A characterization by optimization of the Monge point of a tetrahedron

Abstract. "... nihil omnino in mundo contingint, in quo non maximi minimive ratio quapiam eluceat", translated into "... nothing in all the world will occur in which no maximum or minimum rule is somehow shining forth", used to say L.Euler in 1744. This is confirmed by numerous applications of mathematics in physics, mechanics, economy, etc. In this note, we show that it is also the case for the classical "centers" of a tetrahedron, more specifically for the so-called Monge point (the substitute of the notion of orthocenter for a tetrahedron). To the best of our knowledge, the characterization of the Monge point of a tetrahedron by optimization, that we are going to present, is new. To begin with... What kind of tetrahedron? Let T = ABCD be a tetrahedron in the three dimensional space R 3 (equipped with the usual Euclidean and affine structures); the points A, B, C, D are supposed not to lie in a plane, of course. We begin with two particular types of tetrahedra and, then, with increase in generality, we can classify the tetrahedra into several classes. Here they are: -The regular tetrahedron. This tetrahedron enjoys so many symmetries that it is not very interesting from the optimization viewpoint: all the "centers" usually associated with a tetrahedron (and that we are going to visit again in the next paragraph) coincide. -The trirectangular tetrahedra. They are generalizations to the space of rectangular triangles in the plane. A trirectangular tetrahedron OABC has (two by two) three perpendicular faces OBC, OAB, OAC and a "hypothenuse-face" ABC; such a tetrahedron enjoys a remarkable relationship between areas of its faces (see -The orthocentric tetrahedra. Curiously enough, the four altitudes of a tetrahedron generally do not meet at a point; when this happens, the tetrahedron is called orthocentric. A common characterization of orthocentric tetrahedra is as follows: a tetrahedron is orthocentric if and only if the opposite edges (two by two) are orthogonal. This class of tetrahedra is by far the most studied one in the literature. Regular and trirectangular tetrahedra are indeed orthocentric. -General tetrahedra. Like for triangles, three specific "centers" can be defined for any tetrahedron: the centroid or isobarycenter, the incenter and the circumcenter. We shall see their characterization by optimization, as for some other points, in the next section. As said before, the altitudes do not necessarily meet at a point; moreover, the projection of any vertex on the opposite face does not necessarily coincide with the orthocenter of this face. The notion of orthocenter will be held by a new point: the so-called Monge point

Wheat yield robustness to abiotic and biotic perturbations : methodological framework and agronomic drivers

Face à un contexte croissant d'incertitude, les systèmes agricoles doivent être performants, non seulement dans des conditions moyennes, mais aussi quand les perturbations sont plus importantes. Récemment, de nombreux concepts ont été développés pour étudier la durabilité des systèmes dans des environnements changeants, dont celui de robustesse. Néanmoins, son transfert aux systèmes de grandes cultures reste limité. L'objectif de la thèse est alors de proposer un cadre d'évaluation de la robustesse qui soit opérationnel en conditions de grandes cultures. Nous nous sommes focalisés sur la culture du blé tendre et avons défini la robustesse comme la capacité d'un système agricole à maintenir ses performances de rendement malgré l'apparition de perturbations. Un modèle économétrique défini à l'échelle du système de culture a été développé pour évaluer cette robustesse face à des conditions météorologiques et des niveaux de pression maladies changeants. Il a été appliqué sur 145 exploitations agricoles françaises et 2300 parcelles de blé enquêtées sur la période 2011-2014. Les résultats montrent que la robustesse aux perturbations abiotiques et biotiques varie d'un système de culture à l'autre. Les systèmes les plus robustes (respectivement, les moins robustes) ont pu être identifiés, en tenant compte des niveaux de rendement atteints en conditions moyennes et de la sensibilité de ces rendements à des variations climatiques. Alors que les situations de rendements "élevés" versus "faibles" s'expliquent surtout par des pratiques agronomiques dites d'intensification, ce sont surtout des variables dites de flexibilité qui expliquent la robustesse versus la sensibilité aux perturbations abiotiques. Au-delà de l'intérêt de cette approche pour mieux appréhender la robustesse des systèmes agricoles, la méthodologie développée présente l'avantage de pouvoir être appliquée à d'autres performances et/ou d'autres aléas.Given increasing uncertainties surrounding the future of agriculture, farming systems need to perform well both in average conditions and in situations presenting substantial variations. Different concepts have been developed in recent years to assess the sustainability of agricultural systems within a context of global change, including the concept of robustness. But its empirical operationalization remains a challenge, particularly with regard to arable cropping systems. The aim of this thesis is to propose a methodological framework for assessing robustness which is operational for arable conditions. Focusing on wheat production, the robustness is defined as the ability of a cropping system to maintain yield performances despite the presence of perturbations. An econometric model defined at the cropping system level is used to assess yield robustness in the face of changing weather conditions and fungal disease pressures. It is then applied to data from 145 French wheat-growing farms and 2,300 wheat plots surveyed over the period 2011-2014. The results show that yield robustness varies widely from one cropping system to another. Cropping systems showing the most and the least robustness to abiotic perturbations were identified on the basis of yield performances under both normal and changing weather conditions. While several management intensification and crop rotation practices differentiate high versus low wheat yield cropping systems, it appears to be flexibility practices that distinguish robust versus sensitive cropping systems. Beyond the interest of this approach per se to understand and improve the robustness of agricultural systems, this methodological framework could also be used to assess other performances and/or risks

