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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
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.
  • 2.2K
  • 29 Sep 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.
  • 400
  • 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).
  • 433
  • 08 Oct 2021
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.
  • 12.4K
  • 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.
  • 383
  • 04 Nov 2022
Topic Review
Deformation Theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or vector of small quantities. The infinitesimal conditions are therefore the result of applying the approach of differential calculus to solving a problem with constraints. One might think, in analogy, of a structure that is not completely rigid, and that deforms slightly to accommodate forces applied from the outside; this explains the name. Some characteristic phenomena are: the derivation of first-order equations by treating the ε quantities as having negligible squares; the possibility of isolated solutions, in that varying a solution may not be possible, or does not bring anything new; and the question of whether the infinitesimal constraints actually 'integrate', so that their solution does provide small variations. In some form these considerations have a history of centuries in mathematics, but also in physics and engineering. For example, in the geometry of numbers a class of results called isolation theorems was recognised, with the topological interpretation of an open orbit (of a group action) around a given solution. Perturbation theory also looks at deformations, in general of operators.
  • 578
  • 02 Dec 2022
Topic Review
Design and Experience of Mobile Applications
With the tremendous growth in mobile phones, mobile application development is an important emerging arena. Moreover, various applications fail to serve the purpose of getting the attention of the intended users, which is determined by their User Interface (UI) and User Experience (UX). As a result, developers often find it challenging to meet the users’ expectations. Various aspects of design and the experience of mobile applications using UX/UI are explored. 
  • 721
  • 20 Jul 2022
Topic Review
Dynamic Rectangle
A dynamic rectangle is a right-angled, four-sided figure (a rectangle) with dynamic symmetry, which in this case, means that aspect ratio (width divided by height) is a distinguished value in dynamic symmetry, a proportioning system and natural design methodology described in Jay Hambidge's books. These dynamic rectangles begin with a square, which is extended (using a series of arcs and cross points) to form the desired figure, which can be the golden rectangle (1 : 1.618...), the 2:3 rectangle, the double square (1:2), or a root rectangle (1:√φ, 1:√2, 1:√3, 1:√5, etc.).
  • 277
  • 20 Oct 2022
Topic Review
Euler–Mascheroni Constant
The Euler–Mascheroni constant (also called Euler's constant) is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma (γ). It is defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by [math]\displaystyle{ \log: }[/math] Here, [math]\displaystyle{ \lfloor x\rfloor }[/math] represents the floor function. The numerical value of the Euler–Mascheroni constant, to 50 decimal places, is: 0.57721566490153286060651209008240243104215933593992... 
  • 1.7K
  • 02 Dec 2022
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