Papers

The derivative nonlinear Schrödinger equation is a fundamental model for the propagation of nonlinear dispersive waves in, for example, plasma physics and nonlinear optics. In this work, we consider this model on the one-dimensional torus and study a filtered explicit Fourier integrator for the corresponding periodic problem. After applying a periodic gauge transformation, we consider a frequency-truncated model and its filtered exponential-Euler discretization. The main difficulty comes from the derivative cubic nonlinearity in the periodic setting, since local smoothing is unavailable and resonant interactions are stronger than in the non-periodic case. To address this issue, we develop a discrete Bourgain-space framework adapted to the gauge-transformed equation. For initial data $u_0 \in H^s(\mathbb{T})$ with $1/2 < s \le 5/2$, we prove that the numerical error is of order $\mathcal{O}(\tau^{s/2-1/4})$ in $H^{1/2}(\mathbb{T})$, where $\tau$ denotes the employed time step size. Numerical experiments confirm the predicted convergence behavior and demonstrate the effectiveness of the filtered scheme for rough solutions.

Exact renormalisation techniques are important and powerful, particularly for inflation-generated systems. We review recent results in this direction. We recall the necessary notions for inflation systems and show the renormalisation principle, which allows us to obtain exact values of highly erratic functions, such as window covariograms. We apply these techniques to compute the diffraction pattern of the new monotile tilings with arbitrary precision. We also recall a recent invariant for system with pure-point spectrum, the orbit separation dimension, and its relation to renormalisation. Lastly, we recall results beyond the pure-point spectrum setting and show how renormalisation and Lyapunov exponents can be used to exclude the presence of absolutely continuous part of the spectra.

We study optimal control of a system with multiple decision makers who share a common hidden state and receive fully decentralized observations through identical channels. The dynamics of the hidden state and the cost incurred by the agents depend on the agents' actions only through their empirical distribution. In the limit problem with infinitely many agents, the problem reduces to a single agent control problem where the agent affects the hidden state dynamics via the conditional law of the actions given the past values of the hidden state process. We formulate this problem as a deterministic measure valued control problem over the space of policies and provide a dynamic programming recursion. We first show that for the limiting problem randomization over the control actions is necessary for optimality. However, randomization over the selection of policies (i.e., mixture policies) is not required. We then show that the optimal symmetric policies designed for the infinite population problem are near optimal for the finite population problem. In particular, we establish convergence rates that decay with number of agents as $\frac{1}{\sqrt{N}}$, and grow exponentially with the memory length used in the policy.

For the Zakharov system in four space dimensions, we prove that all solutions inside the potential well of the ground states are global and scattering in the energy space, with no other restriction such as symmetry. The proof has already been reduced by [3] to ruling out the existence of a minimal non-scattering solution that is precompact along some trajectory. This paper carries out the final step in the proof, namely we exclude the possibility of precompact solutions inside the potential well by combining two distinct arguments depending on the motion of trajectory.

We extend the optimal transport theory for signed measures supported on Ahlfors-regular fractal sets (Bwo'Nyahre et al., 2026) to allow a controlled dimensional distortion between source and target. A penalization term $\varepsilon \Phi(d_s(x) - d_t(y))$ – where $\Phi$ is a fixed smooth strictly convex function and $d_s, d_t$ are the local Hausdorff dimensions of the fractal supports – is added to the transport cost on inter-sign regions, with~$\varepsilon \ge 0$ controlling the tolerance for distortion. Under hypotheses H1–H7, we prove: the existence and uniqueness of an optimal transport map~$T^{\varepsilon}$ for every~$\varepsilon > 0$; coupled Monge–Ampère equations with a distortion correction term, generalizing the classical Brenier–Caffarelli equation; a double Legendre–Fenchel characterization of the optimal potentials, giving a complete variational description of the transport in each of the four sign regimes. The double Legendre–Fenchel system (Theorem~4.2) is the central contribution: it shows that the optimal potentials are the unique fixed points of a system of conjugacy equations, one per transport regime, and it provides the foundation for numerical algorithms and asymptotic analysis.

