Friday, March 3, 2017

Annotated Bibliography

Astounding: 1+2+3+4+5+...=-1/12. Dir. Brady Haran. Perf. Ed Copeland and Tony Padilla. Youtube. Numberphile, 9 Jan. 2014. Web. 10 Feb. 2017. <https://www.youtube.com/watch?v=w-I6XTVZXww>.

            This source primarily focuses on zeta functions, and a result from those zeta functions. The main purpose of the video is to show that a zeta function evaluated at negative one results in negative one twelfth. In the video it is explained that this is a rather unusual result because the zeta function suggests that the sum of all positive numbers is a negative fraction. Ed Copeland proves this result with zeta functions, while Tony Padilla offers an easier, algebraic solution. The importance of this is also stressed in the video. This rather unusual result can be found in string theory, the Casimir force, and black holes.
            The source is credible. Both Ed Copeland, and Tony Padilla, are Professors, and experts in their fields. The information in the video is purposed with informing the audience about zeta functions.

The Brachistochrone, with Steven Strogatz. Dir. 3Blue1Brown. Perf. Steven Strogatz. Youtube. 3Blue1Brown, 1 Apr. 2016. Web. 20 Feb. 2017. <https://www.youtube.com/watch?v=Cld0p3a43fU>.
           
            This video is an interview done by a YouTube video maker who goes by the username 3Blue1Brown. In the video he has a conversation with Steven Strogatz, on the Brachistochrone problem. The video first talks about the history of the problem, this includes the mathematicians, and the development of the problem from Johannes Bernoulli. The second part of the video focuses of on a geometrical representation of the solution of the problem concerning cycloids. This proof was given referencing the work of Mark Levi.
            The video format allows for a better representation visually for the viewer. Making it easier to understand. The author and contributors are well qualified to discuss the topic. Steven Strogratz being a mathematical professor at Cornell, and author of several math related books.

Chorin, Alexandre Joel. "Numerical Solution of the Navier-Stokes Equations." Mathematics of Computation (n.d.): n. pag. Berkley. Oct. 1968. Web. 14 Feb. 2017. <https://math.berkeley.edu/~chorin/chorin68.pdf>.
           
This document focuses on two parts, first what the Navier Stokes equations are, and they relate to fluid dynamics. Secondly, it refers to what numerical values can be substituted intot he equations to get real results. The Naiver Stokes equation being unsolvable require some numerical assumptions to useful in physics. The article also references all of The Navier Stokes equation, and breaks them down in to a more depth analysis. The article is 18 pages long, but about half of that is consumed by pictures, diagrams, and equations.
The Article included actual numerical solutions, the interest on mine was not to look into specific examples. So only information regarding the equations themselves was used. The article as a whole comes from a reliable source, but might be out dated as it was published in 1968.

CMI. "Navier-Stokes Equation." Clay Institute. N.p., 23 Feb. 2017. Web. 14 Feb. 2017. <http://www.claymath.org/millennium-problems/navier–stokes-equation>.
           
            This source is the official page of the clay institute regarding the Navier Stokes Smoothness problem. The Clay Institute being the institute that generated the list of millennial problems. The source outlines the contributions of both Navier, and Stokes, to the problem. It outlines the Problem itself, and all its details. It also explains how to solve the problem, in order to collect the prize money associated with solving the problem. The article also offers links to lectures to give a further in depth analysis of the problem.
            This is a good source for the problem, because it has the official problem document, and the rules for solving the millennial problem. Though the source is not packed full of information, it is still a valuable source for researching the problem.
           

CMI. "P vs. NP Problem." Clay Institute. N.p., 23 Feb. 2017. Web. 22 Feb. 2017. <http://www.claymath.org/millennium-problems/p-vs-np-problem>.

            This source is the official page on the P vs. NP problem. The source includes a brief description of the P vs. Np problem, by giving an analogy. The problem itself has to do with how problems are solved using computation. Some problems are rather easy to solve, and some are much easier to check an answer. This is the main difference in computation, and the question asks if there will ever be a way to simplify the difficult to compute problems.  The page also links to the official problem description. The page also links to the official rules of solving the P vs. NP problem.

            The source is the official page of the Clay Institute on the P vs. NP problem, making it accurate. This is source is a good source, because it gives the original problem, and how to solve it in order to claim the million-dollar reward associated with it.

Collatz Conjecture (extra Footage). Dir. Brady Haran. Perf. David Eisenbud. YouTube. N.p., 9 Aug. 2016. Web. 16 Feb. 2017. <https://www.youtube.com/watch?v=O2_h3z1YgEU>.
           
