On the Existence of Completely Saturated Packings and Completely Reduced Coverings

We prove the following conjecture of G. Fejes Toth, G. Kuperberg, and W.Kuperberg: every body K in either n-dimensional Euclidean or n-dimensional hyperbolic space admits a completely saturated packing and a completely reduced covering. Also we prove the following counterintuitive result: for every >0, there is a body K in hyperbolic n-space which admits a completely saturated packing with density less than but which also admits a tiling.     

A Restriction Theorem on the Reduced Product Heisenberg Group

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔΩｂｅａｒｅｇｕｌａｒｃｏｎｅｉｎＲｎ,Φ：Ｃｍ×Ｃｍ→Ｃｎ＝Ｒｎ＋ｉＲｎａｎΩ－ｐｏｓｉｔｉｖｅＨｅｒｍｉｔｉａｎｍａｐ．ＴｈｅＳｉｅｇｅｌｄｏｍａｉｎＤΩ,ΦｏｆｔｙｐｅｔｗｏｉｎＣｎ×ＣｍｉｓｄｅｆｉｎｅｄｂｙＤΩ,Φ＝｛（ｚ,ｗ）∈：Ｃｎ×Ｃｍ：Ｉｍｚ－Φ（ｗ,ｗ）∈Ω｝（１）（ｓｅｅ［６］）．Ｓｐｅｃｉａｌｌｙ,ｗｅａｓｓｕｍｅｔｈａｔｎ＝ｍ,Ω＝｛ｔ＝（ｔ１,ｔ２,…,ｔｎ）∈Ｒｎ：ｔｉ＞０,ｉ＝１,２,…,ｎ｝,Φ（ｕ,ｖ）＝ｕ·ｖ＝（ｕ１ｕ…     

Quasi-Minuscule Quotients and Reduced Words for Reflections

We study the reduced expressions for reflections in Coxeter groups, with particular emphasis on finite Weyl groups. For example, the number of reduced expressions for any reflection can be expressed as the sum of the squares of the number of reduced expressions for certain elements naturally associated to the reflection. In the case of the longest reflection in a Weyl group, we use a theorem of Dale Peterson to provide an explicit formula for the number of reduced expressions. We also show that the reduced expressions for any Weyl group reflection are in bijection with the linear extensions of a natural partial ordering of a subset of the positive roots or co-roots.     

基于卷积神经网络气动力降阶模型的翼型优化方法

针对非线性大扰动翼型气动力优化问题,提出了基于卷积神经网络气动力降阶模型的优化方法.该方法用不同形状参数下翼型的气动力数据作为训练信号,训练卷积神经网络翼型气动力降阶模型.采用该气动力降阶模型,以最大升阻比为目标,对翼型进行优化,结果表明该方法可用于大扰动下翼型气动力的预测和优化.该文同时还讨论了池化法和径向基法的训练...     

RRF rings which are not LRF

Define a ringA to be RRF (respectively LRF) if every right (respectively left)A-module is residually finite. We determine the necessary and sufficient conditions for a formal triangular matrix ring to be RRF (respectively LRF). Using this we give examples of RRF rings which are not LRF.     

Chamber systems which are not geometric

No building of infinite rank is residually connected, among other things.     

Nonamenable products are not treeable

LetX andY be infinite graphs such that the automorphism group ofX is nonamenable and the automorphism group ofY has an infinite orbit. We prove that there is no automorphism-invariant measure on the set of spanning trees in the direct productX×Y. This implies that the minimal spanning forest corresponding to i.i.d. edge-weights in such a product has infinitely many connected components almost surely. Research partially supported by NSF grant DMS-9803597.     

Loops which are semidirect products of groups

We construct loops which are semidirect products of groups of affinities. As their elements in many cases one may take transversal subspaces of an affine space. In particular we obtain in this manner smooth loops having Lie groups of affine real transformations as the groups generated by left translations, whereas the groups generated by right translations are smooth groups of infinite dimension. We also determine the Akivis algebras of these loops. This paper was supported by DAAD.     

Reduced products and sheaves of metric structures

We introduce reduced products in continuous logic, and provide ways to establish satisfaction in the reduced product based on the satisfaction in its fibers, and conversely, on the lines that Horn and Chang did for reduced products in classical logic. We also consider a definition of a sheaf of metric structures, endow its stalks with a metric prestructure, and determine what filters can be used so that a collection of reduced products of arbitrary structures forms a sheaf on a given index space.     

Fibered incidence groups which are not kinematic

An incidence space \((\beta ,\mathfrak{L})\) which is obtained from an affine space \((\beta _a ,\mathfrak{L}_a )\) by omitting a hyperplane is calledstripe space. If \((\beta _a ,\mathfrak{L}_a )\) is desarguesian, then \(\beta \) can be provided with a group operation “ ○ ” such that \((\beta ,\mathfrak{L}, \circ )\) becomes a kinematic space calledstripe group. It will be shown that there are stripe groups \((\beta ,\mathfrak{L}, \circ )\) where the incidence structure \(\mathfrak{L}\) can be replaced by another incidence structure ? such that \((\beta ,\Re , \circ )\) is afibered incidence group which is not kinematic. An application on translation planes concerning the group of affinities is also given.     

Soluble groups which are products of minimax groups

Bertram Huppert zum 60. Geburtstag