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Posynomials
A posynomial, also known as a posinomial in some literature, is a function of the form : f(x_1, x_2, \dots, x_n) = \sum_^K c_k x_1^ \cdots x_n^ where all the coordinates x_i and coefficients c_k are positive real numbers, and the exponents a_ are real numbers. Posynomials are closed under addition, multiplication, and nonnegative scaling. For example, : f(x_1, x_2, x_3) = 2.7 x_1^2x_2^x_3^ + 2x_1^x_3^ is a posynomial. Posynomials are not the same as polynomials in several independent variables. A polynomial's exponents must be non-negative integers, but its independent variables and coefficients can be arbitrary real numbers; on the other hand, a posynomial's exponents can be arbitrary real numbers, but its independent variables and coefficients must be positive real numbers. This terminology was introduced by Richard J. Duffin, Elmor L. Peterson, and Clarence Zener in their seminal book on geometric programming. Posynomials are a special case of signomial A signomial is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signomial
A signomial is an algebraic function of one or more independent variables. It is perhaps most easily thought of as an algebraic extension of multivariable polynomials—an extension that permits exponents to be arbitrary real numbers (rather than just non-negative integers) while requiring the independent variables to be strictly positive (so that division by zero and other inappropriate algebraic operations are not encountered). Formally, a signomial is a function with domain \mathbb_^n which takes values : f(x_1, x_2, \dots, x_n) = \sum_^M \left(c_i \prod_^n x_j^\right) where the coefficients c_i and the exponents a_ are real numbers. Signomials are closed under addition, subtraction, multiplication, and scaling. If we restrict all c_i to be positive, then the function f is a posynomial. Consequently, each signomial is either a posynomial, the negative of a posynomial, or the difference of two posynomials. If, in addition, all exponents a_ are non-negative integers, then the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Programming
A geometric program (GP) is an optimization problem of the form : \begin \mbox & f_0(x) \\ \mbox & f_i(x) \leq 1, \quad i=1, \ldots, m\\ & g_i(x) = 1, \quad i=1, \ldots, p, \end where f_0,\dots,f_m are posynomials and g_1,\dots,g_p are monomials. In the context of geometric programming (unlike standard mathematics), a monomial is a function from \mathbb_^n to \mathbb defined as :x \mapsto c x_1^ x_2^ \cdots x_n^ where c > 0 \ and a_i \in \mathbb . A posynomial is any sum of monomials.S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi. A Tutorial on Geometric Programming'' Retrieved 20 October 2019. Geometric programming is closely related to convex optimization: any GP can be made convex by means of a change of variables. GPs have numerous applications, including component sizing in IC design, aircraft design, maximum likelihood estimation for logistic regression in statistics, and parameter tuning of positive linear systems in control theory. Convex form Geometric programs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. 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). The concept of a function was formalized at the end of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers is denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' join ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Duffin
Richard James Duffin (1909 – October 29, 1996) was an American physicist, known for his contributions to electrical transmission theory and to the development of geometric programming and other areas within operations research. Education and career Duffin obtained a BSc in physics at the University of Illinois, where he was elected to Sigma Xi in 1932. He stayed at Illinois for his PhD, which was advised by Harold Mott-Smith and David Bourgin, producing a thesis entitled ''Galvanomagnetic and Thermomagnetic Phenomena'' (1935). Duffin lectured at Purdue University and Illinois before joining the Carnegie Institute in Washington, D.C. during World War II.Richard J. Duffin from the |
Clarence Zener
Clarence Melvin Zener (December 1, 1905 – July 2, 1993) was the American physicist who first (1934) described the property concerning the breakdown of electrical insulators. These findings were later exploited by Bell Labs in the development of the Zener diode, which was duly named after him. Zener was a theoretical physicist with a background in mathematics who conducted research in a wide range of subjects including: superconductivity, metallurgy, ferromagnetism, elasticity, fracture mechanics, diffusion, and geometric programming. Life Zener was born in Indianapolis, Indiana and earned his PhD in physics under Edwin Kemble at Harvard in 1929. His thesis was titled ''Quantum Mechanics of the Formation of Certain Types of Diatomic Molecules''. In 1957 he received the Bingham Medal for his work in rheology, in 1959 the John Price Wetherill Medal from The Franklin Institute, in 1965 the Albert Souveur Achievement Award, in 1974 the Gold Medal from American Society for Metals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Case
In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case is a type of special case which is arrived at by taking some aspect of the concept to the extreme of what is permitted in the general case. A degenerate case is a special case which is in some way qualitatively different from almost all of the cases allowed. Special case examples include the following: * All squares are rectangles (but not all rectangles are squares); therefore the square is a special case of the rectangle. * Fermat's Last Theorem, that has no solutions in positive integers with , is a special case of Beal's conjecture, that has no primitive solutions in positive integers with , , and all greater than 2, specifically, the case of {{mvar, x {{= y {{= z. See also * Specialization (logic) Specialization or Specializ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
