Multimodal large language models (MLLMs) are increasingly being evaluated for medical applications, where computational constraints often make prompting strategies the only practical alternative to fine-tuning. Such strategies are generally assumed to support diagnostic reasoning, yet their potential failure modes in medical MLLMs remain poorly characterized. We analyze FundusExpert-1B, an open-source ophthalmology MLLM, on a hemorrhage versus drusen discrimination task using the public BRSET dataset, adopted here as a controlled testbed for our analysis. (i) A controlled probe with artificially injected markers confirms that the model retains coarse, region-level spatial grounding. (ii) Compared with zero-shot inference, one-shot textual prompts bias predictions toward the prompted finding. (iii) When an overlaid lesion contour is paired with an inconsistent textual claim, the textual prompt overrides the correct visual cue: overall accuracy drops from 75% to 46% relative to the visual-only condition, and Chain-of-Thought (CoT) reasoning is associated with further degradation rather than self-correction. Although limited to a single model and dataset, our findings suggest that prompting strategies alone may be insufficient for the safe clinical deployment of medical MLLMs.

As artificial intelligence (AI) tools get increasingly deployed for mental healthcare, public trust in these systems remains uncertain. It is unclear how clients perceive AI involvement in counseling interactions, particularly in moments of crisis that require empathy and connection. To address this gap, we analyzed 75,777 crisis counseling conversations from a human-staffed WhatsApp helpline in India to characterize how often clients suspected they were speaking to AI, what triggered those doubts, and how counselors responded. Though no conversations actually involved AI assistance, the proportion of conversations where clients suspected AI use increased from 0.8% in June 2024 to 2.6% in March 2025. Within suspicious conversations, 21.5% of clients stated an explicit preference for humans. Client suspicion primarily arose in the first half of messages (68.3%), and when counselors offered reassurance (e.g. 'I assure you; this is not ai!'), clients continued to press or ended the conversation 17.6% of the time. As AI tools get increasingly integrated into counselor workflows, understanding these dynamics is essential for designing AI systems that preserve the therapeutic relationship between counselors and clients.

Interactive visual analytics workflows are often disrupted by rigid filter panels and context switches that break analysts' cognitive flow. We introduce FluidViews, a web-based framework that elevates filters to first-class, manipulable objects through two novel direct-manipulation interactions. Copy-as-Highlight enables users to duplicate any visual mark into a persistent highlight token for rapid, transient cross-view comparison, while Drag-as-Filter allows analysts to pick up a mark and drop it onto another view to apply context-sensitive filters in place no menus, panels, or modal dialogs required. An optional pop-out micro-view provides on-demand, spatially independent subviews for detailed inspection without disrupting the primary workspace. By embedding these lightweight gestures into coordinated multi-view environments, FluidViews preserves analytic momentum, reduces cognitive overhead, and supports fluid, multi-step exploration across heterogeneous datasets. We describe the system's design and implementation, illustrate its application in exploratory workflows, and discuss how tangible filter objects can transform interactive data exploration.

Human-led truck platooning, where a human-driven truck leads one or more autonomous followers, offers significant benefits in fuel efficiency, safety, and traffic flow. However, its successful deployment hinges on trust between the human driver and the automated systems. Unlike conventional automation, trust in this context is inherently bidirectional: the human must trust the autonomous followers, and the followers must reliably interpret and respond to the human's behavior. While prior research has extensively studied human trust in automation, the reciprocal nature of trust, especially considering the expertise of professional truck drivers, remains underexplored. This paper develops a conceptual framework of bidirectional trust for human-led platooning systems. Drawing on established trust theories (ability, benevolence, integrity) and insights from truck driver psychology, we propose distinct dimensions for human-to-automation trust and automation-to-human trust. To move beyond conceptualization, we introduce a quantitative model that operationalizes the bidirectional dynamics, using the following distance as the key interaction variable to illustrate how trust co-evolves through a feedback loop. Simulation examples demonstrate both positive reinforcement and negative spiral effects. Based on this framework and its quantitative instantiation, we derive design guidelines for autonomous followers to foster appropriate trust calibration, improve safety, and enhance user acceptance. The framework bridges human factors and engineering perspectives, providing a theoretical and preliminary quantitative foundation for future empirical and modeling research.

