diff --git a/books/bookvol10.1.pamphlet b/books/bookvol10.1.pamphlet index 68ebfd2..ee89e7a 100644 --- a/books/bookvol10.1.pamphlet +++ b/books/bookvol10.1.pamphlet @@ -8552,8 +8552,20 @@ Groebner Basis Greatest Common Divisor \chapter{Polynomial Factorization} Polynomial Factorization + \chapter{Cylindrical Algebraic Decomposition} -Cylindrical Algebraic Decomposition +We quote from LaValle \cite{LaVa06}. +The idea is to develop a sequence of projections that drops the dimension +of the semi-algebraic set by one each time. Initially, the set is defined +over ${\bf R}^n$, and after one projection, a semi-algebraic set is obtained +in ${\bf R}^{n-1}$. Eventually, the projection reaches ${\bf R}$, and a +univariate polynomial is obtained for which the zeros are at the critical +places where cell boundaries need to be formed. A cell decomposition of +1-cells (intervals) and 0-cells is formed by partitioning ${\bf R}$. +The sequences is then reversed, and decompositions are formed from ${\bf R}^2$ +up to ${\bf R}^n$. Each iteration starts with a cell decomposition in +${\bf R}^i$ and lifts it to obtain a cylinder of cells in ${\bf R}^{i+1}$. + \chapter{Pade approximant} Pade approximant \chapter{Schwartz-Zippel lemma and testing polynomial identities} diff --git a/books/bookvol10.2.pamphlet b/books/bookvol10.2.pamphlet index b5a1789..1a62084 100644 --- a/books/bookvol10.2.pamphlet +++ b/books/bookvol10.2.pamphlet @@ -35164,6 +35164,42 @@ Ring examples The category of rings with unity, always associative, but not necessarily commutative. +Rings have two operations, '+' and '*' and satisfy 5 axioms: + +1) (associative addition) a + (b + c) = (a + b) + c +2) (commutative addition) a + b = b + a +3) (associative multiplication) a(bc) = (ab)c +4) (distributive mulitplication) a(b + c) = ab + ac; (b + c)a = ba + ca +5) (equation solution) a + x = b has a solution in R + +examples include + + - even integers + - integers + - integers module any positive integer + - rational numbers + - real numbers + - complex numbers + - real quaterions + - a + b*sqrt(2) where a and b are rational + - polynomials in real variables with real coefficients + - real continuous functions in a fixed interval + - all power series in a real variable convergent on an interval + - all subsets of an arbitrary set + - all square matrices with elements from a ring + +Rings are not necessarily commutative. For example, square 2x2 matices: + + +- -+ +- -+ +- -+ + | 0 1 | | 0 0 | = | 1 0 | + | 0 0 | | 1 0 | | 0 0 | + +- -+ +- -+ +- -+ + + +- -+ +- -+ +- -+ + | 0 0 | | 0 1 | = | 0 0 | + | 1 0 | | 0 0 | | 0 1 | + +- -+ +- -+ +- -+ + See Also: o )show Ring @@ -47522,12 +47558,32 @@ IntegralDomain examples ==================================================================== The category of commutative integral domains, i.e. commutative -rings with no zero divisors. +rings with no zero divisors. That means that we can show that +if ab = 0 then either a or b must be 0. + +This does not have to be true. For instance, real 2x2 matrices +have zero divisors since + + A x B = 0 but neither A nor B is 0: + +- -+ +- -+ +- -+ + | 0 1 | | 0 2 | = | 0 0 | + | 0 0 | | 0 0 | | 0 0 | + +- -+ +- -+ +- -+ + + Conditional attributes: canonicalUnitNormal - the canonical field is the same for all associates canonicalsClosed - the product of two canonicals is itself canonical + Ring -> CommutativeRing -> IntegralDomain + +1) (associative addition) a + (b + c) = (a + b) + c +2) (commutative addition) a + b = b + a +3) (associative multiplication) a(bc) = (ab)c +4) (distributive mulitplication) a(b + c) = ab + ac; (b + c)a = ba + ca +5) (equation solution) a + x = b has a solution in R + See Also: o )show IntegralDomain diff --git a/books/bookvolbib.pamphlet b/books/bookvolbib.pamphlet index 5dc52ca..4d47901 100644 --- a/books/bookvolbib.pamphlet +++ b/books/bookvolbib.pamphlet @@ -14758,6 +14758,19 @@ Math. Tables Aids Comput. 10 91--96. (1956) \end{chunk} +\index{LaValle, Steven M.} +\begin{chunk}{axiom.bib} +@book{LaVa06, + author = "LaValle, Steven M.", + title = "Planning Algorithms", + year = "2006", + publisher = "Cambridge University Press" +} +\index{algorithm!cylindrical algebraic decomposition} +\index{cylindrical algebraic decomposition} + +\end{chunk} + \eject %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \chapter{Bibliography} diff --git a/changelog b/changelog index 9d93127..d26ef4a 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,7 @@ +20141004 tpd src/axiom-website/patches.html 20141004.01.tpd.patch +20141004 tpd books/bookvol10.2 add documentation for Ring help file +20141004 tpd books/bookvolbib add LaValle ref for CAD +20141004 tpd books/bookvol10.1 add documentation for CAD 20141002 tpd src/axiom-website/patches.html 20141002.01.tpd.patch 20141002 tpd src/interp/i-analy.lisp fix bug 7260 wrong type on output 20140928 tpd src/axiom-website/patches.html 20140928.01.tpd.patch diff --git a/patch b/patch index 7adea28..10d6bd4 100644 --- a/patch +++ b/patch @@ -1,4 +1,3 @@ -src/interp/i-analy.lisp fix wrong type on output - -fix bug 7260: FINITE() Type: Subdomain(Domain) +books/bookvol10.1, 10.2, bib add documentation +add documentation to help files and general algebra (CAD) diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index 7618a60..1b5da4c 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -4660,6 +4660,8 @@ books/bookvol10.4 document Sylvester matrix