Papers

We propose a score-based stabilization framework for numerical simulation of partial differential equations, in which a learned score model defines a stabilization operator applied to provisional numerical updates. This operator augments standard time-stepping schemes by enforcing structure and physical consistency through a correction that drives iterates toward the manifold of admissible states. We show that the stabilization operator acts as a contraction toward this manifold, yielding a correction mechanism with basin-conditional stability. Numerical experiments on Advection, Korteweg-de Vries (KdV), Nonlinear Schrodinger (NLS), and Burgers' equations demonstrate improved robustness, suppression of nonphysical instabilities, and preservation of qualitative dynamics.

We disprove a conjecture of Duncan asserting that versality of a faithful finite group action can be detected on Sylow subgroups. Our counterexample is an action of $(C_7\rtimes C_3)\times C_2$ on a degree-$2$ Del Pezzo surface.

Urban eco-driving requires the simultaneous planning of speed, acceleration, lane decisions, surrounding-vehicle safety, intersection rules, and vehicle energy use. Existing studies commonly optimize only selected aspects of urban driving, such as longitudinal motion, lane changing, intersection crossing, or powertrain energy management, while treating the remaining decisions separately. Unlike these studies, this paper develops an integrated finite-horizon eco-driving planning framework for urban hybrid vehicles that jointly optimizes traffic behavior and powertrain operation within a unified mixed-integer formulation. The novelty of the proposed method lies in simultaneously modeling longitudinal motion, lane occupancy and changes, car-following and lane-change safety, signalized and unsignalized intersection behavior, and hybrid powertrain operation. The formulation captures lane availability, lane-dependent speed limits, mandatory and emergency lane changes, safe-to-clear signal conditions, downstream clearance, stop-and-yield rules, engine and motor power, battery state of charge, regenerative braking, and fuel consumption. Simulation studies compare the proposed method with rule-following, signal-aware, overtaking-enabled, and kinematic-only optimization baselines. The results demonstrate improved energy performance while maintaining feasible, safe, comfortable, and traffic-rule-compliant urban driving plans.

We develop a no-gelation mechanism that yields global solutions to the spatially homogeneous Boltzmann equation with mass exchange for Grad-cutoff hard potentials $B=E^\gamma b(\xi), 0<\gamma<1$ and regularly varying mass-exchange rates. Without assuming any higher mass moment, we construct a convex superlinear weight assembled from dyadic hinges. Production of the weighted moment by collisions between large particles of comparable mass is absorbed by dissipation through collisions with a uniformly populated reservoir of bounded-mass particles. Regular variation makes the signed increments in these two collision configurations comparable, while mass conservation and $\gamma<1$ yield the vanishing factor $L^{\gamma-1}$. This yields a uniform moment bound of the mass-cutoff approximations on every finite time interval and rules out finite-time mass escape to infinity. Consequently, for every nonnegative initial density with finite physical moments, we obtain a global integral weak solution.

This paper is devoted to the analysis of the entropy stability properties of Active Flux\yolo{-type} scheme for a hyperbolic system equipped with one entropy inequality. This type of scheme evolves two sets of degrees of freedom: point values that are chosen on the boundary of the elements that cover the computational domain, and the average of the solution in these elements. We show that the only thing to do is to get an entropy inequality for the average values, the point values degrees of freedom do not play any role. We construct a monolithic scheme which is bound preserving of \cite{BP_Pampa_VEM}, non oscillatory following \cite{PampaDG}, and entropy diminishing. The entropy condition is implemented in Tadmor's framework\cite{TadmorEntropy}, i.e. for the semi-discrete scheme only. The scheme is tested on the Kurganov-Popov-Petrova test case \cite{KPP} which is known to \yolo{be} sensitive to the satisfaction of an entropy inequality. We show that our entropy correction is effective: if we do not activate the bound-preserving nor the non oscillatory condition, we get the correct solution with some spurious wiggles, as expected. Though the development, implementation and tests are done with the triangle version of the scheme, the same method can be used for polygonal meshes, following \cite{BP_Pampa_VEM}.

