In this article, certain classes of homeomorphisms on the complex plane have been considered. Specifically, we investigate ring and lower Q-homeomorphisms with respect to the p-module, as well as homeomorphisms of finite distortion. The behavior at infinity of these mappings is investigated, and sharp estimates for their lower power $\alpha $-order are obtained.

In this article, we give a boundary rigidity result on some fibered domains. As an application, we obtain the boundary rigidity for holomorphic self-maps on bounded symmetric domains with an interior fixed point.

It is an open problem in additive number theory to compute and understand the full range of sumset sizes of finite sets of integers, that is, the set $ \mathcal R_{\mathbf Z}(h,k) = \{|hA|:A \subseteq \mathbf Z \text { and } |A|=k\}$ for all integers $h \geq 3$ and $k \geq 3$. This article constructs certain infinite families of finite sets of size k, computes their h-fold sumset sizes, and obtains explicit finite arithmetic progressions of sumset sizes in $ \mathcal R_{\mathbf Z}(h,k)$.

In this article, we study the $L_p$ Gauss image problem for C-pseudo-cones, where C is a pointed closed convex cone in $\mathbb {R}^n$ with vertex at the origin and having nonempty interior. We first establish the existence of solutions for a Borel measure supported on compact subsets, and then prove a general existence theorem for $p>0$ via approximation. In addition, we prove a uniqueness theorem for $p<0$ under appropriate assumptions.

Paul Erdős and R. Daniel Mauldin asked a series of questions on certain types of polygons of area $1$, the vertices of which can be found in every planar set of infinite Lebesgue measure. We address two of these questions, one on cyclic quadrilaterals and the other on convex polygons with congruent sides, with, respectively, positive and negative answers.

We consider a mixed Steklov–Dirichlet eigenvalue problem on a smooth bounded domain having a spherical hole. In this article, we take Dirichlet condition on the boundary of the spherical hole and Steklov condition on the other boundary component/s. Under certain symmetry assumptions on multiconnected domains in $\mathbb {R}^{n}$ having a spherical hole, we obtain isoperimetric inequalities for the k-th Steklov–Dirichlet eigenvalues for each $k \in \{2, 3, \dots , n+1\}$. We provide examples to emphasise the fact that the symmetry assumptions, on the family of domains considered, are crucial. We also extend Theorem 3.1 of “Gavitone et al. (2023), An isoperimetric inequality for the first Steklov–Dirichlet Laplacian eigenvalue of convex sets with a spherical hole, Pacific Journal of Mathematics, 320(2): 241–259” not only from Euclidean domains to domains in space forms but also from convex domains to star-shaped domains. In particular, we obtain sharp lower and upper bounds for the first Steklov–Dirichlet eigenvalue on the family of all bounded star-shaped domains on the hemisphere as well as on the hyperbolic space.

Let $\operatorname {GSpin}(V)$ (resp., $\operatorname {GPin}(V)$) be a general Spin group (resp., a general Pin group) associated with a nondegenerate quadratic space V of dimension n over an Archimedean local field F. For a nondegenerate quadratic space W of dimension $n-1$ over F, we also consider $\operatorname {GSpin}(W)$ and $\operatorname {GPin}(W)$. We prove the multiplicity-at-most-one theorem in the Archimedean case for a pair of groups ($\operatorname {GSpin}(V), \operatorname {GSpin}(W)$) and also for a pair of groups ($\operatorname {GPin}(V), \operatorname {GPin}(W)$); namely, we prove that the restriction to $\operatorname {GSpin}(W)$ (resp., $\operatorname {GPin}(W)$) of an irreducible Casselman–Wallach representation of $\operatorname {GSpin}(V)$ (resp., $\operatorname {GPin}(V)$) is multiplicity free.

For each three-dimensional non-Lie Leibniz algebra over the complex numbers, we describe the algebra of polynomial invariants and determine its group of automorphisms. As a consequence, we establish that any two non-nilpotent three-dimensional non-Lie Leibniz algebras can be distinguished by the traces of degrees $\leqslant 2$ and by the dimensions of their automorphism groups.

Delsarte theory, more specifically the study of codes and designs in association schemes, has proved invaluable in studying an increasing assortment of association schemes in recent years. Tools motivated by the study of error-correcting codes in the Hamming scheme and combinatorial t-designs in the Johnson scheme apply equally well in association schemes with irrational eigenvalues. We assume here that we have a commutative association scheme with irrational eigenvalues and wish to study its Delsarte T-designs. We explore when a T-design is also a $T'$-design, where $T'\supseteq T$ is controlled by the orbits of a Galois group related to the splitting field of the association scheme. We then study Delsarte designs in the association schemes of finite groups, with a detailed exploration of the dicyclic groups.

