Computer

A computer is a device that can be instructed to carry out sequences of arithmetic or logical operations automatically via computer programming. Modern computers have the ability to follow generalized sets of called programs; these programs enable computers to perform an wide range of tasks. A "complete" computer including the hardware, the operating system, peripheral equipment required and used for "full" operation can be referred to as a computer system; this term may as well be used for a group of computers that are connected and work together, in particular a computer network or computer cluster. Computers are used as control systems for a wide variety of industrial and consumer devices; this includes simple special purpose devices like microwave ovens and remote controls, factory devices such as industrial robots and computer-aided design, general purpose devices like personal computers and mobile devices such as smartphones. The Internet is run on computers and it connects hundreds of millions of other computers and their users.

Early computers were only conceived as calculating devices. Since ancient times, simple manual devices like the abacus aided people in doing calculations. Early in the Industrial Revolution, some mechanical devices were built to automate long tedious tasks, such as guiding patterns for looms. More sophisticated electrical machines did specialized analog calculations in the early 20th century; the first digital electronic calculating machines were developed during World War II. The speed and versatility of computers have been increasing ever since then. Conventionally, a modern computer consists of at least one processing element a central processing unit, some form of memory; the processing element carries out arithmetic and logical operations, a sequencing and control unit can change the order of operations in response to stored information. Peripheral devices include input devices, output devices, input/output devices that perform both functions. Peripheral devices allow information to be retrieved from an external source and they enable the result of operations to be saved and retrieved.

According to the Oxford English Dictionary, the first known use of the word "computer" was in 1613 in a book called The Yong Mans Gleanings by English writer Richard Braithwait: "I haue read the truest computer of Times, the best Arithmetician that euer breathed, he reduceth thy dayes into a short number." This usage of the term referred to a human computer, a person who carried out calculations or computations. The word continued with the same meaning until the middle of the 20th century. During the latter part of this period women were hired as computers because they could be paid less than their male counterparts. By 1943, most human computers were women. From the end of the 19th century the word began to take on its more familiar meaning, a machine that carries out computations; the Online Etymology Dictionary gives the first attested use of "computer" in the 1640s, meaning "one who calculates". The Online Etymology Dictionary states that the use of the term to mean "'calculating machine' is from 1897."

The Online Etymology Dictionary indicates that the "modern use" of the term, to mean "programmable digital electronic computer" dates from "1945 under this name. Devices have been used to aid computation for thousands of years using one-to-one correspondence with fingers; the earliest counting device was a form of tally stick. Record keeping aids throughout the Fertile Crescent included calculi which represented counts of items livestock or grains, sealed in hollow unbaked clay containers; the use of counting rods is one example. The abacus was used for arithmetic tasks; the Roman abacus was developed from devices used in Babylonia as early as 2400 BC. Since many other forms of reckoning boards or tables have been invented. In a medieval European counting house, a checkered cloth would be placed on a table, markers moved around on it according to certain rules, as an aid to calculating sums of money; the Antikythera mechanism is believed to be the earliest mechanical analog "computer", according to Derek J. de Solla Price.

It was designed to calculate astronomical positions. It was discovered in 1901 in the Antikythera wreck off the Greek island of Antikythera, between Kythera and Crete, has been dated to c. 100 BC. Devices of a level of complexity comparable to that of the Antikythera mechanism would not reappear until a thousand years later. Many mechanical aids to calculation and measurement were constructed for astronomical and navigation use; the planisphere was a star chart invented by Abū Rayhān al-Bīrūnī in the early 11th century. The astrolabe was invented in the Hellenistic world in either the 1st or 2nd centuries BC and is attributed to Hipparchus. A combination of the planisphere and dioptra, the astrolabe was an analog computer capable of working out several different kinds of problems in spherical astronomy. An astrolabe incorporating a mechanical calendar computer and gear-wheels was invented by Abi Bakr of Isfahan, Persia in 1235. Abū Rayhān al-Bīrūnī invented the first mechanical geared lunisolar calendar astrolabe, an early fixed-wired knowledge processing machine with a gear train and gear-wheels, c. 1000 AD.

The sector, a calculating instrument used for solving problems in proportion, trigonometry and division, for various functions, such as squares and cube roots, was developed in

Elliptic function

In complex analysis, an elliptic function is a meromorphic function, periodic in two directions. Just as a periodic function of a real variable is defined by its values on an interval, an elliptic function is determined by its values on a fundamental parallelogram, which repeat in a lattice; such a doubly periodic function cannot be holomorphic, as it would be a bounded entire function, by Liouville's theorem every such function must be constant. In fact, an elliptic function must have at least two poles in a fundamental parallelogram, as it is easy to show using the periodicity that a contour integral around its boundary must vanish, implying that the residues of all simple poles must cancel. Elliptic functions were first discovered by Niels Henrik Abel as inverse functions of elliptic integrals, their theory was improved by Carl Gustav Jacobi. Jacobi's elliptic functions have found numerous applications in physics, were used by Jacobi to prove some results in elementary number theory. A more complete study of elliptic functions was undertaken by Karl Weierstrass, who found a simple elliptic function in terms of which all the others could be expressed.

