Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Nokia Bell Labs is an industrial research and scientific development company owned by Finnish company Nokia. Its headquarters are located in New Jersey. Other laboratories are located around the world. Bell Labs has its origins in the complex past of the Bell System. In the late 19th century, the laboratory began as the Western Electric Engineering Department and was located at 463 West Street in New York City. In 1925, after years of conducting research and development under Western Electric, the Engineering Department was reformed into Bell Telephone Laboratories and under the shared ownership of American Telephone & Telegraph Company and Western Electric. Researchers working at Bell Labs are credited with the development of radio astronomy, the transistor, the laser, the photovoltaic cell, the charge-coupled device, information theory, the Unix operating system, the programming languages C, C++, S. Nine Nobel Prizes have been awarded for work completed at Bell Laboratories. In 1880, when the French government awarded Alexander Graham Bell the Volta Prize of 50,000 francs (approximately US$10,000 at that time for the invention of the telephone, he used the award to fund the Volta Laboratory in Washington, D.
C. in collaboration with Sumner Tainter and Bell's cousin Chichester Bell. The laboratory was variously known as the Volta Bureau, the Bell Carriage House, the Bell Laboratory and the Volta Laboratory, it focused on the analysis and transmission of sound. Bell used his considerable profits from the laboratory for further research and education to permit the " diffusion of knowledge relating to the deaf": resulting in the founding of the Volta Bureau, located at Bell's father's house at 1527 35th Street N. W. in Washington, D. C, its carriage house became their headquarters in 1889. In 1893, Bell constructed a new building close by at 1537 35th Street N. W. to house the lab. This building was declared a National Historic Landmark in 1972. After the invention of the telephone, Bell maintained a distant role with the Bell System as a whole, but continued to pursue his own personal research interests; the Bell Patent Association was formed by Alexander Graham Bell, Thomas Sanders, Gardiner Hubbard when filing the first patents for the telephone in 1876.
Bell Telephone Company, the first telephone company, was formed a year later. It became a part of the American Bell Telephone Company. American Telephone & Telegraph Company and its own subsidiary company, took control of American Bell and the Bell System by 1889. American Bell held a controlling interest in Western Electric whereas AT&T was doing research into the service providers. In 1884, the American Bell Telephone Company created the Mechanical Department from the Electrical and Patent Department formed a year earlier. In 1896, Western Electric bought property at 463 West Street to station their manufacturers and engineers, supplying AT&T with their product; this included everything from telephones, telephone exchange switches, transmission equipment. In 1925, Bell Laboratories was developed to better consolidate the research activities of the Bell System. Ownership was evenly split between Western Electric and AT&T. Throughout the next decade the AT&T Research and Development branch moved into West Street.
Bell Labs carried out consulting work for the Bell Telephone Company, U. S. government work, a few workers were assigned to basic research. The first president of research at Bell Labs was Frank B. Jewett who stayed there until 1940. By the early 1940s, Bell Labs engineers and scientists had begun to move to other locations away from the congestion and environmental distractions of New York City, in 1967 Bell Laboratories headquarters was relocated to Murray Hill, New Jersey. Among the Bell Laboratories locations in New Jersey were Holmdel, Crawford Hill, the Deal Test Site, Lincroft, Long Branch, Neptune, Piscataway, Red Bank and Whippany. Of these, Murray Hill and Crawford Hill remain in existence; the largest grouping of people in the company was in Illinois, at Naperville-Lisle, in the Chicago area, which had the largest concentration of employees prior to 2001. There were groups of employees in Indianapolis, Indiana. Since 2001, many of the former locations closed; the Holmdel site, a 1.9 million square foot structure set on 473 acres, was closed in 2007.
The mirrored-glass building was designed by Eero Saarinen. In August 2013, Somerset Development bought the building, intending to redevelop it into a mixed commercial and residential project. A 2012 article expressed doubt on the success of the newly named Bell Works site however several large tenants had announced plans to move in through 2016 and 2017 Bell Laboratories was, is, regarded by many as the premier research facility of its type, developing a wide range of revolutionary technologies, including radio astronomy, the transistor, the laser, information theory, the operating system Unix, the programming languages C and C++, solar cells, the CCD, floating-gate MOSFET, a whole host of optical and wired communications
California Institute of Technology
The California Institute of Technology is a private doctorate-granting research university in Pasadena, California. Known for its strength in natural science and engineering, Caltech is ranked as one of the world's top-ten universities. Although founded as a preparatory and vocational school by Amos G. Throop in 1891, the college attracted influential scientists such as George Ellery Hale, Arthur Amos Noyes and Robert Andrews Millikan in the early 20th century; the vocational and preparatory schools were disbanded and spun off in 1910 and the college assumed its present name in 1921. In 1934, Caltech was elected to the Association of American Universities and the antecedents of NASA's Jet Propulsion Laboratory, which Caltech continues to manage and operate, were established between 1936 and 1943 under Theodore von Kármán; the university is one among a small group of institutes of technology in the United States, devoted to the instruction of pure and applied sciences. Caltech has six academic divisions with strong emphasis on science and engineering, managing $332 million in 2011 in sponsored research.
