1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times

2.
Encyclopedia of Mathematics
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The Encyclopedia of Mathematics is a large reference work in mathematics. It is available in form and on CD-ROM. The 2002 version contains more than 8,000 entries covering most areas of mathematics at a level. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, the CD-ROM contains animations and three-dimensional objects. Until November 29,2011, a version of the encyclopedia could be browsed online free of charge online This URL now redirects to the new wiki incarnation of the EOM. A new dynamic version of the encyclopedia is now available as a public wiki online and this new wiki is a collaboration between Springer and the European Mathematical Society. This new version of the encyclopedia includes the entire contents of the online version. All entries will be monitored for content accuracy by members of a board selected by the European Mathematical Society. Vinogradov, I. M. Matematicheskaya entsiklopediya, Moscow, Sov, Hazewinkel, M. Encyclopaedia of Mathematics, Kluwer,1994. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.1, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.2, Kluwer,1988. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.3, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.4, Kluwer,1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.5, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.6, Kluwer,1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.7, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.8, Kluwer,1992. Hazewinkel, M. Encyclopaedia of Mathematics, Vol.9, Hazewinkel, M. Encyclopaedia of Mathematics, Vol.10, Kluwer,1994. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement I, Kluwer,1997, Hazewinkel, M. Encyclopaedia of Mathematics, Supplement II, Kluwer,2000. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement III, Kluwer,2002, Hazewinkel, M. Encyclopaedia of Mathematics on CD-ROM, Kluwer,1998. Encyclopedia of Mathematics, public wiki monitored by a board under the management of the European Mathematical Society. List of online encyclopedias Current page of M. Hazewinkel Online Encyclopedia of Mathematics

3.
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

4.
Michiel Hazewinkel
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Born in Amsterdam to Jan Hazewinkel and Geertrude Hendrika Werner, Hazewinkel studied at the University of Amsterdam. After graduation Hazewinkel started his career as Assistant Professor at the University of Amsterdam in 1969. In 1970 he became Associate Professor at the Erasmus University Rotterdam, here he was thesis advisor of Roelof Stroeker, M. van de Vel, Jo Ritzen, and Gerard van der Hoek. From 1973 to 1975 he was also Professor at the Universitaire Instelling Antwerpen, were Marcel van de Vel was his PhD student. At the Centre for Mathematics and Computer CWI in Amsterdam in 1988 he became Professor of Mathematics and head of the Department of Algebra, Analysis, in 1994 Hazewinkel was elected member of the International Academy of Computer Sciences and Systems. Hazewinkel has authored and edited books, and numerous articles. With Michel Demazure and Pierre Gabriel, on invariants, canonical forms and moduli for linear, constant, finite dimensional, dynamical systems. Moduli and canonical forms for linear dynamical systems II, The topological case, on Lie algebras and finite dimensional filtering. Stochastics, a journal of probability and stochastic processes 7. 1–2. Nonexistence of finite-dimensional filters for conditional statistics of the sensor problem. Systems & control letters 3.6, 331–340, the algebra of quasi-symmetric functions is free over the integers

5.
Banach space
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In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn, Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces, the vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. All norms on a vector space are equivalent. Every finite-dimensional normed space over R or C is a Banach space, if X and Y are normed spaces over the same ground field K, the set of all continuous K-linear maps T, X → Y is denoted by B. In infinite-dimensional spaces, not all maps are continuous. For Y a Banach space, the space B is a Banach space with respect to this norm, if X is a Banach space, the space B = B forms a unital Banach algebra, the multiplication operation is given by the composition of linear maps. If X and Y are normed spaces, they are isomorphic normed spaces if there exists a linear bijection T, X → Y such that T, if one of the two spaces X or Y is complete then so is the other space. Two normed spaces X and Y are isometrically isomorphic if in addition, T is an isometry, the Banach–Mazur distance d between two isomorphic but not isometric spaces X and Y gives a measure of how much the two spaces X and Y differ. Every normed space X can be embedded in a Banach space. More precisely, there is a Banach space Y and an isometric mapping T, X → Y such that T is dense in Y. If Z is another Banach space such that there is an isomorphism from X onto a dense subset of Z. This Banach space Y is the completion of the normed space X, the underlying metric space for Y is the same as the metric completion of X, with the vector space operations extended from X to Y. The completion of X is often denoted by X ^, the cartesian product X × Y of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are used, such as ∥ ∥1 = ∥ x ∥ + ∥ y ∥, ∥ ∥ ∞ = max. In this sense, the product X × Y is complete if and only if the two factors are complete. If M is a linear subspace of a normed space X, there is a natural norm on the quotient space X / M

