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.

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).

On functional Nörlund methods

1970 ◽  
Vol 67 (1) ◽  
pp. 47-60 ◽  
Author(s):  
B. Choudhary
Keyword(s):  
The Other ◽  
Integral Transformations ◽  
Other Hand ◽  
Nörlund Means ◽  
The One ◽  
Image Position

Integral transformations analogous to the Nörlund means have been introduced and investigated by Kuttner, Knopp and Vanderburg(6), (5), (4). It is known that with any regular Nörlund mean (N, p) there is associated a functionregular for |z| < 1, and if we have two Nörlund means (N, p) and (N, r), where (N, pr is regular, while the function is regular for |z| ≤ 1 and different) from zero at z = 1, then q(z) = r(z)p(z) belongs to a regular Nörlund mean (N, q). Concerning Nörlund means Peyerimhoff(7) and Miesner (3) have recently obtained the relation between the convergence fields of the Nörlund means (N, p) and (N, r) on the one hand and the convergence field of the Nörlund mean (N, q) on the other hand.

scholarly journals Distribution of rational points on the real line

1973 ◽  
Vol 15 (2) ◽  
pp. 243-256 ◽  
Author(s):  
T. K. Sheng
Keyword(s):  
Algebraic Number ◽  
Rational Number ◽  
Real Line ◽  
Rational Points ◽  
The Other ◽  
Liouville Numbers ◽  
The Real ◽  
Other Hand ◽  
Image Position

It is well known that no rational number is approximable to order higher than 1. Roth [3] showed that an algebraic number is not approximable to order greater than 2. On the other hand it is easy to construct numbers, the Liouville numbers, which are approximable to any order (see [2], p. 162). We are led to the question, “Let Nn(α, β) denote the number of distinct rational points with denominators ≦ n contained in an interval (α, β). What is the behaviour of Nn(α, + 1/n) as α varies on the real line?” We shall prove that and that there are “compressions” and “rarefactions” of rational points on the real line.

Configuration of Quadrivalent Atoms

1929 ◽  
Vol 25 (2) ◽  
pp. 219-221
Author(s):  
T. M. Lowry
Keyword(s):  
The Other ◽  
Tetrahedral Configuration ◽  
X Ray ◽  
Metallic Atom ◽  
Other Hand ◽  
Function Change ◽  
The One ◽  
Image Position ◽  
Octahedral Configuration ◽  
As If

Two alternative views have been expressed in regard to the configuration of quadrivalent atoms. On the one hand le Bel and van't Hoff assigned to quadrivalent carbon a tetrahedral configuration, which has since been confirmed by the X-ray analysis of the diamond. On the other hand, Werner in 1893 adopted an octahedral configuration for radicals of the type MA6, e.g. inand then suggested that “the molecules [MA4]X2 are incomplete molecules [MA6]X2. The radicals [MA4] result from the octahedrally-conceived radicals [MA6] by loss of two groups A, but with no function-change of the acid residue…. They behave as if the bivalent metallic atom in the centre of the octahedron could no longer bind all six of the groups A and lost two of them leaving behind the fragment [MA4]” (p. 303).

A System of Linear Equations with an Infinity of Unknowns

1924 ◽  
Vol 22 (3) ◽  
pp. 282-286
Author(s):  
E. C. Titchmarsh
Keyword(s):  
Present Note ◽  
Linear Equations ◽  
The Other ◽  
System Of Linear Equations ◽  
Other Hand ◽  
Image Position ◽  
Monotonic Character

I have collected in the present note some theorems regarding the solution of a certain system of linear equations with an infinity of unknowns. The general form of the equations isthe numbers a1, a2, … c1, c2, … being given. Equations of this type are of course well known; but in studying them it is generally assumed that the series depend for convergence on the convergence-exponent of the sequences involved, e.g. that and are convergent. No assumptions of this kind are made here, and in fact the series need not be absolutely convergent. On the other hand rather special assumptions are made with regard to the monotonic character of the sequences an and cn.

Adverbial clauses and V3

2019 ◽  
Vol 5 (s3) ◽  
Author(s):  
Eva Csipak
Keyword(s):  
Word Order ◽  
Speech Act ◽  
The Other ◽  
Order Effects ◽  
Matrix Clause ◽  
Other Hand ◽  
Pragmatic Analysis ◽  
The Matrix ◽  
Adverbial Clauses ◽  
Biscuit Conditionals

AbstractThis paper discusses word order effects in German adverbial clauses: often, the matrix clause can exhibit either V2 or V3 word order. I argue that adverbial clauses with V3 word order have an obligatory ‘biscuit’ interpretation and receive a speech act modifying interpretation, as has previously only been argued for ‘biscuit conditionals’. I show that this phenomenon holds more generally. On the other hand, a pragmatic analysis for V2 biscuit conditionals remains necessary.

