The Campbell- Hausdorff Series of Local Analytic Bruck Loops

The class of local analyitic Bruck loops (or equivalently K-loops) is strongly related to locally symmetric spaces. In particular, both have Lie triple systems as their tangent algebra. In this paper, we consider the existence and some properties of the Campbell-Hausdorff series of local analytic Bruck loops (K-loops). This formula can be used to determine the local symmetries of the associated symmetric space.     

Loop Groups and Twin Buildings

In these notes we describe some buildings related to complex Kac–Moody groups. First we describe the spherical building of SL_n() (i.e. the projective geometry PG(ⁿ)) and its Veronese representation. Next we recall the construction of the affine building associated to a discrete valuation on the rational function field (z). Then we describe the same building in terms of complex Laurent polynomials, and introduce the Veronese representation, which is an equivariant embedding of the building into an affine Kac–Moody algebra. Next, we introduce topological twin buildings. These buildings can be used for a proof which is a variant of the proof by Quillen and Mitchell, of Bott periodicity which uses only topological geometry. At the end we indicate very briefly that the whole process works also for affine real almost split Kac–Moody groups.Supported by a Heisenberg fellowship by the Deutsche Forschungsgemeinschaft.     

Natural bound in Kwiecinski's criterion for flatness

Kwiecinski has proved a geometric criterion for flatness: A morphism of germs of analytic spaces is not flat if and only if its fibre power has a vertical component, for some . We show how to bound using Hironaka's local flattener: If is not flat, then has a vertical component, where is the minimal number of generators of the ideal in of the flattener of .

     

锁相环的相位模型与实验电路

讨论了锁相环的相位数学模型,给出了一种适合于数学的锁相环实验电路．     

Multi-component C-KdV Hierarchy of Soliton Equations and Its Multi-component Integrable Coupling System

A new simple loop algebra G M is constructed, which is devoted to establishing an isospectral problem. By making use of Tu scheme, the multi-component C-KdV hierarchy is obtained. Further, an expanding loop algebra FM of the loop algebra G M is presented. Based on FM , the multi-component integrable coupling system of the multi-component C-KdV hierarchy is worked out. The method can be used to other nonlinear evolution equations hierarchy.     

New Simple Method for Obtaining Integrable Hierarchies of Soliton Equations with Multicomponent Potential Functions

A new simple method for obtaining integrable hierarchies of soliton equations is proposed. First of all, a new loop algebra is constructed, whose commutation operation is clear as that in loop algebra . Second, by making use of the Tu scheme, many of integrable hierarchies with multicomponent potential functions can be produced. As a specific application of our method, a multicomponent AKNS hierarchy is obtained. Finally, an expanding loop algebra of the loop algebra is constructed. Taking advantage of above, a type of integrable coupling system of the multicomponent AKNS hierarchy is worked out.     

Ozone monitoring in a Mediterranean forest using diffusive and continuous sampling

Ambient ozone was measured in a forest in Castelporziano (Italy) characterised by the prevailing presence of Holm-oak trees (Quercus ilex L.) from June to November 2003. Two methods for measuring ozone were used: long-term monitoring using diffusive samplers at three heights within the canopy, and continuous monitoring at two heights using the UV method. Results for one week mean ozone levels above and below the canopy from the diffusive samplers were compared to those obtained using the automatic analyser at the same levels. A good correlation between the two sampling techniques was found. Continuous monitoring showed a daily cycle with a midday maximum and a nocturnal minimum. While the forest floor consistently had the lowest ozone concentration, there were no differences during most daytime hours. The midday maximum is clearly due to downward mixing with O₃-rich air from above. The night-time ozone decay within the canopy is the result of dry deposition of O₃ and most likely due to reaction with biogenically produced NO. AOT40 within and above the canopy mostly exceeded the critical levels.     

Some remarks on good sets

It is shown that (1) if a good set has finitely many related components, then they are full, (2) loops correspond one-to-one to extreme points of a convex set. Some other properties of good sets are discussed.     

Iterated monoidal categories

We develop a notion of an n-fold monoidal category and show that it corresponds in a precise way to the notion of an n-fold loop space. Specifically, the group completion of the nerve of such a category is an n-fold loop space, and free n-fold monoidal categories give rise to a finite simplicial operad of the same homotopy type as the classical little cubes operad used to parametrize the higher H-space structure of an n-fold loop space. We also show directly that this operad has the same homotopy type as the n-th Smith filtration of the Barratt-Eccles operad and the n-th filtration of Berger's complete graph operad. Moreover, this operad contains an equivalent preoperad which gives rise to Milgram's small model for when n=2 and is very closely related to Milgram's model of for n>2.     

Commutative Moufang Loops with Finite Conjugacy Classes of Subloops

It is proved that the following conditions are equivalent for an arbitrary commutative Moufang loop :1) the loop is finite over the center;2) every subloop of defines a finite conjugacy class of subloops;3) every associative subloop of defines a finite conjugacy class of subloops;4) every infinite associative subloop of defines a finite conjugacy class of subloops.