Double jumps of minimal degrees over cardinals

1982 ◽  
Vol 47 (2) ◽  
pp. 329-334
Author(s):  
C. T. Chong
Keyword(s):  
Positive Answer ◽  
The Other ◽  
Total Function ◽  
Peculiar Property ◽  
Other Hand ◽  
Power Set

Jockusch and Posner [4] showed that every minimal ω-degree is GL2. This is achieved by exhibiting a function f recursive in 0′ which dominates every function of minimal ω-degree. The function f has the peculiar property that for every s, f(s) is defined after a search (using 0′) over the power set of Ls (Gödel's constructible hierarchy at the level s). It can be seen that a function defined in a similar manner over an infinite successor cardinal k will not be a total function, since for example if k = ρ+, then f(ρ) will not be defined until after all the subsets of ρ have been examined, and this will take at least k steps. The following questions then naturally arise: (i) For successor cardinals k, is there a function dominating every set of minimal k-degree? (ii) For arbitrary cardinals k, is every minimal k-degree GL2 (i.e. b″ = (b ∨ 0′) for b of minimal k-degree)? In this paper we answer (i) in the negative and provide a positive answer to (ii), assuming V = L. We show in fact that if k is a successor cardinal and h ≤k 0′, then there is a function of minimal k-degree below 0′ not dominated by h (Theorem 1). This implies that any refinement of the function f described above will not be able to remove the difficulties encountered. On the other hand, we introduce the notion of ‘strong domination’ to provide a positive answer to (ii) (Theorem 2 and Corollary 1). We end this paper by indicating that for limit cardinals k, there is a function below 0′ dominating every function of minimal k-degree.

A note on direct sums of Friedbergnumberings

1989 ◽  
Vol 54 (3) ◽  
pp. 1009-1010 ◽  
Author(s):  
Martin Kummer
Keyword(s):  
Finite Function ◽  
Direct Sums ◽  
Direct Sum ◽  
Total Function ◽  
Standard Notation ◽  
P 16 ◽  
Standard Approximation

We show that a translator ƒ: ω → ω from a Gödelnumbering φ into a direct sum η of a r.e. family of Friedbergnumberings satisfies ƒ ≰T0′. In particular, η cannot be a Gödelnumbering.In the following we use standard notation (cf.[3]): for i ≥ 1, Pi (respectively, Ri) is the set of partial (total) recursive i-place functions; φ is a Gödel numbering of P1. By φi, s we denote a recursive standard approximation for φi, i.e., φi, s is a finite function, φi, s ⊆ φi, s + 1, φi, s ⊆ φi, φi = ⋃ {φi, s ∣ s ≥ 0}, and a canonical index for φi, s can be computed uniformly in i, s (cf.[3, p. 16 f]).We call v ∈ P2 a numbering of P1 iff {λx.v(i, x)}i ∈ ω = P1; we denote λx.v(i, x) by vi. Let v and γ be numberings of P1. A total function g: ω → ω is called a translator from v into γ iff ∀i: vi = γg(i).

Research of the Thought and Method in Function Reserving Design Based on Individuation Degree Evaluation

2012 ◽  
Vol 544 ◽  
pp. 188-193
Author(s):  
Chun Fu Li ◽  
Ze Qun Ye ◽  
Yu Hui Wang ◽  
Xian Min Feng
Keyword(s):  
Design Procedure ◽  
Total Function ◽  
Volume Production

Thought and method in function reserving design is for realizing the volume-production of individual products, based on the attention of users' individualities. The function tree is built up according to the subdivision of the total function. On this foundation, the method requires all the leaves of the function tree evaluated on their individuation degree by calculating the opening degree, association degree and dependency between the factors and the intimacy and interest degree between function factors and users. According to the result of individual degree evaluation, a rank of individuation degree priority of them is listed to help to decide the proper factors to reserve. By this means, users can participate themselves into the procedure of design, meeting their own individual needs.At last of this paper, the realizability of the thought and method is statemented and a case of lamp design of this method is taken as an example to demonstrate the design procedure.

scholarly journals EXPERIMENTAL NEPHRITIS IN THE FROG

1931 ◽  
Vol 53 (6) ◽  
pp. 763-784 ◽  
Author(s):  
Jean Oliver ◽  
Eshref Shevky
Keyword(s):  
Total Function ◽  
Experimental Nephritis ◽  
Glomerular Lesions

1. A method of testing the frog's kidney by means of perfusion is described. 2. This is made possible by dissociating, as far as possible, from the total function of the organ the functions of its constituent parts. 3. The characteristics by which tubular, glomerular, and combined tubular-glomerular lesions may be recognized are described.

