What exactly are Tensors?

Hans Lundmark wrote a nice short introduction to Tensors in answer to a question at Math.SE, see here.

Missing the Geometry in Algebraic Geometry?

It seems that in the decades following the publication of Hartshorne's Algebraic Geometry, a sufficient number of aspiring mathematicians have run aground in efforts to digest the field, such that for many, an automatic response to a cry for help is to redirect the student to Mumford's "Red Book". I have located a few online alternatives, for those who cannot stomach the cost of yet another mathematics book.

Matt Kerr and Andreas Gathmann (independently) provide lecture notes on Algebraic Geometry that contain a more-than-average amount of motivation through pictures:

Algebraic Geometry III/IV - Matt Kerr
Algebraic Geometry - Andreas Gathmann

Why should one study Algebraic Geometry? Well, to solve problems in Algebraic Geometry, of course! However, in all seriousness, it seems that the advantages of studying Algebraic Geometry far out-weigh the disadvantages.

Unlike most other fields, it seems that the problems that modern Algebraic Geometry has solved, and the correspondences it has uncovered, are yet to trickle down to mathematical expositions at the more elementary level, resulting in a complete blank for the uninducted, at the outset.

arXiv CLI search

Not guaranteed to preserve the order of Boolean operators.

Dynamical Systems, Posets, and Graphs

Suppose we have a set of nodes \(X = \{ x_1, \ldots, x_n \}\), with entries in \(\mathbb{Z}/2\mathbb{Z}\). If we also have a set of unknown "change-of-state" vectors \(V = \{v^*_1, \ldots, v^*_t\}\), we say that \(S = (X, V)\) is an information system.

Whilst our "change-of-state" vectors are initially unknown, we understand that they also have entries in \(\mathbb{Z}/2\mathbb{Z}\), with composition defined as the boolean \( \vee \).

The stabilization of the system can be determined by applying the "change-of-state" vectors, iteratively to \(X\):

\begin{equation*}
X_0 \overset{v^*_1}{\to} X_1 \overset{v^*_2}{\to} \cdots \overset{v^*_n}{\to} X_n = C
\end{equation*}

We say that every state \(X_i\) is an observation of \(S\) and that a system has stabilized when all nodes are equal to \( \mathbf{1} \). This state is denoted as \( C \), as seen above. Of course, we may examine other systems in which there may not exist a finite stabilization state. Also, the "change-of-state" vectors may be composed by some other boolean operator other than \(\vee\).

Let \(A_i = X_i \setminus X_{i-1}\). We say that \(A_i\)  is the set of nodes activated in the \(i\)th observation of the system, and denote the set of all such \(A_i\) as simply \(A\). We define a function \(\tau: A_i \to \mathbb{Z}\) that measures the future increase in activation.

\begin{equation*}
\tau(A_i) = |A_{i + 1}| - |A_i|
\end{equation*}

Following stabilization, we may examine the fibers of \(\tau(A)\). We say that the fibers of \(\tau(A)\) are components of \(S\), whilst \(\tau^{-1}(\{\text{sup}\,\tau(A)\})\) is called the core of \(S\).

Let \(\pi\) be a functor, where \(\pi : \text{Set} \to \text{Grph}\), such that \(\pi(x)\) forms a complete graph with nodes elements of \(x\). If we let \(\pi\) act on the fibers of \(\tau(A)\), we will of course get a set of graphs. Join these graphs according to their partial ordering from \(\tau(A)\), and we will have formed a graph whose connectivity represents the discrete activation of the system \(S\).

Undeniably, this may be a little rough around the edges. The last few steps will surely be clarified with the application of some categorical language. However, the more important issue at hand is, what can we do with all of this?

EDIT: This seems quite related to Network Theory as discussed here: http://johncarlosbaez.wordpress.com/2011/03/04/network-theory-part-1/ . Although, it seems I have some reading to get done, before coming to a conclusion.

Testing!

Testing, \(1\), \(2\), \(e\), \(3\), \(\pi\).