Nick S

So here's what I know: The unit was running (or so I thought) for a while, but the temperature inside was going up, and is now 80 degrees, and the thermostat is set to 73. The inside unit seems to work as per the norm, but the air coming out of the vents is lukewarm, and slow. As I saw in a few posts online, it seems that the problem was likely to be in the outside unit. It is humming (making a standard electrical motor hum), but the outside fan does not run, and the top of the unit (where I presume the fan motor sits) is very warm (read: hot!) to the touch. I don't know if this is something simple to diagnose, but this is everything I have been able to observe at this point.

While I have emailed the landlord to try to get someone out here, is it something very simple to fix in the meantime?

(I am leaving it off for a few hours to see if it cycles back to actually working, but I'm not holding my breath...)

I thought of a fun little exercise in measure theory that turned out to be more interesting than originally anticipated, and I wanted to share.

Prove or provide a counterexample to the following 'conjecture':

Suppose (a_n) is a monotone (increasing) sequence of real numbers contained in the interval (0,1), and suppose that Lim_{n -> infinity} a_n = 1.

Also, suppose (b_n) is a bounded sequence of real numbers.

Then there exists a function f in BV([0,1]) such that f'(a_n) = b_n for all n >= 1, if and only if

Lim_{n -> infinity} b_n

exists.

A recent poll said that roughly 66% of Americans (with +/-3% standard error) support a pathway to citizenship for illegal immigrants.

We all agree that they broke the law by entering this nation without going through the proper channels, and approximately 9600 American citizens per year are killed in altercations involving illegal immigrants each year. Why are so many in favor of rewarding their illegal behavior by putting them at the front of the line for citizenship, and putting people who entered this nation through (their own merits, money, and hard work, and in particular, through) the legal channels of DHS, USCIS, and INS behind these people in the processes?

Should we not be punishing illegal activity in this nation (i.e. enforcing the laws already established), rather than punishing those law-abiding residents of this nation who are seeking citizenship in the legal way?

P.S. This is not a racial argument, nor is this a bigoted argument. It is an argument of law, and the enforcement thereof.

3.4L V6, and while it has been a great car for most of the time I've had it, I'm having some issues.

I just had the rotors, pads, one of the tie rods, and an oil pressure switch replaced. But this is a completely new ball game.

This started a year ago, but only happened once, so I ignored it. It was a very cold winter night, and it was spitting a wet snow/sleet mix, and I was on the highway. The car was running fine, but I started to notice the cabin becoming cool, then freezing cold, to the point that my breath was visible, and frost was forming on the interior of the window from the moisture in my breath. I got home, but had finals, so I was unable to take care of it, but it never did it after that night, so I left well enough alone.

A few weeks ago, I noticed that the turn signals occasionally fail to come on. It's very seldom, but sometimes they just fail. The heat has again decided to fail, but now it is any time there is any precipitation when I am traveling at highway speed. The thermostat is fine, because there is still hot air being generated by the car, but the fan is not blowing it into the cabin, so the passenger cabin becomes very cold (37 degrees out tonight, and the cabin temperature fell below 45 degrees with heat on full). Another thing of note: It has a hard time starting in cold weather, and the battery is not even a year old.

I'm likely taking it in for diagnostics in the morning, but want to be somewhat informed, because I don't want to get taken. What is it likely to be?

I refer, here, to the Fundamental Theorem of Calculus in the context of measure theory. For increasing functions,

Int_[a,b] f' <= f(b) - f(a)

A perfect example of strict inequality is the Lebesgue-Cantor function, defined on [0,1] (The integral is 0, but f(1) = 1, and f(0) = 0, hence the RHS is 1). The question I have is:

What is a necessary condition for equality above? An equivalent question:

What is a necessary condition for a function to be recovered from its derivative?

I know that a sufficient condition is absolute continuity. Is there a weaker necessary condition that may be used to generalize this theorem further?

