In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More each point of an n-dimensional manifold has a neighbourhood, homeomorphic to the Euclidean space of dimension n. In this more precise terminology, a manifold is referred to as an n-manifold. One-dimensional manifolds include circles, but not figure eights. Two-dimensional manifolds are called surfaces. Examples include the plane, the sphere, the torus, which can all be embedded in three dimensional real space, but the Klein bottle and real projective plane, which will always self-intersect when immersed in three-dimensional real space. Although a manifold locally resembles Euclidean space, meaning that every point has a neighbourhood homeomorphic to an open subset of Euclidean space, globally it may not: manifolds in general are not homeomorphic to Euclidean space. For example, the surface of the sphere is not homeomorphic to the Euclidean plane, because it has the global topological property of compactness that Euclidean space lacks, 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 and a transformation is needed to pass from one to the other, called a transition map. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described and understood in terms of the simpler local topological properties of Euclidean space. Manifolds arise as solution sets of systems of equations and as graphs of functions. Manifolds can be equipped with additional structure. One important class of manifolds is the class of differentiable manifolds. A Riemannian metric on a manifold allows 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. A surface is a two dimensional manifold, meaning that it locally resembles the Euclidean plane near each point. For example, the surface of a globe can be described by a collection of maps, which together form an atlas of the globe.
Although no individual map is sufficient to cover the entire surface of the globe, any place in the globe will be in at least one of the charts. Many places will appear in more than one chart. For example, a map of North America will include parts of South America and the Arctic circle; these regions of the globe will be described in full in separate charts, which in turn will contain parts of North America. There is a relation between adjacent charts, called a transition map that allows them to be patched together to cover the whole of the globe. Describing the coordinate charts on surfaces explicitly requires knowledge of functions of two variables, because these patching functions must map a region in the plane to another region of the plane. However, one-dimensional examples of manifolds can be described with functions of a single variable only. Manifolds have applications in computer-graphics and augmented-reality given the need to associate pictures to coordinates. In an augmented reality setting, a picture can be seen as something associated with a coordinate and by using sensors for detecting movements and rotation one can have knowledge of how the picture is oriented and placed in space.
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 the same as a small piece of a line. Consider, for instance, the top part of the unit circle, x2 + y2 = 1, where the y-coordinate is positive. Any point of this arc can be uniquely described by its x-coordinate. So, projection onto the first coordinate is a continuous, invertible, mapping from the upper arc to the open interval: χ t o p = x; such functions along with the open regions they map are called charts. There are charts for the bottom and right parts of the circle: χ b o t t o m = x χ l e f t = y χ r i g h t = y. Together, these parts cover the four charts form an atlas for the circle; the top and right charts, χ t o
In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s.
After contributions from other fields such as number theory and geometry, the group notion was generalized and 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. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory; the modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.
The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884; the third field contributing to group theory was number theory.
Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers; the convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; as of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.
Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004; this project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. 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 an integer. 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
In mathematics, a directed set is a nonempty set A together with a reflexive and transitive binary relation ≤, with the additional property that every pair of elements has an upper bound. In other words, for any a and b in A there must exist c in A with a ≤ b ≤ c; the notion defined above is sometimes called an upward directed set. A downward directed set is defined analogously, meaning when every pair of elements is bounded below; some authors assume. Beware that other authors call a set directed if and only if it is directed both upward and downward. Directed sets are a generalization of nonempty ordered sets; that is, all ordered sets are directed sets. Join semilattices are directed sets as well, but not conversely. Lattices are directed sets both upward and downward. In topology, directed sets are used to define nets, which generalize sequences and unite the various notions of limit used in analysis. Directed sets give rise to direct limits in abstract algebra and category theory. In addition to the definition above, there is an equivalent definition.