Its 124-acre primary campus is located 11 mi northeast of downtown Los Angeles. First-year students are required to live on campus and 95% of undergraduates remain in the on-campus House System at Caltech. Although Caltech has a strong tradition of practical jokes and pranks, student life is governed by an honor code which allows faculty to assign take-home examinations; the Caltech Beavers compete in 13 intercollegiate sports in the NCAA Division III's Southern California Intercollegiate Athletic Conference. As of October 2018, Caltech alumni and researchers include 73 Nobel Laureates, 4 Fields Medalists, 6 Turing Award winners. In addition, there are 53 non-emeritus faculty members who have been elected to one of the United States National Academies, 4 Chief Scientists of the U. S. Air Force and 71 have won the United States National Medal of Technology. Numerous faculty members are associated with the Howard Hughes Medical Institute as well as NASA. According to a 2015 Pomona College study, Caltech ranked number one in the U.
S. for the percentage of its graduates who go on to earn a PhD. Caltech started as a vocational school founded in Pasadena in 1891 by local businessman and politician Amos G. Throop; the school was known successively as Throop University, Throop Polytechnic Institute and Throop College of Technology before acquiring its current name in 1920. The vocational school was disbanded and the preparatory program was split off to form an independent Polytechnic School in 1907. At a time when scientific research in the United States was still in its infancy, George Ellery Hale, a solar astronomer from the University of Chicago, founded the Mount Wilson Observatory in 1904, he joined Throop's board of trustees in 1907, soon began developing it and the whole of Pasadena into a major scientific and cultural destination. He engineered the appointment of James A. B. Scherer, a literary scholar untutored in science but a capable administrator and fund raiser, to Throop's presidency in 1908. Scherer persuaded retired businessman and trustee Charles W. Gates to donate $25,000 in seed money to build Gates Laboratory, the first science building on campus.
In 1910, Throop moved to its current site. Arthur Fleming donated the land for the permanent campus site. Theodore Roosevelt delivered an address at Throop Institute on March 21, 1911, he declared: I want to see institutions like Throop turn out ninety-nine of every hundred students as men who are to do given pieces of industrial work better than any one else can do them. In the same year, a bill was introduced in the California Legislature calling for the establishment of a publicly funded "California Institute of Technology", with an initial budget of a million dollars, ten times the budget of Throop at the time; the board of trustees offered to turn Throop over to the state, but the presidents of Stanford University and the University of California lobbied to defeat the bill, which allowed Throop to develop as the only scientific research-oriented education institute in southern California, public or private, until the onset of the World War II necessitated the broader development of research-based science education.
The promise of Throop attracted physical chemist Arthur Amos Noyes from MIT to develop the institution and assist in establishing it as a center for science and technology. With the onset of World War I, Hale organized the National Research Council to coordinate and support scientific work on military problems. While he supported the idea of federal appropriations for science, he took exception to a federal bill that would have funded engineering research at land-grant colleges, instead sought to raise a $1 million national research fund from private sources. To that end, as Hale wrote in The New York Times: Throop College of Technology, in Pasadena California has afforded a striking illustration of one way in which the Research Council can secure co-operation and advance scientific investigation; this institution, with its able investigators and excellent research laboratories, could be of great service in any broad scheme of cooperation. President S
Robert Endre Tarjan is an American computer scientist and mathematician. He is the discoverer of several graph algorithms, including Tarjan's off-line lowest common ancestors algorithm, co-inventor of both splay trees and Fibonacci heaps. Tarjan is the James S. McDonnell Distinguished University Professor of Computer Science at Princeton University, the Chief Scientist at Intertrust Technologies Corporation, he was born in California. His father was a child psychiatrist specializing in mental retardation, ran a state hospital; as a child, Tarjan read a lot of science fiction, wanted to be an astronomer. He became interested in mathematics after reading Martin Gardner's mathematical games column in Scientific American, he became interested in math in the eighth grade, thanks to a "very stimulating" teacher. While he was in high school, Tarjan got a job, he first worked with real computers while studying astronomy at the Summer Science Program in 1964. Tarjan obtained a Bachelor's degree in mathematics from the California Institute of Technology in 1969.
