1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
Mathematics
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Euclid (holding calipers), Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens.
Mathematics
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Greek mathematician Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the Pythagorean theorem
Mathematics
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Leonardo Fibonacci, the Italian mathematician who established the Hindu–Arabic numeral system to the Western World
Mathematics
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Carl Friedrich Gauss, known as the prince of mathematicians
2.
Statistics
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Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e. g. a scientific, industrial, or social problem, populations can be diverse topics such as all people living in a country or every atom composing a crystal. Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys, statistician Sir Arthur Lyon Bowley defines statistics as Numerical statements of facts in any department of inquiry placed in relation to each other. When census data cannot be collected, statisticians collect data by developing specific experiment designs, representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. In contrast, an observational study does not involve experimental manipulation, inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena. A standard statistical procedure involves the test of the relationship between two data sets, or a data set and a synthetic data drawn from idealized model. A hypothesis is proposed for the relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the hypothesis is done using statistical tests that quantify the sense in which the null can be proven false. Working from a hypothesis, two basic forms of error are recognized, Type I errors and Type II errors. Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis, measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random or systematic, the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics continues to be an area of research, for example on the problem of how to analyze Big data. Statistics is a body of science that pertains to the collection, analysis, interpretation or explanation. Some consider statistics to be a mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty, mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory. In applying statistics to a problem, it is practice to start with a population or process to be studied. Populations can be diverse topics such as all living in a country or every atom composing a crystal. Ideally, statisticians compile data about the entire population and this may be organized by governmental statistical institutes
Statistics
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Scatter plots are used in descriptive statistics to show the observed relationships between different variables.
Statistics
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More probability density is found as one gets closer to the expected (mean) value in a normal distribution. Statistics used in standardized testing assessment are shown. The scales include standard deviations, cumulative percentages, percentile equivalents, Z-scores, T-scores, standard nines, and percentages in standard nines.
Statistics
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Gerolamo Cardano, the earliest pioneer on the mathematics of probability.
Statistics
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Karl Pearson, a founder of mathematical statistics.
3.
Arithmetic mean
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In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the sum of a collection of numbers divided by the number of numbers in the collection. The collection is often a set of results of an experiment, the term arithmetic mean is preferred in some contexts in mathematics and statistics because it helps distinguish it from other means, such as the geometric mean and the harmonic mean. In addition to mathematics and statistics, the mean is used frequently in fields such as economics, sociology, and history. For example, per capita income is the average income of a nations population. While the arithmetic mean is used to report central tendencies, it is not a robust statistic. In a more obscure usage, any sequence of values that form a sequence between two numbers x and y can be called arithmetic means between x and y. The arithmetic mean is the most commonly used and readily understood measure of central tendency, in statistics, the term average refers to any of the measures of central tendency. The arithmetic mean is defined as being equal to the sum of the values of each. For example, let us consider the monthly salary of 10 employees of a firm,2500,2700,2400,2300,2550,2650,2750,2450,2600,2400. The arithmetic mean is 2500 +2700 +2400 +2300 +2550 +2650 +2750 +2450 +2600 +240010 =2530, If the data set is a statistical population, then the mean of that population is called the population mean. If the data set is a sample, we call the statistic resulting from this calculation a sample mean. The arithmetic mean of a variable is denoted by a bar, for example as in x ¯. The arithmetic mean has several properties that make it useful, especially as a measure of central tendency and these include, If numbers x 1, …, x n have mean x ¯, then + ⋯ + =0. The mean is the single number for which the residuals sum to zero. If the arithmetic mean of a population of numbers is desired, the arithmetic mean may be contrasted with the median. The median is defined such that half the values are larger than, and half are smaller than, If elements in the sample data increase arithmetically, when placed in some order, then the median and arithmetic average are equal. For example, consider the data sample 1,2,3,4, the average is 2.5, as is the median. However, when we consider a sample that cannot be arranged so as to increase arithmetically, such as 1,2,4,8,16, in this case, the arithmetic average is 6.2 and the median is 4
Arithmetic mean
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Comparison of mean, median and mode of two log-normal distributions with different skewness.
4.
Median
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The median is the value separating the higher half of a data sample, a population, or a probability distribution, from the lower half. In simple terms, it may be thought of as the value of a data set. For example, in the set, the median is 6. The median is a commonly used measure of the properties of a set in statistics. The basic advantage of the median in describing data compared to the mean is that it is not skewed so much by extremely large or small values, and so it may give a better idea of a typical value. For example, in understanding statistics like household income or assets which vary greatly, Median income, for example, may be a better way to suggest what a typical income is. The median of a finite list of numbers can be found by arranging all the numbers from smallest to greatest, if there is an odd number of numbers, the middle one is picked. For example, consider the set of numbers,1,3,3,6,7,8,9 This set contains seven numbers, the median is the fourth of them, which is 6. If there are a number of observations, then there is no single middle value. For example, in the set,1,2,3,4,5,6,8,9 The median is the mean of the middle two numbers, this is ÷2, which is 4.5. The formula used to find the number of a data set of n numbers is ÷2. This either gives the number or the halfway point between the two middle values. For example, with 14 values, the formula will give 7.5, and you will also be able to find the median using the Stem-and-Leaf Plot. There is no accepted standard notation for the median. In any of these cases, the use of these or other symbols for the needs to be explicitly defined when they are introduced. The median is used primarily for skewed distributions, which it summarizes differently from the arithmetic mean, the median is 2 in this case, and it might be seen as a better indication of central tendency than the arithmetic mean of 4. The widely cited empirical relationship between the locations of the mean and the median for skewed distributions is, however. There are, however, various relationships for the difference between them, see below
Median
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Comparison of mean, median and mode of two log-normal distributions with different skewness.
5.
Mode (statistics)
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The mode is the value that appears most often in a set of data. The mode of a probability distribution is the value x at which its probability mass function takes its maximum value. In other words, it is the value that is most likely to be sampled, the mode of a continuous probability distribution is the value x at which its probability density function has its maximum value, so the mode is at the peak. Like the statistical mean and median, the mode is a way of expressing, in a single number, the numerical value of the mode is the same as that of the mean and median in a normal distribution, and it may be very different in highly skewed distributions. The mode is not necessarily unique to a distribution, since the probability mass function or probability density function may take the same maximum value at several points x1, x2. The most extreme case occurs in uniform distributions, where all values occur equally frequently, when a probability density function has multiple local maxima it is common to refer to all of the local maxima as modes of the distribution. Such a continuous distribution is called multimodal, in symmetric unimodal distributions, such as the normal distribution, the mean, median and mode all coincide. For samples, if it is known that they are drawn from a symmetric distribution, the mode of a sample is the element that occurs most often in the collection. For example, the mode of the sample is 6, given the list of data the mode is not unique - the dataset may be said to be bimodal, while a set with more than two modes may be described as multimodal. For a sample from a distribution, such as, the concept is unusable in its raw form. The mode is then the value where the histogram reaches its peak, the following MATLAB code example computes the mode of a sample, The algorithm requires as a first step to sort the sample in ascending order. It then computes the derivative of the sorted list. Unlike mean and median, the concept of mode makes sense for nominal data. For example, taking a sample of Korean family names, one might find that Kim occurs more often than any other name, then Kim would be the mode of the sample. In any voting system where a plurality victory, a single modal value determines the victor. Unlike median, the concept of mode makes sense for any random variable assuming values from a space, including the real numbers. For example, a distribution of points in the plane will typically have a mean and a mode, the median makes sense when there is a linear order on the possible values. Generalizations of the concept of median to higher-dimensional spaces are the geometric median, for the remainder, the assumption is that we have a real-valued random variable
Mode (statistics)
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Comparison of mean, median and mode of two log-normal distributions with different skewness.
6.
