Order Model Theory Research Articles

A fractional-order model of human liver: Analytic-approximate and numerical solutions comparing with clinical data

In this work, we aim to propose a mathematical formalism for the human liver involving Caputo fractional derivative (CFD) with modified parameters. Regarding the proposed fractional order model (FOM), the positivity and boundedness theory of an explicit solution are established by using the fractional mean-value theorem in the sense of Caputo. In addition, the validation of the FOM is provided to ensure that it complies with the medical interpretation and we also discuss the stability of unique equilibrium point (EP) for the presented model. We apply the Generalized Mittag–Leffler function method (GMLFM) and Predictor–Corrector method (PCM) to solve this model. The GMLFM is used to give the proper results as a series of the time variable to obtain the desired solution at any time and the PCM is implemented to obtain the solution of the FOM over a long time period. The tabled results from this study are compared with the real clinical data and with other methods. The simulations approximate the real clinical data and support the obtained theoretical results. Precisely, the outcomes of the FOM by using the proposed methods coincide with most of the clinical data that means it more accurate. Ultimately, the graphical results indicate the superiority of the proposed FOM over the classical model.

Research on AC Servo Motor Load Disturbance Method

For the decline of the control accuracy of servo system due to the disturbances of load, to design the dimensionality reduction load torque observer, the observed value of load torque after conversion for compensation control. Established the three- loop mathematical model of servo control system, to elicit three- loop control parameter setting formula by the application of optimal second order and third order model theory. Established the simulation model of space vector control AC servo system, by simulation experiment it is verified that three-ring parameters can provide good dynamic performance for system after setting. After the establishment of experimental platform, through the motor load sudden change experiment, it proved the feasibility of the torque compensation control for improving the servo system control precision under load disturbances.

Model theoretic stability and categoricity for complete metric spaces

We deal with the systematic development of stability in the context of approximate elementary submodels of a monster metric space, which is not far, but still very different from first order model theory. In particular, we prove the analogue of Morley’s theorem for classes of complete metric spaces.

Free Interpretation, Quotient Interpretation and Substitution of a Letter with a Term for First Order Languages

Free Interpretation, Quotient Interpretation and Substitution of a Letter with a Term for First Order LanguagesFourth of a series of articles laying down the bases for classical first order model theory. This paper supplies a toolkit of constructions to work with languages and interpretations, and results relating them. The free interpretation of a language, having as a universe the set of terms of the language itself, is defined.The quotient of an interpreteation with respect to an equivalence relation is built, and shown to remain an interpretation when the relation respects it. Both the concepts of quotient and of respecting relation are defined in broadest terms, with respect to objects as general as possible.Along with the trivial symbol substitution generally defined in [11], the more complex substitution of a letter with a term is defined, basing right on the free interpretation just introduced, which is a novel approach, to the author's knowledge. A first important result shown is that the quotient operation commute in some sense with term evaluation and reassignment functors, both introduced in [13] (theorem 3, theorem 15). A second result proved is substitution lemma (theorem 10, corresponding to III.8.3 of [15]). This will be vital for proving satisfiability

First Order Languages: Further Syntax and Semantics

First Order Languages: Further Syntax and SemanticsThird of a series of articles laying down the bases for classical first order model theory. Interpretation of a language in a universe set. Evaluation of a term in a universe. Truth evaluation of an atomic formula. Reassigning the value of a symbol in a given interpretation. Syntax and semantics of a non atomic formula are then defined concurrently (this point is explained in [16], 4.2.1). As a consequence, the evaluation of any w.f.f. string and the relation of logical implication are introduced. Depth of a formula. Definition of satisfaction and entailment (aka entailment or logical implication) relations, see [18] III.3.2 and III.4.1 respectively.

Definition of First Order Language with Arbitrary Alphabet. Syntax of Terms, Atomic Formulas and their Subterms

Definition of First Order Language with Arbitrary Alphabet. Syntax of Terms, Atomic Formulas and their SubtermsSecond of a series of articles laying down the bases for classical first order model theory. A language is defined basically as a tuple made of an integer-valued function (adicity), a symbol of equality and a symbol for the NOR logical connective. The only requests for this tuple to be a language is that the value of the adicity in = is -2 and that its preimage (i.e. the variables set) in 0 is infinite. Existential quantification will be rendered (see [11]) by mere prefixing a formula with a letter. Then the hierarchy among symbols according to their adicity is introduced, taking advantage of attributes and clusters.The strings of symbols of a language are depth-recursively classified as terms using the standard approach (see for example [16], definition 1.1.2); technically, this is done here by deploying the ‘-multiCat' functor and the ‘unambiguous’ attribute previously introduced in [10], and the set of atomic formulas is introduced. The set of all terms is shown to be unambiguous with respect to concatenation; we say that it is a prefix set. This fact is exploited to uniquely define the subterms

