The Ninth Homotopy Class of Spatial 3R Serial Regional Manipulators

2006 ◽  
Vol 129 (4) ◽  
pp. 445-448 ◽  
Author(s):  
Davide Paganelli
Keyword(s):  
Homotopy Class ◽  
Classification Method ◽  
Weak Point ◽  
Homotopy Classes ◽  
Important Notion

Singularities form surfaces in the jointspace of a serial manipulator. Paï and Leu (Paï and Leu, 1992, IEEE Trans. Rob. Autom., 8, pp. 545–559) introduced the important notion of generic manipulator, the singularity surfaces of which are smooth and do not intersect with each other. Burdick (Burdick, 1995, J. Mech. Mach. Theor., 30, pp. 71–89) proposed a homotopy-based classification method for generic 3R manipulators. Through this classification method, it was stated in Wenger, 1998, J. Mech. Des., 120, pp. 327–332 that there exist exactly eight classes of generic 3R manipulators. A counterexample to this classification is provided: a generic 3R manipulator belonging to none of the eight classes identified in (Wenger, 1998, J. Mech. Des., 120, pp. 327–332) is presented. The weak point of the proof given in (J. Mech. Des., 120, pp. 327–332) is highlighted. The counterexample proves the existence of at least nine homotopy classes of generic 3R manipulators. The paper points out two peculiar properties of the manipulator proposed as a counterexample, which are not featured by any manipulator belonging to the eight homotopy classes so far discovered. Eventually, it is proven in this paper that at most four branches of the singularity curve can coexist in the jointspace of a generic 3R manipulator and therefore at most eleven homotopy classes are possible.

Suspension of the Lusternik-Schnirelmann Category

1970 ◽  
Vol 22 (6) ◽  
pp. 1129-1132
Author(s):  
William J. Gilbert
Keyword(s):  
Homotopy Type ◽  
Homotopy Class ◽  
Category Structure ◽  
Topological Spaces ◽  
Homotopy Classes ◽  
Cw Complexes ◽  
Lusternik Schnirelmann ◽  
Image Position ◽  
Simply Connected ◽  
Homotopy Invariants

Let cat be the Lusternik-Schnirelmann category structure as defined by Whitehead [6] and let be the category structure as defined by Ganea [2],We prove thatandIt is known that w ∑ cat X = conil X for connected X. Dually, if X is simply connected,1. We work in the category of based topological spaces with the based homotopy type of CW-complexes and based homotopy classes of maps. We do not distinguish between a map and its homotopy class. Constant maps are denoted by 0 and identity maps by 1.We recall some notions from Peterson's theory of structures [5; 1] which unify the definitions of the numerical homotopy invariants akin to the Lusternik-Schnirelmann category.

scholarly journals Typical representatives of free homotopy classes in multi-punctured plane

2019 ◽  
Vol 11 (03) ◽  
pp. 623-659
Author(s):  
Maxim Arnold ◽  
Yuliy Baryshnikov ◽  
Yuriy Mileyko
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Probability Measure ◽  
Homotopy Class ◽  
Piecewise Linear ◽  
Homotopy Classes ◽  
Simple Method ◽  
Monte Carlo Techniques ◽  
Free Homotopy Class

We show that a uniform probability measure supported on a specific set of piecewise linear loops in a nontrivial free homotopy class in a multi-punctured plane is overwhelmingly concentrated around loops of minimal lengths. Our approach is based on extending Mogulskii’s theorem to closed paths, which is a useful result of independent interest. In addition, we show that the above measure can be sampled using standard Markov Chain Monte Carlo techniques, thus providing a simple method for approximating shortest loops.

