Papers

We study a system of nonlinear Schrodinger equations under the mass-resonance condition and provide a complete description of the asymptotic behavior of small solutions in two space dimensions. Our analysis is based on a detailed study of an associated system of ordinary differential equations governing the asymptotic profile. We establish a phase-amplitude representation for this ODE with arbitrary initial data, where the amplitude is expressed explicitly in terms of Jacobi elliptic functions and the phase is given by elliptic integrals of the third kind. As a consequence, we obtain a fully explicit characterization of the asymptotic profile for the original PDE. In particular, the long-time behavior is described by a modified-scattering-type profile whose profile evolves according to the above integrable structure. While elliptic-function-type asymptotics were previously constructed for special solutions in final-state problems, the present work provides the first complete characterization, in two space dimensions, of asymptotic dynamics associated with arbitrarily small initial data through elliptic-function-type profiles.

The classical Marion's theorem states that the area of the central hexagon obtained by dividing each side of a triangle into three equal parts and connecting the interior division points is exactly $1/10$ of the area of the original triangle. In this paper, the construction is extended to an arbitrary n-dimensional simplex: each edge is divided into three equal parts, and through each n-1 vertices and the two interior division points of the opposite edge, two hyperplanes are drawn. It is shown that the resulting inner polytope has a surprisingly simple volume formula with the volume is $1/C(n,2n+1)$ times the volume of the original simplex. The proof uses a probabilistic interpretation with exponential random variables and reduces the geometric problem to a combinatorial binomial sum.

We construct a boundary-preserving hp interpolation operator on three-dimensional tetrahedral meshes with locally variable polynomial degrees. If the trace of an H^1 function on the prescribed Dirichlet boundary is already a piecewise polynomial trace of the finite element space, the interpolant preserves this trace exactly and satisfies the standard local h_K/p_K approximation estimates. The statement follows the scaling form of Melenk's hp quasi-interpolation for nonsmooth functions. As a consequence, a discrete trace is extended by first applying the continuous trace theorem and then applying the boundary-preserving interpolant; the corresponding discrete harmonic extension is bounded by variational comparison. The proof of the interpolation theorem uses local polynomial trace liftings on tetrahedral boundary layers, nonsingular vertex patches, and a variable-degree tetrahedral projection. These auxiliary liftings are also stable in scaled boundary-layer norms.

Robust optimization offers a tractable approach to balance operating costs and reliability in power systems dominated by weather-dependent renewable uncertainty, but its performance depends critically on the uncertainty set. Standard data-driven approaches often calibrate uncertainty sets to attain predictive coverage, which can produce unnecessarily large sets and costly operating decisions. In contrast, we introduce decision-calibrated prediction sets and embed them as uncertainty sets in robust optimization problems; these are conditional multivariate prediction sets where calibration is defined in terms of the reliability of downstream decisions, rather than in terms of the coverage. First, we learn these conditional prediction sets as sub-level sets of norm-based score functions represented by partially input-convex neural networks, capturing contextual information and multivariate dependence while preserving convexity and tractability in downstream robust formulations. Second, inspired by conformal risk control, we calibrate a score-threshold parameter that sets the volume of the uncertainty set, thereby controlling the expected violations of downstream operational constraints. We apply our approach to 15-minute-ahead reserve scheduling with network-constrained deliverability, which we formulate as a robust DC optimal power flow problem with affine recourse. Numerical experiments show that decision-calibrated sets attain prescribed constraint-satisfaction targets within about three percentage points, whereas standard coverage-based calibration systematically exceeds these targets by more than eleven percentage points, leading to larger sets and higher operating costs.

In this article, we consider the words with cyclic indices. For given $s$, we consider the pair $(\iota,\kappa)$ of indices such that the word of length $s$ from $\iota$ is equal to the word of length $s$ from $\kappa$. We give a characterization of such pairs for a cyclic Fibonacci word, and give the number of them.

In this paper, we investigate critical mass threshold for the Patlak-Keller-Segel-Navier-Stokes system on the two-dimensional whole space and obtain global existence of strong solutions if the initial mass is less than or equal to $8\pi$, regardless of the initial norm of the velocity. One new observation is that the local mass of the density function rearrangement satisfies a good inequality that is independent of velocity; and then an improved maximum principle is applied by choosing a nice auxiliary function.

