The optimal form of the scanning near-field optical microscopy probe

A theoretical approach to determine the optimal form of the near-field optical microscope probe is proposed. An analytical expression of the optimal probe form with subwavelength aperture has been obtained. The advantages of the probe with the optimal form are illustrated using numerical calculations. The conducted calculations show 10 times greater light throughput and the reception possibility of the more compactly localized light at the output probe aperture which could indicate better spatial resolution of the optical images in near-field optical technique using optimal probe.Comment: 12 pages, 6 figure

Structural Theory of Degrees of Unsolvability: Advances and Open Problems

© 2015, Springer Science+Business Media New York. Presented by the Program Committee of the Conference “Mal’tsev Readings

Definable relations in Turing degree structures

In this paper we investigate questions about the definability of classes of n-computably enumerable (c. e.) sets and degrees in the Ershov difference hierarchy. It is proved that the class of all c. e. sets is definable under the set inclusion ⊆ in all finite levels of the difference hierarchy. It is also proved the definability of all m-c. e. degrees in each higher level of the hierarchy. Besides, for each level n, n ≥ 2, of the hierarchy a definable non-trivial subset of n-c. e. degrees is established. © 2014 Allerton Press, Inc

Arithmetic complexity via effective names for random sequences

We investigate enumerability properties for classes of sets which permit recursive, lexicographically increasing approximations, or left-r.e. sets. In addition to pinpointing the complexity of left-r.e. Martin-L\"{o}f, computably, Schnorr, and Kurtz random sets, weakly 1-generics and their complementary classes, we find that there exist characterizations of the third and fourth levels of the arithmetic hierarchy purely in terms of these notions. More generally, there exists an equivalence between arithmetic complexity and existence of numberings for classes of left-r.e. sets with shift-persistent elements. While some classes (such as Martin-L\"{o}f randoms and Kurtz non-randoms) have left-r.e. numberings, there is no canonical, or acceptable, left-r.e. numbering for any class of left-r.e. randoms. Finally, we note some fundamental differences between left-r.e. numberings for sets and reals

Splitting and non-splitting in the difference hierarchy

Copyright © Cambridge University Press 2016In this paper, we investigate splitting and non-splitting properties in the Ershov difference hierarchy, in which area major contributions have been made by Barry Cooper with his students and colleagues. In the first part of the paper, we give a brief survey of his research in this area and discuss a number of related open questions. In the second part of the paper, we consider a splitting of 0′ with some additional properties