Branching-time logic with quantification over branches: The point of view of modal logic

1996 ◽  
Vol 61 (1) ◽  
pp. 1-39 ◽  
Author(s):  
Alberto Zanardo
Keyword(s):  
Modal Logic ◽  
Second Order ◽  
Point Of View ◽  
Branching Time ◽  
First Order ◽  
Kripke Structures ◽  
Time Logic ◽  
The Moment ◽  
Modal Definability ◽  
Or History

AbstractIn Ockhamist branching-time logic [Prior 67], formulas are meant to be evaluated on a specified branch, or history, passing through the moment at hand. The linguistic counterpart of the manifoldness of future is a possibility operator which is read as ‘at some branch, or history (passing through the moment at hand)’. Both the bundled-trees semantics [Burgess 79] and the 〈moment, history〉 semantics [Thomason 84] for the possibility operator involve a quantification over sets of moments. The Ockhamist frames are (3-modal) Kripke structures in which this second-order quantification is represented by a first-order quantification. The aim of the present paper is to investigate the notions of modal definability, validity, and axiomatizability concerning 3-modal frames which can be viewed as generalizations of Ockhamist frames.

Higher-Order Multilevel Solvers for the EHL Line and Point Contact Problem

1994 ◽  
Vol 116 (4) ◽  
pp. 741-750 ◽  
Author(s):  
C. H. Venner
Keyword(s):  
Contact Problem ◽  
Computing Time ◽  
Point Contact ◽  
Higher Order ◽  
Second Order ◽  
Point Of View ◽  
First Order ◽  
Fundamental Point ◽  
Numerical Solver ◽  
Circular Contact

This paper addresses the development of efficient numerical solvers for EHL problems from a rather fundamental point of view. A work-accuracy exchange criterion is derived, that can be interpreted as setting a limit to the price paid in terms of computing time for a solution of a given accuracy. The criterion can serve as a guideline when reviewing or selecting a numerical solver and a discretization. Earlier developed multilevel solvers for the EHL line and circular contact problem are tested against this criterion. This test shows that, to satisfy the criterion a second-order accurate solver is needed for the point contact problem whereas the solver developed earlier used a first-order discretization. This situation arises more often in numerical analysis, i.e., a higher order discretization is desired when a lower order solver already exists. It is explained how in such a case the multigrid methodology provides an easy and straightforward way to obtain the desired higher order of approximation. This higher order is obtained at almost negligible extra work and without loss of stability. The approach was tested out by raising an existing first order multilevel solver for the EHL line contact problem to second order. Subsequently, it was used to obtain a second-order solver for the EHL circular contact problem. Results for both the line and circular contact problem are presented.

scholarly journals Timoshenko Beam Theory: Exact Solution for First-Order Analysis, Second-Order Analysis, and Stability

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Exact Solution ◽  
Closed Form ◽  
Timoshenko Beam ◽  
Beam Theory ◽  
Second Order ◽  
Bernoulli Beam ◽  
Order Analysis ◽  
First Order ◽  
The Moment ◽  
Relationship Of

This paper presents an exact solution to the Timoshenko beam theory (TBT) for first-order analysis, second-order analysis, and stability. The TBT covers cases associated with small deflections based on shear deformation considerations, whereas the Euler–Bernoulli beam theory (EBBT) neglects shear deformations. Thus, the Euler–Bernoulli beam is a special case of the Timoshenko beam. The moment-curvature relationship is one of the governing equations of the EBBT, and closed-form expressions of efforts and deformations are available in the literature. However, neither an equivalent to the moment-curvature relationship of EBBT nor closed-form expressions of efforts and deformations can be found in the TBT. In this paper, a moment-shear force-curvature relationship, the equivalent in TBT of the moment-curvature relationship of EBBT, was presented. Based on this relationship, first-order and second-order analyses were conducted, and closed-form expressions of efforts and deformations were derived for various load cases. Furthermore, beam stability was analyzed and buckling loads were calculated. Finally, first-order and second-order element stiffness matrices were determined.

scholarly journals On a theory of the second order longitudinal spherical aberra­tion for a symmetrical optical system

1920 ◽  
Vol 97 (683) ◽  
pp. 196-198
Keyword(s):  
Optical System ◽  
Spherical Aberration ◽  
Empirical Method ◽  
Second Order ◽  
Point Of View ◽  
Axial Plane ◽  
First Order ◽  
Usual Theory ◽  
Semi Empirical

1. The object of this investigation is to establish a formula for the longitudinal spherical aberration of rays which traverse a symmetrical optical system in an axial plane that shall be capable of fairly easy computation for any combination of lenses, and at the same time shall be accurate to the second order and free from certain important difficulties of convergency which occur in certain neighbourhoods when we attempt to use for the longitudinal aberration the method of aberration of successive orders. From the point of view of the optical designer, the usual theory of aberrations, which, for all practical purposes, is largely restricted to the first order, is known to give an unsatisfactory approximation. In practice, the designer adopts a semi-empirical method of tracing a number of rays through the system by means of the trigonometrical equations, a method which is laborious and lengthy, and which can at best give only incomplete informa­tion and very limited guidance for effecting improvements.

scholarly journals COMPLETENESS OF SECOND-ORDER PROPOSITIONAL S4 AND H IN TOPOLOGICAL SEMANTICS

