1.
Decatur, Georgia
–
Decatur is a city in, and county seat of, DeKalb County, Georgia, United States. With a population of 20,148 in the 2013 census and it is an intown suburb of Atlanta and part of the Atlanta Metropolitan Area, and its public transportation is served by three MARTA rail stations. Decaturs official motto is A city of homes, schools and places of worship, prior to 2000, its motto was A city of homes, schools, and churches. The citizens of Decatur did not want the noise, pollution and growth that would come with such a major terminal, in response, the railroad founded a new city to the west-southwest of Decatur for the terminal. This town later became the city of Atlanta, during the American Civil War, Decatur became a strategic site in Shermans campaign against Atlanta. In July 1864 Union general James B, mcPherson occupied Decatur to cut off the Confederates supply line from Augusta, Georgia. During the Battle of Atlanta on July 22, Confederate cavalry under Major General Joseph Wheeler attacked McPhersons supply wagons, a marker at the Decatur courthouse marks the site of this skirmish. In the last half of the century the metropolitan area of Atlanta expanded into unincorporated DeKalb County. Concurrently many well-to-do and middle class white Americans fled the area to more distant suburbs, the 1960s and 1970s witnessed dramatic drops in property values. Decatur is located at 33°46′17″N 84°17′52″W, according to the United States Census Bureau, the city has a total area of 4.2 square miles, all of it land. The Eastern Continental Divide bisects the city along the CSX trackage right of way, as of the 2010 census, there were 19,335 people,8,599 occupied housing units, and 4,215 families residing in the city. The population density was 4,603.6 people per square mile, there were 9,335 housing units at an average density of 2,222.6 per square mile. The racial makeup of the city was 73. 5% White,20. 2% African American,0. 2% Native American,2. 9% Asian,0. 0% Pacific Islander,0. 6% from other races, and 2. 4% from two or more races. Hispanic or Latino of any race were 3. 2% of the population,3,263 of all households were made up of individuals of those,1,814 had someone living alone who was 65 years of age or older. The average household size was 2.17 and the family size was 2.96. In the city, the population was out with 25. 1% under the age of 19,5. 2% from 20 to 24,32. 9% from 25 to 44,25. 7% from 45 to 64. The median age was 38 years, there are roughly 44 males for every 56 females. The ZIP code 30030 has one of the highest percentages of households with same sex couples in Georgia,9. 20% as of 2000, the median income for a household in the city was $73,602
2.
Princeton, New Jersey
–
As of the 2010 United States Census, the municipalitys population was 28,572, reflecting the former townships population of 16,265, along with the 12,307 in the former borough. Princeton was founded before the American Revolution and is best known as the location of Princeton University, Princeton is roughly equidistant from New York City and Philadelphia. It is close to major highways that serve both cities, and receives major television and radio broadcasts from each. It is also close to Trenton, New Jerseys capital city, the governor of New Jerseys official residence has been in Princeton since 1945, when Morven in the borough became the first Governors mansion. It was later replaced by the larger Drumthwacket, a mansion located in the former Township. Morven became a property of the New Jersey Historical Society. Princeton was ranked 15th of the top 100 towns in the United States to Live, although residents of Princeton traditionally have a strong community-wide identity, the community had been composed of two separate municipalities, a township and a borough. The central borough was completely surrounded by the township, the Borough contained Nassau Street, the main commercial street, most of the University campus, and incorporated most of the urban area until the postwar suburbanization. The Borough and Township had roughly equal populations, the Lenni Lenape Native Americans were the earliest identifiable inhabitants of the Princeton area. Europeans founded their settlement in the part of the 17th century. The first European to find his home in the boundaries of the town was Henry Greenland. He built his house in 1683 along with a tavern, in this drinking hole representatives of West Jersey and East Jersey met to set boundaries for the location of the township. Originally, Princeton was known only as part of nearby Stony Brook, James Leonard first referred to the town as Princetown, when describing the location of his large estate in his diary. The town bore a variety of names subsequently, including, Princetown, Princes Town, although there is no official documentary backing, the town is considered to be named after King William III, Prince William of Orange of the House of Nassau. Another theory suggests that the name came from a large land-owner named Henry Prince, a royal prince seems a more likely eponym for the settlement, as three nearby towns had similar names, Kingston, Queenstown and Princessville. When Richard Stockton, one of the founders of the township, died in 1709 he left his estate to his sons, who helped to expand property, based on the 1880 United States Census, the population of the town comprised 3,209 persons. Local population has expanded from the nineteenth century, according to the 2010 Census, Princeton Borough had 12,307 inhabitants, while Princeton Township had 16,265. Aside from housing the university of the name, the settlement suffered the revolutionary Battle of Princeton on its soil
3.
United States
–
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
4.
University of Chicago
–
The University of Chicago is a private research university in Chicago, Illinois. It holds top-ten positions in national and international rankings and measures. The university currently enrolls approximately 5,700 students in the College, Chicagos physics department helped develop the worlds first man-made, self-sustaining nuclear reaction beneath the viewing stands of universitys Stagg Field. The university is home to the University of Chicago Press. With an estimated date of 2020, the Barack Obama Presidential Center will be housed at the university. Both Harper and future president Robert Maynard Hutchins advocated for Chicagos curriculum to be based upon theoretical and perennial issues rather than on applied sciences, the University of Chicago has many prominent alumni. 92 Nobel laureates have been affiliated with the university as professors, students, faculty, or staff, similarly,34 faculty members and 16 alumni have been awarded the MacArthur “Genius Grant”. Rockefeller on land donated by Marshall Field, while the Rockefeller donation provided money for academic operations and long-term endowment, it was stipulated that such money could not be used for buildings. The original physical campus was financed by donations from wealthy Chicagoans like Silas B, Cobb who provided the funds for the campus first building, Cobb Lecture Hall, and matched Marshall Fields pledge of $100,000. Organized as an independent institution legally, it replaced the first Baptist university of the same name, william Rainey Harper became the modern universitys first president on July 1,1891, and the university opened for classes on October 1,1892. The business school was founded thereafter in 1898, and the law school was founded in 1902, Harper died in 1906, and was replaced by a succession of three presidents whose tenures lasted until 1929. During this period, the Oriental Institute was founded to support, in 1896, the university affiliated with Shimer College in Mount Carroll, Illinois. The agreement provided that either party could terminate the affiliation on proper notice, several University of Chicago professors disliked the program, as it involved uncompensated additional labor on their part, and they believed it cheapened the academic reputation of the university. The program passed into history by 1910, in 1929, the universitys fifth president, Robert Maynard Hutchins, took office, the university underwent many changes during his 24-year tenure. In 1933, Hutchins proposed a plan to merge the University of Chicago. During his term, the University of Chicago Hospitals finished construction, also, the Committee on Social Thought, an institution distinctive of the university, was created. Money that had been raised during the 1920s and financial backing from the Rockefeller Foundation helped the school to survive through the Great Depression, during World War II, the university made important contributions to the Manhattan Project. The university was the site of the first isolation of plutonium and of the creation of the first artificial, in the early 1950s, student applications declined as a result of increasing crime and poverty in the Hyde Park neighborhood
5.
Physics
–
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
6.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
7.
