Sixty years of fractal projections

Sixty years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For many years, the paper attracted very little attention. However, over the past 30 years, Marstrand’s projection theorems have become the prototype for many results in fractal geometry with numerous variants and applications and they continue to motivate leading research.Postprin

Filament Eruptions, Jets, and Space Weather

Previously, from chromospheric H alpha and coronal Xray movies of the Sun's polar coronal holes, it was found that nearly all coronal jets (greater than 90%) are one or the other of two roughly equally common different kinds, different in how they erupt: standard jets and blowout jets (Yamauchi et al 2004, Apl, 605, 5ll: Moore et all 2010, Apj, 720, 757). Here, from inspection of SDO/AIA He II 304 A movies of 54 polar x-ray jets observed in Hinode/XRT movies, we report, as Moore et al (2010) anticipated, that (1) most standard x-ray jets (greater than 80%) show no ejected plasma that is cool enough (T is less than or approximately 10(exp 5K) to be seen in the He II 304 A movies; (2) nearly all blownout X-ray jets (greater than 90%) show obvious ejection of such cool plasma; (3) whereas when cool plasma is ejected in standard X-ray jets, it shows no lateral expansion, the cool plasma ejected in blowout X-ray jets shows strong lateral expansion; and (4) in many blowout X-ray jets, the cool plasma ejection displays the erupting-magnetic-rope form of clasic filament eruptions and is thereby seen to be a miniature filament eruption. The XRT movies also showed most blowout X-ray jets to be larger and brighter, and hence to apparently have more energy, than most standard X-ray jets. These observations (1) confirm the dichotomy of coronal jets, (2) agree with the Shibata model for standard jets, and (3) support the conclusion of Moore et al (2010) that in blowout jets the magnetic-arch base of the jet erupts in the manner of the much larger magnetic arcades in which the core field, the field rooted along the arcade's polarity inversion line, is sheared and twisted (sigmoid), often carries a coolplasma filament, and erupts to blowout the arcade, producing a CME. From Hinode/SOT Ca II movies of the polar limb, Sterling et al (2010, ApJ, 714, L1) found that chromospheric TypeII spicules show a dichotomy of eruption dynamics similar to that found here for the coolplasma component of coronal Xray jets. This favors the idea that TypeII spicules are miniature counterparts of coronal Xray jets. In Moore et al (2011, ApJ, 731, L18), we pointed out that if TypeII spicules are magnetic eruptions that work like coronal Xray jets, they carry an areaaveraged mechanical energy flux of approximately 7x10)(exp 5) erg cm(exp -2) s(exp-1) into the corona in the form of MHD waves and jet outflow, enough to power the heating of the global corona and solar wind. On this basis, from our observations of minifilament eruptions in blowout Xray jets, we infer that magnetic explosions of the type that have erupting filaments in them are the main engines of both (1) the steady solar wind and (2) the CMEs that produce the most severe space weather by blasting out through the corona and solar wind, making solar energetic particle storms, and bashing the Earth's magnetosphere. We conclude that in focusing on prominences and filament eruptions, Einar had his eye on the main bet for understanding what powers all space weather, both the extreme and the normal

Magnetic Untwisting in Most Solar X-Ray Jets

If TypeII spicules are made like Xray jets by granulesize emerging bipoles, then they plausibly power the quiet corona and solar wind

Minkowski dimension for measures

Funding: JMF and KJF are financially supported by an EPSRC Standard Grant (EP/R015104/1) and JMF by a Leverhulme Trust Research Project Grant (RPG-2019-034).The purpose of this article is to introduce and motivate the notion of Minkowski (or box) dimension for measures. The definition is simple and fills a gap in the existing literature on the dimension theory of measures. As the terminology suggests, we show that it can be used to characterise the Minkowski dimension of a compact metric space. We also study its relationship with other concepts in dimension theory.PostprintPeer reviewe

Assouad dimension influences the box and packing dimensions of orthogonal projections

Funding: UK EPSRC Standard Grant (EP/R015104/1) (KJF and JMF). Leverhulme Trust Research Project Grant (RPG-2019-034) (JMF). Royal Society International Exchange grant IES\R1\191195 (KJF and PS). ProjectPICT 2015-3675 (ANPCyT) (PS).We present several applications of the Assouad dimension, and the related quasi-Assouad dimension and Assouad spectrum, to the box and packing dimensions of orthogonal projections of sets. For example, we show that if the (quasi-)Assouad dimension of F ⊆ R n is no greater than m, then the box and packing dimensions of F are preserved under orthogonal projections onto almost all m-dimensional subspaces. We also show that the threshold m for the (quasi-)Assouad dimension is sharp, and bound the dimension of the exceptional set of projections strictly away from the dimension of the Grassmannian.Publisher PDFPeer reviewe

