Reflexive and Spanned Sheaves on {\mathbb{P}^3}

We investigate the reflexive sheaves on ${\mathbb{P}^3}$ spanned in codimension 2 with very low first Chern class c ₁. We also give the sufficient and necessary conditions on numeric data of such sheaves for indecomposabiity. As a by-product we obtain that every reflexive sheaf on ${\mathbb{P}^3}$ spanned in codimension 2 with c ₁ = 2 is spanned.     

The Hilbert functions ofk-configurations in\mathbb{P}^2 and\mathbb{P}^3

In this paper, we proved the set of points which are the vertices of then-gon in $\mathbb{P}^2 $ (n ≥ 3) has the Uniform Position Property and what the graded free resolutions of the ideals ofk-configurations in $\mathbb{P}^3 $ are.     

Classifying ACM sets of points in $${\mathbb{P}^{1} \times \mathbb{P}^{1}}$$ via separators

The purpose of this note is to give a new, short proof of a classification of ACM sets of points in in terms of separators.     

Factoriality condition of some nodal threefolds in {\mathbb{P}^4}

We prove that for n = 8, 9, 10, 11, a nodal hypersurface of degree n in is factorial if it has at most (n − 1)² − 1 nodes. The author is grateful to Ivan Cheltsov for valuable comments and suggestions.     

Proper trajectories of type {\mathbb{C}^{\ast}} of a polynomial vector field on {\mathbb{C}^2}

We prove that if a polynomial vector field on ${\mathbb{C}^2}$ has a proper and non-algebraic trajectory analytically isomorphic to ${\mathbb{C}^{\ast}}$ all its trajectories are proper, and except at most one which is contained in an algebraic curve of type ${\mathbb{C}}$ all of them are of type ${\mathbb{C}^{\ast}}$ . As corollary we obtain an analytic version of Lin?CZa?denberg Theorem for polynomial foliations.     

Extended quasi-homogeneous polynomial system in {\mathbb{R}^{3}}

In this paper we define an extended quasi-homogeneous polynomial system d x/dt = Q = Q ₁ + Q ₂ + ... + Q _δ , where Q _i are some 3-dimensional quasi-homogeneous vectors with weight α and degree i, i = 1, . . . ,δ. Firstly we investigate the limit set of trajectory of this system. Secondly let Q _T be the projective vector field of Q. We show that if δ ≤ 3 and the number of closed orbits of Q _T is known, then an upper bound for the number of isolated closed orbits of the system is obtained. Moreover this upper bound is sharp for δ = 3. As an application, we show that a 3-dimensional polynomial system of degree 3 (resp. 5) admits 26 (resp. 112) isolated closed orbits. Finally, we prove that a 3-dimensional Lotka-Volterra system has no isolated closed orbits in the first octant if it is extended quasi-homogeneous.     

On Quadratic Differentials and Twisted Normal Maps of Surfaces in {\mathbb{S}^{2} \times\mathbb{R}} and {\mathbb{H}^{2} \times\mathbb{R}}

Given a Lie group G with a bi-invariant metric and a compact Lie subgroup K, Bittencourt and Ripoll used the homogeneous structure of quotient spaces to define a Gauss map ${\mathcal{N}:M^{n}\rightarrow{\mathbb{S}}}$ on any hypersupersurface ${M^{n}\looparrowright G/K}$ , where ${{\mathbb{S}}}$ is the unit sphere of the Lie algebra of G. It is proved in Bittencourt and Ripoll (Pacific J Math 224:45–64, 2006) that M ⁿ having constant mean curvature (CMC) is equivalent to ${\mathcal{N}}$ being harmonic, a generalization of a Ruh–Vilms theorem for submanifolds in the Euclidean space. In particular, when n = 2, the induced quadratic differential ${\mathcal{Q}_{\mathcal{N}}:=(\mathcal{N}^{\ast}g)^{2,0}}$ is holomorphic on CMC surfaces of G/K. In this paper, we take ${G/K={\mathbb{S}}^{2}\times{\mathbb{R}}}$ and compare ${\mathcal{Q}_{\mathcal{N}}}$ with the Abresch–Rosenberg differential ${\mathcal{Q}}$ , also holomorphic for CMC surfaces. It is proved that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ , after showing that ${\mathcal{N}}$ is the twisted normal given by (1.5) herein. Then we define the twisted normal for surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ and prove that ${\mathcal{Q}=\mathcal{Q}_{\mathcal{N}}}$ as well. Within the unified model for the two product spaces, we compute the tension field of ${\mathcal{N}}$ and extend to surfaces in ${{\mathbb{H}}^{2}\times{\mathbb{R}}}$ the equivalence between the CMC property and the harmonicity of ${\mathcal{N}.}$      

