Note on a property of matrices for Lewis and Langford's calculi of propositions

1940 ◽  
Vol 5 (4) ◽  
pp. 150-151 ◽  
Author(s):  
James Dugundji
Keyword(s):  
Characteristic Matrix ◽  
Finite Characteristic ◽  
Class 1 ◽  
A Chain ◽  
Image Position

The object of this note is to show that there is no finite characteristic matrix for any one of Lewis and Langford's systems.Theorem I. There is no finite characteristic matrix for any one of the systems S1–S5.Proof. Let M be a matrix with less than n elements for which all the provable formulas in S1, or S2, or S3, or S4, or S5, are satisfied. Let Fn represent the formulawhere ∑ stands for a ∨-chain, and the pi, are variables in any one of the calculi. Using M, there is always at least one summand in Fn where pi and pk have the same value. Therefore, Fn can always be written in the form (a = a)∨ B, and thus will give, for any B, a “designated” value, since the formula (p = p) ∨ q is provable in any one of the systems S1–S5.Give to any one of the systems S1–S5, the following matrix, due to Henle:1. Elements: all possible classes formed from the integers 1, 2, 3, …, n.2. “Designated” element: the class {1, 2, 3, …,n}.3. Boole-Schröder algebra on the elements.4. ◊N = N (N the null class)◊A = {1, 2, …, n} (A any non-null class).5

Note on a problem of Porte

1968 ◽  
Vol 64 (1) ◽  
pp. 1-1 ◽  
Author(s):  
George Rousseau
Keyword(s):  
Propositional Calculus ◽  
The Other ◽  
Characteristic Matrix ◽  
Finite Characteristic ◽  
Other Hand ◽  
Intuitionistic Propositional Calculus ◽  
The Matrix ◽  
Image Position

Porte (1), p. 117, conjectures that the positive implicational propositional calculus has no finite characteristic matrix. The proof of this conjecture is a straightforward modification of Gödel's proof (2) that the intuitionistic propositional calculus has no finite characteristic matrix (see e.g. Church(3), ex. 26.12). Writing (A ∨ B) for ((A ⊃ B) ⊃ B) and Xij for (pj ⊃ pi) (i, j = 1,2,…), we define, for n > l, the formulawhere the terms associate to the left. Since provable formulae take the value n for all systems of values of the variables in the matrix {1,…,n} where x ⊃ y is n when x ≤ y and y otherwise, whereas Gn takes the value n − 1 for the system of values pi = i (i = 1,…,n), it follows that Gn is not provable. On the other hand, since A ⊢ A ∨ B and B ⊢ A ∨ B, it is easily seen that is provable whenever r ≠ s (r, s = 1,…,n). The result follows from these two remarks.

Fast diffusion with loss at infinity

1995 ◽  
Vol 36 (4) ◽  
pp. 438-459 ◽  
Author(s):  
J. R. Philip
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fast Diffusion ◽  
Solution Properties ◽  
Positive Power ◽  
Negative Power ◽  
Exponential Decrease ◽  
Class 1 ◽  
Image Position ◽  
Instantaneous Source

AbstractWe study the equationHere s is not necessarily integral; m is initially unrestricted. Material-conserving instantaneous source solutions of A are reviewed as an entrée to material-losing solutions. Simple physical arguments show that solutions for a finite slug losing material at infinity at a finite nonzero rate can exist only for the following m-ranges: 0 < s < 2, −2s−1 < m ≤ −1; s > 2, −1 < m < −2s−1. The result for s = 1 was known previously. The case s = 2, m = −1, needs further investigation. Three different similarity schemes all lead to the same ordinary differential equation. For 0 < s < 2, parameter γ (0 < γ < ∞) in that equation discriminates between the three classes of solution: class 1 gives the concentration scale decreasing as a negative power of (1 + t/T); 2 gives exponential decrease; and 3 gives decrease as a positive power of (1 − t/T), the solution vanishing at t = T < ∞. Solutions for s = 1, are presented graphically. The variation of concentration and flux profiles with increasing γ is physically explicable in terms of increasing flux at infinity. An indefinitely large number of exact solutions are found for s = 1,γ = 1. These demonstrate the systematic variation of solution properties as m decreases from −1 toward −2 at fixed γ.

The statistical distribution of the length of a rubber molecule

1948 ◽  
Vol 44 (3) ◽  
pp. 342-344 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Normal Distribution ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Statistical Distribution ◽  
Equal Length ◽  
Fixed Angle ◽  
A Chain ◽  
Image Position ◽  
Three Degrees Of Freedom

A rubber molecule containing n + 1 carbon atoms may be represented by a chain of n links of equal length such that successive links are at a fixed angle to each other but are otherwise at random. The statistical distribution of the length of the molecule, that is, the distance between the first and last carbon atoms, has been considered by various authors (Treloar (1) gives references). In particular, if the first atom is kept fixed at the origin of a system of coordinates and the chain is otherwise at random, it has been conjectured that the distribution of the (n + 1)th atom will tend, as n increases, towards a three-dimensional normal distribution of the formwhere σ depends on n. Thus r2 (= x2 + y2 + z2) will be approximately distributed as σ2χ2 with three degrees of freedom.

