ser. B 168 (2024), 208–235] led them to an exact random-cover conjecture for finite metric trees. Let $U[0,r]$ be the uniform probability measure on $[0,r]$. They conjectured that every finite metric tree $T$ of length $L\ge2r$ admits a probability measure on $0$-good ball covers whose expected radius measure is at most $(L/r)U[0,r]$. We prove the conjecture for every finite metric tree. We recast the bootstrapping calculation of Norin and Turcotte as a zero-error replacement certificate. The resulting local scale reduction, together with a three-piece decomposition and a macro-recursion, produces a fractional marked-ball cover with the exact radius budget. We then pass from the fractional cover to random finite covers by a compact rounding argument. For metric-tree balls, Tamir's balancedness theorem and standard balanced-matrix ideality provide the finite-dimensional integrality input. We also prove an arbitrary-budget duality criterion. If $0<R\le L$ and $\beta$ is a finite positive Borel measure on $[0,R]$, then $\beta$ dominates the expected radius measure of a random $0$-good cover if and only if $\sigma(T)\le\int_{[0,R]}\max_{v\in T}\sigma(B_T(v,s))\,d\beta(s)$ for every finite positive Borel measure $\sigma$ on $T$; it is enough to test finite atomic measures. We use this criterion to extend the uniform range to every $r\le L-\operatorname{diam}(T)/2$, determine the exact range for equal-arm metric stars, and derive deterministic bounds, interval rigidity, and a diameter-defect stability estimate.

Recently, we showed that continuous definable maps defined on compact definable sets can be uniformly approximated by continuous definable maps of class $\mathcal{C}^p$ without changing their image. The aim of this paper is to extend the previous result, this time taking into account the (strong) Whitney topology, to continuous locally definable maps defined on locally compact locally definable sets. The argument is an interplay between o-minimal and PL geometry and makes essential use of Pawłucki's desingularization techniques as well as our aforementioned result for the compact case.

We introduce the universality property of a definable $\sigma$-ideal on a Polish space, which, on one hand, can serve as a benchmark for the properness of the associated idealized forcing of positive Borel sets ordered by inclusion, and, on the other hand, unifies many of the results that can be found in Zapletal's book. We show that under mild absoluteness assumptions, it implies properness, various dichotomy theorems, and closure under well-ordered unions in the Solovay model and under $\mathsf{AD}^+$, among other things. All major classes of proper idealized forcings studied in the book have this property. Further, we use this viewpoint to answer a question of Khomskii by showing that the naive idealized forcing for adding an eventually different real or a refining real is not proper below some condition. We also answer a question related to the definability of $\sigma$-ideals generated by Borel sets due to Kanovei, Sabok, and Zapletal.

For a graph $G$, let $h(G)$ be the minimum cardinality of a vertex set meeting every maximum independent set of $G$. We establish two complementary reduction principles for the Bollobás–Erdős–Tuza conjecture: the conjecture for arbitrary graphs is equivalent to its restriction to regular graphs of any fixed positive linear degree, and, within every hereditary graph class, a uniform sublinear bound is equivalent to a sublinear bound on graphs of every fixed positive linear vertex connectivity. We prove the sharp general estimate \[ h(G)\le \left\lfloor\frac{|V(G)|}{2\alpha(G)+\delta(G)-|V(G)|}\right\rfloor \] whenever the denominator is positive, with equality for balanced complete multipartite graphs. Consequently, every $3$-colorable graph of order $n$ with $\kappa(G)\ge\rho n$ and $\rho>1/3$ has a hitting set of size at most $\lfloor(\rho-1/3)^{-1}\rfloor$; direct use of a $3$-coloring improves this to $6$ when $\kappa(G)>4n/9$ and to the sharp bound $3$ when $\kappa(G)>n/2$. For dense regular graphs with independence ratio greater than $1/4$, we obtain a logarithmic bound, while constructions with linear degree and linear independence number show that $h(G)=\Omega(\sqrt n)$ can still occur. We also prove a logarithmic bound for near-regular $3$-colorable graphs and exhibit a critical family at connectivity $n/3$ that explains the limitations of the degree-surplus and degree-ratio methods.

We study the long range behavior of solutions to $i\partial_tu=H_\alpha u+\lambda|u|u$ on $\mathbb R^2$, where $H_\alpha$ is the Friedrichs realization of the Aharonov-Bohm Hamiltonian with a single pole. The logarithmic phase of the long range ansatz may push a profile out of the domain of $H_{\alpha}$. We characterize profiles that stay in the operator domain by the vanishing of boundary traces at 0 of order $\le \frac 12$; at half flux $\alpha=\frac 12$, no nonzero trace survives. However, every profile in the full domain of $H_{\alpha}$ with small $L^\infty$ amplitude determines a unique global solution with a modified final state, with a remainder rate $t^{-b}$ for all $0<b<1/2+\nu_\alpha$, $\nu_\alpha=\min\{\alpha,1-\alpha\}$. For profiles satisfying the vanishing trace condition, the rate improves to every $0<b<1$. This result is sharp in the sense that, if $\alpha\neq \frac 12$, we can construct profiles with an error of size $t^{-1/2-\nu_\alpha}\log t$, ruling out all faster rates. The upper bound comes from a retarded Strichartz estimate for a residual that is not in $L^2$; the lower bound is an explicit calculation via Hankel transforms. For smoother profiles we also compute the first correction, which gives remainder rates with $1<b<2$.

