1.
Fibonacci number
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The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, the sequence described in Liber Abaci began with F1 =1. Fibonacci numbers are related to Lucas numbers L n in that they form a complementary pair of Lucas sequences U n = F n and V n = L n. They are intimately connected with the ratio, for example. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is a journal dedicated to their study. The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody, in the Sanskrit tradition of prosody, there was interest in enumerating all patterns of long syllables that are 2 units of duration, and short syllables that are 1 unit of duration. Counting the different patterns of L and S of a given duration results in the Fibonacci numbers, susantha Goonatilake writes that the development of the Fibonacci sequence is attributed in part to Pingala, later being associated with Virahanka, Gopāla, and Hemachandra. He dates Pingala before 450 BC, however, the clearest exposition of the sequence arises in the work of Virahanka, whose own work is lost, but is available in a quotation by Gopala, Variations of two earlier meters. For example, for four, variations of meters of two three being mixed, five happens, in this way, the process should be followed in all mātrā-vṛttas. The sequence is also discussed by Gopala and by the Jain scholar Hemachandra, outside India, the Fibonacci sequence first appears in the book Liber Abaci by Fibonacci. The puzzle that Fibonacci posed was, how many pairs will there be in one year, at the end of the first month, they mate, but there is still only 1 pair. At the end of the month the female produces a new pair. At the end of the month, the original female produces a second pair. At the end of the month, the original female has produced yet another new pair. At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs plus the number of pairs alive last month and this is the nth Fibonacci number. The name Fibonacci sequence was first used by the 19th-century number theorist Édouard Lucas, the most common such problem is that of counting the number of compositions of 1s and 2s that sum to a given total n, there are Fn+1 ways to do this. For example, if n =5, then Fn+1 = F6 =8 counts the eight compositions, 1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2, all of which sum to 5. The Fibonacci numbers can be found in different ways among the set of strings, or equivalently
2.
Bernoulli distribution
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It can be used to represent a coin toss where 1 and 0 would represent head and tail, respectively. In particular, unfair coins would have p ≠0.5, the Bernoulli distribution is a special case of the binomial distribution where a single experiment/trial is conducted. It is also a case of the two-point distribution, for which the outcome need not be a bit. If X is a variable with this distribution, we have. The probability mass function f of this distribution, over possible outcomes k, is f = { p if k =1,1 − p if k =0 and this can also be expressed as f = p k 1 − k for k ∈. The Bernoulli distribution is a case of the binomial distribution with n =1. The Bernoulli distributions for 0 ≤ p ≤1 form an exponential family, the maximum likelihood estimator of p based on a random sample is the sample mean. When we take the standardized Bernoulli distributed random variable X − E Var we find that this random variable attains q p q with probability p, the Bernoulli distribution is simply B. The categorical distribution is the generalization of the Bernoulli distribution for variables with any constant number of discrete values, the Beta distribution is the conjugate prior of the Bernoulli distribution. The geometric distribution models the number of independent and identical Bernoulli trials needed to get one success, if Y ~ Bernoulli, then has a Rademacher distribution. Bernoulli process Bernoulli sampling Bernoulli trial Binary entropy function Binomial distribution McCullagh, Peter, Nelder, johnson, N. L. Kotz, S. Kemp A. Univariate Discrete Distributions. Binomial distribution, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Weisstein, Eric W. Bernoulli Distribution
3.
