Schiffer’s Problem and an Isoperimetric Inequality for the First Buckling Eigenvalue of Domains on $$\mathbb{S}^{2}$$

We consider the overdetermined eigenvalue problem on a sufficiently regular connected open domain Ω on the 2-sphere :

Poletsky–Stessin Hardy Spaces on Complex Ellipsoids in $$\mathbb {C}^{n}$$

We study Poletsky–Stessin Hardy spaces on complex ellipsoids in \(\mathbb {C}^{n}\). Different from one variable case, classical Hardy spaces are strictly contained in Poletsky–Stessin Hardy spaces on complex ellipsoids so boundary values are not automatically obtained in this case. We have showed that functions belonging to Poletsky–Stessin Hardy spaces have boundary values and they can be approached through admissible approach regions in the complex ellipsoid case. Moreover, we have obtained that polynomials are dense in these spaces. We also considered the composition operators acting on Poletsky–Stessin Hardy spaces on complex ellipsoids and gave conditions for their boundedness and compactness.     

Schatten class Hankel and $$\overline{\partial }$$ ∂ ¯ -Neumann operators on pseudoconvex domains in $$\mathbb {C}^{n}$$ C n

Monatshefte für Mathematik - Let $$\Omega $$ be a $$C^2$$ -smooth bounded pseudoconvex domain in $$\mathbb {C}^n$$ for $$n\ge 2$$ and let $$\varphi $$ be a holomorphic function on $$\Omega $$...     

A short proof of Cartan’s Nullstellensatz for entire functions in $${mathbb{C}^n}$$

Using the fact that the maximal ideals in the polydisk algebra are given by the kernels of point evaluations, we derive a simple formula that gives a solution to the Bézout equation in the space of all entire functions of several complex variables. Thus a short and easy analytic proof of Cartan’s Nullstellensatz is obtained.     

Borsuk’s Partition Problem in $$(mathbb{R}^{n},ell_{p})$$

Mathematical Notes - For a bounded set $$X$$ with diameter $$d_{C}(X)$$ in a finite-dimensional normed space with an origin-symmetric convex body $$C$$ as the unit ball, the Borsuk number of $$X$$...     

ON HEYDE’S THEOREM ON THE GROUP $$\mathbb{R}$$ × $$\mathbb{T}$$

Doklady Mathematics - According to the well-knows Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of...     

Explicit Formulas for the Green’s Function and the Bergman Kernel for Monogenic Functions in Annular Shaped Domains in {\mathbb{R}^{n+1}}

By applying a reflection principle we set up fully explicit representation formulas for the harmonic Green’s function for orthogonal sectors of the annulus of the unit ball of ${\mathbb{R}^n}$ . From the harmonic Green’s function we then can determine the Bergman kernel function of Clifford holomorphic functions by applying an appropriate vector differentiation. As a concrete application we give an explicit analytic representation formula of the solutions to an n-dimensional Dirichlet problem in annular shaped domains that arises in the context of heat conduction.     

An asymptotic version of Dumnicki’s algorithm for linear systems in $${\mathbb{CP}^2}$$

Using Dumnicki’s approach to showing non-specialty of linear systems consisting of plane curves with prescribed multiplicities in sufficiently general points on we develop an asymptotic method to determine lower bounds for Seshadri constants of general points on . With this method we prove the lower bound for 10 general points on .      

Stable spike clusters for the precursor Gierer–Meinhardt system in $$\mathbb {R}^2$$

We consider the Gierer–Meinhardt system with small inhibitor diffusivity, very small activator diffusivity and a precursor inhomogeneity. For any given positive integer k we construct a spike cluster consisting of k spikes which all approach the same nondegenerate local minimum point of the precursor inhomogeneity. We show that this spike cluster can be linearly stable. In particular, we show the existence of spike clusters for spikes located at the vertices of a polygon with or without centre. Further, the cluster without centre is stable for up to three spikes, whereas the cluster with centre is stable for up to six spikes. The main idea underpinning these stable spike clusters is the following: due to the small inhibitor diffusivity the interaction between spikes is repulsive, and the spikes are attracted towards the local minimum point of the precursor inhomogeneity. Combining these two effects can lead to an equilibrium of spike positions within the cluster such that the cluster is linearly stable.     

