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1500 questions
64
votes
3 answers
Is $\lfloor n!/e\rfloor$ always even for $n\in\mathbb N$?
I checked several thousand natural numbers and observed that $\lfloor n!/e\rfloor$ seems to always be an even number. Is it indeed true for all $n\in\mathbb N$? How can we prove it?
Are there any positive irrational numbers $a\ne e$ such that…
july
- 1
64
votes
3 answers
What is difference between a ring and a field?
The ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition.
A field can be thought of as two groups with extra distributivity law.
A ring is more complex: with…
zinking
- 1,119
64
votes
1 answer
Lebesgue measurable set that is not a Borel measurable set
exact duplicate of Lebesgue measurable but not Borel measurable
BUT! can you please translate Miguel's answer and expand it with a formal proof? I'm totally stuck...
In short: Is there a Lebesgue measurable set that is not Borel measurable?
They…
example
- 2,185
64
votes
1 answer
Is there a "greater than about" symbol?
To indicate approximate equality, one can use ≃, ≅, ~, ♎, or ≒.
I need to indicate an approximate inequality. Specifically, I know A is greater than a quantity of approximately B.
Is there a way to succinctly express this mathematically?
foobarbecue
- 753
64
votes
12 answers
$\lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite
How do I prove that $ \displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
Breton
- 1,698
64
votes
1 answer
Direct Sum vs. Direct Product vs. Tensor Product
There are a lot of questions like this all over the site, but I cannot find one that resolved my confusion- what are the formal definitions of direct sums, direct products, and tensor products (in the most general sense), and how are they different?
user247773
64
votes
4 answers
Closed Form for $~\int_0^1\frac{\text{arctanh }x}{\tan\left(\frac\pi2~x\right)}~dx$
Does $$~\displaystyle{\int}_0^1\frac{\text{arctanh }x}{\tan\left(\dfrac\pi2~x\right)}~dx~\simeq~0.4883854771179872995286585433480\ldots~$$ possess a closed form expression ?
This recent post, in conjunction with my age-old interest in…
Lucian
- 49,312
64
votes
13 answers
I have learned that 1/0 is infinity, why isn't it minus infinity?
My brother was teaching me the basics of mathematics and we had some confusion about the positive and negative behavior of Zero. After reading a few post on this we came to know that it depends on the context of its use.
Why do we take 1/0 as…
Rishav Mehta
- 761
64
votes
3 answers
How do I exactly project a vector onto a subspace?
I am trying to understand how - exactly - I go about projecting a vector onto a subspace.
Now, I know enough about linear algebra to know about projections, dot products, spans, etc etc, so I am not sure if I am reading too much into this, or if…
Spacey
- 1,394
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64
votes
17 answers
Is the Law of Large Numbers empirically proven?
Does this reflect the real world and what is the empirical evidence behind this?
Layman here so please avoid abstract math in your response.
The Law of Large Numbers states that the average of the results from multiple trials will tend to converge…
vantage5353
- 767
63
votes
2 answers
What do eigenvalues have to do with pictures?
I am trying to write a program that will perform OCR on a mobile phone, and I recently encountered this article :
Can someone explain this to me ?
Patryk
- 751
63
votes
10 answers
What is the arithmetic mean of no numbers?
I have two programs that both behave nearly identically: they both take in any numbers you give them and can tell you the arithmetic mean and how many numbers were given. However, when you don't give them any numbers, one says the arithmetic mean is…
Ky -
- 1,346
63
votes
15 answers
Why learn to solve differential equations when computers can do it?
I'm getting started learning engineering math. I'm really interested in physics especially quantum mechanics, and I'm coming from a strong CS background.
One question is haunting me.
Why do I need to learn to do complex math operations on paper…
user60462
- 843
63
votes
7 answers
How to explain to the layperson what mathematics is, why it's important, and why it's interesting
A mathematician walks into a party.
No, this is not the beginning of another joke, nor of a graph theory problem, but rather the beginning of a frequent and often frustrating real-life situation. Somebody asks you:
"So, what do you do?"
What do…
Bruno Joyal
- 55,975
63
votes
9 answers
Does it ever make sense NOT to go to the most prestigious graduate school you can get into?
I'm a senior undergrad at a top-ish(say, top 15) math school. I'm a solid, not stellar, student. This year I'm taking the qualifying exam grad courses in algebra and analysis and have been taken aback by the "pressure cooker" atmosphere among grad…
anonymous
- 191