DJBruce

I personally think the easiest way to think about this is by looking at the math. So by Newton's Law of Universal Gravitation for a two bodies of mass [math]m_{1}[/math] and [math]m_{2}[/math] the force between the two bodies is: [math]F_{12}=F_{21}=\frac{Gm_{1}m_{2}}{r^{2}}[/math] So if you consider want to see how fast a body would fall you would say: [math]v_{1}=v_{0}+at=v_{0}+\frac{F}{m_{1}}t=v_{0}+\frac{\frac{Gm_{1}m_{2}}{r^{2}}}{m_{1}}=\frac{Gm_{2}}{r^{2}}t+v_{0}[/math] so we see that regardless of the mass of the object falling the acceleration is independent of the mass of the falling body. This is because although a large body, like the sun has a much larger force acting on it because of its large mass it needs a bigger force to move it. Also it is key to note that Newton's Law is reciprocal in that it states the force of gravity is equal in magnitude for both objects. Think Newtons third law ie: equal and opposite forces. So this means that the force of gravity exerted by the sun on the Earth is the same as the force exerted by the Earth on the Sun. Hope this helps.

This could be a stupid idea, but has a journal ever consider installing a rating system? I was thinking of something all the lines of having groups of registered users that consists of people with some level of credibility, and these users could use a simple rating system- like a 5 star system- to rate a paper they have read. I know this would by no means be a great system, but it might provide administrators with a better understanding of what your peers think of your work. This would subvert the current method of using citations as a level of quality.

I believe he is referring tocosmic inflation. As for your theory you seem to make a lot of "maybe" statements without a lot of explanations, evidence or proof, and for the most part none of them make any sense.

Play lacrosse, ultimate, golf, run, or do anything else athletic. Go boating and hang out with friends. I guess I really don't currently have any real hobbies. For me it is my innate ability to procrastinate what I have to do during the summer, and during school breaks. If you look at it I rarely post when I am way at school.

A matrix is just a rectangular array that is hand for storying and displaying things, simplistically you can think of it as a type of chart since each row and column represent something. The entries of a matrix can be number, variables, or even expressions, and so there are numerous different things you can get out of a matrix depending what your matrix represents. Matrices can be used for things like exploring the structure of networks to finding if a linear function between finite dimensional vector spaces is invertible. That being said matrices are not that special by themselves, but instead become useful when tied with another mathematical field. As for putting the 64 codons into a matrix I am not sure if I see an practical application, however, my guess is that a biologist might be able to come up with something.

Although in hindsight Pawlenty's pardon looks horrible, and the crime the father committed is absolutely horrible I honestly think that had I been in Pawlenty's place back in 1994 I would have issued the pardon. I mean the pardon was for a 19 year old who was convicted of statutory rape after he impregnated his 14 year old girlfriend, when the convicted promised to marry and support the mother and child. Looking at what was known at the time I think that the pardon was not unwarranted. This along with the fact that the pardon was granted unanimously by a three person committee will help Pawlenty perform damage control if this issue comes, however, I doubt that it will come out in the primaries as this happened last November and so far it really hasn't been a network news talking point. That being said I didn't think that Pawlenty was going to get the Republican nomination, and so I this really doesn't change my stance that he will not get the nomination. However, if he does prove me wrong and ends up getting the nomination I bet that this will come out, and at that stage it certainly will not help his campaign, but I think any half decent PR person should be able to handle this fairly easily. For those of you who have not heard the story here are two articles on it: http://blogs.citypages.com/blotter/2010/11/jeremy_giefer_tim_pawlenty.php http://www.rawstory.com/rs/2011/05/23/tim-pawlenty-wiped-a-serial-child-molesters-record-clean-in-2008/

As has been said before pharmaceutical products are expensive in part because of the fact that research and development is incredibly long and expensive process. Successful medications do not only need to generate enough revenue to pay for their own R&D cost, but the costs of the failed drugs before them, and these costs can be huge considering that 3 out of 20 FDA approved pharmaceutical products make enough money to cover their own R&D costs. Add to the fact that this do not consider the numerous chemical and products that never achieve FDA approval, and I think the true cost of things comes more into light. As for medical products like the makes and hosing your were discussing Athena like CaptainPanic hinted at there is much more behind these products than just the actual building material. Since they must to be tested for allergies, effectiveness, safety, and that all medical products are sterilized before being packaged in order to protect patients. http://www.medicinenet.com/script/main/art.asp?articlekey=18892

