Papers

Fluid antenna systems (FASs) offer a promising solution for unmanned aerial vehicle (UAV) air-to-ground (A2G) communications by enabling reconfigurable radiation characteristics. Addressing the limitations of traditional models in capturing the dynamic port configuration of FAS and the near-field nature of UAV communications, this paper proposes a dynamic port-reconfigurable near-field channel model for FAS-assisted UAV-to-mobile user (MU) links. Furthermore, we develop a FAS-adaptive subarray partition scheme utilizing a greedy strategy. By decomposing line-of-sight (LoS) and non-line-of-sight (NLoS) components and integrating UAV motion dynamics with FAS port activation states, the proposed model accurately characterizes the non-uniform spatial distribution of near-field channels. The subarray partition scheme dynamically groups active ports to satisfy near-field conditions while significantly reducing computational complexity, supported by a dynamic update algorithm that efficiently handles subarray adjustments during port switching. To avoid low effective gain and deep-fading ports in dense FAS configurations, a channel gain-based selection strategy is employed to prioritize high-gain ports. We derive and analyze the modeling accuracy and channel capacity, investigating the impact of FAS dimensions, port spacing, active port count, and UAV dynamics on system performance. Finally, the computational complexity of the subarray partition scheme is evaluated, verifying its advantages for real-time applications and providing a theoretical foundation for the design and analysis of FAS in dynamic scenarios.

In recent years, several generalizations of the numerical radius for bounded linear operators on Hilbert spaces have been introduced and extensively studied. These generalizations provide refined tools for investigating operator inequalities and spectral properties. In this paper, we investigate several generalized numerical radii and establish relationships among them. We also establish several relationships among these generalized numerical radii and show that the $(s,t)$-weighted numerical radius can be represented as a rescaled form of the $t$-weighted numerical radius.

We investigate Bohr-type inequalities for bounded analytic functions, particularly in the presence of a parameter or under convex combination structures. Most of our findings are sharp, and we also generalize a number of earlier results.

Large-scale renewable power-to-hydrogen (ReP2H) plants in cold regions suffer from prolonged startup and repeated thermal stress during frequent startup-shutdown operation. The situation becomes worse due to the lack of coordinated heat management among the alkaline electrolysis stacks, balance of plant (BoP), plant thermal utility system (PTUS), and plant building. This paper presents a plant-wide thermal topology and a hierarchical electricity-heat management framework to address the issues. Bidirectional thermal coupling between the stack cluster and PTUS enables preheating, thermal standby, and waste heat recovery, while minute-scale production scheduling is coordinated with second-scale thermal regulation. Case studies based on an 80 MW plant in Northern China show that the proposed framework eliminates cold startups in year round, increases hydrogen yield by 1.50, improves energy and exergy efficiencies by 0.99 and 4.33 percentage points, respectively, and reduces the levelized cost of hydrogen by 3.22%. It also reduces thermal fatigue damage and startup-shutdown-induced voltage degradation.

Multi-path sensing, which aims to extract the geometric attributes of multiple propagation paths, is expected to be a key functionality of 6G. A movable antenna (MA) can enable this functionality by synthesizing an aperture through mechanical motion. However, existing MA-based sensing methods typically rely on exhaustive scanning over the entire movable region, resulting in significant control overhead and sensing latency, which limit their practicality for agile sensing. To address this challenge, this paper develops a prior-guided agile multi-path sensing framework that leverages weak prior angle-of-arrival (AoA) statistics as side information. The proposed framework is built on two key steps. First, the movable plate's three-dimensional orientation is optimized only once to configure a mechanically feasible scan region that enhances path visibility while preserving inter-path discriminability, guided by Fisher information analysis. Second, given the optimal plate orientation, the MA performs only two linear scans, whose non-collinear spatial phase projections are fused with the prior AoA statistics through a maximum a posteriori (MAP)-based estimator to recover the elevation and azimuth AoAs of multiple paths. The estimated AoAs are subsequently used to extract the times-of-arrival (ToAs) by enhancing the target path component while suppressing interference from other paths. With only one orientation adjustment and two linear scans, the proposed framework enables agile multi-path sensing with significantly reduced control overhead and latency, while achieving AoA and ToA estimation accuracy close to the single-path benchmark.

