Need Polynomial Systems Be Doubly-Exponential?

Polynomial Systems, or at least their algorithms, have the reputation of being doubly-exponential in the number of variables [Mayr and Mayer, 1982], [Davenport and Heintz, 1988]. Nevertheless, the Bezout bound tells us that that number of zeros of a zero-dimensional system is singly-exponential in the number of variables. How should this contradiction be reconciled? We first note that [Mayr and Ritscher, 2013] shows that the doubly exponential nature of Gr\"{o}bner bases is with respect to the dimension of the ideal, not the number of variables. This inspires us to consider what can be done for Cylindrical Algebraic Decomposition which produces a doubly-exponential number of polynomials of doubly-exponential degree. We review work from ISSAC 2015 which showed the number of polynomials could be restricted to doubly-exponential in the (complex) dimension using McCallum's theory of reduced projection in the presence of equational constraints. We then discuss preliminary results showing the same for the degree of those polynomials. The results are under primitivity assumptions whose importance we illustrate.Comment: Extended Abstract for ICMS 2016 Presentation. arXiv admin note: text overlap with arXiv:1605.0249

The Potential and Challenges of CAD with Equational Constraints for SC-Square

Cylindrical algebraic decomposition (CAD) is a core algorithm within Symbolic Computation, particularly for quantifier elimination over the reals and polynomial systems solving more generally. It is now finding increased application as a decision procedure for Satisfiability Modulo Theories (SMT) solvers when working with non-linear real arithmetic. We discuss the potentials from increased focus on the logical structure of the input brought by the SMT applications and SC-Square project, particularly the presence of equational constraints. We also highlight the challenges for exploiting these: primitivity restrictions, well-orientedness questions, and the prospect of incrementality.Comment: Accepted into proceedings of MACIS 201

Improved cross-validation for classifiers that make algorithmic choices to minimise runtime without compromising output correctness

Our topic is the use of machine learning to improve software by making choices which do not compromise the correctness of the output, but do affect the time taken to produce such output. We are particularly concerned with computer algebra systems (CASs), and in particular, our experiments are for selecting the variable ordering to use when performing a cylindrical algebraic decomposition of $n$-dimensional real space with respect to the signs of a set of polynomials. In our prior work we explored the different ML models that could be used, and how to identify suitable features of the input polynomials. In the present paper we both repeat our prior experiments on problems which have more variables (and thus exponentially more possible orderings), and examine the metric which our ML classifiers targets. The natural metric is computational runtime, with classifiers trained to pick the ordering which minimises this. However, this leads to the situation were models do not distinguish between any of the non-optimal orderings, whose runtimes may still vary dramatically. In this paper we investigate a modification to the cross-validation algorithms of the classifiers so that they do distinguish these cases, leading to improved results.Comment: 16 pages. Accepted into the Proceedings of MACIS 2019. arXiv admin note: text overlap with arXiv:1906.0145

Enhanced sensitivity to higher ozone in a pathogen-resistant tobacco cultivar

Investigations of the effects of elevated ozone (O3) on the virus–plant system were conducted to inform virus pathogen management strategies better. One susceptible cultivar of tobacco (Nicotiana tabacum L. cv. Yongding) and a resistant cultivar (Nicotiana tabacum L. cv. Vam) to Potato virus Y petiole necrosis strain (PVYN) infection were grown in open-top chambers under ambient and elevated O3 concentrations. Above-ground biomass, foliage chlorophyll, nitrogen and total non-structural carbohydrate (TNCs), soluble protein, total amino acid (TAA) and nicotine content, and peroxidase (POD) activity were measured to estimate the effects of elevated O3 on the impact of PVYN in the two cultivars. Results showed that under ambient O3, the resistant cultivar possessed greater biomass and a lower C/N ratio after infection than the susceptible cultivar; however, under elevated O3, the resistant cultivar lost its biomass advantage but maintained a lower C/N ratio. Variation of foliar POD activity could be explained as a resistance cost which was significantly correlated with biomass and C/N ratio of the tobacco cultivar. Chlorophyll content remained steady in the resistant cultivar but decreased significantly in the susceptible cultivar when stressors were applied. Foliar soluble protein and free amino acid content, which were related to resistance cost changes, are also discussed. This study indicated that a virus-resistant tobacco cultivar showed increased sensitivity to elevated O3 compared to a virus-sensitive cultivar

Recent advances in real geometric reasoning

In the 1930s Tarski showed that real quantifier elimination was possible, and in 1975 Collins gave a remotely practicable method, albeit with doubly-exponential complexity, which was later shown to be inherent. We discuss some of the recent major advances in Collins method: such as an alternative approach based on passing via the complexes, and advances which come closer to "solving the question asked" rather than "solving all problems to do with these polynomials"

Comparing machine learning models to choose the variable ordering for cylindrical algebraic decomposition