At Stanford University, he received his master's degree in computer science in 1971 and a Ph. D. in computer science in 1972. At Stanford, he was supervised by Robert Floyd and Donald Knuth, both prominent computer scientists, his Ph. D. dissertation was An Efficient Planarity Algorithm. Tarjan selected computer science as his area of interest because he believed that computer science was a way of doing mathematics that could have a practical impact. Tarjan has been teaching at Princeton University since 1985, he has held academic positions at Cornell University, University of California, Stanford University, New York University. He has been a fellow of the NEC Research Institute. In April 2013 he joined Microsoft Research Silicon Valley in addition to the position at Princeton. In October 2014 he rejoined Intertrust Technologies as chief scientist. Tarjan has worked at AT&T Bell Labs, Intertrust Technologies and Hewlett Packard. Tarjan is known for his pioneering work on graph theory data structures.
Some of his well-known algorithms include Tarjan's off-line least common ancestors algorithm, Tarjan's connected components algorithm, he was one of five co-authors of the median of medians linear time selection algorithm. The Hopcroft-Tarjan planarity testing algorithm was the first linear-time algorithm for planarity-testing. Tarjan has developed important data structures such as the Fibonacci heap, the splay tree. Another significant contribution was the analysis of the disjoint-set data structure. Tarjan received the Turing Award jointly with John Hopcroft in 1986; the citation for the award states that it was: For fundamental achievements in the design and analysis of algorithms and data structures. Tarjan was elected an ACM Fellow in 1994; the citation for this award states: For seminal advances in the design and analysis of data structures and algorithms. Some of the other awards for Tarjan include: Nevanlinna Prize in Information Science – first recipient National Academy of Sciences Award for Initiatives in Research Paris Kanellakis Award in Theory and Practice, ACM Blaise Pascal Medal in Mathematics and Computer Science, European Academy of Sciences Caltech Distinguished Alumni Award, California Institute of Technology Tarjan holds at least 18 U.
S. patents. These include: J. Bentley, D. Sleator, R. E. Tarjan, U. S. Patent 4,796,003, Data Compaction, 1989 N. Mishra, R. Schreiber, R. E. Tarjan, U. S. Patent 7,818,272, Method for discovery of clusters of objects in an arbitrary undirected graph using a difference between a fraction of internal connections and maximum fraction of connections by an outside object, 2010 B. Pinkas, S. Haber, R. E. Tarjan, T. Sander, U. S. Patent 8220036, Establishing a secure channel with a human user, 2012 Tarjan, Robert E.. Data structures and network algorithms. Philadelphia: Society for Industrial and Applied Mathematics. ISBN 978-0-89871-187-5. OCLC 10120539. Tarjan, Robert E.. Notes on introductory combinatorics. Boston: Birkhauser. ISBN 978-0-8176-3170-3. OCLC 10018128. OCLC entries for Robert E Tarjan Robert E. Tarjan at DBLP Bibliography Server Robert E. Tarjan at DBLP Bibliography Server List of Robert Tarjan's patents on IPEXL's Patent Directory Robert Tarjan's home page at Princeton. Robert Endre Tarjan at the Mathematics Genealogy Project
Gary Miller (computer scientist)
Gary Lee Miller is a professor of Computer Science at Carnegie Mellon University, United States. In 2003 he won the ACM Paris Kanellakis Award for the Miller–Rabin primality test, he was made an ACM Fellow in 2002 and won the Knuth Prize in 2013. Miller received his Ph. D. from the University of California, Berkeley in 1975 under the direction of Manuel Blum. His Ph. D. thesis was titled Riemann's Tests for Primality. Apart from computational number theory and primality testing, he has worked in the areas of computational geometry, scientific computing, parallel algorithms and randomized algorithms. Among his Ph. D. students are Susan Landau, F. Thomson Leighton, Shang-Hua Teng, Jonathan Shewchuk. Gary Miller's web page at Carnegie Mellon. Gary Miller at the Mathematics Genealogy Project. Miller's original paper "Riemann's Hypothesis and Tests for Primality"