We study the space of one-point functions on the torus for a possibly nonrational $C_2$-cofinite vertex operator algebra $V$ by relating it to a trace object of the subcategory of projective objects in the representation category of $V$. We identify the dual of the trace space with symmetric functions on the endomorphism algebra $E$ of a projective generator. Motivated by the Gainutdinov-Runkel conjecture, recently established using different methods by Gui and Zhang, we present a complementary representation-theoretic approach based on Arike-Nagatomo pseudo-traces. In this framework, we prove surjectivity of the Gainutdinov-Runkel map from symmetric functions on $E$ to one-point functions. Under the additional assumption of separated conformal weights modulo $\mathbb{Z}$, we also prove injectivity, using projective-cover techniques inspired by Huang.

We study the Monge–Kantorovich optimal transport problem between two signed measures~$\mu$ and~$\nu$ on convex compact subsets of~$\mathbb{R}^d$, with a positional penalization function~$\lambda(x, y)$ that modulates the cost of inter-sign transport. Using four independent positive measures~$(\pi^{++}, \pi^{+-}, \pi^{-+}, \pi^{--})$ as decision variables, we prove that the admissible set~$\mathcal{A}(\mu, \nu)$ is weakly-$*$ compact and non-empty if and only if $\mu^+(X) = \nu^+(Y)$ and~$\mu^-(X) = \nu^-(Y)$. Strong duality is established via the Kantorovich minimax theorem, yielding a new compatibility condition on~$\lambda$ at the intersection of inter-sign supports. The penalization~$\lambda$ is shown to be Lipschitz and to admit Alexandrov second derivatives almost everywhere. Modified Monge–Ampère equations governing inter-sign transport maps are derived in the Alexandrov sense, with well-posedness characterized by $\sigma \det(D^2_{yx}\Lambda) e > 0$. The classical Brenier equation is recovered in the limit~$\lambda \to 0$.

We obtain several constructions of uniformly positive scalar curvature complete Riemannian metrics on open manifolds. For dimension $n\geq4$, we show that if such a manifold admits a proper Morse function $f$ bounded below, such that $f$ has no critical points of index $\geq n-2$, then it admits a uniformly positive scalar curvature metric. If a manifold of dimension $4 \leq n\leq 7$ admits a positive scalar curvature metric along with a compact exhaustion $\{U_i\}$ such that the boundary of each $U_i$ is minimal, then it also admits a uniformly positive scalar curvature metric. For the same dimensions, we show that if a manifold has product ends and a positive scalar curvature metric with $C$-quadratic decay at infinity for $C>4\pi^2$ with respect to some basepoint, then the existence of a certain mean convex slice far enough from the basepoint implies the existence of a uniformly positive scalar curvature metric on the manifold. In particular, manifolds which have a combination of these behaviours also admit a uniformly positive scalar curvature metric. We also study some applications of these results.

We introduce a new class of numerical integrators for the time integration of non-autonomous linear ordinary differential equations whose coefficient matrix is sparse and evolves within a quadratic matrix Lie group. In contrast to standard Lie group integrators, the proposed methods avoid the evaluation of matrix exponentials acting on vectors and instead rely on solving a sequence of linear systems with sparse coefficient matrices. Moreover, they are well suited for problems arising from unbounded operators, as they inherently produce bounded solutions. We construct optimised schemes of orders four and six and assess their performance on a representative numerical example, demonstrating clear advantages over existing Lie-group integrators.

