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Let $A \subseteq \Z_n$ with $|A| = k$ for some $k \in \Z^+$. We consider the metric space $(\Z_n,\delta)$ in which $\delta$ is the distance metric on $\Z_n$ defined as follows: for every $x,y \in \Z_n$, $\delta(x,y) = |x-y|_n$ where $|z|_n = \min(z,n-z)$ for $z \in \{0,\ldots,n-1\}$. We say that $A$ is \emph{Erd\H{o}s-deep} if, for every $i \in \{1,2,\dots,k-1\}$, there is a positive number $d_i$ satisfying $$|\{\{x,y\} \subseteq A: \delta(x,y)=d_i\}|=i.$$ Erd\H{o}s-deep sets in $\Z_n$ have been previously classified as translates of: $\{0,1,2,4\}$ when $n = 6$; and, modular arithmetic progressions $\{0,g,2g,\cdots,(k-1)g\} \subseteq \Z_n$ for some generator $g$ and size $k$. Erd\H{o}s-deep sets have primarily been considered in metric spaces $(\Z_n,\delta)$ and $(\R^d,\norm{\cdot})$ for $d = 2$, but some exploration for $d > 2$ has been done as well. We introduce the notion of an \emph{Erd\H{o}s-deep family}. Let $\mathcal{F}=\{A_1,A_2,\dots,A_s\}$, where $A_1,\ldots, A_s \subseteq \Z_n$. Then we say $\mathcal{F}$ is Erd\H{o}s-deep if for some $k \in \Z^+$, for every $i \in \{1,2,\dots,k-1\}$ there is exactly one positive number $d_i$ satisfying $$\sum_{j=1}^s |\{\{x,y\} \subseteq A_j: \delta(x,y)=d_i\}|=i,$$ and no such $d_i$ for any $i \ge k$. We provide a complete existence theorem for Erd\H{o}s-deep pairs of arithmetic progressions $A_1,A_2 \subseteq \Z_n$ and also give a conjectured classification for Erd\H{o}s-deep families of three arithmetic progressions. Using an identity on triangular numbers, we show a general construction for larger families whose size $s$ is the square of an integer. This construction suggests the existence of Erd\H{o}s-deep families often relies on such number-theoretic identities. We define an extremal case of the Erd\H{o}s-deep family in $(\Z_n,\delta)$ in which both the distances and multiplicities are in $\{1,\ldots,k-1\}$; such families are called Winograd families. We conjecture that Winograd families of arithmetic progressions do not exist in the metric space $(\Z,|\cdot|)$. Erd\H{o}s-deep sets in $(\Z_n,\delta)$ correspond to a class of interesting musical rhythms. We conclude this work with a variety of musical demonstrations and original compositions using Erd\H{o}s-deep rhythm families as a creative constraint in composing multi-voiced rhythms. / Graduate

The Jazz Community Performance Center explores the intersection between music and architecture. The thesis project poses the question, "How can music form architecture?" The project examines the connection of music and architecture through rhythm, harmony, melody, and form and how these elements can inform and inspire the design of the jazz community performance center. The project also explores how music's emotional and experiential qualities can be translated into a built form. The Jazz Community Performance Center location is adjacent to Union Market in Washington, DC, Ward 5, to complement and extend the vibrancy of Union Market's culture through music. The Union Market fosters a sense of belonging and connection by bringing people together from different races, genders, cultures, and backgrounds. The Jazz Community Perfromance Center will be an extension of the Union Market's purpose of uniting visitors from near and far, but people will gather through jazz music this time. The center will offer diverse jazz music performances, from traditional and classic jazz to more modern and experimental styles. The target demographics for the Jazz Community Performance Center will be music enthusiasts, cultural tourists, students, and deaf communities from Gallaudet University adjacent to the project's location. Its primary goal is to create a Jazz Community Performance Center that serves as a center for jazz education, performance, and culture. The center's mission is to give aspiring and professional musicians a space to perform and exhibit their passion for jazz music. The Jazz Community Performance Center introduces informal and formal indoor jazz performances, practice rooms, a vibration experiential gallery to invite the deaf communities of Gallaudet University, and an outdoor performance space in the plaza's center to grab and connect the community. / Master of Architecture / The Jazz Community Performance Center focuses on designing a center that is inclusive of the community of the union market, which ranges from music enthusiasts, students, cultural tourists, and deaf communities from Gallaudet University adjacent to the project's location. The center will be a hub for jazz culture and education, offering opportunities for aspiring musicians to hone their craft and provide musical performances for audiences. The center's design incorporates acoustic considerations that optimize the audience's listening experience and offer an inviting environment for musicians to perform. The Jazz Community Performance Center includes an informal jazz concert, a formal jazz performance auditorium, an experiential vibration gallery, and an outdoor performance space. The center also consists of a recording studio, practice rooms, a jazz gallery, jazz therapy spaces, and a library with an extensive collection of jazz resources. The project thoroughly examines the history and evolution of jazz music. Moreover, its distinct style and impact on the music industry. Jazz has a rich and diverse history that has evolved over time, which the center seeks to celebrate by hosting performances that showcase various jazz styles and cultures. In addition, the Jazz Performance Center will be an asset to the community, providing inspiration and education for future generations of jazz fans.

