Intensional interpretations of functionals of finite type I

1967 ◽  
Vol 32 (2) ◽  
pp. 198-212 ◽  
Author(s):  
W. W. Tait
Keyword(s):  
Finite Type ◽  
Type I ◽  
Axiom Schema ◽  
Image Position ◽  
Intuitionistic Arithmetic

T0 will denote Gödel's theory T[3] of functionals of finite type (f.t.) with intuitionistic quantification over each f.t. added. T1 will denote T0 together with definition by bar recursion of type o, the axiom schema of bar induction, and the schemaof choice. Precise descriptions of these systems are given below in §4. The main results of this paper are interpretations of T0 in intuitionistic arithmetic U0 and of T1 in intuitionistic analysis is U1. U1 is U0 with quantification over functionals of type (0,0) and the axiom schemata AC00 and of bar induction.

Representation Formulas for Integrable and Entire Functions of Exponential Type I

1988 ◽  
Vol 40 (04) ◽  
pp. 1010-1024 ◽  
Author(s):  
Clément Frappier
Keyword(s):  
Exponential Type ◽  
Entire Functions ◽  
Conjugate Function ◽  
Interpolation Formula ◽  
Type I ◽  
Additional Hypothesis ◽  
Period 2 ◽  
Representation Formulas ◽  
Functions Of Exponential Type ◽  
Image Position

Let Bτ denote the class of entire functions of exponential type τ (>0) bounded on the real axis. For the function f ∊ Bτ we have the interpolation formula [1, p. 143] 1.1 where t, γ are real numbers and is the so called conjugate function of f. Let us put 1.2 The function Gγ,f is a periodic function of α, with period 2. For t = 0 (the general case is obtained by translation) the righthand member of (1) is 2τGγ,f (1). In the following paper we suppose that f satisfies an additional hypothesis of the form f(x) = O(|x|-ε), for some ε > 0, as x → ±∞ and we give an integral representation of Gγ,f(α) which is valid for 0 ≦ α ≦ 2.

scholarly journals Completely bounded isomorphisms of injective von Neumann algebras

1989 ◽  
Vol 32 (2) ◽  
pp. 317-327 ◽  
Author(s):  
Erik Christensen ◽  
Allan M. Sinclair
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Von Neumann Algebras ◽  
Linear Structure ◽  
Type I ◽  
Hausdorff Spaces ◽  
Bounded Degree ◽  
Von Neumann ◽  
Separable Predual ◽  
Image Position

Milutin's Theorem states that if X and Y are uncountable metrizable compact Hausdorff spaces, then C(X) and C(Y) are isomorphic as Banach spaces [15, p. 379]. Thus there is only one isomorphism class of such Banach spaces. There is also an extensive theory of the Banach–Mazur distance between various classes of classical Banach spaces with the deepest results depending on probabilistic and asymptotic estimates [18]. Lindenstrauss, Haagerup and possibly others know that as Banach spaceswhere H is the infinite dimensional separable Hilbert space, R is the injective II 1-factor on H, and ≈ denotes Banach space isomorphism. Haagerup informed us of this result, and suggested considering completely bounded isomorphisms; it is a pleasure to acknowledge his suggestion. We replace Banach space isomorphisms by completely bounded isomorphisms that preserve the linear structure and involution, but not the product. One of the two theorems of this paper is a strengthened version of the above result: if N is an injective von Neumann algebra with separable predual and not finite type I of bounded degree, then N is completely boundedly isomorphic to B(H). The methods used are similar to those in Banach space theory with complete boundedness needing a little care at various points in the argument. Extensive use is made of the conditional expectation available for injective algebras, and the methods do not apply to the interesting problems of completely bounded isomorphisms of non-injective von Neumann algebras (see [4] for a study of the completely bounded approximation property).

scholarly journals A New Evolutionary Interpretation of the IRAS Two-Colour Diagram

1993 ◽  
Vol 155 ◽  
pp. 332-332 ◽  
Author(s):  
P. García-Lario ◽  
A. Manchado ◽  
S.R. Pottasch
Keyword(s):  
Mass Loss ◽  
Loss Rate ◽  
Mass Loss Rate ◽  
Theoretical Models ◽  
Initial Mass ◽  
Type I ◽  
Processed Material ◽  
Ir Stars ◽  
Image Position ◽  
Evolutionary Interpretation

A new evolutionary interpretation of the sequence of colours observed in the IRAS two-colour diagram by AGB and post-AGB stars is given, which is capable of explaining the observational properties of both kind of objects. It is useful to define a parameter λ to define the position of a given star in this “infrared main sequence” (IRMS). Adopting and from the analysis of the expansion velocities, mass loss rates and luminosities observed in a selected sample of non-variable OH/IR stars with no optical counterpart in the Galactic bulge as a function of λ, we conclude that the position in the IRAS two-colour diagram at which a star leaves the IRMS (λmax) only depends on the initial mass Mz of the progenitor star, so that only massive objects can reach the upper end of this sequence. The relation found is: Expansion velocities increase with the initial mass while every point in the IRMS is found to be associated to a certain value of the mass loss rate. This model also predicts the evolution with time of the mass loss rate during the AGB as a function of the initial mass of the progenitor star, and confirms that most known planetary nebulae are the result of the evolution of considerably massive stars (between 2–3 solar masses) which means that the contribution of processed material to the interstellar medium is considerably higher than what theoretical models predict. Type I PNe are the result of the evolution of 3 — 5 M⊙ progenitors while progenitors with Mi ≤ 1.2 M⊙ probably do not give PNe. The model is also in agreement with the narrow distribution of core masses found in central stars of PNe and white dwarfs and with the usual expansion velocities found in OH/IR stars.