Measurement
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Measurement is the assignment of a number to a characteristic of an object or event, which can be compared with other objects or events. The scope and application of a measurement is dependent on the context, however, in other fields such as statistics as well as the social and behavioral sciences, measurements can have multiple levels, which would include nominal, ordinal, interval, and ratio scales. Measurement is a cornerstone of trade, science, technology, historically, many measurement systems existed for the varied fields of human existence to facilitate comparisons in these fields. Often these were achieved by local agreements between trading partners or collaborators, since the 18th century, developments progressed towards unifying, widely accepted standards that resulted in the modern International System of Units. This system reduces all physical measurements to a combination of seven base units. The science of measurement is pursued in the field of metrology, the measurement of a property may be categorized by the following criteria, type, magnitude, unit, and uncertainty. They enable unambiguous comparisons between measurements, the type or level of measurement is a taxonomy for the methodological character of a comparison. For example, two states of a property may be compared by ratio, difference, or ordinal preference, the type is commonly not explicitly expressed, but implicit in the definition of a measurement procedure. The magnitude is the value of the characterization, usually obtained with a suitably chosen measuring instrument. A unit assigns a mathematical weighting factor to the magnitude that is derived as a ratio to the property of a used as standard or a natural physical quantity. An uncertainty represents the random and systemic errors of the measurement procedure, errors are evaluated by methodically repeating measurements and considering the accuracy and precision of the measuring instrument. Measurements most commonly use the International System of Units as a comparison framework, the system defines seven fundamental units, kilogram, metre, candela, second, ampere, kelvin, and mole. Instead, the measurement unit can only ever change through increased accuracy in determining the value of the constant it is tied to and this directly influenced the Michelson–Morley experiment, Michelson and Morley cite Peirce, and improve on his method. With the exception of a few fundamental quantum constants, units of measurement are derived from historical agreements, nothing inherent in nature dictates that an inch has to be a certain length, nor that a mile is a better measure of distance than a kilometre. Over the course of history, however, first for convenience and then for necessity. Laws regulating measurement were originally developed to prevent fraud in commerce.9144 metres, in the United States, the National Institute of Standards and Technology, a division of the United States Department of Commerce, regulates commercial measurements. Before SI units were adopted around the world, the British systems of English units and later imperial units were used in Britain, the Commonwealth. The system came to be known as U. S. customary units in the United States and is still in use there and in a few Caribbean countries. S
Measurement
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A typical tape measure with both metric and US units and two US pennies for comparison
Measurement
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A baby bottle that measures in three measurement systems, Imperial (U.K.), U.S. customary, and metric.
Measurement
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Four measuring devices having metric calibrations
7.
Summation
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In mathematics, summation is the addition of a sequence of numbers, the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a sum, prefix sum. The numbers to be summed may be integers, rational numbers, real numbers, besides numbers, other types of values can be added as well, vectors, matrices, polynomials and, in general, elements of any additive group. For finite sequences of elements, summation always produces a well-defined sum. The summation of a sequence of values is called a series. A value of such a series may often be defined by means of a limit, another notion involving limits of finite sums is integration. The summation of the sequence is an expression whose value is the sum of each of the members of the sequence, in the example,1 +2 +4 +2 =9. Addition is also commutative, so permuting the terms of a sequence does not change its sum. There is no notation for the summation of such explicit sequences. If, however, the terms of the sequence are given by a pattern, possibly of variable length. For the summation of the sequence of integers from 1 to 100. In this case, the reader can guess the pattern. However, for more complicated patterns, one needs to be precise about the used to find successive terms. Using this sigma notation the above summation is written as, ∑ i =1100 i, the value of this summation is 5050. It can be found without performing 99 additions, since it can be shown that ∑ i =1 n i = n 2 for all natural numbers n, more generally, formulae exist for many summations of terms following a regular pattern. By contrast, summation as discussed in this article is called definite summation, when it is necessary to clarify that numbers are added with their signs, the term algebraic sum is used. Mathematical notation uses a symbol that compactly represents summation of many terms, the summation symbol, ∑. The i = m under the symbol means that the index i starts out equal to m
Summation
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The capital sigma
8.
Pythagorean means
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In mathematics, the three classical Pythagorean means are the arithmetic mean, the geometric mean, and the harmonic mean. Averaging, min ≤ M ≤ max These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry, the harmonic and arithmetic means are reciprocal duals of each other for positive arguments while the geometric mean is its own reciprocal dual. There is an ordering to these means min ≤ H M ≤ G M ≤ A M ≤ max with equality holding if and this is a generalization of the inequality of arithmetic and geometric means and a special case of an inequality for generalized means. The proof follows from the arithmetic-geometric mean inequality, A M ≤ max, the study of the Pythagorean means is closely related to the study of majorization and Schur-convex functions. Arithmetic-geometric mean Average Generalized mean Cantrell, David W. Pythagorean Means, nice comparison of Pythagorean means with emphasis on the harmonic mean
Pythagorean means
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A geometric construction of the Quadratic mean and the Pythagorean means (of two numbers a and b). Harmonic mean denoted by H, Geometric by G, Arithmetic by A and Quadratic mean (also known as Root mean square) denoted by Q.
9.
Mean
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In mathematics, mean has several different definitions depending on the context. An analogous formula applies to the case of a probability distribution. Not every probability distribution has a mean, see the Cauchy distribution for an example. Moreover, for some distributions the mean is infinite, for example, the arithmetic mean of a set of numbers x1, x2. Xn is typically denoted by x ¯, pronounced x bar, if the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is termed the sample mean to distinguish it from the population mean. For a finite population, the mean of a property is equal to the arithmetic mean of the given property while considering every member of the population. For example, the mean height is equal to the sum of the heights of every individual divided by the total number of individuals. The sample mean may differ from the mean, especially for small samples. The law of large numbers dictates that the larger the size of the sample, outside of probability and statistics, a wide range of other notions of mean are often used in geometry and analysis, examples are given below. The geometric mean is an average that is useful for sets of numbers that are interpreted according to their product. X ¯ =1 n For example, the mean of five values,4,36,45,50,75 is,1 /5 =243000005 =30. The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, for example speed. AM, GM, and HM satisfy these inequalities, A M ≥ G M ≥ H M Equality holds if, in descriptive statistics, the mean may be confused with the median, mode or mid-range, as any of these may be called an average. The mean of a set of observations is the average of the values, however, for skewed distributions. For example, mean income is typically skewed upwards by a number of people with very large incomes. By contrast, the income is the level at which half the population is below. The mode income is the most likely income, and favors the larger number of people with lower incomes, the mean of a probability distribution is the long-run arithmetic average value of a random variable having that distribution. In this context, it is known as the expected value
Mean
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Comparison of the arithmetic mean, median and mode of two skewed (log-normal) distributions.
10.
Geometric mean
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In mathematics, the geometric mean is a type of mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values. The geometric mean is defined as the nth root of the product of n numbers, i. e. for a set of numbers x1, x2. As another example, the mean of the three numbers 4,1, and 1/32 is the cube root of their product, which is 1/2. A geometric mean is used when comparing different items—finding a single figure of merit for these items—when each item has multiple properties that have different numeric ranges. So, a 20% change in environmental sustainability from 4 to 4.8 has the effect on the geometric mean as a 20% change in financial viability from 60 to 72. The geometric mean can be understood in terms of geometry, the geometric mean of two numbers, a and b, is the length of one side of a square whose area is equal to the area of a rectangle with sides of lengths a and b. The geometric mean applies only to numbers of the same sign, the geometric mean is also one of the three classical Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean. The above figure uses capital pi notation to show a series of multiplications. For example, in a set of four numbers, the product of 1 ×2 ×3 ×4 is 24, note that the exponent 1 / n on the left side is equivalent to the taking nth root. For example,241 /4 =244, the geometric mean of a data set is less than the data sets arithmetic mean unless all members of the data set are equal, in which case the geometric and arithmetic means are equal. This allows the definition of the mean, a mixture of the two which always lies in between. The geometric mean can also be expressed as the exponential of the mean of logarithms. This is sometimes called the log-average and this is less likely to occur with the sum of the logarithms for each number. Instead, the mean is simply 1 n, where n is the number of steps from the initial to final state. If the values are a 0, …, a n and this is the case when presenting computer performance with respect to a reference computer, or when computing a single average index from several heterogeneous sources. In this scenario, using the arithmetic or harmonic mean would change the ranking of the results depending on what is used as a reference. For example, take the following comparison of time of computer programs. However, by presenting appropriately normalized values and using the arithmetic mean, however, this reasoning has been questioned
Geometric mean
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Equal area comparison of the aspect ratios used by Kerns Powers to derive the SMPTE 16:9 standard. TV 4:3/1.33 in red, 1.66 in orange, 16:9/1.7 7 in blue, 1.85 in yellow, Panavision /2.2 in mauve and CinemaScope /2.35 in purple.
11.
Antilog
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In mathematics, the logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed number, in simple cases the logarithm counts factors in multiplication. For example, the base 10 logarithm of 1000 is 3, the logarithm of x to base b, denoted logb, is the unique real number y such that by = x. For example, log2 =6, as 64 =26, the logarithm to base 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the e as its base, its use is widespread in mathematics and physics. The binary logarithm uses base 2 and is used in computer science. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations and they were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the function in the 18th century. Logarithmic scales reduce wide-ranging quantities to tiny scopes, for example, the decibel is a unit quantifying signal power log-ratios and amplitude log-ratios. In chemistry, pH is a measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and they describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting. In the same way as the logarithm reverses exponentiation, the logarithm is the inverse function of the exponential function applied to complex numbers. The discrete logarithm is another variant, it has uses in public-key cryptography, the idea of logarithms is to reverse the operation of exponentiation, that is, raising a number to a power. For example, the power of 2 is 8, because 8 is the product of three factors of 2,23 =2 ×2 ×2 =8. It follows that the logarithm of 8 with respect to base 2 is 3, the third power of some number b is the product of three factors equal to b. More generally, raising b to the power, where n is a natural number, is done by multiplying n factors equal to b. The n-th power of b is written bn, so that b n = b × b × ⋯ × b ⏟ n factors, exponentiation may be extended to by, where b is a positive number and the exponent y is any real number. For example, b−1 is the reciprocal of b, that is, the logarithm of a positive real number x with respect to base b, a positive real number not equal to 1, is the exponent by which b must be raised to yield x
Antilog
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John Napier (1550–1617), the inventor of logarithms
Antilog
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The graph of the logarithm to base 2 crosses the x axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3). For example, log 2 (8) = 3, because 2 3 = 8. The graph gets arbitrarily close to the y axis, but does not meet or intersect it.
