1.
United States
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Forty-eight of the fifty states and the federal district are contiguous and located in North America between Canada and Mexico. The state of Alaska is in the northwest corner of North America, bordered by Canada to the east, the state of Hawaii is an archipelago in the mid-Pacific Ocean. The U. S. territories are scattered about the Pacific Ocean, the geography, climate and wildlife of the country are extremely diverse. At 3.8 million square miles and with over 324 million people, the United States is the worlds third- or fourth-largest country by area, third-largest by land area. It is one of the worlds most ethnically diverse and multicultural nations, paleo-Indians migrated from Asia to the North American mainland at least 15,000 years ago. European colonization began in the 16th century, the United States emerged from 13 British colonies along the East Coast. Numerous disputes between Great Britain and the following the Seven Years War led to the American Revolution. On July 4,1776, during the course of the American Revolutionary War, the war ended in 1783 with recognition of the independence of the United States by Great Britain, representing the first successful war of independence against a European power. The current constitution was adopted in 1788, after the Articles of Confederation, the first ten amendments, collectively named the Bill of Rights, were ratified in 1791 and designed to guarantee many fundamental civil liberties. During the second half of the 19th century, the American Civil War led to the end of slavery in the country. By the end of century, the United States extended into the Pacific Ocean. The Spanish–American War and World War I confirmed the status as a global military power. The end of the Cold War and the dissolution of the Soviet Union in 1991 left the United States as the sole superpower. The U. S. is a member of the United Nations, World Bank, International Monetary Fund, Organization of American States. The United States is a developed country, with the worlds largest economy by nominal GDP. It ranks highly in several measures of performance, including average wage, human development, per capita GDP. While the U. S. economy is considered post-industrial, characterized by the dominance of services and knowledge economy, the United States is a prominent political and cultural force internationally, and a leader in scientific research and technological innovations. In 1507, the German cartographer Martin Waldseemüller produced a map on which he named the lands of the Western Hemisphere America after the Italian explorer and cartographer Amerigo Vespucci
2.
Cornell University
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Cornell University is an American private Ivy League and land-grant doctoral university located in Ithaca, New York. These ideals, unconventional for the time, are captured in Cornells motto, the university also administers two satellite medical campuses, one in New York City and one in Education City, Qatar. Cornell is one of three private land grant universities in the nation and the one in New York. Of its seven colleges, three are state-supported statutory or contract colleges through the State University of New York system, including its agricultural. Of Cornells graduate schools, only the college is state-supported. As a land grant college, Cornell operates a cooperative extension program in every county of New York. The Cornell University Ithaca Campus comprises 745 acres, but is larger when the Cornell Botanic Gardens are considered. Since its founding, Cornell has been a co-educational, non-sectarian institution where admission has not been restricted by religion or race, the student body consists of more than 14,000 undergraduate and 7,000 graduate students from all 50 American states and more than 120 countries. Cornell University was founded on April 27,1865, the New York State Senate authorized the university as the land grant institution. Senator Ezra Cornell offered his farm in Ithaca, New York, as a site, fellow senator and experienced educator Andrew Dickson White agreed to be the first president. During the next three years, White oversaw the construction of the first two buildings and traveled to attract students and faculty, the university was inaugurated on October 7,1868, and 412 men were enrolled the next day. Cornell developed as an innovative institution, applying its research to its own campus as well as to outreach efforts. For example, in 1883 it was one of the first university campuses to use electricity from a dynamo to light the grounds. Cornell has had active alumni since its earliest classes and it was one of the first universities to include alumni-elected representatives on its Board of Trustees. Today the university has more than 4,000 courses, since 2000, Cornell has been expanding its international programs. In 2004, the university opened the Weill Cornell Medical College in Qatar and it has partnerships with institutions in India, Singapore, and the Peoples Republic of China. Former president Jeffrey S. Lehman described the university, with its international profile. On March 9,2004, Cornell and Stanford University laid the cornerstone for a new Bridging the Rift Center to be built, Cornells main campus is on East Hill in Ithaca, New York, overlooking the town and Cayuga Lake
3.
Computer science
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Computer science is the study of the theory, experimentation, and engineering that form the basis for the design and use of computers. An alternate, more succinct definition of science is the study of automating algorithmic processes that scale. A computer scientist specializes in the theory of computation and the design of computational systems and its fields can be divided into a variety of theoretical and practical disciplines. Some fields, such as computational complexity theory, are highly abstract, other fields still focus on challenges in implementing computation. Human–computer interaction considers the challenges in making computers and computations useful, usable, the earliest foundations of what would become computer science predate the invention of the modern digital computer. Machines for calculating fixed numerical tasks such as the abacus have existed since antiquity, further, algorithms for performing computations have existed since antiquity, even before the development of sophisticated computing equipment. Wilhelm Schickard designed and constructed the first working mechanical calculator in 1623, in 1673, Gottfried Leibniz demonstrated a digital mechanical calculator, called the Stepped Reckoner. He may be considered the first computer scientist and information theorist, for, among other reasons and he started developing this machine in 1834, and in less than two years, he had sketched out many of the salient features of the modern computer. A crucial step was the adoption of a card system derived from the Jacquard loom making it infinitely programmable. Around 1885, Herman Hollerith invented the tabulator, which used punched cards to process statistical information, when the machine was finished, some hailed it as Babbages dream come true. During the 1940s, as new and more powerful computing machines were developed, as it became clear that computers could be used for more than just mathematical calculations, the field of computer science broadened to study computation in general. Computer science began to be established as an academic discipline in the 1950s. The worlds first computer science program, the Cambridge Diploma in Computer Science. The first computer science program in the United States was formed at Purdue University in 1962. Since practical computers became available, many applications of computing have become distinct areas of study in their own rights and it is the now well-known IBM brand that formed part of the computer science revolution during this time. IBM released the IBM704 and later the IBM709 computers, still, working with the IBM was frustrating if you had misplaced as much as one letter in one instruction, the program would crash, and you would have to start the whole process over again. During the late 1950s, the science discipline was very much in its developmental stages. Time has seen significant improvements in the usability and effectiveness of computing technology, modern society has seen a significant shift in the users of computer technology, from usage only by experts and professionals, to a near-ubiquitous user base
4.
Physics
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Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
5.
Philip Holmes
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Philip John Holmes is the Eugene Higgins Professor of Mechanical and Aerospace Engineering at Princeton University. As a member of the Mechanical and Aerospace Engineering department, he served as the interim chair until May 2007. Holmes was educated in England at Oxford University where he studied engineering from 1964 to 1967, and at Southampton University and he has made solid contributions to the field of nonlinear dynamics and differential equations. His book on dynamical systems with John Guckenheimer is a landmark in the field and he was elected a Fellow of the American Academy of Arts and Sciences in 1994. In 2001 he was elected an Honorary Member of the Hungarian Academy of Sciences, in 2006 he was elected a Fellow of the American Physical Society, and in 2012 he was elected a Fellow of the American Mathematical Society. He also has published collections of poetry
6.
