It's decided. When I'm ready to diversify, I'm going to build corporate bodies and the fact that it exists within a capitalist system is merely accidental.
@Jakobian Sage has some nice continued fraction features. doc.sagemath.org/html/en/reference/diophantine_approximation/… Here's a short demo. If you give it expressions involving l sqrt() they will be represented exactly. It also accepts expressions involving pi
"expressions involving sqrt()". Sorry, my finger slipped, and it was too late to edit by the time I noticed.
Of course, you can also feed it expressions involving functions like sin, atan, ln, etc, and they'll be stored symbolically (as are rational numbers / expressions), not converted to floating-point. So the continued fractions should be valid to arbitrary precision.
@TedShifrin: As advised, I watched your lecture on polar coordinates. I have a better understanding of the polar coordinates now that I did before. Thanks a lot. I understand now that $S=\{(x,y)\in \mathbb R^2: x^2\le y\le 1, y\ge 0, -1\le x\le 1\}$ will be $\frac\pi 4\le \theta\le 3\frac\pi 4, \color{red}{\tan\theta \sec\theta \le} r\le \csc\theta$ in the polar coordinates plane. The red colored part is true by mapping $y=x^2$ using $r\sin \theta=r^2\cos^2\theta$ which is same as $r=\tan\theta \sec\theta$. Is my understanding correct? Thanks.
Hello everyone. Please guide me on two questions. 1. I have a polynomial.
$p=a_3x^3+a_2x^2+a_1x+a_0$
where:
$a_0=x+y,a_1=x^2+y^2,a_2=xy+x^2,a_3=x^2-y^2$
$-1<x<1,-2<y<2$
Calculate the range of parameter $x$ such that polynomial $p$ has only real solutions for any value of $y$ in the specified range $-2<y<2$. How are such tasks solved?
2. There is an arbitrary function of 4 variables $f(u,v,w,x)$ and:
$u∈[-2\pi,2\pi]$
$v∈[-2\pi,2\pi]$
$w∈[-2\pi,2\pi]$
From these ranges, an arbitrary combination of $uvw$ is taken, and this combination is then substituted into the function $f$, and t…
Alas, the region obtained in r theta plane here is not correct. Ted, but now I have understood how to correctly convert the region $S$ here to the right region in polar coordinates plane with robjohn's help. :-)
Q: A particle moves on a given straight line with a constant speed v. At a certain time it is at a point P on its straight line path. O is a fixed point. Show that (OP×v)is independent of the position P.
My solution:
I considered the axis to be X-axis I.e as 1D motion. Points P & O on it. O to be...
Hey, I'm from the physics side of things and a buddy of mine and I have been thinking about a math "proof" (I use the word very liberally) that I had thought of one day. It can't possibly be correct, and I' worried I'll be crucified for this, but we can't find any holes in it. Thought maybe you guys could point out where the holes in my swiss cheese are
it's a bijection from natural numbers to the reals on [0,1)
we tried applying cantor's diagonalization, but we failed
it goes like this, for all n in naturals, write n.0, then write the digits in reverse. 1 becomes 0.1, 2 becomes 0.2, 1056 becomes 0.6501, etc. This gets you any and every number in [0,1) and is inversible.
@Astyx then, necessarily there must be some integers of indfinite length, otherwise I'd always be able to find an element not in your ennumeration by continuously increasing the length of my chosen outlier
this is specious reasoning. My inability to tell you the number doesn't imply it isn't an integer any more than my inability to tell you Cantor's first name implies he doesn't have one
Anyway, if you want to know what is commonly called an integer, and understand why what you claim are integers are in fact not, I recommend reading something about Peano's axioms, the construction of the integers, the rationals and the reals
Can someone help me with the Conway-chain arrow notation , rules are here ? If $$M=3\rightarrow 3\rightarrow 3\rightarrow 3$$ , how can I compare the numbers , for example , $$M\rightarrow 3\rightarrow 3\rightarrow 3$$ with $$3\rightarrow 3\rightarrow 3\rightarrow 4$$ ?
I was looking at a game called Grepolis, in which there is an event Winter Grepolympia
in the event, each player has an athlete that participates in the contest each athlete has 3 attributes (for example balance, control and speed) (all non-negative integers) the game calculates the score of the partipation based on the attributes of the athlete (and a small bonus/penalty due to luck)
now I am wondering how the formula to compute the score would look like
from pure testing against the game, the community figured out that a 85%-5%-10% distribution for the first contest is the most ideal distribution of the attribute points
for example, 17 balance, 1 control and 2 speed would get the highest score out of any combination of 20 points
given attribute points a, b and c, how could a score be calculated to produce such a favouring of a particular attribute?
I am not really looking for the exact formula the game uses, but I am just wondering how the formula would theoretically work
given that - higher attributes should always result in higher score - better balance (to the intended distribution) should always result in higher score
for example, it isnt something like a*0.85 + b*0.05 + c*0.10 because if they are in isolation, you would just favour the one attribute with the highest result
I think it does meet the criteria as well, any higher number in any of a/b/c would result in higher total and the highest total would be produced by the best even distribution the input numbers would just have to be "normalized" to allow this even distribution
I just realized certain boolean operations hold for multiplicative distributivity. $k (x \land y) = kx \land ky$. $k (x \lor y) = kx \lor ky$.
I could prove this by restricting $k$ to powers of two then using the binary decomposition of any arbitrary integer $k$ and the distributive property to show that it holds.
Ah, my mistake. It only holds for powers of two.
