Currently studying for Real Analysis qualifying exam. This is one problem I haven't been able to figure out though:

Show that L_3 [0,2pi] is a subset of L_2 [0,2pi]. Also, show that ||f||_3<=||f||_2.

Sorry for the poor formatting. Anyway, it is fairly trivial to show that L_3 is a subset of L_2 for any space of finite measure from first principal. I am not sure how to show the inequality, though. Part b) of the problem is showing that l_2 is a subset of l_3, and then show a similar inequality, but I think the method to show the inequality is probably similar in both cases.

the true question might be who is more correct about discovery: the discoverer, or those who attempt to define the methods of discovery

This inequality is backwards (if it were true, then whenever f had finite L^2 norm, it would have finite L^3 norm, so L^2 would be a subset of L^3!) and it appears to be off by a constant, unless you are dividing the Lebesgue measure by 2\pi.

Anyways, one generally has the following inequality on a space (X,\mu) for 0<p<q\leq\infty and a measurable function f:

\|f\|_p \leq \|f\|_q\mu(X)^{1/p-1/q}

The proof is just an easy application of Holder (write |f|^p as |f|^p*1 and use conjugate exponents q/p and q/(q-p))

can't figure this out. int((x^3)(5^x^4)).

on a more general note, what is int(fg)?

i would assume integrate

Substitute u=x^4 and everything should fall into place.

There is no general formula for \int fg. For example, \sin x and 1/x both have easy antiderivatives but (\sin x)/x has no antiderivative using only elementary functions.

doh. so simple lol. i kept looking for the formula for int(fg) and couldn't find it. then i tried to do int (udv) with u=x^3 and dv= 5^x^4 and that was a total mess.

EDIT: I suck at reading

l_2(Z) is equivalent as a Hilbert space to L^2(S^1), by mapping the "standard basis" e_n to the orthonormal functions exp(i*n*\theta)

Under this isomorphism S obviously transforms to the operator on L^2(S^1) given by multiplication by the function exp(i*\theta). The spectrum of multiplication by a function is the range of the function (easily proven directly, or use "spectral permanence" for C* subalgebras).

FYI, this computes the continuous functional calculus applied to S. That is, the commutative C* algebra generated by S is isomorphic to the space of continuous functions on the unit circle.

(An even slicker way is to observe that S generates the group-C* algebra of Z, so its spectrum is the group of characters Hom(Z, S^1) = S^1.)

Good luck on your qual. FWIW, you can get a lot of mileage out of the ideas above, if you really understand them well. Gelfand transform FTW.

looking for some help with

specifically how to use implicit differentiation on the xy term

Well, in implicit differentiation, you (locally) treat y as a function of x. So, how would you differentiate x*y(x)? (use the product rule)

thanks for the help blah_blah

For two integrable functions f, g and where y = f(x), in terms of f', g', g o f', etc. what is the integral of (g o f)(x) dy?

Does anyone have an opinion on which calculator I should get. My current calculator is ancient and I can't convert to radians on it. I was looking at some reviews and the TI-83, 84 or 89 were all recommended, T-83 is £50, 84 is £80 and 89 is £120. My course atm involves calculus, integration and differentiation, it's a year 1 course, and I have more advanced (I assume) coursed in year 2 and year 3, so I'm not too bothered about forking out £120 for a TI-89 is it helps me squeeze out an extra % or two and I'll be using it for years.

ps. One of my courses will involve "The analytical (as opposed to numerical) solution of first-order and of linear, constant-coefficient, second-order ordinary differential equations is discussed, followed by systems of linear and non-linear differential equations and an introduction to methods for solving partial differential equations. The topics in algebra are vector algebra, the theory of matrices and determinants, and eigenvalues and eigenvectors. We develop the elements of the calculus of functions of several variables, including vector calculus and multiple integrals, and make a start on the study of Fourier analysis. Finally, the study of numerical techniques covers the solution of systems of linear algebraic equations, methods for finding eigenvalues and eigenvectors of matrices, and methods for approximating the solution of differential equations."

