Sentence examples for mean value properties from inspiring English sources

The main purpose of this paper is, using the properties of Gauss sums, the estimate for character sums and the analytic method, to study the mean value properties of the two-term Dedekind sums and give an interesting asymptotic formula for (1).

The main purpose of this paper is using the analytic method and the properties of the character sums to study the hybrid mean value properties of G ( χ, n ; p ) and S ( χ, p ), and to give an interesting mean value formula.

Let M ( a, p ) denote the number of all integers 1 ≤ b, c ≤ p − 1 such that b c ≡ a mod p and ( 2, b + c ) = 1, and E ( a, p ) = M ( a, p ) − p − 1 2. Then Zhang [4] also studied the mean value properties of E ( a, p ), and proved that ∑ a = 1 p − 1 E 2 ( a, p ) = 3 4 p 2 + O ( p ⋅ exp ( 3 ln p ln ln p ) ), where exp ( y ) = e y.

For example, Zhang [11] studied the hybrid mean value properties of Cochrane sums and Kloosterman sums and proved that for any prime p > 3, we have the asymptotic formula ∑ h = 1 p − 1 K ( h, 1 ; p ) C ( h, p ) = − 1 2 π 2 p 2 + O ( p ⋅ exp ( 3 ln p ln ln p ) ), where exp ( y ) = e y, K ( m, n ; q ) = ∑ ′ a = 1 q e ( m a + n a ¯ q ).

Conversely, does this mean value property imply thatfis harmonic ?

Another important observation is that, due to the mean value property, means are locally bounded functions.

The two remaining properties (mean value property, elimination principle) are obvious.

First observe that (widetilde{mathscr{M}}) obviously admits the mean value property.

(iii) Mean value property: For all (n in mathbb {N}) and for all ((x,lambda ) in I^{n} times W_{n}(R)) min(x_{1},dots,x_{n}) le mathscr{M} x,lambda)le max(x_{1},dots,x_{n}).

Mean value property: For all (n in mathbb {N}) and for all ((x,lambda ) in I^{n} times W_{n}(R)) min(x_{1},dots,x_{n}) le mathscr{M} x,lambda)le max(x_{1},dots,x_{n}).

The proof of this result is a consequence of a recasting of the classical mean value property for harmonic functions in terms of an identity (5.1) that links the Bochner-artinelli CF form on a sphere with the sphere's Euclidean surface measure.