We briefly present to our readers the late Prof: Galaye Dia in the honour to whom this collection of mathematical texts is published: He mainly focus oh this important role in the development of Mathematics in Africa, his carrier and his achievements both in teaching, research and dissertation supervision.

The present paper gives the solution of a Hartshorne conjecture: Every non singular real algebraic variety is affine. Other related results are provided.

The main aim of this article is to prove the following result.

Theorem 3. Let $K$ be a hilbertian field and $G$ a finite group. Then there exist a Galois extension $L$ of $K$ such that $Gal(L/K)$ is isomorphic to $G$.

We give a parameterization of the algebraic points of given degree over $\mathbb{Q}$ on the Picard curve given by $y^{3}=x^{4}-1$. This result extends a previous result of Klassen and Schaefer(1996) on the set of algebraic points of degree at most $3$ over $\mathbb{Q}$ in the same curve.

Azzalini (1985) introduced a family of skew symmetric distributions. His technique has been applied to skew many continuous distribution defined on the entire real axis. Very few people worked on circular or on distributions defined on finite intervals. In this paper it is considered a sine-skewed cardioid distribution generated by perturbation of symmetric Cardioid distribution. Several basic properties of the sine-skewed cardioid distribution will be presented. Based on the truncated moment some characterizations of this distribution are given.

We consider two statistical problems for inhomogeneous Poisson processes. The first one concerns the parameter estimation and the second one is about the goodness of fit test. We develop the method of moments in parameter estimation problem and describe the asymptotic behavior of the new test based on the method of moments estimator. We show that the constructed estimators are consistent and asymptotically normal and for the test we find the limit distribution of the test statistics under hypothesis and show its consistency.

In this paper we study the kernel estimator for the bidimensional extension of Foster, Greer and Thorbecke class of measures by Duclos et al. (2006a) for the purpose of a dominance approach to multidimensional poverty. The measure they used in their dominance exercise is essentially a generalization, from one to two dimensions, of the FGT index separate poverty aversion parameters for each dimension. The asymptotic normality of the estimator is established. We next indicate how the proposed estimator can generate sequential confidence intervals by a moving kernel process. Our results are extensions of those of Dia (2009) and of Ciss et al. (2016) in one dimension.

In this paper we propose an estimator of Foster, Greer and Thorbecke class of measures indexed by $\alpha>0$ and depending on a poverty line $z>0$ nonparameric methods. The estimator is constructed with adaptive kernel. Uniform almost sure consistency and uniform mean square consistency are established.

We analyze a discretized canonical stochastic volatility model through calibration to synthetic data as well as financial data. To achieve this end we resort to Monte Carlo EM (Chan and Ledolter (1995)) that is, the combination of EM algorithm (Dempster et al. (1977), Wu (1983)) and sequential Monte Carlo methods (Gordon et al. (1993), Doucet et al. (2001), Salmond and Smith (1993)). Finally we consider a slight departure from canonical stochastic volatility model in order to assess the robustness of the MCEM procedure. Simulations studies follow.

The aim of this paper is to establish the asymptotic behavior of the mutual influence of the Gini index and the poverty measures by using the Gaussian fields described in Mergane (2013). The results are given as representation theorems using the Gaussian fields of the unidimensional or the bidimensional functional Brownian bridges. Such representations, when combined with those already available, lead to joint asymptotic distributions with other statistics of interest like growth, welfare and inequality indices and then, unveil interesting results related to the mutual influence between them. The results are also appropriate for studying whether a growth is fair or not, depending on the variation of the inequality measure. Data driven applications are also available. Although the variances may seem complicated at a first sight, their computations which are needed to get confidence intervals of the indices, are possible with the help of R software. Beyond the current results, the provided representations are useful in connection with different ones of other statistics.

In the spirit of recent asymptotic works on the General Poverty Index (GPI) in the field of Welfare Analysis, the asymptotic statistical representation of the non-decomposable Takayama's index, which has failed to be incorporated in the unified GPI approach, is addressed and established here. This representation also allows to extend to it, recent results of statistical decomposability gaps estimations. The theoretical results are applied to real databases. The conclusions of the undertaken applications recommend to use Takayama's index as a practically decomposable one, in virtue of the low decomposability gaps with respect to the large values of the index.