Antilog
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The logarithm keys (lo g for base-10 and ln for base- e) on a typical scientific calculator
Antilog
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A nautilus displaying a logarithmic spiral
12.
Logarithm
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In mathematics, the logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed number, in simple cases the logarithm counts factors in multiplication. For example, the base 10 logarithm of 1000 is 3, the logarithm of x to base b, denoted logb, is the unique real number y such that by = x. For example, log2 =6, as 64 =26, the logarithm to base 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the e as its base, its use is widespread in mathematics and physics. The binary logarithm uses base 2 and is used in computer science. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations and they were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the function in the 18th century. Logarithmic scales reduce wide-ranging quantities to tiny scopes, for example, the decibel is a unit quantifying signal power log-ratios and amplitude log-ratios. In chemistry, pH is a measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and they describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting. In the same way as the logarithm reverses exponentiation, the logarithm is the inverse function of the exponential function applied to complex numbers. The discrete logarithm is another variant, it has uses in public-key cryptography, the idea of logarithms is to reverse the operation of exponentiation, that is, raising a number to a power. For example, the power of 2 is 8, because 8 is the product of three factors of 2,23 =2 ×2 ×2 =8. It follows that the logarithm of 8 with respect to base 2 is 3, the third power of some number b is the product of three factors equal to b. More generally, raising b to the power, where n is a natural number, is done by multiplying n factors equal to b. The n-th power of b is written bn, so that b n = b × b × ⋯ × b ⏟ n factors, exponentiation may be extended to by, where b is a positive number and the exponent y is any real number. For example, b−1 is the reciprocal of b, that is, the logarithm of a positive real number x with respect to base b, a positive real number not equal to 1, is the exponent by which b must be raised to yield x
Logarithm
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John Napier (1550–1617), the inventor of logarithms
Logarithm
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The graph of the logarithm to base 2 crosses the x axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3). For example, log 2 (8) = 3, because 2 3 = 8. The graph gets arbitrarily close to the y axis, but does not meet or intersect it.
Logarithm
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The logarithm keys (lo g for base-10 and ln for base- e) on a typical scientific calculator
Logarithm
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A nautilus displaying a logarithmic spiral
13.
Harmonic mean
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In mathematics, the harmonic mean is one of several kinds of average, and in particular one of the Pythagorean means. Typically, it is appropriate for situations when the average of rates is desired, the harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals. As a simple example, the mean of 1,2. The third formula in the equation expresses the harmonic mean as the reciprocal of the arithmetic mean of the reciprocals. From the following formula, H = n ⋅ ∏ j =1 n x j ∑ i =1 n. it is apparent that the harmonic mean is related to the arithmetic and geometric means. Thus, the harmonic mean cannot be arbitrarily large by changing some values to bigger ones. The harmonic mean is one of the three Pythagorean means, the arithmetic mean is often mistakenly used in places calling for the harmonic mean. In the speed example below for instance, the mean of 50 is incorrect. The harmonic mean is related to the other Pythagorean means, as seen in the formula in the above equation. This can be seen by interpreting the denominator to be the mean of the product of numbers n times. That is, for the first term, we multiply all n numbers except the first, for the second, we multiply all n numbers except the second, and so on. The numerator, excluding the n, which goes with the mean, is the geometric mean to the power n. Thus the nth harmonic mean is related to the nth geometric and arithmetic means, the general formula is H = n A = n A. For the special case of just two numbers, x 1 and x 2, the mean can be written H =2 x 1 x 2 x 1 + x 2. In this special case, the mean is related to the arithmetic mean A = x 1 + x 22. Since G A ≤1 by the inequality of arithmetic and geometric means and it also follows that G = A H, meaning the two numbers geometric mean equals the geometric mean of their arithmetic and harmonic means. Three positive numbers H, G, and A are respectively the harmonic, geometric, W n is associated to the dataset x 1. X n, the harmonic mean is defined by H = ∑ i =1 n w i ∑ i =1 n w i x i = −1
Harmonic mean
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A geometric construction of the three Pythagorean means of two numbers, a and b. The harmonic mean is denoted by H in purple color. The Q denotes a fourth mean, the quadratic mean.
14.
Log-normal distribution
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In probability theory, a log-normal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then Y = ln has a normal distribution, likewise, if Y has a normal distribution, then X = exp has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values, the distribution is occasionally referred to as the Galton distribution or Galtons distribution, after Francis Galton. The log-normal distribution also has associated with other names, such as McAlister, Gibrat. A log-normal process is the realization of the multiplicative product of many independent random variables. This is justified by considering the limit theorem in the log domain. The log-normal distribution is the maximum entropy probability distribution for a random variate X for which the mean and this relationship is true regardless of the base of the logarithmic or exponential function. If log a is normally distributed, then so is log b , likewise, if e X is log-normally distributed, then so is a X, where a is a positive number ≠1. On a logarithmic scale, μ and σ can be called the location parameter, in contrast, the mean, standard deviation, and variance of the non-logarithmized sample values are respectively denoted m, s. d. and v in this article. The two sets of parameters can be related as μ = ln , σ = ln , a random positive variable x is log-normally distributed if the logarithm of x is normally distributed, N =1 σ2 π exp . A change of variables must conserve differential probability, All moments of the log-normal distribution exist and E = e n μ + n 2 σ22 This can be derived by letting z = ln − σ within the integral. However, the expected value E is not defined for any value of the argument t as the defining integral diverges. In consequence the moment generating function is not defined, the last is related to the fact that the lognormal distribution is not uniquely determined by its moments. In consequence, the function of the log-normal distribution cannot be represented as an infinite convergent series. In particular, its Taylor formal series diverges, ∑ n =0 ∞ n n, a relatively simple approximating formula is available in closed form and given by φ ≈ exp 1 + W where W is the Lambert W function. This approximation is derived via a method but it stays sharp all over the domain of convergence of φ. The geometric mean of the distribution is G M = e μ. By analogy with the statistics, one can define a geometric variance, G V a r = e σ2
Log-normal distribution
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Fitted cumulative log-normal distribution to annually maximum 1-day rainfalls, see distribution fitting
Log-normal distribution
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Probability density function
15.
Skewness
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In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive or negative, or even undefined, the qualitative interpretation of the skew is complicated and unintuitive. Skew must not be thought to refer to the direction the curve appears to be leaning, in fact, conversely, positive skew indicates that the tail on the right side is longer or fatter than the left side. In cases where one tail is long but the tail is fat. Further, in multimodal distributions and discrete distributions, skewness is also difficult to interpret, importantly, the skewness does not determine the relationship of mean and median. In cases where it is necessary, data might be transformed to have a normal distribution, consider the two distributions in the figure just below. Within each graph, the values on the side of the distribution taper differently from the values on the left side. A left-skewed distribution usually appears as a right-leaning curve, positive skew, The right tail is longer, the mass of the distribution is concentrated on the left of the figure. A right-skewed distribution usually appears as a left-leaning curve, Skewness in a data series may sometimes be observed not only graphically but by simple inspection of the values. For instance, consider the sequence, whose values are evenly distributed around a central value of 50. If the distribution is symmetric, then the mean is equal to the median, if, in addition, the distribution is unimodal, then the mean = median = mode. This is the case of a coin toss or the series 1,2,3,4, note, however, that the converse is not true in general, i. e. zero skewness does not imply that the mean is equal to the median. Paul T. von Hippel points out, Many textbooks, teach a rule of thumb stating that the mean is right of the median under right skew and this rule fails with surprising frequency. It can fail in multimodal distributions, or in distributions where one tail is long, most commonly, though, the rule fails in discrete distributions where the areas to the left and right of the median are not equal. Such distributions not only contradict the textbook relationship between mean, median, and skew, they contradict the textbook interpretation of the median. It is sometimes referred to as Pearsons moment coefficient of skewness, or simply the moment coefficient of skewness, the last equality expresses skewness in terms of the ratio of the third cumulant κ3 to the 1. 5th power of the second cumulant κ2. This is analogous to the definition of kurtosis as the fourth cumulant normalized by the square of the second cumulant, the skewness is also sometimes denoted Skew. Starting from a standard cumulant expansion around a distribution, one can show that skewness =6 /standard deviation + O
Skewness
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Example distribution with non-zero (positive) skewness. These data are from experiments on wheat grass growth.
