The abcabc-problem for Gabor systems

A Gabor system generated by a window function $\phi$ and a rectangular lattice $a \Z\times \Z/b$ is given by $${\mathcal G}(\phi, a \Z\times \Z/b):=\{e^{-2\pi i n t/b} \phi(t- m a):\ (m, n)\in \Z\times \Z\}.$$ One of fundamental problems in Gabor analysis is to identify window functions $\phi$ and time-frequency shift lattices $a \Z\times \Z/b$ such that the corresponding Gabor system ${\mathcal G}(\phi, a \Z\times \Z/b)$ is a Gabor frame for $L^2(\R)$, the space of all square-integrable functions on the real line $\R$. In this paper, we provide a full classification of triples $(a,b,c)$ for which the Gabor system ${\mathcal G}(\chi_I, a \Z\times \Z/b)$ generated by the ideal window function $\chi_I$ on an interval $I$ of length $c$ is a Gabor frame for $L^2(\R)$. For the classification of such triples $(a, b, c)$ (i.e., the $abc$-problem for Gabor systems), we introduce maximal invariant sets of some piecewise linear transformations and establish the equivalence between Gabor frame property and triviality of maximal invariant sets. We then study dynamic system associated with the piecewise linear transformations and explore various properties of their maximal invariant sets. By performing holes-removal surgery for maximal invariant sets to shrink and augmentation operation for a line with marks to expand, we finally parameterize those triples $(a, b, c)$ for which maximal invariant sets are trivial. The novel techniques involving non-ergodicity of dynamical systems associated with some novel non-contractive and non-measure-preserving transformations lead to our arduous answer to the $abc$-problem for Gabor systems

Spectral measures with arbitrary Hausdorff dimensions

In this paper, we consider spectral properties of Riesz product measures supported on homogeneous Cantor sets and we show the existence of spectral measures with arbitrary Hausdorff dimensions, including non-atomic zero-dimensional spectral measures and one-dimensional singular spectral measures

Investigation of the Effects of Baffles and Nozzles on the Flow Characteristics in a Small Scrubber for Marine Engines

A new version of “Rules for the Prevention of Air Pollution Caused by Ships” published by International Maritime Organization (IMO) on January 1, 2015. The regulation stipulated that the allowable emissions of SOx from ships in the Global Emission Control Area (ECA) reduced from 1% to 0.1%, and the limitation on sulfur oxides in fuel oil used on board ships operating outside designated emission control areas will be reduced to 0.5%. Many technologies have been developed to meet the sulfur oxides limits by shipowners. Many types of devices are installed to reduce emission and to increase power efficiency, such as turbocharger, economizer, SCR, heat recovery system, catalytic converter, and scrubber etc. A turbocharger is a type of turbine-driven forced induction installation used to increase the power and efficiency of an internal combustion engine by intaking more air into the combustion chamber. SCR is a method for converting NOx into N2 and H2O in the presence of a catalyst. An exhaust heat recovery system is a technology that translates thermal losses in the exhaust pipeline into energy to save fuel and reduce CO2 emissions. Economizers are mechanical installations intended to reduce energy consumption or to preheat a fluid. A catalytic converter is a type of exhaust emission control device, and it is designed to convert toxic gases and pollutants into less toxic pollutants from the exhaust gas of an internal combustion engine by catalyzing a redox reaction. Installing a marine exhaust gas scrubber is the best measure with the lowest cost and higher efficiency to convert NOX into N2. Through a series of chemical and physical reactions, scrubber can clean up SOX and particulates produced by diesel engines. A lot of researches have been put into different types of scrubbers. As one of the best measure, scrubber technologies have been developed and commercialized. In this research, it was discussed that the influence of two different types of scrubber structures on pressure drop and flow streamlines. Pressure drop is a target related to the efficiency and power of diesel engine, and the flow streamlines is a target related to the duration for which the exhaust gas and cleaning water are in contact. It was also discussed the influence of the nozzle location and quantity on the cleaning water distribution. The length and number of horizontal and vertical baffles and the number and location of nozzles were calculated and analyzed. The optimal inner structure was found out after fluid analysis of the scrubber for small marine engine, which shows that an inner structure with a baffle length ratio of 0.6 with two baffles was considered optimal in the case of a horizontal scrubber, and a baffle length ratio of 0.7 with two baffles was considered optimal in the case of a vertical scrubber. The result shows that the horizontal location is better than the vertical location, and the case of the six nozzles has the best performance in analyzing the influence of the nozzle location and quantity.1. Introduction 1 1.1 Background 1 1.2 Performance of exhaust gas scrubber 4 1.3 Prospect of scrubber market 6 2. Investigation of the performance of baffles in a small scrubber 8 2.1 Calculation methods and conditions 8 2.1.1 Structure of scrubber 8 2.1.2 Calculation conditions 14 2.2 Analysis of scrubber with horizontal baffles 15 2.2.1 Analyzing baffle's length 15 2.2.2 Analyzing baffle's number 18 2.3 The optimal structure 22 2.4 Analysis of scrubber with vertical baffles 27 2.4.1 Analyzing baffle's length 27 2.4.2 Analyzing baffle's number 30 3. Investigation on the performance of sprays in a small scrubber 34 3.1 Mathematical models and conditions 34 3.1.1 Structure of scrubbers 34 3.1.2 Boundary conditions 38 3.2 Simulation and calculation results 39 3.2.1 Calculation conditions 39 3.2.2 Analyzing positions of nozzles 41 3.2.3 Analyzing numbers of nozzles 55 3.2.4 The optimal structure 65 4. Conclusion 77 Acknowledgment 79 References 81Maste