Special Frobenius Traces in Galois Representations

This thesis studies Frobenius traces in Galois representations from two different directions. In the first problem we explore how often they vanish in Artin-type representations. We give an upper bound for the density of the set of vanishing Frobenius traces in terms of the multiplicities of the irreducible components of the adjoint representation. Towards that, we construct an infinite family of representations of finite groups with an irreducible adjoint action.

In the second problem we partially extend for Hilbert modular forms a result of Coleman and Edixhoven that the Hecke eigenvalues a_p of classical elliptical modular newforms f of weight 2 are never extremal, i.e., a_p is strictly less than 2[square root]p. The generalization currently applies only to prime ideals p of degree one, though we expect it to hold for p of any odd degree. However, an even degree prime can be extremal for f. We prove our result in each of the following instances: when one can move to a Shimura curve defined by a quaternion algebra, when f is a CM form, when the crystalline Frobenius is semi-simple, and when the strong Tate conjecture holds for a product of two Hilbert modular surfaces (or quaternionic Shimura surfaces) over a finite field.

Conserved quantities and the formation of black holes in the Brans-Dicke theory of gravitation

In Part I, we construct a symmetric stress-energy-momentum pseudo-tensor for the gravitational fields of Brans-Dicke theory, and use this to establish rigorously conserved integral expressions for energy-momentum Pⁱ and angular momentum J^ik. Application of the two-dimensional surface integrals to the exact static spherical vacuum solution of Brans leads to an identification of our conserved mass with the active gravitational mass. Application to the distant fields of an arbitrary stationary source reveals that Pⁱ and J^ik have the same physical interpretation as in general relativity. For gravitational waves whose wavelength is small on the scale of the background radius of curvature, averaging over several wavelengths in the Brill-Hartle-Isaacson manner produces a stress-energy-momentum tensor for gravitational radiation which may be used to calculate the changes in Pⁱ and J^ik of their source.

In Part II, we develop strong evidence in favor of a conjecture by Penrose--that, in the Brans-Dicke theory, relativistic gravitational collapse in three dimensions produce black holes identical to those of general relativity. After pointing out that any black hole solution of general relativity also satisfies Brans-Dicke theory, we establish the Schwarzschild and Kerr geometries as the only possible spherical and axially symmetric black hole exteriors, respectively. Also, we show that a Schwarzschild geometry is necessarily formed in the collapse of an uncharged sphere.

Appendices discuss relationships among relativistic gravity theories and an example of a theory in which black holes do not exist.

Rings faithfully represented on their left socle

In 1964 A. W. Goldie [1] posed the problem of determining all rings with identity and minimal condition on left ideals which are faithfully represented on the right side of their left socle. Goldie showed that such a ring which is indecomposable and in which the left and right principal indecomposable ideals have, respectively, unique left and unique right composition series is a complete blocked triangular matrix ring over a skewfield. The general problem suggested above is very difficult. We obtain results under certain natural restrictions which are much weaker than the restrictive assumptions made by Goldie.

We characterize those rings in which the principal indecomposable left ideals each contain a unique minimal left ideal (Theorem (4.2)). It is sufficient to handle indecomposable rings (Lemma (1.4)). Such a ring is also a blocked triangular matrix ring. There exist r positive integers K₁,..., K_r such that the i,j^th block of a typical matrix is a K_i x K_j matrix with arbitrary entries in a subgroup D_ij of the additive group of a fixed skewfield D. Each D_ii is a sub-skewfield of D and D_ri = D for all i. Conversely, every matrix ring which has this form is indecomposable, faithfully represented on the right side of its left socle, and possesses the property that every principal indecomposable left ideal contains a unique minimal left ideal.

The principal indecomposable left ideals may have unique composition series even though the ring does not have minimal condition on right ideals. We characterize this situation by defining a partial ordering ρ on {i, 2,...,r} where we set iρj if D_ij ≠ 0. Every principal indecomposable left ideal has a unique composition series if and only if the diagram of ρ is an inverted tree and every D_ij is a one-dimensional left vector space over D_ii (Theorem (5.4)).

