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We have developed a variant transfer matrix method that is suitable for transport through multiprobe systems. Using this method, we have numerically studied the quantum spin-Hall effect (QSHE) on the 2D graphene with both intrinsic (Vso) and Rashba (Vr) spin-orbit(SO) couplings. The integer QSHE arises in the presence of intrinsic SO interaction and is gradually destroyed by the Rashba SO interaction and disorder fluctuation. We have numerically determined the phase boundaries separating integer QSHE and spin-Hall liquid. We have found that when Vso ≥ 0.2t with t is hopping constant, the energy gap needed for the integer QSHE is the largest satisfying |E| < t. For smaller Vso the energy gap decreases linearly. In the presence of Rashba SO interaction or disorders, the energy gap diminishes. With Rashba SO interaction the integer QSHE is robust at the largest energy within the energy gap while at the smallest energy within the energy gap the integer QSHE is insensitive to the disorders.

In GPS and GLONASS satellite-based positioning, the correct determination of the integer carrier phase ambiguities is the key to achieve precise positioning results. The process of determining the ambiguities, called ambiguity resolution, is usually separated into three steps: (a) estimating the float or real-valued ambiguities, (b) searching the best integer ambiguity set, (c) validating the best ambiguity set. Whilst the theory and methodology for ambiguity estimation and search are well documented, the ambiguity validation issue is still under investigation. Existing ambiguity validation procedures are based on various statistical assumptions and therefore, may have different performances in practical applications. In this paper, the major ambiguity validation procedures are analysed and their performances in both GPS and GLONASS ambiguity resolution are numerically compared. It is concluded that both the probability of estimating ambiguities correctly and confidence levels of the ambiguity discrimination tests are both important indicators of the reliability of ambiguity resolution.

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