Spherical design
Encyclopedia
A spherical design, part of combinatorial design
theory in mathematics
, is a finite set of N points on the d-dimensional unit hypersphere
Sd such that the average value of any polynomial f of degree t or less on the set equals the average value of f on the whole sphere (that is, the integral of f over Sd divided by the area or measure
of Sd). Such a set is often called a spherical t-design to indicate the value of t, which is a fundamental parameter. Spherical t-designs for different values of N and t can be found precomputed at http://www.research.att.com/~njas/sphdesigns.
Spherical designs can be of value in approximation theory
, in statistics
for experimental design (being usable to construct rotatable designs), in combinatorics
, and in geometry
. The main problem is to find examples, given d and t, that are not too large. However, such examples may be hard to come by.
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 POVM
s.
The concept of a spherical design is due to Delsarte, Goethals, and Seidel (1977). The existence and structure of spherical designs with d = 1 (that is, in a circle) was studied in depth by Hong (1982). Shortly thereafter, Seymour and Zaslavsky (1984) proved that such designs exist of all sufficiently large sizes; that is, 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. However, their proof gave no idea of how big N(d,t) is. Good estimates for that were found later on. Besides these "large" sizes, there are many sporadic small spherical designs; many of them are related to finite group action
s on the sphere and are of great interest in themselves.
One application of spherical designs is for whole-sphere data collection.
Spherical t-designs meet the "accurately approximate integrals by sums" criteria for "good" pixelizations of the sphere.
Combinatorial design
Combinatorial design theory is the part of combinatorial mathematics that deals with the existence and construction of systems of finite sets whose intersections have specified numerical properties....
theory in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, is a finite set of N points on the d-dimensional unit hypersphere
Hypersphere
In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any...
Sd such that the average value of any polynomial f of degree t or less on the set equals the average value of f on the whole sphere (that is, the integral of f over Sd divided by the area or measure
Measure
- Legal :* Measure of the Church of England is a law passed by the General Synod and the UK Parliament equivalent of an Act* Measure of the National Assembly for Wales, a law specific to Wales passed by the Welsh Assembly between 2007 and 2011...
of Sd). Such a set is often called a spherical t-design to indicate the value of t, which is a fundamental parameter. Spherical t-designs for different values of N and t can be found precomputed at http://www.research.att.com/~njas/sphdesigns.
Spherical designs can be of value in approximation theory
Approximation theory
In mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby...
, in statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
for experimental design (being usable to construct rotatable designs), in combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
, and in geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
. The main problem is to find examples, given d and t, that are not too large. However, such examples may be hard to come by.
Spherical t-designs have also recently been appropriated in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
in the form of quantum t-designs with various applications to quantum information theory, quantum computing and POVM
POVM
In functional analysis and quantum measurement theory, a POVM is a measure whose values are non-negative self-adjoint operators on a Hilbert space. It is the most general formulation of a measurement in the theory of quantum physics...
s.
The concept of a spherical design is due to Delsarte, Goethals, and Seidel (1977). The existence and structure of spherical designs with d = 1 (that is, in a circle) was studied in depth by Hong (1982). Shortly thereafter, Seymour and Zaslavsky (1984) proved that such designs exist of all sufficiently large sizes; that is, 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. However, their proof gave no idea of how big N(d,t) is. Good estimates for that were found later on. Besides these "large" sizes, there are many sporadic small spherical designs; many of them are related to finite group action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
s on the sphere and are of great interest in themselves.
One application of spherical designs is for whole-sphere data collection.
Spherical t-designs meet the "accurately approximate integrals by sums" criteria for "good" pixelizations of the sphere.