16 Complex Numbers: a Primer
... may be associated to the point (a, b) in the real plane R2 . Thus we may identify the field of complex numbers C with the real Euclidean plane R2 . Every statement about complex numbers immediately gets a geometric counterpart. It is amazing that this (from modern perspective almost obvious) interpr ...
... may be associated to the point (a, b) in the real plane R2 . Thus we may identify the field of complex numbers C with the real Euclidean plane R2 . Every statement about complex numbers immediately gets a geometric counterpart. It is amazing that this (from modern perspective almost obvious) interpr ...
Inclusion-Exclusion Principle
... the answer is clearly not 50 + 33 + 20 = 103. This sum counts twice the numbers that are multiples of 2 and 3 for instance. We must subtract 16 multiples of 6, 10 multiples of 10 and 6 multiples of 15. It seems as if 50 + 33 + 20 − 16 − 10 − 6 = 71 is the final answer, but it is not! The multiples o ...
... the answer is clearly not 50 + 33 + 20 = 103. This sum counts twice the numbers that are multiples of 2 and 3 for instance. We must subtract 16 multiples of 6, 10 multiples of 10 and 6 multiples of 15. It seems as if 50 + 33 + 20 − 16 − 10 − 6 = 71 is the final answer, but it is not! The multiples o ...
An algebraic approach to some models in the KPZ "Universality class"
... stochastic heat equation by adding a non-linear term and Gaussian white noise. The study of such non-linear stochastic differential equation is a very challenging task. One has to worry about the regularity of the solutions etc. However the equation quickly became the default model for random interf ...
... stochastic heat equation by adding a non-linear term and Gaussian white noise. The study of such non-linear stochastic differential equation is a very challenging task. One has to worry about the regularity of the solutions etc. However the equation quickly became the default model for random interf ...
Differentiation and Integration
... Logarithms and Inverse Trig Functions The logarithm function is defined to be the inverse of the exponential function. Since (ez )0 = ez , the argument used at the start of Section 3.8 of Stewart shows that (ln z)0 = 1/z. This tells us how to compute the derivative of the logarithm, but we still don ...
... Logarithms and Inverse Trig Functions The logarithm function is defined to be the inverse of the exponential function. Since (ez )0 = ez , the argument used at the start of Section 3.8 of Stewart shows that (ln z)0 = 1/z. This tells us how to compute the derivative of the logarithm, but we still don ...
PROBLEM SET 7
... The pigeonhole principle is particularly powerful in existence proofs which are not constructive. For example in the previous example we proved the existence of two people with the same number of hairs without specifically identifying these two individuals. Example 2. How many bishops can one put on ...
... The pigeonhole principle is particularly powerful in existence proofs which are not constructive. For example in the previous example we proved the existence of two people with the same number of hairs without specifically identifying these two individuals. Example 2. How many bishops can one put on ...
Delta Function and Optical Catastrophe Models Abstract
... In [Dan3], we defined the integral of a Hyper-real Function. Let f (x ) be a hyper-real function on the interval [a, b ] . The interval may not be bounded. ...
... In [Dan3], we defined the integral of a Hyper-real Function. Let f (x ) be a hyper-real function on the interval [a, b ] . The interval may not be bounded. ...
Towards a Better Notation for Mathematics
... This can trivially be implemented in logic gates (xor and and are sufficient). It is also very intuitive. Multiplication is similarly elegant: multiplying by 10000... is just phaseshifting and any multiplication can be decomposed into adding the result of such multiplications (a ∗ (b + c + d) = a ∗ ...
... This can trivially be implemented in logic gates (xor and and are sufficient). It is also very intuitive. Multiplication is similarly elegant: multiplying by 10000... is just phaseshifting and any multiplication can be decomposed into adding the result of such multiplications (a ∗ (b + c + d) = a ∗ ...
Finite Model Theory
... infinite structures and most of model theory is based on methods that take infiniteness of structures for granted. In that context finite models are anomalies that deserve only marginal attention. Finite model theory arose as an independent field of logic from consideration of problems in theoretica ...
... infinite structures and most of model theory is based on methods that take infiniteness of structures for granted. In that context finite models are anomalies that deserve only marginal attention. Finite model theory arose as an independent field of logic from consideration of problems in theoretica ...
Non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson. He wrote:[...] the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection [...] that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latterRobinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle. Robinson continued:However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits.Robinson continues:It is shown in this book that Leibniz's ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory.In 1973, intuitionist Arend Heyting praised non-standard analysis as ""a standard model of important mathematical research"".