I have finally managed to upload most of the various pieces of

This page will have to be heavily edited because there are now two editions of

What complicates matters a bit is that I have modified the contents in three chapters:

- I have recut Chapter 5 and Chapter 6 differently
- I have removed all the stuff about chain composition from Chapter 8.

Most of the

Following is the previous version of this page. (June 19, 2010 at 2:44 pm EST)

Visitors who remember the previous version of this page will notice that the major organizational difference that was discussed in the Notes From The Mathematical Underground is indeed the one being implemented: instead of introducing a number of concepts on an as-needed basis, the present contents architecture regroups them up-front from a unifying graphical viewpoint. It certainly makes for a very neat, as well as rather standard, architecture, but, as of this day, I am not yet entirely convinced that this is the way to go. It may be that this is too much too soon and possibly too "out of context".

The less important changes, mentioned earlier, corresponding to a more realistic view of what can be accomplished in a fifteen weeks course meeting three hours a week have been retained: the investigation of the

So, here is the table of contents as it has, finally, been implemented. (In the

**Relations And Functions.**Relations, Quantative Rulers, Finite Numbers And Near-Finite Numbers, Quantitative Screens, Functions, Functions Specified by A Global Input-Output Rule, Functions Specified By A Curve.**Towards Local Analysis.**The Fundamental Graphic Problem, Size Of Numbers, Arithmetic Of*large*,*small*And*medium*, Qualitative Rulers, Infinity, Qualitative Analysis.**Graphic Local Analysis.**Local Graphs, Local Language, Place of a Local Graph, ∝-Height Inputs and 0-Height Inputs, Shape of a Local Graph, Local Behavior, Feature-Sign Change Inputs, 0-Slope and 0-Concavity Inputs, Extremum Inputs.**From Local Graphs To Global Graph.**Smoothness, Interpolation, The Essential Question, The Essential Bounded Graph, Essential Notable Inputs.**Regular Positive-Exponent Power Functions.**Input-Output Pairs, Normalized Input-Output Rule, Local Graph Near ?, Types of Local Graphs Near ?, The Essential Question, Essential Bounded Graph, Existence of Notable Inputs, Local Graph near 0, Types of Local Graphs Near 0, Essential Global Graph, Types of Global Graphs.**Negative-Exponent Power Functions.**Input-Output Pairs, Normalized Input-Output Rule, Local Graph Near ?, Types of Local Graphs Near ?, The Essential Question, Essential Bounded Graph, Existence of Notable Inputs, Local Graph near 0, Types of Local Graphs Near 0, Essential Global Graph, Types of Global Graphs.**Exceptional Power Functions.**(Exponent 0 and exponent 1)**Constant Functions**. Global Graph, Existence of Notable Inputs, Local Graph Near ∝, Local Graph Near 0.**Linear Functions**. Global Graph, Existence of Notable Inputs, Local Graph Near ∝, Local Graph Near 0.

**Chains And Operators**Constant Functions, Dilation Functions; Chaining Functions, Chains of Constant Functions, Chains of Dilation Functions, Operators On Functions, Dilation Operators, Add-On Operators.**Affine Functions: Local Analysis**Input-Output Pairs, Normalized Input-Output Rule, Local Rule Near infinity, Local Graph Near infinity, Types of Local Graphs Near infinity, Local Features Near infinity, The Essential Question, Essential Bounded Graph, Existence of Notable Inputs, Local Rule Near a given x_{0}, Approximate Local Rule Near a given x_{0}, Local Graph Near a given x_{0}, Approximate Local Graph Near a given x_{0}, Local Features Near a given x_{0}.**Affine Functions: Global Analysis.**Global Theorems, Qualitative Global Graph, Solving Equation Problems(*), Locating Inputs Whose Output is a given y_{0}, Solving Inequation Problems(*), Locating Inputs Whose Output compares in a given manner with a given y_{0}, Initial Value Problem, Boundary Value Problem.**Quadratic Functions: Local Analysis.**Input-Output Pairs, Normalized Input-Output Rule, Local Rule Near infinity, Local Graph Near infinity, Types of Local Graphs Near infinity, Local Features Near infinity, The Essential Question, Essential Bounded Graph, Existence of Notable Inputs, Addition formula(*), Local Rule Near a given x_{0}, Approximate Local Rule Near a given x_{0}, Local Graph Near a given x_{0}, Approximate Local Graph Near a given x_{0}, Local Features Near a given x_{0}.**Quadratic Functions: Global Analysis.**Global Theorems, Qualitative Global Graph, Locating 0-height Inputs, Locating Inputs Whose Output is a given y_{0}, Locating Inputs Whose Output compares in a given manner with a given y_{0}, Initial Value Problem, Boundary Value Problem.**Cubic Functions: Local Analysis.**Local Rule Near infinity, Local Graph Near infinity, Types of Local Graphs Near infinity, Local Features Near infinity, The Essential Question, Essential Bounded Graph, Existence of Notable Inputs, Addition formula(*), Local Rule Near a given x_{0}, Approximate Local Rules Near a given x_{0}, Local Graph Near a given x_{0}, Approximate Local Graph Near a given x_{0}, Local Features Near a given x_{0}.**Cubic Functions: Global Analysis.**Global Theorems, Qualitative Global Graph, Locating 0-height Inputs, Locating Inputs Whose Output is a given y_{0}, Locating Inputs Whose Output compares in a given manner with a given y_{0},**Rational Degree -- Algebra Reviews(*).**Rational Degree, Binomial Expansions, Division in Descending Order of Exponents, Default Rules for Division, Comparison With Arithmetic, Division in Ascending Order of Exponents, Graphic Difficulties.**Rational Functions: Local Analysis Near Infinity.**Local Input-Output Rule near Near*Imp*, Local Polynomial Approximations, Local Approximate Input-Output Rule Near Infinity, Height-sign Near Infinity, Slope-sign Near Infinity, Concavity-sign Near Infinity, Local Graph Near Infinity.**Rational Functions: Local Analysis Near x**Local Input-Output Rule near Near x_{0}_{0}, Height-sign Near x_{0}, Slope-sign Near x_{0}, Concavity-sign Near x_{0}, Local Graph Near x_{0}**Rational Functions: Global Analysis.**The Essential Question, Locating Infinite-Height inputs (Poles), Offscreen Graph, Feature-sign Change Inputs, Global Graph, Locating 0-Height Inputs (Zeros).

(*) Pulled verbatim from