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1. Parametrized Families of Resolvent Compositions

   https://doi.org/10.1007/s11228-025-00745-7  

   Cornejo, Diego J (March 2025, Set-Valued and Variational Analysis)

   Abstract This paper presents an in-depth analysis of a parametrized version of the resolvent composition, an operation that combines a set-valued operator and a linear operator. We provide new properties and examples, and show that resolvent compositions can be interpreted as parallel compositions of perturbed operators. Additionally, we establish new monotonicity results, even in cases when the initial operator is not monotone. Finally, we derive asymptotic results regarding operator convergence, specifically focusing on graph-convergence and the$$\rho $$ $\rho$-Hausdorff distance. 

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2. Real operator spaces and operator algebras

   https://doi.org/10.4064/sm230329-23-12  

   Blecher, David P (January 2024, Studia Mathematica)

   We verify that a large portion of the theory of complex operator spaces and operator algebras (as represented by the 2004 book by the author and Le Merdy for specificity) transfers to the real case. We point out some of the results that do not work in the real case. We also discuss how the theory and standard constructions interact with the complexification, which is often as important, but sometimes much less obvious. For example, we develop the real case of the theory of operator space multipliers and the operator space centralizer algebra, and discuss how these topics connect with complexifi- cation. This turns out to differ in some important details from the complex case. We also characterize real structure in complex operator spaces and give ‘real’ characterizations of some of the most important objects in the subject. 

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3. Improved multispecies Dougherty collisions

   https://doi.org/10.1017/S0022377822000289  

   Francisquez, Manaure; Juno, James; Hakim, Ammar; Hammett, Gregory W.; Ernst, Darin R. (June 2022, Journal of Plasma Physics)

   The Dougherty model Fokker–Planck operator is extended to describe nonlinear full- $$f$$ (  f is the distribution function) collisions between multiple species in plasmas. Simple relations for cross-species primitive moments are developed which obey conservation laws, and reproduce familiar velocity and temperature relaxation rates. This treatment of multispecies Dougherty collisions, valid for arbitrary mass ratios, avoids unphysical temperatures and satisfies the $$H$$ -theorem ( H is related to the entropy) unlike an analogous Bhatnagar–Gross–Krook operator. Formulas for both a Cartesian velocity space and a gyroaveraged operator are provided for use in Vlasov as well as long-wavelength gyrokinetic models. We present an algorithm for the discontinuous Galerkin discretization of this operator, and provide results from relaxation and Landau damping benchmarks. 

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4. Active Learning of Dynamics for Data-Driven Control Using Koopman Operators

   https://doi.org/10.1109/TRO.2019.2923880  

   Abraham, Ian; Murphey, Todd D. (July 2019, IEEE Transactions on Robotics)

   This paper presents an active learning strategy for robotic systems that takes into account task information, enables fast learning, and allows control to be readily synthesized by taking advantage of the Koopman operator representation. We first motivate the use of representing nonlinear systems as linear Koopman operator systems by illustrating the improved model-based control performance with an actuated Van der Pol system. Information-theoretic methods are then applied to the Koopman operator formulation of dynamical systems where we derive a controller for active learning of robot dynamics. The active learning controller is shown to increase the rate of information about the Koopman operator. In addition, our active learning controller can readily incorporate policies built on the Koopman dynamics, enabling the benefits of fast active learning and improved control. Results using a quadcopter illustrate single-execution active learning and stabilization capabilities during free-fall. The results for active learning are extended for automating Koopman observables and we implement our method on real robotic systems. 

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5. On extremizing sequences for adjoint Fourier restriction to the sphere

   https://doi.org/10.1016/j.aim.2024.109854  

   Flock, Taryn C; Stovall, Betsy (September 2024, Advances in Mathematics)

   In this article, we develop a linear profile decomposition for the $$L^p \to L^q$$ adjoint Fourier restriction operator associated to the sphere, valid for exponent pairs $p\max\{p,\tfrac{d+2}d p'\}$$, or if $$q=\tfrac{d+2}d p'$$ and the operator norm exceeds a certain constant times the operator norm of the parabolic extension operator. 

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