Spherical T-designs Research Articles

On the existence of tight relative 2-designs on binary Hamming association schemes

It is known that there is a close analogy between “Euclidean t-designs vs. spherical t-designs” and “Relative t-designs in binary Hamming association schemes vs. combinatorial t-designs”. In this paper, we want to prove how much we can develop a similar theory in the latter situation, imitating the theory in the former one. We first prove that the weight function is constant on each shell for tight relative t-designs on p shells on a wide class of Q-polynomial association schemes, including Hamming association schemes. In the theory of Euclidean t-designs on 2 concentric spheres (shells), it is known that the structure of coherent configurations is naturally attached. However, it seems difficult to prove this claim in a general context. In the case of tight 2-designs in combinatorial 2-designs, there are great many tight 2-designs, i.e., symmetric 2-designs, while there are very few tight 2e-designs for e≥2. So, as a starting point, we concentrate our study to the existence problem of tight relative 2-designs, in particular on 2 shells, in binary Hamming association schemes H(n,2). We prove that every tight relative 2-design on 2 shells in H(n,2) has the structure of coherent configuration. We determine all the possible parameters of coherent configurations attached to such tight relative 2-designs for n≤30. Moreover for each of them we determine whether there exists such a tight relative 2-design or not, either by constructing them from symmetric 2-designs or Hadamard matrices, or theoretically showing the non-existence. In particular, we show that for n≡6(mod8), there exist such tight relative 2-designs whose weight functions are not constant. These are the first examples of those with non-constant weight.

Optimal asymptotic bounds for spherical designs

In this paper we prove the conjecture of Korevaar and Meyers: for each N cdt d , there exists a spherical t-design in the sphere S d consisting of N points, where cd is a constant depending only on d.

On a generalization of spherical designs

In this paper, we define the concept of a spherical T-design for a finite subset of a sphere. Then, we give some examples of such designs using the Z2-lattice, which is related to Lehmer’s conjecture.

Conformal designs and D.H. Lehmerʼs conjecture

In 1947, Lehmer conjectured that the Ramanujan τ-function τ(m) is non-vanishing for all positive integers m, where τ(m) are the Fourier coefficients of the cusp form Δ of weight 12. It is known that Lehmerʼs conjecture can be reformulated in terms of spherical t-design, by the result of Venkov. In this paper, we show that τ(m)=0 is equivalent to the fact that the homogeneous space of the moonshine vertex operator algebra (V♮)m+1 is a conformal 12-design. Therefore, Lehmerʼs conjecture is now reformulated in terms of conformal t-designs.

On the computation of spherical designs by a new optimization approach based on fast spherical Fourier transforms

Spherical t-designs are point sets $$X_{M}=\{x_{1},\ldots, x_{M}\} \subset {\mathbb{S}^{2}}$$ which provide quadrature rules with equal weights for the sphere which are exact for polynomials up to degree t. In this paper we consider the problem of finding numerical spherical t-designs on the sphere $${\mathbb{S}^{2}}$$ for high polynomial degree $${t \in {\mathbb{N}}}$$. That is, we compute numerically local minimizers of a certain quadrature error A t (X M ). The quadrature error A t was also used for a variational characterization of spherical t-designs by Sloan and Womersley (J Approx Theory 159:308–318, 2009). For the minimization problem we regard several nonlinear optimization methods on manifolds, like Newton and conjugate gradient methods. We show that by means of the nonequispaced fast spherical Fourier transforms we perform gradient and Hessian evaluations in $${\mathcal {O}(t^{2}\log t + M \log^{2}(1/\varepsilon))}$$ arithmetic operations, where $${\varepsilon >0 }$$ is a prescribed accuracy. Using these methods we present numerical spherical t-designs for t ≤ 1,000, even in the case $${M\approx \frac{1}{2} t^{2}}$$.

On spherical designs obtained from Q-polynomial association schemes

We characterize that the image of the embedding of the Q-polynomial association scheme into the first eigenspace by primitive idempotent E1 is a spherical t-design in terms of the Krein numbers. Furthermore, we show that the strengths of P- and Q-polynomial schemes as spherical designs are bounded by a constant. Copyright © 2011 John Wiley & Sons, Ltd. 19:167-177, 2011

Computational existence proofs for spherical t-designs

Spherical t-designs provide quadrature rules for the sphere which are exact for polynomials up to degree t. In this paper, we propose a computational algorithm based on interval arithmetic which, for given t, upon successful completion will have proved the existence of a t-design with (t + 1)2 nodes on the unit sphere $${S^2 \subset \mathbb{R}^3}$$ and will have computed narrow interval enclosures which are known to contain these nodes with mathematical certainty. Since there is no theoretical result which proves the existence of a t-design with (t + 1)2 nodes for arbitrary t, our method contributes to the theory because it was tested successfully for t = 1, 2, . . . , 100. The t-design is usually not unique; our method aims at finding a well-conditioned one. The method relies on computing an interval enclosure for the zero of a highly nonlinear system of dimension (t + 1)2. We therefore develop several special approaches which allow us to use interval arithmetic efficiently in this particular situation. The computations were all done using the MATLAB toolbox INTLAB.

