Understanding it, I realized, required reading other, more elementary books.
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Boost being Lego's more elementary robotics kit offering (vs the veteran Mindstorms platform).
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Or it could be seen as something much more elementary: a reflection of changing times.
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Yet Mitsubishi's meticulousness does not seem to have extended to the more elementary task of security.
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That image of breaking down the machines into modules and modules into even more elementary units.
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Essentially, it looks less disco diva and more elementary schooler who has gotten into the Elmer's and glitter shaker.
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But there may be an even more elementary barrier in Congress to setting the impeachment train in motion: Democrats.
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What could be more elementary than a prohibition against eviscerating a person's human dignity by thrusting him into the shackles of slavery?
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With its latest $34 billion acquisition of Red Hat, IBM may have found something more elementary than "Watson" to save its flagging business.
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But the Collins case suggests an even more elementary restriction: Members of Congress should never be able to sit on a board of directors of a public company.
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Meals don't get more elementary than those sodium-infested bags of goodness; what's more effortless than throwing a solidified pack of noodles and powder into a pot and instantly having a meal ready for consumption on your dinner table?
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The company says it has redesigned the "Everyone Can Code" curriculum with a focus on introducing more elementary and middle school students to coding, while also adding more resources for teachers, a new student guide, and refreshed Swift Coding Club materials.
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Apple expands and updates its 'Everyone Can Code' program The company says it has redesigned the "Everyone Can Code" curriculum with a focus on introducing coding to more elementary and middle school students, while also adding more resources for teachers, a new student guide and refreshed Swift Coding Club materials. 4.
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The big picture: For the past 5 years, Elon Musk and others have warned of a future disaster resulting from unchecked superintelligent AI. But today, much of the field is caught in a rather more elementary tug-of-war over which avenue will imbue AI even with the capacity for basic understanding.
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Setting aside the casual acceptance of the term "girlfriend" for kids this age (and the possessive "his"), as well as the unspecified actions that would have to precede a boy sharing said picture in the first place — a "middle schooler" is much more likely to be in the early teens than the decidedly more elementary-school age of 10.
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"According to several estimates, supercomputers can now—or in the near future—do more elementary operations per second than human brains, so we might already have the necessary hardware to compete with brains," said Jaan Tallinn, a computer programmer, founding member of Skype, and co-founder of the Centre for the Study of Existential Risk, a research center at the University of Cambridge concerned with human extinction scenarios.
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For a more elementary introduction of the formal definition see automata theory.
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A proof based on diophantine approximation was given by . A more elementary variant of Vojta's proof was given by .
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Two more elementary schools and another middle school are planned.Construction , Deercreekschools.org Grove Valley Elementary recently opened in August 2009. All three existing elementary schools have had classrooms added.
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It provides also the structure of the LU decomposition of the Vandermonde matrix. The third one is more elementary and more complicated, using only elementary row and column operations.
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Helmut Hasse's book Vorlesungen über Zahlentheorie was published in 1950, and is different from and more elementary than his book Zahlentheorie. It covers elementary number theory, Dirichlet's theorem, and quadratic fields.
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To compensate for the town's growth, Jamestown has since built two more elementary schools and one more middle school. A community college, Guilford Technical Community College, provides local adults with a higher education.
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The vanishing cycle functor then sits in a distinguished triangle with the nearby cycle functor and a more elementary functor. This formulation has been of continuing influence, in particular in D-module theory.
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There are also coherent motions at much smaller scales such as hairpin vortices and typical eddies, which are typically known as coherent substructures, as in coherent structures which can be broken up into smaller more elementary substructures.
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In the mathematical area of braid theory, the Dehornoy order is a left- invariant total order on the braid group, found by Patrick Dehornoy. Dehornoy's original discovery of the order on the braid group used huge cardinals, but there are now several more elementary constructions of it .
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In quantum field theory, a composite field is a field defined in terms of other more "elementary" fields. It might describe a composite particle (bound state) or it might not. It might be local, or it might be nonlocal. Noether fields are often composite fields and they are local.
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In 1897, the temple school was moved and became a public school, Eniwa Elementary School. Three more elementary schools were opened in the Meiji period. By the early 1940s, Eniwa had eight elementary schools. In 1947, four new junior high schools were created, later amalgamating into two by 1964.
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Similar notions have found applications everywhere in differential geometry. For a more elementary discussion see the article on curvature which discusses the curvature of curves and surfaces in 2 and 3 dimensions, as well as the differential geometry of surfaces. The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.
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The converse that projective manifolds are Hodge manifolds is more elementary and was already known. Kodaira also proved (Kodaira 1963), by recourse to the classification of compact complex surfaces, that every compact Kähler surface is a deformation of a projective Kähler surface. This was later simplified by Buchdahl to remove reliance on the classification (Buchdahl 2008).
