矩阵的多行拟相合分类及其应用

本文把方阵的拟相合概念推广为长方阵的多行拟相合概念，并对最简情形即Ｍ２，４（Ｆｐ）给出分类结果．此概念之一应用出现在有限环的同构分类之中．     

Tableau method for residuated logic

Residuated logic is a generalization of intuitionistic logic, which does not assume the idempotence of the conjunction operator. Such generalized conjunction operators have proved important in expert systems (in the area of Approximate Reasoning) and in some areas of Theoretical Computer Science. Here we generalize the intuitionistic tableau procedure and prove that this generalized tableau method is sound for the semantics (the class of residuated algebras) of residuated propositional calculus (RPC). Since the axioms of RPC are complete for the semantics we may conclude that whenever a formula 0 is tableau provable, it is deducible in RPC. We present two different approaches for constructing residuated algebras which give us countermodels for some formulas φ which are not tableau provable. The first uses the fact that the theory of residuated algebras is equational, to construct quotients of free algebras. The second uses finite algebras. We end by discussing a number of open questions.     

The computation of Lagrange-multiplier estimates for constrained minimization

Almost all efficient algorithms for constrained optimization require the repeated computation of Lagrange-multiplier estimates. In this paper we consider the difficulties in providing accurate estimates and what tests can be made in order to check the validity of the estimates obtained. A variety of formulae for the estimation of Lagrange multipliers are derived and their respective merits discussed. Finally the role of Lagrange multipliers within optimization algorithms is discussed and in addition to other results, it is shown that some algorithms are particularly sensitive to errors in the estimates.     

A generalization of Hamiltonian mechanics

For a symplectic manifold the Poisson bracket on the space of functions is (uniquely) extended to a graded Lie bracket on the space of differential forms modulo exact forms. A large portion of the Hamiltonian formalism is still working.     

On approximations of the de Rham complex and their cohomology

For a commutative algebra R, its de Rham cohomology is an important invariant of R. In the paper, an infinite chain of de Rham-like complexes is introduced where the first member of the chain is the de Rham complex. The complexes are called approximations of the de Rham complex. Their cohomologies are found for polynomial rings and algebras of power series over a field of characteristic zero.     

Developing and refining a framework for mathematical and linguistic complexity in tasks related to rates of change

We present and develop a preliminary framework for describing the relationship between the mathematical and linguistic complexity of instructional tasks used in secondary mathematics. The initial framework was developed through a review of relevant literature. It was refined by examining how 4 ninth grade mathematics teachers of linguistically diverse groups of students described the linguistic and mathematical complexity of a set of tasks from their curriculum unit on linear functions. We close by presenting our refined framework for describing the interaction of linguistic complexity and mathematical complexity in curriculum materials, and discuss potential uses of this framework in the design of more accessible classroom learning environments for linguistically diverse students.     

On Isomorphisms Between Integral Table Algebras and Applications to Finite Groups and Association Schemes

A problem that has been open for a long time is whether an automorphism of the center of the integral group ring of a finite group permutes the class sums. As a generalization, the similar problem for integral table algebras has also been studied in a few papers. In this paper we further the investigation of the problem for integral C-algebras and table algebras. We will first present some necessary and sufficient conditions under which an algebra isomorphism between integral C-algebras is monomial, and as an application, obtain a conceptual proof of the class sum correspondence theorem of group rings of finite groups. Then we prove that an algebra isomorphism compatible with degree maps between standard integral nilpotent table algebras over the ring of algebraic integers is exact. As a direct consequence, we get Hertweck's result [6 Hertweck , M. ( 2008 ). On isomorphisms between centers of integral group rings of finite groups . Proc. Amer. Math. Soc. 136 : 1539 – 1547 .[Crossref], [Web of Science ®] , [Google Scholar]] for finite nilpotent groups. An application to association schemes is also presented.     

Enumerating semifields

In 1960, Kleinfeld published representatives for all of the isomorphism classes of 16 element semifields, [E. Kleinfeld, Techniques for Enumerating Veblen-Wedderburn Systems, J. ACM 7 (1960) 330–337]. It is not entirely clear how Kleinfeld generated some of his results, but it is likely that it was similar to the approach that Walker used in 1962 to generate representative for the isotopism classes of 32 element semifields, [R. J. Walker, Determination of Division Algebras With 32 Elements, Proc. Sympos. Appl. Math. XV (1963) 8385]. This paper introduces an alternative notation for publication which is both simple and practical, and which leads to an alternative method which was used to verify Kleinfeldʼs results.     

Constants and Darboux Polynomials for Tensor Products of Polynomial Algebras with Derivations

Abstract

Let d ₁ : k[X] → k[X] and d ₂ : k[Y] → k[Y] be k-derivations, where k[X] ? k[x ₁,…,x _n ], k[Y] ? k[y ₁,…,y _m ] are polynomial algebras over a field k of characteristic zero. Denote by d ₁ ⊕ d ₂ the unique k-derivation of k[X, Y] such that d| _k[X] = d ₁ and d| _k[Y] = d ₂. We prove that if d ₁ and d ₂ are positively homogeneous and if d ₁ has no nontrivial Darboux polynomials, then every Darboux polynomial of d ₁ ⊕ d ₂ belongs to k[Y] and is a Darboux polynomial of d ₂. We prove a similar fact for the algebra of constants of d ₁ ⊕ d ₂ and present several applications of our results.     

The role of problem representation and feature knowledge in algebraic equation-solving

Domain experts have two major advantages over novices with regard to problem solving: experts more accurately encode deep problem features (feature encoding) and demonstrate better conceptual understanding of critical problem features (feature knowledge). In the current study, we explore the relative contributions of encoding and knowledge of problem features (e.g., negative signs, the equals sign, variables) when beginning algebra students solve simple algebraic equations. Thirty-two students completed problems designed to measure feature encoding, feature knowledge and equation solving. Results indicate that though both feature encoding and feature knowledge were correlated with equation-solving success, only feature knowledge independently predicted success. These results have implications for the design of instruction in algebra, and suggest that helping students to develop feature knowledge within a meaningful conceptual context may improve both encoding and problem-solving performance.