Spherical designs and anticoherent spin states

Anticoherent spin states are quantum states that exhibit maximally nonclassical behaviour in a certain sense. Any spin state whose Majorana representation is a Platonic solid is called a perfect state. By direct calculation, it has been shown that any perfect state is an anticoherent spin state. We show that any spin state whose Majorana representation is both the orbit of a finite subgroup of O(3) and a spherical t-design must be an anticoherent spin state of order t. Since all Platonic solids are spherical designs, this result gives an explanation of the anticoherence of perfect states and explains their observed order. We also show that any spin state whose Majorana representation lies in any single open hemisphere cannot be anticoherent of any order. This result is then used to give further relations between spherical designs and anticoherent spin states. We also pose some questions relating spherical designs and geometric entanglement.

Coherent configurations and triply regular association schemes obtained from spherical designs

Delsarte, Goethals and Seidel showed that if X is a spherical t-design with degree s satisfying t ⩾ 2 s − 2 , X carries the structure of an association scheme. Also Bannai and Bannai showed that the same conclusion holds if X is an antipodal spherical t-design with degree s satisfying t = 2 s − 3 . As a generalization of these results, we prove that a union of spherical designs with a certain property carries the structure of a coherent configuration. We derive triple regularity of tight spherical 4-, 5-, 7-designs, mutually unbiased bases, linked systems of symmetric designs with certain parameters.

Well Conditioned Spherical Designs for Integration and Interpolation on the Two-Sphere

A set $\mathcal{X}_{N}$ of N points on the unit sphere is a spherical t-design if the average value of any polynomial of degree at most t over $\mathcal{X}_{N}$ is equal to the average value of the polynomial over the sphere. This paper considers the characterization and computation of spherical t-designs on the unit sphere $\mathbb{S}^2\subset\mathbb{R}^3$ when $N\geq(t+1)^2$, the dimension of the space $\mathbb{P}_t$ of spherical polynomials of degree at most t. We show how to construct well conditioned spherical designs with $N\geq(t+1)^2$ points by maximizing the determinant of a matrix while satisfying a system of nonlinear constraints. Interval methods are then used to prove the existence of a true spherical t-design very close to the calculated points and to provide a guaranteed interval containing the determinant. The resulting spherical designs have good geometrical properties (separation and mesh norm). We discuss the usefulness of the points for both equal weight numerical integration and polynomial interpolation on the sphere and give an example.

On antipodal Euclidean tight (2e + 1)-designs

Neumaier and Seidel (1988) generalized the concept of spherical designs and defined Euclidean designs in ℝ n . For an integer t, a finite subset X of ℝ n given together with a weight function w is a Euclidean t-design if $$\sum_{i=1}^p\frac{w(X_i)}{|S_i|} \int_{S_i}f(\boldsymbol x)d\sigma_i(\boldsymbol x) =\sum_{\boldsymbol x\in X}w(\boldsymbol x) f(\boldsymbol x)$$ holds for any polynomial f(x) of deg(f)≤ t, where {S i , 1≤ i ≤ p} is the set of all the concentric spheres centered at the origin that intersect with X, X i = X∩ S i , and w:X→ ℝ> 0. (The case of X⊂ S n−1 with w≡ 1 on X corresponds to a spherical t-design.) In this paper we study antipodal Euclidean (2e+1)-designs. We give some new examples of antipodal Euclidean tight 5-designs. We also give the classification of all antipodal Euclidean tight 3-designs, the classification of antipodal Euclidean tight 5-designs supported by 2 concentric spheres.

An evaluation of spherical designs for molecular-like surfaces

The use of spherical harmonics in the molecular sciences is widespread. They have been employed with success in, for instance, the crystallographic fast rotation function, small-angle scattering particle reconstruction, molecular surface visualisation, protein–protein docking, active site analysis and protein function prediction. The calculation of spherical harmonic expansion coefficients requires integration over the full sphere and can be a computationally cumbersome and also numerically sensitive (with respect to the integration weights) procedure. It is shown here how the use of spherical t-designs and pre-computed near-equal weight integration layouts can significantly reduce the computational effort in the determination of spherical harmonic expansion coefficients for molecular surfaces, thus giving rise to a robust and highly efficient algorithm for the construction of molecular-like objects.

SHELLS OF SELFDUAL LATTICES VIEWED AS SPHERICAL DESIGNS

We find out for which t shells of selfdual lattices and of their shadows are spherical t-designs. The method uses theta series of lattices, which are modular forms. We analyze fully cubic and Witt lattices, as well as all selfdual lattices of rank at most 24.