We develop and analyze a Bregman-projected mirror iteration for low-order regularizations of stationary mean-field game (MFG) systems in their natural Banach space setting. For separable Hamiltonians of the form $H(x,p,m)=H_0(x,p)-g(m)$, with quadratic or super-quadratic Hamiltonian growth and linear or super-linear density couplings, we formulate a low-order $\bar\gamma$-Laplacian regularization of the stationary MFG system as a variational inequality on $L^{\bar\beta}(\mathbb T^d)\times W^{1,\bar\gamma}(\mathbb T^d)$. To approximate solutions of this regularized variational inequality, we introduce a Bregman geometry matched to the mixed Lebesgue-Sobolev exponents of the problem and analyze the exact constrained one-step mirror iteration and a frozen two-step variant. Under quantitative monotonicity of the coupling, both exact iterations converge strongly to the unique solution of the regularized variational inequality for each fixed $\varepsilon>0$, under natural summability conditions on the step sizes. Along weakly convergent vanishing-regularization sequences, we also prove strong convergence in the full product space. Numerical experiments in one and two dimensions show first-order accuracy on manufactured solutions and weak dependence of outer iteration counts on the tested grids. At matched effective step sizes, the frozen two-step and one-step implementations agree up to inner-solve accuracy, while the latter requires fewer mirror solves.

This paper studies a Grothendieck-type finite-dimensionality problem for closed linear subspaces embedded in Banach spaces. Let $S_p^{(q)} \subset L_p(M,d\mu)$ be a closed linear subspace of the Banach space $L_p(M,d\mu)$ defined with respect to a probability measure $d\mu$ on $M$. We prove that if $S_p^{(q)}$ is continuously embedded into $L_q(M,d\mu)$ for $q>p$, then its dimension $\dim S_p^{(q)} = N \in \mathbb{N}$ satisfies the estimate \[ \frac{1}{N}\left(\frac{\sqrt{\pi},\Gamma!\left(\frac{N+\tilde q}{2}\right)}{\Gamma!\left(\frac{\tilde q+1}{2}\right)\Gamma!\left(\frac{N}{2}\right)}\right)^{2/\tilde q}\le K_{p,q(m)}^2, \] where $1/\tilde q + 1/q = 1$, $q = 2 + (p-2)2^m > p$ with $p \neq 2$ and $m \in \mathbb{N}$, and $K_{p,q(m)}>0$ is a bounded constant. We also prove that certain closed linear subspaces of $L_p(M,d\mu)$ consisting of continuous functions on $M$ must be finite-dimensional.

Let $G$ be the fundamental group of a compact surface, a finitely generated free group, or more generally a finitely generated right-angled Artin group. We prove that the von Neumann dimension function of $\mathrm{Out}(G)$ is valued in a discrete subgroup of $\mathbb Q$. This is accomplished by establishing the Strong Atiyah Conjecture for a torsion-free subgroup of $\mathrm{Out}(G)$ of finite index. We also prove that for every field $\mathbb K$, there exists a torsion-free subgroup $H \leqslant \mathrm{Out}(G)$ of finite index such that $\mathbb K[H]$ embeds into a division ring, and hence satisfies the Zero Divisor Conjecture. These results are obtained by establishing analogous ones for a suitable open subgroup of $\mathrm{Out}(\mathbf G)$ and its completed group algebra, where $\mathbf G$ denotes the pro-$p$ completion of $G$. In an appendix, the first author shows that an automorphism of a free nilpotent group is inner if and only if it induces an inner automorphism of its pro-$p$ completion.

We present a complete classification of concave kite central configurations in the planar 4-body problem with three equal masses. There are two different types of central configurations when the fourth mass lies inside or outside the triangle formed by the other three. Using a rigorous computer-assisted analytical method and a fixed coordinate system, we show that the central configurations in each case form a one-parameter family and obtain a complete classification of these configurations. In addition, we rigorously show the existence and types of the bifurcation points in the reduced space. We also provide two numerical global bifurcation pictures in the entire planar 4-body configuration space as the mass ratio varies from $0$ to $+\infty$, including symmetric and asymmetric concave central configurations with three equal masses.

The preferential attachment model (PAM) describes the sequential growth of a network based on the "rich-get-richer" principle. Several versions of it have become established for modeling, e.g., citation networks, capturing a power-law degree distribution. Directed versions of the preferential attachment model where the edges are directed from the new to the old vertices have been the subject of extensive research. They have been shown to exhibit remarkable properties such as heavier tails for the limiting graph-normalized PageRank than for the in-degrees. By contrast, for the undirected version, we recently showed that PageRank has similar tails as the degree. In the present paper, we discuss the PageRank asymptotics for a multi-type version of the undirected PAM (here vertices have different colors), complementing previous results of Antunes, Bhamidi, Banerjee and Pipiras on the asymptotics of PageRank on similar directed multi-type or colored PAMs. Our studies are motivated by the aim to go beyond the rigid rule of edge orientation in directed preferential attachment models. As the main result, for the case of a finite set of colors, we show that the power-law hypothesis for PageRank is fulfilled also for the colored undirected PAM, where, by contrast to the directed case, the power-law exponent is color-dependent for some choices of the initial color distribution and the attractiveness function. For the specific case of a two-type model, we discuss implications of our results on fairness in sampling underrepresented nodes from the network.

We study quantum metric space structures on operator algebras arising from free probability, namely those associated to $q$-Gaussians and free Gibbs laws for convex potentials. We note that even for free semicirculars, Voiculescu's dual system does not produce a quantum metric space structure that recovers the weak-$*$ topology on the state space. However, for $q$-Gaussians, we can define a compact quantum metric space using length-like functions by the same method as has already been used for hyperbolic groups, quantum groups of rapid decay, free products, and free graph algebras. Next, motivated by the free transport results for free Gibbs laws, we describe a universal way of defining Lip-norms in terms of a generating set, which behaves well under changes of coordinates. We show using semigroup regularization that this Lip-norm defines a quantum metric space structure for $q$-Gaussians, and then transfer this property to free Gibbs laws for convex potentials using free transport.

In this article, for a convex lattice polytope, we further investigate the summatory function of its Ehrhart polynomial, which is called the summatory Ehrhart polynomial, and introduce its summatory Ehrhart series. We prove several fundamental properties of these invariants. In particular, we derive a summatory analogue of the classical Ehrhart–Macdonald reciprocity law, which establishes a signed functional equation between the polytope and its relative interior via the substitution $t \mapsto -t-1.$

We reduce the embracing exchange distance of bases of oriented matroids to the metric of oriented matroid polyhedra. This allows us to disprove recent conjectures of Caoduro, Khodamoradi, Paat, and Shepherd and of Bérczi and Nádor. On the other hand, we show that any two embracing bases of an oriented matroid of rank $r$ can be transformed into each other in at most $2r^{\log_2(r)+3}$ steps and in at most $r$ steps in a graphic oriented matroid or a Lawrence oriented matroid, thus confirming the conjecture in these cases.

We establish sharp scalar-curvature bounds and rigidity consequences of Gromov's exact-lift two-form method. Let $(M^n,g)$, $n\geq 4$ even, be a closed spin Riemannian manifold carrying a homologically $\hat{A}$-non-singular closed two-form $\omega$ whose lift to the universal cover $X$ is exact. Then $$\inf_M\scal_g\leq -\frac{4n}{n-1}\lambda_0(X).$$ Equality forces $g$ to be Einstein; if $\lambda_0(X)>0$, then $X$ is real hyperbolic, while if $\lambda_0(X)=0$ and $\int_M\omega^{n/2}\neq 0$, then $g$ is flat. The proof combines Gromov's twisted $L^2$-index with a conformal interpretation of the refined Kato equality and a recentering argument. The same method yields untwisted rigidity results when zero belongs to the spectrum of the Dirac operator on the universal cover, with applications to nonvanishing$\widehat A$-genus and enlargeability.