Independence (probability theory)
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In probability theory, two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of occurrence of other. Similarly, two variables are independent if the realization of one does not affect the probability distribution of the other. Two events A and B are independent if their joint probability equals the product of their probabilities, although the derived expressions may seem more intuitive, they are not the preferred definition, as the conditional probabilities may be undefined if P or P are 0. Furthermore, the preferred definition makes clear by symmetry that when A is independent of B, B is also independent of A. A finite set of events is independent if every pair of events is independent—that is, if. A finite set of events is independent if every event is independent of any intersection of the other events—that is, if and only if for every n-element subset. This is called the rule for independent events. Note that it is not a condition involving only the product of all the probabilities of all single events. For more than two events, an independent set of events is pairwise independent, but the converse is not necessarily true. Two random variables X and Y are independent if and only if the elements of the π-system generated by them are independent, that is to say, for every a and b, the events and are independent events. A set of variables is pairwise independent if and only if every pair of random variables is independent. A set of variables is mutually independent if and only if for any finite subset X1, …, X n and any finite sequence of numbers a 1, …, a n. The measure-theoretically inclined may prefer to substitute events for events in the above definition and that definition is exactly equivalent to the one above when the values of the random variables are real numbers. It has the advantage of working also for complex-valued random variables or for random variables taking values in any measurable space. Intuitively, two random variables X and Y are conditionally independent given Z if, once Z is known, for instance, two measurements X and Y of the same underlying quantity Z are not independent, but they are conditionally independent given Z. The formal definition of independence is based on the idea of conditional distributions. If X, Y, and Z are discrete random variables, if X and Y are conditionally independent given Z, then P = P for any x, y and z with P >0. That is, the distribution for X given Y and Z is the same as that given Z alone
4.
Harry Kesten
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Harry Kesten is an American mathematician best known for his work in probability, most notably on random walks on groups and graphs, random matrices, branching processes, and percolation theory. Kesten grew up in the Netherlands, where he moved with his parents in 1933 to escape the Nazis and he received his Ph. D. in 1958 at Cornell University under supervision of Mark Kac. He was an instructor at Princeton University and the Hebrew University before returning to Cornell where he is now Professor Emeritus of mathematics, Kestens work includes many fundamental contributions across almost the whole of probability, including the following highlights. In his 1958 PhD thesis, Kesten studied symmetric random walks on countable groups G generated by a distribution with support G. He showed that the spectral radius equals the exponential rate of the return probabilities. He showed later that this is less than 1 if. The last result is known as Kestens criterion for amenability and he calculated the spectral radius of the d-regular tree, namely 2 d −1. Let Y n = X1 X2 … X n be the product of the first n elements of an ergodic stationary sequence of random k × k matrices. With Furstenberg in 1960, Kesten showed the convergence of n −1 log + ∥ Y n ∥, under the condition E < ∞. Kestens ratio limit theorem states that the number σ n of n-step self-avoiding walks from the origin on the lattice satisfies σ n +2 / σ n → μ2 where μ is the connective constant. This result remains unimproved despite much effort, Kesten and Stigum showed that the correct condition for the convergence of the population size, normalized by its mean, is that E < ∞ where L is a typical family size. Random walk in a random environment, with Kozlov and Spitzer, Kesten proved a deep theorem about random walk in a one-dimensional random environment. They established the laws for the walk across the variety of situations that can arise within the environment. In 1966, Kesten resolved a conjecture of Erdős and Szűsz on the discrepancy of irrational rotations. He studied the discrepancy between the number of rotations by ξ hitting a given interval I, and the length of I, Kesten proved that the growth rate of the arms in d dimensions can be no larger than n 2 /. Kestens most famous work in area is his proof that the critical probability of bond percolation on the square lattice equals 1/2. He followed this with a study of percolation in two dimensions, reported in his book Percolation Theory for Mathematicians. His work on scaling theory and scaling relations has since proved key to the relationship between critical percolation and Schramm-Loewner evolution, Kestens results for this growth model are largely summarized in Aspects of First Passage Percolation
5.
