Simple tangential families and perestroikas of their envelopes

Tangential families are 1-parameter families of rays emanating tangentially from smooth curves. We classify tangential family germs up to Left-Right equivalence: we prove that there are two infinite series and four sporadic simple singularities of tangential family germs (in addition to two stable singularities). We give their normal forms and miniversal tangential deformations (i.e., deformations among tangential families), and we describe the corresponding envelope perestroikas of small codimension. We also discuss envelope singularities of non simple tangential families.Comment: 11 page

Generic singularities of minimax solutions to Hamilton--Jacobi equations

Minimax solutions are weak solutions to Cauchy problems involving Hamilton--Jacobi equations, constructed from generating families quadratic at infinity of their geometric solutions. We give a complete description of minimax solutions and we classify their generic singularities of codimension not greater than 2.Comment: To appear in Journal of Geometry and Physic

Stable tangential families and singularities of their envelopes

We study tangential families, i.e. systems of rays emanating tangentially from given curves. We classify, up to Left-Right equivalence, stable singularities of tangential family germs (under deformations among tangential families) and we study their envelopes. We discuss applications of our results to the case of tangent geodesics of a curve

Legendrian graphs generated by Tangential Families

We construct a Legendrian version of Envelope theory. A tangential family is a 1-parameter family of rays emanating tangentially from a smooth plane curve. The Legendrian graph of the family is the union of the Legendrian lifts of the family curves in the projectivized cotangent bundle $PT^*\R^2$. We study the singularities of Legendrian graphs and their stability under small tangential deformations. We also find normal forms of their projections into the plane. This allows to interprete the beaks perestroika as the apparent contour of a deformation of the Double Whitney Umbrella singularity $A_1^\pm$

On geodesic envelopes and caustics

We give a global description of envelopes of geodesic tangents of regular curves in (not necessarily convex) Riemannian surfaces. We prove that such an envelope is the union of the curve itself, its inflectional geodesics and its tangential caustics (formed by the conjugate points to those of the initial curve along the tangent geodesics). Stable singularities of tangential caustics and geodesic envelopes are discussed. We also prove the (global) stability of tangential caustics of close curves in convex closed surfaces under small deformations of the initial curve and of the ambient metric.Comment: 7 pp. 1 figure. 2nd versio

RISK MANAGEMENT THROUGH INSURANCE AND ENVIRONMENTAL EXTERNALITIES FROM AGRICULTURAL INPUT USE: AN ITALIAN CASE STUDY

The biological nature of agricultural production processes induce a higher degree of uncertainty surrounding the economic performance of farm enterprises. This has contributed to the development and acceptance of forms of public intervention aimed at reducing income variability that have no parallel in other sectors of the economy. In particular, subsidized crop insurance are a widely used tool. The impact of these programs on the decisions of production generates effects on input use, land use and thus, indirectly, environmental outcomes. The importance of this issue has grown in parallel with the growth in importance of the collective role of agriculture sector that has addressed the recent guidelines adopted by many developed countries. To examine the effects of public risk management programs on optimal nitrogen fertilizer use and land allocation to crops, this study carried out an empirical analysis by developing a mathematical programming model of a representative wheat-tomato farm in Apulia southern region of Italy. The model endogenizes nitrogen fertilizer rates and land allocation, as well as the insurance coverage levels, participation in insurance programs and the Environmental Payment (EP). This study utilized direct expected utility maximizing non-linear programming in combination with a simulation approach. Results show that with current crop insurance programs, the optimal nitrogen fertilizer rate slightly increases and the optimal acreage substantially increases for tomato whereas decrease for wheat. Assuming that the environmental negative effects of crop insurance are positively related to nitrogen fertilizer use, this type of public intervention implies negative environmental effects.Uncertainty, Risk Management, Crop Insurance, Input Use Decisions, Environmental Externalities, Mathematical Programming., Agricultural and Food Policy, Environmental Economics and Policy, Research Methods/ Statistical Methods, Risk and Uncertainty, Q10, Q14,

Statistical applications of the multivariate skew-normal distribution

Azzalini & Dalla Valle (1996) have recently discussed the multivariate skew-normal distribution which extends the class of normal distributions by the addition of a shape parameter. The first part of the present paper examines further probabilistic properties of the distribution, with special emphasis on aspects of statistical relevance. Inferential and other statistical issues are discussed in the following part, with applications to some multivariate statistics problems, illustrated by numerical examples. Finally, a further extension is described which introduces a skewing factor of an elliptical density.Comment: full-length version of the published paper, 32 pages, with 7 figures, uses psfra