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In the course of this work they found a more elementary proof of Harish-Chandra's fundamental theorem on the local integrability of characters of Lie groups. With H. Donnelly and I. Singer, he extended Hirzebruch's formula (relating the signature defect at cusps of Hilbert modular surfaces to values of L-functions) from real quadratic fields to all totally real fields.
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If general relativity were considered to be one of the two pillars of modern physics, then quantum theory, the basis of understanding matter from elementary particles to solid state physics, would be the other.An overview of quantum theory can be found in standard textbooks such as ; a more elementary account is given in However, how to reconcile quantum theory with general relativity is still an open question.
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1 (1957), pp. 55–67. showing that Gödelian incompleteness held for formal systems considerably more elementary than that of Kurt Gödel's 1931 landmark paper. The contemporary understanding of Gödel's theorem dates from this 1931 paper. Smullyan later made a compelling case that much of the fascination with Gödel's theorem should be directed at Tarski's theorem, which is much easier to prove and equally disturbing philosophically.
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Combined with what he called "the compulsion of destiny ... come across [in] people all of whose human relationships have the same outcome",Freud, "Beyond" p. 294 and p. 292. such evidence led Freud "to justify the hypothesis of a compulsion to repeat—something that would seem more primitive, more elementary, more instinctual than the pleasure principle which it over-rides".Freud, "Beyond", p. 294.
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Both works offer detailed explanations of each of the eight factors of the path, thus showing how Buddhism is to be practiced as a way of life. For elementary, original source material on the Dhamma, two compilations by the great German scholar-monk Nyanatiloka Thera (Ven. Nyanaponika's teacher) are still unsurpassed. The more elementary is The Word of the Buddha, first compiled in 1906 and subject to some sixteen reprints.
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Its creation was the suggestion of George Alexander Gibson, a professor at University of Glasgow, who wished to remove the more elementary or pedagogical articles from the Proceedings of the Edinburgh Mathematical Society.Edinburgh Mathematical Notes, Cambridge University Press. Retrieved 22 October 2013. Is founding editor was Peter Pinkerton, who at the time headed the mathematics department of George Watson's College and later became rector of the High School of Glasgow.
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Young was born to Carolyn and Robert Young in Houston, Texas. When Young was in high school, his father was a mechanic and his mother was a real estate agent. began his baseball career at St. Thomas More elementary/middle school and later attended national powerhouse Bellaire High School. In 1999, the school went 38-2 and won the Texas high school baseball championship over Duncanville High School by mercy rule.
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Located in the south-west corner of the church, the font's decoration is to Pevsner "far more elementary than that on the font at Winchester". Each face has three medallions, on eleven of which "grotesque figures" are carved, mostly with wings, with a single man carved on the twelfth. The pillars that support the font are not made of Tournai marble but of Purbeck marble, and are renewed.
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In 1893 the German mathematical society invited Hilbert and Minkowski to write reports on the theory of numbers. They agreed that Minkowski would cover the more elementary parts of number theory while Hilbert would cover algebraic number theory. Minkowski eventually abandoned his report, while Hilbert's report was published in 1897. It was reprinted in volume 1 of his collected works, and republished in an English translation in 1998.
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Shell beads and microlithic, as well as Wilton tools were found in this layer. The artefacts became more elementary in character and technique as it is approached the end of this layer. The artefacts in different colours, moreover, other types of raw coloring materials, such as yellow ochre, pencils, pieces of red and brown hematites and ochres, were found. It's supposed that ochres were used to paint the body and walls.
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In the spring of 1978 Kazhdan and Lusztig were studying Springer representations of the Weyl group of an algebraic group on \ell-adic cohomology groups related to unipotent conjugacy classes. They found a new construction of these representations over the complex numbers . The representation had two natural bases, and the transition matrix between these two bases is essentially given by the Kazhdan–Lusztig polynomials. The actual Kazhdan–Lusztig construction of their polynomials is more elementary.
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Therefore, there is no need for the Standard Model Higgs boson and each quark mass is derived from the interaction between each pair of primons by means of three Higgs-like bosons. In his 1989 Nobel Prize acceptance lecture, Hans Dehmelt described a most fundamental elementary particle, with definable properties, which he called the cosmon, as the likely end result of a long but finite chain of increasingly more elementary particles. See also references therein.
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Unlike the above series, is not Cesàro summable nor Abel summable. Those methods work on oscillating divergent series, but they cannot produce a finite answer for a series that diverges to +∞.Hardy p.10 (More detail on the source needed) Most of the more elementary definitions of the sum of a divergent series are stable and linear, and any method that is both stable and linear cannot sum to a finite value; see below.
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The material has a reputation of being hard to read for a number of reasons. More elementary or foundational parts were relegated to the EGA series of Grothendieck and Jean Dieudonné, causing long strings of logical dependencies in the statements. The style is very abstract and makes heavy use of category theory. Moreover, an attempt was made to achieve maximally general statements, while assuming that the reader is aware of the motivations and concrete examples.
