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searching for N-sphere 61 found (218 total)

alternate case: n-sphere

Bingham distribution (400 words) [view diff] exact match in snippet view article find links to article

Christopher Bingham, is an antipodally symmetric probability distribution on the n-sphere. It is a generalization of the Watson distribution and a special case of

normal bundle. The n-sphere Sn is a generalisation of the idea of a circle (1-sphere) and sphere (2-sphere) to higher dimensions. An n-sphere Sn can be constructed

is homotopy equivalent to the n-sphere. It thus has the same homotopy groups and the same homology groups as the n-sphere, and so every homotopy sphere

a homology sphere is an n-manifold X having the homology groups of an n-sphere, for some integer n ≥ 1 {\displaystyle n\geq 1} . That is, H 0 ( X , Z

an addition operation given by connected sum, with 0 the n-sphere. If A + B is the n-sphere, then A + B + A + B + ... is Euclidean space so the Mazur

topological space X may be considered as dual to its complement in the n-sphere, where n is large enough. Its origins lie in Alexander duality theory,

obtained by identifying antipodal points of the unit ⁠ n {\displaystyle n} ⁠-sphere, ⁠ S n {\displaystyle S^{n}} ⁠, in ⁠ R n + 1 {\displaystyle \mathbb {R}

In spherical geometry, a spherical lune (or biangle) is an area on a sphere bounded by two half great circles which meet at antipodal points. It is an

nullhomotopic. For example, if X {\displaystyle X} is an n {\displaystyle n} -sphere, this takes the value 2 {\displaystyle 2} . Lusternik–Schnirelmann theorem

arising as a deformation of the Klein geometry given by the celestial n-sphere, viewed as the homogeneous space O+(n+1,1)/P where P is the stabilizer

if ρ ( n ) {\displaystyle \rho (n)} is the Radon-Hurwitz number of the n-sphere (definition below), then there were at least ρ ( n ) − 1 {\displaystyle

pose the Generalized Poincaré conjecture: is a homotopy n-sphere homeomorphic to the n-sphere? A stronger assumption than simply-connectedness is necessary;

mappings of an ( n + 1 ) {\displaystyle (n+1)} -dimensional space into an n-sphere, was written under the direction of Norman Steenrod. He spent two additional

n-sphere. The development of a conformally flat manifold is a conformal local diffeomorphism from the universal cover of the manifold to the n-sphere

1 + 1/n Sphere n > 1 1/m + 1/n Sphere n > m > 1 1/2 + 1/2n Disk n > 1 1/2m + 1/2n Disk n > m > 1 Elliptic 2 Sphere 2/n Sphere n, n 1/n Sphere 2, 2, n

differential topology. He coined the term exotic sphere, referring to any n-sphere with nonstandard differential structure. Kervaire and Milnor initiated

written as a connected sum of two n-manifolds (neither of which is an n-sphere). An irreducible manifold is thus prime, although the converse does not

; Tamvakis, N. K. (1984), "The decomposition of the n {\displaystyle n} -sphere and the boundaries of plane convex domains", Convexity and graph theory

275–276. T. H. Kiang (1945). "The manifolds of linear elements of an 𝑛-sphere". Bulletin of the American Mathematical Society. 51 (6): 418–428. doi:10

is even. The n-dimensional real projective space is the quotient of the n-sphere by the antipodal map. It follows that its Euler characteristic is exactly

{\displaystyle (n-1)K} ); thus M K n {\displaystyle M_{K}^{n}} is the n-sphere of radius 1 / K {\displaystyle 1/{\sqrt {K}}} if K > 0 {\displaystyle K>0}

S increases the dimension of a space by one: for example, it takes an n-sphere to an (n + 1)-sphere for n ≥ 0. Given a continuous map f : X → Y , {\displaystyle

S increases the dimension of a space by one: for example, it takes an n-sphere to an (n + 1)-sphere for n ≥ 0. Given a continuous map f : X → Y , {\displaystyle

to identify all of the points along the equator of an n {\displaystyle n} -sphere S n {\displaystyle S^{n}} . Doing so results in two copies of the sphere

associated bundle. A sphere bundle is a fiber bundle whose fiber is an n-sphere. Given a vector bundle E {\displaystyle E} with a metric (such as the tangent

-simplex. ∂ Δ n + 1 {\displaystyle \partial \Delta _{n+1}} is a combinatorial n-sphere with its triangulation as the boundary of the n+1-simplex. Given a triangulated

classes of based maps S n → X {\displaystyle S^{n}\to X} from a (pointed) n-sphere S n {\displaystyle S^{n}} to X. As it turns out, for n ≥ 1 {\displaystyle

is known for n ≤ 11, and only conjectural values are known for larger n. Sphere packing on the corners of a hypercube (with Hamming balls, spheres defined

interesting examples of principal bundles. Recall that the n {\displaystyle n} -sphere S n {\displaystyle S^{n}} is a two-fold covering space of real projective

connection. Another example is the conformal sphere. Topologically, it is the n-sphere, but there is no notion of length defined on it, just of angle between

excision, so the composition is a surjection as claimed. Let Sn denote the n-sphere and note that it is (n − 1)-connected so that the groups π n + k ( S n

compactification of n-dimensional Euclidean space Rn is homeomorphic to the n-sphere Sn. As above, the map can be given explicitly as an n-dimensional inverse

homeomorphic to the pair (Sn, Sn−1), where Sn−1 is the equator of the n-sphere. Brown and Mazur received the Veblen Prize for their contributions. Both

they are different, we obtain the Klein bottle (Fig. 2c). Surgery on the n-sphere If n = p + q {\displaystyle n=p+q} , then S n = ∂ D n + 1 ≈ ∂ ( D p + 1

closely linked to Cerf theory. Consider the standard CW-decomposition of the n-sphere, with one zero cell and a single n-cell. From the point of view of smooth

