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Solutions to IVT Problems November 26, 2008

Posted by putnam120 in Math Related, Uncategorized.
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(a) Since f is continuous on a compact set (\left[a,b\right] it attains both its maximum and minimum values, so m\le f(x)\le M for all x\in\left[a,b\right]. Now using these poor estimates we have that m\int_a^b g(x)dx\le\int_a^b f(x)g(x)dx\le M\int_a^b g(x)dx. So if we consider the function V(x)=f(x)\int_a^b g(x)dx we have that V is continuous on the same interval as f. Thus we just apply the intermediate value theorem to V and the desired result follows.

(b) WLOG assume that f(0)=f(1)=0. Let n be given and define new function by L(x)=n\left(f(x+1/n)-f(x)\right). This is just the equation for the slope of the line connecting the points f(x) and f(x+1/n) and is continuous because f is continuous. Now consider the set of points x_i=\frac{i}{n} for i=0,1,\dots,n-1. Now look at the values L takes at these points. If L(x_i)=0 for any i then we have found a solution, so assume that L(x_i)\neq 0 for any chose of i. So WLOG assume that L(x_0)>0, since f(0)=f(1) we know that there must be some i such that L(x_i)<0, call this x_k. Then we there is a x\in\left(x_0,x_k\right) such that L(x)=0.

(c) For this one I will prove something a little stronger. All that we need to assume about f is that it is integrable, and has what I shall call the extreme value property on our interval. Basically this property says that on the interval there exists a point c such that f(c)=\sup f(x)=M and a point d such that f(d)=\inf f(x)=m, where x ranges over all the values in our interval. So clearly an increasing function satisfies these conditions. Now define a function by

V(x)=m\int_a^xg(x)dx+M\int_x^bg(x)dx.

Now V is continuous and V(b)\le\int_a^bf(x)g(x)dx\le V(a), thus by the intermediate value theorem we are done.

Some after thoughts: Problem (a) was pretty straight forward and didn’t take too much insight to actually solve. However, problem (b) was a little more challenging until I visualized what I was being asked to prove. This led me to the idea to look at the slope of the connecting line segment. Finally problem (c), I at first wondered why you were given that f was increasing and not continuous, or even just integrable. As can be seen in my proof continuity imposes more conditions than are necessary, while integrability does not provide you with enough. This led me to think about what makes increasing functions special (well one of the things that makes them special).

Some Problems Involving I.V.T. November 25, 2008

Posted by putnam120 in Math Related, Uncategorized.
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Well I am in a big Analysis mood. I guess that it has something to do with the fact that at this point in my life Analysis is the topic in math that spikes my interest the most. This can be accredited to my Calculus BC teacher from high school (Mrs. Johnson) and my undergraduate Advanced Calculus professor (Dr. Shen). They both did a wonderful job of presenting material and showing applications and fascinating consequences.

Anyway onto the post, I was reading through the intermediate real analysis section of Problem Solving Through Problems. Here are some of the problems I found interesting.

(a) Suppose that f:\left[a,b\right]\to\mathbb{R} is continuous and g:\left[a,b\right]\to\mathbb{R} is integrable and such that g(x)\ge{0} for all x\in\left[a,b\right]. Prove that there is a number c in \left[a,b\right] such that

\int_a^bf(x)g(x)dx=f(c)\int_a^bg(x)dx.

(b) Let f:\left[0,1\right]\to\mathbb{R} be continuous and suppose that f(0)=f(1). Prove that for each positive integer n there is an x in \left[0,1-\frac{1}{n}\right] such that f(x)=f(x+1/n).

(c) Assume the same conditions on f,g as in part (a) except that instead of being continuous f is now assumed to be increasing. Prove that there is a c\in\left[a,b\right] such that

\int_a^bf(x)g(x)dx=f(a)\int_a^cg(x)dx+f(b)\int_c^bg(x)dx.

My thoughts: The solution to (a) is a pretty straight forward application of the intermediate value theorem. For problem (b) I am considering looking at the slope of the line connecting f(x) and f(x+1/n). Finally for (c) I am going to try and generalize it in an appropriate way. My solutions should be posted in the near future.

Hölder’s Inequality November 23, 2008

Posted by putnam120 in Math Related.
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I am only going to do the case where f,g are real functions. The result however, still holds if they are complex.

Statement: Suppose that f,g are integrable functions with respect to \alpha on the interval \left[a,b\right]. Additionally p,q\in\mathbb{R^+} such that \frac 1p+\frac 1q=1. Then we have the following inequality:

\displaystyle\left|\int_a^bfgd\alpha\right|\le\left\{\int_a^b|f|^pd\alpha\right\}^{\frac 1p}\left\{\int_a^b|g|^qd\alpha\right\}^{\frac 1q}.

This can also be stated as ||fg||_1\le ||f||_p||g||_q.