Institute for Advanced Study
–
The IAS is perhaps best known as the academic home of Albert Einstein, John von Neumann and Kurt Gödel, after their immigration to the United States. Although it is close to and collaborates with Princeton University, Rutgers University, Flexners guiding principle in founding the Institute was the pursuit of knowledge for its own sake. There are no programs or experimental facilities at the Institute. Research is never contracted or directed, it is left to each individual researcher to pursue their own goals and it is supported entirely by endowments, grants, and gifts, and is one of the eight American mathematics institutes funded by the National Science Foundation. It is the model for the eight members of the consortium Some Institutes for Advanced Study. The institute consists of four schools–Historical Studies, Mathematics, Natural Sciences, in 2016, the Institute has been in the news for a faculty housing project proposal. While the Institute owns the property on which it wants to build these houses, historians and archaeological evidence confirm the site witnessed Gen. George Washingtons arrival and charge on horseback across the battlefield during the January 3,1777 Battle of Princeton. The Institute was founded in 1930 by Abraham Flexner, together with philanthropists Louis Bamberger, Flexner is generally regarded as one of the most important figures in the history of American medicine and played a major role in the reform of medical education. This led to an interest in education generally and as early as 1890 he had founded a school which had no formal curriculum, exams. It was a success at preparing students for prestigious colleges. The Bamberger siblings wanted to use the proceeds from the sale of their department store in Newark, New Jersey, Flexner convinced them to put their money in the service of more abstract research. In 1932 Veblen resigned from Princeton and became the first professor in the new Institute for Advanced Study and he selected most of the original faculty and also helped the Institute acquire land in Princeton for both the original facility and future expansion. Flexner and Veblen set out to recruit the best mathematicians and physicists they could find, the rise of fascism and the associated anti-semitism forced many prominent mathematicians to flee Europe and some, such as Einstein and Hermann Weyl, found a home at the new institute. Weyl as a condition of accepting insisted that the Institute also appoint the thirty year old Austrian-Hungarian polymath John von Neumann, indeed, the IAS became the key lifeline for scholars fleeing Europe. Einstein was Flexners first coup and shortly after that he was followed by Veblens brilliant student James Alexander, Flexner was fortunate in the luminaries he directly recruited but also in the people that they brought along with them. Thus, by 1934 the fledgeling institute was led by six of the most prominent mathematicians in the world, in 1935 quantum physics pioneer Wolfgang Pauli became a faculty member. With the opening of the Institute for Advanced Study, Princeton replaced Göttingen as the center for mathematics in the twentieth century. Princeton Universitys science departments are less than two miles away and informal ties and collaboration between the two institutions occurred from the beginning and this helped start an incorrect impression that it was part of the University, one that has never been completely eradicated
8.
Princeton University
–
Princeton University is a private Ivy League research university in Princeton, New Jersey, United States. The institution moved to Newark in 1747, then to the current site nine years later, Princeton provides undergraduate and graduate instruction in the humanities, social sciences, natural sciences, and engineering. The university has ties with the Institute for Advanced Study, Princeton Theological Seminary, Princeton has the largest endowment per student in the United States. The university has graduated many notable alumni, two U. S. Presidents,12 U. S. Supreme Court Justices, and numerous living billionaires and foreign heads of state are all counted among Princetons alumni body. New Light Presbyterians founded the College of New Jersey in 1746 in order to train ministers, the college was the educational and religious capital of Scots-Irish America. In 1754, trustees of the College of New Jersey suggested that, in recognition of Governors interest, gov. Jonathan Belcher replied, What a name that would be. In 1756, the moved to Princeton, New Jersey. Its home in Princeton was Nassau Hall, named for the royal House of Orange-Nassau of William III of England, following the untimely deaths of Princetons first five presidents, John Witherspoon became president in 1768 and remained in that office until his death in 1794. During his presidency, Witherspoon shifted the focus from training ministers to preparing a new generation for leadership in the new American nation. To this end, he tightened academic standards and solicited investment in the college, in 1812, the eighth president the College of New Jersey, Ashbel Green, helped establish the Princeton Theological Seminary next door. The plan to extend the theological curriculum met with approval on the part of the authorities at the College of New Jersey. Today, Princeton University and Princeton Theological Seminary maintain separate institutions with ties that include such as cross-registration. Before the construction of Stanhope Hall in 1803, Nassau Hall was the sole building. The cornerstone of the building was laid on September 17,1754, during the summer of 1783, the Continental Congress met in Nassau Hall, making Princeton the countrys capital for four months. The class of 1879 donated twin lion sculptures that flanked the entrance until 1911, Nassau Halls bell rang after the halls construction, however, the fire of 1802 melted it. The bell was then recast and melted again in the fire of 1855, James McCosh took office as the colleges president in 1868 and lifted the institution out of a low period that had been brought about by the American Civil War. McCosh Hall is named in his honor, in 1879, the first thesis for a Doctor of Philosophy Ph. D. was submitted by James F. Williamson, Class of 1877. In 1896, the officially changed its name from the College of New Jersey to Princeton University to honor the town in which it resides
9.
Thesis
–
A thesis or dissertation is a document submitted in support of candidature for an academic degree or professional qualification presenting the authors research and findings. In some contexts, the thesis or a cognate is used for part of a bachelors or masters course, while dissertation is normally applied to a doctorate, while in other contexts. The term graduate thesis is used to refer to both masters theses and doctoral dissertations. The required complexity or quality of research of a thesis or dissertation can vary by country, university, or program, the word dissertation can at times be used to describe a treatise without relation to obtaining an academic degree. The term thesis is used to refer to the general claim of an essay or similar work. The term thesis comes from the Greek θέσις, meaning something put forth, Dissertation comes from the Latin dissertātiō, meaning path. A thesis may be arranged as a thesis by publication or a monograph, with or without appended papers, an ordinary monograph has a title page, an abstract, a table of contents, comprising the various chapters, and a bibliography or a references section. They differ in their structure in accordance with the different areas of study. In a thesis by publication, the chapters constitute an introductory, Dissertations normally report on a research project or study, or an extended analysis of a topic. The structure of the thesis or dissertation explains the purpose, the research literature which impinges on the topic of the study, the methods used. Degree-awarding institutions often define their own style that candidates have to follow when preparing a thesis document. Other applicable international standards include ISO2145 on section numbers, ISO690 on bibliographic references, some older house styles specify that front matter uses a separate page-number sequence from the main text, using Roman numerals. They therefore avoid the traditional separate number sequence for front matter, however, strict standards are not always required. Most Italian universities, for example, have only general requirements on the size and the page formatting. A thesis or dissertation committee is a committee that supervises a students dissertation, the committee members are doctors in their field and have the task of reading the dissertation, making suggestions for changes and improvements, and sitting in on the defense. Sometimes, at least one member of the committee must be a professor in a department that is different from that of the student, all the dissertation referees must already have achieved at least the academic degree that the candidate is trying to reach. At English-speaking Canadian universities, writings presented in fulfillment of undergraduate coursework requirements are normally called papers, a longer paper or essay presented for completion of a 4-year bachelors degree is sometimes called a major paper. High-quality research papers presented as the study of a postgraduate consecutive bachelor with Honours or Baccalaureatus Cum Honore degree are called thesis
10.
