Aneesh - Department Of Mathematics
... 8. Namboodiri, M. N. N.; Remadevi, S. A note on Szegö's theorem. J. Comput. Anal. Appl. 6 (2004), no. 2, 147--152. 9. Namboodiri, M. N. N. Theory of spectral gaps—a short survey. J. Anal. 12 (2004), 69--76. 10. Namboodiri, M. N. N.; Nair, Sindhu G. Collectively compact elementary operators and its a ...
... 8. Namboodiri, M. N. N.; Remadevi, S. A note on Szegö's theorem. J. Comput. Anal. Appl. 6 (2004), no. 2, 147--152. 9. Namboodiri, M. N. N. Theory of spectral gaps—a short survey. J. Anal. 12 (2004), 69--76. 10. Namboodiri, M. N. N.; Nair, Sindhu G. Collectively compact elementary operators and its a ...
beal`s conjecture as global break-through in natural
... with positive integers x, y, z having a common factor and exponent n taking simultaneously the next spectrum of values: n = (k, l, m) , where integers k, l, m at least 3 and n has one independent value for each term. Thus we assume at the beginning that equality (1) exists. Then we can explore some ...
... with positive integers x, y, z having a common factor and exponent n taking simultaneously the next spectrum of values: n = (k, l, m) , where integers k, l, m at least 3 and n has one independent value for each term. Thus we assume at the beginning that equality (1) exists. Then we can explore some ...
equivalents of the compactness theorem for locally finite sets of
... → otherwise xR∗ y iff π(x) = π(y). ...
... → otherwise xR∗ y iff π(x) = π(y). ...
numbers
... called a factor). Of course all numbers have 1 as their divisor. Perfect Numbers Among these natural numbers some are called perfect numbers. What is perfect about them? Number theorists tell you that they are perfect since all the factors (except the number itself) add to give the number itself. Fo ...
... called a factor). Of course all numbers have 1 as their divisor. Perfect Numbers Among these natural numbers some are called perfect numbers. What is perfect about them? Number theorists tell you that they are perfect since all the factors (except the number itself) add to give the number itself. Fo ...
CHAP03 Induction and Finite Series
... In chapter 1 we discussed the basic laws of algebra, from which all the properties of the real numbers can be deduced. Can we prove these basic laws? The answer is, we can, provided we define real numbers and their fundamental operations. We sketched how this might be some, starting with the natural ...
... In chapter 1 we discussed the basic laws of algebra, from which all the properties of the real numbers can be deduced. Can we prove these basic laws? The answer is, we can, provided we define real numbers and their fundamental operations. We sketched how this might be some, starting with the natural ...
On certain positive integer sequences (**)
... D e f i n i t i o n 1. Let k F 2 , l F 1 , m F 2 be positive integers. We say that a positive integer n is a (k , l , m)-number if the sum of digits of n m in its expansion in base k is l times the sum of the digits of the expansion in base k of n . The above sequences respectively represent the ( 2 ...
... D e f i n i t i o n 1. Let k F 2 , l F 1 , m F 2 be positive integers. We say that a positive integer n is a (k , l , m)-number if the sum of digits of n m in its expansion in base k is l times the sum of the digits of the expansion in base k of n . The above sequences respectively represent the ( 2 ...
Homework 4 - Solutions
... x is a unique minimal element). if s1=x, then xRs and we are done with the proof. Let s1 x. Then there exists s2 S, s2 s1, such that s2Rs1, for otherwise s1 is minimal (that is against the fact that x is a unique minimal element). Note that s2 s. Infact if s2 = s, then sRs1, s1Rs. By antisym ...
... x is a unique minimal element). if s1=x, then xRs and we are done with the proof. Let s1 x. Then there exists s2 S, s2 s1, such that s2Rs1, for otherwise s1 is minimal (that is against the fact that x is a unique minimal element). Note that s2 s. Infact if s2 = s, then sRs1, s1Rs. By antisym ...
Real Analysis: Basic Concepts
... In discussing the concept of an open set, it is important to specify the space in which we are considering the set. – For example, the set fx 2 < : 0 < x < 1g is open in <: But the set (x1; x2) 2 <2 : 0 < x1 < 1; x2 = 0 is not open in <2; although, graphically, the two sets “look the same”. Theorem ...
... In discussing the concept of an open set, it is important to specify the space in which we are considering the set. – For example, the set fx 2 < : 0 < x < 1g is open in <: But the set (x1; x2) 2 <2 : 0 < x1 < 1; x2 = 0 is not open in <2; although, graphically, the two sets “look the same”. Theorem ...
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"".