Using recent results of Berman and Boucksom [BB2] we show that for a non-pluripolar compact set K... more Using recent results of Berman and Boucksom [BB2] we show that for a non-pluripolar compact set K ⊂ C d and an admissible weight function w = e −φ any sequence of so-called optimal measures converges weak-* to the equilibrium measure µ K,φ of (weighted) Pluripotential Theory for K, φ.
Journal of Physics A: Mathematical and Theoretical, 2019
For over a decade, there has been intensive work on the numerical and analytic construction of SI... more For over a decade, there has been intensive work on the numerical and analytic construction of SICs (d 2 equiangular lines in C d) as an orbit of the Heisenberg group. The Clifford group, which consists of the unitary matrices which normalise the Heisenberg group, plays a key role in these constructions. All of the known fiducial (generating) vectors for such SICs are eigenvectors of symplectic operations in the Clifford group with canonical order 3. Here we describe the Clifford group and the subgroup of symplectic operations in terms of a natural set of generators. From this, we classify all its elements of canonical order three. In particular, we show (contrary to prior claims) that there are symplectic operations of canonical order 3 for d ≡ 6 mod 9, which are not conjugate to the Zauner matrix. It is as yet unknown whether these give rise to SICs.
Recent Progress in Multivariate Approximation, 2001
For n + 1 points in IR d , in general position, the Kergin polynomial interpolant of C n function... more For n + 1 points in IR d , in general position, the Kergin polynomial interpolant of C n functions may be extended to an interpolant of C d−1 functions. This results in an explicit set of reduced Kergin functionals naturally stratified by their dependence on certain directional derivatives of order k, 0 ≤ k ≤ d − 1. We show that the polynomials dual to the functionals depending on derivatives of order k are multi-ridge functions of d − k variables and moreover, that the polynomials dual to the purely interpolating functionals (k = 0) are always harmonic.
We consider the long standing problem of constructing d 2 equiangular lines in C d , i.e., findin... more We consider the long standing problem of constructing d 2 equiangular lines in C d , i.e., finding a set of d 2 unit vectors (φ j) in C d with | φ j , φ k | = 1 √ d + 1 , j = k. Such 'equally spaced configurations' have appeared in various guises, e.g., as complex spherical 2-designs, equiangular tight frames, isometric embeddings 2 (d) → 4 (d 2), and most recently as SICPOVMs in quantum measurement theory. Analytic solutions are known only for d = 2, 3, 4, 8. Recently, numerical solutions which are the orbit of a discrete Heisenberg group H have been constructed for d ≤ 45. We call these Heisenberg frames. In this paper we study the normaliser of H, which we view as a group of symmetries of the equations that determine a Heisenberg frame. This allows us to simplify the equations for a Heisenberg frame. From these simplified equations we are able construct analytic solutions for d = 5, 7, and make conjectures about the form of a solution when d is odd. Most notably, it appears that solutions for d odd are eigenvectors of some element in the normaliser which has (scalar) order 3.
We discuss three natural pseudodistances and pseudometrics on a bounded domain in IR N based on p... more We discuss three natural pseudodistances and pseudometrics on a bounded domain in IR N based on polynomial inequalities.
Suppose that K ⊂ R d is either the unit ball, the unit sphere or the standard simplex. We show th... more Suppose that K ⊂ R d is either the unit ball, the unit sphere or the standard simplex. We show that there are constants c 1 , c 2 > 0 such that for a set of Fekete points (maximizing the Vandermonde determinant) of degree n, F n ⊂ K, c 1 n min b∈Fn b =a dist(a, b) c 2 n for all a ∈ F n. Here dist(a, b) is a natural distance on K that will be described in the text.
Using recent results of Berman and Boucksom [3] we show that for a non-pluripolar compact set K ⊂... more Using recent results of Berman and Boucksom [3] we show that for a non-pluripolar compact set K ⊂ C d and an admissible weight function w = e −φ any sequence of optimal measures converges weak-* to the equilibrium measure µ K,φ of (weighted) pluripotential theory for K, φ.
The aim of this paper is to investigate symmetry properties of tight frames, with a view to const... more The aim of this paper is to investigate symmetry properties of tight frames, with a view to constructing tight frames of orthogonal polynomials in several variables which share the symmetries of the weight function, and other similar applications. This is achieved by using representation theory to give methods for constructing tight frames as orbits of groups of unitary transformations acting on a given finite-dimensional Hilbert space. Along the way, we show that a tight frame is determined by its Gram matrix and discuss how the symmetries of a tight frame are related to its Gram matrix. We also give a complete classification of those tight frames which arise as orbits of an abelian group of symmetries.
Suppose that K ⊂ IR d is either the unit ball, the unit sphere or the standard simplex. We show t... more Suppose that K ⊂ IR d is either the unit ball, the unit sphere or the standard simplex. We show that there are constants c 1 , c 2 > 0 such that for a set of Fekete points (maximizing the Vandermonde determinant) of degree n, F n ⊂ K, is a natural distance on K that will be described in the text.
