N-dimensional Unit Sphere Research Articles

Submanifolds with constant Möbius scalar curvature in S n

Let Mm be a m-dimensional submanifold in the n-dimensional unit sphere Sn without umbilic point. Two basic invariants of Mm under the Mobius transformation group of Sn are a 1-form Φ called Mobius form and a symmetric (0,2) tensor A called Blaschke tensor. In this paper, we prove the following rigidity theorem: Let Mm be a m-dimensional (m≥3) submanifold with vanishing Mobius form and with constant Mobius scalar curvature R in Sn, denote the trace-free Blaschke tensor by \(\). If \(\), then either ||A||≡0 and Mm is Mobius equivalent to a minimal submanifold with constant scalar curvature in Sn; or \(\) and Mm is Mobius equivalent to \(\) in \(\) for some c≥0 and \(\).

The second variational formula for Willmore submanifolds in Sn

In [17] the third author presented Moebius geometry for sub-manifolds in Sn and calculated the first variational formula of the Willmore functional by using Moebius invariants. In this paper we present the second variational formula for Willmore submanifolds. As an application of these variational formulas we give the standard examples of Willmore hypersurfaces \( \lbrace W_{k}^{m}:= S^{k}(\sqrt {(m-k)/m}) \times S^{m-k}(\sqrt {k/m}), 1 \leq k \leq m-1 \rbrace \) in Sm+1 (which can be obtained by exchanging radii in the Clifford tori \( S^{k}(\sqrt {k/m}) \times S^{m-k}(\sqrt {(m-k)/m)})\) and show that they are stable Willmore hypersurfaces. In case of surfaces in S3, the stability of the Clifford torus \( S^{1}{({1\over \sqrt {2}})}\times S^{1}{({1\over \sqrt {2}})} \) was proved by J. L. Weiner in [18]. We give also some examples of m-dimensional Willmore submanifolds in an n-dimensional unit sphere Sn.

Orientation distribution functions for microstructures of heterogeneous materials (I) — Directional distribution functions and irreducible tensors

In this two-part paper, a thorough investigation is made on Fourier expansions with irreducible tensorial coefficients for orientation distribution functions (oDFs) and crystal orientation distribution functions (CODFs), which are scalar functions defined on the unit sphere and the rotation group, respectively. Recently it has been becoming clearer and clearer that concepts of ODF and CODF play a dominant role in various micromechanically-based approaches to mechanical and physical properties of heterogeneous materials. The theory of group representations shows that a square integrable ODF can be expanded as an absolutely convergent Fourier series of spherical harmonics and these spherical harmonics can further be expressed in terms of irreducible tensors. The fundamental importance of such irreducible tensorial coefficients is that they characterize the macroscopic or overall effect of the orientation distribution of the size, shape, phase, position of the material constitutions and defects. In Part (I), the investigation about the irreducible tensorial Fourier expansions of ODFs defined on the N-dimensional (N-D) unit sphere is carried out. Attention is particularly paid to constructing simple expressions for 2- and 3-D irreducible tensors of any orders in accordance with the convenience of arriving at their restricted forms imposed by various point-group (the synonym of subgroup of the full orthogonal group) symmetries. In the continued work — Part (II), the explicit expression for the irreducible tensorial expansions of CODFs is established. The restricted forms of irreducible tensors and irreducible tensorial Fourier expansions of ODFs and CODFs imposed by various point-group symmetries are derived.

Constructing phylogenies from quartets: elucidation of eutherian superordinal relationships.