Surfaces in $${\mathbb {P}^{4}}$$ fibered in cubics

Any algebraic surface in which is fibered in cubics, so that the generic fibre is a twisted cubic, gives rise to a curve Γ in a suitable compactification X of the space of smooth rational cubics of In this paper the case n = 4 is addressed and the corresponding space X is studied. We apply our results to complete the classification of smooth, rational surfaces in ruled in cubics. This work is within the framework of the national research project “Geometry on Algebraic Varieties” Cofin 2006 of MIUR.     

Complete bounded null curves immersed in {\mathbb {C}^3} and {\rm {SL}(2,\mathbb {C})}

We construct a simply connected complete bounded mean curvature one surface in the hyperbolic 3-space ${\mathcal {H}^3}$ . Such a surface in ${\mathcal {H}^3}$ can be lifted as a complete bounded null curve in ${\rm {SL}(2,\mathbb {C})}$ . Using a transformation between null curves in ${\mathbb {C}^3}$ and null curves in ${\rm {SL}(2,\mathbb {C})}$ , we are able to produce the first examples of complete bounded null curves in ${\mathbb {C}^3}$ . As an application, we can show the existence of a complete bounded minimal surface in ${\mathbb {R}^3}$ whose conjugate minimal surface is also bounded. Moreover, we can show the existence of a complete bounded immersed complex submanifold in ${\mathbb {C}^2}$ .     

Hypersurfaces in nearly Kaehler manifold $$\pmb {\mathbb {S}}^{\varvec{3}}\varvec{\times } \pmb {\mathbb {S}}^{\varvec{3}}$$

In this paper, we initiate the study of contact and minimal hypersurfaces in nearly Kaehler manifold \({\mathbb {S}}^3\times {\mathbb {S}}^3\) with a conformal vector field. There are three almost contact metric structures on a hypersurface of \({\mathbb {S}}^3\times {\mathbb {S}}^3\), and we will give some important properties of them. Besides, we study the influence of the conformal vector field on the almost contact metric structures and use it to characterize the hypersurfaces in \({\mathbb {S}}^3\times {\mathbb {S}}^3\).     

Wiener-Hopf operators on L_{\omega}^{2}(\mathbb{R}^{+})

Let be a weighted space with weight . In this paper we show that for every Wiener-Hopf operator T on and for every a I, there exists a function such that

Counting signed swallowtails of polynomial selfmaps of {\mathbb {R}^3}

For a generic \({f \in C^\infty({\mathbb {R}}^3,{\mathbb {R}}^3)}\), there is a discrete set of swallowtail critical points. At any swallowtail point p there exists a well-oriented coordinate system centred at p, and a coordinate system centred at f(p), such that locally f has the form \({f_\pm(x, y, z) = (\pm xy+x^2 z+x^4, y, z)}\), so one may associate with p a sign \({I(f, p) \in \{\pm 1\}}\). We shall show how to compute the number of swallowtail points having the positive/negative sign, in the case where \({f : {\mathbb {R}}^3 \rightarrow {\mathbb {R}}^3}\) is a polynomial mapping, in terms of signatures of quadratic forms.     

