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Topic Review
(ε, δ)-Definition of Limit
In calculus, the (ε, δ)-definition of limit ("epsilon–delta definition of limit") is a formalization of the notion of limit. The concept is due to Augustin-Louis Cauchy, who never gave a formal (ε, δ) definition of limit in his Cours d'Analyse, but occasionally used ε, δ arguments in proofs. It was first given as a formal definition by Bernard Bolzano in 1817, and the definitive modern statement was ultimately provided by Karl Weierstrass. It provides rigor to the following informal notion: the dependent expression f(x) approaches the value L as the variable x approaches the value c if f(x) can be made as close as desired to L by taking x sufficiently close to c.
  • 1.5K
  • 22 Nov 2022
Biography
Alexander Dewdney
Alexander Keewatin Dewdney (born August 5, 1941) is a Canadian mathematician, computer scientist, author, filmmaker, and conspiracy theorist. Dewdney is the son of Canadian artist and author Selwyn Dewdney, and brother of poet Christopher Dewdney. He was born in London, Ontario. In his student days, Dewdney made a number of influential experimental films, including Malanga, on the poet Gerald
  • 482
  • 12 Dec 2022
Biography
Alexey Parshin
Aleksei (or Alexei) Nikolaevich Parshin (Russian: Алексей Николаевич Паршин; born 7 November 1942 in Sverdlovsk) is a Russian mathematician, specializing in number theory and algebraic geometry. Parshin graduated in 1964 from the Faculty of Mathematics and Mechanics of Moscow State University and then enrolled as a graduate student at the Steklov Institute of Mathematic
  • 415
  • 05 Dec 2022
Topic Review
Areas of Mathematics
Mathematics encompasses a growing variety and depth of subjects over its history, and comprehension of it requires a system to categorize and organize these various subjects into more general areas of mathematics. A number of different classification schemes have arisen, and though they share some similarities, there are differences due in part to the different purposes they serve. A traditional division of mathematics is into pure mathematics; mathematics studied for its intrinsic interest, and applied mathematics; the mathematics that can be directly applied to real-world problems. This division is not always clear and many subjects have been developed as pure mathematics to find unexpected applications later on. Broad divisions, such as discrete mathematics, computational mathematics and so on have emerged more recently. An ideal system of classification permits adding new areas into the organization of previous knowledge, and fitting surprising discoveries and unexpected interactions into the outline. For example, the Langlands program has found unexpected connections between areas previously thought unconnected, at least Galois groups, Riemann surfaces and number theory.
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  • 29 Sep 2022
Biography
Beatrice Aitchison
Beatrice Aitchison (July 18, 1908 – September 22, 1997) was an American mathematician, statistician, and transportation economist who directed the Transport Economics Division of the United States Department of Commerce,[1] and later became the top woman in the United States Postal Service and the first policy-level appointee there.[2] Aitchison's mother was a musician and her father, Clyde
  • 420
  • 01 Dec 2022
Topic Review
Birkhoff's Representation Theorem
In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. The theorem can be interpreted as providing a one-to-one correspondence between distributive lattices and partial orders, between quasi-ordinal knowledge spaces and preorders, or between finite topological spaces and preorders. It is named after Garrett Birkhoff, who published a proof of it in 1937. The name “Birkhoff's representation theorem” has also been applied to two other results of Birkhoff, one from 1935 on the representation of Boolean algebras as families of sets closed under union, intersection, and complement (so-called fields of sets, closely related to the rings of sets used by Birkhoff to represent distributive lattices), and Birkhoff's HSP theorem representing algebras as products of irreducible algebras. Birkhoff's representation theorem has also been called the fundamental theorem for finite distributive lattices.
  • 390
  • 10 Oct 2022
Topic Review
Boundary Conditions, Nonequilibrium Thermodynamics Equations
In this entry, we present a systematical review on boundary conditions (BCs) for partial differential equations (PDEs) from nonequilibrium thermodynamics. From a stability point of view, such PDEs should satisfy the structural stability condition. In particular, they constitute hyperbolic systems, for which the generalized Kreiss condition (UKC) is a sufficient and essentially necessary condition for the well-posedness of the corresponding models (PDEs with BCs).
  • 426
  • 08 Oct 2021
Biography
Brendan McKay
Brendan Damien McKay (born 26 October 1951 in Melbourne, Australia ) is an Emeritus Professor in the Research School of Computer Science at the Australian National University (ANU). He has published extensively in combinatorics. McKay received a Ph.D. in mathematics from the University of Melbourne in 1980, and was appointed Assistant Professor of Computer Science at Vanderbilt University, Nash
  • 355
  • 13 Dec 2022
Topic Review
Complement (Set Theory)
In set theory, the complement of a set A, often denoted by Ac (or A′), is the set of elements not in A. When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set U, the absolute complement of A is the set of elements in U that are not in A. The relative complement of A with respect to a set B, also termed the set difference of B and A, written [math]\displaystyle{ B \setminus A, }[/math] is the set of elements in B that are not in A.
  • 12.2K
  • 28 Nov 2022
Topic Review
Criticism of Non-Standard Analysis
Non-standard analysis and its offshoot, non-standard calculus, have been criticized by several authors, notably Errett Bishop, Paul Halmos, and Alain Connes. These criticisms are analyzed below.
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  • 04 Nov 2022
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