Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, measure, infinite series, and analytic functions. These theories are studied in the context of real and complex numbers. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis, analysis may be distinguished from geometry, however, it can be applied to any space of mathematical objects that has a definition of nearness or specific distances between objects. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, a geometric sum is implicit in Zenos paradox of the dichotomy. The explicit use of infinitesimals appears in Archimedes The Method of Mechanical Theorems, in Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle.
Zu Chongzhi established a method that would be called Cavalieris principle to find the volume of a sphere in the 5th century, the Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolles theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and his followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century. The modern foundations of analysis were established in 17th century Europe. During this period, calculus techniques were applied to approximate discrete problems by continuous ones, in the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the definition of continuity in 1816. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, Cauchy formulated calculus in terms of geometric ideas and infinitesimals.
Thus, his definition of continuity required a change in x to correspond to an infinitesimal change in y. He introduced the concept of the Cauchy sequence, and started the theory of complex analysis. Poisson, Liouville and others studied partial differential equations, the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis. In the middle of the 19th century Riemann introduced his theory of integration, the last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the epsilon-delta definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of numbers without proof. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the size of the set of discontinuities of real functions, monsters began to be investigated
Augustus De Morgan
Augustus De Morgan was a British mathematician and logician. He formulated De Morgans laws and introduced the mathematical induction. Augustus De Morgan was born in Madurai, India in 1806 and his father was Lieut. -Colonel John De Morgan, who held various appointments in the service of the East India Company. His mother, Elizabeth Dodson descended from James Dodson, who computed a table of anti-logarithms, that is, Augustus De Morgan became blind in one eye a month or two after he was born. The family moved to England when Augustus was seven months old, when De Morgan was ten years old, his father died. Mrs. De Morgan resided at various places in the southwest of England and his mathematical talents went unnoticed until he was fourteen, when a family-friend discovered him making an elaborate drawing of a figure in Euclid with ruler and compasses. She explained the aim of Euclid to Augustus, and gave him an initiation into demonstration and he received his secondary education from Mr. Parsons, a fellow of Oriel College, who appreciated classics better than mathematics.
His mother was an active and ardent member of the Church of England, and desired that her son should become a clergyman, I shall use the world Anti-Deism to signify the opinion that there does not exist a Creator who made and sustains the Universe. His college tutor was John Philips Higman, FRS, at college he played the flute for recreation and was prominent in the musical clubs. His love of knowledge for its own sake interfered with training for the great mathematical race, as a consequence he came out fourth wrangler. This entitled him to the degree of Bachelor of Arts, but to take the degree of Master of Arts. To the signing of any such test De Morgan felt a strong objection, in about 1875 theological tests for academic degrees were abolished in the Universities of Oxford and Cambridge. As no career was open to him at his own university, he decided to go to the Bar, and took up residence in London, about this time the movement for founding London University took shape. A body of liberal-minded men resolved to meet the difficulty by establishing in London a University on the principle of religious neutrality, De Morgan, 22 years of age, was appointed professor of mathematics.
His introductory lecture On the study of mathematics is a discourse upon mental education of permanent value, the London University was a new institution, and the relations of the Council of management, the Senate of professors and the body of students were not well defined. A dispute arose between the professor of anatomy and his students, and in consequence of the action taken by the Council, another professor of mathematics was appointed, who drowned a few years later. De Morgan had shown himself a prince of teachers, he was invited to return to his chair and its object was to spread scientific and other knowledge by means of cheap and clearly written treatises by the best writers of the time. One of its most voluminous and effective writers was De Morgan, when De Morgan came to reside in London he found a congenial friend in William Frend, notwithstanding his mathematical heresy about negative quantities
In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between a function and its argument. For real non zero values of x, the exponential integral Ei is defined as Ei = − ∫ − x ∞ e − t t d t, the Risch algorithm shows that Ei is not an elementary function. The definition above can be used for values of x. For complex values of the argument, the definition becomes ambiguous due to points at 0 and ∞. Instead of Ei, the notation is used, E1 = ∫ z ∞ e − t t d t, | A r g | < π. In general, a cut is taken on the negative real axis. For positive values of the part of z, this can be written E1 = ∫1 ∞ e − t z t d t = ∫01 e − z / u u d u, ℜ ≥0. The behaviour of E1 near the cut can be seen by the following relation. Several properties of the exponential integral below, in certain cases, allow one to avoid its explicit evaluation through the definition above. X ≠0 For complex arguments off the real axis. The sum converges for all z, and we take the usual value of the complex logarithm having a branch cut along the negative real axis.