Robustesse du rendement du blé tendre face aux perturbations abiotiques et biotiques : cadre méthodologique et leviers agronomiques

Given increasing uncertainties surrounding the future of agriculture, farming systems need to perform well both in average conditions and in situations presenting substantial variations. Different concepts have been developed in recent years to assess the sustainability of agricultural systems within a context of global change, including the concept of robustness. But its empirical operationalization remains a challenge, particularly with regard to arable cropping systems. The aim of this thesis is to propose a methodological framework for assessing robustness which is operational for arable conditions. Focusing on wheat production, the robustness is defined as the ability of a cropping system to maintain yield performances despite the presence of perturbations. An econometric model defined at the cropping system level is used to assess yield robustness in the face of changing weather conditions and fungal disease pressures. It is then applied to data from 145 French wheat-growing farms and 2,300 wheat plots surveyed over the period 2011-2014. The results show that yield robustness varies widely from one cropping system to another. Cropping systems showing the most and the least robustness to abiotic perturbations were identified on the basis of yield performances under both normal and changing weather conditions. While several management intensification and crop rotation practices differentiate high versus low wheat yield cropping systems, it appears to be flexibility practices that distinguish robust versus sensitive cropping systems. Beyond the interest of this approach per se to understand and improve the robustness of agricultural systems, this methodological framework could also be used to assess other performances and/or risks.Face à un contexte croissant d'incertitude, les systèmes agricoles doivent être performants, non seulement dans des conditions moyennes, mais aussi quand les perturbations sont plus importantes. Récemment, de nombreux concepts ont été développés pour étudier la durabilité des systèmes dans des environnements changeants, dont celui de robustesse. Néanmoins, son transfert aux systèmes de grandes cultures reste limité. L'objectif de la thèse est alors de proposer un cadre d'évaluation de la robustesse qui soit opérationnel en conditions de grandes cultures. Nous nous sommes focalisés sur la culture du blé tendre et avons défini la robustesse comme la capacité d'un système agricole à maintenir ses performances de rendement malgré l'apparition de perturbations. Un modèle économétrique défini à l'échelle du système de culture a été développé pour évaluer cette robustesse face à des conditions météorologiques et des niveaux de pression maladies changeants. Il a été appliqué sur 145 exploitations agricoles françaises et 2300 parcelles de blé enquêtées sur la période 2011-2014. Les résultats montrent que la robustesse aux perturbations abiotiques et biotiques varie d'un système de culture à l'autre. Les systèmes les plus robustes (respectivement, les moins robustes) ont pu être identifiés, en tenant compte des niveaux de rendement atteints en conditions moyennes et de la sensibilité de ces rendements à des variations climatiques. Alors que les situations de rendements "élevés" versus "faibles" s'expliquent surtout par des pratiques agronomiques dites d'intensification, ce sont surtout des variables dites de flexibilité qui expliquent la robustesse versus la sensibilité aux perturbations abiotiques. Au-delà de l'intérêt de cette approche pour mieux appréhender la robustesse des systèmes agricoles, la méthodologie développée présente l'avantage de pouvoir être appliquée à d'autres performances et/ou d'autres aléas

