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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.
  • 15.3K
  • 28 Nov 2022
Topic Review
Fresh Fruit Supply Chain Optimization
The fresh fruit chain has been recognized as a very important and strategic part of the economic development of many countries. The planning framework for production and distribution is highly complex as a result. Mathematical models have been developed over the decades to deal with this complexity. This review focuses on the recent progress in mathematically based decision making to account for uncertainties in the fresh fruit supply chain
  • 5.4K
  • 27 Jul 2021
Topic Review
Two-Dimensional Space
Two-dimensional space (also known as 2D space, 2-space, or bi-dimensional space) is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point). The set [math]\displaystyle{ \mathbb{R}^2 }[/math] of pairs of real numbers with appropriate structure often serves as the canonical example of a two-dimensional Euclidean space. For a generalization of the concept, see dimension. Two-dimensional space can be seen as a projection of the physical universe onto a plane. Usually, it is thought of as a Euclidean space and the two dimensions are called length and width.
  • 5.3K
  • 28 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.
  • 3.4K
  • 29 Sep 2022
Topic Review
Incircle and Excircles of a Triangle
In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides. The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle (at vertex [math]\displaystyle{ A }[/math], for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex [math]\displaystyle{ A }[/math], or the excenter of [math]\displaystyle{ A }[/math]. Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system.:p. 182 All regular polygons have incircles tangent to all sides, but not all polygons do; those that do are tangential polygons. 
  • 3.2K
  • 11 Nov 2022
Biography
James Harris Simons
James Harris "Jim" Simons (/ˈsaɪmənz/; born April 25, 1938) is an American mathematician, billionaire hedge fund manager, and philanthropist. He is known as a quantitative investor and in 1982 founded Renaissance Technologies, a private hedge fund based in New York City . Although Simons retired from the fund in 2009, he remains its non-executive chairman and adviser.[1] He is also known for
  • 2.9K
  • 15 Nov 2022
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.
  • 2.6K
  • 22 Nov 2022
Topic Review
Sine
In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle (the hypotenuse). For an angle [math]\displaystyle{ x }[/math], the sine function is denoted simply as [math]\displaystyle{ \sin x }[/math]. More generally, the definition of sine (and other trigonometric functions) can be extended to any real value in terms of the length of a certain line segment in a unit circle. More modern definitions express the sine as an infinite series, or as the solution of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. The sine function is commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. The function sine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic, and then from Arabic to Latin. The word "sine" (Latin "sinus") comes from a Latin mistranslation by Robert of Chester of the Arabic jiba, which is a transliteration of the Sanskrit word for half the chord, jya-ardha.
  • 2.5K
  • 11 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... 
  • 2.4K
  • 02 Dec 2022
Topic Review
μ-Recursive Function
In mathematical logic and computer science, the general recursive functions (often shortened to recursive functions) or μ-recursive functions are a class of partial functions from natural numbers to natural numbers that are "computable" in an intuitive sense. In computability theory, it is shown that the μ-recursive functions are precisely the functions that can be computed by Turing machines(this is one of the theorems that supports the Church–Turing thesis). The μ-recursive functions are closely related to primitive recursive functions, and their inductive definition (below) builds upon that of the primitive recursive functions. However, not every μ-recursive function is a primitive recursive function—the most famous example is the Ackermann function. Other equivalent classes of functions are the λ-recursive functions and the functions that can be computed by Markov algorithms. The subset of all total recursive functions with values in {0,1} is known in computational complexity theory as the complexity class R.
  • 2.4K
  • 01 Dec 2022
Topic Review
Mathematics of Sudoku
The class of Sudoku puzzles consists of a partially completed row-column grid of cells partitioned into N regions each of size N cells, to be filled in ("solved") using a prescribed set of N distinct symbols (typically the numbers {1, ..., N}), so that each row, column and region contains exactly one of each element of the set. The properties of Sudoku puzzles and their solutions can be investigated using mathematics and algorithms.
  • 2.4K
  • 12 Oct 2022
Topic Review Peer Reviewed
Mechanics and Mathematics in Ancient Greece
This entry presents an overview on how mechanics in Greece was linked to geometry. In ancient Greece, mechanics was about lifting heavy bodies, and mathematics almost coincided with geometry. Mathematics interconnected with mechanics at least from the 5th century BCE and became dominant in the Hellenistic period. The contributions by thinkers such as Aristotle, Euclid, and Archytas on fundamental problems such as that of the lever are sketched. This entry can be the starting point for a deeper investigation on the connections of the two disciplines through the ages until our present day.
  • 2.3K
  • 13 Apr 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].
  • 1.9K
  • 10 Oct 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.
  • 1.9K
  • 26 Oct 2022
Biography
Noor Muhammad
Noor Muhammad (15 April 1951 – 12 April 2004) was a notable Pakistani mathematician and mathematics writer who made an original contribution to the fields of C*-algebra, symmetric topology and pure mathematics. Noor Muhammad joined International Centre for Theoretical Physics on the request of prominent theoretical physics and nobel laureate in Physics Dr. Prof. Abdus Salam. As a post-doct
  • 1.7K
  • 30 Dec 2022
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.
  • 1.7K
  • 17 Oct 2022
Biography
Mikhail Leonidovich Gromov
Mikhail Leonidovich Gromov (also Mikhael Gromov, Michael Gromov or Mischa Gromov; Russian: Михаи́л Леони́дович Гро́мов; born 23 December 1943) is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of IHÉS in France and a Professor of Mathematics at New York University. Gromov has won several prizes, includ
  • 1.4K
  • 29 Nov 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.
  • 1.3K
  • 26 Sep 2021
Biography
Misir Mardanov
Misir Jumayil oglu Mardanov (Azerbaijani: Misir Mərdanov Cumayıl oğlu; born October 3, 1946) is an Azerbaijani academic and politician who is director of the institute of Mathematics and Mechanics of the National Academy of Sciences of Azerbaijan, former Minister of Education of Azerbaijan Republic, Corr.-member of ANAS, doctor of physico-mathematical sciences, professor. Misir Mardanov wa
  • 1.2K
  • 26 Dec 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.
  • 1.1K
  • 08 Nov 2022
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