scholarly journals The Transrational Numbers as an Abstract Data Type

2020 ◽  
Author(s):  
Jan Aldert Bergstra ◽  
John V. Tucker
Keyword(s):  
Rational Numbers ◽  
Data Type ◽  
Abstract Data Type ◽  
Constant Symbol ◽  
Total Function ◽  
Initial Algebra ◽  
Abstract Data ◽  
Initial Algebra Semantics

In an arithmetical structure one can make division a total function by defining 1/0 to be an element of the structure, or by adding a new element, such as an error element also denoted with a new constant symbol, an unsigned infinity or one or both signed infinities, one positive and one negative. We define an enlargement of a field to a transfield, in which division is totalised by setting 1/0 equal to the positive infinite value and -1/0 equal to its opposite, and which also contains an error element to help control their effects. We construct the transrational numbers as a transfield of the field of rational numbers and consider it as an abstract data type. We give it an equational specification under initial algebra semantics.

scholarly journals Groin and hip quandaries

2012 ◽  
Vol 24 (4) ◽  
Author(s):  
H Millson
Keyword(s):  
Hip Joint ◽  
Correct Diagnosis ◽  
Pubic Symphysis ◽  
Kinetic Chain ◽  
Total Function ◽  
Key Factor

There is little consensus on the diagnosis, pathophysiology, investigation and management of groin injuries. A key factor in making the correct diagnosis is to firstly understand the anatomy and likely generators of pain in the region. This requires an understanding of the two joints in the pelvis – the hip joint and the pubic symphysis ‒ which are at the centre of many movements. There are a multitude of varying studies on this topic. However, most importantly, many of the groin/hip pathologies can be averted by thorough and specific prehabilitation, bearing in mind the entire kinetic chain and addressing total function above and below the pelvis.

Self-Adapting Function Principle and Creative Design of BFT Type

2013 ◽  
Vol 397-400 ◽  
pp. 830-832
Author(s):  
Huai Lin Luo ◽  
Ling Yu Zhang ◽  
Quan Yuan
Keyword(s):  
Design Method ◽  
The Self ◽  
Creative Design ◽  
Total Function

The author summed up the total function of the self-adapting system of slow-footed multiple-flexible driving with overloading by making use of the creative design method of function principle, applied the mechanism of self-adapting to creative design this sort equipment of BFT,funded its false, and proposed way of creative design basing on the total functions.This way has super-performance of self-adapting .It has a good foreground

scholarly journals A Modelling Pearl with Sortedness Constraints

10.29007/b4dz ◽  
2018 ◽  
Author(s):  
Nicolas Beldiceanu ◽  
Mats Carlsson ◽  
Pierre Flener ◽  
Xavier Lorca ◽  
Justin Pearson ◽  
...  
Keyword(s):  
Constraint Programming ◽  
Modelling Languages ◽  
Total Function

Some constraint programming solvers and constraint modelling languages feature the SORT(L,P,S) constraint, which holds if S is a nondecreasing rearrangement of the list L, the permutation being made explicit by the optional list P. However, such sortedness constraints do not seem to be used much in practice. We argue that reasons for this neglect are that it is impossible to require the underlying sort to be stable, so that SORT cannot be guaranteed to be a total-function constraint, and that L cannot contain tuples of variables, some of which form the key for the sort. To overcome these limitations, we introduce the StableKeysort constraint, decompose it using existing constraints, and propose a propagator. This new constraint enables a powerful modelling idiom, which we illustrate by elegant and scalable models of two problems that are otherwise hard to encode as constraint programs.

Σ1 definitions with parameters

1986 ◽  
Vol 51 (2) ◽  
pp. 453-461
Author(s):  
T. A. Slaman
Keyword(s):  
Transitive Closure ◽  
Admissible Set ◽  
Selection Scheme ◽  
Ordinal Numbers ◽  
Recursively Enumerable ◽  
Total Function ◽  
Existential Quantification

AbstractLet p be a set. A function Φ is uniformly Σ1(p) in every admissible set if there is a Σ1 formula ϕ in the parameter p so that ϕ defines Φ in every Σ1-admissible set which includes p. A theorem of Van de Wiele states that if Φ is a total function from sets to sets then Φ is uniformly Σ1 in every admissible set if and only if it is E-recursive. A function is ESp-recursive if it can be generated from the schemes for E-recursion together with a selection scheme over the transitive closure of p. The selection scheme is exactly what is needed to insure that the ESP-recursively enumerable predicates are closed under existential quantification over the transitive closure of p. Two theorems are established: a) If the transitive closure of p is countable then a total function on sets is ESp-recursive if and only if it is uniformly Σ1(p) in every admissible set. b) For any p, if Φ is a function on the ordinal numbers then Φ is ESP-recursive if and only if it is uniformly Σ1(p) in every admissible set.

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