AbstractIn recent years, mean field games (MFGs) have garnered considerable attention and emerged as a dynamic and actively researched field across various domains, including economics, social sciences, finance, and transportation. The inverse design and decoding of MFGs offer valuable means to extract information from observed data and gain insights into the intricate underlying dynamics and strategies of these complex physical systems. This paper presents a novel approach to the study of inverse problems in MFGs by analyzing the Cauchy data around their unknown stationary states. This study distinguishes itself from existing inverse problem investigations in three key significant aspects: First, we consider MFG problems in a highly general form. Second, we address the technical challenge of the probability measure constraint by utilizing Cauchy data in our inverse problem study. Third, we enhance existing high‐order linearization methods by introducing a novel approach that involves conducting linearization around non‐trivial stationary states of the MFG system, which are not a priori known. These contributions provide new insights and offer promising avenues for studying inverse problems for MFGs. By unraveling the hidden structure of MFGs, researchers and practitioners can make informed decisions, optimize system performance, and address real‐world challenges more effectively.
We introduce a relativized version of random Kripke's schema and show how it may be applied in the investigation of the expressive power of intuitionistic real algebra by interpreting second-order Heyting arithmetic in it.
Log-atomic numbers are surreal numbers whose iterated logarithms are monomials, and consequently have a trivial expansion as transseries. Presenting surreal numbers as sign sequences, we give the sign sequence formula for log-atomic numbers. To that efect, we relate log-atomic numbers to fixed-points of certain surreal functions.
Purpose: to examine the impact of statins on reducing all-cause mortality among individuals diagnosed with type 2 diabetes. This investigation explored the potential correlations between dosage, drug classification, and usage intensity with the observed outcomes. Methods: The research sample consisted of individuals aged 40 years or older diagnosed with type 2 diabetes. Statin usage was determined as a frequent usage over a minimum of one month subsequent to type 2 diabetes diagnosis, where the average statin dose was ≥28 cumulative defined daily doses per year (cDDD-year). The analysis employed an inverse probability of treatment-weighted Cox hazard model, utilizing statin usage status as a time-varying variable, to evaluate the impact of statin use on all-cause mortality. Results: The incidence of mortality was comparatively lower among the cohort of statin users (n = 50,804 (12.03%)), in contrast to nonusers (n = 118,765 (27.79%)). After adjustments, the hazard ratio (aHR; 95% confidence interval (CI)) for all-cause mortality was estimated to be 0.32 (0.31–0.33). Compared with nonusers, pitavastatin, rosuvastatin, pravastatin, simvastatin, atorvastatin, fluvastatin, and lovastatin users demonstrated significant reductions in all-cause mortality (aHRs (95% CIs) = 0.06 (0.04–0.09), 0.28 (0.27–0.29), 0.29 (0.28–0.31), 0.31 (0.30–0.32), 0.31 (0.30–0.32), 0.36 (0.35–0.38), and 0.48 (0.47–0.50), respectively). In Q1, Q2, Q3, and Q4 of cDDD-year, our multivariate analysis demonstrated significant reductions in all-cause mortality (aHRs (95% CIs) = 0.51 (0.5–0.52), 0.36 (0.35–0.37), 0.24 (0.23–0.25), and 0.13 (0.13–0.14), respectively; p for trend <0.0001). Because it had the lowest aHR (0.32), 0.86 DDD of statin was considered optimal. Conclusions: In patients diagnosed with type 2 diabetes, consistent utilization of statins (≥28 cumulative defined daily doses per year) was shown to have a beneficial effect on all-cause mortality. Moreover, the risk of all-cause mortality decreased as the cumulative defined daily dose per year of statin increased.
We construct a theory definitionally equivalent to first-order Peano arithmetic PA and a non-standard computable model of this theory. The same technique allows us to construct a theory definitionally equivalent to Zermelo-Fraenkel set theory ZF that has a computable model.
Let $K$ be a type-definable infinite field in an NIP theory. If $K$ has characteristic $p > 0$, then $K$ is Artin-Schreier closed (it has no Artin-Schreier extensions). As a consequence, $p$ does not divide the degree of any finite separable extension of $K$. This generalizes a theorem of Kaplan, Scanlon, and Wagner.
In this paper we prove a Robinson consistency theorem for a class of many-sorted hybrid logics as a consequence of an Omitting Types Theorem. An important corollary of this result is an interpolation theorem.
We construct bounded degree acyclic Borel graphs with large Borel chromatic number using a graph arising from Ramsey theory and limits of expander sequences.
We prove that a variety of generalized cardinal characteristics, including meeting numbers, the reaping number, and the dominating number, satisfy an analogue of the Galvin-Hajnal theorem, and hence also of Silver's theorem, at singular cardinals of uncountable cofinality.
We show that if $\mathcal{T}$ is any Hausdorff topology on $ω_{1}$, then any subset of $ω_{1}$ which is homeomorphic to the rationals under $\mathcal{T}$ can be refined to a homeomorphic copy of the rationals on which $\barρ$ is shift-increasing.
We consider existentially closed fields with several orderings, valuations, and $p$-valuations. We show that these structures are NTP$_2$ of finite burden, but usually have the independence property. Moreover, forking agrees with dividing, and forking can be characterized in terms of forking in ACVF, RCF, and $p$CF.
The purpose of this note is to discuss some of the questions raised by Dunn, J. Michael; Moss, Lawrence S.; Wang, Zhenghan in Editors' introduction: the third life of quantum logic: quantum logic inspired by quantum computing.
In this paper we isolate a new criterion for when a given real $x$ is generic over $L$ in terms of $x$'s capability of lifting elementary embeddings of initial segments of $L$.
This short squib looks at how using a broader definition of Gödel numbering to mimic the accessibility relation between possible worlds results in two-world systems that sidestep undecidable sentences as well as the Liar paradox.
We show that a natural, two sorted $\cL_{ω_1,ω}$ theory involving the modular $j$-function is categorical in all uncountable cardinaities. It is also shown that a slight weakening of the adelic Mumford-Tate conjecture for products of elliptic curves is necessary and (along with a couple of other results from arithmetic geometry) sufficient for categoricity.
We characterize the situation of having many normal measures on a measurable cardinal. We show the plausibility of having many normal measures on each compact cardinal.
In this paper using Sperner's lemma for modified partition of a simplex we will constructively prove Brouwer's fixed point theorem for sequentially locally non-constant and uniformly sequentially continuous functions.