We show that for any computably enumerable set A and any equation image set L, if L is low and equation image, then there is a c.e. splitting equation image such that equation image. In Particular, if L is low and n-c.e., then equation image is n-c.e. and hence there is no low maximal n-c.e. degree.

We say that a computably enumerable (c.e.) degree b is a Lachlan nonsplitting base (LNB), if there is a computably enumerable degree a such that a > b, and for any c.e. degrees w,v ≤ a, if a ≤ w or; v or; b then either a ≤ w or; b or a ≤ v or; b. In this paper we investigate the relationship between bounding and nonbounding of Lachlan nonsplitting bases and the high /low hierarchy. We prove that there (...) is a non-Low2 c.e. degree which bounds no Lachlan nonsplitting base. (shrink)

Resumo: Paul Lengrand escreveu extensivamente sobre a filosofia da educação física, ao longo de sua vida. Esses trabalhos foram meticulosamente coletados, categorizados e sintetizados. De acordo com um estudo perceptivo, a postura vitalícia de Paul Lengrand sobre a filosofia da educação física foi significativamente influenciada por seu histórico singular, demandas sociais e interações interpessoais. Além disso, foi revelado que sua visão sobre a filosofia da educação física era uma extensão de seu profundo compromisso com a filosofia da educação, no decorrer (...) da vida, ao mesmo tempo que a aprimorava e refinava. A educação física e o esporte são conteúdos importantes da educação física, ao longo da vida. O esporte não deve ser visto como uma atividade muscular, mas deve ser combinado com a situação atual, moral e arte, e o período deve ser integrado com a educação, durante a vida. Centrada inequivocamente em torno de abraçar o exercício físico em todos os momentos da existência humana, a educação física foi considerada como uma dimensão indispensável da vida que exemplifica a durabilidade, a progressão gradual e a abrangência da filosofia da educação física, ao longo da vida. (shrink)

Lachlan observed that any nonzero d.c.e. degree bounds a nonzero c.e. degree. In this paper, we study the c.e. predecessors of d.c.e. degrees, and prove that given a nonzero d.c.e. degree , there is a c.e. degree below and a high d.c.e. degree such that bounds all the c.e. degrees below . This result gives a unified approach to some seemingly unrelated results. In particular, it has the following two known theorems as corollaries: there is a low c.e. degree isolating (...) a high d.c.e. degree [S. Ishmukhametov, G. Wu, Isolation and the high/low hierarchy, Arch. Math. Logic 41 259–266]; there is a high d.c.e. degree bounding no minimal pairs [C.T. Chong, A. Li, Y. Yang, The existence of high nonbounding degrees in the difference hierarchy, Ann. Pure Appl. Logic 138 31–51]. (shrink)

We show that every nonzero $\Delta _{2}^{0}$ e-degree bounds a minimal pair. On the other hand, there exist $\Sigma _{2}^{0}$ e-degrees which bound no minimal pair.

It is shown that there exists a low2 Harrington non-splitting base-that is, a low2 computably enumerable (c.e.) degree a such that for any c.e. degrees x, y, if $0' = x \vee y$ , then either $0' = x \vee a$ or $0' = y \vee a$ . Contrary to prior expectations, the standard Harrington non-splitting construction is incompatible with the $low_{2}-ness$ requirements to be satisfied, and the proof given involves new techniques with potentially wider application.

We give a corrected proof of an extension of the Robinson Splitting Theorem for the d. c. e. degrees.

A set Asubset of or equal toω is called computably enumerable , if there is an algorithm to enumerate the elements of it. For sets A,Bsubset of or equal toω, we say that A is bounded Turing reducible to reducible to) B if there is a Turing functional, Φ say, with a computable bound of oracle query bits such that A is computed by Φ equipped with an oracle B, written image. Let image be the structure of the c.e. bT-degrees, (...) the c.e. degrees under the bounded Turing reductions. In this paper we study the continuity properties in image. We show that for any c.e. bT-degree image, there is a c.e. bT-degree image such that for any c.e. bT-degree image, image if and only if image. We prove that the analog of the Seetapun local noncappability theorem from the c.e. Turing degrees also holds in image. This theorem demonstrates that every image is noncappable with any nontrivial degree below some image. (shrink)

We study the jump hierarchy of d.c.e. Turing degrees and show that there exists a high d.c.e. degree d which does not bound any minimal pair of d.c.e. degrees.

