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.

The Inscribed Square Problem

This week’s installment of mathematical content will be on the inscribed square problem. This problem is also referred to as the square peg problem, and Toeplitz’ conjecture. This problem though simple, remains unsolved still to today. There is not a million-dollar reward for this problem, but in exchange for the lack of monetary incentive the problem is easy to understand. There have not been a large number of developments in this problem, so the history is brief. In order to compensate a rectangular proof will be mentioned. To format this post, the first part will be the brief history; second will be the problem, and finally the rectangular proof.

It is called the Toeplitz conjecture, because Jewish mathematician, Otto Toeplitz, created it. Otto was born in 1881, and much like his father, and grandfather, he wanted to work in mathematics. He received his doctorate in 1905, and in 1906 started working at Gottingen University. At the time, Gottingen was one of the most prestigious universities in mathematical development. He worked with mathematicians such as Felix Klein, and David Hilbert. During his time there he worked on linear, and quadric functions in high special dimensions. Then in the year 1911 he proposed his conjecture on an inscribed square within a curve.

His conjecture is that on any curve there can be found four points that can construct a square. For example, a circle is probably the easiest to visualize a square inscribed within. In fact, if the square is rotated it can be seen that there can be an infinite number of squares found within a circle. However, this does not only apply to circles, the conjecture says it applies to all Jordan curves. A Jordan curve being a closed curve not having an intersection of any kind. For example, oval shapes, kidney bean shapes, and heart shapes would all be Jordan curves. However, self-intersecting shapes such as a figure eight, or infinity symbol would not be Jordan curves. This also includes all irregular shape variants that do not intersect. This conjecture was purposed a hundred years ago, and still it remains unsolved today. There has not been any progress on the square problem, but it has been proven that this problem can be solved with triangles, and also can be solved with rectangles.

As is the case for most mathematical problems, the solutions tend to be more complicated than the questions. This is the case with the rectangular proof of this problem. Just to be clear, this is not a proof of the inscribed square, but rather an inscribed rectangle. The approach to solving this is to look at a property of rectangles. A rectangle has four points, and two pairs of diagonal lines can be draw within a rectangle. It is important to recognize that the two diagonals have the same length. In fact, if two line segments share a midpoint, and are the same length, a rectangle can be formed from those four corresponding points. So rather than proving that a rectangle can be found, it is easier to prove that two line segments, that share a midpoint, and have the same length, can be found in any Jordan curve. The proof involves mapping the curve in three-dimensional space. Imagine a closed Jordan curve on a horizontal plane, then draw a line connecting two points on the curve. Take the distance between the two points and graph it vertically on the z axis, at the midpoint. Continually doing this over every pair of points on the curve generates a three-dimensional model of the curve. This being hard to visualize, a circle would map to a nice dome shape. At this point it needs to be made clear that the two line segments that are being looked at, need to be different line segments. This is to make sure that a rectangle would be formed, and not just a line. In order to do this the points on the curve must be unordered pairs. This just means that it removes redundancy from our problem. This next part will not be fully explained, but it can be proven. Any closed loop with unordered point pairs maps to a Mobius strip. The outer edge of the Mobius strip, corresponds to the points of the outer edge of the Jordan curve. In order to transform the Mobius strip into the three-dimension curve generated at the midpoints, it requires it to intersect itself. Meaning that there are two pairs of point on the outside edge that correspond in distance and midpoint.

The last part was not easily understandable, with just words. Here is a link to a video that gives a nice visualization to this problem. https://www.youtube.com/watch?v=AmgkSdhK4K8

Sources https://www.youtube.com/watch?v=AmgkSdhK4K8

             http://www.webpages.uidaho.edu/~markn/squares/