1.
Mathematics
–
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

2.
Terence Tao
–
Terence Terry Chi-Shen Tao FAA FRS, is an Australian-American mathematician who has worked in various areas of mathematics. He currently focuses on analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, compressed sensing. As of 2015, he holds the James and Carol Collins chair in mathematics at the University of California, Tao was a co-recipient of the 2006 Fields Medal and the 2014 Breakthrough Prize in Mathematics. Taos father, Dr. Billy Tao, was a pediatrician who was born in Shanghai, Taos mother, Grace, is from Hong Kong, she received a first-class honours degree in physics and mathematics at the University of Hong Kong. She was a school teacher of mathematics and physics in Hong Kong. Billy and Grace met as students at the University of Hong Kong and they then emigrated from Hong Kong to Australia. Tao has two living in Australia, both of whom represented Australia at the International Mathematical Olympiad. Nigel Tao was part of the team at Google Australia that created Google Wave and he now works on the Go programming language. Trevor Tao is an International Master in Chess and he has a double degree in mathematics and music and is an autistic savant. Taos wife, Laura, is an engineer at NASAs Jet Propulsion Laboratory and they live with their son and daughter in Los Angeles, California. Tao exhibited extraordinary mathematical abilities from an age, attending university level mathematics courses at the age of 9. In 1986,1987, and 1988, Tao was the youngest participant to date in the International Mathematical Olympiad, first competing at the age of ten, winning a bronze, silver, and gold medal. He remains the youngest winner of each of the three medals in the Olympiads history, winning the gold medal shortly after his thirteenth birthday, at age 14, Tao attended the Research Science Institute. When he was 15 he published his first assistant paper and he received his bachelors and masters degrees at the age of 16 from Flinders University under Garth Gaudry. In 1992 he won a Fulbright Scholarship to undertake study in the United States. From 1992 to 1996, Tao was a student at Princeton University under the direction of Elias Stein. He joined the faculty of the University of California, Los Angeles in 1996, when he was 24, he was promoted to full professor at UCLA and remains the youngest person ever appointed to that rank by the institution. Within the field of mathematics, Tao is known for his collaboration with Ben J. Green of Oxford University, known for his collaborative mindset, by 2006 Tao had worked with over 30 others in his discoveries, reaching 68 co-authors by October 2015

3.
Riemann zeta function
–
More general representations of ζ for all s are given below. The Riemann zeta function plays a role in analytic number theory and has applications in physics, probability theory. As a function of a variable, Leonhard Euler first introduced and studied it in the first half of the eighteenth century without using complex analysis. The values of the Riemann zeta function at even positive integers were computed by Euler, the first of them, ζ, provides a solution to the Basel problem. In 1979 Apéry proved the irrationality of ζ, the values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, the Riemann zeta function ζ is a function of a complex variable s = σ + it. It can also be defined by the integral ζ =1 Γ ∫0 ∞ x s −1 e x −1 d x where Γ is the gamma function. The Riemann zeta function is defined as the continuation of the function defined for σ >1 by the sum of the preceding series. Leonhard Euler considered the series in 1740 for positive integer values of s. The above series is a prototypical Dirichlet series that converges absolutely to a function for s such that σ >1. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values s ≠1, for s =1 the series is the harmonic series which diverges to +∞, and lim s →1 ζ =1. Thus the Riemann zeta function is a function on the whole complex s-plane. For any positive even integer 2n, ζ = n +1 B2 n 2 n 2, where B2n is the 2nth Bernoulli number. For odd positive integers, no simple expression is known, although these values are thought to be related to the algebraic K-theory of the integers. For nonpositive integers, one has ζ = B n +1 n +1 for n ≥0 In particular, ζ = −12, Similarly to the above, this assigns a finite result to the series 1 +1 +1 +1 + ⋯. ζ ≈ −1.4603545 This is employed in calculating of kinetic boundary layer problems of linear kinetic equations, ζ =1 +12 +13 + ⋯ = ∞, if we approach from numbers larger than 1. Then this is the harmonic series, but its Cauchy principal value lim ε →0 ζ + ζ2 exists which is the Euler–Mascheroni constant γ =0. 5772…. ζ ≈2.612, This is employed in calculating the critical temperature for a Bose–Einstein condensate in a box with periodic boundary conditions, and for spin wave physics in magnetic systems