Let $A$ be a complex abelian variety with $\NS(A)=\Z[L]$, $L$ ample. We show that a vector bundle filtered by line bundles splits uniquely into unipotent blocks, determine when it is ACM with respect to $L$, and show that it is never Ulrich. For a rank-two Ulrich bundle we bound from below the degree of the defect of every saturated line subbundle, compute the Hilbert polynomial of the defect, and show that the bound is attained exactly when an explicit cohomological support locus is nonempty. On abelian surfaces this locus is an ample curve in $\Pic^0(A)$; as a consequence, every rank-two Ulrich bundle is a Serre extension of a fixed numerical type, which we characterize and construct for every ample $L$. For $g\ge3$, we reduce the equality case to the existence of subcanonical codimension-two subschemes with prescribed cohomology.

We study the recovery of a time-dependent potential in a parabolic equation on a conformally transversally anisotropic manifold. The initial state and a lateral Dirichlet datum supported near the front face are used as inputs, while the terminal state and the Neumann trace near the back face are observed. Assuming injectivity of the attenuated geodesic ray transform on the transversal manifold for all sufficiently small constant attenuations, we prove that these partial input-output data uniquely determine the potential. No simplicity assumption is imposed on the transversal manifold. The proof combines a conformal reduction, transversal Gaussian beam quasimodes, boundary Carleman estimates, and geometric optics solutions with prescribed lateral supports.

We present PDHCG-CQP, a GPU-accelerated first-order solver for large-scale conic convex quadratic programming. PDHCG-CQP supports affine constraints and Cartesian products of nonnegative, second-order, rotated second-order, exponential, and three-dimensional power cones. At its core is a restarted averaged primal-dual hybrid gradient (PDHG) method, whose primal update is computed inexactly by solving a conic quadratic proximal subproblem with projected gradient iterations. We establish local linear convergence of the restarted averaged scheme with both exact and inexact primal proximal evaluations under a uniform local quadratic-growth condition on the smoothed primal-dual gap. We further show that this condition holds under strict complementarity by exploiting a rotated second-order-cone lifting together with local primal and dual regularity conditions. Our C/CUDA implementation combines matrix-free linear algebra, batched cone projections, adaptive inner solves, reflected-Halpern acceleration, and fully device-resident KKT residual computations. It also supports multi-GPU execution through a two-dimensional partitioning of the problem data. Extensive experiments on standard and large-scale quadratic programming (QP), convex quadratically constrained quadratic programming (QCQP), second-order cone programming (SOCP), and quasilinear Fisher equilibrium benchmarks demonstrate that PDHCG-CQP achieves state-of-the-art robustness among first-order solvers while scaling efficiently to 8 GPUs and instances with up to $4.4\times10^8$ stored primal coordinates. PDHCG-CQP is open source and available at https://github.com/Lhongpei/PDHCG.

The motion of a projectile released from a simple pendulum is analyzed, with particular emphasis on investigating the optimal release angle that maximizes the horizontal range and the corresponding maximum range. This system serves as a simplified model of the Tarzan jump problem. Using simple analytical methods, the optimal release angle is shown to be characterized by a cubic equation and to increase with the initial velocity. In addition, asymptotic expressions for both the optimal angle and maximum range are derived in the limits of low and high initial velocity.

In this work, we revisit a continuous-time two-compartment pharmacokinetic model of human ethanol metabolism originally proposed by Levitt and Levitt. We first establish the positivity of the solutions, investigate the existence and uniqueness of a positive equilibrium, and analyze its local and global asymptotic stability. As a result, the global dynamics of the original ethanol metabolism model is characterized, thereby complementing and extending the analytical results reported in the original study. Second, building upon the foundational two-compartment model of Levitt and Levitt, we replace the Michaelis–Menten metabolism rate with a general class of nonlinear functions, including both monotone and nonmonotone forms. This generalization allows us to examine a wider range of physiologically motivated scenarios and reveals richer dynamical behavior. We then establish the positivity of the solutions, study the existence and asymptotic stability of equilibria, and analyze the long-term dynamics under different ethanol input conditions. In particular, for a specific nonmonotone family, we establish the occurrence of a nondegenerate saddle-node bifurcation through center-manifold analysis. Furthermore, we prove the absence of nonconstant periodic orbits in the generalized model without assuming monotonicity of the metabolic rate functions. These findings extend the qualitative analysis beyond equilibrium stability and provide further insights into the complex dynamics of human ethanol metabolism. Finally, numerical experiments are conducted to illustrate the theoretical findings. The numerical results provide further support for the theoretical analysis.