Using recent results of Berman and Boucksom [BB2] we show that for a non-pluripolar compact set K... more Using recent results of Berman and Boucksom [BB2] we show that for a non-pluripolar compact set K ⊂ C d and an admissible weight function w = e −φ any sequence of so-called optimal measures converges weak-* to the equilibrium measure µ K,φ of (weighted) Pluripotential Theory for K, φ.
Journal of Physics A: Mathematical and Theoretical, 2019
For over a decade, there has been intensive work on the numerical and analytic construction of SI... more For over a decade, there has been intensive work on the numerical and analytic construction of SICs (d 2 equiangular lines in C d) as an orbit of the Heisenberg group. The Clifford group, which consists of the unitary matrices which normalise the Heisenberg group, plays a key role in these constructions. All of the known fiducial (generating) vectors for such SICs are eigenvectors of symplectic operations in the Clifford group with canonical order 3. Here we describe the Clifford group and the subgroup of symplectic operations in terms of a natural set of generators. From this, we classify all its elements of canonical order three. In particular, we show (contrary to prior claims) that there are symplectic operations of canonical order 3 for d ≡ 6 mod 9, which are not conjugate to the Zauner matrix. It is as yet unknown whether these give rise to SICs.
Recent Progress in Multivariate Approximation, 2001
For n + 1 points in IR d , in general position, the Kergin polynomial interpolant of C n function... more For n + 1 points in IR d , in general position, the Kergin polynomial interpolant of C n functions may be extended to an interpolant of C d−1 functions. This results in an explicit set of reduced Kergin functionals naturally stratified by their dependence on certain directional derivatives of order k, 0 ≤ k ≤ d − 1. We show that the polynomials dual to the functionals depending on derivatives of order k are multi-ridge functions of d − k variables and moreover, that the polynomials dual to the purely interpolating functionals (k = 0) are always harmonic.
We consider the long standing problem of constructing d 2 equiangular lines in C d , i.e., findin... more We consider the long standing problem of constructing d 2 equiangular lines in C d , i.e., finding a set of d 2 unit vectors (φ j) in C d with | φ j , φ k | = 1 √ d + 1 , j = k. Such 'equally spaced configurations' have appeared in various guises, e.g., as complex spherical 2-designs, equiangular tight frames, isometric embeddings 2 (d) → 4 (d 2), and most recently as SICPOVMs in quantum measurement theory. Analytic solutions are known only for d = 2, 3, 4, 8. Recently, numerical solutions which are the orbit of a discrete Heisenberg group H have been constructed for d ≤ 45. We call these Heisenberg frames. In this paper we study the normaliser of H, which we view as a group of symmetries of the equations that determine a Heisenberg frame. This allows us to simplify the equations for a Heisenberg frame. From these simplified equations we are able construct analytic solutions for d = 5, 7, and make conjectures about the form of a solution when d is odd. Most notably, it appears that solutions for d odd are eigenvectors of some element in the normaliser which has (scalar) order 3.
We discuss three natural pseudodistances and pseudometrics on a bounded domain in IR N based on p... more We discuss three natural pseudodistances and pseudometrics on a bounded domain in IR N based on polynomial inequalities.
Suppose that K ⊂ R d is either the unit ball, the unit sphere or the standard simplex. We show th... more Suppose that K ⊂ R d is either the unit ball, the unit sphere or the standard simplex. We show that there are constants c 1 , c 2 > 0 such that for a set of Fekete points (maximizing the Vandermonde determinant) of degree n, F n ⊂ K, c 1 n min b∈Fn b =a dist(a, b) c 2 n for all a ∈ F n. Here dist(a, b) is a natural distance on K that will be described in the text.
Using recent results of Berman and Boucksom [3] we show that for a non-pluripolar compact set K ⊂... more Using recent results of Berman and Boucksom [3] we show that for a non-pluripolar compact set K ⊂ C d and an admissible weight function w = e −φ any sequence of optimal measures converges weak-* to the equilibrium measure µ K,φ of (weighted) pluripotential theory for K, φ.
The aim of this paper is to investigate symmetry properties of tight frames, with a view to const... more The aim of this paper is to investigate symmetry properties of tight frames, with a view to constructing tight frames of orthogonal polynomials in several variables which share the symmetries of the weight function, and other similar applications. This is achieved by using representation theory to give methods for constructing tight frames as orbits of groups of unitary transformations acting on a given finite-dimensional Hilbert space. Along the way, we show that a tight frame is determined by its Gram matrix and discuss how the symmetries of a tight frame are related to its Gram matrix. We also give a complete classification of those tight frames which arise as orbits of an abelian group of symmetries.
Suppose that K ⊂ IR d is either the unit ball, the unit sphere or the standard simplex. We show t... more Suppose that K ⊂ IR d is either the unit ball, the unit sphere or the standard simplex. We show that there are constants c 1 , c 2 > 0 such that for a set of Fekete points (maximizing the Vandermonde determinant) of degree n, F n ⊂ K, is a natural distance on K that will be described in the text.
Uploads
Papers by Shayne Waldron