Arithmetical problems in number fields, abelian varieties and modular forms

Number theory, a fascinating area in mathematics and one of the oldest, has experienced spectacular progress in recent years. The development of a deep theoretical background and the implementation of algorithms have led to new and interesting interrelations with mathematics in general which have paved the way for the emergence of major theorems in the area. This report summarizes the contribution to number theory made by the members of the Seminari de Teoria de Nombres (UB-UAB-UPC) in Barcelona. These results are presented in connection with the state of certain arithmetical problems, and so this monograph seeks to provide readers with a glimpse of some specific lines of current mathematical research.

On the bicanonical map of irregular varieties

From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this paper we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the natural higher-dimensional generalization to this context of curves of genus $2$: varieties birationally equivalent to the theta-divisor of an indecomposable principally polarized abelian variety. The proof is based on the (generalized) Fourier-Mukai transform.

Theta-duality on Prym varieties and a Torelli Theorem

Let $\pi : \widetilde C \to C$ be an unramified double covering of irreducible smooth curves and let $P$ be the attached Prym variety. We prove the scheme-theoretic theta-dual equalities in the Prym variety $T(\widetilde C)=V^2$ and $T(V^2)=\widetilde C$, where $V^2$ is the Brill-Noether locus of $P$ associated to $\pi$ considered by Welters. As an application we prove a Torelli theorem analogous to the fact that the symmetric product $D^{(g)}$ of a curve $D$ of genus $g$ determines the curve.

On the bicanonical map of irregular varieties

From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this paper we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the natural higher-dimensional generalization to this context of curves of genus $2$: varieties birationally equivalent to the theta-divisor of an indecomposable principally polarized abelian variety. The proof is based on the (generalized) Fourier-Mukai transform.

Flexible varieties and automorphism groups

Given an irreducible affine algebraic variety X of dimension n≥2 , we let SAut(X) denote the special automorphism group of X , that is, the subgroup of the full automorphism group Aut(X) generated by all one-parameter unipotent subgroups. We show that if SAut(X) is transitive on the smooth locus X reg , then it is infinitely transitive on X reg . In turn, the transitivity is equivalent to the flexibility of X . The latter means that for every smooth point x∈X reg the tangent space T x X is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X) . We also provide various modifications and applications.

The Finite Embeddability Property for some noncommutative knotted varieties of RL and DRL.

Residuated lattices, although originally considered in the realm of algebra providing a general setting for studying ideals in ring theory, were later shown to form algebraic models for substructural logics. The latter are non-classical logics that include intuitionistic, relevance, many-valued, and linear logic, among others. Most of the important examples of substructural logics are obtained by adding structural rules to the basic logical calculus

Selected topics in algebraic geometry

Cover-title.

Motivic Decomposition of Projective Pseudo-Homogeneous Varieties

Let G be a semi-simple algebraic group over a field k. Projective G-homogeneous varieties are projective varieties over which G acts transitively. The stabilizer or the isotropy subgroup at a point on such a variety is a parabolic subgroup which is always smooth when the characteristic of k is zero. However, when k has positive characteristic, we encounter projective varieties with transitive G-action where the isotropy subgroup need not be smooth. We call these varieties projective pseudo-homogeneous varieties. To every such variety, we can associate a corresponding projective homogeneous variety. In this thesis, we extensively study the Chow motives (with coefficients from a finite connected ring) of projective pseudo-homogeneous varieties for G inner type over k and compare them to the Chow motives of the corresponding projective homogeneous varieties. This is done by proving a generic criterion for the motive of a variety to be isomorphic to the motive of a projective homogeneous variety which works for any characteristic of k. As a corollary, we give some applications and examples of Chow motives that exhibit an interesting phenomenon. We also show that the motives of projective pseudo-homogeneous varieties satisfy properties such as Rost Nilpotence and Krull-Schmidt.

Short-term Physiological Changes In Roots And Leaves Of Sugarcane Varieties Exposed To H2o2 In Root Medium.