This formula can be used to compute E1 with floating point operations for real x between 0 and 2.5, for x >2.5, the result is inaccurate due to cancellation. For example, for x =10 more than 40 terms are required to get an answer correct to three significant figures. However, there is a divergent series approximation that can be obtained by integrating z e z E1 by parts, N which has error of order O and is valid for large values of Re . The relative error of the approximation above is plotted on the figure to the right for various values of N, the number of terms in the truncated sum. From the two series suggested in previous subsections, it follows that E1 behaves like a negative exponential for large values of the argument and like a logarithm for small values. Both Ei and E1 can be more simply using the entire function Ein defined as Ein = ∫0 z d t t = ∑ k =1 ∞ k +1 z k k k
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the function, it counts the number of divisors of an integer. It appears in a number of identities, including relationships on the Riemann zeta function. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities, a related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function. The sum of divisors function σx, for a real or complex number x, is defined as the sum of the xth powers of the positive divisors of n. It can be expressed in sigma notation as σ x = ∑ d ∣ n d x, the notations d, ν and τ are used to denote σ0, or the number-of-divisors function. When x is 1, the function is called the function or sum-of-divisors function. The aliquot sum s of n is the sum of the proper divisors, and equals σ1 − n, the cases x =2 to 5 are listed in A001157 − A001160, x =6 to 24 are listed in A013954 − A013972.
For a non-square integer, n, every divisor, d, of n is paired with divisor n/d of n and σ0 is even, for an integer, one divisor is not paired with a distinct divisor. Similarly, the number σ1 is odd if and only if n is a square or twice a square. For a prime p, σ0 =2 σ0 = n +1 σ1 = p +1 because by definition. Also, where pn# denotes the primorial, σ0 =2 n since n prime factors allow a sequence of binary selection from n terms for each proper divisor formed, clearly,1 < σ0 < n and σ > n for all n >2. The divisor function is multiplicative, but not completely multiplicative and it follows that d is, σ0 = ∏ i =1 r. For example, if n is 24, there are two factors, noting that 24 is the product of 23×31, a1 is 3. Thus we can calculate σ0 as so, σ0 = ∏ i =12 = =4 ⋅2 =8, the eight divisors counted by this formula are 1,2,4,8,3,6,12, and 24. Here s denotes the sum of the divisors of n, that is. This function is the one used to perfect numbers which are the n for which s = n. If s > n n is an abundant number and if s < n n is a deficient number
In theoretical physics, Feynman diagrams are pictorial representations of the mathematical expressions describing the behavior of subatomic particles. The scheme is named after its inventor, American physicist Richard Feynman, the interaction of sub-atomic particles can be complex and difficult to understand intuitively. Feynman diagrams give a simple visualization of what would otherwise be a rather arcane, while the diagrams are applied primarily to quantum field theory, they can be used in other fields, such as solid-state theory. Feynman used Ernst Stueckelbergs interpretation of the positron as if it were an electron moving backward in time, antiparticles are represented as moving backward along the time axis in Feynman diagrams. The calculation of probability amplitudes in theoretical particle physics requires the use of rather large and these integrals do, have a regular structure, and may be represented graphically as Feynman diagrams. A Feynman diagram is a contribution of a class of particle paths.
Within the canonical formulation of field theory, a Feynman diagram represents a term in the Wicks expansion of the perturbative S-matrix. The transition amplitude is given as the matrix element of the S-matrix between the initial and the final states of the quantum system. The amplitude for scattering is the sum of each possible interaction history over all possible intermediate particle states, the number of times the interaction Hamiltonian acts is the order of the perturbation expansion, and the time-dependent perturbation theory for fields is known as the Dyson series. When the intermediate states at times are energy eigenstates the series is called old-fashioned perturbation theory. The Feynman diagrams are much easier to track of than old-fashioned terms. Each Feynman diagram is the sum of exponentially many old-fashioned terms, in a non-relativistic theory, there are no antiparticles and there is no doubling, so each Feynman diagram includes only one term. Feynman gave a prescription for calculating the amplitude for any given diagram from a field theory Lagrangian—the Feynman rules, in addition to their value as a mathematical tool, Feynman diagrams provide deep physical insight into the nature of particle interactions.
