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In Dawes [Algebr. Geom. 12(2025), no. 3, 601–660], families of orthogonal modular varieties $��F�\left(�\Gamma �\right)�$ associated with moduli spaces of compact hyperkähler manifolds of deformation generalized Kummer type (also known as “deformation generalized Kummer varieties”) were studied. The orthogonal modular varieties were defined for an even integer $��2�d�$, corresponding to the degree of polarization of the associated hyperkähler manifolds. It was shown in Dawes [Algebr. Geom. 12(2025), no. 3, 601–660] that the modular varieties are of general type when $��2�d�$ is square-free and sufficiently large. The purpose of this paper is to show that the square-free condition can be removed.

In this paper, we study conditions for a discrete subgroup of the automorphism group of the $�n$-dimensional complex unit ball to be of convergence type or second kind, connecting these classifications to the existence of Green's functions and subharmonic or harmonic functions on its quotient space. Furthermore, we extend the definitions of convergence and divergence types to bounded symmetric domains, introducing a Poincaré series and providing a new criterion for discrete subgroups acting on these domains.

The Rarita-Schwinger–Seiberg-Witten (RS–SW) equations are defined similarly to the classical Seiberg–Witten equations, where a geometric non–Dirac-type operator replaces the Dirac operator called the Rarita–Schwinger operator. In dimension 4, the RS–SW equation was first considered by the second author (Nguyen [J. Geom. Anal. 33(2023), no. 10, 336]). The variational approach will also give us a three-dimensional version of the equations. The RS–SW equations share some features with the multiple-spinor Seiberg–Witten equations, where the moduli space of solutions could be noncompact. In this paper, we prove a compactness theorem regarding the moduli space of solutions of the RS–SW equations defined on 3-manifolds.

In this study, we deal with generalized regularity properties for solutions to $�p$-Laplace equations with degenerate matrix weights. It has been already observed in previous interesting works that gaining Calderón–Zygmund estimates for nonlinear equations with degenerate weights under the so-called $��\log �-�\mathrm{BMO}�$ condition and minimal regularity assumption on the boundary. In this paper, we also follow this direction and extend general gradient estimates for level sets of the gradient of solutions up to more subtle function spaces. In particular, we construct a covering of the super-level sets of the spatial gradient $��|�\nabla �u�|�$ with respect to a large scaling parameter via fractional maximal operators.

We introduce notions of recurrence and transience for graphs over a non-Archimedean ordered field. To achieve this, we establish a connection between these graphs and random walks on directed graphs over the reals. In particular, we give a characterization of the real directed graphs which can arise in such a way. As a main result, we give characterization for recurrence and transience in terms of a quantity related to the capacity.

In this paper, we consider the minimization problem for the first eigenvalue of the fractional Laplacian with respect to the weight functions lying in the rearrangement classes of fixed weight functions. We prove the existence of minimizing weights in the rearrangement classes of weight functions satisfying some assumptions. Also, we provide characterizations of these minimizing weights in terms of the eigenfunctions. Furthermore, we establish various qualitative properties, such as Steiner symmetry, radial symmetry, foliated Schwarz symmetry, etc., of the minimizing weights and corresponding eigenfunctions.

In this paper, we prove the equidistribution of the Hecke eigenvalues of Maass forms over an arbitrary number field at a fixed prime ideal, with respect to the Sato–Tate measure. As an application, we obtain that the proportion of Maass forms that do not satisfy the Ramanujan–Petersson conjecture at a fixed prime ideal is 0.

In this paper, we study the following fourth-order equation of the Kirchhoff type

In this paper, we prove the functional Minkowski inequality for $��(�p�,�\alpha �)�$-torsional rigidity under suitable regularity assumptions, where $��(�p�,�\alpha �)�$-torsional rigidity can be formulated as the weak solution of an elliptic boundary value problem of the $�p$-Laplacian. We also establish the Orlicz Brunn–Minkowski and Orlicz Minkowski inequalities for $��(�p�,�\alpha �)�$-torsional rigidity, which are extensions of the Brunn–Minkowski and Minkowski inequalities for torsional rigidity. Finally, we describe the equivalence between the Orlicz Brunn–Minkowski inequality and the Orlicz Minkowski inequality.