The aim of this study was to evaluate the differential sensitivity of sugarcane genotypes to H2O2 in root medium. As a hypothesis, the drought tolerant genotype would be able to minimize the oxidative damage and maintain the water transport from roots to shoots, reducing the negative effects on photosynthesis. The sugarcane genotypes IACSP94-2094 (drought tolerant) and IACSP94-2101 (drought sensitive) were grown in a growth chamber and exposed to three levels of H2O2 in nutrient solution: control; 3mmolL(-1) and 80mmolL(-1). Leaf gas exchange, photochemical activity, root hydraulic conductance (Lr) and antioxidant metabolism in both roots and leaves were evaluated after 15min of treatment with H2O2. Although, root hydraulic conductance, stomatal aperture, apparent electron transport rate and instantaneous carboxylation efficiency have been reduced by H2O2 in both genotypes, IACSP94-2094 presented higher values of those variables as compared to IACSP94-2101. There was a significant genotypic variation in relation to the physiological responses of sugarcane to increasing H2O2 in root tissues, being root changes associated with modifications in plant shoots. IACSP94-2094 presented a root antioxidant system more effective against H2O2 in root medium, regardless H2O2 concentration. Under low H2O2 concentration, water transport and leaf gas exchange of IACSP94-2094 were less affected as compared to IACSP94-2101. Under high H2O2 concentration, the lower sensitivity of IACSP94-2094 was associated with increases in superoxide dismutase activity in roots and leaves and increases in catalase activity in roots. In conclusion, we propose a general model of sugarcane reaction to H2O2, linking root and shoot physiological responses.

Physiological limitations in two sugarcane varieties under water suppression and after recovering

Increasing water scarcity and depleted water productivity in irrigated soils are inducing farmers to adopt improved varieties, such as those with high-capacity tolerance. The use of tolerant varieties of sugarcane might substantially avoid the decline of productivity under water deficit. This research aimed to evaluate the harmful effects of drought on the physiology of two sugarcane varieties (RB867515 and RB962962) during the initial development. Young plants were subjected to irrigation suspension until total stomata closure, and then rewatered. Significant reduction on stomatal conductance, transpiration, and net photosynthesis were observed. RB867515 showed a faster stomatal closure while RB962962 slowed the effects of drought on the gas exchanges parameters with a faster recovering after rewatering. Accumulation of carbohydrates, amino acids, proline, and protein in the leaves and roots of the stressed plants occurred in both varieties, substantially linked to reduction of the leaf water potential. Due to the severity of stress, this accumulation was not enough to maintain the cell turgor pressure, so relative water content was diminished. Water stress affected the contents of chlorophyll (a, b, and total) in both varieties, but not the levels of carotenoids. There was a significant reduction in dry matter under stress. In conclusion, RB962962 variety endured stressed conditions more than RB867515, since it slowed down the damaging effects of drought on the gas exchanges. In addition, RB962962 presented a faster recovery than RB867515, a feature that qualifies it as a variety capable of enduring short periods of drought without major losses in the initial stage of its development.

Antioxidative responses of cell suspension cultures of two Coffea arabica varieties to low aluminum levels at pH 5.8

The effects of aluminum (Al) on the activities of antioxidant enzymes and ferritin expression were studied in cell suspension cultures of two varieties of Coffea arabica, Mundo Novo and Icatu, in medium with pH at 5.8. The cells were incubated with 300 µM Al3+, and the Al speciation as Al3+ was 1.45% of the mole fraction. The activities of superoxide dismutase (SOD), catalase (CAT), and glutathione S-transferase (GST) were increased in Mundo Novo, whereas glutathione reductase (GR) and guaiacol peroxidase (GPOX) activities remained unchanged. SOD, GR, and GST activities were increased in Icatu, while CAT activity was not changed, and GPOX activity decreased. The expression of two ferritin genes (CaFer1 and CaFer2) were analyzed by Real-Time PCR. Al caused a downregulation of CaFER1 expression and no changes of CaFER2 expression in both varieties. The Western blot showed no alteration in ferritin protein levels in Mundo Novo and a decrease in Icatu. The differential enzymes responses indicate that the response to Al is variety-dependent.

