Bernhard Riemann: A Predecessor to Many


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Bernhard Riemann

Bernhard Riemann

Bernhard Riemann

Without certain figures in history some subjects would still be elementary in content compared to our current knowledge.  If Einstein would have decided not to pursue his studies, the world could very well have an entirely different political structure.

It can be said that each subject is dependent upon the accomplishments of its past pioneers.  If it weren’t for the mathematician Antoine Augustin Cournot, John Forbes Nash, Jr. may not have made the advances in Game theory which nearly redefined a branch of Economics.  This can also be said about Georg Friedrich Bernhard Riemann.

His Life

Riemann grew up in poverty and received his early education by his father and later in his youth by a local teacher.  At sixteen, after moving a few times, he realized that he had a great desire to study mathematical ideas beyond what his school offered.  Riemann approached the director of the school with his request to learn more and was lent books on a variety of different mathematical subjects.  It wasn’t long after that, he was formally enrolled in a university.  Riemann began studies in theology and philology, more than likely by his father’s request, but never passing on attending classes in mathematics.  After his father approved of shifting the focus of his studies to mathematics, Riemann began to shine.

Studying his life’s passion, Riemann was led only to discover the absolute deficiency in his current university’s math program.  He was studying under “The Prince of Mathematics” Carl Gauss (pronounced Goss), who only taught applied mathematics at an elementary level.  This compromised Riemann’s level of expertise, so he made the decision to leave for a new program in Berlin studying under Carl Jacobi, Jakob Steiner, and Johann Dirichlet who had a great influence on his education.

After making another transition back to Johanneum Lüneburg, where he originally started, he began to study under the experimental physicist W. (Wilhelm) Weber.  During this time he documented his thoughts on a standardized mathematical physical view of nature and formed his dissertation: On the Hypotheses which lie at the Bases of Geometry.  His work was ground breaking and resolved the issues his previous professor, Gauss, had concerning physical paradoxes in his extension of Leibniz’s calculus into the domain of complex numbers.  Gauss solved a problem by finding several ways to measure the curvature of a surface by seeing that every point on any type of surface has one point that is the most curved and one that is the least curved.  Moreover, Gauss recognized that the relationship between the most and least curved is a characteristic of functions in the domain of complex numbers.

Riemann expands on Gauss and shows that not only is there such a harmonic relationship, but from the complex functions a complete class of physical manifolds can be expressed.  Riemann saw these complex functions as a way to change a physical manifold into another.  These thoughts he expanded on in his dissertation have been an integral part to the works of many physicists including Albert Einstein.

Riemann had other works which advanced many other branches of mathematics.  Some of his contributions can be seen in topology, calculus, and real analysis.  His contributions to calculus are essential to what is currently being studied.  Newton and Leibniz found that there was a connection between differentiation and integration.  Thus the Fundamental Theorem of Calculus was born and is now the frame work for modern calculus.  Integration had a long history, dating as far back as ancient Egypt (1800 B.C.).  Throughout the years it was only used and studied and needed a more rigorous definition.  It was Riemann who formalized integration by using limits to represent an infinite amount of approximating rectangles.  Ultimately his formalization led to finding the area under a curve and defining a definite integral.

Although the amount of his work that has been published is relatively small compared to others in the field, Riemann’s work was seen as genius.  His mentors have been known to say that they have learned more from him than he from them.  It wasn’t even that he was a child prodigy or even that he studied under someone with an impeccable ability to teach a vast amount of knowledge that defined his life.  It was centered on the fact that he was interested in mathematics and took the initiative to pursue his dreams and goals when other resources weren’t there.


North Korea Collapse: An Analysis


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Poverty in North Korea

Indicative of a North Korea Collapse?
Credit: Roman Harak (Flickr)

North Korea Collapse

It is no surprise that a country as small as North Korea had been fairly unknown to the average American before 2002. They first became a substantial “threat” when they appeared on the American political scene and President Bush announced the organizations that supported terrorism. To the informed, it was not so surprising that, given its history, it became one of the world’s most dangerous countries. But to many throughout the world Kim Jong-Il and his regime came in a seemingly arbitrary manner. The country is small and relatively unknown, and although a neo-communist country, its government is not what was so threatening. The threat lays in the fact that the North is quite possibly on the verge of complete collapse. Consequently, they are more than willing to go to extremes in order to keep their government and country alive and operating. But there is a factor that plays an equal or larger part than the threat of military conflict.

North Korea is bordered by China and South Korea. South Korea has been at odds with the North since 1945. In fact, they have yet to sign a peace settlement, which means, technically, the North and South are still at war. After the Koreas were divided up, the United States played an important part in the reconstruction of the South. Not surprisingly, the United States became – and still is – largely interwoven within the South Korean economy.

The military position and strength of the South combined with the efforts of the United States, the North’s oppressive regime could be ended fairly smoothly by means of conventional warfare. But if this were to happen, China would be obligated to support the North because of their mutual alliance.

Another big problem is that the U.S. has a military that is still trying to recoup from being in so many theaters of war. Consider as well that armed forces enrollment is relatively low compared to historic numbers. Starting another war would only exacerbate the issues of an already fatigued military. If for any reason the North Korean regime would suddenly crumble – and it’s not very far from it, China, South Korea and the United States would all be obligated to raise the funds necessary to reestablish the North as a new nation-state.

In addition to the costs, China and South Korea would face an influx of an inconceivable amount of people seeking refuge and assistance. If this were the situation, South Korea would seek the assistance of the United States, and because the South does not have access to the resources needed to support such a tragedy, it would become the job of the U.S. to step in and coordinate the efforts to restructure the North.

The most prominent concern for China, South Korea, and the U.S. is that the North has kept its citizens oppressed for decades through fear and indoctrination. Therefore, a regime change via a U.S.-funded coup d’état  would be nearly impossible because of the forged love and devotion the North Korean people have for their regime. Moreover, the rising up of those who wish to disband would face the task of recruiting those who are terrified of the repercussions if they are caught in the act. North Korea has been a fairly sanitized country when it comes to traitors. They tend to be fairly ruthless at the slightest indication that someone is acting in a rebellious manor.

North Korea Collapse

The U.S. faces the dilemma to either pay for most of the restructuring if North Korea collapses or risk a nuclear attack if it chooses to invade on the justification of the “G-word” (genocide) – a subject known widely as taboo within the State Department. If the U.S. does not let North Korea collapse by just leaving them alone, then it is quite possible that the North would attempt to intimidate the rest of the world by flaunting their nuclear capabilities and intent to harm. But if the U.S. tries to hurry their demise along, they could, and would, do something to cause severe damage to either the South or to the Unites States. It’s hard to tell how likely this is. But then again, what does North Korea really have to lose?