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1500 questions
176
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:
$$\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x…
Vladimir Reshetnikov
- 32,650
176
votes
6 answers
Deleting any digit yields a prime... is there a name for this?
My son likes his grilled cheese sandwich cut into various numbers, the number depends on his mood. His mother won't indulge his requests, but I often will. Here is the day he wanted 100:
But today he wanted the prime 719, which I obliged. When…
Fixee
- 11,760
176
votes
13 answers
What is the best book to learn probability?
Question is quite straight... I'm not very good in this subject but need to understand at a good level.
Eduardo Xavier
- 148
175
votes
7 answers
What is the difference between Fourier series and Fourier transformation?
What's the difference between Fourier transformations and Fourier Series?
Are they the same, where a transformation is just used when its applied (i.e. not used in pure mathematics)?
Dean
- 1,941
174
votes
6 answers
Why is the Penrose triangle "impossible"?
I remember seeing this shape as a kid in school and at that time it was pretty obvious to me that it was "impossible". Now I looked at it again and I can't see why it is impossible anymore.. Why can't an object like the one represented in the…
user736690
174
votes
17 answers
Intuitive explanation of entropy
I have bumped many times into entropy, but it has never been clear for me why we use this formula:
If $X$ is random variable then its entropy is:
$$H(X) = -\displaystyle\sum_{x} p(x)\log p(x).$$
Why are we using this formula? Where did this formula…
jjepsuomi
- 8,979
174
votes
19 answers
Mathematical ideas that took long to define rigorously
It often happens in mathematics that the answer to a problem is "known" long before anybody knows how to prove it. (Some examples of contemporary interest are among the Millennium Prize problems: E.g. Yang-Mills existence is widely believed to be…
Yly
- 15,791
174
votes
1 answer
Is there a categorical definition of submetry?
(Updated to include effective epimorphism.)
This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use category theory.
Consider the category CpltMet in…
user31373
173
votes
15 answers
General formula for solving quartic (degree $4$) equations
There is a general formula for solving quadratic equations, namely the Quadratic Formula, or the Sridharacharya Formula:
$$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
- 3,621
172
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…
user8465
- 1,823
172
votes
34 answers
Can you provide me historical examples of pure mathematics becoming "useful"?
I am trying to think/know about something, but I don't know if my base premise is plausible. Here we go.
Sometimes when I'm talking with people about pure mathematics, they usually dismiss it because it has no practical utility, but I guess that…
Red Banana
- 24,885
- 21
- 101
- 207
172
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
- 36,898
- 28
- 168
- 346
171
votes
1 answer
Rational roots of polynomials
Can one construct a sequence $(a_k)_{k\geqslant 0}$ of rational numbers such that, for every positive integer $n$ the polynomial $a_nX^n+a_{n-1}X^{n-1}+\cdots +a_0$ has exactly $n$ distinct rational roots ?
If we cannot construct it explicitly, can…
user84673
- 2,057
171
votes
1 answer
Is there a homology theory that counts connected components of a space?
It is well-known that the generators of the zeroth singular homology group $H_0(X)$ of a space $X$ correspond to the path components of $X$.
I have recently learned that for Čech homology the corresponding statement would be that $\check{H}_0(X)$ is…
Dejan Govc
- 17,347
171
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
- 2,154
- 4
- 18
- 19