Welcome to this blog

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Hi, my name is Birzhan Moldagaliyev. In this blog I hope to share my journey in search for patterns in the universe. I believe that Mathematics is a study of general patterns taking place in abstract spaces. My current research involves studying geometries of Cayley graphs corresponding to automatic and Cayley-automatic groups. I intend to share my research progress in this blog.

As for patterns found in the real world, I believe that machine learning is one of the most potent tools to harness such patterns. So I hope to learn some Machine Learning and share my learning experience here.

I hope that you, reader, will find some of the things written here useful. Feel free to comment anything here.

Thanks, Birzhan.

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Popular posts from this blog

On Measures of Size and Complexity

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Introduction People like to measure things. In mathematical analysis there is a notion of measure studied in Measure theory . In Topology we have a notion of topological dimension . In Information Theory we have a notion of entropy . In Computational Complexity Theory we have notions of time and space complexities . So the general idea is to quantify size or complexity of the given object. Then we can talk about whether this object is somehow bigger/more complex than the other object. In this post I would like to say few things about Hausdorff dimension . Motivation Why do we need Hausdorff dimension? There are many reasons. One of the reasons people talk most is its fineness. In other words Hausdorff measure can provide some meaningful information when Lebesgue measure gives none. To see that, observe that there are many sets of measure $0$. A single point is of measure $0$ and a Cantor space is of measure $0$. However, it is clear that those two are completely different objec...
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On Cayley graphs: Part 1

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A Cayley graph can be considered as an attempt to geometrize or visualize groups . Algebraically, a group is just a bunch of letters some of them having an inverse script. By visualizing groups, we may observe some patterns in their structure. So let's see how can we construct Cayley graphs. Take an arbitrary group $G$ with a set of generators $S\subset G$. In other words, every element of $G$ can be written as: \begin{equation} g = s_1^{\epsilon_1}s_2^{\epsilon_2}\ldots s_n^{\epsilon_n},\quad \epsilon_i\in\{\pm1 \} \end{equation} As for $S$, we assume that $S^{-1} = S$, i.e. $s\in S$ implies $s^{-1}\in S$, and $1\not\in S$. In fact these assumptions are not necessary, but tend to make life a bit easier. Let us construct a Cayley graph $\Gamma=(V,E)$ as follows: $V=G$ $(g,h)\in E$ if $gs = h$ for some $s\in S$, or equivalently $g^{-1}h\in S$ We can observe few things straight away about $\Gamma$. It is undirected as $gs = h$ happens when $g = hs^{-1}$. Any two vert...
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From Gales to Covers

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This post continues to study relationship between definitions of Hausdorff dimension obtained through covers and gales. In the last post, we have shown how to switch from covers to gales, which allowed us to conclude that \[ \inf G(X) \le \dim_H(X) \] So now our aim should be verification of reverse inequality, i.e. $\dim_H(X)\le \inf G(X)$. In order to do this, we go back to definitions. Let us start with basics $\mathcal{C} = \{0,1\}^{\mathbb{N}}$ is Cantor space with some subset $X$. Let $d$ be an $s$-gale which succeeds on $X$, i.e. $X\subseteq S^{\infty}[d]$. This means that $\inf G(X)\le s$. Furthermore, for any given $\epsilon>0$, one could choose $s$ such that $\inf G(X)\ge s-\epsilon$. Given such $s$, we claim that $\mathcal{H}^s(X)=0$, i.e. \[ \inf\{\sum_{i=1}^{\infty}diam(C_{w_i})^s \} = 0 \] where $X\subseteq \bigcup C_{w_i}$. This would give us $\dim_H(X)\le s \le \inf G(X) + \epsilon$. By letting $\epsilon\to 0$, the desired inequality follows. So our main...
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