scholarly journals Implicational complexity in intuitionistic arithmetic

1981 ◽  
Vol 46 (2) ◽  
pp. 240-248 ◽  
Author(s):  
Daniel Leivant
Keyword(s):  
Normal Form ◽  
Form Theorem ◽  
Natural Measure ◽  
Normal Form Theorem ◽  
Propositional Constants ◽  
Image Position ◽  
Classical Arithmetic ◽  
Intuitionistic Arithmetic

In classical arithmetic a natural measure for the complexity of relations is provided by the number of quantifier alternations in an equivalent prenex normal form. However, the proof of the Prenex Normal Form Theorem uses the following intuitionistically invalid rules for permuting quantifiers with propositional constants.Each one of these schemas, when added to Intuitionistic (Heyting's) Arithmetic IA, generates full Classical (Peano's) Arithmetic. Schema (3) is of little interest here, since one can obtain a formula intuitionistically equivalent to A ∨ ∀xBx, which is prenex if A and B are:For the two conjuncts on the r.h.s. (1) may be successively applied, since y = 0 is decidable.We shall readily verify that there is no way of similarly going around (1) or (2). This fact calls for counting implication (though not conjunction or disjunction) in measuring in IA the complexity of arithmetic relations. The natural implicational measure for our purpose is the depth of negative nestings of implication, defined as follows. I(F): = 0 if F is atomic; I(F ∧ G) = I(F ∨ G): = max[I(F), I(G)]; I(∀xF) = I(∃xF): = I(F); I(F → G):= max[I(F) + 1, I(G)].

On the independence of Henkin's axioms for fragments of the propositional calculus

1951 ◽  
Vol 16 (1) ◽  
pp. 43-45
Author(s):  
Maurice L'abbé
Keyword(s):  
Propositional Calculus ◽  
General System ◽  
Function Symbol ◽  
The Other ◽  
Axiom Schema ◽  
Other Hand ◽  
Truth Function ◽  
The One ◽  
Image Position ◽  
Made In

A general system of axioms has been given by Henkin for a fragment of the propositional calculus having as primitive symbols, in addition to the usual parentheses, variables, and implication sign ⊃, an arbitrarily given truth function symbol ϕ. This system of axioms, which we shall denote by S(⊃, ϕ), contains the following three axiom schemataplus the 2m further axiom schemata involving the symbol ϕwhere ϕ is an m-placed function symbol. We refer to Henkin's paper, p. 43, for the detailed description of the axiom schemata (4).The remark was made in the above mentioned paper that each of the 2m axiom schemata of (4) is trivially independent of the rest of the axioms of S(⊃, ϕ), and it was conjectured that the axiom schemata (1), (2) and (3) are also independent. In this note, we prove the general independence of the axiom schemata (1) and (2). As for (3), we show on the one hand its independence in the systems S(⊃) and S(⊃, f), and, on the other hand, its dependence in the system S(⊃, ∼). The net result is, therefore, that in any of these systems of axioms S(⊃, ϕ) all the axiom schemata are independent, except possibly the axiom schema (3).

Some models for intuitionistic finite type arithmetic with fan functional

1977 ◽  
Vol 42 (2) ◽  
pp. 194-202 ◽  
Author(s):  
A. S. Troelstra
Keyword(s):  
Finite Type ◽  
Church’S Thesis ◽  
Proper Understanding ◽  
New Model ◽  
Fan Theorem ◽  
Church's Thesis ◽  
Definition Of ◽  
Intuitionistic Arithmetic

In this note we shall assume acquaintance with [T4] and the parts of [T1] which deal with intuitionistic arithmetic in all finite types. The bibliography just continues the bibliography of [T4].The principal purpose of this note is the discussion of two models for intuitionistic finite type arithmetic with fan functional (HAω+ MUC). The first model is needed to correct an oversight in the proof of Theorem 6 [T4, §5]: the model ECF+as defined there cannot be shown to have the required properties inEL+ QF-AC, the reason being that a change in the definition ofW12alone does not suffice—if one wishes to establish closure under the operations of HAωthe definitions ofW1σfor other σ have to be adopted as well. It is difficult to see how to do this directly in a uniform way — but we succeed via a detour, which is described in §2.For a proper understanding, we should perhaps note already here thaton the assumption of the fan theorem, ECF+as defined in [T4] and the new model of this note coincide (since then the definition ofW12[T4, p. 594] is equivalent to the definition forW12in the case of ECF); but inELit is impossible to prove this (and under assumption of Church's thesis the two models differ).

The Summation of a Fourier Integral of Finite Type

1926 ◽  
Vol 23 (4) ◽  
pp. 373-382 ◽  
Author(s):  
S. Pollard
Keyword(s):  
Finite Type ◽  
Generating Function ◽  
Finite Interval ◽  
Fourier Integral ◽  
Image Position

A Fourier integral will be described as of finite type if the range of integration of the integrals by means of which the coefficients in the Fourier integral are defined is a finite interval instead of the usual (−∞ ∞). Thus is of finite type (p, q), such that where ƒ(x) is the generating function of the series.

Sign in / Sign up

Export Citation Format

Share Document