Stability, robustness, vulnerability and resilience of agricultural systems. A review

Global warming and price volatility are increasing uncertainty for the future of agriculture. Therefore, agriculturalsystems must be sustainable not only under average conditions, but also under extreme changes of productivity, economy, environment and social context. Here, we review four concepts: stability, robustness, vulnerability and resilience. Those concepts are commonly used but are sometimes difficult to distinguish due to the lack of clear boundaries. Here, we clarify the role of these concepts in addressing agronomic issues. Our main findings are as follows: (1) agricultural systems face different types of perturbations, from small and usual perturbations to extreme and unpredictable changes; (2) stability, robustness, vulnerability and resilience have been increasingly applied to analyze the agricultural context in order to predict the system response under changing conditions; (3) the four concepts are distinguished by the nature of the system components and by the type of perturbation studied;(4) assessment methods must be tested under contrasted situations; and (5) the major options allowing system adaptation under extreme and unpredictable changes are the increase of diversity and the increase of the adaptive capacity

Does diversity affect dynamics of agricultural system facing perturbations?: -

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Dynamique des systèmes agricole face aux perturbations : review des méthodes d’évaluation et facteurs explicatifs.

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Influence of diversity and intensification level on vulnerability, resilience and robustness of agricultural systems

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Impacts of agricultural land use changes on pesticide use in French agriculture

International audiencePublic policies seeking to regulate pesticide use must be based on a clear identification of the factors influencing such use. Since the agricultural use of pesticides is primarily crop-dependent, agricultural land use change is potentially an important driver of the overall level of pesticide use in a given country. In this paper, we investigate the influence of agricultural land use changes on pesticide use in French agriculture over the period 1989-2013, during which important changes in the Common Agricultural Policy took place. Toward that end, we developed a method allowing the direct effects of agricultural land use changes to be disentangled from other factors affecting the intensity of pesticide use. On the basis of standard protection programs defined by crop protection experts,a fixed pesticide use intensity is estimated for 19 annual and perennial crops representing 90% of French arable land area and the bulk of pesticide use in French agriculture. These coefficients, combined with national agricultural land use statistics, are used to construct an artificial index of pesticide use in France whose variations depend solely on changes in agricultural land use. This index is calculated over the period 1989-2013. Our results indicate that the direct impacts of agricultural land use changes on pesticide use in France have varied depending on the time period considered, reflecting the influence of public regulations, notably the compulsory set-aside policy in force during the 1990s, and market conditions, particularly the context of high prices for cereal grains at the end of the 2000s. Over the six years from 2008 to 2013, this index is roughly constant, indicating that the 17% increase in French pesticide use in 2013 compared to 2008 (as assessed from annual pesticide sales) cannot be even partially attributed to agricultural land use changes. Since 2000, land use changes mainly corresponded to substitutions between crops with similar per-hectare pesticide use intensities, and/or to substitutions with counterbalancing impacts on these intensities. A prospective approach shows that other types of land use changes (e.g. a massive conversion of grassland to arable land or, conversely, a strong diversification of arable crop rotations), could have much higher impacts on pesticide use, with the effect of either offsetting or reinforcing efforts to reduce pesticide use intensity in arable crops. Thus, better coordination is needed between public policies aimed at regulating pesticide use and public policies influencing land use

European Conference on Crop Diversification Does diversity affect dynamics of agricultural system facing perturbations?

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