There has been recent interest in the use of machine learning (ML) approaches within mathematical software to make choices that impact on the computing performance without affecting the mathematical correctness of the result. We address the problem of selecting the variable ordering for cylindrical algebraic decomposition (CAD), an important algorithm in Symbolic Computation. Prior work to apply ML on this problem implemented a Support Vector Machine (SVM) to select between three existing human-made heuristics, which did better than anyone heuristic alone. The present work extends to have ML select the variable ordering directly, and to try a wider variety of ML techniques. We experimented with the NLSAT dataset and the Regular Chains Library CAD function for Maple 2018. For each problem, the variable ordering leading to the shortest computing time was selected as the target class for ML. Features were generated from the polynomial input and used to train the following ML models: k-nearest neighbours (KNN) classifier, multi-layer perceptron (MLP), decision tree (DT) and SVM, as implemented in the Python scikit-learn package. We also compared these with the two leading human constructed heuristics for the problem: Brown's heuristic and sotd. On this dataset all of the ML approaches outperformed the human made heuristics, some by a large margin.Comment: Accepted into CICM 201

Satisfiability Checking and Symbolic Computation

Symbolic Computation and Satisfiability Checking are viewed as individual research areas, but they share common interests in the development, implementation and application of decision procedures for arithmetic theories. Despite these commonalities, the two communities are currently only weakly connected. We introduce a new project SC-square to build a joint community in this area, supported by a newly accepted EU (H2020-FETOPEN-CSA) project of the same name. We aim to strengthen the connection between these communities by creating common platforms, initiating interaction and exchange, identifying common challenges, and developing a common roadmap. This abstract and accompanying poster describes the motivation and aims for the project, and reports on the first activities.Comment: 3 page Extended Abstract to accompany an ISSAC 2016 poster. Poster available at http://www.sc-square.org/SC2-AnnouncementPoster.pd

Correction of beta-thalassemia major by gene transfer in haematopoietic progenitors of pediatric patients

Beta-thalassemia is a common monogenic disorder due to mutations in the beta-globin gene and gene therapy, based on autologous transplantation of genetically corrected haematopoietic stem cells (HSCs), holds the promise to treat patients lacking a compatible bone marrow (BM) donor. We recently showed correction of murine beta-thalassemia by gene transfer in HSCs with the GLOBE lentiviral vector (LV), expressing a transcriptionally regulated human beta-globin gene. Here, we report successful correction of thalassemia major in human cells, by studying a large cohort of pediatric patients of diverse ethnic origin, carriers of different mutations and all candidates to BM transplantation. Extensive characterization of BM-derived CD34(+) cells before and following gene transfer shows the achievement of high frequency of transduction, restoration of haemoglobin A synthesis, rescue from apoptosis and correction of ineffective erythropoiesis. The procedure does not significantly affect the differentiating potential and the relative proportion of haematopoietic progenitors. Analysis of vector integrations shows preferential targeting of transcriptionally active regions, without bias for cancer-related genes. Overall, these results provide a solid rationale for a future clinical translation

A pilot randomised controlled trial investigating a mindfulness-based stress reduction (MBSR) intervention in individuals with pulmonary arterial hypertension (PAH): the PATHWAYS study

Background: Pulmonary arterial hypertension (PAH) is an uncommon condition with progressive heart failure and premature death. Treatment costs up to £120,000 per patient per year, and the psychological burden of PAH is substantial. Mindfulness-based stress reduction (MBSR) is an intervention with the potential to reduce this burden, but to date, it has not been applied to people with pulmonary hypertension. We wished to determine whether a trial of MBSR for people with PAH would be feasible. Methods: A customised gentle MBSR programme of eight sessions was developed for people with physical disability due to PAH, and they were randomised to group-based MBSR or treatment as usual. The completeness of outcome measures including Beck Anxiety Index, Beck Depression Inventory and standard physical assessment at 3 months after randomisation were recorded. Health care utilisation was measured. Attendance at the sessions and the costs involved in delivering the intervention were assessed. Semi-structured interviews were conducted to explore the acceptability of the MBSR intervention and when appropriate the reasons for trial non-participation. Results: Fifty-two patients were recruited, but only 34 were randomised due to patients finding it difficult to travel to sessions. Twenty-two completed all questionnaires and attended all clinics, both routine and additional in order to collect outcomes measures. The MSBR sessions were delivered in Bristol, Cardiff and London, costing, on average, between £2234 (Cardiff) and £4128 (London) per patient to deliver. Attendance at each session averaged between two patients in Bristol and Cardiff and three in London. For those receiving treatment as usual, clinician blinding was achievable. Interviews revealed that people who attended MBSR found it interesting and helpful in managing their symptoms and minimising the psychological component of their disease. Conclusions: We found that attendance at group MBSR was poor in people with chronic PAH within the context of a trial. Achieving better MBSR intervention attendance or use of an Internet-based programme might maximise the benefit of MBSR

Observation of an Exotic S=+1S=+1 Baryon in Exclusive Photoproduction from the Deuteron

In an exclusive measurement of the reaction $\gamma d \to K^+ K^- p n$, a narrow peak that can be attributed to an exotic baryon with strangeness $S=+1$ is seen in the $K^+n$ invariant mass spectrum. The peak is at $1.542\pm 0.005$ GeV/c$^2$ with a measured width of 0.021 GeV/c$^2$ FWHM, which is largely determined by experimental mass resolution. The statistical significance of the peak is $5.2 \pm 0.6 \sigma$. The mass and width of the observed peak are consistent with recent reports of a narrow $S=+1$ baryon by other experimental groups.Comment: 5 pages, 5 figure