Speeding up Cylindrical Algebraic Decomposition by Gr\"obner Bases

Gr\"obner Bases and Cylindrical Algebraic Decomposition are generally thought of as two, rather different, methods of looking at systems of equations and, in the case of Cylindrical Algebraic Decomposition, inequalities. However, even for a mixed system of equalities and inequalities, it is possible to apply Gr\"obner bases to the (conjoined) equalities before invoking CAD. We see that this is, quite often but not always, a beneficial preconditioning of the CAD problem. It is also possible to precondition the (conjoined) inequalities with respect to the equalities, and this can also be useful in many cases.Comment: To appear in Proc. CICM 2012, LNCS 736

A Factorization Algorithm for G-Algebras and Applications

It has been recently discovered by Bell, Heinle and Levandovskyy that a large class of algebras, including the ubiquitous $G$-algebras, are finite factorization domains (FFD for short). Utilizing this result, we contribute an algorithm to find all distinct factorizations of a given element $f \in \mathcal{G}$, where $\mathcal{G}$ is any $G$-algebra, with minor assumptions on the underlying field. Moreover, the property of being an FFD, in combination with the factorization algorithm, enables us to propose an analogous description of the factorized Gr\"obner basis algorithm for $G$-algebras. This algorithm is useful for various applications, e.g. in analysis of solution spaces of systems of linear partial functional equations with polynomial coefficients, coming from $\mathcal{G}$. Additionally, it is possible to include inequality constraints for ideals in the input

Complotype affects the extent of down-regulation by Factor I of the C3b feedback cycle in vitro.

Sera from a large panel of normal subjects were typed for three common polymorphisms, one in C3 (R102G) and two in Factor H (V62I and Y402H), that influence predisposition to age-related macular degeneration and to some forms of kidney disease. Three groups of sera were tested; those that were homozygous for the three risk alleles; those that were heterozygous for all three; and those homozygous for the low-risk alleles. These groups vary in their response to the addition of exogenous Factor I when the alternative complement pathway is activated by zymosan. Both the reduction in the maximum amount of iC3b formed and the rate at which the iC3b is converted to C3dg are affected. For both reactions the at-risk complotype requires higher doses of Factor I to produce similar down-regulation. Because iC3b reacting with the complement receptor CR3 is a major mechanism by which complement activation gives rise to inflammation, the breakdown of iC3b to C3dg can be seen to have major significance for reducing complement-induced inflammation. These findings demonstrate for the first time that sera from subjects with different complement alleles behave as predicted in an in-vitro assay of the down-regulation of the alternative complement pathway by increasing the concentration of Factor I. These results support the hypothesis that exogenous Factor I may be a valuable therapeutic aid for down-regulating hyperactivity of the C3b feedback cycle, thereby providing a treatment for age-related macular degeneration and other inflammatory diseases of later life.This is the author's accepted manuscript and will be under embargo until the 13th of August 2015. This final version is available from Wiley in Clinical & Experimental Immunology at http://onlinelibrary.wiley.com/doi/10.1111/cei.12437/abstract

Introduction of organised mammography screening in tyrol: results of a one-year pilot phase

Background: Efficiency and efficacy of organised mammography screening programs have been proven in large randomised trials. But every local implementation of mammography screening has to check whether the well established quality standards are met. Therefore it was the aim of this study to analyse the most common quality indices after introducing organised mammography screening in Tyrol, Austria, in a smooth transition from the existing system of opportunistic screening. Methods: In June 2007, the system of opportunistic mammography screening in Tyrol was changed to an organised system by introducing a personal invitation system, a training program, a quality assurance program and by setting up a screening database. All procedures are noted in a written protocol. Most EU recommendations for organised mammography screening were followed, except double reading. All women living in Tyrol and covered by social insurance are now invited for a mammography, in age group 40-59 annually and in age group 60-69 biannually. Screening mammography is offered mainly by radiologists in private practice. We report on the results of the first year of piloting organised mammography screening in two counties in Tyrol. Results: 56,432 women were invited. Estimated participation rate was 34.5% at one year of follow-up (and 55.5% at the second year of follow-up); 3.4% of screened women were recalled for further assessment or intermediate screening within six months. Per 1000 mammograms nine biopsies were performed and four breast cancer cases detected (N = 68). Of invasive breast cancer cases 34.4% were ≤ 10 mm in size and 65.6% were node-negative. In total, six interval cancer cases were detected during one year of follow-up; this is 19% of the background incidence rate. Conclusions: In the Tyrolean breast cancer screening program, a smooth transition from a spontaneous to an organised mammography screening system was achieved in a short time and with minimal additional resources. One year after introduction of the screening program, most of the quality indicators recommended by the European guidelines had been reached. However, it will be necessary to introduce double reading, to change the rule for BI-RADS 3, and to concentrate on actions toward improving the participation rate

