Remarks on a paper by Chao-Ping Chen and Feng Qi

In a recent paper, Chao-Ping Chen and Feng Qi (2005) established sharp upper and lower bounds for the sequence . We show that their result follows easily from a theorem of G. N Watson published in 1959. We also show that the main result of Chen and Qi's paper is a special case of a more general inequality which admits a very short proof.

     

Sub- and Superadditive Properties of Fejer's sine Polynomial

Let be Fejér's sine polynomial. We prove the following statements.

1. The inequality holds for all x, y (0, ) with x + y < if and only if 0 and + rß 1.
2. The converse of the above inequality is valid for allx, y (0, ) with x + y < if and only if 0 and + rß 1.
3. For all n N and x, y [0, ] we have . Both bounds are best possible.

2000 Mathematics Subject Classification 42A05, 26D05 (primary),39B62 (secondary).     

Carbon Nanostructures Derived through Hypergolic Reaction of Conductive Polymers with Fuming Nitric Acid at Ambient Conditions

Hypergolic systems rely on organic fuel and a powerful oxidizer that spontaneously ignites upon contact without any external ignition source. Although their main utilization pertains to rocket fuels and propellants, it is only recently that hypergolics has been established from our group as a new general method for the synthesis of different morphologies of carbon nanostructures depending on the hypergolic pair (organic fuel-oxidizer). In search of new pairs, the hypergolic mixture described here contains polyaniline as the organic source of carbon and fuming nitric acid as strong oxidizer. Specifically, the two reagents react rapidly and spontaneously upon contact at ambient conditions to afford carbon nanosheets. Further liquid-phase exfoliation of the nanosheets in dimethylformamide results in dispersed single layers exhibiting strong Tyndall effect. The method can be extended to other conductive polymers, such as polythiophene and polypyrrole, leading to the formation of different type carbon nanostructures (e.g., photolumincent carbon dots). Apart from being a new synthesis pathway towards carbon nanomaterials and a new type of reaction for conductive polymers, the present hypergolic pairs also provide a novel set of rocket bipropellants based on conductive polymers.     

Spherically invariant processes: Their nonlinear structure,discrimination, and estimation

The structure of the nonlinear space of a spherically invariant process is studied and the problem of discriminating between two spherically invariant processes as well as the problem of nonlinear estimation for spherically invariant processes are solved.     

On the Partial Sums of a Fourier Series

We give sharp lower estimates for the partial sums of the Fourier series

Remarks on some completely monotonic functions

Applying the Euler-Maclaurin summation formula, we obtain upper and lower polynomial bounds for the function , x>0, with coefficients the Bernoulli numbers Bk. This enables us to give simpler proofs of some results of H. Alzer and F. Qi et al., concerning complete monotonicity of certain functions involving the gamma function Γ(x), the psi function ψ(x) and the polygamma functions ψ(n)(x), n=1,2,… .     

Inequalities of Fejér-Jackson Type

 We complement, extend, and sharpen some known inequalities for sine sums. Our main result is the following refinement of the classical Fejér-Jackson inequality: For all integers n ⩾ 2 and real numbers x ∈ (0,π) we have

Stable processes: moving averages versus fourier transforms

Summary We study the relationship between Fourier transforms and moving averages of stable processes. The two classes have a large intersection, however as soon as the noise is required to be bounded they become disjoint.Research supported by the Air Force Office of Scientific Research Contract No. F49620 85C 0144 and by the Office of Naval Research Grant No. N00014 86C 0227This work was done in part when the author was visiting the Center for Computational Statistics, George Mason University, Fairfax, VA 22030-4444