New Brunswick, New Jersey
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The City of New Brunswick is a city in Middlesex County, New Jersey, United States. It is the county seat of Middlesex County, and the home of Rutgers University, the city is located on the Northeast Corridor rail line,27 miles southwest of Manhattan, on the southern bank of the Raritan River. The corporate headquarters and production facilities of several global companies are situated in the city, including Johnson & Johnson. New Brunswick is noted for its ethnic diversity, at one time, one quarter of the Hungarian population of New Jersey resided in the city and in the 1930s one out of three city residents was Hungarian. The Hungarian community continues to exist, alongside growing Asian and Hispanic communities that have developed around French Street near Robert Wood Johnson University Hospital and it was first inhabited by the Lenape Native Americans. The first European settlement at the site of New Brunswick was made in 1681, the settlement here was called Prigmores Swamp, then known as Inians Ferry. In 1714, the settlement was given the name New Brunswick, after the city of Braunschweig, in state of Lower Saxony, in Germany. Braunschweig was an influential and powerful city in the Hanseatic League, later in the Holy Roman Empire, and was an administrative seat for the Duchy of Hanover. Shortly after the first settlement of New Brunswick in colonial New Jersey, George, Duke of Brunswick-Lüneburg, alternatively, the city gets its name from King George II of Great Britain, the Duke of Brunswick-Lüneburg. New Brunswick was incorporated as a town in 1736 and chartered as a city in 1784 and it was incorporated into a town in 1798 as part of the Township Act of 1798. It was occupied by the British in the winter of 1776–1777 during the Revolutionary War. The Declaration of Independence received one of its first public readings, by Col. John Neilson, in New Brunswick on July 9,1776, in the days following its promulgation by the Continental Congress. The Trustees of Queens College, founded in 1766, voted to locate the college in New Brunswick, selecting the city over Hackensack, in Bergen County. Classes began in 1771 with one instructor, one sophomore, Matthew Leydt, classes were held through the American Revolution in various taverns and boarding houses, and at a building known as College Hall on George Street, until Old Queens was erected in 1808. It remains the oldest building on the Rutgers University campus, the Queens College Grammar School was established also in 1766, and shared facilities with the College until 1830, when it located in a building across College Avenue from Old Queens. The New Brunswick Theological Seminary, founded in 1784 in New York, moved to New Brunswick in 1810, New Brunswick was incorporated as a city by an act of the New Jersey Legislature on September 1,1784. The existence of an African American community dates back to the late 18th century, the citys Mount Zion African Methodist Episcopal Church, located at 39 Morris Street, was originally established in 1825 at 25 Division Street, making it one of the oldest in New Jersey. New Brunswick began attracting a Hungarian immigrant population around the turn of the 20th century, hungarians were primarily attracted to the city by employment at Johnson & Johnson factories located in the city
7.
University of New Mexico
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The University of New Mexico is a public research university in Albuquerque, New Mexico. It is New Mexicos flagship research institution, the largest post-secondary institution in the state in total enrollment across all campuses as of 2012, founded in 1889, UNM offers bachelors, masters, doctoral, and professional degree programs in a wide variety of fields. Its Albuquerque campus encompasses over 600 acres, and there are campuses in Gallup, Los Alamos, Rio Rancho, Taos. UNM is categorized as an R1 doctoral university in the Carnegie Classification of Institutions of Higher Education, the University of New Mexico was founded on February 28,1889, with the passage of House Bill No. Two years later, Elias S. Stover became the first president of the University, the third president of UNM, William G. Tight, who served from 1901–09, introduced many programs for students and faculty, including the first fraternity and sorority. Tight introduced the Pueblo Revival architecture for which the campus has become known, under David Ross Boyd, the universitys fifth president, the campus was enlarged from 20 to 300 acres and a 200, 000-acre federal land grant was made to the university. In 1922, the university was accredited by the North Central Association of Colleges, under Zimmerman, many new buildings were constructed, student enrollment increased, new departments were added, and greater support was generated for scientific research. Among the new buildings constructed were Zimmerman Library, Scholes Hall, the first student union building and this period also saw the foundation of UNMs branch facilities in Los Alamos and Gallup and the acquisition of the D. H. Lawrence Ranch north of Taos. During the early 1970s, a series of protests were held at the university, on May 5,1970, a protest over the Vietnam War and the Kent State massacre occupied the Student Union Building. The National Guard was ordered to sweep the building and arrest those inside, on May 10,1972, a peaceful sit-in protest near Kirtland Air Force Base led to the arrest of thirty-five people and was pushed back to UNM, leading to eight more arrests. The following day, tear gas was used against hundreds of demonstrators on campus, New programs and schools were created in the 1970s and the university gained control over the hospital on North Campus. At the end of the decade, the university was implicated in a recruiting scandal dubbed Lobogate by the press, subsequent investigation turned up a manufactured college seal from Mercer County Community College in New Jersey, along with blank transcripts and records of previous forgery. Further investigation uncovered alleged incentives like cars and apartments doled out to prime players, the scandal forced Ellenberger to resign and defined the term of William E. Davis, UNMs eleventh president. The university has continued to grow, with expanding enrollment and new facilities, in the 1980s, dramatic expansion occurred at the medical center, business school, and engineering school. The Centennial Library was also constructed, during the 1990s, an Honors College was founded, and the university completed construction of a new bookstore and Dane Smith Hall. The Research Park at South Campus was also expanded, by this point, the university had one of the largest student and faculty populations of Hispanics and Native Americans in the country. A study released in 1995 showed that the number of full-time Hispanic faculty at UNM was four times greater than the national average, the schools of law and business had some of the largest Hispanic student populations of any university in the country. In the first decade of the 2000s, major expansion began on medical facilities on North Campus, the current visitor center, a new engineering center, and George Pearl Hall were constructed
8.
Northwestern University
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Composed of twelve schools and colleges, Northwestern offers 124 undergraduate degrees and 145 graduate and professional degrees. Northwestern was founded in 1851 by John Evans, for whom the City of Evanston is named and its founding purpose was to serve the Northwest Territory, an area that today includes the states of Ohio, Indiana, Illinois, Michigan, Wisconsin and parts of Minnesota. Instruction began in 1855, women were admitted in 1869, today, the main campus is a 240-acre parcel in Evanston, along the shores of Lake Michigan 12 miles north of downtown Chicago. The universitys law, medical, and professional schools are located on a 25-acre campus in Chicagos Streeterville neighborhood, in 2008, the university opened a campus in Education City, Doha, Qatar with programs in journalism and communication. In 2016, Northwestern opened its San Francisco space at 44 Montgomery St. which hosts journalism, engineering, Northwestern is a large research university with a comprehensive doctoral program and it attracts over $650 million in sponsored research each year. Northwestern has the tenth largest university endowment in the United States, in 2017, the university accepted 9. 0% of undergraduate applicants from a pool of 37,255. Northwestern is a member of the Big Ten Conference and remains the only private university in the conference. The Northwestern Wildcats compete in 19 intercollegiate sports in the NCAAs Division I Big Ten Conference, on January 28,1851, the Illinois General Assembly granted a charter to the Trustees of the North-Western University, making it the first chartered university in Illinois. The schools nine founders, all of whom were Methodists, knelt in prayer, John Evans, for whom Evanston is named, bought 379 acres of land along Lake Michigan in 1853, and Philo Judson developed plans for what would become the city of Evanston, Illinois. The first building, Old College, opened on November 5,1855, to raise funds for its construction, Northwestern sold $100 perpetual scholarships entitling the purchaser and his heirs to free tuition. Willard Residential College is named in her honor, Northwestern admitted its first women students in 1869, and the first woman was graduated in 1874. Northwestern fielded its first intercollegiate football team in 1882, later becoming a member of the Big Ten Conference. In the 1870s and 1880s, Northwestern affiliated itself with already existing schools of law, medicine, Northwestern University Pritzker School of Law is the oldest law school in Chicago. The Association of American Universities invited Northwestern to become a member in 1917, in 1933, a proposal to merge Northwestern with the University of Chicago was considered but rejected. Northwestern was also one of the first six universities in the country to establish a Naval Reserve Officers Training Corps in the 1920s, after the golden years of the 1920s, the Great Depression in the United States hit Northwestern hard. Its annual income dropped 25 percent from $4.8 million in 1930-31 to $3.6 million in 1933-34. Investment income shrank, fewer parents could pay full tuition, and annual giving from alumni, the university responded with two salary cuts of 10 percent each for all employees. It imposed a freeze, a building freeze, and slashed appropriations for maintenance, books
9.
