Injectivity of sections of convex harmonic mappings and convolution theorems

In the article the authors consider the class ${\mathcal H}_0$ of sense-preserving harmonic functions $f=h+\overline{g}$ defined in the unit disk $|z|<1$ and normalized so that $h(0)=0=h'(0)-1$ and $g(0)=0=g'(0)$, where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes $\mathcal{P}_H^0(\alpha)$ and $\mathcal{G}_H^0(\beta)$ of functions from ${\mathcal H}_0$ and show that if $f\in \mathcal{P}_H^0(\alpha)$ and $F\in\mathcal{G}_H^0(\beta)$, then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters $\alpha$ and $\beta$ are satisfied. In the second part we study the harmonic sections (partial sums) $$s_{n, n}(f)(z)=s_n(h)(z)+\overline{s_n(g)(z)},$$ where $f=h+\overline{g}\in {\mathcal H}_0$, $s_n(h)$ and $s_n(g)$ denote the $n$-th partial sums of $h$ and $g$, respectively. We prove, among others, that if $f=h+\overline{g}\in{\mathcal H}_0$ is a univalent harmonic convex mapping, then $s_{n, n}(f)$ is univalent and close-to-convex in the disk $|z|< 1/4$ for $n\geq 2$, and $s_{n, n}(f)$ is also convex in the disk $|z|< 1/4$ for $n\geq2$ and $n\neq 3$. Moreover, we show that the section $s_{3,3}(f)$ of $f\in {\mathcal C}_H^0$ is not convex in the disk $|z|<1/4$ but is shown to be convex in a smaller disk.Comment: 16 pages, 3 figures; To appear in Czechoslovak Mathematical Journa

Jitter-aware time-frequency resource allocation and packing algorithm

One of the main components of the next generation protected military satellite communication systems is dynamic bandwidth resource allocation (DBRA). A centralized DBRA algorithm on the satellite dynamically grants terminals time and frequency resources as their traffic demands and channel conditions change, leading to significant increase in the overall system throughput. This paper address one potential issue associated with DBRA for the satellite uplink channel, which is a multi-frequency time-division multiple access (MF-TDMA) channel. As DBRA dynamically assigns time-frequency slots to terminals, there may be uneven temporal gaps in the assignment if special care were not taken. When this happens, even though average rate demands can be met, applications may experience larger than desired delay and jitter, which may reduce the quality of certain voice and video applications. This paper presents a novel algorithm for allocating and packing time-frequency slots in a jitter-aware fashion by using groups of evenly spaced slots. The achievable delay and jitter performance is evaluated using an OPNET simulation.United States. Dept. of Defense (Air Force Contract FA8721-05-C-0002

United States Bureau of Mines Bulletin 173

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