NEW FIXED POINT THEOREMS OF CONDENSING MAPPINGS SATISFYING THE INTERIOR CONDITION IN BANACH SPACES

In this paper, based on a basic result on condensing mappings satisfying the interior condition, some new fixed point theorems of the condensing mappings of this kind are obtained. As a result, the famous Altman＇s theorem, Roth＇s theorem and Petryshyn＇s theorem are extended to condensing mappings satisfying the interior condition.

Structure and Evolution of Decadal Spiciness Variability in the North Pacific during 2004-20,Revealed from Argo Observations

Ocean spiciness is referred to as density-compensated temperature and salinity variations with warm(cool)and salty(fresh)waters having high(low)spiciness,respectively.The structure and evolution of density-compensated(warm/salty or cool/fresh)spiciness anomalies are investigated in the North Pacific thermocline using Argo observations during the period 2004-20.Two well-organized decadal spiciness events are identified through isopycnal surface analyses.One warm/salty spiciness anomaly of about 0.15°C and 0.05 g kg^(−1)temperature and salinity perturbations on the 25 kg m^(−1)isopycnal surface appeared in the eastern subtropical North Pacific at(18°-30°N,120°-150°W)in 2007,which then migrated southwestward along the mean circulation and arrived in the western tropics at(~15°N,145°E-175°W)in 2012-13,with the reduced salinity perturbation of about 0.043 g kg^(−1).Another cool/fresh spiciness anomaly of about−0.2°C and−0.07 g kg^(−1)temperature and salinity perturbations originated from the eastern subtropics at(120°-150°W,~30°N)in 2014 and followed a similar advective pathway during the period from 2014-15 to 2019-20.The subduction pathway can be adequately determined by the mean Montgomery stream function on the isopycnal surface;the propagation direction and speed are in good agreement with the expectation for the role played by advection due to the mean geostrophic current.Moreover,the subducted decadal spiciness anomalies can have negative feedback on sea surface temperature(SST)variability in the western tropical Pacific through the diapycnal processes.The identifications of these density-compensated spiciness anomalies and their propagation pathways provide a clear illustration of the oceanic extratropics-tropics interactions in the North Pacific Ocean.

FIXED POINTS FOR WEAKLY INWARD MAPPINGS IN BANACH SPACES (Ⅱ)

S. Hu and Y. Sun[1] defined the fixed point index for weakly inward mappings, investigated its properties and studied fixed points for such mappings. In this paper, following S. Hu and Y. Sun, we further investigate boundary conditions, under which the fixed point index for i（A, Ω, p） is equal to nonzero, where i（A, Ω, p） is the completely continuous and weakly inward mapping. Correspondingly, we can obtain many new fixed point theorems of the completely continuous and weakly inward mapping, which generalize some famous theorems such as Rothe＇s theorem, Altman＇s theorem, Petryshyn＇s theorem etc. in the case of weakly inward mappings. In addition, our conclusions extend the famous fixed point theorem of cone expansion and compression to the case of weakly inward mappings. Moreover, the main results contain and generalize the corresponding results in the recent work[2].

Even-odd-dependent optical transitions of zigzag monolayer black phosphorus nanoribbons

We analytically study the electronic structure and optical properties of zigzag-edged phosphorene nanoribbons(ZPNRs) using the tight-binding Hamiltonian and Kubo formula. By directly solving the discrete Schrodinger equation, we obtain the energy spectra and wavefunctions for N-ZPNR(where N is the number of transverse zigzag atomic chains) and classify the eigenstates according to the lattice symmetry. Then, we obtain the optical transition selection rule of ZPNRs on the basis of symmetry analysis and analytical expressions of optical transition matrix elements. Under incident light that is linearly polarized along the ribbon, we determine that the optical transition selection rule for N-ZPNR with even-or odd-N is qualitatively different. Specifically, for even-N ZPNRs, the inter-(intra-) band selection rule is ?n =odd(even) because the parity of the wavefunction corresponding to the n-th subband in the conduction(valence) band is(-1)~n[(-1)~((n+1))] owing to the presence of C(2x) symmetry. However, the optical transitions between any subbands are possible owing to the absence of C(2x) symmetry. Our results provide a further understanding on the electronic states and optical properties of ZPNRs, which are useful for explaining the optical experiment data on ZPNR samples.