A task is randomly drawn from a finite set of tasks and is described using a fixed number of bits. All the tasks that share its description must be performed. Upper and lower bounds on the minimum ρ-th moment of the number of performed tasks are derived. The case where a sequence of tasks is produced by a source and n tasks are jointly described using nR bits is considered. If R is larger than the Rényi entropy rate of the source of order 1/(1 + ρ) (provided it exists), then the ρth moment of the ratio of performed tasks to n can be driven to one as n tends to infinity. If R is smaller than the Rényi entropy rate, this moment tends to infinity. The results are generalized to account for the presence of side-information. In this more general setting, the key quantity is a conditional version of Rényi entropy that was introduced by Arimoto. For IID sources two additional extensions are solved, one of a rate-distortion flavor and the other where different tasks may have different nonnegative costs. Finally, a divergence that was identified by Sundaresan as a mismatch penalty in the Massey-Arikan guessing problem is shown to play a similar role here.

Concentrations and bacterioplankton uptake of dissolved free monosaccharides (DFCHO) in relation to bacterioplankton production and phytoplankton biomass were studied in mesotrophic Lake Constance, Germany, from August 1995 to December 1996. Concentrations of total DFCHO, measured by high-performance liquid chromatography (HPLC) and pulsed amperometric detection, ranged from Ͻ20 to 440 nM and exhibited pronounced spatiotemporal variations. Enhanced concentrations of DFCHO were often associated with the breakdown of phytoplankton blooms and periods of low phytoplankton growth. Bacterial uptake rates of DFCHO were also enhanced during such events, as well as at peaks of bacterioplankton production (BP). Ratios of uptake of DFCHO/BP, however, indicated that DFCHO concentrations were relatively more important as substrates during periods of low bacterial growth than during phases of intense bacterial growth. Ratios of bacterial uptake of DFCHO/BP from spring to fall ranged from 0.22 to 0.39 in the layer 0-10 m, from 0.17 to 0.56 at 20 m, and from 0.22 to 0.46 at 50 m, respectively. Glucose, galactose, and mannose dominated the DFCHO pool and constituted ϳ55 to ϳ70 mol%. The analysis of individual turnover times of five DFCHO concentrations from August to November 1995 at 3, 10, and 50 m revealed that the glucose pool turned over most rapidly, followed by galactose and mannose. Glucosamine and fructose always exhibited substantially longer turnover times. Comparisons to previous studies show that DFCHO concentrations are of similar significance for bacterioplankton growth as dissolved free amino acids (DFAA), even though temporal uptake patterns of both substrates are different.The large pool of dissolved organic matter (DOM) in aquatic ecosystems consists of a labile fraction, which is turned over rapidly, i.e., within days or weeks (Münster 1993;Søndergaard and Middelboe 1995;Amon and Benner 1996), and a recalcitrant fraction with a turnover time of years, decades, or longer (Hedges 1992; Münster 1993). The most important components of the labile DOM readily released by phytoplankton and consumed by bacterioplankton are DFAA and combined amino acids (DCAA) and DFCHO and combined monosaccharides (DCCHO;Sundh 1992; Münster 1993;Biddanda and Benner 1997;Simon et al. 1998). Many studies conducted during the last decade have examined the bacterial consumption and turnover of DFAA and DCAA over space and time in various aquatic ecosystems (e.g., Jørgensen 1987;Coffin 1989;Fuhrman 1990; Simon and Rosenstock 1992;Jørgensen et al. 1993;Keil and Kirchman 1993; Rosenstock and Simon 1993;Kroer et al. 1994;Middelboe et al. 1995;Simon 1998). Much less information, however, is available on the turnover and bacterial consumption of dissolved carbohydrates, even though they comprise the largest fraction of the DOM pool so far identified (Benner et al. 1992;Pakulski and Benner 1994) and usually exceed the concentration of DCAA (Münster 1993;Jørgensen et al. 1998). In lacustrine as well as in ma-1 To whom correspondence s...

Abstract-Ahlswede, Cai, and Zhang proved that, in the noisefree limit, the zero-undetected-error capacity is lower-bounded by the Sperner capacity of the channel graph, and they conjectured equality. Here we derive an upper bound that proves the conjecture.

Abstract-The listsize capacity of a discrete memoryless channel is the largest transmission rate for which the expectation-or, more generally, the ρ-th moment-of the number of messages that could have produced the output of the channel approaches one as the blocklength tends to infinity. We show that for channels with feedback this rate is upper-bounded by the maximum of Gallager's E0 function divided by ρ, and that equality holds when the zero-error capacity of the channel is positive. To establish this inequality we prove that feedback does not increase the cutoff rate. Relationships to other notions of channel capacity are explored.

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