We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schrödinger equations with Lévy indices $1 < \alpha < 2$. We consider both non-periodic and periodic cases, and prove that the Cauchy problems are locally well-posed in $H^s$ for $s \geq \frac {2-\alpha}4$. This is shown via a trilinear estimate in Bourgain's $X^{s,b}$ space. We also show that non-periodic equations are ill-posed in $H^s$ for $\frac {2 - 3\alpha}{4(\alpha + 1)} < s < \frac {2-\alpha}4$ in the sense that the flow map is not locally uniformly continuous.

A personal credit evaluation algorithm is proposed by the design of a decision tree with a boosting algorithm, and the classification is carried out. By comparison with the conventional decision tree algorithm, it is shown that the boosting algorithm acts to speed up the processing time. The Classification and Regression Tree (CART) algorithm with the boosting algorithm showed 90.95% accuracy, slightly higher than without boosting, 90.31%. To avoid overfitting of the model on the training set due to unreasonable data set division, we consider cross-validation and illustrate the results with simulation; hypermeters of the model have been applied and the model fitting effect is verified. The proposed decision tree model is fitted optimally with the help of a confusion matrix. In this paper, relevant evaluation indicators are also introduced to evaluate the performance of the proposed model. For the comparison with the conventional methods, accuracy rate, error rate, precision, recall, etc. are also illustrated; we comprehensively evaluate the model performance based on the model accuracy after the 10-fold cross-validation. The results show that the boosting algorithm improves the performance of the model in accuracy and precision when CART is applied, but the model fitting time takes much longer, around 2 min. With the obtained result, it is verified that the performance of the decision tree model is improved under the boosting algorithm. At the same time, we test the performance of the proposed verification model with model fitting, and it could be applied to the prediction model for customers’ decisions on subscription to the fixed deposit business.

We consider the fractional Schrödinger equations with focusing Hartree-type nonlinearities. When the energy is negative, we show that the solution blows up in a finite time. For this purpose, based on Glassey’s argument, we obtain a virial-type inequality.

We establish new Strichartz estimates for orthonormal families of initial data in the case of the wave, Klein–Gordon and fractional Schrödinger equations. Our estimates extend those of Frank–Sabin in the case of the wave and Klein–Gordon equations, and generalize work of Frank et al. and Frank–Sabin for the Schrödinger equation. Due to a certain technical barrier, except for the classical Schrödinger equation, the Strichartz estimates for orthonormal families of initial data have not previously been established up to the sharp summability exponents in the full range of admissible pairs. We obtain the optimal estimates in various notable cases and improve the previous results. The main novelty of this paper is our derivation and use of estimates for weighted oscillatory integrals, which we combine with an approach due to Frank and Sabin. Our weighted oscillatory integral estimates are, in a certain sense, rather delicate endpoint versions of known dispersive estimates with power-type weights of the form $|\xi |^{-\lambda }$ or $(1 + |\xi |^2)^{-\lambda /2}$ , where $\lambda \in \mathbb {R}$ . We achieve optimal decay rates by considering such weights with appropriate $\lambda \in \mathbb {C}$ . For the wave and Klein–Gordon equations, our weighted oscillatory integral estimates are new. For the fractional Schrödinger equation, our results overlap with prior work of Kenig–Ponce–Vega in a certain regime. Our contribution to the theory of weighted oscillatory integrals has also been influenced by earlier work of Carbery–Ziesler, Cowling et al., and Sogge–Stein. Finally, we provide some applications of our new Strichartz estimates for orthonormal families of data to the theory of infinite systems of Hartree type, weighted velocity averaging lemmas for kinetic transport equations, and refined Strichartz estimates for data in Besov spaces.