Euler characteristics and elliptic curves

Let E be a modular elliptic curve over ℚ, without complex multiplication; let p be a prime number where E has good ordinary reduction; and let F∞ be the field obtained by adjoining to ℚ all p-power division points on E. Write G∞ for the Galois group of F∞ over ℚ. Assume that the complex L-series of E over ℚ does not vanish at s = 1. If p ⩾ 5, we make a precise conjecture about the value of the G∞-Euler characteristic of the Selmer group of E over F∞. If one makes a standard conjecture about the behavior of this Selmer group as a module over the Iwasawa algebra, we are able to prove our conjecture. The crucial local calculations in the proof depend on recent joint work of the first author with R. Greenberg.

Adjoint modular Galois representations and their Selmer groups

In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the three-dimensional adjoint representation ad(φ) of a two-dimensional modular Galois representation φ. We start with the p-adic Galois representation φ0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(φ0)) the intersection number of the elliptic curve E and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group Sel(ad(φ0)) from the proof of Wiles of the Shimura–Taniyama conjecture. After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, T, is the weight variable of the universal p-ordinary Hecke algebra, and the second variable is the cyclotomic variable S. In the one-variable case, we let φ denote the p-ordinary Galois representation with values in GL2(Zp[[T]]) lifting φ0, and the characteristic power series of the Selmer group Sel(ad(φ)) is given by a p-adic L-function interpolating L(1, ad(φk)) for weight k + 2 specialization φk of φ. In the two-variable case, we state a main conjecture on the characteristic power series in Zp[[T, S]] of Sel(ad(φ) ⊗ ν−1), where ν is the universal cyclotomic character with values in Zp[[S]]. Finally, we describe our recent results toward the proof of the conjecture and a possible strategy of proving the main conjecture using p-adic Siegel modular forms.

The structure of Selmer groups

The purpose of this article is to describe certain results and conjectures concerning the structure of Galois cohomology groups and Selmer groups, especially for abelian varieties. These results are analogues of a classical theorem of Iwasawa. We formulate a very general version of the Weak Leopoldt Conjecture. One consequence of this conjecture is the nonexistence of proper Λ-submodules of finite index in a certain Galois cohomology group. Under certain hypotheses, one can prove the nonexistence of proper Λ-submodules of finite index in Selmer groups. An example shows that some hypotheses are needed.

On degree 2 Galois representations over

We discuss proofs of some new special cases of Serre’s conjecture on odd, degree 2 representations of Gℚ.

Dynamic friction and the origin of the complexity of earthquake sources.

We study a simple antiplane fault of finite length embedded in a homogeneous isotropic elastic solid to understand the origin of seismic source heterogeneity in the presence of nonlinear rate- and state-dependent friction. All the mechanical properties of the medium and friction are assumed homogeneous. Friction includes a characteristic length that is longer than the grid size so that our models have a well-defined continuum limit. Starting from a heterogeneous initial stress distribution, we apply a slowly increasing uniform stress load far from the fault and we simulate the seismicity for a few 1000 events. The style of seismicity produced by this model is determined by a control parameter associated with the degree of rate dependence of friction. For classical friction models with rate-independent friction, no complexity appears and seismicity is perfectly periodic. For weakly rate-dependent friction, large ruptures are still periodic, but small seismicity becomes increasingly nonstationary. When friction is highly rate-dependent, seismicity becomes nonperiodic and ruptures of all sizes occur inside the fault. Highly rate-dependent friction destabilizes the healing process producing premature healing of slip and partial stress drop. Partial stress drop produces large variations in the state of stress that in turn produce earthquakes of different sizes. Similar results have been found by other authors using the Burridge and Knopoff model. We conjecture that all models in which static stress drop is only a fraction of the dynamic stress drop produce stress heterogeneity.

Seven newly discovered intron positions in the triose-phosphate isomerase gene: evidence for the introns-late theory.

