On models of arithmetic—Answers to two problems raised by H. Gaifman

1975 ◽  
Vol 40 (1) ◽  
pp. 41-47 ◽  
Author(s):  
Alex Wilkie
Keyword(s):  
Diophantine Equation ◽  
Initial Segment ◽  
Original Model ◽  
New Model ◽  
Recursive Functions ◽  
Nonstandard Model ◽  
Definable Functions ◽  
Models Of Arithmetic

In a recent paper [3] H. Gaifman investigated some model theoretic consequences of Matijasevič's theorem [5], and posed some further problems which naturally arise. We provide here partial answers to two of these problems, the results having been previously announced in the postscript of [3].Firstly, it is shown in [3] that if M1 and M2 are models of the Peano axioms P and M1 ⊆ M2, then M1 is closed under the recursive functions of M2. The converse of this statement is false. Moreover, Gaifman asks: Is every initial segment of a model M of P which is closed under the recursive functions of M (or the ∑n-definable functions) also a model of P? We show that this is false and our method gives, en route, another proof of a theorem of Rabin [7] stating the P is not implied by any consistent set of ∑n sentences for any n.Secondly, we partially answer a question posed on p. 129 of [3] by proving (some-what more than) every countable nonstandard model of P has an end extension in which a diophantine equation, not solvable in the original model, has a solution. We can, in fact, take the new model to be isomorphic to the original one. This generalises (apart from the countability restriction) a theorem of Rabin [6].

On the complexity of models of arithmetic

1982 ◽  
Vol 47 (2) ◽  
pp. 403-415 ◽  
Author(s):  
Kenneth McAloon
Keyword(s):  
Standard Model ◽  
Initial Segment ◽  
Peano Arithmetic ◽  
Nonstandard Model ◽  
Recursively Enumerable ◽  
The Standard Model ◽  
Models Of Arithmetic

AbstractLet P0 be the subsystem of Peano arithmetic obtained by restricting induction to bounded quantifier formulas. Let M be a countable, nonstandard model of P0 whose domain we suppose to be the standard integers. Let T be a recursively enumerable extension of Peano arithmetic all of whose existential consequences are satisfied in the standard model. Then there is an initial segment M′ of M which is a model of T such that the complete diagram of M′ is Turing reducible to the atomic diagram of M. Moreover, neither the addition nor the multiplication of M is recursive.

scholarly journals Modification and Validation of the Phosphate Removal Model: A Multicenter Study

2021 ◽  
pp. 1-10
Author(s):  
Weichen Zhang ◽  
Qiuna Du ◽  
Jing Xiao ◽  
Zhaori Bi ◽  
Chen Yu ◽  
...  
Keyword(s):  
Polynomial Regression ◽  
External Validation ◽  
Original Model ◽  
Phosphate Concentration ◽  
Phosphate Removal ◽  
Internal Validation ◽  
Modified Model ◽  
New Model ◽  
Significant Difference ◽  
Removal Model

<b><i>Background:</i></b> Our research group has previously reported a noninvasive model that estimates phosphate removal within a 4-h hemodialysis (HD) treatment. The aim of this study was to modify the original model and validate the accuracy of the new model of phosphate removal for HD and hemodiafiltration (HDF) treatment. <b><i>Methods:</i></b> A total of 109 HD patients from 3 HD centers were enrolled. The actual phosphate removal amount was calculated using the area under the dialysate phosphate concentration time curve. Model modification was executed using second-order multivariable polynomial regression analysis to obtain a new parameter for dialyzer phosphate clearance. Bias, precision, and accuracy were measured in the internal and external validation to determine the performance of the modified model. <b><i>Results:</i></b> Mean age of the enrolled patients was 63 ± 12 years, and 67 (61.5%) were male. Phosphate removal was 19.06 ± 8.12 mmol and 17.38 ± 6.75 mmol in 4-h HD and HDF treatments, respectively, with no significant difference. The modified phosphate removal model was expressed as Tpo<sub>4</sub> = 80.3 × <i>C</i><sub>45</sub> − 0.024 × age + 0.07 × weight + β × clearance − 8.14 (β = 6.231 × 10<sup>−3</sup> × clearance − 1.886 × 10<sup>−5</sup> × clearance<sup>2</sup> – 0.467), where <i>C</i><sub>45</sub> was the phosphate concentration in the spent dialysate measured at the 45th minute of HD and clearance was the phosphate clearance of the dialyzer. Internal validation indicated that the new model was superior to the original model with a significantly smaller bias and higher accuracy. External validation showed that <i>R</i><sup>2</sup>, bias, and accuracy were not significantly different than those of internal validation. <b><i>Conclusions:</i></b> A new model was generated to quantify phosphate removal by 4-h HD and HDF with a dialyzer surface area of 1.3–1.8 m<sup>2</sup>. This modified model would contribute to the evaluation of phosphate balance and individualized therapy of hyperphosphatemia.