So today I was thinking about how over the last 30-40 years there have been a lot of musical groups and artists that have had the staying power to continue to tour and sell out large venues year after year. I mean think of all the big rock groups from the 70's and 80's who still tour or toured in the last few years even if they band wasn't the original members. My question is do you think there are any current groups that you think have this type of staying power? I really cannot think of any current musical acts that will. I mean I can't imagine in 20 to 30 years seeing Lady Gaga, Ke$ha, T-Pain, or most other artists still preforming to large audiences. What groups do you think will still be around in 20-30 years?

No. See DrRocket's link above. This literally makes no sense. Again this is non-sensical. Why do you ignore 2, 3, 5, and all primes greater than 29? See Dr.Rocket's link. Yeah, I saw no formal proofs or even any well stated conjectures or theorems.

My personal opinion is that universities and four year colleges are for providing students the opportunities and resources to use their ability to learn to do just that. I would never expect a college professor to take the time to try and teach me or a class how to learn - if that is even possible, and in fact I would be unhappy if a professor ever did try to this. I am not saying that professors should not facilitate learning, they most certainly should. Meaning they should try and present the material in an effective manner and field well thought out questions from students who are actively grappling with the material. As for the fact that not everyone has these skills coming out of high school all I can say is made not everyone is truly ready for a four year university/college. I mean there are other educational avenues for those who are not prepared for university, and there would be nothing wrong with going to a community or junior college to develop the skills need for a four year institute. Also there is some transition between the high school and university environments, and most institutes and even many professors understand this, and so are willing to assist freshman if they need a little help in the adjustment. Maybe my opinions are shaped by the fact that I had taken a lot of college courses before entering university, and so my transition and expectations are different from others, but I honestly believe that a 4 year college or university is not, and has never been, the place for being taught study skills, and that making it a place for this would greatly devalue the education the level of education these institutes are able to offer. I mean I could not justify cutting back on the quantity or depth of information presented in a class or over a students time at a institute in order to help teach study skills.

Thanks a lot, CharonY!

So I am thinking of buying a NAS server to back up my computers, and serve as an easy way to share files. I would be using this for home use, with probably 5-6 computers running both Mac OS and Windows. I have been looking at both products made Buffalo Technology and Synology, but it seems like everything has mixed reviews. Does anyone have any suggestions on good NAS servers, or comments on bad ones to avoid?

I was planning on just being raptured away last Saturday, but...

