Mathematics of Computer Science, Cybersecurity and Artificial Intelligence

This text is a presentation given during the doctoral sessions of the Ecole Doctorale Mathématiques et Informatique at Cheikh Anta Diop University. Our aim was simply to arouse the curiosity of those present, especially doctoral students in Mathematics, Computer Science, and Telecommunications. We set ourselves the goal of demonstrating the close link between Mathematics and Computer Science. After an overview of the background, we went on to look at some of the research topics that have interested us over the last 10 years, and which have the particularity of having a close link with Computer Science, with Mathematics as the main tool at our disposal.

This chapter is an attempt to popularize science. It relies heavily on the INRIA seminar https://​fidle.​cnrs.​fr/​w3/​. Its objectives are on one hand to present NLP to PhD students and other researchers and on the other hand to take stock of the research done in Senegal in the field of NLP. We first introduce the basic definitions and concepts of NLP such as bags of words, TF-IDF, etc., before presenting new products such as LLMs and multimodal models. Finally, we have published publications obtained over the past five years by our research group in the field of NLP, PhDs, and master’s thesis defended (Kandé et al., icetas.etssm.org; Kandé et al., FWLSA-score: French and Wolof lexicon-based for sentiment analysis, 2019; Kandé et al., Vector space model of text classification based on inertia contribution of document, 2019; Kandé et al., A novel term weighting scheme model, 2018; Kandji and Ndiaye, Design and realization of an NLP application for the massive processing of large volumes of resumes, 2022; Samb et al., Improved bilingual sentiment analysis lexicon using word-level trigram, 2019).

Let \(\mathcal {A}\) be an absolute-valued algebra with nonzero element a such that a and \(a^{2}\) are central, then \(\mathcal {A}\) is pre-Hilbert space and admits an involution. We also show that if \(\mathcal {A}\) is an absolute-valued algebra with nonzero central element satisfying \((x,x^{2},x)=(x^{2},x^{2},x^{2})=0\), then \(\mathcal {A}\) is finite-dimensional, flexible, and isomorphic to either \(\mathbb {R}\), \(\mathbb {C}\), \(\stackrel {\star }{\mathbb {C}}\), \(\mathbb {H}\), \(\stackrel {\star }{\mathbb {H}}\), \(\mathbb {O}\), or \(\stackrel {\star }{\mathbb {O}}\).

Let \(\mathcal {A}\) be an algebraic algebra without divisors of zero of degree \(\neq 8\) with a nonzero idempotent e such that \([e, I(\mathcal {A})]=0\) (resp., e is omnipresent). Then the following assertions are equivalent:

The study of Gröbner–Shirshov bases in a field, where the set of monomials is a semiring, was first extended to valuation rings by Yatma et al. In a subsequent development, S.Diop et al. further generalized these methods to the setting of D-A rings (divisible and annihilable rings), preserving the semiring structure for the set of monomials, whether commutative or not. Using the approach introduced by Diop et al. in the commutative case, we propose a new technique for computing a minimal free resolution for the ideal I as an R module. This extension of the method shows the variety and applicability of the Gröbner–Shirshov basis framework in various algebraic settings.

The Schur complement theory is very important in many areas such as statistics, matrix analysis, numerical analysis, and control theory. It is a powerful tool to discuss many significant results. In this chapter, we establish two inequalities on the eigenvalues of Schur complement of the block Hadamard product, deduce one important corollary, and illustrate them in numerical examples.

Let E be a uniformly convex and uniformly smooth real Banach space and \(E^*\) its dual. Let \(A:E\to E^*\) be a bounded maximal monotone mapping such that \(A^{-1}(0)\neq \emptyset \). We first introduce the algorithm: For given \(x_1\in E\), let \(\{x_n\}\) be generated by the formula: \(x_{n+1}= x_n -\lambda _n J^{-1}Ax_n -\lambda _n\theta _n(x_n-x_1),\,n\geq 1\), where J is the normalized duality mapping from E into \(E^*\) and \(\lambda _n\) and \(\theta _n\) are positive real numbers in \((0,1)\) satisfying suitable conditions. Next, we obtain the strong convergence of the sequence \(\{x_n\}\) to the solution of the equation \(Au=0\) closest to the initial point \(x_1\). Using this result, we deal with the convex minimization problem. Our results improve and unify most of the ones that have been proved in this direction for this important class of nonlinear mappings. Furthermore, our new technique of proof is of independent interest.

