An Arithmetic Function and the k-th Power Part of a Positive Integer

2012 ◽  
Vol 490-495 ◽  
pp. 1941-1944
Author(s):  
Ming Jun Wang
Keyword(s):  
Positive Integer ◽  
Arithmetic Function ◽  
Mean Value ◽  
Asymptotic Formulas ◽  
Elementary Method ◽  
Large Exponent ◽  
Power Part ◽  
The Mean

For any positive integer denotes the largest - power less than or equal to ,and denotes the smallest - power greater than or equal to . Let be a prime, denotes the large exponent of power which divides .In this paper we use elementary method to study the mean value properties of and ,and give two interesting asymptotic formulas.

Mean value of an arithmetic function associated with the Piltz divisor function

2018 ◽  
Vol 13 (03) ◽  
pp. 2050062
Author(s):  
Meselem Karras ◽  
Abdallah Derbal
Keyword(s):  
Multiplicative Function ◽  
Arithmetic Function ◽  
Mean Value ◽  
Divisor Function ◽  
Fixed Integer ◽  
Elementary Methods ◽  
The Mean

Let [Formula: see text] be a fixed integer, we define the multiplicative function [Formula: see text], where [Formula: see text] is the Piltz divisor function and [Formula: see text] is the unitary analogue function of [Formula: see text]. The main purpose of this paper to use elementary methods to study the mean value of the function [Formula: see text].

A Galton–Watson process with a threshold

2016 ◽  
Vol 53 (2) ◽  
pp. 614-621
Author(s):  
K. B. Athreya ◽  
H.-J. Schuh
Keyword(s):  
Positive Integer ◽  
Population Size ◽  
Discrete Time ◽  
Branching Process ◽  
Branching Processes ◽  
Mean Value ◽  
Size Dependent ◽  
The Mean ◽  
Watson Process ◽  
Critical Branching Process

Abstract In this paper we study a special class of size dependent branching processes. We assume that for some positive integer K as long as the population size does not exceed level K, the process evolves as a discrete-time supercritical branching process, and when the population size exceeds level K, it evolves as a subcritical or critical branching process. It is shown that this process does die out in finite time T. The question of when the mean value E(T) is finite or infinite is also addressed.

On the Mean Value of a New Arithmetical Function

2012 ◽  
Vol 155-156 ◽  
pp. 396-400
Author(s):  
Ming Jun Wang
Keyword(s):  
Asymptotic Properties ◽  
General Term ◽  
Mean Value ◽  
Arithmetical Function ◽  
Natural Sequence ◽  
Elementary Method ◽  
Hybrid Functions ◽  
Asymptotic Formulae ◽  
The Mean ◽  
Number Theorist

To study one of the problems that Romania number theorist F. Smarandache has proposed and to generalize it. For any positive integer let denotes the natural sequence where each number is repeated times. Based on the general term formula, the asymptotic properties of this sequence and some hybrid functions are studied using the elementary method, the asymptotic formulae are obtained ,thus enriching the study and application of this sequence.

scholarly journals On the average of the eccentricities of a graph

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1395-1401 ◽  
Author(s):  
Kinkar Das ◽  
Kexiang Xu ◽  
Xia Li ◽  
Haiqiong Liu
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Mean Value ◽  
Extremal Graphs ◽  
The Mean ◽  
Average Eccentricity ◽  
General Graphs ◽  
Simple Connected Graph

Let G = (V; E) be a simple connected graph of order n with m edges. Also let eG(vi) be the eccentricity of a vertex vi in G. We can assume that eG(v1) eG(v2) ? ... ? eG(vn-1) ? eG(vn). The average eccentricity of a graph G is the mean value of eccentricities of vertices of G, avec(G) = 1/n ?n,i=1 eG(vi). Let ? = ?G be the largest positive integer such that eG(vG ) ? avec(G). In this paper, we study the value of G of a graph G. For any tree T of order n, we prove that 2 ? ?T ? n - 1 and we characterize the extremal graphs. Moreover, we prove that for any graph G of order n,2 ? ?G ? n and we characterize the extremal graphs. Finally some Nordhaus-Gaddum type results are obtained on ?G of general graphs G.

