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Heilbronn Set
In mathematics, a Heilbronn set is an infinite set ''S'' of natural numbers for which every real number can be arbitrarily closely approximated by a fraction whose denominator is in ''S''. For any given real number \theta and natural number h, it is easy to find the integer g such that g/h is closest to \theta. For example, for the real number \pi and h=100 we have g=314. If we call the closeness of \theta to g/h the difference between h\theta and g, the closeness is always less than 1/2 (in our example it is 0.15926...). A collection of numbers is a Heilbronn set if for any \theta we can always find a sequence of values for h in the set where the closeness tends to zero. More mathematically let \, \alpha\, denote the distance from \alpha to the nearest integer then \mathcal H is a Heilbronn set if and only if for every real number \theta and every \varepsilon>0 there exists h\in\mathcal H such that \, h\theta\, <\varepsilon. Examples The natural n ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet's Approximation Theorem
In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers \alpha and N , with 1 \leq N , there exist integers p and q such that 1 \leq q \leq N and : \left , q \alpha -p \right , \leq \frac < \frac. Here represents the of . This is a fundamental result in , showing that any real number has a sequence of good rational approximations: in fact an immediate consequence is that for a given irrational α, the inequality : is satisfied by infinitely many integers ''p'' and ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hans Heilbronn
Hans Arnold Heilbronn (8 October 1908 – 28 April 1975) was a mathematician. Education He was born into a German-Jewish family. He was a student at the universities of Berlin, Freiburg and Göttingen, where he met Edmund Landau, who supervised his doctorate. In his thesis, he improved a result of Hoheisel on the size of prime gaps. Life Heilbronn fled Germany for Britain in 1933 due to the rise of Nazism. He arrived in Cambridge, then found accommodation in Manchester and eventually was offered a position at Bristol University, where he stayed for about one and a half years. There he proved that the class number of the number field \mathbb(\sqrt) tends to plus infinity as d does, as well as, in collaboration with Edward Linfoot, that there are at most ten quadratic number fields of the form \mathbb(\sqrt), d a natural number, with class number 1. On invitation of Louis Mordell he moved back to Manchester Manchester () is a city in Greater Manchester, England. It had ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexandru Zaharescu
Alexandru Zaharescu (born June 4, 1961) is a Romanian mathematician. He is a professor in the Department of Mathematics, University of Illinois at Urbana–Champaign, and a Senior Researcher at the Institute of Mathematics of the Romanian Academy. He has two PhDs in mathematics, one from the University of Bucharest in 1991 under the direction of Nicolae Popescu, the other from Princeton University in 1995 under the direction of Peter Sarnak. Zaharescu has numerous publications in highly prestigious journals, and more than 300 in total. Almost all his work is in number theory. Early life Zaharescu was born on June 4, 1961 and grew up in Codlea, Romania. He graduated from high school in Codlea in 1980, and from the University of Bucharest in 1986. He joined the Institute of Mathematics of the Romanian Academy in 1989, and in 1991 he received a PhD from the University of Bucharest. Career Zaharescu received his second PhD from Princeton University in 1995. After that, he has held tem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Van Der Corput Set
In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be equidistributed, or uniformly distributed, if the proportion of terms falling in a subinterval is proportional to the length of that subinterval. Such sequences are studied in Diophantine approximation theory and have applications to Monte Carlo integration. Definition A sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be ''equidistributed'' on a non-degenerate interval 'a'', ''b''if for every subinterval 'c'', ''d''of 'a'', ''b''we have :\lim_= . (Here, the notation , ∩ 'c'', ''d'' denotes the number of elements, out of the first ''n'' elements of the sequence, that are between ''c'' and ''d''.) For example, if a sequence is equidistributed in , 2 since the interval .5, 0.9occupies 1/5 of the length of the interval , 2 as ''n'' becomes large, the proportion of the first ''n'' members of the sequence which fall between 0.5 and 0.9 must approach 1/5. L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Number Theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dirichlet ''L''-functions to give the first proof of Dirichlet's theorem on arithmetic progressions. It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem). Branches of analytic number theory Analytic number theory can be split up into two major parts, divided more by the type of problems they attempt to solve than fundamental differences in technique. *Multiplicative number theory deals with the distribution of the prime numbers, such as estimating the number of primes in an interval, and includes the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. *Additive number th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
