Instruct: Given a question, retrieve relevant passages that can help answer the question.
Query: Who first introduced the concept of mathematical functions?
The term "function" was literally introduced by Gottfried Leibniz, in a 1673 letter, to describe a quantity related to points of a curve, such as a coordinate or curve's slope.[9] Johann Bernoulli started calling expressions made of a single variable "functions." In 1698, he agreed with Leibniz that any quantity formed "in an algebraic and transcendental manner" may be called a function of x.[10] By 1718, he came to regard as a function "any expression made up of a variable and some constants."[11] Alexis Claude Clairaut (in approximately 1734) and Leonhard Euler introduced the familiar notation f ( x ) {\displaystyle {f(x)}} for the value of a function.[12]
The Massieu function was introduced in the 1869 paper “On the Various Functions Characteristic of Fluids” by French engineer Francois Massieu (1832-1896) and is sometimes called Massieu–Gibbs function, Massieu potential, or Gibbs function, or “characteristic function” in its original terminology. The name "Gibbs function" is the eponym of American physicist Willard Gibbs (1839-1903), who cited Massieu in his 1876 "On the Equilibrium of Heterogeneous Substances". Massieu, as discussed in the first footnote to the abstract of Gibbs' "Equilibrium", “appears to have been the first to solve the problem of representing all the properties of a body of invariable composition which are concerned in reversible processes by means of a single function.” Massieu's 1869 paper seems to be the source for the generalized mathematical conception of the energy of a system being equal to summations of the products of pairs of conjugate variables.
The English mathematician John Wallis introduced the expression 1/∞ in his 1655 book "Treatise on the Conic Sections". The symbol, which denotes the reciprocal, or inverse, of ∞, is the symbolic representation of the mathematical concept of an infinitesimal. In his "Treatise on the Conic Sections" Wallis also discusses the concept of a relationship between the symbolic representation of infinitesimal 1/∞ that he introduced and the concept of infinity for which he introduced the symbol ∞. The concept suggests a thought experiment of adding an infinite number of parallelograms of infinitesimal width to form a finite area. This concept was the predecessor to the modern method of integration used in integral calculus. The conceptual origins of the concept of the infinitesimal 1/∞ can be traced as far back as the Greek philosopher Zeno of Elea, whose Zeno's dichotomy paradox was the first mathematical concept to consider the relationship between a finite interval and an interval approaching that of an infinitesimal-sized interval.
"The concept of a function which we are considering now differs essentially from the concepts of a sentential [propositional] and of a designatory function .... Strictly speaking ... [these] do not belong to the domain of logic or mathematics; they denote certain categories of expressions which serve to compose logical and mathematical statements, but they do not denote things treated of in those statements... . The term "function" in its new sense, on the other hand, is an expression of a purely logical character; it designates a certain type of things dealt with in logic and mathematics."[105]
The focus of this area is to introduce the mathematical ideas of people who have generally been excluded from discussions of formal, academic mathematics. The research of the mathematics of these cultures indicates two, slightly contradictory viewpoints. The first supports the objectivity of mathematics and that it is something discovered not constructed. The studies reveal that all cultures have basic counting, sorting and deciphering methods, and that these have arisen independently in different places around the world. This can be used to argue that these mathematical concepts are being discovered rather than created. However, others emphasize that the usefulness of mathematics is what tends to conceal its cultural constructs. Naturally, it is not surprising that extremely practical concepts such as numbers and counting have arisen in all cultures. The universality of these concepts, however, seems harder to sustain as more and more research reveals practices which are typically mathematical, such as counting, ordering, sorting, measuring and weighing, done in radically different ways (see Section 2.1: Numerals and Naming Systems).