Particles interact in every way available, in fact, intermediate virtual particles are allowed to propagate faster than light, the probability of each final state is obtained by summing over all such possibilities. This is closely tied to the integral formulation of quantum mechanics. After renormalization, calculations using Feynman diagrams match experimental results with very high accuracy, Feynman diagram and path integral methods are used in statistical mechanics and can even be applied to classical mechanics. Murray Gell-Mann always referred to Feynman diagrams as Stueckelberg diagrams, after a Swiss physicist, Ernst Stueckelberg, Feynman had to lobby hard for the diagrams, which confused the establishment physicists trained in equations and graphs. In quantum field theories the Feynman diagrams are obtained from Lagrangian by Feynman rules, dimensional regularization writes a Feynman integral as an integral depending on the spacetime dimension d and spacetime points
The law is named after the American linguist George Kingsley Zipf, who popularized it and sought to explain it, though he did not claim to have originated it. The French stenographer Jean-Baptiste Estoup appears to have noticed the regularity before Zipf and it was noted in 1913 by German physicist Felix Auerbach. Zipfs law states that given some corpus of natural language utterances, for example, in the Brown Corpus of American English text, the word the is the most frequently occurring word, and by itself accounts for nearly 7% of all word occurrences. True to Zipfs Law, the word of accounts for slightly over 3. 5% of words, followed by. Only 135 vocabulary items are needed to account for half the Brown Corpus, the appearance of the distribution in rankings of cities by population was first noticed by Felix Auerbach in 1913. When Zipfs law is checked for cities, a better fit has been found with exponent s =1.07, while Zipfs law holds for the upper tail of the distribution, the entire distribution of cities is log-normal and follows Gibrats law.
Both laws are consistent because a log-normal tail can not be distinguished from a Pareto tail. Zipfs law is most easily observed by plotting the data on a log-log graph, for example, the word the would appear at x = log, y = log. It is possible to plot reciprocal rank against frequency or reciprocal frequency or interword interval against rank, the data conform to Zipfs law to the extent that the plot is linear. Formally, let, N be the number of elements, k be their rank and it has been claimed that this representation of Zipfs law is more suitable for statistical testing, and in this way it has been analyzed in more than 30,000 English texts. The goodness-of-fit tests yield that only about 15% of the texts are statistically compatible with this form of Zipfs law, slight variations in the definition of Zipfs law can increase this percentage up to close to 50%. In the example of the frequency of words in the English language, N is the number of words in the English language and, if we use the version of Zipfs law.
F will be the fraction of the time the kth most common word occurs, the law may be written, f =1 k s H N, s where HN, s is the Nth generalized harmonic number. The simplest case of Zipfs law is a 1⁄f function, given a set of Zipfian distributed frequencies, sorted from most common to least common, the second most common frequency will occur ½ as often as the first. The third most common frequency will occur ⅓ as often as the first, the fourth most common frequency will occur ¼ as often as the first. The nth most common frequency will occur 1⁄n as often as the first, this cannot hold exactly, because items must occur an integer number of times, there cannot be 2.5 occurrences of a word. Nevertheless, over fairly wide ranges, and to a good approximation. Mathematically, the sum of all frequencies in a Zipf distribution is equal to the harmonic series
Italy, officially the Italian Republic, is a unitary parliamentary republic in Europe. Located in the heart of the Mediterranean Sea, Italy shares open land borders with France, Austria, San Marino, Italy covers an area of 301,338 km2 and has a largely temperate seasonal climate and Mediterranean climate. Due to its shape, it is referred to in Italy as lo Stivale. With 61 million inhabitants, it is the fourth most populous EU member state, the Italic tribe known as the Latins formed the Roman Kingdom, which eventually became a republic that conquered and assimilated other nearby civilisations. The legacy of the Roman Empire is widespread and can be observed in the distribution of civilian law, republican governments, Christianity. The Renaissance began in Italy and spread to the rest of Europe, bringing a renewed interest in humanism, exploration, Italian culture flourished at this time, producing famous scholars and polymaths such as Leonardo da Vinci, Galileo and Machiavelli. The weakened sovereigns soon fell victim to conquest by European powers such as France and Austria.
