Conjectures about discriminants of Hecke algebras at prime level

We study p-divisibility of discriminants of Hecke algebras associated to spaces of cusp forms of prime level. We make a precise conjecture about the indexes of Hecke algebras in their normalisation which implies (if true) the conjecture that there are no mod p congruences between non-conjugate newforms of weight 2 and level Gamma_0(p).Comment: To appear in ANTS 6 Proceeding

Torsion points on elliptic curves over number fields of small degree

We determine the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$, for $d = 4,5$ and $6$

Unique bid auctions: Equilibrium solutions and experimental evidence

Two types of auction were introduced on the Internet a few years ago and have rapidly been gaining widespread popularity. In both auctions, players compete for an exogenously determined prize by independently choosing an integer in some finite and common strategy space specified by the auctioneer. In the unique lowest (highest) bid auction, the winner of the prize is the player who submits the lowest (highest) bid, provided that it is unique. We construct the symmetric mixed-strategy equilibrium solutions to the two auctions, and then test them in a sequence of experiments that vary the number of bidders and size of the strategy space. Our results show that the aggregate bids, but only a minority of the individual bidders, are accounted for quite accurately by the equilibrium solutions.

Non-commutative Iwasawa theory for modular forms

The aim of the present paper is to give evidence, largely numerical, in support of the non-commutative main conjecture of Iwasawa theory for the motive of a primitive modular form of weight k>2 over the Galois extension of Q obtained by adjoining to Q all p-power roots of unity, and all p-power roots of a fixed integer m>1. The predictions of the main conjecture are rather intricate in this case because there is more than one critical point, and also there is no canonical choice of periods. Nevertheless, our numerical data agrees perfectly with all aspects of the main conjecture, including Kato's mysterious congruence between the cyclotomic Manin p-adic L-function, and the cyclotomic p-adic L-function of a twist of the motive by a certain non-abelian Artin character of the Galois group of this extension.Comment: 40 page

Searching for a Caregiver: The Middle School Principal

The time is well past for the American people—especially those that deal with our school aged children on a day-to-day basis—to see what is happening to our children and by extension to our society. If public education is to become truly effective in this time of alienation—both of race and class—then a more caring, nurturing, and trusting approach to the profession of educational administration must be encouraged and engendered. The purpose of this study is to examine the middle school administrator as a caregiver by examining the perceptions of the role by teachers and the principal. This study focused on the perceptions of the principal as a caregiver in a selected East Tennessee school. This exploratory descriptive case study included thematic development and verification based on data obtained through qualitative means: interviews, observations, and document analysis. The research questions posed at the beginning of this study include: (1) How does this East Tennessee middle school principal respond to the developmental needs of middle school students?; How does this East Tennessee middle school principal respond to the developmental needs of the teachers who support learning for middle school students?; and (3) How does this East Tennessee middle school principal respond to the developmental needs of the middle school as an innovating entity? A theoretical framework based on the work of Brown and Anfara (2002) and Anfara, Roney, Smarkola, Ducette, and Gross (2006) was used to focus the study’s design, and the data collection and analysis, and the reporting of the findings. Subsequently, the conclusions that were developed in this study describe the perceptions of the role of the caring middle school principal. The first major conclusion is that the developmentally responsive middle school principal responds to students and staff with care. The second conclusion is that the developmentally responsive middle school principal actively practices caring leadership. The final conclusion affirms that the developmentally responsive middle school principal uses the team concept to develop in staff and students a sense of ownership of the school and its programs

The Time Required for Nuclear Fission

The present work is concerned with the mean lifetime of compound nuclei formed in the fast-neutron-induced-fission process. A new method for the measurement of these lifetimes is proposed. The present method is based on the recoil of the compound nucleus following neutron capture and the secondary electrons emitted by fission fragments. Previous experimental and theoretical work concerning the secondary electron yields for fission fragments and other charged particles are reviewed. The importance of the higher energy electrons, the so-called delta rays, formed in the primary ionization process is considered. The subsequent secondary ionization produced by these delta rays provirles a natural explanatior of a previous experimental fact that the number of electrons emitted by a fission fragment is larger when the fragment emerges than when it enters the same surface. This viewpoint is extended t0 a consideration of the dependence of the electron yield on the depth beneath the surface at which a fission event takes place. On the basis of a qualitative theory, this yield is expected to increase with depth and approach asymptotically a limiting value which corresponds to the case where the fragment emerges from depths larger than the maximum range of the delta rays. Measurements of this dependence of the yield on the depth were performed with layers of uranium tetrafluoride. In these measurements iv the electrons were detected by accelerating and focusing them on a scintillation detector. A measure of the number of electrons emitted by each fragment was obtained from the height of the pulse from this detector. The general dependence predicted by the theory was observed. These results were used in an attempt to measure the average distance traversed by the recoiling compound nuclei during the time interval between neutron capture and fission. In one arrangement of the apparatus the compound nuclei with a given recoil velocity were expected to move away from the foil surface a distance proportional to their mean lifetime. Later, the arrangement of the apparatus was changed so that the nuclei were expected to move close to the surface. Measurements of the average electron yield were ma.de in each position. From a comparison of these results and the previously determined dependence of the yield on depth, an estimate of the average recoil distance and, therefore, a measure of the mean lifetime could be obtained. Target nuclei of uranium-238 and neptunium-237 were used in this experiment. Neutrons with an assumed fission spectrum were obtained from a reactor. The results of these measurements showed that the mean lifetimes of the compound nuclei of U-239 and Np-238 were, within the sensitivity of the present method, consistent with zero. Upper limits of the lifetimes have been obtained, however, which are considerably lower than those previously obtained. The total mean lifetimes for target nuclei of U-238 and Np-237 were found to be T(U-238)\u3c6 x 10-14 sec and T(Np-237)\u3c4 x 10-14 sec. The partial mean lifetimes for fission and neutron emission were determined from the relation r = rf + rn and the V previously measured ratio of the fission and neutron emission widths. The partial mean lifetimes are as follows: Tf(U-238) \u3c 4 x 10-13 sec; Tn(U-238) \u3c 7 x 10-14 sec; Tf(Np-237) \u3c 9 x 10-14 sec; and Tn(Np--237) \u3c 7 x 10-14 sec

Form and orchestration in the music of Béla Bartók

Thesis (M.M.)--Boston Universit