Quasi–Monte Carlo rules for numerical integration over the unit sphere {\mathbb{S}^2}

We study numerical integration on the unit sphere ${\mathbb{S}^2 \subseteq\mathbb{R}^3}$ using equal weight quadrature rules, where the weights are such that constant functions are integrated exactly. The quadrature points are constructed by lifting a (0, m, 2)-net given in the unit square [0, 1]² to the sphere ${\mathbb{S}^2}$ by means of an area preserving map. A similar approach has previously been suggested by Cui and Freeden [SIAM J Sci Comput 18(2):595–609, 1997]. We prove three results. The first one is that the construction is (almost) optimal with respect to discrepancies based on spherical rectangles. Further we prove that the point set is asymptotically uniformly distributed on ${\mathbb{S}^2}$ . And finally, we prove an upper bound on the spherical cap L ₂-discrepancy of order N ^?1/2(log N)^1/2 (where N denotes the number of points). This improves upon the bound on the spherical cap L ₂-discrepancy of the construction by Lubotzky, Phillips and Sarnak [Commun Pure Appl Math 39(S, suppl):S149–S186, 1986] by a factor of ${\sqrt{\log N}}$ . Numerical results suggest that the (0, m, 2)-nets lifted to the sphere ${\mathbb{S}^2}$ have spherical cap L ₂-discrepancy converging with the optimal order of N ^?3/4.     

Null curves in {\mathbb{C}^3} and Calabi–Yau conjectures

For any open orientable surface M and convex domain ${\Omega\subset \mathbb{C}^3,}$ there exist a Riemann surface N homeomorphic to M and a complete proper null curve F : N → Ω. This result follows from a general existence theorem with many applications. Among them, the followings:

For any convex domain Ω in ${\mathbb{C}^2}$ there exist a Riemann surface N homeomorphic to M and a complete proper holomorphic immersion F : N → Ω. Furthermore, if ${D \subset \mathbb{R}^2}$ is a convex domain and Ω is the solid right cylinder ${\{x \in \mathbb{C}^2 \,|\, \mbox{Re}(x) \in D\},}$ then F can be chosen so that Re(F) : N → D is proper.

There exist a Riemann surface N homeomorphic to M and a complete bounded holomorphic null immersion ${F:N \to {\rm SL}(2, \mathbb{C}).}$

There exists a complete bounded CMC-1 immersion ${X:M \to \mathbb{H}^3.}$

For any convex domain ${\Omega \subset \mathbb{R}^3}$ there exists a complete proper minimal immersion (X _j ) _j=1,2,3 : M → Ω with vanishing flux. Furthermore, if ${D \subset \mathbb{R}^2}$ is a convex domain and ${\Omega=\{(x_j)_{j=1,2,3} \in \mathbb{R}^3 \,|\, (x_1,x_2) \in D\},}$ then X can be chosen so that (X ₁, X ₂) : M → D is proper.

Any of the above surfaces can be chosen with hyperbolic conformal structure.     

Periodic maximal surfaces in the Lorentz–Minkowski space $${\mathbb{L}^3}$$

A maximal surface with isolated singularities in a complete flat Lorentzian 3-manifold

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 \({N \ge 5}\).     

Lagrangian submanifolds in the homogeneous nearly Kähler $${\mathbb {S}}^3 \times {\mathbb {S}}^3$$

In this paper, we investigate Lagrangian submanifolds in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\). We introduce and make use of a triplet of angle functions to describe the geometry of a Lagrangian submanifold in \(\mathbb {S}^3 \times \mathbb {S}^3\). We construct a new example of a flat Lagrangian torus and give a complete classification of all the Lagrangian immersions of spaces of constant sectional curvature. As a corollary of our main result, we obtain that the radius of a round Lagrangian sphere in the homogeneous nearly Kähler \(\mathbb {S}^3 \times \mathbb {S}^3\) can only be \(\frac{2}{\sqrt{3}}\) or \(\frac{4}{\sqrt{3}}\).     

Fredholm and regularity theory of Douglis–Nirenberg elliptic systems on {\mathbb{R}^{N}}

We give a fairly complete exposition of the Fredholm properties of the Douglis–Nirenberg elliptic systems on ${\mathbb{R}^{N}}$ in the classical (unweighted) L ^p Sobolev spaces and under “minimal” assumptions about the coefficients. These assumptions rule out the use of classical pseudodifferential operator theory, although it is indirectly of assistance in places. After generalizing a necessary and sufficient condition for Fredholmness, already known in special cases, various invariance properties are established (index, null space, etc.), with respect to p and the Douglis–Nirenberg numbers. Among other things, this requires getting around the problem that the L ^p spaces are not ordered by inclusion. In turn, with some work, invariance leads to a regularity theory more general than what can be obtained by the method of differential quotients.     