While it is well-known how to compute the cells of a Laguerre tessellation for a given set of weighted generator points, it is not obvious how to invert a Laguerre tessellation. That is, given that one observes a Laguerre tessellation, how can one retrieve the weighted generators corresponding to the observed cells. In this paper, we consider inversion of a class of random Laguerre tessellations known as Poisson-Laguerre tessellations. The weighted generators of observed cells of a Poisson-Laguerre tessellation are of interest because knowledge of these weighted generators is useful for statistical inference of Poisson-Laguerre tessellations. For general Laguerre tessellations we provide a characterization of all configurations of weighted generator points which yield the same Laguerre tessellation. For Poisson-Laguerre tessellations we propose a method for consistent inversion, meaning that as one observes the tessellation through increasing observation windows, a closer approximation of the original weighted generators can be obtained. In a simulation study we examine both performance of the inversion procedure, as well as the use of the obtained approximated weighted generators for nonparametrically estimating the weight distribution function corresponding to a Poisson-Laguerre tessellation.

Overcoming the Kolmogorov barrier for constructing efficient reduced-order models (ROMs) for transport-dominated problems remains a challenge. This has impeded their use as computationally cheap surrogates for partial differential equations (PDEs) associated with optimal control problems. Since such problems require multiple computations of the full PDE, employing their reduced-order surrogates instead could speed up the overall optimal control problem. Motivated by this idea, in this paper we explore the use of a nonlinear model-order reduction technique, namely, the shifted proper orthogonal decomposition (sPOD) in an optimal control context. In doing so, we explore the framework of first-optimize-then-reduce (FOTR) where an optimality system for the full PDE problem is constructed first, followed by approximating the optimality system with reduced-order models. We consider this framework for a linear quadratic optimal control problem constrained by a 1D linear advection equation and compare the computational performance of the sPOD method against the use of the standard POD.

We introduce the class of Special–Dirichlet processes, consisting of càdlàg adapted processes admitting a decomposition \[ X=M+\Gamma, \] where \(M\) is a local martingale and \(\Gamma\) is an adapted càdlàg process with vanishing continuous quadratic variation whose jumps are predictable and \(\mathcal F_{s-}\)-measurable. This class arises naturally from transformations of special semimartingales. Classical results imply that sufficiently regular functions of special semimartingales belong to the broad class of Dirichlet processes. We show that such transformed processes possess substantially more structure: they admit a canonical decomposition in which the predictable jump component is explicitly separated from the martingale component. This yields a refinement of the traditional classification, which previously identified these processes only as Dirichlet processes. We establish uniqueness of the decomposition and prove that the class is stable under a large family of nonsmooth transformations, including primitives of locally bounded functions with at most countably many discontinuities. An explicit Itô-type decomposition is obtained in terms of the martingale jump measure and its compensator. Finally, we investigate stability properties of the canonical decomposition. Under convergence in quadratic variation and Skorokhod \(J_1\)-convergence, we prove stability of both the martingale and singular components after transformation. The proof relies on a threshold isolation principle for jump structures, allowing large jumps to be separated from small-jump contributions and yielding convergence of the transformed decompositions.

We study exact community recovery in the two-community stochastic block model on $n$ vertices under limited and noisy access to network data. The learner may query a noisy neighborhood oracle that reveals each true neighbor of a queried vertex independently with fixed probability and never returns non-neighbors, subject to a finite query budget. We consider both oracle-only access and a combined model where the learner also observes a single subsampled copy of the underlying graph. For oracle-only access, balanced uniform querying gives a sharp non-adaptive benchmark: when each vertex is queried the same integer number of times, the observations reduce to an SBM with attenuated edge probabilities and the Abbe-Bandeira-Hall exact-recovery threshold applies. We show that this benchmark is not adaptively optimal: a two-stage adaptive strategy succeeds with $n+o(n)$ queries in a regime where balanced uniform querying requires $m n$ queries for some $m>1$. With an additional subsampled graph, we prove a sublinear-query adaptivity gap: balanced data-independent uniform querying with a sublinear budget does not improve over the subsampled graph alone, whereas adaptive querying can target a small set of uncertain vertices and achieve exact recovery. Thus adaptive data acquisition can strictly improve the information-theoretic limits of exact recovery.

