Plane curves whose tangent lines at collinear points are concurrent

We are interested in a particular geometry of plane curves in characteristicp>0, which was inspired by Thas's article [13]. We will prove that any plane curve of degree > 2 whose tangent lines at collinear points are concurrent is either a strange curve or projectively equivalent to the Fermat curve of degreeq + 1, whereq is a power ofp.     

On regular tangent cones

The paper contains a sufficient condition for an intersection of regular tangent cones to be a tangent cone. Regular tangent cones and tents for sets given by locally Lipschitz functions are constructed. The cones are described in terms of generalized K-derivatives.     

An infinite class of convex tangent cones

Since the early 1970's, there have been many papers devoted to tangent cones and their applications to optimization. Much of the debate over which tangent cone is best has centered on the properties of Clarke's tangent cone and whether other cones have these properties. In this paper, it is shown that there are an infinite number of tangent cones with some of the nicest properties of Clarke's cone. These properties are convexity, multiple characterizations, and proximal normal formulas. The nature of these cones indicates that the two extremes of this family of cones, the cone of Clarke and the B-tangent cone or the cone of Michel and Penot, warrant further study. The relationship between these new cones and the differentiability of functions is also considered.     

Homogeneous numerical semigroups

We introduce the concept of homogeneous numerical semigroups and show that all homogeneous numerical semigroups with Cohen–Macaulay tangent cones are of homogeneous type. In embedding dimension three, we classify all numerical semigroups of homogeneous type into numerical semigroups with complete intersection tangent cones and the homogeneous ones which are not symmetric with Cohen–Macaulay tangent cones. We also study the behavior of the homogeneous property by gluing and shiftings to construct large families of homogeneous numerical semigroups with Cohen–Macaulay tangent cones. In particular we show that these properties fulfill asymptotically in the shifting classes. Several explicit examples are provided along the paper to illustrate the property.     

On some classes of projectively flat Finsler metrics with constant flag curvature

In this paper, we give the equivalent PDEs for projectively flat Finsler metrics with constant flag curvature defined by a Euclidean metric and two 1-forms. Furthermore, we construct some classes of new projectively flat Finsler metrics with constant flag curvature by solving these equivalent PDEs.     

On the projective differential geometry of a non-holonome surface in ordinary space

Sunto. The paper deals with many projective properties of the neighbourhoods of the third,4th, 5th order of a point on a non-holonome surfaceV ₃ ² inS ₃. Chiefly two remarkable projective correspondences are studied, between the tangent plane and the bundle of directions, with his contact point as centre. Many generalizations are obtained of geometrical loci (so as lines, planes, quadrics) projectively associated with the neighbourhoods of a point on an ordinary (holonome) surface. The last sections are concerned with the extension to theV ₃ ² of the quadric ofMoutard and theSegre's correspondence.      

一类新的二阶组合切导数及其应用

引进了一种新的切锥,讨论它与相依切锥的关系.借助这种新的切锥引进了一类新的二阶组合切导数,并讨论了它与其他二阶切导数的关系.利用这类新的二阶组合切导数,建立了集值优化分别取得Henig有效元和全局有效元的最优性必要条件.     

Projectively Equivalent Riemannian Spaces as Quasi-bi-Hamiltonian Systems

The class of Riemannian spaces admitting projectively, or geodesically, equivalent metrics is very closely related to a certain class of spaces for which the Hamilton–Jacobi equation for geodesics is separable. This fact is established, and its consequences explored, by showing that when a Riemannian space has a projectively equivalent metric its geodesic flow is a quasi-bi-Hamiltonian system. The existence of involutive first integrals of the geodesic flow, quadratic in the momenta, follows by a standard type of argument. When these integrals are independent they generate a Stäckel system.     

一类射影平坦的Finsler度量

本文主要研究由两个Riemann度量和一个1-形式构成的Finsler度量.首先,本文给出这类度量局部射影平坦的等价条件;其次,给出这类度量局部射影平坦且具有常旗曲率的分类情形;最后,构造这类度量局部射影平坦且具有常旗曲率K=-1的例子.     

