On the Assouad dimension of self-similar sets with overlaps

It is known that, unlike the Hausdorff dimension, the Assouad dimension of a self-similar set can exceed the similarity dimension if there are overlaps in the construction. Our main result is the following precise dichotomy for self-similar sets in the line: either the \emph{weak separation property} is satisfied, in which case the Hausdorff and Assouad dimensions coincide; or the \emph{weak separation property} is not satisfied, in which case the Assouad dimension is maximal (equal to one). In the first case we prove that the self-similar set is Ahlfors regular, and in the second case we use the fact that if the \emph{weak separation property} is not satisfied, one can approximate the identity arbitrarily well in the group generated by the similarity mappings, and this allows us to build a \emph{weak tangent} that contains an interval. We also obtain results in higher dimensions and provide illustrative examples showing that the `equality/maximal' dichotomy does not extend to this setting.Comment: 24 pages, 2 figure

Critical current of a Josephson junction containing a conical magnet

We calculate the critical current of a superconductor/ferromagnetic/superconductor (S/FM/S) Josephson junction in which the FM layer has a conical magnetic structure composed of an in-plane rotating antiferromagnetic phase and an out-of-plane ferromagnetic component. In view of the realistic electronic properties and magnetic structures that can be formed when conical magnets such as Ho are grown with a polycrystalline structure in thin-film form by methods such as direct current sputtering and evaporation, we have modeled this situation in the dirty limit with a large magnetic coherence length ($\xi_f$). This means that the electron mean free path is much smaller than the normalized spiral length $\lambda/2\pi$ which in turn is much smaller than $\xi_f$ (with $\lambda$ as the length a complete spiral makes along the growth direction of the FM). In this physically reasonable limit we have employed the linearized Usadel equations: we find that the triplet correlations are short ranged and manifested in the critical current as a rapid oscillation on the scale of $\lambda/2\pi$. These rapid oscillations in the critical current are superimposed on a slower oscillation which is related to the singlet correlations. Both oscillations decay on the scale of $\xi_f$. We derive an analytical solution and also describe a computational method for obtaining the critical current as a function of the conical magnetic layer thickness.Comment: Extended version of the published paper. Additional information about the computational method is included in the appendi

Mitigating Cotton Revenue Risk Through Irrigation, Insurance, and Hedging

This study focuses on managing cotton production and marketing risks using combinations of irrigation levels, put options (as price insurance), and crop insurance. Stochastic cotton yields and prices are used to simulate a whole-farm financial statement for a 1,000 acre furrow irrigated cotton farm in the Texas Lower Rio Grande Valley under 16 combinations of risk management strategies. Analyses for risk-averse decision makers indicate that multiple irrigations are preferred. The benefits to purchasing put options increase with yields, as they are more beneficial when higher yields are expected from applying more irrigation applications. Crop insurance is strongly preferred at lower irrigation levels.cotton, crop insurance, irrigation, options, puts, risk, simulation, stochastic efficiency with respect to a function, Farm Management, Risk and Uncertainty, D81, Q12, Q15,

Exploring Intersections of Race, Gender, Culture, and Power: Collaborative Autoethnography, Freire, and Model for Reform

In a time when the Critical Race Theory is being used as a weapon to eliminate educational learning by ratifying American history at a proliferated rate, this study illustrates the individual experiences of racial inequity across race, gender, culture, and power. This collaborative autoethnography is informed by the Critical Race Theory and Freire’s Pedagogy of the Oppressed. Crenshaw observes about criticism of CRT, “It’s not about a theory. … It’s about an effort to shut down all conversation about the sources and the reproduction of racial inequality” (2021, p. 7). Defending the application of CRT in American classrooms is beyond the scope of this study. This study illustrates that promotion of equality through having honest conversations about inequality