In the excelent book "Advanced calculus" by Hans Sagans there is the following excerpt: In chapter 3 he defined the derivative of a function at an accumulation point of its domain. Next, in the context of several variables, he says that if we tried to define the derivative of a function at an ...

I have a question as I look at the example 8.9(a) in Rudin's Real and Complex Analysis: Let $X$ and $Y$ be the closed unit interval $[0,1]$, let $\{\delta_n\}$ be an increasing sequence of distinct points in $[0,1]$ that converges to $1$, and to each positive integer $n$, let $g_n$ be a real con...

I have a book about adaptive control. This book is fairly new and it contains Model Adaptive Reference Controller/Model Reference Adaptive System with state space. Normally this adaptive controller is often teach out with transfer functions. Which seems to be more advanced than state space mode...

The excircle to side $BC$ of $\triangle ABC$ is tangent to lines $BC,AB$ and $AC$ at $D,E,F$ respectively. Let $P$ be the orthogonal projection of $D$ onto $EF$. Let the midpoint of $EF$ be $M$. Prove $ABCP$ cyclic if and only if $ABCM$ cyclic. I didn't make much progress, except for $AD=AE$ (ob...

Find number of full connected directed graphs where $|V|=5$ (directed $K_5$). solution According to OEIS there are $42$ graphs like that. Let say that we have graph $G$ and we want to find how many there are graphs which are non isomorphic to it and has the same build (I mean directions of edg...

What do folks here think about the tag necklace-and-bracelets? It started in January '14 and currently 9 questions carry this tag. AFAICT those questions are about the application of Burnside's lemma/Polya enumeration of necklaces and bracelets constructed according to some rules. In my opinion...

Burnside lemma and Pólya enumeration theorem examples Observe that the convention at the OEIS is that the term necklace refers to the slots being arranged around a circle with rotational symmetry acting on them. Similarly the term bracelet indicates reflections acting on the slots ...

Let us have a dynamical system $(M, f)$ and some Poincare section, $P_1,..,P_n$. As far as I can understand, similar to what we are doing when we have only a single Poincare section, we can define maps $$f_{ij}(x_k) = x_{k+1}$$ that maps the point on the Poincare section $P_i$ to the correspondi...