Quantitative Derivation of Effective Evolution Equations for the Dynamics of Bose-Einstein Condensates

This thesis proves certain results concerning an important question in non-equilibrium quantum statistical mechanics which is the derivation of effective evolution equations approximating the dynamics of a system of large number of bosons initially at equilibrium (ground state at very low temperatures). The dynamics of such systems are governed by the time-dependent linear many-body Schroedinger equation from which it is typically difficult to extract useful information due to the number of particles being large. We will study quantitatively (i.e. with explicit bounds on the error) how a suitable one particle non-linear Schroedinger equation arises in the mean field limit as number of particles N → ∞ and how the appropriate corrections to the mean field will provide better approximations of the exact dynamics. In the first part of this thesis we consider the evolution of N bosons, where N is large, with two-body interactions of the form N³ᵝv(Nᵝ⋅), 0≤β≤1. The parameter β measures the strength and the range of interactions. We compare the exact evolution with an approximation which considers the evolution of a mean field coupled with an appropriate description of pair excitations, see [18,19] by Grillakis-Machedon-Margetis. We extend the results for 0 ≤ β < 1/3 in [19, 20] to the case of β < 1/2 and obtain an error bound of the form p(t)/Nᵅ, where α>0 and p(t) is a polynomial, which implies a specific rate of convergence as N → ∞. In the second part, utilizing estimates of the type discussed in the first part, we compare the exact evolution with the mean field approximation in the sense of marginals. We prove that the exact evolution is close to the approximate in trace norm for times of the order o(1)√N compared to log(o(1)N) as obtained in Chen-Lee-Schlein [6] for the Hartree evolution. Estimates of similar type are obtained for stronger interactions as well.

Connecting the dots: Examining the role of parental beliefs and preschoolers’ affect and engagement in predicting parent-child number exploration during a meaningful math experience

The current study examined the frequency and quality of how 3- to 4-year-old children and their parents explore the relations between symbolic and non-symbolic quantities in the context of a playful math experience, as well as the role of both parent and child factors in this exploration. Preschool children’s numerical knowledge was assessed while parents completed a survey about the number-related experiences they share with their children at home, and their math-related beliefs. Parent-child dyads were then videotaped playing a modified version of the card game War. Results suggest that parents and children explored quantity explicitly on only half of the cards and card pairs played, and dyads of young children and those with lower number knowledge tended to be most explicit in their quantity exploration. Dyads with older children, on the other hand, often completed their turns without discussing the numbers at all, likely because they were knowledgeable enough about numbers that they could move through the game with ease. However, when dyads did explore the quantities explicitly, they focused on identifying numbers symbolically, used non-symbolic card information interchangeably with symbolic information to make the quantity comparison judgments, and in some instances, emphasized the connection between the symbolic and non-symbolic number representations on the cards. Parents reported that math experiences such as card game play and quantity comparison occurred relatively infrequently at home compared to activities geared towards more foundational practice of number, such as counting out loud and naming numbers. However, parental beliefs were important in predicting both the frequency of at-home math engagement as well as the quality of these experiences. In particular, parents’ specific beliefs about their children’s abilities and interests were associated with the frequency of home math activities, while parents’ math-related ability beliefs and values along with children’s engagement in the card game were associated with the quality of dyads’ number exploration during the card game. Taken together, these findings suggest that card games can be an engaging context for parent-preschooler exploration of numbers in multiple representations, and suggests that parents’ beliefs and children’s level of engagement are important predictors of this exploration.

