A Kakeya maximal function estimate in four dimensions using planebrushes

We obtain an improved Kakeya maximal function estimate in $\mathbb{R}^4$ using a new geometric argument called the planebrush. A planebrush is a higher dimensional analogue of Wolff's hairbrush, which gives effective control on the size of Besicovitch sets when the lines through a typical point concentrate into a plane. When Besicovitch sets do not have this property, the existing trilinear estimates of Guth-Zahl can be used to bound the size of a Besicovitch set. In particular, we establish a maximal function estimate in $\mathbb{R}^4$ at dimension $3.059$. As a consequence, every Besicovitch set in $\mathbb{R}^4$ must have Hausdorff dimension at least $3.059$.Comment: 40 pages 2 figures. v2: revised based on referee's comments. In v1, the Nikishin-Pisier-Stein factorization theorem was stated (and used) incorrectly. This version corrects the problem by introducing several new arguments. The new argument leads to a Kakeya maximal function estimate at dimension 3.059, which is slightly worse than the previously claimed exponent 3.085

An improved bound on the Hausdorff dimension of Besicovitch sets in R3\mathbb{R}^3

We prove that any Besicovitch set in $\mathbb{R}^3$ must have Hausdorff dimension at least $5/2+\epsilon_0$ for some small constant $\epsilon_0>0$. This follows from a more general result about the volume of unions of tubes that satisfy the Wolff axioms. Our proof grapples with a new "almost counter example" to the Kakeya conjecture, which we call the $SL_2$ example; this object resembles a Besicovitch set that has Minkowski dimension 3 but Hausdorff dimension $5/2$. We believe this example may be an interesting object for future study.Comment: 65 pages, 11 figures. v3: Incorporates referee suggestion

Teaching Top-Down Approaches to the Engineering Design Process

One of the main focuses of metacognition is to have students think about their learning. By bringing the engineering design process with educational psychology together, it can lead to a society that thinks more in-depth on various topics. The goal of the internship was to utilize and teach top-down educational and instructional objectives. Throughout the project, we followed a modified progression of the System Architecture Methodology to help other interns think in a top-down fashion. The purpose of this procedure is to create and induce new innovative and creative ideas for different projects. As teaching this methodology was tasked, research and asking questions were skills that developed. In line with the top-down approaches to teaching the curriculum to interns, revising the written curriculum from last year was pursued. Following the top-down methodology, each step in that curriculum turned into a lesson plan. Using the revised Bloom\u27s Taxonomy, writing the objectives will follow cognitive processes as well as a knowledge basis. As students progress their thought processes to create and reach metacognition, they contain deeper learning and higher performances. In conclusion, teaching and learning this methodology can improve students\u27 abilities to think throughout their life

On the discretized sum-product problem

We give a new proof of the discretized ring theorem for sets of real numbers. As a special case, we show that if $A\subset\mathbb{R}$ is a $(\delta,1/2)_1$-set in the sense of Katz and Tao, then either $A+A$ or $A.A$ must have measure at least $|A|^{1-\frac{1}{68}}$Comment: 13 pages, 0 figures. v3: final version, to appear in IMR

Eolian Sand Deposits in Maine

Eolian Sand Deposits in Maine by Joshua Katz - Timson, Schepps & Peters, Inc., Hallowell, Maine Planning Report No.91, August 1990. A report prepared for the Maine Critical Areas Program, State Planning Office. Contents: List of Figures / List of Maps and Tables / Acknowledgements / Introduction / Inland Eolian Sand Deposits / Research Methods / Rating Methods / Significant Criteria / Summary of Results / Significant Eolian Deposits in Maine / Site Descriptions / Recommendations / Glossary / References Citedhttps://digitalcommons.usm.maine.edu/me_collection/1055/thumbnail.jp

