|
Math 55
Math 55 is a two-semester long freshman undergraduate mathematics course at Harvard University founded by Lynn Loomis and Shlomo Sternberg. The official titles of the course are Honors Abstract Algebra (Math 55a) and Honors Real and Complex Analysis (Math 55b). Previously, the official title was Honors Advanced Calculus and Linear Algebra. Description In the past, Harvard University's Department of Mathematics had described Math 55 as "probably the most difficult undergraduate math class in the country." But Math 55 lecturer for 2022 Professor Denis Auroux clarified that "if you’re reasonably good at math, you love it, and you have lots of time to devote to it, then Math 55 is completely fine for you." Formerly, students would begin the year in Math 25 (which was created in 1983 as a lower-level Math 55) and, after three weeks of point-set topology and special topics (for instance, in 1994, ''p''-adic analysis was taught by Wilfried Schmid), students would take a quiz. As of 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Freshman
A freshman, fresher, first year, or frosh, is a person in the first year at an educational institution, usually a secondary school or at the college and university level, but also in other forms of post-secondary educational institutions. Arab world In much of the Arab world, a first-year is called a "Ebtidae" (Pl. Mubtadeen), which is Arabic for "beginner". Brazil In Brazil, students that pass the vestibulares and begin studying in a college or university are called "calouros" or more informally "bixos" ("bixetes" for girls), an alternate spelling of "bicho", which means "animal" (although commonly used to refer to bugs). Calouros are often subject to hazing, which is known as "trote" (lit. "prank") there. The first known hazing episode in Brazil happened in 1831 at the Law School of Olinda and resulted in the death of a student. In 1999, a Chinese Brazilian calouro of the University of São Paulo Medicine School named Edison Tsung Chi Hsueh was found dead at the institutio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Halmos
Paul Richard Halmos ( hu, Halmos Pál; March 3, 1916 – October 2, 2006) was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator theory, ergodic theory, and functional analysis (in particular, Hilbert spaces). He was also recognized as a great mathematical expositor. He has been described as one of The Martians. Early life and education Born in Hungary into a Jewish family, Halmos arrived in the U.S. at 13 years of age. He obtained his B.A. from the University of Illinois, majoring in mathematics, but fulfilling the requirements for both a math and philosophy degree. He took only three years to obtain the degree, and was only 19 when he graduated. He then began a Ph.D. in philosophy, still at the Champaign–Urbana campus; but, after failing his masters' oral exams, he shifted to mathematics, graduating in 1938. Joseph L. Doob supervised his dissertation, titled ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheldon Axler
Sheldon Jay Axler (born November 6, 1949, Philadelphia) is an American mathematician and textbook author. He is a professor of mathematics and the Dean of the College of Science and Engineering at San Francisco State University. He graduated from Miami Palmetto Senior High School in Miami, Florida in 1967. He obtained his AB in mathematics with highest honors at Princeton University (1971) and his PhD in mathematics, under professor Donald Sarason, from the University of California, Berkeley, with the dissertation "Subalgebras of L^" in 1975. As a postdoc, he was a C. L. E. Moore instructor at the Massachusetts Institute of Technology. He taught for many years and became a full professor at Michigan State University. In 1997, Axler moved to San Francisco State University, where he became the chair of the Mathematics Department. Axler received the Lester R. Ford Award for expository writing in 1996 from the Mathematical Association of America for a paper titled "Down with D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus On Manifolds (book)
''Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus'' (1965) by Michael Spivak is a brief, rigorous, and modern textbook of multivariable calculus, differential forms, and integration on manifolds for advanced undergraduates. Description ''Calculus on Manifolds'' is a brief monograph on the theory of vector-valued functions of several real variables (''f'' : R''n''→R''m'') and differentiable manifolds in Euclidean space. In addition to extending the concepts of differentiation (including the inverse and implicit function theorems) and Riemann integration (including Fubini's theorem) to functions of several variables, the book treats the classical theorems of vector calculus, including those of Cauchy–Green, Ostrogradsky–Gauss (divergence theorem), and Kelvin–Stokes, in the language of differential forms on differentiable manifolds embedded in Euclidean space, and as corollaries of the generalized Stokes theorem on manifolds-with-bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Spivak
