Harmonic equiangular tight frames comprised of regular simplices

An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces that are spanned by the ETF's regular simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of optimal packing in a Grassmannian space. We shall see that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix. Next, we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set. It turns out that these relative difference sets lend themselves to a second, related and yet distinct, construction of complex circulant conference matrices. Finally, we provide explicit infinite families of ETFs to which this theory applies

Rural firms, farms and the local economy - a focus on small and medium-sized towns

Small and medium-sized towns have traditionally formed an integral part of the agricultural sector and wider rural economy, acting as a source of farm inputs, a first destination of farm outputs and as a source of consumer goods and services to farm households. In recent years, this relationship has been substantially eroded through processes socio-economic restructuring, including the transformation of agriculture and a decline in other primary industries. Further, a number of endogenous and exogenous drivers have resulted in the uneven development of rural economies throughout Europe, leading not only to disparities but also to decline of small and medium sized towns as thriving economic and service centres. As a result, these settlements have received increasing attention from policy makers aiming to both maintain the traditional socio-economic fabric of rural areas, and to stimulate rural development through territorial, as opposed to sectoral – and namely agricultural – approaches. This paper considers these two issues through an analysis of local economic linkages in and around small and medium-sized towns. Using primary data collected in a study of thirty towns across five European countries, the paper examines the degree to which local firms and farms are integrated into the local economies of such towns relative to other sectors, and identifies the organisational characteristics associated with strong and weak local integration. The implications of the findings are discussed in the context of evolving European rural development policy.

Harmonic Equiangular Tight Frames Comprised of Regular Simplices

An equiangular tight frame (ETF) is a sequence of equal-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs, which are formed by restricting the characters of a finite abelian group to a difference set. Recently, it was shown that some harmonic ETFs are themselves comprised of regular simplices. In this thesis, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces spanned by the ETF\u27s regular simplices form an equiisoclinic tight fusion frame, which is a type of optimal packing in a Grassmannian space. It turns out that such ETFs yield complex circulant conference matrices; this is remarkable since real examples of such matrices are known to not exist. We further show that some of these ETFs yield mutually unbiased simplices, which are a natural generalization of the quantum-information-theoretic concept of mutually unbiased bases. Finally, we provide infinite families of ETFs that have all of these properties

Harmonic Equiangular Tight Frames Comprised of Regular Simplices

An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces that are spanned by the ETF\u27s regular simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of optimal packing in a Grassmannian space. We shall see that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix. Next, we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set. It turns out that these relative difference sets lend themselves to a second, related and yet distinct, construction of complex circulant conference matrices. Finally, we provide explicit infinite families of ETFs to which this theory applies

Harmonic Equiangular Tight Frames Comprised of Regular Simplices

An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces that are spanned by the ETF\u27s regular simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of optimal packing in a Grassmannian space. We shall see that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix. Next, we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set. It turns out that these relative difference sets lend themselves to a second, related and yet distinct, construction of complex circulant conference matrices. Finally, we provide explicit infinite families of ETFs to which this theory applies

Harmonic Equiangular Tight Frames Comprised of Regular Simplices

An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces that are spanned by the ETF\u27s regular simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of optimal packing in a Grassmannian space. We shall see that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix. Next, we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set. It turns out that these relative difference sets lend themselves to a second, related and yet distinct, construction of complex circulant conference matrices. Finally, we provide explicit infinite families of ETFs to which this theory applies

Probing Λ\LambdaCDM cosmology with the Evolutionary Map of the Universe survey

The Evolutionary Map of the Universe (EMU) is an all-sky survey in radio-continuum which uses the Australian SKA Pathfinder (ASKAP). Using galaxy angular power spectrum and the integrated Sachs-Wolfe effect, we study the potential of EMU to constrain models beyond $\Lambda$CDM (i.e., local primordial non-Gaussianity, dynamical dark energy, spatial curvature and deviations from general relativity), for different design sensitivities. We also include a multi-tracer analysis, distinguishing between star-forming galaxies and galaxies with an active galactic nucleus, to further improve EMU's potential. We find that EMU could measure the dark energy equation of state parameters around 35\% more precisely than existing constraints, and that the constraints on $f_{\rm NL}$ and modified gravity parameters will improve up to a factor $\sim2$ with respect to Planck and redshift space distortions measurements. With this work we demonstrate the promising potential of EMU to contribute to our understanding of the Universe.Comment: 15 pages (29 with references and appendices), 6 figures and 10 tables. Matches the published version. Minimal changes from previous versio

Carbon nanotube-based nanorelays for low-power circuit applications

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 83-86).The objective of this research is to reduce static power dissipation by developing a vertically-oriented carbon nanotube-based nanoelectromechanical switch that has no off-state leakage current. This switch, called a nanorelay, is a mechanical switch that uses a carbon nanotube as the active component. The device consists of a line of carbon nanotubes grown on a highly-doped silicon substrate between two contacts that are electrically isolated from the substrate by an insulator. The nanorelay is actuated when a control voltage is applied between the substrate and either one of the contacts. This voltage causes the nanotube to be pulled into and eventually make physical contact with one of the contacts, which allows current to flow through the carbon nanotube. During the off state, a physical gap separates the nanotube from the contact which acts as a near-ideal tunneling barrier to virtually eliminate leakage currents. Since the nanorelay has almost no static power dissipation, it has many potential applications in low-power circuit design. This thesis makes three main contributions. First, a fabrication process to construct nanorelays is presented. Second, potential low-power circuit applications of the nanorelay are explored and implemented in a CMOS test chip. Finally, a test system is developed in order to characterize and quantify the static power savings benefits of using the nanorelay for low-power circuit applications.by Courtney E. Schmitt.S.M

Probing Λ\LambdaCDM cosmology with the Evolutionary Map of the Universe survey

The Evolutionary Map of the Universe (EMU) is an all-sky survey in radio-continuum which uses the Australian SKA Pathfinder (ASKAP). Using galaxy angular power spectrum and the integrated Sachs-Wolfe effect, we study the potential of EMU to constrain models beyond $\Lambda$CDM (i.e., local primordial non-Gaussianity, dynamical dark energy, spatial curvature and deviations from general relativity), for different design sensitivities. We also include a multi-tracer analysis, distinguishing between star-forming galaxies and galaxies with an active galactic nucleus, to further improve EMU's potential. We find that EMU could measure the dark energy equation of state parameters around 35\% more precisely than existing constraints, and that the constraints on $f_{\rm NL}$ and modified gravity parameters will improve up to a factor $\sim2$ with respect to Planck and redshift space distortions measurements. With this work we demonstrate the promising potential of EMU to contribute to our understanding of the Universe.Comment: 15 pages (29 with references and appendices), 6 figures and 10 tables. Matches the published version. Minimal changes from previous versio