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Richard L. Bishop
Richard Lawrence Bishop (August 12, 1931 – December 18, 2019) was an American mathematician who specialized in differential geometry and taught at the University of Illinois at Urbana–Champaign. Bishop went to Case Institute of Technology as an undergraduate, earning a B.S. in 1954. Next he earned his Ph.D. from the Massachusetts Institute of Technology in 1959, and immediately joined the UIUC faculty, where he stayed until his retirement in 1997. His thesis, ''On Imbeddings and Holonomy'', was supervised by Isadore Singer. At UIUC, his doctoral students included future UIUC colleague Stephanie B. Alexander. He is the author of ''Geometry of Manifolds'' (with Richard J. Crittenden, AMS Chelsea Publishing, 1964, translated into Russian 1967 and reprinted 2001) and ''Tensor Analysis on Manifolds'' (with Samuel I. Goldberg, Macmillan, 1968, reprinted by Dover Books on Mathematics, 1980). In 2013, Bishop became one of the inaugural fellows of the American Mathematical Society. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frenet–Serret Formulas
In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space \mathbb^, or the geometric properties of the curve itself irrespective of any motion. More specifically, the formulas describe the derivatives of the so-called tangent, normal, and binormal unit vectors in terms of each other. The formulas are named after the two French mathematicians who independently discovered them: Jean Frédéric Frenet, in his thesis of 1847, and Joseph Alfred Serret, in 1851. Vector notation and linear algebra currently used to write these formulas were not yet available at the time of their discovery. The tangent, normal, and binormal unit vectors, often called T, N, and B, or collectively the Frenet–Serret frame or TNB frame, together form an orthonormal basis spanning \mathbb^ and are defined as follows: * T is the unit vector tangent to the curve, pointing in the direction o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Illinois Urbana-Champaign Faculty
A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States, the designation is reserved for colleges that have a graduate school. The word ''university'' is derived from the Latin ''universitas magistrorum et scholarium'', which roughly means "community of teachers and scholars". The first universities were created in Europe by Catholic Church monks. The University of Bologna (''Università di Bologna''), founded in 1088, is the first university in the sense of: *Being a high degree-awarding institute. *Having independence from the ecclesiastic schools, although conducted by both clergy and non-clergy. *Using the word ''universitas'' (which was coined at its foundation). *Issuing secular and non-secular degrees: grammar, rhetoric, logic, theology, canon law, notarial law.Hunt Janin: "The university i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Massachusetts Institute Of Technology Alumni
Massachusetts (Massachusett: ''Muhsachuweesut Massachusett_writing_systems.html" ;"title="nowiki/> məhswatʃəwiːsət.html" ;"title="Massachusett writing systems">məhswatʃəwiːsət">Massachusett writing systems">məhswatʃəwiːsət'' English: , ), officially the Commonwealth of Massachusetts, is the most populous state in the New England region of the Northeastern United States. It borders on the Atlantic Ocean and Gulf of Maine to the east, Connecticut and Rhode Island to the south, New Hampshire and Vermont to the north, and New York to the west. The state's capital and most populous city, as well as its cultural and financial center, is Boston. Massachusetts is also home to the urban core of Greater Boston, the largest metropolitan area in New England and a region profoundly influential upon American history, academia, and the research economy. Originally dependent on agriculture, fishing, and trade. Massachusetts was transformed into a manufacturing center during th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Case Western Reserve University Alumni
Case or CASE may refer to: Containers * Case (goods), a package of related merchandise * Cartridge case or casing, a firearm cartridge component * Bookcase, a piece of furniture used to store books * Briefcase or attaché case, a narrow box to carry paperwork * Computer case, the enclosure for a PC's main components * Keep case, DVD or CD packaging * Pencil case * Phone case, protective or vanity accessory for mobile phones ** Battery case * Road case or flight case, for fragile equipment in transit * Shipping container or packing case * Suitcase, a large luggage box * Type case, a compartmentalized wooden box for letterpress typesetting Places * Case, Laclede County, Missouri * Case, Warren County, Missouri * Case River, a Kabika tributary in Ontario, Canada * Case Township, Michigan * Case del Conte, Italy People * Case (name), people with the surname (or given name) * Case (singer), American R&B singer-songwriter and producer (Case Woodard) Arts, entertainment, and med ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
21st-century American Mathematicians
