Linear groups as right multiplication groups of quasifields

For quasifields, the concept of parastrophy is slightly weaker than isotopy. Parastrophic quasifields yield isomorphic translation planes but not conversely. We investigate the right multiplication groups of finite quasifields. We classify all quasifields having an exceptional finite transitive linear group as right multiplication group. The classification is up to parastrophy, which turns out to be the same as up to the isomorphism of the corresponding translation planes.     

Non-existence of collineations equivalent to some loop laws

If in a translation plane there is a set of collineations whose existence is necessary and sufficient for the validity of the multiplicative left-inversive law, the weak-inversive law, the anti-automorphicinverse law, or the left Bol law in one of the coordinatizing right quasifields, then the plane is a Moufang plane. The same holds also for the Moufang law, if the coordinatizing quasifield is not the Hall system of order 9.     

Bemerkungen Zu Einer Klasse Kommutativer Inzidenzgruppen

We study a class of commutative incidence groups obtained by a special sort of couplings on affine translation planes over commutative quasifields. We give examples of different couplings on the same plane, leading to non-isomorphic incidence groups by generalizing a construction from Dickson. In some cases all thus obtained groups are algebraically isomorphic.     

Multiplicative Loops of 2-Dimensional Topological Quasifields

We determine the algebraic structure of the multiplicative loops for locally compact 2-dimensional topological connected quasifields. In particular, our attention turns to multiplicative loops which have either a normal subloop of positive dimension or which contain a 1-dimensional compact subgroup. In the last section, we determine explicitly the quasifields which coordinatize locally compact translation planes of dimension 4 admitting an at least 7-dimensional Lie group as collineation group.     

Simplectic spreads and finite semifields

It is well known that associated with a translation plane π there is a family of equivalent spreads. In this paper, we prove that if one of these spreads is symplectic and π is finite, then all the associated spreads are symplectic. Also, using the geometric intepretation of the Knuth’s cubical array, we prove that a symplectic semifield spread of dimension n over its left nucleus is associated via a Knuth operation to a commutative semifield of dimension n over its middle nucleus.      

A note on places of quasifields

A considerable reduction of André's set of axioms for places of quasifields allows us to give a handy and useful characterization of the place rings of these special Cartesian groups. As an application, we present an algebraic proof for the existence of natural places in ordered quasifields.Herrn Professor J. Joussen zum 60. Geburtstag gewidmet     

Isotopy and geotopy for ternary rings of projective planes

We investigate the question of when two ternary rings coordinatize the same finite projective plane. Necessary and sufficient conditions are obtained for right quasifields and for right distributive linear rings whose multiplicative loop is a group.     

A remark on symplectic spreads

We first note that each element of a symplectic spread of PG(2n − 1, 2 ^r ) either intersects a suitable nonsingular quadric in a subspace of dimension n − 2 or is contained in it, then we prove that this property characterises symplectic spreads of PG(2n − 1, 2 ^r ). As an application, we show that a translation plane of order q ⁿ , q even, with kernel containing GF(q), is defined by a symplectic spread if and only if it contains a maximal arc of the type constructed by Thas (Europ J Combin 1:189–192, 1980).     

On some d-dimensional dual hyperovals in

In [H. Taniguchi, On d-dimensional dual hyperovals in PG(2d,2), Innov. Incidence Geom., in press], we construct d-dimensional dual hyperovals in PG(2d,2) from quasifields of characteristic 2. In this note, we show that, if d-dimensional dual hyperovals in PG(2d,2) constructed from nearfields are isomorphic, then those nearfields are isomorphic. Some results on dual hyperovals constructed from quasifields are also proved.     

g-Quasikörper

A generalization of a construction for finite quasifields by Foulser is given, which yields all finite Dickson-nearfields. This constuction is then used to obtain locally finite quasifields as unions of chains of finite g-quasifields.     

The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of the projective plane

By a result of Kedra and Pinsonnault, we know that the topology of groups of symplectomorphisms of symplectic 4-manifolds is complicated in general. However, in all known (very specific) examples, the rational cohomology rings of symplectomorphism groups are finitely generated. In this paper, we compute the rational homotopy Lie algebra of symplectomorphism groups of the 3-point blow-up of the projective plane (with an arbitrary symplectic form) and show that in some cases, depending on the sizes of the blow-ups, it is infinite dimensional. Moreover, we explain how the topology is generated by the toric structures one can put on the manifold. Our method involve the study of the space of almost complex structures compatible with the symplectic structure and it depends on the inflation technique of Lalonde–McDuff.     

