Muḥammad ibn Mūsā al-Khwārizmī Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī[note 1][pronunciation?] (Arabic: عَبْدَالله مُحَمَّد بِن مُوسَى اَلْخْوَارِزْمِي), earlier transliterated as Algoritmi or Algaurizin, (c. 780 – c. 850) was a Persian[1][5] mathematician, astronomer and geographer during the Abbasid Empire, a scholar in the House of Wisdom in Baghdad. Some words reflect the importance of al-Khwarizmi's contributions to mathematics. "Algebra" is derived from al-jabr, one of the two operations he used to solve quadratic equations. Life He was born in a Persian[1][5] family, and his birthplace is given as Chorasmia[9] by Ibn al-Nadim. Few details of al-Khwārizmī's life are known with certainty. Al-Tabari gave his name as Muhammad ibn Musa al-Khwārizmī al-Majousi al-Katarbali (محمد بن موسى الخوارزميّ المجوسـيّ القطربّـليّ). Regarding al-Khwārizmī's religion, Toomer writes: Another epithet given to him by al-Ṭabarī, "al-Majūsī," would seem to indicate that he was an adherent of the old Zoroastrian religion. D. and J.
Multivalued function In mathematics, a multivalued function (short form: multifunction; other names: many-valued function, set-valued function, set-valued map, multi-valued map, multimap, correspondence, carrier) is a left-total relation; that is, every input is associated with at least one output. Examples[edit] Every real number greater than zero has two square roots. The square roots of 4 are in the set {+2,−2}. The square root of 0 is 0.Each complex number except zero has two square roots, three cube roots, and in general n nth roots. As a consequence, arctan(1) is intuitively related to several values: π/4, 5π/4, −3π/4, and so on. The indefinite integral can be considered as a multivalued function. These are all examples of multivalued functions that come about from non-injective functions. Multivalued functions of a complex variable have branch points. Set-valued analysis[edit] Set-valued analysis is the study of sets in the spirit of mathematical analysis and general topology. History[edit]
List of things named after Leonhard Euler In mathematics and physics, there are a large number of topics named in honor of Leonhard Euler, many of which include their own unique function, equation, formula, identity, number (single or sequence), or other mathematical entity. Unfortunately, many of these entities have been given simple and ambiguous names such as Euler's function, Euler's equation, and Euler's formula. Euler's work touched upon so many fields that he is often the earliest written reference on a given matter. Physicists and mathematicians sometimes jest that, in an effort to avoid naming everything after Euler, discoveries and theorems are named after the "first person after Euler to discover it".[1][2] Euler's conjectures[edit] Euler's equations[edit] Euler's formulas[edit] Euler's functions[edit] Euler's identities[edit] Euler's numbers[edit] Euler's theorems[edit] Euler's laws[edit] Other things named after Euler[edit] Topics by field of study[edit] Selected topics from above, grouped by subject. Graph theory[edit]
Bernoulli polynomials In mathematics, the Bernoulli polynomials occur in the study of many special functions and in particular the Riemann zeta function and the Hurwitz zeta function. This is in large part because they are an Appell sequence, i.e. a Sheffer sequence for the ordinary derivative operator. Unlike orthogonal polynomials, the Bernoulli polynomials are remarkable in that the number of crossings of the x-axis in the unit interval does not go up as the degree of the polynomials goes up. In the limit of large degree, the Bernoulli polynomials, appropriately scaled, approach the sine and cosine functions. Bernoulli polynomials Representations[edit] Explicit formula[edit] for n ≥ 0, where bk are the Bernoulli numbers. Generating functions[edit] The generating function for the Bernoulli polynomials is The generating function for the Euler polynomials is Representation by a differential operator[edit] The Bernoulli polynomials are also given by cf. integrals below. Representation by an integral operator[edit] . and
Contributions of Leonhard Euler to mathematics Mathematical notation[edit] Euler introduced much of the mathematical notation in use today, such as the notation f(x) to describe a function and the modern notation for the trigonometric functions. He was the first to use the letter e for the base of the natural logarithm, now also known as Euler's number. The use of the Greek letter to denote the ratio of a circle's circumference to its diameter was also popularized by Euler (although it did not originate with him).[1] He is also credited for inventing the notation i to denote Complex analysis[edit] A geometric interpretation of Euler's formula Euler made important contributions to complex analysis. , the complex exponential function satisfies This has been called "The most remarkable formula in mathematics " by Richard Feynman. [3] Euler's identity is a special case of this: This identity is particularly remarkable as it involves e, , i, 1, and 0, arguably the five most important constants in mathematics. Analysis[edit] for any positive real
Genetic Algorithm Tutorial Genetic Algorithms in Plain English Introduction The aim of this tutorial is to explain genetic algorithms sufficiently for you to be able to use them in your own projects. This tutorial is designed to be read through twice... so don't worry if little of it makes sense the first time you study it. (A reader, Daniel, has kindly translated this tutorial into German. (Another reader, David Lewin, has translated the tutorial into French. First, a Biology Lesson Every organism has a set of rules, a blueprint so to speak, describing how that organism is built up from the tiny building blocks of life. When two organisms mate they share their genes. Life on earth has evolved to be as it is through the processes of natural selection, recombination and mutation. Once upon a time there lived a species of creatures called Hooters. For a while it looked as though the Hooters may be hunted to extinction, for although they liked to eat the moss they could never tell if an eagle was flying above.
