The Riemannian symmetric spaces play an important role in different branches of mathematics. By definition, a (connected) Riemannian manifold M is called symmetric if, to every a ∈ M , there exists an involutory isometric diffeomorphism sa:M → M having a as isolated fixed point in M (or equivalently, if the differential dasa is the negative identity on the the tangent space Ta = TaM of M at a)....

We consider the Euler–Darboux equation with parameters such that their absolute values are equal to $$ \frac{1}{2} . Since Cauchy problem in classical formulation is ill-posed for of parameters, we propose formulations and solutions modified Cauchy-type problems following parameters: (a) \alpha =\beta =\frac{1}{2}; (b) =-\frac{1}{2}\ \mathrm{and}\ \beta (c) =-\frac{1}{2}. In case (a), solved by...

We study the class of functions called monodiffric of the second kind by Rufus Philip Isaacs. They are discrete analogues of holomorphic functions of one or two complex variables. Discrete analogues of the Cauchy–Riemann operator, of domains of holomorphy in one discrete variable, and of the Hartogs phenomenon in two discrete variables are investigated. Two fundamental solutions to the discrete...

Singular complex analytic vector fields on the Riemann surfaces enjoy several geometric properties (singular means that poles and essential singularities are admissible). We describe relations between singular $${\mathbb{X}}$$ smooth $$X$$ . Our approximation route studies three integrability notions for real with plane or sphere. The first notion is related to Cauchy–Riemann equations, we say ...

Formulæ are derived for expressing Cauchy and Hilbert transforms of a function f in terms of Cauchy and Hilbert transforms of f(xr). When r is an integer, this corresponds to evaluating the Cauchy transform of f(xr) at all choices of z1/r. Related formuæ for rational r result in a reduction to a generalized Cauchy transform living on a Riemann surface, which in turn is reducible to the standard...

In this study, the existence and uniqueness of the solution of a class of nonlinear finite-part singular integral equations with Carleman shift preserving orientation has been investigated in the generalized Holder space () Γ ϕ H. 1. Introduction Many important problems of engineering mechanics like elasticity, plasticity, and fracture mechanics and aerodynamics can be reduced to the solution o...