4.
Fibonacci number
–
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

5.
Theoretical physics
–
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to physics, which uses experimental tools to probe these phenomena. The advancement of science depends in general on the interplay between experimental studies and theory, in some cases, theoretical physics adheres to standards of mathematical rigor while giving little weight to experiments and observations. Conversely, Einstein was awarded the Nobel Prize for explaining the photoelectric effect, a physical theory is a model of physical events. It is judged by the extent to which its predictions agree with empirical observations, the quality of a physical theory is also judged on its ability to make new predictions which can be verified by new observations. A physical theory similarly differs from a theory, in the sense that the word theory has a different meaning in mathematical terms. A physical theory involves one or more relationships between various measurable quantities, archimedes realized that a ship floats by displacing its mass of water, Pythagoras understood the relation between the length of a vibrating string and the musical tone it produces. Other examples include entropy as a measure of the uncertainty regarding the positions and motions of unseen particles, Theoretical physics consists of several different approaches. In this regard, theoretical particle physics forms a good example, for instance, phenomenologists might employ empirical formulas to agree with experimental results, often without deep physical understanding. Modelers often appear much like phenomenologists, but try to model speculative theories that have certain desirable features, some attempt to create approximate theories, called effective theories, because fully developed theories may be regarded as unsolvable or too complicated. Other theorists may try to unify, formalise, reinterpret or generalise extant theories, or create completely new ones altogether. Sometimes the vision provided by pure mathematical systems can provide clues to how a system might be modeled, e. g. the notion, due to Riemann and others. Theoretical problems that need computational investigation are often the concern of computational physics, Theoretical advances may consist in setting aside old, incorrect paradigms or may be an alternative model that provides answers that are more accurate or that can be more widely applied. In the latter case, a correspondence principle will be required to recover the previously known result, sometimes though, advances may proceed along different paths. However, an exception to all the above is the wave–particle duality, Physical theories become accepted if they are able to make correct predictions and no incorrect ones. They are also likely to be accepted if they connect a wide range of phenomena. Testing the consequences of a theory is part of the scientific method, Physical theories can be grouped into three categories, mainstream theories, proposed theories and fringe theories. Theoretical physics began at least 2,300 years ago, under the Pre-socratic philosophy, during the Middle Ages and Renaissance, the concept of experimental science, the counterpoint to theory, began with scholars such as Ibn al-Haytham and Francis Bacon

6.
Factorial
–
In mathematics, the factorial of a non-negative integer n, denoted by n. is the product of all positive integers less than or equal to n. =5 ×4 ×3 ×2 ×1 =120, the value of 0. is 1, according to the convention for an empty product. The factorial operation is encountered in areas of mathematics, notably in combinatorics, algebra. Its most basic occurrence is the fact there are n. ways to arrange n distinct objects into a sequence. This fact was known at least as early as the 12th century, fabian Stedman, in 1677, described factorials as applied to change ringing. After describing a recursive approach, Stedman gives a statement of a factorial, Now the nature of these methods is such, the factorial function is formally defined by the product n. = ∏ k =1 n k, or by the relation n. = {1 if n =0. The factorial function can also be defined by using the rule as n. All of the above definitions incorporate the instance 0, =1, in the first case by the convention that the product of no numbers at all is 1. This is convenient because, There is exactly one permutation of zero objects, = n. ×, valid for n >0, extends to n =0. It allows for the expression of many formulae, such as the function, as a power series. It makes many identities in combinatorics valid for all applicable sizes, the number of ways to choose 0 elements from the empty set is =0. More generally, the number of ways to choose n elements among a set of n is = n. n, the factorial function can also be defined for non-integer values using more advanced mathematics, detailed in the section below. This more generalized definition is used by advanced calculators and mathematical software such as Maple or Mathematica, although the factorial function has its roots in combinatorics, formulas involving factorials occur in many areas of mathematics. There are n. different ways of arranging n distinct objects into a sequence, often factorials appear in the denominator of a formula to account for the fact that ordering is to be ignored. A classical example is counting k-combinations from a set with n elements, one can obtain such a combination by choosing a k-permutation, successively selecting and removing an element of the set, k times, for a total of n k _ = n ⋯ possibilities. This however produces the k-combinations in an order that one wishes to ignore, since each k-combination is obtained in k. different ways. This number is known as the coefficient, because it is also the coefficient of Xk in n

7.
Monotonic function
–
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

8.
Geometric series
–
In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series 12 +14 +18 +116 + ⋯ is geometric, Geometric series are among the simplest examples of infinite series with finite sums, although not all of them have this property. Historically, geometric series played an important role in the development of calculus. Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, queueing theory, the terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant. This relationship allows for the representation of a series using only two terms, r and a. The term r is the ratio, and a is the first term of the series. In the case above, where r is one half, the series has the sum one, if r is greater than one or less than minus one the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, if r is equal to one, all of the terms of the series are the same. If r is one the terms take two values alternately. The sum of the oscillates between two values. This is a different type of divergence and again the series has no sum, see for example Grandis series,1 −1 +1 −1 + ···. The sum can be computed using the self-similarity of the series, consider the sum of the following geometric series, s =1 +23 +49 +827 + ⋯. This series has common ratio 2/3, if we multiply through by this common ratio, then the initial 1 becomes a 2/3, the 2/3 becomes a 4/9, and so on,23 s =23 +49 +827 +1681 + ⋯. This new series is the same as the original, except that the first term is missing, subtracting the new series s from the original series s cancels every term in the original but the first, s −23 s =1, so s =3. A similar technique can be used to evaluate any self-similar expression, as n goes to infinity, the absolute value of r must be less than one for the series to converge. When a =1, this can be simplified to 1 + r + r 2 + r 3 + ⋯ =11 − r, the formula also holds for complex r, with the corresponding restriction, the modulus of r is strictly less than one. Since = 1−rn+1 and rn+1 →0 for | r | <1, convergence of geometric series can also be demonstrated by rewriting the series as an equivalent telescoping series. Consider the function, g = r K1 − r