Remarks on derived complete modules and complexes

Let R be a commutative ring and $I\subset R$ a finitely generated ideal. We discuss two definitions of derived I-adically complete (also derived I-torsion) complexes of R-modules, which appear in the literature: the idealistic and the sequential ones. The two definitions are known to be equivalent for a weakly proregular ideal I; we show that they are different otherwise. We argue that the sequential approach works well, but the idealistic one needs to be reinterpreted or properly understood. We also consider I-adically flat R-modules.     

Torsion divisors of plane curves with maximal flexes and Zariski pairs

There is a close relationship between the embedded topology of complex plane curves and the (group-theoretic) arithmetic of elliptic curves. In a recent paper, we studied the topology of some arrangements of curves that include a special smooth component, via the torsion properties induced by the divisors in the special curve associated to the remaining components, which is an arithmetic property. When this special curve has maximal flexes, there is a natural isomorphism between its Jacobian variety and the degree zero part of its Picard group. In this paper, we consider curve arrangements that contain a special smooth component with a maximal flex and exploit these properties to obtain Zariski tuples, which show the interplay between topology, geometry, and arithmetic.      

基于透视变换的直升机桨根扭转角测量方法

现有方法常将相机从下至上拍摄标记物图像，当桨叶扭转幅度较大时，相机主光轴与标记物平面法向量存在明显夹角，导致拍摄图像存在透视畸变现象，从而引发较大测量误差。为解决上述问题，提出一种基于透视变换的直升机桨根扭转角测量方法。首先，将靶标垂直固定在桨叶下方，相机正对桨毂拍摄图像。其次，计算桨叶在低速拉平状态下待测方位与正方位靶标图像中的四个顶点坐标，并求解待测方位的透视变换矩阵，然后利用透视变换矩阵对桨叶高速运动状态下待测方位拍摄的畸变图像进行校正。最后，求得基准图像与校正后图像中的圆心坐标，进而计算出待测方位扭转角。实验结果表明，各方位测量误差均小于0.2°。该方法不但测量精度高，而且具有多方位动态测量的优点，在直升机旋翼桨叶运动参数测量领域具有良好的应用前景。     

Odd-dimensional counterparts of abelian complex and hypercomplex structures

We introduce the notion of abelian almost contact structures on an odd-dimensional real Lie algebra $\mathfrak{g}$. We investigate correspondences with even-dimensional Lie algebras endowed with an abelian complex structure, and with Kähler Lie algebras when $\mathfrak{g}$ carries a compatible inner product. The classification of 5-dimensional Sasakian Lie algebras with abelian structure is obtained. Later, we introduce abelian almost 3-contact structures on real Lie algebras of dimension $4n+3$, obtaining the classification of these Lie algebras in dimension 7. Finally, we deal with the geometry of a Lie group G endowed with a left invariant abelian almost 3-contact metric structure. We determine conditions for G to admit a canonical metric connection with skew torsion, which plays the role of the Bismut connection for hyperKähler with torsion (HKT) structures arising from abelian hypercomplex structures. We provide examples and discuss the parallelism of the torsion of the canonical connection.