Borcherds products and arithmetic intersection theory on Hilbert modular surfaces

We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight 2. Moreover, we determine the arithmetic selfintersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface, and we study Faltings heights of arithmetic Hirzebruch-Zagier divisors.

Galois representations, embedding problems and modular forms

To an odd irreducible 2-dimensional complex linear representation of the absolute Galois group of the field Q of rational numbers, a modular form of weight 1 is associated (modulo Artin's conjecture on the L-series of the representation in the icosahedral case). In addition, linear liftings of 2-dimensional projective Galois representations are related to solutions of certain Galois embedding problems. In this paper we present some recent results on the existence of liftings of projective representations and on the explicit resolution of embedding problems associated to orthogonal Galois representations, and explain how these results can be used to construct modular forms.

Animal Tissue Response To Mid-Term Electronic Modular Artificial Sphincter Implantation

Introduction and objectives: The AMS 800TM is considered the gold standard for sphincter replacement. However, the one-ring design can erode the urethra and lead to severe complications. A mechanism that could alternatively compress successive segments of the urethra would limit such deleterious outcome. We report 12 weeks animal urethral tissue analysis following implantation of a new modular artificial sphincter. METHODS: The device is composed by three parts: the contractile unit, two rings and an integrated microprocessor. The contractile unit is made of Nitinol fibers. The rings are placed around the urethra to control the flow of urine by squeezing the urethra. They work in a sequential alternative mode and are controlled by a microprocessor connected to an external computer. The computer can reveal specific failure of device components. The device was impkanted in eight male sheep. The rings were positioned around the urethra and the control unit was placed 5cm away. The device was working twenty hours per day; it was open 10min. per hour to allow urination. The animals were sacrificed after 12 weeks. The urethra and the tissues surrounding the control unit were macroscopically and microscopically examined. Two transversal sections crossing the sphincter and two transversal sections crossing the urethra alone were obtained and stained with modified Paragon after resin embedding. Urethra was also embedded in paraffin. The first section was stained with safranin-hematoxylin-eosin, the second section was stained with Masson's Trichrome and the remaining eight sections were available for immunolabelling of the macrophages.Results: The chronic study went uneventful. No clinical infection or pain was observed. The computer registered no specific failure in ring function, Nitinol wires and tube connectors. At explantation, except for a slight grade of lymphocytes in two out of eight specimens, no urethral stricture or atrophy could be observed. Immunohistochemistry confirmed the absence of macrophages. Tissue structure and organization of the urethra with and without artificial sphincter were similar. No migration of the device was observed.Conclusions: The study clearly showed no tissue damage or inflammation of the urethra. Electronic design, preservation of urethral vascularisation and adjustability after implantation are the key ideas to improve the actual AUS. Further studies will be carried out to evaluate this potential.

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.

A modular magnetic anastomotic device for minimally invasive digestive anastomosis: proof of concept and preliminary data in the pig model.

BACKGROUND: The aim of our study was to assess the feasibility of minimally invasive digestive anastomosis using a modular flexible magnetic anastomotic device made up of a set of two flexible chains of magnetic elements. The assembly possesses a non-deployed linear configuration which allows it to be introduced through a dedicated small-sized applicator into the bowel where it takes the deployed form. A centering suture allows the mating between the two parts to be controlled in order to include the viscerotomy between the two magnetic rings and the connected viscera. METHODS AND PROCEDURES: Eight pigs were involved in a 2-week survival experimental study. In five colorectal anastomoses, the proximal device was inserted by a percutaneous endoscopic technique, and the colon was divided below the magnet. The distal magnet was delivered transanally to connect with the proximal magnet. In three jejunojejunostomies, the first magnetic chain was injected in its linear configuration through a small enterotomy. Once delivered, the device self-assembled into a ring shape. A second magnet was injected more distally through the same port. The centering sutures were tied together extracorporeally and, using a knot pusher, magnets were connected. Ex vivo strain testing to determine the compression force delivered by the magnetic device, burst pressure of the anastomosis, and histology were performed. RESULTS: Mean operative time including endoscopy was 69.2 ± 21.9 min, and average time to full patency was 5 days for colorectal anastomosis. Operative times for jejunojejunostomies were 125, 80, and 35 min, respectively. The postoperative period was uneventful. Burst pressure of all anastomoses was ≥ 110 mmHg. Mean strain force to detach the devices was 6.1 ± 0.98 and 12.88 ± 1.34 N in colorectal and jejunojejunal connections, respectively. Pathology showed a mild-to-moderate inflammation score. CONCLUSIONS: The modular magnetic system showed enormous potential to create minimally invasive digestive anastomoses, and may represent an alternative to stapled anastomoses, being easy to deliver, effective, and low cost.

Comparative modular analysis of gene expression in vertebrate organs.

