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Maclaurin's Inequality
In mathematics, Maclaurin's inequality, named after Colin Maclaurin, is a refinement of the inequality of arithmetic and geometric means. Let a_1, a_2,\ldots,a_n be non-negative real numbers, and for k=1,2,\ldots,n, define the averages S_k as follows: S_k = \frac. The numerator of this fraction is the elementary symmetric polynomial of degree k in the n variables a_1, a_2,\ldots,a_n, that is, the sum of all products of k of the numbers a_1, a_2,\ldots,a_n with the indices in increasing order. The denominator is the number of terms in the numerator, the binomial coefficient \tbinom n k. Maclaurin's inequality is the following chain of inequalities: S_1 \geq \sqrt \geq \sqrt \geq \cdots \geq \sqrt /math>, with equality if and only if all the a_i are equal. For n=2, this gives the usual inequality of arithmetic and geometric means of two non-negative numbers. Maclaurin's inequality is well illustrated by the case n=4: \begin &\quad \frac \\ pt&\ge \sqrt \\ pt&\ge \sqrt \\ pt&\ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Colin Maclaurin
Colin Maclaurin (; ; February 1698 – 14 June 1746) was a Scottish mathematician who made important contributions to geometry and algebra. He is also known for being a child prodigy and holding the record for being the youngest professor. The Maclaurin series, a special case of the Taylor series, is named after him. Owing to changes in orthography since that time (his name was originally rendered as M'Laurine), his surname is alternatively written MacLaurin. Early life Maclaurin was born in Kilmodan, Argyll. His father, John Maclaurin, minister of Glendaruel, died when Maclaurin was in infancy, and his mother died before he reached nine years of age. He was then educated under the care of his uncle, Daniel Maclaurin, minister of Kilfinan. A child prodigy, he entered university at age 11. Academic career At eleven, Maclaurin, a child prodigy at the time, entered the University of Glasgow. He graduated Master of Arts three years later by defending a thesis on ''the Power ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequality Of Arithmetic And Geometric Means
Inequality may refer to: * Inequality (mathematics), a relation between two quantities when they are different. * Economic inequality, difference in economic well-being between population groups ** Income inequality, an unequal distribution of income ** Wealth inequality, an unequal distribution of wealth ** Spatial inequality, the unequal distribution of income and resources across geographical regions ** International inequality, economic differences between countries * Social inequality, unequal opportunities and rewards for different social positions or statuses within a group ** Gender inequality, unequal treatment or perceptions due to gender ** Racial inequality, social distinctions between racial and ethnic groups within a society * Health inequality, differences in the quality of health and healthcare across populations * Educational inequality, the unequal distribution of academic resources * Environmental inequality, unequal environmental harms between different neighbor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-negative Number
In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. In some contexts, it makes sense to distinguish between a positive and a negative zero. In mathematics and physics, the phrase "change of sign" is associated with exchanging an object for its additive inverse (multiplication with −1, negation), an operation which is not restricted to real numbers. It applies among other objects to vectors, matrices, and complex numbers, which are not prescribed to be only either positive, negative, or zero. The word "sign" is also often used to indicate binary aspects of mathematical or scientific objects, such as odd and even ( sign of a permutation), sense of orientation or rotation ( cw/ccw), one sided limits, and other concepts described in below. Sign of a number Numbers from various number ... [...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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Symmetric Polynomial
