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The Lambert W Function in Finance
/0 Comments/in Mathematics Articles/by Neil ParkerPreamble The classical mathematician practically by instinct views the continuous process as the “real” process, and the discrete process as an approximation to it. The mathematics of finance and certain topics in the modern theory of stochastic processes suggest that, in some cases at least, the opposite is true. Continuous processes are, generally speaking, the…
Why Division by Zero is a Bad Idea
/33 Comments/in Mathematics Articles, Mathematics FAQs/by fresh_42A division by zero is primarily an algebraic question. The reasoning therefore follows the indirect pattern of most algebraic proofs: What if it was allowed? Then we would get a contradiction, and a contradiction is the greatest enemy of mathematical rigor. Many students tried to find a way to divide by zero once in their…
Series in Mathematics: From Zeno to Quantum Theory
/2 Comments/in Mathematics Articles/by fresh_42Introduction Series play a decisive role in many branches of mathematics. They accompanied mathematical developments from Zeno of Elea (##5##-th century BC) and Archimedes of Syracuse (##3##-th century BC), to the fundamental building blocks of calculus from the ##17##-th century on, up to modern Lie theory which is crucial for our understanding of quantum theory….
Differential Equation Systems and Nature
/12 Comments/in Bio/Chem Articles, Mathematics Articles, Physics Articles/by fresh_42Abstract “Mathematics is the native language of nature.” is a phrase that is often used when it comes to explaining why mathematics is all around in natural sciences, especially in physics. What does that mean? A closer look shows us that it primarily means that we describe nature by differential equations, a lot of differential…
Beginners Guide to Precalculus, Calculus and Infinitesimals
/3 Comments/in Mathematics Articles/by Bill HobbaIntroduction I am convinced students learn Calculus far too late. In my view, there has never been a good reason for this. In the US, they go through this sequence of Pre-Algebra, Algebra 1, Geometry, Algebra 2, Precalculus, Calculus 1, and Calculus 2. But is this required? Recently I came across two books that turned…
What Are Numbers?
/7 Comments/in Mathematics Articles/by Bill HobbaIntroduction When doing mathematics, we usually take for granted what natural numbers, integers, and rationals are. They are pretty intuitive. Going from rational numbers to reals is more complicated. The easiest way at the start is probably infinite decimals. Dedekind Cuts can be used to get a bit more fancy. A Dedekind cut is a…
Introduction to the World of Algebras
/0 Comments/in Mathematics Articles/by fresh_42Abstract Richard Pierce describes the intention of his book [2] about associative algebras as his attempt to prove that there is algebra after Galois theory. Whereas Galois theory might not really be on the agenda of physicists, many algebras are: from tensor algebras as the gown for infinitesimal coordinates over Graßmann and Banach algebras for…
What Are Infinitesimals – Simple Version
/5 Comments/in Mathematics Guides/by Bill HobbaIntroduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) can be thrown away because they are negligible. That way, when defining the derivative, for example, you do not run into 0/0, but when…
What Are Infinitesimals – Advanced Version
/0 Comments/in Mathematics Articles/by Bill HobbaIntroduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) can be thrown away because they are negligible. That way, when defining the derivative, for example, you do not run into 0/0, but when…