A directed set is a set A with a preorder such. In this definition, the existence of an upper bound of the empty subset implies. Examples of directed sets include: The set of natural numbers N with the ordinary order ≤ is a directed set. Let D1 and D2 be directed sets; the Cartesian product set D1 × D2 can be made into a directed set by defining ≤ if and only if n1 ≤ m1 and n2 ≤ m2. In analogy to the product order this is the product direction on the Cartesian product, it follows from previous example that the set N × N of pairs of natural numbers can be made into a directed set by defining ≤ if and only if n0 ≤ m0 and n1 ≤ m1. If x0 is a real number, we can turn the set R − into a directed set by writing a ≤ b if and only if |a − x0| ≥ |b − x0|. We say that the reals have been directed towards x0; this is an example of a directed set, not ordered. A example of a ordered set, not directed is the set, in which the only order relations are a ≤ a and b ≤ b. A less trivial example is like the previous example of the "reals directed towards x0" but in which the ordering rule only applies to pairs of elements on the same side of x0.
If T is a topological space and x0 is a point in T, we turn the set of all neighbourhoods of x0 into a directed set by writing U ≤ V if and only if U contains V. For every U: U ≤ U. For every U, V, W: if U ≤ V and V ≤ W we have U ⊇ V and V ⊇ W, which implies U ⊇ W, thus U ≤ W. For every U and V: since x0 ∈ U ∩ V, since both U ⊇ U ∩ V and V ⊇ U ∩ V, we have U ≤ U ∩ V and V ≤ U ∩ V. In a poset P, every lower closure of an element, i.e. every subset of the form where x is a fixed element from P, is directed. Directed sets are a more general concept than semilattices: every join semilattice is a directed set, as the join or least upper bound of two elements is the desired c; the converse does not hold however, witness the directed set ordered bitwise, where has three upper bounds but no least upper bound, cf. picture. The order relation in a directed set is not required to be antisymmetric, therefore directed sets are not always partial orders. However, the term directed set is used in the context of posets.
In this setting, a subset A of a ordered set is called a directed subset if it is a directed set according to the same partial order: in other words, it is not the empty set, every pair of elements has an upper bound. Here the order relation on the elements of A is inherited from P. A directed subset of a poset is not required to be downward closed. While the definition of a directed set is for an "upward-directed" set, it is possible to define a downward-directed set in which every pair of elements has a common lower bound. A subset of a poset is only if its upper closure is a filter. Directed subsets are used in domain theory; these are posets. In this context, directed subsets again provide a generalization of convergent sequences. Filtered category Centered set Linked set General Topology. Gierz, Keimel, et al. Continuous Lattices and Domains, Cambridge University Press. ISBN 0-521-80338-1
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, b is called the imaginary part. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to certain equations. For example, the equation 2 = − 9 has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem; the idea is to extend the real numbers with an indeterminate i, taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1: 2 = 2 = = 9 = − 9, 2 = 2 = 2 = 9 = − 9.
According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers; the 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations. Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i; this means that complex numbers can be added and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can be divided by nonzero complex numbers. Overall, the complex number system is a field. Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
The complex number a + bi can be identified with the point in the complex plane. A complex number whose real part is zero is said to be purely imaginary. A complex number whose imaginary part is zero can be viewed as a real number. Complex numbers can be represented in polar form, which associates each complex number with its distance from the origin and with a particular angle known as the argument of this complex number; the geometric identification of the complex numbers with the complex plane, a Euclidean plane, makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis; the addition of complex numbers is thus depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not available in a vector space. Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1.
