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We give two geometric interpretations for the local type of a line that is highly tangent to a hypersurface in a single point. One interpretation is phrased in terms of the Wronski map, while the other interpretation relates to the fundamental forms of the hypersurface. These local types are the local contributions of a quadratic form-valued Euler number that depends on a choice of orientation.

We prove the existence of a positive radial solution in a unit ball centered at the origin for some classes of Hénon equations involving supercritical nonlinearity. More precisely, we study how Hénon's weight impacts the variable supercritical exponent in the context of the work by do Ó, Ruf, and Ubilla. For this purpose, we combine variational methods with a new Sobolev–Hardy type embedding for radial functions into variable exponent Lebesgue spaces. The suitable use of the Radial Lemma allows us to arrive at the necessary estimate with fewer assumptions than those found in the existing literature.

Let $�E$ be a module of projective dimension 1 over a Noetherian ring $�R$ and consider its Rees algebra $��R�\left(�E�\right)�$. We study this ring as a quotient of the symmetric algebra $��S�\left(�E�\right)�$ and consider the ideal $�A$ defining this quotient. In the case that $��S�\left(�E�\right)�$ is a complete intersection ring, we employ a duality between $�A$ and $��S�\left(�E�\right)�$ in order to study the Rees ring $��R�\left(�E�\right)�$ in multiple settings. In particular, when $�R$ is a complete intersection ring defined by quadrics, we consider its module of Kähler differentials $�{\Omega }_{�R�/�k�}$

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.