Human performance in digital control rooms is strongly influenced by interface characteristics, which shape visual search, cognitive processing, and motor execution. Accurate prediction of operator action time is therefore essential for ergonomic evaluation, interface design, and performance optimization in safety critical systems. However, existing approaches typically rely on extensive experimental data or black box models, limiting their interpretability and practical applicability. This study proposes ATIM (ACTR based Task Interface Model), a theory guided and data calibrated modeling framework that predicts operator action time directly from interface features. The model decomposes total action time into visual, cognitive, motor, and interaction components, integrating principles from visual search theory, Fitts law, and cognitive architecture modeling. Interface characteristics, including target salience and semantic interference, are used as direct inputs, enabling prediction without additional user experiments. A dataset from a digital nuclear control room environment was used for calibration and validation. Task types were identified through data driven clustering, and separate parameter sets were estimated for novice and experienced users. The proposed framework achieved a mean absolute error of 3.17 s and demonstrated strong correlation with observed performance (r = 0.664) on a held-out validation set. The results show that ATIM generalizes well to unseen data while maintaining interpretability, offering a novel tool for ergonomic assessment and interface design in complex human machine systems.

Small modular reactors (SMRs) are increasingly considered for flexible power generation; however, many dynamic studies still neglect the thermodynamic coupling between the primary and secondary loops that is essential for accurate assessment of load-following capability. In this study, we develop a hybrid dynamic framework that couples an equation-based model of the NuScale integral pressurized water reactor, including the reactor, primary loop, and moving-boundary helical-coil once-through steam generator, with a physics-based secondary steam cycle comprising the valve, turbine, condenser, and feedwater pump. This approach enforces mass and energy conservation across the coupled system while preserving physically consistent flow interactions across the domain boundary. The integrated model reproduces nominal design-point conditions and is used to analyze a 5% step load rejection under five control strategies, including a decentralized three-loop control architecture for the valve, feedwater pump, and control rods. The results show that partial control strategies are insufficient for efficient and safe operation, whereas simultaneous action of all three actuators stabilizes steam pressure, limits adverse thermal excursions in the primary loop and maintains acceptable steam generator operating margins during load-following maneuvers. Compared with a conventional linear steam-cycle representation, the coupled framework captures dynamic back-pressure and variable turbine enthalpy drop that are otherwise neglected, leading to different predictions of transient behavior and required steam flow. These findings show that thermodynamically coupled, physics-based steam-cycle models are needed for more accurate assessment of the operational flexibility, efficiency and safety margins of SMRs under realistic load-following conditions.

Faster-than-Nyquist (FTN) signaling is gaining attention as a smart way to pack more data into limited spectrum by intentionally breaking the traditional symbol-spacing rules. This article takes a fresh look at FTN's potential to boost capacity, examining how performance varies across different acceleration factors and signal-to-noise ratio (SNR) definitions. Beyond the theory, we explore what it takes to make FTN work in practice, such as dealing with power amplifier constraints, managing high peak-to-average power, and designing practical coding strategies. We also highlight real-world issues like spectrum sharing, short-packet communication, and receiver complexity. With applications ranging from low-latency links to integrated sensing and satellite systems, FTN offers a compelling path forward for future wireless technologies.

Sparse matrix kernels form the computational backbone of scientific computing, traditionally relying on C/C++ and Fortran implementations that prioritize performance over memory safety. This work evaluates Rust as a systems-level alternative for sparse linear algebra by implementing and benchmarking three core workloads: sparse matrix-vector multiplication (SpMV), Lanczos-based Krylov methods, and matrix-exponential evaluation. We compare native Rust code against established baselines (Intel oneMKL, Eigen, PETSc, and PSBLAS) across a suite of representative matrices. Our results show that Rust's sparse kernels achieve performance comparable to Eigen and PSBLAS, tracking the state-of-the-art for CSC formats, while trailing PETSc's advanced blocked CSR optimizations. By analyzing compile-time monomorphization, SIMD vectorization, and FFI boundaries, we assess the practical impact of Rust's safety model and ecosystem readiness. The study provides concrete, evidence-based guidance for modernizing high-performance numerical software stacks.

