Decidability and undecidability of extensions of second (first) order theory of (generalized) successor

1966 ◽  
Vol 31 (2) ◽  
pp. 169-181 ◽  
Author(s):  
Calvin C. Elgot ◽  
Michael O. Rabin
Keyword(s):  
Order Theory ◽  
Higher Order ◽  
Second Order ◽  
First Order ◽  
First Order Theory ◽  
Function Constant

We study certain first and second order theories which are semantically defined as the sets of all sentences true in certain given structures. Let be a structure where A is a non-empty set, λ is an ordinal, and Pα is an n(α)-ary relation or function4 on A. With we associate a language L appropriate for which may be a first or higher order calculus. L has an n(α)-place predicate or function constant P for each α < λ. We shall study three types of languages: (1) first-order calculi with equality; (2) second-order monadic calculi which contain monadic predicate (set) variables ranging over subsets of A; (3) restricted (weak) second-order calculi which contain monadic predicate variables ranging over finite subsets of A.

scholarly journals Evaporation and combustion of a slowly moving liquid fuel droplet: higher-order theory

1996 ◽  
Vol 307 ◽  
pp. 135-165 ◽  
Author(s):  
M. A. Jog ◽  
P. S. Ayyaswamy ◽  
I. M. Cohen
Keyword(s):  
Flow Field ◽  
Order Theory ◽  
Higher Order ◽  
Second Order ◽  
Spherical Particles ◽  
General Nature ◽  
Fuel Droplet ◽  
First Order ◽  
First Order Theory ◽  
Higher Order Theory

The evaporation and combustion of a single-component fuel droplet which is moving slowly in a hot oxidant atmosphere have been analysed using perturbation methods. Results for the flow field, temperature and species distributions in each phase, inter-facial heat and mass transfer, and the enhancement of the mass burning rate due to the presence of convection have all been developed correct to second order in the translational Reynolds number. This represents an advance over a previous study which analysed the problem to first order in the perturbation parameter. The primary motivation for the development of detailed analytical/numerical solutions correct to second order arises from the need for such a higher-order theory in order to investigate fuel droplet ignition and extinction characteristics in the presence of convective flow. Explanations for such a need, based on order of magnitude arguments, are included in this article. With a moving droplet, the shear at the interface causes circulatory motion inside the droplet. Owing to the large evaporation velocities at the droplet surface that usually accompany drop vaporization and burning, the entire flow field is not in the Stokes regime even for low translational Reynolds numbers. In view of this, the formulation for the continuous phase is developed by imposing slow translatory motion of the droplet as a perturbation to uniform radial flow associated with vigorous evaporation at the surface. Combustion is modelled by the inclusion of a fast chemical reaction in a thin reaction zone represented by the Burke–Schumann flame front. The complete solution for the problem correct to second order is obtained by simultaneously solving a coupled formulation for the dispersed and continuous phases. A noteworthy feature of the higher-order formulation is that both the flow field and transport equations require analysis by coupled singular perturbation procedures. The higher-order theory shows that, for identical conditions, compared with the first-order theory both the flame and the front stagnation point are closer to the surface of the drop, the evaporation is more vigorous, the droplet lifetime is shorter, and the internal vortical motion is asymmetric about the drop equatorial plane. These features are significant for ignition/extinction analyses since the prediction of the location of the point of ignition/extinction will depend upon such details. This article is the first of a two-part study; in the second part, analytical expressions and results obtained here will be incorporated into a detailed investigation of fuel droplet ignition and extinction. In view of the general nature of the formulation considered here, results presented have wider applicability in the general areas of interfacial fluid mechanics and heat/material transport. They are particularly useful in microgravity studies, in atmospheric sciences, in aerosol sciences, and in the prediction of material depletion from spherical particles.

Analysis of three-layer composite plates with a new higher-order layerwise formulation

2014 ◽  
Vol 21 (3) ◽  
pp. 401-404
Author(s):  
Dalal A. Maturi ◽  
Antonio J.M. Ferreira ◽  
Ashraf M. Zenkour ◽  
Daoud S. Mashat
Keyword(s):  
Composite Plates ◽  
Order Theory ◽  
Higher Order ◽  
Basis Functions ◽  
Numerical Accuracy ◽  
First Order ◽  
The Core ◽  
First Order Theory ◽  
Layer Composite ◽  
Vibration Problems

AbstractIn this paper, we combine a new higher-order layerwise formulation and collocation with radial basis functions for predicting the static deformations and free vibration behavior of three-layer composite plates. The skins are modeled via a first-order theory, while the core is modeled by a cubic expansion with the thickness coordinate. Through numerical experiments, the numerical accuracy of this strong-form technique for static and vibration problems is discussed.

