例谈数形结合在高数解题中的应用

通过微分中值的等式证明、求渐近线方程、定积分中的换元法、几何图形的描绘以及曲线积分的计算等例题,说明将代数运算或证明与几何直观相结合给解高等数学问题带来的好处．     

Geometric Algebra in Linear Algebra and Geometry

This article explores the use of geometric algebra in linear and multilinear algebra, and in affine, projective and conformal geometries. Our principal objective is to show how the rich algebraic tools of geometric algebra are fully compatible with and augment the more traditional tools of matrix algebra. The novel concept of an h-twistor makes possible a simple new proof of the striking relationship between conformal transformations in a pseudo-Euclidean space to isometries in a pseudo-Euclidean space of two higher dimensions. The utility of the h-twistor concept, which is a generalization of the idea of a Penrose twistor to a pseudo-Euclidean space of arbitrary signature, is amply demonstrated in a new treatment of the Schwarzian derivative.     

Puzzle geometry and rigidity: The Fibonacci cycle is hyperbolic

We describe a new and robust method to prove rigidity results in complex dynamics. The new ingredient is the geometry of the critical puzzle pieces: under control of geometry and ``complex bounds', two generalized polynomial-like maps which admit a topological conjugacy, quasiconformal outside the filled-in Julia set, are indeed quasiconformally conjugate. The proof uses a new abstract removability-type result for quasiconformal maps, following ideas of Heinonen and Koskela and of Kallunki and Koskela, optimized for applications in complex dynamics. We prove, as the first application of this new method, that, for even criticalities distinct from two, the period two cycle of the Fibonacci renormalization operator is hyperbolic with -dimensional unstable manifold.

     

Hall Numbers and the Composition Algebra of the Kronecker Algebra

We present formulas for the structure constants (Hall numbers) of the Hall algebra associated to the Kronecker algebra. The formulas which in some cases involve the classical Hall polynomials enable us to determine every Hall number. Using again these formulas we construct new PBW-bases with simple structure constants for the composition algebra , making possible the definition of the generic composition algebra via Hall polynomials.Presented by C. Ringel.     

Classroom note: A simple proof of an interesting Fibonacci generalization

It is reasonably well known that the ratios of consecutive terms of a Fibonacci series converge to the golden ratio. This note presents a simple, complete proof of an interesting generalization of this result to a whole family of ‘precious metal ratios’.     

PBW Degeneration: Algebra,Geometry, and Combinatorics

We review the recent progress in the theory of Poincaré–Birkhoff–Witt degenerations of irreducible representations of simple Lie algebras. We describe algebraic, geometric, and combinatorial aspects of the theory.     

Toward an Algebra Acquisition Support System: A Study Based on Using Graphic Calculators in the Classroom

This article discusses the results of a study that focused on using graphic calculators. The algebraic code of the calculator was used to introduce 11- to 12-year-old students to algebraic language as a tool for modeling and solving problems, relating this to their previous arithmetical experience and their evolving use of symbolic language. This study provided empirical evidence for the potential of conceiving algebra as a language and teaching it as a language-in-use, supported by the graphic calculator. The teaching approach was based on Bruner's (1983) research on natural language acquisition. Bruner claimed that natural language is taught and that the adult shapes the environment such that children can learn the rudiments of their mother tongue through its use, without needing to know syntactical rules and definitions. The main aim of this study was to investigate the outcomes of an algebra learning scheme based on Bruner's theory of language acquisition.     

Toward an Algebra Acquisition Support System: A Study Based on Using Graphic Calculators in the Classroom

This article discusses the results of a study that focused on using graphic calculators. The algebraic code of the calculator was used to introduce 11- to 12-year-old students to algebraic language as a tool for modeling and solving problems, relating this to their previous arithmetical experience and their evolving use of symbolic language. This study provided empirical evidence for the potential of conceiving algebra as a language and teaching it as a language-in-use, supported by the graphic calculator. The teaching approach was based on Bruner's (1983) research on natural language acquisition. Bruner claimed that natural language is taught and that the adult shapes the environment such that children can learn the rudiments of their mother tongue through its use, without needing to know syntactical rules and definitions. The main aim of this study was to investigate the outcomes of an algebra learning scheme based on Bruner's theory of language acquisition.     

高等代数与解析几何一体化教学探索——以新疆大学为例

新疆大学数学与系统科学学院在新疆大学入选国家"双一流"建设战略布局院校后,对本科数学教学进行了诸多改革.高等代数与解析几何的改革是其中一个重要部分.根据西部地区实际情况,从教学内容、教学方法和教学模式三方面进行了初步的探索.最后融入思政教育,传递家国情怀,提高西部地区教育水平.