Mathematics

For finite dimensional vector spaces, $e^A$ is well defined because the taylor series converges pointwise via a proof involving operator norms if I recall my 2nd year linear algebra course...

Zophikel

What's the intuition behind Open Set

Secret

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Q: Intuition behind open set in topology

I am reading Munkres Topology Chapter 13, in which some examples of bases of topologies are given. One of the examples compares the two possible bases (a, b) and [a, b) on the real line. I understand that both (a,b) and [a,b) satisfy the definition of basis (intersection between two element has ...

But basically, it generalises from open balls in metric spaces

Zophikel

i'm reviewing the definition on: brilliant.org/wiki/open-sets

Secret

Feb 24 '15 at 15:39, by Chris's sis

@Semiclassical $$\int_0^1 x^{x^x} \ dx$$

Let me just do a quick check of my database...

Jun 1 at 16:04, by Secret

[Random] Experiment No. 1
$$\int_0^1 {}^n x dx$$

Jun 1 at 16:40, by LegionMammal978

@Secret Found an explicit integral formula, turns out it converges to $0.6465031992...$

This is consistent to:

Based on all these findings, I suspect...

$$\lim_{n\to \infty} \int_0^1 {}^nx dx= \int_0^1 \lim_{n\to \infty} {}^nx dx = \int_0^1 e^{-W(-\ln x)}dx$$

Will check these later, or perhaps @Waiting can have a look at these results as she had explored the whole class of tetration integrands before

31

Q: A new interesting pattern to $i↑↑n$ that looks cool (and $z↑↑x$ for $z\in\mathbb C,x\in\mathbb R$)

Many of you may recall "An obvious pattern to $i↑↑n$ that is eluding us all?", an old question of mine, and just recently, I saw this new question that poses a simple extension to tetration at non-integer values: $$a↑↑b=\begin{cases}a^b,&b\in[0,1]\\a^{a↑↑(b-1)},&b\in(1,+\infty)\\\log_a(a↑↑(b+1)),&...

algebra-precalculus complex-numbers recreational-mathematics tetration power-towers

Meanwhile @SimplyBeautifulArt and co. have found the complex case

Now the question is. How far we can push the domain of tetration integrands...

$$\int_V {}^nxd^kx ?$$

or possibly...

$$\int_0^1 \mathop{{\Large E}}_{i=1}^n (x+i)dx$$

Definition of Tetration operator here:

Jun 6 at 16:53, by Secret

Another interesting thing is the following problem: The above feature of right associativity of hyperoperators suggest there exists a finite ascending sequence $(a_i)$ and a finite descending sequence $(b_i)$ such that the following will be true:

$$\mathop{\Large{E}}^n_{i=1}a_i=\mathop{\Large{E}}^n_{i=1}b_i$$

where e.g. $\mathop{\Large{E}}^n_{i=1}a_i=a_1^{a_2^{⋰^{a_n}}}$ and is understood to be right associative.

Simply Beautiful Art

5:58 PM

@Secret I have been pinged?

Secret

yup

Seeing similar results from different people seemed to suggest to me that tetration expressions are easier to converge than lower hyoperations

at least within some interval. I am not sure if that's something in common with tetrations in general. Plan to investigate these later

MiKiDe

Hi everyone!
For a project in mathematics, I have to find a subject in maths (or computer science) related to these theme : "Environment: interface, interaction, homogeneity, break". I havn't find yet... What does that inspire you?
It has to be not too much complicated (I've just finished my first year in university).
It's a really important thing for me to find a subject, so if you can help me I will be forever grateful.

Secret

Sounds like either dynamical systems related maths subjects or biomathematics

Dodsy

is there a way to turn a physics question like "a child sitting in a tree 4 m up throws an apple at a veclocity of 5 m/s [35 degrees upwards]" into a calculus question?

Secret

first, you need to define your coordinates system (e.g. where you set x=0,t=0 etc.). Then you can express velocity in terms of the derivative of position. Using these you should get an ODE, which you can plug the IVP into it to solve

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$Mathematics$ Mathematics