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symmetric polynomials. That is, any symmetric polynomial is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials. There is one elementary symmetric polynomial of degree in variables for each positive integer , and it is formed by adding together all distinct products of distinct variables. Definition The elementary symmetric polynomials in variables , written for , are defined by :\begin e_1 (X_1, X_2, \dots, X_n) &= \sum_ X_a,\\ e_2 (X_1, X_2, \dots, X_n) &= \sum_ X_a X_b,\\ e_3 (X_1, X_2, \dots, X_n) &= \sum_ X_a X_b X_c,\\ \end and so forth, ending with : e_n (X_1, X_2, \dots,X_n) = X_1 X_2 \cdots X_n. In general, for we define : e_k (X_1 , \ldots , X_n )=\s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula : \binom nk = \frac, which using factorial notation can be compactly expressed as : \binom = \frac. For example, the fourth power of is : \begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for gives a triangular array called Pascal's triangle, satisfying the recurrence relation : \binom = \binom + \binom . The binomial coefficients occur in many areas of mathematics, and espe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequality (mathematics)
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. The main types of inequality are less than and greater than (denoted by and , respectively the less-than sign, less-than and greater-than sign, greater-than signs). Notation There are several different notations used to represent different kinds of inequalities: * The notation ''a'' ''b'' means that ''a'' is greater than ''b''. In either case, ''a'' is not equal to ''b''. These relations are known as strict inequalities, meaning that ''a'' is strictly less than or strictly greater than ''b''. Equality is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: * The notation ''a'' ≤ ''b'' or ''a'' ⩽ ''b'' or ''a'' ≦ ''b'' means that ''a'' is less than or equal to ''b'' (or, equivalently, at most ''b'', or no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Inequalities
In mathematics, the Newton inequalities are named after Isaac Newton. Suppose ''a''1, ''a''2, ..., ''a''''n'' are non-negative real numbers and let e_k denote the ''k''th elementary symmetric polynomial in ''a''1, ''a''2, ..., ''a''''n''. Then the elementary symmetric means, given by :S_k = \frac, satisfy the inequality :S_S_ \le S_k^2. Equality holds if and only if all the numbers ''a''''i'' are equal. It can be seen that ''S''1 is the arithmetic mean, and ''S''''n'' is the ''n''-th power of the geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri .... See also * Maclaurin's inequality References * * *D.S. Bernstein ''Matrix Mathematics: Theory, Facts, and Formulas'' (2009 Princeton) p. 55 * * * {{Isaac Newton Isaac Newton ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli's Inequality
In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of 1+x. It is often employed in real analysis. It has several useful variants: Integer exponent * Case 1: (1 + x)^r \geq 1 + rx for every integer r\geq 1 and real number x\geq-1. The inequality is strict if x\neq 0 and r\geq 2. * Case 2: (1 + x)^r \geq 1 + rx for every integer r\geq 0 and every real number x\geq -2. * Case 3: (1 + x)^r \geq 1 + rx for every even integer r\geq 0 and every real number x. Real exponent * (1 + x)^r \geq 1 + rx for every real number r\geq 1 and x\geq -1. The inequality is strict if x\neq 0 and r\neq 1. * (1 + x)^r \leq 1 + rx for every real number 0\leq r\leq 1 and x\geq -1. History Jacob Bernoulli first published the inequality in his treatise "Positiones Arithmeticae de Seriebus Infinitis" (Basel, 1689), where he used the inequality often. According to Joseph E. Hofmann, Über die Exercitatio Geometrica des M. A. Ricci (1963 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Muirhead's Inequality
In mathematics, Muirhead's inequality, named after Robert Franklin Muirhead, also known as the "bunching" method, generalizes the inequality of arithmetic and geometric means. Preliminary definitions ''a''-mean For any real vector :a=(a_1,\dots,a_n) define the "''a''-mean" 'a''of positive real numbers ''x''1, ..., ''x''''n'' by : \frac\sum_\sigma x_^\cdots x_^, where the sum extends over all permutations σ of . When the elements of ''a'' are nonnegative integers, the ''a''-mean can be equivalently defined via the monomial symmetric polynomial m_a(x_1,\dots,x_n) as : = \frac m_a(x_1,\dots,x_n), where ℓ is the number of distinct elements in ''a'', and ''k''1, ..., ''k''ℓ are their multiplicities. Notice that the ''a''-mean as defined above only has the usual properties of a mean (e.g., if the mean of equal numbers is equal to them) if a_1+\cdots+a_n=1. In the general case, one can consider instead , which is called a Muirhead mean.Bullen, P. S. Handbook of means an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