We consider the problem of finding $\delta$-stationary points for $\delta > 0$, i.e., $x \in \mathbb{R}^d$ such that $||\nabla F(x)|| \le \delta$, for smooth, non-convex objectives, where the derivative oracles are not only stochastic but also biased. In the first-order setting, we provide tight lower bounds for finding an $O((\epsilon + B^2)^{1/2})$-stationary point, for $\epsilon > 0$ and where $B$ is a bound on the gradient bias, matching the upper bounds of Ajalloeian and Stich (2020). We then establish bias-dependent lower bounds for algorithms that use higher-order derivative information for finding $O(\epsilon + B_{\max})$-stationary points, where $B_{\max}$ is a bound on the maximum bias for all derivatives. To complement these lower bounds, we develop trust-region based methods that, for certain ranges of bias, provide guarantees that match the corresponding lower bounds. We further improve upon the oracle complexity in high bias settings through a higher-order variance reduction scheme, in particular demonstrating the benefits, in some cases, of using higher-order derivative information, whereas such improvements are known to be unattainable for stochastic unbiased settings.

In this paper we prove “$H=W$" in the context of a Banach function space $X(\Omega)$. Let $\Omega$ be a subset of ${\mathbb R}^n$ and denote by $W^1_X(\Omega)$ the collection of all those $f\in X(\Omega)$ whose distributional derivatives $\partial_jf$ are contained in $X(\Omega)$. Our main result provides a small collection of “universal" hypotheses on $X(\Omega)$ that ensure $W^1_X(\Omega)$ is equal to $H^1_X(\Omega)$, the formal closure of ${Lip}(\Omega)\cap W^1_X(\Omega)$ with respect to the norm \[\|f\|_{W^1_X(\Omega)} = \|f\|_{X(\Omega)} + \|\nabla f\|_{X(\Omega)}.\] The main theorem has two corollaries. The first gives a slightly stronger set of hypotheses for “$H=W$", and the second gives density of $C^\infty_c({\mathbb R}^n)$ in $W^1_X({\mathbb R}^n)$.

In multi-task learning, handling an increasing number of objectives can quickly become challenging, both in terms of the computational resources and the decision maker's capacity to choose appropriate trade-offs. A widely used approach is thus to aggregate the individual losses in a single loss function by a weighted sum. This often fails to capture either the decision maker's preferences as a result of the shape of the Pareto front, or requires multiple adjustments and computations which becomes prohibitively expensive in deep learning applications. To address these issues, we introduce a novel framework, Preference Pareto Exploration (PPE), which enforces the decision maker's preferences while accounting for the geometry of the Pareto set in an interactive exploration process. PPE is based on a predictor-corrector method that performs predictor steps tangential to the manifold of Pareto-optimal solutions, following the decision maker's preference. The subsequent corrector step results in a new trade-off reflecting this preference. To avoid explicit Hessian computations when characterizing the tangent space of the manifold, we employ a Krylov subspace method that relies solely on matrix-vector products. These products can be efficiently obtained via automatic differentiation, ensuring both efficiency and robustness throughout the optimization process. The method's functionality and performance are demonstrated using both toy problems and examples from deep learning.

The celebrated c-cyclical monotonicity property is shown to boil down to the zeroth-order optimality condition for the optimal transport problem. More precisely, we show that optimality is equivalent to the non-negativity of the linear transport cost functional on the radial cone of admissible perturbations. We then utilise this point of view to extend the c-cyclical monotonicity property to the weak optimal transport problem, for which it corresponds to the first-order optimality condition, namely to the non-negativity of the linearisation of the weak transport cost functional near the optimiser. Altogether, this sheds new light on this monotonicity concept. For both classical and weak optimal transport, we show that this property characterises (under suitable assumptions) optimal transport plans. In the classical case, we recover known results of the literature but with revisited proofs.

Given $(A_0, A_1)$ a compatible couple of Banach spaces, we describe the dual of the limiting real interpolation space $(A_0, A_1)^{K}_{ 1,q,b}$ for $0 < q < 1$ and $b$ a slowly varying function. In the process, we use the $J$-spaces $(A_0, A_1)^{J}_{ 1,q,b}$ and we establish an equivalence theorem for $K$ and $J$ spaces of independent interest. We also give examples that recover known results on this topic.

