When I talk about the western mathematical tradition, I think roughly of the mathematics done by the Greeks, and then take up by European countries and then countries with primarily European descendants. Also, I'm not thinking of the mathematicians in particular, but the methods they used, and th...
The Australian Aboriginal counting system was used to send messages on message sticks to neighbouring clans to alert them of, or invite them to, corroborees, set-fights, and ball games. Numbers could clarify the day the meeting was to be held (in a number of "moons") and where (the number of camps' distance away). The messenger would have a message "in his mouth" to go along with the message stick.
A common misconception among non-Aboriginals is that Aboriginals did not have a way to count beyond two or three. However, Alfred Howitt, who studied the peoples of southeastern Australia, disproved...
We often said mathematics is an art, but do we have ideas on how to do it the artistic way. However some will argue that without logic, we will end up with intuitive maths back in the ancient time which is misguided in all directions
but then, having only logic to guide mathematics sounds too limiting, but I don't have any reasonable solutions to this issue
Let $G$ be a simple Lie group. It can have a center, but does it always arise as some quotient of a simple Lie group those center is trivial? I know that $\pi_1(G)$ can be identified with a discrete subgroup of the center $Z(\widetilde{G})$ of the universal covering group and I think the construction should be something like $\widetilde{G}/Z(\widetilde{G})$.
I'm a bit confused. Wikipedia states "Any simple Lie group with trivial center has a universal cover, whose center is the fundamental group of the simple Lie group. The corresponding simple Lie groups with non-trivial center can be obtained as quotients of this universal cover by a subgroup of the center."
So If I take $G$ centerless simple Lie group, then I can construct simple Lie groups with nontrivial centers out of $\widetilde{G}$
But how does the reverse process go?
Maybe take the full $\widetilde{G}/Z(\widetilde{G})$?
@TobiasKildetoft Yeah right, its $PSL_n$ which I want in that case
@abenthy Right, I guess the thing is that the universal cover will not be simple, right?
(I haven't thought nearly as much about these things as I ought to. I usually just go "by usual considerations, we can assume our group to be simple and simply connected and then the results will also hold for arbitrary reductive groups")
It essentially comes from my desire to understand why in working with symmetric spaces of noncompact type, one has $G/K$ with $G$ semisimple Lie group with trivial (sometimes finite) center
What goes wrong when the center is infinite? Why can I reduce to the case that its trivial?
Okay, I was trying to figure out the $\mathbb{Q}$-rank of the algebraic $\mathbb{Q}$-group given by the functor which sends a $\mathbb{Q}$-algebra $A$ to the group $(A \otimes_\mathbb{Q} \mathbb{Q}(\sqrt{3}))^\times$
Is it one and a maximal torus is given by $A \mapsto A^\times \otimes_\mathbb{Q} 1$?
Does every closed subspace of $\Bbb{R}$ have a countable dense set? Let $A \subseteq \Bbb{R}$ be closed. I was thinking $A \cap \Bbb{Q}$ would do the trick, but I first need to prove $\overline{A \cap \Bbb{Q}} = \overline{A} \cap \overline{\Bbb{Q}}$, which I am unsure is true, since the intersection of the closures generally isn't contained in the closure of the intersection.
@MikeMiller Really? That's pretty remarkable. It is a deep and hard to prove fact that every closed subspace of the reals has a countable dense set contained in it?
What I am working on is finding a topological $X$ that has a countable dense subset and a closed subset $A$ that doesn't have a countable dense subset. Obviously what you have said rules out $\Bbb{R}$ having any such examples...So I am not sure which space to look at for an example. Any hints on what space?
@user193319 This is kinda cheating, but given any space $X$ you can turn it into a separable space by adding a single point, by taking the open sets in $X\cup\{\infty\}$ to be those open in $X$ with the bonus point added to them
@MikeMiller It says that if you have an entire relation $R$ on a set (for all $a$ in the set there is a $b$ such that $aRb$) then there is a sequence $(a_n)$ with $a_nRa_{n+1}$ for all $n$. It's a bit stronger than countable choice
I'm reading some primary decomposition stuff, they say that if $\sqrt{(0)}$ is a maximal ideal then the ring is local and $\sqrt{(0)}$ is the unique maximal ideal
> Higher homotopy groups were first defined by Eduard Čech in 1932 (Čech 1932, p. 203). (His first paper was withdrawn on the advice of Pavel Sergeyevich Alexandrov and Heinz Hopf, on the grounds that the groups were commutative so could not be the right generalizations of the fundamental group.)
I was thinking about picking a $b\not\in\sqrt{0}$ and showing it is invertible, consider the ideal $(\sqrt{0},b)$ which must be the whole ring by maximality, so there are some $a,c$ in the ring such that $ab+cx=1$, with $x\in\sqrt{0}$, so $cx=1-ab$ and $0=(1-ab)^k$ for some $k$ such that $x^k=0$, and so $a$ must be the inverse of $b$ (if $1-ab=0$ we get to a contradiction)
Given a $4\times4$ real matrix $A$, let $T:\mathbb R^4\to \mathbb R^4$ be the linear transformation defined by $Tv=Av$, where we think of $\mathbb R^4$ as the set of real $4\times1$ matrices. For which choices of $A$ given below, do the $Image(T)$ and $Image(T^2)$ have respective dimension 2 a...
hey folks, I'm trying to implement a parametric equation but there's something that I couldn't identify yet, perhaps someone here can tell me how its called so I can learn to interpret it myself
basically the image is a link to a function that uses "a" and "b" on x y equations but only K is given as input a k=a/b
not sure how the values of a and b are inferred from it nor how this thing is called so I'm stuck
since I can't even formalize my question properly I decide to give the chat a try, thanks for any help
So recently, I was pondering about historical connections of subjects, such as how concepts, ideas and experiments done in history change the course of humanity as they open new doors to exploration and technologies.
Having been to many panel discussions that covered topics from quantum computin...
Determine the prime integers p, such that $\ mathbb F_p ^ *$ have 2 subgroups, one of order 17, G and order 19, H, such that there exists g in G with g + 1 in H.
Give an example of a function$ f : [a, b] → R$ which is continously differentiable and such that $|f(x) − f(y)| < |x − y|$ for all distinct $x, y ∈ [a, b]$, but such that $|f'(x)| = 1$ for at least one value of $x ∈ [a, b]$.
If you want to do something over $\Bbb C$, complex analysis, Riemann surfaces or even higher dimensional complex manifolds can be useful as motivation (but @Ted is the expert on that)
If you want to learn up to a second year graduate student level though, in only 24 hours, you can save years of your life by just doing "Twenty-four hours of local cohomology".
@NV-US: Interestingly, the converse of the Mean Value Theorem holds "most" of the time, actually. If you're interested, you can find some questions about it on MSE and also get some results by googling.
I liked Matsumura's book more than Eisenbud's, but I wouldn't want to do without either of those, you should look into both and see what suits you more
@TedShifrin in our uni, algebraic curves are treated after at least one year of algebra and another 3/4 year of scheme theory and sheaf cohomology as an application
@studrayght5 To me, it feels like one of those books that you read over a long period of time. Reading a little, finding something you lack, and then correcting that - then returning to the text.
Consider for one car owner the insurance policy with the following clauses:
Deductible: If the loss $X>d$, then the insurer pays only for loss above $d>0$.
Coverage Limit: If the loss $X>l$, then the insurer pays only for loss below $l>d$.
Distorted Distribution: The insurer may base the premiu...