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Alpha Shape
In computational geometry, an alpha shape, or α-shape, is a family of piecewise linear simple curves in the Euclidean plane associated with the shape of a finite set of points. They were first defined by . The alpha-shape associated with a set of points is a generalization of the concept of the convex hull, i.e. every convex hull is an alpha-shape but not every alpha shape is a convex hull. Characterization For each real number ''α'', define the concept of a ''generalized disk of radius'' 1/''α'' as follows: * If ''α'' = 0, it is a closed half-plane; * If ''α'' > 0, it is a closed disk of radius 1/''α''; * If ''α'' < 0, it is the closure of the complement of a disk of radius −1/''α''. Then an edge of the alpha-shape is drawn between two members of the finite point set whenever there exists a generalized disk of radius 1/''α'' containing none of the point set and which has the property that the two points lie o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermi Level
The Fermi level of a solid-state body is the thermodynamic work required to add one electron to the body. It is a thermodynamic quantity usually denoted by ''µ'' or ''E''F for brevity. The Fermi level does not include the work required to remove the electron from wherever it came from. A precise understanding of the Fermi level—how it relates to electronic band structure in determining electronic properties, how it relates to the voltage and flow of charge in an electronic circuit—is essential to an understanding of solid-state physics. In band structure theory, used in solid state physics to analyze the energy levels in a solid, the Fermi level can be considered to be a hypothetical energy level of an electron, such that at thermodynamic equilibrium this energy level would have a ''50% probability of being occupied at any given time''. The position of the Fermi level in relation to the band energy levels is a crucial factor in determining electrical properties. The Fermi le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Washington University
Washington University in St. Louis (WashU or WUSTL) is a private research university with its main campus in St. Louis County, and Clayton, Missouri. Founded in 1853, the university is named after George Washington. Washington University is ranked among the top universities in the United States. The university's 169-acre Danforth Campus is at the center of Washington University and is the academic home to the majority of the university’s undergraduate, graduate, and professional students. The Danforth Campus features predominantly Collegiate Gothic architecture in its academic buildings and is bordered by Forest Park and the cities of St. Louis, Clayton and University City. The university also has a West Campus in Clayton, North Campus in the West End neighborhood of St. Louis, and Medical Campus in the Central West End neighborhood of St. Louis. The Washington University Medical Campus spreads over 17 city blocks and 164 acres. The center is home to the Washington Univers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CGAL
The Computational Geometry Algorithms Library (CGAL) is an open source software library of computational geometry algorithms. While primarily written in C++, Scilab bindings and bindings generated with SWIG (supporting Python and Java for now) are also available. The software is available under dual licensing scheme. When used for other open source software, it is available under open source licenses (LGPL or GPL depending on the component). In other cases commercial license may be purchased, under different options for academic/research and industrial customers. History The CGAL project was founded in 1996, as a consortium of eight research institutions in Europe and Israel: Utrecht University, ETH Zurich, Free University of Berlin, INRIA Sophia Antipolis, Martin-Luther-University Halle-Wittenberg, Max Planck Institute for Informatics Saarbrücken, Johannes Kepler University Linz, and Tel-Aviv University. The original funding for the project came from the ESPRIT project of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Skeleton
In computational geometry and geometric graph theory, a ''β''-skeleton or beta skeleton is an undirected graph defined from a set of points in the Euclidean plane. Two points ''p'' and ''q'' are connected by an edge whenever all the angles ''prq'' are sharper than a threshold determined from the numerical parameter ''β''. Circle-based definition Let ''β'' be a positive real number, and calculate an angle ''θ'' using the formulas :\theta = \begin \sin^ \frac, & \text\beta \ge 1 \\ \pi - \sin^, & \text\beta\le 1\end For any two points ''p'' and ''q'' in the plane, let ''R''''pq'' be the set of points for which angle ''prq'' is greater than ''θ''. Then ''R''''pq'' takes the form of a union of two open disks with diameter ''βd''(''p'',''q'') for ''β'' ≥ 1 and ''θ'' ≤ π/2, and it takes the form of the intersection of two open disks with diameter ''d''(''p'',''q'')/''β'' for ''β'' ≤ 1 and ''θ'' ≥ π/2. When ''β''& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulk Ag, Fermi Surface
