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Erdős–Borwein Constant
The Erdős–Borwein constant, named after Paul Erdős and Peter Borwein, is the sum of the reciprocals of the Mersenne numbers. By definition it is: :E=\sum_^\frac\approx1.606695152415291763\dots Equivalent forms It can be proven that the following forms all sum to the same constant: : E=\sum_^\frac\frac : E=\sum_^\sum_^ \frac : E=1+\sum_^ \frac : E=\sum_^\frac where σ0(''n'') = ''d''(''n'') is the divisor function, a multiplicative function that equals the number of positive divisors of the number ''n''. To prove the equivalence of these sums, note that they all take the form of Lambert series and can thus be resummed as such. Irrationality In 1948, Erdős showed that the constant ''E'' is an irrational number. Later, Borwein provided an alternative proof. Despite its irrationality, the binary representation of the Erdős–Borwein constant may be calculated efficiently. Applications The Erdős–Borwein constant comes up in the average case analysis of the heap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered on discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He was known both for his social practice of mathematics, working with more than 500 collaborators, and for his eccentric lifestyle; ''Time'' magazine called him "The Oddball's Oddba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrational Number
In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being '' incommensurable'', meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio of a circle's circumference to its diameter, Euler's number ''e'', the golden ratio ''φ'', and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Constants
A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory, statistics, and calculus. Some constants arise naturally by a fundamental principle or intrinsic property, such as the ratio between the circumference and diameter of a circle (). Other constants are notable more for historical reasons than for their mathematical properties. The more popular constants have been studied throughout the ages and computed to many decimal places. All named mathematical constants are definable numbers, and usually are also computable numbers ( Chaitin's constant being a significant exception). Basic mathematical constants These are constants which one is likely to enco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Art Of Computer Programming
''The Art of Computer Programming'' (''TAOCP'') is a comprehensive multi-volume monograph written by the computer scientist Donald Knuth presenting programming algorithms and their analysis. it consists of published volumes 1, 2, 3, 4A, and 4B, with more expected to be released in the future. The Volumes 1–5 are intended to represent the central core of computer programming for sequential machines; the subjects of Volumes 6 and 7 are important but more specialized. When Knuth began the project in 1962, he originally conceived of it as a single book with twelve chapters. The first three volumes of what was then expected to be a seven-volume set were published in 1968, 1969, and 1973. Work began in earnest on Volume 4 in 1973, but was suspended in 1977 for work on typesetting prompted by the second edition of Volume 2. Writing of the final copy of Volume 4A began in longhand in 2001, and the first online pre-fascicle, 2A, appeared later in 2001. The first published installment ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heapsort
In computer science, heapsort is an efficient, comparison-based sorting algorithm that reorganizes an input array into a heap (a data structure where each node is greater than its children) and then repeatedly removes the largest node from that heap, placing it at the end of the array in a similar manner to Selection sort. Although somewhat slower in practice on most machines than a well-implemented quicksort, it has the advantages of very simple implementation and a more favorable worst-case runtime. Most real-world quicksort variants include an implementation of heapsort as a fallback should they detect that quicksort is becoming degenerate. Heapsort is an in-place algorithm, but it is not a stable sort. Heapsort was invented by J. W. J. Williams in 1964. The paper also introduced the binary heap as a useful data structure in its own right. In the same year, Robert W. Floyd published an improved version that could sort an array in-place, continuing his earlier research ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Average Case Analysis
In computer science, best, worst, and average cases of a given algorithm express what the resource usage is ''at least'', ''at most'' and ''on average'', respectively. Usually the resource being considered is running time, i.e. time complexity, but could also be memory or some other resource. Best case is the function which performs the minimum number of steps on input data of n elements. Worst case is the function which performs the maximum number of steps on input data of size n. Average case is the function which performs an average number of steps on input data of n elements. In real-time computing, the worst-case execution time is often of particular concern since it is important to know how much time might be needed ''in the worst case'' to guarantee that the algorithm will always finish on time. Average performance and worst-case performance are the most used in algorithm analysis. Less widely found is best-case performance, but it does have uses: for example, where the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Wrench
