On a discrete version of Tanaka's theorem for maximal functions

In this paper we prove a discrete version of Tanaka's Theorem \cite{Ta} for the Hardy-Littlewood maximal operator in dimension $n=1$, both in the non-centered and centered cases. For the discrete non-centered maximal operator $\widetilde{M}$ we prove that, given a function $f: \mathbb{Z} \to \mathbb{R}$ of bounded variation, $$\textrm{Var}(\widetilde{M} f) \leq \textrm{Var}(f),$$ where $\textrm{Var}(f)$ represents the total variation of $f$. For the discrete centered maximal operator $M$ we prove that, given a function $f: \mathbb{Z} \to \mathbb{R}$ such that $f \in \ell^1(\mathbb{Z})$, $$\textrm{Var}(Mf) \leq C \|f\|_{\ell^1(\mathbb{Z})}.$$ This provides a positive solution to a question of Haj{\l}asz and Onninen \cite{HO} in the discrete one-dimensional case.Comment: V4 - Proof of Lemma 3 update

Boundary layer thickness effect on boattail drag

A combined experimental and analytical program was conducted to investigate the effects of boundary layer changes on the flow over high angle boattail nozzles. The tests were run on an isolated axisymmetric sting mounted model. Various boattail geometries were investigated at high subsonic speeds over a range of boundary layer thicknesses. In general, boundary layer effects were small at speeds up to Mach 0.8. However, at higher speeds significant regions of separated flow were present on the boattail. When separation was present large reductions in boattail drag resulted with increasing boundary layer thickness. The analysis predicts both of these trends

Design-thinking, making, and innovating: Fresh tools for the physician\u27s toolbox

Medical school education should foster creativity by enabling students to become \u27makers\u27 who prototype and design. Healthcare professionals and students experience pain points on a daily basis, but are not given the tools, training, or opportunity to help solve them in new, potentially better ways. The student physician of the future will learn these skills through collaborative workshops and having dedicated \u27innovation time.\u27 This pre-clinical curriculum would incorporate skills centered on (1) Digital Technology and Small Electronics (DTSE), (2) Textiles and Medical Materials (TMM), and (3) Rapid Prototyping Technologies (RPT). Complemented by an on-campus makerspace, students will be able to prototype and iterate on their ideas in a fun and accessible space. Designing and making among and between patients and healthcare professionals would change the current dynamic of medical education, empowering students to solve problems in healthcare even at an early stage in their career. By doing so, they will gain empathy, problem-solving abilities, and communication skills that will extend into clinical practice. Our proposed curriculum will equip medical students with the skills, passion, and curiosity to impact the future of healthcare

Numerical calculation of transonic boattail flow

A viscid-inviscid interaction procedure for the calculation of subsonic and transonic flow over a boattail was developed. This method couples a finite-difference inviscid analysis with an integral boundary-layer technique. Results indicate that the effect of the boundary layer is as important as an accurate inviscid method for this type of flow. Theoretical results from the solution of the full transonic-potential equation, including boundary layer effects, agree well with the experimental pressure distribution for a boattail. Use of the small disturbance transonic potential equation yielded results that did not agree well with the experimental results even when boundary-layer effects were included in the calculations

Pronounced grain boundary network evolution in nanocrystalline Cu subjected to large cyclic strains

The grain boundary network of nanocrystalline Cu foils was modified by the systematic application of cyclic loadings and elevated temperatures having a range of magnitudes. Most broadly, the changes to the boundary network were directly correlated to the applied temperature and accumulated strain, including a 300% increase in the twin length fraction. By independently varying each treatment variable, a matrix of grain boundary statistics was built to check the plausibility of hypothesized mechanisms against their expected temperature and stress/strain dependences. These comparisons allow the field of candidate mechanisms to be significantly narrowed. Most importantly, the effect of temperature and strain on twin length fraction were found to be strongly synergistic, with the combined effect being ~150% that of the summed individual contributions. Looking beyond scalar metrics, an analysis of the grain boundary network showed that twin related domain formation favored larger sizes and repeated twin variant selection over the creation of many small domains with diverse variants. Taken together, the evidence indicates that shear-coupled boundary migration twinning is the most likely explanation for grain boundary engineering in nanocrystalline Cu.Comment: 9 figure

A conjectural extension of Hecke’s converse theorem

We formulate a precise conjecture that, if true, extends the converse theorem of Hecke without requiring hypotheses on twists by Dirichlet characters or an Euler product. The main idea is to linearize the Euler product, replacing it by twists by Ramanujan sums. We provide evidence for the conjecture, including proofs of some special cases and under various additional hypotheses