A feasible theory for analysis

1994 ◽  
Vol 59 (3) ◽  
pp. 1001-1011 ◽  
Author(s):  
Fernando Ferreira
Keyword(s):  
Polynomial Time ◽  
Order Theory ◽  
Second Order ◽  
First Order ◽  
Weak König’S Lemma ◽  
König’S Lemma ◽  
Computable Functions

AbstractWe construct a weak second-order theory of arithmetic which includes Weak König's Lemma (WKL) for trees defined by bounded formulae. The provably total functions (with -graphs) of this theory are the polynomial time computable functions. It is shown that the first-order strength of this version of WKL is exactly that of the scheme of collection for bounded formulae.

scholarly journals Polynomial time operations in explicit mathematics

1997 ◽  
Vol 62 (2) ◽  
pp. 575-594 ◽  
Author(s):  
Thomas Strahm
Keyword(s):  
Polynomial Time ◽  
Proof Theory ◽  
Order Theory ◽  
First Order ◽  
First Order Theory ◽  
Computable Functions

AbstractIn this paper we study (self-)applicative theories of operations and binary words in the context of polynomial time computability. We propose a first order theory PTO which allows full self-application and whose provably total functions on = {0, 1}* are exactly the polynomial time computable functions. Our treatment of PTO is proof-theoretic and very much in the spirit of reductive proof theory.

scholarly journals THE PREHISTORY OF THE SUBSYSTEMS OF SECOND-ORDER ARITHMETIC

2017 ◽  
Vol 10 (2) ◽  
pp. 357-396 ◽  
Author(s):  
WALTER DEAN ◽  
SEAN WALSH
Keyword(s):  
Large Scale ◽  
Second Order ◽  
Descriptive Set Theory ◽  
Second Order Arithmetic ◽  
Mathematics Program ◽  
Order Arithmetic ◽  
Transfinite Recursion ◽  
Descriptive Set ◽  
Weak König’S Lemma ◽  
König’S Lemma

AbstractThis paper presents a systematic study of the prehistory of the traditional subsystems of second-order arithmetic that feature prominently in the reverse mathematics program promoted by Friedman and Simpson. We look in particular at: (i) the long arc from Poincaré to Feferman as concerns arithmetic definability and provability, (ii) the interplay between finitism and the formalization of analysis in the lecture notes and publications of Hilbert and Bernays, (iii) the uncertainty as to the constructive status of principles equivalent to Weak König’s Lemma, and (iv) the large-scale intellectual backdrop to arithmetical transfinite recursion in descriptive set theory and its effectivization by Borel, Lusin, Addison, and others.

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.

The Consistent Second-Order Wave Theory and Its Application to a Submerged Spheroid

1970 ◽  
Vol 14 (01) ◽  
pp. 23-50
Author(s):  
Young H. Chey
Keyword(s):  
Free Surface ◽  
Solid Wall ◽  
Three Dimensional ◽  
Wave Theory ◽  
Order Theory ◽  
Wave Resistance ◽  
Second Order ◽  
Surface Phenomena ◽  
First Order ◽  
Theoretical Results

Because of the recognized inadequacy of first-order linearized surface-wave theory, the author has developed, for a three-dimensional body, a new second-order theory which provides a better description of free-surface phenomena. The new theory more accurately satisfies the kinematic boundary condition on the solid wall, and takes into account the nonlinearity of the condition at the free surface. The author applies the new theory to a submerged spheroid, to calculate wave resistance. Experiments were conducted to verify the theory, and their results are compared with the theoretical results. The comparison indicates that the use of the new theory leads to more accurate prediction of wave resistance.

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.

Bubble dynamics in a compressible liquid. Part 2. Second-order theory

1987 ◽  
Vol 185 ◽  
pp. 289-321 ◽  
Author(s):  
A. Lezzi ◽  
A. Prosperetti
Keyword(s):  
Bubble Dynamics ◽  
Characteristic Length ◽  
Geometric Mean ◽  
Order Theory ◽  
Compressible Liquid ◽  
Second Order ◽  
Radial Equation ◽  
Singular Perturbation Method ◽  
First Order ◽  
Liquid Part

The radial dynamics of a spherical bubble in a compressible liquid is studied by means of a rigorous singular-perturbation method to second order in the bubble-wall Mach number. The results of Part 1 (Prosperetti & Lezzi, 1986) are recovered at orders zero and one. At second order the ordinary inner and outer structure of the solution proves inadequate to correctly describe the fields and it is necessary to introduce an intermediate region the characteristic length of which is the geometric mean of the inner and outer lengthscales. The degree of indeterminacy for the radial equation of motion found at first order is significantly increased by going to second order. As in Part 1 we examine several of the possible forms of this equation by comparison with results obtained from the numerical integration of the complete partial-differential-equation formulation. Expressions and results for the pressure and velocity fields in the liquid are also reported.

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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