On the monodromy of the moduli space of Calabi–Yau threefolds coming from eight planes in {\mathbb{P}^3}

It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the moduli space of Calabi–Yau threefolds coming from eight planes in ${\mathbb{P}^3}$ does not have this property. We show furthermore that the monodromy group of a good family is Zariski dense in the corresponding symplectic group. Moreover, we study a natural sublocus which we call hyperelliptic locus, over which the variation of Hodge structures is naturally isomorphic to wedge product of a variation of Hodge structures of weight one. It turns out the hyperelliptic locus does not extend to a Shimura subvariety of type III (Siegel space) within the moduli space. Besides general Hodge theory, representation theory and computational commutative algebra, one of the proofs depends on a new result on the tensor product decomposition of complex polarized variations of Hodge structures.     

On commuting polynomial automorphisms of {\mathbb{C}}^{k} , k ≥ 3

We characterize the polynomial automorphisms of ${\mathbb{C}}^3We characterize the polynomial automorphisms of , which commute with a regular automorphism. We use their meromorphic extension to and consider their dynamics on the hyperplane at infinity. We conjecture the additional hypothesis under which the same characterization is true in all dimensions. We give a partial answer to a question of S. Smale that in our context can be formulated as follows: can any polynomial automorphism of be the uniform limit on compact sets of polynomial automorphisms with trivial centralizer (i.e. )? Partially supported by Progetto MURST di Rilevante Interesse Nazionale Proprietà geometriche delle varietà reali e complesse. Supported by Istituto Nazionale Alta Matematica, “F. Severi”, Roma and G.N.S.A.G.A., Roma.     

Notes on von Neumann–Jordan and James Constants for Absolute Norms on {\mathbb{R}^2}

Let ${\|\cdot\|_{\psi}}$ be the absolute norm on ${\mathbb{R}^2}$ corresponding to a convex function ${\psi}$ on [0, 1] and ${C_{\text{NJ}}(\|\cdot\|_{\psi})}$ its von Neumann–Jordan constant. It is known that ${\max \{M_1^2, M_2^2\} \leq C_{\text{NJ}}(\| \cdot \|_{\psi}) \leq M_1^2 M_2^2}$ , where ${M_1 = \max_{0 \leq t \leq 1} \psi(t)/ \psi_2(t)}$ , ${M_2 = \max_{0\leq t \leq 1} \psi_2(t)/ \psi(t)}$ and ${\psi_2}$ is the corresponding function to the ? ₂-norm. In this paper, we shall present a necessary and sufficient condition for the above right side inequality to attain equality. A corollary, which is valid for the complex case, will cover a couple of previous results. Similar results for the James constant will be presented.     

{\mathbb{Z}_2}-bordism and the Borsuk–Ulam Theorem

The purpose of this work is to classify, for given integers \({m,\, n\geq 1}\), the bordism class of a closed smooth \({m}\)-manifold \({X^m}\) with a free smooth involution \({\tau}\) with respect to the validity of the Borsuk–Ulam property that for every continuous map \({\phi : X^m \to \mathbb{R}^n}\) there exists a point \({x\in X^m}\) such that \({\phi (x)=\phi (\tau (x))}\). We will classify a given free \({\mathbb{Z}_2}\)-bordism class \({\alpha}\) according to the three possible cases that (a) all representatives \({(X^m, \tau)}\) of \({\alpha}\) satisfy the Borsuk–Ulam property; (b) there are representatives \({({X_{1}^{m}}, \tau_1)}\) and \({({X_{2}^{m}}, \tau_2)}\) of \({\alpha}\) such that \({({X_{1}^{m}}, \tau_1)}\) satisfies the Borsuk–Ulam property but \({({X_{2}^{m}}, \tau_2)}\) does not; (c) no representative \({(X^m, \tau)}\) of \({\alpha}\) satisfies the Borsuk–Ulam property.     

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.     