A Simple Higher-Order Theory for Laminated Composite Plates

1984 ◽  
Vol 51 (4) ◽  
pp. 745-752 ◽  
Author(s):  
J. N. Reddy
Keyword(s):  
Shear Deformation ◽  
Deformation Theory ◽  
Shear Deformation Theory ◽  
Laminated Composite ◽  
Composite Plates ◽  
Order Theory ◽  
Higher Order ◽  
Laminated Composite Plates ◽  
First Order ◽  
First Order Theory

A higher-order shear deformation theory of laminated composite plates is developed. The theory contains the same dependent unknowns as in the first-order shear deformation theory of Whitney and Pagano [6], but accounts for parabolic distribution of the transverse shear strains through the thickness of the plate. Exact closed-form solutions of symmetric cross-ply laminates are obtained and the results are compared with three-dimensional elasticity solutions and first-order shear deformation theory solutions. The present theory predicts the deflections and stresses more accurately when compared to the first-order theory.

An Experimental Study of Deviations from the Linear Transfer Function

1974 ◽  
Vol 32 ◽  
pp. 400-401
Author(s):  
William A. Voter ◽  
Harold P. Erickson
Keyword(s):  
Experimental Study ◽  
Image Formation ◽  
Order Theory ◽  
Second Order ◽  
Diffraction Spot ◽  
Order Effects ◽  
First Order ◽  
First Order Theory ◽  
Second Order Effects ◽  
The Fourier Transform

In a previous experimental study of image formation using a thin (20 nm) negatively stained catalase crystal, it was found that a linear or first order theory of image formation would explain almost entirely the changes in the Fourier transform of the image as a function of defocus. In this case it was concluded that the image is a valid picture of the object density. For thicker, higher contrast objects the first order theory may not be valid. Second order effects could generate false diffraction spots which would lead to spurious and artifactual image details. These second order effects would appear as deviations of the diffraction spot amplitudes from the first order theory. Small deviations were in fact noted in the study of the thin crystals, but there was insufficient data for a quantitative analysis.

Numerical Calculation of Second-Order Wave Resistance

1977 ◽  
Vol 21 (02) ◽  
pp. 94-106
Author(s):  
Young S. Hong
Keyword(s):  
Numerical Calculation ◽  
Steady Motion ◽  
Order Theory ◽  
Iteration Scheme ◽  
Wave Resistance ◽  
Second Order ◽  
Lagrangian Coordinates ◽  
Restricted Range ◽  
First Order ◽  
First Order Theory

The wave resistance due to the steady motion of a ship was formulated in Lagrangian coordinates by Wehausen [1].2 By introduction of an iteration scheme the solutions for the first order and second order3 were obtained. The draft/length ratio was assumed small in order to simplify numerical computation. In this work Wehausen's formulas are used to compute the resistance numerically. A few models are selected and the wave resistance is calculated. These results are compared with other methods and experiments. Generally speaking, the second-order resistance shows better agreement with experiment than first-order theory in only a restricted range of Froude number, say 0.25 to 0.35, and even here not uniformly. For larger Froude numbers it underestimates seriously.

scholarly journals HOL-λσ: an intentional first-order expression of higher-order logic

2001 ◽  
Vol 11 (1) ◽  
pp. 21-45 ◽  
Author(s):  
GILLES DOWEK ◽  
THERESE HARDIN ◽  
CLAUDE KIRCHNER
Keyword(s):  
Order Theory ◽  
Higher Order ◽  
Search Method ◽  
Order Logic ◽  
Automated Deduction ◽  
Higher Order Logic ◽  
First Order ◽  
Proof Search ◽  
First Order Theory ◽  
Explicit Substitutions

We give a first-order presentation of higher-order logic based on explicit substitutions. This presentation is intentionally equivalent to the usual presentation of higher-order logic based on λ-calculus, that is, a proposition can be proved without the extensionality axioms in one theory if and only if it can be in the other. We show that the Extended Narrowing and Resolution first-order proof-search method can be applied to this theory. In this way we get a step-by-step simulation of higher-order resolution. Hence, expressing higher-order logic as a first-order theory and applying a first-order proof search method is a relevant alternative to a direct implementation. In particular, the well-studied improvements of proof search for first-order logic could be reused at no cost for higher-order automated deduction. Moreover, as we stay in a first-order setting, extensions, such as equational higher-order resolution, may be easier to handle.