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Circadian rhythms are nearly ubiquitous and are responsible for timing biological processes and allowing for anticipation of regular changes in the environment. The internal clocks of most organisms have a period very close to 24 hours with little variation. Spiders, however, do not seem to follow this pattern. Both the fastest (18 hours) and slowest (29 hours) naturally-occurring clocks are found in spiders, and variation within a species can be orders of magnitude larger than that of previously studied animals. Circadian rhythms are assumed to be adaptive, yet little is known about their heritability in arthropods. Heritability is defined as the amount of phenotypic variation that can be attributed to genetic variation passed down from parent to offspring. Phenotype can be influenced by many complex factors including environmental effects, dominance of genetic sequences, and gene interactions. Because of these influences, the phenotypic characteristics of an individual can vary greatly, and it is often difficult to precisely identify what is truly heritable. Using spiders as a model system, we can exploit the extreme variation in circadian rhythms to investigate the potential contribution from heritability. Strong heritability would suggest that wide variation in circadian rhythms likely reflects high genetic variability in the species. Alternatively, the environment may have a greater contribution in this variation relative to the effects of heritability. To test this, we chose Parasteatoda tepidariorum, a common cobweb spider with a relatively short circadian period of 21.7 hours and intraspecific variation of more than 4 hours. To estimate the heritability of circadian rhythm, adult females were gathered with accompanying egg cases, and juveniles were raised from those cases. Six fundamental parameters of circadian rhythms were measured from the locomotor activity of adults and juveniles. Of those six, only one parameter differed between adults and juveniles: the onset of locomotor activity during the first five days when light cycles were present (Mann-Whitney U= 1814, p= 0.04). When all six circadian parameters were compared by regression of adults to respective offspring, none showed significant correlation. This indicates that variation in circadian rhythms was likely not caused by parental genetics, and that environmental factors, such as artificial light at night, may be the source of the extreme circadian rhythms seen in spiders. Another possible cause for this variation may be the presence of weak molecular circadian oscillators that are more sensitive to environmental factors than those in most other circadian systems.

Circadian rhythms are outputs of the internal clock that regulates the daily functions of almost all living organisms. Circadian rhythms are typically 24 hours because they are synchronized by external cues such as the natural light/dark cycles of the environment. When external cues are removed, the circadian rhythm “free-runs,” thus revealing the organism’s endogenous circadian period. Recent studies have found that the trashline orbweaving spiders Cyclosa turbinata and Allocyclosa bifurca have abnormally short circadian rhythms of approximately 19 and 18 hours, respectively. Trashline orbweavers construct a line of debris made of prey carcasses in the center of their web and then remain undetectable by being cryptic within their trashline. Despite similar circadian rhythms and web-building behaviors, recent genetic findings indicate that these species actually are not closely related. In fact, both genetic and morphological data now suggest A. bifurca is more closely related to Cyrtophora citricola, the Tropical Tent-web spider. This would suggest that trashline behavior and exceptionally short circadian clocks evolved independently in C. turbinataand A. bifurca. This study analyzed the circadian rhythm of C. citricola and compared it to the circadian rhythm of A. bifurca. If C. citricola has an abnormally short clock like A. bifurca, this would indicate that the evolution of the short clock preceded the divergence of these species’ lineages. However, if C. citricola has a more typical clock, this would suggest that the unusually short clock evolved in the A. bifurca lineage and may be more ecologically linked to the trashline behavior. Thirty-two female C. citricola were collected in Southern Florida and had their locomotor activity measured over four days of 12:12 light/dark cycles followed by complete darkness to determine their circadian free-running periods (FRP). Cyrtophora citricola was found to have a more typical FRP of 24.0 + 0.43 hours. Despite being closely related, C. citricola and A. bifurca differ significantly in their circadian rhythms, suggesting that short circadian rhythms may be ecologically linked with trashline behavior.

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