I'm developing a song synthesizer (like Yamaha's Vocaloid). I decided to mimic the acoustics of human speech. I modeled the space between articulators as a solid of revolution, and came up with the following PDE:
$$
(PA)_{tt} = c^2 (P_x A)_x
$$
where $P$ is air pressure (difference to the atmosph...
Sorry for this silly question, but I want to approximate $2^{73}$. The exact value that I got was 9 444 732 965 739 290 427 392. Is it $9.45 \times 10^{19}$?
Hi @Ted. I have two (complex) 1-submanifolds of $\mathbb{CP}^1\times\mathbb{CP}^1$, whose Poincare duals are represented by $(p_1, q_1)$ and $(p_2, q_2)$ in $\mathbb{Z}\oplus\mathbb{Z}$. Suppose they are transverse. I want to count their intersection.
I know its given by the cup product $H^2(\mathbb{CP}^1 \times\mathbb{CP}^1) \times H^2(\mathbb{CP}^1 \times\mathbb{CP}^1) \to H^4(\mathbb{CP}^1\times\mathbb{CP}^1)$. But I don't what map it induces on $(\mathbb{Z}\oplus\mathbb{Z})\times(\mathbb{Z}\oplus\mathbb{Z}) \to \mathbb{Z}$.
reminds me of teaching exponential decay out of a book where all of the exercises were poorly written. they'd ask, how much americium or whatever is left after X time of decay. the answer was less than the mass of a proton. and all of the exercises were like that.
i began to think maybe it was on purpose.
i remember pointing it out to the class. you literally cannot have this much americium. i don't know why it's written out to three significant figures. blank stares.
Arzela Ascoli theorem says that if X is a compact metric space, and F is a subset of C(X), then: F is compact if and only if F is closed, uniformly bounded and equicontinuous.
From this, how do we prove this corollary: Let X be a compact metric space. If (f_n) is a uniformly bounded, equicontinuous sequence in C(X), then some subsequence of (f_n) converges uniformly on X
Can anybody please help me on how to evaluate infinite products? The definition of the product that I am using is similar to definition of series : we say f(1).f(2).f(3)... f(k)f(k+1)... converges to a finite value L (non zero value) if the limit f(1).f(2)... f(n) converges to L as n tends to infty and then we write f(1).f(2).f(3)... f(k)f(k+1)... =L
In particular, I want to find the infinite product for f(n) = (1+x^(2n-1))
as n tends to infty.
The necessary condition for convergence of the product is clearly $f(n)\to 1$ as n tends to infty
And this is to be solved using limit definition. It is not expected to use continuity, differentiability etc. But I would like to see that as well. It can also further be deduced that $|x| $ should be less than 1
@leslietownes "Now equal angles stand on equal circumferences when they are at the centers, therefore the circumference $BK$ equals the circumference $EF$. by I.26" which is side and two angles congruency.
translating that into normal reasoning is tough. i think the idea is that if you have two congruent circles with centers O, P, and an angle at O and and angle at P that are congruent, then they cut off equal circumferences. the fact that the circles have the same radius seems important to this.
@feynhat You can see this with cohomology by pulling back Kähler forms from the factors, or you can see it geometrically. If $L_1 = \Bbb P^1\times \{q\}$ and $L_2 = \{p\}\times \Bbb P^1$, then you can see intersections geometrically: $L_1\cdot L_2 = 1$ and $L_1\cdot L_1 = L_2 \cdot L_2 = 0$ (just "wiggle" the submanifold by taking a different fiber, and then they are clearly disjoint).
In linear algebra, a circulant matrix is a square matrix in which each row vector is rotated one element to the right relative to the preceding row vector. It is a particular kind of Toeplitz matrix.
In numerical analysis, circulant matrices are important because they are diagonalized by a discrete Fourier transform, and hence linear equations that contain them may be quickly solved using a fast Fourier transform. They can be interpreted analytically as the integral kernel of a convolution operator on the cyclic group
C
n...
