The elimination of contextually defined predicates in a modal system

1950 ◽  
Vol 15 (2) ◽  
pp. 92-92
Author(s):  
Ruth Barcan Marcus
Keyword(s):  
Fourth Order ◽  
Functional Calculus ◽  
Elimination Rule ◽  
Modal System ◽  
Image Position

In a recent note Bergmann states that “nonextensional languages may contain sentences from which contextually denned predicates are not eliminable.” A restatement of his argument is as follows. Suppose L is a functional calculus of fourth order. Included among the definitions occurring in L isLet ϕ occur in A (where A represents a well-formed formula of L) without an argument. The elimination of ϕ from A in accordance with (1) requiresThe proof of (2) depends on the principle of extensionalityHence Bergmann's conclusion “no such general elimination rule can be constructed for nonextensional languages.”

26.—The Strong Limit-2 Case of Fourth-order Differential Equations

1974 ◽  
Vol 71 (4) ◽  
pp. 297-304
Author(s):  
V. Krishna Kumar
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Order Equation ◽  
Complex Numbers ◽  
Strong Limit ◽  
Fourth Order Equation ◽  
Linearly Independent ◽  
Image Position ◽  
Adjoint Operators ◽  
Bounded Below

SynopsisThe fourth-order equation considered isConditions are given on the coefficients r, p and q which ensure that this differential equation (*) is in the strong limit-2 case at ∞, i.e. is limit-2 at ∞. This implies that (*) has exactly two linearly independent solutions which are in the integrable-square space ℒ2(0, ∞) for all complex numbers λ with im [λ] ≠ 0. Additionally the conditions imply that self-adjoint operators generated by M[·] in ℒ2(0, ∞) are semi-bounded below. The results obtained are applied to the case when the coefficients r, p and q are powers of x ∈ [0, ∞).

Some nonexistence theorems for semilinear fourth-order equations

2018 ◽  
Vol 149 (03) ◽  
pp. 761-779 ◽  
Author(s):  
M. Á. Burgos-Pérez ◽  
J. García-Melián ◽  
A. Quaas
Keyword(s):  
Fourth Order ◽  
Exterior Domains ◽  
Order Problem ◽  
Previous Condition ◽  
Fourth Order Problem ◽  
Radially Symmetric Solutions ◽  
Image Position ◽  
Fourth Order Equations

AbstractIn this paper, we analyse the semilinear fourth-order problem ( − Δ)2 u = g(u) in exterior domains of ℝN. Assuming the function g is nondecreasing and continuous in [0, + ∞) and positive in (0, + ∞), we show that positive classical supersolutions u of the problem which additionally verify − Δu > 0 exist if and only if N ≥ 5 and $$\int_0^\delta \displaystyle{{g(s)}\over{s^{(({2(N-2)})/({N-4}))}}} {\rm d}s \lt + \infty$$ for some δ > 0. When only radially symmetric solutions are taken into account, we also show that the monotonicity of g is not needed in this result. Finally, we consider the same problem posed in ℝN and show that if g is additionally convex and lies above a power greater than one at infinity, then all positive supersolutions u automatically verify − Δu > 0 in ℝN, and they do not exist when the previous condition fails.

A Two-Point Boundary Problem for Ordinary Self-Adjoint Differential Equations of Fourth Order

1954 ◽  
Vol 6 ◽  
pp. 416-419 ◽  
Author(s):  
H. M. Sternberg ◽  
R. L. Sternberg
Keyword(s):  
Boundary Problem ◽  
Fourth Order ◽  
Symmetric Matrix ◽  
Positive Definite ◽  
Matrix Functions ◽  
Point Boundary ◽  
Vector Identity ◽  
Class C ◽  
Image Position ◽  
Homogeneous Vector

The purpose of this note is to establish Theorem A below for the two-point homogeneous vector boundary problemwhere the Pi(x) are given real m × m symmetric matrix functions of x with P0(x) positive definite and Pi(x) of class C2−i on an infinite interval [a, ∞), and where by a solution of (1.1) — (1.2) for a ≤ x1 < x2 < ∞ we understand a real m-dimensional column vector u = u(x) of class C2 on [a, ∞) which is such that Pi(x)u(2−i) is of class C2−i on [a, ∞) and which satisfies (1.1) — (1.2) with the former a vector identity on [a, ∞).

scholarly journals Asymptotic formulae for linear equations

1972 ◽  
Vol 13 (2) ◽  
pp. 147-152 ◽  
Author(s):  
Don B. Hinton
Keyword(s):  
Asymptotic Behaviour ◽  
Fourth Order ◽  
Linear Equations ◽  
Order Equation ◽  
Third Order ◽  
Fourth Order Equation ◽  
The Third ◽  
Asymptotic Formulae ◽  
Image Position

Numerous formulae have been given which exhibit the asymptotic behaviour as t → ∞solutions ofwhere F(t) is essentially positive and Several of these results have been unified by a theorem of F. V. Atkinson [1]. It is the purpose of this paper to establish results, analogous to the theorem of Atkinson, for the third order equationand for the fourth order equation

XXIII.—On an Extension to an Integro-differential Inequality of Hardy, Littlewood and Polya

