Any time you exponentiate a Lie algebra, you get a Lie group. Isn't that thing a Lie algebra? I guess the infinite sums cause issues. But direct sum should mean you only take a finite sum ...
Oh, there may be some theorem about dealing with completions, but I don't know it.
@TedShifrin that's the reason why I like walking so much. Thinking while walking is not that dangerous (of course you shouldn't be completely absent-minded)
I know what the character group of a group is, it's just the group of homomorphism to $\Bbb C^\times$, but that probably doesn't help you with hopf algebras
I'm trying to establish a base case for an induction problem $\forall n \in \mathcal{N}, (n+1)\cdot(n+2)\cdots 2n = 2^n\cdot 1\cdot 3\cdot 5\cdots (2n-1)$. How would I interpret the case $n=0$? Would it just be $2\cdot0 = 2^0$ (which isn't true)?
If I have a sequence $S = f(c_1S_1 - c_2S_2)$ such that $f(x) = \begin{cases} x & x \ge 0 \\ \frac x 2 & x < 0 \end{cases}$, is there a term for this type of sequence?
$S_1$ and $S_2$ are other sequences.
Or at least is there a term for the $f$ function? I've seen it before used as a transfer function in a neural network, but if it has a name, that would help out.
The subset {5, 1-2i} of R-vector space C is linearly independent because 5a+b(1-2i)=0, so (5a+b)-2bi=0 holds only when a=b=0 because a and b must be reals, right?
I have to choose electives for my course and "advanced number theory" and "algebraic number theory" both are open. Could you suggest which should i take? I like number theory in general, but i am not sure which should i take. Advaned N.T comprises of fermat, wilson theorms, congruence theory, dirchlet prime number theorom etc... whereas Algebraic N.T consists of algebraic number fields, nonetherian rings,
dedekind domians, dirchlet unit theorom etc. I have some exposure to advanced N.T but almost none to Algebraic N.T. Which would be better, like, if i were to do Phd in N.T (say).
It does sound weird. I mean, Wilson's theorem is one I give as an exercise in the third week of Algebra, and Fermat is a special case of Euler which is covered in week 2.
How can we check if the operator L of the C-vector space C^2 defined by $L(z_1, z_2)=((1+i)z_1+(2-i)z_2, (2+i)z_1+3z_2)$ is hermitian? @TobiasKildetoft
I'm a bit confused by the definition of measurable. Why is it chosen to be such that it's measurable if one can remove parts of the set to make it arbitrarily small? How does that exclude bad sets?
The Cantor set isn't measurable, right? What is it with this that doesn't make it measurable according to this definition?
"A subset $E$ of $\Bbb{R}^n$ is called Lebesgue measurable if for any $\epsilon>0$ there exists an open set $\mathcal{O}$ where $E\subset\mathcal{O}$ such that $m_\ast(\mathcal{O}-E)\leq\epsilon.$
Let $W$ the subspace of $\mathbb{R}^3$ that is spanned by $w_1=(1,-1,0)$, $w_2=(1,1,2)$ and $p=(x,y,z)$ the projection of the vector $v=(-1,1,2)$ onto $W$. What is the number $2x+7y+3z$ equal to?
Can someone mention some book, or lecture note or link ... that defines $a^p$ (where $a>0$ and $p$ any real), and proves properties like $(a^p)^q=a^{pq}$ or $0\le a\le b\implies a^p\le b^p$?
I know that Rudin in PMA defines, and asks some properties as exercise, but he uses many more in his text.
More specifically I'm looking at chapter 7, section "Completions in the nonarchimedean case" of Milne's ANT notes. He writes "Let $|\cdot|$ be a discrete nonarchimedean absolute value on $K$, and let $\pi$ be an element of $K$ with largest value $<1$ (therefore $\pi$ generates the maximal ideal $\mathfrak m$ in the valuation ring $A$). Such a $\pi$ is called a local uniformizing parameter."
Idea
Let's try to express objects in terms of their line elements and line element vector rather than the co-ordinate system. Let the object be a line in three dimensions:
Line
At the origin we have:
$$ d \vec s|_{(0,0,0)} = dx \hat i + dy \hat j + dz \hat k $$
$$ d \vec s|_{(a,a,a)} = dx ...
Hello! I need help regarding a probability question
Find the probability that the square of any integer has last digit as 1,5
I dont know what will be total number of outcomes will it be all digits from 0 to 9 or will it be all probable digits which can be the last digit of a number
Oh, by the way, it seems that I found a dorm room for Bonn! (but just for the first term, this is still pretty good though, finding a flatshare should be much easier when I know the city, people and can actually visit the flats)
We add wine to some dishes (especially when making a risotto) at the beginning of the cooking process but it's supposed to evaporate and just leave some flavour
Idea
Let's try to express objects in terms of their line elements and line element vector rather than the co-ordinate system. Let the object be a line in three dimensions:
Line
At the origin we have:
$$ d \vec s|_{(0,0,0)} = dx \hat i + dy \hat j + dz \hat k $$
$$ d \vec s|_{(a,a,a)} = dx ...