Probabilistic Boolean circuits have recently been proposed as a string-diagrammatic foundation for finite probabilistic programming. In this paper, we present a complete set of axioms for their semantics in terms of Markov kernels. Our approach is based on two intermediate results: completeness for \emph{partial} Boolean circuits and completeness for probabilistic Boolean tapes, a diagrammatic language for rig categories.

We study monadic dependence of binary relational structures including at least one antisymmetric relation. Our cornerstone result gives sufficient conditions for proving that a structure is monadically dependent by only considering some of its reducts, assuming they are structurally well-behaved and compatible enough. As an application, we consider some reorientation rules preserving monadic dependence of binary structures, as well as replacement of one antisymmetric relation with bounded independence number by another. Then, we apply our main (technical) result to the study of twin-width in two ways. First, generalizing the fact that twin-width boundedness is equivalent to being expandable by a linear order into a monadically dependent class, we prove that it is also equivalent to being expandable by an oriented graph with bounded independence number (for instance by a poset with bounded width or by a tournament), and that FO-model checking is fixed-parameter tractable on such an expansion. Second, we show delineation by twin-width for some new classes of oriented graphs, including oriented split graphs and local tournaments. In all these cases, we also obtain fixed-parameter tractability of FO-model checking.

This paper proposes a novel approach to design of Nonlinear Model Predictive Control (NMPC) schemes based on Finite-Gain Stability (FGS) concepts. The proposed formulation considers the case where the plant is affected by unknown but bounded disturbances, which renders difficult the classical Lyapunov-based analysis/design. Based on FGS conditions for a closed-loop system, we develop a systematic NMPC design methodology, allowing us to choose the relevant NMPC parameters that lead to closed-loop FGS and provide a satisfactory tracking performance, also for the case of time-varying reference signals. A simulated example is presented to demonstrate the effectiveness of our framework, concerned with lateral/longitudinal control of an automated vehicle.

This paper studies theory-guided advanced regulatory control (ARC) synthesis for cooling-limited exothermic semi-batch reactors, whose productivity and thermal safety are governed by changing active constraints. Industrial ARC uses feedback loops, cascades, selectors, feedforward/override logic, and valve-position elements, but signal selection, pairing, interconnection, and tuning remain heuristic. Nonlinear model predictive control (NMPC) gives a systematic constrained-operation workflow, but requires a maintained nonlinear model, state estimator, and online optimizer. We combine finite-horizon minimum-time optimality with local safety analysis to develop a systematic analysis-to-architecture ARC synthesis workflow for cooling-limited semi-batch reactors. Under stated assumptions, the workflow translates boundary-seeking optimality into a cooling-demand valve-position-control (VPC) architecture and translates local safety requirements into near-boundary tuning rules. On a reduced benchmark and an industrial-scale polymerization, ARC is nominally competitive with an implemented nominal-model output-feedback nonlinear model predictive control (OF-NMPC) benchmark using extended Kalman filter (EKF) state estimation. In the studied adverse parameter mismatch and unmodeled fault scenarios, ARC keeps temperature-limit violation at 0%, whereas OF-NMPC either violates the limit or fails to complete the batch.

