Related papers: The central curve in linear programming
A novel and deterministic algorithm is presented to detect whether two given rational plane curves are related by means of a similarity, which is a central question in Pattern Recognition. As a by-product it finds all such similarities, and…
For applications in computing, Bezier curves are pervasive and are defined by a piecewise linear curve L which is embedded in R^3 and yields a smooth polynomial curve C embedded in R^3. It is of interest to understand when L and C have the…
Delsarte, Goethals, and Seidel (1977) used the linear programming method in order to find bounds for the size of spherical codes endowed with prescribed inner products between distinct points in the code. In this paper, we develop the…
This paper presents rigorous forward error bounds for linear conic optimization problems. The error bounds are formulated in a quite general framework; the underlying vector spaces are not required to be finite-dimensional, and the convex…
Principal curves are natural generalizations of principal lines arising as first principal components in the Principal Component Analysis. They can be characterized from a stochastic point of view as so-called self-consistent curves based…
We develop a natural generalization to the notion of the central path -- a notion that lies at the heart of interior-point methods for convex optimization. The generalization is accomplished via the "derivative cones" of a "hyperbolicity…
We apply polynomial techniques (linear programming) to obtain lower and upper bounds on the covering radius of spherical designs as function of their dimension, strength, and cardinality. In terms of inner products we improve the lower…
Identifying optimal basic feasible solutions to linear programming problems is a critical task for mixed integer programming and other applications. The crossover method, which aims at deriving an optimal extreme point from a suboptimal…
Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject…
Linear algebra's main concerns are sets of vectors, linear functions, subspaces, linear systems, matrices and concepts about those, such as whether the solution of linear system exists or is unique; a set of vectors is linearly independent…
We define the type of a plane curve as the initial degree of the corresponding Bourbaki ideal. Then we show that this invariant behaves well with respect to the union of curves. Curves of type $0$ are precisely the free curves, while curves…
Learning curves are a concept from social sciences that has been adopted in the context of machine learning to assess the performance of a learning algorithm with respect to a certain resource, e.g., the number of training examples or the…
The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to…
This paper demonstrates that parallel vector curves are piecewise cubic rational curves in 3D piecewise linear vector fields. Parallel vector curves -- loci of points where two vector fields are parallel -- have been widely used to extract…
The `linear orbit' of a plane curve of degree d is its orbit in the projective space of dimension d(d+3)/2 parametrizing such curves under the natural action of PGL(3). In this paper we compute the degree of the closure of the linear orbits…
A linear series on a curve C in $P^3$ is "primary" when it does not contain the series cut by planes. We provide a lower bound for the degree of these series, in terms of deg(C), g(C) and of the number $s = min{i: h^0(I_C(i))\neq 0}$; as a…
The `linear orbit' of a plane curve of degree d is its orbit in P^{d(d+3)/2} under the natural action of PGL(3). We classify curves with positive dimensional stabilizer, and we compute the degree of the closure of the linear orbits of…
We study the parameter space of a family of planar maps, which are linear on each of the right and left half-planes. We consider the set of parameters for which every orbit recurs to the boundary between half-planes. These parameters…
Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically,…
Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar…