1.
Network topology
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Network topology is the arrangement of the various elements of a computer network. Essentially, it is the structure of a network and may be depicted physically or logically. Distances between nodes, physical interconnections, transmission rates, or signal types may differ between two networks, yet their topologies may be identical, an example is a local area network. Conversely, mapping the data flow between the components determines the topology of the network. Two basic categories of network topologies exist, physical topologies and logical topologies, the cabling layout used to link devices is the physical topology of the network. This refers to the layout of cabling, the locations of nodes, a networks logical topology is not necessarily the same as its physical topology. For example, the twisted pair Ethernet using repeater hubs was a logical bus topology carried on a physical star topology. Token ring is a ring topology, but is wired as a physical star from the media access unit. Logical topologies are often associated with media access control methods. Some networks are able to change their logical topology through configuration changes to their routers. The study of network topology recognizes eight basic topologies, point-to-point, bus, star, ring or circular, mesh, tree, hybrid, the simplest topology with a dedicated link between two endpoints. Easiest to understand, of the variations of point-to-point topology, is a point-to-point communications channel that appears, to the user, a childs tin can telephone is one example of a physical dedicated channel. Using circuit-switching or packet-switching technologies, a point-to-point circuit can be set up dynamically, switched point-to-point topologies are the basic model of conventional telephony. The value of a permanent point-to-point network is unimpeded communications between the two endpoints, the value of an on-demand point-to-point connection is proportional to the number of potential pairs of subscribers and has been expressed as Metcalfes Law. In local area networks where bus topology is used, each node is connected to a single cable and this central cable is the backbone of the network and is known as the bus. A signal from the travels in both directions to all machines connected on the bus cable until it finds the intended recipient. If the machine address does not match the address for the data. Alternatively, if the matches the machine address, the data is accepted

2.
Klein bottle
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Informally, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. Other related non-orientable objects include the Möbius strip and the projective plane. Whereas a Möbius strip is a surface boundary, a Klein bottle has no boundary. The Klein bottle was first described in 1882 by the German mathematician Felix Klein and it may have been originally named the Kleinsche Fläche and then misinterpreted as Kleinsche Flasche, which ultimately may have led to the adoption of this term in the German language as well. The following square is a polygon of the Klein bottle. The idea is to glue together the corresponding coloured edges so that the arrows match, note that this is an abstract gluing in the sense that trying to realize this in three dimensions results in a self-intersecting Klein bottle. To construct the Klein Bottle, glue the red arrows of the square together, to glue the ends of the cylinder together so that the arrows on the circles match, you must pass one end through the side of the cylinder. Note that this creates a circle of self-intersection - this is an immersion of the Klein bottle in three dimensions and this immersion is useful for visualizing many properties of the Klein bottle. For example, the Klein bottle has no boundary, where the surface stops abruptly, the common physical model of a Klein bottle is a similar construction. The Science Museum in London has on display a collection of hand-blown glass Klein bottles, the bottles date from 1995 and were made for the museum by Alan Bennett. The Klein bottle, proper, does not self-intersect, nonetheless, there is a way to visualize the Klein bottle as being contained in four dimensions. By adding a dimension to the three-dimensional space, the self-intersection can be eliminated. Gently push a piece of the containing the intersection along the fourth dimension. A useful analogy is to consider a curve on the plane. Suppose for clarification that we adopt time as that fourth dimension, consider how the figure could be constructed in xyzt-space. The accompanying illustration shows one useful evolution of the figure, at t=0 the wall sprouts from a bud somewhere near the intersection point. After the figure has grown for a while, the earliest section of the wall begins to recede, disappearing like the Cheshire Cat but leaving its ever-expanding smile behind. By the time the front gets to where the bud had been, there’s nothing there to intersect

3.
Shape of the universe
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The shape of the universe is the local and global geometry of the Universe, in terms of both curvature and topology. The shape of the universe is related to general relativity which describes how spacetime is curved and bent by mass, cosmologists distinguish between the observable universe and the global universe. The observable universe consists of the part of the universe that can, in principle, simply connected space or multiply connected. There are certain logical connections among these properties, for example, a universe with positive curvature is necessarily finite. Although it is assumed in the literature that a flat or negatively curved universe is infinite. Theorists have been trying to construct a mathematical model of the shape of the universe. In formal terms, this is a 3-manifold model corresponding to the section of the 4-dimensional space-time of the universe. The model most theorists currently use is the Friedmann–Lemaître–Robertson–Walker model, ideally, one can continue to look back all the way to the Big Bang, in practice, however, the farthest away one can look is the cosmic microwave background, as anything past that was opaque. Experimental investigations show that the universe is very close to isotropic. If the observable universe encompasses the entire universe, we may be able to determine the structure of the entire universe by observation. The universe may be small in dimensions and not in others. For example, if the universe is a closed loop, one would expect to see multiple images of an object in the sky. The curvature of space is a description of length relationships in spatial coordinates. In mathematics, any geometry has three possible curvatures, so the geometry of the universe has the three possible curvatures. Flat Positively curved Negatively curved An example of a flat curvature would be any Euclidean geometry, curved geometries are in the domain of Non-Euclidean geometry. An example of a curved surface would be the surface of a sphere such as the Earth. A triangle drawn from the equator to a pole will result in at least two angles being 90°, making the sum of the 3 angles greater than 180°, an example of a negative curved surface would be the shape of a saddle or mountain pass. A triangle drawn on a saddle shape will result in the sum of the angles adding up to less than 180° due to the curving away as the triangle moves away from the center