The new Sine Skewed Cardioid (ssc) distribution been just introduced and characterized by Ahsanullah (2018). Here, we study the asymptotic properties of its tails by determining its extreme value domain, the characteristic function, the moments and likelihood estimators of the two parameters, the asymptotic normality of the moments estimators and the random generation of data from the ssc distribution. Finally, we proceed to a simulation study to show the performance of the random generation method and the quality of the moments estimation of the parameters.

The aim of this paper is to prove the following theorem.

Theorem 34. Let $X$ be a locally Hausdorff compact space, $\mu$ a Radon Nykodym on $X$ and $(f_{n})$ be a sequence of measurable functions (with respect to $\mu$) belonging to $\mathcal{L}^{p}(X,\mu)$ which converges in measure to a measurable function. Let $\={g}$ stand for the equivalence class of a measurable function $g$ with the equivalence relation $\mathcal R$ induced by the v-a.e equality and $\mathcal{L}^{p}(X,\mu)$ be the quotient by $\mathcal R$. Then the following conditions are equivalent.

• The function $\={f}$ belongs to $\mathcal{L}^{p}$ and $(\={f})_{n \ge 0}$ weakly converges to $\={f}$ in $\mathbb{L}^{p}$.

• The sequence $(\={f})_{n \ge 0}$ weakly converges in $\mathbb{L}^{p}$.

• The sequence is $(\={f})_{n \ge 0}$ is bounded $\mathbb{L}^{p}$.

Let $E$ be a uniformly convex real Banach space with uniformly Gâteaux differentiable norm and $E^*$ its dual space and let $A : E \rightarrow E^*$ be a bounded and uniformly monotone mapping such that $A^{-1}(0)\not= \varnothing$. In this paper, we introduce an new explicit iterative algorithm that converges strongly to the unique zeros of $A$. The results proved here are applied to the convex optimization problem.

We solve the $\partial\bar{\partial}$-problem for extendable currents defined on a half space of $\mathbb{C}^{n}$.

The Riemannian extension of torsion free affine manifolds $(M, \nabla)$ is an important method to produce pseudo-Riemannian manifolds. It is known that, if the manifold $(M, \nabla)$ is a torsion-free affine two-dimensional manifold with skew symmetric tensor Ricci, then $(M, \nabla)$ is affine Osserman manifold. In higher dimensions the skew symmetric of the tensor Ricci is a necessary but not sufficient condition for a affine connection to be Osserman. In this paper we construct affine Osserman connection with Ricci flat but not flat and example of Osserman pseudo-Riemannian metric of signature $(3,3)$ is exhibited.

In this paper, an identification method of a real connected set $\Sigma \subset \mathbb{R}^2$ is presented in the case of irrigation Dirichlet problem. We use conformal mapping tools in Complex Analysis to approximate the unknown $\Sigma$. Some numerical tests are given as illustrations.

We provide general central limit theorems (CLT's) for associated random variables following the approaches used by Newman (1980) and Olivera et al.(2012). Given some assumptions, a Lynderberg-Feller-Levy type theorem is stated. We next specify different particular CTL versions of associated sequences based on moment conditions. Comparisons and application are addressed too.

We establish homogenization results for semilinear partial differential equations (PDEs) on the half space according to the reflecting Neumann boundary condition and highly oscillating coefficients. Our method is based on the martingale property of the stochastic exponential and Girsanov transformation.

In this contribution, periodic and almost periodic stationary processes are respectively studied in the frequency domain. A relation of proximity is clearly established between one of the spectral tools associated with these processes: the associated random measure. This is a way to consider, for such processes, that the filters resulting from Principal Components Analysis in the frequency domain are close.

Recently, several strong limit theorems for the oscillation moduli of the empirical process have been given in the iid-case. We show that, with very slight differences, those strong results are also obtained for some representation of the reduced empirical process based on the (nonoverlapping) k-spacings generated by a sequence of independent random variables (rv’s) uniformly distributed on $(0, 1)$. This yields weak limits for the mentioned process. Our study includes the case where the step $k$ is unbounded. The results are mainly derived from several properties concerning the increments of gamma functions with parameters $k$ and one.