Bulk can refer to: Industry * Bulk cargo * Bulk liquids * Bulk mail * Bulk material handling * Bulk pack, packaged bulk materials/products * Bulk purchasing * Baking * Bulk fermentation, the period after mixing when dough is left alone to ferment in bulk, meaning before division to final weights. Physics *Bulk density *Bulk modulus *In brane cosmology and M-theory (see also the AdS/CFT correspondence), the bulk is a hypothetical higher-dimensional space within which the eleven dimensions of our universe (the three dimensions we can see, plus time, plus the seven extra dimensions that we can't see but M-theory theorizes are all around us) may exist. People *Mike Waters, known as Bulk, British professional wrestler with the UK Pitbulls Fiction *Bulk and Skull, a pair of characters in the ''Power Rangers'' universe Places *Bulk, Lancashire, area of Lancaster, England *Bülk Lighthouse Bülk Lighthouse (german: Leuchtturm Bülk) is on the westernmost headland of the Kiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brillouin Zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice is broken up into Brillouin zones. The boundaries of this cell are given by planes related to points on the reciprocal lattice. The importance of the Brillouin zone stems from the description of waves in a periodic medium given by Bloch's theorem, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone. The first Brillouin zone is the locus of points in reciprocal space that are closer to the origin of the reciprocal lattice than they are to any other reciprocal lattice points (see the derivation of the Wigner–Seitz cell). Another definition is as the set of points in ''k''-space that can be reached from the origin without crossing any Bragg plane. Equivalently, this is the Vor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green's Function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differential operator, then * the Green's function G is the solution of the equation \operatorname G = \delta, where \delta is Dirac's delta function; * the solution of the initial-value problem \operatorname y = f is the convolution (G \ast f). Through the superposition principle, given a linear ordinary differential equation (ODE), \operatorname y = f, one can first solve \operatorname G = \delta_s, for each , and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of . Green's functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green's functions are s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermi Surface
In condensed matter physics, the Fermi surface is the surface in reciprocal space which separates occupied from unoccupied electron states at zero temperature. The shape of the Fermi surface is derived from the periodicity and symmetry of the crystalline lattice and from the occupation of electronic energy bands. The existence of a Fermi surface is a direct consequence of the Pauli exclusion principle, which allows a maximum of one electron per quantum state. The study of the Fermi surfaces of materials is called fermiology. Theory Consider a spin-less ideal Fermi gas of N particles. According to Fermi–Dirac statistics, the mean occupation number of a state with energy \epsilon_i is given by :\langle n_i\rangle =\frac, where, *\left\langle n_i\right\rangle is the mean occupation number of the i^ state *\epsilon_i is the kinetic energy of the i^ state *\mu is the chemical potential (at zero temperature, this is the maximum kinetic energy the particle can have, i.e. Fermi ene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Geometry
Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational geometry. While modern computational geometry is a recent development, it is one of the oldest fields of computing with a history stretching back to antiquity. Analysis of algorithms, Computational complexity is central to computational geometry, with great practical significance if algorithms are used on very large datasets containing tens or hundreds of millions of points. For such sets, the difference between O(''n''2) and O(''n'' log ''n'') may be the difference between days and seconds of computation. The main impetus for the development of computational geometry as a discipline was progress in computer graphics and computer-aided design and manufacturing (Computer-aided design, CAD/Compu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Equivalent
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial from an abstract simplicial complex, the former is often called a geometric simplicial complex.'', Section 4.3'' Definitions A simplicial complex \mathcal is a set of simplices that satisfies the following conditions: :1. Every face of a simplex from \mathcal is also in \mathcal. :2. The non-empty intersection of any two simplices \sigma_1, \sigma_2 \in \mathcal is a face of both \sigma_1 and \sigma_2. See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry. A simplicial ''k''-complex \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
