关于交换半环上一类矩阵的研究

对半环上可逆矩阵的概念进行推广,给出了[e]-可逆矩阵的定义。通过探讨可逆矩阵与[e]-可逆矩阵之间的内在联系,给出了交换半环上[e]-可逆矩阵的等价刻画。同时,对交换半环上[e]-可逆矩阵的全体关于矩阵乘法构成的半群进行研究,给出了此类矩阵半群的分解定理,并证明了此类矩阵半群均存在极大子群,且所有极大子群的并是Clifford半群。     

Clifford and Graßmann Hopf algebras via the BIGEBRA package for Maple

Hopf algebraic structures will replace groups and group representations as the leading paradigm in forthcoming times. K-theory, co-homology, entanglement, statistics, representation categories, quantized or twisted structures as well as more geometric topics of invariant theory, e.g., the Graßmann-Cayley bracket algebra, are all covered by the Hopf algebraic framework. The new branch of experimental mathematics allows one to easily enter these fields through direct calculations using symbolic manipulation and computer algebra system (CAS). We discuss problems which were solved when building the BIGEBRA package for Maple and CLIFFORD to handle tensor products, Graßmann and Clifford algebras, coalgebras and Hopf algebras. Recent results showing the usefulness of CAS for investigating new and involved mathematics provide us with examples. An outlook on further developments is given.     

Vectorial Equations Solving for Mechanical Geometry Theorem Proving

In this paper a new method is proposed for mechanical geometry theorem proving. It combines vectorial equations solving in Clifford algebra formalism with Wu"s method. The proofs produced have significantly enhanced geometric meaning and fewer nongeometric nondegeneracy conditions.     

基于Clifford半群上共轭搜索问题的密钥建立协议

本文采用更广泛的半群作为平台,推广了Iris Anshel等提出的代数密钥建立协议模型。在定义了Clifford半群上的多重同时共轭搜索问题（MSCSP）后,给出了基于此问题的密钥建立协议。在理论上证明了若Clifford半群上的多重同时共轭搜索问题（MSCSP）是困难的,那么可以利用MSCSP来构造密钥建立协议。从而说明利用半群作为平台构建密钥建立协议是可能的。本文也提供了一种新的利用辫群的思路,即考虑利用辫群上的强半格（许多辨群按照一定规则形成的无交并）构成的Clifford半群来构建密码协议,以弥补单个辫群可能存在的安全缺陷。     

A Generalized Theoretical Model for the Relationship Between Critical Micelle Concentrations,Pressure, and Temperature for Surfactants

Directional solvent extraction (DSE) has been gaining interest as a water treatment technology in recent years. DSE utilizes the process of micellization for the purposes of species separation between water and complex chemical systems. In this article, we develop a conformal geometric algebra-based formulation that models surfactants, their solubilities, and critical micelle concentration (CMC), with relation to temperature and pressure. Molecules are represented as spatially distributed networks embedded in R ^4,1 space, and the mathematical characterizations of these molecules are shown to be effective in modelling CMC as a function of temperature and pressure. One of the contributions of this work is the utilization of this formulation to develop a governing expression, in the form of a three-dimensional relationship, between CMC, pressure, and temperature for a general surfactant. In prior works, the CMC–temperature plane and CMC–pressure plane expressions have been extensively documented for sodium alkyl sulfates. In this work, we extend the formulation to model the CMC of decanoic acid, sodium octyl sulfate, sodium decyl sulfate, sodium dodecyl sulfate, and sodium tetradecyl sulfate. Using this theoretical model, a relationship between CMC and the directional solubility of water in a surfactant is determined. Directional solubility is related to temperature and pressure, and on this basis, we devise a directional solubility–pressure–temperature expression for an arbitrary surfactant to improve the state of the art for DSE. From this expression, we propose a novel isothermal DSE process for water treatment.     

Linearly ordered semigroups for fuzzy set theory

The fuzzy set theory initiated by Zadeh (Information Control 8:338–353, 1965) was based on the real unit interval [0,1] for support of membership functions with the natural product for intersection operation. This paper proposes to extend this definition by using the more general linearly ordered semigroup structure. As Moisil (Essais sur les Logiques non Chrysippiennes. Académie des Sciences de Roumanie, Bucarest, 1972, p. 162) proposed to define Lukasiewicz logics on an abelian ordered group for truth values set, we give a simple negative answer to the question on the possibility to build a Many-valued logic on a finite abelian ordered group. In a constructive way characteristic properties are step by step deduced from the corresponding set theory to the semigroup order structure. Some results of Clifford on topological semigroups (Clifford, A.H., Proc. Amer. Math. Soc. 9:682–687, 1958; Clifford, A.H., Trans. Amer. Math. Soc. 88:80–98, 1958), Paalman de Miranda work on I-semigroups (Paalman de Miranda, A.B., Topological Semigroups. Mathematical Centre Tracts, Amsterdam, 1964) and Schweitzer, Sklar on T-norms (Schweizer, B., Sklar, A., Publ. Math. Debrecen 10:69–81, 1963; Schweizer, B., Sklar, A., Pacific J. Math. 10:313–334, 1960; Schweizer, B., Sklar, A., Publ. Math. Debrecen 8:169–186, 1961) are revisited in this framework. As a simple consequence of Faucett theorems (Proc. Amer. Math. Soc. 6:741–747, 1955), we prove how canonical properties from the fuzzy set theory point of view lead to the Zadeh choice thus giving another proof of the representation theorem of T-norms. This structural approach shall give a new perspective to tackle the question of G. Moisil about the definition of discrete Many-valued logics as approximation of fuzzy continuous ones.      

An embedded, FPGA-based computer graphics coprocessor with native geometric algebra support

The representation of geometric objects and their transformation are the two key aspects in computer graphics applications. Traditionally, computer-intensive matrix calculations are involved in modeling and rendering three-dimensional (3D) scenery. Geometric algebra (aka Clifford algebra) is attracting attention as a natural way to model geometric facts and as a powerful analytical tool for symbolic calculations. In this paper, the architecture of Clifford coprocessor (CliffoSor) is introduced. CliffoSor is an embedded parallel coprocessing core that offers direct hardware support to Clifford algebra operators. A prototype implementation on a programmable gate array (FPGA) board is detailed. Initial test results show the potential to achieve a 20× speedup for 3D vector rotations, a 12× speedup for Clifford sums and differences, and more than a 4× speedup for Clifford products, compared to the analogous operations in GAIGEN, a standard geometric algebra library generator for general-purpose processors. An execution analysis of a raytracing application is also presented.     

Obituary: Willard Clifford Olson (1899-1978).

Willard C. Olson, former dean of the School of Education of the University of Michigan, died at the Huron Valley Lodge and Nursing Home, not far from his Ann Arbor residence, October 30, 1978. His survivors include his wife, the former Violet Hunt Pickard, a daughter, Mrs. Mary Ann Hayes, five grandchildren, three great-grandchildren, a brother, and a sister. His death followed a long siege of Parkinson's disease. (PsycINFO Database Record (c) 2010 APA, all rights reserved)