Solo and Chamber Music of Pulitzer Prize-Winning Composers

The Pulitzer Prize in Music, established in 1943, is one of America's most prestigious awards. It has been awarded to fifty-three composers for a "distinguished musical composition of significant dimension by an American that has had its first performance in the United States during the year." Composers who have won the Pulitzer Prize are considered to be at the pinnacle of their creativity and have provided the musical world with classical music compositions worthy of future notice. By tracing the history of Pulitzer Prize-winning composers and their compositions, researchers and musicians enhance their understanding of the historical evolution of American music, and its impact on American culture. Although the clarinet music of some of these composers is rarely performed today, their names will be forever linked to the Pulitzer, and because of that, their compositions will enjoy a certain sense of immortality. Of the fifty-four composers who have won the award, forty-seven have written for the clarinet in a solo or chamber music setting (five or less instruments). Just as each Pulitzer Prize-winning composition is a snapshot of the state of American music at that time, these works trace the history of American clarinet musical development, and therefore, they are valuable additions to the clarinet repertoire and worthy of performance. This dissertation project consists of two recitals featuring the solo and chamber clarinet music of sixteen Pulitzer Prize-winning composers, extended program notes containing information on each composer's life, their music, the Pulitzer Prize-winning composition and the recital selection, and a complete list of all Pulitzer Prize-winning composers and their solo and chamber clarinet music. Featured Composers Dominick Argento, To Be Sung Upon the Water Leslie Bassett, Soliloquies William Bolcom, Little Suite of Four Dances Aaron Copland, As it Fell Upon a Day John Corigliano, Soliloquy Norman Dello Joio, Concertante Morton Gould, Benny's Gig Charles Ives, Largo Douglas Moore, Quintet for Clarinet and Strings George Perle, Three Sonatas Quincy Porter, Quintet for Clarinet and Strings Mel Powell, Clarinade Shulamit Ran, Private Game Joseph Schwantner, Entropy Leo Sowerby, Sonata Ernst Toch, Adagio elegiaco

Sikh Sabad Kirtan as a Musical Construction of Memory

The performance of devotional music in India has been an active, sonic conduit where spiritual identities are shaped and forged, and both history and mythology lived out and remembered daily. For the followers of Sikhism, congregational hymn singing has been the vehicle through which text, melody and ritual act as repositories of memory, elevating memory to a place where historical and social events can be reenacted and memorialized on levels of spiritual significance. This dissertation investigates the musical process of Shabad Kirtan, Sikh hymn singing, in a Sikh musical service as a powerful vehicle to forge a sense of identification between individual and the group. As an intimate part of Sikh life from birth to death, the repertoire of Shabad Kirtan draws from a rich mosaic of classical and folk genres as well as performance styles, acting as a musical and cognitive archive. Through a detailed analysis of the Asa Di Var service, Shabad Kirtan is explored as a phenomenological experience where time, place and occasion interact as a meaningful unit through which the congregation creates and recreates themselves, invoking deep memories and emotional experiences. Supported by explanatory tables, diagrams and musical transcriptions, the sonic movements of the service show how the Divine Word as Shabad is not only embodied through the Guru Granth Sahib, but also encountered through the human enactment of the service, aurally, viscerally and phenomenologically.

FOLLOWING THE TRAIL OF THE SNAKE: A LIFE HISTORY OF COBRA MANSA "COBRINHA" MESTRE OF CAPOEIRA

This dissertation is a cultural biography of Mestre Cobra Mansa, a mestre of the Afro-Brazilian martial art of capoeira angola. The intention of this work is to track Mestre Cobrinha's life history and accomplishments from his beginning as an impoverished child in Rio to becoming a mestre of the tradition-its movements, music, history, ritual and philosophy. A highly skilled performer and researcher, he has become a cultural ambassador of the tradition in Brazil and abroad. Following the Trail of the Snake is an interdisciplinary work that integrates the research methods of ethnomusicology (oral history, interview, participant observation, musical and performance analysis and transcription) with a revised life history methodology to uncover the multiple cultures that inform the life of a mestre of capoeira. A reflexive auto-ethnography of the author opens a dialog between the experiences and developmental steps of both research partners' lives. Written in the intersection of ethnomusicology, studies of capoeira, social studies and music education, the academic dissertation format is performed as a roda of capoeira aiming to be respectful of the original context of performance. The result is a provocative ethnographic narrative that includes visual texts from the performative aspects of the tradition (music and movement), aural transcriptions of Mestre Cobra Mansa's storytelling and a myriad of writing techniques to accompany the reader in a multi-dimensional journey of multicultural understanding. The study follows Cinezio Feliciano Pe anha in his childhood struggle for survival as a street performer in Rio de Janeiro. Several key moves provided him with the opportunity to rebuild his life and to grow into a recognized mestre of the capoeira angola martial art as Mestre Cobra Mansa ("Tame Snake" in Portuguese). His dedicated work enabled him to contribute to the revival of the capoeira angola tradition during the 1980's in Bahia. After his move to the United States in the early 1990's, Mestre Cobrinha founded the International Capoeira Angola Foundation, which today has expanded to 28 groups around the world. Mestre Cobra returned home to Brazil to initiate projects that seek to develop a new sense of community from all that he has learned and been able to accomplish in his life through the performance and study of capoeira angola.