Irving Segal
–
Irving Ezra Segal was an American mathematician known for work on theoretical quantum mechanics. He shares credit for what is referred to as the Segal–Shale–Weil representation. Early in his career Segal became known for his developments in quantum theory and in functional and harmonic analysis. Irving Ezra Segal was born in the Bronx in 1918 to Jewish parents, in 1934 was admitted to Princeton University at the age of 16. He was then admitted to Yale, and in three years time had completed his doctorate, receiving his PhD in 1940. Segal taught at Harvard University, then he joined the Institute for Advanced Study in Princeton on a Guggenheim Memorial Fellowship, working from 1941–43 with Albert Einstein and Von Neumann. During World War II Segal served in the U. S. Army conducting research in ballistics at the Aberdeen Proving Ground in Maryland and he joined the mathematics department at the University of Chicago in 1948 where he served until 1960. In 1960 he joined the department at M. I. T. where he remained as a professor until his death in 1998. He won three Guggenheim Fellowships, in 1947,1951 and 1967, and received the Humboldt Award in 1981 and he was an Invited Speaker of the ICM in 1966 in Moscow and in 1970 in Nice. He was elected to the National Academy of Sciences in 1973, Segal died in Lexington, Massachusetts in 1998. Edward Nelsons obituary article about Segal concludes, metaplectic group Symplectic group Symplectic spinor bundle Segal, I. Robertson, Edmund F. Irving Segal, MacTutor History of Mathematics archive, Irving Segal at the Mathematics Genealogy Project
11.
Mathematical physics
–
Mathematical physics refers to development of mathematical methods for application to problems in physics. It is a branch of applied mathematics, but deals with physical problems, there are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. The rigorous, abstract and advanced re-formulation of Newtonian mechanics adopting the Lagrangian mechanics, both formulations are embodied in analytical mechanics. These approaches and ideas can be and, in fact, have extended to other areas of physics as statistical mechanics, continuum mechanics, classical field theory. Moreover, they have provided several examples and basic ideas in differential geometry, the theory of partial differential equations are perhaps most closely associated with mathematical physics. These were developed intensively from the half of the eighteenth century until the 1930s. Physical applications of these developments include hydrodynamics, celestial mechanics, continuum mechanics, elasticity theory, acoustics, thermodynamics, electricity, magnetism, and aerodynamics. The theory of atomic spectra developed almost concurrently with the fields of linear algebra. Nonrelativistic quantum mechanics includes Schrödinger operators, and it has connections to atomic, Quantum information theory is another subspecialty. The special and general theories of relativity require a different type of mathematics. This was group theory, which played an important role in quantum field theory and differential geometry. This was, however, gradually supplemented by topology and functional analysis in the description of cosmological as well as quantum field theory phenomena. In this area both homological algebra and category theory are important nowadays, statistical mechanics forms a separate field, which includes the theory of phase transitions. It relies upon the Hamiltonian mechanics and it is related with the more mathematical ergodic theory. There are increasing interactions between combinatorics and physics, in statistical physics. The usage of the mathematical physics is sometimes idiosyncratic. Certain parts of mathematics that arose from the development of physics are not, in fact, considered parts of mathematical physics. The term mathematical physics is sometimes used to research aimed at studying and solving problems inspired by physics or thought experiments within a mathematically rigorous framework
12.
Consistency
–
In classical deductive logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms, the semantic definition states that a theory is consistent if and only if it has a model, i. e. there exists an interpretation under which all formulas in the theory are true. This is the used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states a theory T is consistent if and only if there is no formula ϕ such that both ϕ and its negation ¬ ϕ are elements of the set T. Let A be set of closed sentences and ⟨ A ⟩ the set of closed sentences provable from A under some formal deductive system, the set of axioms A is consistent when ⟨ A ⟩ is. If there exists a system for which these semantic and syntactic definitions are equivalent for any theory formulated in a particular deductive logic. Stronger logics, such as logic, are not complete. A consistency proof is a proof that a particular theory is consistent. The early development of mathematical theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilberts program. Hilberts program was impacted by incompleteness theorems, which showed that sufficiently strong proof theories cannot prove their own consistency. Although consistency can be proved by means of theory, it is often done in a purely syntactical way. The cut-elimination implies the consistency of the calculus, since there is obviously no cut-free proof of falsity, in theories of arithmetic, such as Peano arithmetic, there is an intricate relationship between the consistency of the theory and its completeness. A theory is complete if, for every formula φ in its language, Presburger arithmetic is an axiom system for the natural numbers under addition. It is both consistent and complete, Gödels incompleteness theorems show that any sufficiently strong recursively enumerable theory of arithmetic cannot be both complete and consistent. Gödels theorem applies to the theories of Peano arithmetic and Primitive recursive arithmetic, moreover, Gödels second incompleteness theorem shows that the consistency of sufficiently strong recursively enumerable theories of arithmetic can be tested in a particular way. Thus the consistency of a strong, recursively enumerable, consistent theory of arithmetic can never be proven in that system itself. The same result is true for recursively enumerable theories that can describe a strong enough fragment of arithmetic—including set theories such as Zermelo–Fraenkel set theory and these set theories cannot prove their own Gödel sentence—provided that they are consistent, which is generally believed. Because consistency of ZF is not provable in ZF, the weaker notion relative consistency is interesting in set theory
13.
Philosophy of mathematics
–
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics, the logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts. The terms philosophy of mathematics and mathematical philosophy are frequently used interchangeably, the latter, however, may be used to refer to several other areas of study. Another refers to the philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Recurrent themes include, What is the role of Mankind in developing mathematics, what are the sources of mathematical subject matter. What is the status of mathematical entities. What does it mean to refer to a mathematical object, what is the character of a mathematical proposition. What is the relation between logic and mathematics, what is the role of hermeneutics in mathematics. What kinds of play a role in mathematics. What are the objectives of mathematical inquiry, what gives mathematics its hold on experience. What are the human traits behind mathematics, what is the source and nature of mathematical truth. What is the relationship between the world of mathematics and the material universe. The origin of mathematics is subject to argument, whether the birth of mathematics was a random happening or induced by necessity duly contingent upon other subjects, say for example physics, is still a matter of prolific debates. Many thinkers have contributed their ideas concerning the nature of mathematics, there are traditions of mathematical philosophy in both Western philosophy and Eastern philosophy. Greek philosophy on mathematics was strongly influenced by their study of geometry, for example, at one time, the Greeks held the opinion that 1 was not a number, but rather a unit of arbitrary length. A number was defined as a multitude, therefore,3, for example, represented a certain multitude of units, and was thus not truly a number. At another point, an argument was made that 2 was not a number. These earlier Greek ideas of numbers were later upended by the discovery of the irrationality of the root of two
14.