We derive a fully 3-dimensional Summation-By-Parts scheme for a class of linear wave equations on hyperboloidal slices that meet future null infinity on a Minkowski background. The scheme is derived in spherical polar coordinates, with a major strength being that it is provably stable and allows having grid points at the origin and on the $z$-axis, despite coordinate singularities, and at infinity, by introducing compactification followed by rescaling. Reducing it to the standard Cauchy problem, or on finite spacelike slices with an outer boundary, will follow a similar procedure. Interesting relations are obtained between the rescaling and compactification factors that simplify the equations, and the conditions on constraint addition terms are discovered to maintain symmetric hyperbolicity. Numerical implementation is achieved using finite-difference methods at second-order accuracy, which can be generalized to higher-order or spectral accuracies as well. Dissipation operators are given a more abstract treatment, which makes it possible to define them everywhere in the domain, including at the boundary points, in curvilinear coordinates, such that they satisfy the dissipative property (DP) in our energy norms. These generalizations reduce to the well-known Kreiss-Oliger dissipation operators whenever defined on a Cartesian grid in the bulk and satisfy the DP in the standard $L^2$-norms. We also propose new norm convergence tests that produce more accurate outputs. Promising results are obtained, giving hope for application to fully nonlinear systems, like the Einstein Field Equations, and extracting the resulting gravitational waves free of systematic errors or gauge ambiguities.

We introduce Convex Distance Operator Transport (CDOT), the first convex optimal transport framework that aligns distributions across heterogeneous domains by jointly preserving feature correspondence and intrinsic geometric structure. Specifically, CDOT employs an operator-based regularization that aligns aggregated distance structures by introducing distance and conditional expectation operators. Consequently, the proposed regularization improves the robustness to local geometric variations. We further prove that the resulting CDOT discrepancy is a valid pseudometric on the space of attributed compact metric-measure spaces. In addition, we characterize the relationship between CDOT and Gromov–Wasserstein (GW) through a new notion of dispersion gap, formally elucidating the geometric source of non-convexity in GW compared to the convexity of CDOT. In the finite-sample regime, we derive a non-asymptotic risk bound decomposed into optimization and statistical errors, establishing risk consistency under a globally convergent Frank–Wolfe algorithm. Experiments on synthetic point clouds, brain connectomes, and graph classification benchmarks demonstrate better performance over existing methods, with stable and reliable behavior in practice.

We explicitly construct dual bimodules in a semistrict monoidal 2-category, using Frobenius algebra structure. The main result shows that a coherent dual of the underlying object can be promoted to a coherent dual of the bimodule, with zigzag 2-isomorphisms additionally require special Frobenius structures. We also prove that every special Frobenius algebra in $\mathbf{2Vect}$ is rigid, via a categorified Casimir object argument, and discuss the relationship between the Frobenius, rigid, special Frobenius, and separable algebra hierarchies.

A fundamental open problem in the homological theory of Banach spaces is the calculation of the injective dimension of the Banach space $c_0$. We make a contribution to the study of this problem by proving that, if the Continuum Hypothesis ($\mathsf{CH}$) holds, then the injective dimension of $c_0$ is at least 3. In the course of proving this result, we introduce the notion of an almost disjoint family on a topological space $X$, generalizing the classical notion of almost disjoint families of subsets of $\mathbb{N}$, which we feel is of interest in its own right. We prove that, if $\mathfrak{b} = 2^{\aleph_0}$, then there exists an almost disjoint family of cardinality $2^{\aleph_1}$ on the Čech-Stone remainder of $\mathbb{N}$.

In this paper, we study self-intersection local times for a stochastic process $x(u(\cdot),t)$, where $u$ is a Gaussian process of the form $u(t)=\int^t_0k(t,s)\mathrm{d}{w(s)}$, $k$ is a deterministic kernel of the Volterra type, $w$ is a Wiener process, and $x$ is a solution to the equation with interaction. Equations with interaction are a class of interacting particle system described by stochastic differential equations whose coefficients depend on a random measure (initial distribution of particles) transformed by the flow of solutions. Considering the occupation measure of $u$ as the initial condition for the equation with interaction allows us to define a stochastic flow with interaction driven by self-intersection local times of the process $u$. The study of such stochastic differential equations whose coefficients carry information about the geometric properties of curves is new. They previously appeared only for deterministic differential equations and smooth curves, where the geometric characteristics typically considered are length, curvature, and so on. In this paper, we prove the existence of multiple self-intersection local times for the process $x(u(\cdot),t)$ and establish a “change of variable formula" that allows us to describe self-intersection local times for the process $x(u(\cdot),t)$ in terms of the weighted self-intersection local times for the process $u.$ We describe the corresponding asymptotics of the self-intersection local times for $x(u(\cdot),t)$ for large $t$. Moreover, the existence of weighted self-intersection local times is established for a large class of unbounded weights, which is of independent interest.