We show that every nonzero Δ20, e-degree bounds a minimal pair. On the other hand, there exist Σ20, e-degrees which bound no minimal pair.

The Major Sub-degree Problem of A. H. Lachlan (first posed in 1967) has become a long-standing open question concerning the structure of the computably enumerable (c.e.) degrees. Its solution has important implications for Turing definability and for the ongoing programme of fully characterising the theory of the c.e. Turing degrees. A c.e. degree a is a major subdegree of a c.e. degree b > a if for any c.e. degree x, ${{\bf 0' = b \lor x}}$ if and only if (...) ${{\bf 0' = a \lor x}}$ . In this paper, we show that every c.e. degree b ≠ 0 or 0′ has a major sub-degree, answering Lachlan’s question affirmatively. (shrink)

We prove that for any computably enumerable degree c, if it is cappable in the computably enumerable degrees, then there is a d.c.e. degree d such that c d = 0′ and c ∩ d = 0. Consequently, a computably enumerable degree is cappable if and only if it can be complemented by a nonzero d.c.e. degree. This gives a new characterization of the cappable degrees.

A computably enumerable degree a is called nonbounding, if it bounds no minimal pair, and plus cupping, if every nonzero c.e. degree x below a is cuppable. Let NB and PC be the sets of all nonbounding and plus cupping c.e. degrees, respectively. Both NB and PC are well understood, but it has not been possible so far to distinguish between the two classes. In the present paper, we investigate the relationship between the classes NB and PC, and show that (...) there exists a minimal pair which join to a plus cupping degree, so that PC ⊈ NB. This gives a first known difference between NB and PC. (shrink)

It will be shown that there exist computably enumerable degrees a and b such that a $>$ b, and for any computably enumerable degree u, if u $\leq$ a and u is cappable, then u $<$ b.

In this paper, we prove that there exist computably enumerable degrees a and b such that $\mathbf{a} > \mathbf{b}$ and for any degree x, if x ≤ a and x is a minimal degree, then $\mathbf{x}.

In this paper, we first prove that there exist computably enumerable (c.e.) degrees a and b such that ${\bf a\not\leq b}$ , and for any c.e. degree u, if ${\bf u\leq a}$ and u is cappable, then ${\bf u\leq b}$ , so refuting a conjecture of Lempp (in Slaman [1996]); secondly, we prove that: (A. Li and D. Wang) there is no uniform construction to build nonzero cappable degree below a nonzero c.e. degree, that is, there is no computable function (...) $f$ such that for all $e\in\omega,$ (i) $W_{f(e)}\leq_{\rm T}W_e$ , (ii) $W_{f(e)}$ has a cappable degree, and (iii) $W_{f(e)}\not\leq_{\rm T}\emptyset$ unless $W_e\leq_{\rm T}\emptyset.$. (shrink)

The system reachability set is calculated by covering all possible behaviours of the system through a finite number of simulation steps to ensure that the system trajectory stays within a set safety region. In this paper, the theory of the game method is applied to the design of the controller, a very small controller is designed, and good control results are obtained by simulation. The system gradually shows a divergent trend and cannot achieve stable control. A multihop channel reservation Medium (...) Access Control protocol based on a parallel mechanism is proposed. The multihop channel reservation mechanism is proposed based on periodic node sleep, and the node uses the reservation frame to make the reservation of the channel and transmits the data according to the reservation information; the parallel mechanism is used to make the reservation information and data transmission simultaneously. (shrink)

We say that a computably enumerable (c. e.) degree a is plus-cupping, if for every c.e. degree x with $0 < x \leq a$ , there is a c. e. degree $y \not= 0'$ such that $x \vee y = 0/\'$ . We say that a is n-plus-cupping. if for every c. e. degree x, if $0 < x \leq a$ , then there is a $low_n$ c. e. degree 1 such that $x \vee l = 0'$ . Let PC (...) and $PC_n$ be the set of all plus-cupping, and n-plus-cupping c. e. degrees respectively. Then $PC_{1} \subseteq PC_{2} \subseteq PC_3 = PC$ . In this paper we show that $PC_{1} \subset PC_2$ , so giving a nontrivial hierarchy for the plus cupping degrees. The theorem also extends the result of Li, Wu and Zhang [14] showing that $LC_{1} \subset LC_2$ , as well as extending the Harrington plus-cupping theorem [8]. (shrink)