This chapter generalizes the Rickart (resp., Baer) property on semirings and semimodules. We introduce weak Rickart (resp., Baer ) semimodules and then identify i-Rickart semimodules as a specific subclass of the former. Basic links between the different Baer and Rickart semimodules are discussed. A characterization of the Rickart semimodules by their endomorphism semiring is provided.

In this chapter, \((A, (I_n)_{n \in \mathbb {N}})\) is a filtered noncommutative ring, S is a saturated multiplicative subset of A satisfying the left Ore conditions, and \((M, (M_n)_{n \in \mathbb {N}})\) is a filtered left A-module.

The main results in this chapter are the following theorems:

The notion of lifting module is well studied in rings and module theory. Recently, many concepts in rings and modules were introduced in semirings and semimodules such us radical of semiring, projective covers of semimodules, and superfluous subsemimodules. In this chapter, we introduce the notion of s-lifting semimodules, and we study their properties.

Considering a field \(\mathbb {K}\) of characteristic 0, the n-variate commutative polynomial ring \(\mathbb {K}[x_1,\ldots ,x_n]\) over \(\mathbb {K}\), the n-variate noncommutative polynomial ring \(\mathbb {K}\langle X_1,\ldots ,X_n\rangle \) over \(\mathbb {K}\), and \(\gamma :\mathbb {K}\langle X_1,\ldots ,X_n\rangle \longrightarrow \mathbb {K}[x_1,\ldots ,x_n]\) the application sending \(X_i\) to \(x_i\), Eisenbud et al. proved that for any ideal \(\mathcal {I}\) of \(\mathbb {K}[x_1,\ldots ,x_n]\), the ideal \(\mathcal {J}=\gamma ^{-1}(\mathcal {I})\) has a finite Gröbner basis.

Y. Diop and D. Sow dealt with the opposite problem and proved that any noncommutative ideal which contains all commutators and has a finite Gröbner basis is a preimage of a commutative ideal by \(\gamma \).

In this work, we prove that this application \(\gamma \) can be replaced by any surjective homomorphism. Thus we generalize the two results previously cited.

Let x be a positive real number and b an integer greater than or equal to 2. In this chapter, we will construct from the expansion of x in base b an another real number y such that:

Missing observations constitute one of the most important issues in data analysis in applied research studies. The magnitude and their structure impact parameters estimation in the modeling with important consequences for decision-making. This chapter aims to evaluate the efficiency of imputation methods combined with the backpropagation algorithm in a nonlinear regression context. The evaluation is conducted through a simulation study including sample sizes (50, 100, 200, 300, and 400) with different missing data rates (10, 20, 30 40, and 50%) and three missingness mechanisms (MCAR, MAR, and MNAR). Four imputation methods (Last Observation Carried Forward, Random Forest, Amelia, and MICE) were used to impute datasets before making prediction with backpropagation algorithm. 3-MLP model was used by varying the activation functions (Logistic-Linear, Logistic-Exponential, TanH-Linear, and TanH-Exponentiel), the number of nodes in the hidden layer (3–15), and the learning rate (20–70%). Analysis of the performance criteria (\(R^2, r\), and RMSE) of the network revealed good performances when it is trained with TanH-Linear functions, 11 nodes in the hidden layer, and a learning rate of 50%. MICE and Random Forest were the most appropriate for data imputation. These methods can support up to 50% of missing rate with an optimal sample size of 200.

