8,657 research outputs found
On a generalization of distance sets
A subset in the -dimensional Euclidean space is called a -distance
set if there are exactly distinct distances between two distinct points in
and a subset is called a locally -distance set if for any point
in , there are at most distinct distances between and other points
in .
Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the
cardinalities of -distance sets on a sphere in 1977. In the same way, we are
able to give the same bound for locally -distance sets on a sphere. In the
first part of this paper, we prove that if is a locally -distance set
attaining the Fisher type upper bound, then determining a weight function ,
is a tight weighted spherical -design. This result implies that
locally -distance sets attaining the Fisher type upper bound are
-distance sets. In the second part, we give a new absolute bound for the
cardinalities of -distance sets on a sphere. This upper bound is useful for
-distance sets for which the linear programming bound is not applicable. In
the third part, we discuss about locally two-distance sets in Euclidean spaces.
We give an upper bound for the cardinalities of locally two-distance sets in
Euclidean spaces. Moreover, we prove that the existence of a spherical
two-distance set in -space which attains the Fisher type upper bound is
equivalent to the existence of a locally two-distance set but not a
two-distance set in -space with more than points. We also
classify optimal (largest possible) locally two-distance sets for dimensions
less than eight. In addition, we determine the maximum cardinalities of locally
two-distance sets on a sphere for dimensions less than forty.Comment: 17 pages, 1 figur
A new Euclidean tight 6-design
We give a new example of Euclidean tight 6-design in .Comment: 9 page
Spherical two-distance sets and related topics in harmonic analysis
This dissertation is devoted to the study of applications of
harmonic analysis. The maximum size of spherical few-distance sets
had been studied by Delsarte at al. in the 1970s. In particular,
the maximum size of spherical two-distance sets in
had been known for except by linear programming
methods in 2008. Our contribution is to extend the known results
of the maximum size of spherical two-distance sets in
when , and . The maximum size of equiangular lines in had
been known for all except and
since 1973. We use the semidefinite programming method to
find the maximum size for equiangular line sets in
when and .
We suggest a method of constructing spherical two-distance sets
that also form tight frames. We derive new structural properties
of the Gram matrix of a two-distance set that also forms a tight
frame for . One of the main results in this part is
a new correspondence between two-distance tight frames and certain
strongly regular graphs. This allows us to use spectral properties
of strongly regular graphs to construct two-distance tight
frames. Several new examples are obtained using this
characterization.
Bannai, Okuda, and Tagami proved that a tight spherical designs of
harmonic index 4 exists if and only if there exists an equiangular
line set with the angle in the Euclidean
space of dimension for each integer . We
show nonexistence of tight spherical designs of harmonic index
on with by a modification of the semidefinite
programming method. We also derive new relative bounds for
equiangular line sets. These new relative bounds are usually
tighter than previous relative bounds by Lemmens and Seidel
Cubature formulas, geometrical designs, reproducing kernels, and Markov operators
Cubature formulas and geometrical designs are described in terms of
reproducing kernels for Hilbert spaces of functions on the one hand, and Markov
operators associated to orthogonal group representations on the other hand. In
this way, several known results for spheres in Euclidean spaces, involving
cubature formulas for polynomial functions and spherical designs, are shown to
generalize to large classes of finite measure spaces and
appropriate spaces of functions inside . The last section
points out how spherical designs are related to a class of reflection groups
which are (in general dense) subgroups of orthogonal groups
Note on cubature formulae and designs obtained from group orbits
In 1960, Sobolev proved that for a finite reflection group G, a G-invariant
cubature formula is of degree t if and only if it is exact for all G-invariant
polynomials of degree at most t. In this paper, we find some observations on
invariant cubature formulas and Euclidean designs in connection with the
Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998)
on necessary and sufficient conditions for the existence of cubature formulas
with some strong symmetry. The new proof is shorter and simpler compared to the
original one by Xu, and moreover gives a general interpretation of the
analytically-written conditions of Xu's theorems. Second, we extend a theorem
by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean
designs, and thereby classify tight Euclidean designs obtained from unions of
the orbits of the corner vectors. This result generalizes a theorem of Bajnok
(2007) which classifies tight Euclidean designs invariant under the Weyl group
of type B to other finite reflection groups.Comment: 18 pages, no figur
Complex spherical codes with two inner products
A finite set in a complex sphere is called a complex spherical -code
if the number of inner products between two distinct vectors in is equal to
. In this paper, we characterize the tight complex spherical -codes by
doubly regular tournaments, or skew Hadamard matrices. We also give certain
maximal 2-codes relating to skew-symmetric -optimal designs. To prove them,
we show the smallest embedding dimension of a tournament into a complex sphere
by the multiplicity of the smallest or second-smallest eigenvalue of the Seidel
matrix.Comment: 10 pages, to appear in European Journal of Combinatoric
Tight informationally complete quantum measurements
We introduce a class of informationally complete positive-operator-valued
measures which are, in analogy with a tight frame, "as close as possible" to
orthonormal bases for the space of quantum states. These measures are
distinguished by an exceptionally simple state-reconstruction formula which
allows "painless" quantum state tomography. Complete sets of mutually unbiased
bases and symmetric informationally complete positive-operator-valued measures
are both members of this class, the latter being the unique minimal rank-one
members. Recast as ensembles of pure quantum states, the rank-one members are
in fact equivalent to weighted 2-designs in complex projective space. These
measures are shown to be optimal for quantum cloning and linear quantum state
tomography.Comment: 20 pages. Final versio
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