16.
Rotation (mathematics)
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Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a space that preserves at least one point. It can describe, for example, the motion of a body around a fixed point. A clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude, mathematically, a rotation is a map. All rotations about a fixed point form a group under composition called the rotation group, for example, in two dimensions rotating a body clockwise about a point keeping the axes fixed is equivalent to rotating the axes counterclockwise about the same point while the body is kept fixed. These two types of rotation are called active and passive transformations, the rotation group is a Lie group of rotations about a fixed point. This fixed point is called the center of rotation and is identified with the origin. The rotation group is a point stabilizer in a group of motions. For a particular rotation, The axis of rotation is a line of its fixed points and they exist only in n >2. The plane of rotation is a plane that is invariant under the rotation, unlike the axis, its points are not fixed themselves. The axis and the plane of a rotation are orthogonal, a representation of rotations is a particular formalism, either algebraic or geometric, used to parametrize a rotation map. This meaning is somehow inverse to the meaning in the group theory, rotations of spaces of points and of respective vector spaces are not always clearly distinguished. The former are sometimes referred to as affine rotations, whereas the latter are vector rotations, see the article below for details. A motion of a Euclidean space is the same as its isometry, but a rotation also has to preserve the orientation structure. The improper rotation term refers to isometries that reverse the orientation, in the language of group theory the distinction is expressed as direct vs indirect isometries in the Euclidean group, where the former comprise the identity component. Any direct Euclidean motion can be represented as a composition of a rotation about the fixed point, there are no non-trivial rotations in one dimension. In two dimensions, only a single angle is needed to specify a rotation about the origin – the angle of rotation that specifies an element of the circle group. The rotation is acting to rotate an object counterclockwise through an angle θ about the origin, composition of rotations sums their angles modulo 1 turn, which implies that all two-dimensional rotations about the same point commute
Rotation (mathematics)
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Rotation of an object in two dimensions around a point O.
17.
Quadratic mean
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In statistics and its applications, the root mean square is defined as the square root of mean square. The RMS is also known as the mean and is a particular case of the generalized mean with exponent 2. RMS can also be defined for a continuously varying function in terms of an integral of the squares of the values during a cycle. For a cyclically alternating electric current, RMS is equal to the value of the current that would produce the same average power dissipation in a resistive load. In Estimation theory the mean square error of an estimator is a measure of the imperfection of the fit of the estimator to the data. The RMS value of a set of values is the root of the arithmetic mean of the squares of the values. In Physics, the RMS current is the value of the current that dissipates power in a resistor. In the case of a set of n values, the RMS x r m s =1 n, the RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a function or signal can be approximated by taking the RMS of a sequence of equally spaced samples. Additionally, the RMS value of various waveforms can also be determined without calculus, in the case of the RMS statistic of a random process, the expected value is used instead of the mean. If the waveform is a sine wave, the relationships between amplitudes and RMS are fixed and known, as they are for any continuous periodic wave. However, this is not true for a waveform which may or may not be periodic or continuous. For a zero-mean sine wave, the relationship between RMS and peak-to-peak amplitude is, Peak-to-peak =22 × R M S ≈2.8 × R M S, for other waveforms the relationships are not the same as they are for sine waves. Waveforms made by summing known simple waveforms have an RMS that is the root of the sum of squares of the component RMS values, if the component waveforms are orthogonal. R M S Total = R M S12 + R M S22 + ⋯ + R M S n 2 A special case of this, another special case, useful in statistics, is given in #Relationship to other statistics. Electrical engineers often need to know the power, P, dissipated by an electrical resistance and it is easy to do the calculation when there is a constant current, I, through the resistance. Average power can also be using the same method that in the case of a time-varying voltage, V, with RMS value VRMS. This equation can be used for any waveform, such as a sinusoidal or sawtooth waveform
Quadratic mean
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Sine, square, triangle, and sawtooth waveforms
18.
Generalized mean
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In mathematics, generalized means are a family of functions for aggregating sets of numbers, that include as special cases the Pythagorean means. The generalized mean is known as power mean or Hölder mean. If p is a real number, and x 1, …, x n are positive real numbers, then the generalized mean or power mean with exponent p of these positive real numbers is. Note the relationship to the p-norm, the generalized mean always lies between the smallest and largest of the x values. The generalized mean is a function of its arguments, permuting the arguments of a generalized mean does not change its value. Like most means, the mean is a homogeneous function of its arguments x1. Like the quasi-arithmetic means, the computation of the mean can be split into computations of equal sized sub-blocks. M p = M p In general, if p < q, then M p ≤ M q, the inequality is true for real values of p and q, as well as positive and negative infinity values. It follows from the fact that, for all p, ∂ ∂ p M p ≥0 which can be proved using Jensens inequality. In particular, for p in, the generalized mean inequality implies the Pythagorean means inequality as well as the inequality of arithmetic and geometric means. F is a function, so it does have a second derivative, f ″ = x q p −2 which is strictly positive within the domain of f, since q > p. The power mean could be generalized further to the generalized f-mean, the power mean is obtained for f = xp. A power mean serves a non-linear moving average which is shifted towards small signal values for small p, given an efficient implementation of a moving arithmetic mean called smooth one can implement a moving power mean according to the following Haskell code. For big p it can serve an envelope detector on a rectified signal, for small p it can serve an baseline detector on a mass spectrum
Generalized mean
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A visual depiction of some of the specified cases for with, harmonic mean, geometric mean, arithmetic mean and quadratic mean.
19.
Interquartile range
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In other words, the IQR is the 1st quartile subtracted from the 3rd quartile, these quartiles can be clearly seen on a box plot on the data. It is an estimator, defined as the 25% trimmed range. The interquartile range is a measure of variability, based on dividing a set into quartiles. Quartiles divide a rank-ordered data set into four equal parts, the values that separate parts are called the first, second, and third quartiles, and they are denoted by Q1, Q2, and Q3, respectively. Unlike total range, the range has a breakdown point of 25%. The IQR is used to box plots, simple graphical representations of a probability distribution. For a symmetric distribution, half the IQR equals the median absolute deviation, the median is the corresponding measure of central tendency. The IQR can be used to identify outliers, the quartile deviation or semi-interquartile range is defined as half the IQR. If P is normally distributed, then the score of the first quartile, z1, is -0.67. However, a distribution can be trivially perturbed to maintain its Q1 and Q2 std. scores at 0.67 and -0.67. A better test of normality, such as Q-Q plot would be indicated here, the interquartile range is often used to find outliers in data. Outliers here are defined as observations that fall below Q1 −1.5 IQR or above Q3 +1.5 IQR, in a boxplot, the highest and lowest occurring value within this limit are indicated by whiskers of the box and any outliers as individual points. Midhinge Interdecile range Robust measures of scale
Interquartile range
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Boxplot (with an interquartile range) and a probability density function (pdf) of a Normal N(0,σ 2) Population
20.
Continuous function
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In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function, a continuous function with a continuous inverse function is called a homeomorphism. Continuity of functions is one of the concepts of topology. The introductory portion of this focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the general case of functions between two metric spaces. In order theory, especially in theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article, as an example, consider the function h, which describes the height of a growing flower at time t. By contrast, if M denotes the amount of money in an account at time t, then the function jumps at each point in time when money is deposited or withdrawn. A form of the definition of continuity was first given by Bernard Bolzano in 1817. Cauchy defined infinitely small quantities in terms of quantities. The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s but the work wasnt published until the 1930s, all three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of continuity in 1872. This is not a definition of continuity since the function f =1 x is continuous on its whole domain of R ∖ A function is continuous at a point if it does not have a hole or jump. A “hole” or “jump” in the graph of a function if the value of the function at a point c differs from its limiting value along points that are nearby. Such a point is called a discontinuity, a function is then continuous if it has no holes or jumps, that is, if it is continuous at every point of its domain. Otherwise, a function is discontinuous, at the points where the value of the function differs from its limiting value, there are several ways to make this definition mathematically rigorous. These definitions are equivalent to one another, so the most convenient definition can be used to determine whether a function is continuous or not. In the definitions below, f, I → R. is a function defined on a subset I of the set R of real numbers and this subset I is referred to as the domain of f
Continuous function
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Illustration of the ε-δ-definition: for ε=0.5, c=2, the value δ=0.5 satisfies the condition of the definition.
21.