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Since 1990, there has been continual construction on the campus, which has expanded to . The land was cleared and four dorms, a high school, a soccer field and track, an auditorium, a lunch pavilion, more elementary buildings, and a two-court gym have been built. In October 2001, 144 students and 60 teachers with American nationality were dismissed after anti-American protests. In 2002, the name of the school was changed to Mountainview International Christian School.
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A classroom-based education program was begun in 1981 and was expanded to more elementary and high school grade levels in 1984. A program called "Opera Highlights" became part of the curriculum of the University of Hawaii's College of Continuing Education in 1983. HOT now has several educational programs, most notably its "Opera For Everyone" (OFE) program. OFE allows students to participate in all aspects of an opera production from technical crew to performing to simply watching an opera.
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Also \sin \theta is algebraic since it equals the algebraic number \cos(\theta-\pi /2). Finally, \tan \theta, where again \theta is a rational multiple of , is algebraic as being the ratio of two algebraic numbers. In a more elementary way, this can also be seen by equating the imaginary parts of the two sides of the expansion of the de Moivre equation to each other and dividing through by \cos^n \theta to obtain a polynomial equation in \tan \theta.
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In general, the composition of two permutations is not commutative, that is, \pi\sigma eq \sigma\pi. As a bijection from a set to itself, a permutation is a function that performs a rearrangement of a set, and is not a rearrangement itself. An older and more elementary viewpoint is that permutations are the rearrangements themselves. To distinguish between these two, the identifiers active and passive are sometimes prefixed to the term permutation, whereas in older terminology substitutions and permutations are used.
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An experimental method for locating the center of mass is to suspend the object from two locations and to drop plumb lines from the suspension points. The intersection of the two lines is the center of mass. The shape of an object might already be mathematically determined, but it may be too complex to use a known formula. In this case, one can subdivide the complex shape into simpler, more elementary shapes, whose centers of mass are easy to find.
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The publishes an academic journal, the Proceedings of the Edinburgh Mathematical Society, published by Cambridge University Press (ISSN 0013-0915.) The Proceedings were first published in 1884, and is issued three times a year. It covering a range of pure and applied mathematics. Between 1909 and 1961, the Society also published the Edinburgh Mathematical Notes, on the suggestion of George Alexander Gibson, a professor at the University of Glasgow, who wished to remove the more elementary or pedagogical articles from the Proceedings.
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Döderlein's most elaborate work as a philologist was marred by over-subtlety, and lacked method and clearness. He is best known by his Lateinische Synonymen und Etymologien (1826-1838), and his Homerisches Glossarium (1850-1858). To the same class belong his Lateinische Wortbildung (1838), Handbuch der lateinischen Synonymik (1839), and the Handbuch der lateinischen Etymologie (1841), besides various works of a more elementary kind intended for the use of schools and gymnasia. Most of the works named have been translated into English.
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The first-published of these was the Janua Linguarum Reserata (The Gate of Tongues Unlocked), issued in 1631. This was followed later by a more elementary text, the Vestibulum, and a more advanced one, the Atrium, and other texts. In 1658 the Orbis Pictus was published, probably the most renowned and most widely circulated of school textbooks. It was also the first successful application of illustrations to the work of teaching, though not, as often stated, the first illustrated book for children.
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Clarke grew up in St. Louis, Missouri and began playing soccer as a youth at St. Thomas More Elementary School. He attended McBride High School, but when the school closed between his junior and senior year, he moved to Normandy Senior High School, graduating in 1972. Following his graduation from high school, Clarke entered St. Louis University where he played on the Billiken's soccer team from 1972 to 1975. At the time, St. Louis was the dominant college soccer team.
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Croatia had double more elementary schools than Serbia. Croatian and Vojvodina had 4910 km of railway track compared to 1187 km in Central Serbia. Persecutions of the Muslims by the Serbs resulted in their massive emigration to Turkey soon after the foundation of Kingdom of Serbs, Croats and Slovenes in 1918, where Serbia was the leading and privileged nation. The same happened to several hundred thousand Muslims soon after the Second World War. On November 28, 1920 elections to the Constitutional Assembly were held.
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In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them. For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory.
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Throughout the 1950s and 1960s, Ursuline Academy also operated a candy store on its premises (with sweets manufactured by one of the sisters). Expansion of the Catholic educational system in the city of Great Falls in the 1950s and 1960s led to the creation of many more elementary and junior high schools with larger, modern facilities. Enrollment at Ursuline Academy dropped steadily, and the elementary school closed in 1966. The high school closed in 1973, and the Great Falls Public Schools purchased the building in 1975.
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SNOMED was designed as a comprehensive nomenclature of clinical medicine for the purpose of accurately storing and/or retrieving records of clinical care in human and veterinary medicine. The metaphor used by Roger A. Côté, the first editorial chair, was that SNOMED would become the periodic table of elements of medicine because of its definitional organization beyond the hierarchical design. Indeed, diseases and procedures were ordered hierarchically and are further referenced back to more elementary terms (see Reference Ontology and Multi-Axial Design, below).
