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Understanding the surface temperature and interior structure of cold-to-temperate sub-Neptunes is critical for assessing their habitability, yet direct observations are challenging. In this study, we investigate the impact of water condensation on the atmospheric compositions of sub-Neptunes, focusing on the implications for JWST spectroscopic observations. By modeling the atmospheric photochemistry of two canonical sub-Neptunes, K2-18 b and LHS 1140 b, both with and without water condensation and with and without thick atmospheres, we demonstrate that water condensation can significantly affect the predicted atmospheric compositions. This effect is driven by oxygen depletion from the condensation of water vapor and primarily manifests as an increase in the C/O ratio within the photochemically active regions of the atmosphere. This change in composition particularly affects planets with thin H2-dominated atmospheres, leading to a transition in dominant nitrogen and carbon carriers from N2 and oxygen-rich species like CO/CO2 towards heavier hydrocarbons and nitriles. While our models do not fully account for the loss mechanisms of these higher-order species, such molecules can go on to form more refractory molecules or hazes. Planets with thin H2-rich atmospheres undergoing significant water condensation are thus likely to exhibit very hazy atmospheres. The relatively flat JWST spectra observed for LHS 1140 b could be consistent with such a scenario, suggesting a shallow surface with extensive water condensation or a high atmospheric C/O ratio. Conversely, the JWST observations of K2-18 b are better aligned with a volatile-rich mini-Neptune with a thick atmosphere., Comment: 21 pages, 7 figures

In this paper, we investigate the convergence behavior of the Accelerated Newton Proximal Extragradient (A-NPE) method when employing inexact Hessian information. The exact A-NPE method was the pioneer near-optimal second-order approach, exhibiting an oracle complexity of $\Tilde{O}(\epsilon^{-2/7})$ for convex optimization. Despite its theoretical optimality, there has been insufficient attention given to the study of its inexact version and efficient implementation. We introduce the inexact A-NPE method (IA-NPE), which is shown to maintain the near-optimal oracle complexity. In particular, we design a dynamic approach to balance the computational cost of constructing the Hessian matrix and the progress of the convergence. Moreover, we show the robustness of the line-search procedure, which is a subroutine in IA-NPE, in the face of the inexactness of the Hessian. These nice properties enable the implementation of highly effective machine learning techniques like sub-sampling and various heuristics in the method. Extensive numerical results illustrate that IA-NPE compares favorably with state-of-the-art second-order methods, including Newton's method with cubic regularization and Trust-Region methods.

We study the well-posedness of a system of forward-backward stochastic differential equations (FBSDEs) corresponding to a degenerate mean field type control problem, when the diffusion coefficient depends on the state together with its measure and also the control. Degenerate mean field type control problems are rarely studied in the literature. Our method is based on a lifting approach which embeds the control problem and the associated FBSDEs in Wasserstein spaces into certain Hilbert spaces. We use a continuation method to establish the solvability of the FBSDEs and that of the G\^ateaux derivatives of this FBSDEs. We then explore the regularity of the value function in time and in measure argument, and we also show that it is the unique classical solution of the associated Bellman equation. We also study the higher regularity of the linear functional derivative of the value function, by then, we obtain the classical solution of the mean field type master equation.

Ultra-cool dwarf stars are abundant, long-lived, and uniquely suited to enable the atmospheric study of transiting terrestrial companions with JWST. Amongst them, the most prominent is the M8.5V star TRAPPIST-1 and its seven planets. While JWST Cycle 1 observations have started to yield preliminary insights into the planets, they have also revealed that their atmospheric exploration requires a better understanding of their host star. Here, we propose a roadmap to characterize the TRAPPIST-1 system -- and others like it -- in an efficient and robust manner. We notably recommend that -- although more challenging to schedule -- multi-transit windows be prioritized to mitigate the effects of stellar activity and gather up to twice more transits per JWST hour spent. We conclude that, for such systems, planets cannot be studied in isolation by small programs, but rather need large-scale, jointly space- and ground-based initiatives to fully exploit the capabilities of JWST for the exploration of terrestrial planets.

In this paper, we study the maximum principle of mean field type control problems when the volatility function depends on the state and its measure and also the control, by using our recently developed method. Our method is to embed the mean field type control problem into a Hilbert space to bypass the evolution in the Wasserstein space. We here give a necessary condition and a sufficient condition for these control problems in Hilbert spaces, and we also derive a system of forward-backward stochastic differential equations.

The prime graph $\Gamma(G)$ of a finite group $G$ (also known as the Gruenberg-Kegel graph) has as its vertices the prime divisors of $|G|$, and $p\text-q$ is an edge in $\Gamma(G)$ if and only if $G$ has an element of order $pq$. Since their inception in the 1970s these graphs have been studied extensively; however, completely classifying the possible prime graphs for larger families of groups remains a difficult problem. For solvable groups such a classification was found in 2015. In this paper we go beyond solvable groups for the first time and characterize prime graphs of a more general class of groups we call pseudo-solvable. These are groups whose composition factors are either cyclic or $A_5$. The classification is based on two conditions: the vertices $\{2,3,5\}$ form a triangle in $\overline\Gamma(G)$ or $\{p,3,5\}$ form a triangle for some prime $p\neq 2$.