Formalism (philosophy of mathematics)
–
In playing this game one can prove that the Pythagorean theorem is valid because the string representing the Pythagorean theorem can be constructed using only the stated rules. According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other subject matter — in fact. They are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation, Formalism is associated with rigorous method. In common use, a means the out-turn of the effort towards formalisation of a given limited area. In other words, matters can be formally discussed once captured in a formal system, complete formalisation is in the domain of computer science. Formalism stresses axiomatic proofs using theorems, specifically associated with David Hilbert, a formalist is an individual who belongs to the school of formalism, which is a certain mathematical-philosophical doctrine descending from Hilbert. Formalists are relatively tolerant and inviting to new approaches to logic, non-standard number systems, new set theories, the more games they study, the better. However, in all three of these examples, motivation is drawn from existing mathematical or philosophical concerns, the games are usually not arbitrary. Because of their connection with computer science, this idea is also advocated by mathematical intuitionists and constructivists in the computability tradition. Another version of formalism is known as deductivism. In deductivism, the Pythagorean theorem is not an absolute truth, under deductivism, the same view is held to be true for all other statements of formal logic and mathematics. Thus, formalism need not mean that these deductive sciences are nothing more than meaningless symbolic games and it is usually hoped that there exists some interpretation in which the rules of the game hold. Taking the deductivist view allows the working mathematician to suspend judgement on the philosophical questions. Many formalists would say that in practice, the systems to be studied are suggested by the demands of the particular science. A major early proponent of formalism was David Hilbert, whose program was intended to be a complete, Hilbert aimed to show the consistency of mathematical systems from the assumption that the finitary arithmetic was consistent. The way that Hilbert tried to show that a system was consistent was by formalizing it using a particular language. In order to formalize a system, you must first choose a language in which you can express. This language must include five components, It must include such as x
15.
Platonism (mathematics)
–
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics, the logical and structural nature of mathematics itself makes this study both broad and unique among its philosophical counterparts. The terms philosophy of mathematics and mathematical philosophy are frequently used interchangeably, the latter, however, may be used to refer to several other areas of study. Another refers to the philosophy of an individual practitioner or a like-minded community of practicing mathematicians. Recurrent themes include, What is the role of Mankind in developing mathematics, what are the sources of mathematical subject matter. What is the status of mathematical entities. What does it mean to refer to a mathematical object, what is the character of a mathematical proposition. What is the relation between logic and mathematics, what is the role of hermeneutics in mathematics. What kinds of play a role in mathematics. What are the objectives of mathematical inquiry, what gives mathematics its hold on experience. What are the human traits behind mathematics, what is the source and nature of mathematical truth. What is the relationship between the world of mathematics and the material universe. The origin of mathematics is subject to argument, whether the birth of mathematics was a random happening or induced by necessity duly contingent upon other subjects, say for example physics, is still a matter of prolific debates. Many thinkers have contributed their ideas concerning the nature of mathematics, there are traditions of mathematical philosophy in both Western philosophy and Eastern philosophy. Greek philosophy on mathematics was strongly influenced by their study of geometry, for example, at one time, the Greeks held the opinion that 1 was not a number, but rather a unit of arbitrary length. A number was defined as a multitude, therefore,3, for example, represented a certain multitude of units, and was thus not truly a number. At another point, an argument was made that 2 was not a number. These earlier Greek ideas of numbers were later upended by the discovery of the irrationality of the root of two
16.
Italy
–
Italy, officially the Italian Republic, is a unitary parliamentary republic in Europe. Located in the heart of the Mediterranean Sea, Italy shares open land borders with France, Switzerland, Austria, Slovenia, San Marino, Italy covers an area of 301,338 km2 and has a largely temperate seasonal climate and Mediterranean climate. Due to its shape, it is referred to in Italy as lo Stivale. With 61 million inhabitants, it is the fourth most populous EU member state, the Italic tribe known as the Latins formed the Roman Kingdom, which eventually became a republic that conquered and assimilated other nearby civilisations. The legacy of the Roman Empire is widespread and can be observed in the distribution of civilian law, republican governments, Christianity. The Renaissance began in Italy and spread to the rest of Europe, bringing a renewed interest in humanism, science, exploration, Italian culture flourished at this time, producing famous scholars, artists and polymaths such as Leonardo da Vinci, Galileo, Michelangelo and Machiavelli. The weakened sovereigns soon fell victim to conquest by European powers such as France, Spain and Austria. Despite being one of the victors in World War I, Italy entered a period of economic crisis and social turmoil. The subsequent participation in World War II on the Axis side ended in defeat, economic destruction. Today, Italy has the third largest economy in the Eurozone and it has a very high level of human development and is ranked sixth in the world for life expectancy. The country plays a prominent role in regional and global economic, military, cultural and diplomatic affairs, as a reflection of its cultural wealth, Italy is home to 51 World Heritage Sites, the most in the world, and is the fifth most visited country. The assumptions on the etymology of the name Italia are very numerous, according to one of the more common explanations, the term Italia, from Latin, Italia, was borrowed through Greek from the Oscan Víteliú, meaning land of young cattle. The bull was a symbol of the southern Italic tribes and was often depicted goring the Roman wolf as a defiant symbol of free Italy during the Social War. Greek historian Dionysius of Halicarnassus states this account together with the legend that Italy was named after Italus, mentioned also by Aristotle and Thucydides. The name Italia originally applied only to a part of what is now Southern Italy – according to Antiochus of Syracuse, but by his time Oenotria and Italy had become synonymous, and the name also applied to most of Lucania as well. The Greeks gradually came to apply the name Italia to a larger region, excavations throughout Italy revealed a Neanderthal presence dating back to the Palaeolithic period, some 200,000 years ago, modern Humans arrived about 40,000 years ago. Other ancient Italian peoples of undetermined language families but of possible origins include the Rhaetian people and Cammuni. Also the Phoenicians established colonies on the coasts of Sardinia and Sicily, the Roman legacy has deeply influenced the Western civilisation, shaping most of the modern world
17.
Benito Mussolini
–
Benito Amilcare Andrea Mussolini was an Italian politician, journalist, and leader of the National Fascist Party, ruling the country as Prime Minister from 1922 to 1943. He ruled constitutionally until 1925, when he dropped all pretense of democracy, known as Il Duce, Mussolini was the founder of Italian Fascism. In 1912 Mussolini was the member of the National Directorate of the Italian Socialist Party. Mussolini was expelled from the PSI for withdrawing his support for the stance on neutrality in World War I. He served in the Royal Italian Army during the war until he was wounded and discharged in 1917, Mussolini denounced the PSI, his views now centering on nationalism instead of socialism, and later founded the fascist movement. Following the March on Rome in October 1922 he became the youngest Prime Minister in Italian history until the appointment of Matteo Renzi in February 2014, within five years he had established dictatorial authority by both legal and extraordinary means, aspiring to create a totalitarian state. Mussolini remained in power until he was deposed by King Victor Emmanuel III in 1943, a few months later, he became the leader of the Italian Social Republic, a German client regime in northern Italy, he held this post until his death in 1945. Mussolini had sought to delay a major war in Europe until at least 1942, however, Germany invaded Poland on 1 September 1939, resulting in declarations of war by France and the United Kingdom and starting World War II. In the summer of 1941 Mussolini sent Italian forces to participate in the invasion of the Soviet Union, and war with the United States followed in December. On 24 July 1943, soon after the start of the Allied invasion of Italy, the Grand Council of Fascism voted against him, on 12 September 1943, Mussolini was rescued from prison in the Gran Sasso raid by German special forces. In late April 1945, with total defeat looming, Mussolini attempted to escape north and his body was then taken to Milan, where it was hung upside down at a service station for public viewing and to provide confirmation of his demise. Mussolini was born in Dovia di Predappio, a town in the province of Forlì in Romagna on 29 July 1883. During the Fascist era, Predappio was dubbed Duces town, pilgrims went to Predappio and Forlì, to see the birthplace of Mussolini. His father, Alessandro Mussolini, was a blacksmith and a Socialist, while his mother, Benito was the eldest of his parents three children. His siblings Arnaldo and Edvige followed, as a young boy, Mussolini would spend some time helping his father in his smithy. His fathers political outlook combined views of anarchist figures like Carlo Cafiero and Mikhail Bakunin, the military authoritarianism of Garibaldi, in 1902, at the anniversary of Garibaldis death, Benito Mussolini made a public speech in praise of the republican nationalist. The conflict between his parents about religion meant that, unlike most Italians, Mussolini was not baptized at birth, as a compromise with his mother, Mussolini was sent to a boarding school run by Salesian monks. After joining a new school, Mussolini achieved good grades, in 1902, Mussolini emigrated to Switzerland, partly to avoid military service
18.
