On time graphs

For n points on the real line, joining each pair of points such that their difference is less than a certain positive constant, we have a time graph. In this paper we characterize time graphs and enumerate them.     

Normal forms and invariants for 2-dimensional almost-Riemannian structures

2-Dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, there are three types of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not, and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket.In this paper we consider the problem of finding normal forms and functional invariants at each type of point. We also require that functional invariants are “complete” in the sense that they permit to recognize locally isometric structures.The problem happens to be equivalent to the one of finding a smooth canonical parameterized curve passing through the point and being transversal to the distribution.For Riemannian points such that the gradient of the Gaussian curvature K is different from zero, we use the level set of K as support of the parameterized curve. For Riemannian points such that the gradient of the curvature vanishes (and under additional generic conditions), we use a curve which is found by looking for crests and valleys of the curvature. For Grushin points we use the set where the vector fields are parallel.Tangency points are the most complicated to deal with. The cut locus from the tangency point is not a good candidate as canonical parameterized curve since it is known to be non-smooth. Thus, we analyse the cut locus from the singular set and we prove that it is not smooth either. A good candidate appears to be a curve which is found by looking for crests and valleys of the Gaussian curvature. We prove that the support of such a curve is uniquely determined and has a canonical parametrization.     

Limit Weierstrass points on nodal reducible curves

In the 1980s D. Eisenbud and J. Harris posed the following question: ``What are the limits of Weierstrass points in families of curves degenerating to stable curves not of compact type?' In the present article, we give a partial answer to this question. We consider the case where the limit curve has components intersecting at points in general position and where the degeneration occurs along a general direction. For this case we compute the limits of Weierstrass points of any order. However, for the usual Weierstrass points, of order one, we need to suppose that all of the components of the limit curve intersect each other.

     

Probabilistic analysis of some euclidean clustering problems

We are given n points distributed randomly in a compact region D of R^m. We consider various optimisation problems associated with partitioning this set of points into k subsets. For each problem we demonstrate lower bounds which are satisfied with high probability. For the case where D is a hypercube we use a partitioning technique to give deterministic upper bounds and to construct algorithms which with high probability can be made arbitrarily accurate in polynomial time for a given required accuracy.     

Rest points of generalized dynamical systems

In this paper we consider generalized dynamical systems whose integral vortex (that is, the set of all trajectories of the system starting at a given point) is an acyclic set in the corresponding space of curves. For such systems we apply the theory of fixed points for multi-valued maps in order to prove the existence of rest points. In this way we obtain new existence theorems for rest points of generalized dynamical systems. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 28–36, January, 1999.     

Structure of equilibria inN-person non-cooperative games

Here we study the structure of Nash equilibrium points forN-person games. For two-person games we observe that exchangeability and convexity of the set of equilibrium points are synonymous. This is shown to be false even for three-person games. For completely mixed games we get the necessary inequality constraints on the number of pure strategies for the players. Whereas the equilibrium point is unique for completely mixed two-person games, we show that it is not true for three-person completely mixed game without some side conditions such as convexity on the equilibrium set. It is a curious fact that for the special three-person completely mixed game with two pure strategies for each player, the equilibrium point is unique; the proof of this involves some combinatorial arguments.     

A generic approach to approximate efficiency and applications to vector optimization with set-valued maps

In this paper we focus on approximate minimal points of a set in Hausdorff locally convex spaces. Our aim is to develop a general framework from which it is possible to deduce important properties of these points by applying simple results. For this purpose we introduce a new concept of ε-efficient point based on set-valued mappings and we obtain existence results and properties on the behavior of these approximate efficient points when ε is fixed and by considering that ε tends to zero. Finally, the obtained results are applied to vector optimization problems with set-valued mappings.     

Algebraic geometric codes on anticanonical surfaces

Algebraic geometric codes (or AG codes) provide a way to correct errors that occur during the transmission of digital information. AG codes on curves have been studied extensively, but much less work has been done for AG codes on higher dimensional varieties. In particular, we seek good bounds for the minimum distance.We study AG codes on anticanonical surfaces coming from blow-ups of P² at points on a line and points on the union of two lines. We can compute the dimension of such codes exactly due to known results. For certain families of these codes, we prove an exact result on the minimum distance. For other families, we obtain lower bounds on the minimum distance.     

Extensions of the Appelrot classes for the generalized gyrostat in a double force field

For the integrable system on e(3, 2) found by Sokolov and Tsiganov we obtain explicit equations of some invariant 4-dimensional manifolds on which the induced systems are almost everywhere Hamiltonian with two degrees of freedom. These subsystems generalize the famous Appelrot classes of critical motions of the Kowalevski top. For each subsystem we point out a commutative pair of independent integrals, describe the sets of degeneration of the induced symplectic structure. With the help of the obtained invariant relations, for each subsystem we calculate the outer type of its points considered as critical points of the initial system with three degrees of freedom.     

Exceptional sets of hypergeometric series

For a family of transcendental hypergeometric series, we determine explicitly the set of algebraic points at which the series takes algebraic values (the so-called exceptional set). This answers a question of Siegel in special cases. For this, we first prove identities, each one relating locally one hypergeometric series to modular functions. In some cases, the identity and the theory of complex multiplication allow the determination of an infinite subset of the exceptional set. These subsets are shown to be the whole sets in using a consequence of Wüstholz's Analytic Subgroup Theorem together with mapping properties of Schwarz triangle functions. Further consequences of the identities are explicit evaluations of hypergeometric series at algebraic points. Some of them provide examples for Kroneckers Jugendtraum.     

