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Feb
1
asked What is the complexity of the arithmetic operations in base $b$?
Feb
1
comment Is this iteration involving primes known?
You have the iteration $$x_{n+1} = x_n + p_n + 1$$ Where $(p_n)$ is some sequence of primes (this is what you call $y$ in your question). Therefore $$x_n = x_0 + n + \sum_{k=0}^n p_n$$ You're asking if for any/for all prime $x_0$, we can find a sequence of primes $(p_n)$ such that every $x_n$ is prime.
Jan
26
comment Find depth of three node tree
Does every node of the tree have three children?
Jan
25
comment How to find irrational numbers between rationals. (And is my method correct?)
Yes, your method is correct. Basically, given $a<b$, we know that if $a^2<r<b^2$, then $a<\sqrt r < b$, so the problem is to show that there is always at least one $r$ which is not a rational square, which you've done in probably the simplest way possible. I prefer your construction to the ones in the answers, which are only "simpler" if you don't think about the amount of machinery that has to go into defining what it means to multiply irrationals by rationals, what it means for one real number to be greater than another, etc. Your method is simpler and more direct.
Jan
22
comment Distributive Law and how it works
Every question of the form "Why X?" should be answered by "Why do you expect not-X?". Why should it be the case that the relationship between addition and multiplication is symmetric in this way? Are you saying that just because A bears relationship R to B, that therefore it's reasonable to expect that B should bear relationship R to A? Do you find it surprising that cats eat mice, and yet mice don't eat cats?
Jan
22
comment ODE Maximal solution
It indeed somewhat trivial, which is why the proofs of this theorem are never all that complicated, conceptually. Formally they may require a bit of set-theoretic machinery, but that's to be expected when you're working with infinite sets of functions like this. These proofs by Zorn's Lemma or what have you are just giving names to your intuitions about why this is "obviously" true. Similarly, it seems "trivial" that from an infinite collection of sets you can construct a set which chooses one element from each set, but the formal details can lead you down a very deep rabbit hole.
Jan
15
comment convergence of continued nested function
@user55622 That isn't a proof of convergence, though.
Jan
15
answered convergence of continued nested function
Jan
15
accepted How can I prove $x_{n+1} = e^{-x_n}$ is convergent?
Jan
14
answered How do we deduce that it is the zero function?
Jan
14
comment Book for studying Linear Algebra
Possible duplicate of What is a good book to study linear algebra?
Jan
12
comment Limit of real logarithm
The kind of intuition you need to nurture with these problems is the ability to unpack the expressions one function at a time, and to imagine zooming in on the limit point. So $\cos(x)$, that's gonna be tending towards $1$. Now zoom into the graph of the logarithm at $x=1$. It looks like a straight line with slope one, in other words, for $x$ close to one, we basically have $\ln x = x - 1$. So the numerator's pretty much just $\cos x - 1$. And so on.
Jan
12
comment whether the set is dense?
Don't you see that the integral of a function is a continuous mapping from functions to numbers? That is, if you have a function and you make a "small" change to the function (pushing its graph up or down in once place by a tiny amount), it's integral shouldn't change too much?
Jan
11
comment After removing any part the rest can be split evenly. Consequences?
Surely it's the case that in general, if $(b_i)$ is a basis for $\ker A$ over $\mathbb Q$, where $A$ has rational entries, then $(b_i)$ is also a basis for $\ker A$ over $\mathbb R$, no? In which case Claim 2 immediately implies Claim 3.
Jan
11
answered logarithms properties
Jan
11
comment How to find out if a number is a hundred or thousand?
Just use a while loop to continually divide by ten until the number because less than 1, and count how many divisions it took.
Jan
10
comment Why Riemann hypothesis and not Riemann's conjecture
Probably "Riemann's hypothesis" is just older terminology.
Jan
10
comment Showing that an iterative method solves a particular system
@Variable The idea is to show that when $\theta$ is chosen to make it converge for all $x_0$, it converges to a solution to $Ax=b$.
Jan
9
comment Showing that an iterative method solves a particular system
@Variable An old exercise sheet from a numerical analysis class which covered the Jacobi, Gauss-Seidel, and SOR methods. Maybe we really were just intended to see that the solution to $Ax=b$ is $(1, 1)$ and then plug that into the fixed point equation.
Jan
9
comment Proof that $A^{-1}=adj(A)/|A|$
@SanchayanDutta As you do more and more of your own mathematics, this sort of thing bothers you less and less. When working on a difficult problem, very often the solution will just pop into your head after a long time thinking about it, seemingly out of the blue. Other times you find a solution in a more natural way, but then a radical and more elegant reformulation of it comes into your head out of nowhere. If that can happen to you, just imagine the kinds of crazy ideas that would appear when the best minds in history spend their lives working on problems over thousands of years.