And I suppose a certain subset of x and y for any k
Hi professor Ted, I'm able to work my way through exercise 11.28 in Apostol's vol 2. The exercises involve double integrals to be solved by converting to iterated integrals in r theta plane. I saw both the videos you suggested and that helped a lot. Thanks a lot professor :).
You're welcome :) I'm surprised you didn't learn all that stuff years ago in your regular multivariable course, but it takes practice.
In actual applications, knowing how to set up the circle $(x-a)^2+y^2=a^2$ in polar coordinates is quite important. (Similarly in spherical coordinates.)
I also noted how $x^2+y^2=a^2, a\ne 0$ is a straight line in r theta but it's surprising how circle of radius a centred at (a,0) is not mapped to a straight line :).
Professor I may have told this earlier that I come from engineering background. We did have multivariable calculus but it was probably assumed that everyone knew polar coordinates. By knowing I mean that $(x,y)$ becomes $(r\cos\theta, r\sin\theta)$.
That was all about polars that I knew. :(
But now I have much better understanding of polars :).
@Derivative So your Faa di Bruno pointer did help somewhat, I reduced it to something relating to Bell polynomials so I can continue with that now. Thanks for your assistance
I propose that in addition to surjective, injective, bijective, etc. we must also allow for objection and subjection between sets where objection between a set A and a set B is when A asserts dominance over B and subjection when B crawls over and submits itself to A.
ted it's a funny thing. i do think a little bit goes a long way. if taken to an extreme, it turns the already troubling job of learning math into the more vexatious task of having to tolerate the author.
although most authors are more tolerable than i am, so what do i know.
Studies show that high functioning groups often have a 'clown' member who may not be the greatest direct contributor but enables the group to interact in a more productive way. Appropriate humour is always good.
For example, the relationship between the irrationals and rationals is objective from the irrationals because it beats the rationals into submission as opposed to the rationals striving to be irrational ($\forall x\in \Bbb{Q}, \exists \lim_{x\to y} = z, z \in \Bbb{R\backslash\Bbb{Q}}$) which would make the relationiship subjective from rationals to irrationals.
i was once scolded at work for writing a memo that suggested a person's name was probably misspelled, i couldn't resolve it. we had documents both ways. so i put "Smith/Smyth" everywhere after identifying the problem. someone thought i was trying to be funny. i just didn't want to get a phone call some day asking why we spent money trying to track down a wrongly spelled name.
you often have stuff in my job where people get their name on a patent and then fall off the face of the earth. especially if they're in other countries.
astyx: some people don't like "hint" answers, i think that is the high level thing that is going on. it is a difference of opinion but not a community standard.
leslie: that was my guess as well. I was surprised because the two votes arrived quickly, which seemed to imply some kind of agreement that it deserved to be deleted
but you don't see the votes copper? that's really weird
Hi. I have a quick question. Is a class of groups closed under quotients iff it is closed under homomorphic images? I think it follows from the first isomorphism theorem . . .
astyx: i sometimes get a random downvote with no explanation, i assume someone just doesn't like my style. which is fine, i probably don't like their style. we all get to click on buttons.
i think i've done about one or two downvotes in 10 years. for something that was abusive. maybe i'm not pulling my weight by not downvoting other stuff.
downvoting without comment seems a little rude unless the problem is obvious.
you sometimes see fairly well-posed problems being downvoted i think mainly because they are what the downvoters regard as stupid questions. similarly a crummy question will get upvotes and answers if it is 'advanced' enough to be interesting.
there was a good supreme court case in the 1980s. there was a cap on what an attorney could charge in some kind of proceeding involving, i think, veteran's benefits, of about $100. this meant essentially that nobody could get legal assistance for those claims. it was upheld on the rationale that congress might have wanted these proceedings to be non-adversarial. the $100 figure dated from the civil war era when it was a reasonable amount of money. the majority did not engage with this at all.
they could've made the usual move which was 'if congress doesn't like this they can change it,' but they adopted this counterfactual narrative instead.
'if congress doesn't like this they could change it' is actually a very persuasive narrative in a world where congress actually does things. in a world where it doesn't, it is harder to stomach.
@leslietownes I have a copy of Rose's, "A Course on Group Theory" handy. Whereabouts in the book should I check? There's nothing obvious about it in the index.
shaun: this is called 'calling my bluff.' my copy is gone, i think. i think it occurs later in the book, where he starts to do things relating to classification of groups. it's not near the beginning.
Birkhoff's HSP theorem states that if a class of algebras (for a given type) is closed under products, subalgebras and homomorphic images ($\iff$ quotient), then it is actually defined by some equations.
I was wondering what could be said about other combinations of these operators $H,S,P$, in ...
Does there exist anything that might be described as a temporal density function that relates the distance over time from one point to another in the form $(x(t), y(t))$?
If there were such a thing, and we had an algebra surrounding it, we could compute inverses of functions by way of their arc length parametrization and temporal density function.
Indeed. But there's probably also language issues. I would not say "remove." I would say explicitly: Multiply the first equation by blah. Now subtract this new equation from the second equation.
Meanwhile, I'm fighting this battle. The picture seems to be completely out of whack. My computations, which I've checked, seem to put the bottom vertex of his "rectangle" below the $x$-axis.
Unrelated: is there a name for the tensor algebra quotiented by commutative relations (like you would do to get the exterior algebra, except you impose commutativity instead of anticommutativity)
Birkhoff's HSP theorem states that if a class of algebras (for a given type) is closed under products, subalgebras and homomorphic images ($\iff$ quotient), then it is actually defined by some equations.
I was wondering what could be said about other combinations of these operators $H,S,P$, in ...
