Mathematical optimizationMathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.
Taylor seriesIn mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century.
Mathematical economicsMathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, or other computational methods. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity.
Persian alphabetThe Persian alphabet (Alefbâye Fârsi), also known as the Perso-Arabic script, is the right-to-left alphabet used for the Persian language. It is a variation of the Arabic alphabet with four additional letters added: پ چ ژ گ. It was the basis of many Arabic-based scripts used in Central and South Asia. It is used for the Iranian and Dari standard varieties of Persian; and is one of two official writing systems for the Persian language, alongside the Cyrillic-based Tajik alphabet.
Special functionsSpecial functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by consensus, and thus lacks a general formal definition, but the list of mathematical functions contains functions that are commonly accepted as special. Many special functions appear as solutions of differential equations or integrals of elementary functions.
DerivativeIn mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
Mathematical modelA mathematical model is an abstract description of a concrete system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in applied mathematics and in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering), as well as in non-physical systems such as the social sciences (such as economics, psychology, sociology, political science).
Hyperbolic functionsIn mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively. Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry.
Brook TaylorBrook Taylor (18 August 1685 – 29 December 1731) was an English mathematician best known for creating Taylor's theorem and the Taylor series, which are important for their use in mathematical analysis. Brook Taylor was born in Edmonton (former Middlesex). Taylor was the son of John Taylor, MP of Patrixbourne, Kent and Olivia Tempest, the daughter of Sir Nicholas Tempest, Baronet of Durham. He entered St John's College, Cambridge, as a fellow-commoner in 1701, and took degrees in LL.B. in 1709 and LL.D. in 1714.
Persian languagePersian (ˈpɜrʒən or ˈpɜrʃən ), also known by its endonym Farsi (فارسی, , fɒːɾˈsiː), is a Western Iranian language belonging to the Iranian branch of the Indo-Iranian subdivision of the Indo-European languages. Persian is a pluricentric language predominantly spoken and used officially within Iran, Afghanistan, and Tajikistan in three mutually intelligible standard varieties, namely Iranian Persian (officially known as Persian), Dari Persian (officially known as Dari since 1964) and Tajiki Persian (officially known as Tajik since 1999).
Mathematical analysisAnalysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).
Taylor's theoremIn calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation.
Arabic alphabetThe Arabic alphabet (الْأَبْجَدِيَّة الْعَرَبِيَّة, ALA ʔælʔæbʒædijːæ-lʕɑrɑbijːæ or الْحُرُوف الْعَرَبِيَّة, ALA), or Arabic abjad, is the Arabic script as it is codified for writing Arabic. It is written from right to left in a cursive style and includes 28 letters. Most letters have contextual letterforms. The Arabic alphabet is considered an abjad, meaning it only uses consonants, but it is now considered an "impure abjad". As with other impure abjads, such as the Hebrew alphabet, scribes later devised means of indicating vowel sounds by separate vowel diacritics.
Function (mathematics)In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity).
AlephAleph (or alef or alif, transliterated ʾ) is the first letter of the Semitic abjads, including Phoenician 𐤀, Hebrew א, Aramaic 𐡀, Syriac ܐ, Arabic ʾ ا, and North Arabian 𐪑. It also appears as South Arabian 𐩱 and Ge'ez አ. These letters are believed to have derived from an Egyptian hieroglyph depicting an ox's head to describe the initial sound of *ʾalp, the West Semitic word for ox (compare Biblical Hebrew ʾelef, "ox"). The Phoenician variant gave rise to the Greek alpha (Α), being re-interpreted to express not the glottal consonant but the accompanying vowel, and hence the Latin A and Cyrillic А.
Linear approximationIn mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations. Given a twice continuously differentiable function of one real variable, Taylor's theorem for the case states that where is the remainder term.
Millennium Prize ProblemsThe Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US$1 million prize for the first correct solution to each problem. The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Riemann hypothesis, Yang–Mills existence and mass gap, and the Poincaré conjecture at the Millennium Meeting held on May 24, 2000.
Hilbert's problemsHilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 at the Sorbonne. The complete list of 23 problems was published later, in English translation in 1902 by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society.
AfghanistanAfghanistan, officially the Islamic Emirate of Afghanistan, is a landlocked country located at the crossroads of Central Asia and South Asia. Referred to as the Heart of Asia, it is bordered by Pakistan to the east and south, Iran to the west, Turkmenistan to the northwest, Uzbekistan to the north, Tajikistan to the northeast, and China to the northeast and east. Occupying of land, the country is predominantly mountainous with plains in the north and the southwest, which are separated by the Hindu Kush mountain range.
Arabic diacriticsArabic script has numerous diacritics, which include consonant pointing known as ALA (إِعْجَام), and supplementary diacritics known as ALA (تَشْكِيل). The latter include the vowel marks termed ALA (حَرَكَات; singular: حَرَكَة, ALA). The Arabic script is a modified abjad, where short consonants and long vowels are represented by letters but short vowels and consonant length are not generally indicated in writing. ALA is optional to represent missing vowels and consonant length.
