By using the bifurcation theory of dynamical systems, we study the coupled Higgs field equation and the existence of new solitary wave solutions, and uncountably infinite many periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. All exact explicit parametric representations of the ...

We prove that if ZFC is consistent so is ZFC + “for any sequence (An) of subsets of a Polish space 〈X, τ〉 there exists a separable metrizable topology τ ′ on X with B(X, τ) ⊆ B(X, τ ′), MGR(X, τ ′) ∩ B(X, τ) = MGR(X, τ) ∩B(X, τ) and An Borel in τ ′ for all n.” This is a category analogue of a theorem of Carlson on the possibility of extending Lebesgue measure to any countable collection of sets...

I. The basic example A0 (Figure 1). A regular normed projection of our basic example of a mildly wild non-Wilder arc is shown in Figure 1. (Using the methods of [4], one could easily give a precise description.) (A) Ao is not L.P.U. at p. The invariants of [7] will be used to show that the penetration index PiA0, p) of A0 at p is equal to 4. (For a definition of the penetration index see [l] an...

By using the theory of planar dynamical systems to a coupled non-linear wave equation, the existence of solitary wave solutions and uncountably infinite, many smooth and non-smooth, periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. 2003 Published by Elsevier Science Ltd.