# All Questions

3 views

### What series of 'hyperpolyhedrons' do exist? Is there an effective way to derive their cross-sections by 3-d subspace?

There are two obvious series of 'hyperpolyhedrons'. 'Hyperoctahedron' with vertices $(\pm1,0...0), (0,\pm1,0,...0)...(0,...0,\pm1)$ and each vertex connected by an edge with each other vertex ...
7 views

### Problem in Linear algebra?

Let $V=V_0\oplus V_1$ is a $Z_2$-graded inner product vector space and $T:V\longrightarrow V$ is an odd linear map i.e, $T(V_0)\subseteq V_1 \,\text{and} \, T(V_1)\subseteq V_0$. If $T$ be ...
28 views

### No. of maps satisfying ϕ(ab)=ϕ(a)+ϕ(b)

How to show that there are finitely many maps $\phi:\mathbb{N}\cup\{0\}\to \mathbb{N}\cup\{0\}$ with the property that $\phi(ab)=\phi(a)+\phi(b)$ for all $a,b\in \mathbb{N}\cup\{0\}$.
9 views

### Diagonalization of a matrix with change of basis

I was trying to diagonalize a not really nice matrix doing first a change of basis but I noticed that the two characteristic polynomials I get are different. Original matrix and its characteristic ...
8 views

### A direct limit of pullbacks

Let an $R$-module $C$ be a direct limit of finitely presented $R$-modules $C_i$, and we have a short exact sequence as follows: $$0→A↪B\stackrel{\pi}→C→0.\qquad (S)$$ From each $C_i$ to the direct ...
10 views

### Problem on product space of Sorgenfrey line.

Let $(\mathbb R,\tau)$ be Sorgenfrey line, $(\mathbb R^2,\tau_1):=(\mathbb R,\tau)\times (\mathbb R,\tau)$. Let $L = \{(x, y) : x, y\in\mathbb R^2, x + y = 0\}$. Show that the line L is closed in ...
9 views

11 views

### Ordered field - Adding inequalities (Looking for a proof)

In an ordered field, it is allowed to add two inequalities. Where can I find a detailed proof?
37 views

15 views

### How to calculate lower & upper quartiles?

I'm sure this has been asked many times before but it's confusing me a lot so hopefully someone can help! I am given this data set: 0.28 0.30 0.42 0.59 0.71 1.67 1.82 2.39 4.71 4.79 4.89 5.00 5.00 ...
3 views

### Regression with rounded dependent variable

I am looking at the following model: $c$ is a fixed vector in $\mathbb{R}_+^n$ and for any $x \in \mathbb{R}_+^n$ we obtain a value $y =[c^Tx]$, i.e. rounding $c^Tx$ to the nearest integer. I want ...
61 views

### Nested Radicals with multiplication

I think this one goes to section of nested radicals, I was trying to solve if for a couple of days now. Maybe you have some nice solution to this one. $$\sqrt{1\sqrt{2\sqrt{3\sqrt{4\sqrt{...}}}}}$$ ...
96 views

### Determine the cardinal of the following infinite set?

Me and my friends were debating the following question in set theory, it has been some time since we studied this subject and we don't remember if there are rules that determine the answer for our ...
63 views

### Evaluation of $\sum n a^n$ using telescoping property

Show that the series $$\sum\limits_{n=1}^{\infty}n a^n = \frac{a}{(a-1)^2}$$ for $|a|<1$ using the telescoping property. I know how to do this using other methods. But the exercise ...
37 views

### Calculate $\lim_{n\to\infty} \sum_{k=1}^{n} \Big(\frac{k}{n}\Big)^{\alpha k}$

Let $\alpha$ be a positive number. Calculate $$\lim_{n\to\infty} \sum_{k=1}^{n} \Big(\frac{k}{n}\Big)^{\alpha k}$$ Edit: I have deleted my attempt, it didn't seem to lead me anywhere and I discovered ...
15 views

### Inequality involving products

One is given two intervals $I_{a-\epsilon,b+\epsilon}$, $I_{a,b}$ of $\mathbb{R}^n$, and is asked to show that $\lambda(I_{a-\epsilon,b+\epsilon}) - \lambda(I_{a,b}) \leq c\epsilon$ for some constant ...
Let $G$ be a group and $A$ & $B$ be two sets s.t. $G$ acts transitively on each of $A$ & $B$. Choose some $\alpha$ and $\beta$ in $A$ & $B$ respectively then prove that if $G=G_\alpha ... 0answers 12 views ### In an abelian category,every morphism can be written as composition of epi and mono. Following Weibel's book on homological algebra, he states without proof that every morphism$f\colon A \to B$can be written as composition of an epimorphism followed by a monomorphism. After many ... 2answers 44 views ### Compute$0.y_1y_2y_1y_2y_1y_2…\$
Any tips? I'm taking $$Sn = \frac{y_1}{10} + \frac{y_2}{100} +\frac{y_1}{1000} +$$ and then I find $$\frac{S_n}{100}=...$$ I do $$Sn - \frac{S_n}{100}$$ and go on from there but I can't find the ...