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 A note on locking materials and gradient polyconvexity

2018 ◽  
Vol 28 (12) ◽  
pp. 2367-2401 ◽  
Author(s):  
Barbora Benešová ◽  
Martin Kružík ◽  
Anja Schlömerkemper
Keyword(s):  
Deformation Gradient ◽  
Variational Problems ◽  
Energy Functional ◽  
Second Grade ◽  
Young Measures ◽  
Elastic Materials ◽  
Lipschitz Maps ◽  
Existence Of Minimizers ◽  
Second Derivatives ◽  
Derivatives Of

We use gradient Young measures generated by Lipschitz maps to define a relaxation of integral functionals which are allowed to attain the value [Formula: see text] and can model ideal locking in elasticity as defined by Prager in 1957. Furthermore, we show the existence of minimizers for variational problems for elastic materials with energy densities that can be expressed in terms of a function being continuous in the deformation gradient and convex in the gradient of the cofactor (and possibly also the gradient of the determinant) of the corresponding deformation gradient. We call the related energy functional gradient polyconvex. Thus, instead of considering second derivatives of the deformation gradient as in second-grade materials, only a weaker higher integrability is imposed. Although the second-order gradient of the deformation is not included in our model, gradient polyconvex functionals allow for an implicit uniform positive lower bound on the determinant of the deformation gradient on the closure of the domain representing the elastic body. Consequently, the material does not allow for extreme local compression.

Sobolev maps with integer degree and applications to Skyrme’s problem

1992 ◽  
Vol 436 (1896) ◽  
pp. 197-201 ◽  
Keyword(s):  
Finite Energy ◽  
Natural Class ◽  
Sobolev Maps

Let ɸ : R 3 → S 3 ⊂ R 4 , ∣ A ( ɸ )∣ 2 ═ Ʃ 3 α,β═1 │∂ ɸ /∂ x α ∧ ∂ ɸ /∂ x β ∣ 2 and let k ϵ Z . Skyrme's problem consists in minimizing the energy ε( ɸ ) : ═ ∫ R 3 ∣∇ ɸ ∣ 2 + ∣ A ( ɸ )∣ 2 d x among maps with degree k ═ d ( ɸ ) : ═ 1/2π 2 ∫ R 3 det ( ɸ , ∇ ɸ ) d x . We show that for all ɸ with finite energy d ( ɸ ) is an integer and then obtain existence of a minimizer of ε in the natural class of maps with finite energy.

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.

scholarly journals Regularity of Weak Solution of Variational Problems Modeling the Cosserat Micropolar Elasticity

2020 ◽  
Author(s):  
Yimei Li ◽  
Changyou Wang
Keyword(s):  
Weak Solution ◽  
Harmonic Maps ◽  
Lagrange Equation ◽  
Variational Problems ◽  
Energy Functional ◽  
Geometrically Nonlinear ◽  
Micropolar Elasticity ◽  
One Dimensional ◽  
The Poisson Equation ◽  
System Coupling

Abstract In this paper, we consider weak solutions of the Euler–Lagrange equation to a variational energy functional modeling the geometrically nonlinear Cosserat micropolar elasticity of continua in dimension three, which is a system coupling between the Poisson equation and the equation of $p$-harmonic maps ($2\le p\le 3$). We show that if a weak solution is stationary, then its singular set is discrete for $2<p<3$ and has zero one-dimensional Hausdorff measure for $p=2$. If, in addition, it is a stable-stationary weak solution, then it is regular everywhere when $p\in [2, \frac{32}{15}]$.

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).

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