# All Questions

504 views

### How to do a very long division: continued fraction for tan

I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in ...
136 views

### Generating function of a convolution

If I need to find generating function of such a sum: $\sum_{k=0}^{n} (n-k) a_k$, can I write $\sum_{k=0}^{n} (n-k) a_k = \sum_{n\ge 0} nx^n \cdot \sum_{n\ge 0} a_nx^n \cdot \frac{1}{1-x}$ and then ...
101 views

### Brownian motions identical distributions

Let $(B_t)_t$ be a standard Brownian motion, and $$A = \sup\{t\leq 1\mid B_t =0 \},\qquad B = \inf\{ t\geq 1\mid B_t =0 \}.$$ I would like to show that $A$ and $B^{-1}$ are identically distributed ...
71 views

### Fréchet mean for a general shape space

I am posting this question in order to gain a better understand of what the Fréchet mean is for a generalised shape space. So firstly I gather that the Fréchet mean of a probabilty measure $\mu$ on a ...
1k views

### Prove the supremum of the set of affine functions is convex

Let $\langle f_i \rangle _{i \in I}$ be a family of affine functions on a convex and compact set $\Omega \subset \mathbb{R^d}$ such that $f_i = a_i.x +b_i$ for $x \in \Omega$. Prove that f, defined by ...
115 views

### tangential and normal projection of a vector in the ambient vector field of a sphere

I'm having unexpected trouble to perform this computation: Let $M=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2+z^2=3\}$ and $v_p = (1,0,0)_{(1,1,1)}$ be a vector from the ambient vector field on $M$. How do I ...
287 views

### Finite sum of products of binomial coefficients and quadratic polynomial

How can I calculate the value of such a sum? $\sum_{k=0}^{n} (2k^2-3k+1){n\choose k}$ Should I split it into three sums? But then I don't know what to do with $k^2{n\choose k}$. I know that ...
54 views

69 views

### What is the mathematics of UML?

What is the mathematics of Unified Modeling Language (UML)? The concepts introduced in UML such as classes, associations, association classes, subclasses smell mathematical. Is there a mathematical ...
1k views

### How many $n$-digit palindromes are there?

How can one count the number of all $n$-digit palindromes? Is there any recurrence for that? Thanks. I'm not sure if my reasoning is right, but I thought that for n=1 we have 10 such numbers ...
182 views

### Proving continuity using the topological definition?

Let $f$ be a function from $\mathbb{R}$ to $\mathbb{R}$, and suppose that $f(x) = 0$, $\forall x \in \mathbb{R}$, except when $x=c$, for a fixed $c \in \mathbb{R}$. Now, $f$ is clearly discontinuous ...
Regard this statement $x \ge 0$. According to my teacher, by negating this statement, it will become $x < 0$. Why is this so; why does the $\ge$ morph into $<$, and not into $\le$?