4.
Topology
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, important topological properties include connectedness and compactness. Topology developed as a field of study out of geometry and set theory, through analysis of such as space, dimension. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs, Leonhard Eulers Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the fields first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, by the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to geometry and together they make up the geometric theory of differentiable manifolds. Geometric topology primarily studies manifolds and their embeddings in other manifolds, a particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots, Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler and his 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron and this led to his polyhedron formula, V − E + F =2. Some authorities regard this analysis as the first theorem, signalling the birth of topology, further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term Topologie in Vorstudien zur Topologie, written in his native German, in 1847, the term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Their work was corrected, consolidated and greatly extended by Henri Poincaré, in 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a case of a general topological space. In 1914, Felix Hausdorff coined the term topological space and gave the definition for what is now called a Hausdorff space, currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski

5.
Link (knot theory)
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In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component, links and knots are studied in a branch of mathematics called knot theory. Implicit in this definition is that there is a reference link, usually called the unlink. For example, a co-dimension two link in 3-dimensional space is a subspace of 3-dimensional Euclidean space whose connected components are homeomorphic to circles, the simplest nontrivial example of a link with more than one component is called the Hopf link, which consists of two circles linked together once. The circles in the Borromean rings are collectively linked despite the fact that no two of them are directly linked, the Borromean rings thus form a Brunnian link and in fact constitute the simplest such link. The notion of a link can be generalized in a number of ways, frequently the word link is used to describe any submanifold of the sphere S n diffeomorphic to a disjoint union of a finite number of spheres, S j. If M is disconnected, the embedding is called a link, if M is connected, it is called a knot. While links are defined as embeddings of circles, it is interesting and especially technically useful to consider embedded intervals. The type of a tangle is the manifold X, together with an embedding of ∂ X. The condition that the boundary of X lies in R × says that intervals either connect two lines or connect two points on one of the lines, but imposes no conditions on the circles. Tangles include links, braids, and others besides – for example, in this context, a braid is defined as a tangle which is always going down – whose derivative always has a non-zero component in the vertical direction. In particular, it must consist solely of intervals, and not double back on itself, however, a string link is a tangle consisting of only intervals, with the ends of each strand required to lie at. – i. e. connecting the integers, and ending in the order that they began, if this has ℓ components. A string link need not be a braid – it may double back on itself, a braid that is also a string link is called a pure braid, and corresponds with the usual such notion. The key technical value of tangles and string links is that they have algebraic structure, the tensor structure is given by juxtaposition of tangles – putting one tangle to the right of the other. For a fixed ℓ, isotopy classes of ℓ-component string links form a monoid, however, concordance classes of string links do have inverses, where inverse is given by flipping the string link upside down, and thus form a group. Linking number Hyperbolic link Unlink Link group

