A singular integral identity for surface measure

We prove that the integral of a certain Riesz-type kernel over $(n-1)$-rectifiable sets in $\mathbb{R}^n$ is constant, from which a formula for surface measure immediately follows. Geometric interpretations are given, and the solution to a geometric variational problem characterizing convex domains follows as a corollary, strengthening a recent inequality of Steinerberger.Comment: 9 pages, 3 figure

On the packing dimension of exceptional sets of projections

Let $A \subseteq \mathbb{R}^n$ be analytic. An exceptional set of projections for $A$ is a set of $k$-dimensional subspaces of $\mathbb{R}^n$ onto which the orthogonal projection of $A$ has "unexpectedly low" Hausdorff dimension. The famous projection theorems of Mattila (1975) and Falconer (1982) place upper bounds on the Hausdorff dimensions of exceptional sets, considered as subsets of the Grassmannian $\mathbf{Gr}(n,k)$. A 2015 result of Orponen bounds the packing dimension of the exceptional set in the case that $n = 2$, $k = 1$, and $A$ is self-similar or homogeneous. Our purpose is to extend Orponen's result to the case of arbitrary $0 < k < n$.Comment: 15 pages, 1 figur

False discovery rate regression: an application to neural synchrony detection in primary visual cortex

Many approaches for multiple testing begin with the assumption that all tests in a given study should be combined into a global false-discovery-rate analysis. But this may be inappropriate for many of today's large-scale screening problems, where auxiliary information about each test is often available, and where a combined analysis can lead to poorly calibrated error rates within different subsets of the experiment. To address this issue, we introduce an approach called false-discovery-rate regression that directly uses this auxiliary information to inform the outcome of each test. The method can be motivated by a two-groups model in which covariates are allowed to influence the local false discovery rate, or equivalently, the posterior probability that a given observation is a signal. This poses many subtle issues at the interface between inference and computation, and we investigate several variations of the overall approach. Simulation evidence suggests that: (1) when covariate effects are present, FDR regression improves power for a fixed false-discovery rate; and (2) when covariate effects are absent, the method is robust, in the sense that it does not lead to inflated error rates. We apply the method to neural recordings from primary visual cortex. The goal is to detect pairs of neurons that exhibit fine-time-scale interactions, in the sense that they fire together more often than expected due to chance. Our method detects roughly 50% more synchronous pairs versus a standard FDR-controlling analysis. The companion R package FDRreg implements all methods described in the paper

Phytoplankton Community and Algal Toxicity at a Recurring Bloom in Sullivan Bay, Kabetogama Lake, Minnesota, USA

Kabetogama Lake in Voyageurs National Park, Minnesota, USA suffers from recurring late summer algal blooms that often contain toxin-producing cyanobacteria. Previous research identified the toxin microcystin in blooms, but we wanted to better understand how the algal and cyanobacterial community changed throughout an open water season and how changes in community structure were related to toxin production. Therefore, we sampled one recurring bloom location throughout the entire open water season. The uniqueness of this study is the absence of urban and agricultural nutrient sources, the remote location, and the collection of samples before any visible blooms were present. Through quantitative polymerase chain reaction (qPCR), we discovered that toxin-forming cyanobacteria were present before visible blooms and toxins not previously detected in this region (anatoxin-a and saxitoxin) were present, indicating that sampling for additional toxins and sampling earlier in the season may be necessary to assess ecosystems and human health risk

