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S. B. Damelin

Publications and source records attributed to S. B. Damelin.

11 recordsLinked to original sources
arXiv2026

A Note on an Analytic Approach to the Problem of Finite Matroid Representability, The Cardinality of Sets of k-Independent Vectors over Finite Fields and the Maximum Distance Separable Conjecture

We introduce various quantities that can be defined for an arbitrary finite matroid, and show that certain conditions on these quantities imply that a matroid is not representable over $\mathbb{F}_q$ where $q$ is a prime power. Mostly, for a finite matroid of rank $r$, we examine the proportion of size-$(r-k)$ subsets that are dependent, and give bounds, in terms of the cardinality of the matroid and $q$, for this proportion, below which the matroid is not representable over $\mathbb{F}_q$. We also explore connections between the defined quantities and demonstrate that they can be used to prove that random matrices have high proportions of subsets of columns independent. Our study relates to the results of our papers [4,5,11] dealing with the cardinality of sets of $k$-independent vectors over $\mathbb{F}_q$ and the Maximal Distance Separation Conjecture over $\mathbb{F}_q$.

math.CO↗
arXiv2024

On Smooth Whitney Extensions of almost isometries with small distortion, Interpolation and Alignment in $\Bbb R^D$-Part 1

In this paper, we study the following problem: Let $D\geq 2$ and let $E\subset \mathbb R^D$ be finite satisfying certain conditions. Suppose that we are given a map $ϕ:E\to \mathbb R^D$ with $ϕ$ a small distortion on $E$. How can one decide whether $ϕ$ extends to a smooth small distortion $Φ:\mathbb R^D\to \mathbb R^D$ which agrees with $ϕ$ on $E$. We also ask how to decide if in addition $Φ$ can be approximated well by certain rigid and non-rigid motions from $\mathbb R^D\to \mathbb R^D$. Since $E$ is a finite set, this question is basic to interpolation and alignment of data in $\mathbb R^D$. The work in this paper appears in the research memoir [10].

math.MG↗
arXiv2023

Whitney extensions and orthonormal expansions

The Whitney near extension problem for finite sets in $\mathbb R^d,\, d\geq 2$ asks the following: Let $ϕ:E\to \mathbb R^d$ be a near distortion on a finite set $E\subset \mathbb R^d$ with certain geometry. How to decide whether $ϕ$ extends to a smooth, one to one and onto near distortion $Φ:\mathbb R^d\to \mathbb R^d$ which agrees with $ϕ$ on $E$ and with Euclidean motions in $\mathbb R^d$. The Whitney near extension problem for compact sets $E\subset U$ in open subsets $U$ of $\mathbb R^n,\, n\geq 1$ asks the following: Let $U\subset R^n$ be open and let $E\subset U$ be a compact set. Let $ϕ:U\to \mathbb R^n$ be a smooth near isometry. How to decide if there exists a smooth one-to-one and onto near isometry $Φ:\mathbb R^n\to \mathbb R^n$ which extends $ϕ$ on $E$ and agrees with Euclidean motions on $\mathbb R^n$. The classical Whitney extension problem asks the following: Let $ϕ:E\to \mathbb R$ be a map defined on an arbitrary set $E\subset \mathbb R^n$. How can one decide whether $ϕ$ extends to a map $Φ:\mathbb R^n\to \mathbb R$ which agrees with $ϕ$ on $E$ and is in $C^m(\mathbb R^n),\, m\geq 1$, the space of functions from $\mathbb R^n$ to $\mathbb R$ whose derivatives of order $m$ are continuous and bounded. In this paper, we survey some of our work on the near Whitney extension problem [2] in $\mathbb R^n$. Thereafter, we survey some of our work on weighted $L_p(\mathbb R),\, 1<p\leq \infty$ convergence of orthonormal expansions in $\mathbb R$ [3] and present a result of [13]. The motivation for doing this is motivated by interesting connections between Whitney extension theorems, Taylor series and Fourier expansions. Finally, we raise various open questions to study.

math.CA↗
arXiv2022

An Analytic and Numerical Analysis of Weighted Singular Cauchy Integrals with Exponential Weights on $\mathbb R$