Santa Fe, New Mexico
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Santa Fe is the capital of the state of New Mexico. It is the fourth-largest city in the state and is the seat of Santa Fe County and this area was occupied for at least several hundred years by indigenous peoples who built villages. The city of Santa Fe, founded by Spanish colonists in 1610, is known as the oldest state capital city in the United States, Santa Fe had a population of 69,204 in 2012. It is the city of a Metropolitan Statistical Area which encompasses all of Santa Fe County and is part of the larger Albuquerque–Santa Fe–Las Vegas combined statistical area. The citys full name when founded was La Villa Real de la Santa Fe de San Francisco de Asís, the area of Santa Fe was originally occupied by indigenous Tanoan peoples, who lived in numerous Pueblo villages along the Rio Grande. One of the earliest known settlements in what today is downtown Santa Fe came sometime after 900, the river had a year-round flow until the 1700s. By the 20th century the Santa Fe River was a seasonal waterway, as of 2007, the river was recognized as the most endangered river in the United States, according to the conservation group American Rivers. Don Juan de Oñate led the first European effort to colonize the region in 1598, under Juan de Oñate and his son, the capital of the province was the settlement of San Juan de los Caballeros north of Santa Fe near modern Ohkay Owingeh Pueblo. In 1610, he designated it as the capital of the province, Santa Fe remained Spains provincial seat until the outbreak of the Mexican War of Independence in 1810. It was considered important to fur traders based in present-day Saint Louis, when the area was still under Spanish rule, the Chouteau brothers of Saint Louis gained a monopoly on the fur trade, before the United States acquired Missouri under the Louisiana Purchase of 1803. The fur trade contributed to the wealth of St. Louis, the citys status as the capital of the Mexican territory of Santa Fe de Nuevo México was formalized in the 1824 Constitution after Mexico achieved independence from Spain. When the Republic of Texas seceded from Mexico in 1836, it claimed Santa Fe as part of the portion of Texas along the Rio Grande. In 1841, a military and trading expedition set out from Austin. Known as the Texan Santa Fe Expedition, the force was prepared and was easily captured by the Mexican army. In 1846, the United States declared war on Mexico, brigadier General Stephen W. Kearny led the main body of his Army of the West of some 1,700 soldiers into Santa Fe to claim it and the whole New Mexico Territory for the United States. By 1848 the U. S. officially gained New Mexico through the Treaty of Guadalupe Hidalgo, colonel Alexander William Doniphan, under the command of Kearny, recovered ammunition from Santa Fe labeled Spain 1776. This showed that New Mexico had received munitions and other support under Mexican rule, some American visitors at first saw little promise in the remote town. One traveller in 1849 wrote, I can hardly imagine how Santa Fe is supported, the country around it is barren
10.
Three-body problem
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The three-body problem is a special case of the n-body problem. Historically, the first specific three-body problem to receive extended study was the one involving the Moon, the Earth, in an extended modern sense, a three-body problem is a class of problems in classical or quantum mechanics that model the motion of three particles. The gravitational problem of three bodies in its traditional sense dates in substance from 1687, when Isaac Newton published his Principia. The physical problem was addressed by Amerigo Vespucci and subsequently by Galileo Galilei, however the accuracy of the lunar theory was low, due to the perturbing effect of the Sun and planets on the motion of the Moon around the Earth. They submitted their competing first analyses to the Académie Royale des Sciences in 1747 and it was in connection with these researches, in Paris, in the 1740s, that the name three-body problem began to be commonly used. An account published in 1761 by Jean le Rond dAlembert indicates that the name was first used in 1747, in 1887, mathematicians Heinrich Bruns and Henri Poincaré showed that there is no general analytical solution for the three-body problem given by algebraic expressions and integrals. The motion of three bodies is generally non-repeating, except in special cases, a prominent example of the classical three-body problem is the movement of a planet with a satellite around a star. In this case, the problem is simplified to two instances of the two-body problem, however, the effect of the star on the movement of the satellite around the planet can be considered as a perturbation. While a spaceflight involving a gravity assist tends to be at least a problem, once far away from the Earth when Earths gravity becomes negligible. The general statement for the three body problem is as follows, in the circular restricted three-body problem, two massive bodies move in circular orbits around their common center of mass, and the third mass is negligible with respect to the other two. It can be useful to consider the effective potential, in 1767 Leonhard Euler found three families of periodic solutions in which the three masses are collinear at each instant. In 1772 Lagrange found a family of solutions in which the three form an equilateral triangle at each instant. Together, these form the central configurations for the three-body problem. These solutions are valid for any mass ratios, and the move on Keplerian ellipses. These five families are the only solutions for which there are explicit analytic formulae. In 1893 Meissel stated what is called the Pythagorean three-body problem. Burrau further investigated this problem in 1913, in 1967 Victor Szebehely and coworkers established eventual escape for this problem using numerical integration, while at the same time finding a nearby periodic solution. In 1911, United States scientist William Duncan MacMillan found one special solution, in 1961, Russian mathematician Sitnikov improved this solution
11.
Classical mechanics
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In physics, classical mechanics is one of the two major sub-fields of mechanics, along with quantum mechanics. Classical mechanics is concerned with the set of physical laws describing the motion of bodies under the influence of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology. Classical mechanics describes the motion of objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars. Within classical mechanics are fields of study that describe the behavior of solids, liquids and gases, Classical mechanics also provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light. When both quantum and classical mechanics cannot apply, such as at the level with high speeds. Since these aspects of physics were developed long before the emergence of quantum physics and relativity, however, a number of modern sources do include relativistic mechanics, which in their view represents classical mechanics in its most developed and accurate form. Later, more abstract and general methods were developed, leading to reformulations of classical mechanics known as Lagrangian mechanics and these advances were largely made in the 18th and 19th centuries, and they extend substantially beyond Newtons work, particularly through their use of analytical mechanics. The following introduces the concepts of classical mechanics. For simplicity, it often models real-world objects as point particles, the motion of a point particle is characterized by a small number of parameters, its position, mass, and the forces applied to it. Each of these parameters is discussed in turn, in reality, the kind of objects that classical mechanics can describe always have a non-zero size. Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the degrees of freedom. However, the results for point particles can be used to such objects by treating them as composite objects. The center of mass of a composite object behaves like a point particle, Classical mechanics uses common-sense notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as where an object is in space, non-relativistic mechanics also assumes that forces act instantaneously. The position of a point particle is defined with respect to a fixed reference point in space called the origin O, in space. A simple coordinate system might describe the position of a point P by means of a designated as r. In general, the point particle need not be stationary relative to O, such that r is a function of t, the time
12.
General relativity
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General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. General relativity generalizes special relativity and Newtons law of gravitation, providing a unified description of gravity as a geometric property of space and time. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter, the relation is specified by the Einstein field equations, a system of partial differential equations. Examples of such differences include gravitational time dilation, gravitational lensing, the redshift of light. The predictions of relativity have been confirmed in all observations. Although general relativity is not the only theory of gravity. Einsteins theory has important astrophysical implications, for example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive stars. The bending of light by gravity can lead to the phenomenon of gravitational lensing, General relativity also predicts the existence of gravitational waves, which have since been observed directly by physics collaboration LIGO. In addition, general relativity is the basis of current cosmological models of an expanding universe. Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his new relativistic framework. In 1907, beginning with a thought experiment involving an observer in free fall. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations. These equations specify how the geometry of space and time is influenced by whatever matter and radiation are present, the Einstein field equations are nonlinear and very difficult to solve. Einstein used approximation methods in working out initial predictions of the theory, but as early as 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric. This solution laid the groundwork for the description of the stages of gravitational collapse. In 1917, Einstein applied his theory to the universe as a whole, in line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to match that observational presumption. By 1929, however, the work of Hubble and others had shown that our universe is expanding and this is readily described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant. Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which our universe has evolved from an extremely hot, Einstein later declared the cosmological constant the biggest blunder of his life
13.