2018 ◽  
Vol 11 (3) ◽  
pp. 507-518
Author(s):  
PHILIP KREMER
Keyword(s):  
Modal Logic ◽  
Intuitionistic Logic ◽  
Second Order ◽  
Topological Semantics ◽  
First Order ◽  
Propositional Modal Logic ◽  
Propositional Quantifiers ◽  
Modal Logic S4 ◽  
Natural Way

AbstractWe add propositional quantifiers to the propositional modal logic S4 and to the propositional intuitionistic logic H, introducing axiom schemes that are the natural analogs to axiom schemes typically used for first-order quantifiers in classical and intuitionistic logic. We show that the resulting logics are sound and complete for a topological semantics extending, in a natural way, the topological semantics for S4 and for H.

Stability of a Beddington–DeAngelis type predator–prey model with trichotomous noises

2016 ◽  
Vol 30 (17) ◽  
pp. 1650102 ◽  
Author(s):  
Yanfei Jin ◽  
Siyong Niu
Keyword(s):  
Correlation Time ◽  
Gaussian White Noise ◽  
Second Order ◽  
Predator Prey ◽  
First Order ◽  
Predator Prey Model ◽  
Order Solution ◽  
Prey Model ◽  
The Stability ◽  
The Moment

The stability analysis of a Beddington–DeAngelis (B–D) type predator–prey model driven by symmetric trichotomous noises is presented in this paper. Using the Shapiro–Loginov formula, the first-order and second-order solution moments of the system are obtained. The moment stability conditions of the B–D predator–prey model are given by using Routh–Hurwitz criterion. It is found that the stabilities of the first-order and second-order solution moments depend on the noise intensities and correlation time of noise. The first-order and second-order moments are stable when the correlation time of noise is increased. That is, the trichotomous noise plays a constructive role in stabilizing the solution moment with regard to Gaussian white noise. Finally, some numerical results are performed to support the theoretical analyses.

First-order definability in modal logic

1975 ◽  
Vol 40 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. I. Goldblatt
Keyword(s):  
Modal Logic ◽  
Binary Relation ◽  
Second Order ◽  
Order Conditions ◽  
Kripke Models ◽  
Order Condition ◽  
First Order ◽  
Axiom Systems ◽  
Set Of Points ◽  
First Order Definability

In the early days of the development of Kripke-style semantics for modal logic a great deal of effort was devoted to showing that particular axiom systems were characterised by a class of models describable by a first-order condition on a binary relation. For a time the approach seemed all encompassing, but recent work by Thomason [6] and Fine [2] has shown it to be somewhat limited—there are logics not determined by any class of Kripke models at all. In fact it now seems that modal logic is basically second-order in nature, in that any system may be analysed in terms of structures having a nominated class of second-order individuals (subsets) that serve as interpretations of propositional variables (cf. [7]). The question has thus arisen as to how much of modal logic can be handled in a first-order way, and precisely which modal sentences are determined by first-order conditions on their models. In this paper we present a model-theoretic characterisation of this class of sentences, and show that it does not include the much discussed LMp → MLp.Definition 1. A modal frame ℱ = 〈W, R〉 consists of a set W on which a binary relation R is defined. A valuation V on ℱ is a function that associates with each propositional variable p a subset V(p) of W (the set of points at which p is “true”).

Application of the Method of Second Order Sign Behavior To the Study of Non-Spatial Delayed Response in Rhesus Monkeys

Behaviour ◽  
1969 ◽  
Vol 35 (1-2) ◽  
pp. 128-136 ◽  
Author(s):  
Bruno Cardu
Keyword(s):  
Rhesus Monkeys ◽  
Short Term Memory ◽  
Memory Span ◽  
Second Order ◽  
Delayed Response ◽  
Short Term ◽  
Term Memory ◽  
First Order ◽  
The Moment ◽  
Sign Behavior

AbstractThe behavior of seven rhesus monkeys on a test of non-spatial delayed response based on the method of second order sign behavior is reported. Four stimuli were used: two first order stimuli presented individually (two sounds or two lights) and two second order stimuli presented simultaneously (two objects). Subjects first learned to associate one of the objects to each of the two first order stimuli. An interval between the termination of the first signal and the moment of choice was then introduced; hence the subjects' short-term memory could be estimated. All subjects succeeded in this task; the limits of the memory span ranged from 20 to 45 seconds.

scholarly journals All the existences that there are

Disputatio ◽  
2012 ◽  
Vol 4 (32) ◽  
pp. 361-383 ◽  
Author(s):  
Alberto Voltolini
Keyword(s):  
Second Order ◽  
Point Of View ◽  
Order Property ◽  
First Order

Abstract In this paper, I will defend the claim that there are three existence properties: the second-order property of being instantiated, a substantive first-order property (or better a group of such properties) and a formal, hence universal, first-order property. I will first try to show what these properties are and why we need all of them for ontological purposes. Moreover, I will try to show why a Meinong-like option that positively endorses both the former and the latter first-order property is the correct view in ontology. Finally, I will add some methodological remarks as to why this debate has to be articulated from the point of view of reality, i.e., by speaking of properties, rather than from the point of view of language, i.e., by speaking of predicates (for such properties).

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