A Structure Theorem for \displaystyle\mathbb{P}^{{1}} — Spec k-bimodules

Let k be an algebraically closed field. Using the Eilenberg–Watts theorem over schemes (Nyman, J Pure Appl Algebra 214:1922–1954, 2010), we determine the structure of k-linear right exact direct limit and coherence preserving functors from the category of quasi-coherent sheaves on $\mathbb{P}^{1}_{k}$ to the category of vector spaces over k. As a consequence, we characterize those functors which are integral transforms.     

A Fatou–Bieberbach domain in \mathbb {C}^2 which is not Runge

Since a paper by Rosay and Rudin (Trans. Am. Math. Soc. 310, 47–86, 1988) there has been an open question whether all Fatou–Bieberbach domains are Runge. We give an example of a Fatou–Bieberbach domain Ω in which is not Runge. The domain Ω provides (yet) a negative answer to a problem of Bremermann. Supported by Schweizerische Nationalfonds grant 200021-116165/1.     

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.     

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.     

Time-periodic solutions of the compressible Navier–Stokes equations in {\mathbb{R}^{4}}

This paper deals with the existence of time-periodic solutions to the compressible Navier–Stokes equations effected by general form external force in \({\mathbb{R}^{N}}\) with \({N = 4}\). Using a fixed point method, we establish the existence and uniqueness of time-periodic solutions. This paper extends Ma, UKai, Yang’s result [5], in which, the existence is obtained when the space dimension \({N \ge 5}\).     

On a Schrödinger equation with periodic potential and critical growth in \mathbb{R}^{2}

The main purpose of this paper is to establish the existence of a solution of the semilinear Schr?dinger equation

Extremal Self-Dual Codes over \mathbb{F} 2 × \mathbb{F} 2

In this paper, it is shown that extremal (Hermitian) self-dual codes over ₂ × ₂ exist only for lengths 1, 2, 3, 4, 5, 8 and 10. All extremal self-dual codes over ₂ × ₂ are found. In particular, it is shown that there is a unique extremal self-dual code up to equivalence for lengths 8 and 10. Optimal self-dual codes are also investigated. A classification is given for binary [12, 7, 4] codes with dual distance 4, binary [13, 7, 4] codes with dual distance 4 and binary [13, 8, 4] codes with dual distance 4.     

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.     

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.     

On some nonlinear elliptic problems for p–Laplacian in \mathbb {R}^N

In this paper, we consider a nonlinear elliptic problem involving the p-Laplacian with perturbation terms in the whole . Via variational arguments, we obtain existence and regularity of nontrivial solutions. The research of the first and second authors is supported by grant num, #28/12 from the Al-Imam University, Riyadh, KSA.     

( m,n)-Equidista nt Sets i n $\ mathbb{R}^{k},\ mathbb{S}^{k}$ , a nd ℙk

We call a metric space X (m,n)-equidistant if, when A⊆X has exactly m points, there are exactly n points in X each of which is equidistant from (the points of) A. We prove that, for k≥2, the Euclidean space ℝ ^k contains an (m,1)-equidistant set if and only if k≥m. Although the sphere is (3,2)-equidistant, and ℝ⁴ contain no (4,2)-equidistant sets. We discuss related results about projective spaces, and state a conjecture about analogous to the Double Midset Conjecture.     

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...     

Symplectic Twistor Operator on \mathbb R ^{2n} and the Segal–Shale–Weil Representation

The aim of our article is the study of solution space of the symplectic twistor operator $T_s$ in symplectic spin geometry on standard symplectic space $(\mathbb R ^{2n},\omega )$ , which is the symplectic analogue of the twistor operator in (pseudo) Riemannian spin geometry. In particular, we observe a substantial difference between the case $n=1$ of real dimension $2$ and the case of $\mathbb R ^{2n}, n>1$ . For $n>1$ , the solution space of $T_s$ is isomorphic to the Segal–Shale–Weil representation.     

On $$\mathbb {R}$$-Linear Problem and Truncated Wiener–Hopf Equation

Siberian Advances in Mathematics - We consider the $$\mathbb {R}$$-linear problem (also known as the Markushevich problem and the generalized Riemann boundary value problem) and the convolution...