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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.
Group (mathematics)
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In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure and it allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, after contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right, to explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. A theory has developed for finite groups, which culminated with the classification of finite simple groups. Since the mid-1980s, geometric group theory, which studies finitely generated groups as objects, has become a particularly active area in group theory. One of the most familiar groups is the set of integers Z which consists of the numbers, −4, −3, −2, −1,0,1,2,3,4. The following properties of integer addition serve as a model for the group axioms given in the definition below. For any two integers a and b, the sum a + b is also an integer and that is, addition of integers always yields an integer. This property is known as closure under addition, for all integers a, b and c, + c = a +. Expressed in words, adding a to b first, and then adding the result to c gives the final result as adding a to the sum of b and c. If a is any integer, then 0 + a = a +0 = a, zero is called the identity element of addition because adding it to any integer returns the same integer. For every integer a, there is a b such that a + b = b + a =0. The integer b is called the element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a class sharing similar structural aspects. To appropriately understand these structures as a collective, the abstract definition is developed
3.
Cycle graph (algebra)
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In group theory, a sub-field of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. A cycle is the set of powers of a group element a, where an. The element a is said to generate the cycle, in a finite group, some non-zero power of a must be the group identity, e, the lowest such power is the order of the cycle, the number of distinct elements in it. Cycles can overlap, or they can have no element in common, the cycle graph displays each interesting cycle as a polygon. If a generates a cycle of order 6, then a6 = e, then the set of powers of a2, is a cycle, but this is really no new information. Similarly, a5 generates the same cycle as a itself, so, only the primitive cycles need be considered, namely those that are not subsets of another cycle. Each of these is generated by some primitive element, a, take one point for each element of the original group. For each primitive element, connect e to a, a to a2, an−1 to an, etc. until e is reached. The result is the cycle graph, when a2 = e, a has order 2, and is connected to e by two edges. Except when the intent is to emphasize the two edges of the cycle, it is drawn as a single line between the two elements. As an example of a cycle graph, consider the dihedral group Dih4. The multiplication table for this group is shown on the left, notice the cycle e, a, a2, a3. It can be seen from the table that successive powers of a behave this way. In other words,2 = a2,3 = a and this behavior is true for any cycle in any group – a cycle may be traversed in either direction. Cycles that contain a number of elements implicitly have cycles that are not shown in the graph. For the group Dih4 above, we want to draw a line between a2 and e since 2 = e, but since a2 is part of a larger cycle. There can be ambiguity when two cycles share an element that is not the identity element, consider for example, the simple quaternion group, whose cycle graph is shown on the right. Each of the elements in the row when multiplied by itself gives −1
4.
Dihedral group of order 6
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In mathematics, the smallest non-abelian group has 6 elements. It is a group with notation D3 and the symmetric group of degree 3. This page illustrates many group concepts using this group as example, in two dimensions, the group D3 is the symmetry group of an equilateral triangle. In contrast with the case of a square or other polygon, all permutations of the vertices can be achieved by rotation, let a be the action swap the first block and the second block, and let b be the action swap the second block and the third block. In multiplicative form, we traditionally write xy for the combined action first do y, then do x, so that ab is the action RGB ↦ RBG ↦ BRG, i. e. take the last block and move it to the front. Note that the action aa has the effect RGB ↦ GRB ↦ RGB, leaving the blocks as they were, similarly, bb = e, = e, and = = e, so each of the above actions has an inverse. By inspection, we can determine associativity and closure, note for example that a = a = aba. The group is non-abelian since, for example, ab ≠ ba, since it is built up from the basic actions a and b, we say that the set generates it. Note that the second means that the group is a Coxeter group. With the generators a and b, we define the additional shorthands c, = aba, d, = ab, in the form of a Cayley table, the group operations now read, Note that non-equal non-identity elements only commute if they are each others inverse. Therefore the group is centerless, i. e. the center of the group consists only of the identity element, therefore, if we apply, then, and then the inverse of, which is also, the resulting permutation is. Note that conjugate group elements always have the order. From Lagranges theorem we know that any subgroup of a group with 6 elements must have order 2 or 3. The existence of subgroups of order 2 and 3 is also a consequence of Cauchys theorem, the first-mentioned is, the alternating group A3. The left cosets and the cosets of A3 coincide and consist of A3. The left cosets of are, The right cosets of are, Thus A3 is normal, the quotient group G / A3 is isomorphic with C2. G = A3 ⋊ H, a product, where H is a subgroup of two elements, and one of the three swaps. This decomposition is also a consequence of the Schur–Zassenhaus theorem, in terms of permutations the two group elements of G / A3 are the set of even permutations and the set of odd permutations
5.