For what it's worth.

In my experience, a TI-89 is well worth it. If you know how to use it well, it can make tests involving a calculator a breeze.

I remember using it on the AP Calculus exam I took in high school. The calculator section used was almost trivial with proper use of the TI-89. Whenever the testmakers thought they were being clever by using functions that a pocket calculator couldn't integrate easily, I simply taylor expanded those functions and got an answer with arbitrary approximation.

So, in conclusion, I think it's worth the extra few dollars.

Stuck with this exercise in my book:

Q. Find f''(3) where f(x) = 45 ln x

I know that f'(x) = 45/x so 45/5 = 9

I do not know how to find a second order derivative for f(x)= ln x

The answer in the back of the book is:

f''(x) = - 45 / x^2

But how do they/I get to that? I can do 2nd order derivatives of f(x)=x^n but haven't come across 2nd order derivative of f(x) = ln x at all.

Longliveyork: thanks for the tip.

Go from here.

Been thinking about this while watching Italy - Slovakia and figured that 1/x is also x ^-1, so f''(x) is -1 x ^ -2 = -1/x^2. So that f''(3) = -45/3^2 = -5

Sometimes I feel like maths is owning me pretty hard

The problem you seem to be having based on your previous post is that you are thinking about memorization WAY too much. You said you know how to take the second derivative of x^n, but not ln x. However, you know how to take the first derivative of ln x, and you know how to take the first derivative of x^n, which 1/x is an example of. Taking the second derivative is nothing more than taking one derivative, then taking another derivative.

Could you explain what you mean by 'thinking about memorisation WAY too much'? I'm not sure what I'm doing wrong, but I'm starting to struggle slightly and it's only basic calculus. I think it's that I forget what I've previously learned, such as x^-1 = 1/x and because of that I can't solve problems that rely heavily on previously learned material. I'll probably have to go back to basic algebra and plug some holes in my knowledge because it's really starting to show.

Was lurking this homework question and I have a question;

Why cant you use the quotient rule (even though you dont really have f(x) and g(x) because f(x) is just a constant?

45/x , g(x) = x , f = 45 g'x = 1 , f' = 0

((f' * g(x) - f * g'(x)) / (g(x)^2)) = (0*x - 45*1 ) / (x^2 ) =

-45/(x^2 )

Even though its a detour, just wondering if this is allowed?

Yes. You can. I advise students not to, because they seem to have problems with f '=0 interacting with quotient rule for whatever reason.

Students also often ask "can I use X on problem Y?" Well, go back to your calculus book and check the hypotheses for the quotient rule: IIRC, you need f(x) and g(x) differentiable and g(x) nonzero. Certainly f(x) = 1 and g(x) = x satisfy this for x not 0, so you can use the quotient rule.

One thing students rarely do is check the hypotheses on the theorems (or "rules") they learn. Understanding these leads to a better understanding of mathematics in general (as you can start to ask things like "well, what goes wrong when condition X is not satisfied?").

Ok today was first day of advanced engineering math class im taking this summer, been a year and a half since I have taken differential equations so I have forgoten a lot. we were given

u''(t)=f(t)
u(0)=0
u(L)=0

L=10
f(t)=e^(-t^2)


Now I'm not even totally sure what all I need to solve for this prof wasnt to specific, do need to find the kernel function I know

Any help is greatly appreciated, class doesn't have a book or anything and im stuck atm.

I didn't work through the solution but I think this page is your friend: dip.sun.ac.za/courses/TWB834/TFtechniques1.pdf

I know about the cantor set and how to answer this questions when a=1, but not in this instance. help is greatly appreciated.

Suppose 0<a<1 and f is the characteristic function of C_a ("fat"cantor set). Show there exists a K contained in C_a with m(K)>0 (lebesgue measure) such that f is discontinuous at each point of K.