@EM4 He had a hyperextended knee, after game 4 against the Atlanta Hawks; but two MRIs performed the evening of the injury revealed absolutely no structural damage. He sat out game 5 and game 6, with a likely return if game 7 was required, but Middleton and Holiday secured to more wins for the Bucks, who tromped the Atlanta Hawks 4-2 in best of seven.
We had no clue after his injury (The Greek Freak (term of endearment) Gianis), but there's more depth in the team than doubters realized.
@EM4 Although I'm talking big, I have a thing about not being too overconfident. Sort of a Karma thing. As always, may the best team win (but I pray the best team is the Bucks! ;D)
@EM4 Indeed! I think the "final four" were, for the most part, teams that haven't enjoyed a Championship for some time. I always love underdogs. And some teams have been strong, season wise, but couldn't quite pull it off in playoffs.
Let the outcomes in the sample space be represented by two letter words, in which the first letter is ‘s’ if the first question on the exam is seen and ‘u’ otherwise, and the second letter similarly describes the second question. Thus the sample space is
S = {su, ss, us, uu}.
We are give that at least one of the questions on the exam is unseen, i.e. the event
A = {su, us, uu}
has taken place.
Thus we need to find the conditional probability of the event B = {su, us}, that one of the questions on the exam is seen, given A.
@FitzWatson Did you read my computation? You didn't need to write everything out, because I had all of that there. I just switched whether the student saw the seen/unseen question (so that is all you needed to say).
but yes, your computation is correct.
Had they said "old" and "new" rather than "seen" and "unseen" it would have been less confusing.
36% that both are seen, 16% that both are unseen, 48% that one and not the other has been seen. Knock out the 16% that both are seen, that gives $\frac47$ that one is seen and one is unseen.
it's a simple switch... "16% that both are unseen" is switched to "16% that both are seen".
@robjohn But, this computation assumes that we ourselves ask the person setting the question paper (now referred to as paper-setter) "is atleast one of your questions unseen?". However, the question says that we "somehow find out that one of the questions is unseen". Could this not also mean that the paper-setter randomly chooses to talk about one of the questions? In that case the probability would change to 0.6
translate "we somehow find out that one of the questions is unseen" as "given that one of the questions is unseen"
translate "the probability that the other question is seen" as "the probability that at least one question is seen"
given that one of the questions is unseen, what is the probability that at least one question is seen?
The only information we have is that one of the questions is unseen.
"Could this not also mean that the paper-setter randomly chooses to talk about one of the questions?" it could, but the only thing we know is that they said that that question was unseen.
However it is we find out, we know that one of the questions is unseen.
I have seen some "Anti-Pi Rant" videos where they say that Pi being a boring number. There argument is that "Sure pi is an irrational number , but so is root 2 , root 3 , the golden ratio phi etc".My confusion is that , isn't pi also a transcendental number too ? I heard that to this day , very few numbers were proven to be transcendental.
@Prithubiswas In my opinion, I find that argument completely ridiculous; pi's irrationality is hardly it's most extraordinary feature!! I'd say that the variety of problems in which it appears on the solution to makes it extraordinary. As you say, it's also transcendental which is a very interesting property.
@EdwardEvans I guess pi is a number which is proven to be transendental and can be visualized as the ratio of a circle's circumference to its diameter.
Not sure if other transendental numbers have nice visualizations.
hi, im sort of lost on how to show the highlighted part of this imgur.com/a/DkhLq4l , or really what exactly the highlighted part is saying (the stuff above the highlighted part is fairly standard)
it seems like its saying that if $f : [a,b] \rightarrow [0,1 - \epsilon]$ is a $C^1$ function, such that $f(a) = 0$ and $f(b) = 1 - \epsilon$, then there are intervals $I_1, I_2 , ... I_k$ that are disjoint, with left endpoints in increasing order according to indexing, and $f \restriction I_j$ is monotone increasing, and $sup_{t \in I_j} f(t) = inf_{t \in I_{j+1}} f(t)$, and $inf_{t \in I_1} f(t) = 0$, $sup_{t \in I_k} f(t) = 1- \epsilon$
for some reason the poincare metric as these notes state it doesn't have the factor, but evidently it should because on wikipedia the area associated to $\rho$ is supposed to be $4 / (1 - |z|^2)^2 dx dy$
probably a dumb question, if $\rho_2 : \Omega_2 \rightarrow [0,+\infty)$ is $C^1$ and $\Omega_2 \subset \mathbb{C}$, is connected and open, and $f : \Omega_1 \rightarrow \Omega_2$ is holomorphic ($\Omega_1$ is also a connected open), why is $(\rho_2 \circ f ) |f'|$ necessarily $C^1$?
