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.
Euclidean plane isometry
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In geometry, a Euclidean plane isometry is an isometry of the Euclidean plane, or more informally, a way of transforming the plane that preserves geometrical properties such as length. There are four types, translations, rotations, reflections, the set of Euclidean plane isometries forms a group under composition, the Euclidean group in two dimensions. It is generated by reflections in lines, and every element of the Euclidean group is the composite of at most three distinct reflections, informally, a Euclidean plane isometry is any way of transforming the plane without deforming it. For example, suppose that the Euclidean plane is represented by a sheet of transparent plastic sitting on a desk, examples of isometries include, Shifting the sheet one inch to the right. Rotating the sheet by ten degrees around some marked point, turning the sheet over to look at it from behind. Notice that if a picture is drawn on one side of the sheet, then turning the sheet over. These are examples of translations, rotations, and reflections respectively, there is one further type of isometry, called a glide reflection. However, folding, cutting, or melting the sheet are not considered isometries, neither are less drastic alterations like bending, stretching, or twisting. An isometry of the Euclidean plane is a transformation of the plane. That is, it is a map M, R2 → R2 such that for any points p and q in the plane, d = d and it can be shown that there are four types of Euclidean plane isometries. The line L is called the axis or the associated mirror. The combination of rotations about the origin and reflections about a line through the origin is obtained with all orthogonal matrices forming orthogonal group O, in the case of a determinant of −1 we have, R0, θ =. Which is a reflection in the x-axis followed by a rotation by an angle θ, or equivalently, reflection in a parallel line corresponds to adding a vector perpendicular to it. A translation can be seen as a composite of two parallel reflections, rotations, denoted by Rc, θ, where c is a point in the plane, and θ is the angle of rotation. In terms of coordinates, rotations are most easily expressed by breaking them up into two operations, first, a rotation around the origin is given by R0, θ =. These matrices are the matrices, with determinant 1. They form the orthogonal group SO. A rotation around c can be accomplished by first translating c to the origin, then performing the rotation around the origin, and finally translating the origin back to c

3.
Surface (topology)
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In topology and differential geometry, a surface is a two-dimensional manifold, and, as such, may be an abstract surface not embedded in any Euclidean space. For example, the Klein bottle is a surface, which cannot be represented in the three-dimensional Euclidean space without introducing self-intersections, in mathematics, a surface is a geometrical shape that resembles to a deformed plane. The most familiar examples arise as boundaries of solid objects in ordinary three-dimensional Euclidean space R3, the exact definition of a surface may depend on the context. Typically, in geometry, a surface may cross itself, while, in topology and differential geometry. A surface is a space, this means that a moving point on a surface may move in two directions. In other words, around almost every point, there is a patch on which a two-dimensional coordinate system is defined. For example, the surface of the Earth resembles a two-dimensional sphere, the concept of surface is widely used in physics, engineering, computer graphics, and many other disciplines, primarily in representing the surfaces of physical objects. For example, in analyzing the properties of an airplane. A surface is a space in which every point has an open neighbourhood homeomorphic to some open subset of the Euclidean plane E2. Such a neighborhood, together with the corresponding homeomorphism, is known as a chart and it is through this chart that the neighborhood inherits the standard coordinates on the Euclidean plane. These coordinates are known as coordinates and these homeomorphisms lead us to describe surfaces as being locally Euclidean. In most writings on the subject, it is assumed, explicitly or implicitly, that as a topological space a surface is also nonempty, second countable. It is also assumed that the surfaces under consideration are connected. The rest of this article will assume, unless specified otherwise, that a surface is nonempty, Hausdorff, second countable and these homeomorphisms are also known as charts. The boundary of the upper half-plane is the x-axis, a point on the surface mapped via a chart to the x-axis is termed a boundary point. The collection of points is known as the boundary of the surface which is necessarily a one-manifold, that is. On the other hand, a point mapped to above the x-axis is an interior point, the collection of interior points is the interior of the surface which is always non-empty. The closed disk is an example of a surface with boundary

4.
Classification of finite simple groups
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In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four broad classes described below. These groups can be seen as the building blocks of all finite groups. The Jordan–Hölder theorem is a precise way of stating this fact about finite groups. The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors. Gorenstein, Lyons, and Solomon are gradually publishing a simplified and revised version of the proof, the classification theorem has applications in many branches of mathematics, as questions about the structure of finite groups can sometimes be reduced to questions about finite simple groups. Thanks to the theorem, such questions can sometimes be answered by checking each family of simple groups. Daniel Gorenstein announced in 1983 that the simple groups had all been classified. The completed proof of the classification was announced by Aschbacher after Aschbacher, the simple groups of small 2-rank include, Groups of 2-rank 0, in other words groups of odd order, which are all solvable by the Feit–Thompson theorem. Alperin showed that the Sylow subgroup must be dihedral, quasidihedral, wreathed, Groups of sectional 2-rank at most 4, classified by the Gorenstein–Harada theorem. All groups not of small 2 rank can be split into two classes, groups of component type and groups of characteristic 2 type. A group is said to be of component type if for some centralizer C of an involution and these are more or less the groups of Lie type of odd characteristic of large rank, and alternating groups, together with some sporadic groups. A major step in case is to eliminate the obstruction of the core of an involution. This is accomplished by the B-theorem, which states that every component of C/O is the image of a component of C. The idea is that groups have a centralizer of an involution with a component that is a smaller quasisimple group. So to classify these groups one takes every central extension of known finite simple group. A group is of characteristic 2 type if the generalized Fitting subgroup F* of every 2-local subgroup Y is a 2-group. As the name suggests these are roughly the groups of Lie type over fields of characteristic 2, the rank 1 groups are the thin groups, classified by Aschbacher, and the rank 2 ones are the notorious quasithin groups, classified by Aschbacher and Smith. These correspond roughly to groups of Lie type of ranks 1 or 2 over fields of characteristic 2

5.
Jordan normal form
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Such a matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal, and with identical diagonal entries to the left and below them. This condition is satisfied if K is algebraically closed. The diagonal entries of the form are the eigenvalues. If the operator is given by a square matrix M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix, the Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for normal matrices, is a special case of the Jordan normal form. The Jordan normal form is named after Camille Jordan, some textbooks have the ones on the subdiagonal, i. e. immediately below the main diagonal instead of on the superdiagonal. The eigenvalues are still on the main diagonal, an n × n matrix A is diagonalizable if and only if the sum of the dimensions of the eigenspaces is n. Or, equivalently, if and only if A has n linearly independent eigenvectors, consider the following matrix, A =. Including multiplicity, the eigenvalues of A are λ =1,2,4,4, the dimension of the eigenspace corresponding to the eigenvalue 4 is 1, so A is not diagonalizable. However, there is an invertible matrix P such that A = PJP−1, the matrix J is almost diagonal. This is the Jordan normal form of A, the section Example below fills in the details of the computation. In general, a complex matrix A is similar to a block diagonal matrix J = where each block Ji is a square matrix of the form J i =. So there exists an invertible matrix P such that P−1AP = J is such that the only non-zero entries of J are on the diagonal, J is called the Jordan normal form of A. Each Ji is called a Jordan block of A, in a given Jordan block, every entry on the superdiagonal is 1. Assuming this result, we can deduce the properties, Counting multiplicity. Given an eigenvalue λi, its geometric multiplicity is the dimension of Ker, the sum of the sizes of all Jordan blocks corresponding to an eigenvalue λi is its algebraic multiplicity. A is diagonalizable if and only if, for every eigenvalue λ of A, the Jordan block corresponding to λ is of the form λ I + N, where N is a nilpotent matrix defined as Nij = δi, j−1