Besides their practical use in the evaluation of integrals and the explicit solution of certain differential equations, they have deep connections with elliptic curves and modular forms. Formally, an elliptic function is a function f meromorphic on ℂ for which there exist two non-zero complex numbers ω1 and ω2 with ω1/ω2 ∉ ℝ, such that f = f and f = f for all z ∈ ℂ. Denoting the "lattice of periods" by Λ =, it follows that f = f for all ω ∈ Λ. There are those of Weierstrass. Although Jacobi's elliptic functions are older and more directly relevant to applications, modern authors follow Weierstrass when presenting the elementary theory, because his functions are simpler, any elliptic function can be expressed in terms of them. With the definition of elliptic functions given above the Weierstrass elliptic function ℘ is constructed in the most obvious way: given a lattice Λ as above, put ℘ = 1 z 2 + ∑ ω ∈ Λ ∖ This function is invariant with respect to the transformation z ↦ z + ω for any ω ∈ Λ.

The addition of the −1/ω2 terms is necessary to make the sum converge. The technical condition to ensure that an infinite sum such as this converges to a meromorphic function is that on any compact set, after omitting the finitely many terms having poles in that set, the remaining series converges normally. On any compact disk defined by |z| ≤ R, for any |ω| > 2R, one has | 1 2 − 1 ω 2 | = | 2 ω z − z 2 ω 2 2 | = | z ω 3 2 | ≤ 10 R | ω | 3 and it can be shown that the sum ∑ ω ≠ 0 1 | ω | 3 converges regardless of Λ. By writing ℘ as a Laurent series and explicitly comparing terms, one may verify that it satisfies the relation 2 = 4 3 − g 2 ℘ − g 3 where g 2 = 60 ∑ ω ∈ Λ ∖ 1 ω 4 {\displaystyle g_=60\sum _{\omeg

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

Astronomy

Astronomy is a natural science that studies celestial objects and phenomena. It applies mathematics and chemistry in an effort to explain the origin of those objects and phenomena and their evolution. Objects of interest include planets, stars, nebulae and comets. More all phenomena that originate outside Earth's atmosphere are within the purview of astronomy. A related but distinct subject is physical cosmology, the study of the Universe as a whole. Astronomy is one of the oldest of the natural sciences; the early civilizations in recorded history, such as the Babylonians, Indians, Nubians, Chinese and many ancient indigenous peoples of the Americas, performed methodical observations of the night sky. Astronomy has included disciplines as diverse as astrometry, celestial navigation, observational astronomy, the making of calendars, but professional astronomy is now considered to be synonymous with astrophysics. Professional astronomy is split into theoretical branches. Observational astronomy is focused on acquiring data from observations of astronomical objects, analyzed using basic principles of physics.

Theoretical astronomy is oriented toward the development of computer or analytical models to describe astronomical objects and phenomena. The two fields complement each other, with theoretical astronomy seeking to explain observational results and observations being used to confirm theoretical results. Astronomy is one of the few sciences in which amateurs still play an active role in the discovery and observation of transient events. Amateur astronomers have made and contributed to many important astronomical discoveries, such as finding new comets. Astronomy means "law of the stars". Astronomy should not be confused with astrology, the belief system which claims that human affairs are correlated with the positions of celestial objects. Although the two fields share a common origin, they are now distinct. Both of the terms "astronomy" and "astrophysics" may be used to refer to the same subject. Based on strict dictionary definitions, "astronomy" refers to "the study of objects and matter outside the Earth's atmosphere and of their physical and chemical properties," while "astrophysics" refers to the branch of astronomy dealing with "the behavior, physical properties, dynamic processes of celestial objects and phenomena."

In some cases, as in the introduction of the introductory textbook The Physical Universe by Frank Shu, "astronomy" may be used to describe the qualitative study of the subject, whereas "astrophysics" is used to describe the physics-oriented version of the subject. However, since most modern astronomical research deals with subjects related to physics, modern astronomy could be called astrophysics; some fields, such as astrometry, are purely astronomy rather than astrophysics. Various departments in which scientists carry out research on this subject may use "astronomy" and "astrophysics" depending on whether the department is affiliated with a physics department, many professional astronomers have physics rather than astronomy degrees; some titles of the leading scientific journals in this field include The Astronomical Journal, The Astrophysical Journal, Astronomy and Astrophysics. In early historic times, astronomy only consisted of the observation and predictions of the motions of objects visible to the naked eye.

