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1500 questions
124
votes
1 answer

Continuous projections on $\ell_1$ with norm $>1$

I was trying to find papers and articles about non-contractive continuous projections on $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case $S=\mathbb{N}$. I've found one quite general…
Norbert
  • 58,398
124
votes
11 answers

Closed form for $ \int_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$

I've been looking at $$\int\limits_0^\infty {\frac{{{x^n}}}{{1 + {x^m}}}dx }$$ It seems that it always evaluates in terms of $\sin X$ and $\pi$, where $X$ is to be determined. For example: $$\displaystyle \int\limits_0^\infty {\frac{{{x^1}}}{{1 +…
Pedro
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123
votes
2 answers

Making Friends around a Circular Table

I have $n$ people seated around a circular table, initially in arbitrary order. At each step, I choose two people and switch their seats. What is the minimum number of steps required such that every person has sat either to the right or to the left…
Vincent Tjeng
  • 3,324
123
votes
10 answers

Proof of Frullani's theorem

How can I prove the Theorem of Frullani? I did not even know all the hypothesis that $f$ must satisfy, but I think that this are Let $\,f:\left[ {0,\infty } \right) \to \mathbb R$ be a a continuously differentiable function such that $$ \mathop…
August
  • 3,633
123
votes
15 answers

Infiniteness of non-twin primes.

Well, we all know the twin prime conjecture. There are infinitely many primes $p$, such that $p+2$ is also prime. Well, I actually got asked in a discrete mathematics course, to prove that there are infinitely many primes $p$ such that $p + 2$ is…
Tomas Wolf
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123
votes
5 answers

What is the term for a factorial type operation, but with summation instead of products?

(Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems) I'm aware of Sigma notation, but is there a function/name for e.g. $$ 4 + 3 + 2 + 1 \longrightarrow 10…
barfoon
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123
votes
1 answer

Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?

A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization of the groups in which this property holds? If this question is too broad,…
Alexander Gruber
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123
votes
7 answers

What are the issues in modern set theory?

This is spurred by the comments to my answer here. I'm unfamiliar with set theory beyond Cohen's proof of the independence of the continuum hypothesis from ZFC. In particular, I haven't witnessed any real interaction between set-theoretic issues…
Paul VanKoughnett
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123
votes
15 answers

Why rationalize the denominator?

In grade school we learn to rationalize denominators of fractions when possible. We are taught that $\frac{\sqrt{2}}{2}$ is simpler than $\frac{1}{\sqrt{2}}$. An answer on this site says that "there is a bias against roots in the denominator of a…
Reinstate Monica
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122
votes
14 answers

Can you give an example of a complex math problem that is easy to solve?

I am working on a project presentation and would like to illustrate that it is often difficult or impossible to estimate how long a task would take. I’d like to make the point by presenting three math problems (proofs, probably) that on the surface…
Judy
  • 1,331
122
votes
8 answers

Lebesgue integral basics

I'm having trouble finding a good explanation of the Lebesgue integral. As per the definition, it is the expectation of a random variable. Then how does it model the area under the curve? Let's take for example a function $f(x) = x^2$. How do we…
user957
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122
votes
16 answers

Is 10 closer to infinity than 1?

This may be considered a philosophical question but is the number "10" closer to infinity than the number "1"?
termsofservice
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122
votes
0 answers

A question about divisibility of sum of two consecutive primes

I was curious about the sum of two consecutive primes and after proving that the sum for the odd primes always has at least 3 prime divisors, I came up with this question: Find the least natural number $k$ so that there will be only a finite number…
CODE
  • 5,119
122
votes
7 answers

Strategies for Effective Self-Study

I have a long-term goal of acquiring graduate-level knowledge in Analysis, Algebra and Geometry/Topology. Once that is achieved, I am interested in applying this knowledge to both pure and applied mathematics. In particular, I am interested in…
ItsNotObvious
  • 11,263
122
votes
15 answers

Expected Number of Coin Tosses to Get Five Consecutive Heads

A fair coin is tossed repeatedly until 5 consecutive heads occurs. What is the expected number of coin tosses?
leava_sinus
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