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
The development of calculus was at the forefront of 18th century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Understanding the infinite was the major focus of Euler's research. While some of Euler's proofs may not have been acceptable under modern standards of rigor, his ideas were responsible for many great advances. First of all, Euler introduced the concept of a function, and introduced the use of the exponential function and logarithms in analytic proofs
In mathematics, a moment is a specific quantitative measure of the shape of a function. It is used in both mechanics and statistics. If the function represents physical density, then the zeroth moment is the total mass, the first moment divided by the total mass is the center of mass, and the second moment is the rotational inertia. If the function is a probability distribution, then the zeroth moment is the total probability (i.e. one), the first moment is the mean, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics.
George Green (14 July 1793 – 31 May 1841) was a British mathematical physicist who wrote "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism" (Green, 1828). The essay introduced several important concepts, among them a theorem similar to the modern Green's theorem, the idea of potential functions as currently used in physics, and the concept of what are now called Green's functions. Green was the first person to create a mathematical theory of electricity and magnetism and his theory formed the foundation for the work of other scientists such as James Clerk Maxwell, William Thomson, and others. His work on potential theory ran parallel to that of Carl Friedrich Gauss.
In mathematics, a function was 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). The concept of function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.
The concept of "negligibility" can find its trace back to sound models of analysis. Though the concepts of "continuity" and "infinitesimal" became important in mathematics during Newton and Leibniz's time (1680s), they were not well-defined until the late 1810s. The first reasonably rigorous definition of "continuity" in mathematical analysis was due to Bernard Bolzano, who wrote in 1817 the modern definition of continuity. Later Cauchy, Weierstrass and Heine also defined as follows (with all numbers in the real number domain formula_5):
In mathematics, a Dirichlet algebra is a particular type of algebra associated to a compact Hausdorff space "X". It is a closed subalgebra of "C"("X"), the uniform algebra of bounded continuous functions on "X", whose real parts are dense in the algebra of bounded continuous real functions on "X". The concept was introduced by .
According to Boris Golubov, "BV" functions of a single variable were first introduced by Camille Jordan, in the paper dealing with the convergence of Fourier series. The first successful step in the generalization of this concept to functions of several variables was due to Leonida Tonelli, who introduced a class of "continuous" "BV" functions in 1926 , to extend his direct method for finding solutions to problems in the calculus of variations in more than one variable. Ten years after, in , Lamberto Cesari "changed the continuity requirement" in Tonelli's definition "to a less restrictive integrability requirement", obtaining for the first time the class of functions of bounded variation of several variables in its full generality: as Jordan did before him, he applied the concept to resolve of a problem concerning the convergence of Fourier series, but for functions of "two variables". After him, several authors applied "BV" functions to study Fourier series in several variables, geometric measure theory, calculus of variations, and mathematical physics. Renato Caccioppoli and Ennio de Giorgi used them to define measure of nonsmooth boundaries of sets (see the entry ""Caccioppoli set"" for further information). Olga Arsenievna Oleinik introduced her view of generalized solutions for nonlinear partial differential equations as functions from the space "BV" in the paper , and was able to construct a generalized solution of bounded variation of a first order partial differential equation in the paper : few years later, Edward D. Conway and Joel A. Smoller applied "BV"-functions to the study of a single nonlinear hyperbolic partial differential equation of first order in the paper , proving that the solution of the Cauchy problem for such equations is a function of bounded variation, provided the initial value belongs to the same class. Aizik Isaakovich Vol'pert developed extensively a calculus for "BV" functions: in the paper he proved the chain rule for BV functions and in the book he, jointly with his pupil Sergei Ivanovich Hudjaev, explored extensively the properties of "BV" functions and their application. His chain rule formula was later extended by Luigi Ambrosio and Gianni Dal Maso in the paper .
George Green wrote "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism" in 1828. The essay introduced several important concepts, among them a theorem similar to the modern Green's theorem, the idea of potential functions as currently used in physics, and the concept of what are now called Green's functions. George Green was the first person to create a mathematical theory of electricity and magnetism and his theory formed the foundation for the work of other scientists such as James Clerk Maxwell, William Thomson, and others.