Some theorems on R-maximal sets and major subsets of recursively enumerable sets

1971 ◽  
Vol 36 (2) ◽  
pp. 193-215 ◽  
Author(s):  
Manuel Lerman
Keyword(s):  
Turing Degrees ◽  
Elementary Equivalence ◽  
Recursive Functions ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Nonstandard Models ◽  
Relational Systems ◽  
The Relationship ◽  
Models Of Arithmetic

In [5], we studied the relational systems /Ā obtained from the recursive functions of one variable by identifying two such functions if they are equal for all but finitely many х ∈ Ā, where Ā is an r-cohesive set. The relational systems /Ā with addition and multiplication defined pointwise on them, were once thought to be potential candidates for nonstandard models of arithmetic. This, however, turned out not to be the case, as was shown by Feferman, Scott, and Tennenbaum [1]. We showed, letting A and B be r-maximal sets, and letting denote the complement of X, that /Ā and are elementarily equivalent (/Ā ≡ ) if there are r-maximal supersets C and D of A and B respectively such that C and D have the same many-one degree (C =mD). In fact, if A and B are maximal sets, /Ā ≡ if, and only if, A =mB. We wish to study the relationship between the elementary equivalence of /Ā and , and the Turing degrees of A and B.

Incorporation of PBRM1, BAP1, TP53 mutation status into the Memorial Sloan Kettering Cancer Center (MSKCC) risk model: A genomically annotated tool to improve stratification of patients (pts) with advanced renal cell carcinoma (RCC).

2018 ◽  
Vol 36 (6_suppl) ◽  
pp. 639-639 ◽  
Author(s):  
Martin Henner Voss ◽  
Yuan Cheng ◽  
Mahtab Marker ◽  
Fengshen Kuo ◽  
Toni K. Choueiri ◽  
...  
Keyword(s):  
Clinical Trial ◽  
Prognostic Value ◽  
Risk Model ◽  
Original Model ◽  
Phase Iii ◽  
Training Set ◽  
First Line ◽  
Mutation Status ◽  
New Model ◽  
Validation Set

639 Background: The MSKCC risk model, an established prognostic tool fo r metastatic RCC, integrates clinical + laboratory data, but is ignorant to tumor genomics. Mutations in BAP1, PBRM1, TP53, cumulatively found in over 50% of pts, have prognostic value in RCC. We sought to study the use of integrating mutation status into the MSKCC model using two large clinical trial datasets. Methods: Pts had received first line sunitinib or pazopanib on the phase III COMPARZ (training set, n = 357) or the phase II RECORD3 trial (validation set, n = 130). Genes were evaluated by next generation sequencing using archival tissue. Association of mutation status and overall survival (OS) was tested by multivariate Cox regression analysis (MVA) in the training set. An annotated model was constructed combining the original clinical variables and mutation status for the 3 genes. We compared risk group assignment and concordance index (c-index) for the original vs. new model in training and validation set. Results: Mutation status for each gene: BAP1, TP53 and PBRM1 independently correlated with OS on MVA (p≤0.0035). Comparing the original (clinical only) to the annotated (clinical + genomics) model, risk categories changed in 139 pts (39%). The C-index was improved with integration of genomic information (0.595 original model - > 0.628 new model). The independent validation cohort confirmed improvement of c-index for predicting OS with integration of genomic data (c-index 0.622 original model - > 0.641 new model). Conclusions: Mutation status for BAP1, PBRM1, and TP53 has prognostic value in pts with advanced RCC. The annotated risk model alters risk status in over 1/3 of pts and improves accuracy of estimating outcomes in patients receiving first-line therapy. Clinical trial information: NCT00720941. [Table: see text]

