Remarks on the operator-norm convergence of the Trotter product formula

We revise the operator-norm convergence of the Trotter product formula for a pair {A,B} of generators of semigroups on a Banach space. Operator-norm convergence holds true if the dominating operator A generates a holomorphic contraction semigroup and B is a A-infinitesimally small generator of a contraction semigroup, in particular, if B is a bounded operator. Inspired by studies of evolution semigroups it is shown in the present paper that the operator-norm convergence generally fails even for bounded operators B if A is not a holomorphic generator. Moreover, it is shown that operator norm convergence of the Trotter product formula can be arbitrary slow.Comment: 12 page

Convergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems

In the present paper we advocate the Howland-Evans approach to solution of the abstract non-autonomous Cauchy problem (non-ACP) in a separable Banach space X. The main idea is to reformulate this problem as an autonomous Cauchy problem (ACP) in a new Banach space Lp(I;X), p 2 [1;1), consisting of X-valued functions on the time-interval I. The fundamental observation is a one-to-one correspondence between solution operators (propagators) for a non-ACP and the corresponding evolution semigroups for ACP in Lp(I;X). We show that the latter also allows to apply a full power of the operatortheoretical methods to scrutinise the non-ACP including the proof of the Trotter product approximation formulae with operator-norm estimate of the rate of convergence. The paper extends and improves some recent results in this direction in particular for Hilbert spaces

Convergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems

In the present paper we advocate the Howland-Evans approach to solution of the abstract non-autonomous Cauchy problem (non-ACP) in a separable Banach space X. The main idea is to reformulate this problem as an autonomous Cauchy problem (ACP) in a new Banach space Lp(J,X), consisting of X-valued functions on the time-interval J. The fundamental observation is a one-to-one correspondence between solution operators (propagators) for a non-ACP and the corresponding evolution semigroups for ACP in Lp(J,X). We show that the latter also allows to apply a full power of the operator-theoretical methods to scrutinise the non-ACP including the proof of the Trotter product approximation formulae with operator-norm estimate of the rate of convergence. The paper extends and improves some recent results in this direction in particular for Hilbert spaces

An Energy-Efficient Power Allocation Game with Selfish Channel State Reporting in Cellular Networks

International audienceEnergy-efficient ressource allocation is a powerful approach to reduce the operation costs and environmental footprint of cellular networks. With energy-efficient resource allocation, mobile users and base station have different objectives. While the base station strives for an energy-efficient operation of the complete cell, each user aims to maximize its own data rate. To obtain this individual benefit, users may selfishly adjust their \ac{CSI} reports, reducing the cell's energy efficiency. To analyze this conflict of interest, we formalize energy-efficient power allocation as a utility maximization problem and present a simple algorithm that performs close to the optimum. By formulating selfish CSI reporting as a game, we prove the existence of an unique equilibrium and characterize energy efficiency with true and selfish CSI in closed form. Our numerical results show that, surprisingly, energy-efficient power allocation in small cells is more robust against selfish CSI than cells with large transmit powers. This and further design rules show that our paper provides valuable theoretical insight to energy-efficient networks when CSI reports cannot be trusted

Prevalence and characteristics of methicillin-resistant coagulase-negative staphylococci from livestock, chicken carcasses, bulk tank milk, minced meat, and contact persons

BACKGROUND: Methicillin-resistant coagulase-negative staphylococci (MR-CNS) are of increasing importance to animal and public health. In veterinary medicine and along the meat and milk production line, only limited data were so far available on MR-CNS characteristics. The aim of the present study was to evaluate the prevalence of MR-CNS, to identify the detected staphylococci to species level, and to assess the antibiotic resistance profiles of isolated MR-CNS strains. RESULTS: After two-step enrichment and growth on chromogenic agar, MR-CNS were detected in 48.2% of samples from livestock and chicken carcasses, 46.4% of samples from bulk tank milk and minced meat, and 49.3% of human samples. Using matrix-assisted laser desorption ionization-time of flight mass spectrometry (MALDI-TOF MS), 414 selected MR-CNS strains belonged to seven different species (S. sciuri, 32.6%; S. fleurettii, 25.1%; S. haemolyticus, 17.4%; S. epidermidis, 14.5%, S. lentus, 9.2%; S. warneri, 0.7%; S. cohnii, 0.5%). S. sciuri and S. fleurettii thereby predominated in livestock, BTM and minced meat samples, whereas S. epidermidis and S. haemolyticus predominated in human samples. In addition to beta-lactam resistance, 33-49% of all 414 strains were resistant to certain non-beta-lactam antibiotics (ciproflaxacin, clindamycin, erythromycin, tetracycline). CONCLUSIONS: A high prevalence of MR-CNS was found in livestock production. This is of concern in view of potential spread of mecA to S. aureus (MRSA). Multiresistant CNS strains might become an emerging problem for veterinary medicine. For species identification of MR-CNS isolated from different origins, MALDI-TOF MS proved to be a fast and reliable tool and is suitable for screening of large sample amounts