Evolution Equations for Special Lagrangian 3-Folds in C3

This is the third in a series of papers constructing explicit examples of special Lagrangian submanifolds in C ^m . The previous paper (Math. Ann. 320 (2001), 757–797), defined the idea of evolution data, which includes an (m – 1)-submanifold P in R ⁿ , and constructed a family of special Lagrangian m-folds N in C ^m , which are swept out by the image of P under a 1-parameter family of affine maps _t : R ⁿ C ^m , satisfying a first-order o.d.e. in t. In this paper we use the same idea to construct special Lagrangian 3-folds in C³. We find a one-to-one correspondence between sets of evolution data with m = 3 and homogeneous symplectic 2-manifolds P. This enables us to write down several interesting sets of evolution data, and so to construct corresponding families of special Lagrangian 3-folds in C³.Our main results are a number of new families of special Lagrangian 3-foldsin C³, which we write very explicitly in parametric form. Generically these are nonsingular as immersed 3-submanifolds, and diffeomorphic to R³ or ¹× R². Some of the 3-folds are singular, and we describe their singularities, which we believe are of a new kind.We hope these 3-folds will be helpful in understanding singularities ofcompact special Lagrangian 3-folds in Calabi–Yau 3-folds. This will beimportant in resolving the SYZ conjecture in Mirror Symmetry.     

A projection theorem and tangential boundary behavior of potentials

Let be the Weinstein operator on the half space, . Suppose there is a sequence of Borel sets such that a certain tangential projection of onto forms a pairwise disjoint subset of the boundary. Let be a finite test measure on the boundary for a specific non-isotropic Hausdorff measure. The measure is carried back to a measure on a subset of by the projection. We give an upper bound for the Weinstein potential corresponding to the measure in terms of a universal constant and a Weinstein subharmonic function. We use this upper bound to deduce a result concerning tangential behavior of Weinstein potentials at the boundary with the exception of sets on the boundary of vanishing non-isotropic Hausdorff measure.

     

Generalized Bicyclic Semigroups and Jones Semigroups

In this paper, we consider the generalized bicyclic semigroups B_n = a, b | aⁿb = 1 and the Jones semigroups A_n = a, b | aⁿ⁺¹b = a. They are the generalizations of the bicyclic semigroup B = a, b | ab = 1 and its analogous semigroup A = a, b | a²b = a discovered by P.R., Jones in 1987. The word problem for these kinds of semigroups is solved. It is proved that, for n 2, B_n are bisimple right inverse but not inverse semigroups and that the semigroup C = a, b | a²b = a, ab² = b is the smallest idempotent-free homomorphic image of A_n. Moreover, we also prove that A_n and A_m are mutually embeddable but not isomorphic with each other if n m. As a consequence, different kind of -nontrivial [0-]simple semigroups without idempotents are discussed.AMS 1991 Subject Classification: primary 20M10 secondary 20M05.Supported by NNSF of China (19671063) and KSRF of Sichuan Education Committee ([1999]127).     

A Note on Shiffman's Theorems

Shiffman proved his famous first theorem, that if A R³ is a compact minimal annulus bounded by two convex Jordan curves in parallel (say horizontal) planes, then A is foliated by strictly convex horizontal Jordan curves. In this article we use Perron's method to construct minimal annuli which have a planar end and are bounded by two convex Jordan curves in horizontal planes, but the horizontal level sets of the surfaces are not all convex Jordan curves or straight lines. These surfaces show that unlike his second and third theorems, Shiffman's first theorem is not generalizable without further qualification.     

电流控阈技术及三值电流型 CMOS施密特电路

本文以开关信号理论为指导 ,对电流型 CM OS电路中如何实现阈值控制进行研究. 建立实现阈 值控制电路的电流传输开关运算 ,并用于指导电流型 CMOS施密特电路的开关级设计 .用 PSPICE程序 模拟证明了所设计的电路具有理想的施密特电路特性.     

Are covering (enveloping) morphisms minimal?

We prove that for certain classes of modules such that direct sums of -covers ( -envelopes) are -covers ( -envelopes), -covering ( -enveloping) homomorphisms are always right (left) minimal. As a particular case we see that over noetherian rings, essential monomorphisms are left minimal. The same type of results are given when direct products of -covers are -covers. Finally we prove that over commutative noetherian rings, any direct product of flat covers of modules of finite length is a flat cover.     

NorTRACK product tracking system—development and implementation

This paper presents the experience gained by developers and users with implementation and operation of NorTRACK^TM, a real-time computerized product tracking system. A Programmable Logic Controller (PLC) collects and transfers data in real time to NorTRACK’s Oracle^TM database on a Windows NT^TM server network. After extensive development and Beta testing at MDS Nordion’s Canadian Irradiation Centre in Montreal, Canada, NorTRACK was installed in January 1997 with a new irradiation facility in Ethicon Endo-Surgery Inc.’s Albuquerque plant in the United States. NorTRACK communicates with the irradiator control and safety system, the plant's central manufacturing database, an innovative pallet staging and tote loading robot, and an automated dosimetry reading system. This integrated system allows the sterilization facility to monitor the irradiator operation and the flow of many products, through varied processing modes, continuously and reliably. As a result of operating with NorTRACK, both MDS Nordion’s CIC facility and the Endo-Surgery manufacturing site, are beginning to realize unique benefits in their respective operations. MDS Nordion is also initiating several future product enhancements and additional productivity modules. This paper describes the NorTRACK system, the various stages of the development project and Beta tests, and the experience of the users to date in their operations.     

A new general algorithmic method in reaction syntheses using linear algebra

We discuss here a new general linear algebraic method (both model and algorithm) for describing and generating (among others) minimal reactions and also minimal mechanisms in stoichiometry, or dimensionless groups in physics as well. (Further applications in process network syntheses will be discussed in .) With some minor modifications of the input this method can be extended for several related questions: for generating direct and overall reactions, direct (steady state) mechanisms, for finding the possible resulting (overall) reactions among all possible mechanisms, etc.Computational results in section 4 show the speed of our algorithm.We give also mathematical background and results in sections 3, 5 and 6. However, we do not restrict ourselves to mathematics only, we also talk on the language of chemistry, too.The theoretical results in sections 3.2, 3.3, 5 and the computational examples in section 4 are completely new, further theoretical results will appear in and in .