Lemma: If u,v\ge{0} then uv\le\frac{u^p}{p}+\frac{v^q}{q}, where we have the same conditions on p,q as before.

Proof of lemma: Just apply Jensen’s Inequality to e^x and the fact that e^{\ln x}=x. \mathbb{Q.E.D.}

Proof: Without loss of generality we can assume that ||f||_p=||g||_q=1, if not we can just divide f,g by the appropriate constants and make it so. Now from the lemma we have that \forall x\in\left[a,b\right] \displaystyle |f(x)g(x)|\le \frac{|f(x)|^p}{p}+\frac{|g(x)|^q}{q} we then integrate both sides of the inequality and the result follows. \mathbb{Q.E.D.}

Integral Test November 22, 2008

Posted by putnam120 in Math Related.
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Well it’s been a while since I have posted a math related post. So I am going to do this one on one of the problems from our analysis homework.  Basically we were asked to prove the integral test, not too difficult but definitely something that should be done.

Statement: Assume that f(x)\ge 0 and that f decreases monotonically on \left[1,\infty\right). Then \displaystyle\int_1^\infty f(x)dx converges if and only if \displaystyle\sum_{n=1}^\infty f(n) converges.

Aside: When I submitted this to my professor for grading in proved the theorem in both direction. Here I am going to try and combine them, thus saving time on my part.

Proof: Consider the interval \left[m,n\right] where m,n are integers with m<n. Additionally let P be the partition \left\{m,m+1,\dots ,n\right\}. Now because f is monotonically decreasing and f\ge 0 we have

(1)   \displaystyle 0\le\sum_{k=m+1}^nf(k)=L(P,f)\le\sum_{k=m}^{n-1}f(k)=U(P,f)\le\sum_{k=m}^nf(k)

Let \epsilon>0. If \displaystyle\sum_{n=1}^\infty f(n) converges then there exists an $N$ such that \displaystyle\sum_{k=m}^n f(k)<\epsilon whenever m,n>N. Similarly if \int_1^\infty f(x)dx converges we have that there exists W such that \int_m^nf(x)dx<\epsilon whenever m,n>W. The theorem follows from combining these facts with (1). \mathbb{Q.E.D.}

I would like to mention that I have left out some of the small details, such as proving that the integral actually does exist if the sum converges.

IMPORTANT NEWS November 22, 2008

Posted by putnam120 in Life Events, Math Related.
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Well for unknown reasons (well not totally unknown) I can no longer use LaTex on this blog. I did look up ways to get this back but ran into a few problems. Most of the solutions only worked for Unix like systems, and I would like to be able to use LaTex on the blog even if I was on a Windows machine. Also most of these solutions would have required me to add another version of Tex to my linux system and I didn’t really want to do that. There was however, one solution that would work with all systems (supposedly). All I had to do was edit the source code for a GreaseMonkey script, but after doing that I was still running into the same issues as before so I quickly gave up on that dead end.

I will still be using this blog but whenever I want to post anything mathematically related I shall post a link to my WordPress blog in the post. The reason I am using WordPress is becasuse it has LaTex built in and thus less work on my part.

Welcome Home Steve Spurrier November 16, 2008

Posted by putnam120 in Uncategorized.
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Well I finally went to a UF football game. I was pretty much forced into it by Veronica and Juliana. Really it wasn’t a bad choice for a first game really, UF was playing USC (South Carolina, not Southern Cal), so I got to see Steve Spurrier.

We had pretty good seats for the game, right behind the band. I admit it was quite the experience, though not all that people have talked it up to being. The game itself was not really worth it since it was a blow out (56-6 UF) and really who want’s to watch that?

The half time show…oh how I miss half time at FAMU games, and that is all I have to say about that.

Before:

After:

Amendment 2 November 6, 2008

Posted by putnam120 in Uncategorized.
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Sometimes I am just shocked at how closed minded and “selfish” Americans (in this particular case Floridians) can be. Look at the title, if you don’t know what I’m talking about follow this link.

Below is what one of my friends had to say about this topic. I would have to agree with him 100%, yes even the willingness to leave America and live somewhere else.

It is not often enough that something happens in this country that makes me ashamed to be a part of it. However, there has been a growing trend which has stopped me dead in my tracks. With the recent Florida elections, I have witnessed first hand the bigotry that continues to permeate our society.

I am speaking of course about the Florida Constitutional Amendment number 2. This amendment passed on November 4th, and it prohibited homosexual marriage and stripped rights from a domestic partnership. For those of you who are ignorant on the subject, domestic partnership has not been an exclusively or even predominantly homosexual union. It is merely what the name says. Two beings entering in a legal partnership in order to relieve stress of financial situations among many other things. Now, domestic partnerships across Florida have been reduced to a mere shadow of the benefits a man and a woman may have by being legally married.

