X charts with estimated control limits are commonly used in practice and treated as if the in-control process parameters were known. However, the former can behave quite differently from the latter. To understand the ...X charts with estimated control limits are commonly used in practice and treated as if the in-control process parameters were known. However, the former can behave quite differently from the latter. To understand the differences, it is necessary to study the run length distribution (RLD), its mean (ARL) and standard deviation (SDRL) of the X charts when the control limits are estimated. However, ARL and SDRL are integrals over an infinite region with a boundless integrand, the finiteness has not been proved in literature. In this paper, we show the finiteness and uniform integrability of ARL and SDRL. Furthermore, we numerically evaluate the ARL, SDRL and the RLD using number theory method. A numerical study is conducted to assess the performance of the proposed method and the results are compared with those given by Quesenberry and Chen.展开更多
At the energy regimes close to Planck scales, the usual structure of Lorentz symmetry fails to address certain fundamental issues and eventually breaks down, thus paving the way for an alternative road map. It is thus...At the energy regimes close to Planck scales, the usual structure of Lorentz symmetry fails to address certain fundamental issues and eventually breaks down, thus paving the way for an alternative road map. It is thus argued that some subgroup of proper Lorentz group could stand consistent and might possibly help us to circumvent this problem.It is this subgroup that goes by the name of Very Special Relativity(VSR). Apart from violating rotational symmetry,VSR is believed to preserve the very tenets of special relativity. The gaugeon formalism due to type-I Yokoyama and type-II Izawa are found to be invariant under BRST symmetry. In this paper, we analyze the scope of this invariance in the scheme of VSR. Furthermore, we will obtain VSR modified Lagrangian density using path integral derivation. We will explore the consistency of VSR with regard to these theories.展开更多
基金This research is is partially supported by the National Natural Science Foundation of China.
文摘X charts with estimated control limits are commonly used in practice and treated as if the in-control process parameters were known. However, the former can behave quite differently from the latter. To understand the differences, it is necessary to study the run length distribution (RLD), its mean (ARL) and standard deviation (SDRL) of the X charts when the control limits are estimated. However, ARL and SDRL are integrals over an infinite region with a boundless integrand, the finiteness has not been proved in literature. In this paper, we show the finiteness and uniform integrability of ARL and SDRL. Furthermore, we numerically evaluate the ARL, SDRL and the RLD using number theory method. A numerical study is conducted to assess the performance of the proposed method and the results are compared with those given by Quesenberry and Chen.
文摘At the energy regimes close to Planck scales, the usual structure of Lorentz symmetry fails to address certain fundamental issues and eventually breaks down, thus paving the way for an alternative road map. It is thus argued that some subgroup of proper Lorentz group could stand consistent and might possibly help us to circumvent this problem.It is this subgroup that goes by the name of Very Special Relativity(VSR). Apart from violating rotational symmetry,VSR is believed to preserve the very tenets of special relativity. The gaugeon formalism due to type-I Yokoyama and type-II Izawa are found to be invariant under BRST symmetry. In this paper, we analyze the scope of this invariance in the scheme of VSR. Furthermore, we will obtain VSR modified Lagrangian density using path integral derivation. We will explore the consistency of VSR with regard to these theories.