Monotonicity
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In mathematics, a monotonic function is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was generalized to the more abstract setting of order theory. In calculus, a function f defined on a subset of the numbers with real values is called monotonic if. That is, as per Fig.1, a function that increases monotonically does not exclusively have to increase, a function is called monotonically increasing, if for all x and y such that x ≤ y one has f ≤ f, so f preserves the order. Likewise, a function is called monotonically decreasing if, whenever x ≤ y, then f ≥ f, if the order ≤ in the definition of monotonicity is replaced by the strict order <, then one obtains a stronger requirement. A function with this property is called strictly increasing, again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing. The terms non-decreasing and non-increasing should not be confused with the negative qualifications not decreasing, for example, the function of figure 3 first falls, then rises, then falls again. It is therefore not decreasing and not increasing, but it is neither non-decreasing nor non-increasing, the term monotonic transformation can also possibly cause some confusion because it refers to a transformation by a strictly increasing function. Notably, this is the case in economics with respect to the properties of a utility function being preserved across a monotonic transform. A function f is said to be absolutely monotonic over an interval if the derivatives of all orders of f are nonnegative or all nonpositive at all points on the interval, F can only have jump discontinuities, f can only have countably many discontinuities in its domain. The discontinuities, however, do not necessarily consist of isolated points and these properties are the reason why monotonic functions are useful in technical work in analysis. In addition, this result cannot be improved to countable, see Cantor function, if f is a monotonic function defined on an interval, then f is Riemann integrable. An important application of functions is in probability theory. If X is a variable, its cumulative distribution function F X = Prob is a monotonically increasing function. A function is unimodal if it is monotonically increasing up to some point, when f is a strictly monotonic function, then f is injective on its domain, and if T is the range of f, then there is an inverse function on T for f. A map f, X → Y is said to be if each of its fibers is connected i. e. for each element y in Y the set f−1 is connected. A subset G of X × X∗ is said to be a set if for every pair. G is said to be monotone if it is maximal among all monotone sets in the sense of set inclusion
Monotonicity
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Figure 1. A monotonically increasing function. It is strictly increasing on the left and right while just non-decreasing in the middle.
22.
Permutation
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These differ from combinations, which are selections of some members of a set where order is disregarded. For example, written as tuples, there are six permutations of the set, namely and these are all the possible orderings of this three element set. As another example, an anagram of a word, all of whose letters are different, is a permutation of its letters, in this example, the letters are already ordered in the original word and the anagram is a reordering of the letters. The study of permutations of finite sets is a topic in the field of combinatorics, Permutations occur, in more or less prominent ways, in almost every area of mathematics. For similar reasons permutations arise in the study of sorting algorithms in computer science, the number of permutations of n distinct objects is n factorial, usually written as n. which means the product of all positive integers less than or equal to n. In algebra and particularly in group theory, a permutation of a set S is defined as a bijection from S to itself and that is, it is a function from S to S for which every element occurs exactly once as an image value. This is related to the rearrangement of the elements of S in which each element s is replaced by the corresponding f, the collection of such permutations form a group called the symmetric group of S. The key to this structure is the fact that the composition of two permutations results in another rearrangement. Permutations may act on structured objects by rearranging their components, or by certain replacements of symbols, in elementary combinatorics, the k-permutations, or partial permutations, are the ordered arrangements of k distinct elements selected from a set. When k is equal to the size of the set, these are the permutations of the set, fabian Stedman in 1677 described factorials when explaining the number of permutations of bells in change ringing. Starting from two bells, first, two must be admitted to be varied in two ways which he illustrates by showing 12 and 21 and he then explains that with three bells there are three times two figures to be produced out of three which again is illustrated. His explanation involves cast away 3, and 1.2 will remain, cast away 2, and 1.3 will remain, cast away 1, and 2.3 will remain. He then moves on to four bells and repeats the casting away argument showing that there will be four different sets of three, effectively this is an recursive process. He continues with five bells using the casting method and tabulates the resulting 120 combinations. At this point he gives up and remarks, Now the nature of these methods is such, in modern mathematics there are many similar situations in which understanding a problem requires studying certain permutations related to it. There are two equivalent common ways of regarding permutations, sometimes called the active and passive forms, or in older terminology substitutions and permutations, which form is preferable depends on the type of questions being asked in a given discipline. The active way to regard permutations of a set S is to them as the bijections from S to itself. Thus, the permutations are thought of as functions which can be composed with each other, forming groups of permutations
Permutation
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In the 15 puzzle the goal is to get the squares in ascending order. Initial positions which have an odd number of inversions are impossible to solve.
Permutation
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In the popular puzzle Rubik's cube invented in 1974 by Ernő Rubik, each turn of the puzzle faces creates a permutation of the surface colors.
Permutation
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Biologist and statistician Ronald Fisher
23.
Moving average
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In statistics, a moving average is a calculation to analyze data points by creating series of averages of different subsets of the full data set. It is also called a mean or rolling mean and is a type of finite impulse response filter. Variations include, simple, and cumulative, or weighted forms, given a series of numbers and a fixed subset size, the first element of the moving average is obtained by taking the average of the initial fixed subset of the number series. Then the subset is modified by shifting forward, that is, excluding the first number of the series, a moving average is commonly used with time series data to smooth out short-term fluctuations and highlight longer-term trends or cycles. The threshold between short-term and long-term depends on the application, and the parameters of the average will be set accordingly. For example, it is used in technical analysis of financial data, like stock prices. It is also used in economics to examine gross domestic product, mathematically, a moving average is a type of convolution and so it can be viewed as an example of a low-pass filter used in signal processing. When used with non-time series data, a moving average filters higher frequency components without any connection to time. Viewed simplistically it can be regarded as smoothing the data, in financial applications a simple moving average is the unweighted mean of the previous n data. However, in science and engineering the mean is taken from an equal number of data on either side of a central value. This ensures that variations in the mean are aligned with the variations in the rather than being shifted in time. An example of an equally weighted running mean for a n-day sample of closing price is the mean of the previous n days closing prices. In financial terms moving-average levels can be interpreted as support in a falling market, if the data used are not centered around the mean, a simple moving average lags behind the latest datum point by half the sample width. An SMA can also be influenced by old datum points dropping out or new data coming in. One characteristic of the SMA is that if the data have a periodic fluctuation, but a perfectly regular cycle is rarely encountered. For a number of applications, it is advantageous to avoid the shifting induced by using only past data, hence a central moving average can be computed, using data equally spaced on either side of the point in the series where the mean is calculated. This requires using an odd number of points in the sample window. A major drawback of the SMA is that it lets through a significant amount of the shorter than the window length
Moving average
Moving average
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For other uses, see Moving average (disambiguation).
24.
Time series
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A time series is a series of data points indexed in time order. Most commonly, a series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data, examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average. Time series are very frequently plotted via line charts, Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values, Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in there is no natural ordering of the observations. Time series analysis is also distinct from data analysis where the observations typically relate to geographical locations. A stochastic model for a series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. Methods for time series analysis may be divided into two classes, frequency-domain methods and time-domain methods, the former include spectral analysis and wavelet analysis, the latter include auto-correlation and cross-correlation analysis. In the time domain, correlation and analysis can be made in a filter-like manner using scaled correlation, additionally, time series analysis techniques may be divided into parametric and non-parametric methods. The parametric approaches assume that the stationary stochastic process has a certain structure which can be described using a small number of parameters. In these approaches, the task is to estimate the parameters of the model describes the stochastic process. By contrast, non-parametric approaches explicitly estimate the covariance or the spectrum of the process without assuming that the process has any particular structure, Methods of time series analysis may also be divided into linear and non-linear, and univariate and multivariate. A time series is one type of Panel data, Panel data is the general class, a multidimensional data set, whereas a time series data set is a one-dimensional panel. A data set may exhibit characteristics of both data and time series data. One way to tell is to ask what makes one data record unique from the other records, if the answer is the time data field, then this is a time series data set candidate. If determining a unique record requires a data field and an additional identifier which is unrelated to time. If the differentiation lies on the identifier, then the data set is a cross-sectional data set candidate
Time series
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Time series: random data plus trend, with best-fit line and different applied filters
25.
Digital filter
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In signal processing, a digital filter is a system that performs mathematical operations on a sampled, discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the major type of electronic filter, the analog filter. Finally a digital-to-analog converter to complete the output stage, program Instructions running on the microprocessor implement the digital filter by performing the necessary mathematical operations on the numbers received from the ADC. Digital filters may be more expensive than an equivalent analog filter due to their increased complexity, Digital filters are commonplace and an essential element of everyday electronics such as radios, cellphones, and AV receivers. A digital filter is characterized by its function, or equivalently. Mathematical analysis of the function can describe how it will respond to any input. As such, designing a filter consists of developing specifications appropriate to the problem, see Z-transforms LCCD equation for further discussion of this transfer function. A variety of techniques may be employed to analyze the behaviour of a given digital filter. Many of these techniques may also be employed in designs. Typically, one characterizes filters by calculating how they respond to a simple input such as an impulse. One can then extend this information to compute the filters response to more complex signals, the impulse response, often denoted h or h k, is a measurement of how a filter will respond to the Kronecker delta function. For example, given an equation, one would set x 0 =1 and x k =0 for k ≠0. The impulse response is a characterization of the filters behaviour, Digital filters are typically considered in two categories, infinite impulse response and finite impulse response. In discrete-time systems, the filter is often implemented by converting the transfer function to a linear constant-coefficient difference equation via the Z-transform. The discrete frequency-domain transfer function is written as the ratio of two polynomials, applying the filter to an input in this form is equivalent to a Direct Form I or II realization, depending on the exact order of evaluation. The design of filters is a deceptively complex topic. Although filters are easily understood and calculated, the challenges of their design. There are two categories of digital filter, the filter and the nonrecursive filter
Digital filter
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A general finite impulse response filter with n stages, each with an independent delay, d i, and amplification gain, a i.