Cuboctahedron
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In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, as such, it is a quasiregular polyhedron, i. e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. Its dual polyhedron is the rhombic dodecahedron, the cuboctahedron was probably known to Plato, Herons Definitiones quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Heptaparallelohedron Fuller applied the name Dymaxion to this shape, used in a version of the Dymaxion map. He also called it the Vector Equilibrium and he called a cuboctahedron consisting of rigid struts connected by flexible vertices a jitterbug. With Oh symmetry, order 48, it is a cube or rectified octahedron With Td symmetry, order 24. With D3d symmetry, order 12, it is a triangular gyrobicupola. The area A and the volume V of the cuboctahedron of edge length a are, the cuboctahedron has four special orthogonal projections, centered on a vertex, an edge, and the two types of faces, triangular and square. The last two correspond to the B2 and A2 Coxeter planes, the skew projections show a square and hexagon passing through the center of the cuboctahedron. The cuboctahedron can also be represented as a tiling. This projection is conformal, preserving angles but not areas or lengths, straight lines on the sphere are projected as circular arcs on the plane. The cuboctahedrons 12 vertices can represent the vectors of the simple Lie group A3. With the addition of 6 vertices of the octahedron, these represent the 18 root vectors of the simple Lie group B3. The cuboctahedron can be dissected into two triangular cupolas by a common hexagon passing through the center of the cuboctahedron, if these two triangular cupolas are twisted so triangles and squares line up, Johnson solid J27, the triangular orthobicupola, is created. The cuboctahedron can also be dissected into 6 square pyramids and 8 tetrahedra meeting at a central point and this dissection is expressed in the alternated cubic honeycomb where pairs of square pyramids are combined into octahedra. A cuboctahedron can be obtained by taking a cross section of a four-dimensional 16-cell. Its first stellation is the compound of a cube and its dual octahedron, the cuboctahedron is a rectified cube and also a rectified octahedron. It is also a cantellated tetrahedron, with this construction it is given the Wythoff symbol,33 |2
14.
Polyomino
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A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares and it may be regarded as a finite subset of the regular square tiling with a connected interior. Polyominoes are classified according to how many cells they have, Polyominoes have been used in popular puzzles since at least 1907, and the enumeration of pentominoes is dated to antiquity. Many results with the pieces of 1 to 6 squares were first published in Fairy Chess Review between the years 1937 to 1957, under the name of dissection problems. The name polyomino was invented by Solomon W. Golomb in 1953, related to polyominoes are polyiamonds, formed from equilateral triangles, polyhexes, formed from regular hexagons, and other plane polyforms. Polyominoes have been generalized to higher dimensions by joining cubes to form polycubes, like many puzzles in recreational mathematics, polyominoes raise many combinatorial problems. The most basic is enumerating polyominoes of a given size, no formula has been found except for special classes of polyominoes. A number of estimates are known, and there are algorithms for calculating them, Polyominoes with holes are inconvenient for some purposes, such as tiling problems. In some contexts polyominoes with holes are excluded, allowing only simply connected polyominoes, there are three common ways of distinguishing polyominoes for enumeration, free polyominoes are distinct when none is a rigid transformation of another. Translating, rotating, reflecting, or glide reflecting a free polyomino does not change its shape, one-sided polyominoes are distinct when none is a translation or rotation of another. Translating or rotating a one-sided polyomino does not change its shape, fixed polyominoes are distinct when none is a translation of another. Translating a fixed polyomino will not change its shape, the following table shows the numbers of polyominoes of various types with n cells. As of 2004, Iwan Jensen has enumerated the fixed polyominoes up to n =56, free polyominoes have been enumerated up to n =28 by Tomás Oliveira e Silva. The dihedral group D4 is the group of symmetries of a square and this group contains four rotations and four reflections. It is generated by alternating reflections about the x-axis and about a diagonal, one free polyomino corresponds to at most 8 fixed polyominoes, which are its images under the symmetries of D4. However, those images are not necessarily distinct, the more symmetry a free polyomino has, therefore, a free polyomino that is invariant under some or all non-trivial symmetries of D4 may correspond to only 4,2 or 1 fixed polyominoes. Mathematically, free polyominoes are equivalence classes of fixed polyominoes under the group D4, the following table shows the numbers of polyominoes with n squares, sorted by symmetry groups. Each polyomino of order n+1 can be obtained by adding a square to a polyomino of order n and this leads to algorithms for generating polyominoes inductively
15.
NP-completeness
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In computational complexity theory, a decision problem is NP-complete when it is both in NP and NP-hard. The set of NP-complete problems is often denoted by NP-C or NPC, the abbreviation NP refers to nondeterministic polynomial time. That is, the required to solve the problem using any currently known algorithm increases very quickly as the size of the problem grows. As a consequence, determining whether or not it is possible to solve problems quickly. NP-complete problems are addressed by using heuristic methods and approximation algorithms. A problem p in NP is NP-complete if every problem in NP can be transformed into p in polynomial time. NP-complete problems are studied because the ability to quickly verify solutions to a problem seems to correlate with the ability to solve that problem. It is not known whether every problem in NP can be quickly solved—this is called the P versus NP problem, because of this, it is often said that NP-complete problems are harder or more difficult than NP problems in general. A decision problem C is NP-complete if, C is in NP, C can be shown to be in NP by demonstrating that a candidate solution to C can be verified in polynomial time. Note that a problem satisfying condition 2 is said to be NP-hard, a consequence of this definition is that if we had a polynomial time algorithm for C, we could solve all problems in NP in polynomial time. The concept of NP-completeness was introduced in 1971, though the term NP-complete was introduced later, at 1971 STOC conference, there was a fierce debate among the computer scientists about whether NP-complete problems could be solved in polynomial time on a deterministic Turing machine. This is known as the question of whether P=NP, nobody has yet been able to determine conclusively whether NP-complete problems are in fact solvable in polynomial time, making this one of the great unsolved problems of mathematics. The Clay Mathematics Institute is offering a US $1 million reward to anyone who has a proof that P=NP or that P≠NP. Cook–Levin theorem states that the Boolean satisfiability problem is NP-complete, in 1972, Richard Karp proved that several other problems were also NP-complete, thus there is a class of NP-complete problems. For more details refer to Introduction to the Design and Analysis of Algorithms by Anany Levitin, an interesting example is the graph isomorphism problem, the graph theory problem of determining whether a graph isomorphism exists between two graphs. Two graphs are isomorphic if one can be transformed into the other simply by renaming vertices, consider these two problems, Graph Isomorphism, Is graph G1 isomorphic to graph G2. Subgraph Isomorphism, Is graph G1 isomorphic to a subgraph of graph G2, the Subgraph Isomorphism problem is NP-complete. The graph isomorphism problem is suspected to be neither in P nor NP-complete and this is an example of a problem that is thought to be hard, but is not thought to be NP-complete
16.
Power law
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For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four. Few empirical distributions fit a power law for all their values, acoustic attenuation follows frequency power-laws within wide frequency bands for many complex media. Allometric scaling laws for relationships between biological variables are among the best known power-law functions in nature, one attribute of power laws is their scale invariance. Given a relation f = a x − k, scaling the argument x by a constant factor c causes only a proportionate scaling of the function itself and that is, f = a − k = c − k f ∝ f. That is, scaling by a constant c simply multiplies the original power-law relation by the constant c − k, thus, it follows that all power laws with a particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others. This behavior is what produces the linear relationship when logarithms are taken of both f and x, and the straight-line on the plot is often called the signature of a power law. With real data, such straightness is a necessary, but not sufficient, in fact, there are many ways to generate finite amounts of data that mimic this signature behavior, but, in their asymptotic limit, are not true power laws. Thus, accurately fitting and validating power-law models is an area of research in statistics. This can be seen in the thought experiment, imagine a room with your friends. Now imagine the worlds richest person entering the room, with an income of about 1 billion US$. What happens to the income in the room. Income is distributed according to a known as the Pareto distribution. On the one hand, this makes it incorrect to apply traditional statistics that are based on variance, on the other hand, this also allows for cost-efficient interventions. For example, given that car exhaust is distributed according to a power-law among cars it would be sufficient to eliminate those very few cars from the road to reduce total exhaust substantially. For instance, the behavior of water and CO2 at their boiling points fall in the universality class because they have identical critical exponents. In fact, almost all material phase transitions are described by a set of universality classes. Similar observations have made, though not as comprehensively, for various self-organized critical systems. Formally, this sharing of dynamics is referred to as universality, scientific interest in power-law relations stems partly from the ease with which certain general classes of mechanisms generate them
17.
