Related papers: Gradient trajectories for plane singular metrics I…

A family of plane oriented continuous paths depending on a fixed real positive number $R$ is considered. For any point $x$ on the path, the previous points lie out of any circle of radius $R$ having at $x$ interior normal in a suitable…

The problem of inverting a system in presence of a series-defined output is analyzed. Inverse models are derived that consist of a set of algebraic equations. The inversion is performed explicitly for an output trajectory functional, which…

The central purpose of this article is to establish new inverse and implicit function theorems for differentiable maps with isolated critical points. One of the key ingredients is a discovery of the fact that differentiable maps with…

We develop a systematic study of the equations of motion in the first order gravity with matter fields for degenerate metrics. Like the Hilbert-Palatini action functional for pure gravity, the action functionals for matter fields used are…

We define a number of natural (from geometric and combinatorial points of view) deformation spaces of valuations on finite graphs, and study functions over these deformation spaces. These functions include both direct metric invariants…

We study singularity confinement phenomena in examples of delay-differential Painlev\'e equations, which involve shifts and derivatives with respect to a single independent variable. We propose a geometric interpretation of our results in…

We give an example of a dominant rational selfmap of the projective plane whose dynamical degree is a transcendental number.

In this book we collect many different and useful facts around gradient descent method. First of all we consider gradient descent with inexact oracle. We build a general model of optimized function that include composite optimization…

We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…

The notion of singular one-parameter deformation of a Lie algebra is introduced. It is shown that the complex infinite-dimensional Lie algebra of polynomial vector fields in C with trivial 1-jet at the origin has such singular deformation.

In this short note, we construct an example of spiraling conformal geodesic in Euclidean signature in dimension $3$, answering the question posed by Helmuth Friedrich and Paul Tod, if such objects exists. Our example is not real analytic,…

Shadowing trajectories are model trajectories consistent with a sequence of observations of a system, given a distribution of observational noise. The existence of such trajectories is a desirable property of any forecast model. Gradient…

Fractional derivatives are a well-studied generalization of integer order derivatives. Naturally, for optimization, it is of interest to understand the convergence properties of gradient descent using fractional derivatives. Convergence…

The metric for gravitational plane waves has very high symmetry (two spacelike commuting Killing vectors). For this high symmetry, a simple renormalization of the lapse function is found which allows the constraint algebra for canonical…

We report an unexpected reverse spiral turn in the final stage of the motion of rolling rings. It is well known that spinning disks rotate in the same direction of their initial spin until they stop. While a spinning ring starts its motion…

Singular and sectional hyperbolic sets are the objects of the extension of the classical Smale Hyperbolic Theory to flows having invariant sets with singularities accumulated by regular orbits within the set. It is by now well-known that…

This paper shows that a wide class of effective learning rules -- those that improve a scalar performance measure over a given time window -- can be rewritten as natural gradient descent with respect to a suitably defined loss function and…

We consider spacetime metrics with a given (but quite generic) dependence on a dimensionful parameter such that in the 0 and infinity limits of that parameter the metric becomes singular. We study the isometry groups of the original…

We study tangent sets of strictly self-affine sets in the plane. If a set in this class satisfies the strong separation condition and projects to a line segment for sufficiently many directions, then for each generic point there exists a…

In this paper, we propose a metric on the space of finite sets of trajectories for assessing multi-target tracking algorithms in a mathematically sound way. The main use of the metric is to compare estimates of trajectories from different…