Despite being one of the victors in World War I, Italy entered a period of economic crisis and social turmoil. The subsequent participation in World War II on the Axis side ended in defeat, economic destruction. Today, Italy has the third largest economy in the Eurozone and it has a very high level of human development and is ranked sixth in the world for life expectancy. The country plays a prominent role in regional and global economic, military and diplomatic affairs, as a reflection of its cultural wealth, Italy is home to 51 World Heritage Sites, the most in the world, and is the fifth most visited country. The assumptions on the etymology of the name Italia are very numerous, according to one of the more common explanations, the term Italia, from Latin, was borrowed through Greek from the Oscan Víteliú, meaning land of young cattle. The bull was a symbol of the southern Italic tribes and was often depicted goring the Roman wolf as a defiant symbol of free Italy during the Social War. Greek historian Dionysius of Halicarnassus states this account together with the legend that Italy was named after Italus, mentioned by Aristotle and Thucydides.
The name Italia originally applied only to a part of what is now Southern Italy – according to Antiochus of Syracuse, but by his time Oenotria and Italy had become synonymous, and the name applied to most of Lucania as well. The Greeks gradually came to apply the name Italia to a larger region, excavations throughout Italy revealed a Neanderthal presence dating back to the Palaeolithic period, some 200,000 years ago, modern Humans arrived about 40,000 years ago. Other ancient Italian peoples of undetermined language families but of possible origins include the Rhaetian people and Cammuni. Also the Phoenicians established colonies on the coasts of Sardinia and Sicily, the Roman legacy has deeply influenced the Western civilisation, shaping most of the modern world
The base-2 system is a positional notation with a radix of 2. Because of its implementation in digital electronic circuitry using logic gates. Each digit is referred to as a bit, the modern binary number system was devised by Gottfried Leibniz in 1679 and appears in his article Explication de lArithmétique Binaire. Systems related to binary numbers have appeared earlier in multiple cultures including ancient Egypt, Leibniz was specifically inspired by the Chinese I Ching. The scribes of ancient Egypt used two different systems for their fractions, Egyptian fractions and Horus-Eye fractions, the method used for ancient Egyptian multiplication is closely related to binary numbers. This method can be seen in use, for instance, in the Rhind Mathematical Papyrus, the I Ching dates from the 9th century BC in China. The binary notation in the I Ching is used to interpret its quaternary divination technique and it is based on taoistic duality of yin and yang. Eight trigrams and a set of 64 hexagrams, analogous to the three-bit and six-bit binary numerals, were in use at least as early as the Zhou Dynasty of ancient China.
The Song Dynasty scholar Shao Yong rearranged the hexagrams in a format that resembles modern binary numbers, the Indian scholar Pingala developed a binary system for describing prosody. He used binary numbers in the form of short and long syllables, Pingalas Hindu classic titled Chandaḥśāstra describes the formation of a matrix in order to give a unique value to each meter. The binary representations in Pingalas system increases towards the right, the residents of the island of Mangareva in French Polynesia were using a hybrid binary-decimal system before 1450. Slit drums with binary tones are used to encode messages across Africa, sets of binary combinations similar to the I Ching have been used in traditional African divination systems such as Ifá as well as in medieval Western geomancy. The base-2 system utilized in geomancy had long been applied in sub-Saharan Africa. Leibnizs system uses 0 and 1, like the modern binary numeral system, Leibniz was first introduced to the I Ching through his contact with the French Jesuit Joachim Bouvet, who visited China in 1685 as a missionary.
Leibniz saw the I Ching hexagrams as an affirmation of the universality of his own beliefs as a Christian. Binary numerals were central to Leibnizs theology and he believed that binary numbers were symbolic of the Christian idea of creatio ex nihilo or creation out of nothing. Is not easy to impart to the pagans, is the ex nihilo through Gods almighty power. In 1854, British mathematician George Boole published a paper detailing an algebraic system of logic that would become known as Boolean algebra
Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table.
