On the Approximation of the Quantum Gates using Lattices

A central question in Quantum Computing is how matrices in $SU(2)$ can be approximated by products over a small set of "generators". A topology will be defined on $SU(2)$ so as to introduce the notion of a covering exponent \cite{letter}, which compares the length of products required to covering $SU(2)$ with $\varepsilon$ balls against the Haar measure of $\varepsilon$ balls. An efficient universal set over $PSU(2)$ will be constructed using the Pauli matrices, using the metric of the covering exponent. Then, the relationship between $SU(2)$ and $S^3$ will be manipulated to correlate angles between points on $S^3$ and give a conjecture on the maximum of angles between points on a lattice. It will be shown how this conjecture can be used to compute the covering exponent, and how it can be generalized to universal sets in $SU(2)$.Comment: This is an updated version of arxiv.org:1506.0578

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.Comment: Revised 21 July, 2017. Revised 12 January, 2018. To appear in International Journal of Mathematics and Mathematical Science

On the Whitney distortion extension problem for Cm(Rn)C^m(\mathbb R^n) and C∞(Rn)C^{\infty}(\mathbb R^n) and its applications to interpolation and alignment of data in Rn\mathbb R^n

Let $n,m\geq 1$, $U\subset\mathbb R^n$ open. In this paper we provide a sharp solution to the following Whitney distortion extension problems: (a) Let $\phi:U\to \mathbb R^n$ be a $C^m$ map. If $E\subset U$ is compact (with some geometry) and the restriction of $\phi$ to $E$ is an almost isometry with small distortion, how to decide when there exists a $C^m(\mathbb R^n)$ one-to-one and onto almost isometry $\Phi:\mathbb R^n\to \mathbb R^n$ with small distortion which agrees with $\phi$ in a neighborhood of $E$ and a Euclidean motion $A:\mathbb R^n\to \mathbb R^n$ away from $E$. (b) Let $\phi:U\to \mathbb R^n$ be $C^{\infty}$ map. If $E\subset U$ is compact (with some geometry) and the restriction of $\phi$ to $E$ is an almost isometry with small distortion, how to decide when there exists a $C^{\infty}(\mathbb R^n)$ one-to-one and onto almost isometry $\Phi:\mathbb R^n\to \mathbb R^n$ with small distortion which agrees with $\phi$ in a neighborhood of $E$ and a Euclidean motion $A:\mathbb R^n\to \mathbb R^n$ away from $E$. Our results complement those of [14,15,20] where there, $E$ is a finite set. In this case, the problem above is also a problem of interpolation and alignment of data in $\mathbb R^n$.Comment: This is part three of four papers with C. Fefferman (arXiv:1411.2451, arXiv:1411.2468, involve-v5-n2-p03-s.pdf) dealing with the problem of Whitney type extensions of $\delta>0$ distortions from certain compact sets $E\subset \Bbb R^n$ to $\varepsilon>0$ distorted diffeomorphisms on $\Bbb R^n

On Whitney extensions, Whitney extensions of small distortions and Laguerre polynomials

The Whitney extension problem asks the following: Let $\phi:E\to \mathbb R$ be a map defined on an arbitrary set $E\subset \mathbb R^d, d\geq 2$. How to decide whether $\phi$ extends to a map $\Phi:\mathbb R^d\to \mathbb R$ which agrees with $\phi$ on $E$ and is in $C^m(\mathbb R^d),\, m\geq 1$, the space of functions from $\mathbb R^d$ to $\mathbb R$ whose derivatives of order $m$ are continuous and bounded. In this paper, we present some of the work in our monograph [D] related to Whitney extensions of small distortions from $\mathbb R^d\to \mathbb R^d$. An application to alignment problems of data in $\mathbb R^d$ is given. Whitney's extension theorem, as studied by Hassler Whitney [W],is a partial converse to Taylor's theorem. We explain this and provide a relation of Whitney extensions to certain Laguerre polynomial orthonormal expansions taken from [JP].Comment: arXiv admin note: text overlap with arXiv:2103.0974

An Analytic and Probabilistic Approach to the Problem of Matroid Representibility

We introduce various quantities that can be defined for an arbitrary matroid, and show that certain conditions on these quantities imply that a matroid is not representable over $\mathbb{F}_q$. Mostly, for a 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 $q$ a prime power, 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

A Bounded mean oscillation (BMO) theorem for small distorted diffeomorphisms from RD\mathbb R^D to RD\mathbb R^D and PDE

This announcement considers the following problem. We produce a bounded mean oscillation theorem for small distorted diffeomorphisms from $\mathbb R^D$ to $\mathbb R^D$. A revision of this announcement is in the memoir preprint: arXiv:2103.09748, [1], submitted for consideration for publication.Comment: This paper appears as arXiv:1610.08138 which was submitted as a new work by accident. Thus withdrawal is appropriat

Supporting Student System Modelling Practice Through Curriculum and Technology Design

Developing and using models to make sense of phenomena or to design solutions to problems is a key science and engineering practice. Classroom use of technology-based tools can promote the development of students’ modelling practice, systems thinking, and causal reasoning by providing opportunities to develop and use models to explore phenomena. In previous work, we presented four aspects of system modelling that emerged during our development and initial testing of an online system modelling tool. In this study, we provide an in-depth examination and detailed evidence of 10th grade students engaging in those four aspects during a classroom enactment of a system modelling unit. We look at the choices students made when constructing their models, whether they described evidence and reasoning for those choices, and whether they described the behavior of their models in connection with model usefulness in explaining and making predictions about the phenomena of interest. We conclude with a set of recommendations for designing curricular materials that leverage digital tools to facilitate the iterative constructing, using, evaluating, and revising of models