Fault-tolerant broadcasting in dense Gaussian networks is recovered by re-rooting the broadcast at a new source at maximum graph distance from the faulty nodes. This paper extends the re-rooting framework by replacing its boundary-search source-selection step with a quotient-lattice-aware algebraic construction. The first contribution is a constant-time counting method for valid new sources, formulated as an intersection of two diameter-$k$ boundary sets in the Gaussian quotient. The exact count is obtained by a fixed union of side-pair intervals over nine quotient-lattice copies, giving a closed-form procedure without scanning the network or boundary. The second contribution is a shifted direct selector for two arbitrary faulty nodes. Given faulty nodes $A$ and $B$, the problem is translated to $C=\operatorname{mod}_{G_k}(B-A)$, and the selector finds $P$ satisfying $d(P,0)=d(P,C)=k$. For each of nine quotient-lattice shifts, sixteen signed linear systems are checked. Nonparallel systems are solved via Cramer's rule; parallel systems are handled by interval-endpoint selection. At most $9\times16=144$ shifted sign cases are evaluated, giving $O(1)$ selection under the word-RAM model. Validation reports zero count mismatches over $26{,}623$ tested nodes, $500{,}000$ valid outputs over $500{,}000$ sampled fault pairs, and $40{,}000$ successful re-rooted broadcast trials. The shifted selector achieves a $5.92\times$ speedup over boundary search at $k=200$, remaining stable as $k$ increases. These results make new-source selection algebraic, bounded, and independent of network size.

Fault-tolerant broadcasting in dense Eisenstein--Jacobi networks requires efficient recovery when faulty nodes disrupt the original broadcast structure. A re-rooting-based method guarantees that, for any two faulty nodes, a valid new source exists at maximum graph distance from both faults. However, identifying such a source without scanning the network or testing all boundary candidates remains an open practical problem. This paper presents a closed-form, constant-time algorithm for counting and selecting a valid new source in dense Eisenstein--Jacobi networks under two node faults. The two-fault problem is reduced to a boundary-intersection problem involving the origin and a difference node. The distance-$t$ boundary, where $t$ is the network diameter, is partitioned into six directed sides of the Eisenstein--Jacobi hexagon. Since the network is a quotient structure, intersection equations are solved modulo the defining lattice, requiring evaluation of seven quotient-lattice shifts across all $6\times 6$ side pairs, yielding at most $252$ algebraic systems. The first algorithm counts all valid new sources for faults at $0$ and $A$. The second algorithm selects one valid new source for arbitrary fault pairs by solving translated side-pair systems, verifying each candidate, and shifting back. Each system is either a non-parallel $2\times 2$ linear system with at most one candidate, or a parallel system whose feasible candidates form an integer interval. Both algorithms run in $O(1)$ time under the fixed-word arithmetic model. Computational validation over $500{,}000$ sampled fault pairs and $40{,}000$ re-rooting trials confirms correctness: the selector always returns a valid new source, and the recovered broadcast reaches all non-faulty nodes.

Dense Eisenstein--Jacobi networks are degree-six algebraic interconnection topologies with regular structure, vertex symmetry, small diameter, and efficient communication algorithms. These properties make them suitable for parallel and on-chip communication systems in which collective operations such as one-to-all broadcasting are frequent. Existing optimal broadcasting algorithms for dense hexagonal/Eisenstein--Jacobi networks assume fault-free operation. However, a faulty internal forwarding node may interrupt message propagation and prevent complete delivery. This paper proposes a lightweight re-rooting-based fault-tolerant broadcasting method for dense Eisenstein--Jacobi networks. The main idea is to relocate the effective broadcast source to a new source node such that each faulty node is located at graph distance equal to the network diameter from the new source. Consequently, faulty nodes become leaf-level nodes in the broadcast process and are not required to forward the message. We present source-selection algorithms for one- and two-node failures and prove that for any pair of faulty nodes in a dense Eisenstein--Jacobi network there exists a common distance-diameter node that can serve as a valid re-rooted source. The source-selection procedure requires linear time in the network diameter. Equivalently, since $N=3t^2+3t+1$, the selection cost is $O(\sqrt{N})$ in the number of nodes. Since the standard one-to-all broadcast completes in one diameter time and the relocation phase is also bounded by one diameter, the proposed method completes in at most twice the network diameter. We also show that the two-fault guarantee does not generally extend to arbitrary three-fault configurations by giving an explicit counterexample. The proposed approach improves broadcast reliability without constructing redundant spanning trees, backup paths, or additional broadcast structures.

