A NOTE ON THE ERDŐS–GRAHAM THEOREM

2018 ◽  
Vol 97 (3) ◽  
pp. 363-366
Author(s):  
WENHUI WANG ◽  
MIN TANG
Keyword(s):  
Large Integer ◽  
Consecutive Integers ◽  
The Difference ◽  
Asymptotic Basis

Let ${\mathcal{A}}=\{a_{1}<a_{2}<\cdots \,\}$ be a set of nonnegative integers. Put $D({\mathcal{A}})=\gcd \{a_{k+1}-a_{k}:k=1,2,\ldots \}$. The set ${\mathcal{A}}$ is an asymptotic basis if there exists $h$ such that every sufficiently large integer is a sum of at most $h$ (not necessarily distinct) elements of ${\mathcal{A}}$. We prove that if the difference of consecutive integers of ${\mathcal{A}}$ is bounded, then ${\mathcal{A}}$ is an asymptotic basis if and only if there exists an integer $a\in {\mathcal{A}}$ such that $(a,D({\mathcal{A}}))=1$.

scholarly journals On the difference of values of the kernel function at consecutive integers

2003 ◽  
Vol 2003 (67) ◽  
pp. 4249-4262
Author(s):  
Jean-Marie De Koninck ◽  
Florian Luca
Keyword(s):  
Positive Integer ◽  
Kernel Function ◽  
Large Family ◽  
Fixed Integer ◽  
Consecutive Integers ◽  
The Difference

For each positive integern, setγ(n)=Πp|np. Given a fixed integerk≠±1, we establish that if theABC-conjecture holds, then the equationγ(n+1)−γ(n)=khas only finitely many solutions. In the particular casesk=±1, we provide a large family of solutions for each of the corresponding equations.

A construction of maximal asymptotic nonbases

2018 ◽  
Vol 14 (04) ◽  
pp. 919-923 ◽  
Author(s):  
Dengrong Ling
Keyword(s):  
Large Integer ◽  
Asymptotic Basis

Let [Formula: see text] denote the set of all nonnegative integers and [Formula: see text] be a subset of [Formula: see text]. The set [Formula: see text] is called an asymptotic basis of order [Formula: see text] if every sufficiently large integer can be written as the sum of two elements of [Formula: see text]. Otherwise, [Formula: see text] is called an asymptotic nonbasis of order [Formula: see text]. Let [Formula: see text] denote the number of representations of [Formula: see text] in the form [Formula: see text], where [Formula: see text] and [Formula: see text]. An asymptotic nonbasis [Formula: see text] of order [Formula: see text] is called a maximal asymptotic nonbasis of order [Formula: see text] if [Formula: see text] is an asymptotic basis of order [Formula: see text] for every [Formula: see text]. In this paper, a maximal asymptotic nonbasis [Formula: see text] is constructed satisfying [Formula: see text] for all [Formula: see text] and [Formula: see text] as [Formula: see text], where [Formula: see text] is an increasing sequence of [Formula: see text].

A Short Note on the Difference Between Inverses of Consecutive Integers Modulo p

Integers ◽  
2009 ◽  
Vol 9 (6) ◽  
Author(s):  
Tsz Ho Chan
Keyword(s):  
Short Note ◽  
Modulo P ◽  
Consecutive Integers ◽  
The Difference

AbstractIn a previous paper, the author studied the distribution of differences between the multiplicative inverses of consecutive integers modulo

scholarly journals Periodic Solutions of Certain Differential Equations with Piecewise Constant Argument

2015 ◽  
Vol 2015 ◽  
pp. 1-17
Author(s):  
James Guyker
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Difference Equations ◽  
Sufficient Conditions ◽  
Piecewise Constant ◽  
Piecewise Constant Argument ◽  
Consecutive Integers ◽  
The Difference ◽  
Existence Criteria ◽  
Constant Argument

Existence criteria are derived for the eventually periodic solutions of a class of differential equations with piecewise constant argument whose solutions at consecutive integers satisfy nonlinear recurrence relations. The proof characterizes the initial values of periodic solutions in terms of the coefficients of the resulting difference equations. Sufficient conditions for the unboundedness, boundedness, and symmetry of general solutions also follow from the corresponding properties of the difference equations.