New York City
–
The City of New York, often called New York City or simply New York, is the most populous city in the United States. With an estimated 2015 population of 8,550,405 distributed over an area of about 302.6 square miles. Located at the tip of the state of New York. Home to the headquarters of the United Nations, New York is an important center for international diplomacy and has described as the cultural and financial capital of the world. Situated on one of the worlds largest natural harbors, New York City consists of five boroughs, the five boroughs – Brooklyn, Queens, Manhattan, The Bronx, and Staten Island – were consolidated into a single city in 1898. In 2013, the MSA produced a gross metropolitan product of nearly US$1.39 trillion, in 2012, the CSA generated a GMP of over US$1.55 trillion. NYCs MSA and CSA GDP are higher than all but 11 and 12 countries, New York City traces its origin to its 1624 founding in Lower Manhattan as a trading post by colonists of the Dutch Republic and was named New Amsterdam in 1626. The city and its surroundings came under English control in 1664 and were renamed New York after King Charles II of England granted the lands to his brother, New York served as the capital of the United States from 1785 until 1790. It has been the countrys largest city since 1790, the Statue of Liberty greeted millions of immigrants as they came to the Americas by ship in the late 19th and early 20th centuries and is a symbol of the United States and its democracy. In the 21st century, New York has emerged as a node of creativity and entrepreneurship, social tolerance. Several sources have ranked New York the most photographed city in the world, the names of many of the citys bridges, tapered skyscrapers, and parks are known around the world. Manhattans real estate market is among the most expensive in the world, Manhattans Chinatown incorporates the highest concentration of Chinese people in the Western Hemisphere, with multiple signature Chinatowns developing across the city. Providing continuous 24/7 service, the New York City Subway is one of the most extensive metro systems worldwide, with 472 stations in operation. Over 120 colleges and universities are located in New York City, including Columbia University, New York University, and Rockefeller University, during the Wisconsinan glaciation, the New York City region was situated at the edge of a large ice sheet over 1,000 feet in depth. The ice sheet scraped away large amounts of soil, leaving the bedrock that serves as the foundation for much of New York City today. Later on, movement of the ice sheet would contribute to the separation of what are now Long Island and Staten Island. The first documented visit by a European was in 1524 by Giovanni da Verrazzano, a Florentine explorer in the service of the French crown and he claimed the area for France and named it Nouvelle Angoulême. Heavy ice kept him from further exploration, and he returned to Spain in August and he proceeded to sail up what the Dutch would name the North River, named first by Hudson as the Mauritius after Maurice, Prince of Orange
19.
World War II
–
World War II, also known as the Second World War, was a global war that lasted from 1939 to 1945, although related conflicts began earlier. It involved the vast majority of the worlds countries—including all of the great powers—eventually forming two opposing alliances, the Allies and the Axis. It was the most widespread war in history, and directly involved more than 100 million people from over 30 countries. Marked by mass deaths of civilians, including the Holocaust and the bombing of industrial and population centres. These made World War II the deadliest conflict in human history, from late 1939 to early 1941, in a series of campaigns and treaties, Germany conquered or controlled much of continental Europe, and formed the Axis alliance with Italy and Japan. Under the Molotov–Ribbentrop Pact of August 1939, Germany and the Soviet Union partitioned and annexed territories of their European neighbours, Poland, Finland, Romania and the Baltic states. In December 1941, Japan attacked the United States and European colonies in the Pacific Ocean, and quickly conquered much of the Western Pacific. The Axis advance halted in 1942 when Japan lost the critical Battle of Midway, near Hawaii, in 1944, the Western Allies invaded German-occupied France, while the Soviet Union regained all of its territorial losses and invaded Germany and its allies. During 1944 and 1945 the Japanese suffered major reverses in mainland Asia in South Central China and Burma, while the Allies crippled the Japanese Navy, thus ended the war in Asia, cementing the total victory of the Allies. World War II altered the political alignment and social structure of the world, the United Nations was established to foster international co-operation and prevent future conflicts. The victorious great powers—the United States, the Soviet Union, China, the United Kingdom, the Soviet Union and the United States emerged as rival superpowers, setting the stage for the Cold War, which lasted for the next 46 years. Meanwhile, the influence of European great powers waned, while the decolonisation of Asia, most countries whose industries had been damaged moved towards economic recovery. Political integration, especially in Europe, emerged as an effort to end pre-war enmities, the start of the war in Europe is generally held to be 1 September 1939, beginning with the German invasion of Poland, Britain and France declared war on Germany two days later. The dates for the beginning of war in the Pacific include the start of the Second Sino-Japanese War on 7 July 1937, or even the Japanese invasion of Manchuria on 19 September 1931. Others follow the British historian A. J. P. Taylor, who held that the Sino-Japanese War and war in Europe and its colonies occurred simultaneously and this article uses the conventional dating. Other starting dates sometimes used for World War II include the Italian invasion of Abyssinia on 3 October 1935. The British historian Antony Beevor views the beginning of World War II as the Battles of Khalkhin Gol fought between Japan and the forces of Mongolia and the Soviet Union from May to September 1939, the exact date of the wars end is also not universally agreed upon. It was generally accepted at the time that the war ended with the armistice of 14 August 1945, rather than the formal surrender of Japan
20.
Russian language
–
Russian is an East Slavic language and an official language in Russia, Belarus, Kazakhstan, Kyrgyzstan and many minor or unrecognised territories. Russian belongs to the family of Indo-European languages and is one of the four living members of the East Slavic languages, written examples of Old East Slavonic are attested from the 10th century and beyond. It is the most geographically widespread language of Eurasia and the most widely spoken of the Slavic languages and it is also the largest native language in Europe, with 144 million native speakers in Russia, Ukraine and Belarus. Russian is the eighth most spoken language in the world by number of native speakers, the language is one of the six official languages of the United Nations. Russian is also the second most widespread language on the Internet after English, Russian distinguishes between consonant phonemes with palatal secondary articulation and those without, the so-called soft and hard sounds. This distinction is found between pairs of almost all consonants and is one of the most distinguishing features of the language, another important aspect is the reduction of unstressed vowels. Russian is a Slavic language of the Indo-European family and it is a lineal descendant of the language used in Kievan Rus. From the point of view of the language, its closest relatives are Ukrainian, Belarusian, and Rusyn. An East Slavic Old Novgorod dialect, although vanished during the 15th or 16th century, is considered to have played a significant role in the formation of modern Russian. In the 19th century, the language was often called Great Russian to distinguish it from Belarusian, then called White Russian and Ukrainian, however, the East Slavic forms have tended to be used exclusively in the various dialects that are experiencing a rapid decline. In some cases, both the East Slavic and the Church Slavonic forms are in use, with different meanings. For details, see Russian phonology and History of the Russian language and it is also regarded by the United States Intelligence Community as a hard target language, due to both its difficulty to master for English speakers and its critical role in American world policy. The standard form of Russian is generally regarded as the modern Russian literary language, mikhail Lomonosov first compiled a normalizing grammar book in 1755, in 1783 the Russian Academys first explanatory Russian dictionary appeared. By the mid-20th century, such dialects were forced out with the introduction of the education system that was established by the Soviet government. Despite the formalization of Standard Russian, some nonstandard dialectal features are observed in colloquial speech. Thus, the Russian language is the 6th largest in the world by number of speakers, after English, Mandarin, Hindi/Urdu, Spanish, Russian is one of the six official languages of the United Nations. Education in Russian is still a choice for both Russian as a second language and native speakers in Russia as well as many of the former Soviet republics. Russian is still seen as an important language for children to learn in most of the former Soviet republics, samuel P. Huntington wrote in the Clash of Civilizations, During the heyday of the Soviet Union, Russian was the lingua franca from Prague to Hanoi
21.
