scholarly journals On a Problem of Erdös and Szekeres

1961 ◽  
Vol 4 (1) ◽  
pp. 7-12 ◽  
Author(s):  
F. V. Atkinson
Keyword(s):  
Lower Bound ◽  
Order Of Magnitude ◽  
Image Position ◽  
Positive Integers

Write12where the maximum is over all real θ, and the lower bound is over all sets of positive integers a1 ≤ a2 ≤ … ≤ an. The problem of the order of magnitude of f(n) was posed by Erdös and Szekeres [1], side by side with a number of other interesting questions. Writing g(n) = log f(n), it is obvious that g(n) is sub-additive, in the sense that g(m+n) ≤ g(m) + g(n), and also that g(1) = log 2, so that g(n) ≤ n log 2.

scholarly journals Monochromatic sequences whose gaps belong to {d, 2d, …, md}

1998 ◽  
Vol 58 (1) ◽  
pp. 93-101 ◽  
Author(s):  
Bruce M. Landman
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Order Of Magnitude ◽  
Positive Integers

For m and k positive integers, define a k-term hm-progression to be a sequence of positive integers {x1,…,xk} such that for some positive integer d, xi + 1 − xi ∈ {d, 2d,…, md} for i = 1,…, k - 1. Let hm(k) denote the least positive integer n such that for every 2-colouring of {1, 2, …, n} there is a monochromatic hm-progression of length k. Thus, h1(k) = w(k), the classical van der Waerden number. We show that, for 1 ≤ r ≤ m, hm(m + r) ≤ 2c(m + r − 1) + 1, where c = ⌈m/(m − r)⌉. We also give a lower bound for hm(k) that has order of magnitude 2k2/m. A precise formula for hm(k) is obtained for all m and k such that k ≤ 3m/2.

On Linear Independence of a Certain Multivariate Infinite Product

2008 ◽  
Vol 51 (1) ◽  
pp. 32-46 ◽  
Author(s):  
Stephen Choi ◽  
Ping Zhou
Keyword(s):  
Lower Bound ◽  
Vector Space ◽  
Linear Independence ◽  
Infinite Product ◽  
Infinite Products ◽  
Image Position ◽  
Positive Integers

AbstractLet q,m,M ≥ 2 be positive integers and r1, r2, … , rm be positive rationals and consider the following M multivariate infinite productsfor i = 0, 1, … ,M –1. In this article, we study the linear independence of these infinite products. In particular, we obtain a lower bound for the dimension of the vector space ℚF0+ℚF1+· · ·+ℚFM–1+ℚ over ℚ and show that among these M infinite products, F0, F1, … , FM–1, at least ∼ M/m(m + 1) of them are irrational for fixed m and M → ∞.

A Lower Bound for the Scholz-Brauer Problem

1969 ◽  
Vol 21 ◽  
pp. 675-683 ◽  
Author(s):  
Kenneth B. Stolarsky
Keyword(s):  
Lower Bound ◽  
Precise Definition ◽  
Addition Chains ◽  
Definition Of ◽  
Image Position ◽  
Positive Integers

In (6)Scholz asked if the inequality1.1held for all positive integers q, where l(n)is the number of multiplications required to raise xto the nth power (a precise definition of l(n)in terms of addition chains is given in § 2). Soon afterwards, Brauer (2) showed, among other things, that l(n) ∼(log n)/(log2). This suggests the problem of Calculating1.2It can be deduced from (2) that θ≦ 1. If θ <1, (1.1) follows immediately for infinitely many q.My main result,Theorem 5 of § 4, merely shows that θ is slightly larger than ⅓.Actually, I know of no case where (1.1) is not in fact an equality; a tedious calculation verifies this for 1 ≦ q≦ 8.

scholarly journals On rational approximations of the exponential function at rational points

1974 ◽  
Vol 10 (3) ◽  
pp. 325-335 ◽  
Author(s):  
Kurt Mahler
Keyword(s):  
Lower Bound ◽  
Exponential Function ◽  
Rational Points ◽  
Rational Approximations ◽  
Image Position ◽  
Positive Integers ◽  
Special Case ◽  
Effective Result