The extreme points of some classes of polynomials

1985 ◽  
Vol 101 (1-2) ◽  
pp. 99-110 ◽  
Author(s):  
D. A. Brannan ◽  
J. G. Clunie
Keyword(s):  
Extreme Points ◽  
The Other ◽  
Other Hand ◽  
Image Position

SynopsisWe study the extreme points of two classes of polynomials of degree at most n:It turns out that f ∈ Ext if and only if Re f(eiθ) has exactly 2n zeros in [0, 2π). On the other hand, if f∈Hn and 1−|f(eiθ)|2 has 2n zeros in [0, 2π), then either f ∈ Ext Hn or else f(z) = α + βzn where |α|+|β| = l and αβ≠0; if 1−|f(eiθ)|2 has 2m zeros, 2n, then f may or may not belong to Ext Hn.

The No-Three-In-Line Problem

1968 ◽  
Vol 11 (4) ◽  
pp. 527-531 ◽  
Author(s):  
Richard K. Guy ◽  
Patrick A. Kelly
Keyword(s):  
Finite Number ◽  
The Other ◽  
Probabilistic Argument ◽  
Number Of Solutions ◽  
Other Hand ◽  
Image Position

Let Sn be the set of n2 points with integer coordinates n (x, y), 1 ≤ x, y <n. Let fn be the maximum cardinal of a subset T of Sn such that no three points of T are collinear. Clearly fn < 2n.For 2 ≤ n ≤ 10 it is known ([2], [3] for n = 8, [ 1] for n = 10, also [4], [6]) that fn = 2n, and that this bound is attained in 1, 1, 4, 5, 11, 22, 57, 51 and 156 distinct configurations for these nine values of n. On the other hand, P. Erdös [7] has pointed out that if n is prime, fn ≥ n, since the n points (x, x2) reduced modulo n have no three collinear. We give a probabilistic argument to support the conjecture that there is only a finite number of solutions to the no-three-in-line problem. More specifically, we conjecture that

scholarly journals Thermo-Mechanical Fatigue of Grey Cast Iron for Cylinder Heads - Effect of Niobium, Molybdenum and Solidification Time

2018 ◽  
Vol 925 ◽  
pp. 377-384 ◽  
Author(s):  
Peter Skoglund ◽  
Jessica Elfsberg ◽  
Nulifer Ipek ◽  
Lucian Vasile Diaconu ◽  
Mari Larsson ◽  
...  
Keyword(s):  
Mechanical Performance ◽  
Cyclic Stress ◽  
The Other ◽  
Solidification Time ◽  
Alloying Elements ◽  
Static Properties ◽  
Mechanical Fatigue ◽  
Other Hand ◽  
The Matrix ◽  
Grey Cast

Grey iron alloyed with molybdenum and niobium in seven different compositions has been casted using three, in industrial components viable, solidification times which resulted in 21 different samples. The samples have been investigated with respect to microstructure, static properties and thermo-mechanical fatigue performance. It was found that the solidification time is very important for both the static and thermo-mechanical performance. If the solidification time is long the properties are controlled entirely by the large graphite flakes and there is no influence of the alloying elements. On the other hand if the solidification time can be kept short the need for alloying elements may be removed. For the shorter solidification times an influence from the matrix and thus the alloying elements can be seen. It was found that molybdenum enhances TMF-life while no such effect was found for niobium. Niobium, on the other hand, has a larger effect on static strength than molybdenum and also on the cyclic stress in the thermo-mechanical fatigue experiments.

scholarly journals A supramolecular proposal of lignin structure and its relation with the wood properties

2009 ◽  
Vol 81 (1) ◽  
pp. 137-142 ◽  
Author(s):  
Heber S. Abreu ◽  
João V.F. Latorraca ◽  
Regina P.W. Pereira ◽  
Maria Beatriz O. Monteiro ◽  
Fábio A. Abreu ◽  
...  
Keyword(s):  
Wood Properties ◽  
The Other ◽  
Virtual Model ◽  
Structure Form ◽  
Wood Technology ◽  
Model Based ◽  
Other Hand ◽  
Science Discussion ◽  
The Matrix ◽  
Matrix Compound

In spite of the great importance of cellulose the lignin is considered the second most abundant substance of the wood. However, little attention has been given it, mainly to wood properties. The lignin as well as other structural compounds (cellulose and hemicelluloses), has obviously an important role on the wood properties, probably due its composition and existent bonds. In general lignins have β-O-4 (Alkyl Aril Ether) as majoritary bond. This bond in a continued structure form big molecules with spiral conformation as virtual model. Based on this idea, lignins that have high/low β-O-4 content may have differentiated spiraled structures,suggesting different behaviors on the wood properties,which shows that the lignins (Guaicyl:Syringyl (GS)) of angiosperms, for example, which have higher β-O-4 content would present higher spiral conformation than gymnosperms lignins(HG). On the other hand HG lignins have chance of being more anchored on the matrix compound than GS lignins. In this context, the β-O-4 bonds of lignins possibly affect the wood properties, therefore, it is considered relevant for wood technology science discussion.

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