Distance matrices of subsets of the Hamming cube

@article{Doust2020DistanceMO,
  title={Distance matrices of subsets of the Hamming cube},
  author={Ian Doust and G. Neil Robertson and Alan Stoneham and Anthony Weston},
  journal={arXiv: Functional Analysis},
  year={2020}
}
Graham and Winkler derived a formula for the determinant of the distance matrix of a full-dimensional set of $n + 1$ points $\{ x_{0}, x_{1}, \ldots , x_{n} \}$ in the Hamming cube $H_{n} = ( \{ 0,1 \}^{n}, \ell_{1} )$. In this article we derive a formula for the determinant of the distance matrix $D$ of an arbitrary set of $m + 1$ points $\{ x_{0}, x_{1}, \ldots , x_{m} \}$ in $H_{n}$. It follows from this more general formula that $\det (D) \not= 0$ if and only if the vectors $x_{0}, x_{1… Expand
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References

SHOWING 1-10 OF 31 REFERENCES
Distance matrices of subsets of the Hamming cube
  • Indag. Math. 32
  • 2021
On the gap of finite metric spaces of p-negative type
Let (X,d) be a metric space of p-negative type. Recently I. Doust and A. Weston introduced a quantification of the p-negative type property, the so called gap {\Gamma} of X. This talk introduces someExpand
Finite quasihypermetric spaces
AbstractLet (X, d) be a compact metric space and let $$ \mathcal{M} $$(X) denote the space of all finite signed Borel measures on X. Define I: $$ \mathcal{M} $$(X) → ℝ by I(μ) = ∫X∫Xd(x,Expand
The maximal generalised roundness of finite metric spaces
We provide two simplifications of Sanchez's formula for the maximum generalised roundness of a finite metric space.
Estimating the gap of finite metric spaces of strict p-negative type
  • R. Wolf
  • Mathematics
  • Linear Algebra and its Applications
  • 2018
Let (X,d) be a finite metric space. This paper first discusses the spectrum of the p-distance matrix of a finite metric space of p-negative type and then gives upper and lower bounds for the soExpand
Euclidean Distance Matrices: Essential theory, algorithms, and applications
TLDR
The fundamental properties of EDMs, such as rank or (non)definiteness, are reviewed, and it is shown how the various EDM properties can be used to design algorithms for completing and denoising distance data. Expand
Euclidean Distance Geometry and Applications
TLDR
The theory of Euclidean distance geometry and its most important applications are surveyed, with special emphasis on molecular conformation problems. Expand
Supremal p-negative type of vertex transitive graphs
Abstract We study the supremal p-negative type of connected vertex transitive graphs. The analysis provides a way to characterize subsets of the Hamming cube { 0 , 1 } n ⊂ l 1 ( n ) ( n ⩾ 1 ) thatExpand
On the supremal $p$-negative type of a finite metric space
We study the supremal $p$-negative type of finite metric spaces. An explicit expression for the supremal $p$-negative type $\wp (X,d)$ of a finite metric space $(X,d)$ is given in terms itsExpand
DISTANCE GEOMETRY IN QUASIHYPERMETRIC SPACES. I
Let ( X , d ) be a compact metric space and let ℳ( X ) denote the space of all finite signed Borel measures on  X . Define I :ℳ( X )→ℝ by and set M ( X )=sup I ( μ ), where μ ranges over theExpand
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