Loading...
• ###### optimal spherical codes
• These common structures are layers of optimal spherical codes in most cases, a surprising fact given the significant faceting of the icosahedron and dodecahedron. (pnas.org)
• ###### displaystyle
• For d ≤ 2 {\displaystyle d\leq 2} the critical temperature occurs at absolute zero, resulting in no phase transition for the spherical model. (wikipedia.org)
• Then, for any x ∈ Sn−1 and any A ⊆ Sn−1, θ n ( { g ∈ O ( n ) ∣ g ( x ) ∈ A } ) = σ n − 1 ( A ) . {\displaystyle \theta ^{n}(\{g\in \mathrm {O} (n)\mid g(x)\in A\})=\sigma ^{n-1}(A).} In the case that Sn is a topological group (that is, when n is 0, 1 or 3), spherical measure σn coincides with (normalized) Haar measure on Sn. (wikipedia.org)
• the remaining factor can be regarded as a function of the spherical angular coordinates θ {\displaystyle \theta } and φ {\displaystyle \varphi } only, or equivalently of the orientational unit vector r {\displaystyle {\mathbf {r} }} specified by these angles. (wikipedia.org)
• ###### angular
• Spherical Harmonic involves an imagined universe based on the Hilbert space described by the angular wave functions that solve the Laplace Equation. (wikipedia.org)
• The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. (wikipedia.org)
• ###### coordinates
• While spherical polar coordinates are one orthogonal coordinate system for expressing vectors and tensors using polar and azimuthal angles and radial distance, the spherical basis are constructed from the standard basis and use complex numbers. (wikipedia.org)
• Subsequently, in his 1782 memoire, Laplace investigated these coefficients using spherical coordinates to represent the angle γ between x1 and x. (wikipedia.org)
• ###### trigonometry
• However, Gerolamo Cardano noted a century later that much of its material on spherical trigonometry was taken from the twelfth-century work of the Andalusi scholar Jabir ibn Aflah. (wikipedia.org)
• ###### platonic
• Here, we report densest found clusters of the Platonic solids in spherical confinement, for up to N = 60 constituent polyhedral particles. (pnas.org)
• ###### astronomy
• The earliest reliably documented mention of the spherical Earth concept dates from around the 6th century BC when it appeared in ancient Greek philosophy but remained a matter of speculation until the 3rd century BC, when Hellenistic astronomy established the spherical shape of the Earth as a physical given. (wikipedia.org)
• The primary elements of spherical astronomy are coordinate systems and time. (wikipedia.org)
• ###### approximation
• It is just too complicated to try to figure out the topography of a horse and how it affects the animal's coefficient of friction , when a spherical approximation will do just as well for learning the concepts. (everything2.com)
• ###### angle
• It can be bisected into two lunes of half the angle, or it can be bisected by an equatorial line into two right spherical triangles. (wikipedia.org)
• If AB is a semidisk that forms a ball when completely revolved about the z-axis, revolving AB only through a given α produces a spherical wedge of the same angle α. (wikipedia.org)
• ###### friction
• a friction reducing member located between the spherical ball and the frame of the vehicle. (google.com)
• ###### orthogonal
• Spherical measure has a nice relationship to Haar measure on the orthogonal group. (wikipedia.org)
• ###### concentric
• After the 4th century BCE this was gradually replaced by a Greek scientific cosmology of a spherical earth surrounded by multiple concentric heavens. (wikipedia.org)
• ###### polyhedral
• In organic chemistry, spherical aromaticity is formally used to describe an unusually stable nature of some spherical compounds such as fullerenes, polyhedral boranes. (wikipedia.org)
• Our dense packings display a wide variety of symmetries and structures, and indicate that the presence of the spherical container suppresses packing effects due to polyhedral shape. (pnas.org)
• ###### earliest
• Though the earliest written mention of a spherical Earth comes from ancient Greek sources, there is no account of how the sphericity of the Earth was discovered. (wikipedia.org)
• Ball joint / Heim joint Race (bearing) Self-aligning ball bearing Shaker tilting chair one of the earliest patented applications of this technology Spherical roller bearing Rolt, L.T.C. (1986) . (wikipedia.org)
• ###### particles
• Spherical nucleic acids are structures that are made by taking a nanoparticle template and using chemistry to arrange short strands of DNA or RNA on the surface of those particles. (news-medical.net)
• ###### quantum mechanics
• Spherical t-designs have also recently been appropriated in quantum mechanics in the form of quantum t-designs with various applications to quantum information theory, quantum computing and POVMs. (wikipedia.org)
• ###### outer
• A spherical bearing by itself consists of an outer ring and an inner ring and a locking feature that makes the inner ring captive within the outer ring in the axial direction only. (wikipedia.org)