You can work through the simplified basics of RSA with pen and paper, however, the level of mathematics required to full understand it may be beyond a high school student, but I'll see if I can explain it to you. So RSA is a public key encryption system. This means that there are two keys, one which is public, and one which is private. The idea behind this method that the public key is used to encrypt a message, and then the private key is used to decrypt the message. So for example, if I wished to send you the message "BuyBuyBuy" I would use your public key to encrypt the message, and send the cyphertext to you, and you would then use your private key to decrypt it. So the first thing that must be done for RSA is that a key must be select. First select two prime numbers [math] p, q[/math] where [math]p\neq q[/math]. Normally these primes are quite large, however to make things easier lets pick [math]p=7, q=11[/math]. Now we calculate [math]n=pq[/math] so in this example [math]n=7*11=77[/math] Now we calculate [math]\phi(n)=(p-1)(q-1)[/math]. If you don't know what this is it's Euler's Toitent function, which calculates the number of positive integers coprime to [math]n[/math]. So since [math]p, q[/math] are prime there is only one number that is not coprime to them ie: 1, and so [math]\phi(p)=(p-1)[/math] and math]\phi(q)=(q-1)[/math]. So for our example math]\phi(77)=(7-1)(11-1)=6*10=60.[/math]. Now an integer [math]a[/math] is select such that [math]1<a<\phi(n)[/math] and [math]gcd(a,\phi(n))=1[/math]. So we just need to select a integer between 1 and [math]\phi(n)[/math] such that it does not divide [math]\phi(n)[/math]. So for our example we need an integer between 1 and 60, which does not divide 60. To make this easy lets pick 13 since it does not divide 60. We now have the public key! The public key we release to the public is (a,n), which in our case would be (13,77). Now we need to find the private key. To do this we simply want to find the multiplicative inverse of [math]amod(\phi(n)[/math]. This means you want to find the number [math]x[/math] such that [math]ax\equiv 1mod(\phi(n))[/math]. So for our problem we want to find [math]x[/math] such that [math]13x\equiv 1mod(60)[/math], and in this case [math]x=37[/math]. If you have never seen modular arithmetic you might want to read a little about it just so you understand the basics of what we are doing here. So now the private key that is kept secret is (x,n), which in our case is (37,77). So now we have almost everything we need to encrypt messages. The only thing left is to create a agreed upon way of converting text into numbers. This is called a padding scheme, and using a good padding scheme can increase the security of the encryption, but really the only required thing is that the scheme is agreed upon, ie: both parties know how the scheme works, and that each integer we select must be between 1 and n. So for example, we could agree that we will simply say that A=2, B=3, C=4....Z=27. Now that we have a public key, a private key, and a padding scheme we can send messages. So lets say I wanted to sent you the message "BUY". So first I would use the padding scheme to convert this to numbers ie: BUY= 3 22 26. Now I am going to use your public key to encrypt the message. To encrypt an integer, [math]m[/math] we evaluate [math]m^{a}mod(n)=t[/math]. So for this example I would do: [math]3^{13}mod(77)=38[/math] [math]22^{13}mod(77)=22[/math] [math]26^{13}mod(77)=75[/math] So now I would send you the encrypted message 38 22 75. Now you need to decrypt the message. To decrypt [math]t[/math] we would use our public key to do [math]t^{x}mod(77)[/math]. So for our example you would do: [math]38^{37}mod(77)=3[/math] [math]22^{37}mod(77)=22[/math] [math]75^{37}mod(77)=26[/math] So know you would use the padding scheme to evaluate 3 22 26 to mean B U Y. So thats basically how RSA encryption works. Of course there is a lot of mathematics behind this showing why it will always work, and why it is so secure, but my guess is that it might be beyond high school. However, the example above can be done with basically only a little knowledge of modular arithmetic. Hope this helped!

Any reason you keep posting random problems? I know you say they aren't homework, but it certainly seems like they might be.

Your proof looks correct, but its way to long and induction is completely unnecessary. Take [math]x \in \mathbb{Z}[/math] [math]x[/math] can be written as: [math]x=\sum_{i=0}^{n}a_{i}10^{i}, a_{i}\in\{0,1,2,3,4,5,6,7,8,9\}[/math] Note [math]10\equiv 1mod(9)[/math] Therefore, [math]\sum_{i=0}^{n}a_{i}10^{i}\equiv \sum_{i=0}^{n}a_{i} mod(9) \Rightarrow \sum_{i=0}^{n}a_{i}10^{i}-\sum_{i=0}^{n}a_{i} \equiv 0mod(9)[/math] So like Dr.Rocket said this should be a one liner since really only the last line of mine is truly needed. Also you have to realize that when you post a question with simply, "Give a rigorous proof of the above statement!" it makes it seem like you are a student asking for someone to do their homework. I mean if you frequent any forum you will see numerous posts like this that do come from people looking for answers. If you just want to post problems you find interesting make it clear that it is not homework because it will mean people will be more likely to respond to your challenge since they will be sure its not homework.

So lets take a quadratic equation: [math]f(x)=ax^{2}+bx+c[/math] So the quadratic formula tells us that the roots for this equation will be given by: [math]\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}[/math] So you'll see that the discriminate is just equal to: the term under the radical in quadratic equation. So lets look at the three cases for the discriminate: -[math]b^{2}-4ac=0[/math] If the discriminate is 0 then we see we get: [math]\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}=\frac{-b\pm \sqrt{0}}{2a}=\frac{-b\pm 0}{2a}=\frac{-b}{2a}[/math] So we see that there is only one real root! -[math]b^{2}-4ac>0[/math] If the discriminate is greater than 0 we know then that: [math]\sqrt{b^{2}-4ac}=M[/math] Where [math]M[/math] is just some positive real number. Therefore the quadratic equation becomes: [math]\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}=\frac{-b\pm M}{2a}[/math] So we know we have two real roots of this quadratic namely [math]\frac{-b+M}{2a}[/math] and [math]\frac{-b-M}{2a}[/math]. -[math]b^{2}-4ac<0[/math] So since the discriminate is negative we know that when we take the square root of a negative number we will get a non-real number, and so if [math]\sqrt{b^{2}-4ac}[/math] is complex we know that the the quadratic equation will give us non-real solutions, and therefore this quadratic will have 0 real roots.