We develop a framework for group-covariant quantum measurements in which measurement-induced transitions between irreducible representation sectors are described by a stochastic process on representation space. Starting from the Peter-Weyl decomposition, we construct covariant measurement operators from irreducible tensor operators and show that, for symmetry-invariant states, the measurement channel reduces to a Markov process on the representation graph. We further show that analyticity of the measurement operator constrains the detector spectrum through Sugiura's theorem, motivating a class of exponentially decaying detector models. Specializing to SU(2), we obtain the transition kernel in closed form, establish reversibility and the associated invariant measure, and derive a continuum Fokker-Planck description of the induced dynamics. Analytical predictions for the drift and diffusion coefficients are found to agree with numerical simulations. Our results provide a stochastic description of repeated covariant quantum measurements on representation space.

We prove that harmonic functions with Dirichlet boundary data in Barron space, a function class tailored to wide ReLU networks with a single hidden layer and suitably bounded weights, are generally neither Lipschitz continuous nor in the Sobolev class $H^2$. A fortiori, they are not in any function class in which the norm controls the Lipschitz constant, which rules out not only Barron space regularity, but also regularity in function classes for deeper ReLU networks with bounded coefficients. They can, however, be approximated to accuracy $\sim \varepsilon$ by Barron functions of low norm $\sim |\log\varepsilon|$ in various Lebesgue and Sobolev norms (with at most two derivatives). The positive result holds on very simple domains: Half-spaces in arbitrary dimension and rectangular domains in two dimensions. As an application of this regularity theory, we obtain a priori error estimates for Deep Ritz neural PDE solvers.

We answer several questions in the literature concerning the Grothendieck property of ideals of the Banach lattice $\ell_\infty$ that contain $c_0$. Any such ideal can be represented as a space $c_{0,\mathcal I}$ for $\mathcal I$ an ideal over the natural numbers. We provide a characterization of when $c_{0,\mathcal I}$ is a Grothendieck space in terms of finitely additive measures over $\mathcal P(\omega)$ and elements of $\mathcal I$. Using this characterization we show that there are analytic ideals $\mathcal I$ such that $c_{0,\mathcal I}$ is Grothendieck. In the opposite direction we show that for any AD family $\mathcal A$, $c_{0,\mathcal I(\mathcal A)}$ and $C(K_\mathcal A)$ are not Grothendieck spaces, and that for most of the Borel ideals present in the literature, $c_{0,\mathcal I}$ is not Grothendieck. In particular, the family of ideals that do not have the Grothendieck property is cofinal in the Rudin-Keisler order, so the Grothendieck property is not downward closed in the Katětov order. Continuing the work in \cite{Sobota-Zuchowski, Zuchowski}, we also provide similar results on the Nikodym property of the Boolean subalgebras of $\mathcal P(\omega)$ generated by the ideal $\mathcal I$.

Modern power grids are increasingly vulnerable to coordinated cyber-attacks, particularly false data injection attacks (FDIAs) that can evade conventional residual-based detectors. While most existing detection methods rely on instantaneous measurements, coordinated dynamic attacks can remain stealthy at each time step while introducing structured temporal deviations. This paper develops a joint framework for modeling and detecting such attacks in multi-area power systems. A time-aggregated attack model is first formulated to capture temporal evolution and inter-area coordination. For detection, a kernel-embedded functional subspace detection (KEFSD) method is proposed, which models residual trajectories in a reproducing kernel Hilbert space (RKHS) and employs RKHS-constrained functional principal component analysis (PCA) to identify anomalous temporal patterns. Simulation results on a modified IEEE 14-bus system demonstrates the proposed method achieves improved detection performance compared to the conventional residual based 2-norm detector.

The alignment of Small Language Models (SLMs) in the 70–500M parameter range using reinforcement learning is often considered unstable, though the underlying failure mechanisms have not been systematically investigated. In the State-of-the-Art (SOTA) research, fifteen (model, corpus) configurations were trained using Proximal Policy Optimization (PPO). The experiments included Pythia-70M, 160M, 410M and SmolLM2-135M, 360M on the TinyStories, CNN/DailyMail, and Wikitext-103 corpora. Three reproducible failure modes were identified in small-scale language models: silent LoRA parameter freezing in standard PEFT/TRL pipelines, numerical overflow in importance ratios when using bfloat16, and catastrophic policy collapse due to reward-model error. These issues were addressed using a merge-and-reinitialize adapter technique, float32 precision during PPO updates, and a three-layer safety mechanism comprising reward whitening, importance-ratio guarding, and weight rollback. In this paper, a capacity-headroom hypothesis is proposed, which states that PPO performance at the SLM scale depends on both a fluent supervised model ($\text{PPL}<20$) and a discriminative reward signal, rather than on the number of model parameters. The proposed system converged stably in all experiments and improved preference win rate over the SFT baseline in configurations with a fluent prior and an informative reward signal. Furthermore, it outperformed instruction-tuned baselines while requiring significantly less training data. All checkpoints, preference datasets, and training scripts are publicly released$^{\S}$.

