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Let K be a quadratic imaginary number field, let Kp^(infinity) the top of its p-class field tower for p an odd prime, and let G=Gal(Kp^(infinity)/K). It is known, due to a tremendous collection of work ranging from the principal results of class field theory to the famous Golod-Shafarevich inequality, that G is finite if the p-rank of the class group of K is 0 or 1, and is infinite if this rank is at least 3. This leaves the rank 2 case as the only remaining unsolved case. In this case, while finiteness is still a mystery, much is still known about G: It is a 2-generated, 2-related pro-p-group equipped with an involution that acts as the inverse modulo commutators, and is of one of three possible Zassenhaus types (defined in the paper). If such a group is finite, we will call it an interesting p-tower group. We further the knowledge on such groups by showing that one particular Zassenhaus type can occur as an interesting p-tower group only if the group has order at least p^24 (Proposition 8.1), and by proving a succinct cohomological condition (Proposition 4.7) for a p-tower group to be infinite. More generally, we prove a Golod-Shafarevich equality (Theorem 5.2), refining the famous Golod-Shafarevich inequality, and obtaining as a corollary a strict strengthening of previous Golod-Shafarevich inequalities (Corollary 5.5). Of interest is that this equality applies not only to finite p-groups but also to p-adic analytic pro-p-groups, a class of groups of particular relevance due to their prominent appearance in the Fontaine-Mazur conjecture. This refined version admits as a consequence that the sizes of the first few modular dimension subgroups of an interesting p-tower group G are completely determined by p and its Zassenhaus type, and we compute these sizes. As another application, we prove a new formula (Corollary 5.3) for the Fp-dimensions of the successive quotients of dimension subgroups of free pro-p-groups.

Music has been studied in numerous ways. Yet, there is no research which examines the link between acoustic properties of music and social processes. Three frameworks guide my research. The Homology thesis which states that elites prefer elite culture (Bourdieu, 1984); the Omnivore thesis which argues that elites are omnivores and prefer many different types of cultural objects as compared to univores (Peterson and Simkus, 1992), and the reconciliation thesis which states that elites prefer many different types of cultural objects, but they do so in ways that are patterned (Bryson, 1996). Representative samples of songs from 18 music genres form the General Social Survey are collected and acoustic features are extracted. Sixty-one percent (61%) of the songs were correctly classified into their respective genres. The misclassifications of genres represent genres which share acoustic information in meaningful ways. Network analysis reveals a structure with three spaces, popular, elite and folk. Acoustic features are then grouped into three fundamental dimensions: timbre, rhythm, loudness. Crisp genres are grouped according to four dimensions: timbre, rhythm, loudness and genre boundary. Correspondence Analysis is used to plot the musical aversions of genre sets by class and status groups. Findings indicate that respondents with increased levels of economic capital reject genre sets with strong boundaries. This study then develops 3 socioacoustic profiles which represents a respondent's orientation towards sound. I then estimate the effects of education, income, occupational prestige, age, gender, race, religious preference, political identity, racial intolerance, political intolerance, being an omnivore and being a univore on each socioacoustic fuzzy set. Results indicate that socioacoustic profiles sets associated with elite spaces are associated with high occupational prestige, increased age, being female, not being Black, not being a univore, being politically tolerant and being racially intolerant. Contrary to the omnivore thesis, being an omnivore was not a strong predictor in each profile set. Findings support the reconciliation thesis and highlight how moving beyond class evidences differing bases of cultural structuration.

The subject being discussed constantly at present is the age of criminals which is becoming younger and younger. This is a great problem of our society because the future of a society depends not only on the economic results but also on the proper social potential of the people who act. A survey shows that a large number of teenagers under sixteen who do not learn is crucial to their inappropriate and criminal activity. School, whose main aim is to educate an independent and creative personality, can contribute to the solution of this problem because only keeping a child at school enables us to reduce the threat of doing inappropriate activity. The current paper discusses the causes crucial to the schoolchildren who can be involved into a risk group. It also analyses the possible contribution of a tutor as a coordinator of the education process to the positive change in the society through the positive changes of the risk group schoolchildren. To be able to foresee the possibilities for the activity of a tutor, two surveys were carried out. The first one was a general survey in which 667 schoolchildren of a risk group and their tutors were questioned by the group of post graduates. The second survey, in which 262 schoolchildren of Vilnius Pilaite Secondary School had to answer the questions presented, was fulfilled personally by the author. On the basis of the research carried out and the analysis of the practical activity it can be maintained that each member of a... [to full text]

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