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Perhaps the most successful of these sensory prostheses is the cochlear implant which has restored hearing abilities to the deaf. Visual prosthesis for restoring visual capabilities of blind persons is still in more elementary stages of development. Motor prosthesics are devices involved with electrical stimulation of biological neural muscular system that can substitute for control mechanisms of the brain or spinal cord. Smart prostheses can be designed to replace missing limbs controlled by neural signals by transplanting nerves from the stump of an amputee to muscles.
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Union districts are formed by joining two or more elementary districts. School districts are governed by an elected school board (sometimes called a "board of education" or "board of trustees"), which manages the schools within its jurisdiction. There are also county special service schools and regional occupational programs provide vocational and technical education. Historically, school districts were organized at the primary level (Kindergarten through 8th grade, approximately ages 5–13), and the secondary (high school) level (9th through 12th grade, approximately ages 14–17).
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Paul Edward Haggis was born in London, Ontario, the son of Mary Yvonne (née Metcalf) and Ted Haggis, an Olympic sprinter. He was raised as a Catholic, but considered himself an atheist in early adulthood. The Gallery Theatre in London was owned by his parents, and Haggis gained experience in the field through work at the theatre. Haggis attended St. Thomas More Elementary School, and after being inspired by Alfred Hitchcock and Jean-Luc Godard, proceeded to study art at H. B. Beal Secondary School.
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Elementary particles included in the Standard Model In particle physics, an elementary particle or fundamental particle is a -->subatomic particle with no substructure, i.e. it is not composed of other particles. Particles currently thought to be elementary include the fundamental fermions (quarks, leptons, antiquarks, and antileptons), which generally are "matter particles" and "antimatter particles", as well as the fundamental bosons (gauge bosons and the Higgs boson), which generally are "force particles" that mediate interactions among fermions. A particle containing two or more elementary particles is called a composite particle.
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In theoretical physics, composite gravity refers to models that attempted to derive general relativity in a framework where the graviton is constructed as a composite bound state of more elementary particles, usually fermions. A theorem by Steven Weinberg and Edward Witten shows that this is not possible in Lorentz covariant theories: massless particles with spin greater than one are forbidden. The AdS/CFT correspondence may be viewed as a loophole in their argument. However, in this case not only the graviton is emergent; a whole spacetime dimension is emergent, too.
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288, 308. The fourth was the so-called "destiny neurosis", manifested in 'the life-histories of men and women ... [as] an essential character-trait which remains always the same and which is compelled to find expression in a repetition of the same experience'.Freud, Beyond. p. 293. All such activities appeared to Freud to contradict the organism's search for pleasure, and therefore 'to justify the hypothesis of a compulsion to repeat—something that seems more primitive, more elementary, more instinctual than the pleasure principle which it over-rides':Freud, Beyond. p. 294.
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While this depiction is valid, the more strongly contributing process is the sequential excitation of the activator by two or more different sensitizer ions. The upconversion process is said to be cooperative when there are one or more elementary steps (sensitization or luminescence) in the process which involve multiple lanthanide ions. In cooperative sensitization process, two ions in their excited state simultaneously decay to their ground states, generating a higher energy photon. Similarly, in cooperative luminescence, two excited state ions transfer their energy to a neighboring ion in one elementary step.
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A sonatina is literally a small sonata. As a musical term, sonatina has no single strict definition; it is rather a title applied by the composer to a piece that is in basic sonata form, but is shorter and lighter in character, or technically more elementary, than a typical sonata.Collins Music Encyclopedia (1959: William Collins & Co. Ltd.): sonatina The term has been in use at least since the late baroque; there is a one-page, one-movement harpsichord piece by Handel called "Sonatina".Oxford Companion to Music by Percy A. Scholes (1938, 1978 et al.).
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Book VI, known only through translation from the Arabic, contains 33 propositions, the least of any book. It also has large lacunae, or gaps in the text, due to damage or corruption in the previous texts. The topic is relatively clear and uncontroversial. Preface 1 states that it is “equal and similar sections of cones.” Apollonius extends the concepts of congruence and similarity presented by Euclid for more elementary figures, such as triangles, quadrilaterals, to conic sections. Preface 6 mentions “sections and segments” that are “equal and unequal” as well as “similar and dissimilar,” and adds some constructional information.
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One of Peuerbach's best known works is his Theoricae Novae Planetarum (written in 1454, published by Regiomontanus in 1472)."The Early Manuscripts of Georg von Peuerbach's Theoricae Novae Planetarum" It began as a series of lectures transcribed by Regiomontanus. The Theoricae Novae Planetarum was an attempt to present Ptolemaic astronomy in a more elementary and comprehensible way. The book was very successful, replacing the older Theorica Planetarum Communis, sometimes attributed to a certain Gerardus Cremonensis, as the standard university text on astronomy and was studied by many later-influential astronomers including Nicolaus Copernicus and Johannes Kepler.
