On a Problem of P. Erdös

1972 ◽  
Vol 15 (2) ◽  
pp. 309-310 ◽  
Author(s):  
I. Ruzsa
Keyword(s):  
Absolute Constant ◽  
Infinite Sequence ◽  
Affirmative Answer ◽  
Powers Of 2 ◽  
Image Position

P. Erdös asked the following problem: Does there exist an infinite sequence of integers a1<…satisfying for every x≥11so that every integer is of the form 2k+ai [1]. The analogous questions can easily be answered affirmatively if the powers of 2 are replaced by the rth power.In this note we give a simple affirmative answer to the problem of Erdôs. Let c2 be a sufficiently small absolute constant. Our sequence A consists of all the integers of the form2

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.

Arcs with increasing chords

1972 ◽  
Vol 72 (2) ◽  
pp. 205-207 ◽  
Author(s):  
D. G. Larman ◽  
P. McMullen
Keyword(s):  
Absolute Constant ◽  
Complex Analysis ◽  
Private Communication ◽  
Arc Length ◽  
Euclidean Length ◽  
Jordan Arc ◽  
Image Position

Let f:[0, 1]→R2 be a Jordan arc, and for t, u ∈ [0, 1] let d(t, u) = d(f(t), f(u)) denote the Euclidean length of the chord between f(t) and f(u), and l(t, u) = l(f(t), f(u)) the corresponding arc-length, when this is defined. We say that f has the increasing chord property if d(t2, t3) ≤ d(t1, t4) whenever 0 ≤ t1 ≤ t2 ≤ t3 ≤ t4 ≤ 1. In connexion with a problem in complex analysis, K. Binmore has asked (private communication, see (1)) whether there exists an absolute constant K such that.

On the Riemann zeta-function

1932 ◽  
Vol 28 (3) ◽  
pp. 273-274 ◽  
Author(s):  
E. C. Titchmarsh
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Absolute Constant ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Image Position

It was proved by Littlewood that, for every large positive T, ζ (s) has a zero β + iγ satisfyingwhere A is an absolute constant.

Elements of Order Coxeter Number +1 in Chevalley Groups

1982 ◽  
Vol 34 (4) ◽  
pp. 945-951 ◽  
Author(s):  
Bomshik Chang
Keyword(s):  
Weyl Group ◽  
Chevalley Group ◽  
Trivial Solution ◽  
Affirmative Answer ◽  
Chevalley Groups ◽  
General Solutions ◽  
Coxeter Number ◽  
Image Position

Following the notation and the definitions in [1], let L(K) be the Chevalley group of type L over a field K, W the Weyl group of L and h the Coxeter number, i.e., the order of Coxeter elements of W. In a letter to the author, John McKay asked the following question: If h + 1 is a prime, is there an element of order h + 1 in L(C)? In this note we give an affirmative answer to this question by constructing an element of order h + 1 (prime or otherwise) in the subgroup Lz = 〈xτ(1)|r ∈ Φ〉 of L(K), for any K.Our problem has an immediate solution when L = An. In this case h = n + 1 and the (n + l) × (n + l) matrixhas order 2(h + 1) in SLn+1(K). This seemingly trivial solution turns out to be a prototype of general solutions in the following sense.

Perturbations of AF-Algebras

1979 ◽  
Vol 31 (5) ◽  
pp. 1012-1016 ◽  
Author(s):  
John Phillips ◽  
Iain Raeburn
Keyword(s):  
Hilbert Space ◽  
Unit Ball ◽  
Von Neumann Algebra ◽  
Unitary Operator ◽  
Affirmative Answer ◽  
Positive Answer ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Image Position ◽  
Af Algebras

Let A and B be C*-algebras acting on a Hilbert space H, and letwhere A1 is the unit ball in A and d(a, B1) denotes the distance of a from B1. We shall consider the following problem: if ‖A – B‖ is sufficiently small, does it follow that there is a unitary operator u such that uAu* = B?Such questions were first considered by Kadison and Kastler in [9], and have received considerable attention. In particular in the case where A is an approximately finite-dimensional (or hyperfinite) von Neumann algebra, the question has an affirmative answer (cf [3], [8], [12]). We shall show that in the case where A and B are approximately finite-dimensional C*-algebras (AF-algebras) the problem also has a positive answer.

On Functional Properties of Incomplete Gaussian Sums

1991 ◽  
Vol 43 (1) ◽  
pp. 182-212 ◽  
Author(s):  
K. I. Oskolkov
Keyword(s):  
Functional Properties ◽  
Positive Constant ◽  
Absolute Constant ◽  
Special Function ◽  
Absolute Positive Constant ◽  
Image Position ◽  
Gaussian Sums

AbstractThe following special function of two real variables x2 and x1 is considered: and its connections with the incomplete Gaussian sums where ω are intervals of length |ω| ≤1. In particular, it is proved that for each fixed x2 and uniformly in X2 the function H(x2, x1) is of weakly bounded 2-variation in the variable x1 over the period [0, 1]. In terms of the sums W this means that for collections Ω = {ωk}, consisting of nonoverlapping intervals ωk ∪ [0,1) the following estimate is valid: where card denotes the number of elements, and c is an absolute positive constant. The exact value of the best absolute constant к in the estimate (which is due to G. H. Hardy and J. E. Littlewood) is discussed.