An Arithmetical Function and the Divisor Product Sequences

2012 ◽  
Vol 524-527 ◽  
pp. 3834-3837
Author(s):  
Ming Shun Yang ◽  
Ya Juan Tu
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Mean Value ◽  
Arithmetical Function ◽  
Elementary Methods ◽  
The Mean

Let n be any positive integer, Pd(n)denotes the produce of all positive divisors of n. Let p be a prime, ap(n)denotes the largest exponent (of power p) such that divisible by n. In this paper, we shall use the elementary methods to study the mean value properties of ap(Pd(n)), and give an interesting asymptotic formula for it.

scholarly journals On restricted divisor sums associated to the Chowla–Walum conjecture

2021 ◽  
pp. 1-7
Author(s):  
Olivier Bordellès
Keyword(s):  
Mean Value ◽  
Asymptotic Formulas ◽  
Recent Estimate ◽  
The Mean ◽  
Divisor Sums

We first study the mean value of certain restricted divisor sums involving the Chowla–Walum sums, improving in particular a recent estimate given by Iannucci. The aim of the second part of this work is the generalization of the previous study, by restricting the range of the divisors in the studied divisor sums, extending the Chowla–Walum conjecture, proving a small part of this extended conjecture and generalizing the asymptotic formulas previously obtained in the first part.

On the Mean Value of Euler Function in a Special Set

2014 ◽  
Vol 536-537 ◽  
pp. 907-910
Author(s):  
Lan Qi
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Mean Value ◽  
Euler Function ◽  
The Mean

Let n be a positive integer and k≥2, bk(n) denotes the k-th power complement fuction, we define a new set A.This paper is mainly to study the mean value properties of the Euler function in set A,and give an interesting asymptotic formula.

Long-period modulated superstructures in Pd-12.5 at.%Ce and Pd-15 at.%Ce alloys

1990 ◽  
Vol 48 (4) ◽  
pp. 164-165
Author(s):  
Noriyuki Kuwano ◽  
Masaru Itakura ◽  
Kensuke Oki
Keyword(s):  
Electron Microscopy ◽  
Mean Value ◽  
Heavy Elements ◽  
Arc Furnace ◽  
X Ray ◽  
Long Period ◽  
Ultra High Purity ◽  
Heat Treated ◽  
Atomic Scattering ◽  
The Mean

Pd-Ce alloys exhibit various anomalies in physical properties due to mixed valences of Ce, and the anomalies are thought to be strongly related with the crystal structures. Since Pd and Ce are both heavy elements, relative magnitudes of (fcc-fpd) are so small compared with <f> that superlattice reflections, even if any, sometimes cannot be detected in conventional x-ray powder patterns, where fee and fpd are atomic scattering factors of Ce and Pd, and <f> the mean value in the crystal. However, superlattices in Pd-Ce alloys can be analyzed by electron microscopy, thanks to the high detectability of electron diffraction. In this work, we investigated modulated superstructures in alloys with 12.5 and 15.0 at.%Ce.Ingots of Pd-Ce alloys were prepared in an arc furnace under atmosphere of ultra high purity argon. The disc specimens cut out from the ingots were heat-treated in vacuum and electrothinned to electron transparency by a jet method.

Partition Coefficient Ratios and Tumour Perfusion Studied with 85mKr and 133Xe

1987 ◽  
Vol 26 (06) ◽  
pp. 253-257
Author(s):  
M. Mäntylä ◽  
J. Perkkiö ◽  
J. Heikkonen
Keyword(s):  
Partition Coefficient ◽  
Partition Coefficients ◽  
Regional Blood Flow ◽  
Blood Perfusion ◽  
Compartmental Analysis ◽  
Malignant Tumour ◽  
Mean Value ◽  
Perfusion Rate ◽  
Biological Variables ◽  
The Mean

The relative partition coefficients of krypton and xenon, and the regional blood flow in 27 superficial malignant tumour nodules in 22 patients with diagnosed tumours were measured using the 85mKr- and 133Xe-clearance method. In order to minimize the effect of biological variables on the measurements the radionuclides were injected simultaneously into the tumour. The distribution of the radiotracers was assumed to be in equilibrium at the beginning of the experiment. The blood perfusion was calculated by fitting a two-exponential function to the measuring points. The mean value of the perfusion rate calculated from the xenon results was 13 ± 10 ml/(100 g-min) [range 3 to 38 ml/(100 g-min)] and from the krypton results 19 ± 11 ml/(100 g-min) [range 5 to 45 ml/(100 g-min)]. These values were obtained, if the partition coefficients are equal to one. The equations obtained by using compartmental analysis were used for the calculation of the relative partition coefficient of krypton and xenon. The partition coefficient of krypton was found to be slightly smaller than that of xenon, which may be due to its smaller molecular weight.

Sign in / Sign up

Export Citation Format

Share Document