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Founded as the SABIS Foxborough Regional School, a member of the SABIS international network of private charter schools, Foxborough Regional Charter School, or FRCS, opened at the start of the 1998 school year with 582 students in grades Kindergarten through Eight. To accommodate their growing student population's need for a practical cafeteria and gym, the school built a cafetorium in 1999, which served as a gymnasium during gym class, and a cafeteria during lunch hours. In 2000, a new wing was built to allow for more elementary students. In 2001 a temporary, 6-classroom modular building was installed for the middle school students.
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The main conjecture of Iwasawa theory was formulated as an assertion that two methods of defining p-adic L-functions (by module theory, by interpolation) should coincide, as far as that was well-defined. This was proved by for Q, and for all totally real number fields by . These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (the Herbrand–Ribet theorem). Karl Rubin found a more elementary proof of the Mazur–Wiles theorem by using Thaine's method and Kolyvagin's Euler systems, described in and , and later proved other generalizations of the main conjecture for imaginary quadratic fields.
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The next three years were quite difficult until in 1830 he got a job that assured him of solid financial security for the rest of his life. Thanks to this job, he could focus on writing his two major works: Series of Progressive Lessons (1831) and Second Series of Progressive Lessons (1832). The first series of the Lessons were more elementary in character, and designed for the use of beginners; the second series, on the other hand, went deeply into all the known openings. Here, for the first time we find the Evans Gambit, which is named after its inventor, Capt. Evans.
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Undergraduate Texts in Mathematics (UTM) is a series of undergraduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are small yellow books of a standard size. The books in this series tend to be written at a more elementary level than the similar Graduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. There is no Springer-Verlag numbering of the books like in the Graduate Texts in Mathematics series.
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For all k digits, we need time O(k^3 n^2 \log(B)). The only internal storage needed is r, which is O(k) digits on the kth iteration. That this algorithm does not have bounded memory usage puts an upper bound on the number of digits which can be computed mentally, unlike the more elementary algorithms of arithmetic. Unfortunately, any bounded memory state machine with periodic inputs can only produce periodic outputs, so there are no such algorithms which can compute irrational numbers from rational ones, and thus no bounded memory root extraction algorithms.
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Formally, a complex projective space is the space of complex lines through the origin of an (n+1)-dimensional complex vector space. The space is denoted variously as P(Cn+1), Pn(C) or CPn. When , the complex projective space CP1 is the Riemann sphere, and when , CP2 is the complex projective plane (see there for a more elementary discussion). Complex projective space was first introduced by as an instance of what was then known as the "geometry of position", a notion originally due to Lazare Carnot, a kind of synthetic geometry that included other projective geometries as well.
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Iwasawa formulated the main conjecture of Iwasawa theory as an assertion that two methods of defining p-adic L-functions (by module theory, by interpolation) should coincide, as far as that was well-defined. This was proved by for \Q and for all totally real number fields by . These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (the so-called Herbrand–Ribet theorem). Karl Rubin found a more elementary proof of the Mazur-Wiles theorem by using Kolyvagin's Euler systems, described in and , and later proved other generalizations of the main conjecture for imaginary quadratic fields.
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In the University of Göttingen, Kohlrausch documented his practical experiments resulting in the book Leitfaden der praktischen Physik (Guidelines to Practical Physics), which was published in 1870 as the first book of its type in Germany. It contained not only descriptions of experiments, experimental setups and measuring techniques, but also tables of physical quantities. It was issued in many editions (the 9th enlarged and revised edition of 1901 being entitled Lehrbuch der praktischen Physik; a more elementary work based on it being entitled Kleiner Leitfaden der praktischen Physik) and translated into English. It was considered the standard work on physical laboratory methods and measurements.
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See , page 263–277: "In a sense, al- Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers". A debate now exists whether who (in the general sense) is more entitled to be known as "the father of algebra". Those who support Diophantus point to the fact that the algebra found in Al-Jabr is slightly more elementary than the algebra found in Arithmetica and that Arithmetica is syncopated while Al-Jabr is fully rhetorical.
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In 1987, Huisken adapted his methods to consider an alternative "mean curvature"-driven flow for closed hypersurfaces in Euclidean space, in which the volume enclosed by the surface is kept constant; the result is directly analogous. Similarly to Hamilton's result, Huisken's results can be viewed as providing proofs that any smooth closed convex hypersurface of Euclidean space is diffeomorphic to a sphere, and is the boundary of a region which is diffeomorphic to a ball. However, both of these results can be proved by more elementary means using the Gauss map. In 1986, Huisken extended the calculations in his proof to consider hypersurfaces in general Riemannian manifolds.
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Moreover, at this same period, no one could report what > had taken place in the more distant past, since the Chinese themselves only > had a fragmentary knowledge of that. One should not forget that, in China > itself, autochthonous mathematics was not rediscovered on a large scale > prior to the last quarter of the 18th century. Correspondingly, scholars paid less attention to mathematics; pre-eminent mathematicians such as Gu Yingxiang and Tang Shunzhi appear to have been ignorant of the Tian yuan shu (Increase multiply) method. Without oral interlocutors to explicate them, the texts rapidly became incomprehensible; worse yet, most problems could be solved with more elementary methods.
