After a review of the notions of Hausdorff and Fourier dimensions from fractal geometry and Fourier analysis and the properties of local times of Brownian motion, we study the Fourier structure of Brownian level sets. We show that if δa(X) is the Dirac measure of one-dimensional Brownian motion X at the level a, that is the measure defined by the Brownian local time La at level a, and μ is its restriction to the random interval [0, L−1 a (1)], then the Fourier transform of μ is such that, with positive probability, for all 0 ≤ β < 1/2, the function u → |u|β|μ(u)|2, (u ∈ R), is bounded. This growth rate is the best possible. Consequently, each Brownian level set, reduced to a compact interval, is with positive probability, a Salem set of dimension 1/2. We also show that the zero set of X reduced to the interval [0, L−1 0 (1)] is, almost surely, a Salem set. Finally, we show that the restriction μ of δ0(X) to the deterministic interval [0, 1] is such that its Fourier transform satisfies E (|ˆμ(u)|2) ≤ C|u|−1/2, u 6= 0 and C > 0. Key words: Hausdorff dimension, Fourier dimension, Salem sets, Brownian motion, local times, level sets, Fourier transform, inverse local times. / Decision Sciences / PhD. (Operations Research)

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ABSTRACT Although Cotton Mather, as the official chronicler of the 1692 Salem witch trials, is infamously associated with those events, and excerpts from his apologia on Salem, Wonders of the Invisible World, are widely anthologized today, no annotated critical edition of the entire work has appeared in print since the nineteenth century. This present edition of Wonders seeks to remedy this lacuna in modern scholarship. In Wonders, Mather applies both his views on witchcraft and on millennialism to events at Salem. This edition to Mather's Wonders presents this seventeenth-century text beside an integrated theory of the initial causes of the Salem witch panic. The juxtaposition of the probable natural causes of Salem's bewitchment with Mather's implausible explanations exposes the disingenuousness of his writing about Salem. My theory of what happened at Salem includes the probability that a group of conspirators led by the Rev. Samuel Parris deliberately orchestrated the "witchcraft" and that a plant, the thorn apple, used in Algonquian initiation rites, caused the initial symptoms of bewitchment (39-189). Furthermore, key spectral evidence used at the Salem witch trials and recorded by Mather in Wonders appears to have been generated by intense nightmares, commonly thought at the time to be witch visitations, resulting from what is today termed sleep paralysis (215-310). This dissertation provides a detailed look at some of the testimony given in the Salem court records and in Wonders of the Invisible World as it relates to the interpretation in folklore of the phenomenology of nightmares associated with sleep paralysis. The third chapter of this dissertation focuses extensively on Mather's text as a disingenuous response to the Salem witch trials (320-456). The final section of chapter three posits a "Scythian" or Eurasian connection between Swedish and Salem witchcraft. Similarities in shamanic practices among respective indigenous populations of Lapland, Eurasia, Asia, and New England, caused the devil's involvement in both the visible and invisible worlds to appear more than theoretical to writers like Jose Acosta, Johannes Scheffer, Nicholas Fuller, Joseph Mede, Anthony Horneck, and Cotton Mather, inducing Mather to include a lengthy abstract of the Swedish account in Wonders (404-449).

ABSTRACT Although Cotton Mather, as the official chronicler of the 1692 Salem witch trials, is infamously associated with those events, and excerpts from his apologia on Salem, Wonders of the Invisible World, are widely anthologized today, no annotated critical edition of the entire work has appeared in print since the nineteenth century. This present edition of Wonders seeks to remedy this lacuna in modern scholarship. In Wonders, Mather applies both his views on witchcraft and on millennialism to events at Salem. This edition to Mather's Wonders presents this seventeenth-century text beside an integrated theory of the initial causes of the Salem witch panic. The juxtaposition of the probable natural causes of Salem's bewitchment with Mather's implausible explanations exposes the disingenuousness of his writing about Salem. My theory of what happened at Salem includes the probability that a group of conspirators led by the Rev. Samuel Parris deliberately orchestrated the "witchcraft" and that a plant, the thorn apple, used in Algonquian initiation rites, caused the initial symptoms of bewitchment (39-189). Furthermore, key spectral evidence used at the Salem witch trials and recorded by Mather in Wonders appears to have been generated by intense nightmares, commonly thought at the time to be witch visitations, resulting from what is today termed sleep paralysis (215-310). This dissertation provides a detailed look at some of the testimony given in the Salem court records and in Wonders of the Invisible World as it relates to the interpretation in folklore of the phenomenology of nightmares associated with sleep paralysis. The third chapter of this dissertation focuses extensively on Mather's text as a disingenuous response to the Salem witch trials (320-456). The final section of chapter three posits a "Scythian" or Eurasian connection between Swedish and Salem witchcraft. Similarities in shamanic practices among respective indigenous populations of Lapland, Eurasia, Asia, and New England, caused the devil's involvement in both the visible and invisible worlds to appear more than theoretical to writers like Jose Acosta, Johannes Scheffer, Nicholas Fuller, Joseph Mede, Anthony Horneck, and Cotton Mather, inducing Mather to include a lengthy abstract of the Swedish account in Wonders (404-449).

Indiana University-Purdue University Indianapolis (IUPUI) / Springing up in the last quarter of the nineteenth century, Indiana women's study clubs provided generations of women with the opportunity to improve their educations in a friendly environment. They also brought culture to their communities by hosting art exhibits, musical entertainments, and lectures, building libraries and museums, and participating in community improvement endeavors. The activities of urban clubs in big cities have been documented in histories of the women's club movement, but small towns have recieved little attention even through they were vital parts of their communities. This study considers the characteristics, organization, study practices, and civic improvement work of three small-town Indiana women's clubs in the Gilded Age and Progressive Era. The Zerelda Reading Club (Warsaw) studied a wide variety of subjects, while the Ladies' Piano Club (Salem) and Florentine Club (Lebanon) limited their studies to art and music, respectively. All three clubs participated in community improvement efforts that helped their towns achieve urban amenities. The Zerelda Reading Club helped to establish a ladies' rest room, the Ladies' Piano Club worked with other community organizations to build a Carnegie public library, and the Florentine Club raised money to beautify Oak Hill Cemetery. Forming in decades of tremendous growth in popularity of club activity, the organization of all three clubs shows influences of those associations already in existence. This study argues that the individual circumstances of members and their communities resulted in the organization of three women's clubs that prospered under the guidance of extant clubs, but served their members and their communities by adapting activities to suit local needs.