John William Wrench, Jr. (October 13, 1911 – February 27, 2009) was an American mathematician who worked primarily in numerical analysis. He was a pioneer in using computers for mathematical calculations, and is noted for work done with Daniel Shanks to calculate the mathematical constant pi to 100,000 decimal places. Life and education Wrench was born on October 13, 1911, in Westfield, New York, and grew up in Hamburg, New York. He received a BA summa cum laude in mathematics in 1933 and an MA in mathematics in 1935, both from the University at Buffalo. He received his PhD in mathematics in 1938 from Yale University. His thesis was titled ''The derivation of arctangent relations''. Wrench died on February 27, 2009, of pneumonia in Frederick, Maryland. Career Wrench started his career teaching at George Washington University, but switched to doing research for the United States Navy during World War II. His specialty for the Navy was developing high-speed computation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Numeral System
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" ( zero) and "1" ( one). A ''binary number'' may also refer to a rational number that has a finite representation in the binary numeral system, that is, the quotient of an integer by a power of two. The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation. History The modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harrio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proceedings Of The Cambridge Philosophical Society
''Mathematical Proceedings of the Cambridge Philosophical Society'' is a mathematical journal published by Cambridge University Press for the Cambridge Philosophical Society. It aims to publish original research papers from a wide range of pure and applied mathematics. The journal, titled ''Proceedings of the Cambridge Philosophical Society'' before 1975, has been published since 1843. Abstracting and indexing The journal is abstracted and indexed in *MathSciNet *Science Citation Index Expanded *Scopus *ZbMATH Open See also *Cambridge Philosophical Society The Cambridge Philosophical Society (CPS) is a scientific society at the University of Cambridge. It was founded in 1819. The name derives from the medieval use of the word philosophy to denote any research undertaken outside the fields of law ... External linksofficial website References Academic journals associated with learned and professional societies Cambridge University Press academic journals Mathematics e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Constant
A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an Letter (alphabet), alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as and pi, occurring in such diverse contexts as geometry, number theory, statistics, and calculus. Some constants arise naturally by a fundamental principle or intrinsic property, such as the ratio between the circumference and diameter of a circle (). Other constants are notable more for historical reasons than for their mathematical properties. The more popular constants have been studied throughout the ages and computed to many decimal places. All named mathematical constants are Definable real number, definable numbers, and usually are also computable numbers (Chaitin's constant being a significant exception). Basic mathematical constants These a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Borwein
Peter Benjamin Borwein (born St. Andrews, Scotland, May 10, 1953 – 23 August 2020) was a Canadian mathematician and a professor at Simon Fraser University. He is known as a co-author of the paper which presented the Bailey–Borwein–Plouffe algorithm (discovered by Simon Plouffe) for computing π. First interest in mathematics Borwein was born into a Jewish family. He became interested in number theory and classical analysis during his second year of university. He had not previously been interested in math, although his father was the head of the University of Western Ontario's mathematics department and his mother is associate dean of medicine there. Borwein and his two siblings majored in mathematics. Academic career After completing a Bachelor of Science in Honours Math at the University of Western Ontario in 1974, he went on to complete an MSc and Ph.D. at the University of British Columbia. He joined the Department of Mathematics at Dalhousie University. Wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas Clausen (mathematician)
Thomas Clausen (16 January 1801, Snogbæk, Sottrup Municipality, Duchy of Schleswig – 23 May 1885, Tartu, Imperial Russia) was a Danish mathematician and astronomer. Life Clausen learned mathematics at home. In 1820, he became a trainee at the Munich Optical Institute and in 1824, at the Altona Observatory after he showed Heinrich Christian Schumacher his paper on calculating longitude by the occultation of stars by the moon. In 1828, he discovered Clausen's formula. He eventually returned to Munich, where he conceived and published his best known works on mathematics. In 1832, he discovered the Clausen function. In 1842, Clausen was hired by the staff of the Tartu Observatory, becoming its director in 1866–1872. Works by Clausen include studies on the stability of Solar System, comet movement, ABC telegraph code and calculation of 250 decimals of pi (later, only 248 were confirmed to be correct). In 1840, he discovered the Von Staudt–Clausen theorem. Also in 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