Multiplicity of solutions to GW-type approximations

We show that the equations underlying the $GW$ approximation have a large number of solutions. This raises the question: which is the physical solution? We provide two theorems which explain why the methods currently in use do, in fact, find the correct solution. These theorems are general enough to cover a large class of similar algorithms. An efficient algorithm for including self-consistent vertex corrections well beyond $GW$ is also described and further used in numerical validation of the two theorems

On elliptic solutions of the quintic complex one-dimensional Ginzburg-Landau equation

The Conte-Musette method has been modified for the search of only elliptic solutions to systems of differential equations. A key idea of this a priory restriction is to simplify calculations by means of the use of a few Laurent series solutions instead of one and the use of the residue theorem. The application of our approach to the quintic complex one-dimensional Ginzburg-Landau equation (CGLE5) allows to find elliptic solutions in the wave form. We also find restrictions on coefficients, which are necessary conditions for the existence of elliptic solutions for the CGLE5. Using the investigation of the CGLE5 as an example, we demonstrate that to find elliptic solutions the analysis of a system of differential equations is more preferable than the analysis of the equivalent single differential equation.Comment: LaTeX, 21 page

Automated Generation of User Guidance by Combining Computation and Deduction

Herewith, a fairly old concept is published for the first time and named "Lucas Interpretation". This has been implemented in a prototype, which has been proved useful in educational practice and has gained academic relevance with an emerging generation of educational mathematics assistants (EMA) based on Computer Theorem Proving (CTP). Automated Theorem Proving (ATP), i.e. deduction, is the most reliable technology used to check user input. However ATP is inherently weak in automatically generating solutions for arbitrary problems in applied mathematics. This weakness is crucial for EMAs: when ATP checks user input as incorrect and the learner gets stuck then the system should be able to suggest possible next steps. The key idea of Lucas Interpretation is to compute the steps of a calculation following a program written in a novel CTP-based programming language, i.e. computation provides the next steps. User guidance is generated by combining deduction and computation: the latter is performed by a specific language interpreter, which works like a debugger and hands over control to the learner at breakpoints, i.e. tactics generating the steps of calculation. The interpreter also builds up logical contexts providing ATP with the data required for checking user input, thus combining computation and deduction. The paper describes the concepts underlying Lucas Interpretation so that open questions can adequately be addressed, and prerequisites for further work are provided.Comment: In Proceedings THedu'11, arXiv:1202.453

Reducing Bit-Vector Polynomials to SAT using Gröbner Bases

We address the satisfiability of systems of polynomial equations over bit-vectors. Instead of conventional bit-blasting, we exploit word-level inference to translate these systems into non-linear pseudo-boolean constraints. We derive the pseudo-booleans by simulating bit assignments through the addition of (linear) polynomials and applying a strong form of propagation by computing Gröbner bases. By handling bit assignments symbolically, the number of Gröbner basis calculations, along with the number of assignments, is reduced. The final Gröbner basis yields expressions for the bit-vectors in terms of the symbolic bits, together with non-linear pseudo-boolean constraints on the symbolic variables, modulo a power of two. The pseudo-booleans can be solved by translation into classical linear pseudo-boolean constraints (without a modulo) or by encoding them as propositional formulae, for which a novel translation process is described

EFFECT OF TIME STEP AND DATA AGGREGATION ON CROSS CORRELATION OF RESIDENTIAL DEMANDS

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