A novel class of Random Forests (RFs), namely Mondrian Forests (MFs), which are an ensemble of Mondrian Trees, achieves competitive performance relatively to classical Breiman RFs. They have attractive properties like performing Bayesian inference at the tree level and being trainable online. However, they perform poorly in the presence of less or low predictive power features. Thus, we propose to extend MF by using label information during splits in order to make them more accurate and robust. We showed an increase in performance when using labels during splits on four datasets where we notice a big improvement on a dataset containing many non-predictive features which is very important as feature relevancy is unknown at first. Additionally, this extension yields equal or superior performance relatively to classical RFs.

The antipersonnel and anti-tank landmines create a lot of human and material damage in the Sahel countries affected by terrorism. Explosive mine detection methods are based on tools handled by human operators and target industrial metal mines. These methods are risky and limited because the types of mines most commonly used in the Sahelian context are mainly homemade and are encased in various local materials such as metal, plastic, glass, ceramic, or wood. This chapter presents a solution for remote sensing of artisanal mines buried in the ground using infrared thermography. A DJI Phantom 4 Quadcopter equipped with a FLIR thermal camera and a GNSS sensor performs an automatic low-level flyover of the potentially mined road. Thermal images of the road are collected with an overlap rate of 80% and referenced with the GNSS sensor. Photogrammetry algorithms are used to process the thermal images to detect and locate anomalies related to the presence of buried mines. Despite the limitations due to environmental influences, the model showed a detection rate of 75% during flights at an altitude of 6 m and a speed of 3 m/s. The experimental results show a good correlation between the thermal contrast of the mathematical model and the cooler areas containing a mine-related chemical substance.

Blockchain technology is widely used across various domains for its security and distributed ledger capabilities. To secure transactions, most blockchain platforms such as Ethereum employ the Elliptic Curve Digital Signature Algorithm (ECDSA).

However, the use of ECDSA can pose risks, such as the inadvertent exposure of the private key in case of errors, thus facilitating obtaining corresponding signatures for various documents. To address this issue, a solution emerges: the integration of the Edwards-curve Digital Signature Algorithm (EdDSA). By opting for EdDSA to generate transaction signatures, several advantages emerge, such as increased speed, optimal performance, and enhanced independence in random number generation. Indeed, this innovative proposition significantly bolsters security compared to the conventional use of ECDSA, marking a substantial advancement within the Ethereum ecosystem.

Furthermore, we have implemented both algorithms to sign and verify Ethereum transactions to make a performance comparison. The implementation is carried out in Python on an Intel Core i3 processor with 8 GB of RAM and a 64-bit operating system.

With the rise of new information technologies around the world, many distributed applications and web applications have emerged, so it is important to make them secure. Despite the emphasis placed by software security experts on the need to build secure web applications, the number of new vulnerabilities found in web applications is growing. Machine Learning (ML) and Deep Learning (DL) through their vulnerability prediction approach are increasingly being offered for source code analysis, providing a powerful way to make web applications less vulnerable. Many ML- and DL-based approaches have been proposed to automatically detect, locate, and repair software vulnerabilities. Although ML-based are more effective than vulnerability analysis tools based on static source code analysis by security experts, accurately identifying types of vulnerabilities and estimations severity remains challenging. The graphical representation of source code, the best vulnerability differentiation, and the support of a large corpus of vulnerabilities are not at least. This thesis aims to study the prediction of vulnerabilities in web applications from source codes using ML and DL techniques. A comprehensive review of the literature on the different approaches proposed for the prediction of vulnerabilities of web applications will allow us to identify the current state of research and challenges in this field, thus positioning us well to make a significant contribution in the prediction of vulnerabilities of web applications using the techniques of ML and DL.

This review addresses the integration of Business Process Management (BPM) and Process Mining (PM) in the context of Industry 4.0’s digital transformation. It highlights how BPM enhances business process efficiency, reducing costs and boosting profit, while PM, leveraging data from event logs, offers insights into actual process execution, bridging data and process science. Despite their complementary nature, the extent of BPM and PM integration remains underexplored. This review synthesizes existing literature and identifies future research directions, aiming to inform researchers and practitioners in these evolving fields.