Related papers: Degree-optimal moving frames for rational curves
Parameterized algebraic curves and surfaces are widely used in geometric modeling and their manipulation is an important task in the processing of geometric models. In particular, the determination of the intersection loci between points,…
In this paper, we present a motion planning framework for multi-modal vehicle dynamics. Our proposed algorithm employs transcription of the optimization objective function, vehicle dynamics, and state and control constraints into sparse…
Let F : P^N --> P^N be a dominant rational map. The dynamical degree of F is the quantity d_F = lim (deg F^n)^(1/n). When F is defined over a number field, we define the arithmetic degree of an algebraic point P to be a_F(P) = limsup…
We give formulas and effective sharp bounds for the degree of multi-graded rational maps and provide some effective and computable criteria for birationality in terms of their algebraic and geometric properties. We also extend the Jacobian…
Motivated by the dynamical sampling problem, we study frames in an infinite dimensional Hilbert space generated by the iterates of a bounded operator T, also known as dynamical frames. We first characterize the operators that generate…
Since the 1960's the finite element method emerged as a powerful tool for the numerical simulation of countless physical phenomena or processes in applied sciences. One of the reasons for this undeniable success is the great versatility of…
Issues in modal tracking in the presence of crossings and crossing avoidances between eigenvalue traces are solved via the theory of point groups. The von Neumann-Wigner theorem is used as a key factor in predictively determining mode…
We give new positive and negative results (some conditional) on speeding up computational algebraic geometry over the reals: (1) A new and sharper upper bound on the number of connected components of a semialgebraic set. Our bound is novel…
The Zariski closure of the central path which interior point algorithms track in convex optimization problems such as linear, quadratic, and semidefinite programs is an algebraic curve. The degree of this curve has been studied in relation…
This study presents a set of algorithms that deal with trajectory planning of rational single-loop mechanisms with one degree of freedom (DoF). Benefiting from a dual quaternion representation of a rational motion, a formula for direct…
We consider the disjoint bilinear programming problem in which one of the disjoint subsets has the structure of an acute-angled polytope. An optimality criterion for such a problem is formulated and proved, and based on this, a polynomial…
We determine all modular curves $X_0(N)$ with infinitely many quartic points. To do this, we define a pairing that induces a quadratic form representing all possible degrees of a rational morphism from $X_0(N)$ to a positive rank elliptic…
Convex algebraic geometry concerns the interplay between optimization theory and real algebraic geometry. Its objects of study include convex semialgebraic sets that arise in semidefinite programming and from sums of squares. This article…
We find the complete equivalence group of a class of (1+1)-dimensional second-order evolution equations, which is infinite-dimensional. The equivariant moving frame methodology is invoked to construct, in the regular case of the…
In this paper we construct parameterizations of elliptic curves over the rationals which have many consecutive integral multiples. Using these parameterizations, we perform searches in GMP and Magma to find curves with points of small…
We present a new perspective on the celebrated Sinkhorn algorithm by showing that is a special case of incremental/stochastic mirror descent. In order to see this, one should simply plug Kullback-Leibler divergence in both mirror map and…
Given their flexibility and encouraging performance, deep-learning models are becoming standard for motion prediction in autonomous driving. However, with great flexibility comes a lack of interpretability and possible violations of…
We introduce the Riemannian Motion Policy (RMP), a new mathematical object for modular motion generation. An RMP is a second-order dynamical system (acceleration field or motion policy) coupled with a corresponding Riemannian metric. The…
Recent work has shown a variety of ways in which machine learning can be used to accelerate the solution of constrained optimization problems. Increasing demand for real-time decision-making capabilities in applications such as artificial…
The dynamic formulation of optimal transport has attracted growing interests in scientific computing and machine learning, and its computation requires to solve a PDE-constrained optimization problem. The classical Eulerian discretization…