The Winding Number

1995 ◽  
pp. 35-47
Author(s):  
William Fulton
Keyword(s):  
Winding Number

scholarly journals Simulations of a bubble wall interacting with an electroweak plasma

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Zong-Gang Mou ◽  
Paul M. Saffin ◽  
Anders Tranberg
Keyword(s):  
Large Scale ◽  
Baryon Asymmetry ◽  
Winding Number ◽  
Bubble Wall ◽  
Electroweak Phase Transition ◽  
First Order ◽  
The Standard Model ◽  
Technical Details ◽  
Real Time Simulations ◽  
Chern Simons

Abstract We perform large-scale real-time simulations of a bubble wall sweeping through an out-of-equilibrium plasma. The scenario we have in mind is the electroweak phase transition, which may be first order in extensions of the Standard Model, and produce such bubbles. The process may be responsible for baryogenesis and can generate a background of primordial cosmological gravitational waves. We study thermodynamic features of the plasma near the advancing wall, the generation of Chern-Simons number/Higgs winding number and consider the potential for CP-violation at the wall generating a baryon asymmetry. A number of technical details necessary for a proper numerical implementation are developed.

scholarly journals Winding Numbers, Unwinding Numbers, and the Lambert W Function

2021 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Complex Plane ◽  
Complex Number ◽  
Lambert W Function ◽  
Complex Numbers ◽  
Winding Number ◽  
Line Integral ◽  
Complex Functions

AbstractThe unwinding number of a complex number was introduced to process automatic computations involving complex numbers and multi-valued complex functions, and has been successfully applied to computations involving branches of the Lambert W function. In this partly expository note we discuss the unwinding number from a purely topological perspective, and link it to the classical winding number of a curve in the complex plane. We also use the unwinding number to give a representation of the branches $$W_k$$ W k of the Lambert W function as a line integral.

scholarly journals A theory of giant vortices

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Alexander A. Penin ◽  
Quinten Weller
Keyword(s):  
Asymptotic Expansion ◽  
Analytic Form ◽  
Scalar Fields ◽  
Winding Number ◽  
Asymptotic Results ◽  
Convergence Region ◽  
Ginzburg Landau ◽  
Landau Model ◽  
Neutral Scalar ◽  
Vortex Solutions

Abstract We elaborate a theory of giant vortices [1] based on an asymptotic expansion in inverse powers of their winding number n. The theory is applied to the analysis of vortex solutions in the abelian Higgs (Ginzburg-Landau) model. Specific properties of the giant vortices for charged and neutral scalar fields as well as different integrable limits of the scalar self-coupling are discussed. Asymptotic results and the finite-n corrections to the vortex solutions are derived in analytic form and the convergence region of the expansion is determined.

scholarly journals COMBED 3-MANIFOLDS WITH CONCAVE BOUNDARY, FRAMED LINKS, AND PSEUDO-LEGENDRIAN LINKS

2001 ◽  
Vol 10 (01) ◽  
pp. 1-35 ◽  
Author(s):  
RICCARDO BENEDETTI ◽  
CARLO PETRONIO
Keyword(s):  
Vector Field ◽  
Finite Order ◽  
Singular Vector ◽  
Contact Structures ◽  
Winding Number ◽  
Legendrian Links ◽  
Concave Boundary

We provide combinatorial realizations, according to the usual objects/moves scheme, of the following three topological categories: (1) pairs (M, v) where M is a 3-manifold (up to diffeomorphism) and v is a (non-singular vector) field, up to homotopy; here possibly ∂M≠∅, and v may be tangent to ∂M, but only in a concave fashion, and homotopy should preserve tangency type; (2) framed links L in M, up to framed isotopy; (3) triples (M, v, L), with (M, v) as above and L transversal to v, up to pseudo-Legendrian isotopy (transverality-preserving simultaneous homotopy of v and isotopy of L). All realizations are based on the notion of branched standard spine, and build on results previously obtained, Links are encoded by means of diagrams on branched spines, where the diagram is C 1 with respect to the branching. Several motivations for being interested in combinatorial realizations of the topological categories considered in this paper are given in the introduction. The encoding of links is suitable for the comparison of the framed and the pseudo-Legendrian categories, and some applications are given in connection with contact structures, torsion and finite-order invariants. An estension of Trace's notion of winding number of a knot diagram is introduced and discussed.