{\displaystyle E_{b}} is an n-dimensional real vector space. We can form an n-sphere bundle Sph ⁡ ( E ) → B {\displaystyle \operatorname {Sph} (E)\to B} by

Trees on the website of the music club Knust in Hamburg, Germany (German) N-SPHERE November 2010 issu: Interview with Justin Jones and Simon Huw Jones, Sfere

{\displaystyle \partial M} is non-empty and M {\displaystyle M} is compact. The n-sphere is the double of the n-ball. In this context, the two balls would be the

Borsuk–Ulam theorem, which says that a continuous map from the n {\displaystyle n} -sphere to Euclidean n {\displaystyle n} -space will always map some pair of antipodal

groups of such a space are trivial. As the example of the n {\displaystyle n} -sphere S n {\displaystyle \mathbf {S} ^{n}} shows, there is, in general, no hope

music and culture magazine. The October 2010 issue of visual arts magazine n-sphere featured Malboeuf's art on the cover and in the lead article, accompanied

Circular segment — the analogous 2D object Solid angle — contains formula for n-sphere caps Spherical segment Spherical sector Spherical wedge Polyanin, Andrei

the circle is null-cobordant since it bounds a disk. More generally, a n-sphere is null-cobordant since it bounds a (n + 1)-disk. Also, every orientable

infinite symmetric product S P ( S n ) {\displaystyle SP(S^{n})} of a n-sphere is a K ( Z , n ) {\displaystyle K(\mathbb {Z} ,n)} . More generally S P

step from n to n + 1, generate a vector v uniformly distributed on the n-sphere Sn, embed the n × n matrix in the next larger size with last column (0

⁠ S 2 {\displaystyle S^{2}} ⁠, or more generally the n {\displaystyle n} -sphere S n {\displaystyle S^{n}} should be obvious from ordinary intuition: spheres

circumscribed n-sphere, with radius r {\displaystyle r} . Then the (n + 1)-simplex made of the vertices of the n-simplex and the center of the n-sphere is degenerate

that varies continuously. For example, a circle or, more generally, an n-sphere can be oriented; i.e., orientable. On the other hand, a Möbius strip (a

△ S + 2 ) , {\displaystyle -\triangle _{S}(\triangle _{S}+2),} on the n-sphere. Via the Cayley transform this operator is conformally equivalent to the

{\displaystyle g_{0}} is the standard metric on the n {\displaystyle n} -sphere S n {\displaystyle S^{n}} . It follows that if we define σ ( M ) = sup

suspension S 2 X {\displaystyle S^{2}X} is a topological n {\displaystyle n} -sphere. Choosing a triangulation t : | S | → S 2 X {\displaystyle t:|{\mathcal

n-dimensional disk) to a single point. This space is homeomorphic to the n-sphere Sn. On the other hand, the pushout p t ⊔ S n − 1 p t {\displaystyle pt\sqcup

dimension n – specifically the monoid of smooth structures on the standard n-sphere – is isomorphic to the group Θ n {\displaystyle \Theta _{n}} of h-cobordism

quotient is defined by "straightening the angles". If the base manifold is an n-sphere S n {\displaystyle S^{n}} , then by iterating this procedure over several

classification of the mappings of a finite complex into a topological group or an n-sphere". Annals of Mathematics. 50 (1): 158–173. doi:10.2307/1969359. JSTOR 1969359

commonly denoted by Ωn0Sn, where ΩnSn is the n-fold loop space of the n-sphere Sn, and similarly Map0(S∞,S∞) is denoted by Ω∞0S∞. Therefore the Barratt–Priddy

1 is therefore isomorphic to the projectivized cotangent bundle of the n-sphere. This identification is not invariant under Lie transformations: in Lie

− 1 } {\displaystyle \in \{+1,0,-1\}} is respectively isometric to the n-sphere S n {\displaystyle S^{n}} , the n-Euclidean space E n {\displaystyle E^{n}}

If S n {\displaystyle \mathbb {S} _{n}} denotes the n {\displaystyle n} -sphere, then dim ⁡ X ≤ n {\displaystyle \dim X\leq n} if, and only if, for every

1007/BF01297505. Whitney, Hassler (1937). "The maps of an n-complex into an n-sphere" (PDF). Duke Mathematical Journal. 3: 51–55. Hopf, Heinz (1933-12-01).

classification of exotic spheres: the monoid of smooth structures on the n-sphere is the collection of oriented smooth n-manifolds which are homeomorphic