Generalized Coupled Algebraic Riccati Equations for Discrete-time Markov Jump with Multiplicative Noise Systems

In this paper we consider the existence of the maximal and mean square stabilizing solutions for a set of generalized coupled algebraic Riccati equations (GCARE for short) associated to the infinite-horizon stochastic optimal control problem of discrete-time Markov jump with multiplicative noise linear systems. The weighting matrices of the state and control for the quadratic part are allowed to be indefinite. We present a sufficient condition, based only on some positive semi-definite and kernel restrictions on some matrices, under which there exists the maximal solution and a necessary and sufficient condition under which there exists the mean square stabilizing solution fir the GCARE. We also present a solution for the discounted and long run average cost problems when the performance criterion is assumed be composed by a linear combination of an indefinite quadratic part and a linear part in the state and control variables. The paper is concluded with a numerical example for pension fund with regime switching.

Effect of sugarcane varieties on the development of Mahanarva fimbriolata (Hemiptera: Cercopidae)

The sugarcane spittlebug, Mahanarva fimbriolata (Hemiptera: Cercopidae), is considered the most important pest of sugarcane harvested without the burning of trash, or green cane, in Brazil. The objective of this work was to compare the biology of M. fimbriolata on six sugarcane varieties: SP79-1011, SP80-1816, SP80-1842, SP81-3250, RB72454, and RB835486. The experiments were conducted at a temperature of 25 +/- 1 degrees C, RH of 70 +/- 10%, and a photoperiod of 14:10 [L:D]. Variety RB72454 outperformed the rest, reducing the nymphal population that fed on its roots by 50%. With regard to adults, variety SP81-3250 allowed greater mean longevity of males (38 days) and females (51 days), greater mean oviposition period (46 days), and higher mean fecundity (1215 eggs/female); these parameters were statistically different from those obtained with other varieties. For the eggs, there was no significant effect of variety on developmental time or viability. Consequently, the variety SP81-3250 should be avoided in areas predisposed to the occurrence of M. fimbriolata.

Algebraic Bethe ansatz method for the exact calculation of energy spectra and form factors: applications to models of Bose-Einstein condensates and metallic nanograins

In this review we demonstrate how the algebraic Bethe ansatz is used for the calculation of the-energy spectra and form factors (operator matrix elements in the basis of Hamiltonian eigenstates) in exactly solvable quantum systems. As examples we apply the theory to several models of current interest in the study of Bose-Einstein condensates, which have been successfully created using ultracold dilute atomic gases. The first model we introduce describes Josephson tunnelling between two coupled Bose-Einstein condensates. It can be used not only for the study of tunnelling between condensates of atomic gases, but for solid state Josephson junctions and coupled Cooper pair boxes. The theory is also applicable to models of atomic-molecular Bose-Einstein condensates, with two examples given and analysed. Additionally, these same two models are relevant to studies in quantum optics; Finally, we discuss the model of Bardeen, Cooper and Schrieffer in this framework, which is appropriate for systems of ultracold fermionic atomic gases, as well as being applicable for the description of superconducting correlations in metallic grains with nanoscale dimensions.; In applying all the above models to. physical situations, the need for an exact analysis of small-scale systems is established due to large quantum fluctuations which render mean-field approaches inaccurate.

Superconducting Correlations in Metallic Nanoparticles: Exact Solution of the BCS Model by the Algebraic Bethe Ansatz

Superconducting pairing of electrons in nanoscale metallic particles with discrete energy levels and a fixed number of electrons is described by the reduced Bardeen, Cooper, and Schrieffer model Hamiltonian. We show that this model is integrable by the algebraic Bethe ansatz. The eigenstates, spectrum, conserved operators, integrals of motion, and norms of wave functions are obtained. Furthermore, the quantum inverse problem is solved, meaning that form factors and correlation functions can be explicitly evaluated. Closed form expressions are given for the form factors and correlation functions that describe superconducting pairing.