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Consider a Cayley graph of some big finite group. Consider random walk on such a graph - think of it as drunk driver. Fix some number $N$ which is much smaller than group size. Question 1: How to cho …

Consider some compact Riemannian manifold $M$. Fix some point $p$. Consider a "sub-sphere of radius $r$" - i.e. set of points on distance $r$ from $p$. Consider growth function $g(r)$ to be volume of …

Consider some manifold $M$ say compact smooth. Let $b_i$ be its Betti numbers (non-zero), i.e. its cohomology dimensions. Assume $M$ can be subsequently fibered by many manifolds, i.e. there is $ M_ …

Standard setup. Consider a group and choose generators. Word-metric (or in the other words - distance on the Cayley graph of the group+generators) - converts a group into a metric space, which is top …

I wonder several questions about Cayley graphs of finite Heisenberg groups H3(Z/nZ). Question 1: do we know the diameter dependence on "n", at least for the standard choice of generators ? What about …

Context Working with some biological datasets it was puzzling to see the patterns like Figure 2 (right) below. The first feeling was, that it corresponds to some biological effects like correlations b …

Consider a graph, choose some "p: 0<p<1" (probability to infect the neighbor node). Choose some random number "K" of nodes which are "infected" initially. So we can generate epidemic model on a graph …

In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation groups are considered and cen …

Consider a vector $p =(p_1,\dots,p_n)$, $p_i>0$, $\sum p_i = 1$ and a matrix $M_{ij}$, which is doubly stochastic: $\sum_i M_{ij} = 1, \sum_j M_{ij} = 1, M_{ij} > 0$. Question 1 Just apply matrix M …

Latent Dirichlet allocation - is quite a popular topic in data-mining. Wikepedia mentions thousands citations in few years. Question 0 Can one give some digest for a math minded person of the key i …

Math behind might be interesting. Quite recent bloggingg activity might have interesting math model. The point is that bloggers compete for subscribers and at the same time cooperate gaining subscribe …

$\newcommand{\Fun}{F_\text{un}}$There was lots of "Fun with $\Fun$" (field with one element) in recent years. One of the points is that it provides bridge between geometrical and combinatorial questi …

Let's start from a little bit far. Basic probability theory - chain rule reads: $$ P(AB) = P(A)P(B|A)$$ Example: consider n+m balls, where n - white balls, m - black balls, consider A - first cho …

Question: How the maximal height grows for random Tetris like blocks falling down ? Numeric simulation (see below) shows leading term is linear with some constant depending on shapes of blocks allowe …

Consider general Brownian bridge W(0)=0; W(T) = a. (Here "general" means: $W(T)\ne 0$). What is the probability W(t) >= b, for all $ t \in [0, T] $ ? Is there close simple formula in terms of a …