Related papers: Limit linear systems and applications

A set S of 2n+1 points in the plane is said to be in general position if no three points of S are collinear and no four are concyclic. A circle is called halving with respect to S if it has three points of S on its circumference, n-1 points…

Linear systems on Lie groups are a natural generalization of linear system on Euclidian spaces. For such systems, this paper studies the properties of the maximal sets of approximate controllability.

Let $P$ be a set of $n$ points in the real plane contained in an algebraic curve $C$ of degree $d$. We prove that the number of distinct distances determined by $P$ is at least $c_d n^{4/3}$, unless $C$ contains a line or a circle. We also…

We illustrate with several new applications the power and elegance of the Bendixson Dulac theorem to obtain upper bounds of the number of limit cycles for several families of planar vector fields. In some cases we propose to use a function…

We consider ruled surfaces with finite multiplicity. We study behaviors of the striction curves and the singularities of the ruled surfaces. We also give geometric meanings of invariants related to the ruled surfaces.

We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on…

A system of N points, each having mass m, and a central mass M forming a planar central configuration, is considered. The equations of motion of a test particle are given and compared using different coordinates. For large values of N, even…

In this paper, we study the restrictions on the number $m$ of conic-line curves in special pencils. The most general result we obtain is the relation between upper bounds on $m$ and the number $p$ of concurrent lines in these pencils. We…

We prove that, if \mu<\lfloor n/2\rfloor, then every rational plane curve of degree n and class \mu is a limit of parametrizations of the same degree and class \mu+1. This property was conjectured in D.Cox, T.Sederberg,F.Chen's paper: "The…

It is proved that a parameterized curve in a metric space $X$ is absolutely continuous if and only if its composition with any Lipschitz function on $X$ is absolutely continuous.

Linear-constraint loops are programs whose transition relation is specified by a system of linear inequalities. The termination problem asks, given a loop, whether it admits an infinite computation. Decidability of termination remains open…

We study combinatorial configurations with the associated point and line graphs being strongly regular. Examples not belonging to known classes such as partial geometries and their generalizations or elliptic semiplanes are constructed.…

A closed plane meander of order $n$ is a closed self-avoiding curve intersecting an infinite line $2n$ times. Meanders are considered distinct up to any smooth deformation leaving the line fixed. We have developed an improved algorithm,…

We prove that the orbit closure of the determinant is not normal. A similar result is obtained for the orbit closure of the permanent multiplied by a power of a linear form.

It is widely recognized that no tractable necessary and sufficient conditions exist for determining whether a system is, in general, differentially flat. However, specific cases do provide such conditions. For instance, driftless systems…

We consider linear systems on toric varieties of any dimension, with invariant base points, giving a characterization of special linear systems. We then make a new conjecture for linear systems on rational surfaces.

Linear systems on Lie groups are a natural generalization of linear system on Euclidian spaces. When the state space is a solvable connected Lie group, controllability of the linear system is assured if the ad-rank condition holds.

Recently, there has been great interest in the application of composition laws to problems in enumerative geometry. Using the moduli space of stable maps, we compute the number of irreducible, reduced, nodal, degree-$d$ genus-$2$ plane…

We enumerate the singular algebraic curves in a complete linear system on a smooth projective surface. The system must be suitably ample in a rather precise sense. The curves may have up to eight nodes, or a triple point of a given type and…

Integer iteration rules such as n |-> {a n + b, c n +d} are studied as minimal examples of the general process of multicomputation. Despite the simplicity of such rules, their multiway graphs can be complex, exhibiting, for example,…