The gene encoding the glycolytic enzyme triose-phosphate isomerase (TPI; EC 5.3.1.1) has been central to the long-standing controversy on the origin and evolutionary significance of spliceosomal introns by virtue of its pivotal support for the introns-early view, or exon theory of genes. Putative correlations between intron positions and TPI protein structure have led to the conjecture that the gene was assembled by exon shuffling, and five TPI intron positions are old by the criterion of being conserved between animals and plants. We have sequenced TPI genes from three diverse eukaryotes--the basidiomycete Coprinus cinereus, the nematode Caenorhabditis elegans, and the insect Heliothis virescens--and have found introns at seven novel positions that disrupt previously recognized gene/protein structure correlations. The set of 21 TPI introns now known is consistent with a random model of intron insertion. Twelve of the 21 TPI introns appear to be of recent origin since each is present in but a single examined species. These results, together with their implication that as more TPI genes are sequenced more intron positions will be found, render TPI untenable as a paradigm for the introns-early theory and, instead, support the introns-late view that spliceosomal introns have been inserted into preexisting genes during eukaryotic evolution.

Equações de quarta ordem na modelagem de oscilações de pontes

Equações diferenciais de quarta ordem aparecem naturalmente na modelagem de oscilações de estruturas elásticas, como aquelas observadas em pontes pênseis. São considerados dois modelos que descrevem as oscilações no tabuleiro de uma ponte. No modelo unidimensional estudamos blow up em espaço finito de soluções de uma classe de equações diferenciais de quarta ordem. Os resultados apresentados solucionam uma conjectura apresentada em [F. Gazzola and R. Pavani. Wide oscillation finite time blow up for solutions to nonlinear fourth order differential equations. Arch. Ration. Mech. Anal., 207(2):717752, 2013] e implicam a não existência de ondas viajantes com baixa velocidade de propagação em uma viga. No modelo bidimensional analisamos uma equação não local para uma placa longa e fina, suportada nas extremidades menores, livre nas demais e sujeita a protensão. Provamos existência e unicidade de solução fraca e estudamos o seu comportamento assintótico sob amortecimento viscoso. Estudamos ainda a estabilidade de modos simples de oscilação, os quais são classificados como longitudinais ou torcionais.

Desenvolvimento da modelagem de turbulência e interação fluido-estrutura para as vibrações induzidas por vórtices de cilindro rígido.

Esta tese apresenta o desenvolvimento e aplicação de modelos de turbulência, transição laminar-turbulenta e de interações fluido-estrutura ao escoamento externo em cilindro rígido estacionário e em vibrações induzidas por vórtices. Tais desenvolvimentos foram realizados no código ReFRESCO, baseado em técnicas de dinâmica de fluidos computacional (CFD). Realizou-se um estudo quanto ao desempenho do modelo k- SST em extensa faixa de números de Reynolds, segundo o qual se identificaram as deficiências de modelagem para este escoamento. A modelagem adaptativa das escalas (SAS) e o modelo de transição por correlações locais (LCTM), ambos combinados ao SST, melhoraram a aderência aos resultados experimentais para este escoamento, em uma contribuição original deste trabalho. A aplicação de técnicas de verificação e validação possibilitou a estimação de incertezas e erros para os modelos e números de Reynolds e também de identificada como outra contribuição deste trabalho. A combinação da modelagem em SST, SAS e LCTM com movimentos impostos de realizada para números de Reynolds moderados, diferentes frequências e amplitudes de vibração, algo que poucas publicações abordam em detalhes. Com relação aos movimentos livres, este trabalho traz contribuições com a aplicação dos modelos SST e SAS ao estudo de vibrações induzidas por vórtices em dois graus de liberdade, baixa razão de massa e números de Reynolds moderados, mais altos do que normalmente observados na literatura. Por fim, a investigação da importância relativa de efeitos da turbulência aos casos de movimentos livres e impostos, com relação ao caso de cilindro estacionário, comprovou a conjetura formulada na parte inicial deste trabalho, no que tange à escolha do modelo de turbulência em determinadas aplicações. Tal escolha mostrou-se menos decisiva no caso do cilindro em movimento imposto e ainda menos nos movimentos livres, em comparação ao caso estacionário, uma vez que a resposta em movimentos do corpo filtra grande parte dos efeitos turbulentos de ordem superior. Esta observação mostra-se relevante, uma vez que pode permitir simplificações na modelagem e aplicação de ferramentas de CFD em uma classe importante de projetos de engenharia.