Preliminaries to Classical First Order Model Theory

Mathematics Department G. Castelnuovo, Sapienza University of Rome, Piazzale Aldo Moro 5, 00185 Roma, Italy

Some remarks on the bearing of model theory on the theory of theories

The present paper offers some remarks on the significance of first order model theory for our understanding of theories, and more generally, for our understanding of the “structuralist” accounts of the nature of theoretical knowledge that we associate with Russell, Ramsey and Carnap. What is unique about the presentation is the prominence it assigns to Craig’s Interpolation Lemma, some of its corollaries, and the manner of their demonstration. They form the underlying logical basis of the analysis.

On d-finiteness in continuous structures

We observe that certain classical results of first order model theory fail in the context of continuous first order logic. We argue that this happens since finite tuples in a continuous structure may behave as infinite tuples in classical model theory. Th

On the maximality of logics with approximations

In this paper we analyze some aspects of the question of using methods from model theory to study structures of functional analysis.By a well known result of P. Lindström, one cannot extend the expressive power of first order logic and yet preserve its most outstanding model theoretic characteristics (e.g., compactness and the Löwenheim-Skolem theorem). However, one may consider extending the scope of first order in a different sense, specifically, by expanding the class of structures that are regarded as models (e.g., including Banach algebras or other structures of functional analysis), and ask whether the resulting extensions of first order model theory preserve some of its desirable characteristics.A formal framework for the study of structures based on Banach spaces from the perspective of model theory was first introduced by C. W. Henson in [8] and [6]. Notions of syntax and semantics for these structures were defined, and it was shown that using them one obtains a model theoretic apparatus that satisfies many of the fundamental properties of first order model theory. For instance, one has compactness, Löwenheim-Skolem, and omitting types theorems. Further aspects of the theory, namely, the fundamentals of stability and forking, were first introduced in [10] and [9].The classes of mathematical structures formally encompassed by this framework are normed linear spaces, possibly expanded with additional structure, e.g., operations, real-valued relations, and constants. This notion subsumes wide classes of structures from functional analysis. However, the restriction that the universe of a structure be a normed space is not necessary. (This restriction has a historical, rather than technical origin; specifically, the development of the theory was originally motivated by questions in Banach space geometry.) Analogous techniques can be applied if the universe is a metric space. Now, when the underlying metric topology is discrete, the resulting model theory coincides with first order model theory, so this logic extends first order in the sense described above. Furthermore, without any cost in the mathematical complexity, one can also work in multi-sorted contexts, so, for instance, one sort could be an operator algebra while another is. say, a metric space.

Coordinatisation and canonical bases in simple theories

In this paper we discuss several generalization of theorems from stability theory to simple theories. Cherlin and Hrushovski, in [2] develop a substitute for canonical bases in finite rank, ω-categorical supersimple theories. Motivated by methods there, we prove the existence of canonical bases (in a suitable sense) for types in any simple theory. This is done in Section 2. In general these canonical bases will (as far as we know) exist only as “hyperimaginaries”, namely objects of the form a/E where a is a possibly infinite tuple and E a type-definable equivalence relation. (In the supersimple, ω-categorical case, these reduce to ordinary imaginaries.) So in Section 1 we develop the general theory of hyperimaginaries and show how first order model theory (including the theory of forking) generalises to hyperimaginaries. We go on, in Section 3 to show the existence and ubiquity of regular types in supersimple theories, ω-categorical simple structures and modularity is discussed in Section 4. It is also shown here how the general machinery of simplicity simplifies some of the general theory of smoothly approximable (or Lie-coordinatizable) structures from [2].Throughout this paper we will work in a large, saturated model M of a complete theory T. All types, sets and sequences will have size smaller than the size of M. We will assume that the reader is familiar with the basics of forking in simple theories as laid out in [4] and [6]. For basic stability-theoretic results concerning regular types, orthogonality etc., see [1] or [9].

