Articles of soft question

Intuitive idea of adapted processes with respect to “information”

My text says the following, “If the process $X_{n}$ is adapted to the filtration $\mathcal{F}_{n}$ the value $X_{n}(\omega)$ is known to us at time $n$.” This sounds wierd to me since it is all still random stuff, nothing is said about a realisation of $X_{n}$ up until $n$ for instance. At time $n$ we ONLY […]

Can we start studying number theory before abstract algebra?

I have started studying elementary number theory independently. I was wondering that is it necessary to first finish abstract algebra

What can we say about the prime factors of $​^{10}10+23$?

In a video on ultrafinitism I saw a claim that the number $​^{10}10+23$ does not have prime factorization. While I don’t accept the premise of ultrafinitism, I got curious, what can we say about the prime factors of this number? $​^{10}10$ refers to the hyperoperation tetration. In other words, the number is equal to $10^{10^{10^{10^{10^{10^{10^{10^{10^{10}}}}}}}}}$, […]

Is mathematics considered a science

Possible Duplicate: what is the definition of Mathematics ? I would like to know if mathematics is considered a science? I’ve searched the internet and asked many people for insight to no avail. I thought science was based on observation, experimentation and falsification. I don’t understand how mathematics can satisfy all the above. Especially, when […]

What is a real-valued random variable?

This question arose when someone (and surely not the least!) commented that something like $\left(X\mid Y=y\right)$ , i.e. $X$ under condition $Y=y$, where $X$ and $Y$ are real-valued random variables and $P\left\{ Y=y\right\}>0 $, is not a well defined random variable. To see if he is right I need the definition of real-valued random variable. […]

Why do you need to specify that a coin is fair?

This sounds like the kind of etherial question that generally gets dropped from stack exchange sites, but I don’t know of a better venue to ask so I’m hoping this question will help other folks with a similar dilemma. I recently posted this question: Probability of selecting a combination of two variables. I have a […]

What Does the Associative Property Mean Intuitively Across All Notational Schemes?

You can find descriptions of associativity as intuitively meaning that the order of operations performed does not matter, e. g. such as that of Wikipedia. However, if you write what associativity means in terms of a formula in either prefix notation, that is that for some binary operation X, XXxyz=XxXyz, or suffix notation xyzXX=xyXzX, the […]

summing series using circles inside curves

After watching the infinity elephants video http://www.youtube.com/watch?v=DK5Z709J2eo and seeing how a geometric series could be represented by drawing a circle between a pair of lines, then the largest circle that would fit in the gap etc. I wondered about using curves instead of lines to find other types of series. For example, circles with radii […]

Are there ways of telling whether or not an integral will have a closed expression?

I was looking at some of the great integral posts that have graced this website and I am wondering if there is a way to tell whether or not a definite integral will have a closed form. If there is no “one-size-fits-all” criterion, are there any general rules? (e.g. integrals of polynomial functions will have […]

Can I research in complex analysis, PDE and differential geometry without exposure to mathematical physics?

I love mathematics, but physics is far away from my interest. I see that recent mathematics research is strongly connected to mathematical physics which is something doesn’t interest me! I love mathematical entities and structures, stuff like relativity and quantum mechanics are not even close to my area of interest, so I wonder if I […]