Disturbing arithmetic

1985 ◽  
Vol 50 (2) ◽  
pp. 375-379 ◽  
Author(s):  
Thomas J. Grilliot
Keyword(s):  
Fine Structure ◽  
Prime Number ◽  
Long Range ◽  
Set Theory ◽  
New Model ◽  
New Models ◽  
Models Of Arithmetic ◽  
Two Alternatives

One long-range objective of logic is to find models of arithmetic with noteworthy properties, perhaps properties that imply some long-standing number theoretic conjectures. In areas of mathematics such as algebra or set theory, new models are often made by extending old models, that is, by adjoining new elements to already existing models. Usually the extension retains most of the characteristics of the old model with at least one exception that makes the new model interesting. However, such a scheme is difficult in the area of arithmetic. Many interesting properties of the fine structure of arithmetic are diophantine and hence unchangeable in extensions. For instance, one cannot change a prime number into a composite one by adjoining new elements.One could possibly get around this diophantine difficulty in one of two ways. One way is to change the usual language of addition and multiplication to an equivalent language that does not transmit so much information to extensions. For instance, multiplication is definable from the squaring function, as one sees from the identity 2xy = (x + y)2 − x2 − y2, and the squaring function in turn is definable either from the unary square predicate (as one sees from the fact that n = m2 if n and n + 2m + 1 are successive squares) or from the divisor relation (as one sees from the fact that n = m2 if n is the smallest number such that m divides n and m + 1 divides n + m). Either of these two alternatives to multiplication might make for interesting extensions.

scholarly journals Improving the Prediction of Maturity From Anthropometric Variables Using a Maturity Ratio

2018 ◽  
Vol 30 (2) ◽  
pp. 296-307 ◽  
Author(s):  
Job Fransen ◽  
Stephen Bush ◽  
Stephen Woodcock ◽  
Andrew Novak ◽  
Dieter Deprez ◽  
...  
Keyword(s):  
Akaike Information Criterion ◽  
Nonlinear Models ◽  
Information Criterion ◽  
Peak Height ◽  
Prediction Equation ◽  
Original Model ◽  
Height Velocity ◽  
Peak Height Velocity ◽  
New Model ◽  
Youth Athletes

Purpose:This study aimed to improve the prediction accuracy of age at peak height velocity (APHV) from anthropometric assessment using nonlinear models and a maturity ratio rather than a maturity offset.Methods:The dataset used to develop the original prediction equations was used to test a new prediction model, utilizing the maturity ratio and a polynomial prediction equation. This model was then applied to a sample of male youth academy soccer players (n = 1330) to validate the new model in youth athletes.Results:A new equation was developed to estimate APHV more accurately than the original model (new model: Akaike information criterion: −6062.1,R2 = 90.82%; original model: Akaike information criterion = 3048.7,R2 = 88.88%) within a general population of boys, particularly with relatively high/low APHVs. This study has also highlighted the successful application of the new model to estimate APHV using anthropometric variables in youth athletes, thereby supporting the use of this model in sports talent identification and development.Conclusion:This study argues that this newly developed equation should become standard practice for the estimation of maturity from anthropometric variables in boys from both a general and an athletic population.