A classical theorem due to Beurling-Lax-Halmos characterizes the invariant subspaces of the unilateral shift as ranges of isometric multiplication operators acting on the Hardy space. The class of Beurling-Lax-Halmos (BLH) pairs of reproducing kernels is introduced, consisting of those pairs that admit an analogue of this theorem. The BLH class is, under varying hypotheses, characterized in several equivalent ways: dilation-theoretically, via a complete Leech interpolation property, and through a sums-of-squares-inspired Agler-style decomposition. These characterizations unify and extend a variety of related results in the literature. Examples are given to illustrate the theory, the hypotheses and compare and contrast with the recent developments in the study of complete Pick pairs. In particular, while it is anticipated, perhaps under some mild assumptions, that the class of complete Pick pairs of kernels is contained in the class of BLH pairs of kernels, it is shown that the reverse inclusion fails in a strong sense.

This paper introduces shadowing theory based backward stability assessment of data driven computational analysis of discrete dynamical systems in the framework of the Koopman composition operator and the Extended Dynamic Mode Decomposition. Data driven spectral analysis of the dynamics using the approximate (computed) eigenpairs of the Koopman operator is cast in terms of the backward error analysis. The individual residuals of the computed eigenpairs are aggregated in a backward perturbation of the Koopman operator, so that the computed eigenpairs are exact for a perturbed operator that is not a composition operator. Then, the impact of the perturbation in the operator is carried over to the map and the initial condition of the original system, showing that the computed approximations correspond exactly to a pseudo–trajectory of the system. In the final step, the pseudo–trajectory is shadowed by an exact trajectory of the system. This interpretation of errors is in particular suitable in data driven scenarios where the data is contaminated by noise. New insights into the numerical shadowing are provided.

All Killing tensor fields on the spaces of constant curvature and on the complex projective space are decomposable, that is, can be represented as the sum of symmetric tensor products of Killing vector fields (equivalently, every polynomial integral of the geodesic flow is a polynomial in the linear integrals). This is no longer true for quadratic Killing tensor fields on the quaternionic projective spaces $\mathbb{H} P^n, \, n \ge 3$, and on the Cayley projective plane $\mathbb{O} P^2$. We prove that for the real Grassmannians and for the spaces $\mathrm{SL}(n)/\mathrm{SO}(n)$, all quadratic Killing tensor fields are decomposable.

We investigate the relationship between the space of ends of a unimodular random hyperbolic manifold and the dynamics of its geodesic flow. We show that having two ends of infinite volume implies recurrence, while having infinitely many such ends implies positive drift and entropy. We also provide a transience criterion which applies to deterministic hyperbolic manifolds. Our method relies on studying Delaunay graphs over point processes and on the analytic notion of capacity.

Convex subsets of R^n carry two first-order structures: barycentric (affine) structure, with operations C_l(p,q)=(1-l)p+lq for l in [0,1], and betweenness (projective) structure, with ternary relation B(a,x,b) meaning x lies in [a,b]. Isomorphism means affine equivalence in the first and, for n at least 2 and sets open or closed, projective equivalence in the second. We ask when elementary equivalence already determines the body. Main theorem: two compact convex bodies of any dimension, with no regularity hypotheses, are elementarily equivalent in the barycentric language if and only if they are affinely equivalent. The proof rests on a definable compact family of gauges: simplices stationary for barycentric coordinates, with volume bounded below via the anticomplementary simplex. For the betweenness language we develop an interior von Staudt calculus, all quantifiers ranging over the body, making harmonic conjugacy and rational cross-ratio comparisons first-order; a relativization scheme then propagates planar separations to R^n. Consequences: the closed unit ball is separated from sum x_i^4 <= 1 for every n at least 2; and in the plane, projective categoricity holds outright for convex polygons and for bodies with real-analytic, positively curved, non-conic boundary, the latter via a definable finite projective invariant, the conic-cluster set: the points whose every boundary arc contains six co-conic extreme points. A general reduction isolates what remains of the projective conjecture: recovery of boundary coordinates in dimension at least three, and a definable compact gauge, obstructed exactly by non-compact projective symmetry, as on the quadric. For closed noncompact bodies the asymptotic structure is itself elementary: the dimensions of the recession cone and of the lineality space are determined by the betweenness theory, separating the solid cylinder from the slab.