Saint Petersburg
–
Saint Petersburg is Russias second-largest city after Moscow, with five million inhabitants in 2012, and an important Russian port on the Baltic Sea. It is politically incorporated as a federal subject, situated on the Neva River, at the head of the Gulf of Finland on the Baltic Sea, it was founded by Tsar Peter the Great on May 271703. In 1914, the name was changed from Saint Petersburg to Petrograd, in 1924 to Leningrad, between 1713 and 1728 and 1732–1918, Saint Petersburg was the capital of imperial Russia. In 1918, the government bodies moved to Moscow. Saint Petersburg is one of the cities of Russia, as well as its cultural capital. The Historic Centre of Saint Petersburg and Related Groups of Monuments constitute a UNESCO World Heritage Site, Saint Petersburg is home to The Hermitage, one of the largest art museums in the world. A large number of consulates, international corporations, banks. Swedish colonists built Nyenskans, a fortress, at the mouth of the Neva River in 1611, in a then called Ingermanland. A small town called Nyen grew up around it, Peter the Great was interested in seafaring and maritime affairs, and he intended to have Russia gain a seaport in order to be able to trade with other maritime nations. He needed a better seaport than Arkhangelsk, which was on the White Sea to the north, on May 1703121703, during the Great Northern War, Peter the Great captured Nyenskans, and soon replaced the fortress. On May 271703, closer to the estuary 5 km inland from the gulf), on Zayachy Island, he laid down the Peter and Paul Fortress, which became the first brick and stone building of the new city. The city was built by conscripted peasants from all over Russia, tens of thousands of serfs died building the city. Later, the city became the centre of the Saint Petersburg Governorate, Peter moved the capital from Moscow to Saint Petersburg in 1712,9 years before the Treaty of Nystad of 1721 ended the war, he referred to Saint Petersburg as the capital as early as 1704. During its first few years, the city developed around Trinity Square on the bank of the Neva, near the Peter. However, Saint Petersburg soon started to be built out according to a plan, by 1716 the Swiss Italian Domenico Trezzini had elaborated a project whereby the city centre would be located on Vasilyevsky Island and shaped by a rectangular grid of canals. The project was not completed, but is evident in the layout of the streets, in 1716, Peter the Great appointed French Jean-Baptiste Alexandre Le Blond as the chief architect of Saint Petersburg. In 1724 the Academy of Sciences, University and Academic Gymnasium were established in Saint Petersburg by Peter the Great, in 1725, Peter died at the age of fifty-two. His endeavours to modernize Russia had met opposition from the Russian nobility—resulting in several attempts on his life
22.
Prisoner of war
–
A prisoner of war is a person, whether combatant or non-combatant, who is held in custody by a belligerent power during or immediately after an armed conflict. The earliest recorded usage of the prisoner of war dates to 1660. The first Roman gladiators were prisoners of war and were named according to their ethnic roots such as Samnite, Thracian, typically, little distinction was made between enemy combatants and enemy civilians, although women and children were more likely to be spared. Sometimes, the purpose of a battle, if not a war, was to capture women, a known as raptio. Typically women had no rights, and were legally as chattel. For this he was eventually canonized, during Childerics siege and blockade of Paris in 464, the nun Geneviève pleaded with the Frankish king for the welfare of prisoners of war and met with a favourable response. Later, Clovis I liberated captives after Genevieve urged him to do so, many French prisoners of war were killed during the Battle of Agincourt in 1415. In the later Middle Ages, a number of religious wars aimed to not only defeat, in Christian Europe, the extermination of heretics was considered desirable. Examples include the 13th century Albigensian Crusade and the Northern Crusades, likewise, the inhabitants of conquered cities were frequently massacred during the Crusades against the Muslims in the 11th and 12th centuries. Noblemen could hope to be ransomed, their families would have to send to their captors large sums of wealth commensurate with the status of the captive. In feudal Japan there was no custom of ransoming prisoners of war, in Termez, on the Oxus, all the people, both men and women, were driven out onto the plain, and divided in accordance with their usual custom, then they were all slain. The Aztecs were constantly at war with neighbouring tribes and groups, for the re-consecration of Great Pyramid of Tenochtitlan in 1487, between 10,000 and 80,400 persons were sacrificed. During the early Muslim conquests, Muslims routinely captured large number of prisoners, aside from those who converted, most were ransomed or enslaved. Christians who were captured during the Crusades, were either killed or sold into slavery if they could not pay a ransom. The freeing of prisoners was highly recommended as a charitable act, there also evolved the right of parole, French for discourse, in which a captured officer surrendered his sword and gave his word as a gentleman in exchange for privileges. If he swore not to escape, he could gain better accommodations, if he swore to cease hostilities against the nation who held him captive, he could be repatriated or exchanged but could not serve against his former captors in a military capacity. Early historical narratives of captured colonial Europeans, including perspectives of literate women captured by the peoples of North America. The writings of Mary Rowlandson, captured in the fighting of King Philips War, are an example
23.
American Mathematical Society
–
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
24.
Quantum field theory
–
QFT treats particles as excited states of the underlying physical field, so these are called field quanta. In quantum field theory, quantum mechanical interactions among particles are described by interaction terms among the corresponding underlying quantum fields and these interactions are conveniently visualized by Feynman diagrams, which are a formal tool of relativistically covariant perturbation theory, serving to evaluate particle processes. The first achievement of quantum theory, namely quantum electrodynamics, is still the paradigmatic example of a successful quantum field theory. Ordinarily, quantum mechanics cannot give an account of photons which constitute the prime case of relativistic particles, since photons have rest mass zero, and correspondingly travel in the vacuum at the speed c, a non-relativistic theory such as ordinary QM cannot give even an approximate description. Photons are implicit in the emission and absorption processes which have to be postulated, for instance, the formalism of QFT is needed for an explicit description of photons. In fact most topics in the development of quantum theory were related to the interaction of radiation and matter. However, quantum mechanics as formulated by Dirac, Heisenberg, and Schrödinger in 1926–27 started from atomic spectra, as soon as the conceptual framework of quantum mechanics was developed, a small group of theoreticians tried to extend quantum methods to electromagnetic fields. A good example is the paper by Born, Jordan & Heisenberg. The basic idea was that in QFT the electromagnetic field should be represented by matrices in the way that position. The ideas of QM were thus extended to systems having a number of degrees of freedom. The inception of QFT is usually considered to be Diracs famous 1927 paper on The quantum theory of the emission and absorption of radiation, here Dirac coined the name quantum electrodynamics for the part of QFT that was developed first. Employing the theory of the harmonic oscillator, Dirac gave a theoretical description of how photons appear in the quantization of the electromagnetic radiation field. Later, Diracs procedure became a model for the quantization of fields as well. These first approaches to QFT were further developed during the three years. P. Jordan introduced creation and annihilation operators for fields obeying Fermi–Dirac statistics and these differ from the corresponding operators for Bose–Einstein statistics in that the former satisfy anti-commutation relations while the latter satisfy commutation relations. The methods of QFT could be applied to derive equations resulting from the treatment of particles, e. g. the Dirac equation, the Klein–Gordon equation. Schweber points out that the idea and procedure of second quantization goes back to Jordan, in a number of papers from 1927, some difficult problems concerning commutation relations, statistics, and Lorentz invariance were eventually solved. The first comprehensive account of a theory of quantum fields, in particular
25.
