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.     

CP~n中具有循环调和序列的平坦极小环面

研究具有循环调和序列的平坦极小浸入ψ:T2→CPn。证明了存在T2的一个有限覆盖p:T2→T2和一个全迷向的平坦极小浸入φ:T2→CPn使得ψ。p=A。φ,其中A:CPn→CPn:[v]→[vA]是线性全纯同胚。     

关于代换系统拓扑序列熵的注记

Goodman证明了对两符号等长代换系统, 如果代换规则中0和1对应的词只有一个位置不同,那么对应的代换系统为null的, 即此系统沿着任意正整数序列的序列熵均为0. 在本文中, 我们针对系统的结构特征, 通过考察因子系统, 给出了此经典结果的另外一种证明. 同时, 对此类代换系统沿着给定序列的复杂性, 我们得到了比Goodman更为精确的估计.     

Optical logic redux

Twenty years ago IBM physicist Robert Keyes published a paper entitled “Optical Logic—in the light of computer technology.” It caused an instant furor in the fledgling optical logic community. Now, 20 years after that devastating critique, the field of optical logic has grown enormously. There are literally thousands of papers. Many of them are collected in a bibliography given here. Was Keyes’ critique wrong? Have opticists simply ignored what Keyes pointed out? Have new developments made some of his remarks not quite so relevant? We argue here that

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Keyes was and still is mostly correct, but that may change in a few years

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Many researchers have indeed simply ignored what he said

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New developments in both optical logic and its applications open niches for optical logic that Keyes did not (and probably could not) anticipate

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New and anticipated developments in electronics may increase the role for optics