26.
Digital signal processing
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Digital signal processing is the use of digital processing, such as by computers, to perform a wide variety of signal processing operations. The signals processed in this manner are a sequence of numbers that represent samples of a variable in a domain such as time, space. Digital signal processing and analog signal processing are subfields of signal processing, digital signal processing can involve linear or nonlinear operations. Nonlinear signal processing is closely related to system identification and can be implemented in the time, frequency. DSP is applicable to both streaming data and static data, the increasing use of computers has resulted in the increased use of, and need for, digital signal processing. To digitally analyze and manipulate an analog signal, it must be digitized with an analog-to-digital converter, sampling is usually carried out in two stages, discretization and quantization. Discretization means that the signal is divided into equal intervals of time, quantization means each amplitude measurement is approximated by a value from a finite set. Rounding real numbers to integers is an example, the Nyquist–Shannon sampling theorem states that a signal can be exactly reconstructed from its samples if the sampling frequency is greater than twice the highest frequency of the signal. In practice, the frequency is often significantly higher than twice that required by the signals limited bandwidth. Theoretical DSP analyses and derivations are typically performed on discrete-time signal models with no amplitude inaccuracies, numerical methods require a quantized signal, such as those produced by an analog-to-digital converter. The processed result might be a frequency spectrum or a set of statistics, but often it is another quantized signal that is converted back to analog form by a digital-to-analog converter. In DSP, engineers usually study digital signals in one of the domains, time domain, spatial domain, frequency domain. They choose the domain in which to process a signal by making an assumption as to which domain best represents the essential characteristics of the signal. The most common processing approach in the time or space domain is enhancement of the signal through a method called filtering. Digital filtering generally consists of linear transformation of a number of surrounding samples around the current sample of the input or output signal. There are various ways to characterize filters, for example, A linear filter is a transformation of input samples. A causal filter uses only samples of the input or output signals. A non-causal filter can usually be changed into a filter by adding a delay to it
Digital signal processing
27.
Estimation
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Estimation is the process of finding an estimate, or approximation, which is a value that is usable for some purpose even if input data may be incomplete, uncertain, or unstable. The value is nonetheless usable because it is derived from the best information available, typically, estimation involves using the value of a statistic derived from a sample to estimate the value of a corresponding population parameter. The sample provides information that can be projected, through formal or informal processes. An estimate that out to be incorrect will be an overestimate if the estimate exceeded the actual result. Estimation is often done by sampling, which is counting a number of examples something. An example of estimation would be determining how many candies of a given size are in a glass jar, estimates can similarly be generated by projecting results from polls or surveys onto the entire population. In making an estimate, the goal is often most useful to generate a range of outcomes that is precise enough to be useful. Such a projection, intended to pick the single value that is believed to be closest to the value, is called a point estimate. A corresponding concept is an estimate, which captures a much larger range of possibilities. For example, if one were asked to estimate the percentage of people who like candy, such an estimate would provide no guidance, however, to somebody who is trying to determine how many candies to buy for a party to be attended by a hundred people. In statistics, an estimator is the name for the rule by which an estimate is calculated from data. This process is used in processing, for approximating an unobserved signal on the basis of an observed signal containing noise. For estimation of yet-to-be observed quantities, forecasting and prediction are applied, estimation is important in business and economics, because too many variables exist to figure out how large-scale activities will develop. An informal estimate when little information is available is called a guesstimate, the estimated sign, ℮, is used to designate that package contents are close to the nominal contents
Estimation
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The exact number of candies in this jar cannot be determined by looking at it, because most of the candies are not visible. The amount can be estimated by presuming that the portion of the jar that cannot be seen contains an amount equivalent to the amount contained in the same volume for the portion that can be seen.
28.
Oxford English Dictionary
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The Oxford English Dictionary is a descriptive dictionary of the English language, published by the Oxford University Press. The second edition came to 21,728 pages in 20 volumes, in 1895, the title The Oxford English Dictionary was first used unofficially on the covers of the series, and in 1928 the full dictionary was republished in ten bound volumes. In 1933, the title The Oxford English Dictionary fully replaced the name in all occurrences in its reprinting as twelve volumes with a one-volume supplement. More supplements came over the years until 1989, when the edition was published. Since 2000, an edition of the dictionary has been underway. The first electronic version of the dictionary was available in 1988. The online version has been available since 2000, and as of April 2014 was receiving two million hits per month. The third edition of the dictionary will probably appear in electronic form, Nigel Portwood, chief executive of Oxford University Press. As a historical dictionary, the Oxford English Dictionary explains words by showing their development rather than merely their present-day usages, therefore, it shows definitions in the order that the sense of the word began being used, including word meanings which are no longer used. The format of the OEDs entries has influenced numerous other historical lexicography projects and this influenced later volumes of this and other lexicographical works. As of 30 November 2005, the Oxford English Dictionary contained approximately 301,100 main entries, the dictionarys latest, complete print edition was printed in 20 volumes, comprising 291,500 entries in 21,730 pages. The longest entry in the OED2 was for the verb set, as entries began to be revised for the OED3 in sequence starting from M, the longest entry became make in 2000, then put in 2007, then run in 2011. Despite its impressive size, the OED is neither the worlds largest nor the earliest exhaustive dictionary of a language, the Dutch dictionary Woordenboek der Nederlandsche Taal is the worlds largest dictionary, has similar aims to the OED and took twice as long to complete. Another earlier large dictionary is the Grimm brothers dictionary of the German language, begun in 1838, the official dictionary of Spanish is the Diccionario de la lengua española, and its first edition was published in 1780. The Kangxi dictionary of Chinese was published in 1716, trench suggested that a new, truly comprehensive dictionary was needed. On 7 January 1858, the Society formally adopted the idea of a new dictionary. Volunteer readers would be assigned particular books, copying passages illustrating word usage onto quotation slips, later the same year, the Society agreed to the project in principle, with the title A New English Dictionary on Historical Principles. He withdrew and Herbert Coleridge became the first editor, on 12 May 1860, Coleridges dictionary plan was published and research was started
Oxford English Dictionary
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Seven of the twenty volumes of the printed version of the second edition of the OED
Oxford English Dictionary
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Frederick Furnivall, 1825–1910
Oxford English Dictionary
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James Murray in the Scriptorium at Banbury Road
Oxford English Dictionary
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The 78 Banbury Road, Oxford, house, erstwhile residence of James Murray, Editor of the Oxford English Dictionary
29.
General average
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In the exigencies of hazards faced at sea, crew members often have precious little time in which to determine precisely whose cargo they are jettisoning. While general average traces its origins in ancient maritime law, still it remains part of the admiralty law of most countries, a form of what is now called general average was included in the Lex Rhodia, the Rhodes Maritime Code of circa 800 BC. The first codification of general average was the York Antwerp Rules of 1890, american companies accepted it in 1949. General average requires three elements which are stated by Justice Grier in Barnard v. Adams, 1st. This attempt to avoid the imminent common peril must be successful, the York-Antwerp Rules remain in effect, having been modified and updated several times since their 1890 introduction. Despite advances in transport technology, general average continues to come into play. The M/V MSC Sabrina declared general average in effect after grounding in the Saint Lawrence river on 8 March 2008, the owners of the Hanjin Osaka declared general average following an explosion in the ships engine room on 8 January 2012. Cooke, Julian, ed. Lowndes & Rudolf, The Law of General Average, definition of General Average, Duhaimes Law Dictionary Carver, Thomas Gilbert
General average
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The owners of Hanjin Osaka, seen here transiting the Panama Canal in 2008, declared general average following an explosion in the ship's engine room on 8 January 2012.
30.
