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In this paper, we consider a class of Choquard equations with Hardy–Littlewood–Sobolev lower or upper critical exponent in the whole space R N . We combine an argument of L. Jeanjean and H. Tanaka (see (Proc. Am. Math. Soc. 131:2399–2408, 2003) with a concentration–compactness argument, and then we obtain the existence of ground state solutions, which extends and complements the earlier results. [ABSTRACT FROM AUTHOR]

We develop a dynamic generalized conditional gradient method (DGCG) for dynamic inverse problems with optimal transport regularization. We consider the framework introduced in Bredies and Fanzon (ESAIM: M2AN 54:2351–2382, 2020), where the objective functional is comprised of a fidelity term, penalizing the pointwise in time discrepancy between the observation and the unknown in time-varying Hilbert spaces, and a regularizer keeping track of the dynamics, given by the Benamou–Brenier energy constrained via the homogeneous continuity equation. Employing the characterization of the extremal points of the Benamou–Brenier energy (Bredies et al. in Bull Lond Math Soc 53(5):1436–1452, 2021), we define the atoms of the problem as measures concentrated on absolutely continuous curves in the domain. We propose a dynamic generalization of a conditional gradient method that consists of iteratively adding suitably chosen atoms to the current sparse iterate, and subsequently optimizing the coefficients in the resulting linear combination. We prove that the method converges with a sublinear rate to a minimizer of the objective functional. Additionally, we propose heuristic strategies and acceleration steps that allow to implement the algorithm efficiently. Finally, we provide numerical examples that demonstrate the effectiveness of our algorithm and model in reconstructing heavily undersampled dynamic data, together with the presence of noise. [ABSTRACT FROM AUTHOR]

We explore extensions of tropical methods to arithmetic enumerative problems such as A 1 -enumeration with values in the Grothendieck–Witt ring and rationality over Henselian valued fields, using bitangents to plane quartics as a test case. We consider quartic curves over valued fields whose tropicalizations are smooth and satisfy a mild genericity condition. We then express obstructions to rationality of bitangents and their points of tangency in terms of twisting of edges of the tropicalization; the latter depends only on the tropicalization and the initial coefficients of the defining equation modulo squares. We also show that the GW -multiplicity of a tropical bitangent, i.e. the multiplicity with which its lifts contribute to the A 1 -enumeration of bitangents as defined by Larson and Vogt (Res Math Sci 8(26):1–21, 2021), can be computed from the tropicalization of the quartic together with the initial coefficients of the defining equation. As an application, we show that the four lifts of most tropical bitangent classes contribute 2 H , twice the class of the hyperbolic plane, to the A 1 -enumeration. These results rely on a degeneration theorem relating the Grothendieck–Witt ring of a Henselian valued field to the Grothendieck–Witt ring of its residue field, in residue characteristic not equal to two. [ABSTRACT FROM AUTHOR]

This work illustrates how one could apply Lie point symmetries for finding the analytical solution to first-order mechanical systems. Although the classical Lie method constitutes a powerful tool for solving differential equations, an obstacle appears in the case of systems of first-order equations because they admit an infinite number of symmetries, and it is not possible to compute them by following a systematic procedure. To overcome this difficulty, we follow the idea exposed in Nucci (J. Math. Phys. 37: 1772–1775, 1996), Nucci (Electr. J. Diff. Eqn. 12: 87–101, 2005), Nucci (J. Math. Phys. 42: 746, 2001) consisting of transforming the original system to an equivalent system in which one of the equations is of second–order. The presented approach is applied to Hathaway's circular pursuit problem, leading to the analytical solution of the system expressed in terms of the general solution to an Abel equation. [ABSTRACT FROM AUTHOR]

In this work, we consider a quasilinear system of viscoelastic equations with degenerate damping and general source terms. According to some suitable hypothesis, we study the blow-up of solutions. This is the general case of the recent results of Boulaaras' works (Bull. Malays. Math. Sci. Soc. 43:725–755, 2020) and (Appl. Anal. 99:1724–1748, 2020). [ABSTRACT FROM AUTHOR]