Let p, q, u, and v be any four positive integers, and let further δ be a number in the interval 0 < δ ≤ 2. In this note an effective lower bound for q will be obtained which insures that In the special case when u = v = 1, it was shown by J. Popken, Math. Z. 29 (1929), 525–541, that Here c and C are two positive absolute constants which, however, were not determined explicity. A similarly non-effective result was given in my paper, J. reine angew. Math. 166 (1932), 118–150.The method of this note depends again on the classical formulae by Hermite which I applied also op. cit.

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

Isolated minima of the product of n linear forms

1953 ◽  
Vol 49 (1) ◽  
pp. 59-62 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Linear Forms ◽  
Image Position

Letbe n linear forms with real coefficients and determinant Δ = ∥ aij∥ ≠ 0; and denote by M(X) the lower bound of | X1X2 … Xn| over all integer sets (u) ≠ (0). It is well known that γn, the upper bound of M(X)/|Δ| over all sets of forms Xi, is finite, and the value of γn has been determined when n = 2 and n = 3.

scholarly journals Monochromatic solutions to equations with unit fractions

1991 ◽  
Vol 43 (3) ◽  
pp. 387-392 ◽  
Author(s):  
Tom C. Brown ◽  
Voijtech Rödl
Keyword(s):  
Solutions To Equations ◽  
Unit Fractions ◽  
Image Position ◽  
Positive Integers

Our main result is that if G(x1, …, xn) = 0 is a system of homogeneous equations such that for every partition of the positive integers into finitely many classes there are distinct y1,…, yn in one class such that G(y1, …, yn) = 0, then, for every partition of the positive integers into finitely many classes there are distinct Z1, …, Zn in one class such thatIn particular, we show that if the positive integers are split into r classes, then for every n ≥ 2 there are distinct positive integers x1, x1, …, xn in one class such thatWe also show that if [1, n6 − (n2 − n)2] is partitioned into two classes, then some class contains x0, x1, …, xn such that(Here, x0, x2, …, xn are not necessarily distinct.)

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.

Central Limit Theorems for Interchangeable Processes

1958 ◽  
Vol 10 ◽  
pp. 222-229 ◽  
Author(s):  
J. R. Blum ◽  
H. Chernoff ◽  
M. Rosenblatt ◽  
H. Teicher
Keyword(s):  
Stochastic Process ◽  
Probability Measure ◽  
Central Limit ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Probability Measures ◽  
Image Position ◽  
Positive Integers

Let {Xn} (n = 1, 2 , …) be a stochastic process. The random variables comprising it or the process itself will be said to be interchangeable if, for any choice of distinct positive integers i 1, i 2, H 3 … , ik, the joint distribution of depends merely on k and is independent of the integers i 1, i 2, … , i k. It was shown by De Finetti (3) that the probability measure for any interchangeable process is a mixture of probability measures of processes each consisting of independent and identically distributed random variables.

Minimal Regular Graphs of Girths Eight and Twelve

1966 ◽  
Vol 18 ◽  
pp. 1091-1094 ◽  
Author(s):  
Clark T. Benson
Keyword(s):  
Group Theory ◽  
Lower Bound ◽  
Regular Graph ◽  
Regular Graphs ◽  
Image Position

In (3) Tutte showed that the order of a regular graph of degree d and even girth g > 4 is greater than or equal toHere the girth of a graph is the length of the shortest circuit. It was shown in (2) that this lower bound cannot be attained for regular graphs of degree > 2 for g ≠ 6, 8, or 12. When this lower bound is attained, the graph is called minimal. In a group-theoretic setting a similar situation arose and it was noticed by Gleason that minimal regular graphs of girth 12 could be constructed from certain groups. Here we construct these graphs making only incidental use of group theory. Also we give what is believed to be an easier construction of minimal regular graphs of girth 8 than is given in (2). These results are contained in the following two theorems.

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