Stochastic process
–
In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a collection of random variables. Stochastic processes are used as mathematical models of systems and phenomena that appear to vary in a random manner. Furthermore, seemingly random changes in financial markets have motivated the use of stochastic processes in finance. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such processes include the Wiener process or Brownian motion process, used by Louis Bachelier to study price changes on the Paris Bourse. Erlang to study the number phone calls occurring in a period of time. The term random function is used to refer to a stochastic or random process. The terms stochastic process and random process are used interchangeably, often no specific mathematical space for the set that indexes the random variables. But often these two terms are used when the variables are indexed by the integers or an interval of the real line. If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, the values of a stochastic process are not always numbers and can be vectors or other mathematical objects. The theory of processes is considered to be an important contribution to mathematics. The set used to index the random variables is called the index set, historically, the index set was some subset of the real line, such as the natural numbers, giving the index set the interpretation of time. Each random variable in the collection takes values from the same space known as the state space. This state space can be, for example, the integers, an increment is the amount that a stochastic process changes between two index values, often interpreted as two points in time. A stochastic process can have many outcomes, due its randomness, and an outcome of a stochastic process is called, among other names. A stochastic process can be classified in different ways, for example, by its space, its index set. One common way of classification is by the cardinality of the index set, if the index set is some interval of the real line, then time is said to be continuous. The two types of processes are respectively referred to as discrete-time and continuous-time stochastic processes
26.
Quantum mechanics
–
Quantum mechanics, including quantum field theory, is a branch of physics which is the fundamental theory of nature at small scales and low energies of atoms and subatomic particles. Classical physics, the physics existing before quantum mechanics, derives from quantum mechanics as an approximation valid only at large scales, early quantum theory was profoundly reconceived in the mid-1920s. The reconceived theory is formulated in various specially developed mathematical formalisms, in one of them, a mathematical function, the wave function, provides information about the probability amplitude of position, momentum, and other physical properties of a particle. In 1803, Thomas Young, an English polymath, performed the famous experiment that he later described in a paper titled On the nature of light. This experiment played a role in the general acceptance of the wave theory of light. In 1838, Michael Faraday discovered cathode rays, Plancks hypothesis that energy is radiated and absorbed in discrete quanta precisely matched the observed patterns of black-body radiation. In 1896, Wilhelm Wien empirically determined a distribution law of black-body radiation, ludwig Boltzmann independently arrived at this result by considerations of Maxwells equations. However, it was only at high frequencies and underestimated the radiance at low frequencies. Later, Planck corrected this model using Boltzmanns statistical interpretation of thermodynamics and proposed what is now called Plancks law, following Max Plancks solution in 1900 to the black-body radiation problem, Albert Einstein offered a quantum-based theory to explain the photoelectric effect. Among the first to study quantum phenomena in nature were Arthur Compton, C. V. Raman, robert Andrews Millikan studied the photoelectric effect experimentally, and Albert Einstein developed a theory for it. In 1913, Peter Debye extended Niels Bohrs theory of structure, introducing elliptical orbits. This phase is known as old quantum theory, according to Planck, each energy element is proportional to its frequency, E = h ν, where h is Plancks constant. Planck cautiously insisted that this was simply an aspect of the processes of absorption and emission of radiation and had nothing to do with the reality of the radiation itself. In fact, he considered his quantum hypothesis a mathematical trick to get the right rather than a sizable discovery. He won the 1921 Nobel Prize in Physics for this work, lower energy/frequency means increased time and vice versa, photons of differing frequencies all deliver the same amount of action, but do so in varying time intervals. High frequency waves are damaging to human tissue because they deliver their action packets concentrated in time, the Copenhagen interpretation of Niels Bohr became widely accepted. In the mid-1920s, developments in mechanics led to its becoming the standard formulation for atomic physics. In the summer of 1925, Bohr and Heisenberg published results that closed the old quantum theory, out of deference to their particle-like behavior in certain processes and measurements, light quanta came to be called photons
27.
Probability theory
–
Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena. It is not possible to predict precisely results of random events, two representative mathematical results describing such patterns are the law of large numbers and the central limit theorem. As a mathematical foundation for statistics, probability theory is essential to human activities that involve quantitative analysis of large sets of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, a great discovery of twentieth century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics. Christiaan Huygens published a book on the subject in 1657 and in the 19th century, initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory and this culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of space, introduced by Richard von Mises. This became the mostly undisputed axiomatic basis for modern probability theory, most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The more mathematically advanced measure theory-based treatment of probability covers the discrete, continuous, consider an experiment that can produce a number of outcomes. The set of all outcomes is called the space of the experiment. The power set of the space is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results, one collection of possible results corresponds to getting an odd number. Thus, the subset is an element of the set of the sample space of die rolls. In this case, is the event that the die falls on some odd number, If the results that actually occur fall in a given event, that event is said to have occurred. Probability is a way of assigning every event a value between zero and one, with the requirement that the event made up of all possible results be assigned a value of one, the probability that any one of the events, or will occur is 5/6. This is the same as saying that the probability of event is 5/6 and this event encompasses the possibility of any number except five being rolled. The mutually exclusive event has a probability of 1/6, and the event has a probability of 1, discrete probability theory deals with events that occur in countable sample spaces. Modern definition, The modern definition starts with a finite or countable set called the sample space, which relates to the set of all possible outcomes in classical sense, denoted by Ω
28.
Non-standard analysis
–
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals, Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson and he wrote, the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the stages of the Differential and Integral Calculus. Robinson continued, However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort, as a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits. The key to our method is provided by the analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory. In 1973, intuitionist Arend Heyting praised non-standard analysis as a model of important mathematical research. A non-zero element of an ordered field F is infinitesimal if and only if its value is smaller than any element of F of the form 1 n, for n. Ordered fields that have infinitesimal elements are also called non-Archimedean, more generally, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle. A field which satisfies the principle for real numbers is a hyperreal field. Robinsons original approach was based on these models of the field of real numbers. His classic foundational book on the subject Non-standard Analysis was published in 1966 and is still in print, on page 88, Robinson writes, The existence of non-standard models of arithmetic was discovered by Thoralf Skolem. Skolems method foreshadows the ultrapower construction Several technical issues must be addressed to develop a calculus of infinitesimals, for example, it is not enough to construct an ordered field with infinitesimals. See the article on numbers for a discussion of some of the relevant ideas. In this section we outline one of the simplest approaches to defining a hyperreal field ∗ R, let R be the field of real numbers, and let N be the semiring of natural numbers. Denote by R N the set of sequences of real numbers, a field ∗ R is defined as a suitable quotient of R N, as follows. Take a nonprincipal ultrafilter F ⊂ P, in particular, F contains the Fréchet filter. There are at least three reasons to consider non-standard analysis, historical, pedagogical, and technical, much of the earliest development of the infinitesimal calculus by Newton and Leibniz was formulated using expressions such as infinitesimal number and vanishing quantity
29.
