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We prove that for each positive integer n, the V n-equivalence classes of ribbon knot types form a subgroup R n , of index two, of the free abelian group V n constructed by the author and Stanford. As a corollary , any non-ribbon knot whose Arf invariant is trivial cannot be distinguished from ribbon knots by finitely many independent Vassiliev invariants.(More)
The third-integer coupling resonance at ν(x)-2ν(z)=ℓ, known as the Walkinshaw resonance, is important in high-power accelerators. We find that, when the betatron tunes ramp through a Walkinshaw resonance the fractional emittance growth (FEG) is a universal function of the effective resonance strength: G(1,-2,ℓ)√[ε(xi)]|Δ(ν(x)-2ν(z))/Δn|(-1/2), where(More)
The muon collider would extend limitations of the e + e ? colliders and provide new physics potentials with a possible discovery of the heavy Higgs bosons. At the maximum energy of 2 TeV the projected luminosity is of the order of 10 35 cm ?2 s ?1. The colliding + ? bunches have to be focused to a very small transverse size of 2.8 m which is accomplished by(More)
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