scholarly journals Asymptotics for multilinear averages of multiplicative functions

2016 ◽  
Vol 161 (1) ◽  
pp. 87-101 ◽  
Author(s):  
NIKOS FRANTZIKINAKIS ◽  
BERNARD HOST
Keyword(s):  
Unit Disc ◽  
Convergence Result ◽  
Mean Value ◽  
Arithmetic Mean ◽  
Mean Value Theorem ◽  
Two Dimensional ◽  
Multiplicative Functions ◽  
Structural Result ◽  
Convergence Results ◽  
The Mean

AbstractA celebrated result of Halász describes the asymptotic behavior of the arithmetic mean of an arbitrary multiplicative function with values on the unit disc. We extend this result to multilinear averages of multiplicative functions providing similar asymptotics, thus verifying a two dimensional variant of a conjecture of Elliott. As a consequence, we get several convergence results for such multilinear expressions, one of which generalises a well known convergence result of Wirsing. The key ingredients are a recent structural result for multiplicative functions with values on the unit disc proved by the authors and the mean value theorem of Halász.

scholarly journals The Mean Value Theorem for Integrals Method for Estimating Two-Dimensional Renewal Functions

2020 ◽  
Vol 4 (1) ◽  
pp. 49
Author(s):  
Leopoldus Ricky Sasongko ◽  
Bambang Susanto
Keyword(s):  
Weibull Model ◽  
Mean Value ◽  
The Other ◽  
Mean Value Theorem ◽  
Two Dimensional ◽  
Failure Model ◽  
Expected Number ◽  
Warranty Period ◽  
Bivariate Data ◽  
The Mean

An important aspect in the provision of a two-dimensional warranty is the expected number of failures of a component during the two-dimensional warranty period. The purpose of this paper is to present a new method to obtain the expected number of failures of a nonrepairable compo­nent from the two-dimensional renewal functions as the so­lution of two-dimensional renewal integral equations through the Mean Value Theorem for Integrals (MeVTI) method. The two-dimensional renewal integral equation involves Lu-Bhattacharyya’s bivariate Weibull model as a two-dimensional failure model. It turns out that the estimation of the expected number of failures using the MeVTI method is close to that of the other method, Riemann-Stieljies method. The bivariate data behaviour of the failures of an automobile component is also studied in this paper.

scholarly journals On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1303
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Faraidun Kadir Hamasalh
Keyword(s):  
Time Scale ◽  
Initial Value Problems ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Discrete Fractional Calculus ◽  
Initial Value ◽  
Forward Difference ◽  
The Mean ◽  
Monotonicity Analysis ◽  
Monotonicity Results

Monotonicity analysis of delta fractional sums and differences of order υ∈(0,1] on the time scale hZ are presented in this study. For this analysis, two models of discrete fractional calculus, Riemann–Liouville and Caputo, are considered. There is a relationship between the delta Riemann–Liouville fractional h-difference and delta Caputo fractional h-differences, which we find in this study. Therefore, after we solve one, we can apply the same method to the other one due to their correlation. We show that y(z) is υ-increasing on Ma+υh,h, where the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and then, we can show that y(z) is υ-increasing on Ma+υh,h, where the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) for each z∈Ma+h,h. Conversely, if y(a+υh) is greater or equal to zero and y(z) is increasing on Ma+υh,h, we show that the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) on Ma,h. Furthermore, we consider some related results for strictly increasing, decreasing, and strictly decreasing cases. Finally, the fractional forward difference initial value problems and their solutions are investigated to test the mean value theorem on the time scale hZ utilizing the monotonicity results.

scholarly journals On a sum analogous to Dedekind sum and its mean square value formula

2002 ◽  
Vol 32 (1) ◽  
pp. 47-55 ◽  
Author(s):  
Zhang Wenpeng
Keyword(s):  
Asymptotic Property ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Mean Square ◽  
Dedekind Sum ◽  
The Mean

The main purpose of this paper is using the mean value theorem of DirichletL-functions to study the asymptotic property of a sum analogous to Dedekind sum, and give an interesting mean square value formula.

The mean-value theorem for elliptic operators on stratified sets

2007 ◽  
Vol 81 (3-4) ◽  
pp. 365-372
Author(s):  
S. N. Oshchepkova ◽  
O. M. Penkin
Keyword(s):  
Mean Value ◽  
Elliptic Operators ◽  
Mean Value Theorem ◽  
The Mean
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