Set theory The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known. Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. History[edit] Mathematical topics typically emerge and evolve through interactions among many researchers. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Cantor's work initially polarized the mathematicians of his day. Basic concepts and notation[edit] Some ontology[edit]
Arity In logic, mathematics, and computer science, the arity Examples[edit] The term "arity" is rarely employed in everyday usage. A nullary function takes no arguments.A unary function takes one argument.A binary function takes two arguments.A ternary function takes three arguments.An n-ary function takes n arguments. Nullary[edit] Unary[edit] Binary[edit] Most operators encountered in programming are of the binary form. Ternary[edit] with arbitrary precision. n-ary[edit] From a mathematical point of view, a function of n arguments can always be considered as a function of one single argument which is an element of some product space. The same is true for programming languages, where functions taking several arguments could always be defined as functions taking a single argument of some composite type such as a tuple, or in languages with higher-order functions, by currying. Variable arity[edit] In computer science, a function accepting a variable number of arguments is called variadic.
Cartesian product Cartesian product of the sets and The simplest case of a Cartesian product is the Cartesian square, which returns a set from two sets. A Cartesian product of n sets can be represented by an array of n dimensions, where each element is an n-tuple. The Cartesian product is named after René Descartes,[1] whose formulation of analytic geometry gave rise to the concept. Examples[edit] A deck of cards[edit] An illustrative example is the standard 52-card deck. Ranks × Suits returns a set of the form {(A, ♠), (A, ♥), (A, ♦), (A, ♣), (K, ♠), ..., (3, ♣), (2, ♠), (2, ♥), (2, ♦), (2, ♣)}. Suits × Ranks returns a set of the form {(♠, A), (♠, K), (♠, Q), (♠, J), (♠, 10), ..., (♣, 6), (♣, 5), (♣, 4), (♣, 3), (♣, 2)}. A two-dimensional coordinate system[edit] An example in analytic geometry is the Cartesian plane. Most common implementation (set theory)[edit] A formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair. . , where For example: Similarly
Signature (logic) In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes. Signatures play the same role in mathematics as type signatures in computer programming. They are rarely made explicit in more philosophical treatments of logic. Formally, a (single-sorted) signature can be defined as a triple σ = (Sfunc, Srel, ar), where Sfunc and Srel are disjoint sets not containing any other basic logical symbols, called respectively function symbols (examples: +, ×, 0, 1) andrelation symbols or predicates (examples: ≤, ∈), and a function ar: Sfunc Srel → which assigns a non-negative integer called arity to every function or relation symbol. A signature with no function symbols is called a relational signature, and a signature with no relation symbols is called an algebraic signature. Symbol types S.
Model theory This article is about the mathematical discipline. For the informal notion in other parts of mathematics and science, see Mathematical model. Model theory recognises and is intimately concerned with a duality: It examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language. To quote the first page of Chang and Keisler (1990):[1] universal algebra + logic = model theory. Model theory developed rapidly during the 1990s, and a more modern definition is provided by Wilfrid Hodges (1997): although model theorists are also interested in the study of fields. In a similar way to proof theory, model theory is situated in an area of interdisciplinarity among mathematics, philosophy, and computer science. Branches of model theory[edit] This article focuses on finitary first order model theory of infinite structures. During the last several decades applied model theory has repeatedly merged with the more pure stability theory. and or