Minkowski dimension for measures

We introduce a Minkowski dimension for measures and show that it can be used to characterise the Minkowski dimension of a compact metric space. We also study its relationship with other concepts in dimension theory.Comment: 1 figure; the proof of Theorem 3.3 in v1 contains a gap; we improved the main result and polished the presentatio

Lq-spectra of measures on planar non-conformal attractors

Funding: KJF and JMF were supported by an EPSRC Standard Grant (EP/R015104/1). JMF was also supported by a Leverhulme Trust Research Project Grant (RPG-2019-034). LDL was supported by an EPSRC Doctoral Training Grant.We study the Lq-spectrum of measures in the plane generated by certain nonlinear maps. In particular we consider attractors of iterated function systems consisting of maps whose components are C1+α and for which the Jacobian is a lower triangular matrix at every point subject to a natural domination condition on the entries. We calculate the Lq-spectrum of Bernoulli measures supported on such sets by using an appropriately defined analogue of the singular value function and an appropriate pressure function.PreprintPostprintPeer reviewe

The fractal structure of elliptical polynomial spirals

We investigate fractal aspects of elliptical polynomial spirals; that is, planar spirals with differing polynomial rates of decay in the two axis directions. We give a full dimensional analysis of these spirals, computing explicitly their intermediate, box-counting and Assouad-type dimensions. An exciting feature is that these spirals exhibit two phase transitions within the Assouad spectrum, the first natural class of fractals known to have this property. We go on to use this dimensional information to obtain bounds for the H\"older regularity of maps that can deform one spiral into another, generalising the 'winding problem' of when spirals are bi-Lipschitz equivalent to a line segment. A novel feature is the use of fractional Brownian motion and dimension profiles to bound the H\"older exponents.Comment: 19 pages, 7 figure

Technical development and feasibility of a reusable vest to integrate cardiovascular magnetic resonance with electrocardiographic imaging

Abstract Background Electrocardiographic imaging (ECGI) generates electrophysiological (EP) biomarkers while cardiovascular magnetic resonance (CMR) imaging provides data about myocardial structure, function and tissue substrate. Combining this information in one examination is desirable but requires an affordable, reusable, and high-throughput solution. We therefore developed the CMR-ECGI vest and carried out this technical development study to assess its feasibility and repeatability in vivo. Methods CMR was prospectively performed at 3T on participants after collecting surface potentials using the locally designed and fabricated 256-lead ECGI vest. Epicardial maps were reconstructed to generate local EP parameters such as activation time (AT), repolarization time (RT) and activation recovery intervals (ARI). 20 intra- and inter-observer and 8 scan re-scan repeatability tests. Results 77 participants were recruited: 27 young healthy volunteers (HV, 38.9 ± 8.5 years, 35% male) and 50 older persons (77.0 ± 0.1 years, 52% male). CMR-ECGI was achieved in all participants using the same reusable, washable vest without complications. Intra- and inter-observer variability was low (correlation coefficients [rs] across unipolar electrograms = 0.99 and 0.98 respectively) and scan re-scan repeatability was high (rs between 0.81 and 0.93). Compared to young HV, older persons had significantly longer RT (296.8 vs 289.3 ms, p = 0.002), ARI (249.8 vs 235.1 ms, p = 0.002) and local gradients of AT, RT and ARI (0.40 vs 0.34 ms/mm, p = 0,01; 0.92 vs 0.77 ms/mm, p = 0.03; and 1.12 vs 0.92 ms/mm, p = 0.01 respectively). Conclusion Our high-throughput CMR-ECGI solution is feasible and shows good reproducibility in younger and older participants. This new technology is now scalable for high throughput research to provide novel insights into arrhythmogenesis and potentially pave the way for more personalised risk stratification.British Heart Foundation special project grant to GC (SP/20/2/34841). Society for Cardiovascular Magnetic Resonance Seed Grant award to GC and FD (2021). Medical Research Council (MC_UU_00019/1, MC_UU_00019/2, MC_UU_00019/3).Peer ReviewedPostprint (published version