Given a graph $G$, a dominating set is a subset $X\subseteq V(G)$ such that $N[X]=V(G)$. The domatic number of $G$, denoted ${\rm dom}(G)$, is the maximum size of a partition of $V(G)$ into dominating sets. In analogy with the lower bound of the chromatic number by the clique number, the domatic number satisfies the upper bound ${\rm dom}(G)\le \delta(G)+1$ where $\delta(G)$ is the minimum degree of $G$. Therefore, as an analogue of the notion of $\chi$-bounded graph classes, we say that a class of graphs $\mathscr{G}$ is DOM-bounded if there exists a positive unbounded function $f_{\mathscr{G}}$ such that for every $G\in \mathscr{G}$, we have ${\rm dom}(G) \ge f_{\mathscr{G}}(\delta(G))$. We propose the following conjecture for graphs forbidding a fixed induced subgraph, analogous to the Gyárfás–Sumner Conjecture for $\chi$-bounded graph classes: for every connected graph $H$, the class of $H$-free graphs is DOM-bounded if and only if $H$ is a tree of diameter at most $3$. We reduce the case of disconnected graphs to the connected setting and show that the conditions on $H$ are necessary. We show that star-free graphs of minimum degree at least $\delta$ have domatic number $\Omega(\delta /\log \delta)$, which is best possible up to a constant factor. We also identify a subclass of star-free graphs for which the domatic number is linear in $\delta$: line graphs of bounded rank hypergraphs. In support of our conjecture in the case of double stars, we prove that $P_4$-free graphs (i.e. cographs) of minimum degree $\delta$ have domatic number at least $1 + \frac{\delta}{2}$, which is best possible.

The aim of the paper is to survey results on Jordan types for pairs of nilpotent commuting matrices. We also review recent proof of the Box Conjecture on Jordan types that have equal dense orbit in the nilpotent commutator.

For an internal category $\mathbb{C}$ in a cartesian category $\mathcal{C}$ we define, naturally in objects $X$ of $\mathcal{C}$, $Prin_{\mathbb{C}}(X)$. This is a category whose objects are principal $c \mathbb{C}$-bundles over $X$ and whose morphisms are principal $c(\mathbb{C}^{\uparrow})$-bundles. Here $c(\_)$ denotes taking the core groupoid of a category (same objects but only isomorphisms as morphisms) and $\mathbb{C}^{\uparrow}$ is the arrow category of $\mathbb{C}$ (objects are morphisms, morphisms are commuting squares). We show that $X \mapsto Prin_{\mathbb{C}}(X)$ is a stack of categories and call stacks of this sort lax-geometric. We then provide two sufficient conditions for a stack to be lax-geometric and use them to prove that the pseudo-functor $X \mapsto \mathbf{LK}_{Sh(X)}$ on the category of locales $\mathbf{Loc}$ is a lax-geometric stack. Here $\mathbf{LK}_{Sh(X)}$ is the category of locally compact locales in the topos of sheaves over $X$, $Sh(X)$. Therefore there exists a localic category $\mathbb{C}_{\mathfrak{LK}}$ such that $\mathbf{LK}_{Sh(X)} \simeq Prin_{\mathbb{C}_{\mathfrak{LK}}}(X)$ naturally for every locale $X$. Keywords: Topos, locale, principal bundle, internal category and groupoid, category theory, geometric logic, stacks.

We study two families of quandles arising from Coxeter quandles. One is the quandle defined by Andruskiewitsch-Graña, which is the set of roots with binary operation defined by using the negatives of reflections. We observe that this is realized as a conjugation quandle in a Pin group. The other, which we call a rotational $D_n$ quandle, is the set of some right angle rotations in the Coxeter group of type $D_n$ with binary operation given by conjugation. We determine their inner automorphism groups, and observe that they are quite similar.

This paper introduces the $n$-th order multifractional stable motion ($n$-MFSM), a novel stochastic process that simultaneously unifies three key modelling features: heavy-tailed distributions ($\alpha$-stable with $\alpha\in(1,2]$), time-varying local regularity via a functional Hurst parameter $H(t)\in(n-1,n)$, and extended scaling behaviour of order $n\geq1$. No existing framework combines all three. We establish rigorous existence via $L^\alpha$-integrability analysis, derive both moving-average and harmonizable representations with explicit constants, prove local asymptotic self-similarity with complete identification of the limit process, determine the exact pointwise Hölder regularity $\alpha_X(t)=H(t)-1/\alpha$, and characterize long-range dependence through codifference asymptotics. In particular, we obtain the precise decay exponent $(\alpha-1)H_+ + H(s)-n$ and the LRD criterion $(\alpha-1)H_++H(s)<n$, which generalizes the classical condition $H(s)+H_+<1$ for first-order Gaussian multifractional processes and reduces to $\alpha H-1$ for LFSM with constant $H$.

