This article makes the key observation that when using cylindrical algebraic decomposition (CAD) to solve a problem with respect to a set of polynomials, it is not always the signs of those polynomials that are of paramount importance but rather the truth values of certain quantifier free formulae involving them. This motivates our definition of a Truth Table Invariant CAD (TTICAD). We generalise the theory of equational constraints to design an algorithm which will efficiently construct a TTICAD for a wide class of problems, producing stronger results than when using equational constraints alone. The algorithm is implemented fully in Maple and we present promising results from experimentation.

When using cylindrical algebraic decomposition (CAD) to solve a problem with respect to a set of polynomials, it is likely not the signs of those polynomials that are of paramount importance but rather the truth values of certain quantifier free formulae involving them. This observation motivates our article and definition of a Truth Table Invariant CAD (TTICAD).In ISSAC 2013 the current authors presented an algorithm that can efficiently and directly construct a TTICAD for a list of formulae in which each has an equational constraint. This was achieved by generalising McCallum's theory of reduced projection operators. In this paper we present an extended version of our theory which can be applied to an arbitrary list of formulae, achieving savings if at least one has an equational constraint. We also explain how the theory of reduced projection operators can allow for further improvements to the lifting phase of CAD algorithms, even in the context of a single equational constraint.The algorithm is implemented fully in Maple and we present both promising results from experimentation and a complexity analysis showing the benefits of our contributions.

Cylindrical algebraic decomposition (CAD) is an important tool for the study of real algebraic geometry with many applications both within mathematics and elsewhere. It is known to have doubly exponential complexity in the number of variables in the worst case, but the actual computation time can vary greatly. It is possible to offer different formulations for a given problem leading to great differences in tractability. In this paper we suggest a new measure for CAD complexity which takes into account the real geometry of the problem. This leads to new heuristics for choosing: the variable ordering for a CAD problem, a designated equational constraint, and formulations for truth-table invariant CADs (TTICADs). We then consider the possibility of using Groebner bases to precondition TTICAD and when such formulations constitute the creation of a new problem.Comment: To appear in: Proceedings of Conferences on Intelligent Computer Mathematics (CICM '13) - Calculemus stran

We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve y 4 = x 5 + λ 4 x 4 + λ 3 x 3 + λ 2 x 2 + λ 1 x + λ 0 . We construct Abelian functions using the multivariate σ-function associated to the curve, generalising the theory of the Weierstrass ℘-function.We demonstrate that such functions can give a solution to the KP-equation, outlining how a general class of solutions could be generated using a wider class of curves. We also present the associated partial differential equations satisfied by the functions, the solution of the Jacobi Inversion Problem, a power series expansion for σ(u) and a new addition formula.

-The social media network phenomenon leads to a massive amount of valuable data that is available online and easy to access. Many users share images, videos, comments, reviews, news and opinions on different social networks sites, with Twitter being one of the most popular ones. Data collected from Twitter is highly unstructured, and extracting useful information from tweets is a challenging task. Twitter has a huge number of Arabic users who mostly post and write their tweets using the Arabic language. While there has been a lot of research on sentiment analysis in English, the amount of researches and datasets in Arabic language is limited. This paper introduces an Arabic language dataset which is about opinions on health services and has been collected from Twitter. The paper will first detail the process of collecting the data from Twitter and also the process of filtering, pre-processing and annotating the Arabic text in order to build a big sentiment analysis dataset in Arabic. Several Machine Learning algorithms (Naïve Bayes, Support Vector Machine and Logistic Regression) alongside Deep and Convolutional Neural Networks were utilized in our experiments of sentiment analysis on our health dataset.

Abstract. Cylindrical algebraic decomposition(CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. When using CAD, there is often a choice for the ordering placed on the variables. This can be important, with some problems infeasible with one variable ordering but easy with another. Machine learning is the process of fitting a computer model to a complex function based on properties learned from measured data. In this paper we use machine learning (specifically a support vector machine) to select between heuristics for choosing a variable ordering, outperforming each of the separate heuristics.

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When building a cylindrical algebraic decomposition (CAD) savings can be made in the presence of an equational constraint (EC): an equation logically implied by a formula.The present paper is concerned with how to use multiple ECs, propagating those in the input throughout the projection set. We improve on the approach of McCallum in ISSAC 2001 by using the reduced projection theory to make savings in the lifting phase (both to the polynomials we lift with and the cells lifted over). We demonstrate the benefits with worked examples and a complexity analysis.