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Fixed LaTeX conversion errors.

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This commit is contained in:
gheber 2016-03-14 19:07:50 -05:00
parent cdff20c1a5
commit d0b79e1e34
1 changed files with 466 additions and 61 deletions
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527
examples/Chapter13.ipynb
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@ -10,7 +10,7 @@
"\n", "\n",
"Let us recall the definition of the *free group* generated by a set ${\\cal A}$. Consider the set ${\\cal A}\\times \\lbrace +1, -1 \\rbrace$ and make, as long as there is no ambiguity, the identification $a_i= (a_i,1)$ and $a_i^{-1}= (a_i,-1)$ for all $a_i\\in{\\cal A}$.\n", "Let us recall the definition of the *free group* generated by a set ${\\cal A}$. Consider the set ${\\cal A}\\times \\lbrace +1, -1 \\rbrace$ and make, as long as there is no ambiguity, the identification $a_i= (a_i,1)$ and $a_i^{-1}= (a_i,-1)$ for all $a_i\\in{\\cal A}$.\n",
"\n", "\n",
"**Definition.** $a_i$ and its formal inverse $a_i^{-1}$ are {\\em letters}.\n", "**Definition.** $a_i$ and its formal inverse $a_i^{-1}$ are *letters*.\n",
"\n", "\n",
"**Definition.** A word is a *finite* sequence of letters, possibly repeated, without any pair of the form $a_ja_j^{-1}$ and $a_k^{-1}a_k$. The word is said *reduced*.\n", "**Definition.** A word is a *finite* sequence of letters, possibly repeated, without any pair of the form $a_ja_j^{-1}$ and $a_k^{-1}a_k$. The word is said *reduced*.\n",
"\n", "\n",
@ -148,44 +148,88 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 1,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"<<Loop>>"
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(loop3)" "(loop3)"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 2,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"<<Loop[A\\2][4-2 B\\-3][3-2 C]>>"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(loop3 0 'a 2 20 'b -3 12 'c 1)" "(loop3 0 'a 2 20 'b -3 12 'c 1)"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 3,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"<<Loop[A\\2][4-2 B\\-3][3-2 C][A\\3]>>"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(loop3 0 'a 2 20 'b -3 12 'c 1 0 'a 3)" "(loop3 0 'a 2 20 'b -3 12 'c 1 0 'a 3)"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 4,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"<<Loop[2-1-0 P\\2][5-4-3-2-1 Q\\5]>>"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(loop3 (dgop-ext-int '(2 1 0)) 'p 2 \n", "(loop3 (dgop-ext-int '(2 1 0)) 'p 2 \n",
" (dgop-ext-int '(5 4 3 2 1)) 'q 5)" " (dgop-ext-int '(5 4 3 2 1)) 'q 5)"
@ -201,22 +245,71 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 5,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"[K1 Simplicial-Set]"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(def moore23 (moore 2 3))" "(def moore23 (moore 2 3))"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 6,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"\n",
"Dimension = 0:\n",
"\n",
" Vertices : (*)\n",
"\n",
"Dimension = 1:\n",
"\n",
"Dimension = 2:\n",
"\n",
"Dimension = 3:\n",
"\n",
" Simplex : M3\n",
"\n",
" Faces : (<AbSm 1-0 *> <AbSm 1-0 *> <AbSm 1-0 *> <AbSm 1-0 *>)\n",
"\n",
"Dimension = 4:\n",
"\n",
" Simplex : N4\n",
"\n",
" Faces : (<AbSm - M3> <AbSm 2-1-0 *> <AbSm - M3> <AbSm 2-1-0 *> <AbSm 2-1-0 *>)"
]
},
{
"data": {
"text/plain": [
"NIL"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(show-structure moore23 4)" "(show-structure moore23 4)"
] ]
@ -230,66 +323,132 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 7,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"[K6 Simplicial-Group]"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(def o-moore23 (loop-space moore23))" "(def o-moore23 (loop-space moore23))"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 8,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"<AbSm 2-1-0 <<Loop>>>"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(face o-moore23 0 4 +null-loop+)" "(face o-moore23 0 4 +null-loop+)"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 9,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"<AbSm - <<Loop[M3\\2]>>>"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(face o-moore23 0 3 (loop3 0 'N4 2))" "(face o-moore23 0 3 (loop3 0 'N4 2))"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 10,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"<AbSm 1-0 <<Loop>>>"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(face o-moore23 1 3 (loop3 0 'N4 2))" "(face o-moore23 1 3 (loop3 0 'N4 2))"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 11,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"<AbSm - <<Loop[N4\\2]>>>"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(face o-moore23 1 4 (loop3 1 'N4 2))" "(face o-moore23 1 4 (loop3 1 'N4 2))"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 12,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"<AbSm 0 <<Loop>>>"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(face o-moore23 0 2 (loop3 0 'M3 2))" "(face o-moore23 0 2 (loop3 0 'M3 2))"
] ]
@ -303,11 +462,46 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 13,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"The object is a STANDARD-OBJECT of type CAT::SIMPLICIAL-SET.\n",
"0. CMPR: #<FUNCTION S-CMPR>\n",
"1. BASIS: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-BASIS) {1005791FCB}>\n",
"2. BSGN: *\n",
"3. DFFR: [K19 Morphism (degree -1): K18 -> K18]\n",
"4. GRMD: [K18 Simplicial-Set]\n",
"5. EFHM: \"unbound\"\n",
"6. IDNM: 18\n",
"7. ORGN: (BUILD-FINITE-SS (* 2 S2 (* * *) 3 S3 (S2 S2 S2 S2)))\n",
"8. CPRD: [K22 Morphism (degree 0): K18 -> K20]\n",
"9. FACE: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-FACE) {1005791FEB}>\n",
"\n",
"> \n",
"Checking the 0-simplices...\n",
"Checking the 1-simplices...\n",
"Checking the 2-simplices...\n",
"Checking the 3-simplices..."
