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1500 questions
118
votes
8 answers
Understanding of the theorem that all norms are equivalent in finite dimensional vector spaces
The following is a well-known result in functional analysis: If the vector space $X$ is finite dimensional, all norms are equivalent. Here is the standard proof in one textbook. First, pick a norm for $X$, say $$\|x\|_1=\sum_{i=1}^n|\alpha_i|$$…
user9464
118
votes
6 answers
How often does it happen that the oldest person alive dies?
Today, we are brought the sad news that Europe's oldest woman died. A little over a week ago the oldest person in the U.S. unfortunately died. Yesterday, the Netherlands' oldest man died peacefully. The Gerontology Research Group keeps records:…
Řídící
- 3,288
118
votes
8 answers
Are there real world applications of finite group theory?
I would like to know whether there are examples where finite group theory can be directly applied to solve real world problems outside of mathematics. (Sufficiently applied mathematics such as cryptography, coding theory, or statistics still count.)…
Alexander Gruber
- 28,325
118
votes
5 answers
A multiplication algorithm found in a book by Paul Erdős: how does it work?
I am trying to understand the following problem from Erdős and Surányi's Topics in the theory of numbers (Springer), chapter 1 ("Divisibility, the Fundamental Theorem of Number Theory"): We can multiply two (positive integer) numbers together in the…
iadvd
- 8,969
118
votes
4 answers
$X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed
Let $X$ be a topological space. The diagonal of $X \times X$ is the subset $$D = \{(x,x)\in X\times X\mid x \in X\}.$$ Show that $X$ is Hausdorff if and only if $D$ is closed in $X \times X$. First, I tried to show that $X \times X \setminus D$ is…
Br09
- 2,320
117
votes
7 answers
Geometric interpretation of $\det(A^T) = \det(A)$
$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns , could someone give a geometric interpretation of the property?
dfg
- 4,121
117
votes
0 answers
What is the largest volume of a polyhedron whose skeleton has total length 1? Is it the regular triangular prism?
Say that the perimeter of a polyhedron is the sum of its edge lengths. What is the maximum volume of a polyhedron with a unit perimeter? A reasonable first guess would be the regular tetrahedron of side length $1/6$ , with volume…
RavenclawPrefect
- 19,199
- 3
- 43
- 112
117
votes
7 answers
Why is it important for a matrix to be square?
I am currently trying to self-study linear algebra. I've noticed that a lot of the definitions for terms (like eigenvectors, characteristic polynomials, determinants, and so on) require a square matrix instead of just any real-valued matrix. For…
Beneschan
- 1,113
117
votes
9 answers
What are Some Tricks to Remember Fatou's Lemma?
For a sequence of non-negative measurable functions $f_n$, Fatou's lemma is a statement about the inequality $$\int \liminf_{n\rightarrow \infty} f_n \mathrm{d}\mu \leq \liminf_{n\rightarrow \infty}(\int f_n \mathrm{d} \mu)$$ or alternatively (for…
Learner
- 7,618
117
votes
19 answers
Is there a "positive" definition for irrational numbers?
An irrational number is a real number that cannot be expressed as a ratio of integers. Is it possible to formulate a "positive" definition for irrational numbers? A few examples: An irrational number is a real number that can be expressed as... An…
barak manos
- 43,717
117
votes
3 answers
What are the differences between class, set, family, and collection?
In school, I have always seen sets. I was watching a video the other day about functors, and they started talking about a set being a collection, but not vice-versa. I also heard people talking about classes. What is their relation? Some background…
Asinomás
- 107,927
117
votes
8 answers
Much less than, what does that mean?
What exactly does $\ll$ mean? I am familiar that this symbol means much less than. ...but what exactly does "much less than" mean? (Or the corollary, $\gg$) On Wikipedia , the example they use is that $1\ll 9999999999$ But my thought on that is that…
user285523
117
votes
6 answers
What did Alan Turing mean when he said he didn't fully understand dy/dx?
Alan Turing's notebook has recently been sold at an auction house in London. In it he says this: Written out: The Leibniz notation $\frac{\mathrm{d}y}{\mathrm{d}x}$ I find extremely difficult to understand in spite of it having been the one I…
Matta
- 1,698
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116
votes
11 answers
How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$
Please help me to find a closed form for this integral: $$I=\int_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx\tag1$$ I suspect it might exist because there are similar integrals having closed forms: $$\begin{align}\int_0^1\frac{\ln^3(1-x)\ln x}x\mathrm…
116
votes
5 answers
How to sum this series for $\pi/2$ directly?
The sum of the series $$ \frac{\pi}{2}=\sum_{k=0}^\infty\frac{k!}{(2k+1)!!}\tag{1} $$ can be derived by accelerating the Gregory Series $$ \frac{\pi}{4}=\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}\tag{2} $$ using Euler's Series Transformation . Mathematica…
robjohn
- 355,927