6.
Stereotype space
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In functional analysis and related areas of mathematics stereotype spaces are topological vector spaces defined by a special variant of reflexivity condition. Each pseudocomplete barreled space X is stereotype, a metrizable locally convex space X is stereotype if and only if X is complete. Each infinite dimensional normed space X considered with the X ⋆ -weak topology is not stereotype, there exist stereotype spaces which are not Mackey spaces. Some simple connections between the properties of a stereotype space X and those of its dual space X ⋆ are expressed in the following list of regularities, the first results on this type of reflexivity of topological vector spaces were obtained by M. F. Smith in 1952. Further investigations were conducted by B. S. Brudovskii, W. C, waterhouse, K. Brauner, S. S. Akbarov, and E. T. Shavgulidze. Each locally convex space X can be transformed into a space with the help of the standard operations of pseudocompletion and pseudosaturation defined by the following two propositions. If X is a locally convex space, then its pseudosaturation X △ is stereotype. Dually, if X is a locally convex space, then its pseudocompletion X ▽ is stereotype. For arbitrary locally convex space X the spaces X △ ▽ and X ▽ △ are stereotype and it defines two natural tensor products X ⊛ Y, = Hom ⋆, X ⊙ Y, = Hom. This condition is weaker than the existence of the Schauder basis, the following proposition holds, If two stereotype spaces X and Y have the stereotype approximation property, then the spaces Hom, X ⊛ Y and X ⊙ Y have the stereotype approximation property as well. In particular, if X has the approximation property, then the same is true for X ⋆. This allows to reduce the list of counterexamples in comparison with the Banach theory, the arising theory of stereotype algebras allows to simplify constructions in the duality theories for non-commutative groups. In particular, the group algebras in these theories become Hopf algebras in the algebraic sense. Schaefer, Helmuth H. Topological vector spaces, Robertson, A. P. Robertson, W. J. Topological vector spaces. The Pontrjagin duality theorem in linear spaces, on k- and c-reflexivity of locally convex vector spaces. Brauner, K. Duals of Fréchet spaces and a generalization of the Banach-Dieudonné theorem, Akbarov, S. S. Pontryagin duality in the theory of topological vector spaces and in topological algebra. Akbarov, S. S. Holomorphic functions of exponential type, envelopes and refinements in categories, with applications to functional analysis. On two classes of spaces reflexive in the sense of Pontryagin, Akbarov, S. S. Pontryagin duality and topological algebras

7.
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

8.
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

9.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space

10.
PlanetMath
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PlanetMath is a free, collaborative, online mathematics encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, intended to be comprehensive, the project is currently hosted by the University of Waterloo. The site is owned by a US-based nonprofit corporation, PlanetMath. org, the main PlanetMath focus is on encyclopedic entries, and some forum discussions. In addition, the project hosts data about books, expositions, a system for semi-private messaging among users is also in place. Developing software recommendations for improved content authoring and editorial functions, PlanetMath content is licensed under the copyleft Creative Commons Attribution/Share-Alike License. All content is written in LaTeX, a typesetting system popular among mathematicians because of its support of the needs of mathematical typesetting. The software running PlanetMath is written in Perl and runs on Linux and it is known as Noösphere and has been released under the free BSD License. As of March 13,2013 PlanethMath has retired Noösphere and runs now on a software called Planetary, encyclopedic content and bibliographic materials related to physics, mathematics and mathematical physics are developed by PlanetPhysics. The site, launched in 2005, uses similar software, but a significantly different moderation model with emphasis on current research in physics and peer review

11.
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

12.
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