This article establishes the completeness of an axiomatization for the robust safety of dynamical systems with polynomial differential equations on bounded time horizons. Safety properties of robust systems are uniformly reduced to a sound axiomatization of polynomial invariants, resulting in reliable logical proofs of correctness. Approximate decidability results are also established: there is a computable algorithm such that, given any perturbation parameter $δ$, it either produces a symbolic proof of robust safety (hence correctly decides the dynamical system to be robustly safe), or correctly decides that the system is not robustly safe under a perturbation of level $δ$. In contrast to earlier works, this article crucially leverages results from subanalytic geometry to retain a level of exactness, thereby establishing positive results of provability/decidability allowing for arbitrary bounded (semialgebraic) initial/post conditions even without positive separation at their (topological) boundaries. This enables the generation of proofs of inductive safety beyond finite time horizons for general hybrid dynamical systems.

The Mixed-Radix One-Time Pad (MR-OTP) extends the classical OTP to heterogeneous alphabets while preserving perfect secrecy. We provide a practical, bias-free method to convert raw binary entropy from a QKD source into uniform mixed-radix keys by identifying Horner's method and its inverse as the natural mapping between binary integers and mixed-radix tuples. We show that naive modular reduction induces bias and prove that rejection sampling restores uniformity with optimal expected cost. We establish end-to-end information-theoretic security for single and multi-session pipelines, quantify efficiency gains, present a batched extractor, and give unconditional and conditional results on the Base Recovery Problem.

We present sufficient conditions for the semi-global exponential stability of nonlinear systems whose dynamics have both slow and fast time variations. Unlike most existing results, the fast variation is non-periodic, thereby allowing a wider class of systems, especially switched systems with fast (non-periodic) switching and those with quasi-periodic variations; we therefore rely on general averaging to construct an average system. It is assumed that the average system admits a time-invariant equilibrium that is globally exponentially stable when the slow variation is frozen, i.e., remaining at a fixed value. This slow variation is allowed to be discontinuous in time, provided its total variation (flows and jumps) is bounded. The main result is illustrated using a nonlinear switched system with slow-fast non-periodic switching.

A convex scene communicates more than shape: the pattern of which groups of objects are mutually apart and which cross is a discrete relational payload. We treat the apartness table of a finite family of convex bodies, a separoid, as a source signal; a renderable convex scene as its encoder; and the rendered image as a noisy visual channel from which the apartness structure is decoded. For disjoint index sets $A,B$, the source bit records whether $\operatorname{conv}(\bigcup_{a\in A} C_a)$ and $\operatorname{conv}(\bigcup_{b\in B} C_b)$ are disjoint. Within this view, apartness-preserving rendering becomes a rate--distortion problem: the rate is a differentiable geometric code length for the carrier scene, while the distortion is closure-aware and weights maximal separations and minimal Radon partitions by the number of consequences they control. A differentiable support-function realization turns separability into a soft directional margin and represents each separation by a distribution over witnessing directions, yielding a variational lower bound on apartness mutual information $I(Σ;Y)$ and an information-theoretic account of view selection. Experiments on planar convex scenes show that scenes are recovered from the apartness table alone at 99.9% bit accuracy, with the certificate skeleton already determining the full table; coordinate quantization gives a clean operational rate--distortion frontier where certificate distortion is more stringent than Hamming error; and rendered $48\times48$ images transmit about 0.72 of the apartness-graph entropy under mild noise. Increasing the viewpoint-robustness term widens separating cones with only a modest geometry-rate surcharge. The result is a certificate-aware rendering objective for scenes whose purpose is to make relational convex structure recoverable rather than merely pixel-faithful.

Recent work studied the problem of finding clusters and denoising pairwise distances from noisy distances of points sampled on a manifold. We study the same problems in more general metric measure spaces under \lowerphiregularity{}. We give an algorithm that extracts large localized clusters around every sampled point and uses them to denoise distances to any fixed accuracy, with near-linear running time in the dense fixed-accuracy regime. We also show how to achieve much higher accuracy with a non-efficient algorithm. This suggests that unlike the Riemannian case, denoising to higher accuracy in more general metric spaces has a statistical-computational gap.