In this paper, we investigate a class of sparse optimization problems in which both the objective and constraint functions are Fréchet differentiable and possess locally Lipschitz continuous gradient mappings. More precisely, by utilizing the limiting (Mordukhovich) second-order subdifferential of the associated Lagrangian function, we establish new second-order necessary and sufficient optimality conditions for local optimal solutions. The obtained results are derived under mild assumptions and extend several existing results in the literature. In addition, we apply our theoretical developments to sparse multiobjective optimization problems and derive second-order sufficient optimality conditions for efficient solutions. Several examples are also presented to demonstrate the applicability and effectiveness of the proposed results.

The Ramsey number $r(H)$ of a graph $H$ is the minimum positive integer $n$ such that every red/blue edge-coloring of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ of $H$ is the minimum number of monochromatic copies of $H$ over all red/blue edge-colorings of $K_{r(H)}$. Let $P_t$ and $C_t$ be a path and a cycle on $t$ vertices, respectively. In this paper, by using combinatorial and local random construction, we show that $$m(C_{2t})\le t^{-\gamma+o(1)}\frac{(2t-1)!}{2}, \qquad m(P_{2t+1})\le t^{-\gamma+o(1)}\frac{t}{2}(2t)!,$$ and $$m(P_{2t})\leq \left(\frac{7}{8}+o(1)\right)\frac{(2t)!}{2},$$ for sufficiently large $t$, where $\gamma=1/(1+\sqrt{2})$. These results disprove two conjectures on the threshold Ramsey multiplicity for even cycles and paths, due to Conlon, Fox, Sudakov, and Wei.

The Admixture Model describes genetic marker data by representing each individual's genome as a mixture of contributions from $K$ ancestral populations, with the individual admixture vector summarizing the corresponding ancestry proportions. In population and forensic genetics, a key question is whether an individual's genome supports a predominantly single-ancestry interpretation or whether an admixed interpretation is more appropriate. We propose a statistical test for single-population ancestry in the supervised Admixture Model, where ancestral allele frequencies are treated as known. The test assesses whether the largest admixture component exceeds a practitioner-chosen dominance threshold, giving precise meaning to the notion of a sufficiently strong single-population contribution. To calibrate the test, we develop a constrained parametric bootstrap procedure that generates data under a null-constrained maximum likelihood estimator, accounting for the constrained hypothesis structure, the marker-wise heterogeneity and small sample sizes. Under standard regularity conditions, we prove that the proposed test has asymptotic level $\alpha$ and is consistent, ensuring control of false single-ancestry declarations while reliably detecting dominant ancestry components. Simulation studies demonstrate good finite-sample performance across different numbers of ancestral populations, marker-panel sizes, dominance thresholds, and allele-frequency distributions. We further illustrate the practical utility of the method using data from the 1000 Genomes Project. The proposed framework delivers interpretable, threshold-based ancestry assessment with rigorous error control, and extends constrained bootstrap methodology to the independent but non-identically distributed setting of genetic marker data.

We study the spectral theory of infinite Toeplitz matrices $T = (a_{k - l})$ under the assumption that $(a_k)$ and $(a_{-k})$ are completely monotone sequences. We derive expressions for generalised eigenvectors and prove a generalised eigenvector expansion of $T$. Even if the matrix $T$ is not normal, our expressions involve only eigenvalues and eigenvectors with real entries.

We provide a brief primer for the idea behind formalising hierarchical causality in the context of complex systems. Here actors are not simply agents. Actors instantiate causation classes. Agents implement local dynamics in given levels or organisation in a given system. Hierarchical causality then describes how actor-level roles constrain, select, and organise agent-level behaviour across levels. The system then necessarily requires three additional structures. First, causation classes to abstract a given form of causal influence that an actor instantiates. Second, aggregation operators to move across the levels. Third, discrete event-time maps are required because the system comprises events, and the relation between local event counts and any global clock must be specified. Our formulation here is purposefully simple and discrete.