1972 ◽  
Vol 69 (4) ◽  
pp. 295-333 ◽  
Author(s):  
W. N. Everitt
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Differential Inequality ◽  
Order Differential Equation ◽  
Order Equation ◽  
Second Order ◽  
Image Position ◽  
Fourth Order Differential Equation

SynopsisThis paper considers an extension of the following inequality given in the book Inequalities by Hardy, Littlewood and Polya; let f be real-valued, twice differentiable on [0, ∞) and such that f and f are both in the space fn, ∞), then f′ is in L,2(0, ∞) andThe extension consists in replacing f′ by M[f] wherechoosing f so that f and M[f] are in L2(0, ∞) and then seeking to determine if there is an inequality of the formwhere K is a positive number independent of f.The analysis involves a fourth-order differential equation and the second-order equation associated with M.A number of examples are discussed to illustrate the theorems obtained and to show that the extended inequality (*) may or may not hold.

New foundations for Lewis modal systems

1957 ◽  
Vol 22 (2) ◽  
pp. 176-186 ◽  
Author(s):  
E. J. Lemmon
Keyword(s):  
Modal Logic ◽  
Propositional Calculus ◽  
Modal System ◽  
New Foundations ◽  
Image Position ◽  
Classical Propositional Calculus ◽  
Modal Systems

The main aims of this paper are firstly to present new and simpler postulate sets for certain well-known systems of modal logic, and secondly, in the light of these results, to suggest some new or newly formulated calculi, capable of interpretation as systems of epistemic or deontic modalities. The symbolism throughout is that of [9] (see especially Part III, Chapter I). In what follows, by a Lewis modal system is meant a system which (i) contains the full classical propositional calculus, (ii) is contained in the Lewis system S5, (iii) admits of the substitutability of tautologous equivalents, (iv) possesses as theses the four formulae:We shall also say that a system Σ1 is stricter than a system Σ2, if both are Lewis modal systems and Σ1 is contained in Σ2 but Σ2 is not contained in Σ1; and we shall call Σ1absolutely strict, if it possesses an infinity of irreducible modalities. Thus, the five systems of Lewis in [5], S1, S2, S3, S4, and S5, are all Lewis modal systems by this definition; they are in an order of decreasing strictness from S1 to S5; and S1 and S2 alone are absolutely strict.

Jacobi-type polynomials under an indefinite inner product

1981 ◽  
Vol 90 (1-2) ◽  
pp. 147-153 ◽  
Author(s):  
Angelo B. Mingarelli ◽  
Allan M. Krall
Keyword(s):  
Differential Operator ◽  
Fourth Order ◽  
Krein Space ◽  
Inner Product ◽  
Order Differential Operator ◽  
Indefinite Inner Product ◽  
Kreĭn Space ◽  
Image Position ◽  
Fourth Order Differential Operator

SynopsisThe polynomials which are orthogonal with respect towhen α> – 1, M>0 are considered when α<–1 and/or M<0. The Cauchy regularization of 〈·, ·〉 provides orthogonality and generates a Pontrjagin (Krein) space spanned by the polynomials. The polynomials are eigenfunctions associated with a self-adjoint, fourth order differential operator.

scholarly journals Properties of the extremal solution for a fourth-order elliptic problem

2012 ◽  
Vol 142 (5) ◽  
pp. 1051-1069 ◽  
Author(s):  
Baishun Lai ◽  
Zhuoran Du
Keyword(s):  
Weak Solution ◽  
Fourth Order ◽  
Elliptic Problem ◽  
Open Problem ◽  
Extremal Solution ◽  
Normal Vector ◽  
Unique Weak Solution ◽  
Unit Normal Vector ◽  
Order Elliptic Problem ◽  
Image Position

Let λ* > 0 denote the largest possible value of λ such that the systemhas a solution, where $\mathbb{B}$ is the unit ball in ℝn centred at the origin, p > 1 and n is the exterior unit normal vector. We show that for λ = λ* this problem possesses a unique weak solution u*, called the extremal solution. We prove that u* is singular when n ≥ 13 for p large enough and actually solve part of the open problem which Dávila et al. left unsolved.

On certain fourth-order integral inequalities

1979 ◽  
Vol 83 (3-4) ◽  
pp. 205-211 ◽  
Author(s):  
A. Russell
Keyword(s):  
Fourth Order ◽  
Integral Inequalities ◽  
Image Position

SynopsisThe integral inequalities with which this paper is concerned areandwhereK(μ) andK1(μ) are positive numbers which depend on μ.It is shown that each of these inequalities is valid only when μ = 0. Results relating to the corresponding inequalities on the extended interval (−∞, ∞) are also given.

Proofs of non-deducibility in intuitionistic functional calculus

1948 ◽  
Vol 13 (4) ◽  
pp. 204-207 ◽  
Author(s):  
Andkzej Mostowski
Keyword(s):  
Functional Calculus ◽  
Image Position ◽  
General Method

It has been proved by S. C. Kleene and David Nelson that the formulais intuitionistically non-deducible, i.e., non-deducible within the intuitionistic functional calculus.The aim of this note is to outline a general method which permits us to establish the intuitionistic non-deducibility of many formulas and in particular of the formula (1).

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