Canonical decomposition of projective and affine killing vectors on the tangent bundle

For an affine connection on the tangent bundle T(M) obtained by lifting an affine connection on M, the structure of vector fields on T(M) which generate local one-parameter groups of projective and affine collineations is described. On the T(M) of a complete irreducible Riemann manifold, every projective collineation is affine. On the T(M) of a projectively Euclidean space, every affine collineation preserves the fibration of T(M), and on the T(M) of a projectively non-Duclidean space which is maximally homogeneous (in the sense of affine collineations) there exist affine collineations permuting the fibers of T(M).Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 247–258, February, 1976.     

Non‐archimedean stratifications of tangent cones

We study the impact of a kind of non‐archimedean stratifications (t‐stratifications) on tangent cones of definable sets in real closed fields. We prove that such stratifications induce stratifications of the same nature on the tangent cone of a definable set at a fixed point. As a consequence, the archimedean counterpart of a t‐stratification is shown to induce Whitney stratifications on the tangent cones of a semi‐algebraic set. Extensions of these results are proposed for real closed fields with further structure.     

Characterization of seminuclear sets in a finite projective plane

LetC be a set ofq + a points in the desarguesian projective plane of orderq, such that each point ofC is on exactly 1 tangent, and onea+ 1-secant (a>1). Then eitherq=a + 2 andC consists of the symmetric difference of two lines, with one further point removed from each line, orq=2a + 3 andC is projectively equivalent to the set of points {(0,1,s),(s, 0, 1),(1,s, 0): -s is not a square inGF(q)}.     

On uniqueness of tangent cones for Einstein manifolds

We show that for any Ricci-flat manifold with Euclidean volume growth the tangent cone at infinity is unique if one tangent cone has a smooth cross-section. Similarly, for any noncollapsing limit of Einstein manifolds with uniformly bounded Einstein constants, we show that local tangent cones are unique if one tangent cone has a smooth cross-section.     

On generalized trade-off directions in nonconvex multiobjective optimization

Trade-off information related to Pareto optimal solutions is important in multiobjective optimization problems with conflicting objectives. Recently, the concept of trade-off directions has been introduced for convex problems. These trade-offs are characterized with the help of tangent cones. Generalized trade-off directions for nonconvex problems can be defined by replacing convex tangent cones with nonconvex contingent cones. Here we study how the convex concepts and results can be generalized into a nonconvex case. Giving up convexity naturally means that we need local instead of global analysis. Received: December 2000 / Accepted: October 2001?Published online February 14, 2002     

Intersection cuts for convex mixed integer programs from translated cones

We develop a general framework for linear intersection cuts for convex integer programs with full-dimensional feasible regions by studying integer points of their translated tangent cones, generalizing the idea of Balas (1971). For proper (i.e, full-dimensional, closed, convex, pointed) translated cones with fractional vertices, we show that under certain mild conditions all intersection cuts are indeed valid for the integer hull, and a large class of valid inequalities for the integer hull are intersection cuts, computable via polyhedral approximations. We also give necessary conditions for a class of valid inequalities to be tangent halfspaces of the integer hull of proper translated cones. We also show that valid inequalities for non-pointed regular translated cones can be derived as intersection cuts for associated proper translated cones under some mild assumptions.     

On optimality conditions for maximizations with respect to cones

For a Pareto maximization problem defined in infinite dimensions in terms of cones, relationships among several types of maximal elements are noted and optimality conditions are developed in terms of tangent cones.     

On the singularities of weakly normal varieties

Among the weakly normal varieties (in the sense of Andreotti and Bombieri, [1]) are of particular interest those varieties such that the normalization morphism is unramified outside a subvariety of codimension not less than 2. We describe the singularities of these varieties (called here WN1) by means of analytic equations, tangent cones, analytic branches and we show that any irreducible projective variety is birationally equivalent to a WN1 hypersur face and that a Gorenstein variety is weakly normal if and only if it is WN1.This research was done when the authors were members of G.N.S.A.G.A. of the C.N.R.     

Local approximation of uniformly regular Carnot-Carathéodory quasispaces by their tangent cones

We prove a local approximation theorem for the Carnot-Carathéodory quasimetrics on uniformly regular (equiregular) Carnot-Carathéodory spaces. Using this theorem, we study convergence of the Carnot-Carathéodory quasispaces to their tangent cones. In particular, we prove a Mitchell type theorem on convergence of an equiregular Carnot-Carathéodory quasispace with distinguished point to its tangent cone.