Key polynomials for simple extensions of valued fields

Let $\iota:K\hookrightarrow L\cong K(x)$ be a simple transcendental extension of valued fields, where $K$ is equipped with a valuation $\nu$ of rank 1. That is, we assume given a rank 1 valuation $\nu$ of $K$ and its extension $\nu'$ to $L$. Let $(R_\nu,M_\nu,k_\nu)$ denote the valuation ring of $\nu$. The purpose of this paper is to present a refined version of MacLane's theory of key polynomials, similar to those considered by M. Vaqui\'e, and reminiscent of related objects studied by Abhyankar and Moh (approximate roots) and T.C. Kuo. Namely, we associate to $\iota$ a countable well ordered set $$\mathbf{Q}=\{Q_i\}_{i\in\Lambda}\subset K[x];$$ the $Q_i$ are called {\bf key polynomials}. Key polynomials $Q_i$ which have no immediate predecessor are called {\bf limit key polynomials}. Let $\beta_i=\nu'(Q_i)$. We give an explicit description of the limit key polynomials (which may be viewed as a generalization of the Artin--Schreier polynomials). We also give an upper bound on the order type of the set of key polynomials. Namely, we show that if $\operatorname{char}\ k_\nu=0$ then the set of key polynomials has order type at most $\omega$, while in the case $\operatorname{char}\ k_\nu=p>0$ this order type is bounded above by $\omega\times\omega$, where $\omega$ stands for the first infinite ordinal.Comment: arXiv admin note: substantial text overlap with arXiv:math/060519

An integrated model for computer assisted diagnosis, treatment and design of insoles for the diabetic foot

The incidence of diabetes has increased significantly in recent decades. In Germany, there is an estimate of three million diabetics and this number is growing at a rate of about 2 percent per year. In the U.S.A., the American Diabetes Association estimates that thirteen million people suffer from this condition, representing 5.2 percent of the entire population and every year, some 35 thousand patients have a lower limb amputated. In Latin America, it has been reported that the prevalence of diabetes is of the order of 14 to 20 percent, according to a research conducted by the WHO's Ad Hoc Diabetes Reporting Group. In Colombia, a study by Ashner et al concludes that the prevalence is 7 percent in both sexes

Ballistic Localization in Quasi-1D Waveguides with Rough Surfaces

Structure of eigenstates in a periodic quasi-1D waveguide with a rough surface is studied both analytically and numerically. We have found a large number of "regular" eigenstates for any high energy. They result in a very slow convergence to the classical limit in which the eigenstates are expected to be completely ergodic. As a consequence, localization properties of eigenstates originated from unperturbed transverse channels with low indexes, are strongly localized (delocalized) in the momentum (coordinate) representation. These eigenstates were found to have a quite unexpeted form that manifests a kind of "repulsion" from the rough surface. Our results indicate that standard statistical approaches for ballistic localization in such waveguides seem to be unappropriate.Comment: 5 pages, 4 figure

Photocurable Bioink for the Inkjet 3D Pharming of Hydrophilic Drugs.

Novel strategies are required to manufacture customized oral solid dosage forms for personalized medicine applications. 3D Pharming, the direct printing of pharmaceutical tablets, is an attractive strategy, since it allows for the rapid production of solid dosage forms containing custom drug dosages. This study reports on the design and characterization of a biocompatible photocurable pharmaceutical polymer for inkjet 3D printing that is suitable for hydrophilic active pharmaceutical ingredients (API). Specifically, hyaluronic acid was functionalized with norbornene moieties that, in the presence of poly(ethylene) glycol dithiol, Eosin Y as a photoinitiator, and a visible light source, undergoes a rapid step-growth polymerization reaction through thiol-ene chemistry. The engineered bioink was loaded with Ropinirole HCL, dispensed through a piezoelectric nozzle onto a blank preform tablet, and polymerized. Drug release analysis of the tablet resulted in 60% release within 15 min of tablet dissolution. The study confirms the potential of inkjet printing for the rapid production of tablets through the deposition of a photocurable bioink designed for hydrophilic APIs

The Odyssey Approach for Optimizing Federated SPARQL Queries

Answering queries over a federation of SPARQL endpoints requires combining data from more than one data source. Optimizing queries in such scenarios is particularly challenging not only because of (i) the large variety of possible query execution plans that correctly answer the query but also because (ii) there is only limited access to statistics about schema and instance data of remote sources. To overcome these challenges, most federated query engines rely on heuristics to reduce the space of possible query execution plans or on dynamic programming strategies to produce optimal plans. Nevertheless, these plans may still exhibit a high number of intermediate results or high execution times because of heuristics and inaccurate cost estimations. In this paper, we present Odyssey, an approach that uses statistics that allow for a more accurate cost estimation for federated queries and therefore enables Odyssey to produce better query execution plans. Our experimental results show that Odyssey produces query execution plans that are better in terms of data transfer and execution time than state-of-the-art optimizers. Our experiments using the FedBench benchmark show execution time gains of at least 25 times on average.Comment: 16 pages, 10 figure

Present Limits on the Precision of SM Predictions for Jet Energies

We investigate the impact of theoretical uncertainties on the accuracy of measurements involving hadronic jets. The analysis is performed using events with a Z boson and a single jet observed in $p\bar{p}$ collisions at $\sqrt{s}$ = 1.96 TeV in 4.6 $\mathrm{fb^{-1}}$ of data from the Collider Detector at Fermilab (CDF). The transverse momenta (\pt) of the jet and the boson should balance each other due to momentum conservation in the plane transverse to the direction of the $p$ and $\bar{p}$ beams. We evaluate the dependence of the measured \pt-balance on theoretical uncertainties associated with initial and final state radiation, choice of renormalization and factorization scales, parton distribution functions, jet-parton matching, calculations of matrix elements, and parton showering. We find that the uncertainty caused by parton showering at large angles is the largest amongst the listed uncertainties. The proposed method can be re-applied at the LHC experiments to investigate and evaluate the uncertainties on the predicted jet energies. The distributions produced at the CDF environment are intended for comparison to those from modern event generators and new tunes of parton showering.Comment: Submitted to Nucl. Instr. and Meth.