Michael David Spivak (25 May 19401 October 2020)Biographical sketch in Notices of the AMS', Vol. 32, 1985, p. 576. was an American mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. Spivak was the author of the five-volume ''A Comprehensive Introduction to Differential Geometry''. Biography Spivak was born in Queens, New York (state), New York. He received an Bachelor of Arts, A.B. from Harvard University in 1960, while in 1964 he received a Doctor of Philosophy, Ph.D. from Princeton University under the supervision of John Milnor, with thesis ''On Spaces Satisfying Poincaré Duality''. In 1985 Spivak received the Leroy P. Steele Prize. Spivak lectured on elementary physics. Spivak's book, ''Physics for Mathematicians: Mechanics I'' (published December 6, 2010), contains the material that these lectures stemmed from and more. Spivak was also the designer of the MathTime Professional 2 fonts (which are wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walter Rudin
Walter may refer to: People * Walter (name), both a surname and a given name * Little Walter, American blues harmonica player Marion Walter Jacobs (1930–1968) * Gunther (wrestler), Austrian professional wrestler and trainer Walter Hahn (born 1987), who previously wrestled as "Walter" * Walter, standard author abbreviation for Thomas Walter (botanist) ( – 1789) Companies * American Chocolate, later called Walter, an American automobile manufactured from 1902 to 1906 * Walter Energy, a metallurgical coal producer for the global steel industry * Walter Aircraft Engines, Czech manufacturer of aero-engines Films and television * ''Walter'' (1982 film), a British television drama film * Walter Vetrivel, a 1993 Tamil crime drama film * ''Walter'' (2014 film), a British television crime drama * ''Walter'' (2015 film), an American comedy-drama film * ''Walter'' (2020 film), an Indian crime drama film * ''W*A*L*T*E*R'', a 1984 pilot for a spin-off of the TV series ''M*A*S*H'' * ''W ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiable Manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normed Vector Spaces
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real (physical) world. A norm is a real-valued function defined on the vector space that is commonly denoted x\mapsto \, x\, , and has the following properties: #It is nonnegative, meaning that \, x\, \geq 0 for every vector x. #It is positive on nonzero vectors, that is, \, x\, = 0 \text x = 0. # For every vector x, and every scalar \alpha, \, \alpha x\, = , \alpha, \, \, x\, . # The triangle inequality holds; that is, for every vectors x and y, \, x+y\, \leq \, x\, + \, y\, . A norm induces a distance, called its , by the formula d(x,y) = \, y-x\, . which makes any normed vector space into a metric space and a topological vector space. If this metric space is complete then the normed space is a Banach space. Every normed vec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Maturity
In mathematics, mathematical maturity is an informal term often used to refer to the quality of having a general understanding and mastery of the way mathematicians operate and communicate. It pertains to a mixture of mathematical experience and insight that cannot be directly taught. Instead, it comes from repeated exposure to mathematical concepts. It is a gauge of mathematics students' erudition in mathematical structure In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additional ...s and methods, and can overlap with other related concepts such as mathematical intuition and mathematical competence. The topic is occasionally also addressed in literature in its own right. Definitions Mathematical maturity has been defined in several different ways by various authors, and is often tied to ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Rham Cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties. On any smooth manifold, every exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of "holes" in the manifold, and the de Rham cohomology groups comprise a set of topological invariants of smooth manifolds that precisely quantify this relationship. Definition The de Rham complex is the cochain complex of differential forms on some smooth manifold , with the exterior derivative as the differential: :0 \to \Omega^0(M)\ \stackrel\ \Omega^1(M)\ \stackrel\ \Omega^2(M)\ \stackrel\ \Omega^3(M) \to \cd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism compos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.jpg "Click for more on -> Freshman")
.jpg "Click for more on -> Sheldon Axler")
