More Like this

1. A Riemannian Smoothing Steepest Descent Method for Non-Lipschitz Optimization on Embedded Submanifolds of Rn

   https://doi.org/10.1287/moor.2022.0286  

   Zhang, Chao; Chen, Xiaojun; Ma, Shiqian (August 2023, Mathematics of Operations Research)

   In this paper, we study the generalized subdifferentials and the Riemannian gradient subconsistency that are the basis for non-Lipschitz optimization on embedded submanifolds of [Formula: see text]. We then propose a Riemannian smoothing steepest descent method for non-Lipschitz optimization on complete embedded submanifolds of [Formula: see text]. We prove that any accumulation point of the sequence generated by the Riemannian smoothing steepest descent method is a stationary point associated with the smoothing function employed in the method, which is necessary for the local optimality of the original non-Lipschitz problem. We also prove that any accumulation point of the sequence generated by our method that satisfies the Riemannian gradient subconsistency is a limiting stationary point of the original non-Lipschitz problem. Numerical experiments are conducted to demonstrate the advantages of Riemannian [Formula: see text] [Formula: see text] optimization over Riemannian [Formula: see text] optimization for finding sparse solutions and the effectiveness of the proposed method. Funding: C. Zhang was supported in part by the National Natural Science Foundation of China [Grant 12171027] and the Natural Science Foundation of Beijing [Grant 1202021]. X. Chen was supported in part by the Hong Kong Research Council [Grant PolyU15300219]. S. Ma was supported in part by the National Science Foundation [Grants DMS-2243650 and CCF-2308597], the UC Davis Center for Data Science and Artificial Intelligence Research Innovative Data Science Seed Funding Program, and a startup fund from Rice University. 

   more » « less
2. 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. 

   more » « less
3. Inequalities involving Aharonov–Bohm magnetic potentials in dimensions 2 and 3

   https://doi.org/10.1142/S0129055X21500069  

   Bonheure, Denis; Dolbeault, Jean; Esteban, Maria J.; Laptev, Ari; Loss, Michael (November 2020, Reviews in Mathematical Physics)

   null (Ed.)

   This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions 2 and 3. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in the presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions [Formula: see text] and [Formula: see text]. 

   more » « less
4. 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. 

   more » « less
5. Measurement of the Taylor scale in a magnetized turbulent laboratory plasma wind-tunnel

   https://doi.org/10.1063/5.0073207  

   Cartagena-Sanchez, C. A.; Carlson, J. M.; Schaffner, D. A. (March 2022, Physics of Plasmas)

   The fluid Taylor scale is measured in the Bryn Mawr Experiment (BMX) of the Bryn Mawr Plasma Laboratory and examined as a potential dissipation scale of magnetic turbulence within the plasma. We present the first laboratory measurements of the Taylor scale of a turbulent magnetized plasma through multi-point correlations of broadband magnetic fluctuations. From spatial and temporal correlations, respectively, the measured Taylor scales are [Formula: see text] and [Formula: see text]. These measurements are on the same order of magnitude as estimated ion dissipation scales within the BMX plasma with ion inertial scales between [Formula: see text] and ion gyroscales between [Formula: see text]. From these measurements, a magnetic Reynolds number can be computed. Since Taylor scale values are determined using multi-point correlations and a Richardson extrapolation technique, an estimate of the magnetic Reynolds number can be found without the added complication of specifying a model of microscopic diffusivity, a parameter often difficult to obtain experimentally. 

   more » « less