Degree distribution
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The degree of a node in a network is the number of connections or edges the node has to other nodes. The degree distribution P of a network is defined to be the fraction of nodes in the network with degree k. Thus if there are n nodes in total in a network and nk of them have degree k, the degree distribution is very important in studying both real networks, such as the Internet and social networks, and theoretical networks. The simplest network model, for example, the random graph, most networks in the real world, however, have degree distributions very different from this. Most are highly right-skewed, meaning that a majority of nodes have low degree but a small number. Some networks, notably the Internet, the wide web, and some social networks are found to have degree distributions that approximately follow a power law, P ~ k−γ. Such networks are called scale-free networks and have attracted attention for their structural and dynamical properties. Graph theory Complex network Scale-free network Random graph Structural cut-off Albert, R. Barabasi, dorogovtsev, S. Mendes, J. F. F. The structure and function of complex networks, Complex Networks, Structure, Robustness and Function
18.
Traceroute
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In computing, traceroute is a computer network diagnostic tool for displaying the route and measuring transit delays of packets across an Internet Protocol network. Traceroute proceeds unless all sent packets are lost more than twice, then the connection is lost, ping, on the other hand, only computes the final round-trip times from the destination point. The command traceroute is available on modern operating systems. On Apple macOS, it is available by opening the menu Network Utilities and selecting Traceroute, on other Unix systems, such as FreeBSD or Linux, it is also available as a command line tool. On Microsoft Windows, it is named tracert, Windows NT-based operating systems also provide PathPing, with similar functionality. For Internet Protocol Version 6 the tool sometimes has the name traceroute6 or tracert6, in Linux, traceroute by default sends a sequence of User Datagram Protocol packets addressed to a destination host, ICMP Echo Request or TCP SYN packets can also be used. In Windows, traceroute sends ICMP echo requests instead of UDP packets, the time-to-live value, also known as hop limit, is used in determining the intermediate routers being traversed towards the destination. Routers decrement TTL values of packets by one when routing and discard packets whose TTL value has reached zero, common default values for the initial TTL are 128 and 64. Traceroute works by sending packets with gradually increasing TTL value, starting with TTL value of one, the first router receives the packet, decrements the TTL value and drops the packet because it then has TTL value zero. The router sends an ICMP Time Exceeded message back to the source, the next set of packets are given a TTL value of two, so the first router forwards the packets, but the second router drops them and replies with ICMP Time Exceeded. The timestamp values returned for each router along the path are the delay values, the sender expects a reply within a specified number of seconds. If a packet is not acknowledged within the interval, an asterisk is displayed. The Internet Protocol does not require packets to take the route towards a particular destination. If the host at hop #N does not reply, the hop is skipped in the output, on Unix-like operating systems, traceroute employs User Datagram Protocol datagrams by default, with destination port numbers ranging from 33434 to 33534. The traceroute utility usually has an option to instead use ICMP Echo Request packets, like the Windows utility tracert does, or to use TCP SYN packets. If a network has a firewall and operates both Windows and Unix-like systems, more than one protocol must be enabled inbound through the firewall for traceroute to work, some traceroute implementations use TCP packets, such as tcptraceroute and layer four traceroute. PathPing is a utility introduced with Windows NT that combines ping, MTR is an enhanced version of ICMP traceroute available for Unix-like and Windows systems. The various implementations of traceroute all rely on ICMP Time Exceeded packets being sent to the source, on Linux, tracepath is a utility similar to traceroute, with the primary difference of not requiring superuser privileges
19.
Hierarchical clustering
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In data mining and statistics, hierarchical clustering is a method of cluster analysis which seeks to build a hierarchy of clusters. Divisive, This is a top down approach, all start in one cluster. In general, the merges and splits are determined in a greedy manner, the results of hierarchical clustering are usually presented in a dendrogram. In the general case, the complexity of agglomerative clustering is O, divisive clustering with an exhaustive search is O, which is even worse. However, for special cases, optimal efficient agglomerative methods ) are known, SLINK for single-linkage. In order to decide which clusters should be combined, or where a cluster should be split, the choice of an appropriate metric will influence the shape of the clusters, as some elements may be close to one another according to one distance and farther away according to another. Some commonly used metrics for hierarchical clustering are, For text or other non-numeric data, the linkage criterion determines the distance between sets of observations as a function of the pairwise distances between observations. Some commonly used linkage criteria between two sets of observations A and B are, where d is the chosen metric, other linkage criteria include, The sum of all intra-cluster variance. The decrease in variance for the cluster being merged, the probability that candidate clusters spawn from the same distribution function. The product of in-degree and out-degree on a k-nearest-neighbour graph, the increment of some cluster descriptor after merging two clusters. Hierarchical clustering has the advantage that any valid measure of distance can be used. In fact, the observations themselves are not required, all that is used is a matrix of distances, for example, suppose this data is to be clustered, and the Euclidean distance is the distance metric. Cutting the tree at a height will give a partitioning clustering at a selected precision. In this example, cutting after the row of the dendrogram will yield clusters. Cutting after the row will yield clusters, which is a coarser clustering. The hierarchical clustering dendrogram would be as such, This method builds the hierarchy from the elements by progressively merging clusters. In our example, we have six elements and, the first step is to determine which elements to merge in a cluster. Usually, we want to take the two closest elements, according to the chosen distance, optionally, one can also construct a distance matrix at this stage, where the number in the i-th row j-th column is the distance between the i-th and j-th elements
20.
Phase transition
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The term phase transition is most commonly used to describe transitions between solid, liquid and gaseous states of matter, and, in rare cases, plasma. A phase of a system and the states of matter have uniform physical properties. For example, a liquid may become gas upon heating to the boiling point, the measurement of the external conditions at which the transformation occurs is termed the phase transition. Phase transitions are common in nature and used today in many technologies, the same process, but beginning with a solid instead of a liquid is called a eutectoid transformation. A peritectic transformation, in which a two component single phase solid is heated and transforms into a phase and a liquid phase. A spinodal decomposition, in which a phase is cooled. Transition to a mesophase between solid and liquid, such as one of the crystal phases. The transition between the ferromagnetic and paramagnetic phases of materials at the Curie point. The transition between differently ordered, commensurate or incommensurate, magnetic structures, such as in cerium antimonide, the martensitic transformation which occurs as one of the many phase transformations in carbon steel and stands as a model for displacive phase transformations. Changes in the structure such as between ferrite and austenite of iron. Order-disorder transitions such as in alpha-titanium aluminides, the dependence of the adsorption geometry on coverage and temperature, such as for hydrogen on iron. The emergence of superconductivity in certain metals and ceramics when cooled below a critical temperature, the superfluid transition in liquid helium is an example of this. The breaking of symmetries in the laws of physics during the history of the universe as its temperature cooled. Isotope fractionation occurs during a transition, the ratio of light to heavy isotopes in the involved molecules changes. When water vapor condenses, the heavier water isotopes become enriched in the liquid phase while the lighter isotopes tend toward the vapor phase, Phase transitions occur when the thermodynamic free energy of a system is non-analytic for some choice of thermodynamic variables. This condition generally stems from the interactions of a number of particles in a system. It is important to note that phase transitions can occur and are defined for non-thermodynamic systems, examples include, quantum phase transitions, dynamic phase transitions, and topological phase transitions. In these types of other parameters take the place of temperature
21.