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The Boolean set operations can be implemented in terms of more elementary operations (`pop`, `clear`, and `add`), but specialized algorithms may yield lower asymptotic time bounds. If sets are implemented as sorted lists, for example, the naive algorithm for `union(S,T)` will take time proportional to the length m of S times the length n of T; whereas a variant of the list merging algorithm will do the job in time proportional to m+n. Moreover, there are specialized set data structures (such as the union-find data structure) that are optimized for one or more of these operations, at the expense of others.
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They allow us to give much more depth to the study of moduli spaces by endowing them with additional features that were not present in the previous, more elementary works. After World War II the subject was developed further in this analytic vein, in particular by Lars Ahlfors and Lipman Bers. The theory continues to be active, with numerous studies of the complex structure of Teichmüller space (introduced by Bers). The geometric vein in the study of Teichmüller space was revived following the work of William Thurston in the late seventies, who introduced a geometric compactification which he used in his study of the mapping class group of a surface.
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The transcendence of and are direct corollaries of this theorem. Suppose is a non-zero algebraic number; then is a linearly independent set over the rationals, and therefore by the first formulation of the theorem is an algebraically independent set; or in other words is transcendental. In particular, is transcendental. (A more elementary proof that is transcendental is outlined in the article on transcendental numbers.) Alternatively, by the second formulation of the theorem, if is a non-zero algebraic number, then is a set of distinct algebraic numbers, and so the set is linearly independent over the algebraic numbers and in particular cannot be algebraic and so it is transcendental.
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Ritmico, non troppo allegro The term sonatina has no single strict definition, but is rather a title applied by the composer to a piece in basic sonata form which is shorter and lighter in character, or technically more elementary, than a typical sonata. The Rondo was used as a test piece in the 1928 Daily Express Piano Competition, which was won by Cyril Smith. It had been recorded by William Murdoch as a guide to competitors. Lewis Foreman has written, "that Ireland even then recognised the piano not only for its romantic and singing qualities, but also - almost Bartók like - as a percussion instrument".
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Chiral symmetry breaking is most apparent in the mass generation of nucleons from more elementary light quarks, accounting for approximately 99% of their combined mass as a baryon. It thus accounts for most of the mass of all visible matter.Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics, (Oxford 1984) ; For example, in the proton, of mass mp ≈ 938 MeV, the valence quarks, two up quarks with mu ≈ 2.3 MeV and one down quark with md ≈ 4.8 MeV, only contribute about 9.4 MeV to the proton's mass. The source of the bulk of the proton's mass is quantum chromodynamics binding energy, which arises out of QCD chiral symmetry breaking.
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Oak Park's educational history began with the Clinton School, a one-room schoolhouse on property donated by Barney Clinton in the early 20th century. As the population grew rapidly, Clinton School was expanded and more elementary schools were built, particularly beginning in the 1950s. Clinton School was made a junior high school and another was built in the mid-1960s, then named for the poet Robert Frost. At that time, one school in Oak Park had a special education department for children with learning disabilities: Lessenger Elementary School on Albany St. at Sunset St. Consequently, many families with such special children gravitated to the neighborhood surrounding Lessenger, creating a "cluster" of such families rarely found elsewhere.
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The original plan was to start the integration at the elementary schools, but as more elementary school parents spoke their opinions the plan was altered to begin integration in the fall of 1957 at Central High School, then to the junior high by 1960 and to the elementary level by 1963. The plan also included that any student whose race was a minority in their school could ask for a transfer. Despite the plan General Faubus called the National Guard to turn students away from the school. On February 8, 1956 the court made the Blossom plan a "court mandated operation," and it was carried out with the help of President Eisenhower.
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Two years later, the Master of Arts in Education was introduced in 1938. During the Japanese occupation of the Philippines in 1941, the school operation was interrupted when the Japanese Army requisitioned some of the school's buildings. When schooling resumed in 1943, the classes were crammed in the remaining buildings and some neighbor houses. In 1944, Bachelor of Music was offered with various majors through the years: piano, organ, violin, marimba, voice, and others. After the war, new buildings were constructed as student population grew more: Elementary Building (1947), Paraclete Auditorium (1948), Canteen (1949), College Building (1956), College Building-Annex (1961–62), the College Library Annex (1964), New Elementary Building (1966), and the College Cafeteria (1970).
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In mathematics, the Daniell integral is a type of integration that generalizes the concept of more elementary versions such as the Riemann integral to which students are typically first introduced. One of the main difficulties with the traditional formulation of the Lebesgue integral is that it requires the initial development of a workable measure theory before any useful results for the integral can be obtained. However, an alternative approach is available, developed by that does not suffer from this deficiency, and has a few significant advantages over the traditional formulation, especially as the integral is generalized into higher-dimensional spaces and further generalizations such as the Stieltjes integral. The basic idea involves the axiomatization of the integral.