A Note on the Tight Spherical 7-Design in {\mathbb R}^{23} and 5-Design in {\mathbb R}^{7*}

In this note we prove the uniqueness of the tight spherical 7-design in consisting of 4600 vectors and with automorphism group 2 × Co₂ as well as the uniqueness of the tight spherical 5-design in on 112 vectors and with automorphism group 2 × Sp₆(2).To the memory of Jaap Seidel     

Hyperbolic Laplace Operator and the Weinstein Equation in {\mathbb{R}^3}

We study the Weinstein equation $$\Delta u - \frac{k}{{x}_{2}} \frac{\partial}{\partial{x}_{2}} + \frac{l}{x^{2}_{2}}u = 0$$ , on the upper half space ${\mathbb{R}^3_{+} = \{ (x_{0}, x_{1}, x_{2}) \in \mathbb{R}^{3} | x_2 > 0\}}$ in case ${4l \leq (k + 1)^{2}}$ . If l =  0 then the operator ${x^{2k}_{2} (\Delta - \frac{k}{x_{2}} \frac{\partial}{\partial{x}_{2}})}$ is the Laplace- Beltrami operator of the Riemannian metric ${ds^2 = x^{-2k}_{2} (\sum^{2}_{i = 0} dx^{2}_{i})}$ . The general case ${\mathbb{R}^{n}_{+}}$ has been studied earlier by the authors, but the results are improved in case ${\mathbb{R}^3_{+}}$ . If k =  1 then the Riemannian metric is the hyperbolic distance of Poincaré upper half-space. The Weinstein equation is connected to the axially symmetric potentials. We compute solutions of the Weinstein equation depending only on the hyperbolic distance and x ₂. The solutions of the Weinstein equation form a socalled Brelot harmonic space and therefore it is known that they satisfy the mean value properties with respect to the harmonic measure. However, without using the theory of Brelot harmonic spaces, we present the explicit mean value properties which give a formula for a harmonic measure evaluated in the center point of the hyperbolic ball. Earlier these results were proved only for k =  1 and l =  0 or k =  1 and l =  1. We also compute the fundamental solutions. The main tools are the hyperbolic metric and its invariance properties. In the consecutive papers, these results are applied to find explicit kernels for k-hypermonogenic functions that are higher dimensional generalizations of complex holomorphic functions.     

Reachable sets for contact sub-Lorentzian structures on {\mathbb{R}^3} . Application to control affine systems on {\mathbb{R}^3} with a scalar input

In this paper, we investigate the structure of reachable sets for general contact sub-Lorentzian metrics on $ {\mathbb{R}^3} $ . In some particular cases, the presented method leads to explicit formulas for functions describing reachable sets. We also compute the image under exponential mapping and prove that the sub-Lorentzian distance is continuous for the mentioned structures. All presented results concerning reachable sets can be directly applied to generic control affine systems in $ {\mathbb{R}^3} $ with a scalar input u and constraints |u|??????.     

Die Modulräume^3 M_{\mathbb{P}^3 }^{st} \left( { - 2, 3, c_3 } \right)

Ohne Zusammenfassung     

Maximal q-Plurisubharmonic Functions in {\mathbb{C}^{n}}

In this paper we will study maximal q-plurisubharmonic functions in ${\mathbb{C}^n}$ . At the same time, we define a notion above weakly q-plurisubharmonic functions and describe the relation between these functions and maximal q-plurisubharmonic functions.     

Foliations and polynomial diffeomorphisms of $${\mathbb{R}^{3}}$$

Let be a C ² map and let Spec(Y) denote the set of eigenvalues of the derivative DY _p , when p varies in . We begin proving that if, for some ϵ > 0, then the foliation with made up by the level surfaces {k = constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek’s Jacobian Conjecture for polynomial maps of The first author was supported by CNPq-Brazil Grant 306992/2003-5. The first and second author were supported by FAPESP-Brazil Grant 03/03107-9.