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1. Is the Finite-Time Lyapunov Exponent Field a Koopman Eigenfunction?

   https://doi.org/10.3390/math9212731  

   Bollt, Erik M.; Ross, Shane D. (November 2021, Mathematics)

   null (Ed.)

   This work serves as a bridge between two approaches to analysis of dynamical systems: the local, geometric analysis, and the global operator theoretic Koopman analysis. We explicitly construct vector fields where the instantaneous Lyapunov exponent field is a Koopman eigenfunction. Restricting ourselves to polynomial vector fields to make this construction easier, we find that such vector fields do exist, and we explore whether such vector fields have a special structure, thus making a link between the geometric theory and the transfer operator theory. 

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2. How students reason with derivatives of vector field diagrams

   Topdemir, Zeynep; Thompson, John R; Loverude, Michael E (October 2023, The Learning and Teaching of Calculus Across Disciplines – Proceedings of the Second Calculus Conference)

   Dreyfus, T; González-Martín, A S; Nardi, E; Monaghan, J; Thompson, P W (Ed.)

   Physics students are introduced to vector fields in introductory courses, typically in the contexts of electric and magnetic fields. Vector calculus provides several ways to describe how vector fields vary in space including the gradient, divergence, and curl. Physics majors use vector calculus extensively in a junior-level electricity and magnetism (E&M) course. Our focus here is exploring student reasoning with the partial derivatives that constitute divergence and curl in vector field representations, adding to the current understanding of how students reason with derivatives. 

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3. $$ L^{p} $$ compactness criteria with an application to variational convergence of some nonlocal energy functionals

   https://doi.org/10.3934/mine.2023097  

   Du, Qiang; Mengesha, Tadele; Tian, Xiaochuan (January 2023, Mathematics in Engineering)

   Motivated by some variational problems from a nonlocal model of mechanics, this work presents a set of sufficient conditions that guarantee a compact inclusion in the function space of $$ L^{p} $$ vector fields defined on a domain $$ \Omega $$ that is either a bounded domain in $$ \mathbb{R}^{d} $$ or $$ \mathbb{R}^{d} $$ itself. The criteria are nonlocal and are given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. Moreover, these criteria for vector fields are also different from those given for scalar fields in that the conditions are based on nonlocal interactions involving only parts of the components of the vector fields. The $$ L^{p} $$ compactness criteria are utilized in demonstrating the convergence of minimizers of parameterized nonlocal energy functionals. 

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4. Connections between nonlocal operators: from vector calculus identities to a fractional Helmholtz decomposition. F

   https://doi.org/10.2172/1855046  

   D'Elia, M.; Gulian, M.; Mengesha, T.; Scott, J. M. (December 2022, Fractional Calculus Applied Analysis)

   Nonlocal vector calculus, which is based on the nonlocal forms of gradient, divergence, and Laplace operators in multiple dimensions, has shown promising applications in fields such as hydrology, mechanics, and image processing. In this work, we study the analytical underpinnings of these operators. We rigorously treat compositions of nonlocal operators, prove nonlocal vector calculus identities, and connect weighted and unweighted variational frameworks. We combine these results to obtain a weighted fractional Helmholtz decomposition which is valid for sufficiently smooth vector fields. Our approach identifies the function spaces in which the stated identities and decompositions hold, providing a rigorous foundation to the nonlocal vector calculus identities that can serve as tools for nonlocal modeling in higher dimensions. 

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5. Weighted Radon transforms of vector fields, with applications to magnetoacoustoelectric tomography

   https://doi.org/10.1088/1361-6420/acd07a  

   Kunyansky, L; McDugald, E; Shearer, B (May 2023, Inverse Problems)

   Abstract Currently, theory of ray transforms of vector and tensor fields is well developed, but the Radon transforms of such fields have not been fully analyzed. We thus consider linearly weighted and unweighted longitudinal and transversal Radon transforms of vector fields. As usual, we use the standard Helmholtz decomposition of smooth and fast decreasing vector fields over the whole space. We show that such a decomposition produces potential and solenoidal components decreasing at infinity fast enough to guarantee the existence of the unweighted longitudinal and transversal Radon transforms of these components. It is known that reconstruction of an arbitrary vector field from only longitudinal or only transversal transforms is impossible. However, for the cases when both linearly weighted and unweighted transforms of either one of the types are known, we derive explicit inversion formulas for the full reconstruction of the field. Our interest in the inversion of such transforms stems from a certain inverse problem arising in magnetoacoustoelectric tomography (MAET). The connection between the weighted Radon transforms and MAET is exhibited in the paper. Finally, we demonstrate performance and noise sensitivity of the new inversion formulas in numerical simulations. 

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