We investigate the Cauchy problem for the focusing modified Korteweg–de Vries (mKdV) equation with finite-genus algebro-geometric quasi-periodic initial data. By applying the nonlinear steepest-descent method of Deift–Zhou to the associated Riemann–Hilbert (RH) problem, we derive the long-time asymptotics of the solution in the critical regime where complex stationary phase points coalesce with the endpoints of the finite-genus branch cuts. The collision is resolved via a local Painlevé XXXIV parametrix, and the discrete spectrum (breathers) is incorporated into the analysis. The resulting expansion is valid uniformly up to an error of order $\mathcal{O}(t^{-1/2})$. In this critical region, the leading-order term comprises the finite-genus algebro-geometric background together with breathers, whose parameters are slowly modulated by the background solution.

The separation of multicomponent signals with crossing instantaneous frequency (IF) curves remains a fundamental challenge in time-frequency analysis. Although the synchrosqueezed wavelet-chirplet transform (SWCT) enhances time-frequency readability by introducing a chirprate variable, its effectiveness is constrained by the underlying assumption of local linear chirp. Consequently, this method does not perform well when analyzing signals characterized by strong frequency modulation. This paper extends the SWCT framework by relaxing the linear chirp assumption. We model signal components as having polynomial phase behavior over short intervals and derive compact expressions for high-order IF and chirprate reassignment operators. The proposed high-order synchrosqueezed wavelet-chirplet transform (HSWCT) enables accurate estimation of both IF and chirprate, and supports robust mode retrieval even with intersecting IF curves. Another key contribution is a rigorous mathematical analysis of the approximation errors of arbitrary-order reassignment operators for IF and chirprate estimation. When the chirprate vanishes, HSWCT simplifies to the traditional high-order synchrosqueezed wavelet transform. To our best knowledge, no theoretical analysis exists in the literature on the approximation of arbitrary-order SST IF reassignment operators to the IF. As a by-product of this work, our established theorem provides such an analysis, thereby filling a gap in the theoretical framework of high-order SSTs.

We study local coordination on finite graphs, equivalently, the ability of randomized local protocols with independent private randomness to produce highly correlated outputs at neighboring vertices. Hutchcroft, Rospuskova, and Tamuz (2026) showed that if such local coordination can be achieved with low cost, then the graph admits a low-cost leader profile, or equivalently an appropriate bounded-radius decomposition, with a square-root loss in the cost. We substantially improve this bound for unbiased binary outputs. If the average disagreement is $\varepsilon$, we construct a leader profile of cost $O(\varepsilon\log(1/\varepsilon))$. The key ingredient is an information-theoretic influence measure: after fixing an ordering of the independent information sources, we assign to each binary output the increments of mutual information revealed along that ordering. For two outputs that disagree with probability $p$, the corresponding measures are within binary entropy $h(p)$ in total variation. Combined with a grand coupling theorem, this yields the improved bound. We also show that a universal linear bound is impossible. We construct graphs with binary local profiles of cost $O(\varepsilon)$ for which every leader profile has cost $$ \Omega\!\left(\varepsilon\sqrt{\log(1/\varepsilon)}\right). $$ The construction uses weighted majority functions and can be implemented as an action-symmetric Nash equilibrium, giving a negative answer to the question of HRT whether leader equilibria are always optimal up to a constant factor. Finally, we show that the binary assumption is essential: for general real-valued outputs, the square-root dependence of the HRT theorem is sharp up to a universal constant.

We consider the problem of detecting a Return to Baseline (RtB) in high-frequency monitoring data preceding and following an intervention, where the aim is to identify the time at which the data-generating distribution realigns with its pre-intervention distribution. We propose a sequential, distribution-free testing procedure that does not rely on specifying a null model and provides anytime-valid error control. The method relies on ideas from universal inference to define a discrepancy measure that is aggregated into a non-negative super-martingale, and is then empirically cal- ibrated to form an e-process. The calibration is performed using the baseline data, and is thus subject-specific. We establish finite-sample bounds for the calibration error (under a flexible non-parametric assumption), discuss the impact of tuning parameters and computational complexity, and illustrate through simulations and a clinical case study that the procedure accurately detects RtB from monitoring data.

In this paper, we establish a propagation of chaos result for mean-field mean reflected backward stochastic differential equations (BSDEs), where both the generator and constraint depend on the distribution of the solution. When the generator does not rely on $z$, under mild Lipschitz and integrability conditions, we prove existence and uniqueness of the solution to the interacting particle system for general reflections. We are able to consider the case where the generator depends on $z$ when the reflection is linear. In both cases, we obtain the convergence rate of solution to the interacting particle system towards the solution to the mean-field mean reflected BSDEs.