We show (Theorem (2.2)) that every ring A of the type we are studying is a unique subdirect sum of less complex rings A₁,...,A_s of the same type. Namely, each A_i has only one isomorphism class of minimal left ideals and the minimal left ideals of different A_i are non-isomorphic as left A-modules. We give (Theorem (2.1)) necessary and sufficient conditions for a ring which is a subdirect sum of rings A_i having these properties to be faithfully represented on the right side of its left socle. We show ((4.F), p. 42) that up to technical trivia the rings A_i are matrix rings of the form

[...]. Each Q_j comes from the faithful irreducible matrix representation of a certain skewfield over a fixed skewfield D. The bottom row is filled in by arbitrary elements of D.

In Part V we construct an interesting class of rings faithfully represented on their left socle from a given partial ordering on a finite set, given skewfields, and given additive groups. This class of rings contains the ones in which every principal indecomposable left ideal has a unique minimal left ideal. We identify the uniquely determined subdirect summands mentioned above in terms of the given partial ordering (Proposition (5.2)). We conjecture that this technique serves to construct all the rings which are a unique subdirect sum of rings each having the property that every principal-indecomposable left ideal contains a unique minimal left ideal.

On the distribution of the signs of the conjugates of the cyclotomic units in the maximal real subfield of the qth cyclotomic field, q A prime

Let F = Ǫ(ζ + ζ ^–1) be the maximal real subfield of the cyclotomic field Ǫ(ζ) where ζ is a primitive qth root of unity and q is an odd rational prime. The numbers u₁=-1, u_k=(ζ^k-ζ^-k)/(ζ-ζ^-1), k=2,…,p, p=(q-1)/2, are units in F and are called the cyclotomic units. In this thesis the sign distribution of the conjugates in F of the cyclotomic units is studied.

Let G(F/Ǫ) denote the Galoi's group of F over Ǫ, and let V denote the units in F. For each σϵ G(F/Ǫ) and μϵV define a mapping sgn_σ: V→GF(2) by sgn_σ(μ) = 1 iff σ(μ) ˂ 0 and sgn_σ(μ) = 0 iff σ(μ) ˃ 0. Let {σ₁, ... , σ_p} be a fixed ordering of G(F/Ǫ). The matrix M_q=(sgn_σj(v_i) ) , i, j = 1, ... , p is called the matrix of cyclotomic signatures. The rank of this matrix determines the sign distribution of the conjugates of the cyclotomic units. The matrix of cyclotomic signatures is associated with an ideal in the ring GF(2) [x] / (x^p+ 1) in such a way that the rank of the matrix equals the GF(2)-dimension of the ideal. It is shown that if p = (q-1)/ 2 is a prime and if 2 is a primitive root mod p, then M_q is non-singular. Also let p be arbitrary, let ℓ be a primitive root mod q and let L = {i | 0 ≤ i ≤ p-1, the least positive residue of defined by ℓⁱ mod q is greater than p}. Let H_q(x) ϵ GF(2)[x] be defined by H_q(x) = g. c. d. ((Σ xⁱ/I ϵ L) (x+1) + 1, x^p + 1). It is shown that the rank of M_q equals the difference p - degree H_q(x).

Further results are obtained by using the reciprocity theorem of class field theory. The reciprocity maps for a certain abelian extension of F and for the infinite primes in F are associated with the signs of conjugates. The product formula for the reciprocity maps is used to associate the signs of conjugates with the reciprocity maps at the primes which lie above (2). The case when (2) is a prime in F is studied in detail. Let T denote the group of totally positive units in F. Let U be the group generated by the cyclotomic units. Assume that (2) is a prime in F and that p is odd. Let F₍₂₎ denote the completion of F at (2) and let V₍₂₎ denote the units in F₍₂₎. The following statements are shown to be equivalent. 1) The matrix of cyclotomic signatures is non-singular. 2) U∩T = U². 3) U∩F²₍₂₎ = U². 4) V₍₂₎/ V₍₂₎² = ˂v₁ V₍₂₎²˃ ʘ…ʘ˂v_p V₍₂₎²˃ ʘ ˂3V₍₂₎²˃.