This paper concerns an analytic and numerical analysis of a class of weighted singular Cauchy integrals with exponential weights $w:=\exp(-Q)$ with finite moments and with smooth external fields $Q:\mathbb R\to [0,\infty)$, with varying smooth convex rate of increase for large argument. Our analysis relies in part on weighted polynomial interpolation at the zeros of orthonormal polynomials with respect to $w^2$. We also study bounds for the first derivatives of a class of functions of the second kind for $w^2$.

math.CA↗
arXiv2018

A Note on a Quantitative Form of the Solovay-Kitaev Theorem

The problem of finding good approximations of arbitrary 1-qubit gates is identical to that of finding a dense group generated by a universal subset of $SU(2)$ to approximate an arbitrary element of $SU(2)$. The Solovay-Kitaev Theorem is a well-known theorem that guarantees the existence of a finite sequence of 1-qubit quantum gates approximating an arbitrary unitary matrix in $SU(2)$ within specified accuracy $\varepsilon > 0$. In this note we study a quantitative description of this theorem in the following sense. We will work with a universal gate set $T$, a subset of $SU(2)$ such that the group generated by the elements of $T$ is dense in $SU(2)$. For $\varepsilon > 0$ small enough, we define $t_{\varepsilon}$ as the minimum reduced word length such that every point of $SU(2)$ lies within a ball of radius $\varepsilon$ centered at the points in the dense subgroup generated by $T$. For a measure of efficiency on T, which we denote $K(T)$, we prove the following theorem: Fix a $δ$ in $[0, \frac{2}{3}]$. Choose $f: (0, \infty) \rightarrow (1, \infty)$ satisfying $\lim_{\varepsilon\to 0+}\dfrac{\log(f(t_{\varepsilon}))}{t_{\varepsilon}}$ exists with value $0$. Assume that the inequality $\varepsilon \leqslant f(t_{\varepsilon})\cdot 5^{\frac{-t_{\varepsilon}}{6-3δ}}$ holds. Then $K(T) \leqslant 2-δ$. Our conjecture implies the following: Let $ν(5^{t_{\varepsilon}})$ denote the set of integer solutions of the quadratic form: $x_1^2+x_2^2+x_3^2+x_4^2=5^{t_{\varepsilon}}$. Let $M\equiv M_{S^3}(\mathcal{N})$ denote the covering radius of the points $\mathcal{N}=ν(5^{t_{\varepsilon}})\cupν(5^{t_{\varepsilon}-1})$ on the sphere $S^{3}$ in $\mathbb{R}^{4}$. Then $M \sim f(\log N)N^{\frac{-1}{6-3δ}}$. Here $N\equiv N(\varepsilon)=6\cdot5^{t_{\varepsilon}}-2$.

math.QA↗
arXiv2018

On surface completion and image inpainting by biharmonic functions: Numerical aspects

Numerical experiments with smooth surface extension and image inpainting using harmonic and biharmonic functions are carried out. The boundary data used for constructing biharmonic functions are the values of the Laplacian and normal derivatives of the functions on the boundary. Finite difference schemes for solving these harmonic functions are discussed in detail.

math.AP↗
arXiv2017

A Koksma-Hlawka-Potential Identity on the $d$ Dimensional Sphere and its Applications to Discrepancy

Let $d\geq 2$ be an integer, $S^d\subset {\mathbb R}^{d+1}$ the unit sphere and $σ$ a finite signed measure whose positive and negative parts are supported on $S^d$ with finite energy. In this paper, we derive an error estimate for the quantity $\left|\int_{S^d}fdσ\right|$, for a class of harmonic functions $f:\mathbb R^{d+1}\to \mathbb R$. Our error estimate involves 2 sided bounds for a Newtonian potential with respect to $σ$ away from its support. In particular, our main result allows us to study quadrature errors, for scatterings on the sphere with given mesh norm.

math.CA↗
arXiv2016

The Truncated & Supplemented Pascal Matrix and Applications

In this paper, we introduce the $k\times n$ (with $k\leq n$) truncated, supplemented Pascal matrix which has the property that any $k$ columns form a linearly independent set. This property is also present in Reed-Solomon codes; however, Reed-Solomon codes are completely dense, whereas the truncated, supplemented Pascal matrix has multiple zeros. If the maximal-distance separable code conjecture is correct, then our matrix has the maximal number of columns (with the aformentioned property) that the conjecture allows. This matrix has applications in coding, network coding, and matroid theory.

math.CO↗