Solutions to the Riemann-Hilbert problems with irregular singularities naturally associated to semisimple Frobenius manifold structures on Hurwitz spaces (moduli spaces of meromorphic functions on Riemann surfaces) are constructed. The solutions are given in terms of meromorphic bidifferentials defined on the underlying Riemann surface. The relationship between different classes of Frobenius ma...

It is a classical result that flat coordinates for a Hurwitz Frobenius manifold can be obtained as periods of a differential along cycles on the domain curve. We generalise this construction to primary invariants of the Hurwitz Frobenius manifolds. We show that they can be obtained as periods of multidifferentials along the same cycles. The multidifferentials are obtained via the topological re...

We use algebraic methods to compute the simple Hurwitz numbers for arbitrary source and target Riemann surfaces. For an elliptic curve target, we reproduce the results previously obtained by string theorists. Motivated by the Gromov-Witten potentials, we find a general generating function for the simple Hurwitz numbers in terms of the representation theory of the symmetric group Sn. We also fin...

We calculate the cycle class of the Hurwitz divisor D2 on Mg for g = 2k given by the degree k+1 covers of P with simple ramification points, two of which lie in the same fibre. We also study some aspects of the geometry of the natural map from the Hurwitz space H2k,k+1 to the moduli space M2k .

In this paper we determine the orbits of the braid group Bn action on Gn when G is a dihedral group and for any T ∈ Gn. We prove that the following invariants serve as necessary and sufficient conditions for Hurwitz equivalence. They are: the product of its entries, the subgroup generated by its entries, and the number of times each conjugacy class (in the subgroup generated by its entries) is ...

We give detailed discussion of a procedure for determining the robust D-stability of a 4×4 real matrix. The procedure begins from the Hurwitz stability criterion. The procedure is applied to two numerical examples.

For the two sixth-order polynomials a(s) and b(s), Hurwitz stability of their convex combination is necessary and sufficient for the existence of a polynomial c(s) such that c(s)/a(s) and c(s)/b(s) are both strictly positive real. Our reasoning method is constructive, and is insightful and helpful in solving the general robust strictly positive real synthesis problem.

We describe the pairs (p, n) such that n-dimensional affine space is fibered by pairwise skew p-dimensional affine subspaces. The problem is closely related with the theorem of Adams on vector fields on spheres and the Hurwitz-Radon theory of composition of quadratic forms.