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On the average of the sum-of-a-divisors function

Shi-Chao Chen; Yong-Gao Chen — 2004

Colloquium Mathematicae

We prove an Ω result on the average of the sum of the divisors of n which are relatively coprime to any given integer a. This generalizes the earlier result for a prime proved by Adhikari, Coppola and Mukhopadhyay.

Smoothness for the collision local time of two multidimensional bifractional Brownian motions

Guangjun Shen; Litan Yan; Chao Chen — 2012

Czechoslovak Mathematical Journal

Let $B^{H_{i}, K_{i}} = {B_{t}^{H_{i}, K_{i}}, t \geq 0}$ , $i = 1, 2$ be two independent, $d$ -dimensional bifractional Brownian motions with respective indices $H_{i} \in (0, 1)$ and $K_{i} \in (0, 1]$ . Assume $d \geq 2$ . One of the main motivations of this paper is to investigate smoothness of the collision local time $ℓ_{T} = \int_{0}^{T} δ (B_{s}^{H_{1}, K_{1}} - B_{s}^{H_{2}, K_{2}}) d s, T > 0,$ where $δ$ denotes the Dirac delta function. By an elementary method we show that $ℓ_{T}$ is smooth in the sense of Meyer-Watanabe if and only if $min {H_{1} K_{1}, H_{2} K_{2}} < 1 / (d + 2)$ .

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Currently displaying 1 – 17 of 17

On the average of the sum-of-a-divisors function

Smoothness for the collision local time of two multidimensional bifractional Brownian motions

Some existence and bifurcation results for solutions of nonlinear Sturm-Liouville eigenvalue problems.

Proof of the best bounds in Wallis' inequality.

Monotonicity and convexity for the gamma function.

Maslov index for homoclinic orbits of hamiltonian systems

Variational and penalization methods for studying connecting orbits of Hamiltonian systems.

Some properties of Palais-Smale sequences with applications to elliptic boundary-value problems.

Existence and multiplicity results for homoclinic orbits of Hamiltonian systems.

Inequalities for the gamma function.

Notes on double inequalities of Mathieu's series.

Dynamical analysis of DTNN with impulsive effect.

Analysis of crack propagation path on the anisotropic bi-material rock.

On Shafer and Carlson inequalities.

Generalizations of the Ky Fan inequality.

Note on weighted Carleman-type inequality.

A logarithmically completely monotonic function involving the Gamma functions.