Boolean satisfiability problem
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In computer science, the Boolean Satisfiability Problem is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called satisfiable, on the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is unsatisfiable. For example, the formula a AND NOT b is satisfiable because one can find the values a = TRUE and b = FALSE, in contrast, a AND NOT a is unsatisfiable. SAT is one of the first problems that was proven to be NP-complete and this means that all problems in the complexity class NP, which includes a wide range of natural decision and optimization problems, are at most as difficult to solve as SAT. g. Artificial intelligence, circuit design, and automatic theorem proving, a propositional logic formula, also called Boolean expression, is built from variables, operators AND, OR, NOT, and parentheses. A formula is said to be if it can be made TRUE by assigning appropriate logical values to its variables. The Boolean satisfiability problem is, given a formula, to whether it is satisfiable. This decision problem is of importance in various areas of computer science, including theoretical computer science, complexity theory, algorithmics, cryptography. There are several cases of the Boolean satisfiability problem in which the formulas are required to have a particular structure. A literal is either a variable, then called positive literal, or the negation of a variable, a clause is a disjunction of literals. A clause is called a Horn clause if it contains at most one positive literal, a formula is in conjunctive normal form if it is a conjunction of clauses. The formula is satisfiable, choosing x1 = FALSE, x2 = FALSE, and x3 arbitrarily, since ∧ ∧ ¬FALSE evaluates to ∧ ∧ TRUE, and in turn to TRUE ∧ TRUE ∧ TRUE. In contrast, the CNF formula a ∧ ¬a, consisting of two clauses of one literal, is unsatisfiable, since for a=TRUE and a=FALSE it evaluates to TRUE ∧ ¬TRUE and FALSE ∧ ¬FALSE, different sets of allowed boolean operators lead to different problem versions. As an example, R is a clause, and R ∧ R ∧ R is a generalized conjunctive normal form. This formula is used below, with R being the operator that is TRUE just if exactly one of its arguments is. Using the laws of Boolean algebra, every propositional logic formula can be transformed into an equivalent conjunctive normal form, for example, transforming the formula ∨ ∨. ∨ into conjunctive normal form yields ∧ ∧ ∧ ∧, ∧ ∧ ∧ ∧, while the former is a disjunction of n conjunctions of 2 variables, the latter consists of 2n clauses of n variables
22.
Search for extraterrestrial intelligence
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There are great challenges in searching the universe for signs of intelligent life, including their identification and interpretation. As various SETI projects have progressed, their claims have been criticized by researchers as being too euphoric. Scientific investigation of the phenomenon began shortly after the advent of radio in the early 1900s. Focused international efforts to answer a variety of questions have been going on since the 1980s. In 2015, Stephen Hawking and Russian billionaire Yuri Milner announced a well-funded effort, There have been many earlier searches for extraterrestrial intelligence within the Solar System. In 1896, Nikola Tesla suggested that a version of his wireless electrical transmission system could be used to contact beings on Mars. On August 21–23,1924, Mars entered an opposition closer to Earth than at any time in the century before or the next 80 years. In the United States, a National Radio Silence Day was promoted during a 36-hour period from August 21–23, with all radios quiet for five minutes on the hour, every hour. The program was led by David Peck Todd with the assistance of Admiral Edward W. Eberle, with William F. Friedman. A1959 paper by Philip Morrison and Giuseppe Cocconi first pointed out the possibility of searching the microwave spectrum, and proposed frequencies and a set of initial targets. In 1960, Cornell University astronomer Frank Drake performed the first modern SETI experiment, named Project Ozma, a 400 kilohertz band around the marker frequency was scanned, using a single-channel receiver with a bandwidth of 100 hertz. The Soviet scientists took a strong interest in SETI during the 1960s, in the March 1955 issue of Scientific American, John D. Kraus described an idea to scan the cosmos for natural radio signals using a flat-plane radio telescope equipped with a parabolic reflector. Within two years, his concept was approved for construction by Ohio State University, with a total of US$71,000 in grants from the National Science Foundation, construction began on an 8-hectare plot in Delaware, Ohio. This Ohio State University Radio Observatory telescope was called Big Ear, later, it began the worlds first continuous SETI program, called the Ohio State University SETI program. In 1971, NASA funded a SETI study that involved Drake, Bernard M. Oliver of Hewlett-Packard Corporation, the resulting report proposed the construction of an Earth-based radio telescope array with 1,500 dishes known as Project Cyclops. The price tag for the Cyclops array was US$10 billion, Cyclops was not built, but the report formed the basis of much SETI work that followed. The Ohio State SETI program gained fame on August 15,1977, when Jerry Ehman and he quickly circled the indication on a printout and scribbled the exclamation Wow. in the margin. In 1980, Carl Sagan, Bruce Murray, and Louis Friedman founded the U. S, Planetary Society, partly as a vehicle for SETI studies
23.
Quantum algorithm
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In quantum computing, a quantum algorithm is an algorithm which runs on a realistic model of quantum computation, the most commonly used model being the quantum circuit model of computation. A classical algorithm is a sequence of instructions, or a step-by-step procedure for solving a problem. Similarly, an algorithm is a step-by-step procedure, where each of the steps can be performed on a quantum computer. Problems which are undecidable using classical computers remain undecidable using quantum computers, what makes quantum algorithms interesting is that they might be able to solve some problems faster than classical algorithms. The most well known algorithms are Shors algorithm for factoring, Shors algorithms runs exponentially faster than the best known classical algorithm for factoring, the general number field sieve. Grovers algorithm runs quadratically faster than the best possible classical algorithm for the same task, Quantum algorithms are usually described, in the commonly used circuit model of quantum computation, by a quantum circuit which acts on some input qubits and terminates with a measurement. A quantum circuit consists of quantum gates which act on at most a fixed number of qubits. Quantum algorithms may also be stated in other models of quantum computation, Quantum algorithms can be categorized by the main techniques used by the algorithm. Quantum algorithms may also be grouped by the type of problem solved, the quantum Fourier transform is the quantum analogue of the discrete Fourier transform, and is used in several quantum algorithms. The Hadamard transform is also an example of a quantum Fourier transform over a vector space over the field F2. The quantum Fourier transform can be implemented on a quantum computer using only a polynomial number of quantum gates. The algorithm determines whether a function f is constant or balanced. Simons algorithm solves a black-box problem exponentially faster than any classical algorithm and this algorithm, which achieves an exponential speedup over all classical algorithms that we consider efficient, was the motivation for Shors factoring algorithm. The quantum phase estimation algorithm is used to determine the eigenphase of an eigenvector of a unitary gate given a state proportional to the eigenvector. The algorithm is used as a subroutine in other algorithms. Shors algorithm solves the discrete problem and the integer factorization problem in polynomial time. These problems are not known to be in P or NP-complete and it is also one of the few quantum algorithms that solves a non–black-box problem in polynomial time where the best known classical algorithms run in super-polynomial time. There are efficient quantum algorithms known for the Abelian hidden subgroup problem, the more general hidden subgroup problem, where the group isnt necessarily abelian, is a generalization of the previously mentioned problems and graph isomorphism and certain lattice problems
24.
Graph isomorphism
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This kind of bijection is commonly described as edge-preserving bijection, in accordance with the general notion of isomorphism being a structure-preserving bijection. If an isomorphism exists between two graphs, then the graphs are called isomorphic and denoted as G ≃ H, a set of graphs isomorphic to each other is called an isomorphism class of graphs. The two graphs shown below are isomorphic, despite their different looking drawings, in the above definition, graphs are understood to be undirected non-labeled non-weighted graphs. with the following exception. For labeled graphs, two definitions of isomorphism are in use, under one definition, an isomorphism is a vertex bijection which is both edge-preserving and label-preserving. For example, the K2 graph with the two vertices labelled with 1 and 2 has a single automorphism under the first definition, in such cases two labeled graphs are sometimes said to be isomorphic if the corresponding underlying unlabeled graphs are isomorphic. For example, if a graph has exactly one cycle, then all graphs in its isomorphism class also have exactly one cycle, on the other hand, in the common case when the vertices of a graph are the integers 1,2. N, then the expression ∑ v ∈ V v ⋅ deg v may be different for two isomorphic graphs, the Whitney graph isomorphism theorem, shown by H. The Whitney graph theorem can be extended to hypergraphs, while graph isomorphism may be studied in a classical mathematical way, as exemplified by the Whitney theorem, it is recognized that it is a problem to be tackled with an algorithmic approach. The computational problem of determining whether two finite graphs are isomorphic is called the isomorphism problem. Its practical applications include primarily cheminformatics, mathematical chemistry, and electronic design automation, the graph isomorphism problem is one of few standard problems in computational complexity theory belonging to NP, but not known to belong to either of its well-known subsets, P and NP-complete. It is one of two, out of 12 total, problems listed in Garey & Johnson whose complexity remains unresolved. It is however known that if the problem is NP-complete then the hierarchy collapses to a finite level. In November 2015, László Babai, a mathematician and computer scientist at the University of Chicago and this work has not yet been vetted. In January 2017, Babai shortly retracted the quasi-polynomiality claim and stated a sub-exponential time time complexity bound instead and he restored the original claim five days later. Its generalization, the isomorphism problem, is known to be NP-complete. Graph homomorphism Graph automorphism problem Graph canonization Garey, Michael R. Johnson, computers and Intractability, A Guide to the Theory of NP-Completeness, W. H. Freeman, ISBN 0-7167-1045-5
25.
