Recently, the rapid development of automotive industries has given rise to large multidimensional datasets both in the production sites and after-sale services. Fault diagnostic systems are one of the services that the automotive industries provide. As a consequence of the rapid development of cars features, traditional rule-based diagnostic systems became very limited. Therefore, more sophisticated AI approaches need to be investigated towards more efficient solutions. In this paper, we focus on utilising deep learning so as to build a diagnostic system that is able to estimate the required services in an efficient and effective way. We propose a new model, called Deep Symptoms-Based Model Deep-SBM, as an approach to predict a wide range of faults by relying on the deep learning technique. The new proposed model is validated through a set of experiments in order to demonstrate how the underlying model runs and its impact on improving the overall performance metrics. We have applied the Deep-SBM on a real historical diagnostic data provided by Cognitran Ltd. The performance of the Deep-SBM was compared against the state-of-the-art approaches and better result has been reported in terms of accuracy, precision, recall, and F-Score. Based on the obtained results, some further directions are suggested in this context. The final goal is having fault prediction data collected online relying on IoT.Index Terms-AI, deep learning, deep neural network, vehicle fault diagnosis, Internet of Things (IoT).

We investigate linear relations in Mordell-Weil groups of abelian varieties over finitely generated fields over Q. Based on important and classical results for abelian varieties over these fields and on lifts of abelian varieties to suitable abelian schemes, we prove theorems concerning the reduction maps on torsion and non-torsion elements in Mordell-Weil groups of these varieties. These theorems and the arithmetic of abelian schemes and their endomorphism algebras are our key tools in the solutions of linear relation problems we work with in the last chapter of this paper.

We investigate possible orders of reductions of a point in the Mordell-Weil groups of certain abelian varieties and in direct products of the multiplicative group of a number field. We express the result obtained in terms of divisibility sequences.Let B be an abelian group with finite torsion subgroup and r v : B → B v be an infinite family of group homomorphisms whose targets B v are finite abelian groups. We will use the following notation:tors the torsion subgroup of B e the exponent of B tors ord T the order of a torsion point T ∈ B ord v P the order of a point P mod v.We impose the following assumption on the family r v : B → B v :• For every point P ∈ B of infinite order and for almost every natural number n there exists v such that ord v P = n.Theorem.a point of infinite order such that the points P 1 , . . . , P k are pairwise linearly dependent over Z then for every sufficiently large integer n there exists v such that ord v (P 1 , . . . , P k ) = en.Let B be the Mordell-Weil group of an elliptic curve E over a number field K. If v is a prime ideal in O K of good reduction then r v : B → B v is the reduction map E(K) → E v (k v ). In this case the assumption we have imposed holds; moreover, for all but finitely many P there exists such a prime v for each n > 0. This was proved by J. Silverman [3] for K = Q and then by J. Cheon and S. Hahn [1] for arbitrary number fields K.

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