Confronting Forfeiture

Phil Parhamovich was pulled over on I-80 in Wyoming. He had changed lanes improperly, and his seat belt was not fastened. Seven hundred seventy-five dollars would have been a reasonable fine. Instead, Parhamovich nearly lost $91,800. The money represented his life savings; he was on his way to Wisconsin, where he planned to buy a music studio. During the traffic stop, though, under intensive questioning, he lied. When police suggested that the money was indicative of drug crimes and led Parhamovich to believe, incorrectly, that simply carrying so much cash was illegal, he claimed it belonged to a friend. The money was seized under suspicion that it—not Parhamovich—had been involved in a drug crime. Since Parhamovich had denied the money was his, he could not claim it. The fictional friend, of course, also could not claim the money. As a result, the State of Wyoming maintained that the money was abandoned. The State maintained this, in part, because of Parhamovich’s statement that the money was not his. This statement is hearsay, but would be admissible, at least prima facie, as an opposing party statement. At first glance, the Crawford doctrine might seem to supersede the opposing party statement exclusion. This doctrine prohibits the admission of testimonial hearsay against criminal defendants. However, the Crawford doctrine applies only in criminal cases, and civil asset forfeiture proceedings are not considered criminal. This Note argues that the historical sources cited in Crawford and later cases suggest that the concerns regarding testimonial hearsay also apply in certain civil contexts, and that Crawford protections should be extended to citizens, like Phil Parhamovich, facing asset forfeiture. Civil asset forfeiture is a prime candidate: it shares important characteristics of criminal proceedings and imposes comparable penalties. Even if the Sixth Amendment is inapplicable in such proceedings, the logic of Crawford suggests that, as a policy matter, confrontation should be available in civil asset forfeiture proceedings. This could be done through legislation or through amendment of the Federal Rules of Evidence (FREs)

Oral tradition in linguistics

Historical linguists and Indo-Europeanists are not known for their attention to Oral Tradition. Nevertheless, oral tradition--with lowercase letters--plays a critical role in linguistic scholarship, one so basic, indeed, that it is rarely acknowledged as such. In order to understand how this is so, let us examine the two words "oral" and "tradition."Not

From Protest Song to Social Song: Music and Politics in Colombia, 1966-2016

Based on archival and ethnographic fieldwork in the cities of Bogotá and Medellín, this dissertation documents the development of canción protesta (protest song) in Colombia in the 1960s, and tracks its evolution in subsequent decades. At the turn of the 1970s, songwriters affiliated with a grassroots canción protesta movement in Bogotá used music as a vehicle for disseminating leftist political ideology and extolling the revolutionary guerrilla groups that had formed in the Colombian countryside in the preceding years. However, canción protesta was an urban phenomenon that emerged in tandem with other countercultural currents, and their confluence in the late 1960s facilitated the rise of a commercial variant of protest song in the 1970s, the reception of which was politically mixed. By the 1980s, many activist-musicians were breaking away from what they viewed as the crudely propagandist song texts of the prior decade. To make sense of protest song’s shifting guises, I situate them in broader discourses of and about resistance, emphasizing the ways in which the resistant dimensions of oppositional music have been discursively articulated in changing political contexts. During the 1990s, the category of canción social (social song) began to replace the term canción protesta in public discourse. While canción social generally denotes the same music that was formerly labelled as canción protesta, it embraces a wider range of artists and carries different associations. One of the central arguments of the dissertation is that the terminological shift from canción protesta to canción social represents a profound transformation, over the course of five decades of civil conflict, in Colombian society’s relationship with the idea of pursuing political change through armed resistance

An improved bound on the Hausdorff dimension of Besicovitch sets in ℝ^3

We prove that every Besicovitch set in ℝ^3 must have Hausdorff dimension at least 5/2 + ϵ_0 for some small constant ϵ_0 > 0. This follows from a more general result about the volume of unions of tubes that satisfies the Wolff axioms. Our proof grapples with a new “almost counterexample” to the Kakeya conjecture, which we call the SL_2 example; this object resembles a Besicovitch set that has Minkowski dimension 3 but Hausdorff dimension 5/2. We believe this example may be an interesting object for future study