Parabolic equations with the second order Cauchy conditions on the boundary

The paper studies some ill-posed boundary value problems on semi-plane for parabolic equations with homogenuous Cauchy condition at initial time and with the second order Cauchy condition on the boundary of the semi-plane. A class of inputs that allows some regularity is suggested and described explicitly in frequency domain. This class is everywhere dense in the space of square integrable functions.Comment: 7 page

On the number of solutions of a transcendental equation arising in the theory of gravitational lensing

The equation in the title describes the number of bright images of a point source under lensing by an elliptic object with isothermal density. We prove that this equation has at most 6 solutions. Any number of solutions from 1 to 6 can actually occur.Comment: 26 pages, 12 figure

Successive Coefficients of Univalent Functions

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135332/1/jlms0448.pd

An Arclength Problem for Close‐to‐Convex Functions

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135569/1/jlms0181.pd

Predictability of band-limited, high-frequency, and mixed processes in the presence of ideal low-pass filters

Pathwise predictability of continuous time processes is studied in deterministic setting. We discuss uniform prediction in some weak sense with respect to certain classes of inputs. More precisely, we study possibility of approximation of convolution integrals over future time by integrals over past time. We found that all band-limited processes are predictable in this sense, as well as high-frequency processes with zero energy at low frequencies. It follows that a process of mixed type still can be predicted if an ideal low-pass filter exists for this process.Comment: 10 page