Mathematics Stack Exchange
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1500 questions
65
votes
3 answers

Optimal strategy for cutting a sausage?

You are a student, assigned to work in the cafeteria today, and it is your duty to divide the available food between all students. The food today is a sausage of 1m length, and you need to cut it into as many pieces as students come for lunch,…
Stenzel
  • 633
65
votes
7 answers

How to find a basis for the intersection of two vector spaces in $\mathbb{R}^n$?

What is the general way of finding the basis for intersection of two vector spaces in $\mathbb{R}^n$? Suppose I'm given the bases of two vector spaces U and W: $$ \mathrm{Base}(U)= \left\{ \left(1,1,0,-1\right), \left(0,1,3,1\right) \right\} $$ $$…
Cu7l4ss
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65
votes
6 answers

Create unique number from 2 numbers

is there some way to create unique number from 2 positive integer numbers? Result must be unique even for these pairs: 2 and 30, 1 and 15, 4 and 60. In general, if I take 2 random numbers result must be unique(or with very high probability…
drizzt
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65
votes
3 answers

Functions that are Riemann integrable but not Lebesgue integrable

I know there are functions which are Riemann integrable but not Lebesgue integrable, for instance, $$\int_{\mathbb{R}} \frac{\sin(x)}{x} \mathrm{d}x$$ Is Riemann integrable and it is easily shown that its value is $\pi $, nevertheless, it is not…
user438666
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65
votes
16 answers

What exactly is calculus?

I've researched this topic a lot, but couldn't find a proper answer to this, and I can't wait a year to learn it at school, so my question is: What exactly is calculus? I know who invented it, the Leibniz controversy, etc., but I'm not exactly…
Hugh Chalmers
  • 947
65
votes
9 answers

Riemann hypothesis: is Bender-Brody-Müller Hamiltonian a new line of attack?

There is a beautiful paper in Physical Review Letters [PRL 118, 130201 (2017), DOI:10.1103/PhysRevLett.118.130201] by Carl Bender, Dorje Brody, and Markus Müller (BBM) on a Hamiltonian approach to the Riemann Hypothesis. The paper is surprisingly…
Slava Kashcheyevs
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65
votes
1 answer

How to find a total order with constrained comparisons

There are $25$ horses with different speeds. My goal is to rank all of them, by using only runs with $5$ horses, and taking partial rankings. How many runs do I need, at minimum, to complete my task? As a partial answer, I know that is possible to…
Jack D'Aurizio
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65
votes
9 answers

Linear Algebra Versus Functional Analysis

As it is mentioned in the answer by Sheldon Axler in this post, we usually restrict linear algebra to finite dimensional linear spaces and study the infinite dimensional ones in functional analysis. I am wondering that what are those parts of the…
Hosein Rahnama
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65
votes
7 answers

Why is the Continuum Hypothesis (not) true?

I'm making my way through Thomas W Hungerfords's seminal text "Abstract Algebra 2nd Edition w/ Sets, Logics and Categories" where he makes the statement that the Continuum Hypothesis (There does not exist a set with a cardinality less than the reals…
zetavolt
  • 883
65
votes
3 answers

Does multiplying all a number's roots together give a product of infinity?

This is a recreational mathematics question that I thought up, and I can't see if the answer has been addressed either. Take a positive, real number greater than 1, and multiply all its roots together. The square root, multiplied by the cube root,…
glowing-fish
  • 623
65
votes
3 answers

Motivation to understand double dual space

I am helping my brother with Linear Algebra. I am not able to motivate him to understand what double dual space is. Is there a nice way of explaining the concept? Thanks for your advices, examples and theories.
Theorem
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65
votes
6 answers

What does strength refer to in mathematics?

My professors are always saying, "This theorem is strong" or "There is a way to make a much stronger version of this result" or things like that. In my mind, a strong theorem is able to tell you a lot of important information about something, but…
Zachary F
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65
votes
1 answer

Generating correlated random numbers: Why does Cholesky decomposition work?

Let's say I want to generate correlated random variables. I understand that I can use Cholesky decomposition of the correlation matrix to obtain the correlated values. If $C$ is the correlation matrix, then we can do the cholesky…
Flux Capacitor
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65
votes
3 answers

Algebraic Topology Challenge: Homology of an Infinite Wedge of Spheres

So the following comes to me from an old algebraic topology final that got the best of me. I wasn't able to prove it due to a lack of technical confidence, and my topology has only deteriorated since then. But, I'm hoping maybe someone can figure…
squiggles
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65
votes
6 answers

If we randomly select 25 integers between 1 and 100, how many consecutive integers should we expect?

Question: Suppose we have one hundred seats, numbered 1 through 100. We randomly select 25 of these seats. What is the expected number of selected pairs of seats that are consecutive? (To clarify: we would count two consecutive selected seats as a…
David Steinberg
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