Polynomial rings and weak second-order logic

1985 ◽  
Vol 50 (4) ◽  
pp. 953-972 ◽  
Author(s):  
Anne Bauval
Keyword(s):  
Zero Divisor ◽  
Order Theory ◽  
Second Order ◽  
Order Logic ◽  
Polynomial Rings ◽  
Commutative Field ◽  
First Order ◽  
First Order Theory ◽  
Order Language ◽  
Second Order Logic

This article is a rewriting of my Ph.D. Thesis, supervised by Professor G. Sabbagh, and incorporates a suggestion from Professor B. Poizat. My main result can be crudely summarized (but see below for detailed statements) by the equality: first-order theory of F[Xi]i∈I = weak second-order theory of F.§I.1. Conventions. The letter F will always denote a commutative field, and I a nonempty set. A field or a ring (A; +, ·) will often be written A for short. We shall use symbols which are definable in all our models, and in the structure of natural numbers (N; +, ·):— the constant 0, defined by the formula Z(x): ∀y (x + y = y);— the constant 1, defined by the formula U(x): ∀y (x · y = y);— the operation ∹ x − y = z ↔ x = y + z;— the relation of division: x ∣ y ↔ ∃ z(x · z = y).A domain is a commutative ring with unity and without any zero divisor.By “… → …” we mean “… is definable in …, uniformly in any model M of L”.All our constructions will be uniform, unless otherwise mentioned.§I.2. Weak second-order models and languages. First of all, we have to define the models Pf(M), Sf(M), Sf′(M) and HF(M) associated to a model M = {A; ℐ) of a first-order language L [CK, pp. 18–20]. Let L1 be the extension of L obtained by adjunction of a second list of variables (denoted by capital letters), and of a membership symbol ∈. Pf(M) is the model (A, Pf(A); ℐ, ∈) of L1, (where Pf(A) is the set of finite subsets of A. Let L2 be the extension of L obtained by adjunction of a second list of variables, a membership symbol ∈, and a concatenation symbol ◠.

Complex language, complex thought?

2016 ◽  
Vol 33 ◽  
pp. 28-40
Author(s):  
Suzanne T.M. Bogaerds-Hazenberg ◽  
Petra Hendriks
Keyword(s):  
Theory Of Mind ◽  
Order Theory ◽  
Second Order ◽  
False Belief ◽  
Production Task ◽  
False Belief Task ◽  
First Order ◽  
First Order Theory ◽  
Complex Thought

Abstract It has been argued (e.g., by De Villiers and colleagues) that the acquisition of sentence embedding is necessary for the development of first-order Theory of Mind (ToM): the ability to attribute beliefs to others. This raises the question whether the acquisition of double embedded sentences is related to, and perhaps even necessary for, the development of second-order ToM: the ability to attribute beliefs about beliefs to others. This study tested 55 children (aged 7-10) on their ToM understanding in a false-belief task and on their elicited production of sentence embeddings. We found that second-order ToM passers produced mainly double embeddings, whereas first-order ToM passers produced mainly single embeddings. Furthermore, a better performance on second-order ToM predicted a higher rate of double embeddings and a lower rate of single embeddings in the production task. We conclude that children’s ability to produce double embeddings is related to their development of second-order ToM.

Application des miroirs électrostatiques à l'élimination de l'effet chromatique dans les spectromètres magnétiques. Première partie: théorie linéaire

1969 ◽  
Vol 47 (3) ◽  
pp. 331-340 ◽  
Author(s):  
Marcel Baril
Keyword(s):  
Matrix Algebra ◽  
Order Term ◽  
Order Theory ◽  
Second Order ◽  
Energy Dispersive ◽  
Powerful Method ◽  
First Order ◽  
First Order Theory ◽  
Electrostatic Mirror ◽  
Premiere Partie

Combining an energy-dispersive element with a magnetic prism results in an achromatic mass dispersive instrument, if parameters are chosen appropriately. A plane electrostatic mirror has been chosen as the energy-dispersive element. Trajectories are described in terms of lateral, angular, and energy variations about the principal trajectory. Achromatism and conjugate plane conditions have been calculated by the powerful method of matrix algebra. The first order theory is given in this article (part one), the second order term will be studied in part two which will be published later.

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