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In this work, we propose to define a new space that will be called locally Cartesian space (Saidou’Spaces). It is a space defined by a new topology whose fundamental system of neighborhood are compact cartesian subsets, theses subsets were already defined during our previous work, “Geometrical introduction to Cartesian geometry in complex shapes see [15]. It will have the same properties of a Banach space. Then, we will give the steps of a cartesianization method of complex shapes or region in this space. The objective sought by this technique is to find a cartesian form (denoted Car(A)), that approximates any form A, see Figure1 below. It should be noted that cartesianization will be a generalization of the polygonization whish remains the object of research for several researchers in the spaces R2 Ref [12] [13] [6].

Actually, additive manufacturing (AM) is considered as a major class of complex parts manufacturing technologies. Including a wide range of materials, a huge set of physico-chemical phenomena are involved and adapted to control and master the variety of materials processing. On the other side, AM still knows a low productivity rate, due to different reasons that are mainly related to the material-process interaction control. In this paper, the authors propose a novel 2D scanning strategy that could be adapted to AM processes such as Laser Beam Powder Bed Fusion of Metals (PBF-LB/M) and polymers (PBF-LB/P), Electron Beam Powder Bed Fusion for metals (PBF-EB/M), and Material Extrusion-based (MEx) (ISO/ASTM 52,900 standards). The novelty presented corresponds to a Skeleton Based Perpendicularly (SBP) scanning strategy that aims to reduce the scanning lengths, and thus the production time and processing energy. The competitiveness of the new technique is mainly discussed according to the hatch space distance and the dimensions of a rectangular shape that was selected for the proof of concept of this new scanning strategy. In other words, it is proposed to investigate the competitiveness of the new scanning technique compared to four classical scanning strategies that are widely used in AM in term of process productivity. A detailed benchmark analysis has been applied to the following strategies: chess, stripe, spiral, and contour scanning. An analytical mathematical modeling was developed leading to the evaluation of the performance of SBP scanning compared to the scanning benchmark strategies by means of two proposed geometrical indices: the “gain of length” and the “specific gain of length per surface unit”. These were exploited in two separate study cases. The simulation showed that the SBP scanning length exhibits an increasing quadratic dependence on rectangle dimensions and a decreasing hyperbolic behavior according to hatch space distance. The lengths of the benchmark scanning strategies also presented a hyperbolic decreasing behavior according to hatch space distance. After that, it was proved that the SBP strategy is absolutely competitive compared to chess and stripe scanning; the competitiveness fluctuates around 95%, but it is highly concentrated around 100%. Otherwise, for the contour and stripe strategies, it has been shown that the competitiveness is strongly affected by the hatch space distance and by the dimensions of the shape being scanned. A particular behavior of the feasibility percentage of decision variable combinations, was detected as power laws or polynomials of the hatch space distance in the case of SBP/spiral comparison. In this case, the competitiveness (feasibility) ranged from 20 to 98% while it ranged from 0 to 75% in the case of SBP/contour comparison. The results of this study could also constitute a major contribution to related scientific and technical fields concerned with optimal area or volume control.

Convex non linear optimization problems may not have a solution in innite dimension spaces. The aim of this paper is to formulate some new results in this topic by using "technical regularization" of the objective function. The rst result shows that a non linear convex proper lower semi-continuous function, on a Banach space which have the Radon-Nikodym property, could be minimized by using a small regularization. while the second one shows that this regularization can be chosen as small as required. In addition, application tracks are presented and illustrated by elementary examples.

In this paper we will establish a new relationship between differentiability and optimization of convex functions. We consider convex lower semi continuous functions in Banach spaces and we characterize respectively the relation between Fr´echet (Gaˆteaux) differentiability and *strong minimum (*weak minimum) in the dual.

Optimization of geometric structures and shapes is currently an important part of applied research. the aim of this work is to use some mathematical methods to give tools for mounting geometric structures characterized by a self-similarity (fractals for example). The natural continuation of this montage is the transition to an automation of the 3D printing of objects. The mathematical approach used is based on iterated functions system (IFS). Indeed, based on an appropriate IFS, we define an operator W, as a superposition of all functions of the system, starting with a contraction of the geometric form representing the basic structure. The model we are trying to develop aims to create the equation of cellular structure and to apply the developed technique to a certain multiplicity number to obtain the desired structure.

Open source and closed software security has been debated for decades, vulnerabilities reported for both types of software has been under scrutiny for years. In this study, a descriptive and correlation study for two selected systems is conducted using SPSS 19. The results show that the severity score average of Linux kernel vulnerabilities is lower by %30 than the severity score average of vulnerabilities of windows server 2008, and the time required to patch published vulnerabilities is relatively faster for commercial than open source software. However, a positive correlation between Windows server 2008's variables reveal the irony that severity scores and patch delays are moving in the same direction. The contribution of this article is to shed light on the perception about the security divide between open source and closed software under study that is, although the commercial companies respond faster, however, not according to the severity of vulnerabilities disclosed.

The Cartesian word or “Cartesianity” was born with the philosophy of Descart (1596 - 1650). He was at the base of a doctrine based on rationalism, that it is means the search for truth by reason. Among others, Sigmend Freud had also approached this notion of psychological point to study the enigma of thoughts in humans. Other aspects of the Cartesian word have been used in mathematical geometry, namely cartesian coordinates and Cartesian referentials. As you know, studying a shape with curved and enclosed borders is more complicated than working on shapes with linear borders without curvature. In the way, we will introduce to the Cartesian geometry and characterize he Cartesian shapes.

Nous mettons en évidence des propriétés géométriques de fonctions convexes semi-continues inférieurement et nous caractérisons les espaces de Banach ayant la propriété de Radon-Nikodym par des propriétés géométriques des fonctions convexes qui y sont définies.