Modular matrices from universal wave function overlaps in Gutzwiller-projected parton wave functions

We implement the universal wave function overlap (UWFO) method to extract modular $S$ and $T$ matrices for topological orders in Gutzwiller-projected parton wave functions (GPWFs). The modular $S$ and $T$ matrices generate a projective representation of $SL(2,\mathbb{Z})$ on the degenerate-ground-state Hilbert space on a torus and may fully characterize the 2+1D topological orders, i.e. the quasi-particle statistics and chiral central charge (up to $E_8$ bosonic quantum Hall states). We used the variational Monte Carlo method to computed the $S$ and $T$ matrices of the chiral spin liquid (CSL) constructed by the GPWF on the square lattice, and confirm that the CSL carries the same topological order as the $\nu=\frac{1}{2}$ bosonic Laughlin state. We find that the non-universal exponents in UWFO can be small and direct numerical computation is able to be applied on relatively large systems. We also discuss the UWFO method for GPWFs on other Bravais lattices in two and three dimensions by using the Monte Carlo method. UWFO may be a powerful method to calculate the topological order in GPWFs.Comment: 5 pages with 3 figure

DPCA: Dimensionality Reduction for Discriminative Analytics of Multiple Large-Scale Datasets

Principal component analysis (PCA) has well-documented merits for data extraction and dimensionality reduction. PCA deals with a single dataset at a time, and it is challenged when it comes to analyzing multiple datasets. Yet in certain setups, one wishes to extract the most significant information of one dataset relative to other datasets. Specifically, the interest may be on identifying, namely extracting features that are specific to a single target dataset but not the others. This paper develops a novel approach for such so-termed discriminative data analysis, and establishes its optimality in the least-squares (LS) sense under suitable data modeling assumptions. The criterion reveals linear combinations of variables by maximizing the ratio of the variance of the target data to that of the remainders. The novel approach solves a generalized eigenvalue problem by performing SVD just once. Numerical tests using synthetic and real datasets showcase the merits of the proposed approach relative to its competing alternatives.Comment: 5 pages, 2 figure

Use of Simultaneous Inference Under Order Restriction, Stepdown Testing Procedure and Stage-wise Sequential Optimal Design in Clinical Dose Study

This dissertation discusses the design approaches of adaptive dose escalation study and the analysis methods of dose study data, and the relationship between the study design approach and data analysis methods.A general max-min approach to construct simultaneous confidence Intervals for the monotone means of correlated and normally istributed random samples is proposed to analyze correlated dose response data. The approach provides an accurate, flexible and computationally easy way to obtain critical values of simultaneous confidence intervals under monotone order restriction.Stepdown testing procedure for analyzing dose study data is examined and an modified stepdown testing approach is proposed to incorporate the adaptive sampling nature of the study data. An approximate mixture normal distribution of the dose response is proposed to analyze the binary outcome with small sample size at the first stage of the adaptive design.Finally, an optimal stage-wise adptive clinical dose study design is proposed to be applied in dose escalation study with binary outcome and correlated dose response. The study design criteria is defined as a weighted average power to identify all effective dose levels. A back induction algorithm is used to obtain the design parameters. The values of optimal design parameters vary when different analysis methods are used to analyze the study data.Simulation studies are performed to illustrate the two proposed analysis methods and the proposed optimal design approach