Wavelet Galerkin method for fractional elliptic differential equations

Under the guidance of the general theory developed for classical partial differential equations (PDEs), we investigate the Riesz bases of wavelets in the spaces where fractional PDEs usually work, and their applications in numerically solving fractional elliptic differential equations (FEDEs). The technique issues are solved and the detailed algorithm descriptions are provided. Compared with the ordinary Galerkin methods, the wavelet Galerkin method we propose for FEDEs has the striking benefit of efficiency, since the condition numbers of the corresponding stiffness matrixes are small and uniformly bounded; and the Toeplitz structure of the matrix still can be used to reduce cost. Numerical results and comparison with the ordinary Galerkin methods are presented to demonstrate the advantages of the wavelet Galerkin method we provide.Comment: 20 pages, 0 figure

Excited state dynamics in low-dimensional perovskite nanocrystals

Two-dimensional (2D) perovskites have emerged as promising building blocks for optoelectronic applications. To fabricate high-performance devices, the relation between the material structures and their function in terms of excited state and charge carrier dynamics need to be well understood. In this thesis, we investigate photophysics of 2D lead halide perovskite with different compositions to reveal the relation between lattice distortion and electronic properties. Firstly, we found that the threshold of the Goldschmidt tolerance factor is relaxed and thereby the range of possible composition for forming stable 2D perovskite is extended compared to 3D perovskite. In addition, lattice distortion is greater when containing large cations inside octahedral cages and the formed 2D perovskite has a larger band gap and higher trap state density. The link between lattice distortion and modification of electronic structure shows a potential approach to designing and developing high-performance PV materials. To further evaluate the influence of local lattice distortion on the electronic properties of 2D perovskite, we analyze fluorescence signals from different facets of samples with different spacers. We found that free carriers dominate in the in-plane facet (IF) while self-trapped excitons (STE) are the main emitters from the facet perpendicular to the 2D layer (PF). The strain accumulated along the 2D layers leads to enhanced carrier-phonon coupling and facilitates STE formation in PF, while in IF the separated flexible spacers contribute to releasing the strain accumulation. To directly characterize the electronic structure at different areas of a 2D perovskite single crystal, electrons emitted from Pb 5d and I 4d core levels are mapped at the edge and the bulk areas by using X-ray photoemission electron microscopy. The observed asymmetric shifts of the emission spectra of 2D perovskite indicate different degrees of lattice distortion at the edge and the bulk areas since the internal strain accumulation is released at the edge area. The different shift in Pb 5d core level emission between edge and bulk areas at 2D perovskites with different layer thickness confirms the contribution of spacers in releasing accumulated strain. In addition, we investigated the ultrafast hot carrier (HC) relaxation dynamics in 2D perovskite single crystals by employing transient absorption (TA) spectroscopy and time-resolved two-photon photoemission (TR-2PPE) spectroscopy. With TR-2PPE, the distribution of hot electrons and their dynamics in the conduction band can be directly visualized. The different cooling rates of HC observed in the two techniques reflect the spatial sensitivity of relaxation dynamics across the 2D perovskite single crystal. We believe the comprehensive study on HC relaxation in 2D perovskite provides an effective approach to compare the potential of different materials in hot carrier solar cell (HCSC) applications and can extend thermoelectric applications based on 2D perovskites. We also investigated the influence of transition metal doping on electronic and phononic features of three-dimensional perovskite by studying the HC relaxation processes in Mn2+-doped and undoped CsPbI3 nanocrystals (NCs). The Mn2+ doping leads to the enlarged phononic gap between longitudinal optical (LO) – acoustic phonons, enhanced carrier-LO phonon coupling strength, and additional Mn orbitals within the original bands of the undoped sample, which are beneficial for establishing a hot quasi-equilibrium to recycle the energy from HC relaxation to reheat cold carriers. The results present a methodology to optimize HC dynamics by element doping and are meaningful for guiding the future development of HCSC applications

TRINOMIAL-TREE DISTRIBUTION OF A NONLINEAR POSITIVE STOCHASTIC INTEREST RATE MODEL WITH CONNECTIONS TO THE POTENTIAL APPROACH AND ITS APPLICATION TO COMPUTING CORPORATE DEFAULT RISK