Hillel Furstenberg
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He is known for his application of probability theory and ergodic theory methods to other areas of mathematics, including number theory and Lie groups. Hillel Furstenberg was born in Germany, in 1935, and the family emigrated to the United States in 1939 and he attended Marsha Stern Talmudical Academy and then Yeshiva University, where he concluded his BA and MSc studies in 1955. He obtained his Ph. D. under Salomon Bochner at Princeton University in 1958, after several years at the University of Minnesota he became a Professor of Mathematics at the Hebrew University of Jerusalem in 1965. He gained attention at a stage in his career for producing an innovative topological proof of the infinitude of prime numbers. He proved unique ergodicity of horocycle flows on compact hyperbolic Riemann surfaces in the early 1970s, in 1977, he gave an ergodic theory reformulation, and subsequently proof, of Szemerédis theorem. The Furstenberg boundary and Furstenberg compactification of a symmetric space are named after him. 1993 – Furstenberg received the Israel Prize, for exact sciences,1993 – Furstenberg received the Harvey Prize from Technion. 2006/7 – He received the Wolf Prize in Mathematics, Furstenberg, Harry, Stationary processes and prediction theory, Princeton, N. J. Furstenberg, Harry, Recurrence in ergodic theory and combinatorial number theory, Princeton, compactification Ratners theorems List of Israel Prize recipients OConnor, John J. Robertson, Edmund F. Hillel Furstenberg, MacTutor History of Mathematics archive, University of St Andrews. Mathematics Genealogy page Press release Israel Academy of Sciences and Humanities
6.
Exponential growth
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Exponential decay occurs in the same way when the growth rate is negative. In the case of a domain of definition with equal intervals, it is also called geometric growth or geometric decay. In either exponential growth or exponential decay, the ratio of the rate of change of the quantity to its current size remains constant over time. The formula for growth of a variable x at the growth rate r. This formula is transparent when the exponents are converted to multiplication, in this way, each increase in the exponent by a full interval can be seen to increase the previous total by another five percent. Since the time variable, which is the input to function, occurs as the exponent. Biology The number of microorganisms in a culture will increase exponentially until an essential nutrient is exhausted, typically the first organism splits into two daughter organisms, who then each split to form four, who split to form eight, and so on. Because exponential growth indicates constant growth rate, it is assumed that exponentially growing cells are at a steady-state. However, cells can grow exponentially at a constant rate while remodelling their metabolism, a virus typically will spread exponentially at first, if no artificial immunization is available. Each infected person can infect multiple new people, human population, if the number of births and deaths per person per year were to remain at current levels. This means that the time of the American population is approximately 50 years. Physics Avalanche breakdown within a dielectric material, a free electron becomes sufficiently accelerated by an externally applied electrical field that it frees up additional electrons as it collides with atoms or molecules of the dielectric media. These secondary electrons also are accelerated, creating larger numbers of free electrons, the resulting exponential growth of electrons and ions may rapidly lead to complete dielectric breakdown of the material. Each uranium nucleus that undergoes fission produces multiple neutrons, each of which can be absorbed by adjacent uranium atoms, due to the exponential rate of increase, at any point in the chain reaction 99% of the energy will have been released in the last 4.6 generations. It is an approximation to think of the first 53 generations as a latency period leading up to the actual explosion. Economics Economic growth is expressed in terms, implying exponential growth. For example, U. S. GDP per capita has grown at a rate of approximately two percent since World War 2. Finance Compound interest at a constant interest rate provides exponential growth of the capital, pyramid schemes or Ponzi schemes also show this type of growth resulting in high profits for a few initial investors and losses among great numbers of investors
7.