Draught animal
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A working animal is an animal, usually domesticated, that is kept by humans and trained to perform tasks. They may be members of the family, such as guide dogs or other assistance dogs, or they may be animals trained to provide tractive force. The latter types of animals are called animals or beasts of burden. Most working animals are either service animals or draft animals and they may also be used for milking or herding, jobs that require human training to encourage the animal to cooperate. Some, at the end of their lives, may also be used for meat or other products such as leather. The history of working animals may predate agriculture, with used by our hunter-gatherer ancestors. Around the world, millions of animals work in relationship with their owners, domesticated species are often bred to be suitable for different uses and conditions, especially horses and working dogs. Working animals are raised on farms, though some are still captured from the wild, such as dolphins. People have found uses for a variety of abilities found in animals. The strength of horses, elephants and oxen is used in pulling carts, the keen sense of smell of dogs is used to search for drugs and explosives as well helping to find game while hunting and to search for missing or trapped people. Several animals including camels, donkeys, horses and dogs are used for transport, either riding or to pull wagons, other animals including dogs and monkeys provide assistance to blind or disabled people. Conversely, not all domesticated animals are working animals, for example, while cats may perform work catching mice, it is an instinctive behavior, not one that can be trained by human intervention. Other domesticated animals, such as sheep, or rabbits, may have uses for meat, hides and wool. Finally, small pets such as most birds or hamsters are generally incapable of performing work other than that of providing simple companionship. Some animals are used due to physical strength in tasks such as ploughing or logging. Such animals are grouped as a draught or draft animal, others may be used as pack animals, for animal-powered transport, the movement of people and goods. People ride some animals directly as mounts, Alternatively, one or more animals in harness may be used to pull vehicles and they include equines such as horses, ponies, donkeys, and mules, elephants, yaks, and camels. Dromedary camels in arid areas of Australia, North Africa and the Middle East, on occasion, reindeer, though usually driven, may be ridden
Draught animal
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A bullock team hauling wool in New South Wales
Draught animal
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Traditional Farming Methods using Oxen
Draught animal
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The horse-drawn winch of a former limestone quarry (France)
Draught animal
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A pack llama
31.
Domesday Book
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Domesday Book is a manuscript record of the Great Survey of much of England and parts of Wales completed in 1086 by order of King William the Conqueror. The Anglo-Saxon Chronicle states, Then, at the midwinter, was the king in Glocester with his council. After this had the king a large meeting, and very deep consultation with his council, about this land, how it was occupied and it was written in Medieval Latin, was highly abbreviated, and included some vernacular native terms without Latin equivalents. The assessors reckoning of a mans holdings and their values, as recorded in Domesday Book, was dispositive, the name Domesday Book came into use in the 12th century. As Richard FitzNeal wrote in the Dialogus de Scaccario, for as the sentence of that strict and terrible last account cannot be evaded by any skilful subterfuge and its sentence cannot be quashed or set aside with impunity. That is why we have called the book the Book of Judgement, because its decisions, like those of the Last Judgement, are unalterable. The manuscript is held at The National Archives at Kew, London, in 2011, the Open Domesday site made the manuscript available online. The book is a primary source for modern historians and historical economists. Domesday Book encompasses two independent works, Little Domesday and Great Domesday, no surveys were made of the City of London, Winchester, or some other towns, probably due to their tax-exempt status. Most of Cumberland and Westmorland are missing, the omission of the other counties and towns is not fully explained, although in particular Cumberland and Westmorland had yet to be fully conquered. Little Domesday – so named because its format is smaller than its companions – is the more detailed survey. It may have represented the first attempt, resulting in a decision to avoid such level of detail in Great Domesday, some of the largest such magnates held several hundred fees, in a few cases in more than one county. For example, the chapter of the Domesday Book Devonshire section concerning Baldwin the Sheriff lists 176 holdings held in-chief by him, as a review of taxes owed, it was highly unpopular. Each countys list opened with the demesne lands. It should be borne in mind that under the system the king was the only true owner of land in England. He was thus the ultimate overlord and even the greatest magnate could do no more than hold land from him as a tenant under one of the contracts of feudal land tenure. In some counties, one or more principal towns formed the subject of a separate section and this principle applies more specially to the larger volume, in the smaller one, the system is more confused, the execution less perfect. Domesday names a total of 13,418 places and these include fragments of custumals, records of the military service due, of markets, mints, and so forth
Domesday Book
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Domesday Book: an engraving published in 1900. Great Domesday (the larger volume) and Little Domesday (the smaller volume), in their 1869 bindings, lying on their older " Tudor " bindings.
Domesday Book
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Great Domesday in its " Tudor " binding: a wood-engraving of the 1860s
Domesday Book
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Domesday chest, the German-style iron-bound chest of c.1500 in which Domesday Book was kept in the 17th and 18th centuries
Domesday Book
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Entries for Croydon and Cheam, Surrey, in the 1783 edition of Domesday Book
32.
Expected value
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In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the experiment it represents. For example, the value in rolling a six-sided die is 3.5. Less roughly, the law of large states that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions approaches infinity. The expected value is known as the expectation, mathematical expectation, EV, average, mean value, mean. More practically, the value of a discrete random variable is the probability-weighted average of all possible values. In other words, each value the random variable can assume is multiplied by its probability of occurring. The same principle applies to a random variable, except that an integral of the variable with respect to its probability density replaces the sum. The expected value does not exist for random variables having some distributions with large tails, for random variables such as these, the long-tails of the distribution prevent the sum/integral from converging. The expected value is a key aspect of how one characterizes a probability distribution, by contrast, the variance is a measure of dispersion of the possible values of the random variable around the expected value. The variance itself is defined in terms of two expectations, it is the value of the squared deviation of the variables value from the variables expected value. The expected value plays important roles in a variety of contexts, in regression analysis, one desires a formula in terms of observed data that will give a good estimate of the parameter giving the effect of some explanatory variable upon a dependent variable. The formula will give different estimates using different samples of data, a formula is typically considered good in this context if it is an unbiased estimator—that is, if the expected value of the estimate can be shown to equal the true value of the desired parameter. In decision theory, and in particular in choice under uncertainty, one example of using expected value in reaching optimal decisions is the Gordon–Loeb model of information security investment. According to the model, one can conclude that the amount a firm spends to protect information should generally be only a fraction of the expected loss. Suppose random variable X can take value x1 with probability p1, value x2 with probability p2, then the expectation of this random variable X is defined as E = x 1 p 1 + x 2 p 2 + ⋯ + x k p k. If all outcomes xi are equally likely, then the weighted average turns into the simple average and this is intuitive, the expected value of a random variable is the average of all values it can take, thus the expected value is what one expects to happen on average. If the outcomes xi are not equally probable, then the simple average must be replaced with the weighted average, the intuition however remains the same, the expected value of X is what one expects to happen on average. Let X represent the outcome of a roll of a fair six-sided die, more specifically, X will be the number of pips showing on the top face of the die after the toss
Expected value
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An illustration of the convergence of sequence averages of rolls of a die to the expected value of 3.5 as the number of rolls (trials) grows.
33.
Central limit theorem
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If this procedure is performed many times, the central limit theorem says that the computed values of the average will be distributed according to the normal distribution. The central limit theorem has a number of variants, in its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations, in more general usage, a central limit theorem is any of a set of weak-convergence theorems in probability theory. When the variance of the i. i. d, Variables is finite, the attractor distribution is the normal distribution. In contrast, the sum of a number of i. i. d, Random variables with power law tail distributions decreasing as | x |−α −1 where 0 < α <2 will tend to an alpha-stable distribution with stability parameter of α as the number of variables grows. Suppose we are interested in the sample average S n, = X1 + ⋯ + X n n of these random variables, by the law of large numbers, the sample averages converge in probability and almost surely to the expected value µ as n → ∞. The classical central limit theorem describes the size and the form of the stochastic fluctuations around the deterministic number µ during this convergence. For large enough n, the distribution of Sn is close to the distribution with mean µ. The usefulness of the theorem is that the distribution of √n approaches normality regardless of the shape of the distribution of the individual Xi, formally, the theorem can be stated as follows, Lindeberg–Lévy CLT. Suppose is a sequence of i. i. d, Random variables with E = µ and Var = σ2 < ∞. Then as n approaches infinity, the random variables √n converge in distribution to a normal N, n → d N. Note that the convergence is uniform in z in the sense that lim n → ∞ sup z ∈ R | Pr − Φ | =0, the theorem is named after Russian mathematician Aleksandr Lyapunov. In this variant of the limit theorem the random variables Xi have to be independent. The theorem also requires that random variables | Xi | have moments of order. Suppose is a sequence of independent random variables, each with finite expected value μi, in practice it is usually easiest to check Lyapunov’s condition for δ =1. If a sequence of random variables satisfies Lyapunov’s condition, then it also satisfies Lindeberg’s condition, the converse implication, however, does not hold. In the same setting and with the notation as above. Suppose that for every ε >0 lim n → ∞1 s n 2 ∑ i =1 n E =0 where 1 is the indicator function
Central limit theorem
Central limit theorem
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A distribution being "smoothed out" by summation, showing original density of distribution and three subsequent summations; see Illustration of the central limit theorem for further details.
34.
