Embeddings of locally compact abelian p-groups in Hawaiian groups

We show that locally compact abelian p-groups can be embedded in the first Hawaiian group on a compact path connected subspace of the Euclidean space of dimension four. This result gives a new geometric interpretation for the classification of locally compact abelian groups which are rich in commuting closed subgroups. It is then possible to introduce the idea of an algebraic topology for topologically modular locally compact groups via the geometry of the Hawaiian earring. Among other things, we find applications for locally compact groups which are just noncompact.

The fundamental group of quotients of products of some topological spaces by a finite group - A generalization of a Theorem of Bauer-Catanese-Grunewald-Pignatelli: Le groupe fondamental de quotients de produits de certains espaces topologiques par ungroupe fini — Généralisation d’un théorème de Bauer–Catanese–Grunewald–Pignatelli

We provide a description of the fundamental group of the quotient of a product of topological spaces X i, each admitting a universal cover, by a finite group G, provided that there is only a finite number of path-connected components in X g i for every g ∈ G. This generalizes previous work of Bauer-Catanese-Grunewald-Pignatelli and Dedieu-Perroni. Nous fournissons une description du groupe fondamental du quotient d’un produitd’espaces topologiques Xi , chacun admettant un revêtement universel, par un groupe fini G,pourvu qu’il n’existe qu’un nombre ni de composantes connexes par arcs dans Xgi pour chaque g ∈ G. Cela généralise des résultats antérieurs de Bauer–Catanese–Grunewald–Pignatelli et deDedieu–Perroni.

Manipulator Kinematics and Dynamics On Differentiable Manifolds: Part II Dynamics

The manipulator differentiable manifold formulation presented in Part I of this paper is used to create algorithms for forward and inverse kinematics on maximal, singularity free, path connected manifold components. Existence of forward and inverse configuration mappings in manifold components is extended to obtain forward and inverse velocity mappings. Computational algorithms for forward and inverse configuration and velocity analysis on a time grid are derived for each of the four categories of manipulator treated. Manifold parameterizations derived in Part I are used to transform variational equations of motion in Cartesian generalized coordinates to second order ordinary differential equations of manipulator dynamics, in terms of both input and output coordinates, avoiding ad-hoc derivation of equations of motion. This process is illustrated in evaluating terms required for equations of motion of the four model manipulators analyzed in Part I. It is shown that computation involved in forward and inverse kinematics and in evaluation of equations of manipulator dynamics can be carried out in real-time on modern microprocessors, supporting on-line implementation of modern methods of manipulator control.

Regularity and global structure for Hamilton–Jacobi equations with convex Hamiltonian

We consider the multidimensional Hamilton–Jacobi (HJ) equation [Formula: see text] with [Formula: see text] being a constant and for bounded [Formula: see text] initial data. When [Formula: see text], this is the typical case of interest with a uniformly convex Hamiltonian. When [Formula: see text], this is the famous Eikonal equation from geometric optics, the Hamiltonian being Lipschitz continuous with homogeneity [Formula: see text]. We intend to fill the gap in between these two cases. When [Formula: see text], the Hamiltonian [Formula: see text] is not uniformly convex and is only [Formula: see text] in any neighborhood of [Formula: see text], which causes new difficulties. In particular, points on characteristics emanating from points with vanishing gradient of the initial data could be “bad” points, so the singular set is more complicated than what is observed in the case [Formula: see text]. We establish here the regularity of solutions and the global structure of the singular set from a topological standpoint: the solution inherits the regularity of the initial data in the complement of the singular set and there is a one-to-one correspondence between the connected components of the singular set and the path-connected components of the set [Formula: see text].

Necessary and sufficient conditions for pairwise majority decisions on path-connected domains

In this paper, we consider choice functions that are unanimous, anonymous, symmetric, and group strategy-proof and consider domains that are single-peaked on some tree. We prove the following three results in this setting. First, there exists a unanimous, anonymous, symmetric, and group strategy-proof choice function on a path-connected domain if and only if the domain is single-peaked on a tree and the number of agents is odd. Second, a choice function is unanimous, anonymous, symmetric, and group strategy-proof on a single-peaked domain on a tree if and only if it is the pairwise majority rule (also known as the tree-median rule) and the number of agents is odd. Third, there exists a unanimous, anonymous, symmetric, and strategy-proof choice function on a strongly path-connected domain if and only if the domain is single-peaked on a tree and the number of agents is odd. As a corollary of these results, we obtain that there exists no unanimous, anonymous, symmetric, and group strategy-proof choice function on a path-connected domain if the number of agents is even.

Connected Fundamental Groups and Homotopy Contacts in Fibered Topological (C, R) Space

The algebraic as well as geometric topological constructions of manifold embeddings and homotopy offer interesting insights about spaces and symmetry. This paper proposes the construction of 2-quasinormed variants of locally dense p-normed 2-spheres within a non-uniformly scalable quasinormed topological (C, R) space. The fibered space is dense and the 2-spheres are equivalent to the category of 3-dimensional manifolds or three-manifolds with simply connected boundary surfaces. However, the disjoint and proper embeddings of covering three-manifolds within the convex subspaces generates separations of p-normed 2-spheres. The 2-quasinormed variants of p-normed 2-spheres are compact and path-connected varieties within the dense space. The path-connection is further extended by introducing the concept of bi-connectedness, preserving Urysohn separation of closed subspaces. The local fundamental groups are constructed from the discrete variety of path-homotopies, which are interior to the respective 2-spheres. The simple connected boundaries of p-normed 2-spheres generate finite and countable sets of homotopy contacts of the fundamental groups. Interestingly, a compact fibre can prepare a homotopy loop in the fundamental group within the fibered topological (C, R) space. It is shown that the holomorphic condition is a requirement in the topological (C, R) space to preserve a convex path-component. However, the topological projections of p-normed 2-spheres on the disjoint holomorphic complex subspaces retain the path-connection property irrespective of the projective points on real subspace. The local fundamental groups of discrete-loop variety support the formation of a homotopically Hausdorff (C, R) space.

A Monoparametric Family of Piecewise Linear Systems to Generate Scroll Attractors via Path-Connected Set of Polynomials

In this work, we present a monoparametric family of piecewise linear systems to generate multiscroll attractors through a polynomial family defined by path curves that connect to the roots. The idea is to define path curves where the roots of a polynomial can take values by determining an initial and a final polynomial. As a consequence, structural stability and bifurcation of the system can be obtained. Structural stability is obtained by preserving the same stability of the initial and final polynomials. However, the system bifurcates by changing the stability of the final polynomial with respect to the initial polynomial. The aim is achieved by the design of a piecewise linear controller that is applied to affine linear systems. Our results are mathematically proved and numerical examples are also provided to illustrate the approach.