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1500 questions
177
votes
6 answers
A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$
A friend of mine sent me a math contest problem that I am not able to solve (he does not know a solution either). So, I thought I might ask you for help. Prove: $$\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x…
Vladimir Reshetnikov
- 32,880
176
votes
17 answers
Intuitive explanation of entropy
I have bumped many times into entropy, but it has never been clear for me why we use this formula: If $X$ is random variable then its entropy is: $$H(X) = -\displaystyle\sum_{x} p(x)\log p(x).$$ Why are we using this formula? Where did this formula…
jjepsuomi
- 9,049
176
votes
6 answers
Deleting any digit yields a prime... is there a name for this?
My son likes his grilled cheese sandwich cut into various numbers, the number depends on his mood. His mother won't indulge his requests, but I often will. Here is the day he wanted 100: But today he wanted the prime 719, which I obliged. When…
Fixee
- 11,840
176
votes
19 answers
Mathematical ideas that took long to define rigorously
It often happens in mathematics that the answer to a problem is "known" long before anybody knows how to prove it. (Some examples of contemporary interest are among the Millennium Prize problems: E.g. Yang-Mills existence is widely believed to be…
Yly
- 16,181
176
votes
7 answers
What is the difference between Fourier series and Fourier transformation?
What's the difference between Fourier transformations and Fourier Series? Are they the same, where a transformation is just used when its applied (i.e. not used in pure mathematics)?
Dean
- 1,951
175
votes
9 answers
Why do people use "it is easy to prove"?
Math is not generally what I am doing, but I have to read some literature and articles in dynamic systems and complexity theory. What I noticed is that authors tend to use (quite frequently) the phrase "it is easy to see/prove/verify/..." in the…
oleksii
- 903
175
votes
6 answers
Why is the Penrose triangle "impossible"?
I remember seeing this shape as a kid in school and at that time it was pretty obvious to me that it was "impossible". Now I looked at it again and I can't see why it is impossible anymore.. Why can't an object like the one represented in the…
user736690
174
votes
15 answers
General formula for solving quartic (degree $4$) equations
There is a general formula for solving quadratic equations, namely the Quadratic Formula, or the Sridharacharya Formula: $$x = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a } $$ For cubic equations of the form $ax^3+bx^2+cx+d=0$ , there is a set of three…
John Gietzen
- 3,631
174
votes
3 answers
Examples of bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$
Could any one give an example of a bijective map from $\mathbb{R}^3\rightarrow \mathbb{R}$? Thank you.
Myshkin
- 37,144
- 28
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- 351
174
votes
1 answer
Is there a categorical definition of submetry?
(Updated to include effective epimorphism.) This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use category theory. Consider the category CpltMet in…
user31373
173
votes
4 answers
The square roots of different primes are linearly independent over the field of rationals
I need to find a way of proving that the square roots of a finite set of different primes are linearly independent over the field of rationals. I've tried to solve the problem using elementary algebra and also using the theory of field extensions,…
user8465
- 1,833
173
votes
1 answer
How to determine with certainty that a function has no elementary antiderivative?
Given an expression such as $f(x) = x^x$, is it possible to provide a thorough and rigorous proof that there is no function $F(x)$ (expressible in terms of known algebraic and transcendental functions) such that $ \frac{d}{dx}F(x) = f(x)$? In other…
hesson
- 2,174
173
votes
1 answer
What's the significance of Tate's thesis?
I've just sat through several lectures that proved most of the results in Tate's thesis: the self-duality of the adeles, the construction of "zeta functions" by integration, and the proof of the functional equation. However, while I was able to…
Akhil Mathew
- 32,550
172
votes
11 answers
Do we know if there exist true mathematical statements that can not be proven?
Given the set of standard axioms (I'm not asking for proof of those), do we know for sure that a proof exists for all unproven theorems? For example, I believe the Goldbach Conjecture is not proven even though we "consider" it true. Phrased another…
Jeremy
- 1,611
172
votes
7 answers
Find a real function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$?
I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success: Find a function $f: \mathbb{R} \to \mathbb{R}$ satisfying $f(f(x)) = -x$ for…
Gamma Function
- 5,802