In this dissertation, I consider a new nonlinear stochastic interest rate model that is adapted from a stochastic population growth model and exhibits the desirable properties of positivity of interest rates and mean reversion. We show that in the constant parameter case this model falls within the paradigm of the Rogers approach for generating positive interest rate models.Moreover, motivated by a procedure initiated by Hull and White, we alsooer a variant of the model with a time-dependent parameter that allows calibration of the model to a specified initial term structure when a trinomial-tree method is implemented to obtain discrete approximations of the distributions of the interest-rate process. Although nonlinear, our model has a closed form solution, which facilitates the generation of sample paths by standard numericalmethods. This allows us to carry out the trinomial-tree method to obtainapproximate distribution of the interest-rate process and compare that result to the approximate distributions obtained by Monte Carlo simulation. We incorporated the positive interest rate to derive the firm's default probability, which thereby extends Qian's work from a linear interest rate model to a non-linear interest rate model. In the research, first comparing to Qian's method, we used the rst passage time method based on the Fortet integral equation to derive the firm's default probability as driven by the Vasicek interest rate model.As an alternative, we also proposed the coupled trinomial tree method to derive the default probability. With the comparison of the numerical results among the three methods, we successfully extended the coupled trinomial tree algorithm for default probability from the linear model to a nonlinear model and obtained reasonably consistent results

To investigate the magnetic-field-induced distortion of NSs through GRB X-ray plateaus

Magnetic field may distort neutron stars (NSs), but the effect has not been robustly tested through gravitational-wave observation yet due to the absence of a fast rotating Galactic magnetar. The central objects of Gamma-ray bursts (GRBs) could be millisecond magnetars. Under the magnetar scenario on the X-ray plateaus of GRB afterglows,the spindown evolution modulated by the gravitational-wave radiation may be inferred from some special samples, so that the magnetically-induced distorting can be further estimated. According to two samples, GRB 060807 and GRB 070521, we found that the correlation between the effective ellipticity, $\varepsilon_{\rm B,eff}$, and effective dipole magnetic field strength on a neutron star (NS) surface, $B_{\rm eff}$, is $\varepsilon_{\rm B,eff}\sim 10^{-4} (\frac{B_{\rm eff}}{10^{14}\;\rm G})^{2}$. This result demands that $B_{\rm eff}\sim 0.01 B_{\rm t}$ with $B_{\rm t}$ being the internal toroidal magnetic field strength of NSs. We suggested that the torque generated during few unsymmetrical massive-star collapses may induce differential rotations in proto-NSs to amplify the internal toroidal fields.Comment: 8 pages, 2 figures, RAA submitte

MicroRNA390-Directed TAS3 Cleavage Leads to the Production of tasiRNA-ARF3/4 During Somatic Embryogenesis in Dimocarpus longan Lour

Trans-acting short-interfering RNAs (tasiRNAs) originate from TAS3 families through microRNA (miRNA) 390-guided cleavage of primary transcripts and target auxin response factors (ARF3/-4), which are involved in the normal development of lateral roots and flowers in plants. However, their roles in embryo development are still unclear. Here, the pathway miR390-TAS3-ARF3/-4 was identified systematically for the first time during somatic embryo development in Dimocarpus longan. We identified the miR390 primary transcript and promoter. The promoter contained cis-acting elements responsive to stimuli such as light, salicylic acid, anaerobic induction, fungal elicitor, circadian control and heat stress. The longan TAS3 transcript, containing two miR390-binding sites, was isolated; the miR390- guided cleavage site located near the 3' end of the TAS3 transcript was verified. Eight TAS3-tasiRNAs with the 21-nucleotide phase were found among longan small RNA data, further confirming that miR390-directed TAS3 cleavage leads to the production of tasiRNA in longan. Among them, TAS3_5'D5+ and 5'D6+ tasiRNAs were highly abundant, and verified to target ARF3 and -4, implying that miR390-guided TAS3 cleavage with 21-nucleotide phase leading to the production of tasiRNA-ARF is conserved in plants. Pri-miR390 was highly expressed in friable-embryogenic callus (EC), and less expressed in incomplete compact pro-embryogenic cultures,while miR390 showed its lowest expression in EC and highest expression in torpedo-shaped embryo. DlTAS3 and DlARF4 both exhibited their lowest expressions in EC, and reached their peaks in the globular embryos stage, which were mainly inversely proportional to the expression of miR390, especially at the GE to CE stages. While DlARF3 showed little variation from the EC to torpedo-shaped embryos stages, and exhibited its lowest expression in the cotyledonary embryos stage. There was a general lack of correlation between the expressions of DlARF3 and miR390. In addition, miR390, DlTAS3, DlARF3, and -4 were up-regulated by 2,4-D in a concentration-dependent manner. They were also preferentially expressed in roots, pulp, and seeds of ‘Sijimi’ longan, implying their extended roles in the development of longan roots and fruit. This study provided insights into a possible role of miR390-tasiRNAs-ARF in plant somatic embryo development