Related papers: Gradient trajectories for plane singular metrics I…

Let (X,O) be a real analytic isolated surface singularity at the origin o of a real analytic manifold M equipped with a real analytic metric g. Given a real analytic function f:(M,O) --> (R,0) singular at O, we prove that the gradient…

We explicitly construct parameter transformations between gradient flows in metric spaces, called curves of maximal slope, having different exponents when the associated function satisfies a suitable convexity condition. These…

We consider a class of piecewise smooth one-dimensional maps with critical points and singularities (possibly with infinite derivative). Under mild summability conditions on the growth of the derivative on critical orbits, we prove the…

We construct a planar homogeneous self-similar measure, with strong separation, dense rotations and dimension greater than $1$, such that there exist lines for which dimension conservation does not hold and the projection of the measure is…

Accelerated gradient descent iterations are widely used in optimization. It is known that, in the continuous-time limit, these iterations converge to a second-order differential equation which we refer to as the accelerated gradient flow.…

This paper investigates the properties of trajectories in harmonic oscillator systems equipped with a point, absolutely continuous, or singular measure. As demonstrated in [30], infinite-dimensional linear flows of countable oscillator…

We establish sufficient conditions for existence of curves minimizing length as measured with respect to a degenerate metric on the plane while enclosing a specified amount of Euclidean area. Non-existence of minimizers can occur and…

In the article, we exhibit a series of new examples of rigid plane curves, that is, curves, whose collection of singularities determines them almost uniquely up to a projective transformation of the plane.

In the study of discrete dynamical systems, we typically start with a function from a space into itself, and ask questions about the properties of sequences of iterates of the function. In this paper we reverse the direction of this study.…

Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the…

We show under some natural smoothness assumptions that pure in-plane drill rotations as deformation mappings of a $C^2$-smooth regular shell surface to another one parametrized over the same domain are impossible provided that the rotations…

This article establishes several remarkably simple identities relating certain metric invariants of level curves of real and complex functions. In particular, we relate lengths of level curves to their curvature and to the gradient field of…

Using backpropagation to compute gradients of objective functions for optimization has remained a mainstay of machine learning. Backpropagation, or reverse-mode differentiation, is a special case within the general family of automatic…

Topological bases of behaviour of trajectories for autonomous differential systems of the second order on the projective phase plane are stated. By means of Poincare's circles the projective atlas of trajectories is constructed.…

Planar curves with periodically varying curvature arise in the natural sciences as the result of a wide variety of periodic processes. The total curvature of a periodic arc in such curves constrains their symmetry. It is shown how the total…

Near full-null degenerate singular points of analytic vector fields, asymptotic behaviors of orbits are not given by eigenvectors but totally decided by nonlinearities. Especially, in the case of high full-null degeneracy, i.e., the lowest…

In this work, we introduce a novel gradient descent-based approach for optimizing control systems, leveraging a new representation of stable closed-loop dynamics as a function of two matrices i.e. the step size or direction matrix and value…

We study scaling function geometry. We show the existence of the scaling function of a geometrically finite one-dimensional mapping. This scaling function is discontinuous. We prove that the scaling function and the asymmetries at the…

We study multifractal properties in time evolution of a single particle subject to repeated measurements. For quantum systems, we consider circuit models consisting of local unitary gates and local projective measurements. For classical…

Let M be a smooth connected compact surface, P be either the real line R^1 or the circle S^1, and f:M-->P be a smooth mapping. In a previous series of papers for the case when f is a Morse map the author calculated the homotopy types of…