Spherical designs and finite group representations (some results of E. Bannai)

We reprove several results of Bannai concerning spherical t-designs and finite subgroups of orthogonal groups. These include criteria in terms of harmonic representations of subgroups of O( n) for the corresponding orbits to be t-designs ( t=0,1,2,3,…) in S n−1 . We also discuss a conjecture of Bannai, dating from 1984, according to which t is bounded independently of the dimension n (for n≥3) for such designs.

Spherical t-designs and the Bernstein Theorem

In this paper we study the construction problem of spherical t-designs as a generalization of the Chebyshev problem of Gauss type quadrature formula. It is known that spherical t-designs can be constructed if we can construct interval t-designs for the weight function (1−x)d−22. We shall show the exsitence and non-existence theorems of some interval t-desings.

Groupes engendrés par des réflexions, designs sphériques et réseau de Leech

For a finite subset X⊂ R n of unit vectors, G X denotes the group generated by reflections r x fixing hyperplanes orthogonal to x∈ X. When X is a spherical t-design and π ( k) har the representation of G X in the harmonic polynomials in n variables of degree k, the spectrum of the Markov operator 1 |X| ∑ x∈Xπ (k) har (r x) is analyzed. If k is small enough, this operator is a scalar multiple of the identity. When X is the spherical 11-design of short vectors in a Leech lattice, it is shown that the infinite group G X contains the finite Conway group Co and is a quotient of a remarkable Coxeter group.

Directional methods for structural reliability analysis

Directional simulation reduces the dimension of the limit state probability integral by identifying a set of directions for integration, integrating either in closed-form or by approximation in those directions, and estimating the probability as a weighted average of the directional integrals. Most existing methods identify these directions by a set of points distributed on the unit hypersphere. The accuracy of the directional simulation depends on how the points are identified. When the limit state is highly nonlinear, or the inherent failure probability is small, a very large number of points may be required, and the method can become inefficient. This paper introduces several new approaches for identifying directions for evaluating the probability integral — Spherical t-design, Spiral Points, and Fekete Points — and compares the failure probabilities with those determined in a number of examples in previously published work. Once these points have been identified for a probability integral of given dimension, they can be used repeatedly for other probability integrals of the same dimension in a fashion analogous to Gauss Quadrature.

Spherical Designs and Generalized Sum-Free Sets in AbelianGroups

We extend the concepts of sum-free sets and Sidon-sets of combinatorial number theory with the aim to provide explicit constructions for spherical designs. We call a subset S of the (additive) abelian group G t-free if for all non-negative integers k and l with k+l \leq t, the sum of k (not necessarily distinct) elements of S does not equal the sum of l (not necessarily distinct) elements of S unless k=l and the two sums contain the same terms. Here we shall give asymptotic bounds for the size of a largest t-free set in \small{{ Z}}_n, and for t \leq 3 discuss how t-free sets in \small{{ Z}}_n can be used to construct spherical t-designs.

Constructions of Spherical 3-Designs

Spherical t-designs are Chebyshev-type averaging sets on the d-sphere which are exact for polynomials of degree at most t. This concept was introduced in 1977 by Delsarte, Goethals, and Seidel, who also found the minimum possible size of such designs, in particular, that the number of points in a 3-design on S d must be at least . In this paper we give explicit constructions for spherical 3-designs on S d consisting of n points for d=1 and ; d=2 and ; d=3 and ; d=4 and ; and odd or even. We also provide some evidence that 3-designs of other sizes do not exist. We will introduce and apply a concept from additive number theory generalizing the classical Sidon-sequences. Namely, we study sets of integers S for which the congruence mod n, where and , only holds in the trivial cases. We call such sets Sidon-type sets of strength t, and denote their maximum cardinality by s(n, t). We find a lower bound for s(n, 3), and show how Sidon-type sets of strength 3 can be used to construct spherical 3-designs. We also conjecture that our lower bound gives the true value of s(n, 3) (this has been verified for n≤125).

Some constructions of spherical 5-designs

Spherical designs were introduced by Delsarte, Goethals, and Seidel in 1977. A spherical t-design in, R n is a finite set X ⊂ S n−1 with the property that for every polynomial p with degree ⩽ t, the average value of p on X equals the average value of p on S n−1 . This paper contains some existence and nonexistence results, mainly for spherical 5-designs in R 3. Delsarte, Goethals, and Seidel proved that if X is a spherical 5-design in R 3, then | X| ⩾ 12 and if | Xz. sfnc; = 12, then X consists of the vertices of a regular icosahedron. We show that such designs exist with cardinality 16, 18, 20, 22, 24, and every integer ⩾ 26. If X is a spherical 5-design in R n , then | X| ⩾ n( n + 1); if | X| = n( n + 1), then X has been called tight. Tight spherical 5-designs in R n are known to exist only for n = 2, 3, 7, 23 and possibly n = u 2 − 2 for odd u ⩾ 7. Any tight spherical 5-design in R n must consist of n(n + 1) 2 antipodal pairs of points. We show that for n ⩾ 3, there are no spherical 5-designs in R n consisting of n(n + 1) 2 + 1 antipodal pairs of points.