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1. Games with filters I

   https://doi.org/10.1142/S021906132450003X  

   Foreman, Matthew; Magidor, Menachem; Zeman, Martin (August 2023, Journal of Mathematical Logic)

   Chong, Chita; Feng, Qi; Slaman, Theodore_A; Woodin, W_Hugh (Ed.)

   This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call Welch games. Player II having a winning strategy in the Welch game of length [Formula: see text] on [Formula: see text] is equivalent to weak compactness. Winning the game of length [Formula: see text] is equivalent to [Formula: see text] being measurable. We show that for games of intermediate length [Formula: see text], II winning implies the existence of precipitous ideals with [Formula: see text]-closed, [Formula: see text]-dense trees. The second part shows the first is not vacuous. For each [Formula: see text] between [Formula: see text] and [Formula: see text], it gives a model where II wins the games of length [Formula: see text], but not [Formula: see text]. The technique also gives models where for all [Formula: see text] there are [Formula: see text]-complete, normal, [Formula: see text]-distributive ideals having dense sets that are [Formula: see text]-closed, but not [Formula: see text]-closed. 

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2. The Littlewood–Paley decomposition for periodic functions and applications to the Boussinesq equations

   https://doi.org/10.1142/s0219530519500234  

   Dai, Yichen; Hu, Weiwei; Wu, Jiahong; Xiao, Bei (July 2020, Analysis and Applications)

   The Littlewood–Paley decomposition for functions defined on the whole space [Formula: see text] and related Besov space techniques have become indispensable tools in the study of many partial differential equations (PDEs) with [Formula: see text] as the spatial domain. This paper intends to develop parallel tools for the periodic domain [Formula: see text]. Taking advantage of the boundedness and convergence theory on the square-cutoff Fourier partial sum, we define the Littlewood–Paley decomposition for periodic functions via the square cutoff. We remark that the Littlewood–Paley projections defined via the circular cutoff in [Formula: see text] with [Formula: see text] in the literature do not behave well on the Lebesgue space [Formula: see text] except for [Formula: see text]. We develop a complete set of tools associated with this decomposition, which would be very useful in the study of PDEs defined on [Formula: see text]. As an application of the tools developed here, we study the periodic weak solutions of the [Formula: see text]-dimensional Boussinesq equations with the fractional dissipation [Formula: see text] and without thermal diffusion. We obtain two main results. The first assesses the global existence of [Formula: see text]-weak solutions for any [Formula: see text] and the existence and uniqueness of the [Formula: see text]-weak solutions when [Formula: see text] for [Formula: see text]. The second establishes the zero thermal diffusion limit with an explicit convergence rate. 

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3. A trace inequality of Ando, Hiai, and Okubo and a monotonicity property of the Golden–Thompson inequality

   https://doi.org/10.1063/5.0091111  

   Carlen, Eric A.; Lieb, Elliott H. (June 2022, Journal of Mathematical Physics)

   The Golden–Thompson trace inequality, which states that Tr  e H+ K ≤ Tr  e H e K , has proved to be very useful in quantum statistical mechanics. Golden used it to show that the classical free energy is less than the quantum one. Here, we make this G–T inequality more explicit by proving that for some operators, notably the operators of interest in quantum mechanics, H = Δ or [Formula: see text] and K = potential, Tr  e H+(1− u) K e uK is a monotone increasing function of the parameter u for 0 ≤ u ≤ 1. Our proof utilizes an inequality of Ando, Hiai, and Okubo (AHO): Tr  X s Y t X 1− s Y 1− t ≤ Tr  XY for positive operators X, Y and for [Formula: see text], and [Formula: see text]. The obvious conjecture that this inequality should hold up to s + t ≤ 1 was proved false by Plevnik [Indian J. Pure Appl. Math. 47, 491–500 (2016)]. We give a different proof of AHO and also give more counterexamples in the [Formula: see text] range. More importantly, we show that the inequality conjectured in AHO does indeed hold in the full range if X, Y have a certain positivity property—one that does hold for quantum mechanical operators, thus enabling us to prove our G–T monotonicity theorem. 

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4. Primal-Dual Extrapolation Methods for Monotone Inclusions Under Local Lipschitz Continuity

   https://doi.org/10.1287/moor.2024.0407  

   Lu, Zhaosong; Mei, Sanyou (October 2024, Mathematics of Operations Research)

   In this paper, we consider a class of monotone inclusion (MI) problems of finding a zero of the sum of two monotone operators, in which one operator is maximal monotone, whereas the other is locally Lipschitz continuous. We propose primal-dual (PD) extrapolation methods to solve them using a point and operator extrapolation technique, whose parameters are chosen by a backtracking line search scheme. The proposed methods enjoy an operation complexity of [Formula: see text] and [Formula: see text], measured by the number of fundamental operations consisting only of evaluations of one operator and resolvent of the other operator, for finding an ε-residual solution of strongly and nonstrongly MI problems, respectively. The latter complexity significantly improves the previously best operation complexity [Formula: see text]. As a byproduct, complexity results of the primal-dual extrapolation methods are also obtained for finding an ε-KKT or ε-residual solution of convex conic optimization, conic constrained saddle point, and variational inequality problems under local Lipschitz continuity. We provide preliminary numerical results to demonstrate the performance of the proposed methods. Funding: This work was partially supported by the National Science Foundation [Grant IIS-2211491], the Office of Naval Research [Grant N00014-24-1-2702], and the Air Force Office of Scientific Research [Grant FA9550-24-1-0343]. 

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5. Fractional Hardy-type and trace theorems for nonlocal function spaces with heterogeneous localization

   https://doi.org/10.1142/S0219530521500329  

   Du, Qiang; Mengesha, Tadele; Tian, Xiaochuan (May 2022, Analysis and Applications)

   This work aims to prove a Hardy-type inequality and a trace theorem for a class of function spaces on smooth domains with a nonlocal character. Functions in these spaces are allowed to be as rough as an [Formula: see text]-function inside the domain of definition but as smooth as a [Formula: see text]-function near the boundary. This feature is captured by a norm that is characterized by a nonlocal interaction kernel defined heterogeneously with a special localization feature on the boundary. Thus, the trace theorem we obtain here can be viewed as an improvement and refinement of the classical trace theorem for fractional Sobolev spaces [Formula: see text]. Similarly, the Hardy-type inequalities we establish for functions that vanish on the boundary show that functions in this generalized space have the same decay rate to the boundary as functions in the smaller space [Formula: see text]. The results we prove extend existing results shown in the Hilbert space setting with p = 2. A Poincaré-type inequality we establish for the function space under consideration together with the new trace theorem allows formulating and proving well-posedness of a nonlinear nonlocal variational problem with conventional local boundary condition. 

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