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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

2.
Hilbert space
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The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of algebra and calculus from the two-dimensional Euclidean plane. A Hilbert space is a vector space possessing the structure of an inner product that allows length. Furthermore, Hilbert spaces are complete, there are limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces, the earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis —and ergodic theory, john von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis, geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space, at a deeper level, perpendicular projection onto a subspace plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be specified by its coordinates with respect to a set of coordinate axes. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of the space of sequences that are square-summable. The latter space is often in the literature referred to as the Hilbert space. One of the most familiar examples of a Hilbert space is the Euclidean space consisting of vectors, denoted by ℝ3. The dot product takes two vectors x and y, and produces a real number x·y, If x and y are represented in Cartesian coordinates, then the dot product is defined by ⋅ = x 1 y 1 + x 2 y 2 + x 3 y 3. The dot product satisfies the properties, It is symmetric in x and y, x · y = y · x. It is linear in its first argument, · y = ax1 · y + bx2 · y for any scalars a, b, and vectors x1, x2, and y. It is positive definite, for all x, x · x ≥0, with equality if. An operation on pairs of vectors that, like the dot product, a vector space equipped with such an inner product is known as a inner product space. Every finite-dimensional inner product space is also a Hilbert space, multivariable calculus in Euclidean space relies on the ability to compute limits, and to have useful criteria for concluding that limits exist

3.
Sequence
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In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members, the number of elements is called the length of the sequence. Unlike a set, order matters, and exactly the elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the numbers or the set of the first n natural numbers. The position of an element in a sequence is its rank or index and it depends on the context or of a specific convention, if the first element has index 0 or 1. For example, is a sequence of letters with the letter M first, also, the sequence, which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, the empty sequence is included in most notions of sequence, but may be excluded depending on the context. A sequence can be thought of as a list of elements with a particular order, Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations, Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers. There are a number of ways to denote a sequence, some of which are useful for specific types of sequences. One way to specify a sequence is to list the elements, for example, the first four odd numbers form the sequence. This notation can be used for sequences as well. For instance, the sequence of positive odd integers can be written. Listing is most useful for sequences with a pattern that can be easily discerned from the first few elements. Other ways to denote a sequence are discussed after the examples, the prime numbers are the natural numbers bigger than 1, that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence, the prime numbers are widely used in mathematics and specifically in number theory. The Fibonacci numbers are the integer sequence whose elements are the sum of the two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is, for a large list of examples of integer sequences, see On-Line Encyclopedia of Integer Sequences

4.
Inner product space
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In linear algebra, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a quantity known as the inner product of the vectors. Inner products allow the introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors. They also provide the means of defining orthogonality between vectors, inner product spaces generalize Euclidean spaces to vector spaces of any dimension, and are studied in functional analysis. An inner product induces a associated norm, thus an inner product space is also a normed vector space. A complete space with a product is called a Hilbert space. An space with a product is called a pre-Hilbert space, since its completion with respect to the norm induced by the inner product is a Hilbert space. Inner product spaces over the field of numbers are sometimes referred to as unitary spaces. In this article, the field of scalars denoted F is either the field of real numbers R or the field of complex numbers C, formally, an inner product space is a vector space V over the field F together with an inner product, i. e. Some authors, especially in physics and matrix algebra, prefer to define the inner product, then the first argument becomes conjugate linear, rather than the second. In those disciplines we would write the product ⟨ x, y ⟩ as ⟨ y | x ⟩, respectively y † x. Here the kets and columns are identified with the vectors of V and this reverse order is now occasionally followed in the more abstract literature, taking ⟨ x, y ⟩ to be conjugate linear in x rather than y. A few instead find a ground by recognizing both ⟨ ⋅, ⋅ ⟩ and ⟨ ⋅ | ⋅ ⟩ as distinct notations differing only in which argument is conjugate linear. There are various reasons why it is necessary to restrict the basefield to R and C in the definition. Briefly, the basefield has to contain an ordered subfield in order for non-negativity to make sense, the basefield has to have additional structure, such as a distinguished automorphism. More generally any quadratically closed subfield of R or C will suffice for this purpose, however in these cases when it is a proper subfield even finite-dimensional inner product spaces will fail to be metrically complete. In contrast all finite-dimensional inner product spaces over R or C, such as used in quantum computation, are automatically metrically complete. In some cases we need to consider non-negative semi-definite sesquilinear forms and this means that ⟨ x, x ⟩ is only required to be non-negative