This paper develops an accurate and effective combination of second order backward differentiation time discretization (BDF2) with modular, 2-step nudging-based data assimilation \begin{align} \text{Forecast step: } \quad &\frac{3\widetilde{v}^{n+2}-4v^{n+1}+v^n}{2\Delta t}+\widetilde{v}^{n+2} \cdot \nabla \widetilde{v}^{n+2} - \nu \Delta \widetilde{v}^{n+2} + \nabla q^{n+2}=f(x) \notag \\ &\nabla \cdot \widetilde{v}^{n+2} = 0 \notag \\ \text{Analysis step: } \quad &\frac{3v^{n+2}-3\widetilde{v}^{n+2}}{2\Delta t}-\chi I_H(u(t^{n+2})-v^{n+2})=0. \notag \end{align} If $I_H=I_H^2$, the analysis step can be made explicit, taking the form \begin{align} v^{n+2}=\widetilde{v}^{n+2}+\frac{2\Delta t\chi}{3+2\Delta t\chi}I_H(u^{n+2}-\widetilde{v}^{n+2}). \notag \end{align} This implies the analysis step has the stability property of an implicit step and lower complexity than an explicit analysis step. Stability and error estimates for the BDF2 scheme are presented along with their proofs. Numerical experiments are conducted to assess the performance of BDF2 modular assimilation algorithm. The results of the experiments support the conclusion that modular data assimilation has comparable accuracy to standard, fully coupled data assimilation while greatly reducing computational complexity and cost.

In this work we consider an Aztec diamond model split into two unequal regions which are asymptotically fixed in size. Each region is weighted with a distinct two-periodic weighting. We refer to this model as the t-split two-periodic Aztec diamond, to signify its difference from the previous work title Split Two-Periodic Aztec Diamond, where the model was split into two equal regions. We derive an integral expression for the correlation kernel of the model and give a partial description of the scaling limit behavior, along with a conjecture for the remainder. We refer to the larger and smaller sides of the model as the dominant and non-dominant sides, and to the location of the weight change as the interface. The dominant side exhibits a limit shape that depends only on its own weighting and is identical to that of the two-periodic Aztec diamond, while the non-dominant side appears to have a novel limit shape that depends on both weightings and the location of the interface. Lastly, we consider the complete limit shape in the case where the dominant side two-periodic parameter goes to 0.

We investigate distributive lattices and Heyting algebras definable in o-minimal structures. We give a complete description of one-dimensional distributive lattices definable in o-minimal structures expanding a real-closed field, and prove a definable analogue of Birkhoff representation. As applications, we obtain a sharp lower bound on the dimension of definable Heyting algebras which contain infinite free subalgebras, and determine all one-variable equations in the language of Heyting algebras whose solution set can constitute a maximal-dimension proper subset of an infinite algebra.

We prove that, in every degree, the rational Spin bordism classes represented by manifolds admitting metrics with positive Ricci curvature span exactly the kernel of the $\hat A$-genus. More precisely, for \[ R=\Omega_*^{Spin}\otimes\mathbb{Q},\qquad J=\ker(\hat A:R\longrightarrow\mathbb{Q}[u]),\] the $\mathbb{Q}$-span of bordism classes of Ricci-positive Spin manifolds equals $J$ in each degree. This answers, in the differentiable rational Spin category, a question about rational bordism obstructions to positive Ricci curvature which was raised in the context of complex elliptic genera. The proof uses smooth complete intersections of an odd number $\ell$ of quadrics \[ Y_{m,\ell}\subset \mathbb{CP}^{2m+\ell}, \qquad \ell=1,\, 3,\, \ldots,\, 2m-1. \] These manifolds have real dimension $4m$, are Spin and Fano, and therefore admit metrics with positive Ricci curvature. A first-order thickening of the $\hat A$-genus induces $m-1$ linear functionals on $(J/J^2)_{4m}$. Their values on the classes $[Y_{m,\ell}]$ are governed by polynomials $P_{m,q}(\ell)$ of strictly increasing degrees $q+1=1$, $2$, $\ldots$, $m-1$. This gives full rank by a polynomial-interpolation argument.