On the existence of exponential polynomials with prefixed gaps

This paper shows that the conjecture of Lapidus and Van Frankenhuysen on the set of dimensions of fractality associated with a nonlattice fractal string is true in the important special case of a generic nonlattice self-similar string, but in general is false. The proof and the counterexample of this have been given by virtue of a result on exponential polynomials P(z), with real frequencies linearly independent over the rationals, that establishes a bound for the number of gaps of RP, the closure of the set of the real projections of its zeros, and the reason for which these gaps are produced.

On the Complex Dimensions of Nonlattice Fractal Strings in Connection with Dirichlet Polynomials

In this paper we give a new characterization of the closure of the set of the real parts of the zeros of a particular class of Dirichlet polynomials that is associated with the set of dimensions of fractality of certain fractal strings. We show, for some representative cases of nonlattice Dirichlet polynomials, that the real parts of their zeros are dense in their associated critical intervals, confirming the conjecture and the numerical experiments made by M. Lapidus and M. van Frankenhuysen in several papers.

Democratic Contestation, Accountability, and Citizen Satisfaction at the Regional Level. CES Germany & Europe Working Papers, No. 04.2, 2004

Democratic theory tells us that competition between political parties fosters more responsive government by disciplining elected leaders. Yet party competition may not always attain the levels desirable for holding leaders accountable, notably at the sub-national level. This paper hypothesizes that variations in competition-induced accountability affect regional, or state, government behavior, and that this variation is reflected in citizen satisfaction with regional government performance. The hypothesis is confirmed using survey data from sixty-eight German state election studies. Specifically, a widening of the gap between the two main parties of each state is shown to affect subsequent individual-level satisfaction negatively. This finding presents a conjecture that should be generalizable to other countries with strong sub-national units.

Why Do Some International Institutions Contain Strong Dispute Settlement Provisions: New Evidence from Preferential Trade Agreements

To understand why some international institutions have stronger dispute settlement mechanisms (DSMs) than others, we investigate the dispute settlement provisions of nearly 600 preferential trade agreements (PTAs), which possess several desirable case-selection features and are evoked more than is realized. We broaden the study of dispute settlement design beyond “legalization” and instead reorient theorizing around a multi-faceted conceptualization of the strength of DSMs. We posit that strong DSMs are first and foremost a rational response to features of agreements that require stronger dispute settlement, such as depth and large memberships. Multivariate empirical tests using a new data set on PTA design confirm these expectations and reveal that depth – the amount of policy change specified in an agreement – is the most powerful and consistent predictor of DSM strength, providing empirical support to a long-posited but controversial conjecture. Yet power also plays a sizeable role, since agreements among asymmetric members are more likely to have strong DSMs due to their mutual appeal, as are those involving the United States. Important regional differences also emerge, as PTAs across the Americas are designed with strong dispute settlement, as are Asian PTAs, which contradicts the conventional wisdom about Asian values and legalization. Our findings demonstrate that rationalism explains much of international institutional design, yet it can be enhanced by also incorporating power-based and regional explanations.

Annibal dans les Alpes,

On cover: Ouvrage renfermant dix-sept cartes et six photographies. Paris, A. Picard, 1902.

Aspects of Markov Chains and Particle Systems

Thesis (Ph.D.)--University of Washington, 2016-06

Irreducible cyclic presentations of the trivial group

We produce families of irreducible cyclic presentations of the trivial group. These families comprehensively answer questions about such presentations asked by Dunwoody and by Edjvet, Hammond, and Thomas. Our theorems are purely theoretical, but their derivation is based on practical computations. We explain how we chose the computations and how we deduced the theorems.