Classical grasp quality metrics assume one deterministic friction coefficient and therefore cannot assess whether a grasp maintains force closure across plausible friction values. We present FIRMGrasp, a family of grasp quality metrics that incorporates friction uncertainty through Conditional Value-at-Risk (CVaR). At confidence level $\beta$, we evaluate the force-closure margin at the mean of the adverse friction tail. This evaluation defines the risk-adjusted margin $\varepsilon^{(\beta)}$, the inscribed-ball radius of the corresponding grasp wrench space. We prove that $\varepsilon^{(\beta)}$ varies monotonically with $\beta$, remains differentiable in the grasp parameters, and certifies that any grasp with $\varepsilon^{(\beta)} > 0$ achieves force closure with probability at least $\beta$. Across 1,599 LEAP Hand and Allegro Hand grasps, $\varepsilon^{(\beta)}$ identifies friction-sensitive grasps that receive high nominal Ferrari-Canny scores, and 53% of the nominally force-closed grasps lose closure in the adverse friction tail. The nominal margin ranks a successful grasp above a failed grasp with probabilities of only 0.53 in the shake test and 0.67 in the pick test, whereas $\varepsilon^{(\beta)}$ achieves 0.63 and 0.78. At an adverse friction coefficient of 0.2, 70% of grasps with positive $\varepsilon^{(\beta)}$ withstand a simulated lift and lateral pull, compared with 25% of grasps with positive nominal margin and nonpositive $\varepsilon^{(\beta)}$. We also synthesize grasps with positive $\varepsilon^{(\beta)}$ for the RealHand L6 and LEAP Hand, both of which retain the object during adverse-friction lifts. In MuJoCo trials with the RealHand L6, 95% of grasps that establish contact and have positive $\varepsilon^{(\beta)}$ retain the object at the same adverse friction coefficient.

A Derivative-Free Optimization (DFO) model is developed and analyzed for solving inverse structured symmetric matrix problems for which the eigenvalues are specified. Some (zero and nonzero) entries are preassigned and cannot be changed, while others should be nonzero but their values are not given. The rest of the entries are completely free. The obtained matrix must meet these criteria and have the specified eigenvalues. This specialized inverse eigenvalue problem is relevant to various applications and is linked to determining the graph, with weights on the undirected edges, of the matrix associated with its sparse pattern. Our novel optimization model requires computing the eigenvalues of a symmetric matrix to evaluate the non-differentiable objective function. We apply deterministic DFO schemes, specifically the global variant GLODS of the well-known family of directional direct search (DDS) methods. We discuss its convergence properties which are based on the fact that the objective function of our model is Lipschitz continuous. Additionally, we explore the potential benefits of using several well-established heuristic strategies to solve the proposed optimization model. We present preliminary numerical results to illustrate and compare the performance of the considered deterministic and heuristic DFO options in various possible scenarios.

For $n$ points in the plane and $\varepsilon>0$, we construct a $(1+\varepsilon)$-spanner with $O(n/\varepsilon)$ edges in which every edge has $\tilde{O}(1/\varepsilon^3)$ crossings, the total number of crossings is $\tilde{O}(n/\varepsilon^4)$, and the ratio between the lengths of any two crossing edges is $O(1/\varepsilon^2)$. Our spanner construction substantially improves on the previous upper bound for the number of crossings in a $(1+\varepsilon)$-spanner, and it is the first spanner construction that ensures $O(1)$ crossings per edge for any constant $\varepsilon>0$. In contrast, we construct $n$ points in the plane for which every $(1+\varepsilon)$-spanner has $\Omega(n/\varepsilon^3)$ crossings, $n$ points for which every $(1+\varepsilon)$-spanner has an edge with $\Omega(1/\varepsilon^{5/2})$ crossings, and 4 points for which every $(1+\varepsilon)$-spanner contains two crossing edges where one is $\Omega(1/\varepsilon)$ times longer than the other.