We develop additive-energy estimates and weighted Young inequalities for finite sets on bounded-degree real algebraic varieties. For an irreducible $m$-dimensional variety $V$, let $\sigma(V)=2m-\dim\overline{V-V}^{\mathrm{Zar}}$ and $\alpha(V)=\max{2,1+\frac{2\sigma(V)}{m}}$. For every $a\in[\alpha(V),3)$ we define a finite-degree translation-partition flag parameter $\Lambda_{a,R}(X;V)$ and prove $E(X)\ll \Lambda_{a,R}(X;V)^{3-a}|X|^{a+\varepsilon}$. This recovers the line-concentration theorem of Jing and Wu for algebraic surfaces in $\mathbb{R}^3$. For codimension-two quadratic threefolds ${(u,Q_1(u),Q_2(u))\in\mathbb{R}^3}\subset\mathbb{R}^5$ with positive-definite $Q_1$ and simple generalized spectrum, we prove the sharp estimate $E(X)\ll_{\varepsilon}|X|^{2+\varepsilon}$ without a flag loss. Hereditary versions of these estimates imply weighted $L^4$ restriction bounds and off-diagonal Young inequalities; at the near-diagonal threshold the sharp region is $1\le p,q\le 2$ and $p^{-1}+q^{-1}\ge 1$. We also prove a sharp turning-complexity extension of the Cushman-Demeter-Wu theorem: $J_3(P)\ll_{\varepsilon}\kappa(P)^2|P|^{3+\varepsilon}$, with matching examples at every power scale.

Erdős Problem 1154 asks whether every number in $[0,1]$ occurs as the Hausdorff dimension of a subring or subfield of $\mathbb{R}$. We prove that a subring of $\mathbb{R}$ closed under binary interleaving has Hausdorff dimension either zero or one. Consequently, every Turing ideal and its associated subfield satisfy the same dichotomy.

On this article we provide a relation between two inherently different semantic structures for intuitionistic modal logic. We start by recalling the Heyting Algebras, then defining the language and the axioms for the iS4h intuitionistic calculus. We then proceed to analyse two structures discussed on [4] and their relation. Finally, we discuss about a structure defined on [3, Chapter 6], establishing properties in order for it to be sound with iS4h, and finally conclude that it is semantically equivalent to the other two structures defined by [4].

In this paper we give a complete characterization of the supporting functionals at any point on the unit sphere of Orlicz-Lorentz spaces $\Lambda_{\varphi, \omega}$. Departing from traditional approaches, we establish our results without assuming that the Orlicz function $\varphi$ is an N–function. These results provide a basis for studying the extremal structures of Orlicz-Lorentz spaces.

Given an oriented graph $\vec{G}$ and a subset of vertices $X \subseteq V(\vec{G})$, the inversion of $X$ is the operation that reverses the orientation of every arc with both endpoints in $X$. For a simple graph $G$, the inversion diameter $\operatorname{diam}(I(G))$ is the maximum distance between two orientations of $G$ under inversions of vertex sets. We prove the sharp bound \[ \operatorname{diam}(I(G))\le 2\chi_a(G)-2, \] where $\chi_a(G)$ is the acyclic chromatic number. Consequently, every planar graph has inversion diameter at most $8$, improving the previously known bound $12$. Using strong-degeneracy arguments, we also obtain upper bounds $7$, $5$, and $4$ for planar graphs of girth at least $4$, $5$, and $6$, respectively.

The real classical shadows protocol of West et al. replaces the unitary (Clifford) ensemble of the Huang–Kueng–Preskill scheme by the orthogonal (real Clifford) ensemble, and for symmetric observables achieves strictly smaller estimator variances: a factor approaching two for global evolution and an exponential factor $(3/2)^k$ for $k$-local real Pauli observables. Real hardware, however, never implements the ideal evolution. Building on the noisy classical shadows framework of Koh and Grewal, we give a complete theory of the real classical shadows protocol in the presence of a known completely positive trace-preserving noise channel acting after the orthogonal evolution. We derive the noisy global and local orthogonal shadow channels from first principles using the Weingarten calculus of the orthogonal group, prove that each is a depolarizing channel acting on the symmetric (respectively locally symmetric) component of its input, and derive from it the exact single-shot variance in closed form, together with the associated shadow seminorm, two-sided bounds on it, and the resulting sample-complexity guarantees. We prove that the noiseless sample-complexity advantages survive intact under noise. Because the variances are exact rather than bounded, the ratio is controlled by a single dimensionless parameter, which gives a closed-form criterion for when the factor of two is attainable: both the second-moment and the variance ratio reach it exactly when the observable's norm profile grows, and the noise enters that limit only through a factor lying within $2/(d+2)$ of two, so the advantage is uniform in the noise. For rank-one targets it is provably unattainable, saturating strictly below two. The local real-Pauli advantage remains $(3/2)^k$. We treat complex measurement bases through a reality parameter and a transposed-noise scalar, recovering unitary shadows in the appropriate limit.

The goal of this article is to present a computation of the Galois cohomology of the group $G = \mathrm{PGL}_2$ over rings of Laurent polynomials. This computation recovers the main result of \cite{CGP} in this case by a new method, based on the analysis of actions on appropriate geometric objects (products of trees and, in general, of affine buildings), which was already used in \cite{ARR} to give a new proof of the theorem of Raghunathan-Ramanathan \cite{RR} concerning Galois cohomology over polynomial rings. Going beyond Galois cohomology, this method also enables one to determine the finite subgroups in the group of points over relevant polynomial rings. In view of these and other potential applications of the method in different situations (in particular, in the study of algebraic groups over the coordinate rings of general affine curves), we have attempted to make our exposition largely self-contained and accessible to broad mathematical audience.