# All Questions

98 views

### Last step of the proof of the Chinese remainder theorem.

For the Chinese Remainder Theorem for rings we have: $$A/(I\cap J) \cong A/I\times A/J$$ So far I have proven that there is a ring homomorphism from $\phi :A \rightarrow A/I\times A/J$ and the ...
188 views

### Gamma Function Contour Integration

So, I've been trying to prove the following integral related to the gamma function, and I'm really banging my head against the wall over this: ...
209 views

### Is complex symplectic structure equivalent to a quaternionic structure?

Let $S$ be a vector space over $\mathbb{C}$ of dimension $\operatorname{dim} S =2n$, let me denote by $S_\mathbb{R}$ real form of $S$ i.e. vector space over $\mathbb{R}$ of dimension $4n$. Is it true ...
97 views

### Functions of the form $\int_a^x f(t) dt$ that are commonly used.

I am a graduate student and teaching assistant, and I am teaching Calc 1 for the first time. In a few weeks I will be covering the Fundamental Theorem of Calculus. I'm using James Stewart's Calculus ...
435 views

### Convert a boolean function into K-map

I would like to know how can I convert the following boolean function into a truth table and accordingly construct the k-map $$F = A'B'C'+B'CD'+A'BCD'+AB'C'$$ thanks in advance :)
293 views

### Bisection Method Question, Multiple Roots

I understand how to do the bisection method and how to do it with a point of intersection. My question is should this not actually have multiple points of intersection? and if you're not given any ...
811 views

### Is $\sum_{n=1}^{\infty} \frac{ \sqrt{n + 1} (n + 1)! }{ e^{n + 5} }$convergent or divergent

I have worked on this my answer is L= Div and B Consider the series $\displaystyle \sum_{n=1}^{\infty} a_n$ where $$a_n = \frac{ \sqrt{n + 1} (n + 1)! }{ e^{n + 5} }$$ In this problem you must ...
115 views

### How many trials until I each my desired outcome

I'll just get straight to the point; I want to know how many independent trials are required to get an outcome(?) of 10. The situation is that I have an 80% success rate and I increment x by 1, ...
81 views

### Showing a subset is uncountable [closed]

How do I show if $A \subseteq B$, and $A$ is uncountable then $B$ is uncountable?
135 views

### Cardinality of the union of all repeated Cartesian products of N with itself [duplicate]

Here is a silly question, but I am a silly person. Consider the: Natural Numbers. Natural Numbers X Natural Numbers. Natural Numbers X Natural Numbers X Natural Numbers ... Now take the union of all ...
48 views

### Finding the $\log$ of a matrix by contour integration

My teacher presented this way of determining the logarithm of a matrix $\Omega$ in class today: $$\log \Omega = \frac1{2\pi i}\oint_{\Gamma} (\zeta I - \Omega)^{-1} \log \zeta \,d\zeta.$$ Does ...
57 views

### Why does the limit of a subsequence lie in a closed set?

I am reading through a proof where I encountered the following: Let $K_0\subset K$, where $K$ is a compact metric space, and $K_0$ is closed. If $x_{n}\subset K$ has a convergent subsequence, ...
306 views

### Paths on a Rubik's cube

Here is the Question i'm trying to solve: An ant is initially positioned at one corner of the Rubik's cube and wishes to go to the farthest corner of the block from its initial position. Assuming ...
28 views

59 views

### Convergence of an alternating series (from exam Q)

State whether $\sum_{n=1}^{\infty} a_n$ converges or diverges and prove your result: iv) $$a_{2^n + r} = (-1)^n(2^n + r)^{-1}\text{ for }0 \leq r \leq 2^n -1\text{ and }n \geq 0$$ I tried ...
97 views

### GAP-Character table

I the following link I have found the character table of $S_8$ which is computed with the program GAP. http://groupprops.subwiki.org/wiki/Linear_representation_theory_of_symmetric_groups But I ...
114 views

### On $C_c^{\infty}$ being dense in $L^p$

We had the theorem about $C_c^{\infty}$ being dense in $L^p$, which, as I understand, means that if we already have an $L^p$ function, there is a $C_c^{\infty}$ function arbitrary close to it with ...
### Find the abs max and min values of $f(x,y,z)=x-2y+3z$ on the ellipsoid $\frac{x^2}{6}+\frac{y^2}{3}+ \frac{z^2}{2}-1=0$
How do I do the following Find the abs max and min values of $f(x,y,z)=x-2y+3z$ on the ellipsoid $\frac{x^2}{6}+\frac{y^2}{3}+ \frac{z^2}{2}-1=0$ I found my two gradient ofcourse. $<1,-2,3>$ ...