We investigate core quandles and idempotents in quandle rings of core quandles. We answer several questions on the rank of core quandles and nontrivial idempotents in quandle rings. We also present solutions to two questions raised in a recent paper about non-trivial idempotents in quandle rings $\mathbb{Z}[R_5]$ and $\mathbb{Z}[C_5]$, where $\mathbb{Z}$, $R_5$ and $C_5$ are the ring of integers, the dihedral quandle of order 5, and the commutative quandle of order 5, respectively. We then study units in extended quandle rings of a trivial quandle and the Joyce quandle, and nilpotent elements in extended quandles rings of a trivial quandle, where the ground ring is an integral domain. As a consequence, we show that the quandle ring and the extended quandle ring of a trivial quandle are not nil clean rings. We also explore prime rings and semi-prime rings among quandle rings. We introduce zero-divisor graphs of quandle rings and find an intriguing mirror symmetry among the in-degree and out-degree of the vertices. Moreover, we find a bivariate polynomial in the ring $(\mathbb{Z}_{2n+1}[Q])[X,Y]$ that determines the commutative quandle of order $2n+1$, where $2n+1$ is prime.

Low Earth orbit (LEO) satellite networks are envisioned as a promising solution for providing ubiquitous connectivity and narrowing the digital divide. The extensive footprint of LEO satellite constellations enables broad coverage, resulting in spatially non-uniform traffic demand across the serviced areas. Meanwhile, stringent on-board power constraints make power-intensive transmission architectures less attractive and motivate energy-efficient transmission strategies that effectively exploit scarce satellite network resources. To this end, this paper proposes a cooperative transmission framework that jointly accounts for non-uniform traffic demand and network-wide power consumption. Each LEO satellite integrates hybrid precoding (HPC), radio frequency (RF) chain activation, and hardware quantization, while user-equipment (UE)-centric satellite clusters are organized using statistical channel state information (sCSI) and traffic demands. A framework for joint optimization of cooperative transmission architecture and resource allocation is designed to maximize demand-aware energy efficiency (EE), resulting in a mixed-integer nonlinear program (MINLP) for which finding a globally optimal solution is generally intractable. Accordingly, a two-stage algorithm is developed under a distributed linear precoding structure, in which a modified cross-entropy (CE) method searches over discrete variables, while fractional programming is employed for transmit power allocation. Numerical results indicate that the proposed framework outperforms benchmark schemes while accounting for traffic demands and EE.

In this work, we consider a class of continuous-time autonomous dynamical systems that model various important phenomena and processes encountered in real-world situations. We construct a second-order nonstandard finite difference (NSFD) method that simultaneously preserves two essential properties of the dynamical systems for all finite step sizes, namely the positivity of the solutions, the set of equilibrium points and their asymptotic stability. This NSFD method is constructed based on an appropriate choice of nonstandard denominator functions and a weighted discretization of the right-hand side functions. Under easily-verified conditions, the denominator functions guarantee second-order convergence, whereas the weights ensure the dynamic consistency. By taking advantage of the specific structure of the right-hand side functions, a simple discretization is utilized instead of the nonlocal discretization approaches commonly used in previous works. This simplifies the construction of the proposed NSFD method and, in particular, makes its asymptotic stability analysis easier. As an illustration and an important application, we apply the constructed second-order NSFD method to a well-known two-stage structured species model with recruitment. Consequently, a simple second-order NSFD scheme for the considered two-stage structured species model is derived, improving upon a first-order NSFD scheme constructed in a previous work. Numerical experiments demonstrate the advantages of the second-order NSFD scheme over a standard second-order numerical method, namely, the explicit trapezoidal method. The proposed NSFD method is simple and can be applied to a broad class of dynamical system models arising in both theory and applications. Moreover, it can be readily combined with the Richardson extrapolation technique to improve its accuracy.

