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Braid Groups
In mathematics, the braid group on strands (denoted B_n), also known as the Artin braid group, is the group whose elements are equivalence classes of -braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see ). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see ); and in monodromy invariants of algebraic geometry. Introduction In this introduction let ; the generalization to other values of will be straightforward. Consider two sets of four items lying on a table, with the items in each set being arranged in a vertical line, and such that one set sits next to the other. (In the illustrations below, these are the black dots.) Using four strands, each item of the first set is connected with an item of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
5 Strand Braiding Technique
5 (five) is a number, numeral (linguistics), numeral and numerical digit, digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. Humans, and many other animals, have 5 Digit (anatomy), digits on their Limb (anatomy), limbs. Mathematics 5 is a Fermat prime, a Mersenne prime exponent, as well as a Fibonacci number. 5 is the first congruent number, as well as the length of the hypotenuse of the smallest integer-sided right triangle, making part of the smallest Pythagorean triple (3, 4, 5). 5 is the first safe prime and the first good prime. 11 forms the first pair of sexy primes with 5. 5 is the second Fermat number, Fermat prime, of a total of five known Fermat primes. 5 is also the first of three known Wilson primes (5, 13, 563). Geometry A shape with five sides is called a pentagon. The pentagon is the first regular polygon that does not Tessellation, tile the Plane (geometry), plane with copies of itself. It is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Braid S3
A braid (also referred to as a plait; ) is a complex structure or pattern formed by interlacing three or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-stranded structure. More complex patterns can be constructed from an arbitrary number of strands to create a wider range of structures (such as a fishtail braid, a five-stranded braid, rope braid, a French braid and a waterfall braid). The structure is usually long and narrow with each component strand functionally equivalent in zigzagging forward through the overlapping mass of the others. It can be compared with the process of weaving, which usually involves two separate perpendicular groups of strands ( warp and weft). Historically, the materials used have depended on the indigenous plants and animals available in the local area. During the Industrial Revolution, mechanized braiding equipment was invented to increase production. The braiding ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Physics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary (macroscopic and Microscopic scale, (optical) microscopic) scale, but is not sufficient for describing them at very small submicroscopic (atomic and subatomic) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales. Quantum systems have Bound state, bound states that are Quantization (physics), quantized to Discrete mathematics, discrete values of energy, momentum, angular momentum, and ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nielsen–Thurston Classification
In mathematics, Thurston's classification theorem characterizes homeomorphisms of a compact orientable surface. William Thurston's theorem completes the work initiated by . Given a homeomorphism ''f'' : ''S'' → ''S'', there is a map ''g'' isotopic to ''f'' such that at least one of the following holds: * ''g'' is periodic, i.e. some power of ''g'' is the identity; * ''g'' preserves some finite union of disjoint simple closed curves on ''S'' (in this case, ''g'' is called ''reducible''); or * ''g'' is pseudo-Anosov. The case where ''S'' is a torus (i.e., a surface whose genus is one) is handled separately (see torus bundle) and was known before Thurston's work. If the genus of ''S'' is two or greater, then ''S'' is naturally hyperbolic, and the tools of Teichmüller theory become useful. In what follows, we assume ''S'' has genus at least two, as this is the case Thurston considered. (Note, however, that the cases where ''S'' has boundary or is not orie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Entropy
In mathematics, the topological entropy of a topological dynamical system is a nonnegative extended real number that is a measure of the complexity of the system. Topological entropy was first introduced in 1965 by Adler, Konheim and McAndrew. Their definition was modelled after the definition of the Kolmogorov–Sinai, or metric entropy. Later, Dinaburg and Rufus Bowen gave a different, weaker definition reminiscent of the Hausdorff dimension. The second definition clarified the meaning of the topological entropy: for a system given by an iterated function, the topological entropy represents the exponential growth rate of the number of distinguishable orbits of the iterates. An important variational principle relates the notions of topological and measure-theoretic entropy. Definition A topological dynamical system consists of a Hausdorff topological space ''X'' (usually assumed to be compact) and a continuous self-map ''f'' : ''X'' → ''X''. Its topolog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaotic Mixing
In chaos theory and fluid dynamics, chaotic mixing is a process by which flow tracers develop into complex fractals under the action of a fluid flow. The flow is characterized by an exponential growth of fluid filaments. Even very simple flows, such as the blinking vortex, or finitely resolved wind fields can generate exceptionally complex patterns from initially simple tracer fields. The phenomenon is still not well understood and is the subject of much current research. Context of chaotic advection Fluid flows Two basic mechanisms are responsible for fluid mixing (physics), mixing: diffusion and advection. In liquids, molecular diffusion alone is hardly efficient for mixing. Advection, that is the transport of matter by a flow, is required for better mixing. The fluid flow obeys fundamental equations of fluid dynamics (such as the conservation of mass and the conservation of momentum) called Navier–Stokes equations. These equations are written for the Eulerian Flow veloci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fluid Mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasma (physics), plasmas) and the forces on them. Originally applied to water (hydromechanics), it found applications in a wide range of disciplines, including mechanical engineering, mechanical, aerospace engineering, aerospace, civil engineering, civil, chemical engineering, chemical, and biomedical engineering, as well as geophysics, oceanography, meteorology, astrophysics, and biology. It can be divided into ''fluid statics'', the study of various fluids at rest; and ''fluid dynamics'', the study of the effect of forces on fluid motion. It is a branch of ''continuum mechanics'', a subject which models matter without using the information that it is made out of atoms; that is, it models matter from a macroscopic viewpoint rather than from microscopic. Fluid mechanics, especially fluid dynamics, is an active field of research, typically mathematically complex. Many problems a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Element
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is a right inverse of . (An identity element is an element such that and for all and for which the left-hand sides are defined.) When the operation is associative, if an element has both a left inverse and a right inverse, then these two inverses are equal and unique; they are called the ''inverse element'' or simply the ''inverse''. Often an adjective is added for specifying the operation, such as in additive inverse, multiplicative inverse, and functional inverse. In this case (associative operation), an invertible element is an element that has an inverse. In a ring, an ''invertible element'', also called a unit, is an element that is invertible under multiplication (this is not ambiguous, as every element is invertible under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as group (mathematics), groups and ring (mathematics), rings. The term ''identity element'' is often shortened to ''identity'' (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with. Definitions Let be a set equipped with a binary operation ∗. Then an element of is called a if for all in , and a if for all in . If is both a left identity and a right identity, then it is called a , or simply an . An identity with respect to addition is called an Additive identity, (often denoted as 0) and an identity with respect to m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Braid S3 Inv Squared
A braid (also referred to as a plait; ) is a complex structure or pattern formed by interlacing three or more strands of flexible material such as textile yarns, wire, or hair. The simplest and most common version is a flat, solid, three-stranded structure. More complex patterns can be constructed from an arbitrary number of strands to create a wider range of structures (such as a fishtail braid, a five-stranded braid, rope braid, a French braid and a waterfall braid). The structure is usually long and narrow with each component strand functionally equivalent in zigzagging forward through the overlapping mass of the others. It can be compared with the process of weaving, which usually involves two separate perpendicular groups of strands (Warp (weaving), warp and weft). Historically, the materials used have depended on the indigenous plants and animals available in the local area. During the Industrial Revolution, mechanized braiding equipment was invented to increase production ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