On the Geometry of Lagrangian Submanifolds

We prove that a Lagrangian submanifold passes through each point of a symplectic manifold in the direction of arbitrary Lagrangian plane at this point. Generally speaking, such a Lagrangian submanifold is not unique; nevertheless, the set of all such submanifolds in Hermitian extension of a symplectic manifold of dimension greater than 4 for arbitrary initial data contains a totally geodesic submanifold (which we call the s-Lagrangian submanifold) iff this symplectic manifold is a complex space form. We show that each Lagrangian submanifold in a complex space form of holomorphic sectional curvature equal to c is a space of constant curvature c/4. We apply these results to the geometry of principal toroidal bundles.     

Towards the multiplicative behavior of the K-theoretical McKay correspondence

When the quotient of a symplectic vector space by the action of a finite subgroup of symplectic automorphisms admits as a crepant projective resolution of singularities the Hilbert scheme of regular orbits of Nakamura, then there is a natural isomorphism between the Grothendieck group of this resolution and the representation ring of the group, given by the Bridgeland-King-Reid map. However, this isomorphism is not compatible with the ring structures. For the Hilbert scheme of points on the affine plane, we study the multiplicative behavior of this map.     

SYMPLECTIC SCHEMES OF VORTEX SYSTEM IN HALF PLAIN

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关于辛算法稳定性的若干注记

我们讨论辛算法的线性稳定性和非线性稳定性,从动力系统和计算的角度论述了研究辛算法的这两类稳定性问题的重要性,分析总结了相关重要结果.我们给出了解析方法的明确定义,证明了稳定函数是亚纯函数的解析辛方法是绝对线性稳定的.绝对线性稳定的辛方法既有解析方法（如Runge-Kutta辛方法）,也有非解析方法（如基于常数变易公式对线性部分进行指数积分而对非线性部分使用其它数值积分的方法）.我们特别回顾并讨论了R.I.McLachlan,S.K.Gray和S.Blanes,F.Casas,A.Murua等关于分裂算法的线性稳定性结果,如通过选取适当的稳定多项式函数构造具有最优线性稳定性的任意高阶分裂辛算法和高效共轭校正辛算法,这类经优化后的方法应用于诸如高振荡系统和波动方程等线性方程或者线性主导的弱非线性方程具有良好的数值稳定性.我们通过分析辛算法在保持椭圆平衡点的稳定性,能量面的指数长时间慢扩散和KAM不变环面的保持等三个方面阐述了辛算法的非线性稳定性,总结了相关已有结果.最后在向后误差分析基础上,基于一个自由度的非线性振子和同宿轨分析法讨论了辛算法的非线性稳定性,提出了一个新的非线性稳定性概念,目的是为辛算法提供一个实际可用的非线性稳定性判别法.     

Smooth Projective Planes

Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to . We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plane curves are smooth collineations, and prove a variety of results analogous to the theory of classical projective planes. *Thanks to Robert Bryant and John Franks.     

Formal power series over Cartesian groups and their spaces of orderings

We introduce formal power series over Cartesian groups on arbitrary, ordered loops, and show that, under a weak additional hypothesis, their spaces of orderings (in the sense of M. Marshall) are as in the classical case. In particular, we obtain that any classical space of orderings can be realized over proper quasifields, proper nearfields, and proper semifields.Dedicated to Professor H. Salzmann on the occasion of his 60th birthday     

二维弹性平面问题中任意边界条件下应力分布的封闭解

应用辛方法研究了正交各向异性二维平面（x,z）弹性问题,在任意边界和不考虑梁假设条件下的解析应力分布解．辛方法通过将位移和应力作为对偶量推导得到一组辛的偏微分方程组,并且应用变量分离法对方程组进行了求解．同动力学中的问题比较,将弹性问题中的x轴模拟成时间轴,这样z轴成为唯一一个独立的坐标轴．问题中的Hamilton矩阵的指数展开具有辛的特征．在齐次问题求解中,通过边界条件和边界上的积分求得级数中的未知数．齐次解中包括减阶的零特征值的特征向量（零本征向量）和完好的非零本征值的特征向量（非零本征向量）．零本征值的Jordan链给出了经典的Saint Venant解,反映了平均的整体行为像刚体位移、刚体旋转和弯曲等．另外,非零本征向量反映的是指数衰减的局部解,它们通常在Saint Venant原理下被忽略．文中给出了完整的算例,并且和已有结果进行了对比．