Related papers: Computing Linear Matrix Representations of Helton-…
In this paper, we study the rectifying curves in multiplicative Euclidean space of dimension 3, i.e., those curves for which the position vector always lies in its rectifying plane. Since the definition of rectifying curve is affine and not…
Recently it was proposed that the theory space of effective field theories with consistent UV completions can be described as a positive geometry, termed the EFThedron. In this paper we demonstrate that at the core, the geometry is given by…
We study multivariate normal models that are described by linear constraints on the inverse of the covariance matrix. Maximum likelihood estimation for such models leads to the problem of maximizing the determinant function over a…
Hyperbolic homogeneous polynomials with real coefficients, i.e., hyperbolic real projective hypersurfaces, and their determinantal representations, play a key role in the emerging field of convex algebraic geometry. In this paper we…
We propose an alternative variational principle whose critical point is the algebraic plane curve associated to a matrix model (the spectral curve, i.e. the large $N$ limit of the resolvent). More generally, we consider a variational…
Given any finite set of nonnegative integers, there exists a closed convex set whose facial dimension signature coincides with this set of integers, that is, the dimensions of its nonempty faces comprise exactly this set of integers. In…
We solve the so far open problem of constructing all spatial rational curves with rational arc length functions. More precisely, we present three different methods for this construction. The first method adapts a recent approach of (Kalkan…
We present a matrix-based algorithm for deciding if the parametrization of a curve or a surface is invertible or not, and for computing the inverse of the parametrization if it exists.
We construct C-algebras for a class of surfaces that are inverse images of certain polynomials of arbitrary degree. By using the directed graph associated to a matrix, the representation theory can be understood in terms of ``loop'' and…
We use algebraic geometry to study matrix rigidity, and more generally, the complexity of computing a matrix-vector product, continuing a study initiated by Kumar, et. al. We (i) exhibit many non-obvious equations testing for (border)…
In this article, we study rectifying curves in arbitrary dimensional Euclidean space. A curve is said to be a rectifying curve if, in all points of the curve, the orthogonal complement of its normal vector contains a fixed point. We…
The article proposes an n-dimensional mathematical model of the visual representation of a linear programming problem. This model makes it possible to use artificial neural networks to solve multidimensional linear optimization problems,…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
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…
A programming tactic involving polyhedra is reported that has been widely applied in the polyhedral analysis of (constraint) logic programs. The method enables the computations of convex hulls that are required for polyhedral analysis to be…
Knowing the symmetries of a polyhedron can be very useful for the analysis of its structure as well as for practical polyhedral computations. In this note, we study symmetry groups preserving the linear, projective and combinatorial…
A computer-algebra aided method is carried out, for determining geometric objects associated to differential operators that satisfy the elliptic ansatz. This results in examples of Lame curves with double reduction and in the explicit…
We solve a randomized version of the following open question: is there a strictly convex, bounded curve \gamma in the plane such that the number of rational points on \gamma, with denominator $n$, approaches infinity with $n$? Although this…
In this paper we show that every combinatorial problem has an exact explicit equation that returns its solution. We present a method to obtain an equation that solves exactly any combinatorial problem, both inversion, constraint…
We develop a transfer matrix formalism to visualize the framing of discrete piecewise linear curves in three dimensional space. Our approach is based on the concept of an intrinsically discrete curve, which enables us to more effectively…