Small Solutions of the Congruence ax2 + by2 ≡ c(mod k)

1969 ◽  
Vol 12 (3) ◽  
pp. 311-320 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Asymptotic Formula ◽  
Absolute Constant ◽  
Log P ◽  
Positive Absolute Constant ◽  
Image Position

In 1957, Mordell [3] provedTheorem. If p is an odd prime there exist non-negative integers x, y ≤ A p3/4 log p, where A is a positive absolute constant, such that(1.1)provided (abc, p) = 1.Recently Smith [5] has obtained a sharp asymptotic formula for the sum where r(n) denotes the number of representations of n as the sum of two squares.

scholarly journals Almost Isoperimetric Subsets of the Discrete Cube

2011 ◽  
Vol 20 (3) ◽  
pp. 363-380 ◽  
Author(s):  
DAVID ELLIS
Keyword(s):  
Absolute Constant ◽  
Image Position

We show that a set A ⊂ {0, 1}n with edge-boundary of size at most can be made into a subcube by at most (2ε/log2(1/ε))|A| additions and deletions, provided ε is less than an absolute constant.We deduce that if A ⊂ {0, 1}n has size 2t for some t ∈ ℕ, and cannot be made into a subcube by fewer than δ|A| additions and deletions, then its edge-boundary has size at least provided δ is less than an absolute constant. This is sharp whenever δ = 1/2j for some j ∈ {1, 2, . . ., t}.

The sets of Dirichlet non-improvable numbers versus well-approximable numbers

2019 ◽  
Vol 40 (12) ◽  
pp. 3217-3235 ◽  
Author(s):  
AYREENA BAKHTAWAR ◽  
PHILIP BOS ◽  
MUMTAZ HUSSAIN
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Affirmative Answer ◽  
Partial Quotient ◽  
Dimensional Hausdorff Measure ◽  
Image Position

Let $\unicode[STIX]{x1D6F9}:[1,\infty )\rightarrow \mathbb{R}_{+}$ be a non-decreasing function, $a_{n}(x)$ the $n$th partial quotient of $x$ and $q_{n}(x)$ the denominator of the $n$th convergent. The set of $\unicode[STIX]{x1D6F9}$-Dirichlet non-improvable numbers, $$\begin{eqnarray}G(\unicode[STIX]{x1D6F9}):=\{x\in [0,1):a_{n}(x)a_{n+1}(x)>\unicode[STIX]{x1D6F9}(q_{n}(x))\text{ for infinitely many }n\in \mathbb{N}\},\end{eqnarray}$$ is related with the classical set of $1/q^{2}\unicode[STIX]{x1D6F9}(q)$-approximable numbers ${\mathcal{K}}(\unicode[STIX]{x1D6F9})$ in the sense that ${\mathcal{K}}(3\unicode[STIX]{x1D6F9})\subset G(\unicode[STIX]{x1D6F9})$. Both of these sets enjoy the same $s$-dimensional Hausdorff measure criterion for $s\in (0,1)$. We prove that the set $G(\unicode[STIX]{x1D6F9})\setminus {\mathcal{K}}(3\unicode[STIX]{x1D6F9})$ is uncountable by proving that its Hausdorff dimension is the same as that for the sets ${\mathcal{K}}(\unicode[STIX]{x1D6F9})$ and $G(\unicode[STIX]{x1D6F9})$. This gives an affirmative answer to a question raised by Hussain et al [Hausdorff measure of sets of Dirichlet non-improvable numbers. Mathematika 64(2) (2018), 502–518].

Additive functions of intervals and Hausdorff measure

1946 ◽  
Vol 42 (1) ◽  
pp. 15-23 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Hausdorff Measure ◽  
Convex Sets ◽  
Infinite Sequence ◽  
Small Positive Number ◽  
Additive Functions ◽  
Image Position ◽  
Bounded Sets ◽  
Increasing Function

Consider bounded sets of points in a Euclidean space Rq of q dimensions. Let h(t) be a continuous increasing function, positive for t>0, and such that h(0) = 0. Then the Hausdroff measure h–mE of a set E in Rq, relative to the function h(t), is defined as follows. Let ε be a small positive number and suppose E is covered by a finite or enumerably infinite sequence of convex sets {Ui} (open or closed) of diameters di less than or equal to ε. Write h–mεE = greatest lower bound for any such sequence {Ui}. Then h–mεE is non-decreasing as ε tends to zero. We define

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