Kind of a silly question, but suppose you have two ODEs, y' = f(x) and y' = g(x). Fix a starting point, and flow time t along the flow of the first, then time t along the second, then time t along the first, and so forth, alternating. As $t\to 0$, does the resulting "flow" converge to something?
not sure how the alternating 'flow' is intended to interact with t going to zero. in general i would not expect solutions or solution families to different differential equations to have much to do with one another.
Yeah, I didn't think so. I had one person show me demos where they said it converged to the average of the flows.
But like I would have thought something like that would be easy to see if it were the case, but I can't put my finger on how to even formalize this alternating flow concept as t goes to 0.
it's not even clear that if you grab a solution to y' = f(x) at 0 and use it to guide you from x = 0 to x = t that y' = g(x) will even be well posed at x = t.
@leslietownes Maybe I don't understand what you were saying. It's imagined like a dynamical system: pick a point x_0, flow for a fixed time t along the flow (integral curve) of f(x) containing x_0 to a point x_1. Then flow x_1 along the flow (integral curve) of g(x) containing x_1 to a point x_2. Then just continue doing this, alternating to get a sequence of points x_0, x_1, x_2, ... .
yeah, and then somehow x_0, x_1, x_2, ... , x_n are confined to an interval [0,t] and we push t to 0? like we're partitioning an increasingly smaller interval into the same number of pieces?
@TedShifrin I agree, I don't see how it is obvious what this actually ends up being. They are basically running little programs to plot these points x_0, x_1, x_2, ... and then if they make t small enough, it appears as a curve which they claim is lining up with the integral curve of y' = (f + g)/2 containing x_0.
I felt it was ill-posed. They were not doing the flow even in their experiment. They were just doing linear approximations of the flows (along the tangent lines).
i guess, i would ask for more precision about what limiting process one has in mind. sure, if your limiting process amounts to averaging f and g, then you might get an average of f and g. i don't think that there's enough detail in what is envisioned to even figure that out.
koro often one fixes a handful of operations at a time. it's rare to consider, say, a multitude of ring structures (going to infinity) on the same underlying set.
@copper.hat Yes. The solution to the ODE is just the family of integrals $\{F + C \mid C\in\mathbb R\}$ for a primitive $F$ of $f$.
@Koro sadly I can't tell you much more. But it's kind of cheating your idea, from what little I do understand. It's like having infinitely many algebraic objects (and hence infinitely many operations).
@copper.hat I am not sure I know what you mean by that. The geometric idea is to view it as a dynamical system where you develop a sequence of points on the plane x_n by starting with an x_0, then flowing along F for a fixed time t to a point x_1. Then flow along G for the same fixed time t to x_2, and then alternate back and forth.
your function looks like int 0...t_1 f + int t_1..t_2 g + int t_2..t_3 f + int t_3...t g
[0,t_1], [t_1, t_2], [t_2, t_3], etc being a partition of the time interval. its length and how many subintervals are in it and how it goes to zero, kind of unknown.
not to stuff words in the mouth of an unruly irishman who is severely deprived of his 'caffeine' (we all know what that means). but that's my best guess.
so @anakhro, for 'nice' functions it seems reasonable that some sort of average could be obtained with suitable conditions on the 'time switching'.
there is a vaguely related issue in control systems called relaxed controls. the idea being that a sort of relaxed/limiting system is obtained by wildly (my term) switching around the controls.
here you would have a system like $y'=u \cdot f+ (1-u) \cdot g$ where $u$ 'switches between $0$ and $1$.
Maybe I misunderstand, but A and B being a disjoint union of the intervals on which you are flowing along respectively the flow of f and of g, and the constant of integration being chosen suitably on that interval?
yes. when you do perform integrals you're implicitly choosing a constant of integration, chosen by the endpoints of the intervals you are integrating over.
@copper.hat so you'd think that just defaulting to $t = 1/n$, and then taking the limit as $n\to \infty$ doesn't necessarily get you the average? That you'd have to be more thoughtful than that?
i think at a minimum, you need to specify a length of the interval that we are finding this solution for, and then some parittion o the interval. keeping track of both how t goes to 0 and n goes to infinity.
