# All Questions

64 views

### What is $\sum_{i=1}^m\sum_{j=i+1}^n \frac{1}{j}$

What is $$\sum_{i=1}^m\sum_{j=i+1}^n \frac{1}{j}$$? What I did I know that \begin{align*} &\sum_{i=1}^m\sum_{j=i+1}^n \frac{1}{j}\\ &=\sum_{i=1}^m\sum_{j=i+1}^n \frac{1}{j}+\sum_{i=1}^m\sum_{...
88 views

What are some suggestions for texts on introductory combinatorics and its applications? I would prefer if the applications would be to other branches of mathematics rather than outside of mathematics. ...
2k views

### what is the cardinality of a Null set?

Does the cardinality of a Null set is same as the cardinality of a set containing single element? If a set A contains Null set as its subset, then the null set is taken into account to calculate the ...
37 views

### How to recognize this sequence pattern?

What's the pattern for the following sequence -1,-1,1,1,-1,-1,1,1... And what is it's series? I tried (-1)^(n^3+1) and many things with no successful results.. And i didn't find any similar question ...
113 views

44 views

### Suppose $x$ and $y$ are in the same orbit. Show that $G_x$ and $G_y$ are conjugate subgroups.

Let $X$ be a $G$-set and $x,y \in X$. Let $G_x$, $G_y$ be the stabilizers of $x$ and $y$ respectively. Suppose $x$ and $y$ are in the same orbit. Show that $G_x$ and $G_y$ are conjugate subgroups. ...
15 views

### Independence of two r.v.

Let $\{ X_n, n\ge \}$ be iid with $P[X_1=1]=p=1-P[X_1=0]$. What is the probability that the pattern $1,0,1$ appears infinitely often? \ Hint: Let \begin{align*} A_k=[X_k=1,X_{k+1}=0, X_{k+2}=1] \end{...
40 views

### search for a theorem related to $\sum\limits_{n=1}^{\infty}n^2\exp(-n^2)<\int_{1}^{\infty}\nu^2\exp(-\nu^2)d\nu$

I need to use the following inequality: $$\sum_{n=1}^{\infty}n^2\exp(-n^2)<\int_{1}^{\infty}\nu^2\exp(-\nu^2)d\nu\tag{1}$$ what is the name of such a theorem?
40 views

### $X_i = \{ (x, y) \in \mathbb R^2 : x^2 + y^2 = i^2\}$. Show that the collection of sets $X_i$ for $i \in I$ form a partition of $\mathbb R^2$

$X_i = \{ (x, y) \in \mathbb R^2 : x^2 + y^2 = i^2\}$. Show that the collection of sets $X_i$ for $i \in I$ form a partition of $\mathbb R^2$. Do we use bijection? Is there any other way to solve it ...
58 views

### characteristic function of complement of non-Borel set

Let $C$ be the Cantor set and $M \subseteq C$ a subset, $M \notin \mathcal B(\mathbb R)$. Obviously the characteristic function of $M, \chi_M:[0,1] \rightarrow \mathbb R$, is not (Borel-) measurable. ...
27 views

### CHECK: Show that $N_G(aHa^{-1})=aN_G(H)a^{-1}$ for $a \in G$.

Show that $N_G(aHa^{-1})=aN_G(H)a^{-1}$ for $a \in G$. Hint: When working this problem, I found that in showing the two sets equal I had to be extremely careful. Try not to make big jumps. At one ...
239 views

### Find simple path with the largest sum of weights

In the longest path problem, given a weighted graph G and a starting node v0, find the simple path starting with v0 with the largest sum of weights. Show that the longest path problem is NP-hard in ...
100 views

### Conditional expectation on exchangeable variables

Let $X=(X_1,X_2,...,X_n)$ be an exchangeable $n$-dimensional vector of random variables with values in $\mathbb{R}$, let $\mathfrak{G}$ be the $\sigma$-algebra of permutation-invariant (symmetric) ...
101 views

### Could we calculate pi using an iterative series

I know that, as a hobbyist mathematician, this is generally a term we can use to express pi \begin{equation*} \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} +...
90 views

### Cardinality with Cartesian Cross Product problem

A and B are finite sets Prove that |AxB| = |A||B|. I need a solution/hint. I suspect that the answer has to do with the fact that the we can say that |A| = |B| and then from that say = |AxB|. I ...
167 views

### Minimum value of a positive definite binary quadratic form along integers

Is there a formula for the least non-zero value of $$f(x,y):=ax^2+bxy+cy^2$$ as $x,y$ assume integer values? Here $a,b,c$ are integers with $a,d>0$ and $b^2-4ac<0$.
29 views

### Show that if $\lim_{x\rightarrow\infty}f(x)=a \ \text{and} \ \lim_{x\rightarrow\infty}f'(x)=b$ then $b=0$

Consider: $$f:[c,\infty] \rightarrow\mathbb{R}$$ Show that if:$$\lim_{x\rightarrow\infty}f(x)=a \ \text{and} \ \lim_{x\rightarrow\infty}f'(x)=b$$ Then: $$b=0$$
128 views

### What does it mean for an integral to “vanish”?

I had a question; What does it mean for an integral to vanish in complex analysis? There is supposedly something, which says if the integral "vanishes," the sum of the residues is 0. But what does ...
41 views

### Find $n$ such that $(\mathbb{Z}/n\mathbb{Z})^\times \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$.

In the ring $(\mathbb{Z}_n, +, \times)$, we know that only $\phi(n)$ of the elements have multiplicative inverses which form the multiplicative group $\mathbb{Z}_n^\times$. However, since $\phi(n)$ ...
40 views

### Is this approach wrong?

Where have i gone wrong? $\frac{dq}{dt}=1000\cdot 2^{t}$ $dq=1000\cdot 2^{t}\cdot dt$ $\int\ln dq=\int\ln2000t\cdot dt$ $\ln\int dq=\int\ln2000t\cdot dt$ $\ln(q)+c_1=t\cdot\ln(2000t)-2000t+c_2$ ...
Let $M$ be a square matrix. By transposing rows and columns, can we get any permutation of the entries of $M?$ If we can't, which permutations are generated?