This paper presents an axiomatization of Ludwig von Mises' praxeology in many-sorted first-order logic, isolating the foundational layer. We introduce a formal language with five sorts ({\sf Actors}, {\sf Actions}, {\sf Ends}, {\sf Things}, {\sf Times}) and six primitive relations ({\em Acts}, {\em Avail}, {\em EndOf}, {\em Use}, a preference order, and a time order), together with a base axiom system organised into three layers: the structure of action itself, the actor's preference order together with its revelation in choice, and material scarcity. The base system captures purposeful action in its bare praxeological form. Working entirely within the base system we derive the core classical Misesian propositions as Hilbert-style theorems: the asymmetry of revealed preference, the existence of opportunity cost, the structural scarcity of time, the subjectivity of opportunity cost, the law of diminishing marginal utility, and the increasing marginal disutility of labour. Where a theorem requires structure beyond the praxeological core -- as with diminishing marginal utility -- the additional premises are made explicit; identifying these hidden premises is one of the methodological payoffs of the approach. A self-contained {\em Lean} companion encodes the language as {\em Lean} type classes and constructs concrete models -- a three-period Robinson Crusoe economy and its infinite-time extension -- whose acceptance by the type-checker is a constructive consistency proof of the full base theory.

This paper studies the propagation of fake news through a stochastic rumor spreading model based on Markov chains. Inspired by classical epidemiological SIR models, we consider a generalization of the Daley-Kendall framework for rumours that incorporates fact-checkers, following the Ignorant/Spreader/Checker/Stifler model introduced in Piqueira (2020). The model analyzes the influence of checkers on fake news dynamics. Numerical simulations are used to illustrate the behavior of the system and the impact of fact-checkers.

Flight recovery, aircraft rerouting, and passenger reallocation are critical in airline recovery. To preserve their interdependence that is neglected by the regular sequential recovery, we consider these recovery phases from an integration perspective. In addition, we incorporate cruise speed control to enhance the recovery performance. While using flight copies is a common modelling method in airline disruption management, the resulting integrated mathematical model is challenging to solve in real time due to the large number of flight copies, especially when considering cruise speed control. This paper introduces a new sparse-dense flight copy approach and proposes an innovative interactive mechanism that alternately adjusts aircraft routes on the sparse flight copy-based network and reallocates passenger itineraries on the dense flight copy-based network. Under the interactive mechanism, the involved sparse and dense networks are much smaller than those in the conventional flight copy approach. To implement such a mechanism, we develop an integrated flight, aircraft, and passenger recovery model (IFAPRM) and propose a customized Benders decomposition (CBD) to solve the model. Besides, we further propose some acceleration techniques to speed up the CBD method, including an effective feasibility certificate, scale management, and valid inequalities. Computational experiments on real-world data demonstrate that the sparse-dense flight copy-based interactive mechanism outperforms the conventional flight copy approach. In essence, the proposed interactive mechanism, along with its corresponding modelling method, algorithm, and acceleration techniques, provides a comprehensive methodology and a general decision-support framework for integrated rescheduling problems in complex operations, with potential applications in logistics, transportation, and beyond.

In this article, we study isometric cohomogeneity-one actions on symmetric spaces of mixed type, i.e., those whose universal cover splits as a nontrivial product of symmetric spaces of compact, noncompact, and Euclidean types. We provide a new family of "diagonal" cohomogeneity-one actions on symmetric spaces of the form $\mathbb{R}^n \times M_-$, where $M_-$ is of noncompact type. We show that, with the exception of this family, any cohomogeneity-one action on a symmetric space decomposes as a product of isometric actions on its compact, Euclidean, and noncompact factors. This fully reduces the classification problem for cohomogeneity-one actions to symmetric spaces of a single type.

If $Λ$ is a non-amenable bi-exact group and $Λ\hookrightarrow Γ$ is an existential embedding, then each of the intersections $Λ\cap g Λg^{-1}$ for $g$ a member of $Γ\backslash Λ$ is amenable. This in conjunction with work of Bekka and Kalantar demonstrates that in this situation, the weak equivalence class of the quasi-regular representation $λ_{Γ/Λ}$ determines $Λ$ up to conjugacy among the self-commensurating subgroups of $Γ$.