5.
Vector projection
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The vector projection of a vector a on a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. It is a parallel to b, defined as a 1 = a 1 b ^ where a 1 is a scalar, called the scalar projection of a onto b. The scalar projection is equal to the length of the vector projection, the vector component or vector resolute of a perpendicular to b, sometimes also called the vector rejection of a from b, is the orthogonal projection of a onto the plane orthogonal to b. Both the projection a1 and rejection a2 of a vector a are vectors, and their sum is equal to a, typically, a vector projection is denoted in a bold font, and the corresponding scalar projection with normal font. In some cases, especially in handwriting, the projection is also denoted using a diacritic above or below the letter. The vector projection of a on b and the corresponding rejection are sometimes denoted by a∥b and a⊥b, the scalar projection of a on b is a scalar equal to a 1 = | a | cos θ where θ is the angle between a and b. A scalar projection can be used as a factor to compute the corresponding vector projection. The vector projection of a on b is a vector whose magnitude is the projection of a on b. The latter formula is more efficient than the former. By definition, a 2 = a − a 1 Hence, the scalar projection a on b is a scalar which has a negative sign if 90 < θ ≤180 degrees. It coincides with the length |c| of the vector projection if the angle is smaller than 90°, more exactly, a1 = |a1| if 0 ≤ θ ≤90 degrees, a1 = −|a1| if 90 < θ ≤180 degrees. The vector projection of a on b is a vector a1 which is null or parallel to b. More exactly, a1 =0 if θ = 90°, a1 and b have the direction if 0 ≤ θ <90 degrees, a1. The vector rejection of a on b is a vector a2 which is null or orthogonal to b. More exactly, a2 =0 if θ =0 degrees or θ =180 degrees, a2 is orthogonal to b if 0 < θ <180 degrees and it is also used in the Separating axis theorem to detect whether two convex shapes intersect. In some cases, the inner product coincides with the dot product, whenever they dont coincide, the inner product is used instead of the dot product in the formal definitions of projection and rejection. The projection of a vector on a plane is its orthogonal projection on that plane, the rejection of a vector from a plane is its orthogonal projection on a straight line which is orthogonal to that plane. The first is parallel to the plane, the second is orthogonal, for a given vector and plane, the sum of projection and rejection is equal to the original vector

6.
Inequality (mathematics)
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In mathematics, an inequality is a relation that holds between two values when they are different. The notation a ≠ b means that a is not equal to b and it does not say that one is greater than the other, or even that they can be compared in size. If the values in question are elements of a set, such as the integers or the real numbers. The notation a < b means that a is less than b, the notation a > b means that a is greater than b. In either case, a is not equal to b and these relations are known as strict inequalities. The notation a < b may also be read as a is less than b. The notation a ≥ b means that a is greater than or equal to b, not less than can also be represented by the symbol for less than bisected by a vertical line, not. In engineering sciences, a formal use of the notation is to state that one quantity is much greater than another. The notation a ≪ b means that a is less than b. The notation a ≫ b means that a is greater than b. Inequalities are governed by the following properties, all of these properties also hold if all of the non-strict inequalities are replaced by their corresponding strict inequalities and monotonic functions are limited to strictly monotonic functions. The transitive property of inequality states, For any real numbers a, b, c, If a ≥ b and b ≥ c, If a ≤ b and b ≤ c, then a ≤ c. If either of the premises is an inequality, then the conclusion is a strict inequality. E. g. if a ≥ b and b > c, then a > c An equality is of course a special case of a non-strict inequality. E. g. if a = b and b > c, then a > c The relations ≤ and ≥ are each others converse, For any real numbers a and b, If a ≤ b, then b ≥ a. If a ≥ b, then a + c ≥ b + c, If a ≤ b and c >0, then ac ≤ bc and a/c ≤ b/c. If c is negative, then multiplying or dividing by c inverts the inequality, If a ≥ b and c <0, then ac ≤ bc, If a ≤ b and c <0, then ac ≥ bc and a/c ≥ b/c. More generally, this applies for a field, see below