• ###### parameters
• The kissing number problem may be stated as the problem of finding the maximal N for a given n for which a spherical code with parameters (n,N,1/2) exists. (wikipedia.org)
• For simple designs one can sometimes calculate parameters that minimize spherical aberration. (wikipedia.org)
• ###### given
• that is, given positive integers n and t, there is a number N(d,t) such that for every N ≥ N(d,t) there exists a spherical t-design of N points in dimension d. (wikipedia.org)
• ###### vector
• It was rigorously proved by Kac and C. J. Thompson that the spherical model is a limiting case of the N-vector model. (wikipedia.org)
• ###### finite
• For d greater than 2, the spherical model exhibits the typical ferromagnetic behaviour, with a finite Curie temperature where ferromagnetism ceases. (wikipedia.org)
• ###### shape
• It is often considered to be an imperfection of telescopes and other instruments which makes their focusing less than ideal due to the spherical shape of lenses and mirrors. (wikipedia.org)
• The reason that's important is that it forces the orientation and gives the architecture both its spherical shape and also the properties that I've been mentioning. (news-medical.net)
• ###### rotation
• For higher ranks, one may use either the commutator, or rotation definition of a spherical tensor. (wikipedia.org)
• ###### model
• Spherical fusion categories give rise to a family of three-dimensional topological state sum models (a particular formulation of a topological quantum field theory), the Turaev-Viro model, or rather Turaev-Viro-Barrett-Westbury model. (wikipedia.org)
• The spherical model in statistical mechanics is a model of ferromagnetism similar to the Ising model, which was solved in 1952 by T. H. Berlin and M. Kac. (wikipedia.org)
• The critical behaviour of the spherical model was derived in the completely general circumstances that the dimension d may be a real non-integer dimension. (wikipedia.org)
• M. Kac and C. J. Thompson, Spherical model and the infinite spin dimensionality limit, Physica Norvegica, 5(3-4):163-168, 1971. (wikipedia.org)
• To modern historians, these details confirm the truth of the Phoenicians' report and even open the possibility that the Phoenicians knew about the spherical model. (wikipedia.org)
• ###### often
• On another level, it is about the emotional toll that war exacts on those who survive it, a theme Asaro often explores in the Saga of the Skolian Empire, as exemplified by the subplot in Spherical Harmonic involving the attempts of Selei and her husband to come to terms with his shattering experiences during the war. (wikipedia.org)
• Such a set is often called a spherical t-design to indicate the value of t, which is a fundamental parameter. (wikipedia.org)
• ###### Greek
• Pythagoras Early Greek philosophers alluded to a spherical Earth, though with some ambiguity. (wikipedia.org)
• ###### commonly
• The spherical horse in simple harmonic motion is a canonical reference to a basic lack of real world practicality commonly encountered in physics , particularly as taught in the classroom . (everything2.com)
• A spherical bear in simple harmonic motion is another commonly encountered creature. (everything2.com)
• ###### existence
• The existence and structure of spherical designs with d = 1 (that is, in a circle) were studied in depth by Hong (1982). (wikipedia.org)
• ###### surface
• The propulsion mechanism has a spherical ball disposed within the socket and adapted to frictionally engage the ground surface. (google.com)
• ###### Theory
• Paolo Bravi and Guido Pezzini, The spherical systems of the wonderful reductive subgroups, J. Lie Theory 25 (2015), 105-123. (wikipedia.org)
• Michel Brion, "Introduction to actions of algebraic groups" Knop, Friedrich (2014), "Localization of spherical varieties", Algebra & Number Theory, 8: 703-728, doi:10.2140/ant.2014.8.703 Losev, Ivan (2006). (wikipedia.org)
• ###### made
• Because of spherical aberration, telescopes shorter than f/10 are usually made with non-spherical mirrors or with correcting lenses. (wikipedia.org)
• ###### concept
• The concept of a spherical Earth displaced earlier beliefs in a flat Earth: In early Mesopotamian mythology, the world was portrayed as a flat disk floating in the ocean with a hemispherical sky-dome above, and this forms the premise for early world maps like those of Anaximander and Hecataeus of Miletus. (wikipedia.org)
• ###### science
• Spherical Harmonic is a science fiction novel from the Saga of the Skolian Empire by Catherine Asaro. (wikipedia.org)
• ###### sometimes
• This article will focus on the spherical (sometimes oval) shaped bacteria. (brighthub.com)
• ###### area
• A typical value of refractive index for crown glass is 1.5 (see list), which indicates that only about 43% of the area (67% of diameter) of a spherical lens is useful. (wikipedia.org)