ABSTRACT: BACKGROUND: The degree of conservation of gene expression between homologous organs largely remains an open question. Several recent studies reported some evidence in favor of such conservation. Most studies compute organs' similarity across all orthologous genes, whereas the expression level of many genes are not informative about organ specificity. RESULTS: Here, we use a modularization algorithm to overcome this limitation through the identification of inter-species co-modules of organs and genes. We identify such co-modules using mouse and human microarray expression data. They are functionally coherent both in terms of genes and of organs from both organisms. We show that a large proportion of genes belonging to the same co-module are orthologous between mouse and human. Moreover, their zebrafish orthologs also tend to be expressed in the corresponding homologous organs. Notable exceptions to the general pattern of conservation are the testis and the olfactory bulb. Interestingly, some co-modules consist of single organs, while others combine several functionally related organs. For instance, amygdala, cerebral cortex, hypothalamus and spinal cord form a clearly discernible unit of expression, both in mouse and human. CONCLUSIONS: Our study provides a new framework for comparative analysis which will be applicable also to other sets of large-scale phenotypic data collected across different species.

Nickel incorporation in Fe(II, III) hydroxysulfate Green Rust: effect on crystal lattice spacing and oxidation products

Ni(II)-Fe(II)-Fe(III) layered double hydroxides (LDH) or Ni-containing sulfate green rust (GR2) samples were prepared from Ni(II), Fe(II) and Fe(III) sulfate salts and analyzed with X ray diffraction. Nickel is readily incorporated in the GR2 structure and forms a solid solution between GR2 and a Ni(II)-Fe(III) LDH. There is a correlation between the unit cell a-value and the fraction of Ni(II) incorporated into the Ni(II)-GR2 structure. Since there is strong evidence that the divalent/trivalent cation ratio in GR2 is fixed at 2, it is possible in principle to determine the extent of divalent cation substitution for Fe(II) in GR2 from the unit cell a-value. Oxidation forms a mixture of minerals but the LDH structure is retained if at least 20 % of the divalent cations in the initial solution are Ni(II). It appears that Ni(II) is incorporated in a stable LDH structure. This may be important for two reasons, first for understanding the formation of LDHs, which are anion exchangers, in the natural environment. Secondly, this is important for understanding the fate of transition metals in the environment, particularly in the presence of reduced Fe compounds.

Max-convex decompositions for cooperative TU games

We show that any cooperative TU game is the maximum of a finite collection of convex games. This max-convex decomposition can be refined by using convex games with non-negative dividends for all coalitions of at least two players. As a consequence of the above results we show that the class of modular games is a set of generators of the distributive lattice of all cooperative TU games. Finally, we characterize zero-monotonic games using a strong max-convex decomposition

Lattice-dynamical study of the premartensitic state of the Cu-Al-Be alloys

Neutron-scattering techniques have been used to study the premartensitic state of a family of Cu-Al-Be alloys, which transform from the bcc phase to an 18R martensitic structure. We find that the phonon modes of the TA2[110] branch have very low energies with anomalous temperature dependence. A slight anomaly at q=2/3 was observed; this anomaly, however, does not change significantly with temperature. No elastic peaks, related to the martensite structure, were found in the premartensitic state of these alloys. The results are compared with measurements, performed under the same instrumental conditions, on two Cu-Al-Ni and a Cu-Zn-Al martensitic alloy.

Interaction of a discrete breather with a lattice junction

We study the scattering of a moving discrete breather (DB) on a junction in a Fermi-Pasta-Ulam chain consisting of two segments with different masses of the particles. We consider four distinct cases: (i) a light-heavy (abrupt) junction in which the DB impinges on the junction from the segment with lighter mass, (ii) a heavy-light junction, (iii) an up mass ramp in which the mass in the heavier segment increases continuously as one moves away from the junction point, and (iv) a down mass ramp. Depending on the mass difference and DB characteristics (frequency and velocity), the DB can either reflect from, or transmit through, or get trapped at the junction or on the ramp. For the heavy-light junction, the DB can even split at the junction into a reflected and a transmitted DB. The latter is found to subsequently split into two or more DBs. For the down mass ramp the DB gets accelerated in several stages, with accompanying radiation (phonons). These results are rationalized by calculating the Peierls-Nabarro barrier for the various cases. We also point out implications of our results in realistic situations such as electron-phonon coupled chains.

From lattice-gas models to nonlinear diffusion models: A derivation of macroscopic equations and fluctuations

We derive nonlinear diffusion equations and equations containing corrections due to fluctuations for a coarse-grained concentration field. To deal with diffusion coefficients with an explicit dependence on the concentration values, we generalize the Van Kampen method of expansion of the master equation to field variables. We apply these results to the derivation of equations of phase-separation dynamics and interfacial growth instabilities.