M. C. Câmara

Generalised factorisation for a class of Jones form matrix functions

A systematic approach is proposed for the generalised factorisation of certain non-rational n × n matrix functions. The first main result consists in a transformation of a meromorphic into a generalised factorisation by algebraic means. It closes a gap between the classical Wiener-Hopf procedure and the operator theoretic method of generalised factorisation. Secondly, as examples we consider certain matrix functions of Jones form or of N -part form, which are equivalent to each other, in a sense. The factorisation procedure is complete and explicit, based only on the factorisation of scalar functions, of rational matrix functions and upon linear algebra. Applications in elastodynamic diffraction theory are treated in detail and in a most effective way.

Explicit Wiener–Hopf factorization and nonlinear Riemann–Hilbert problems

A method for explicit Wiener–Hopf factorization of 2 × 2 matrix-valued functions is presented together with an abstract definition of a class of functions, C ( Q 1 , Q 2 ), to which it applies. The method involves the reduction of the original factorization problem to certain nonlinear scalar Riemann–Hilbert problems, which are easier to solve. The class C ( Q 1 , Q 2 ) contains a wide range of classes dealt with in the literature, including the well-known Daniele–Khrapkov class. The structure of the factors in the factorization of any element of the class C ( Q 1 , Q 2 ) is studied and a relation between the two columns of the factors, which gives one of the columns in terms of the other through a linear transformation, is established. This greatly simplifies the complete determination of the factors and gives relevant information on the nature of the factorization. Two examples suggested by applications are completely worked out.

Almost Periodic Factorization of Some Triangular Matrix Functions

The paper is devoted to matrices of the form \( G(x) = \left[ {\begin{array}{*{20}c} {e^{i\lambda x} } & 0 \\ {f(x)} & {e^{ - i\lambda x} } \\ \end{array} } \right], \), with almost periodic off-diagonal entry f. Some new cases are found, in terms of the Bohr-Fourier spectrum of f, in which G is factorable. Formulas for the partial indices are derived and, under additional constraints, the factorization itself is constructed explicitly. Some a priori conditions on the Bohr-Fourier spectra of the factorization factors (provided that a canonical factorization exists) are also given.

Factorizations, Riemann–Hilbert problems and the corona theorem

The solvability of the Riemann–Hilbert boundary value problem on the real line is described in the case when its matrix coefficient admits a Wiener–Hopf-type factorization with bounded outer factors, but rather general diagonal elements of its middle factor. This covers, in particular, the almost periodic setting, when the factorization multiples belong to the algebra generated by the functions eλ(x ): =e iλx , λ ∈ R. Connections with the corona problem are discussed. Based on those, constructive factorization criteria are derived for several types of triangular 2 × 2

Wiener-Hopf factorization for a class of oscillatory symbols

Two classes of 2×2 matrix symbols involving oscillatory functions are considered, one of which consists of triangular matrices. An equivalence theorem is obtained, concerning the solution of Riemann-Hilbert problems associated with each of them. Conditions for existence of canonical generalized factorization are established, as well as boundedness conditions for the factors. Explicit formulas are derived for the factors, showing in particular that only one of the columns needs to be calculated. The results are applied to solving a corona problem.

Wiener-Hopf factorization for a group of exponentials of nilpotent matrices

Abstract A complete study of the generalized factorization for a group of 2×2 matrix functions of the form G=I+γN , where γ∈ C ( R ) , I denotes the 2×2 identity matrix and N represents a rational nilpotent matrix function, is presented. A closely related class involving the same matrix N is also studied. The canonical and non-canonical factorizations are considered and explicit formulas are obtained for the partial indices and the factors in such factorizations. It is shown in particular that only one of the columns in the factors needs to be determined, as a solution to a homogeneous linear Riemann–Hilbert problem, the other column being expressed in terms of the first. Necessary and sufficient conditions for existence of a canonical factorization within the same class are established, as well as explicit formulas for the factors in this case.

Wiener–Hopf factorization of a generalized Daniele–Khrapkov class of 2×2 matrix symbols

The Wiener–Hopf factorization of a class of 2×2 symbols including matrices of Daniele–Khrapkov type is studied. The partial indices and the factors are determined, both in the canonical and non-canonical cases. A non-linear method is used which reduces the solution of a homogeneous Riemann–Hilbert problem to a non-linear scalar equation. Copyright

Generalized factorization for a class ofn×n matrix functions — Partial indices and explicit formulas

The generalized factorization of a class of continuous non-rationaln×n matrix-functions is studied. The partial indices are determined and, in the case of existence of a canonical factorization, explicit formulas for the factors are obtained.

A Riemann-Hilbert approach to rotating attractors

A bstractWe construct rotating extremal black hole and attractor solutions in gravity theories by solving a Riemann-Hilbert problem associated with the Breitenlohner-Maison linear system. By employing a vectorial Riemann-Hilbert factorization method we explicitly factorize the corresponding monodromy matrices, which have second order poles in the spectral parameter. In the underrotating case we identify elements of the Geroch group which implement Harrison-type transformations which map the attractor geometries to interpolating rotating black hole solutions. The factorization method we use yields an explicit solution to the linear system, from which we do not only obtain the spacetime solution, but also an explicit expression for the master potential encoding the potentials of the infinitely many conserved currents which make this sector of gravity integrable.

A nonlinear approach to generalized factorization of matrix functions

The generalized factorization of some classes of 2 s 2 matrix symbols is determined by reduction to the study of certain scalar non-linear Riemann-Hilbert problems. This method is applied to several types of matrix functions, whose factorization is explicitly obtained.