7.
Series (mathematics)
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In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a sequence has defined first and last terms. To emphasize that there are a number of terms, a series is often called an infinite series. In order to make the notion of an infinite sum mathematically rigorous, given an infinite sequence, the associated series is the expression obtained by adding all those terms together, a 1 + a 2 + a 3 + ⋯. These can be written compactly as ∑ i =1 ∞ a i, by using the summation symbol ∑. The sequence can be composed of any kind of object for which addition is defined. A series is evaluated by examining the finite sums of the first n terms of a sequence, called the nth partial sum of the sequence, and taking the limit as n approaches infinity. If this limit does not exist, the infinite sum cannot be assigned a value, and, in this case, the series is said to be divergent. On the other hand, if the partial sums tend to a limit when the number of terms increases indefinitely, then the series is said to be convergent, and the limit is called the sum of the series. An example is the series from Zenos dichotomy and its mathematical representation, ∑ n =1 ∞12 n =12 +14 +18 + ⋯. The study of series is a part of mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures, in addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For any sequence of numbers, real numbers, complex numbers, functions thereof. By definition the series ∑ n =0 ∞ a n converges to a limit L if and this definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k. When the index set is the natural numbers I = N, a series indexed on the natural numbers is an ordered formal sum and so we rewrite ∑ n ∈ N as ∑ n =0 ∞ in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers ∑ n =0 ∞ a n = a 0 + a 1 + a 2 + ⋯. When the semigroup G is also a space, then the series ∑ n =0 ∞ a n converges to an element L ∈ G if. This definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k, a series ∑an is said to converge or to be convergent when the sequence SN of partial sums has a finite limit

8.
Limit of a sequence
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In mathematics, the limit of a sequence is the value that the terms of a sequence tend to. If such a limit exists, the sequence is called convergent, a sequence which does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of analysis ultimately rests, limits can be defined in any metric or topological space, but are usually first encountered in the real numbers. The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes, leucippus, Democritus, Antiphon, Eudoxus and Archimedes developed the method of exhaustion, which uses an infinite sequence of approximations to determine an area or a volume. Archimedes succeeded in summing what is now called a geometric series, Newton dealt with series in his works on Analysis with infinite series, Method of fluxions and infinite series and Tractatus de Quadratura Curvarum. In the latter work, Newton considers the binomial expansion of n which he then linearizes by taking limits, at the end of the century, Lagrange in his Théorie des fonctions analytiques opined that the lack of rigour precluded further development in calculus. Gauss in his etude of hypergeometric series for the first time rigorously investigated under which conditions a series converged to a limit, the modern definition of a limit was given by Bernhard Bolzano and by Karl Weierstrass in the 1870s. In the real numbers, a number L is the limit of the if the numbers in the sequence become closer and closer to L. If x n = c for some constant c, then x n → c, if x n =1 n, then x n →0. If x n =1 / n when n is even, given any real number, one may easily construct a sequence that converges to that number by taking decimal approximations. For example, the sequence 0.3,0.33,0.333,0.3333, note that the decimal representation 0.3333. is the limit of the previous sequence, defined by 0.3333. ≜ lim n → ∞ ∑ i =1 n 310 i, finding the limit of a sequence is not always obvious. Two examples are lim n → ∞ n and the Arithmetic–geometric mean, the squeeze theorem is often useful in such cases. In other words, for measure of closeness ϵ, the sequences terms are eventually that close to the limit. The sequence is said to converge to or tend to the limit x, symbolically, this is, ∀ ϵ >0 ∃ N ∈ R ∀ n ∈ N. If a sequence converges to some limit, then it is convergent, limits of sequences behave well with respect to the usual arithmetic operations. For any continuous function f, if x n → x then f → f, in fact, any real-valued function f is continuous if and only if it preserves the limits of sequences. Some other important properties of limits of sequences include the following