Abstract classes with few models have ‘homogeneous-universal’ models

This paper is concerned with a class K of models and an abstract notion of submodel ≤. Experience in first order model theory has shown the desirability of finding a ‘monster model’ to serve as a universal domain for K. In the original constructions of Jónsson and Fraïssé, K was a universal class and ordinary substructure played the role of ≤. Working with a cardinal λ satisfying λ<λ = λ guarantees appropriate downward Löwenheim-Skolem theorems; the existence and uniqueness of a homogeneous-universal model appears to depend centrally on the amalgamation property. We make this apparent dependence more precise in this paper.The major innovation of this paper is the introduction of a weaker notion (chain homogeneous-universal) to replace the natural notion of (K, <)-homogeneous-universal model. Modulo a weak extension of ZFC (provable if V = L), we show (Corollary 5.24) that a class K obeying certain minimal restrictions satisfies a fundamental dichotomy. For arbitrarily large λ, either K has the maximal number of models in power λ or K has a unique chain homogeneous-universal model of power λ. We show (5.25) in a class with amalgamation this dichotomy holds for the notion of K-homogeneous-universal model in the more normal sense.The methods here allow us to improve our earlier results [5] in two other ways: certain requirements on all chains of a given length are replaced by requiring winning strategies in certain games; the notion of a canonically prime model is avoided. A full understanding of these extensions requires consideration of the earlier papers but we summarize them quickly here.

Co-elementary equivalence for compact Hausdorff spaces and compact abelian groups

The notions of elementary equivalence and elementary mapping in first order model theory have category-theoretic reflections in many well-known topological settings. We study the dualized notions in the categories of compact Hausdorff spaces and compact abelian groups.

On nonstandard models in higher order logic

There are two concepts of standard/nonstandard models in simple type theory.The first concept—we might call it the pragmatical one—interprets type theory as a first order logic with countably many sorts of variables: the variables for the urelements of type 0,…, the n-ary relational variables of type (τ1, …, τn) with arguments of type (τ1,…,τn), respectively. If A ≠ ∅ then 〈Aτ〉 is called a model of type logic, if A0 = A and . 〈Aτ〉 is called full if, for every τ = (τ1,…,τn), . The variables for the urelements range over the elements of A and the variables of type (τ1,…, τn) range over those subsets of which are elements of . The theory Th(〈Aτ〉) is the set of all closed formulas in the language which hold in 〈Aτ〉 under natural interpretation of the constants. If 〈Bτ〉 is a model of Th(〈Aτ〉), then there exists a sequence 〈fτ〉 of functions fτ: Aτ → Bτ such that 〈fτ〉 is an elementary embedding from 〈Aτ〉 into 〈Bτ〉. 〈Bτ〉 is called a nonstandard model of 〈Aτ〉, if f0 is not surjective. Otherwise 〈Bτ〉 is called a standard model of 〈Aτ〉.This first concept of model theory in type logic seems to be preferable for applications in model theory, for example in nonstandard analysis, since all nice properties of first order model theory (completeness, compactness, and so on) are preserved.

Some model theory of modules. III. On infiniteness of sets definable in modules

This is the third and last part of an investigation in several topics of first order model theory of modules which I began in Some model theory of modules. I. On total transcendence of modules (this Journal, vol. 48 (1983), pp. 570–574) and continued in Some model theory of modules. II. On stability and categoricity of flat modules (this Journal, vol. 48 (1983), pp. 970–985). Throughout I refer to these papers as “Part I” and “Part II”.Although these parts are only loosely connected, I will tacitly use the notation and preliminaries already introduced in the preceding ones. Further details are given in §0. Concerning Part II, the reader is assumed to be familiar with most of §1, a very small part of §4, and the criterion for elementary equivalence of modules over regular rings given in §3 (Lemma 20) of that part.Using those tools in this paper I consider completeness of the (elementary) theory of all modules (and of some of its extensions) (§2), and eliminability of cardinality quantifiers in the elementary theories of modules (§3).§2 is almost entirely devoted to a new short proof based on the technique developed in Part II of a theorem of Tukavkin which seems to be the first relevant result concerning the completeness of the theory of all modules (after the simple observation that any theory of infinite vector spaces is complete).

Partial algebras—survey of a unifying approach towards a two-valued model theory for partial algebras

In this survey, with no proofs included, we collect some material scattered through recent papers and a planned monograph, which shows that partial algebras do have a two-valued first order model theory which is simpler and nicer than one might have expected it to be. In section 1 we comment and present some basic definitions. In section 2 a correct and complete two-valued first order logic is developed. In section 3 the three main concepts of “varieties” are presented, while sections 4 and 5 contain some additional axiomatizability results and some applications, respectively. Section 6 contains some additional remarks.