The rank of M_q was computed for 5≤q≤929 and the results appear in tables. On the basis of these results and additional calculations the following conjecture is made: If q and p = (q -1)/ 2 are both primes, then M_q is non-singular.

Distância, velocidade, proximidade e redes: fabricando conceitos em educação a distância

Apresentamos neste trabalho uma investigação acerca de algumas das condições que compõem as aplicações da chamada Educação a Distância no Brasil. Interessa-nos, em particular, descobrir as potencialidades desta modalidade de ensino-aprendizagem que se dão através do uso de computadores atrelados à Internet. Porém, optamos por adotar uma perspectiva imanente de investigação, que engloba uma multiplicidade de questões que atravessam a Educação como um todo e não somente o seu modo de operação à distância. Para tanto, nós realizamos uma espécie de cartografia mezzo-histórica do ensinar-aprender à distância no nosso país com o intuito de mostrar como é longa a distância entre as primeiras experiências em Educação a Distância e o cenário atual propício a estas práticas. Discutimos a apropriação de técnicas, tecnologias e usos que ocorrem em meio a linhas políticas, econômicas, pedagógicas e, sobretudo, éticas, constituindo algumas das condições cruciais à invenção e captura deste modo de fazer educacional. Analisamos alguns dos possíveis impactos na Educação a Distância vigente em nosso país que são efeito do ressentimento oriundo do contínuo e inacabável processo de Globalização, que afeta diferentes segmentos da nossa sociedade, não deixando de forma alguma ilesos os processos de ensino-aprendizagem contemporâneos. Frente a esta conjectura, nós avançamos na tentativa de verificar até que ponto é possível extrair potências da suposta fragilidade imposta pela política neoliberal nos dias de hoje, amparados pelas pistas deixadas pelo trabalho de Deleuze e de Foucault no movimento que vai das Sociedades Disciplinares às de Controle misturadas às alternativas apontadas pelo que se convém denominar Capitalismo Cognitivo. As condições de experiência e de produção de sentido(s) possíveis em processos de Educação a Distância são problematizadas através da fabricação dos conceitos de distância, proximidade e velocidade, por um percurso que nos posiciona entre as duas margens mais comuns na discussão sobre esta referida modalidade de ensino-aprendizagem, que são compostas de um lado por entusiastas e de outro pelos algozes críticos. Apresentamos ainda neste estudo algumas noções de rede, considerando as suas possíveis conceituações, seja como parafernália técnica que viabiliza conexões informáticas, permitindo a comunicação na Internet, seja como modo de mobilização de coletivos que buscam a multiplicação de sentidos em suas práticas. Apostamos na capacidade inerente às redes de aumentar consideravelmente a potência de ação dos coletivos envolvidos com práticas de ensino-aprendizagem à distância, através da implementação de estratégias e táticas micropolíticas, conforme inspiração na análise acerca do processo de invenção do cotidiano registrado na obra de Michel de Certeau.

Gillnet selectivity for juvenile blacktip sharks (Carcharhinus limbatus)

Gillnet mesh selectivity parameters were estimated for juvenile blacktip sharks (Carcharhinus limbatus) by using length data from an experimental fishery-independent gillnet survey in the northeastern Gulf of Mexico. Length data for 1720 blacktip sharks were collected over 17 years (1994–2010) with seven mesh sizes ranging from 7.6 to 20.3 cm. Four selectivity models, a normal model assuming fixed spread, a normal model assuming that spread is proportional to mesh size, a lognormal model, and a gamma model were fitted to the data by using the SELECT (share each length’s catch total) method. Each model was run twice under separate assumptions of 1) equal fishing intensity; and 2) fishing intensity proportional to mesh size. The normal, fixed-spread selectivity curve where fishing intensity is assumed to be proportional to mesh size provided the best fit to the data according to model deviance estimates and was chosen as the best model. Results indicate that juvenile blacktip sharks are susceptible as bycatch in some commercial gillnet fisheries.