T-quaternion rings are a special class of commutative unital subrings of the quaternion rings over commutative rings. Permutation Polynomials (PP) are typically defined over finite fields, such as $\mathbb{F}_{q}$, where $q$ is a prime power. This article examines permutation polynomials over finite T-quaternion rings, focusing specifically on linear, quadratic, and cubic cases. The study begins with permutation polynomials over the rings $\mathbb{Z}_{n}$, before extending the discussion to T-quaternion rings.

High-dimensional Principal Component Analysis (PCA) has become an essential tool in modern data analysis, offering dimensionality reduction and feature extraction. However, the presence of missing data introduces significant challenges, distorting the performance of PCA and complicating statistical inference. In this paper, we study the asymptotic behavior of PCA under a spiked population model with missing observations, leveraging recent advances in random matrix theory. We demonstrate that while the spiked sample eigenvalues exhibit asymptotic normality, the limiting parameters differ substantially from those in the complete data case, reflecting the non?trivial influence of the missing data mechanism. As an application of our results, we propose a test to evaluate the independent structure of a spiked population.

For every $n\geq1$ and $\gamma>-1$, we construct $f\in L^1(\mathbb B_n,dv_\gamma)$ whose Toeplitz form extends to a bounded, noncompact operator on the weighted Bergman spaces $A^2_\gamma(\mathbb B_n)$ and whose Berezin transform vanishes at the boundary. Boundary vanishing of the Berezin transform therefore does not imply compactness for Toeplitz operators with integrable symbols, in contrast with operators in the Toeplitz algebra generated by bounded symbols; this answers a question of Bauer and Isralowitz. The construction approximates rank-one operators in norm by Toeplitz operators with smooth, compactly supported symbols and transports suitably separated blocks toward the boundary. On the unweighted disk, a diagonal version produces a real symbol with the same vanishing and noncompactness properties, for which the operator is positive and Zorboska's two-sided localization condition holds at $p=3$. Her hypothesis $p>3$ is shown to be sharp.

Bicycle (two-block circulant) quantum low-density parity-check (LDPC) codes include some of the best known small quantum codes, yet their design has relied on group-algebra formulations in which the dimension and distance are accessible only through matrix computation. We show that in the cyclic case the construction collapses into the polynomial ring $\F_2[x]/(x^{l}-1)$: self-orthogonality is automatic, the quantum dimension is read off from a polynomial gcd, and the minimum distance is certified exactly through the Calderbank correspondence to additive codes over $\F_4$, turning code search into an algebraically pre-filtered enumeration that reaches parameter regimes poorly covered by existing tables. A computer search based on this framework recovers the short codes $[[42,12,4]]_2$ and $[[62,12,4]]_2$ and produces a family of codes with competitive figure of merit $kd^2/n$, including $[[66,20,7]]_2$ with $kd^2/n=14.85$, above the bivariate bicycle code $[[144,12,12]]_2$ ($kd^2/n=12$) at less than half the block length, together with $[[46,2,8]]_2$, $[[66,2,9]]_2$, $[[66,4,8]]_2$, $[[66,6,8]]_2$ and, at $n=90$, $[[90,16,6]]_2$, $[[90,18,6]]_2$, $[[90,20,6]]_2$. An exhaustive census at $n=48$ delineates the boundary of this picture: we exhibit a $[[48,10,6]]_2$ code from a minimal $48$-element group (the Aydin–Tamo–Barg realization uses $72$ elements), and prove that distance $5$ forces a stabilizer-rank loss, which excludes $[[48,10,5]]_2$ from the weight-$8$ symmetric coset family. The framework thus opens a systematic route to bicycle-type quantum LDPC codes beyond the reach of group-theoretic searches, and identifies exactly where genuinely coset-theoretic phenomena begin.

