An extension of Asgeirsson's mean value theorem for solutions of the ultra-hyperbolic equation in dimension four

On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis

Mathematics ◽

10.3390/math9111303 ◽

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.

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On a hybrid version of the Vinogradov mean value theorem

Acta Mathematica Hungarica ◽

10.1007/s10474-020-01111-9 ◽

2021 ◽

Vol 163 (1) ◽

pp. 1-17

Author(s):

C. Chen ◽

I. E. Shparlinski

Keyword(s):

Mean Value ◽

Mean Value Theorem

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Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

Mathematische Zeitschrift ◽

10.1007/s00209-021-02695-w ◽

2021 ◽

Author(s):

Tim Browning ◽

Shuntaro Yamagishi

Keyword(s):

Circle Method ◽

Rational Points ◽

Mean Value ◽

Mean Value Theorem ◽

Waring’S Problem ◽

Waring Problem ◽

Waring's Problem ◽

Main Conjecture ◽

Higher Dimensional ◽

Thin Set

AbstractWe study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.

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A Simple Auxiliary Function for the Mean Value Theorem

College Mathematics Journal ◽

10.1080/07468342.1989.11973252 ◽

1989 ◽

Vol 20 (4) ◽

pp. 323-323

Author(s):

Herb Silverman

Keyword(s):

Auxiliary Function ◽

Mean Value ◽

Mean Value Theorem ◽

The Mean

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A Simple Auxiliary Function for the Mean Value Theorem

College Mathematics Journal ◽

10.2307/2686855 ◽

1989 ◽

Vol 20 (4) ◽

pp. 323 ◽

Cited By ~ 1

Author(s):

Herb Silverman

Keyword(s):

Auxiliary Function ◽

Mean Value ◽

Mean Value Theorem ◽

The Mean

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

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171202006877 ◽

2002 ◽

Vol 32 (1) ◽

pp. 47-55 ◽

Cited By ~ 3

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.

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