The Origin of the Moon

1962 ◽  
Vol 14 ◽  
pp. 149-155 ◽  
Author(s):  
E. L. Ruskol
Keyword(s):  
Chemical Composition ◽  
Solar System ◽  
The Moon ◽  
The Earth ◽  
The Difference ◽  
Origin Of The Moon

The difference between average densities of the Moon and Earth was interpreted in the preceding report by Professor H. Urey as indicating a difference in their chemical composition. Therefore, Urey assumes the Moon's formation to have taken place far away from the Earth, under conditions differing substantially from the conditions of Earth's formation. In such a case, the Earth should have captured the Moon. As is admitted by Professor Urey himself, such a capture is a very improbable event. In addition, an assumption that the “lunar” dimensions were representative of protoplanetary bodies in the entire solar system encounters great difficulties.

Influence of Cell Wall Composition on the Fossilisation of Bacteria and the Implications for the Search for Early Life Forms

1997 ◽  
Vol 161 ◽  
pp. 491-504 ◽  
Author(s):  
Frances Westall
Keyword(s):  
Cell Wall ◽  
Life Forms ◽  
Cation Binding ◽  
Positive Bacterium ◽  
Gram Positive ◽  
Gram Positive Bacteria ◽  
Transmission Electron ◽  
Hydrothermal Sediments ◽  
Identification Of Bacteria ◽  
The Difference

AbstractThe oldest cell-like structures on Earth are preserved in silicified lagoonal, shallow sea or hydrothermal sediments, such as some Archean formations in Western Australia and South Africa. Previous studies concentrated on the search for organic fossils in Archean rocks. Observations of silicified bacteria (as silica minerals) are scarce for both the Precambrian and the Phanerozoic, but reports of mineral bacteria finds, in general, are increasing. The problems associated with the identification of authentic fossil bacteria and, if possible, closer identification of bacteria type can, in part, be overcome by experimental fossilisation studies. These have shown that not all bacteria fossilise in the same way and, indeed, some seem to be very resistent to fossilisation. This paper deals with a transmission electron microscope investigation of the silicification of four species of bacteria commonly found in the environment. The Gram positiveBacillus laterosporusand its spore produced a robust, durable crust upon silicification, whereas the Gram negativePseudomonas fluorescens, Ps. vesicularis, andPs. acidovoranspresented delicately preserved walls. The greater amount of peptidoglycan, containing abundant metal cation binding sites, in the cell wall of the Gram positive bacterium, probably accounts for the difference in the mode of fossilisation. The Gram positive bacteria are, therefore, probably most likely to be preserved in the terrestrial and extraterrestrial rock record.

Structuring, Motions and Shapes of Corona Emission Line Profiles

1994 ◽  
Vol 144 ◽  
pp. 421-426
Author(s):  
N. F. Tyagun
Keyword(s):  
Emission Line ◽  
Filling Factor ◽  
Direct Relationship ◽  
Coronal Plasma ◽  
Line Of Sight ◽  
Line Profiles ◽  
The Difference ◽  
Yellow Line ◽  
Red Line

AbstractThe interrelationship of half-widths and intensities for the red, green and yellow lines is considered. This is a direct relationship for the green and yellow line and an inverse one for the red line. The difference in the relationships of half-widths and intensities for different lines appears to be due to substantially dissimilar structuring and to a set of line-of-sight motions in ”hot“ and ”cold“ corona regions.When diagnosing the coronal plasma, one cannot neglect the filling factor - each line has such a factor of its own.

Difference Fourier Analysis of Glucose Embedded and Frozen Hydrated Purple Membrane

1982 ◽  
Vol 40 ◽  
pp. 74-75
Author(s):  
Jules S. Jaffe ◽  
Robert M. Glaeser
Keyword(s):  
Purple Membrane ◽  
Data Set ◽  
High Resolution Data ◽  
X Ray ◽  
X Ray Crystallography ◽  
Fourier Techniques ◽  
Versus Protein ◽  
The Difference ◽  
Difference Fourier ◽  
Ideal Method

Although difference Fourier techniques are standard in X-ray crystallography it has only been very recently that electron crystallographers have been able to take advantage of this method. We have combined a high resolution data set for frozen glucose embedded Purple Membrane (PM) with a data set collected from PM prepared in the frozen hydrated state in order to visualize any differences in structure due to the different methods of preparation. The increased contrast between protein-ice versus protein-glucose may prove to be an advantage of the frozen hydrated technique for visualizing those parts of bacteriorhodopsin that are embedded in glucose. In addition, surface groups of the protein may be disordered in glucose and ordered in the frozen state. The sensitivity of the difference Fourier technique to small changes in structure provides an ideal method for testing this hypothesis.

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