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Harmonic Mean
In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is the most appropriate average for ratios and rate (mathematics), rates such as speeds, and is normally only used for positive arguments. The harmonic mean is the multiplicative inverse, reciprocal of the arithmetic mean of the reciprocals of the numbers, that is, the generalized f-mean with f(x) = \frac. For example, the harmonic mean of 1, 4, and 4 is :\left(\frac\right)^ = \frac = \frac = 2\,. Definition The harmonic mean ''H'' of the positive real numbers x_1, x_2, \ldots, x_n is :H(x_1, x_2, \ldots, x_n) = \frac = \frac. It is the reciprocal of the arithmetic mean of the reciprocals, and vice versa: :\begin H(x_1, x_2, \ldots, x_n) &= \frac, \\ A(x_1, x_2, \ldots, x_n) &= \frac, \end where the arithmetic mean is A(x_1, x_2, \ldots, x_n) = \tfrac1n \sum_^n x_i. The harmonic mean is a Schur-concave function, and is greater than or equal to the minimum of its arguments: for positive a ... [...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] |
Alloy
An alloy is a mixture of chemical elements of which in most cases at least one is a metal, metallic element, although it is also sometimes used for mixtures of elements; herein only metallic alloys are described. Metallic alloys often have properties that differ from those of the pure elements from which they are made. The vast majority of metals used for commercial purposes are alloyed to improve their properties or behavior, such as increased strength, hardness or corrosion resistance. Metals may also be alloyed to reduce their overall cost, for instance alloys of gold and Copper(II) sulfate, copper. A typical example of an alloy is SAE 304 stainless steel, 304 grade stainless steel which is commonly used for kitchen utensils, pans, knives and forks. Sometime also known as 18/8, it as an alloy consisting broadly of 74% iron, 18% chromium and 8% nickel. The chromium and nickel alloying elements add strength and hardness to the majority iron element, but their main function is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weighted Arithmetic Mean
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics. If all the weights are equal, then the weighted mean is the same as the arithmetic mean. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counterintuitive properties, as captured for instance in Simpson's paradox. Examples Basic example Given two school with 20 students, one with 30 test grades in each class as follows: :Morning class = :Afternoon class = The mean for the morning class is 80 and the mean of the afternoon class is 90. The unweighted mean of the two means is 85. However, this does not account for the difference in numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weighted Harmonic Mean
In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is the most appropriate average for ratios and rates such as speeds, and is normally only used for positive arguments. The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the numbers, that is, the generalized f-mean with f(x) = \frac. For example, the harmonic mean of 1, 4, and 4 is :\left(\frac\right)^ = \frac = \frac = 2\,. Definition The harmonic mean ''H'' of the positive real numbers x_1, x_2, \ldots, x_n is :H(x_1, x_2, \ldots, x_n) = \frac = \frac. It is the reciprocal of the arithmetic mean of the reciprocals, and vice versa: :\begin H(x_1, x_2, \ldots, x_n) &= \frac, \\ A(x_1, x_2, \ldots, x_n) &= \frac, \end where the arithmetic mean is A(x_1, x_2, \ldots, x_n) = \tfrac1n \sum_^n x_i. The harmonic mean is a Schur-concave function, and is greater than or equal to the minimum of its arguments: for positive arguments, \min(x_1 \ldots x_n) \le H(x_1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Analysis
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing Limit (mathematics), limiting behavior. As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as becomes very large, the term becomes insignificant compared to . The function is said to be "''asymptotically equivalent'' to , as ". This is often written symbolically as , which is read as " is asymptotic to ". An example of an important asymptotic result is the prime number theorem. Let denote the prime-counting function (which is not directly related to the constant pi), i.e. is the number of prime numbers that are less than or equal to . Then the theorem states that \pi(x)\sim\frac. Asymptotic analysis is commonly used in computer science as part of the analysis of algorithms and is often expressed there in terms of big O notation. Definition Formally, given functions and , we define a binary relation f( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorization, factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow primality test, method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number Theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function). The first such distribution found is , where is the prime-counting function (the number of primes less than or equal to ''N'') and is the natural logarithm of . This means that for large enough , the probability that a random integer not greater than is prime is very close to . Consequently, a random integer with at most digits (for large enough ) is about half as likely to be prime as a random integer with at most digits. For example, among the positive integers of at most 1000 digits, about on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weight Function
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is a weighted sum or weighted average. Weight functions occur frequently in statistics and analysis, and are closely related to the concept of a measure. Weight functions can be employed in both discrete and continuous settings. They can be used to construct systems of calculus called "weighted calculus" and "meta-calculus".Jane Grossma''Meta-Calculus: Differential and Integral'' , 1981. Discrete weights General definition In the discrete setting, a weight function w \colon A \to \R^+ is a positive function defined on a discrete set A, which is typically finite or countable. The weight function w(a) := 1 corresponds to the ''unweighted'' situation in which all elements have equal weight. One can then apply this weight to various con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crux Mathematicorum
''Crux Mathematicorum'' is a scientific journal of mathematics published by the Canadian Mathematical Society. It contains mathematical problems for secondary school and undergraduate students. Its editor-in-chief is Kseniya Garaschuk. The journal was established in 1975, under the name ''Eureka'', by the Carleton-Ottawa Mathematics Association, with Léo Sauvé as its first editor-in-chief An editor-in-chief (EIC), also known as lead editor or chief editor, is a publication's editorial leader who has final responsibility for its operations and policies. The editor-in-chief heads all departments of the organization and is held accoun .... It took the name ''Crux Mathematicorum'' with its fourth volume, in 1978, to avoid confusion with another journal '' Eureka'' published by the Cambridge University Mathematical Society. The Canadian Mathematical Society took over the journal in 1985, and soon afterwards G.W. (Bill) Sands became its new editor. Bruce L. R. Shawyer took over as e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
