Let's say you have a top-dimensional left-invariant form ω on a Lie group G, and that this ω determines the orientation on G. Denote by L_g:G\rightarrow G the left-multiplication map, which is a diffeomorphism. I'm pretty sure that L_g is then orientation-preserving, and I'm pretty sure this is a proof: since ω is left-invariant, (L_g)*ω = ω, thus L_g preserved the guy we picked to give our orientation, so it's orientation-preserving. Is this the right way to prove this?

(In case you're wondering, I'm on my way to proving something about Haar measure.)

I learned about the properties of logarithms today and I'm struggling with two problems, not sure if I'm doing them right or wrong, but it definitely does not look right:

Use the properties of logarithms to write the expression as a sum, difference, and/or constant multiple of logarithms:

ln z(z-1)^2

here's what I did:

2 ln z(z-1)

annnd the second one:

ln \/x^2
----
\/y^3

what I did:

1/2(2 ln x - 3 ln y)

and I get lost from there :|

thanks for the help!

So my analysis final is coming up on Friday, and I've been going over some homework problems.


(1) A compact operator is self-adjoint if and only if all of its eigenvalues are real.

This, I think, is false. One direction is certainly true (self-adjoint --> real e-vals), but the other, I suspect is not. However, I've been trying to come up with a compact operator with real eigenvalues that is not self-adjoint... to no avail. Thoughts?

(2) If H is a Hilbert space, A is in B(H), and AT = TA for every compact operator T, then A is a multiple of the identity.

Not sure where to begin. I know that if T is compact and normal, then Aφ(T) = φ(T)A, but that doesn't seem helpful.... Any help would be great.


I imagine I might have another question or two tomorrow or Wednesday. Thanks a lot!

Hello! First post here.
My finite math final is at 7am tomorrow morning and my teacher wouldn't cover this question in class simply because it's too long. "It's a three by three. Let's do a two by two instead. Shorter."

Help!

We're supposed to now use the gauss jordan elimination method, which I do know how to do, but I can't seem to make this one work.

Thanks!

**EDIT Wed Dec 9 00:28:16 UTC 2009 **

The notification system has been fixed in the new release... we are currently processing the queue, which is upwards of 12m jobs... please bear with us while our workers chew through this large queue and get your notification / emails out. Some may come more quickly than others due to weights on the notifications themselves, but we are hoping in the next 24 hours to have all the queues cleared and all notifications delivered that had been queued up over the past few days.

Again I apologize for this inconvenience, but we are almost out of the woods as soon as we are done clear cutting some of the forest ;)

Thanks,

** END EDIT**

Hey Guys,

Unfortunately with our last release, and its instability, we were forced to roll back releases. Unfortunately in doing so, it would seem that our notification system has been broken somehow. Our engineers are working on this issue as quickly as possible. We hope to have a patch within the next day, so we can deploy our code and fix the notification system at the same time. Please *bear* with us ;)

Currently all notifications are being queued up so they can be processed as soon as the fix is pushed and verified to be working correctly.

Thank you,

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Hiya,

I've been working on this question:



I've fallen down at the bit where I need to show u_n is uniformly continuous on E_R. To attempt to explain what I'm thinking right now:Am I even attempting the right approach and just messing how to look at the M_ns, or are the two parts of the question not supposed to be related? Or am I wrong in some other way?

In any case, I'm completely stuck - guidance would be greatly appreciated since I've made no headway on this for quite some time now.

Thanks for reading!

I am horrible at finding the period and sometimes the shift which leads to problems finding the key points.

One of the problems:

y = -cos1/3x

this is what I put... am I on the right track?

Amplitude: 1
Period: 2pi/3
Phase shitf: none
Vertical shift: none

I have to figure out the keys points but I don't know if this is correct.

And, what is the period, shift and key points for y = 2cot(x/2) and y = 2sec(x+pi/2)? I am so lost. I was doing fine in pre-calculus until this unit showed up. I'd appreciate any help! Thank you!

So I have a question. I'm looking for a family of subsets of ω with the family having cardinality 2^ℵ₀ such that they are almost disjoint (That is, for any two members, the intersection is finite). The first example that came to mind was collections of real numbers. At least, taking the set of all lines through the origin. This set has cardinality 2^ℵ₀, and is almost disjoint. I'm just wondering is there a way I can restrict this to ω?