Zachary's karate club
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The Zachary Karate Club is a well-known social network of a university karate club described in An Information Flow Model for Conflict and Fission in Small Groups paper by Wayne W. Zachary. A social network of a club was studied by Wayne W. Zachary for a period of three years from 1970 to 1972. The network captures 34 members of a club, documenting 78 pairwise links between members who interacted outside the club. During the study a conflict arose between the administrator John A and instructor Mr. Hi, which led to the split of the club into two. Half of the members formed a new club around Mr. Hi, basing on collected data Zachary assigned correctly all but one member of the club to the groups they actually joined after the split. Before the split each side tried to recruit adherents of another party, thus, communication flow had a special importance and the initial group would likely split at the borders of the network. Zachary correctly predicted each members decision except member #9, who went with Mr. Hi instead of John A, the data set for Zacharys karate club is in open access on the internet. The data can be summarized as list of integer pairs, each integer represents one karate club member and a pair indicates the two members interacted. The data set is summarized below and also in the adjoining image, node 1 stands for the instructor, node 34 for the club administrator / president. The first scientist to be awarded was Cristopher Moore in 2013, network Scientists with Karate Trophies K-Means Clustering with Python Tutorial using Zacharys Karate Club dataset
26.
American Physical Society
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Not be be confused with the American Physical Society which was absorbed by the Royal Physical Society of Edinburgh in 1796. The American Physical Society is the second largest organization of physicists. The Society publishes more than a dozen scientific journals, including the prestigious Physical Review and Physical Review Letters, APS is a member society of the American Institute of Physics. The American Physical Society was founded on May 20,1899 and they proclaimed the mission of the new Society to be to advance and diffuse the knowledge of physics, and in one way or another the APS has been at that task ever since. In the early years, virtually the sole activity of the APS was to hold scientific meetings, in 1913, the APS took over the operation of the Physical Review, which had been founded in 1893 at Cornell University, and journal publication became its second major activity. The Physical Review was followed by Reviews of Modern Physics in 1929, rev. has subdivided into five separate sections as the fields of physics proliferated and the number of submissions grew. In more recent years, the activities of the Society have broadened considerably, in addition, the Society conducts extensive programs in education, science outreach, and media relations. APS has 14 divisions and 11 topical groups covering all areas of physics research, there are 6 forums that reflect the interest of its 50,000 members in broader issues, and 9 sections organized by geographical region. In 1999, APS Physics celebrated its centennial with the biggest-ever physics meeting in Atlanta, einstein@Home, one of the projects APS initiated during World Year of Physics, is an ongoing and popular distributed computing project. During the summer of 2005, the society conducted an electronic poll, the poll became the motivation for a proposal of a name change promised in the leadership election that year. However, because of issues, the planned name change was eventually abandoned by the APS Executive Board. To promote public recognition of APS as a society, while retaining the name American Physical Society. General use of APS Physics to refer to APS or the American Physical Society is encouraged, the new APS Physics logo was designed by Kerry G. Johnson. At least now when you are in an elevator at an APS meeting and someone looks at your badge, the American Physical Society publishes 13 international research journals and an open-access on-line news and commentary website Physics. Physical Review Letters Reviews of Modern Physics Physical Review A, Atomic, molecular, Physical Review B, Condensed matter and materials physics. Physical Review D, Particles, fields, gravitation, and cosmology, Physical Review E, Statistical, nonlinear, and soft matter physics. Physical Review X, Open access, pure, applied, Physical Review Applied, Experimental and theoretical applications of physics. Physical Review Accelerators and Beams, Open access, accelerator science, Physical Review Physics Education Research, Open access, experimental and theoretical research on physics education
27.
American Mathematical Society
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The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. It was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, john Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, the result was the Bulletin of the New York Mathematical Society, with Fiske as editor-in-chief. The de facto journal, as intended, was influential in increasing membership, the popularity of the Bulletin soon led to Transactions of the American Mathematical Society and Proceedings of the American Mathematical Society, which were also de facto journals. In 1891 Charlotte Scott became the first woman to join the society, the society reorganized under its present name and became a national society in 1894, and that year Scott served as the first woman on the first Council of the American Mathematical Society. In 1951, the headquarters moved from New York City to Providence. The society later added an office in Ann Arbor, Michigan in 1984, in 1954 the society called for the creation of a new teaching degree, a Doctor of Arts in Mathematics, similar to a PhD but without a research thesis. Mary W. Gray challenged that situation by sitting in on the Council meeting in Atlantic City, when she was told she had to leave, she refused saying she would wait until the police came. After that time, Council meetings were open to observers and the process of democratization of the Society had begun, julia Robinson was the first female president of the American Mathematical Society but was unable to complete her term as she was suffering from leukemia. In 1988 the Journal of the American Mathematical Society was created, the 2013 Joint Mathematics Meeting in San Diego drew over 6,600 attendees. Each of the four sections of the AMS hold meetings in the spring. The society also co-sponsors meetings with other mathematical societies. The AMS selects a class of Fellows who have made outstanding contributions to the advancement of mathematics. The AMS publishes Mathematical Reviews, a database of reviews of mathematical publications, various journals, in 1997 the AMS acquired the Chelsea Publishing Company, which it continues to use as an imprint. Blogs, Blog on Blogs e-Mentoring Network in the Mathematical Sciences AMS Graduate Student Blog PhD + Epsilon On the Market Some prizes are awarded jointly with other mathematical organizations. The AMS is led by the President, who is elected for a two-year term, morrey, Jr. Oscar Zariski Nathan Jacobson Saunders Mac Lane Lipman Bers R. H. Andrews Eric M. Friedlander David Vogan Robert L
28.
PubMed Identifier
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PubMed is a free search engine accessing primarily the MEDLINE database of references and abstracts on life sciences and biomedical topics. The United States National Library of Medicine at the National Institutes of Health maintains the database as part of the Entrez system of information retrieval, from 1971 to 1997, MEDLINE online access to the MEDLARS Online computerized database primarily had been through institutional facilities, such as university libraries. PubMed, first released in January 1996, ushered in the era of private, free, home-, the PubMed system was offered free to the public in June 1997, when MEDLINE searches via the Web were demonstrated, in a ceremony, by Vice President Al Gore. Information about the journals indexed in MEDLINE, and available through PubMed, is found in the NLM Catalog. As of 5 January 2017, PubMed has more than 26.8 million records going back to 1966, selectively to the year 1865, and very selectively to 1809, about 500,000 new records are added each year. As of the date,13.1 million of PubMeds records are listed with their abstracts. In 2016, NLM changed the system so that publishers will be able to directly correct typos. Simple searches on PubMed can be carried out by entering key aspects of a subject into PubMeds search window, when a journal article is indexed, numerous article parameters are extracted and stored as structured information. Such parameters are, Article Type, Secondary identifiers, Language, publication type parameter enables many special features. As these clinical girish can generate small sets of robust studies with considerable precision, since July 2005, the MEDLINE article indexing process extracts important identifiers from the article abstract and puts those in a field called Secondary Identifier. The secondary identifier field is to store numbers to various databases of molecular sequence data, gene expression or chemical compounds. For clinical trials, PubMed extracts trial IDs for the two largest trial registries, ClinicalTrials. gov and the International Standard Randomized Controlled Trial Number Register, a reference which is judged particularly relevant can be marked and related articles can be identified. If relevant, several studies can be selected and related articles to all of them can be generated using the Find related data option, the related articles are then listed in order of relatedness. To create these lists of related articles, PubMed compares words from the title and abstract of each citation, as well as the MeSH headings assigned, using a powerful word-weighted algorithm. The related articles function has been judged to be so precise that some researchers suggest it can be used instead of a full search, a strong feature of PubMed is its ability to automatically link to MeSH terms and subheadings. Examples would be, bad breath links to halitosis, heart attack to myocardial infarction, where appropriate, these MeSH terms are automatically expanded, that is, include more specific terms. Terms like nursing are automatically linked to Nursing or Nursing and this important feature makes PubMed searches automatically more sensitive and avoids false-negative hits by compensating for the diversity of medical terminology. The My NCBI area can be accessed from any computer with web-access, an earlier version of My NCBI was called PubMed Cubby
29.