Polygonizations of point sets in the plane

We examine the different ways a set ofn points in the plane can be connected to form a simple polygon. Such a connection is called apolygonization of the points. For some point sets the number of polygonizations is exponential in the number of points. For this reason we restrict our attention to star-shaped polygons whose kernels have nonempty interiors; these are callednondegenerate star-shaped polygons.We develop an algorithm and data structure for determining the nondegenerate star-shaped polygonizations of a set ofn points in the plane. We do this by first constructing an arrangement of line segments from the point set. The regions in the arrangement correspond to the kernels of the nondegenerate star-shaped polygons whose vertices are the originaln points. To obtain the data structure representing this arrangement, we show how to modify data structures for arrangements of lines in the plane. This data structure can be computed inO(n ⁴) time and space. By visiting the regions in this data structure in a carefully chosen order, we can compute the polygon associated with each region inO(n) time, yielding a total computation time ofO(n ⁵) to compute a complete list ofO(n ⁴) nondegenerate star-shaped polygonizations of the set ofn points.     

Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve

Periodic points are points on Veech surfaces, whose orbit under the group of affine diffeomorphisms is finite. We characterize those points as being torsion points if the Veech surfaces is suitably mapped to its Jacobian or an appropriate factor thereof. For a primitive Veech surface in genus two we show that the only periodic points are the Weierstraß points and the singularities. Our main tool is the Hodge-theoretic characterization of Teichmüller curves. We deduce from it a finiteness result for the Mordell-Weil group of the family of Jacobians over a Teichmüller curve. The link to the classification of periodic points is provided by interpreting them as sections of the family of curves over a covering of the Teichmüller curve.     

Pareto efficiency in locally convex spaces II: Stability results

In this second part of our work on Pareto efficiency we study the stability of the set of efficient points under perturbations of the data. For that purpose we use a general notion of convergence of sets which is known as the Kuratowski-Mosco convergence. We also determine the properties of Eff_K(S) as a function of the set S and of the dominance cone K. Finally our work is extended to stochastic Pareto efficiency and we derive several stability results, we determine the measurability of the efficiency multifunction and we obtain a new characterization of the efficient points of the aggregate set.     

The timelike cut locus and conjugate points in Lorentz 2-step nilpotent Lie groups

We investigate the timelike cut locus and the locus of conjugate points in Lorentz 2-step nilpotent Lie groups. For these groups with a timelike center, we give some criteria for the existence of conjugate points along timelike geodesics. We show for instance that a nonsingular timelike geodesic which is translated by an element of the group has a conjugate point. For those that we will call of GH-type, and for those which are globally hyperbolic with a timelike center and a one-dimensional derived subgroup, we show that if in addition the derived subgroup is spacelike, then nonsingular timelike geodesics maximize distance up to and including the first conjugate point. Other related results and corollaries are also obtained.Mathematics Subject Classification (2000): 53C50, 53C22, 22E25     

Equidistribution over function fields

We prove equidistribution of a generic net of small points in a projective variety X over a function field K. For an algebraic dynamical system over K, we generalize this equidistribution theorem to a small generic net of subvarieties. For number fields, these results were proved by Yuan and we transfer here his methods to function fields. If X is a closed subvariety of an abelian variety, then we can describe the equidistribution measure explicitly in terms of convex geometry.     

On the Distribution of Alternation Points in Uniform Polynomial Approximation of Entire Functions

We consider the distribution of alternation points in best real polynomial approximation of a functionf∈C[−1, 1]. For entire functionsfwe look for structural properties of f that will imply asymptotic equidistribution of the corresponding alternation points.     

REGULARITY OF THE FREE BOUNDARY FOR SOME ELLIPTIC AND PARABOLIC PROBLEMS. I

We study the free boundary of solutions to some obstacle problems in the elliptic and parabolic cases. In the one-phase Stefan problem, the parabolic case, we prove that the points where the zero set has no density lie in a Lipschitz surface in space and time.

For some fully nonlinear elliptic equations of second order, we get similar results.

Furthermore, we prove the C ¹ regularity for singular points with some (n ? 1)-dimensional density.     

Computer-based proof of the existence of superconvergence points in the finite element method; superconvergence of the derivatives in finite element solutions of Laplace's,Poisson's,and the elasticity equations

In this article we address the problem of the existence of superconvergence points for finite element solutions of systems of linear elliptic equations. Our approach is quite different from all other studies of superconvergence. We prove that the existence of superconvergence points can be guaranteed by a numerical algorithm, which employs a finite number of operations (provided that there is no roundoff-error). By employing this approach, we can reproduce all known results on superconvergence of finite element solutions for linear elliptic problems and we can obtain many new results. Here, in particular, we address the problem of the superconvergence points for the gradient of finite element solutions of Laplace's and Poisson's equations and we show that the sets of superconvergence points are very different for these two cases. We also study the superconvergence of the components of the gradient of the displacement, the strain and stress for finite element solutions of the equations of elasticity. For Laplace's and Poisson's equations (resp. the equations of elasticity), we consider meshes of triangular as well as square elements of degree p, 1 ? p ? 7 (resp. 1 ? p ? 4). For the meshes of triangular elements we investigate the effect of the geometry of the mesh by considering four mesh patterns that typically occur in practical meshes, while in the case of square elements, we study the effect of the element type (tensor-product, serendipity, or other). © 1996 John Wiley & Sons, Inc.     

A dynamical method for the calculation of Nash-equilibria in n-person games

In this paper, we are concerned with the calculation of Nash-equilibria in non-cooperative n-person games. For this purpose, we construct a continuous mapping of the Cartesian product of the strategy sets of the players into itself such that the fixed points of this mapping are Nash-equilibria. This gives rise to an iteration method for the calculation of fixed points of this mapping which leads to Nash-equilibria, if it converges. As important special cases Bi-matrix games and evolution matrix games are considered.