On cohesive powers of linear orders

Cohesive powers of computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let $\omega$, $\zeta$, and $\eta$ denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of $\omega$. If $\mathcal{L}$ is a computable copy of $\omega$ that is computably isomorphic to the standard presentation of $\omega$, then every cohesive power of $\mathcal{L}$ has order-type $\omega + \zeta\eta$. However, there are computable copies of $\omega$, necessarily not computably isomorphic to the standard presentation, having cohesive powers not elementarily equivalent to $\omega + \zeta\eta$. For example, we show that there is a computable copy of $\omega$ with a cohesive power of order-type $\omega + \eta$. Our most general result is that if $X \subseteq \mathbb{N} \setminus \{0\}$ is either a $\Sigma_2$ set or a $\Pi_2$ set, thought of as a set of finite order-types, then there is a computable copy of $\omega$ with a cohesive power of order-type $\omega + \sigma(X \cup \{\omega + \zeta\eta + \omega^*\})$, where $\sigma(X \cup \{\omega + \zeta\eta + \omega^*\})$ denotes the shuffle of the order-types in $X$ and the order-type $\omega + \zeta\eta + \omega^*$. Furthermore, if $X$ is finite and non-empty, then there is a computable copy of $\omega$ with a cohesive power of order-type $\omega + \sigma(X)$

Strong jump inversion

© The Author(s) 2018. We say that a structure A admits strong jump inversion provided that for every oracle X, if X' computes D(C)'for some C ≅ A, then X computes D(B) for some B ≅ A. Jockusch and Soare (1991, APAL, 52, 39-64) showed that there are low linear orderings without computable copies, but Downey and Jockusch (1994, PAMS, 122, 871-880) showed that every Boolean algebra admits strong jump inversion. More recently, D. Marker and R. Miller (2017, JSL, 82, 1-25) have shown that all countable models of DCF0 (the theory of differentially closed fields of characteristic 0) admit strong jump inversion. We establish a general result with sufficient conditions for a structure A to admit strong jump inversion. Our conditions involve an enumeration of B1-types, where these are made up of formulas that are Boolean combinations of existential formulas. Our general result applies to some familiar kinds of structures, including some classes of linear orderings and trees. We do not get the result of Downey and Jockusch for arbitrary Boolean algebras, but we do get a result for Boolean algebras with no 1-atom, with some extra information on the complexity of the isomorphism. Our general result gives the result of Marker and Miller. In order to apply our general result, we produce a computable enumeration of the types realized in models of DCF0. This also yields the fact that the saturated model of DCF0 has a decidable copy

Surface hardening alloy VT6 of electric explosion and by electron beam

The aim is to study the phase composition, structure and properties of the surface layer of the VT6 titanium alloy, subjected to combined treatment, consisting of alloying by the plasma of an electric explosion of a graphite fiber with a charge of the SiC powder and subsequent exposure by a high-intense electron beam. As a result of such treatment, a multiphase surface layer with a submicron and nanosize structure forms with the microhardness manifold exceeding its value in the sample volume are presented

МОДИФИКАЦИЯ ПОВЕРХНОСТИ СПЛАВА ВТ6 ПЛАЗМОЙ ЭЛЕКТРИЧЕСКОГО ВЗРЫВА ПРОВОДЯЩЕГО МАТЕРИАЛА И ОБЛУЧЕНИЕМ ЭЛЕКТРОННЫМ ПУЧКОМ

The results of investigations of phase composition, structure, and properties of VT6 titanium alloy surface layer subjected to combined processing, which consists of alloying by plasma of electric explosion of carbon-graphite fiber with SiC powder specimen and subsequent high-intensity electron-beam irradiation, are given. As a result of such a treatment, a multiphase surface layer with submicro- and nanosized structure the microhardness of which many times exceeds the value in the sample bulk.Приведены результаты исследования фазового состава, структуры и свойств поверхностного слоя титанового сплава ВТ6, подвергнутого комбинированной обработке, заключающейся в легировании плазмой электрического взрыва углеграфитового волокна с навеской порошка SiC и последующем облучении высокоинтенсивным электронным пучком. В результате такой обработки формируется многофазный поверхностный слой с субмикро- и наноразмерной структурой, микротвердость которого многократно превышают ее величину в объеме образца

Relativized Degree Spectra

Abstract. We present a relativized version of the notion of a degree spectrum of a structure with respect to finitely many abstract structures. We study the connection to the notion of joint spectrum. We prove that some properties of the degree spectrum as a minimal pair theorem and the existence of quasi-minimal degrees are true for the relative spectrum. Key words: enumeration degrees; forcing; degree spectra; recursive Σ + k formulae.

A Jump Inversion Theorem for the Degree Spectra

A jump inversion theorem for the degree spectra is presented. For a structure A which degree spectrum is a subset of the jump spectrum of a structure B, a structure C is constructed as a Marker’s extension of A such that the jump spectrum of C is exactly the degree spectrum of A and the degree spectrum of C is a subset of the degree spectrum of B