Higher-order functions are closely related to first-class functions in that higher-order functions and first-class functions both allow functions as arguments and results of other functions. The distinction between the two is subtle: "higher-order" describes a mathematical concept of functions that operate on other functions, while "first-class" is a computer science term that describes programming language entities that have no restriction on their use (thus first-class functions can appear anywhere in the program that other first-class entities like numbers can, including as arguments to other functions and as their return values).
In non-standard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal formula_1, the unique real formula_2 infinitely close to it, i.e. formula_3 is infinitesimal. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de Fermat, as well as Leibniz's Transcendental law of homogeneity.
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be quite vague.
Foundations of mathematics can be conceived as the study of the basic mathematical concepts (set, function, geometrical figure, number, etc.) and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics (formulas, theories and their models giving a meaning to formulas, definitions, proofs, algorithms, etc.) also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges.
Cournot was mainly a mathematician, but did have some influence over economics. His theories on monopolies and duopolies are still famous. In 1838 the book "Researches on the Mathematical Principles of the Theory of Wealth" was published, in which he used the application of the formulas and symbols of mathematics in economic analysis. This book was highly criticized and not very successful during Cournot's lifetime. He did attempted to rewrite it twice. However, it still has influence in economics today. Today many economists believe this book to be the point of departure for modern economic analysis. Cournot introduced the ideas of functions and probability into economic analysis. He derived the first formula for the rule of supply and demand as a function of price and in fact was the first to draw supply and demand curves on a graph, anticipating the work of Alfred Marshall by roughly thirty years. The Cournot duopoly model developed in his book also introduced the concept of a (pure strategy) Nash equilibrium, the Reaction function and best-response dynamics.
In applied mathematics, the reflecting function formula_1 of a differential system formula_2 connects the past state formula_3 of the system with the future state formula_4 of the system by the formula formula_5 The concept of the reflecting function was introduced by Uladzimir Ivanavich Mironenka.
She began her doctorate studies under the supervision of who was the first person to introduce relativity to Spain. Her thesis was titled: "The concept of function, continuous and semicontinuous functions, and their properties". She was awarded the "Extraordinary Price of Doctorate" for her doctoral thesis and became the first Spanish woman to earn a PhD in mathematics.
The Hitchin functional is a mathematical concept with applications in string theory that was introduced by the British mathematician Nigel Hitchin. and are the original articles of the Hitchin functional.
The idea of function, starting in the 17th century, was fundamental to the new infinitesimal calculus (see History of the function concept). At that time, only real-valued functions of a real variable were considered, and all functions were assumed to be smooth. But the definition was soon extended to functions of several variables and to function of a complex variable. In the second half of 19th century, the mathematically rigorous definition of a function was introduced, and functions with arbitrary domains and codomains were defined.
While pure mathematics has existed as an activity since at least Ancient Greece, the concept was elaborated upon around the year 1900, after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable, and Russell's paradox). This introduced the need of renewing the concept of mathematical rigor and rewriting all mathematics accordingly, with a systematic use of axiomatic methods. This led many mathematicians to focus on mathematics for its own sake, that is, pure mathematics.
Givi has taught 32 different courses during his career in a wide range of topics including thermal-fluids, systems & controls, vibrations & shock, machine design, engineering mathematics, numerical methods, and data analysis. He was the first scientist who introduced the concept of filtered density function for accurate prediction of turbulent combustion. His current research interests include high performance computing, turbulence, combustion, computational fluid dynamics, data-driven modeling, and stochastic processes.
In intuitionism, the term "explicit construction" is not cleanly defined, and that has led to criticisms. Attempts have been made to use the concepts of Turing machine or computable function to fill this gap, leading to the claim that only questions regarding the behavior of finite algorithms are meaningful and should be investigated in mathematics. This has led to the study of the computable numbers, first introduced by Alan Turing. Not surprisingly, then, this approach to mathematics is sometimes associated with theoretical computer science.