6.
Three utilities problem
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It is part of the mathematical field of topological graph theory which studies the embedding of graphs on surfaces. In more formal terms, the problem asks whether the complete bipartite graph K3,3 is planar. This graph is often referred to as the utility graph in reference to the problem, a review of the problems history is given by Kullman who states that most published references to the problem characterize it as very ancient. In the earliest publication found by Kullman, Henry Dudeney names it water, gas, however, Dudeney states that the problem is as old as the hills. much older than electric lighting, or even gas. Dudeney also published the same puzzle previously, in The Strand Magazine in 1913, another early version of the problem involves connecting three houses to three wells. K3,3 makes an appearance as a mathematical graph in Henneberg. As it is presented, the solution to the utility puzzle is no -- meaning. The problem may be formalized mathematically as asking whether the complete bipartite graph K3,3 is planar, kazimierz Kuratowski stated in 1930 that K3,3 is nonplanar, from which it follows that the problem has no solution. Kullman, however, states that Interestingly enough, Kuratowski did not publish a proof that non-planar. One proof of the impossibility of finding a planar embedding of K3,3 uses a case involving the Jordan curve theorem. In this solution, one examines different possibilities for the locations of the vertices with respect to the 4-cycles of the graph, in the utility graph, E =9 and 2V −4 =8, violating this inequality, so the utility graph cannot be planar. K3,3 is equivalent to the circulant graph Ci6 and it is toroidal, which means it can be embedded on a torus. In terms of the three cottage problem this means the problem can be solved by punching two holes through the plane and connecting them with a tube and this changes the topological properties of the surface and using the tube we can connect the three cottages without crossing lines. An equivalent statement is that the genus of the utility graph is one. A surface of one is equivalent to a torus. Another way of changing the rules of the puzzle is to allow utility lines to pass through the cottages or utilities, the utility graph K3,3 may be drawn with only one crossing, but not with zero crossings, so its crossing number is one. The utility graph K3,3 is the -cage, the smallest triangle-free cubic graph, like all other complete bipartite graphs, it is a well-covered graph, meaning that every maximal independent set has the same size. In this graph, the only two independent sets are the two sides of the bipartition, and obviously they are equal

7.
Geometric topology
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In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. This was the origin of simple homotopy theory, manifolds differ radically in behavior in high and low dimension. High-dimensional topology refers to manifolds of dimension 5 and above, or in terms, embeddings in codimension 3. Low-dimensional topology is concerned with questions in dimensions up to 4, thus the topological classification of 4-manifolds is in principle easy, and the key questions are, does a topological manifold admit a differentiable structure, and if so, how many. Notably, the case of dimension 4 is the last open case of the generalized Poincaré conjecture. The distinction is because surgery theory works in dimension 5 and above, in dimension 4 and below, surgery theory does not work, and other phenomena occur. Indeed, one approach to discussing low-dimensional manifolds is to ask what would surgery theory predict to be true, were it to work, – and then understand low-dimensional phenomena as deviations from this. The precise reason for the difference at dimension 5 is because the Whitney embedding theorem, in surgery theory, the key step is in the middle dimension, and thus when the middle dimension has codimension more than 2, the Whitney trick works. The key consequence of this is Smales h-cobordism theorem, which works in dimension 5 and above, the limit of this tower yields a topological but not differentiable map, hence surgery works topologically but not differentiably in dimension 4. A manifold is orientable if it has a consistent choice of orientation, in this setting, various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Thus an i-handle is the analogue of an i-cell. Handle decompositions of manifolds arise naturally via Morse theory, the modification of handle structures is closely linked to Cerf theory. Local flatness is a property of a submanifold in a manifold of larger dimension. In the category of manifolds, locally flat submanifolds play a role similar to that of embedded submanifolds in the category of smooth manifolds. Suppose a d dimensional manifold N is embedded into an n dimensional manifold M and that is, there exists a homeomorphism U → R n such that the image of U ∩ N coincides with R d. Brown and Mazur received the Veblen Prize for their independent proofs of this theorem, low-dimensional topology includes, Surface s 3-manifolds 4-manifolds each have their own theory, where there are some connections. Knot theory is the study of mathematical knots, while inspired by knots which appear in daily life in shoelaces and rope, a mathematicians knot differs in that the ends are joined together so that it cannot be undone. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, to gain further insight, mathematicians have generalized the knot concept in several ways

8.
Topological defect
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See also topological excitations and the base concepts, topology, topological manifold, differential equations, quantum mechanics and condensed matter physics. A topological defect can be proven to exist because the boundary conditions entail the existence of distinct solutions. Topological defects occur in partial differential equations and are believed to drive phase transitions in condensed matter physics, topological defects in lambda transition universality class systems including, screw/edge-dislocations in liquid crystals, magnetic flux tubes in superconductors, and vortices in superfluids. Topological defects, of the type, are extremely high-energy phenomena. Observation of proposed topological defects that formed during the formation could theoretically be observed without significant energy expenditure. In the Big Bang theory, the universe cools from an initial hot, certain grand unified theories predict the formation of stable topological defects in the early universe, during these phase transitions. Depending on the nature of symmetry breakdown, various solitons are believed to have formed in the universe according to the Kibble-Zurek mechanism. The well-known topological defects are, Cosmic strings are one-dimensional lines that form when an axial or cylindrical symmetry is broken, domain walls, two-dimensional membranes that form when a discrete symmetry is broken at a phase transition. These walls resemble the walls of a foam, dividing the universe into discrete cells. Monopoles, cube-like defects that form when a spherical symmetry is broken, are predicted to have magnetic charge, textures form when larger, more complicated symmetry groups are completely broken. They are not as localized as the defects, and are unstable. Skyrmions Extra dimensions and higher dimensions, other more complex hybrids of these defect types are also possible. The matter composing these boundaries is in the an ordered phase, defects have also been found in biochemistry, notably in the process of protein folding. The homotopy theory of defects uses the group of the order parameter space of a medium to discuss the existence, stability. Suppose R is the parameter space for a medium. Let H be the subgroup of G for the medium. Then, the parameter space can be written as the Lie group quotient R=G/H. If G is a cover for G/H then, it can be shown that πn =πn-1

