In group or ring theory we have the first isomorphism theorem which gives an isomorphism between two structures involving kernel and image of the homomorphism. The Rank Nullity theorem also has a kind of similarly if we write it like this: $$\operatorname{dim}(V)-\operatorname{dim}(\operatorname{...

Let $G$ be the multiplicative group of complex numbers of modulus $1$ and $G_n,n\in \Bbb N$ the subgroup consisting of the $n$-th roots of unity. For positive integers $m$ and $n$, show that $ G/G_m$ and $G/G_n$ are isomorphic groups. Below is my attempt at the problem: Consider the ...

Problem Let $G$ be a simple graph such that for each vertex $v$ of $G$, $\operatorname{deg}(v)\geq 3$. Then show that there is a cycle in $G$ of size not multiple of $3$. Attempt Let $P$ be a maximal path of $G$ and $v$ is one of its end point. Now $\operatorname{deg}(v)\geq 3$, so there are tw...

Suppose that there is a deterministic relation $y_t=ax_t$ where $x_t,y_t$ are real sequences or real functions and $a$ a constant. But only $X_t=x_t+e_t$ and $Y_t+u_t$ can be observed, with $e_t, u_t$ being zero mean i.i.d. random variables. How can I estimate the parameter $a$ using $X_t$ and $Y...

I'm doing some modelling on social science data using René Thom's cusp catastrophe. The canonical form for this is $${V=x^4}+ax^2+bx$$ which produces the standard cusp surface when its critical points are plotted: However, my data are better fit by the variation on the cusp catastrophe below: $$...

When I use catastrophe here, I mean a system exhibiting a finite number of bifurcations and by chaos, I mean a system exhibiting a (very) large number of bifurcations. I do know that catastrophe theory is based on Thom's theorem and chaos theory on qualitative analysis but I can't get over the fa...

I am studying Thom's theorem in catastrophe theory and am having a hard time understanding what the "generating functions" actually do. How exactly are they used to classify generic caustics? The literature I am being exposed to is mostly physics-related so doesn't have very rigorous mathematical...

I'm looking for a technical introduction to catastrophe theory, preferably something short. I have a good background so graduate level texts are welcome. Thanks in advance.

The cusp catastrophe corresponds to the equation $$F(x,a,b)=x^4+ax^2+bx$$ where $a, b$ are the control parameters. The following diagram of cusp catastrophe shows the curves that satisfy $\frac{dF}{dx}=0$ for the parameters $a,b$ drawn for parameter $b$ continuously varied, for several values of...

I have seen a lot the Arnold's classification of singular surfaces by the simple Lie groups. I have even asked the author of a book that used this classification about its origin and his answer was that when he wrote it someone explained him the connection, but he had long forgotten it. Is there...

Coming back on the system I already mentioned in another post, this time I am working on some bifurcation analysis of a 2D System. The system is defined by the following equations. I am assuming $\tau_a >1$ to be kept fixed. \begin{equation} \begin{aligned} \dot d_{1} &= - d_1 - e_1 + \varepsilon...