Kernels with only a finite number of characteristic values

1971 ◽  
Vol 70 (2) ◽  
pp. 257-262
Author(s):  
Dale W. Swann
Keyword(s):  
Finite Number ◽  
Higher Order ◽  
Complex Numbers ◽  
Finite Characteristic ◽  
Characteristic Values ◽  
Image Position ◽  
Complex Valued

Let K(s, t) be a complex-valued L2 kernel on the square ⋜ s, t ⋜ by which we meanand let {λν}, perhaps empty, be the set of finite characteristic values (f.c.v.) of K(s, t), i.e. complex numbers with which there are associated non-trivial L2 functions øν(s) satisfyingFor such kernels, the iterated kernels,are well-defined (1), as are the higher order tracesCarleman(2) showed that the f.c.v. of K are the zeros of the modified Fredhoim determinantthe latter expression holding only for |λ| sufficiently small (3). The δn in (3) may be calculated, at least in theory, by well-known formulae involving the higher order traces (1). For our later analysis of this case, we define and , respectively, as the minimum and maximum moduli of the zeros of , the nth section of D*(K, λ).

Extensions of the Lewis system S5

1951 ◽  
Vol 16 (2) ◽  
pp. 112-120 ◽  
Author(s):  
Schiller Joe Scroggs
Keyword(s):  
Broad Class ◽  
Characteristic Matrix ◽  
Normal Extension ◽  
Strict Implication ◽  
Finite Characteristic ◽  
Material Implication ◽  
Sentential Calculus ◽  
The Third ◽  
Simple Class ◽  
Definition Of

Dugundji has proved that none of the Lewis systems of modal logic, S1 through S5, has a finite characteristic matrix. The question arises whether there exist proper extensions of S5 which have no finite characteristic matrix. By an extension of a sentential calculus S, we usually refer to any system S′ such that every formula provable in S is provable in S′. An extension S′ of S is called proper if it is not identical with S. The answer to the question is trivially affirmative in case we make no additional restrictions on the class of extensions. Thus the extension of S5 obtained by adding to the provable formulas the additional formula p has no finite characteristic matrix (indeed, it has no characteristic matrix at all), but this extension is not closed under substitution—the formula q is not provable in it. McKinsey and Tarski have defined normal extensions of S4* by imposing three conditions. Normal extensions must be closed under substitution, must preserve the rule of detachment under material implication, and must also preserve the rule that if α is provable then ~◊~α is provable. McKinsey and Tarski also gave an example of an extension of S4 which satisfies the first two of these conditions but not the third. One of the results of this paper is that every extension of S5 which satisfies the first two of these conditions also satisfies the third, and hence the above definition of normal extension is redundant for S5. We shall therefore limit the extensions discussed in this paper to those which are closed under substitution and which preserve the rule of detachment under material implication. These extensions we shall call quasi-normal. The class of quasi-normal extensions of S5 is a very broad class and actually includes all extensions which are likely to prove interesting. It is easily shown that quasi-normal extensions of S5 preserve the rules of replacement, adjunction, and detachment under strict implication. It is the purpose of this paper to prove that every proper quasi-normal extension of S5 has a finite characteristic matrix and that every quasi-normal extension of S5 is a normal extension of S5 and to describe a simple class of characteristic matrices for S5.

A ratio limit theorem for (sub) Markov chains on {1,2, …} with bounded jumps

1995 ◽  
Vol 27 (3) ◽  
pp. 652-691 ◽  
Author(s):  
Harry Kesten
Keyword(s):  
Harmonic Function ◽  
Invariant Measure ◽  
Probability Measure ◽  
Markov Chains ◽  
Positive Matrices ◽  
Ratio Limit Theorems ◽  
Ratio Limit ◽  
A Chain ◽  
Image Position ◽  
Quasi Stationary Distribution

We consider positive matrices Q, indexed by {1,2, …}. Assume that there exists a constant 1 L < ∞ and sequences u1< u2< · ·· and d1d2< · ·· such that Q(i, j) = 0 whenever i < ur < ur + L < j or i > dr + L > dr > j for some r. If Q satisfies some additional uniform irreducibility and aperiodicity assumptions, then for s > 0, Q has at most one positive s-harmonic function and at most one s-invariant measure µ. We use this result to show that if Q is also substochastic, then it has the strong ratio limit property, that is for a suitable R and some R–1-harmonic function f and R–1-invariant measure µ. Under additional conditions µ can be taken as a probability measure on {1,2, …} and exists. An example shows that this limit may fail to exist if Q does not satisfy the restrictions imposed above, even though Q may have a minimal normalized quasi-stationary distribution (i.e. a probability measure µ for which R–1µ = µQ).The results have an immediate interpretation for Markov chains on {0,1,2, …} with 0 as an absorbing state. They give ratio limit theorems for such a chain, conditioned on not yet being absorbed at 0 by time n.

