Euler's totient functionIn number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as or , and may also be called Euler's phi function. In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1. The integers k of this form are sometimes referred to as totatives of n. For example, the totatives of n = 9 are the six numbers 1, 2, 4, 5, 7 and 8.
Shiing-Shen ChernShiing-Shen Chern (tʃɜrn; , tʂhən.ɕiŋ.ʂən; October 28, 1911 – December 3, 2004) was a Chinese-American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geometry" and is widely regarded as a leader in geometry and one of the greatest mathematicians of the twentieth century, winning numerous awards and recognition including the Wolf Prize and the inaugural Shaw Prize.
Mathematical notationMathematical notation consists of using symbols for representing operations, unspecified numbers, relations, and any other mathematical objects and assembling them into expressions and formulas. Mathematical notation is widely used in mathematics, science, and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way. For example, Albert Einstein's equation is the quantitative representation in mathematical notation of the mass–energy equivalence.
Geometric quantizationIn mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in.
Pontryagin classIn mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four. Given a real vector bundle E over M, its k-th Pontryagin class is defined as where: denotes the -th Chern class of the complexification of E, is the -cohomology group of M with integer coefficients. The rational Pontryagin class is defined to be the image of in , the -cohomology group of M with rational coefficients.
Euclid's theoremEuclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proved by Euclid in his work Elements. There are several proofs of the theorem. Euclid offered a proof published in his work Elements (Book IX, Proposition 20), which is paraphrased here. Consider any finite list of prime numbers p1, p2, ..., pn. It will be shown that at least one additional prime number not in this list exists. Let P be the product of all the prime numbers in the list: P = p1p2.
Benjamin FranklinBenjamin Franklin ( April 17, 1790) was an American polymath who was active as a writer, scientist, inventor, statesman, diplomat, printer, publisher, and political philosopher. Among the leading intellectuals of his time, Franklin was one of the Founding Fathers of the United States, a drafter and signer of the Declaration of Independence, and the first postmaster general. As a scientist, he was a major figure in the American Enlightenment and the history of physics for his studies of electricity, and for charting and naming the Gulf Stream current.
