Using a Heliospheric Upwinding eXtrapolation Technique to Magnetically Connect Different Regions of the Heliosphere

Understanding how coronal structure propagates and evolves from the Sun and into the heliosphere has been thoroughly explored using sophisticated MHD models. From these, we have a reasonably good working understanding of the dynamical processes that shape the formation and evolution of stream interaction regions and rarefactions, including their locations, orientations, and structure. However, given the technical expertize required to produce, maintain, and run global MHD models, their use has been relatively restricted. In this study, we refine a simple Heliospheric eXtrapolation Technique (HUX) to include not only forward mapping from the Sun to 1 AU (or elsewhere), but backward mapping toward the Sun. We demonstrate that this technique can provide substantially more accurate mappings than the standard, and often applied “ballistic” approximation. We also use machine learning (ML) methods to explore whether the HUX approximation to the momentum equation can be refined without loss of simplicity, finding that it likely provides the optimum balance. We suggest that HUX can be used, in conjunction with coronal models (PFSS or MHD) to more accurately connect measurements made at 1 AU, Stereo-A, Parker Solar Probe, and Solar Orbiter with their solar sources. In particular, the HUX technique: 1) provides a substantial improvement over the “ballistic” approximation for connecting to the source longitude of streams; 2) is almost as accurate, but considerably easier to implement than MHD models; and 3) can be applied as a general tool to magnetically connect different regions of the inner heliosphere together, as well as providing a simple 3-D reconstruction.

Local Muckenhoupt class for variable exponents

This work extends the theory of Rychkov, who developed the theory of $A_{p}^{\mathrm{loc}}$ A p loc weights. It also extends the work by Cruz-Uribe SFO, Fiorenza, and Neugebauer. The class $A_{p(\cdot )}^{\mathrm{loc}}$ A p ( ⋅ ) loc is defined. The weighted inequality for the local Hardy–Littlewood maximal operator on Lebesgue spaces with variable exponents is proven. Cruz-Uribe SFO, Fiorenza, and Neugebauer considered the Muckenhoupt class for Lebesgue spaces with variable exponents. However, due to the setting of variable exponents, a new method for extending weights is needed. The proposed extension method differs from that by Rychkov. A passage to the vector-valued inequality is realized by means of the extrapolation technique. This technique is an adaptation of the work by Cruz-Uribe and Wang. Additionally, a theory of extrapolation adapted to our class of weights is also obtained.

Determination of Maximum Accumulated Oxygen Deficit Using Backward Extrapolation

This study aimed to compare the Maximum Accumulated Oxygen Deficit determined by the conventional method (MAODC) with that determined by the backward extrapolation technique (MAODEXTR) in runners. Fourteen runners underwent a maximal incremental test for determination of iVO2MAX, ten submaximal efforts (50–95% of iVO2MAX for 7 min). During the submaximal efforts oxygen consumption (VO2) values were obtained conventionally and through the backward extrapolation technique (~ 3 s after the end of each effort). A supramaximal effort (110% of iVO2MAX) (tLimC) and five supramaximal bouts (tLimEXTR) were performed. MAODC and MAODEXTR were determined from the difference between the VO2 accumulated during tLimC and tLimEXTR and the predicted values. The tLimC was lower than tLimEXTR (164.06±36.32 s, 200.23±63.78 s, p<0.05). No significant differences were found between absolute and relative MAODC and MAODEXTR values, however, low intraclass correlations (0.26 and 0.24), high typical errors (2.03 L and 24 mL∙kg−1) were observed, and coefficients of variation (46 and 48%), respectively. The graphical analysis of the differences showed agreement and correlation between the methods (r=0.86 and 0.85). Thus, it can be concluded that the MAODEXTR is not a valid method for estimating the anaerobic capacity of runners, moreover, unreliable.