All Questions
0
votes
0answers
2 views
Distribution of first hitting time of a non-monotone bivariate process to a zone
I have a trouble about the computation of the distribution of the first hitting time of a non-monotone bivariate process to a zone. I hope to have your helps.
Let denote $Z_t=Z_{r,t}+iZ_{i,t}$ is a ...
0
votes
0answers
13 views
Prove $z = \frac {5^{125 }- 1}{5^{25} - 1} $ is composite.
We need to prove that $z = \frac {5^{125 }- 1}{5^{25} - 1} $ isn't a prime number . This form comes out to be the sum of a geometric progression with the first term $1$ and ratio $5^{25}$, with five ...
1
vote
0answers
7 views
Limit values of a continuous function with a parameter
Denote $S=(0,1]\times[0,1]$ a square without the left side. Does there exist a function $f$ smooth enough on $S$, say $f\in C^1(S)$ or $C(S)$ s.t.
I) $f$ is not bounded in any neighbourhood of any ...
0
votes
0answers
3 views
General type variety has nef canonical bundle
Let $X$ be a compact projective variety of general type, then $K_X$ is always nef divisor ?
0
votes
2answers
16 views
Suppose 6 people are queueing for a bus. If all orderings are equally likely, what is probability that Jack is in front of Jill?
Suppose 6 people are queueing for a bus.
If all orderings are equally likely, what is probability that Jack is in front of
Jill?
I know that the total number of permutations is 6! which is 720. I am ...
0
votes
0answers
5 views
dimension of kernel of differential operator restricted to polynomials
I have the differential operator:
$$\frac{d}{dx} +2x \cdot$$
operating on $P$, the polynomials.
I want to know the dimension of the kernel of $\frac{d}{dx} +2x\cdot$.
My argument is that the ...
0
votes
0answers
3 views
Limit of slowly varying function
If $L_X$ and $L_Y$ are slowly varying function why is limit
$$
\lim_{x\to\infty}\frac{L_X(x)L_Y(x)}{L_X(x)+L_Y(x) }=0?
$$
Thanks!
0
votes
0answers
7 views
Maximal Ideals in $K[y]$
Let $K=F[x]/(p(x))$, where $F$ is a field, $p(x)\in F[x]$ is an irreducible polynomial.
What are all the maximal ideals in $K[y]$?
Attempt:
I have an attempt, but not entirely sure it is correct.
...
0
votes
1answer
10 views
Boundedness of $\frac{1}{n}+\frac{1}{n-x}+\frac{1}{n-2x}+\dots+\frac{1}{n-(n-1)x}$
Let $0\leq x\le 1$, and $n$ a positive integer. Define the function
$$f(n,x)=\frac{1}{n}+\frac{1}{n-x}+\frac{1}{n-2x}+\dots+\frac{1}{n-(n-1)x}.$$
For which $x$ is the sum bounded by a constant for ...
0
votes
0answers
15 views
Proof $\mathbb{Z}[i]/(2- i) \cong \mathbb{Z_5}$
My worked example sheet states
Prove $\mathbb{Z}[i]/(2- i) \cong \mathbb{Z_5}$ using the following steps:
Let $G = \mathbb{Z}[i]/(2- i)$
$G = \{ \bar{x} + \bar{y}\bar{i} \mid x,y \in \...
0
votes
1answer
7 views
$\oint \frac{dz}{e^z(z^2-1)^2}$, around $|z|=2$
Background Wunsch Complex Analysis chapter 4.5 #23
Does someone see my mistake? Textbook answer: $-\pi i e^{-1}$
Using $2\pi i (f_{z1}'(i)+f_{z2}'(-i))$
My attempt:
$$2\pi i \left(-\left[\frac{e^z(z+...
0
votes
0answers
9 views
Discretize Integral
Suppose I want to evaluate
\begin{align}
\int^{T}_0{e^{-t}x(t)dt}.
\end{align}
However, I can only approximate the continuous function $\mathbb{R}_+ \ni t \mapsto x(t)$ by a discrete sequence $\{x_t : ...
-3
votes
0answers
20 views
To find a limit without using l'Hopitals rule or series
How do I find the following limit?
$$\lim_{x\to0}\frac{\sin(\tan(x)) - \tan(\sin(x))}{2x\cos(\tan(x))-2x\cos(\sin(x))+x^5}$$
1
vote
1answer
18 views
Prove that all elements occur in the same number of distinct sets given these conditions
Anyone know how to answer/attempt this question?
There are $n \geq 2 $ sets each containing 10 elements each. Any 2 sets contain 1 element in common and each 2 elements are only in the same set ...
1
vote
0answers
5 views
Formulating dimensionality reduction problem
Paper titled : Probabilistic Visualisation of High-Dimensional Binary Data
download link
http://papers.nips.cc/paper/1561-probabilistic-visualisation-of-high-dimensional-binary-data.pdf
explains ...