Well since you are dealing with a family of curves which you are flowing along, surely the constant by which the curves differ by is important to the set up.
if $f$ is slowly varying and you integrate over $[0, {1 \over n}] \cup [{2 \over n}, {3 \over n}] \cup ...$ then I would expect to get ${1 \over 2} \int f$.
anak something like int 0..t_1 f(t) dt + int t_1...t_2 g(t) dt + int t_2...t_3 f(t) dt would automatically glue these things together with the right +C's.
kind of sucks to write out the general formula for that, let alone analyze it as some upper bound of something goes to 0 and n goes to infinity, but we can't have everything.
have i posted the one were he was invited to oxford to give a talk about the distinction between prose & poetry. probably needs some uk/irish vocab to get the immediate impact plus the degree to which he stood on local sensibilities :-)
i like him, he has no respect but gives lots of dignity
wow, i really do degenerate the tone of any conversation i take part in...
my poor son wonders what he did to deserve such a useless lump of dad
i sent her a picture from my uncle's suitcase (he was a parish priest in livingston outside edinburgh): Medical Fact: Flies spread DISEASE. Button yours.
no wonder the victorian prudes at mse were emailing me
somewhere i have a bunch of manuals my grandfather was given when he landed in north africa in WW2. they are chock full of offensive stereotypes but also contain useful information about social disease.
technically not a landfall in cobh, but gimme a little leeway so to speak
when words appear in brain all mathematical ability, rare as it is, disappears
as an engineer, i was always a bit appalled by some of the modeling performed. odes are reasonable, but pdes involve all sorts of infinities and a depth of knowledge that any practitioner (as in solving, finite elements, etc) cannot possible know.
presumably you could reach the convex hull of the separate solutions. i love convex stuff.
been a while, but there used to be control systems that were loosely based on averaging of some sort. More so you could replace a linear actuator (which cost $$$) by a simple on off switch.
My guess is no too, but a quarter of my students claimed this in a homework assignment, and I don't want to embarrass myself by deducting points if this is true after all x') Also, a fellow TA of mine (who did a master in stochastics) gave full points for this argument...
my wife has a masters degree in statistics but she mainly knows, in considerable depth. the specific models that she uses and not what people would regard as general theory. probability theory in general is just a nightmare world.
i would be interested in the answer to that question.
The object of a proof is to convince your audience. If a student's argument fails to convince you, it's not a crime to say "this is not clear to me", and deduct points. If you wanted to be open to an explanation, you can also offer "if you have an elementary argument for this, drop by my office hours and show me".
that's something that seems missing, any idea of how Y would relate to 1.
when i taught analysis i would give people a fairly generous set of axioms. i said, if you can't prove it in terms of these axioms, with each line citing something on the formula sheet, i'm certainly willing to listen, but don't be surprised if i take points off. it's not enough to be right. spell it out for me. pretend i'm dumb. (i'm dumb!)
Let's say that $Y=0$ whenever $X>1$ and $Y=X$ whenever $X\le 1$. These are not independent. If $X$ is uniformly distributed on $[0,2]$, then $P(X<Y) = 1/2$, but $P(X<Y|X>1) = 0$. Now how do we decide independence?
we used to do a dish like that. my wife learned she had a potato sensitivity, so now it's just my daughter and i who eat that dish. we use a recipe from one of my high school classmates who immigrated from lebanon.
you don't see people publishing papers on something where the kernel is ghastly and only makes sense for some apparently randomly selected subset of the function space.
it seems to be nightshades in general although she can be OK with them when limited in quantity. it's not an on-or-off thing as much as not eating too much of any one thing in a week.
i have a similar thing with cheese. i can eat about half of a pizza's worth of cheese, in about a week's time, but if i eat more than that i break out in hives. so i have to monitor the situation.
@leslietownes let's say you take the Mellin Transform (kernel is $x^{s-1}$) of $f(x).$ What do you do with the Mellin Transform of $f(x)?$ Or does it really depend on context
i only studied the mellin transform in connection with louis de branges's largely failed attempts to prove the riemann hypothesis. he proved some deep results and also glossed over really superficial details.
please use f(x). if you can avoid an integral transform, do so.
when they test orchestral halls for acoustic stuff, they do so by firing a starting pistol or its equivalent and recording the response. the response to the impulse tells you how the room is shaped. it gives you the kernel.