The Burnside ring of a finite category, introduced by Webb, generalizes the classical Burnside ring of a finite group. However, unlike the classical case, the Burnside ring of a finite category has finite rank if and only if the category is equivalent to a groupoid. In this article, we introduce a new invariant associated with a finite category $\mathcal{C}$, called the simple Burnside ring of $\mathcal{C}$ and denoted by $B^S(\mathcal{C})$. This construction is obtained from simple $\mathcal{C}$-sets and generalizes the classical Burnside ring of a finite group. Moreover, the ring $B^S(\mathcal{C})$ always has finite rank. We develop the basic theory of simple $\mathcal{C}$-sets and study several structural properties of the ring $B^S(\mathcal{C})$. In particular, we determine all ring homomorphisms from $B^S(\mathcal{C})$ to $\mathbb{Z}$, describe its prime spectrum, and obtain a decomposition theorem expressing $B^S(\mathcal{C})$ as a product of simple Burnside rings of strongly connected subcategories.

Clustering is a fundamental problem in statistics, with applications across many scientific disciplines. In many modern applications involving clustering, the primary dataset (the target data) is accompanied by related datasets (the source data). Transferring information from such sources may improve clustering accuracy in the target, making transfer learning for clustering practically important. Despite recent progress, the conditions under which source data improve target clustering remain unclear in high-dimensional settings, even for the canonical Gaussian mixture model. In this paper, we study the clustering problem in a two-community Gaussian mixture model where relatedness is captured by the geometric alignment of the target and source cluster means. We develop a minimax-optimal transfer-assisted clustering procedure and characterize, up to logarithmic factors, the phase transition for consistent target clustering in terms of the signal-to-noise ratios, sample sizes, ambient dimension, and degree of alignment between the datasets. The technique is also extended to adaptively choose between the target-only or the source assisted clustering depending on the target signal strength. Furthermore, we also extend our techniques to accommodate multiple communities and and multiple source datasets. Extensive simulations and an analysis of a human lung single-cell RNA-sequencing atlas demonstrate the practical effectiveness of our methods.

We simultaneously investigate the inviscid and Low Mach number limits on expanding domains for the $3D$ degenerate compressible Navier-Stokes equations, whose viscosity depends on density. Starting from ill-prepared data, we show the limit system is the incompressible Euler system. Our result extends a previous result of Feireisl et al. concerning the constant viscosity Navier-Stokes equations, and is the first to establish the inviscid and incompressible limits at the same time for density-dependent viscosity.

Due to the insecurity of McEliece cryptosystems instantiated with Generalized Reed-Solomon codes, there have been several proposals of McEliece type systems that replace the permutation matrix by a matrix $M$ with larger row and column weight. In many of them, the secret key is still a GRS code. There have been successful attacks on some of those schemes with row and column weight between $1$ and $1 + R$, where $R$ is the rate of the code. The case of weight two and larger has been left open in these works. Subsequently, several authors proposed schemes with weight exactly two and with even higher weight. We provide distinguishers for the public codes appearing in these cryptosystems in the high rate regime. In addition, we give a framework to turn a good enough distinguisher into a key-recovery attack. In the case where the matrix $M$ has row and column weight $2$, we can successfully attack the scheme in the high rate regime using a cube code distinguisher.

We first derive a formula for the mean curvature and the squared norm of the shape operator of equifocal hypersurfaces in simply-connected irreducible symmetric spaces of compact type. The formulas are given explicitly in terms of the tangential focal data of the equifocal hypersurfaces. Third, we study the backward mean curvature flow for equifocal hypersurfaces. The long-time existence of this flow for an equifocal hypersurface was established by Liu and Radeschi. We analyze the time evolution of the mean curvature and the squared norm of the shape operator along the long-time solution, thereby we generalize the result of Liu and Terng for isoparametric hypersurfaces in the sphere. Our analysis also gives an extension of Liu-Terng conjecture on the backward mean curvature flows in the sphere to the simply-connected irreducible symmetric space of compact type.

