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2d
comment Which is the largest power of natural number that can be evaluated by computers?
@parkhyeyoo: Then just add another disk to increase the storage. However at some point in time we will run out of material to build disks from. However I just suggest to switch to a different representation: Just store numbers as triple (base, mantissa, exponent), and even much larger powers can be stored :-)
2d
comment Which is the largest power of natural number that can be evaluated by computers?
Since there is no need to store the number in RAM, and multi-terabyte disks are common these days, you can even go much farther than that (note that to do multiplications, you only need to have a small part of the number in RAM at any point in time).
2d
comment Which is the largest power of natural number that can be evaluated by computers?
But, in base $10$, a power of $10$ is an 1 followed by a sequence of 0s. So one could argue that the highest power of ten the following code can calculate is limited only by the time until the program is stopped or the computer or the output fails: putchar('1'); while(1) putchar('0');
Jun
18
comment Why are we defining the norms on certain vector spaces the way they are?
You're right; I've removed that norm (it's a vector norm, but not a matrix norm). Your alternative is the operator-1-norm, therefore it's not a good idea to call it an $\infty$ norm.
Jun
8
comment Open and Closed Sets?
Constant functions are entire functions. But for constant functions and non-empty $S$, $K$ contains a single element, and is thus closed in $\mathbb R$ and discrete in $\mathbb C$, but open neither in $\mathbb R$ nor $\mathbb C$. Moreover, the empty set is an open bounded subset of $\mathbb C$, and for $S=\emptyset$ also $K=\emptyset$, which is both open and closed in $\mathbb R$, and open and discrete in $\mathbb C$. Your answer is correct in the case that $f$ is not constant and $S$ is not empty.
Jun
8
comment What is the smallest unknown natural number?
Obviously it's the answer to the question "What is the smallest unknown natural number?" The answer to this question is unknown, by definition, and it is known to be a natural number, also by definition. And while in general, for two unknown numbers you cannot say which one is smaller, for this specific problem we know, again by definition, that it is the smallest.
May
16
comment Properties of Intersection of Sets
@GregoryGrant: I know what $2^A$ means, basically because I learned it here on math.SE. Before that, I didn't (but I definitely knew what a power set is, and knew the notation $P(A)$ for it).
May
16
comment Properties of Intersection of Sets
@GregoryGrant: Then you would think wrong.
May
16
comment Properties of Intersection of Sets
@GregoryGrant: Not necessarily more people, just another set of people.
May
15
comment What function satisfies $f(x)+f(−x)=f(x^2)$?
Actually the question doesn't specify the domain. So also the function $f:\{-1,1\}\to\{0,1\}$ with $f(-1)=0$ and $f(1)=1$ satisfies the given requirements.
May
15
comment Notation problem in integration: write $dx$ or ${\mathrm{d}}x$?
@ChristianBlatter: What's wrong with $\sin n\pi$? I'm no mathematician, and I don't see anything wrong with it.
May
14
comment What should be the intuition when working with compactness?
This is a link-only answer. Link-only answers are frowned upon because the link can (and indeed, over sufficient time is almost guaranteed to) become invalid. For example, the file you link to is obviously on the personal homepage of someone at UCLA with the local login name "tao" (the form of the URL is a dead giveaway for that). So what do you think happens to this link if that person some day moves to another university (or retires), and his user account on UCLA's servers gets deleted?
May
14
comment Prove that $\partial A$ is a cutset of connected $X$ if $\operatorname{Int}(A)$ and $\operatorname{Int}(X - A)$ are nonempty
Hint: $\partial A = \partial(X-A)$
May
6
comment Probability of winning the lottery the more you play it?
"If you play once, you have a 50% chance of winning. If you play twice, you have doubled your chance of winning it once." So if I play twice, I have a chance of $2\times 50\% = 100\%$ of winning? That doesn't seem right.
May
6
comment Simplest proof that some number is transcendental?
Transcendental numbers are "more common" than algebraic numbers, because almost all numbers are transcendental. That's because there are only countably many algebraic numbers, but uncountably many reals.
May
6
comment How to determine whether a polytope is self-tessellating?
One obvious way to create self-tessellating objects of higher dimension is extrusion of self-tessellating objects of lower dimension.
May
4
comment How to prove $\mathcal{L}^2[(0,1)]$ is a Hilbert Space
Can you please give an explicit example for the non-positive definiteness of the inner product?
May
3
comment Can we have a one-one function from [0,1] to the set of irrational numbers?
@user21820: What exactly do you think I have to justify? That $\pi$ is transcendental is a well-known fact; see e.g. Wikipedia.
May
2
comment Is $f:\mathbb{Q^*} \rightarrow \mathbb{Q}$ by $f(\frac{a}{b}) = \frac{\max{(a,b)}}{\min{(a,b)}}$ a function?
What are the possible values of $a$ and $b$? In particular, can $b$ be negative?
May
2
comment Is there fundamental goal of mathematics?
Does this qualify?