International Standard Serial Number
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An International Standard Serial Number is an eight-digit serial number used to uniquely identify a serial publication. The ISSN is especially helpful in distinguishing between serials with the same title, ISSN are used in ordering, cataloging, interlibrary loans, and other practices in connection with serial literature. The ISSN system was first drafted as an International Organization for Standardization international standard in 1971, ISO subcommittee TC 46/SC9 is responsible for maintaining the standard. When a serial with the content is published in more than one media type. For example, many serials are published both in print and electronic media, the ISSN system refers to these types as print ISSN and electronic ISSN, respectively. The format of the ISSN is an eight digit code, divided by a hyphen into two four-digit numbers, as an integer number, it can be represented by the first seven digits. The last code digit, which may be 0-9 or an X, is a check digit. Formally, the form of the ISSN code can be expressed as follows, NNNN-NNNC where N is in the set, a digit character. The ISSN of the journal Hearing Research, for example, is 0378-5955, where the final 5 is the check digit, for calculations, an upper case X in the check digit position indicates a check digit of 10. To confirm the check digit, calculate the sum of all eight digits of the ISSN multiplied by its position in the number, the modulus 11 of the sum must be 0. There is an online ISSN checker that can validate an ISSN, ISSN codes are assigned by a network of ISSN National Centres, usually located at national libraries and coordinated by the ISSN International Centre based in Paris. The International Centre is an organization created in 1974 through an agreement between UNESCO and the French government. The International Centre maintains a database of all ISSNs assigned worldwide, at the end of 2016, the ISSN Register contained records for 1,943,572 items. ISSN and ISBN codes are similar in concept, where ISBNs are assigned to individual books, an ISBN might be assigned for particular issues of a serial, in addition to the ISSN code for the serial as a whole. An ISSN, unlike the ISBN code, is an identifier associated with a serial title. For this reason a new ISSN is assigned to a serial each time it undergoes a major title change, separate ISSNs are needed for serials in different media. Thus, the print and electronic versions of a serial need separate ISSNs. Also, a CD-ROM version and a web version of a serial require different ISSNs since two different media are involved, however, the same ISSN can be used for different file formats of the same online serial
International Standard Serial Number
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ISSN encoded in an EAN-13 barcode with sequence variant 0 and issue number 5
35.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
International Standard Book Number
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A 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar code
36.
Oxford University Press
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Oxford University Press is the largest university press in the world, and the second oldest after Cambridge University Press. It is a department of the University of Oxford and is governed by a group of 15 academics appointed by the known as the delegates of the press. They are headed by the secretary to the delegates, who serves as OUPs chief executive, Oxford University has used a similar system to oversee OUP since the 17th century. The university became involved in the print trade around 1480, and grew into a printer of Bibles, prayer books. OUP took on the project became the Oxford English Dictionary in the late 19th century. Moves into international markets led to OUP opening its own offices outside the United Kingdom, by contracting out its printing and binding operations, the modern OUP publishes some 6,000 new titles around the world each year. OUP was first exempted from United States corporation tax in 1972, as a department of a charity, OUP is exempt from income tax and corporate tax in most countries, but may pay sales and other commercial taxes on its products. The OUP today transfers 30% of its surplus to the rest of the university. OUP is the largest university press in the world by the number of publications, publishing more than 6,000 new books every year, the Oxford University Press Museum is located on Great Clarendon Street, Oxford. Visits must be booked in advance and are led by a member of the archive staff, displays include a 19th-century printing press, the OUP buildings, and the printing and history of the Oxford Almanack, Alice in Wonderland and the Oxford English Dictionary. The first printer associated with Oxford University was Theoderic Rood, the first book printed in Oxford, in 1478, an edition of Rufinuss Expositio in symbolum apostolorum, was printed by another, anonymous, printer. Famously, this was mis-dated in Roman numerals as 1468, thus apparently pre-dating Caxton, roods printing included John Ankywylls Compendium totius grammaticae, which set new standards for teaching of Latin grammar. After Rood, printing connected with the university remained sporadic for over half a century, the chancellor, Robert Dudley, 1st Earl of Leicester, pleaded Oxfords case. Some royal assent was obtained, since the printer Joseph Barnes began work, Oxfords chancellor, Archbishop William Laud, consolidated the legal status of the universitys printing in the 1630s. Laud envisaged a unified press of world repute, Oxford would establish it on university property, govern its operations, employ its staff, determine its printed work, and benefit from its proceeds. To that end, he petitioned Charles I for rights that would enable Oxford to compete with the Stationers Company and the Kings Printer and these were brought together in Oxfords Great Charter in 1636, which gave the university the right to print all manner of books. Laud also obtained the privilege from the Crown of printing the King James or Authorized Version of Scripture at Oxford and this privilege created substantial returns in the next 250 years, although initially it was held in abeyance. The Stationers Company was deeply alarmed by the threat to its trade, under this, the Stationers paid an annual rent for the university not to exercise its full printing rights – money Oxford used to purchase new printing equipment for smaller purposes
Oxford University Press
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Oxford University Press on Walton Street.
Oxford University Press
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2008 conference booth
37.
John Ray
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John Ray was an English naturalist widely regarded as one of the earliest of the English parson-naturalists. Until 1670, he wrote his name as John Wray, from then on, he used Ray, after having ascertained that such had been the practice of his family before him. He published important works on botany, zoology, and natural theology and his classification of plants in his Historia Plantarum, was an important step towards modern taxonomy. He was the first to give a definition of the term species. John Ray was born in the village of Black Notley in Essex and he is said to have been born in the smithy, his father having been the village blacksmith. He was sent at the age of sixteen to Cambridge University, studying at Trinity College and his tutor at Trinity was James Duport, and his intimate friend and fellow-pupil the celebrated Isaac Barrow. Ray was chosen fellow of Trinity in 1649, and later major fellow. Among these sermons were his discourses on The wisdom of God manifested in the works of the creation, Ray was also highly regarded as a tutor. He communicated his own passion for history to several pupils. His religious views were generally in accord with those imposed under the restoration of Charles II of England, from this time onwards he seems to have depended chiefly on the bounty of his pupil Francis Willughby, who made Ray his constant companion while he lived. Willughby arranged that after his death, Ray would have 6 shillings a year for educating Willughbys two sons, from this tour Ray and Willughby returned laden with collections, on which they meant to base complete systematic descriptions of the animal and vegetable kingdoms. The plants gathered on his British tours had already described in his Catalogus plantarum Angliae. In 1667 Ray was elected Fellow of the Royal Society, and in 1669 he, in 1671, he presented the research of Francis Jessop on formic acid to the Royal Society. In this volume, he moved on from the naming and cataloguing of species like his successor Carl Linnaeus, instead, Ray considered species lives and how nature worked as a whole, giving facts that are arguments for Gods will expressed in His creation of all visible and invisible. Ray gave a description of dendrochronology, explaining for the ash tree how to find its age from its tree-rings. In 1673 Ray married Margaret Oakley of Launton, in 1676 he went to Middleton Hall near Tamworth, finally, in 1679, he removed to Black Notley, where he afterwards remained. His life there was quiet and uneventful, although he had poor health, Ray kept writing books and corresponded widely on scientific matters. He lived, in spite of his infirmities, to the age of seventy-seven, Ray was the first person to produce a biological definition of species, in his 1686 History of plants
John Ray
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John Ray
John Ray
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Woodcut (1693)
38.
Average
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In colloquial language, an average is the sum of a list of numbers divided by the number of numbers in the list. In mathematics and statistics, this would be called the arithmetic mean, in statistics, mean, median, and mode are all known as measures of central tendency. The most common type of average is the arithmetic mean, one may find that A = /2 =5. Switching the order of 2 and 8 to read 8 and 2 does not change the value obtained for A. The mean 5 is not less than the minimum 2 nor greater than the maximum 8. If we increase the number of terms in the list to 2,8, and 11, one finds that A = /3 =7. Along with the arithmetic mean above, the mean and the harmonic mean are known collectively as the Pythagorean means. The geometric mean of n numbers is obtained by multiplying them all together. See Inequality of arithmetic and geometric means, thus for the above harmonic mean example, AM =50, GM ≈49, and HM =48 km/h. The mode, the median, and the mid-range are often used in addition to the mean as estimates of central tendency in descriptive statistics, the most frequently occurring number in a list is called the mode. For example, the mode of the list is 3 and it may happen that there are two or more numbers which occur equally often and more often than any other number. In this case there is no agreed definition of mode, some authors say they are all modes and some say there is no mode. The median is the number of the group when they are ranked in order. Thus to find the median, order the list according to its elements magnitude, if exactly one value is left, it is the median, if two values, the median is the arithmetic mean of these two. This method takes the list 1,7,3,13, then the 1 and 13 are removed to obtain the list 3,7. Since there are two elements in this remaining list, the median is their arithmetic mean, /2 =5, the table of mathematical symbols explains the symbols used below. Other more sophisticated averages are, trimean, trimedian, and normalized mean, one can create ones own average metric using the generalized f-mean, y = f −1 where f is any invertible function. The harmonic mean is an example of this using f = 1/x, however, this method for generating means is not general enough to capture all averages
Average
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Comparison of arithmetic mean, median and mode of two log-normal distributions with different skewness.