We have a theorem due to Lebesgue that states the following:

If f: [a, b] --> R is an increasing function, then

i) f' exists almost everywhere

ii) f' >= 0 where it exists

iii) there exists a measurable function g such that g = f' almost everywhere

iv) Int_{a,b} f' <= f(b) - f(a)

What I wonder is:

What (if any) additional condition(s) is(are) required to imply that there exists c in (a,b) such that f' is continuous at c?

Further, for any epsilon > 0, is there a function h, continuous on [a,b], such that

Int_{a,b} |h-f'| < epsilon?

Sorry that this question is a bit vague, but I don't know a better way to state it.

I'll preface this by saying I'm a fan of Kieth Olbermann's social commentaries on Countdown. He comes across as quite liberal, but at the same time I've seen him make some very candid, and very harsh comments about people on the left as well as on the right.

This is a question about marriage. I am not homosexual, but I hold no personal philosophy of superiority as a heterosexual male. I instead simply see that we are all citizens of the world, each trying to carve our own paths to happiness. The passage of Proposition 8 has prompted many questions everywhere, but I'd just like to know people's take on what Mr. Olbermann had to say.

http://video.msn.com/video.aspx?mkt=en-us&vid=1d93...

Personally, I believe it to be, at best, reprehensible, and at most, inconceivable, that the heterosexual populus of this nation feels justified in considering the marriage of two same-sex individuals as destroying the sanctity of marriage. Is marriage truly sacred, when 50% end without "till death do us part" actually coming in? Is it possible that we are being unfair in making sure that same-sex couples are unable to celebrate their love in the same manner in which heterosexual couples do?

Yes, I understand that the marriage of a man and a woman is dictated in the Bible, but it also says quite explicitly in the New Testament "Love your neighbor as yourself." In other words, "do unto others as you would have them do unto you". I have to agree with Mr. Olbermann here. What if heterosexual couples were no longer allowed to marry?

Just something to think about...

This isn't a question of right or wrong...It's a question of love, and what empowers us with the authority to dictate who can or cannot express that love, and exchange symbols of that love in the eye of the state? Remember, before answering, that we are to recognize, and be dictated by, separation of church and state.

Also remember to be respectful of others, as this is a sensitive issue in society. One could argue this is the racism of our generation, and I would agree.

I have a butyr aldehyde treated thermal oxide (SiO2) surface on a silicon wafer, and am trying to figure out a way to get E. coli to stick to the surface. I have thought of how to keep them alive. When incubating the disc, I just need to keep them in a minimal growth medium. When washing with PBST, I should likely actually mix PBST and the minimal growth medium to keep them from going dormant during the wash.

To bond them to the surface, would a simple antibody print on the disc work? I am considering trying to use a polyclonal anti-rabbit antibody from Abcam, but am unsure if that will be enough to grab onto the bacterium (the antibody is roughly 1/1000 the size of the bacterium), and don't want to make a hasty purchase if it definitely won't work ($300/mg of antibody).

Is there another direction I should be looking to go? Poly-lysine is not an option, unfortunately, as I am trying to observe a phenomenon that is very height sensitive...

Any ideas?

This summer, I'm likely going to be doing a study on E. Coli. I know most strains are fairly harmless (O157:H7 not being one of them, but I do not believe this is one I'll be working with), and that it will grow in temperatures around 37C, perhaps up to 49C, but what are the ideal conditions for growth and culture of this bacteria? Can it be grown in environments maintained below 98.6F? Can it survive any room temperature cultures?

I'm sorry if this has been asked before, but I'm completely new to microbiology, biochemistry is not a strong suit for me.

Thanks, everyone.

Nick

Let R+ = {x | x in R, x > 0}, where R is the field of real numbers.

