Mathematics

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1500 questions
74
votes
5 answers

Integers $n$ such that $i(i+1)(i+2) \cdots (i+n)$ is real or pure imaginary

A couple of days ago I happened to come across [1], where the curious fact that $i(i-1)(i-2)(i-3)=-10$ appears ($i$ is the imaginary unit). This led me to the following question: Problem 1: Is $3$ the only positive integer value of $n$ such that…
Dave L. Renfro
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74
votes
6 answers

Probability for the length of the longest run in $n$ Bernoulli trials

Suppose a biased coin (probability of head being $p$) was flipped $n$ times. I would like to find the probability that the length of the longest run of heads, say $\ell_n$, exceeds a given number $m$, i.e. $\mathbb{P}(\ell_n > m)$. It suffices to…
Sasha
  • 71,954
74
votes
10 answers

Proving that $1$- and $2D$ simple symmetric random walks return to the origin with probability $1$

How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the origin with probability $1$ ? Edit : note that while…
Isaac
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74
votes
4 answers

Finite Groups with exactly $n$ conjugacy classes $(n=2,3,...)$

I am looking to classify (up to isomorphism) those finite groups $G$ with exactly 2 conjugacy classes. If $G$ is abelian, then each element forms its own conjugacy class, so only the cyclic group of order 2 works here. If $G$ is not abelian, I am…
RHP
  • 2,663
74
votes
3 answers

Distance/Similarity between two matrices

I'm in the process of writing an application which identifies the closest matrix from a set of square matrices $M$ to a given square matrix $A$. The closest can be defined as the most similar. I think finding the distance between two given matrices…
Synex
  • 1,005
74
votes
2 answers

Geometric intuition for the tensor product of vector spaces

First of all, I am very comfortable with the tensor product of vector spaces. I am also very familiar with the well-known generalizations, in particular the theory of monoidal categories. I have gained quite some intuition for tensor products and…
Martin Brandenburg
  • 183,887
74
votes
13 answers

What would have been our number system if humans had more than 10 fingers? Try to solve this puzzle.

Try to solve this puzzle: The first expedition to Mars found only the ruins of a civilization. From the artifacts and pictures, the explorers deduced that the creatures who produced this civilization were four-legged beings with a tentatcle that…
Vishnu Vivek
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74
votes
4 answers

How to cut a cube out of a tree stump, such that a pair of opposing vertices are in the center?

I saw this picture of a cube cut out of a tree stump. I've been trying to craft the same thing out of a tree stump, but I found it hard to figure out how to do it. One of the opposing vertices pair is on the center of the tree stump: I've been…
Aaron Cheuk
  • 711
74
votes
3 answers

How does a calculator calculate the sine, cosine, tangent using just a number?

Sine $\theta$ = opposite/hypotenuse Cosine $\theta$ = adjacent/hypotenuse Tangent $\theta$ = opposite/adjacent In order to calculate the sine or the cosine or the tangent I need to know $3$ sides of a right triangle. $2$ for each corresponding…
themhz
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74
votes
4 answers

Why it is important to write a function as sum of even and odd functions?

For the function $f(x)$ we can write it as sum of even and odd functions: $$f(x)=\underbrace{\frac{f(x)+f(-x)}{2}}_{\text{Even}}+\underbrace{\frac{f(x)-f(-x)}{2}}_{\text{Odd}}$$ My question is why it is important for us to write a function as sum of…
User
  • 8,563
74
votes
2 answers

Are there any valid continuous Sudoku grids?

A standard Sudoku is a $9\times 9$ grid filled with digits such that every row, column, and $3\times 3$ box contains all the integers from $1$ to $9$ . I am thinking about a generalization of Sudoku which I call "continuous Sudoku", which consists…
ZKG
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74
votes
2 answers

What is the difference between the Jacobian, Hessian and the Gradient?

I know there is a lot of topic regarding this on the internet, and trust me, I've googled it. But things are getting more and more confused for me. From my understanding, The gradient is the slope of the most rapid descent. Modifying your position…
Pluviophile
  • 991
74
votes
5 answers

Intuition for the Importance of Modular Forms

I am learning about modular forms for the first time this term and am just starting to wrap my head around what might be the big picture of things. I was wondering if the following interpretation of why modular forms are important is correct a)…
Alex Youcis
  • 57,478
74
votes
10 answers

List of problem books in undergraduate and graduate mathematics

I would like to know some good problem books in various branches of undergraduate and graduate mathematics like group theory, galois theory, commutative algebra, real analysis, complex analysis, topology etc. The books should contain solution to…
Mohan
  • 15,664
74
votes
5 answers

Show that $\int_{0}^{\pi/2}\frac {\log^2\sin x\log^2\cos x}{\cos x\sin x}\mathrm{d}x=\frac14\left( 2\zeta (5)-\zeta(2)\zeta (3)\right)$

Show that : $$ \int_{0}^{\Large\frac\pi2} {\ln^{2}\left(\vphantom{\large A}\cos\left(x\right)\right) \ln^{2}\left(\vphantom{\large A}\sin\left(x\right)\right) \over \cos\left(x\right)\sin\left(x\right)}\,{\rm d}x ={1 \over 4}\,…
Ryan
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