We investigate Khintchine-type inequalities for the weighted sums $S=\sum_ka_kX_k$ of independent copies of a symmetric random variable $X$. We show how log-monotonicity of the sequence $r_k(X)=k! \mathbb{E}[X^{2k}]/(2k)!$ implies sharp comparisons between the $L_p$ and $L_2$ norms of $S$ for every even integer $p\geq 2$, extending classic Khintchine-type inequalities and yielding new results in the log-convex setting. We also investigate the stability of our inequalities. Our first stability inequality sharpens the classic inequality by a deviation of the coefficient vector from the coordinate extremizers, while the second quantifies deviation from the Gaussian limit. Our results recover recent stability inequalities for random signs and apply to a broad class of distributions, including type-$\mathscr{L}$ random variables, ultra sub-Gaussian random variables and Gaussian mixtures.

We prove norm inflation for the cubic hyperbolic nonlinear Schrödinger equation in $H^s(\mathbb T^2)$ for every $s\in(-\infty,0)\cup(0,\frac12]$. The scaling-critical point $s=0$ is excluded by conservation of the $L^2$ norm. The strong ill-posedness below and above the scaling-critical point arises from two completely different mechanisms. Particularly in the scaling-subcritical regime, this dynamical instability stems from the hyperbolic nature. Together with the local well-posedness result in \cite{WangHNLS}, this gives a sharp dichotomy away from the mass space $L^2(\mathbb T^2)$: local well-posedness holds for $s>\frac12$, whereas norm inflation occurs for all $s\le \frac12$ with $s\ne0$.

We shed light on two alternative stick-breaking constructions of the normalized inverse Gaussian (NIG) random discrete distribution which appear to have been overlooked so far in the Bayesian nonparametric setting. The first is derived from a result in Aldous and Pitman (1998) for the conditional Brownian excursion partition, mixing over the local time at zero up to time one. The second arises as a particular case of a result in James (2013) for priors obtained by a random spatial and temporal change of the normalized generalized Gamma subordinator. Both constructions are in terms of straightforward transformations of standard random variables and can be easily generalized to provide the stick-breaking construction of any element, respectively, in a) the family of mixed Poisson-Kingman models driven by the $1/2$ stable Lévy measure and b) the family of Poisson-Gamma processes driven by the Inverse Gaussian subordinator.

We study the multiple key-cast problem in network coding under active node-based adversaries. In multiple key-cast, a source generates independent secret keys to be securely and reliably delivered to designated terminal subsets. The network adversary can observe \(\ell_o\) nodes, inject additive or overwrite errors into \(\ell_e\) nodes, and simultaneously observe and corrupt \(\ell_{oe}\) nodes, while having full knowledge of the topology and coding operations. Adversarial models of similar nature, however, where corruption and eavesdropping is done on edges instead of nodes, have seen previous studies in the context of secure multicast network-coding. The work at hand builds on and extends these studies to address the challenges in node-based adversaries in the context of (multiple) key distribution. For single-source networks where every node is d-vertex connected from the source, we show that perfectly secure multiple key-cast under additive and overwrite error models is asymptotically achievable at the key-capacity of \(d-\ell_o-\ell_e-2\ell_{oe}\). We then extend our analysis to networks where only terminal nodes satisfy this connectivity requirement, while intermediate nodes may be only partially connected. For these topologies, we develop coding schemes that achieve secure and reliable multiple key-cast capacities determined by the source vertex-connectivity and additional structural properties of the network. Finally, we show that our results generalize to multi-source settings, ensuring perfect secrecy even if the adversary observes all but one source node, and establish that our constructions apply directly to secure multicast network coding and to network secret-sharing scenarios. As part of our studies, we improve the security guarantee of a central scheme in [Zhang et al., IEEE Trans. Comm., 2023] addressing parallel-edge networks, from weak-security to perfect-security.

We prove that if $p/q$ is a continued fraction convergent of $1/e$ with $q\geq 3$, then, for the secretary problem with $q$ applicants, the optimal number of initially rejected applicants is $p$.