Maximal branching processes and ‘long-range percolation’

1970 ◽  
Vol 7 (01) ◽  
pp. 89-98
Author(s):  
John Lamperti
Keyword(s):  
Probability Distribution Function ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
Transition Matrices ◽  
A Chain ◽  
Image Position ◽  
The Right

In the first part of this paper, we will consider a class of Markov chains on the non-negative integers which resemble the Galton-Watson branching process, but with one major difference. If there are k individuals in the nth “generation”, and are independent random variables representing their respective numbers of offspring, then the (n + 1)th generation will contain max individuals rather than as in the branching case. Equivalently, the transition matrices Pij of the chains we will study are to be of the form where F(.) is the probability distribution function of a non-negative, integervalued random variable. The right-hand side of (1) is thus the probability that the maximum of i independent random variables distributed by F has the value j. Such a chain will be called a “maximal branching process”.

Solutions d'une classe de systèmes symétriques

1977 ◽  
Vol 77 (1-2) ◽  
pp. 87-95
Author(s):  
Said Benachour
Keyword(s):  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Singular Systems ◽  
Characteristic Matrix ◽  
Constant Rank ◽  
Open Set ◽  
Image Position ◽  
Uniqueness Of A Solution

SynopsisLet Ω be an open set of ℝm; we consider some symmetric singular systems:We study the existence and uniqueness of a solution of the problem Lu = f plus boundary conditions when the characteristic matrix is of constant rank only on the boundary.

A Caeretan Hydria in Dunedin

1955 ◽  
Vol 75 ◽  
pp. 1-6
Author(s):  
J. K. Anderson
Keyword(s):  
Good Deal ◽  
The Body ◽  
A Chain ◽  
Image Position

The vase here described was recently presented to the Otago Museum in commemoration of the distinguished services of Dr. H. D. Skinner, for many years Director of the Museum. It was formerly on the Rome market. It is restored from fragments, and missing pieces of the neck, mouth, and shoulder have been replaced by plaster. The joints and plaster restorations have been carefully painted over, and there has been a good deal of repainting where the glaze was worn. On the mouth, neck, and shoulder the restorations, though extensive, merely fill gaps in a well-defined pattern, and can therefore be passed over without a detailed description. The repainting of the figures on the body of the vase will be described at greater length below. The clay is a fine, clear red, rather lighter than the usual colour of Attic. The principal dimensions of the vase are as follows (measurements in metres):The body is ovoid, with high, flat shoulders. It is separated from the wide flaring foot by a low, raised ridge. A similar ridge separates the shoulder from the neck, which is cylindrical with slightly concave sides. The lip flares widely. The side handles are small and slope slightly upward; they are attached just above the widest part of the vase and below the sharpest curve of the shoulder. The vertical handle is divided by three deep, vertical grooves. The inside of the mouth and the upper surface of the foot are ornamented with rounded tongues of black glaze. These were painted alternately red and white, but the paint, which was applied on top of the black glaze, is now much worn. On the lower part of the body are short black rays; above these is a rather wider zone with a chain of five-petalled lotuses linked to five-leaved palmettes.

AN IMAGE ANALYSIS OF MULTIPLE‐LAYER RESISTIVITY PROBLEMS

Geophysics ◽  
1959 ◽  
Vol 24 (3) ◽  
pp. 485-509 ◽  
Author(s):  
Irwin Roman
Keyword(s):  
Recursion Formula ◽  
Test Point ◽  
Reflection Factor ◽  
Primary Sequence ◽  
Method Of Images ◽  
Original Source ◽  
Exterior Regions ◽  
A Chain ◽  
Image Position ◽  
Upper Boundary

The Kelvin method of images is expressible by a transflection at a boundary. The original source is augmented by a supplement and a complement. The supplement contributes to the potential on the same side of the boundary as the source, but it lies at the optical image position of the source in the boundary. The complement lies at the position of the source but contributes to the potential on the opposite side of the boundary. For two or more boundaries, there are two exterior regions and one or more interior regions. For a source in the top layer, a primary sequence starts with a downward transflection and a secondary sequence with an upward transflection. To each primary sequence of transflections there corresponds a secondary sequence with an upward transflection at the upper boundary ahead of it. The exterior images are not transflected again. Successive transflections occur at adjacent boundaries, suggesting a link of two transflections. To a sequence of links, called a chain, there corresponds an associated sequence, obtained by dropping the last transflection. Exterior images follow from interior, associated from chain, and secondary from primary. Thus, only primary, interior, chain images need to be traced. Each potential is the sum of terms of the form m/r where m is the strength of a specific image, r is the distance of that image from the test point, and the sum includes all images contributing to that potential. The addition of each boundary introduces images and potentials that must be added to those existing prior to the introduction, but it does not otherwise alter them. For the three‐boundary problem, the separate image strengths are determined by simple multiplication after a kernel polynomial is calculated. The latter is a finite polynomial in the reflection‐factor at the middle boundary and can be tabulated. For the images of a specific potential and depth group, the strengths satisfy a recursion formula that serves as a check on direct evaluations.

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