Some properties of Hopf-type constructions

1995 ◽  
Vol 117 (2) ◽  
pp. 287-301 ◽  
Author(s):  
Martin Arkowitz ◽  
Paul Silberbush
Keyword(s):  
Homotopy Type ◽  
Homotopy Class ◽  
Smash Product ◽  
Homotopy Classes ◽  
One To One

If f: X × Y → Z is a map, then the classical Hopf construction associates to f a map hf: X * Y → ΣZ, where X * Y is the join of X and Y and ΣZ the suspension of Z. Since X * Y has the homotopy type of Σ(X Λ Y), the suspension of the smash product of X and Y, the homotopy class of hf can be regarded as an element Hf ↦ [Σ(X Λ Y), ΣZ]. Now elements of [Σ(X Λ Y), ] are in one to one correspondence with homotopy classes in the group [σ(X Λ Y), ΣZ] which are trivial on the suspension of the wedge Σ(X ≷ Y).

scholarly journals GAUGE THEORY OF FADDEEV–SKYRME FUNCTIONALS

2010 ◽  
Vol 12 (05) ◽  
pp. 871-908
Author(s):  
SERGIY KOSHKIN
Keyword(s):  
Homotopy Class ◽  
Homogeneous Spaces ◽  
Variational Problems ◽  
Finite Energy ◽  
Energy Functional ◽  
Target Space ◽  
Homotopy Classes ◽  
Existence Of Minimizers ◽  
Sobolev Maps ◽  
Geometric Variational Problems

We study geometric variational problems for a class of nonlinear σ-models in quantum field theory. Mathematically, one needs to minimize an energy functional on homotopy classes of maps from closed 3-manifolds into compact homogeneous spaces G/H. The minimizers are known as Hopfions and exhibit localized knot-like structure. Our main results include proving existence of Hopfions as finite energy Sobolev maps in each (generalized) homotopy class when the target space is a symmetric space. For more general spaces, we obtain a weaker result on existence of minimizers in each 2-homotopy class.Our approach is based on representing maps into G/H by equivalence classes of flat connections. The equivalence is given by gauge symmetry on pullbacks of G → G/H bundles. We work out a gauge calculus for connections under this symmetry, and use it to eliminate non-compactness from the minimization problem by fixing the gauge.

scholarly journals Multistring based matrices

2020 ◽  
Vol 29 (06) ◽  
pp. 2050038
Author(s):  
David R. Freund
Keyword(s):  
Homotopy Class ◽  
Open Problem ◽  
Chord Diagram ◽  
Homotopy Classes ◽  
Chord Diagrams ◽  
Text String ◽  
Reidemeister Moves

A virtual[Formula: see text]-string is a chord diagram with [Formula: see text] core circles and a collection of arrows between core circles. We consider virtual [Formula: see text]-strings up to virtual homotopy, compositions of flat virtual Reidemeister moves on chord diagrams. Given a virtual 1-string [Formula: see text], Turaev associated a based matrix that encodes invariants of the virtual homotopy class of [Formula: see text]. We generalize Turaev’s method to associate a multistring based matrix to a virtual [Formula: see text]-string, addressing an open problem of Turaev and constructing similar invariants for virtual homotopy classes of virtual [Formula: see text]-strings.

scholarly journals Robustness Study of Generic and Non-Generic 3R Positioning Manipulators

2005 ◽  
Author(s):  
Stephane Caro ◽  
Philippe Wenger ◽  
Fouad Bennis
Keyword(s):  
Homotopy Class ◽  
Geometric Parameters ◽  
Homotopy Classes

This paper presents a robustness study of 3R manipulators and aims at answering the following question: are generic manipulators more robust than non-generic manipulators? We exploit several properties specific to 3R manipulators such as singularities, cuspidality, homotopy classes, and path feasibility, in order to find some correlations between genericity and robustness concepts. For instance, we show that generic manipulators, close to non-generic ones in the space of geometric parameters, are not robust with respect to their homotopy class and to the feasibility of paths. Moreover, we notice that the dexterity and the accuracy of 3R manipulator do not depend on genericity.

scholarly journals Realization of homotopy classes of torus homeomorphisms by the simplest structurally stable diffeomorphisms

2021 ◽  
Vol 23 (2) ◽  
pp. 171-184
Author(s):  
Andrei I. Morozov
Keyword(s):  
Finite Order ◽  
Canonical Form ◽  
Homotopy Class ◽  
Canonical Forms ◽  
Anosov Diffeomorphism ◽  
Homotopy Classes ◽  
Orientable Surfaces ◽  
Structurally Stable

