We use mean-variance hedging in discrete time in order to value an insurance liability. The prediction of the insurance liability is decomposed into claims development results, that is, yearly deteriorations in its conditional expected values until the liability is finally settled. We assume the existence of a tradeable derivative with binary pay-off written on the claims development result and...

In this paper, we use analytic method to study the mean value properties of Smarandache-Type Multiplicative Functions Km(n), and give its asymptotic formula. Finally, the convolution method is used to improve the error term. Suppose m ≥ 2 is a fixed positive integer. If n = p α1 1 p α2 2 ...p α k k , we define K m (n) = p β1 1 p β2 which is a Smarandache-type multiplicative function. Yang Cundi...

It is proved in this paper that for any point on an elliptic curve, the mean value of x-coordinates of its n-division points is the same as its x-coordinate.

We obtain estimates for Vinogradov’s integral which for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring’s problem holds for sums of s kth powers of natural numbers whenever s > 2k + 2k − 3.

Let f be a function on a bounded domain Ω ⊆ R and δ be a positive function on Ω such that B(x, δ(x)) ⊆ Ω. Let σ(f)(x) be the average of f over the ball B(x, δ(x)). The restricted mean-value theorems discuss the conditions on f, δ, and Ω under which σ(f) = f implies that f is harmonic. In this paper, we study the stability of harmonic functions with respect to the map σ. One expects that, in gen...

For k > 1, r 6= 0 and large x, let π 2r (x) denote the number of prime pairs (p, p+2r) with p ≤ x. By the Bateman–Horn conjecture the function π 2r(x) should be asymptotic to (2/k)C k 2rli2(x), with certain specific constants C 2r . Heuristic arguments lead to the conjecture that these constants have mean value one, just like the Hardy–Littlewood constants C2r for prime pairs (p, p + 2r). The c...

R. Feng, and H. Wu recently established a certain mean-value formula for the x-coordinates of the n-division points on an elliptic curve given in Weierstrass form (A mean value formula for elliptic curves, 2010, available at http://eprint.iacr.org/2009/586.pdf ). We prove a similar result for both the x and y-coordinates on a twisted Edwards elliptic curve.

For simplicity, we use the following convention: X is a non empty set, S is a σ-field of subsets of X, M is a σ-measure on S, f , g are partial functions from X to R, and E is an element of S. One can prove the following three propositions: (1) If for every element x of X such that x ∈ dom f holds f(x) ≤ g(x), then g − f is non-negative. (2) For every set Y and for every partial function f from...

We give conditions on the functions σ and u on R such that if u is given by the convolution of σ and u, then u is harmonic on R.