For example, 2 + 3i is a complex number. This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials; the relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i, which hold for all integers k. The real number a is called the real part of the complex number a + bi. To emphasize, the imaginary part does not include a factor i and b, not bi, is the imaginary part. Formally, the complex numbers are defined as the quotient ring of the polynomia
The Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane, its name arises because it was discovered by Jakob Steiner when he was in Rome in 1844. The simplest construction is as the image of a sphere centered at the origin under the map f =; this gives an implicit formula of x 2 y 2 + y 2 z 2 + z 2 x 2 − r 2 x y z = 0. Taking a parametrization of the sphere in terms of longitude and latitude, gives parametric equations for the Roman surface as follows: x = r2 cos θ cos φ sin φ y = r2 sin θ cos φ sin φ z = r2 cos θ sin θ cos2 φ; the origin is a triple point, each of the xy-, yz-, xz-planes are tangential to the surface there. The other places of self-intersection are double points, defining segments along each coordinate axis which terminate in six pinch points; the entire surface has tetrahedral symmetry. It is a particular type of Steiner surface, that is, a 3-dimensional linear projection of the Veronese surface.
For simplicity we consider only the case r = 1. Given the sphere defined by the points such that x 2 + y 2 + z 2 = 1, we apply to these points the transformation T defined by T = =, say, but we have U 2 V 2 + V 2 W 2 + W 2 U 2 = z 2 x 2 y 4 + x 2 y 2 z 4 + y 2 z 2 x 4 = = = = U V W, so U 2 V 2 + V 2 W 2 + W 2 U 2 − U V W = 0 as desired. Conversely, suppose we are given satisfying U 2 V 2 + V 2 W 2 + W 2 U 2 − U V W = 0. We prove that there exists such that x 2 + y 2 + z 2 = 1, for which U = x y, V = y z, W = z x, with one exception: In case 3.b. Below, we show. 1. In the case where none of U, V, W is 0, we can set x = W U V, y = U V W, z = V W U, it is easy to use to confirm that holds for x, y, z defined this way. 2. Suppose that W is 0. From this implies U 2 V 2
In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901. Unlike the Roman surface and the cross-cap, it has no other singularities than self-intersections. Boy's surface is discussed in Jean-Pierre Petit's Topo the world. Boy's surface was first parametrized explicitly by Bernard Morin in 1978. See below for another parametrization, discovered by Rob Kusner and Robert Bryant. Boy's surface is one of the two possible immersions of the real projective plane which have only a single triple point. To make a Boy's surface: Start with a sphere. Remove a cap. Attach one end of each of three strips to alternate sixths of the edge left by removing the cap. Bend each strip and attach the other end of each strip to the sixth opposite the first end, so that the inside of the sphere at one end is connected to the outside at the other. Make the strips skirt the middle rather than go through it. Join the loose edges of the strips; the joins intersect the strips.
Boy's surface has 3-fold symmetry. This means that it has an axis of discrete rotational symmetry: any 120° turn about this axis will leave the surface looking the same; the Boy's surface can be cut into three mutually congruent pieces. The Mathematical Research Institute of Oberwolfach has a large model of a Boy's surface outside the entrance and donated by Mercedes-Benz in January 1991; this model minimizes the Willmore energy of the surface. It consists of steel strips which represent the image of a polar coordinate grid under a parameterization given by Robert Bryant and Rob Kusner; the meridians become i.e. twisted by 180 degrees. All but one of the strips corresponding to circles of latitude are untwisted, while the one corresponding to the boundary of the unit circle is a Möbius strip twisted by three times 180 degrees — as is the emblem of the institute. Boy's surface can be used as a half-way model. A half-way model is an immersion of the sphere with the property that a rotation interchanges inside and outside, so can be employed to evert a sphere.
Boy's and Morin's surfaces begin a sequence of half-way models with higher symmetry first proposed by George Francis, indexed by the integers 2p. Kusner's parametrization yields all these. Boy's surface can be parametrized in several ways. One parametrization, discovered by Rob Kusner and Robert Bryant, is the following: given a complex number w whose magnitude is less than or equal to one, let g 1 = − 3 2 Im g 2 = − 3 2 Re g 3 = Im − 1 2 so that = 1 g 1 2 + g 2 2 + g 3 2 where x, y, z are the desired Cartesian coordinates of a point on the Boy's surface. If one performs an inversion of this parametrization centered on the