In this paper we continue the study of prime graphs of finite solvable groups. The prime graph, or Gruenberg-Kegel graph, of a finite group G has vertices consisting of the prime divisors of the order of G and an edge from primes p to q if and only if G contains an element of order pq. Since the discovery of a simple, purely graph theoretical characterization of the prime graphs of solvable groups in 2015 these graphs have been studied in more detail from a graph theoretic angle. In this paper we explore several new aspects of these graphs. We characterize regular reseminant graphs and study the automorphisms of reseminant graphs for arbitrary base graphs. We then study minimal prime graphs on larger vertex sets by a novel regular graph construction for base graphs and by proving results on prime graph properties under graph products. Lastly, we present the first new way, different from vertex duplication, to obtain a new minimal prime graph from a given minimal prime graph., Comment: 22 pages, 7 figures

In this paper, we consider mean field games (MFGs) with a major and $N$ minor agents. We first consider the limiting problem and allow the state coefficients to vary with the conditional distribution in a nonlinear way. We use the sufficient Pontryagin principle for optimality to transform the limiting control problem into a system of two coupled conditional distribution dependent forward backward stochastic differential equations (FBSDEs), and prove the existence and uniqueness of solutions of the FBSDEs when the dependence between major agent and minor agents is sufficiently weak and the convexity parameter of the running cost of minor agents on the control is sufficiently large. A weak monotonicity property is required for minor agents' cost functions and the proof is based on a continuation method in coefficients. We then consider the equilibrium property of MFG with major and minor agents and use the solution of the limit problem to construct an $\mathcal{O}(N^{-\frac{1}{2}})$-Nash equilibrium for MFG with a major and $N$ minor agents.

In this paper, we study a combined system of a Fokker-Planck (FP) equation for $m^{t,\mu}$ with initial $(t,\mu)\in[0,T]\times L^2(\mathbb{R}^d)$, and a stochastic differential equation for $X^{t,x,\mu}$ with initial $(t,x)\in[0,T]\times \mathbb{R}^d$, whose coefficients depend on the solution of FP equation. We develop a combined probabilistic and analytical method to explore the regularity of the functional $V(t,x,\mu)=\mathbb{E}[\Phi(X^{t,x,\mu}_T,m^{t,\mu}(T,\cdot))]$. Our main result states that, under a non-degenerate condition and appropriate regularity assumptions on the coefficients, the function $V$ is the unique classical solution of a nonlocal partial differential equation of mean-field type. The proof depends heavily on the differential properties of the flow $\mu\mapsto (m^{t,\mu}, X^{t,x,\mu})$ over $\mu\in L^2(\mathbb{R}^d)$. We also give an example to illustrate the role of our main result. Finally, we give a discussion on the $L^1$ choice case in the initial $\mu$., Comment: 47 pages

In this paper, we study fully coupled nonlocal second order quasilinear forward-backward partial differential equations (FBPDEs), which arise from solution of the mean field game (MFG) suggested by Lasry and Lions [Japan. J. Math. 2 (2007), p. 237 (Remark iv)]. We show the existence of solutions $(u,m)\in C^{1+\frac{1}{4},2+\frac{1}{2}}([0,T]\times\mathbb{R}^n)\times C^{\frac{1}{2}}([0,T],\mathcal{P}_1(\mathbb{R}^n))$, and also the uniqueness under an additional monotonicity condition. Then, we improve the regularity of our weak solution $m$ to get a classical solution under appropriate regularity assumptions on coefficients. The FBPDEs can be used to investigate a system of mean field equations (MFEs), where the backward one is a Hamilton-Jacobi-Bellman equation and the forward one is a Fokker-Planck equation. Moreover, we prove a verification theorem and give an optimal strategy of the associated MFG via the solution of MFEs. Finally, we address the linear-quadratic problems.

In this paper, we consider the mean field game with a common noise and allow the state coefficients to vary with the conditional distribution in a nonlinear way. We assume that the cost function satisfies a convexity and a weak monotonicity property. We use the sufficient Pontryagin principle for optimality to transform the mean field control problem into existence and uniqueness of solution of conditional distribution dependent forward-backward stochastic differential equation (FBSDE). We prove the existence and uniqueness of solution of the conditional distribution dependent FBSDE when the dependence of the state on the conditional distribution is sufficiently small, or when the convexity parameter of the running cost on the control is sufficiently large. Two different methods are developed. The first method is based on a continuation of the coefficients, which is developed for FBSDE by Hu and Peng \cite{YH2}. We apply the method to conditional distribution dependent FBSDE. The second method is to show the existence result on a small time interval by Banach fixed point theorem and then extend the local solution to the whole time interval.