# Tagged Questions

**8**

votes

**2**answers

481 views

### Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...

**1**

vote

**1**answer

208 views

### Asymptotic behaviour of a mean

Fix $x>0$ and $c\in\mathbb{N}$. Let $f(x):=\frac{c}{4c-2+2x^2}$ and
$$m_N(x):=\frac{1}{N} \sum_{i=0}^{f(x)N} \log(\frac{c N}{2}-i(2c-1))$$
I'm pretty sure $m_N(x)\to\infty$ as $N\to\infty$.
I ...

**3**

votes

**1**answer

160 views

### bounding the probability that a polynomial is near 0

Given a polynomial $p(x_1,\ldots, x_k)$ in $k$ variables with maximum degree $n$, and $x_1,\ldots, x_k \in [0,1]$. Suppose $\max_{x \in [0,1]^k} p(x) = 1$, can we get an upper bound on the probability ...

**6**

votes

**3**answers

864 views

### Is the function $e^{x^2/2} \Phi(x)$ monotone increasing?

Hello,
Here is an interesting problem. It looks elementary, but it has taken me some efforts without solving it. Let
$$
h(x) = e^{x^2/2} \Phi(x),\qquad \text{with}\quad \Phi(x):=\int_{-\infty}^x ...

**9**

votes

**2**answers

546 views

### “Probabilistic ultrafilters?”

A naive question.
Let $S$ be a set and let $[0,1]^S$ the set of functions from $S$ to the closed interval $[0,1]$.
Suppose given some function $P \colon [0,1]^S \to [0,1]$ satisfying the following ...

**16**

votes

**2**answers

1k views

### There is mathematics behind the 1989 Tour de France !

The $1989$ Tour was won by Greg Lemond (USA, $1961$ - ), who beat Laurent Fignon (France, $1960$ - $2010$) by $8''$. Yes, eight seconds! The closest tour in history.
Let me recall a few rules ...

**13**

votes

**1**answer

1k views

### Probability of a Point on a Unit Sphere lying within a Cube

Suppose we have a (n-1 dimensional) Unit Sphere centered at the origin: $$ \sum_{i=1}^{n}{x_i}^2 = 1$$
What is the probability that a randomly selected point on the sphere, $ (x_1,x_2,x_3,...,x_n)$, ...

**6**

votes

**1**answer

2k views

### Big picture concerning Ito integral, Stratonovich integral and standard results in probability theory

I am confused and don't get the big picture concerning the connection between
Ito integral
Stratonovich integral
Standard results in probability theory concerning skewed distributions.
Example: ...

**3**

votes

**1**answer

773 views

### “Nice” Solution to repeated integral

I have a problem wherein I have defined a function $I_r(t) = \int e^{(2r-1)at} \int e^{(2r-3)at} \cdots \int e^{at} dt\cdots dt$, and $I_r(0) = 0$, for $r = 1,2,3,\ldots$.
I find that $e^{-ar^2t} ...

**3**

votes

**2**answers

274 views

### How to fill a simplex with almost disjoint cuboids?

There is an algorithm that give us cuboids in $\mathbb{R}^3$, say $Q_1,Q_2,\ldots$, such that $\cup_{i=1}^{\infty} Q_i$ is the simplex with vertices $(0,0,0), (1,0,0) , (0,1,0), (0,0,1)$, and the ...

**7**

votes

**5**answers

751 views

### How can I sample uniformly from a surface?

Given an equation of a parametric surface, is there a general way to sample of points uniformly distributed on that surface?
I'm interested in this problem for purposes of visualisation - rather than ...