Resonances and Torsion Numbers of Driven Dissipative Nonlinear Oscillators

1986 ◽  
Vol 41 (4) ◽  
pp. 605-614 ◽  
Author(s):  
Ulrich Parlitz ◽  
Werner Lauterborn
Keyword(s):  
Nonlinear Oscillators ◽  
Winding Number ◽  
Local Flow ◽  
One Dimensional ◽  
Bifurcation Set ◽  
Local Bifurcations ◽  
Closed Orbits

The torsion of the local flow around closed orbits and its relation to the superstructure in the bifurcation set of strictly dissipative nonlinear oscillators is investigated. The torsion number describing the twisting behaviour of the flow turns out to be a suitable invariant for the classification of local bifurcations and resonances in those systems. Furthermore, the notions of winding number and resonance are generalized to arbitrary one-dimensional dissipative oscillators.

scholarly journals Physical and Topological Properties of Circular DNA

1966 ◽  
Vol 49 (6) ◽  
pp. 103-125 ◽  
Author(s):  
Jerome Vinograd ◽  
Jacob Lebowitz'
Keyword(s):  
Room Temperature ◽  
Melting Behavior ◽  
Neutral Salt ◽  
Winding Number ◽  
Complementary Strand ◽  
Circular Dna ◽  
Tumor Viruses ◽  
Weakly Closed ◽  
A Site ◽  
Melting Experiments

Several types of circular DNA molecules are now known. These are classified as single-stranded rings, covalently closed duplex rings, and weakly bonded duplex rings containing an interruption in one or both strands. Single rings are exemplified by the viral DNA from ϕX174 bacteriophage. Duplex rings appear to exist in a twisted configuration in neutral salt solutions at room temperature. Examples of such molecules are the DNA's from the papova group of tumor viruses and certain intracellular forms of ϕX and λ-DNA. These DNA's have several common properties which derive from the topological requirement that the winding number in such molecules is invariant. They sediment abnormally rapidly in alkaline (denaturing) solvents because of the topological barrier to unwinding. For the same basic reason these DNA's are thermodynamically more stable than the strand separable DNA's in thermal and alkaline melting experiments. The introduction of one single strand scission has a profound effect on the properties of closed circular duplex DNA's. In neutral solutions a scission appears to generate a swivel in the complementary strand at a site in the helix opposite to the scission. The twists are then released and a slower sedimenting, weakly closed circular duplex is formed. Such circular duplexes exhibit normal melting behavior, and in alkali dissociate to form circular and linear single strands which sediment at different velocities. Weakly closed circular duplexes containing an interruption in each strand are formed by intramolecular cyclization of viral λ-DNA. A third kind of weakly closed circular duplex is formed by reannealing single strands derived from circularly permuted T2 DNA. These reconstituted duplexes again contain an interruption in each strand though not necessarily regularly spaced with respect to each other.

scholarly journals Skyrmion Lattices in Chiral Metal–organic Frameworks

2020 ◽  
Author(s):  
Emma Wolpert ◽  
François-Xavier Coudert ◽  
Andrew Goodwin
Keyword(s):  
Simple Model ◽  
Data Storage ◽  
Guest Molecule ◽  
Metal Organic Frameworks ◽  
Orientational Order ◽  
Phase Behaviour ◽  
Smart Devices ◽  
Winding Number ◽  
Metal Organic ◽  
Dynamics Simulations

<p>Skyrmions are knot-like topologically-protected objects of use in data storage and low-energy smart devices. They can be generated by applying a magnetic field to certain chiral ferromagnets, with the knotted state involving a curling of the underlying magnetisation to give a nonzero winding number. Here we explore the possibility that chiral metal–organic frameworks (MOFs) might in principle host skyrmionic phases—realised not through the winding of magnetic spins but through that of guest molecule orientations. We propose a simple model for the interactions governing guest orientational order in chiral MOFs, with uniaxial strain acting as conjugate field. Using Monte Carlo simulations we show that this model gives a rich phase behaviour that includes molecular skyrmion crystals. <i>Ab initio</i> molecular dynamics simulations carried out for a candidate chiral MOF of tractable complexity demonstrate that our simple model effectively captures its underlying energetics. Our results suggest that skyrmionic states may indeed be realisable in MOFs and related porous media and may even arise spontaneously in thin-film samples.</p>

scholarly journals Computation of the winding number diffusion rate due to the cosmological sphaleron

1996 ◽  
Vol 54 (8) ◽  
pp. 5014-5030 ◽  
Author(s):  
Mikhail S. Volkov
Keyword(s):  
Diffusion Rate ◽  
Winding Number
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