Contrasting abundance and residency patterns of two sympatric populations of transient killer whales (Orcinus orca) in the northern Gulf of Alaska

Two sympatric populations of “transient” (mammal-eating) killer whales were photo-identified over 27 years (1984–2010) in Prince William Sound and Kenai Fjords, coastal waters of the northern Gulf of Alaska (GOA). A total of 88 individuals were identified during 203 encounters with “AT1” transients (22 individuals) and 91 encounters with “GOA” transients (66 individuals). The median number of individuals identified annually was similar for both populations (AT1=7; GOA=8), but mark-recapture estimates showed the AT1 whales to have much higher fidelity to the study area, whereas the GOA whales had a higher exchange of individuals. Apparent survival estimates were generally high for both populations, but there was a significant reduction in the survival of AT1 transients after the Exxon Valdez oil spill in 1989, with an abrupt decline in estimated abundance from a high of 22 in 1989 to a low of seven whales at the end of 2010. There was no detectable decline in GOA population abundance or survival over the same period, but abundance ranged from just 6 to 18 whales annually. Resighting data from adjacent coastal waters and movement tracks from satellite tags further indicated that the GOA whales are part of a larger population with a more extensive range, whereas AT1 whales are resident to the study area.

Bomb radiocarbon and tag-recapture dating of sandbar shark (Carcharhinus plumbeus)

The sandbar shark (Carcharhinus plumbeus) was the cornerstone species of western North Atlantic and Gulf of Mexico large coastal shark fisheries until 2008 when they were allocated to a research-only fishery. Despite decades of fishing on this species, important life history parameters, such as age and growth, have not been well known. Some validated age and growth information exists for sandbar shark, but more comprehensive life history information is needed. The complementary application of bomb radiocarbon and tag-recapture dating was used in this study to determine valid age-estimation criteria and longevity estimates for this species. These two methods indicated that current age interpretations based on counts of growth bands in vertebrae are accurate to 10 or 12 years. Beyond these years, we could not determine with certainty when such an underestimation of age begins; however, bomb radiocarbon and tag-recapture data indicated that large adult sharks were considerably older than the estimates derived from counts of growth bands. Three adult sandbar sharks were 20 to 26 years old based on bomb radiocarbon results and were a 5- to 11-year increase over the previous age estimates for these sharks. In support of these findings, the tag-recapture data provided results that were consistent with bomb radiocarbon dating and further supported a longevity that exceeds 30 years for this species.

Validation of a method for estimating realized annual fecundity in a multiple spawner, the long-snouted seahorse (Hippocampus guttulatus), using underwater visual census

The long-snouted seahorse (Hippocampus guttulatus) (Cuvier, 1829), was used to validate the pre-dictive accuracy of three progressively realistic models for estimating the realized annual fecundity of asyn-chronous, indeterminate, multiple spawners. Underwater surveys and catch data were used to estimate the duration of the reproductive season, female spawning frequency, male brooding frequency, and batch fecun-dity. The most realistic model, a generalization of the spawning fraction method, produced unbiased estimates of male brooding frequency (mean ±standard deviation [SD]=4.2 ±1.6 broods/year). Mean batch fecundity and realized annual fecundity were 213.9 (±110.9) and 903.6 (±522.4), respectively. However, females prepared significantly more clutches than the number of broods produced by males. Thus, methods that infer spawning frequency from patterns in female egg production may lead to significant overestimates of realized annual fecundity. The spawning fraction method is broadly applicable to many taxa that exhibit parental care and can be applied nondestructively to species for which conservation is a concern.

The climate and eastern ocean systems project

A brief description of the NMFS/ORSTOM/ILCARM climate and Eastern Ocean systems (CEOS) project is given. CEOS will study the four major eastern boundary current regions (Peru/Chile, California, Northwest and Southwest Africa) and attempt to separate local short term changes of their resources and/or dynamics from long-term, climatic global changes. Expected products range from a large, widely accessible oceanographic / atmospheric database to various documents that will present key results as well as improved contacts and stronger analytical capabilities in cooperating national institutions.