Dynamics and Stability of Non-Smooth Dynamical Systems with Two Switches

One of the most common hypotheses on the theory of non-smooth dynamical systems is a regular surface as switching manifold, at which case there is at least well-defined and established Filippov dynamics. However, systems with singular switching manifolds still lack such well-established dynamics, although present in many relevant models of phenomena where multiple switches or multiple abrupt changes occur. At this work, we leverage a methodology that, through blow-ups and singular perturbation, allows the extension of Filippov dynamics to the singular case. Specifically, tridimensional systems whose switching manifold consists of an algebraic manifold with transversal self-intersection are considered. This configuration, known as double discontinuity, represents systems with two switches and whose singular part consists of a straight line, where ordinary Filippov dynamics is not directly applicable. For the general, non-linear case, beyond defining the so-called fundamental dynamics over the singular part, general theorems on its qualitative behavior are provided. For the affine case, however, theorems fully describing the fundamental dynamics are obtained. Finally, this fine-grained control over the dynamics is leveraged to derive Peixoto-like theorems characterizing semi-local structural stability.

On a New Distance Approximation for an Implicit Polynomial Manifold Fitting

The application of a new approximate point-to-algebraic manifold distance formula is suggested to the geometric approach to curve fitting and surface reconstruction using implicit polynomial manifolds. A brief overview of the fitting methods features for implicit algebraic manifolds is given. To illustrate the possibilities of a new approximate point-to-manifold distance formula, the equidistant curves of the exact distance, Samson’s distance and the present formula are given. A four-step algorithm for implicit algebraic manifold fitting is proposed, using one of the algebraic fitting methods at the initial step, the present approximate formula for the distance finding to calculate the geometric criterion of approximation quality and an optimization method for updating the value of the vector of coefficients of the manifold. The first results of the proposed algorithm on test data are briefly characterized. In conclusion, the tasks and directions for further research are described.

Period Domains and Period Mappings

This chapter seeks to develop a working understanding of the notions of period domain and period mapping, as well as familiarity with basic examples thereof. It first reviews the notion of a polarized Hodge structure H of weight n over the integers, for which the motivating example is the primitive cohomology in dimension n of a projective algebraic manifold of the same dimension. Next, the chapter presents lectures on period domains and monodromy, as well as elliptic curves. Hereafter, the chapter provides an example of period mappings, before considering Hodge structures of weight. After expounding on Poincaré residues, this chapter establishes some properties of the period mapping for hypersurfaces and the Jacobian ideal and the local Torelli theorem. Finally, the chapter studies the distance-decreasing properties and integral manifolds of the horizontal distribution.

Singularity Variety of a 3SPS-1S Spherical Parallel Manipulator

In this paper, we study the manifold of configurations of a 3SPS-1S spherical parallel manipulator. This manifold is obtained as the intersection of quadrics in the hypersphere defined by quaternion coordinates and is called its constraint manifold. We then formulate Jacobian for this manipulator and consider its singular. This is a quartic algebraic manifold called the singularity variety of the parallel manipulator. A survey of the architectures that can be defined for the 3SPS-1S spherical parallel manipulators yield a number of special cases, in particular the architectures with coincident base or moving pivots yields singularity varieties that factor into two quadric surfaces.

An ℓ-th root of a test configuration of exponent ℓ

Let (X, L) be a polarized algebraic manifold. Then for every test configuration μ = (X, L,Ψ) for (X, L) of exponent ℓ, we obtain an ℓ-th root (κ, D) of μ and Gm-equivariant desingularizations ι : ^X → X and η : ^X → Y, both isomorphic on^X \^X 0, such thatwhereκ= (Y, Q, η) is a test configuration for (X, L) of exponent 1, and D is an effective Q-divisor on^X such that ℓD is an integral divisor with support in the fiber X0. Then (κ, D) can be chosen in such a way thatwhere C1 and C2 are positive real constants independent of the choice of μ and ℓ. This plays an important role in our forthcoming papers on the existence of constant scalar curvature Kähler metrics (cf. [6]) and also on the compactified moduli space of test configurations (cf. [5],[7]).

Integrability conditions of geodesic flow on homogeneous Monge manifolds

We prove a meromorphic integrability criterion for the geodesic flow of an algebraic manifold of the form ${z}^{p} - f({x}_{1} , \ldots , {x}_{n} )= 0$ with the induced metric of ${ \mathbb{C} }^{n+ 1} $ and $f$ a homogeneous rational function, using a parallel between the properties of such algebraic manifolds and homogeneous potentials. We then apply this criterion to the manifolds of the form $z= {\lambda }_{1} { x}_{1}^{k} + \cdots + {\lambda }_{n} { x}_{n}^{k} $, $k\in { \mathbb{Z} }^{+ } $, and ${x}^{n} {y}^{m} {z}^{l} = 1, n, m, l\in \mathbb{Z} $, and prove that their geodesic flow is not integrable except for some given exceptional cases.

Visual Synthesis of RRR- and RPR-Legged Planar Parallel Manipulators Using Constraint Manifold Geometry

In this paper, we present our ongoing work on development of a visual design tool for designing planar parallel manipulators that satisfy a given rational motion. Although in this paper, we have restricted ourselves to RRR- and RPR-type legs, the approach presented here is general enough to accommodate other leg topologies. The basic idea is to represent the kinematic constraint of such parallel manipulators as an algebraic manifold and the given motion as a one-parameter curve in the image space of planar displacements. The algebraic manifold is projected in the three-dimensional space and a simple set of relationships are obtained that couple the geometry of the projected manifold to the design parameters of the parallel manipulators. Simple geometric transformations in the projected space allow a user to visually contain the image curve inside the manifold, thus satisfying the kinematic constraints. This interactive process, at the end, gives the dimensions of the links of the legs and the location of the fixed and moving frames. This is an extension of our previous work on the dimensional synthesis of planar 6R closed chains.