Symmetric group
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Since there are n. possible permutation operations that can be performed on a tuple composed of n symbols, it follows that the order of the symmetric group Sn is n. For the remainder of this article, symmetric group will mean a group on a finite set. The symmetric group is important to diverse areas of such as Galois theory, invariant theory, the representation theory of Lie groups. Cayleys theorem states that every group G is isomorphic to a subgroup of the group on G. The symmetric group on a finite set X is the group elements are all bijective functions from X to X. For finite sets, permutations and bijective functions refer to the same operation, the symmetric group of degree n is the symmetric group on the set X =. The symmetric group on a set X is denoted in various ways including SX,
6.
Pythagorean theorem
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In mathematics, the Pythagorean theorem, also known as Pythagorass theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the two sides. There is some evidence that Babylonian mathematicians understood the formula, although little of it indicates an application within a mathematical framework, Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases. The theorem has been given numerous proofs – possibly the most for any mathematical theorem and they are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. The Pythagorean theorem was known long before Pythagoras, but he may well have been the first to prove it, in any event, the proof attributed to him is very simple, and is called a proof by rearrangement. The two large squares shown in the figure each contain four triangles, and the only difference between the two large squares is that the triangles are arranged differently. Therefore, the space within each of the two large squares must have equal area. Equating the area of the white space yields the Pythagorean theorem and that Pythagoras originated this very simple proof is sometimes inferred from the writings of the later Greek philosopher and mathematician Proclus. Several other proofs of this theorem are described below, but this is known as the Pythagorean one, If the length of both a and b are known, then c can be calculated as c = a 2 + b 2. If the length of the c and of one side are known. The Pythagorean equation relates the sides of a triangle in a simple way. Another corollary of the theorem is that in any triangle, the hypotenuse is greater than any one of the other sides. A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. This theorem may have more known proofs than any other, the book The Pythagorean Proposition contains 370 proofs, Let ABC represent a right triangle, with the right angle located at C, as shown on the figure. Draw the altitude from point C, and call H its intersection with the side AB, point H divides the length of the hypotenuse c into parts d and e. By a similar reasoning, the triangle CBH is also similar to ABC, the proof of similarity of the triangles requires the triangle postulate, the sum of the angles in a triangle is two right angles, and is equivalent to the parallel postulate. Similarity of the leads to the equality of ratios of corresponding sides. The first result equates the cosines of the angles θ, whereas the second result equates their sines, the role of this proof in history is the subject of much speculation
7.
Lie group
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In mathematics, a Lie group /ˈliː/ is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of transformation groups. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie’s student Arthur Tresse, an extension of Galois theory to the case of continuous symmetry groups was one of Lies principal motivations. Lie groups are smooth manifolds and as such can be studied using differential calculus. Lie groups play an role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various geometries by specifying an appropriate transformation group that leaves certain geometric properties invariant and this idea later led to the notion of a G-structure, where G is a Lie group of local symmetries of a manifold. On a global level, whenever a Lie group acts on an object, such as a Riemannian or a symplectic manifold. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry, Linear actions of Lie groups are especially important, and are studied in representation theory. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, a real Lie group is a group that is also a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps. Smoothness of the group multiplication μ, G × G → G μ = x y means that μ is a mapping of the product manifold G×G into G. These two requirements can be combined to the requirement that the mapping ↦ x −1 y be a smooth mapping of the product manifold into G. The 2×2 real invertible matrices form a group under multiplication, denoted by GL or by GL2 and this is a four-dimensional noncompact real Lie group. This group is disconnected, it has two connected components corresponding to the positive and negative values of the determinant, the rotation matrices form a subgroup of GL, denoted by SO. It is a Lie group in its own right, specifically, using the rotation angle φ as a parameter, this group can be parametrized as follows, SO =. Addition of the angles corresponds to multiplication of the elements of SO, thus both multiplication and inversion are differentiable maps. The orthogonal group also forms an example of a Lie group. All of the examples of Lie groups fall within the class of classical groups. Hilberts fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples, if the underlying manifold is allowed to be infinite-dimensional, then one arrives at the notion of an infinite-dimensional Lie group
8.
