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1. Offline Congestion Games: How Feedback Type Affects Data Coverage Requirement

   Jiang, Haozhe; Cui, Qiwen; Xiong, Zhihan; Fazel, Maryam; Du, Simon S. (February 2023, International Conference on Learning Representations)

   This paper investigates when one can efficiently recover an approximate Nash Equilibrium (NE) in offline congestion games. The existing dataset coverage assumption in offline general-sum games inevitably incurs a dependency on the number of actions, which can be exponentially large in congestion games. We consider three different types of feedback with decreasing revealed information. Starting from the facility-level (a.k.a., semi-bandit) feedback, we propose a novel one-unit deviation coverage condition and show a pessimism-type algorithm that can recover an approximate NE. For the agent-level (a.k.a., bandit) feedback setting, interestingly, we show the one-unit deviation coverage condition is not sufficient. On the other hand, we convert the game to multi-agent linear bandits and show that with a generalized data coverage assumption in offline linear bandits, we can efficiently recover the approximate NE. Lastly, we consider a novel type of feedback, the game-level feedback where only the total reward from all agents is revealed. Again, we show the coverage assumption for the agent-level feedback setting is insufficient in the game-level feedback setting, and with a stronger version of the data coverage assumption for linear bandits, we can recover an approximate NE. Together, our results constitute the first study of offline congestion games and imply formal separations between different types of feedback. 

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2. A spatially varying robin interface condition for fluid‐structure coupled simulations

   https://doi.org/10.1002/nme.6386  

   Cao, Shunxiang; Wang, Guangyao; Wang, Kevin G. (August 2020, International Journal for Numerical Methods in Engineering)

   Summary We present a spatially varying Robin interface condition for solving fluid‐structure interaction problems involving incompressible fluid flows and nonuniform flexible structures. Recent studies have shown that for uniform structures with constant material and geometric properties, a constant one‐parameter Robin interface condition can improve the stability and accuracy of partitioned numerical solution procedures. In this work, we generalize the parameter to a spatially varying function that depends on the structure's local material and geometric properties, without varying the exact solution of the coupled fluid‐structure system. We present an algorithm to implement the Robin interface condition in an embedded boundary method for coupling a projection‐based incompressible viscous flow solver with a nonlinear finite element structural solver. We demonstrate the numerical effects of the spatially varying Robin interface condition using two example problems: a simplified model problem featuring a nonuniform Euler‐Bernoulli beam interacting with an inviscid flow and a generalized Turek‐Hron problem featuring a nonuniform, highly flexible beam interacting with a viscous laminar flow. Both cases show that a spatially varying Robin interface condition can clearly improve numerical accuracy (by up to two orders of magnitude in one instance) for the same computational cost. Using the second example problem, we also demonstrate and compare two models for determining the local value of the combination function in the Robin interface condition. 

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3. Investigation of Constant Volume and Constant Flux Initial Conditions on Bidensity Particle-Laden Slurries on an Incline

   https://doi.org/10.33697/ajur.2019.029  

   Diaz, Dominic; Bojorquez, Jessica; Crasto, Joshua; Koulikova, Margaret; Latib, Tameez; Prins, Aviva; Shapiro, Andrew; Ye, Clover; Arnold, David; Falcon, Claudia; et al (December 2019, American Journal of Undergraduate Research)

   Particle-laden slurries are pervasive in both natural and industrial settings, whenever particles are suspended or transported in a fluid. Previous literature has investigated the case of a single species of negatively buoyant particles suspended in a viscous fluid. On an incline, three distinct regimes emerge depending on the particle concentration and inclination angle: settled (where particles settle and there is a pure fluid front), well-mixed (where particle concentration is constant throughout), and ridged (where a particle-rich ridge leads the flow). Recently, the same three regimes were also found for constant volume two species bidensity slurries. We extend the literature on bidensity slurries by presenting results on constant volume and a new type of initial condition: constant flux, where slurry is pumped onto the incline at a constant rate. We present front positions of the slurries and compare them to theoretical predictions. In addition, height profiles (film thicknesses) are also presented for the constant flux case, showing the distinct behavior of the ridged regime. We find that for constant flux conditions the settled regime forms for small particle volume fractions and inclination angles while the ridged regime forms for large corresponding values. Intermediate values of these two parameters are shown to produce a well-mixed regime.KEYWORDS: Thin Films; Particle-Laden Flow; Multiphase Fluids; Interfacial Flows; Particle Segregation 

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4. Behavior of absorbing and generating $p$ -Robin eigenvalues in bounded and exterior domains

   https://doi.org/10.1016/j.na.2025.113943  

   Bundrock, Lukas; Giorgi, Tiziana; Smits, Robert (January 2026, Nonlinear Analysis)

   We establish rigorous quantitative inequalities for the first eigenvalue of the generalized 𝑝-Robin problem, for both the classical diffusion absorption case, where the Robin boundary parameter 𝛼 is positive, and the superconducting generation regime (𝛼 < 0), where the boundary acts as a source. In bounded domains, we use a unified approach to derive a precise asymptotic behavior for all 𝑝 and all small real 𝛼, improving existing results in various directions, including requiring weaker boundary regularity for the case of the classical 2-Robin problem, studied in the fundamental work by René Sperb. In exterior domains, we characterize the existence of eigenvalues, establish general inequalities and asymptotics as 𝛼 → 0 for the first eigenvalue of the exterior of a ball, and obtain some sharp geometric inequalities for convex domains in two dimensions. 

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5. Stratified Type Theory

   https://doi.org/10.1007/978-3-031-91118-7_10  

   Chan, Jonathan; Weirich, Stephanie (January 2025, Springer Nature Switzerland)

   A hierarchy of type universes is a rudimentary ingredient in the type theories of many proof assistants to prevent the logical inconsistency resulting from combining dependent functions and the type-in-type axiom. In this work, we argue that a universe hierarchy is not the only option for universes in type theory. Taking inspiration from Leivant’s Stratified System F, we introduce Stratified Type Theory (StraTT), where rather than stratifying universes by levels, we stratify typing judgements and restrict the domain of dependent functions to strictly lower levels. Even with type-in-type, this restriction suffices to enforce consistency. In StraTT, we consider a number of extensions beyond just stratified dependent functions. First, the subsystem subStraTT employs McBride’s crude-but-effective stratification (also known as displacement) as a simple form of level polymorphism where global definitions with concrete levels can be displaced uniformly to any higher level. Second, to recover some expressivity lost due to the restriction on dependent function domains, the full StraTT includes a separate nondependent function type with a floating domain whose level matches that of the overall function type. Finally, we have implemented a prototype type checker for StraTT extended with datatypes and inference for level and displacement annotations, along with a small core library. We have proven subStraTT to be consistent and StraTT to be type safe, but consistency of the full remains an open problem, largely due to the interaction between floating functions and cumulativity of judgements. Nevertheless, we StraTT believe to be consistent, and as evidence have verified the ill-typedness of some well-known type-theoretic paradoxes using our implementation. 

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