My current understanding of the Riesz representation theorem is that it is useful since it tells you what all bounded linear functionals on Lp look like. They look like the integral of fg where g is some function in Lq. So, I was trying to think of an example of a bounded linear functional on an Lp space (1 <= p < infinity) that didn't look like the integral of fg where g is some function in Lq, and then find it's "riesz representation" --

I had a hard time thinking of a bounded linear functional, mostly because I'm still unsteady about the concept of "bounded" in this context. But I'm pretty sure that for f(x) in L2, T(f(x)) = f(1) would be a bounded linear functional on L2. T(af(x)+bh(x)) = af(1)+bh(1)= T(af(x)) +T(bh(x)), so it is linear. Now is it bounded? That is where I'm stuck.

||T|| = sup |f(1)|/||f||2

I don't know what to make of the numerator. It's smaller than the denominator? So ||T|| <= 1 and it is bounded (?) That means there must be a function g in L2 (I choose L2 since conjugate indicies are confusing otherwise) with T(f(x)) = integral [fg]. OK. What is g? That is where I'm stuck again.

I just want to see this theory "in action" or maybe I have misunderstood its purpose.

So for reasons that would take too long to explain, I'd like to make a bunch of single words into eps files. I found a pretty decent way to do this with LaTeXiT for the Mac, which will even let me do XeTeX so I can use the font I want. However, it seems like this is the kind of thing there is probably a pre-existing macro or package for, somewhere. It would really be cute if I could wrap some text in a command and get Xe/LaTeX to think it's an eps, so I could do something like \includegraphics{\MagicEpsThing{some text}}. Does this exist? Thanks in advance!

I need a function that maps the real numbers to pairs of real numbers (one to one mapping):

R-> R*R

The cardinality of the sets is the same (according to wikipedia), so such a mapping should exist. Does anyone know a way to do this mapping?


EDIT: I'm looking for one-to-one AND onto
EDIT: Hah, found the right terminology: I'm looking for a "Bijection"

Can anyone point me to a more rigorous NA textbook (that covers all the essentials: interpolation, forward/backwards/central differences, finite differences, solutions to linear/nonlinear systems, approximation theory, etc.)?

I currently have Burden's, but its exercises are horribly basic and mostly involve using the algorithm provided. I'm kind of wanting things involving more proofs, showing where problems can occur (like in stability, computational time, etc.), you know, more "interesting" things.

And of course something that actually proves the various theorems/postulates in it (or at least what it can). Just something more rigorous and complete.

It can be aimed at either undergrads or grads, I'm not too picky on difficulty.

Doing Differential Equations and I got stuck on this problem, I can't figure out what I'm doing wrong.

( problem and work I've done here )

**EDIT Thu Dec 3 23:24:15 UTC 2009 **

Hey Everyone, we are about to run the last alter job that we need to on our database servers. This will effect userpics / scrapbook / vgift images for the next few hours. Have no fear, your images aren't lost, there is just a really intensive process running on the servers which store the information for mogilefs. Thank you for your understanding and all the LJ love...

Hey LJers,

I just wanted to let you all know that we are going to be performing some mogilefs maintenance over the next few days. We will be upgrading our current version to latest stable as well as changing some db config information to better handle the amount of files we are currently hosting. This shouldn't cause a big impact on site stability, but you may see some minor delays with userpic / scrapbook images appearing or other requests associated with our mogilefs. We would love to not have that happen, but unfortunately with some of the steps we need to take we have to cause a delay with images. I figured this was a better solution than taking down all of LiveJournal because well lets face it, we all need our daily LJ fix ;)

Thanks,

Hello all,

I'm trying to show that H^1([0,1]) is a proper, dense subset of L^2([0,1]).

First of all, I want to find some L^2 function that is not in H^1, but I really don't know many L^2 functions off the top of my head, and finding one not in H^1 has proved even more challenging.

And I genuinely have no idea how to show H^1 is dense in L^2. I mean, I know that if I can show the orthogonal complement of H^1 is {0}, then that'll do it.....but I need some help getting started.


Thanks!

I went to a guest lecture the other day and I was very pleased that I was able to follow... oh... almost half of it! At any rate, he was talking about (among other things) how to select a random matrix with determinant 1. (so a random member of SL(N, Z))

He said there were two ways of doing this.

1. "the number theory way" picking it "out of a hat"

2. using a generating set and random walks on a graph (where did the graph come from??) to make "words" with the generating set and then these words are going to form a random matrix.

I could not understand the 2nd way-- has anyone heard of this? Why was he using a graph? making a random graph and choosing a "random walk" seems like making one problem in to another for no real reason and the whole process was lost on me. Anyone know what this is about?

I was wondering how can I use exponential/log functions to find out when the population of a town is cut in half. I was thinking to use the pe^rt formula but I'm not sure... Can anyone help me asap!!