Manifold
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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of a manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. One-dimensional manifolds include lines and circles, but not figure eights, two-dimensional manifolds are also called surfaces. Although a manifold locally resembles Euclidean space, globally it may not, for example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of map projections of the region into the Euclidean plane. When a region appears in two neighbouring charts, the two representations do not coincide exactly and a transformation is needed to pass from one to the other, Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. One important class of manifolds is the class of differentiable manifolds and this differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured, symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity. After a line, the circle is the simplest example of a topological manifold, Topology ignores bending, so a small piece of a circle is treated exactly the same as a small piece of a line. Consider, for instance, the top part of the circle, x2 + y2 =1. Any point of this arc can be described by its x-coordinate. So, projection onto the first coordinate is a continuous, and invertible, mapping from the arc to the open interval. Such functions along with the regions they map are called charts. Similarly, there are charts for the bottom, left, and right parts of the circle, together, these parts cover the whole circle and the four charts form an atlas for the circle. The top and right charts, χtop and χright respectively, overlap in their domain, Each map this part into the interval, though differently. Let a be any number in, then, T = χ r i g h t = χ r i g h t =1 − a 2 Such a function is called a transition map. The top, bottom, left, and right charts show that the circle is a manifold, charts need not be geometric projections, and the number of charts is a matter of some choice. These two charts provide a second atlas for the circle, with t =1 s Each chart omits a single point, either for s or for t and it can be proved that it is not possible to cover the full circle with a single chart. Viewed using calculus, the transition function T is simply a function between open intervals, which gives a meaning to the statement that T is differentiable
9.
Symmetry group
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In abstract algebra, the symmetry group of an object is the group of all transformations under which the object is invariant with composition as the group operation. For a space with a metric, it is a subgroup of the group of the space concerned. If not stated otherwise, this article considers symmetry groups in Euclidean geometry, the objects may be geometric figures, images, and patterns, such as a wallpaper pattern. The definition can be more precise by specifying what is meant by image or pattern. For symmetry of objects, one may also want to take their physical composition into account. The group of isometries of space induces an action on objects in it. The symmetry group is also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries under which the figure is invariant. The subgroup of orientation-preserving isometries that leave the figure invariant is called its symmetry group. The proper symmetry group of an object is equal to its symmetry group if. The proper symmetry group is then a subgroup of the orthogonal group SO. A discrete symmetry group is a group such that for every point of the space the set of images of the point under the isometries in the symmetry group is a discrete set. There are also continuous symmetry groups, which contain rotations of arbitrarily small angles or translations of arbitrarily small distances, the group of all symmetries of a sphere O is an example of this, and in general such continuous symmetry groups are studied as Lie groups. With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups, for example, two 3D figures have mirror symmetry, but with respect to different mirror planes. Two 3D figures have 3-fold rotational symmetry, but with respect to different axes, two 2D patterns have translational symmetry, each in one direction, the two translation vectors have the same length but a different direction. When considering isometry groups, one may restrict oneself to those where for all points the set of images under the isometries is topologically closed. This includes all discrete isometry groups and also involved in continuous symmetries. A figure with this group is non-drawable and up to arbitrarily fine detail homogeneous. The group generated by all translations, this group cannot be the group of a pattern, it would be homogeneous