James Crutchfield
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James Crutchfield was a St. Louis barrelhouse blues singer, piano player and songwriter whose career spanned seven decades. His repertoire consisted of original and classic blues and boogie-woogie and depression-era popular songs, known as the King of Barrelhouse Blues, his better-known songs include I Believe You Need a Shot and My Baby Cooks My Breakfast. There is no record of James Crutchfields birth, My mama never knowd what day it was, she never knowd what month it was, lotta folks back in them days never even knowd that much, but my mama always did. She told me I was born in 12, in Baton Rouge, Crutchfield said his mother, Sarah, was a Geechee - a descendant of slaves of the Georgia/Carolina sea islands and said he much resembled her. An only child, James and his mother, a worker, migrated through Louisiana and East Texas with the cotton and sugarcane seasons, moving often. His earliest memories were of the coming home from World War I and the silent westerns of William S. Hart. Around 1920, his married and settled in Bogalusa, Louisiana. In his early teens, while employed as the janitor in a theater, also around this time, curious about the exact day of his birth, he went to the Baton Rouge library and told the story his mother had told him to an intrigued librarian. Together they looked through the 1912 newspapers and found that indeed, there had been a flood then, from that time on, he considered that date as his birthday. In 1927, working as an employee for a local railroad. The railroad settled out of court for twenty thousand dollars, part of the money was used to buy his mother a house in Baton Rouge and the rest, given his now diminished opportunities for employment, was used to subsidize his fledgling musical career. The establishments that served the lumber and levee camps typically stayed open all day and night and provided food, drink, another early influence was Papa Lord God, a Texan, Oh Papa Lord God, he was bad, man, he was baaad. Crutchfield worked as accompanist to Joe Pullum in the early 1930s and performed with him in Texas and Louisiana and he was to play Pullums hit Black Gal for the rest of his life. Shortly after the end of World War II, Crutchfield performed with Elmore James and Boyd Gilmore around the Goodman, in 1948, Crutchfield moved to St. Louis, Missouri, a city with a venerable blues piano tradition dating back to the ragtime era. He worked in the Gaslight Square entertainment district at various venues and he was found there by Paul Affeldt), on a tip from policeman Charlie OBrien, and recorded a few days later along with Speckled Red. Several of the songs were released in the Barrelhouse Blues. Six selections can be heard on the compilation album Biddle Street Barrelhousin, the decline of Gaslight Square in the late 1960s was also the decline of Crutchfields musical career. He was professionally inactive in the 1970s and worked as a cook at the State Hospital for a number of years, in the early 1980s he was collecting and selling junk tires and running an illegal gambling operation
30.
Discrete and Computational Geometry
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Discrete & Computational Geometry is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard M. Pollack, the journal is indexed by Mathematical Reviews, Zentralblatt MATH, Science Citation Index, and Current Contents/Engineering, Computing and Technology. Its 2009 impact factor is 0.935
31.
ArXiv
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In many fields of mathematics and physics, almost all scientific papers are self-archived on the arXiv repository. Begun on August 14,1991, arXiv. org passed the half-million article milestone on October 3,2008, by 2014 the submission rate had grown to more than 8,000 per month. The arXiv was made possible by the low-bandwidth TeX file format, around 1990, Joanne Cohn began emailing physics preprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity. Additional modes of access were added, FTP in 1991, Gopher in 1992. The term e-print was quickly adopted to describe the articles and its original domain name was xxx. lanl. gov. Due to LANLs lack of interest in the rapidly expanding technology, in 1999 Ginsparg changed institutions to Cornell University and it is now hosted principally by Cornell, with 8 mirrors around the world. Its existence was one of the factors that led to the current movement in scientific publishing known as open access. Mathematicians and scientists regularly upload their papers to arXiv. org for worldwide access, Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv. The annual budget for arXiv is approximately $826,000 for 2013 to 2017, funded jointly by Cornell University Library, annual donations were envisaged to vary in size between $2,300 to $4,000, based on each institution’s usage. As of 14 January 2014,174 institutions have pledged support for the period 2013–2017 on this basis, in September 2011, Cornell University Library took overall administrative and financial responsibility for arXivs operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it was supposed to be a three-hour tour, however, Ginsparg remains on the arXiv Scientific Advisory Board and on the arXiv Physics Advisory Committee. The lists of moderators for many sections of the arXiv are publicly available, additionally, an endorsement system was introduced in 2004 as part of an effort to ensure content that is relevant and of interest to current research in the specified disciplines. Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors, new authors from recognized academic institutions generally receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for allegedly restricting scientific inquiry, perelman appears content to forgo the traditional peer-reviewed journal process, stating, If anybody is interested in my way of solving the problem, its all there – let them go and read about it. The arXiv generally re-classifies these works, e. g. in General mathematics, papers can be submitted in any of several formats, including LaTeX, and PDF printed from a word processor other than TeX or LaTeX. The submission is rejected by the software if generating the final PDF file fails, if any image file is too large. ArXiv now allows one to store and modify an incomplete submission, the time stamp on the article is set when the submission is finalized
32.
Nature (journal)
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Nature is an English multidisciplinary scientific journal, first published on 4 November 1869. Nature claims a readership of about 3 million unique readers per month. The journal has a circulation of around 53,000. There are also sections on books and arts, the remainder of the journal consists mostly of research papers, which are often dense and highly technical. There are many fields of research in which important new advances, the papers that have been published in this journal are internationally acclaimed for maintaining high research standards. In 2007 Nature received the Princess of Asturias Award for Communications, the enormous progress in science and mathematics during the 19th century was recorded in journals written mostly in German or French, as well as in English. Britain underwent enormous technological and industrial changes and advances particularly in the half of the 19th century. In addition, during this period, the number of popular science periodicals doubled from the 1850s to the 1860s. According to the editors of these popular science magazines, the publications were designed to serve as organs of science, in essence, Nature, first created in 1869, was not the first magazine of its kind in Britain. While Recreative Science had attempted to more physical sciences such as astronomy and archaeology. Two other journals produced in England prior to the development of Nature were the Quarterly Journal of Science and Scientific Opinion, established in 1864 and 1868 and these similar journals all ultimately failed. The Popular Science Review survived longest, lasting 20 years and ending its publication in 1881, Recreative Science ceased publication as the Student, the Quarterly Journal, after undergoing a number of editorial changes, ceased publication in 1885. The Reader terminated in 1867, and finally, Scientific Opinion lasted a mere 2 years, janet Browne has proposed that far more than any other science journal of the period, Nature was conceived, born, and raised to serve polemic purpose. Perhaps it was in part its scientific liberality that made Nature a longer-lasting success than its predecessors and this is what Lockyers journal did from the start. Norman Lockyer, the founder of Nature, was a professor at Imperial College and he was succeeded as editor in 1919 by Sir Richard Gregory. Gregory helped to establish Nature in the scientific community. During the years 1945 to 1973, editorship of Nature changed three times, first in 1945 to A. J. V. Gale and L. J. F. Brimble, then to John Maddox in 1965, and finally to David Davies in 1973. In 1980, Maddox returned as editor and retained his position until 1995, philip Campbell has since become Editor-in-chief of all Nature publications