We introduce a robust OR polynomial framework for composing positive semidefinite certificates across OR constraints. We demonstrate the power of this method in two applications. The first is on acute-free families. A set $\mathcal F=\{(x_i^{(1)},\ldots,x_i^{(r)})\}_{i=1}^M \subseteq (S^{n-1})^r$ is $r$-way acute-free if, for every $i\neq j$, there is a coordinate $t\in[r]$ such that $\langle x_i^{(t)},x_j^{(t)}\rangle\leq 0$. We write $M_r(n)$ for the maximum size of such a set, and $M_r^{\pm}(n)$ for the hypercube restriction. On the hypercube, $r$-way acute-free sets are independent sets for some strong power graph $G_n^{\boxtimes r}$. The Lovász theta number $\vartheta(G_n)$ is multiplicative but exponentially loose, whereas the Schrijver number $\vartheta'(G_n)$ gives the correct order, but is not multiplicative. We bypass this obstruction by proving a general quasi-tensorization result for the Schrijver number. That is, for every collection of graphs $G_1,\ldots, G_r$ satisfying $\vartheta'(G_i)\geq 2$, there is an absolute constant $C$ such that $\vartheta'(G_1\boxtimes \cdots \boxtimes G_r) \leq \prod_{i=1}^r \vartheta'(G_i)^{C\log r \log \vartheta'(G_i)}$. Applying this result gives that $M_r^{\pm}(n) \le M_r(n) \le (2n)^{C_0 r\log r\log(2n)}$ for some absolute constant $C_0$. The second application is on multicolor Ramsey numbers. The $r$-color Ramsey number $R_r(k)$ is the minimum $n$ such that every $r$-coloring of the edges of the complete graph on $n$ vertices contains a monochromatic copy of $K_k$. In a breakthrough result, Balister et al. [arXiv:2410.17197] showed that $R_r(k)\le \exp(-\Omega(k/r^{12}))r^{rk}$ via a geometric lemma. By improving the $r$ dependency in their geometric lemma via the OR polynomial framework, we prove that $R_r(k)\le \exp(-\Omega(k/(r^9(\log r)^6)))r^{rk}$.

We prove that the vertices of any $n$-vertex graph can be partitioned into pieces of radius $r = O(\log n)$ such that the sum of the sizes of their closed neighborhoods is at most $4n$. This answers a recent question of Bukh and Dubroff and directly yields an improvement to their upper bound on the optimal cover time of the $\epsilon$-steered random walk. We also demonstrate that our bound on $r$ is best possible up to a constant factor for graphs with strong vertex expansion.

We prove that a contractible $3$-manifold admitting a complete Riemannian metric of nonnegative scalar curvature is diffeomorphic to $\R^3$. We also prove that, if the interior of a compact handlebody of genus $\gamma$ admits such a metric, then $\gamma\leq1$. This answers two open questions posed by Wang and Gromov, respectively.

Muscle redundancy can be formulated as a constrained selection on a time-varying convex set of feasible activations. We introduce Torque Fiber Proximal Dynamics (TFPD), where activation evolves as the Euclidean projection of the previous state onto a convex polytope defined by torque equality and physiological bounds. TFPD is equivalent to a backward-Euler discretization of a sweeping process and a variational inequality with a maximal monotone normal cone operator. Within this framework, antagonist recruitment emerges naturally under a projection-based control hypothesis as a structural consequence of active-set transitions induced by the geometry, not as an imposed cost. We derive sufficient KKT-based conditions for antagonist activation under generic moment-arm asymmetry, linking boundary projection, strict complementarity, and torque coupling. We validate TFPD on a three-muscle elbow model using exact quadratic programming, benchmark it against five classical methods and a random baseline, and achieve a Pearson correlation of 0.68-0.73 with triceps EMG envelopes across ten subjects without any tunable cost-weight parameters. Sensitivity analysis and a three-dimensional activation trajectory visualization demonstrate robustness. TFPD connects neuromuscular coordination to proximal point theory, variational inequalities, and projected dynamical systems.