A two-cardinal theorem for homogeneous sets and the elimination of Malitz quantifiers

Introduction. The aim of the present paper is to give a first approximation to the problem of finding natural conditions for a theory in the first order language to admit the elimination of certain Malitz quantifiers in the same sense as the substructure completeness theorem does for elimination of elementary quantifiers. In other words, we look for the classes of first order theories which remain essentially the same when we add certain Malitz quantifiers. Thus, the present paper is a contribution to first order model theory, and not to logics with additional quantifiers. Our investigations were inspired by the second author's mostly unpublished work (cf. [TU1] and [TU2]) concerning the eliminability of the quantifiers there are K.a many (= Malitz quantifiers for 1-tuples) which we intended to extend to the general case. Although we did not completely succeed in our design, we obtained some partial results which we present in the following order. ? 1 contains definitions, conventions, central properties related to the quantifiers, and an eliminability condition for Ramsey quantifiers (= Malitz quantifiers in the N0-interpretation). ?2 is devoted to a two cardinal theorem for maximally homogeneous sets (without any reference to quantifiers). ?3 is an application of ?2 to the eliminability problem. ?4 provides a negative answer to a question of Baldwin and Kueker [BK, Question 4]. Open questions are scattered about the paper. Our results were obtained independently of [BK] in September-October 1979. For further historical remarks, see below.

An introduction to recursively saturated and resplendent models

The notions of recursively saturated and resplendent models grew out of the study of admissible sets with urelements and admissible fragments of Lω1ω, but, when applied to ordinary first order model theory, give us new tools for research and exposition. We will discuss their history in §3.The notion of saturated model has proven to be important in model theory. Its most important property for applications is that if , are saturated and of the same cardinality then = iff ≅ . See, e.g., Chang-Keisler [3]. The main drawback is that saturated models exist only under unusual assumptions of set theory. For example, if 2κ = κ+ then every theory T of L has a saturated model of power κ+. (Similarly, if κ is strongly inaccessible, then every T has a saturated model of power κ.) On the other hand, a theory T like Peano arithmetic, with types, cannot have a saturated model in any power κ with ω ≤ κ ≤ .One method for circumventing these problems of existence (or rather non-existence) is the use of “special” models (cf. [3]). If κ = Σλ<κ2λ, κ < ω, then every theory T of L has a special model of power κ. Such cardinals are large and, themselves, rather special. There are definite aesthetic objections to the use of these large, singular models to prove results about first order logic.

An introduction to recursively saturated and resplendent models

The notions of recursively saturated and resplendent models grew out of the study of admissible sets with urelements and admissible fragments of Lω1ω, but, when applied to ordinary first order model theory, give us new tools for research and exposition. We will discuss their history in §3.The notion of saturated model has proven to be important in model theory. Its most important property for applications is that if , are saturated and of the same cardinality then = iff ≅ . See, e.g., Chang-Keisler [3]. The main drawback is that saturated models exist only under unusual assumptions of set theory. For example, if 2κ = κ+ then every theory T of L has a saturated model of power κ+. (Similarly, if κ is strongly inaccessible, then every T has a saturated model of power κ.) On the other hand, a theory T like Peano arithmetic, with types, cannot have a saturated model in any power κ with ω ≤ κ ≤ .One method for circumventing these problems of existence (or rather non-existence) is the use of “special” models (cf. [3]). If κ = Σλ<κ2λ, κ < ω, then every theory T of L has a special model of power κ. Such cardinals are large and, themselves, rather special. There are definite aesthetic objections to the use of these large, singular models to prove results about first order logic.

Infinitary analogs of theorems from first order model theory

The material presented here belongs to the model theory of the Lκ, λ languages. Our results are either infinitary analogs of important theorems in finitary model theory, or else show that such analogs do not exist.For example, it is well known that whenever i, and , have the same true Lω, ω sentences (i.e., are elementarily equivalent) for i = 1, 2, then the cardinal sums 1 + 2 and + have the same true Lω, ω sentences, and the direct products 1 · 2 and · have the same true Lω, ω sentences [3]. We show that this is true when ‘Lω, ω’ is replaced by ‘Lκ, λ’ if and only if κ is strongly inaccessible. For Lω1, ω this settles a question, posed by Lopez-Escobar [7].In §3 we give a complete description of the expressive power of those sentences of Lκ, λ in which the identity symbol is the only relation symbol which occurs. This extends a result by Hanf [4].