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Fixed a few typos.

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Gerd Heber 2017-02-12 18:51:32 -06:00
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examples/07_Simplicial_Sets.ipynb
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@ -18,15 +18,15 @@
"\n",
"$$\n",
"\\begin{aligned}\n",
"\\partial_i^{q-1}\\partial_j^q & = & \\partial_{j-1}^{q-1}\\partial_i^q, & i < j \\cr\n",
"\\eta_i^{q+1}\\eta_j^q & = & \\eta_j^{q+1}\\eta_{i-1}^q, & i > j \\cr\n",
"\\partial_i^{q+1}\\eta_j^q & = & \\eta_{j-1}^{q-1}\\partial_i^q, & i < j \\cr\n",
"\\partial_i^{q+1}\\eta_i^q & = & \\partial_{i+1}^{q+1}\\eta_i^q & = \\mbox{Identity}, \\cr\n",
"\\partial_i^{q+1}\\eta_j^q & = & \\eta_j^{q-1}\\partial_{i-1}^q, & i > j+1.\\cr \n",
"\\partial_i^{q-1}\\partial_j^q & = & \\partial_{j-1}^{q-1}\\partial_i^q, & & i < j \\cr\n",
"\\eta_i^{q+1}\\eta_j^q & = & \\eta_j^{q+1}\\eta_{i-1}^q, & & i > j \\cr\n",
"\\partial_i^{q+1}\\eta_j^q & = & \\eta_{j-1}^{q-1}\\partial_i^q, & & i < j \\cr\n",
"\\partial_i^{q+1}\\eta_i^q & = & \\partial_{i+1}^{q+1}\\eta_i^q & = \\mbox{Identity}, & \\cr\n",
"\\partial_i^{q+1}\\eta_j^q & = & \\eta_j^{q-1}\\partial_{i-1}^q, & & i > j+1.\\cr \n",
"\\end{aligned}\n",
"$$\n",
"\n",
"An element of $K^q$ is an (*abstract*) $q$-simplex of $K$ and the functions $\\partial$ and $\\eta$ are respectively the *face operators* and the *degeneracy operators*. Formally, their action on a simplex is very simple. For instance, if a $q$--simplex is an ordered set of vertices, like $\\lbrace v_0, v_1, \\ldots,v_i,\\ldots, v_q \\rbrace$, the rules are the following (we omit the indication of the dimension of the simplex):\n",
"An element of $K^q$ is an (*abstract*) $q$-simplex of $K$ and the functions $\\partial$ and $\\eta$ are respectively the *face operators* and the *degeneracy operators*. Formally, their action on a simplex is very simple. For instance, if a $q$-simplex is an ordered set of vertices, like $\\lbrace v_0, v_1, \\ldots,v_i,\\ldots, v_q \\rbrace$, the rules are the following (we omit the indication of the dimension of the simplex):\n",
"\n",
"$$\n",
"\\begin{aligned}\n",
@ -78,9 +78,9 @@
"## Notion of abstract simplex in this Lisp implementation\n",
"\n",
"In this implementation, on account of the essential following property: *any simplex\n",
"may be expressed in an unique way as a possibly iterated degeneracy of a non--degenerate simplex*, we call an **abstract simplex**, (in short **absm**), a pair consisting of:\n",
"may be expressed in an unique way as a possibly iterated degeneracy of a non-degenerate simplex*, we call an **abstract simplex**, (in short **absm**), a pair consisting of:\n",
"- a (possibly iterated) degeneracy operator,\n",
"- a \"geometric\" simplex, i.e. a non--degenerate simplex.\n",
"- a \"geometric\" simplex, i.e. a non-degenerate simplex.\n",
"For instance, if $\\sigma$ is a non-degenerate simplex, and $\\sigma'$ is the degenerate simplex $\\eta_3\\eta_1\\sigma$, the corresponding abstract simplices are respectively $\\lbrace \\emptyset, \\sigma \\rbrace$ and $\\lbrace \\eta_3\\eta_1, \\sigma \\rbrace$.\n",
"\n",
"An abstract simplex is represented internally in the system by the following lisp object:\n",
@ -89,7 +89,7 @@
"```\n",
"where,\n",
"\n",
"1. *dgop* is a non-negative integer coding a *strictly decreasing integer list*. This strictly decreasing list of integers represents a sequence of $\\eta$ operators and is coded as a **unique** integer as follows. The iterated degeneracy operator $\\eta_{i_k} \\ldots \\eta_{i_1}$ is coded as the integer $\\sum_{j = 1}^k 2^{i_j}$; in other words we use a binary number where the binary digits 1 correspond to the indices of the $\\eta_i$'s. This is made possible by the \\emph{strictly} decreasing property of the indices. For example the degeneracy operator $\\eta_6 \\eta 3 \\eta_0$ is coded by the integer 73, for $73 = 2^6 + 2^3 + 2^0$. This leads to a compact coding, replacing the numerous operations about degeneracy operators by efficient handlings of bits of fixnums. In particular, 0 codes the void composition of degeneracy operator, that is, the identity operator. Two functions are provided to help the user for the transformation in both directions.\n",
"1. *dgop* is a non-negative integer coding a *strictly decreasing integer list*. This strictly decreasing list of integers represents a sequence of $\\eta$ operators and is coded as a **unique** integer as follows. The iterated degeneracy operator $\\eta_{i_k} \\ldots \\eta_{i_1}$ is coded as the integer $\\sum_{j = 1}^k 2^{i_j}$; in other words we use a binary number where the binary digits 1 correspond to the indices of the $\\eta_i$'s. This is made possible by the *strictly* decreasing property of the indices. For example the degeneracy operator $\\eta_6 \\eta_3 \\eta_0$ is coded by the integer 73, for $73 = 2^6 + 2^3 + 2^0$. This leads to a compact coding, replacing the numerous operations about degeneracy operators by efficient handlings of bits of fixnums. In particular, 0 codes the void composition of degeneracy operator, that is, the identity operator. Two functions are provided to help the user for the transformation in both directions.\n",
"2. *gmsm* is a geometric simplex, i.e. any kind of lisp object modelizing a non-degenerate simplex. For practical reason, a type `GMSM`, (any kind of lisp object), has been defined.\n",
"\n",
"The simplest constructor is the macro `(absm (dgop-ext-int` *dgop-list*`)` *gmsm*`)`, where *dgop-list* is a strictly decreasing integer list. A structure class `ABSM` has been defined. The associated printing method prints such an object under the form:\n",