Abstract. According to Thurston’s classification, the set of homotopy classes of orientation-preserving homeomorphisms of orientable surfaces is split into four disjoint subsets. A homotopy class from each subset is characterized by the existence of a homeomorphism called Thurston’s canonical form, namely: a periodic homeomorphism, a reducible nonperiodic homeomorphism of algebraically finite order, a reducible homeomorphism that is not a homeomorphism of an algebraically finite order, and a pseudo-Anosov homeomorphism. Thurston’s canonical forms are not structurally stable diffeomorphisms. Therefore, the problem naturally arises of constructing the simplest (in a certain sense) structurally stable diffeomorphisms in each homotopy class. In this paper, the problem posed is solved for torus homeomorphisms. In each homotopy class, structurally stable representatives are analytically constructed, namely, a gradient-like diffeomorphism, a Morse-Smale diffeomorphism with an orientable heteroclinic, and an Anosov diffeomorphism, which is a particular case of a pseudo-Anosov diffeomorphism.

Homotopical Nilpotency of Loop-Spaces

1969 ◽  
Vol 21 ◽  
pp. 479-484 ◽  
Author(s):  
C S. Hoo
Keyword(s):  
Homotopy Class ◽  
Base Point ◽  
Loop Spaces ◽  
Homotopy Classes ◽  
Base Points ◽  
Cw Complexes

In this paper we shall work in the category of countable CW-complexes with base point and base point preserving maps. All homotopies shall also respect base points. For simplicity, we shall frequently use the same symbol for a map and its homotopy class. Given spaces X, Y, we denote the set of homotopy classes of maps from X to Y by [X, Y]. We have an isomorphism τ: [∑X, Y] → [X, Ω Y] taking each map to its adjoint, where ∑ is the suspension functor and Ω is the loop functor. We shall denote τ(1 ∑x) by e′ and τ-1(1Ωx) by e.

Non-singular bilinear maps and stable homotopy classes of spheres

1977 ◽  
Vol 82 (3) ◽  
pp. 419-425 ◽  
Author(s):  
Kee Yuen Lam
Keyword(s):  
Homotopy Class ◽  
Stable Homotopy ◽  
Background Information ◽  
Bilinear Maps ◽  
Bilinear Map ◽  
Homotopy Classes ◽  
Image Position

A bilinear map ø Ra x Rb → Rc is non-singular if ø (x, y) = 0 implies x = 0 or y = 0. For background information on such maps see (4, 5, 6, 14). If we apply the ‘Hopf construction’ to ø, we get a mapdefined by 2ø(x, y)) for all x ∈ Ra, y ∈ Rb satisfying ∥x∥2 + ∥y∥2 = 1. Homotopically, Jø coincides with the map obtained by first restricting and normalizing ø to , and then applying the standard Hopf construction ((13), p. 112). In any case, one gets an element [Jø] in , which in turn determines a stable homotopy class of spheres {Jø} in the d-stem , where d = a + b − c −1. An element in which equals {Jø} for some non-singular bilinear map ø will be called bilinearly representable. The first purpose of this paper is to prove

Cyclic Maps From Suspensions to Suspensions

1972 ◽  
Vol 24 (5) ◽  
pp. 789-791 ◽  
Author(s):  
C. S. Hoo
Keyword(s):  
Homotopy Type ◽  
Homotopy Class ◽  
Homotopy Classes ◽  
Whitehead Product ◽  
Base Points ◽  
Cyclic Map ◽  
Cyclic Maps

In [7] Varadarajan denned the notion of a cyclic map f : A → X. The collection of all homotopy classes of such cyclic maps forms the Gottlieb subset G(A, X) of [A, X]. If A = S1 this reduces to the group G(X, X0) of Gottlieb [5]. We show that a cyclic map f maps ΩA into the centre of ΩX in the sense of Ganea [4]. If A and X are both suspensions, we then show that if f : A → X maps ΩA into the centre of ΩX, then f is cyclic. Thus for maps from suspensions to suspensions, Varadarajan's cyclic maps are just those maps considered by Ganea. We also define G (Σ4, ΣX) in terms of the generalized Whitehead product [1], This gives the computations for G(Sn+k, Sn) in terms of Whitehead products in π2n+k(Sn).We work in the category of spaces with base points and having the homotopy type of countable CW-complexes. All maps and homotopies are with respect to base points. For simplicity, we shall frequently use the same symbol for a map and its homotopy class.

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