This manuscript investigates the Cauchy problem for an incompressible fluid flow governed by a dual-scale hereditary memory, representing a viscoelastic variant of the classical Navier-Stokes equations that captures anomalous momentum transport. The non-local dissipation breaks exact global scale invariance, dictating a pseudo-differential analysis within the Hörmander class $S^{-2}_{1,0}$ where the spatial gradient induces a fractional temporal penalty. We rigorously establish $L^q-L^p$ decay estimates and identify the critical Lebesgue threshold $p_c = N (\frac{1+\alpha_\infty}{1-\alpha_\infty})$. In the supercritical regime $1 < p < p_c$, we bypass the loss of spatial localization by mapping the frequency-modulated bilinear flow directly into Fourier space; by utilizing Bernstein's inequalities, we prove instantaneous norm inflation at the origin and confirm intrinsic ill-posedness. Conversely, in the topological limit $p \to \infty$, we demonstrate that the dual-scale memory structurally prevents the collapse traditionally observed for classical fluids within the maximal critical Besov space $\dot{B}^{-1}_{\infty, \infty}$. By exploiting an asymmetric interpolation within Bony's para-differential calculus, we prove that the temporal smoothing overpowers the high-high convective resonant cascade. This delicate analytical balance confines ill-posedness to the non-separable high-frequency tail of the Besov topology, thereby establishing global-in-time Hadamard well-posedness for small initial data possessing high-frequency adherence within $\dot{B}^{-\kappa}_{\infty, \infty}(\mathbb{R}^N)$, a well-posedness regime strictly broader than the separable little Besov closure $\dot{b}^{-\kappa}_{\infty, \infty}(\mathbb{R}^N)$, where $\kappa = \frac{1-\alpha_\infty}{1+\alpha_\infty}$.

It is a surprising, modestly known fact that when given a single observation from a normal distribution with unknown mean and unknown variance, valid and non-trivial confidence intervals for the mean can be constructed. These intervals are presented in papers fully formed, providing limited intuition for how they arise or how to generalize them. We show that these intervals can be constructed in a principled way using two separate Bayesian reasonings. In the first, for any continuous symmetric location-scale family (under mild regularity conditions) with $n=1$ observation, we derive priors which produce $(1 - \alpha)100\%$ credible intervals that are, asymptotically in the confidence level $\alpha \rightarrow 0$, valid $(1 - \alpha)100\%$ confidence intervals. In the second, we show that the $n=1$ frequentist intervals can be seen as $t$-intervals augmented with a prior value, and that these augmented $t$-intervals are equivalent to inverted frequentist tests using a Bayes factor (using appropriate priors) as a test statistic. For $n \geq 2$, our credible interval approach does not maintain the confidence level. However, for $n \geq 2$, our augmented $t$-intervals produce valid confidence intervals with lower expected squared width in parts of the parameter space than the Student $t$-intervals, indicating improvements when prior knowledge is available. We demonstrate these methods on an $n = 3$ dataset of hyperbolic excess velocities of interstellar objects.

This article is primarily expository, but it also contains several new observations and reformulations concerning the Möbius function, probabilistic models, dynamical systems, and the Riemann hypothesis. Its starting question is classical: in what sense can the Möbius function be said to behave randomly? We begin with Denjoy's random-walk heuristic and place it in the context of later work on random multiplicative functions, short intervals, and Möbius pseudorandomness. We then develop two smoothing forms of the classical Mertens criterion for the Riemann hypothesis. The first uses the discrete Laplace transform \[ \Phi(t)=\sum_{n\geq1}\mu(n)e^{-nt} \] and identifies RH with the condition \[ \Phi\in L^p(0,\infty) \qquad\text{for every }1\leq p<2. \] The second uses normalized Möbius Fourier polynomials and local moments on arcs of length comparable with \(1/N\). The paper also revisits the author's earlier criterion for RH in terms of discrete measures, as presented in Broughan's account of analytic equivalents of RH. Denjoy's heuristic is made precise in a simple independent coefficient model, and this model is carefully distinguished from the modern theory of random multiplicative functions. The resulting coefficient space also gives a measure–category contrast: the relevant \(L^p\)-property has full measure but is topologically meagre. A further point of the paper is dynamical: the multiplicative semigroup of positive integers acts naturally on the coefficient space, and the Möbius sequence is a distinguished arithmetic point for this action. The aim throughout is explanatory, while keeping these new observations visible: to show how arithmetic, probability, Fourier analysis, Mellin transforms, and dynamics fit together, and to indicate what modern results add to the older random-walk picture.