The stabilizer for..-qubit symmetric states

The stabilizer group for an n-qubit state ｜Φ〉 is the set of all invertible local operators（ILO） g = g1g2…gn,gi 2 GL（2,C） such that ｜Φ〉= g｜Φ〉. Recently, Gour et al. [Gour G, Kraus B and Wallach N R 2017 J. Math. Phys. 58092204] presented that almost all n-qubit states jyi own a trivial stabilizer group when n≥5. In this article, we consider the case when the stabilizer group of an n-qubit symmetric pure state jyi is trivial. First we show that the stabilizer group for an n-qubit symmetric pure state ｜Φ〉 is nontrivial when n≤4. Then we present a class of n-qubit symmetric states ｜Ψ〉 with a trivial stabilizer group when n≥5. Finally, we propose a conjecture and prove that an n-qubit symmetric pure state owns a trivial stabilizer group when its diversity number is bigger than 5 under the conjecture we make, which confirms the main result of Gour et al. partly.

Stability and stabilization of discrete T-S fuzzy time-delay system based on maximal overlapped-rules group

The problems of stability and stabilization for the discrete Takagi-Sugeno（T-S） fuzzy time-delay system are investigated.By constructing a discrete piecewise Lyapunov-Krasovskii function（PLKF） in each maximal overlapped-rules group（MORG）,a new sufficient stability condition for the open-loop discrete T-S fuzzy time-delay system is proposed and proved.Then the systematic design of the fuzzy controller is investigated via the parallel distributed compensation control scheme,and a new stabilization condition for the closed-loop discrete T-S fuzzy time-delay system is proposed.The above two sufficient conditions only require finding common matrices in each MORG.Compared with the common Lyapunov-Krasovskii function（CLKF） approach and the fuzzy Lyapunov-Krasovskii function（FLKF） approach,these proposed sufficient conditions can not only overcome the defect of finding common matrices in the whole feasible region but also largely reduce the number of linear matrix inequalities to be solved.Finally,simulation examples show that the proposed PLKF approach is effective.

Comparison in phytoplankton diversity-productivity-community stability between river-type reservoir and lake-type reservoir

Relationship between biodiversity and ecosystem function is one of the core issues in ecological research.Phytoplankton,as the main producer of aquatic ecosystem,its diversity,productivity,and community stability are of great signifi cance to reveal ecosystem function.There are signifi cant diff erences in hydrodynamics,water retention time,and phytoplankton community structure between river-type reservoir and newly built lake-type reservoir.The comparative analysis of phytoplankton community stability between the two types of reservoir has not been reported.Jiuquwan Reservoir(river-type)and Taihu Reservoir(lake-type),the two reservoirs in the Dongjiang River source area of Zhujiang(Pearl)River Basin,were selected for comparison in terms of multi-year operation vs.new impoundment,river-type vs.lake-type,and shallow water vs.sub-deep water reservoirs.Samples were collected in dry season(December 2019),normal season(March 2020),and wet season(August 2020),on which the phytoplankton diversity and productivity of the two reservoirs,and the relationship and diff erence of community stability were examined.Results show that(1)the number of phytoplankton species in Jiuquwan Reservoir decreased comparing that before algal bloom and the restoration treatment,while that in Taihu Reservoir increased compared with that before the impoundment of the reservoir.There was no signifi cant diff erence in functional groups and species number between the two reservoirs(P>0.05);(2)the biological stability,diversity,productivity,and resource utilization effi ciency of newly built lake-type reservoir were higher than those of multi-year river-type reservoir.In addition,the utilization effi ciency of phytoplankton resources was the highest in wet season in both reservoirs.The increases in biodiversity,richness,and evenness promoted the stability of the community,while increases in productivity and resource utilization effi ciency weakened the stability of the community;(3)community stability was aff ected by both biotic and abiotic factors,and hydrodynamic index was the main factor.This study is helpful to understand the relationship and diff erences in phytoplankton diversity,productivity and community stability in diff erent types of reservoirs,and provides a guidance for maintaining the stability of reservoir water ecosystem and protecting the biodiversity.The relationships between phytoplankton diversity,productivity,and community stability will be investigated in depth,for which a long-term observation will be conducted on the impact of environmental factors and diversity on the local biostability in diff erent types of reservoirs.

Syntheses, Structure and Properties of a Strontium(Ⅱ) Complex with 5-Sulfoisophthalic Acid Monosodium

A new Sr（Ⅱ） complex, [Sr（Hsip）（H2O）] （1, NaH2sip = 5-sulfoisophthalic acid monosodium）, has been synthesized by solvothermal reaction of SrCO3 and NaH2sip at 120 ℃, and characterized by single-crystal X-ray diffraction studies, elemental analysis and FT-IR. Single- crystal X-ray diffraction reveals that compound 1 has a 3D architecture, and there is only one crystallographically independent Sr（Ⅱ） ion in 1. The coordination geometry of Sr（Ⅱ） is a distorted tetragonai anti-prism. The whole Hsip2- ligand performs aμ6-coordination model. In the solid state, complex 1 shows luminescence with the maximum emission intensity at 417 nm upon excitation at 320 nm. Thermal stability of complex 1 was also investigated.

Local symmetries of finite type hypersurfaces in C^2 Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

The first part of this paper gives a complete description of local automorphism groups for Levi degenerate hypersurfaces of finite type in C2. It is also proved that, with the exception of hypersurfaces of the form v = |z|k, local automorphisms are always determined by their 1-jets. Using this result, the second part describes special normal forms which by an additional normalization eliminate the nonlinear symmetries of the model and allows to decide effectively about local equivalence of two hypersurfaces given in this normal form.