Hierarchies over recursive well-orderings

1964 ◽  
Vol 29 (4) ◽  
pp. 183-190 ◽  
Author(s):  
Herbert Enderton ◽  
David Luckham
Keyword(s):  
Uniqueness Property ◽  
Ordinal Notations ◽  
Degrees Of Unsolvability

In the original example of a transfinite hierarchy of degrees of unsolvability, a predicate Ha is associated with each a in the set 0 of ordinal notations. (See Kleene [K1], Spector [S1], and the references there to Davis and Mostowski.) The predicates are defined by means of induction over the partial well-ordering relation ≤0 on the set 0 of notations for the recursive ordinals. The usefulness of this hierarchy of predicates is enhanced by Spector's proof [S1] of the “uniqueness” property: viz. that the degree of Ha depends only on the ordinal |a| for which a is a notation.

A hierarchy for the 1-section of any type two object

1974 ◽  
Vol 39 (1) ◽  
pp. 88-94 ◽  
Author(s):  
S. S. Wainer
Keyword(s):  
Affirmative Answer ◽  
The Other ◽  
Uniqueness Property ◽  
Hyperarithmetical Hierarchy ◽  
Other Hand ◽  
Primitive Recursion ◽  
Ordinal Notations ◽  
Definition Of ◽  
Image Position ◽  
Primitive Recursive

Shoenfield [6] constructed a hierarchy for any type two object F in which 2E is recursive by generalizing the hyperarithmetical hierarchy, using a jump operation jF defined by(A similar hierarchy was constructed independently by Hinman [3].) In order to construct a hierarchy for an arbitrary type two F we must first associate with F an operator which, on one hand will always be recursive in F, and on the other hand will generate all the functions recursive in F when iterated over a simultaneously generated set of ordinal notations OF. Clearly the use of 2E in the above definition of jF(f) can be avoided if, instead of diagonalizing F over all functions recursive in f, we only diagonalize F over, say, the functions primitive recursive in f. If furthermore we code in a function which enumerates all functions primitive recursive in f then the resulting operation will certainly generate a primitive recursively expanding hierarchy of functions recursive in F. The problem that remains is whether this hierarchy will exhaust the 1-section of F. But this reduces to an effectiyized version of the following problem: If g is recursive in some level of the hierarchy, is g primitive recursive in some higher level ? An affirmative answer is suggested by the completeness results of Feferman [2], and our main theorem below will be proved by combining his ideas with those of Shoenfield [6]. The result is a hierarchy which applies to all type two objects, and which replaces the notion of recursion in F by the simpler notion of primitive recursion in certain functions generated by F. Unfortunately, in contrast with the Shoenfield hierarchy, the hierarchy developed here cannot always be expected to have the uniqueness property (even w.r.t. ≤T), and for this reason the proof of our main theorem is rather more complicated than the corresponding proof in [6].

Stability of stochastic differential equations driven by multifractional Brownian motion

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Oussama El Barrimi ◽  
Youssef Ouknine
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stationary Process ◽  
Strong Stability ◽  
Pathwise Uniqueness ◽  
Uniqueness Property ◽  
Selection Theorem ◽  
Multifractional Brownian Motion ◽  
Stability Results

Abstract Our aim in this paper is to establish some strong stability results for solutions of stochastic differential equations driven by a Riemann–Liouville multifractional Brownian motion. The latter is defined as a Gaussian non-stationary process with a Hurst parameter as a function of time. The results are obtained assuming that the pathwise uniqueness property holds and using Skorokhod’s selection theorem.

Two variable implicational calculi of prescribed many-one degrees of unsolvability

1976 ◽  
Vol 41 (1) ◽  
pp. 39-44 ◽  
Author(s):  
Charles E. Hughes
Keyword(s):  
Decision Problems ◽  
Constructive Proof ◽  
Distinct Variable ◽  
The Family ◽  
Propositional Calculi ◽  
Degrees Of Unsolvability

AbstractA constructive proof is given which shows that every nonrecursive r.e. many-one degree is represented by the family of decision problems for partial implicational propositional calculi whose well-formed formulas contain at most two distinct variable symbols.

A. A. Fridman. Stépéni nérazréšimosti problémy toždéstva v konéčno oprédélénnyh gruppah. Doklady Akadémii Nauk SSSR, vol. 147 (1962), pp. 805–808. - A. A. Fridman. Degrees of insolvability of the word problem in finitely defined groups. English translation of the preceding by Sue Ann Walker. Soviet mathematics, vol. 3 no. 6 (1962), pp. 1733–1737. - C. R. J. Clapham. Finitely presented groups with word problems of arbitrary degrees of insolubility. Proceedings of the London Mathematical Society, ser. 3 vol. 14 (1964), pp. 633–676. - William W. Boone. Finitely presented group whose word problem has the same degree as that of an arbitrarily given Thue system (an application of methods of Britton). Proceedings of the National Academy of Sciences, vol. 53 (1965), pp. 265–269. - William W. Boone. Word problems and recursively enumerable degrees of unsolvability. A first paper on Thue systems. Annals of mathematics, ser. 2 vol. 83 (1966), pp. 520–571. - William W. Boone. Word problems and recursively enumerable degrees of unsolvability. A sequel on finitely presented groups. Annals of mathematics, ser. 2 vol. 84 (1966), pp. 49–84.

1968 ◽  
Vol 33 (2) ◽  
pp. 296-297
Author(s):  
J. C. Shepherdson
Keyword(s):  
Word Problem ◽  
Word Problems ◽  
Nauk Sssr ◽  
London Mathematical Society ◽  
National Academy Of Sciences ◽  
Doklady Akademii Nauk Sssr ◽  
Finitely Presented Groups ◽  
Recursively Enumerable ◽  
Finitely Presented ◽  
Degrees Of Unsolvability

Local uniqueness of certain geodesics related to Heegaard splittings

2018 ◽  
Vol 27 (09) ◽  
pp. 1842003
Author(s):  
Liang Liang ◽  
Fengling Li ◽  
Fengchun Lei ◽  
Jie Wu
Keyword(s):  
Heegaard Splitting ◽  
Sufficient Condition ◽  
Heegaard Splittings ◽  
Local Uniqueness ◽  
Uniqueness Property

Suppose [Formula: see text] is a Heegaard splitting and [Formula: see text] is an essential separating disk in [Formula: see text] such that a component of [Formula: see text] is homeomorphic to [Formula: see text], [Formula: see text]. In this paper, we prove that if there is a locally complicated simplicial path in [Formula: see text] connecting [Formula: see text] to [Formula: see text], then the geodesic connecting [Formula: see text] to [Formula: see text] is unique. Moreover, we give a sufficient condition such that [Formula: see text] is keen and the geodesic between any pair of essential disks on the opposite sides has local uniqueness property.

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