I can see that its $C^1$ on $\Omega_1 \setminus {f'}^{-1}(0)$
@porridgemathematics here is a good way to remember this. the constant curvature +1 metric on C is 4/(1+|z|^2)^2 dzdzbar; this is what you get when you pullback the spherical metric on CP^1 by stereoprojection
the 4 is there because the equator of the sphere gets mapped to |z|=1
near that, the scaling factor is 4/(1+1^2)^2 = 1, as it should be, since stereoprojection is an isometry near the equator
sometimes results of this flavor are referred to as 'choquet theory' after a frenchman who represented functionals as integrals over the extreme points of a convex set.
i tihnk it's that paragraph between (3) and (4). they are identifying things in some random function algebra with their unique extensions to things that restrict to H.
sorry if my thoughts seem scattered, my daughter has been yelling at me.
she's having quite a morning.
this isn't choquet theory after all but i do like choquet theory. peter lax's functional analysis text has a good chapter on it.
my daughter's now yelling at the cat, which is a useful change from me.
i wonder if otimes really means tensor there. i haven't checked the inequality but if they say it, it must be true.
i rhetorically asked my daughter where the cat was. "i think she's under the couch" was the response. this turned out to be correct. she then started yelling the cat's name over and over and throwing toys under the couch.
First, observe there is a one-one correspondence between $M$ and $A$, hence the author uses the SAME LETTER to denote functions in $M$ and their unique extensions to $K$, which are in $A$.
(8) holds for $f \in C(H)$, so it also holds for $f\in M$
$M$ is a subspace of $C(H)$
Now, the integral is over $H$
So we might as well consider the extensions of the functions in $M$, to $K$, which are in $A$
because it doesn't matter whether you integrate $f\in A$ or $f\in M$, the result is the same as long as the integral is over $H$
Since (8) holds for $f\in M$, it holds for $f\in A$ as well
Am I the only one who disagrees with this answer that says that the function $f(x)=1/x$ is strictly decreasing? The definition of "strictly decreasing" that I'm familiar with is that if $a$ and $b$ are in the domain of $f$, then $a<b \implies f(a)>f(b)$. In this case, it seems that we can take $a=-1$ and $b=2$ to show that the function is in fact not strictly decreasing.
i don't know what the right authorial choice is here. if you don't use the same letter, you're putting funny hats and things on stuff to make these tiny differences. if you do use the same letter, the same letter can mean different things.
joe, you are not the only one who disagrees with that answer.
the only right definition of 'strictly decreasing' is the order theoretic one which you state. it does not distinguish between intervals on which a function might or might not be continuous.
points of discontinuity complicate all of the calculus-theoretic characterizations of order properties.
@leslietownes: I thought I was losing my sanity for a moment there. Thanks for clarifying. I'm tempted to write up my own answer, even though that post is almost two years old
i'm trying to think of other instances where this comes up. changes of variable in definite integrals, maybe. the formula in calculus books tends only to work in complete generality if the change is a monotone, one-to-one function, although it's usually used beyond that and god help you if there are discontinuities.
it would be fair to say that 1/x is locally decreasing or something like that. that is all you get from the derivative being negative everywhere.
computer algebra packages often f--- up inverse trigonometric functions and definite integration because of the discontinuities that are associated with that.
