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

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.

Minimal Size of Counters for (Real-Time) Multicounter Automata

2021 ◽  
Vol 181 (2-3) ◽  
pp. 99-127
Author(s):  
Viliam Geffert ◽  
Zuzana Bednárová
Keyword(s):  
Finite Number ◽  
Real Time ◽  
The Other ◽  
Binary Representation ◽  
Minimal Size ◽  
Nondeterministic Automaton ◽  
Other Hand ◽  
Weak Space ◽  
Fixed Constant

We show that, for automata using a finite number of counters, the minimal space that is required for accepting a nonregular language is (log n)ɛ. This is required for weak space bounds on the size of their counters, for real-time and one-way, and for nondeterministic and alternating versions of these automata. The same holds for two-way automata, independent of whether they work with strong or weak space bounds, and of whether they are deterministic, nondeterministic, or alternating. (Here ɛ denotes an arbitrarily small—but fixed—constant; the “space” refers to the values stored in the counters, rather than to the lengths of their binary representation.) On the other hand, we show that the minimal space required for accepting a nonregular language is nɛ for multicounter automata with strong space bounds, both for real-time and one-way versions, independent of whether they are deterministic, nondeterministic, or alternating, and also for real-time and one-way deterministic multicounter automata with weak space bounds. All these bounds are optimal both for unary and general nonregular languages. However, for automata equipped with only one counter, it was known that one-way nondeterministic automata cannot recognize any unary nonregular languages at all, even if the size of the counter is not restricted, while, with weak space bound log n, we present a real-time nondeterministic automaton recognizing a binary nonregular language here.

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.

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.

On Bernstein's Inequality

1979 ◽  
Vol 31 (2) ◽  
pp. 347-353 ◽  
Author(s):  
A. Giroux ◽  
Q. I. Rahman ◽  
G. Schmeisser
Keyword(s):  
The Other ◽  
Bernstein’S Inequality ◽  
Other Hand ◽  
Famous Result ◽  
Bernstein's Inequality ◽  
Image Position

1. Introduction and statement of results. If pn(z) is a polynomial of degree at most n, then according to a famous result known as Bernstein's inequality (for references see [4])(1)Here equality holds if and only if pn(z) has all its zeros at the origin and so it is natural to seek for improvements under appropriate assumptions on the zeros of pn(z). Thus, for example, it was conjectured by P. Erdös and later proved by Lax [2] that if pn(z) does not vanish in │z│ < 1, then (1) can be replaced by(2)On the other hand, Turán [5] showed that if pn(z) is a polynomial of degree n having all its zeros in │z│ ≦ 1, then(3)

Imaging of tilted, extended, thin phase-objects

1978 ◽  
Vol 36 (1) ◽  
pp. 216-217
Author(s):  
P. Schiske
Keyword(s):  
Wave Field ◽  
The Other ◽  
Single Exposure ◽  
Coordinate Axis ◽  
Other Hand ◽  
Spatial Frequencies ◽  
Geometrical Data ◽  
Image Position ◽  
Transfer Properties ◽  
Different Parts

In images of extended strongly tilted specimens, different parts of the object are subject to different defocus. This makes it possible to obtain, by a single exposure, an overall view of the transfer properties of the objective [1,2]; on the other hand, in reconstruction work it requires the use of correcting procedures [3]. The importance of both points seems to justify formal investigation--restricted here to thin phase objects. The geometrical data are shown in Fig. 1; the axis of tilt is normal to the plane of the drawing and is used as coordinate axis X2. Representing the scattered amplitude by u(k1,k2) where spatial frequencies k by δ = λk represent the inclination δ of the different components of the wave field a(x1,x2,z), one hasPutting z = x1 one finds for the amplitude immediately beyond the object(1)

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).

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.

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