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1. Quotient maps and configuration spaces of hard disks

   https://doi.org/10.1007/s10035-022-01235-5  

   Ericok, Ozan B.; Mason, Jeremy K. (May 2022, Granular Matter)

   AbstractHard disks systems are often considered as prototypes for simple fluids. In a statistical mechanics context, the hard disk configuration space is generally quotiented by the action of various symmetry groups. The changes in the topological and geometric properties of the configuration spaces effected by such quotient maps are studied for small numbers of disks on a square and hexagonal torus. A metric is defined on the configuration space and the various quotient spaces that respects the desired symmetries. This is used to construct explicit triangulations of the configuration spaces as$$\alpha$$ $\alpha$-complexes. Critical points of the hard disk potential on a configuration space are associated with changes in the topology of the accessible part of the configuration space as a function of disk radius, are conjectured to be related to the configurational entropy of glassy systems, and could reveal the origins of phase transitions in other systems. The number of critical points and their topological and geometric properties are found to depend on the symmetries by which the configuration space is quotiented. Graphic abstract 

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2. 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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3. The Zaremba problem in two-dimensional Lipschitz graph domains

   https://doi.org/10.1090/tran/9425  

   Carro, María; Luque, Teresa; Naibo, Virginia (July 2025, Transactions of the American Mathematical Society)

   We study the Zaremba problem, or mixed problem associated to the Laplace operator, in two-dimensional Lipschitz graph domains with mixed Dirichlet and Neumann boundary data in Lebesgue and Lorentz spaces. We obtain an explicit value $r$such that the Zaremba problem is solvable in $L^p$for $1>p>r$and in the Lorentz space $L^{r,1}$. Applications include those where the domain is a cone with vertex at the origin and, more generally, a Schwarz–Christoffel domain. The techniques employed are based on results of the Zaremba problem in the upper half-plane, the use of conformal maps and the theory of solutions to the Neumann problem. For the case when the domain is the upper half-plane, we work in the weighted setting, establishing conditions on the weights for the existence of solutions and estimates for the non-tangential maximal function of the gradient of the solution. In particular, in the $L^2$-unweighted case, where known examples show that the gradient of the solution may fail to be squared-integrable, we prove restricted weak-type estimates. 

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4. Cartan actions of higher rank abelian groups and their classification

   https://doi.org/10.1090/jams/1033  

   Spatzier, Ralf; Vinhage, Kurt (July 2024, Journal of the American Mathematical Society)

   We study ${\mathbb{R}}^k\times <\#comment/>{\mathbb{Z}}^{\mathrm{\ell <\#comment/>}}$actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We completely classify such actions as built from low-dimensional Anosov flows and diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for this class. This is achieved by introducing a new tool, the action of a dynamically defined topological group describing paths in coarse Lyapunov foliations, and understanding its generators and relations. We obtain applications to the Zimmer program. 

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5. Sparse optimization problems in fractional order Sobolev spaces

   https://doi.org/10.1088/1361-6420/acbe5e  

   Antil, Harbir; Wachsmuth, Daniel (March 2023, Inverse Problems)

   Abstract We consider optimization problems in the fractional order Sobolev spaces with sparsity promoting objective functionals containingL^p-pseudonorms, $p\in (0,1)$. Existence of solutions is proven. By means of a smoothing scheme, we obtain first-order optimality conditions, which contain an equation with the fractional Laplace operator. An algorithm based on this smoothing scheme is developed. Weak limit points of iterates are shown to satisfy a stationarity system that is slightly weaker than that given by the necessary condition. 

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