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David R. Williams1, Mark B. McClellan and Alice M. Rivlin, "Beyond The Affordable Care Act: Achieving Real Improvements In Americans' Health" Health Affairs August 2010 29: 81481–88 Secondly, there is research, including a 2005 study by Dr. Charles H. Hillmam of The University of Illinois at Urbana-Champaign that concludes that fitness is globally related to academic achievement.Hillman et al. 2005; CDE, 2001 The opportunities, challenges, and risks that No Child Left Behind poses for science education in elementary and middle schools—worldwide competition insists on rapidly improving science education. Adding science assessments to the NCLB requirements may ultimately result in science being taught in more elementary schools and by more teachers than ever before.
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A projective plane satisfying Pappus's theorem universally is called a Pappian plane. Alternative, not necessarily associative, division algebras like the octonions correspond to Moufang planes. There is no known purely geometric proof of the purely geometric statement that Desargues' theorem implies Pappus' theorem in a finite projective plane (finite Desarguesian planes are Pappian). (The converse is true in any projective plane and is provable geometrically, but finiteness is essential in this statement as there are infinite Desarguesian planes which are not Pappian.) The most common proof uses coordinates in a division ring and Wedderburn's theorem that finite division rings must be commutative; give a proof that uses only more "elementary" algebraic facts about division rings.
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Practitioners of kung fu refer to two separate forms of personal force: Li (Traditional Chinese: 力) refers to the more elementary use of tangible physical (or "external") force, such as that produced by muscles. Neijing (Traditional Chinese:內勁) or Neigong (Traditional Chinese: 內功), in contrast, refer to "internal" forces produced via advanced mental control over psychic energy (the qi). The degree of Li force one can employ in kung fu depends on several variables such as resilience of muscles, strength of bones, speed and timing of attack and so on. An effective way to enhance the Li force is to exercise one's muscles and bones by applying increasing pressure on them (weight training, gym exercises, etc.).
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Minkowski himself considered Einstein's theory as a generalization of Lorentz's and credited Einstein for completely stating the relativity of time, but he criticized his predecessors for not fully developing the relativity of space. However, modern historians of science argue that Minkowski's claim for priority was unjustified, because Minkowski (like Wien or Abraham) adhered to the electromagnetic world-picture and apparently did not fully understand the difference between Lorentz's electron theory and Einstein's kinematics.Miller (1981), Ch. 7.4.6Walter (1999b), Ch. 3 In 1908, Einstein and Laub rejected the four-dimensional electrodynamics of Minkowski as overly complicated "learned superfluousness" and published a "more elementary", non-four-dimensional derivation of the basic-equations for moving bodies.
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The Bayview, a historically predominant black neighborhood, is home to more elementary school-age students than any other neighborhood in the city and combined with the Mission and Excelsior, houses a quarter of all students in the district. Schools in the Bayview have suffered from declining enrollment for the past two decades. Out of the 6,000 students who live in the Bayview, more than 70% choose to attend school outside of their neighborhood. In 2016, in attendance with Jonathan Garcia, Adonal Foyle and Theo Ellington, Willie L. Brown middle school in Bayview-Hunter's Point commemorated the unveiling of the new Golden State Warrior outside basketball court at the school, donated by the Warriors Community Foundation.
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This can be seen by noting that any such diffeomorphism can be lifted to a diffeomorphism f of the reals satisfying [f(x + 1) = f(x) + 1]; this space is convex and hence path-connected. A smooth, eventually constant path to the identity gives a second more elementary way of extending a diffeomorphism from the circle to the open unit disc (a special case of the Alexander trick). Moreover, the diffeomorphism group of the circle has the homotopy-type of the orthogonal group O(2). The corresponding extension problem for diffeomorphisms of higher-dimensional spheres Sn−1 was much studied in the 1950s and 1960s, with notable contributions from René Thom, John Milnor and Stephen Smale.
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The part saying p divides Bp−n if Gn is not trivial is due to Jacques Herbrand. The converse, that if p divides Bp−n then Gn is not trivial is due to Kenneth Ribet, and is considerably more difficult. By class field theory, this can only be true if there is an unramified extension of the field of pth roots of unity by a cyclic extension of degree p which behaves in the specified way under the action of Σ; Ribet proves this by actually constructing such an extension using methods in the theory of modular forms. A more elementary proof of Ribet's converse to Herbrand's theorem, a consequence of the theory of Euler systems, can be found in Washington's book.
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Classical images of separate particles fail to model known charge distributions in very small nuclei. A more accurate image is that the spatial distribution of nucleons in a helium nucleus is much closer to the helium electron cloud shown here, although on a far smaller scale, than to the fanciful nucleus image. The nucleus of an atom consists of neutrons and protons, which in turn are the manifestation of more elementary particles, called quarks, that are held in association by the nuclear strong force in certain stable combinations of hadrons, called baryons. The nuclear strong force extends far enough from each baryon so as to bind the neutrons and protons together against the repulsive electrical force between the positively charged protons.
