Symmetric polynomials, fundamental theorem of symmetric functions and an application

Let \(R\) be a commutative ring and \(R[x_1, x_2 , \ldots , x_n]\) the polynomial ring in \(n\) indeterminates over \(R\). A polynomial \(f(x)\) in it is said to be a symmetric polynomial (or symmetric function) of \(x_1, \ldots , x_n\) if it is unchanged by any permutation of the indeterminates. Consider the polynomials:

\begin{eqnarray*} e_1(x_1, \ldots , x_n) & = & x_1 + x_2 + \cdots + x_n \\ e_2(x_1, \ldots , x_n) & = & x_1x_2 + x_1x_3 + \cdots + x_1x_n + x_2x_3 + \cdots + x_{n-1}x_n \\ & \vdots & \\ e_i(x_1, \ldots , x_n) & = & \sum_{1\leq j_1 < j_2 < \cdots < j_i \leq n} x_{j_1} \dotsm x_{j_i} \\ & \vdots & \\ e_n(x_1, \ldots , x_n) & = & x_1x_2 \dotsm x_n \end{eqnarray*}

All of \(f(t ,x_1, \ldots, x_n)\) and \(e_i(x_1, \ldots , x_n)\) are symmetric functions of \(x_1, \ldots , x_n\). The \(e_i\)'s are called elementary symmetric polynomials of \(x_1, \ldots , x_n\). Sums, products and scalar multiples of symmetric polynomials are again symmetric polynomials. A polynomial \(g(e_1, \ldots , e_n)\) of \(e_1, \ldots , e_n\) becomes a symmetric polynomial \(g(x_1, \ldots x_n)\) when \(e_i\)'s are written in terms of \(x_i\)'s. The fundamental theorem on symmetric functions is the converse of this statement, credited to Gauss2, which will here be more precisely stated and proved.
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Determinants of Matrices over Commutative Rings

Let \(R\) be a commutative ring. Fix an integer \(n \geq 1\). Let \(\mathbf{A} = (a_{ij})\) be the \(n \times n\) square matrix

$$\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{pmatrix}$$

with entry \(a_{ij}\in R\) and \(1 \leq i,j \leq n\) is at the intersection of the \(i^\mathrm{th}\) row and \(j^\mathrm{th}\) column. We will define a function \(\det(\mathbf{A}) =\det (a_{ij})\) of this matrix \(\mathbf{A}\), which will also be denoted by

$$\begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{vmatrix}$$

We will also state and prove the various familiar properties of the determinant in this setting.
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The long road to building a static blog with pelican

In this post it is documented how this blog was constructed as a static site, meaning, the webpages are served from storage and not generated dynamically upon each visit using scripts (written for example in php) and databases (e. g., MySQL).


Why a static site?


The obvious and ubiquitous software choice for blogging is wordpress, a dynamic engine, which IMHO, is in several ways superior to other dynamic blogging services such as Google Blogger, Livejournal, Tumblr, etc., if nothing else for control of content.

It would have been natural to stop there and take that route. However in a more insidious manifestation of technophilia, a static blog was decided upon. There are many reasons, which all basically comes down to simplicity, control, cost savings, and, because I can do it.

The environment issue

In wordpress, everything is web-based, and the content storage, database and scripts are all intermixed. Editing must take place in the wordpress editor or in custom html/javascript/css/php. This is all too much detail for the writer to be bothered about, if s/he wants any degree of customization. Once the design is fixed, s/he should be able to focus on the content without any distractions, and if design is to be changed, it should be doable without disturbing the content files.
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