Classification of solutions to the higher order Liouville’s equation on {\mathbb{R}^{2m}}

We classify the solutions to the equation (−Δ) ^m u = (2m − 1)!e ^2mu on giving rise to a metric with finite total Q-curvature in terms of analytic and geometric properties. The analytic conditions involve the growth rate of u and the asymptotic behaviour of Δu at infinity. As a consequence we give a geometric characterization in terms of the scalar curvature of the metric at infinity, and we observe that the pull-back of this metric to S ^2m via the stereographic projection can be extended to a smooth Riemannian metric if and only if it is round.     

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.     

Carnot–Carathéodory metrics in unbounded subdomains of {{\mathbb{C}}^2}

We introduce a new class of unbounded model subdomains of \({\mathbb{C}^2}\) for the \({\Box_b}\) problem. Unlike previous finite type models, these domains need not be bounded by algebraic varieties. In this paper we obtain precise global estimates for the Carnot–Carathéodory metric induced on the boundary of such domains by the real and imaginary parts of the CR vector field.     

Classification of hypersurfaces with constant M?bius curvature in {\mathbb{S}^{m+1}}

Let ${x: M^{m} \rightarrow \mathbb{S}^{m+1}}$ be an m-dimensional umbilic-free hypersurface in an (m?+?1)-dimensional unit sphere ${\mathbb{S}^{m+1}}$ , with standard metric I?= dx · dx. Let II be the second fundamental form of isometric immersion x. Define the positive function ${\rho=\sqrt{\frac{m}{m-1}}\|II-\frac{1}{m}tr(II)I\|}$ . Then positive definite (0,2) tensor ${\mathbf{g}=\rho^{2}I}$ is invariant under conformal transformations of ${\mathbb{S}^{m+1}}$ and is called M?bius metric. The curvature induced by the metric g is called M?bius curvature. The purpose of this paper is to classify the hypersurfaces with constant M?bius curvature.     

Existence and concentration of positive solutions for semilinear Schrödinger–Poisson systems in {\mathbb{R}^{3}}

In this paper, we study the existence and concentration of positive ground state solutions for the semilinear Schrödinger–Poisson system $$\left\{\begin{array}{ll}-\varepsilon^{2}\Delta u + a(x)u + \lambda\phi(x)u = b(x)f(u), & x \in \mathbb{R}^{3},\\-\varepsilon^{2}\Delta\phi = u^{2}, \ u \in H^{1}(\mathbb{R}^{3}), &x \in \mathbb{R}^{3},\end{array}\right.$$ where ε > 0 is a small parameter and λ ≠ 0 is a real parameter, f is a continuous superlinear and subcritical nonlinearity. Suppose that a(x) has at least one minimum and b(x) has at least one maximum. We first prove the existence of least energy solution (u _ε , φ _ε ) for λ ≠ 0 and ε > 0 sufficiently small. Then we show that u _ε converges to the least energy solution of the associated limit problem and concentrates to some set. At the same time, some properties for the least energy solution are also considered. Finally, we obtain some sufficient conditions for the nonexistence of positive ground state solutions.     

On the Fučik point spectrum for Schrödinger operators on {\mathbb{R}}^{N}

We investigate the Fučik point spectrum of the Schr?dinger operator when the potential V_λ has a steep potential well for sufficiently large parameter λ > 0. It is allowed that S_λ has essential spectrum with finitely many eigenvalues below the infimum of . We construct the first nontrivial curve in the Fučik point spectrum by minimax methods and show some qualitative properties of the curve and the corresponding eigenfunctions. As applications we establish some results on existence of multiple solutions for nonlinear Schr?dinger equations with jumping nonlinearity.      

Explicit Inversion Formulas for the X–Ray Transform in {\mathbb{Z}^n}

In the present paper, we state and prove explicit inversion formulas for the X–Ray transform in the lattice ${\mathbb{Z}^n}$ by using certain arithmetical geometrical techniques.     

Dual Split Quaternions and Chasles’ Theorem in 3-Dimensional Minkowski Space {\mathbb{E}^{3}_{1}}

In 1831, Michel Chasles proved the existence of a fixed line under a general displacement in ${\mathbb{R}^3}$ . The fixed line called the screw axis of displacement was obtained by McCharthy in [10]. The purpose of this paper is to develop the method which is given for the pure rotation in [14], and thus to obtain the screw axis of spatial displacement in 3-dimensional Minkowski space. Firstly, we give a relation between dual vectors and lines in ${\mathbb{E}^{3}_{1}}$ , characterize the screw axis. Also, we discuss the dual split quaternion representation of a spatial displacement.