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1959 also saw the merging of the Franklin Local School District and the Brady Lake School District into the Kent City Schools, which added two more elementary schools: Franklin Elementary and Emma Williard (Brady Lake) Elementary. The 1960s saw the last elementary schools built in Kent with the opening of Holden Elementary in 1965 on the city's south side and Walls Elementary on the east side in 1966. Most of South School was razed in 1966 following the completion of Holden except for the building's gym which was leased to the Kent Parks and Recreation Department and used as the Kent Recreation Center. Enrollment growth through the 1960s and into the 1970s resulted in additions at Walls School, Davey Junior High School, and Roosevelt High School.
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In the 1960s and '70s, the ever-growing list of strongly interacting particles -- mesons and baryons -- made it clear to physicists that none of these particles is elementary. Geoffrey Chew and others went so far as to question the distinction between composite and elementary particles, advocating a "nuclear democracy" in which the idea that some particles were more elementary than others was discarded. Instead, they sought to derive as much information as possible about the strong interaction from plausible assumptions about the S-matrix, which describes what happens when particles of any sort collide, an approach advocated by Werner Heisenberg two decades earlier. The reason the program had any hope of success was because of crossing, the principle that the forces between particles are determined by particle exchange.
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The Ore condition, which (if true) allows a ring of fractions to be defined, and the Ore extension, a non- commutative analogue of rings of polynomials, are part of this work. In more elementary number theory, Ore's harmonic numbers are the numbers whose divisors have an integer harmonic mean. As a teacher, Ore is notable for teaching mathematics to two doctoral students who would make contributions to science and mathematics, Grace Hopper, who would eventually become a United States rear admiral and computer scientist, who was a pioneer in the development of the first computers, and Marshall Hall, Jr., an American mathematician who did important research in group theory and combinatorics. In 1930 the Collected Works of Richard Dedekind were published in three volumes, jointly edited by Ore and Emmy Noether.
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The computation of the homology groups of the product of two topological spaces involves the tensor product; but only in the simplest cases, such as a torus, is it directly calculated in that fashion (see Künneth theorem). The topological phenomena were subtle enough to need better foundational concepts; technically speaking, the Tor functors had to be defined. The material to organise was quite extensive, including also ideas going back to Hermann Grassmann, the ideas from the theory of differential forms that had led to de Rham cohomology, as well as more elementary ideas such as the wedge product that generalises the cross product. The resulting rather severe write-up of the topic (by Bourbaki) entirely rejected one approach in vector calculus (the quaternion route, that is, in the general case, the relation with Lie groups).
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For nilpotent groups the theory simplifies much from the general case, and stays similar to the case of Abelian groups. All lattices in a nilpotent Lie group are uniform, and if N is a connected simply connected nilpotent Lie group (equivalently it does not contain a nontrivial compact subgroup) then a discrete subgroup is a lattice if and only if it is not contained in a proper connected subgroup (this generalises the fact that a discrete subgroup in a vector space is a lattice if and only if it spans the vector space). A nilpotent Lie group contains a lattice if and only if it can be defined over the rationals, that is if and only if its structure constants are rational numbers. More precisely, in a nilpotent group satisfying this condition lattices correspond via the exponential map to lattices (in the more elementary sense of Lattice (group)) in the Lie algebra.
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All parts of the nervous system have the same elementary property; sensibility. Thus sensibility belongs as much to the lower centres of the spinal cord as to the brain, the former, more elementary, form contributing to the subconscious region of mental life, while the higher functions of the nervous system, which make up our conscious mental life, are more complex modifications of this fundamental property of nerve substance. The nervous organism acts as a whole, particular mental operations cannot be referred to definite regions of the brain, and the hypothesis of nervous activity by an isolated pathway from one nerve cell to another is altogether illusory. By insisting on the complete coincidence between the regions of nerve action and sentience, that these are but different aspects of one thing, he was able to attack the doctrine of animal and human automatism which affirms that feeling or consciousness is merely an incidental concomitant of nerve action in no way essential to the chain of physical events.
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One can define a set of disjoint planes in this way from the circles of the packing, and a second set of disjoint planes defined by the circles that circumscribe each triangular gap between three of the circles in the packing. These two sets of planes meet at right angles, and form the generators of a reflection group whose fundamental domain can be viewed as a hyperbolic manifold. By Mostow rigidity, the hyperbolic structure of this domain is uniquely determined, up to isometry of the hyperbolic space; these isometries, when viewed in terms of their actions on the Euclidean plane on the boundary of the half-plane model, translate to Möbius transformations. There is also a more elementary proof of the same uniqueness property, based on the maximum principle and on the observation that, in the triangle connecting the centers of three mutually tangent circles, the angle formed at the center of one of the circles is monotone decreasing in its radius and monotone increasing in the two other radii.
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