In this article, we introduce Orlicz spaces on $ \mathbb Z^n \times \mathbb T^n $ and Orlicz modulation spaces on $\mathbb Z^n$, and study inclusion relations, convolution relations, and duality of these spaces. We show that the Orlicz modulation space $M^{\Phi }(\mathbb Z^n)$ is close to the modulation space $M^{2}(\mathbb Z^n)$ for some particular Young function $\Phi $. Then, we study localization operators on $\mathbb Z^n$. In particular, using appropriate classes for symbols, we prove that these operators are bounded on Orlicz modulation spaces on $\mathbb Z^n$, compact and in the Schatten–von Neumann classes.

Let K be an imaginary quadratic field, and let $E/\mathbb {Q}$ be an elliptic curve with complex multiplication by $\mathcal {O}_K$. Let $K_\infty /K$ be the anticyclotomic $\mathbb {Z}_p$-extension of K and $K_n$ be the intermediate layers. Under additional assumptions on Kobayashi’s signed Selmer groups, we prove an asymptotic formula for .

Understanding how spectral quantities localize on manifolds is a central theme in geometric spectral theory and index theory. Within this framework, the BFK formula, obtained by Burghelea, Friedlander, and Kappeler in 1992, describes how the zeta-regularized determinant of an elliptic operator decomposes as the underlying manifold is cut into pieces. In this article, we present a novel proof of this result. Inspired by the work of Brüning and Lesch on the eta invariant of Dirac operators, we derive the BFK formula by interpolating continuously between boundary conditions and understanding the variation of the determinant along this deformation.

We define the type of a plane curve as the initial degree of the corresponding Bourbaki ideal. Then, we show that this invariant behaves well with respect to the union of curves. Curves of type $0$ are precisely the free curves, while curves of type $1$ are the plus-one generated curves. In this article, we first show that line arrangements and conic-line arrangements can exhibit all the theoretically possible types. In the second part, we study the properties of the curves of type $2$ and construct families of line arrangements and conic-line arrangements of this type.

In this article, we define a new class of operators called weak U-Dunford–Pettis, which generalizes the U-Dunford–Pettis, weak Dunford–Pettis, and order Dunford–Pettis classes, then we also give a characterization for this class, which we compare with some lattice properties, we then set out the conditions under which this class coincides with the U-Dunford–Pettis class, the weak Dunford–Pettis class, and the order Dunford–Pettis class.

The usual division algorithms on ${\mathbb {Z}}$ and ${\mathbb {Z}}[i]$ measure the size of remainders using the algebraic norm. These rings are Euclidean with respect to several functions. The pointwise minimum of all Euclidean functions $f: R \setminus \{0\} \rightarrow {\mathbb {N}}$ on a Euclidean domain R is itself a Euclidean function, called the minimal Euclidean function and denoted by $\phi _R$. To the author’s knowledge, the integers, ${\mathbb {Z}}$ and the Gaussians, ${\mathbb {Z}}[i]$ are the only rings of integers of number fields for which we have a formula to compute their minimal Euclidean functions, $\phi _{{\mathbb {Z}}}$ and $\phi _{{\mathbb {Z}}[i]}$. This article presents the first division algorithm (that the author knows of) for ${\mathbb {Z}}[i]$ relative to $\phi _{{\mathbb {Z}}[i]}$, empowering readers to perform the Euclidean algorithm on ${\mathbb {Z}}[i]$ using its minimal Euclidean function.

We show that a normed linear space is isometrically isomorphic to an inner product space if and only if it is a strongly n-point homogeneous metric space for any (or every) $n \geqslant 3$. The counterpart for $n=2$ is the Banach–Mazur problem.

Quantales can be regarded as a combination of complete lattices and semigroups. Unital quantales constitute a significant subclass within quantale theory, which play a crucial role in the theoretical framework of quantale research. It is well known that every complete lattice can support a quantale. However, the question of whether every complete lattice can support a unital quantale has not been considered before. In this article, we first give some counter-examples to indicate that the answer to the above question is negative, and then investigate the complete lattices of supporting unital quantales.

In this article, we present an elementary proof of the Cartan decomposition theorem for the group $ \mathrm {Sp}(1, n) $. As an application, we determine the largest regular domain for discrete quaternionic hyperbolic groups acting on $ \mathbb {HP}^n $. Furthermore, we demonstrate that Bers’ simultaneous uniformization and Köebe’s retrosection theorem fail to hold for higher-dimensional quaternionic Kleinian groups.

In the article “Irreducible modules of modular Lie superalgebras and super version of the first Kac-Weisfeiler conjecture, Canad. Math. Bull. 67 (2024), no. 3, 554–573.” The statement in Theorem 4.7 is improper, which is fixed here. Theorem 4.7 is an isolated result in the article. This correction does not influence any arguments and any main results after that in the original article.