This video is a continuation of the Collatz conjecture video by Numberphile. In it Professor David Eisenbud talks about a generalization of the Collatz conjecture. The Collatz conjecture being 3n+1. Eisenbud talks about a general form an+b. In doing so it shows how if mathematicians cannot solve a problem, they approach a similar problem. The video also talks about why the problem does not have a large monetary reward associated with solving the problem. The problem is an isolated part of math that has not developed any important connection, unlike the seven millennial problems which do have many connections, and large rewards.
Professor David Eisenbud is professor of mathematics, more specifically topology, but still qualified to talk about the Collatz Conjecture. The purpose of the video is to inform the average person of the mathematics behind the Collatz conjecture.

Hall, Nancy. "Navier-Stokes Equations." Glenn Resaerch Center. N.p., 5 May 2015. Web. 14 Feb. 2017. <https://www.grc.nasa.gov/www/k-12/airplane/nseqs.html>.

            This website talks about the Navier Stokes equations and their importance to physics. The Navier Stoke equations are the representations of Newton’s second law applied to fluid dynamics. The equations take in to account the three dimensions, and included momentums in all three dimensions of space.
            The article as a whole is informative, but rather difficult to navigate through information. The purpose of the article is to inform the reader about 5 of the Navier Stokes equations, and how to use them appropriately.

Riemann Hypothesis. Dir. Brady Haran. Perf. Professor Edward Frenkel. YouTube. Numberphile, 11 Mar. 2014. Web. 10 Feb. 2017. <https://www.youtube.com/watch?v=d6c6uIyieoo>.

            This video is a presentation of the Riemann hypothesis by Professor Edward Frenkle. Frenkle goes into the graph of The zeta function, and the critical points on it. He talks about how Riemann developed a symmetry equation relation negative x values to the domain of the zeta function. He also goes in depth into the critical strip, and zero distribution of the zeta function. Finally, Frenkle talks about the importance of the zeta function, and its relation to prime number distribution.
The purpose of the video is to explain to the average individual the Riemann Hypothesis, and its importance to math as a whole. Professor Frenkle is a qualified mathematician to discuss the Riemann Hypothesis.

Sondow, Jonathan and Weisstein, Eric W. "Riemann Zeta Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RiemannZetaFunction.html  

            This article focuses on the working of the zeta function, and its graphic representation. The zeta function is a function to sums up infinite series as a function x and the imaginary plane, and outputs a value associated with that point. This was originally studied by Euler, and then further generalized by Riemann. Riemann worked out a symmetric formula that allowed for the extension of the domain to outside the positive values of x.
The information is accurate, and informative. The format allows for a different take on the visualization of the function. This new visualization is a much more interesting approach to understanding the zeta function as a whole.

Uncrackable? The Collatz Conjecture-Numberphile. Dir. Brady Haran. Perf. Professor David Eisenbud. YouTube. N.p., 8 Aug. 2016. Web. 16 Feb. 2017. <https://www.youtube.com/watch?v=5mFpVDpKX70>.

            This video primarily discusses the Collatz Conjecture. The collatz conjecture is a conjecture that concerns how numbers break down. For any even value of n divide it by two. If n is odd multiple, it by three and add one to it. This process is carried out until the number reaches 1. The conjecture is that all whole numbers if following these rules will converge to 1.
            The video is designed to be easy for anyone to understand. Professor Eisenbud does an excellent job explaining the problem, giving many examples, and answering many questions.
Unknown. "Sum of Three Cubes." Mathpages. N.p., n.d. Web. 22 Feb. 2017. <http://www.mathpages.com/home/kmath071.htm>.

            The article is focused on Diophantus equations. The Diophantus equation A cubed + B cubed = C cubed is known to have no solutions. This was proven with the solving of Fermat’s Last Theorem by Andrew Wiles. The question that arises from the Diophantus equation is, can any number be written as the sum of three cubes. The answer is no, but can it be proven for every number that it can or cannot be written as a sum of three cubes? That is the real question. The answer seems to be no. For example, 33 is unknown, whether it can or cannot be written as a sum of three cubes. This part of number theory still needs some work done on it.
The source is informative, but is more of rambling then a coherent source of information. The information lines up with other sources, but seems simplified. The authors purpose is more in making a blog then informing the audiences about mathematics.

Weisstein, Eric W. "Collatz Problem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CollatzProblem.html

            This article talks about the Collatz conjecture and its findings. The Collatz conjecture was proposed by L. Collatz in 1937. The conjecture talks about numbers that follow two rules. If the number is even half it, if the number is odd times it by three and add one. This sequence carries on until the number reaches 1. That is the conjecture at least, that the sequence will reach one. So far there is no evidence to suggest otherwise. This is a big part of number theory, because if this problem can be solved it opens up gateways to more general solutions.

This source is well articulated, and offers good visualization with graphics. The purpose of the article is informative, because there is not a side taken on whether or not the problem can be solved, the purpose is informative, and not biased.

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