 # Calabi–Yau manifold A 2D slice of the 6D Calabi-Yau quintic manifold. Calabi–Yau manifolds are complex manifolds that are higher-dimensional analogues of K3 surfaces. They are sometimes defined as compact Kähler manifolds whose canonical bundle is trivial, though many other similar but inequivalent definitions are sometimes used. They were named "Calabi–Yau spaces" by Candelas et al. (1985) after E. Calabi (1954, 1957) who first studied them, and S. T. Definitions There are many different inequivalent definitions of a Calabi–Yau manifold used by different authors. A Calabi–Yau n-fold or Calabi–Yau manifold of (complex) dimension n is sometimes defined as a compact n-dimensional Kähler manifold M satisfying one of the following equivalent conditions: These conditions imply that the first integral Chern class c1(M) of M vanishes, but the converse is not true. In particular if a compact Kähler manifold is simply connected then the weak definition above is equivalent to the stronger definition. G2 manifold Related:  wikipediaPhysicsà revoir modèles

Cayley graph The Cayley graph of the free group on two generators a and b Definition Suppose that is a generating set. is a colored directed graph constructed as follows:  Each element of is assigned a vertex: the vertex set of is identified with Each generator of is assigned a color .For any the vertices corresponding to the elements and are joined by a directed edge of colour Thus the edge set consists of pairs of the form with providing the color. In geometric group theory, the set is usually assumed to be finite, symmetric (i.e. Examples Cayley graph of the dihedral group Dih4 on two generators a and b On two generators of Dih4, which are both self-inverse A Cayley graph of the dihedral group D4 on two generators a and b is depicted to the left. A different Cayley graph of Dih4 is shown on the right. b is still the horizontal reflection and represented by blue lines; c is a diagonal reflection and represented by green lines. Part of a Cayley graph of the Heisenberg group. . .

Sign (semiotics) There are two major theories about the way in which signs acquire the ability to transfer information; both theories understand the defining property of the sign as being a relation between a number of elements. In the tradition of semiotics developed by Ferdinand de Saussure the sign relation is dyadic, consisting only of a form of the sign (the signifier) and its meaning (the signified). Saussure saw this relation as being essentially arbitrary motivated only by social convention. According to Saussure (1857–1913), a sign is composed of the signifier (signifiant), and the signified (signifié). A famous thesis by Saussure states that the relationship between a sign and the real-world thing it denotes is an arbitrary one. Charles Sanders Peirce (1839–1914) proposed a different theory. A sign is something which depends on an object in a way that enables (and, in a sense, determines) an interpretation, an interpretant, to depend on the object as the sign depends on the object.

Cotton–Mouton effect In physical optics, the Cotton–Mouton effect refers to birefringence in a liquid in the presence of a constant transverse magnetic field. It is a similar but stronger effect than the Voigt effect (in which the medium is a gas instead of a liquid). The electric analog is the Kerr effect. It was discovered in 1907 by Aimé Cotton and Henri Mouton, working in collaboration. When a linearly polarized wave propagates perpendicular to magnetic field (e.g. in a magnetized plasma), it can become elliptized. Cotton effect

Kaluza–Klein theory This article is about gravitation and electromagnetism. For the mathematical generalization of K theory, see KK-theory. In 1926, Oskar Klein gave Kaluza's classical 5-dimensional theory a quantum interpretation, to accord with the then-recent discoveries of Heisenberg and Schroedinger. Klein introduced the hypothesis that the fifth dimension was curled up and microscopic, to explain the cylinder condition. Klein also calculated a scale for the fifth dimension based on the quantum of charge. It wasn't until the 1940s that the classical theory was completed, and the full field equations including the scalar field were obtained by three independent research groups: Thiry, working in France on his dissertation under Lichnerowicz; Jordan, Ludwig, and Müller in Germany, with critical input from Pauli and Fierz; and Scherrer  working alone in Switzerland. The Kaluza Hypothesis , where roman indices span 5 dimensions. . where the index where or

Hopf fibration The Hopf fibration can be visualized using a stereographic projection of S3 to R3 and then compressing R3 to a ball. This image shows points on S2 and their corresponding fibers with the same color. Pairwise linked keyrings mimic part of the Hopf fibration. In the mathematical field of topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle. Technically, Hopf found a many-to-one continuous function (or "map") from the 3-sphere onto the 2-sphere such that each distinct point of the 2-sphere comes from a distinct circle of the 3-sphere (Hopf 1931). This fiber bundle structure is denoted meaning that the fiber space S1 (a circle) is embedded in the total space S3 (the 3-sphere), and p : S3 → S2 (Hopf's map) projects S3 onto the base space S2 (the ordinary 2-sphere).

Birefringence A calcite crystal laid upon a graph paper with blue lines showing the double refraction Doubly refracted image as seen through a calcite crystal, seen through a rotating polarizing filter illustrating the opposite polarization states of the two images. Explanation The simplest (and most common) type of birefringence is that of materials with uniaxial anisotropy. That is, the structure of the material is such that it has an axis of symmetry with all perpendicular directions optically equivalent. What's more, the extraordinary ray is an inhomogeneous wave whose power flow (given by the Poynting vector) is not exactly parallel to the wave vector. When the light propagates either along or orthogonal to the optic axis, such a lateral shift does not occur. Sources of optical birefringence While birefringence is usually obtained using an anisotropic crystal, it can result from an optically isotropic material in a few ways: Examples of uniaxial birefringent materials . and

String theory String theory was first studied in the late 1960s as a theory of the strong nuclear force before being abandoned in favor of the theory of quantum chromodynamics. Subsequently, it was realized that the very properties that made string theory unsuitable as a theory of nuclear physics made it a promising candidate for a quantum theory of gravity. Five consistent versions of string theory were developed until it was realized in the mid-1990s that they were different limits of a conjectured single 11-dimensional theory now known as M-theory. Many theoretical physicists, including Stephen Hawking, Edward Witten and Juan Maldacena, believe that string theory is a step towards the correct fundamental description of nature: it accommodates a consistent combination of quantum field theory and general relativity, agrees with insights in quantum gravity (such as the holographic principle and black hole thermodynamics) and has passed many non-trivial checks of its internal consistency.

