Springer basic sets and modular Springer correspondence for classical types

We define the notion of basic set data for finite groups (building on the notion of basic set, but including an order on the irreducible characters as part of the structure), and we prove that the Springer correspondence provides basic set data for Weyl groups. Then we use this to determine explicitly the modular Springer correspondence for classical types (defined over a base field of odd characteristic $p$, and with coefficients in a field of odd characteristic $?\ne p$): the modular case is obtained as a restriction of the ordinary case to a basic set. In order to do so, we compare the order on bipartitions introduced by Dipper and James with the order induced by the Springer correspondence. We provide a quick proof, by sorting characters according to the dimension of the corresponding Springer fibre, an invariant which is directly computable from symbols.     

A modular integer GCD algorithm

This paper describes the first algorithm to compute the greatest common divisor (GCD) of two n-bit integers using a modular representation for intermediate values U, V and also for the result. It is based on a reduction step, similar to one used in the accelerated algorithm [T. Jebelean, A generalization of the binary GCD algorithm, in: ISSAC '93: International Symposium on Symbolic and Algebraic Computation, Kiev, Ukraine, 1993, pp. 111–116; K. Weber, The accelerated integer GCD algorithm, ACM Trans. Math. Softw. 21 (1995) 111–122] when U and V are close to the same size, that replaces U by (U−bV)/p, where p is one of the prime moduli and b is the unique integer in the interval (−p/2,p/2) such that . When the algorithm is executed on a bit common CRCW PRAM with O(nlognlogloglogn) processors, it takes O(n) time in the worst case. A heuristic model of the average case yields O(n/logn) time on the same number of processors.     

Rankin-Cohen brackets on pseudodifferential operators

Pseudodifferential operators that are invariant under the action of a discrete subgroup Γ of SL(2,R) correspond to certain sequences of modular forms for Γ. Rankin-Cohen brackets are noncommutative products of modular forms expressed in terms of derivatives of modular forms. We introduce an analog of the heat operator on the space of pseudodifferential operators and use this to construct bilinear operators on that space which may be considered as Rankin-Cohen brackets. We also discuss generalized Rankin-Cohen brackets on modular forms and use these to construct certain types of modular forms.     

Metamorphic Algorithm of Self-reconfigurable Modular Robotic System

Introduction Aself reconfigurableroboticsystemconsistsofman simpleidenticalmechatronicmodules.Thesemodulesca connectanddisconnectfromonetoanothertochangethe overalltopology.Thesystemcanautonomouslyandynamicallyadaptitsshapetosuittheneedsofthetask hand,e.g.manipulation,locomotion,andcreationo staticstructures.Thegeneralphilosophyofsel reconfigurableroboticsystemistosimplifythedesignanreducemanufacturingcostandenhancefunctionalityan versatilitythroughalargernumberofidenticalmodules.I addition…     

An error-free algorithm to solve a linear system of polynomial equations

The paper presents an error-free algorithm to solve a system of linear equations with polynomial coefficients. Modular arithmetic in residual polynomial class and in residual numeric class is employed. The algorithm is iterative and well suited for implementation for computers with vector operations and fast and error-free convolutors.     

应用模块法杂凑语言的上下文无关文法

编译程序的编制过程中,人们首先接触是语言,而上下文无关文法是作为程序设计语言重要的语法描述工具。因此,只有将语言用上下文无关文法表示才能使用现有的技术,从而用计算机自动实现。通过应用模块法将语言进行模块化划分,并在模块内实现杂凑,能够成功地、方便地实现语言到上下文无关文法的转换。     

CM newforms with rational coefficients

We classify newforms with rational Fourier coefficients and complex multiplication for fixed weight up to twisting. Under the extended Riemann hypothesis for odd real Dirichlet characters, these newforms are finite in number. We produce tables for weights 3 and 4, where finiteness holds unconditionally. I am indepted to K. Hulek for his continuous interest and encouragement. Partial support by the DFG Schwerpunkt 1094 “Globale Methoden in der komplexen Geometrie” is gratefully acknowledged. My thanks go also to the referee for helpful comments. Part of the revising took place while I enjoyed the hospitality of the Dipartimento di Matematica “Frederico Enriques” of Milano University. Funding from the network Arithmetic Algebraic Geometry, a Marie Curie Research Training Network, is gratefully acknowledged. I particularly thank M. Bertolini and B. van Geemen. The final version was prepared while I was funded by DFG under grant Schu 2266/2-2.     

Modular optimization of enantiopure epoxide-derived P,S-ligands for rhodium-catalyzed hydrogenation of dehydroamino acids

The optimization of P,S-ligands derived from enantiopure (2S,3S)-phenylglycidol for asymmetric rhodium-catalyzed hydrogenation of dehydroamino esters is described. The exceptionally high modular character of the (2S,3S)-phenylglycidol platform is demonstrated by systematic modification of the ether and thioether moieties, the skeletal aryl substituent and the stereo and regiochemistry of the ligands. An experimentally useful method for the monitoring of hydrogenation reactions is also described and used for obtaining relevant data of the catalytic systems studied.     

模n－王后问题中的部分n－解（Ⅲ）

记Ｍ（ｎ）为模。棋盘上互不攻击的皇后的最大个数。在本文中，我们证明了当ｇｃｄ（ｎ，６）＝１时，Ｍ（ｎ）＝ｎ；当ｇｃｄ（ｎ，１２）＝２时，Ｍ（ｎ）＝ｎ－１；当ｇｅｄ（ｎ，１２）＝３，４，６，１２时，Ｍ（ｎ）－ｎ－２。最后，给出了Ｍ（ｎ）＝ｎ的三个等价命题。