This does generalize to any desired base, and the proof is relatively straightforward. Have you made any attempts at it? I feel like this is a homework question so I won't give you all the work -especially if you haven't shown any work towards the problem- but I'll give you a few tips. -What do we really mean when we write a number in base-10? -What does it mean that something is divisible by 9? Answer these and the proof should be fairly straight forward. However, if you still need help post your work so we can see what you might be stuck on.

No, the Constitution does not delineate between various types of speech. However, the Supreme Court has ruled that certain classes of speech such as Commercial Speech are subject to different levels of protection, but the selling of speech that is protect by the Constitution is still full protect. For example, if you did not want to extend freedom of speech to commercial then hypothetically the government could censor art work if it was sold at an art gallery. Personally, I don't think that whether something is sold or not should affect whether that speech is protected or not. I would say that definitively it would not be prostitution. The thing to note is that the couple is not directly selling sex. They are just selling a movie, which would assuming it passes the Miller test be consider protected under the Constitution. Ethically, I see very few differences between prostitution and pornography, however, laws are not founded upon ethics alone, and I believe that the complete legalization of prostitution poses numerous issues, which do not arise with pornography. So I do completely understand how although ethically they might be similar we outlaw prostitution because we feel that the issues it brings are not worth dealing with.

I think that it is important to distinguish between policy and personal opinion. President Obama recently changed the militaries official policy so that open homosexuality would be allow, and he has succeeded in changing the military's official policy. However, an organization's official policy does not define what the members of the organization do or believe, and it does not even necessarily define the atmosphere of the group as a whole. President Obama can change policy, but he -like all presidents- has very little power to change the culture of a group of people with out dramatic actions. So I would not consider him failing for being unable to force those in the army to create an atmosphere where open homosexuality is accepted. The official policy of the federal government since the 1870's is that African American's are equal in their civil liberties, however, we all know that these polices did not mean that African Americans were not discriminated against.

[math]x^{2}+y^{2}=20[/math] is a quadratic form, ie: a polynomial of degree 2 in multiply variables. So it does fit in a chapter about quadratics. As for what this graph should look like: http://www.wolframalpha.com/input/?i=x%5E2%2By%5E2%3D20

Bertrand Russel did eventually change his stance to one of relative pacifism, and to some extend did support World War 2. Also it is questionable how much the general population knew about the Holocaust as it happened. Marat, one thing thing about your characterization of World War I is that during the lead up to the start of the war a few countries especially had incredibly vocal and widespread support for going to war. In fact some historians have speculated that had Kaiser Wilhelm II not declared war that he may have face such a large public backlash that Germany might have descended into revolution. So I am not sure if I would characterize World War I to be strictly the work of European elites. As for rigney's original question. I am not that surprised that soldiers did not discuss politics because I can see that in state of war discussing politics has few advantages, and in fact probably has more disadvantages. When soldiers discuss politics I feel that they would more than likely question either why they are fighting or how the war is being managed. Discussing either one of these things would in my opinion lower morale and/or make fighting more difficult. When in a time of war either one of these can prove deadly to soldiers, and my guess is that they realize this and so do not bring politics up. Also the strict nature of the military does probably to some extent deter people from discussing things like politics. http://en.wikipedia.org/wiki/Bertrand_Russell Ecksteins' Rites of Spring: The Great War and the Birth of the Modern Age

It is similar to summation notation, but instead of adding each terms you are going to multiply them together. So for example, [math]\prod_{k=1}^n \frac{1}{x_{k}}=1*\frac{1}{2}*\frac{1}{3}*...*\frac{1}{n} [/math]