Poincaré disk model Metric If u and v are two vectors in real n-dimensional vector space Rn with the usual Euclidean norm, both of which have norm less than 1, then we may define an isometric invariant by where denotes the usual Euclidean norm. Such a distance function is defined for any two vectors of norm less than one, and makes the set of such vectors into a metric space which is a model of hyperbolic space of constant curvature −1. The associated metric tensor of the Poincaré disk model is given by where the xi are the Cartesian coordinates of the ambient Euclidean space. Relation to the hyperboloid model The hyperboloid model can be seen as the equation of t2=x2+y2+1 can can be used to construct a Poincaré disk model as a perspective projection viewed from (t=-1,x=0,y=0), projecting the upper half hyperboloid onto an (x,y) unit disk at t=0. The Poincaré disk model, as well as the Klein model, are related to the hyperboloid model projectively. Angles Artistic realizations M.

Kakapo The Kakapo is critically endangered; as of March 2014, with an additional six from the first hatchings since 2011, the total known population is only 130 living individuals, as reported by the Kakapo Recovery program, most of which have been given names. Because of Polynesian and European colonisation and the introduction of predators such as cats, rats, ferrets, and stoats, the Kakapo was almost wiped out. Conservation efforts began in the 1890s, but they were not very successful until the implementation of the Kakapo Recovery Plan in the 1980s. As of April 2012, surviving Kakapo are kept on three predator-free islands, Codfish (Whenua Hou), Anchor and Little Barrier islands, where they are closely monitored. Two large Fiordland islands, Resolution and Secretary, have been the subject of large-scale ecological restoration activities to prepare self-sustaining ecosystems with suitable habitat for the Kakapo. Taxonomy, systematics and naming Description

Doughnut theory of the universe Bloom Toroidal Model of the Universe The doughnut theory of the universe is an informal description of the theory that the shape of the universe is a three-dimensional torus. The name comes from the shape of a doughnut, whose surface has the topology of a two-dimensional torus. The foundation for the doughnut theory started with Bell Lab’s discovery of cosmic microwave background (CMB). With the information provided from the study of CMB, Dr. Supporting evidence Dr. Cosmic Background Explorer (COBE) The Cosmic Background Explorer was an explorer satellite launched in 1989 by NASA that used a Far Infrared Absolute Spectrometer (FIRAS) to measure the radiation of the universe. Led by researchers John C. Wilkinson Microwave Anisotropy Probe (WMAP) WMAP cosmic microwave background map The Wilkinson Microwave Anisotropy Probe (WMAP) was launched in 2001 as NASA’s second explorer satellite intended to map the precise distribution of CMB across the universe. Notes References

M-theory M-theory is a theory in physics that unifies all consistent versions of superstring theory. The existence of such a theory was first conjectured by Edward Witten at the string theory conference at the University of Southern California in the summer of 1995. Witten's announcement initiated a flurry of research activity known as the second superstring revolution. Background Quantum gravity and strings One of the deepest problems in modern physics is the problem of quantum gravity. Number of dimensions In everyday life, there are three familiar dimensions of space (up/down, left/right, and forward/backward), and there is one dimension of time (later/earlier). Despite the obvious relevance of four-dimensional spacetime for describing the physical world, there are several reasons why physicists often consider theories in other dimensions. Dualities Main articles: S-duality and T-duality A diagram of string theory dualities. and winding number in the dual description. .

Convergence of Fourier series Determination of convergence requires the comprehension of pointwise convergence, uniform convergence, absolute convergence, Lp spaces, summability methods and the Cesàro mean. Preliminaries Consider ƒ an integrable function on the interval [0,2π]. are defined by the formula It is common to describe the connection between ƒ and its Fourier series by The notation ~ here means that the sum represents the function in some sense. The question we will be interested in is: Do the functions (which are functions of the variable t we omitted in the notation) converge to ƒ and in which sense? Before continuing the Dirichlet kernel needs to be introduced. , inserting it into the formula for and doing some algebra will give that where ∗ stands for the periodic convolution and is the Dirichlet kernel which has an explicit formula, The Dirichlet kernel is not a positive kernel, and in fact, its norm diverges, namely a fact that will play a crucial role in the discussion. If for and and therefore, if Suppose

Breccia Upper Triassic breccia from York County, Pennsylvania. Tertiary breccia at Resting Springs Pass, Mojave Desert, California. The word is a loan from Italian, and in that language indicates either loose gravel or stone made by cemented gravel. Types Sedimentary Sedimentary breccias are a type of clastic sedimentary rock which are made of angular to subangular, randomly oriented clasts of other sedimentary rocks. The other derivation of sedimentary breccia is as angular, poorly sorted, immature fragments of rocks in a finer grained groundmass which are produced by mass wasting. In the field, it may at times be difficult to distinguish between a debris flow sedimentary breccia and a colluvial breccia, especially if one is working entirely from drilling information. Sedimentary breccias can be described as rudaceous. Collapse A collapse breccia forms where there has been a collapse of rock, typically in a karst landscape. Tectonic Fault Igneous Volcanic

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