In some locations, early cultures assembled massive artifacts that had some astronomical purpose. In addition to their ceremonial uses, these observatories could be employed to determine the seasons, an important factor in knowing when to plant crops and in understanding the length of the year. Before tools such as the telescope were invented, early study of the stars was conducted using the naked eye; as civilizations developed, most notably in Mesopotamia, Persia, China and Central America, astronomical observatories were assembled and ideas on the nature of the Universe began to develop. Most early astronomy consisted of mapping the positions of the stars and planets, a science now referred to as astrometry. From these observations, early ideas about the motions of the planets were formed, the nature of the Sun and the Earth in the Universe were explored philosophically; the Earth was believed to be the center of the Universe with the Sun, the Moon and the stars rotating around it. This is known as the geocentric model of the Ptolemaic system, named after Ptolemy.

A important early development was the beginning of mathematical and scientific astronomy, which began among the Babylonians, who laid the foundations for the astronomical traditions that developed in many other civilizations. The Babylonians discovered. Following the Babylonians, significant advances in astronomy were made in ancient Greece and the Hellenistic world. Greek astronomy is characterized from the start by seeking a rational, physical explanation for celestial phenomena. In the 3rd century BC, Aristarchus of Samos estimated the size and distance of the Moon and Sun, he proposed a model of the Solar System where the Earth and planets rotated around the Sun, now called the heliocentric model. In the 2nd century BC, Hipparchus discovered precession, calculated the size and distance of the Moon and inven

Mathematical table

Mathematical tables are lists of numbers showing the results of calculation with varying arguments. Before calculators were cheap and plentiful, people would use such tables to simplify and drastically speed up computation. Tables of logarithms and trigonometric functions were common in science textbooks. Specialized tables were published for applications such as astronomy, celestial navigation and statistics. To compute the sine function of 75 degrees, 9 minutes, 50 seconds using a table of trigonometric functions such as the Bernegger table from 1619 illustrated here, one might round up to 75 degrees, 10 minutes and find the 10 minute entry on the 75 degree page, shown above-right, 0.9666746. However, this answer is only accurate to four decimal places. If one wanted greater accuracy, one could interpolate linearly as follows: From the Bernegger table: sin = 0.9666746 sin = 0.9666001The difference between these values is 0.0000745. Since there are 60 seconds in a minute of arc, we multiply the difference by 50/60 to get a correction of *0.0000745 ≈ 0.0000621.

For tables with greater precision, higher order interpolation may be needed to get full accuracy. In the era before electronic computers, interpolating table data in this manner was the only practical way to get high accuracy values of mathematical functions needed for applications such as navigation and surveying. To understand the importance of accuracy in applications like navigation note that at sea level one minute of arc along the Earth's equator or a meridian equals one nautical mile; the first tables of trigonometric functions known to be made were by Hipparchus and Menelaus, but both have been lost. Along with the surviving table of Ptolemy, they were all tables of chords and not of half-chords, i.e. the sine function. The table produced by the Indian mathematician Āryabhaṭa is considered the first sine table constructed. Āryabhaṭa's table remained the standard sine table of ancient India. There were continuous attempts to improve the accuracy of this table, culminating in the discovery of the power series expansions of the sine and cosine functions by Madhava of Sangamagrama, the tabulation of a sine table by Madhava with values accurate to seven or eight decimal places.

Tables of common logarithms were used until the invention of computers and electronic calculators to do rapid multiplications and exponentiations, including the extraction of nth roots. Mechanical special-purpose computers known as difference engines were proposed in the 19th century to tabulate polynomial approximations of logarithmic functions – i.e. to compute large logarithmic tables. This was motivated by errors in logarithmic tables made by the human computers of the time. Early digital computers were developed during World War II in part to produce specialized mathematical tables for aiming artillery. From 1972 onwards, with the launch and growing use of scientific calculators, most mathematical tables went out of use. One of the last major efforts to construct such tables was the Mathematical Tables Project, started in 1938 as a project of the Works Progress Administration, employing 450 out-of-work clerks to tabulate higher mathematical functions, lasted through World War II. Tables of special functions are still used.

Creating tables stored in random-access memory is a common code optimization technique in computer programming, where the use of such tables speeds up calculations in those cases where a table lookup is faster than the corresponding calculations. In essence, one trades computing speed for the computer memory space required to store the tables. Tables containing common logarithms were extensively used in computations prior to the advent of computers and calculators because logarithms convert problems of multiplication and division into much easier addition and subtraction problems. Base-10 logarithms have an additional property, unique and useful: The common logarithm of numbers greater than one that differ only by a factor of a power of ten all have the same fractional part, known as the mantissa. Tables of common logarithms included only the mantissas; the fractional part of the common logarithm of numbers greater than zero but less than one is just 1 minus the mantissa of the same number with the decimal point shifted to the right of the first non-zero digit.

But same mantissa could be used for numbers less than one by offsetting the characteristic. Thus a single table of common logarithms can be used for the entire range of positive decimal numbers. See common logarithm for details on the use of characteristics and mantissas. Michael Stifel published Arithmetica integra in Nuremberg in 1544 which contains a table of integers and powers of 2, considered an early version of a logarithmic table; the method of logarithms was publicly propounded by John Napier in 1614, i