We give a complete characterization of the nef, effective, pseudo-effective, and semi-positive cones on Hopf manifolds. In particular, by computing the Ueda classes on non-diagonal Hopf surfaces, we show that the invariant elliptic curve is nef but not semi-positive. Via Bott–Chern cohomology calculations, we construct new examples of nef but non-semi-positive line bundles on Hopf manifolds of arbitrary dimension. We also prove that a minimal compact complex surface \(X\) containing an elliptic curve \(C\) and a nef line bundle \(L\) such that \((X,C,L)\) has finite generalized Ueda type is either a Hopf surface or Serre's example.

We investigate positive weak solutions of the critical $p$-Laplace equation $$ -\Delta_p u = u^{p^*-1} h(u), \qquad 1 < p < n, $$ where $h$ is a positive, bounded, continuous, and nonincreasing function. Our first main result is a complete classification of normalized, bounded, positive entire solutions for every equation in a compact family determined by $h$: Any such solution must coincide with an Aubin–Talenti profile. Moreover, the existence of an Aubin–Talenti profile as a solution implies that $h$ is constant on the entire interval of values attained by that profile. Subsequently, applying this classification result, we establish the following scale-invariant Schoen-type estimate $$ \left(\sup_{B_R} u\right)\left(\inf_{B_{2R}} u\right)^{p-1} \le C R^{p-n} $$ for nonnegative weak solutions defined in $B_{3R}$. As a direct corollary, we apply the profound result of Ciraolo–Gatti \cite{CG2026} to obtain a fully unrestricted Liouville theorem: Every positive entire solution must coincide with an Aubin–Talenti profile. For the purely critical equation, we also show that the corresponding Liouville classification is equivalent to a Schoen-type Harnack inequality. The arguments in this work give quasilinear versions of the Kelvin transform and the method of moving spheres, which was previously available only in the semilinear setting, and also yield alternative proofs of the classical results. The technique developed here can likely be extended to a wider class of Liouville-type problems for critical equations.

Sample $n$ i.i.d. points from a triangle, according to some continuous density function. Given two vertices $A,B$ of the triangle, what is the maximum number of samples that form a convex chain with initial point $A$ and terminal point $B$? We show that to leading order, the answer is $cn^{1/3}$, generalizing a result of Ambrus and Bárány that considered uniformly distributed points. Furthermore, we express the constant $c$ using a variational formula whose maximizer (if unique) gives the limiting curve formed by the longest convex chain. By comparison, for $n$ i.i.d. samples from the unit square, the length of the longest monotone chain is asymptotically $c'n^{1/2}$. Despite the difference in scale, our formula is nicely connected to one established for $c'$ by Deuschel and Zeitouni.

An efficient aggregation-based multigrid approach is developed in this work to solve linear algebraic systems generated by discretizing the Landau-Lifshitz equation in micromagnetics. The discretization couples the first-order backward differentiation formula with first-order extrapolation, and the underlying equation describes magnetization dynamics within micromagnetic simulations. For these linear systems, standard iterative methods and conventional multigrid solvers intended for symmetric problems suffer from deteriorated convergence under mesh refinement and poor computational performance. Existing algorithms struggle to handle the system's intrinsic features, namely its sparse spectral properties and non-symmetric matrix structure. To overcome these limitations, a multigrid framework is constructed, where smoothing and coarse-grid correction operators are customized for the target linear system. Numerical tests confirm the robustness and mesh-independent convergence of the resulting method. Compared with well-established Krylov-subspace solvers and conventional multigrid techniques, the aggregation-based multigrid solver cuts down iteration counts and overall computational cost considerably, while producing faithful representations of magnetization dynamics.