Field (physics)
–
In physics, a field is a physical quantity, typically a number or tensor, that has a value for each point in space and time. For example, on a map, the surface wind velocity is described by assigning a vector to each point on a map. Each vector represents the speed and direction of the movement of air at that point, as another example, an electric field can be thought of as a condition in space emanating from an electric charge and extending throughout the whole of space. When a test electric charge is placed in this electric field, physicists have found the notion of a field to be of such practical utility for the analysis of forces that they have come to think of a force as due to a field. In the modern framework of the theory of fields, even without referring to a test particle, a field occupies space, contains energy. This led physicists to consider electromagnetic fields to be a physical entity, the fact that the electromagnetic field can possess momentum and energy makes it very real. A particle makes a field, and a field acts on another particle, in practice, the strength of most fields has been found to diminish with distance to the point of being undetectable. One consequence is that the Earths gravitational field quickly becomes undetectable on cosmic scales, a field has a unique tensorial character in every point where it is defined, i. e. a field cannot be a scalar field somewhere and a vector field somewhere else. For example, the Newtonian gravitational field is a vector field, moreover, within each category, a field can be either a classical field or a quantum field, depending on whether it is characterized by numbers or quantum operators respectively. In fact in this theory an equivalent representation of field is a field particle, to Isaac Newton his law of universal gravitation simply expressed the gravitational force that acted between any pair of massive objects. In the eighteenth century, a new quantity was devised to simplify the bookkeeping of all these gravitational forces and this quantity, the gravitational field, gave at each point in space the total gravitational force which would be felt by an object with unit mass at that point. The development of the independent concept of a field began in the nineteenth century with the development of the theory of electromagnetism. In the early stages, André-Marie Ampère and Charles-Augustin de Coulomb could manage with Newton-style laws that expressed the forces between pairs of electric charges or electric currents. However, it became more natural to take the field approach and express these laws in terms of electric and magnetic fields. The independent nature of the field became more apparent with James Clerk Maxwells discovery that waves in these fields propagated at a finite speed, Maxwell, at first, did not adopt the modern concept of a field as fundamental quantity that could independently exist. Instead, he supposed that the field expressed the deformation of some underlying medium—the luminiferous aether—much like the tension in a rubber membrane. If that were the case, the velocity of the electromagnetic waves should depend upon the velocity of the observer with respect to the aether. Despite much effort, no evidence of such an effect was ever found
30.
Four color theorem
–
Two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. Despite the motivation from coloring political maps of countries, the theorem is not of particular interest to mapmakers, according to an article by the math historian Kenneth May, “Maps utilizing only four colors are rare, and those that do usually require only three. Books on cartography and the history of mapmaking do not mention the four-color property, a number of false proofs and false counterexamples have appeared since the first statement of the four color theorem in 1852. Martin Gardner wrote an account of what was known at the time about the four color theorem in his September 1960 Mathematical Games column in Scientific American magazine. In 1975 Gardner revisited the topic by publishing a map said to be a counter-example in his infamous April fools hoax column of April 1975, the four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer, Appel and Hakens approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem. Appel and Haken used a computer program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps, showing this required hundreds of pages of hand analysis. Appel and Haken concluded that no smallest counterexamples exist because any must contain, yet do not contain and this contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their proof was not accepted by all mathematicians because the proof was infeasible for a human to check by hand. Since then the proof has gained acceptance, although doubts remain. To dispel remaining doubt about the Appel–Haken proof, a proof using the same ideas and still relying on computers was published in 1997 by Robertson, Sanders, Seymour. Additionally, in 2005, the theorem was proved by Georges Gonthier with general purpose theorem proving software, the intuitive statement of the four color theorem, i. e. First, all corners, points that belong to three or more countries, must be ignored. In addition, bizarre maps can require more than four colors, second, for the purpose of the theorem, every country has to be a connected region, or contiguous. In the real world, this is not true, because all the territory of a particular country must be the same color, four colors may not be sufficient. For instance, consider a simplified map, In this map and this map then requires five colors, since the two A regions together are contiguous with four other regions, each of which is contiguous with all the others. A similar construction also applies if a color is used for all bodies of water. For maps in which more than one country may have multiple disconnected regions, a simpler statement of the theorem uses graph theory
31.
Martin Gardner
–
He was considered a leading authority on Lewis Carroll. The Annotated Alice, which incorporated the text of Carrolls two Alice books, was his most successful work and sold over a million copies and he had a lifelong interest in magic and illusion and was regarded as one of the most important magicians of the twentieth century. He was considered the dean of American puzzlers and he was a prolific and versatile author, publishing more than 100 books. Gardner was one of the foremost anti-pseudoscience polemicists of the 20th century and his book Fads and Fallacies in the Name of Science, published in 1957, became a classic and seminal work of the skeptical movement. In 1976 he joined with fellow skeptics to found CSICOP, an organization promoting scientific inquiry, Gardner, son of a petroleum geologist, grew up in and around Tulsa, Oklahoma. His lifelong interest in puzzles started in his boyhood when his father gave him a copy of Sam Loyds Cyclopedia of 5000 Puzzles, Tricks and he attended the University of Chicago, where he earned his bachelors degree in philosophy in 1936. Early jobs included reporter on the Tulsa Tribune, writer at the University of Chicago Office of Press Relations, during World War II, he served for four years in the U. S. Navy as a yeoman on board the destroyer escort USS Pope in the Atlantic. His ship was still in the Atlantic when the war came to an end with the surrender of Japan in August 1945, after the war, Gardner returned to the University of Chicago. He attended graduate school for a year there, but he did not earn an advanced degree, in 1950 he wrote an article in the Antioch Review entitled The Hermit Scientist. His paper-folding puzzles at that magazine led to his first work at Scientific American, appropriately enough—given his interest in logic and mathematics—they lived on Euclid Avenue. The year 1960 saw the edition of his best-selling book ever. In 1979, Gardner retired from Scientific American and he and his wife Charlotte moved to Hendersonville and he also revised some of his older books such as Origami, Eleusis, and the Soma Cube. Charlotte died in 2000 and two years later Gardner returned to Norman, Oklahoma, where his son, James Gardner, was a professor of education at the University of Oklahoma and he died there on May 22,2010. An autobiography — Undiluted Hocus-Pocus, The Autobiography of Martin Gardner — was published posthumously, the main-belt asteroid 2587 Gardner discovered by Edward L. G. Bowell at Anderson Mesa Station in 1980 is named after Martin Gardner. Martin Gardner had a impact on mathematics in the second half of the 20th century. His column was called Mathematical Games but it was more than that. His writing introduced many readers to real mathematics for the first time in their lives, the column lasted for 25 years and was read avidly by the generation of mathematicians and physicists who grew up in the years 1956 to 1981. It was the inspiration for many of them to become mathematicians or scientists themselves