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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
Localization of a Ring
In commutative algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units (invertible elements) in R*. Further one wants R* to be the 'best possible' or 'most general' way to do this—in the usual fashion this should be expressed by a universal property. The localization of R by S is usually denoted by S −1R; however other notations are used in some important special cases. If S is the set of the non zero elements of an integral domain, then the localization is the field of fractions and thus usually denoted Frac(R). If S is the complement of a prime ideal I the localization is denoted by RI, and Rf is used to denote the localization by the powers of an element f. The two latter cases are fundamental in algebraic geometry and scheme theory. In particular the definition of an affine scheme is based on the properties of these two kinds of localizations.
  • 1.4K
  • 26 Oct 2022
Topic Review
First-Order Arithmetic
In set theory and mathematical logic, first-order arithmetic is a collection of axiomatic systems formalising natural and subsets of the natural numbers. It is a choice for axiomatic theory as a basis for many mathematics, but not all. The primary first-order axiom is Peano arithmetic, created by Giuseppe Peano: Peano arithmetic has a proof-theoretic ordinal of [math]\displaystyle{ \varepsilon_0 = \varphi(1, 0) = \psi_0(\Omega) }[/math].
  • 1.4K
  • 10 Oct 2022
Topic Review
Journal Axioms
Axioms (ISSN 2075-1680) is an international, peer-reviewed, open access journal of mathematics, mathematical logic and mathematical physics, published quarterly online by MDPI. It's now indexed within SCIE (Web of Science), Scopus, dblp, and other databases.
  • 1.0K
  • 26 Sep 2021
Topic Review
Symmetric Difference
In mathematics, the symmetric difference, also known as the disjunctive union, of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by or or For example, the symmetric difference of the sets [math]\displaystyle{ \{1,2,3\} }[/math] and [math]\displaystyle{ \{3,4\} }[/math] is [math]\displaystyle{ \{1,2,4\} }[/math]. The power set of any set becomes an abelian group under the operation of symmetric difference, with the empty set as the neutral element of the group and every element in this group being its own inverse. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring and intersection as the multiplication of the ring.
  • 946
  • 17 Oct 2022
Topic Review
Hybrid Number
A hybrid number is a generalization of complex numbers [math]\displaystyle{ \left(a+\mathbf{i}b, \mathbf{i}^{2}=-1\right) }[/math], split-complex numbers (or "hyperbolic number") [math]\displaystyle{ \left(a+\mathbf{h}b, \mathbf{h}^2=1\right) }[/math] and dual numbers [math]\displaystyle{ \left(a+\mathbf{\varepsilon} b, \mathbf{\varepsilon}^2 = 0\right) }[/math]. Hybrid numbers form a noncommutative ring. Complex, hyperbolic and dual numbers are well known two-dimensional number systems. It is well known that, the set of complex numbers, hyperbolic numbers and dual numbers are respectively. The algebra of hybrid numbers is a noncommutative algebra which unifies all three number systems calls them hybrid numbers., , . A hybrid number is a number created with any combination of the complex, hyperbolic and dual numbers satisfying the relation Because these numbers are a composition of dual, complex and hyperbolic numbers, Ozdemir calls them hybrid numbers . A commutative two-dimensional unital algebra generated by a 2 by 2 matrix is isomorphic to either complex, dual or hyperbolic numbers . Due to the set of hybrid numbers is a two-dimensional commutative algebra spanned by 1 and [math]\displaystyle{ \mathbf{i}b+c\mathbf{\varepsilon }+d\mathbf{h} }[/math], it is isomorphic to one of the complex, dual or hyperbolic numbers. Especially in the last century, a lot of researchers deal with the geometric and physical applications of these numbers. Just as the geometry of the Euclidean plane can be described with complex numbers, the geometry of the Minkowski plane and Galilean plane can be described with hyperbolic numbers. The group of Euclidean rotations SO(2) is isomorphic to the group U(1) of unit complex numbers. The geometrical meaning of multiplying by [math]\displaystyle{ e^{\mathbf{i}\theta}=\cos \theta +\mathbf{i}\sin \theta }[/math] means a rotation of the plane. , . The group of Lorentzian rotations [math]\displaystyle{ SO(1,1) }[/math] is isomorphic to the group of unit spacelike hyperbolic numbers. This rotation can be viewed as hyperbolic rotation. Thus, multiplying by [math]\displaystyle{ e^{\mathbf{h}\theta }=\cosh \theta +\mathbf{h} \sinh \theta }[/math] means a map of hyperbolic numbers into itself which preserves the Lorentzian metric. , , , The Galilean rotations can be interpreted with dual numbers. The concept of a rotation in the dual number plane is equivalent to a vertical shear mapping since [math]\displaystyle{ \left( 1+x\mathbf{\varepsilon }\right) \left( 1+y\mathbf{\varepsilon }\right) =1+\left( x+y\right) \mathbf{\varepsilon } }[/math]. The Euler formula for dual numbers is [math]\displaystyle{ e^{\mathbf{\varepsilon }\theta }=1+\mathbf{\varepsilon }\theta }[/math]. Multiplying by [math]\displaystyle{ e^{\mathbf{\varepsilon \theta }} }[/math] is a map of dual numbers into itself which preserves the Galilean metric. This rotation can be named as parabolic rotation , , , , , . File:Planar rotations.tif In abstract algebra, the complex, the hyperbolic and the dual numbers can be described as the quotient of the polynomial ring [math]\displaystyle{ \mathbb{R}[x] }[/math] by the ideal generated by the polynomials [math]\displaystyle{ x^2+1, }[/math], [math]\displaystyle{ x^2-1 }[/math] and [math]\displaystyle{ x^{2} }[/math] respectively. That is, Matrix represantations of the units [math]\displaystyle{ \mathbf{i} }[/math], [math]\displaystyle{ \mathbf{\varepsilon } }[/math], [math]\displaystyle{ \mathbf{h} }[/math] are respectively.
  • 765
  • 08 Nov 2022
Topic Review
Mersenne Conjectures
In mathematics, the Mersenne conjectures concern the characterization of prime numbers of a form called Mersenne primes, meaning prime numbers that are a power of two minus one.
  • 723
  • 10 Oct 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
Right Triangle
A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry. The side opposite the right angle is called the hypotenuse (side c in the figure). The sides adjacent to the right angle are called legs (or catheti, singular: cathetus). Side a may be identified as the side adjacent to angle B and opposed to (or opposite) angle A, while side b is the side adjacent to angle A and opposed to angle B. If the lengths of all three sides of a right triangle are integers, the triangle is said to be a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple.
  • 704
  • 17 Oct 2022
Topic Review
Technology for Science Education
The COVID-19 confinement has represented both opportunities and losses for education. Rarely before has any other period moved the human spirit into such discipline or submission—depending on one’s personal and emotional points of view. Both extremes have been widely influenced by external factors on each individual’s life path. Education in the sciences and engineering has encountered more issues than other disciplines due to specialized mathematical handwriting, experimental demonstrations, abstract complexity, and lab practices. 
  • 605
  • 22 Sep 2021
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