We study a restarting Josephus process in which the participants retain their linear order and counting restarts at the current leftmost survivor after every deletion. For step size $m$, put $q=m-1$, and let $F_n(q)$ denote the initial position of the survivor. Reverse insertion gives $F_1(q)=1$ and $F_k(q)=F_{k-1}(q)+\mathbf{1}_{\{q\bmod k<F_{k-1}(q)\}}$. Writing $L_n=\operatorname{lcm}(1,\ldots,n)$, we establish three results for the compatible residue system in this recurrence. First, the full period group of $F_n$ is exactly $L_n\mathbb{Z}$. Second, $F_n$ is surjective onto $\{1,\ldots,n\}$. The proof is constructive and unconditional but computer-assisted: a Chinese-remainder construction and explicit prime estimates reduce it to a finite exact certificate. Third, if $\widetilde Q_n$ is uniform modulo $L_n$, then $(F_n(\widetilde Q_n)-1)/(n-1)$ converges to a symmetric, nondegenerate law on $[0,1]$. A common Haar coupling yields almost-sure and $L^r$ convergence for every $1\le r<\infty$, together with an $O(n^{-1/4})$ bound in $W_1$. Logarithmic boundary-mass estimates rule out every symmetric beta law. We also formulate endpoint dominance as an open problem, prove strict dominance over the two nearest internal positions for every $n\ge4$, exclude prime levels as minimal counterexamples, and verify the claim exactly through $n=49$.

To comply with stringent data privacy regulations, federated unlearning (FU) has emerged as a critical paradigm. However, its implementation over wireless networks introduces severe communication latency and reliability challenges due to iterative calibration requirements and physical-layer channel uncertainties. In this paper, we investigate the problem of delay minimization for federated unlearning networks (FUN). Specifically, we establish a comprehensive system model that jointly incorporates the convergence behavior of the FUN algorithm, local device computation dynamics, and a worst-case robust transmission model operating under bounded channel state information (CSI) error. To solve the resulting non-convex joint resource allocation problem, we propose an efficient iterative algorithm. By exploiting the monotonicity and convexity properties of the system constraints, the problem is decomposed via a uniform scan over the local accuracy parameter, within which the optimal delay, bandwidth, power, and computation frequency are determined utilizing nested bisection and golden-section searches. Both theoretical analysis and extensive numerical results demonstrate that the proposed algorithm achieves polynomial complexity and significantly reduces the overall unlearning completion time compared to conventional baseline schemes.

We develop a rigorous mathematical framework for the governance of systems of K self-adapting generative AI models under the principles of Model Risk Management (MRM). When multiple models share a meta-learning coupling through an interaction matrix, the per-agent Lyapunov analysis that underpins standard MRM is provably insufficient: individual agents can each satisfy their declared stability bounds while the joint system is in a regime of emergent ensemble-level drift. We formalize this gap through the Joint Lyapunov Proof (JLP)—a cryptographic and stochastic protocol that attests, without revealing proprietary weights, that the aggregate dynamics satisfy MRM Ongoing Monitoring standard at every validation epoch. Our main contributions are the following. We give a complete characterization of the infinitesimal generator of the joint quadratic Lyapunov function. We derive the exact critical coupling threshold above which the system loses mean-square stability. We prove a Noise-Floor Theorem and identify the correct target for zero-knowledge attestation. A per-epoch Succinct Non-Interactive Argument of Knowledge (SNARK) on the live weights is derived. All theoretical claims are validated against five numerical studies using a multi-agent softmax system.

Sum-rank-metric codes have recently attracted considerable attention of many researchers, due to their applications in network coding, space-time codes and distributed storage. MSRD codes are those good codes attaining the Singleton bound in the sum-rank metric. However, MSRD codes do not exist for some dimensions. Motivated by this fact, we introduce the notion of quasi-MSRD (QMSRD) codes and provide some properties of them. What is more, we find that not every QMSRD code has a QMSRD dual code, so we give the definition of dually QMSRD codes, whose dual codes and themselves are both QMSRD. Finally, we characterize such codes, derive their support, rank-list and sum-rank distributions as well as compute their generalized sum-rank weights.

