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An Introduction to Isomorphic Mathematical Analysis System

Yuan Liu

Abstract

This paper is to build a primitive framework for a new possible extended system of real mathematical analysis - the Isomorphic Mathematical Analysis System (IMAS). It is based on some new concepts: e.g. isomorphic frame, dual-variable-isomorphic function, and isomorphic coordinate system. More concepts are introduced to constitute the whole IMAS framework. This IMAS attempts to provide current MA with new implements, which include isomorphic frame, thereby effectively putting the MA to work as well in some unevenly distributed coordinate spaces, realized by the isomorphic coordinate system. With these implements, many existing MA concepts are extended and incorporated into the new IMAS.

An Introduction to Isomorphic Mathematical Analysis System

Abstract

This paper is to build a primitive framework for a new possible extended system of real mathematical analysis - the Isomorphic Mathematical Analysis System (IMAS). It is based on some new concepts: e.g. isomorphic frame, dual-variable-isomorphic function, and isomorphic coordinate system. More concepts are introduced to constitute the whole IMAS framework. This IMAS attempts to provide current MA with new implements, which include isomorphic frame, thereby effectively putting the MA to work as well in some unevenly distributed coordinate spaces, realized by the isomorphic coordinate system. With these implements, many existing MA concepts are extended and incorporated into the new IMAS.
Paper Structure (143 sections, 36 theorems, 130 equations, 10 figures)

This paper contains 143 sections, 36 theorems, 130 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The Basics of IMAS
  3. Isomorphic frame
  4. Definition
  5. Embedding in isomorphic frame
  6. Bonding on isomorphic frame
  7. About embedding and bonding
  8. Isomorphic number and isomorphic variable
  9. Dual-variable-isomorphic function
  10. Definition
  11. Sub-classing of DVI function
  12. Anti-dual-variable-isomorphic function
  13. V-scaleshift: Vertical scale and shift of a function
  14. H-scaleshift: Horizontal scale and shift of a function
  15. HV-scaleshift: Horizontal and vertical scale and shift of a function
  16. ...and 128 more sections

Key Result

Lemma 1.1

Let $X_1,...,X_n,U_1,...,U_n\subseteq\mathbb{R}$, $X=X_1\times ...\times X_n$, $~U=U_1\times ...\times U_n$ and $g_i\colon X_i\to U_i(i=1,...,n)$ be $n$ bijections. The ordered set $\{g_1,...,g_n\}$ that can map $\forall x=(x_1,...,x_n)\in X$ to $u=(g_1(x_1),...,g_n(x_n))\in U$ is a bijection, if i

Figures (10)

  • Figure :
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  • Figure :
  • Figure :
  • ...and 5 more figures

Theorems & Definitions (116)

  • Lemma 1.1
  • proof
  • Definition 1.2
  • Theorem 1.4
  • Theorem 1.6
  • Remark 1.1
  • Definition 1.9
  • Remark 1.2
  • Definition 1.10
  • Theorem 1.11
  • ...and 106 more