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It is well-known that every c.e. Turing degree is the degree of categoricity of a rigid structure. In this work we study the possibility of extension of this result to properly 2-c.e. degrees. We found a condition such that if a $$\Delta^{0}_{2}$$ -degree is a degree of categoricity of a rigid structure and satisfies this condition then it must be c.e. Also we show that degrees of non-categoricity are dense in the c.e. degrees.

In this work we consider isolation from side in different degree structures, in particular, in the 2-computably enumerable wtt-degrees and in low Turing degrees. Intuitively, a 2-computably enumerable degree is isolated from side if all computably enumerable degrees from its lower cone are bounded from above by some computably enumerable degree which is incomparable with the given one. It is proved that any properly 2-computably enumerable wtt-degree is isolated from side by some computable enumerable wtt-degree. Also it is shown that the same result holds for the low 2-computable enumerable Turing degrees.

In chronic infections caused by retroviruses, "antigen antibody" immune complexes (AAC) appear in body fluids. In the course of the studies to clarify the structure of the AAC we proved that immune complexes in cattle leukemia could contain the virus's viral DNA. The role of circulating immune complexes in the pathogenesis of the bovine leukemia virus (BLV) infection is not sufficiently studied. AAC containing the viral DNA may be formed during the death of infected B-lymphocytes by linking non-invasive viral DNA to surface IgMs with high DNA affinity. Determining the appearance of AAC with the viral DNA in biological fluids depending on the stages of the infectious process is an important criterion in antileukemia measures. The article describes the results of research to determine circulating immune complexes containing the provirus DNA in the milk of cows infected with BLV as compared with hematological changes at different stages of the infection process. It is reported that the viral DNA in the AAC of the milk of cows infected with BLV is detected much earlier than hematological changes that allow to consider them leukemia patients. Of the 12 cows studied, by the end of the experiment, six cows had AAC containing the viral DNA in their milk samples. And only one cow with the detection of the viral DNA in the AAC of milk showed parallel hematological changes typical for leukemia. The use of milk samples for molecular genetic analysis to identify BLV in AAC is quite a promising area for assessing the epizootic situation on farms and is very informative in determining the animal status.

A standard tool for classifying the complexity of equivalence relations on $$\omega$$ is provided by computable reducibility. This reducibility gives rise to a rich degree structure. The paper studies equivalence relations, which induce minimal degrees with respect to computable reducibility. Let $$\Gamma$$ be one of the following classes: $$\Sigma^{0}_{\alpha}$$ , $$\Pi^{0}_{\alpha}$$ , $$\Sigma^{1}_{n}$$ , or $$\Pi^{1}_{n}$$ , where $$\alpha\geq 2$$ is a computable ordinal and $$n$$ is a non-zero natural number. We prove that there are infinitely many pairwise incomparable minimal equivalence relations that are properly in $$\Gamma$$ .

The article presents the results of studying the dynamics of the formation of antibodies and immune complexes, reveals the prospects for improving the early diagnosis of cattle leukaemia. Studies were conducted for 6 months on an experimental group of animals consisting of 12 cows. The titers of free and bound antibodies in blood serum and milk were determined by enzyme-linked immunosorbent assay. The results of studies showing that changes in titers of anti-leukaemia antibodies in the blood serum of cows naturally infected with BLV (bovine leukaemia virus) are significantly different from experimental infection data are adduced. In cows infected with BLV, there is no definite relationship between antibody titers in milk and in blood serum. With sufficiently high titers of serum antibodies, antibody titers in milk can be minimal for the same cows; conversely, with low titers of serum antibodies, there can be high antibody titers in milk. In the titers of antibodies free and bound in the immune complexes in blood serum with the development of the disease, a certain dependence is traced. With a decrease in titers of free antibodies, in most cases, an increase in titers of “bound” antibodies is observed, i.e., the degree of formation of circulating immune complexes (CECs) increases. There is no clearly defined dependence in the dynamics of changes in titers free and bound in immune complexes of antibodies in milk samples. They can remain at the same level for a long time, both at low and rather high levels.

We investigate the problem of characterizing proper levels of the fine hierarchy (up to Turing equivalence). It is known that the fine hierarchy exhausts arithmetical sets and contains as a small fragment finite levels of Ershov hierarchies (relativized to ∅n, n < ω), which are known to be proper. Our main result is finding a least new (i.e., distinct from the levels of the relativized Ershov hierarchies) proper level. We also show that not all new levels are proper.

Questions of definability of computably enumerable degrees in the difference hierarchy (degrees of sets from finite levels of the Ershov difference hierarchy) are studied. Several approaches to the solution of this problem are outlined.