9.
Topological skeleton
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In shape analysis, skeleton of a shape is a thin version of that shape that is equidistant to its boundaries. The skeleton usually emphasizes geometrical and topological properties of the shape, such as its connectivity, topology, length, direction, together with the distance of its points to the shape boundary, the skeleton can also serve as a representation of the shape. Skeletons have several different mathematical definitions in the literature. Various different variants of skeleton can also be found, including straight skeletons, morphological skeletons, in the technical literature, the concepts of skeleton and medial axis are used interchangeably by some authors, while some other authors regard them as related, but not the same. Similarly, the concepts of skeletonization and thinning are also regarded as identical by some, within the life sciences skeletons found extensive use to characterize protein folding and plant morphology on various biological scales. If one sets fire at all points on the boundary of that grass field simultaneously, then the skeleton is the set of quench points and this intuitive description is the starting point for a number of more precise definitions. A disk B is said to maximal in a set A if B ⊆ A, one way of defining the skeleton of a shape A is as the set of centers of all maximal disks in A. The skeleton of a shape A can also be defined as the set of centers of the discs that touch the boundary of A in two or more locations. This definition assures that the points are equidistant from the shape boundary and is mathematically equivalent to Blums medial axis transform. Many definitions of skeleton make use of the concept of distance function, using the distance function is very attractive because its computation is relatively fast. One of the definitions of skeleton using the function is as the ridges of the distance function. There is a common mis-statement in the literature that the skeleton consists of points which are locally maximum in the distance transform and this is simply not the case, as even cursory comparison of a distance transform and the resulting skeleton will show. Points with no upstream segments in the distance function, the upstream of a point x is the segment starting at x which follows the maximal gradient path. Pruning algorithms are used to remove these branches. Shape Modeling and Applications, pp. 63–71, doi,10. 1109/SMI.2008.4547951, ISBN 978-1-4244-2260-9. Blum, Harry, A Transformation for Extracting New Descriptors of Shape, in Wathen-Dunn, W. Models for the Perception of Speech and Visual Form, Cambridge, MA, MIT Press, pp. 362–380. Bucksch, Alexander, A practical introduction to skeletons for the plant sciences, Applications in Plant Sciences,2, cychosz, Joseph, Graphics gems IV, San Diego, CA, USA, Academic Press Professional, Inc. pp. 465–473, ISBN 0-12-336155-9. An Introduction to Morphological Image Processing, ISBN 0-8194-0845-X, fundamentals of Digital Image Processing, ISBN 0-13-336165-9

10.
Topology optimization
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TO is different from shape optimization in the sense that the design can attain any shape within the design space, instead of dealing with predefined configurations. The conventional TO formulation uses a finite element method to evaluate the design performance, the design is optimised using nonlinear programming techniques such as the optimality criteria algorithm, the method of moving asymptotes and genetic algorithms. Currently, engineers mostly use TO at the level of a design process. Due to the forms that naturally occur, the result is often difficult to manufacture. For that reason the result emerging from TO is often fine-tuned for manufacturability, adding constraints to the formulation in order to increase the manufacturability is an active field of research. In some cases results from topology optimization can be manufactured using additive manufacturing. M The problem statement includes the following, An objective function and this function indicates the quality of the design and is to be minimised. A classic example is the goal of generating a stiff structure by minimisation of compliance, the material distribution as a problem variable. This is described by the density of the material at each location ρ, Material is either present, indicated by a 1, or absent, indicated by a 0. This indicates the volume within which the design can exist. Assembly and packaging requirements, human and tool accessibility are some of the factors that need to be considered in identifying this space. With the definition of the space, regions or components in the model that cannot be modified during the course of the optimization are considered as non-design regions. M constraints G j ≤0 a characteristic that the solution must satisfy, examples are the maximum amount of material to be distributed or maximum stress values. Evaluating u often includes solving a differential equation and this is most commonly done using the finite element method since these equations do not have a, known, analytical solution. There are various implementation methodologies that have used to solve TO problems. Solving TO problems in a sense is done by discretizing the design domain into finite elements. The material densities inside these elements are treated as the the problem variables. In this case material density of one indicates the presence of material, due to the attainable topological complexity of the design being dependent of the amount of elements, a large amount is preferred