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1500 questions
69
votes
3 answers

Path-connected and locally connected space that is not locally path-connected

I'm trying to classify the various topological concepts about connectedness. According to 3 assertions ((Locally) path-connectedness implies (locally) connectedness. Connectedness together with locally path-connectedness implies…
Qian
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69
votes
2 answers

Why is $\mathbb{Z}[\sqrt{-n}], n\ge 3$ not a UFD?

I'm considering the ring $\mathbb{Z}[\sqrt{-n}]$, where $n\ge 3$ and square free. I want to see why it's not a UFD. I defined a norm for the ring by $|a+b\sqrt{-n}|=a^2+nb^2$. Using this I was able to show that $2$, $\sqrt{-n}$ and $1+\sqrt{-n}$…
Danielle Intal
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69
votes
5 answers

Does every prime divide some Fibonacci number?

I am tring to show that $\forall a \in \Bbb P\; \exists n\in\Bbb N : a|F_n$, where $F$ is the fibonacci sequence defined as $\{F_n\}:F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2}$ $(n=2,3,...)$. How can I do this? Originally, I was trying to show that…
Chanhee Jeong
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69
votes
6 answers

Functions which are Continuous, but not Bicontinuous

What are some examples of functions which are continuous, but whose inverse is not continuous? nb: I changed the question after a few comments, so some of the below no longer make sense. Sorry.
isomorphismes
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69
votes
11 answers

What is the difference between only if and iff?

I have read this question. I am now stuck with the difference between "if and only if" and "only if". Please help me out. Thanks
user2857
69
votes
7 answers

$\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}}$ approximation

Is there any trick to evaluate this or this is an approximation, I mean I am not allowed to use calculator. $$\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}}$$
user2378
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69
votes
5 answers

Integral $\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx$

Is there a closed form for the integral $$\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx.$$ I do not have a strong reason to be sure it exists, but I would be very interested to see an approach to…
Frida Mauer
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69
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2 answers

Differential forms on fuzzy manifolds

This post will take a bit to set up properly, but it is an easy read (and most likely easy to answer); in any event, please bear with me. Question In the usual setting of open subsets of $\mathbb{R}^n$, differential forms are defined as…
Isaac Sledge
  • 699
69
votes
4 answers

Difference between NFA and DFA

In very simple terms please, all resources I'm finding are talking about tuples and stuff and I just need a simple explanation that I can remember easily because I keep getting them mixed up.
Ogen
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69
votes
4 answers

Can there be two distinct, continuous functions that are equal at all rationals?

Akhil showed that the Cardinality of set of real continuous functions is the same as the continuum, using as a step the observation that continuous functions that agree at rational points must agree everywhere, since the rationals are dense in the…
Charles Stewart
  • 4,597
69
votes
15 answers

Prove if $n^2$ is even, then $n$ is even.

I am just learning maths, and would like someone to verify my proof. Suppose $n$ is an integer, and that $n^2$ is even. If we add $n$ to $n^2$, we have $n^2 + n = n(n+1)$, and it follows that $n(n+1)$ is even. Since $n^2$ is even, $n$ is even. Is…
user79612
  • 575
69
votes
5 answers

Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$

Please help me to find a closed form for the following integral: $$\int_0^1\log\left(\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\right)\,{\mathrm d}x.$$ I was told it could be calculated in a closed form.
Laila Podlesny
  • 13,085
69
votes
4 answers

How much does symbolic integration mean to mathematics?

(Before reading, I apologize for my poor English ability.) I have enjoyed calculating some symbolic integrals as a hobby, and this has been one of the main source of my interest towards the vast world of mathematics. For instance, the integral…
Sangchul Lee
  • 167,468
69
votes
6 answers

"Gaps" or "holes" in rational number system

In Rudin's Principles of Mathematical Analysis 1.1, he first shows that there is no rational number $p$ with $p^2=2$. Then he creates two sets: $A$ is the set of all positive rationals $p$ such that $p^2<2$, and $B$ consists of all positive…
Larry
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69
votes
2 answers

Order of general- and special linear groups over finite fields.

Let $\mathbb{F}_3$ be the field with three elements. Let $n\geq 1$. How many elements do the following groups have? $\text{GL}_n(\mathbb{F}_3)$ $\text{SL}_n(\mathbb{F}_3)$ Here GL is the general linear group, the group of invertible n×n matrices,…
user9656
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