The 1st century was the century spanning AD 1 ( I) through AD 100 ( C) according to the Julian calendar. It is often written as the or to distinguish it from the 1st century BC (or BCE) which preceded it. The 1st century is considered part of the Classical era, epoch, or historical period. The 1st century also saw the appearance of Christianity. During this period, Europe, North Africa and the Near East fell under increasing domination by the Roman Empire, which continued expanding, most notably conquering Britain under the emperor Claudius ( AD 43). The reforms introduced by Augustus during his long reign stabilized the empire after the turmoil of the previous century's civil wars. Later in the century the Julio-Claudian dynasty, which had been founded by Augustus, came to an end with the suicide of Nero in AD 68. There followed the famous Year of Four Emperors, a brief period of civil war and instability, which was finally brought to an end by Vespasian, ninth Roman empero ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2019 Deaths
This is a list of deaths of notable people, organised by year. New deaths articles are added to their respective month (e.g., Deaths in ) and then linked here. 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991 1990 1989 1988 1987 See also * Lists of deaths by day The following pages, corresponding to the Gregorian calendar, list the historical events, births, deaths, and holidays and observances of the specified day of the year: Footnotes See also * Leap year * List of calendars * List of non-standard ... * Deaths by year {{DEFAULTSORT:deaths by year ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1931 Births
Events January * January 2 – South Dakota native Ernest Lawrence invents the cyclotron, used to accelerate particles to study nuclear physics. * January 4 – German pilot Elly Beinhorn begins her flight to Africa. * January 22 – Sir Isaac Isaacs is sworn in as the first Australian-born Governor-General of Australia. * January 25 – Mohandas Gandhi is again released from imprisonment in India. * January 27 – Pierre Laval forms a government in France. February * February 4 – Soviet leader Joseph Stalin gives a speech calling for rapid industrialization, arguing that only strong industrialized countries will win wars, while "weak" nations are "beaten". Stalin states: "We are fifty or a hundred years behind the advanced countries. We must make good this distance in ten years. Either we do it, or they will crush us." The first five-year plan in the Soviet Union is intensified, for the industrialization and collectivization of agriculture. * February 10 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Warped Geometry
In mathematics and physics, in particular differential geometry and general relativity, a warped geometry is a Riemannian or Lorentzian manifold whose metric tensor can be written in form :ds^2 = g_(y) \, dy^a \, dy^b + f(y) g_(x) \, dx^i \, dx^j. The geometry almost decomposes into a Cartesian product of the ''y'' geometry and the ''x'' geometry – except that the ''x'' part is warped, i.e. it is rescaled by a scalar function of the other coordinates ''y''. For this reason, the metric of a warped geometry is often called a warped product metric. Warped geometries are useful in that separation of variables can be used when solving partial differential equations over them. Examples Warped geometries acquire their full meaning when we substitute the variable ''y'' for ''t'', time and ''x'', for ''s'', space. Then the ''f''(''y'') factor of the spatial dimension becomes the effect of time that in words of Einstein "curves space". How it curves space will define one or other sol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sectional Curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a point ''p'' of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σ''p'' as a tangent plane at ''p'', obtained from geodesics which start at ''p'' in the directions of σ''p'' (in other words, the image of σ''p'' under the exponential map at ''p''). The sectional curvature is a real-valued function on the 2-Grassmannian fiber bundle, bundle over the manifold. The sectional curvature determines the Riemann curvature tensor, curvature tensor completely. Definition Given a Riemannian manifold and two linearly independent tangent vectors at the same point, ''u'' and ''v'', we can define :K(u,v)= Here ''R'' is the Riemann curvature tensor, defined here by the convention R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barrett O'Neill
Barrett O'Neill (1924– 16 June 2011) was an American mathematician. He is known for contributions to differential geometry, including two widely-used textbooks on its foundational theory. He was the author of eighteen research articles, the last of which was published in 1973. He received his Ph.D. in mathematics in 1951 from the Massachusetts Institute of Technology. His doctoral advisor was Witold Hurewicz. His dissertation thesis was titled ''Some Fixed Point Theorems'' He has worked as a professor of mathematics at UCLA, where he supervised the PhDs of eight doctoral students. He made a foundational contribution to the theory of Riemannian submersions, showing how geometric quantities on the total space and on the base are related to one another. "O'Neill's formula" refers to the relation between the sectional curvatures. O'Neill's calculations simplified earlier work by other authors, and have become standard textbook material.Peter Petersen. ''Riemannian geometry.'' Third ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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