The Mathematical Work of S.C.Kleene

1995 ◽  
Vol 1 (1) ◽  
pp. 9-43 ◽  
Author(s):  
J.R. Shoenfield
Keyword(s):  
Recursion Theory ◽  
Mathematical Work ◽  
Church’S Thesis ◽  
Recursive Functions ◽  
Close Relationship ◽  
Church's Thesis ◽  
Mathematical Career ◽  
Definable Functions ◽  
Class Of Functions ◽  
Computable Functions

§1. The origins of recursion theory. In dedicating a book to Steve Kleene, I referred to him as the person who made recursion theory into a theory. Recursion theory was begun by Kleene's teacher at Princeton, Alonzo Church, who first defined the class of recursive functions; first maintained that this class was the class of computable functions (a claim which has come to be known as Church's Thesis); and first used this fact to solve negatively some classical problems on the existence of algorithms. However, it was Kleene who, in his thesis and in his subsequent attempts to convince himself of Church's Thesis, developed a general theory of the behavior of the recursive functions. He continued to develop this theory and extend it to new situations throughout his mathematical career. Indeed, all of the research which he did had a close relationship to recursive functions.Church's Thesis arose in an accidental way. In his investigations of a system of logic which he had invented, Church became interested in a class of functions which he called the λ-definable functions. Initially, Church knew that the successor function and the addition function were λ-definable, but not much else. During 1932, Kleene gradually showed1 that this class of functions was quite extensive; and these results became an important part of his thesis 1935a (completed in June of 1933).

SPECIFIC HEATS OF POTASSIUM IODIDE BY A MODIFIED SHELL MODEL

1967 ◽  
Vol 45 (5) ◽  
pp. 1885-1899 ◽  
Author(s):  
R. G. Deo ◽  
B. Dayal
Keyword(s):  
Dielectric Properties ◽  
Shell Model ◽  
Potassium Iodide ◽  
Original Model ◽  
Ionic Crystals ◽  
Many Body ◽  
New Model ◽  
Specific Heats ◽  
Dependent Potential ◽  
The Many

The shell model of Woods et al. has been modified to include the effect of the many-body interaction in the ionic crystals in a simple phenomenological way. This is based on the work of Lowdin, Lundqvist, and Verma and Dayal. In the new model which the authors refer to as the modified shell model, it is assumed that the volume-dependent potential is due to the interaction between the cores of the ions only. This introduces additional parameters in the model which have been derived from the elastic and dielectric properties of the crystal. The variation of the specific heat with temperature of potassium iodide has been studied by the original model of Woods et al. as well as by the new model. It is seen that the theoretical specific heats and dispersion curves given by both the models are in fair agreement with the experimental results. However, the shell model fails to account for the Cauchy discrepancy in this crystal, whereas the new model removes this difficulty.

Flipping properties in arithmetic

1982 ◽  
Vol 47 (2) ◽  
pp. 416-422 ◽  
Author(s):  
L. A. S. Kirby
Keyword(s):  
Standard Model ◽  
Set Theory ◽  
Initial Segment ◽  
Peano Arithmetic ◽  
Combinatorial Properties ◽  
Nonstandard Model ◽  
Standard Part ◽  
The Standard Model ◽  
Nonstandard Models ◽  
Image Position

Flipping properties were introduced in set theory by Abramson, Harrington, Kleinberg and Zwicker [1]. Here we consider them in the context of arithmetic and link them with combinatorial properties of initial segments of nonstandard models studied in [3]. As a corollary we obtain independence resutls involving flipping properties.We follow the notation of the author and Paris in [3] and [2], and assume some knowledge of [3]. M will denote a countable nonstandard model of P (Peano arithmetic) and I will be a proper initial segment of M. We denote by N the standard model or the standard part of M. X ↑ I will mean that X is unbounded in I. If X ⊆ M is coded in M and M ≺ K, let X(K) be the subset of K coded in K by the element which codes X in M. So X(K) ⋂ M = X.Recall that M ≺IK (K is an I-extension of M) if M ≺ K and for some c∈K,In [3] regular and strong initial segments are defined, and among other things it is shown that I is regular if and only if there exists an I-extension of M.

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