We study a nearest-neighbor non-Hermitian spin chain obtained from one member of an exact non-braided scalene Yang–Baxter triple. Its local Hamiltonian density violates both the difference-form Reshetikhin condition and its general non-difference counterpart, obstructing its realization by a differentiable homogeneous regular solution of the ordinary Yang–Baxter equation. The transfer matrices constructed from the regular member do not commute among themselves at distinct spectral parameters. Nevertheless, the scalene Yang–Baxter relation implies cross-commutativity with another transfer matrix constructed from the third member of the scalene triple. We evaluate the latter for arbitrary chain length and show that, on even periodic chains, it is a finite generating function of a non-obvious staggered nilpotent symmetry. The resulting conserved hierarchy belongs entirely to the algebra generated by this single symmetry and hence does not constitute an extensive family of algebraically independent charges. Nevertheless, this example demonstrates that scalene Yang–Baxter triples can act as an algebraic symmetry-discovery mechanism beyond the ordinary self-commuting transfer-matrix framework.

An extension theory for nearrings along the lines of the Schreier extension for groups and the Everett extension for rings is given. The semidirect sum of two nearrings is a special case of this theory. Already there are many examples of purpose built nearring constructions in the theory of nearrings which fall under this semidirect sum construction.

We study the nested permutation classes $Av(4123,4231,4312)\subset Av(4123,4312)$. We relate the corresponding generating functions $D^{(2)}(z)$, $D^{(3)}(z)$ to $q$-difference equations. Solving these equations allows to show that the generation functions are not D-finite, and that $[z^n]D^{(3)}(z)\sim 0.0189672325071988\ldots\, (4.46840899160393\ldots)^n$, $[z^n]D^{(2)}(z)\sim0.0849833632890843\ldots\, (3+2\sqrt2)^n n^{-3/2}$.

Forward uncertainty propagation in complex physical systems can induce structured covariance across field-valued outputs. For a probabilistic surrogate, the total predictive covariance comprises the covariance of conditional means across input realizations and the average conditional predictive covariance. Probabilistic DeepONet (Prob-DeepONet) provides lightweight uncertainty quantification by predicting pointwise Gaussian means and variances in a single forward pass, but its conditional predictive covariance is restricted to a diagonal form. To represent cross-location conditional dependence without explicitly parameterizing a full high-dimensional covariance matrix, we develop a two-step mean-variance DeepONet (two-step MV-DeepONet) through two principal modifications. First, two-step training is used to decouple output-basis learning from the input-to-coefficient mapping, together with basis orthogonalization and subspace rotation. Second, Gaussian probabilistic modeling is transferred from the high-dimensional physical output space to the low-dimensional rotated coefficient space. Mapping these probabilistic coefficients through the shared basis induces a generally non-diagonal conditional predictive covariance in the physical output space while retaining single-pass inference. A Frobenius-norm error decomposition and corresponding upper bound identify low-rank covariance compressibility, trunk-subspace approximation, finite-sample statistical error, and coefficient-space covariance estimation as the principal factors governing covariance recovery. Numerical experiments on three representative problems governed by partial differential equations (PDEs) and a hypersonic blunt-body aerothermal problem show improved generalization, more structured uncertainty bands, and accurate recovery of off-diagonal correlation patterns compared with Prob-DeepONet.

Model risk management (MRM) guidance assumes a static model lifecycle, in which models are developed, independently validated, and implemented without further autonomous modification. Continually self-adapting generative AI systems — models that update their own weights during production deployment — fundamentally violate this assumption and render point-in-time validation inadequate. This paper addresses the resulting governance problem in two parts. Part I develops a rigorous telemetry architecture for such models, operating simultaneously in discrete and continuous time. We establish a Minimal Sufficient Statistic for audit purposes, construct a tamper-evident Merkle chain for discrete weight sequences, derive the appropriate continuous-time generalization via the Ito formula, and propose event-driven logging via KL divergence stopping times that is both computationally tractable and meaningful for validation. Part II asks what happens when the model provider is adversarial. A firm deploying such a model has strong incentives to conceal learning updates that would trigger mandatory validation review. We formalize this as the Model Hiding Problem and provide a systematic taxonomy of six distinct attack strategies against the Part I architecture, spanning discrete and continuous time, with a formal countermeasure for each. Together the two parts establish a dual-regime architecture in which continuous telemetry is necessary but not sufficient, narrowing—but never replacing—periodic invasive audit. The framework is model-architecture-agnostic and is designed to satisfy the three pillars of traditional MRM.