]
},
{
"data": {
"text/plain": [
"[K18 Simplicial-Set]"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(def ss2 (build-finite-ss '( * \n", "(def ss2 (build-finite-ss '( * \n",
" 2 s2 (* * *) \n", " 2 s2 (* * *) \n",
@ -316,110 +510,220 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 14,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"[K23 Simplicial-Group]"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(def o-ss2 (loop-space ss2))" "(def o-ss2 (loop-space ss2))"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 15,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"<AbSm - <<Loop>>>"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(face o-ss2 0 1 (loop3 0 's2 1))" "(face o-ss2 0 1 (loop3 0 's2 1))"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 16,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"<AbSm - <<Loop[S2\\2]>>>"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(face o-ss2 0 2 (loop3 0 's3 2))" "(face o-ss2 0 2 (loop3 0 's3 2))"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 17,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"<AbSm - <<Loop[S2\\2]>>>"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(face o-ss2 1 2 (loop3 0 's3 2))" "(face o-ss2 1 2 (loop3 0 's3 2))"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 18,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"<AbSm - <<Loop[S2\\2]>>>"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(face o-ss2 1 2 (loop3 0 's3 2))" "(face o-ss2 1 2 (loop3 0 's3 2))"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 19,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"[K40 Simplicial-Group]"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(def db (gdeltab))" "(def db (gdeltab))"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 20,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"<AbSm 1-0 <<Loop>>>"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(face db 3 3 (loop3 12 7 3))" "(face db 3 3 (loop3 12 7 3))"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 21,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"<AbSm 0 <<Loop[7\\3]>>>"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(face db 3 3 (loop3 5 7 3))" "(face db 3 3 (loop3 5 7 3))"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 22,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"<AbSm - <<Loop[1 7\\3][0 7\\3][1-0 3\\-1][1-0 5][1-0 3\\-1][1-0 5][1-0 3\\-1][1-0 5]>>>"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(face db 3 3 (loop3 6 7 3 5 7 3 3 7 3))" "(face db 3 3 (loop3 6 7 3 5 7 3 3 7 3))"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 23,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"<AbSm - <<Loop[7\\3][0 3\\-1][0 5][0 3\\-1][0 5][0 3\\-1][0 5][0 3\\-1][0 5][0 3\\-1][0 5][0 3\\-1][0 5]>>>"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(face db 3 2 *)" "(face db 3 2 *)"
] ]
@ -433,11 +737,24 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 24,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"\n",
"----------------------------------------------------------------------{CMBN 1}\n",
"------------------------------------------------------------------------------\n"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(? db (? db 3 (loop3 (dgop-ext-int '(3 2)) (dlop-ext-int '(0 1 2)) 3\n", "(? db (? db 3 (loop3 (dgop-ext-int '(3 2)) (dlop-ext-int '(0 1 2)) 3\n",
" (dgop-ext-int '(3 1)) (dlop-ext-int '(0 1 2)) 3\n", " (dgop-ext-int '(3 1)) (dlop-ext-int '(0 1 2)) 3\n",
@ -453,66 +770,154 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 25,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"[K52 Simplicial-Set]"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(def moore-22 (moore 2 2)) " "(def moore-22 (moore 2 2)) "
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 26,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"(<AbSm - *>) \n",
"(<AbSm 0 *>) \n",
"(<AbSm - M2> <AbSm 1-0 *>) \n",
"(<AbSm - N3> <AbSm 0 M2> <AbSm 1 M2> <AbSm 2 M2> <AbSm 2-1-0 *>) "
]
},
{
"data": {
"text/plain": [
"NIL"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(dotimes (i 4) (print (basis moore-22 i :dgnr)))" "(dotimes (i 4) (print (basis moore-22 i :dgnr)))"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 27,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"[K57 Simplicial-Group]"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(def o-moore-22 (loop-space moore-22))" "(def o-moore-22 (loop-space moore-22))"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 28,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"\n",
"----------------------------------------------------------------------{CMBN 3}\n",
"<1 * <<Loop[1-0 M2][2 N3]>>>\n",
"<-1 * <<Loop[2-0 M2][1 N3]>>>\n",
"<1 * <<Loop[2-1 M2][0 N3]>>>\n",
"------------------------------------------------------------------------------\n"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(aprd o-moore-22 3 (tnpr 1 (loop3 0 'm2 1) 2 (loop3 0 'n3 1)))" "(aprd o-moore-22 3 (tnpr 1 (loop3 0 'm2 1) 2 (loop3 0 'n3 1)))"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 29,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"[K74 Chain-Complex]"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(def bar-om (bar o-moore-22))" "(def bar-om (bar o-moore-22))"
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": null, "execution_count": 30,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
"outputs": [], "outputs": [
{
"data": {
"text/plain": [
"\n",
"----------------------------------------------------------------------{CMBN 4}\n",
"<1 * <<Abar[4 <<Loop[1-0 M2][2 N3]>>]>>>\n",
"<-1 * <<Abar[4 <<Loop[2-0 M2][1 N3]>>]>>>\n",
"<1 * <<Abar[4 <<Loop[2-1 M2][0 N3]>>]>>>\n",
"<-2 * <<Abar[2 <<Loop[M2]>>][2 <<Loop[M2]>>]>>>\n",
"------------------------------------------------------------------------------\n"
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [ "source": [
"(? bar-om 5 (abar 2 (loop3 0 'm2 1) 3 (loop3 0 'n3 1)))" "(? bar-om 5 (abar 2 (loop3 0 'm2 1) 3 (loop3 0 'n3 1)))"
] ]