In this paper we study cycles in multiset permutations and parking functions. As combinatorial objects, multiset permutations are essential building blocks for mappings and permutations, while parking functions lie between mappings and permutations. We take both algebraic and analytic views in our investigation and present exact as well as asymptotic results. We point to a surprising correspondence between two statistics on multiset permutations, terminal closers and cyclic points, shedding light on the combinatorial structure.

Bluetooth Low Energy (BLE) direction-finding is promising for indoor industrial localization, but its accuracy degrades in multipath environments where reflections and scattering bias angle estimates. Although line-of-sight (LOS) and non-line-of-sight (NLOS) detection is well studied for wide-band radios, BLE direction-finding still lacks narrow-band channel-feature representations, scalable kernel-based feature transformations, and dedicated datasets for data-driven, lightweight channel classification. To address this gap, the work introduces a controlled BLE measurement setup that generates labeled LOS/NLOS data in two distinct propagation environments. A quality-driven machine learning (ML)-based pipeline is then developed for BLE Constant Tone Extension (CTE) In-phase-Quadrature (IQ) features. First, robust quantile-based standardization is applied to reduce the influence of outliers and heavy-tailed effects. The standardized features are then analyzed using Principal Component Analysis (PCA) and Adaptive Kernel Density Estimation (AKDE) to verify scenario-dependent statistics and reveal LOS/NLOS separability. Next, Nyström Kernel Approximation (NKA) constructs low-rank nonlinear feature maps followed by a lightweight Support Vector Classifier (SVC) head for LOS/NLOS detection. This classifier is compared with Random Forest (RF) and Multilayer Perceptron (MLP) models. Results show that NKA improves accuracy by about 7-14% relative to the raw baseline. Although the MLP achieves higher absolute accuracy, the Nyström–SVC approach offers a more favorable trade-off between training complexity, inference cost, and memory footprint. Finally, several pipeline-calibrated posterior probabilities are utilized for cost-aware threshold selection and efficient real-time LOS/NLOS detection in resource-constrained localization systems.

The Bures–Helstrom metric is the minimal monotone Riemannian metric on the state space of a qubit. With the quantum Fisher normalization used here, it identifies the Bloch ball with a geodesic hemisphere of the unit round three–sphere. We describe its Ricci flow explicitly. In a general rotationally symmetric gauge the flow is a coupled system for the radial lapse and warping factor; a single scalar equation appears only after a Hamilton–DeTurck gauge choice. In the corresponding moving DeTurck frame the squared warping function $\Psi=\Phi^2$ satisfies the linear forced heat equation \begin{equation*} D_t\Psi=\Psi_{ss}-2, \end{equation*} while the fixed-lapse coordinate form contains the associated transport term. Since the Bures–Helstrom metric is Einstein, the geometric flow itself is the homothetic shrinker \begin{equation*} g(t)=(1-4t)g_{\mathrm{BH}}, \end{equation*} with scalar curvature $6/(1-4t)$ and extinction time $T=1/4$. Thus the metric remains inside the monotone cone for all $t<T$ and leaves the cone of nondegenerate Riemannian metrics only through the collapsed limit. We also record the volume–normalized flow, for which the Bures–Helstrom metric is a fixed point. Its linearization is the shifted round–sphere Laplacian $\Delta_{\mathbb S^3}+3$, with spectrum \begin{equation*} \sigma_\ell=-(\ell-1)(\ell+3), \end{equation*} and spectral gap $5$ after removal of the scaling mode.