D.I.Y. Clean Hood

The DIY Clean Hood is a low-cost, sterile, and accessible scientific workspace intended for installation in the BioElectroFluidics Lab of California Polytechnic State University’s (San Luis Obispo) Biomedical Engineering Department (BMED), under the sponsorship of Dr. Benjamin Hawkins, PhD. As normal Clean Hood, Biosafety Cabinets, and the like are generally too expensive for a university, a competitive solution to an expensive problem can assist research students and professors alike continue their own work with an inexpensive yet effective environment. Specific design elements that the customer requirements entailed for the project include a low-particle-count air filtration system, positive pressure air flow inside the vessel, and compatibility with common cleaning agents. These critical details for the DIY Clean Hood are to ensure that the cell cultures being cultivated and studied are free from any foreign contaminants or agents that could compromise the product. Unlike the original, mislabeled identification of the project as a “DIY Biosafety Cabinet”, it is not the responsibility of the Clean Hood to protect either the environment or the user of the DIY Clean Hood. Despite the non-hazardous conditions of the cells being manipulated, proper design components and features ensure sterility and effectiveness. Other notable design elements of the DIY Clean Hood include a 15 degree angled, swinging sash opening, an air filtration system utilizing a HEPA (High-Efficiency Particulate Air) filter, an installed UV light for an additional sterilization option, and a wide opening for comfortable mobility while using the Clean Hood. The project was a recipient of the Biomedical Engineering Department’s Hannah-Forbes Grant, which allows the DIY Clean Hood project an additional $500 towards any necessary purchases and bringing the total budget to$700. Due to the onset of the COVID-19 pandemic, the project required its focus to shift from a manufacturing and qualification testing standpoint to a more design and technically-centered frame, as several factors prevented the project from proceeding originally as planned. These included, but were not limited to, the closure of manufacturing and assembly facilities on the Cal Poly campus, anticipated delays in material acquisition due to non-essential items, social distancing of team members, and limited alternative build options. This decision was agreed upon in correspondence with project sponsor Dr. Hawkins, Engineering Design overseer Dr. Michael Whitt, and the members of the DIY Clean Hood Team. As a result, the DIY Clean Hood prepared a final, detailed design for the product to ensure all customer requirements were met in approaches the team thought would provide the best performance and usability; the BioElectroFluidics lab will find a team of their own in late 2020 to build the device using the enclosed detailed designs, and qualify the product with the DIY Clean Hood’s testing protocols

Development of explosive welding techniques for fabrication of regeneratively cooled thrust chambers for large rocket engine requirements Final report, 28 Jun. 1967 - 15 Sep. 1970

Explosive welding techniques in fabricating regeneratively cooled thrust chambers for large rocket engine requirements including ultrasonic inspection, metallography, and burst testin

Complex-Distance Potential Theory and Hyperbolic Equations

An extension of potential theory in R^n is obtained by continuing the Euclidean distance function holomorphically to C^n. The resulting Newtonian potential is generated by an extended source distribution D(z) in C^n whose restriction to R^n is the delta function. This provides a natural model for extended particles in physics. In C^n, interpreted as complex spacetime, D(z) acts as a propagator generating solutions of the wave equation from their initial values. This gives a new connection between elliptic and hyperbolic equations that does not assume analyticity of the Cauchy data. Generalized to Clifford analysis, it induces a similar connection between solutions of elliptic and hyperbolic Dirac equations. There is a natural application to the time-dependent, inhomogeneous Dirac and Maxwell equations, and the `electromagnetic wavelets' introduced previously are an example.Comment: 25 pages, submited to Proceedings of 5th Intern. Conf. on Clifford Algebras, Ixtapa, June 24 - July 4, 199

Using shared goal setting to improve access and equity: a mixed methods study of the Good Goals intervention

Background: Access and equity in children’s therapy services may be improved by directing clinicians’ use of resources toward specific goals that are important to patients. A practice-change intervention (titled ‘Good Goals’) was designed to achieve this. This study investigated uptake, adoption, and possible effects of that intervention in children’s occupational therapy services. Methods: Mixed methods case studies (n = 3 services, including 46 therapists and 558 children) were conducted. The intervention was delivered over 25 weeks through face-to-face training, team workbooks, and ‘tools for change’. Data were collected before, during, and after the intervention on a range of factors using interviews, a focus group, case note analysis, routine data, document analysis, and researchers’ observations. Results: Factors related to uptake and adoptions were: mode of intervention delivery, competing demands on therapists’ time, and leadership by service manager. Service managers and therapists reported that the intervention: helped therapists establish a shared rationale for clinical decisions; increased clarity in service provision; and improved interactions with families and schools. During the study period, therapists’ behaviours changed: identifying goals, odds ratio 2.4 (95% CI 1.5 to 3.8); agreeing goals, 3.5 (2.4 to 5.1); evaluating progress, 2.0 (1.1 to 3.5). Children’s LoT decreased by two months [95% CI −8 to +4 months] across the services. Cost per therapist trained ranged from £1,003 to £1,277, depending upon service size and therapists’ salary bands. Conclusions: Good Goals is a promising quality improvement intervention that can be delivered and adopted in practice and may have benefits. Further research is required to evaluate its: (i) impact on patient outcomes, effectiveness, cost-effectiveness, and (ii) transferability to other clinical contexts