This ignorance is gut wrenching. Imagine a similar amendment, one which prohibited interracial marriage. That’s right Florida, let’s all vote on whether our constitution says I can marry a black woman. Such a proposition would be met with instant hostility and cries of overwhelming racism. Hell, lets take it one step further. Black people can’t marry. Period. There is no alternative to give them equal status of being “married”. By the simple fact that they were born black, they are denied the rights which are extended to another race.

If you think there is a fundamental difference between those two examples and amendment 2, you are blinder than I ever thought my friends could be.

There is no excuse for this. There is no justification. This, and the progression of 29 other US states is fueled by nothing but sheer bigotry towards homosexual people. In the process, states such as Florida have crippled many non-homosexual partnerships, all in the name of preserving the American family.

I cannot and I will not tolerate this. My desire to complete my education keeps me in the US, but I vow that if such blind hatred continues to be entwined into the very constitutions of the states that make up this nation, I will be leaving. I love America, and I have always thought how lucky I was to live in a nation that was for the most part, better off than most others. But this is not an issue I will compromise.

Normally, I would be open for debate on an issue, but I’m afraid I am making an exception to this. I am not gay, and honestly this decision will probably never affect my personal life. It is the principle itself which is evil. If you think this could have been defended by saying that the domestic partnership was abusable, then I counter with allowing homosexuals an equal equivalent to marriage. If you counter with marriage only being between a man and a woman by decree of God, I will counter with an absolute declaration of your stupidity.

My only salvation is to hope that in the future, children will read about this is textbooks, much as we read about the injustices against black people in America and their faces will be filled with shock. Shock that at one point in the not so distant past, their country, their grandfathers and grandmothers, could have believed in such bullshit.

Also I would like to mention that I am not gay and just like my friend this will most likely not affect me in my life. But it is the principle of the matter, who gave the government to power to say that marriage is the union of one man and one women? Some people will say something like “…it’s in the bible…”, well I am still waiting for someone to actually show it to me. Also since when was the bible the doctrine that governed this country? The bible also says to turn the other cheek…and yet we are in a war over a particular even that happened a few years ago. So don’t tell me that we are a “Christian nation”, because being a Christian is about more than just reading the bible, you also have to follow the lifestyle, and so far this country isn’t doing a very good job. Yes I know that homosexuality is looked down upon by the bible (this I have actually read and/or been shown) but so are the 2 commandments Jesus gave us (for crying out loud he cut the 10 down to 2, at least have the decency to try and follow them both). Here they are, “The first commandment is this: Hear, O Israel: The Lord our God is the only Lord. Love the Lord your God with all your hear, with all your soul, and with all your mind, and with all your strength. The second is this: Love your neighbor as yourself. There is no other commandment greater than these.” (Mark 12:29 if you need a reference). Now I don’t know about the first one, but this Amendment surely goes against the second.

I shall leave you with the following quote, and if you don’t see how it relates then I truly do feel sorry for you.

If mankind minus one were of one opinion, then mankind is no more justified in silencing the one than the one – if he had the power – would be justified in silencing mankind.
- John Stuart Mill

Minor Incidents November 2, 2008

Posted by putnam120 in Life Events.
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Well I have decided to tag a Computer Science minor to go along with my Mathematics major. After this semester is over I will only have 4 more CS classes to take before I have completed the minor, this basically comes down to 1.3333… CS classes per semester. The classes I have to take are; Applications of Discrete Structures, Intro. to Computer Organization, Data Structures and Algorithms, and Operating Systems. Due to prerequisites I will be taking the fist two next semester, then Data Structures fall of next year, and finally OS my last semester (because this seems to be the most difficult of the 4 and I would rather not to have to worry about it while working on graduate school applications).

I came to this decision after realizing that there weren’t any outside of major classes that interested me, other than Economics, Finance, or CS. Honestly, I find all these disciplines to be just as interesting and rewarding. However, what made me choose CS was the kinds of people that would be in the classes. In my experience most of the people in the Economics or Finance classes don’t seem to have any interest in the subject and are only there because they have to take it for the major. In addition they constantly complain about the work and other consequential details about the class. While the CS majors are very adamant about their classes and willing to learn. Also there isn’t as much complaining about the class (unless it is to say that they aren’t learning enough or that it’s not challenging enough).

In addition to the above issues there are some other personal “problems”. Now that I think about it problems really isn’t the most appropriate phrase to use. Here is the general overview: A little while ago (2-3months) stopped talking to one of my friends because I felt used. A few days ago I was pretty bored and lonely (most everyone I knew was out of town) and I considered giving them a call. After thinking about it I didn’t mainly because I would feel that I was using them, and that’s just something I refuse to do, even to someone I no longer care about. Well I care about them but it is basically at the same level I care about a complete stranger. The day after this happened I talked to one of my friends and basically what I remember from the conversation was this, “…you will constantly be filtering people in and out of your life. It sucks at time but it is just something you have to learn to accept.”

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