The Riemann problem on a finite-sheeted Riemann surface of infinite genus

Let R be the Riemann surface of the function u(z) specified by the equation u n = P(z) with n ∈ ℕ, n ≥ 2, and z ∈ ℂ, where P(z) is an entire function with infinitely many simple zeros. On R, the Riemann boundary-value problem for an arbitrary piecewise smooth contour Γ is considered. Necessary and sufficient conditions for its solvability are obtained, and its explicit solution is constructed. ©2000 Kluwer Academic/Plenum Publishers

Vekua integral operators on Riemann surfaces

On an arbitrary (in general, noncompact) Riemann surface R, we study integral operators T and II analogous to the operators introduced by Vekua in his theory of generalized analytic functions. By way of application, we obtain necessary and sufficient conditions for the solvability of the nonhomogeneous Cauchy-Riemann equation ∂̄f = F in the class of functions f exhibiting Λ 0-behavior in the vicinity of the ideal boundary of the surface R. ©2001 Plenum Publishing Corporation

The Riemann boundary-value problem on an n-sheeted surface free of limit points of projections of branch points onto ℂ

We obtain solvability conditions and explicit solutions for the Riemann boundary-value problem on an n-sheeted surface in the case when projections of branch points on the complex plane condense only at infinity. © 2010 Allerton Press, Inc

A uniqueness theorem for linear elliptic equations with dominating derivative with respect to z

© 2016, Pleiades Publishing, Ltd.The interior uniqueness theorem for analytic functions was generalized by M.B. Balk to the case of polyanalytic functions of order n. He proved that, if the zeros of a polyanalytic function have an accumulation point of order n, then this function is identically zero. M.F. Zuev generalized this result to the case of metaanalytic functions. In this paper, we generalize the interior uniqueness theorem to solutions of linear homogeneous elliptic differential equations of order n with analytic coefficients whose senior derivative is the n-th power of the Cauchy–Riemann operator

Uniqueness theorem for linear elliptic equation of the second order with constant coefficients

© 2017, Allerton Press, Inc.The interior uniqueness theorem for analytic functions was generalized by M. B. Balk to the case of polyanalytic functions of order n. He proved that if the zeros of a polyanalytic function have an accumulation point of order n, then this function is identically zero. In this paper the interior uniqueness theorem is generalized to the solution to a linear homogeneous second order differential equation of elliptic type with constant coefficients

Boundary value problem for a second-order elliptic equation in the exterior of an ellipse

© 2017, Pleiades Publishing, Ltd. We consider a boundary value problem for a second-order linear elliptic differential equation with constant coefficients in a domain that is the exterior of an ellipse. The boundary conditions of the problem contain the values of the function itself and its normal derivative. We give a constructive solution of the problem and find the number of solvability conditions for the inhomogeneous problem as well as the number of linearly independent solutions of the homogeneous problem. We prove the boundary uniqueness theorem for the solutions of this equation

On boundary uniqueness theorems for a linear elliptic equation with constant coefficients

For the solutions of an elliptic equation with constant coefficients, we prove uniqueness theorems that generalize the classical boundary uniqueness theorems for analytic functions. © 2014 Pleiades Publishing, Ltd

Rare-earth-containing magnetic liquid crystals

Rare-earth-containing metallomesogens with 4-alkoxy-N-alkyl-2- hydroxybenzaldimine ligands are reported. The stoichiometry of the complexes is [Ln(LH)3(NO3)3], where Ln is the trivalent rare-earth ion (Y, La, and Pr to Lu, except Pm) and LH is the Schiff base. The Schiff base ligands are in the zwitterionic form and coordinate through the phenolic oxygen only. The three nitrate groups coordinate in a bidentate fashion. The X-ray single- crystal structures of the nonmesogenic homologous complexes [Ln(LH)3(NO3)3], where Ln = Nd(III), Tb(III), and Dy(III) and LH = CH3OC6H3(2-OH)CH=NC4H9, are described. Although the Schiff base ligands do not exhibit a mesophase, the metal complexes do (SmA phase). The mesogenic rare-earth complexes were studied by NMR, IR, EPR, magnetic susceptibility measurements, X-ray diffraction, and molecular modeling. The metal complexes in the mesophase have a very large magnetic anisotropy, so that these magnetic liquid crystals can easily be aligned by an external magnetic field

Vekua integral operators on Riemann surfaces

On an arbitrary (in general, noncompact) Riemann surface R, we study integral operators T and II analogous to the operators introduced by Vekua in his theory of generalized analytic functions. By way of application, we obtain necessary and sufficient conditions for the solvability of the nonhomogeneous Cauchy-Riemann equation ∂̄f = F in the class of functions f exhibiting Λ 0-behavior in the vicinity of the ideal boundary of the surface R. ©2001 Plenum Publishing Corporation

On boundary uniqueness theorems for a linear elliptic equation with constant coefficients

For the solutions of an elliptic equation with constant coefficients, we prove uniqueness theorems that generalize the classical boundary uniqueness theorems for analytic functions. © 2014 Pleiades Publishing, Ltd