In this work we present two new approaches for constructing phylogenetic trees. The input is a list of weighted quartets over n taxa. Each quartet is a subtree on four taxa, and its weight represents a confidence level for the specific topology. The goal is to construct a binary tree with n leaves such that the total weight of the satisfied quartets is maximized (an NP hard problem). The first approach we present is based on geometric ideas. Using semidefinite programming, we embed the n points on the n-dimensional unit sphere, while maximizing an objective function. This function depends on Euclidean distances between the four points and reflects the quartet topology. Given the embedding, we construct a binary tree by performing geometric clustering. This process is similar to the traditional neighbor joining, with the difference that the update phase retains geometric meaning: When two neighbors are joined together, their common ancestor is taken to be the center of mass of the original points. The geometric algorithm runs in poly(n) time, but there are no guarantees on the quality of its output. In contrast, our second algorithm is based on dynamic programming, and it is guaranteed to find the optimal tree (with respect to the given quartets). Its running time is a modest exponential, so it can be implemented for modest values of n. We have implemented both algorithms and ran them on real data for n = 15 taxa (14 mammalian orders and an outgroup taxon). The two resulting trees improve previously published trees and seem to be of biological relevance. On this dataset, the geometric algorithm produced a tree whose score is 98.2% of the optimal value on this input set (72.1% vs. 73.4%). This gives rise to the hope that the geometric approach will prove viable even for larger cases where the exponential, dynamic programming approach is no longer feasible.

Multiple solutions in hollow geometries in the theory of thermal ignition

Following Burnell et al., the dimensionless form of the steady state heat balance equation for material undergoing an exothermic reaction over the symmetrical N-dimensional unit sphere is u″(r) + (N − 1)r −1u′(r) + λe −( 1 u ) = 0 . The parameter λ is held constant so that the solution structure is dependent on the bifurcation parameter U appearing in the boundary condition u(1) = U. In previous papers we discussed the existence of multiple solutions for both class A (slab, infinite cylinder, and sphere) and nonclass A geometries and showed that multiple solutions (of multiplicity greater that three) occur for 2 < N < 12. In this paper, we present numerical results for some hollow geometries which indicate that similar solution structure to that of previous cases is preserved and that the multiplicity of solutions does not always depend on the size of the hollow region.

Scalar curvatures of conformal metrics on Sn

In this paper we consider the following problem: Given a smooth function K on the n-dimensional unit sphere Sn(n ≥ 3) with its canonical metric g0, is it possible to find a pointwise conformal metric which has K as its scalar curvature? This problem was presented by J. L. Kazdan and F. W. Warner. The associated problem for Gaussian curvature in dimension 2 had been presented by L. Nirenberg several years before.

On Ramsey sets in spheres

We prove that for every vertex set L of a simplex of circumradius 1 in R m , every r and every ϵ > 0, one can find N such that for any r-coloring of a sphere S of radius (1 + ϵ) in R N there is a monochromatic congruent copy of L in S. We also show that for almost all such configurations L, the “blowup by (1 + ϵ)” is really necessary; i.e., for any N, one can color the N-dimensional unit sphere by a certain fixed number of colors such that it contains no monochromatic congruent copy of L . The exceptional configurations for which the coloring does not work have a simple description.

Maximizing the Surface Area of an N-Dimensional Unit Sphere

Maximizing the Surface Area of an N-Dimensional Unit Sphere

A Helly type theorem on the sphere

This paper establishes a Helly type theorem for convex sets in S n {S_n} , the n-dimensional unit sphere.

An extreme covering of 4-space by spheres

Let Λ be a lattice in n-dimensional Euclidean space En. For any lattice there is a unique minimal positive number μ such that if spheres of radius μ are placed at the points of the lattice then the entire space is covered, i.e. every point in En lies in at least one of the spheres. The density of this covering is defined to be θn(Λ) = Jnμn/d(Λ), where Jn is the volume of an n-dimensional unit sphere and d(Λ) is the determinant of the lattice.

The Closest Packing of Spherical Caps in n Dimensions

Let Sn denote the “surface” of an n-dimensional unit sphere in Euclidean space of n dimensions. We may suppose that the sphere is centred at the origin of coordinates O, so that the points P(x1, x2, …, xn) of Sn satisfyWe suppose that n≥2.