Projective Compactification of $$\mathbb {R}^{1,1}$$ and Its Möbius Geometry

We examine the semi-Riemannian manifold \(\mathbb {R}^{1,1}\), which is realized as the split complex plane, and its conformal compactification as an analogue of the complex plane and the Riemann sphere. We also consider conformal maps on the compactification and study some of their basic properties.     

Erratum to: Borsuk’s Partition Problem in $$(mathbb{R}^{n},ell_{p})$$

Mathematical Notes - An Erratum to this paper has been published: https://doi.org/10.1134/S0001434622010321     

Equivalent moduli of continuity, Bloch’s theorem for pluriharmonic mappings in mathbb{B}{kern-8pt}mathbb{B}^{boldsymbol n}

In this paper, we first establish a Schwarz?CPick type theorem for pluriharmonic mappings and then we apply it to discuss the equivalent norms on Lipschitz-type spaces. Finally, we obtain several Landau??s and Bloch??s type theorems for pluriharmonic mappings.     

Optimal software-implemented Itoh–Tsujii inversion for $$\mathbb {F}_{2^{m}}$$

Field inversion in \(\mathbb {F}_{2^{m}}\) dominates the cost of modern software implementations of certain elliptic curve cryptographic operations, such as point encoding/hashing into elliptic curves (Brown et al. in: Submission to NIST, 2008; Brown in: IACR Cryptology ePrint Archive 2008:12, 2008; Aranha et al. in: Cryptology ePrint Archive, Report 2014/486, 2014) Itoh–Tsujii inversion using a polynomial basis and precomputed table-based multi-squaring has been demonstrated to be highly effective for software implementations (Taverne et al. in: CHES 2011, 2011; Oliveira et al. in: J Cryptogr Eng 4(1):3–17, 2014; Aranha et al. in: Cryptology ePrint Archive, Report 2014/486, 2014), but the performance and memory use depend critically on the choice of addition chain and multi-squaring tables, which in prior work have been determined only by suboptimal ad-hoc methods and manual selection. We thoroughly investigated the performance/memory tradeoff for table-based linear transforms used for efficient multi-squaring. Based upon the results of that investigation, we devised a comprehensive cost model for Itoh–Tsujii inversion and a corresponding optimization procedure that is empirically fast and provably finds globally-optimal solutions. We tested this method on eight binary fields commonly used for elliptic curve cryptography; our method found lower-cost solutions than the ad-hoc methods used previously, and for the first time enables a principled exploration of the time/memory tradeoff of inversion implementations.     

The Intrinsic π-Operator on Domain Manifolds in $${\mathbb{C}^{n+1}}$$

The main aim of this article is to study the hypercomplex π-operator over \mathbbCⁿ⁺¹{\mathbb{C}^{n+1}} via real, compact, n + 1-dimensional manifolds called domain manifolds. We introduce an intrinsic Dirac operator for such types of domain manifolds and define an intrinsic π-operator, study its mapping properties and introduce a Clifford–Beltrami equation in this context.     

Axler–Zheng Type Theorem on a Class of Domains in {\mathbb{C}^n}

We prove a version of Axler–Zheng’s Theorem on smooth bounded pseudoconvex domains in ${\mathbb{C}^n}$ on which the ${\overline{\partial}}$ -Neumann operator is compact.     

Equidistribution of saddle periodic points for Hénon-type automorphisms of $$\mathbb {C}^k$$

In this paper, we prove the equidistribution of saddle periodic points for Hénon-type automorphisms of \(\mathbb {C}^k\) with respect to its equilibrium measure. A general strategy to obtain equidistribution properties in any dimension is presented. It is based on our recent theory of densities for positive closed currents. Several fine properties of dynamical currents are also proved.