scholarly journals A Generalisation of a Result of Abel with an Application to Tree Enumerations

1965 ◽  
Vol 14 (3) ◽  
pp. 239-241 ◽  
Author(s):  
M. M. Robertson
Keyword(s):  
Real Numbers ◽  
Image Position ◽  
Positive Integers

We prove the following theorem, which was established by Abel (1) for the case u = 1.Theorem. If u, k are positive integers and x, 1, , u, are real numbers, thenwhere the sum is taken over all distinct ordered solutions (s1, , su) in non-negative integers of the equation .

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

scholarly journals A Cyclic Inequality and an Extension of it. II

1962 ◽  
Vol 13 (2) ◽  
pp. 143-152 ◽  
Author(s):  
P. H. Diananda
Keyword(s):  
Real Numbers ◽  
Positive Real ◽  
Cyclic Inequality ◽  
Image Position ◽  
Positive Integers

Throughout this paper, unless otherwise stated, n and L stand for positive integers and α, t, x, x1, x2, … for positive real numbers. Letwhereand

The definition of measure in function space

1950 ◽  
Vol 46 (1) ◽  
pp. 19-27
Author(s):  
H. R. Pitt
Keyword(s):  
Function Space ◽  
Real Variable ◽  
Fundamental Result ◽  
Real Numbers ◽  
Real Functions ◽  
Definition Of ◽  
Image Position ◽  
The Right ◽  
Positive Integers

A fundamental result in the theory of measure in the space Ω of real functions x(t) of a real variable t is the following theorem of Kolmogoroff:Theorem 1. Suppose that functions F(t1, …, tn; b1 …, bn) = F(t; b) are defined for positive integers n and real numbers t1, …, tn, b1, …, bn, and have the following properties:(1·1) For every fixedt1, …, tn, F(t; b) has non-negative differenceswith respect to the variables bl, b2,…, bn, and is continuous on the right with respect to each of them;if (i1, …, in) is any permutation of (1, 2, …, n). Then a measure P(X) can be defined in a Borel system of subsets of Ω in such a way that the set of functions satisfyingis measurable for any realbi, tiand has measure F(t; b).

Generalization of the Hausdorff Moment Problem

1981 ◽  
Vol 33 (4) ◽  
pp. 946-960 ◽  
Author(s):  
David Borwein ◽  
Amnon Jakimovski
Keyword(s):  
Moment Problem ◽  
Sufficient Conditions ◽  
Natural Generalization ◽  
Necessary And Sufficient Conditions ◽  
Real Numbers ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Hausdorff Moment Problem ◽  
Positive Integers ◽  
Classical Moment Problem

Suppose throughout that {kn} is a sequence of positive integers, thatthat k0 = 1 if l0 = 1, and that {un(r)}; (r = 0, 1, …, kn – 1, n = 0, 1, …) is a sequence of real numbers. We shall be concerned with the problem of establishing necessary and sufficient conditions for there to be a function a satisfying(1)and certain additional conditions. The case l0 = 0, kn = 1 for n = 0, 1, … of the problem is the version of the classical moment problem considered originally by Hausdorff [5], [6], [7]; the above formulation will emerge as a natural generalization thereof.

On Some Diophantine Inequalities Involving the Exponential Function

1965 ◽  
Vol 17 ◽  
pp. 616-626 ◽  
Author(s):  
A. Baker
Keyword(s):  
Real Number ◽  
Exponential Function ◽  
Diophantine Inequalities ◽  
Real Numbers ◽  
Image Position ◽  
Positive Integers

It is well known that for any real number θ there are infinitely many positive integers n such thatHere ||a|| denotes the distance of a from the nearest integer, taken positively. Indeed, since ||a|| < 1, this implies more generally that if θ1, θ2, . . . , θk are any real numbers, then there are infinitely many positive integers n such that

scholarly journals On a conjecture of Mahler

1977 ◽  
Vol 16 (1) ◽  
pp. 125-129 ◽  
Author(s):  
V.C. Dumir ◽  
R.J. Hans-Gill
Keyword(s):  
Real Numbers ◽  
Image Position ◽  
Positive Integers ◽  
Infinite Set

Let R be the field of real numbers. For α in R, let ‖α‖ be the distance of α from the nearest integer. The following conjecture of Kurt Mahler [Bull. Austral. Math. Soc. 14 (1976), 463–465] is proved.Let m, n be two positive integers n ≥ 2m. Let S be a finite or infinite set of positive integers with the following properties:(Q1) S contains the integers m, m+1, …, n−m;(Q2) every element of S satisfiesThen

The lattice properties of asymmetric hyperbolic regions

1948 ◽  
Vol 44 (1) ◽  
pp. 1-7 ◽  
Author(s):  
J. W. S. Cassels
Keyword(s):  
Infinite Number ◽  
Continued Fractions ◽  
Classical Result ◽  
Real Numbers ◽  
Image Position ◽  
Positive Integers

Let θ > 0 and α ≠ 0 be real numbers, and let θ be irrational. Khintchine has shown, by the use of continued fractions, that there is an infinite number of pairs of positive integers (p, q) which satisfy the inequalityfor any given K > 5−½; and, more recently, Jogin has shown the same is still true with K = 5−½. The condition that p and q shall be positive is, of course, essential, as otherwise there is the classical result K = ¼ due to Minkowski.

scholarly journals A theorem on diophantine approximations

1976 ◽  
Vol 14 (3) ◽  
pp. 463-465 ◽  
Author(s):  
Kurt Mahler
Keyword(s):  
Real Numbers ◽  
Diophantine Approximations ◽  
Image Position ◽  
Positive Integers

If S is a set of positive integers which contains 1, 2, …, n–1, but not n or any multiple of n, where n ≥ 2, thenHere R is the field of real numbers, and ∥α∥ denotes the distance of α from the nearest integer.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

scholarly journals An Inhomogeneous Minimum For Non-Convex Star-Regions With Hexagonal Symmetry

1955 ◽  
Vol 7 ◽  
pp. 337-346 ◽  
Author(s):  
R. P. Bambah ◽  
K. Rogers
Keyword(s):  
Quadratic Form ◽  
Binary Quadratic Form ◽  
Elementary Geometry ◽  
Hexagonal Symmetry ◽  
Real Numbers ◽  
Inhomogeneous Minimum ◽  
Image Position ◽  
Indefinite Binary Quadratic Form

1. Introduction. Several authors have proved theorems of the following type:Let x0, y0 be any real numbers. Then for certain functions f(x, y), there exist numbers x, y such that1.1 x ≡ x0, y ≡ y0 (mod 1),and1.2 .The first result of this type, but with replaced by min , was given by Barnes (3) for the case when the function is an indefinite binary quadratic form. A generalisation of this was proved by elementary geometry by K. Rogers (6).

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