The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational.
Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke of so-called polygonal or figurate numbers
In probability theory and statistics, the Weibull distribution /ˈveɪbʊl/ is a continuous probability distribution. Its complementary cumulative distribution function is an exponential function. The Weibull distribution is related to a number of probability distributions, in particular. If the quantity X is a time-to-failure, the Weibull distribution gives a distribution for which the rate is proportional to a power of time. The shape parameter, k, is that power plus one and this happens if there is significant infant mortality, or defective items failing early and the failure rate decreasing over time as the defective items are weeded out of the population. This might suggest random external events are causing mortality, or failure, the Weibull distribution reduces to an exponential distribution, A value of k >1 indicates that the failure rate increases with time. This happens if there is a process, or parts that are more likely to fail as time goes on. In the context of the diffusion of innovations, this means positive word of mouth, the function is first concave, convex with an inflexion point at / e 1 / k, k >1.
In the field of science, the shape parameter k of a distribution of strengths is known as the Weibull modulus. In the context of diffusion of innovations, the Weibull distribution is a pure imitation/rejection model, in medical statistics a different parameterization is used. The shape parameter k is the same as above and the parameter is b = λ. For x ≥0 the hazard function is h = b k x k −1, a third parameterization is sometimes used. In this the shape parameter k is the same as above, the form of the density function of the Weibull distribution changes drastically with the value of k. For 0 < k <1, the density function tends to ∞ as x approaches zero from above and is strictly decreasing, for k =1, the density function tends to 1/λ as x approaches zero from above and is strictly decreasing. For k >1, the density function tends to zero as x approaches zero from above, increases until its mode, for k =2 the density has a finite positive slope at x =0. As k goes to infinity, the Weibull distribution converges to a Dirac delta distribution centered at x = λ, the skewness and coefficient of variation depend only on the shape parameter.
The cumulative distribution function for the Weibull distribution is F =1 − e − k for x ≥0, the quantile function for the Weibull distribution is Q = λ1 / k for 0 ≤ p <1. The failure rate h is given by h = k λ k −1, the moment generating function of the logarithm of a Weibull distributed random variable is given by E = λ t Γ where Γ is the gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers. That is, if n is an integer, Γ =. The gamma function is defined for all numbers except the non-positive integers. The gamma function can be seen as a solution to the interpolation problem. The simple formula for the factorial, x. =1 ×2 × … × x, a good solution to this is the gamma function. There are infinitely many continuous extensions of the factorial to non-integers, the gamma function is the most useful solution in practice, being analytic, and it can be characterized in several ways. The Bohr–Mollerup theorem proves that these properties, together with the assumption that f be logarithmically convex, uniquely determine f for positive, from there, the gamma function can be extended to all real and complex values by using the unique analytic continuation of f. Also see Eulers infinite product definition below where the properties f =1 and f = x f together with the requirement that limn→+∞. nx / f =1 uniquely define the same function.
The notation Γ is due to Legendre, if the real part of the complex number z is positive, the integral Γ = ∫0 ∞ x z −1 e − x d x converges absolutely, and is known as the Euler integral of the second kind. The identity Γ = Γ z can be used to extend the integral formulation for Γ to a meromorphic function defined for all complex numbers z. It is this version that is commonly referred to as the gamma function. When seeking to approximate z. for a number z, it turns out that it is effective to first compute n. for some large integer n. And use the relation m. = m. backwards n times. Furthermore, this approximation is exact in the limit as n goes to infinity, for a fixed integer m, it is the case that lim n → + ∞ n. m. =1, and we can ask that the formula is obeyed when the arbitrary integer m is replaced by an arbitrary complex number z lim n → + ∞ n. z. =1. Multiplying both sides by z. gives z. = lim n → + ∞ n. z, Z = lim n → + ∞1 ⋯ n ⋯ z = ∏ n =1 + ∞. Similarly for the function, the definition as an infinite product due to Euler is valid for all complex numbers z except the non-positive integers.
By this construction, the function is the unique function that simultaneously satisfies Γ =1, Γ = z Γ for all complex numbers z except the non-positive integers