Introduction to Discrete Structures Instructional Material
... group. One of the girls will be the birthday celebrant. The rest are visitors. It is a European party and it is customary to shake the hands of the house owner before entering their premises. Let the class observe how many handshakes the birthday celebrant did when inviting her visitors in her house ...
... group. One of the girls will be the birthday celebrant. The rest are visitors. It is a European party and it is customary to shake the hands of the house owner before entering their premises. Let the class observe how many handshakes the birthday celebrant did when inviting her visitors in her house ...
COMPLEX NUMBERS
... √ the idea of a complex number, a quantity consisting of a real (or integer) number and a multiple of −1. Complex numbers have become an essential part of pure and applied mathematics. It is unfortunate that such numbers are called ’imaginary’ since they are no more or less real than any other type ...
... √ the idea of a complex number, a quantity consisting of a real (or integer) number and a multiple of −1. Complex numbers have become an essential part of pure and applied mathematics. It is unfortunate that such numbers are called ’imaginary’ since they are no more or less real than any other type ...
CONGRUENCE PROPERTIES OF VALUES OF L
... L(FD , s) = n=1 χD (n)a(n)n−s . These L-functions have analytic continuations to C and satisfy well known functional equations. If Λ(F, s) = (2π)−s Γ(s)M s/2 L(F, s), then Λ(F, s) = ² · Λ(F, 2k − s), where ² = ±1 is the so-called sign of the functional equation, and the quadratic twists satisfy Λ(FD ...
... L(FD , s) = n=1 χD (n)a(n)n−s . These L-functions have analytic continuations to C and satisfy well known functional equations. If Λ(F, s) = (2π)−s Γ(s)M s/2 L(F, s), then Λ(F, s) = ² · Λ(F, 2k − s), where ² = ±1 is the so-called sign of the functional equation, and the quadratic twists satisfy Λ(FD ...
THE PRIME FACTORS OF CONSECUTIVE, INTEGERS II by P
... Grimm observed that there are integers u and v, u < v,P(u) _ P(v) so that there is no prime between u and v e .g . u = 24, v = 27 . It is easy to find many other such examples, but we cannot prove that there are infinitely many such pairs u i , v i and we cannot get good upper or lower bounds for v ...
... Grimm observed that there are integers u and v, u < v,P(u) _ P(v) so that there is no prime between u and v e .g . u = 24, v = 27 . It is easy to find many other such examples, but we cannot prove that there are infinitely many such pairs u i , v i and we cannot get good upper or lower bounds for v ...
CARLOS AUGUSTO DI PRISCO The notion of infinite appears in
... The notion of infinite appears in mathematics in many different ways. The notion of limit or endless processes of approximations have been considered since ancient times, but it was in the decade of 1870 that the systematic study of infinite collections as completed totalities was initiated by Georg ...
... The notion of infinite appears in mathematics in many different ways. The notion of limit or endless processes of approximations have been considered since ancient times, but it was in the decade of 1870 that the systematic study of infinite collections as completed totalities was initiated by Georg ...
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"".