Suppose A is a subset of R+, and define 1/A = {1/y | y is in A}

Note that 1/A is also a subset of R+

Further, let C be the field of complex numbers and suppose

f: A --> C

is such that f is integrable on A, and further, f is integrable on 1/A (integrable in this context could be restricted to general Riemann integration, but Lebesgue integration may be more useful)

Question:

Is there a way to evaluate the following expression:

∫ (A) f(x) f(1/x) dx

I know it should be defined, because, letting g(x) = f(1/x), as f is integrable on 1/A, then g is integrable on A, so the product fg is integrable on A...

I'm wondering if there's a way to evaluate it.

As a particular case, consider the integral:

∫ (pi to 3pi) e^((e^(ix))/2) e^((e^(-ix))/2) dx

(yes, this is ∫ (pi to 3pi) e^(cos(x)) dx, but I wanted to see if there was a trick to the more general case, as I'm guessing this integral is done by contour integration)

As a follow-up to

http://answers.yahoo.com/question/index;_ylt=Arkcq...

If, as scythian said, we allow x=1/2n, it does indeed get rid of the t^(x-1/2n) term, but it also affects the exponential term, and changes the nature of the integral.

For, if we let x=1/2n, then the integral:

(1/2n)Γ(1/2n) = Γ((2n+1)/2n) = ∫ (0 to ∞) e^(-x^2n) dx

= ∫ (0 to ∞) e^(-(1/2n)^2n) dx

but e^(-(1/2n)^(2n)) is a constant for any fixed n, so the integral above is not defined.

Can anyone tweak this proof? Thank you.

This isn't for a class, but it's simply something interesting I thought of.

Take a Gaussian:

Int_(-infinity, infinity) Exp[-x^2] dx

Now of course this evaluates to Sqrt(pi)

What if we let y = x^2 and took the Gaussian in y? It goes to 2*Gamma[5/4]...y=x^3? 2*Gamma[7/6]. This pattern follows all the way to at least n = 100, for y = x^n...

Now, this is something I'd like to know a starting point on (I know this should probably be done using induction, and I have the n=1 case done, as Sqrt(pi) = 2*Gamma[3/2]; What I need a starting point on is how to carry out the inductive step)

For any natural number n, prove that

Int_(-infinity, infinity) Exp[-x^(2n)] dx = 2 * Gamma[(2*n+1)/(2*n)]

Further, is there a finite natural number n > 1 such that this result is expressible in simple closed form (i.e. an exact numerical expression, and not in terms of the Gamma function)

In regards to the response by JCS here:

http://answers.yahoo.com/question/index;_ylt=AkPiG...

In particular, where he says:

"dxdy = dx∧dy (it's a 2-form)

Where the "∧" is an algebraic operation called the exterior product; again, it's a generalization of the more familiar exterior product on R^3."

Why is it not instead the following, at least for a 2-form?

dxdy = |dx∧dy|

And in more general terms, are n-forms going to be something like

dx1dx2...dxn = |dx∧dx2∧...∧dxn|

And how is this generating Stokes theorem from de Rham's Theorem?

I have read the proof of de Rham's Theorem, after working out a satisfactory justification of it, but was unable to figure out the connection between the differential form in question and the Stokes Theorem in R^3...

Also, I would appreciate hints on arriving at Gauss' Theorem and Green's Theorem from de Rham's, but in particular, Stokes Theorem is the focus.

Thank you very much...

i am asking about 2 main things...

how would one go about proving that the vector product is invariant under an orthogonal transformation?

also, how would one go about showing that the arc length, curvature, and torsion of a parametrized curve are invariant under rigid motions?

-------------------------

for the first one, i had the idea that you could consider the vector product of Ax with Ax (A an orthogonal transformation)...and then just take a basis of A, say v1, v2, v3, where v1, v2, v3 are just rows of A. then you could express the vector product as a determinant of a matrix whose entries are just inner products...then i suppose it might fall out naturally...

but for the second one, clearly the arclength is invariant wrt translation and orthogonal transformations, but how can i express that mathematically? is it the same as considering invariance of the norm of the tangent vector over translation and orthogonal transformations? it seems correct, but almost too easy...