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1500 questions
72
votes
18 answers
Why is radian so common in maths?
I have learned about the correspondence of radians and degrees so 360° degrees equals $2\pi$ radians. Now we mostly use radians (integrals and so on)
My question: Is it just mathematical convention that radians are much more used in higher maths…
Slater
- 795
72
votes
11 answers
Have there been efforts to introduce non Greek or Latin alphabets into mathematics?
As a physics student, often I find when doing blackboard problems, the lecturer will struggle to find a good variable name for a variable e.g. "Oh, I cannot use B for this matrix, that's the magnetic field".
Even ignoring the many letters used for…
Mark Allen
- 833
72
votes
2 answers
Continuity and the Axiom of Choice
In my introductory Analysis course, we learned two definitions of continuity.
$(1)$ A function $f:E \to \mathbb{C}$ is continuous at $a$ if every sequence $(z_n) \in E$ such that $z_n \to a$ satisfies $f(z_n) \to f(a)$.
$(2)$ A function $f:E \to…
John Gowers
- 24,959
72
votes
4 answers
Is the product of symmetric positive semidefinite matrices positive definite?
I see on Wikipedia that the product of two commuting symmetric positive definite matrices is also positive definite. Does the same result hold for the product of two positive semidefinite matrices?
My proof of the positive definite case falls apart…
nullUser
- 27,877
72
votes
4 answers
Does every set have a group structure?
I know that there is no vector space having precisely $6$ elements. Does every set have a group structure?
user23505
71
votes
1 answer
Fractal behavior along the boundary of convergence?
The complex power series $$\sum_{n=1}^{\infty}\frac{z^{n^2}}{n^2}$$ has radius $1$ (Ratio Test) and is absolutely convergent along $|z|=1$. Recalling something that my calculus professor (Ray Mayer, emeritus of Reed College) showed me 15 years ago,…
2'5 9'2
- 54,717
71
votes
9 answers
Best Algebraic Topology book/Alternative to Allen Hatcher free book?
Allen Hatcher seems impossible and this is set as the course text?
So was wondering is there a better book than this? It's pretty cheap book compared to other books on amazon and is free online.
Any good intro to Algebraic topology books?
I can…
simplicity
- 3,694
71
votes
6 answers
Expressing the determinant of a sum of two matrices?
Can $\det(A + B)$ expressed in terms of $\det(A), \det(B), n$
where $A,B$ are $n\times n$ matrices?
I made the edit to allow $n$ to be factored in.
Sidharth Ghoshal
- 16,771
71
votes
5 answers
What does "curly (curved) less than" sign $\succcurlyeq$ mean?
I am reading Boyd & Vandenberghe's Convex Optimization. The authors use curved greater than or equal to (\succcurlyeq)
$$f(x^*) \succcurlyeq \alpha$$
and curved less than or equal to (\preccurlyeq)
$$f(x^*) \preccurlyeq \alpha$$
Can someone explain…
Dinesh K.
- 811
71
votes
5 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
- 70,631
71
votes
1 answer
Difference between Analytic and Holomorphic function
A function $f : \mathbb{C} \rightarrow \mathbb{C}$ is said to be holomorphic in an open set $A \subset \mathbb{C}$ if it is differentiable at each point of the set $A$.
The function $f : \mathbb{C} \rightarrow \mathbb{C}$ is said to be analytic if…
Supriyo
- 6,119
- 7
- 32
- 60
71
votes
2 answers
What is the difference between projected gradient descent and ordinary gradient descent?
I just read about projected gradient descent but I did not see the intuition to use Projected one instead of normal gradient descent. Would you tell me the reason and preferable situations of projected gradient descent? What does that projection…
erogol
- 1,097
71
votes
4 answers
Symbol for elementwise multiplication of vectors
This is a notation question. Assume one is given two vector $\mathbf{a}$ and $\mathbf{b}$, and one constructs a third vector $\mathbf{c}$ whose elements are given by
$$c_k=a_k b_k$$
Is there any standard notation for this simple operation?
Is…
D R
- 1,158
71
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
- 965
71
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
- 163,620