@leslietownes: It seems that things are stickier than I thought. I just looked at Spivak and Wolfram MathWorld, and they only define increasing/decreasing on intervals
@leslietownes: Haha, thanks. Unfortunately, I just checked Rudin, and even he (who generally offers very general definitions) only defines increasing/decreasing on intervals :(
i do think the natural home of the definition is a function from one linearly ordered space to another. this would not incorporate concepts like connectedness.
but if RUDIN disagrees with me maybe i am wrong about this.
rudin did a lot of things extremely well, and for some reason, people rely on him for other things too. i've insulted rudin's treatment of various things many times.
i can say without a shred of doubt that i have learnt the most math from his books. his books are very effective if you're self studying and like the painful approach of figuring most things out on your own
i think his treatments of multivariable calculus and measure theory in principles of mathematical analysis are a disaster. and the orgy of point-set topology in chapter 2 is disgusting. thankfully nobody ever gets too far into the book.
Wiki :="In mathematical logic, a theory is complete if, for every closed formula in the theory's language, that formula or its negation is demonstrable" can both formula and its negation allowed to be demonstrable?
@Prithubiswas in general, i'd rather not comment. completeness alone is probably not sufficient to make a comment. but if you also have soundness, then: math.stackexchange.com/questions/105575/…
@leslietownes: I have reluctantly written up an answer about increasing vs. decreasing that reflects the (wrong) definition that is given in textbooks. Any feedback?
The veronese surface has the property that any curve on it can be obtained by the intersection of a hypersurface with the veronese surface. Is there a name for surfaces which have this property?
it is a thing, albeit perhaps not a well-defined one
Krantz & Parks have a book called Geometric Integration Theory, which is actually just a GMT book. Whitney has a book called Geometric Integration Theory, which is Topology/Geometry with a heavy focus on integration, measurable forms and stuff like that.
@Thorgott I had Krantz for Graduate Real Analysis and Complex Analysis and an independent study in Fourier Series (ala Katznelson). Those were some of the best classes I had at UCLA.
@TedShifrin I am sorry.
I cooked for 30 people (BBQ) yesterday. I am resting today.
@TedShifrin I only cooked the main courses people brought: chicken, steak, burgers, hot dogs, brats, etc. Luckily, no fish this year. The salads were shared and the drinks and dessert was provided by the host
@Joe that gets at another issue, decreasing, strictly decreasing, nonincreasing. i am very fine with 'decreasing' to mean a > b implies f(a) \leq f(b) because it reflects a lot of conventions from functional analysis (where 'positive' means 'nonnegative').
@leslietownes: Indeed, I did mention that in the answer because I wanted to reduce confusion, not enhance it. I was surprised that the OP actually saw my answer and accepted it, but I'm glad that they did, because that question had over $1000$ views, and I didn't want any more people to be misled...
I have quick question about the meaning of tie here please, "How many ways are there for a horse race with three horses to finish if ties are possible"?
What is meant by ties possible or not possible here please?
my great grandfather litigated an insurance case about a horse being set on fire by a bolt of lightning. it made it to the supreme court. (long pause) of new hampshire.
my daughter's really yelling at the cat. i have this issue every day. at what point does this become my problem?
i never know.
earlier the cat handled the situation by escaping to a nearby closet. now, for some inexplicable reason, she's sitting there and frowning at this verbal abuse.
i bought my advisor some bordeaux and i heard from people who were able to make it that they served it at his funeral. that wasn't paul marcus, it was "vino!" by the safeway.
he had written some books and i said that thanks to him, everyone had discovered their love of reading . . . amazon's return policy. that was the nicest joke.
my brother made a slightly negative comment at my mom's service about god and choice of pouring rain on the day. amusing, but slightly awkward reception.
Suppose $\int_{0}^{1}f(x)*e^{n*x}dx=0$ for all natural numbers and 0. f must equal 0. I've tried proving it with Weierstrass theorem. But I am out of ideas. Anyone have any?
@JakeFreeman If $f$ is the characteristic function of a non empty set of measure zero then the integral is zero but $f$ is not. There is no point in continuing without some hypotheses.
