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.

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.

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

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}.

On the Radius of Curvature for Convex Analytic Functions

1970 ◽  
Vol 22 (3) ◽  
pp. 486-491 ◽  
Author(s):  
Paul Eenigenburg
Keyword(s):  
Real Axis ◽  
Jordan Curve ◽  
Convex Functions ◽  
Analytic Functions ◽  
Tangent Line ◽  
Positive Real Axis ◽  
Radius Of Curvature ◽  
Arc Length ◽  
Positive Real ◽  
Image Position

Definition 1.1. Let be analytic for |z| < 1. If ƒ is univalent, we say that ƒ belongs to the class S.Definition 1.2. Let ƒ ∈ S, 0 ≦ α < 1. Then ƒ belongs to the class of convex functions of order α, denoted by Kα, provided(1)and if > 0 is given, there exists Z0, |Z0| < 1, such thatLet ƒ ∈ Kα and consider the Jordan curve ϒτ = ƒ(|z| = r), 0 < r < 1. Let s(r, θ) measure the arc length along ϒτ; and let ϕ(r, θ) measure the angle (in the anti-clockwise sense) that the tangent line to ϒτ at ƒ(reiθ) makes with the positive real axis.

Length and area inequalities for the derivative of a bounded and holomorphic function

1984 ◽  
Vol 30 (3) ◽  
pp. 457-462
Author(s):  
Shinji Yamashita
Keyword(s):  
Holomorphic Function ◽  
Euclidean Length ◽  
Length And Area ◽  
Image Position

The Schwarz-Pick lemma,for f analytic and bounded, |f|<1, in the disk |z|<1, is refined:where Φ(z, r) is a quantity determined by the non-Euclidean area of the image ofand ψ(z, r) is that determined by the non-Euclidean length of the image of the boundary of D(z, r). The multiplicities in both images by f are not counted.

On the Growth of Blaschke Products

1969 ◽  
Vol 21 ◽  
pp. 595-601 ◽  
Author(s):  
G. R. MacLane ◽  
L. A. Rubel
Keyword(s):  
Angular Distribution ◽  
Analytic Function ◽  
Blaschke Product ◽  
Bounded Function ◽  
Related Question ◽  
Complex Numbers ◽  
Private Communication ◽  
Blaschke Products ◽  
The Subject ◽  
Image Position

It is well known that the distribution of the zeros of an analytic function affects its rate of growth. The literature is too extensive to indicate here. We only point out (1, p. 27; 2; 3; 5), where the angular distribution of the zeros plays a role, as it will in this paper. In private communication, A. Zygmund has raised the following related question, which is the subject of our investigation here.Let {zn}, n = 1, 2, 3, …, be a sequence of non-zero complex numbers of modulus less than 1, such that ∑(1 – |zn|) < ∞, and consider the Blaschke product1Let2What are the sequences {zn} for which I(r) is a bounded function of r?

scholarly journals On the centre of spherical curvature of a curve

1937 ◽  
Vol 30 ◽  
pp. xi-xii
Author(s):  
C. E. Weatherburn
Keyword(s):  
Position Vector ◽  
Perpendicular Bisector ◽  
Arc Length ◽  
Normal Plane ◽  
Adjacent Point ◽  
Current Point ◽  
Image Position ◽  
Spherical Curvature ◽  
Unit Vectors ◽  
Polar Line

The position of the centre S of spherical curvature at a point P of a given curve C may be found in the following manner, regarding S as the limiting position of the centre of a sphere through four adjacent points P, P1, P2, P3 on the curve, as these points tend to coincidence at P. The centre of a sphere through P and P1 lies on the plane which is the perpendicular bisector of the chord PP1 and so on. Thus the centre of spherical curvature is the limiting position of the intersection of three normal planes at adjacent points. Let s be the arc-length of the curve C, r the position vector of the point P, and t, n, b unit vectors in the directions of the tangent, principal normal and binormal at P. Then if s is the position vector of the current point on the normal plane at P, the equation of this plane isSince r and t are functions of s, the limiting position of the line of intersection of the normal planes at P and an adjacent point (i.e. the polar line) is determined by (1) and the equation obtained by differentiating this with respect to s, viz.

scholarly journals ROTH’S THEOREM FOR FOUR VARIABLES AND ADDITIVE STRUCTURES IN SUMS OF SPARSE SETS

2016 ◽  
Vol 4 ◽  
Author(s):  
TOMASZ SCHOEN ◽  
OLOF SISASK
Keyword(s):  
Absolute Constant ◽  
Related Result ◽  
Arithmetic Progressions ◽  
High Dimensional ◽  
Nontrivial Solutions ◽  
Affine Subspaces ◽  
Image Position ◽  
The Right

We show that if $A\subset \{1,\ldots ,N\}$ does not contain any nontrivial solutions to the equation $x+y+z=3w$, then $$\begin{eqnarray}|A|\leqslant \frac{N}{\exp (c(\log N)^{1/7})},\end{eqnarray}$$ where $c>0$ is some absolute constant. In view of Behrend’s construction, this bound is of the right shape: the exponent $1/7$ cannot be replaced by any constant larger than $1/2$. We also establish a related result, which says that sumsets $A+A+A$ contain long arithmetic progressions if $A\subset \{1,\ldots ,N\}$, or high-dimensional affine subspaces if $A\subset \mathbb{F}_{q}^{n}$, even if $A$ has density of the shape above.

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