All the available results on the approximation of the k-spacings process to Gaussian processes have only used one approach, that is the Shorack and Pyke’s one. Here, it is shown that this approach cannot yield a rate better than $(N/ \log\log N)^{−\frac{1}{4}}(\log N)^{\frac{1}{2}}$. Strong and weak bounds for that rate are specified both where k is fixed and where $k \rightarrow + \infty$. A Glivenko-Cantelli Theorem is given while Stute’s result for the increments of the empirical process based on independent and identically distributed random variables is extended to the spacings process. One of the Mason- Wellner-Shorack cases is also obtained.

Let $U_{1}$, $U_{2}$, ... be a sequence of independent and uniformly distributed random variables on $(0,1)$ defined on the same probability space. Let $U_{1,n} \le ...\le U_{n,n}$ be the order statistics of the sample $U_{1}$, $U_{2}$,...$U_{n}$ of size $n \geq 1$. Let $(k(n))_{n\geq 1}$ be a sequence of integers such that $1\leq k(n) \leq n$ and $k(n) \longrightarrow +\infty$. We prove that $nU_{k(n),n}/k(n) \longrightarrow 1$ a.s as $n \longrightarrow +\infty$. We take the opportunity to make a simple Round-up on the validity of different type of convergences when the sequence of random variables is replaced by another sequence preserving parts of the probability law of the original sequence.

Motivated by the fact that Floquet theory and averaging methods used to study the stability of linear periodic systems in continuous time, we formulate and analyze the dynamics of a nonlinear and non-autonomous system of ordinary differential equations describing the dynamics of an invasive reproductive plant: the Typha. Its two modes of reproduction namely; sexual (via seeds) and asexual (via rizhomes) are included into the hybrid system which combines the features of classical continuous time and discrete time systems. Stability of the null equilibrium is investigated via the basic reproduction rate Ro of the model in the absence of Typha is computed. For $R_{0,\alpha} < 1$ the a useful too which can be applied to analyze the stability of models with seasonality. The theory of averaging is based on the construction of approximate solutions essentially first-order differential equation with rapidly oscillating ordinary. A condition of stability of the trivial equilibrium of the switching system is given. Numerical simulations to support the analytical results are provided.

In this paper, we propose a fast numerical method to solve a two dimensional non stationary Navier-Stokes problem: the projection method. The advantages of this method are, on the one hand it can be applied all types of flows (laminar, viscous and turbulent) and on the other hand, its convergence analysis is faster, compared to classical numerical methods (Newton method, Jacobian Fixed method).

In this paper, we propose a numerical method to study phononic and photonic crystal problems. The proposed method is based on topological optimization tools. In fact, after modeling crystal photonics and phononic problems, we use topological optimization tools to build a numerical method for getting optimal design. Here the optimal design is the one in which all frequencies near the reference frequency corresponding to the reference wave length a can pass.

This paper is devoted to numerical simulations of sand transport problems submitted to the tide near the seabed. We characterized at first a corrector result from two-scale convergence. We aim also to do numerical simulation for short term dynamical of dunes presented in Faye et al. (2011) and Thiam et al. (2018). We consider a two-scale numerical approach based on finite element method. The stability of the scheme is established and finally, we present some numerical results.

This paper is devoted to numerical simulations of sand transport problem submitted to the tide near the seabed. We consider a two-scale numerical approach based on finite element method. The stability of the scheme is solved and finally, we present some numerical results.

We give an (up-to-date) overview of existing encoding and hash functions into (hyper)elliptic curves. We also design an indifferentiable hash function into the Jacobian of certain families of hyperelliptic curves of genus $g\leq 5$ using the unified formulas of Seck and Diarra (AFRICACRYPT-2018).

In that paper, we explore the notion of equality through four simple but fundamental objects of mathematical structure: numbers, functions in particular Boolean functions, sets, and the algorithmic assignment. (The paper is in French.)