On-Line Encyclopedia of Integer Sequences
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The On-Line Encyclopedia of Integer Sequences, also cited simply as Sloanes, is an online database of integer sequences. It was created and maintained by Neil Sloane while a researcher at AT&T Labs, Sloane continues to be involved in the OEIS in his role as President of the OEIS Foundation. OEIS records information on integer sequences of interest to professional mathematicians and amateurs, and is widely cited. As of 30 December 2016 it contains nearly 280,000 sequences, the database is searchable by keyword and by subsequence. Neil Sloane started collecting integer sequences as a student in 1965 to support his work in combinatorics. The database was at first stored on punched cards and he published selections from the database in book form twice, A Handbook of Integer Sequences, containing 2,372 sequences in lexicographic order and assigned numbers from 1 to 2372. The Encyclopedia of Integer Sequences with Simon Plouffe, containing 5,488 sequences and these books were well received and, especially after the second publication, mathematicians supplied Sloane with a steady flow of new sequences. The collection became unmanageable in book form, and when the database had reached 16,000 entries Sloane decided to go online—first as an e-mail service, as a spin-off from the database work, Sloane founded the Journal of Integer Sequences in 1998. The database continues to grow at a rate of some 10,000 entries a year, Sloane has personally managed his sequences for almost 40 years, but starting in 2002, a board of associate editors and volunteers has helped maintain the database. In 2004, Sloane celebrated the addition of the 100, 000th sequence to the database, A100000, in 2006, the user interface was overhauled and more advanced search capabilities were added. In 2010 an OEIS wiki at OEIS. org was created to simplify the collaboration of the OEIS editors and contributors, besides integer sequences, the OEIS also catalogs sequences of fractions, the digits of transcendental numbers, complex numbers and so on by transforming them into integer sequences. Sequences of rationals are represented by two sequences, the sequence of numerators and the sequence of denominators, important irrational numbers such as π =3.1415926535897. are catalogued under representative integer sequences such as decimal expansions, binary expansions, or continued fraction expansions. The OEIS was limited to plain ASCII text until 2011, yet it still uses a form of conventional mathematical notation. Greek letters are represented by their full names, e. g. mu for μ. Every sequence is identified by the letter A followed by six digits, sometimes referred to without the leading zeros, individual terms of sequences are separated by commas. Digit groups are not separated by commas, periods, or spaces, a represents the nth term of the sequence. Zero is often used to represent non-existent sequence elements, for example, A104157 enumerates the smallest prime of n² consecutive primes to form an n×n magic square of least magic constant, or 0 if no such magic square exists. The value of a is 2, a is 1480028129, but there is no such 2×2 magic square, so a is 0
8.
Mathematical constant
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A mathematical constant is a special number, usually a real number, that is significantly interesting in some way. Constants arise in areas of mathematics, with constants such as e and π occurring in such diverse contexts as geometry, number theory. The more popular constants have been studied throughout the ages and computed to many decimal places, all mathematical constants are definable numbers and usually are also computable numbers. These are constants which one is likely to encounter during pre-college education in many countries, however, its ubiquity is not limited to pure mathematics. It appears in many formulas in physics, and several physical constants are most naturally defined with π or its reciprocal factored out and it is debatable, however, if such appearances are fundamental in any sense. For example, the textbook nonrelativistic ground state wave function of the atom is ψ =11 /2 e − r / a 0. This formula contains a π, but it is unclear if that is fundamental in a physical sense, furthermore, this formula gives only an approximate description of physical reality, as it omits spin, relativity, and the quantal nature of the electromagnetic field itself. The numeric value of π is approximately 3.1415926535, memorizing increasingly precise digits of π is a world record pursuit. The constant e also has applications to probability theory, where it arises in a way not obviously related to exponential growth, suppose a slot machine with a one in n probability of winning is played n times. Then, for large n the probability that nothing will be won is approximately 1/e, another application of e, discovered in part by Jacob Bernoulli along with French mathematician Pierre Raymond de Montmort, is in the problem of derangements, also known as the hat check problem. Here n guests are invited to a party, and at the door each guest checks his hat with the butler who then places them into labelled boxes, the butler does not know the name of the guests, and so must put them into boxes selected at random. The problem of de Montmort is, what is the probability that none of the hats gets put into the right box, the answer is p n =1 −11. + ⋯ + n 1 n. and as n tends to infinity, the numeric value of e is approximately 2.7182818284. The square root of 2, often known as root 2, radical 2, or Pythagorass constant, and written as √2, is the algebraic number that. It is more called the principal square root of 2. Geometrically the square root of 2 is the length of a diagonal across a square sides of one unit of length. It was probably the first number known to be irrational and its numerical value truncated to 65 decimal places is,1.41421356237309504880168872420969807856967187537694807317667973799. The quick approximation 99/70 for the root of two is frequently used
![{\sqrt[ {n}]{|f_{n}|}}\to 1.13198824\dots {\text{ as }}n\to \infty .](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcbfac47cb65a95204a4e8a4bdf998a63ecbb6a8)
. doi:10.1016/j.jnt.2006.01.002.
