Time-dependent Hermite-Galerkin spectral method and its applications

A time-dependent Hermite-Galerkin spectral method (THGSM) is investigated in this paper for the nonlinear convection-diffusion equations in the unbounded domains. The time-dependent scaling factor and translating factor are introduced in the definition of the generalized Hermite functions (GHF). As a consequence, the THGSM based on these GHF has many advantages, not only in theorethical proofs, but also in numerical implementations. The stability and spectral convergence of our proposed method have been established in this paper. The Korteweg-de Vries-Burgers (KdVB) equation and its special cases, including the heat equation and the Burgers' equation, as the examples, have been numerically solved by our method. The numerical results are presented, and it surpasses the existing methods in accuracy. Our theoretical proof of the spectral convergence has been supported by the numerical results.Comment: 16 pages, 7 table

Whole genome single nucleotide polymorphism genotyping of Staphylococcus aureus

Next-generation sequencing technology enables routine detection of bacterial pathogens for clinical diagnostics and genetic research. Whole genome sequencing has been of importance in the epidemiologic analysis of bacterial pathogens. However, few whole genome sequencing-based genotyping pipelines are available for practical applications. Here, we present the whole genome sequencing-based single nucleotide polymorphism (SNP) genotyping method and apply to the evolutionary analysis of methicillin-resistant Staphylococcus aureus. The SNP genotyping method calls genome variants using next-generation sequencing reads of whole genomes and calculates the pair-wise Jaccard distances of the genome variants. The method may reveal the high-resolution whole genome SNP profiles and the structural variants of different isolates of methicillin-resistant S. aureus (MRSA) and methicillin-susceptible S. aureus (MSSA) strains. The phylogenetic analysis of whole genomes and particular regions may monitor and track the evolution and the transmission dynamic of bacterial pathogens. The computer programs of the whole genome sequencing-based SNP genotyping method are available to the public at https://github.com/cyinbox/NGS.Comment: 6 figures, 2 table

Hermite spectral method to 1D forward Kolmogorov equation and its application to nonlinear filtering problems

In this paper, we investigate the Hermite spectral method (HSM) to numerically solve the forward Kolmogorov equation (FKE). A useful guideline of choosing the scaling factor of the generalized Hermite functions is given in this paper. It greatly improves the resolution of HSM. The convergence rate of HSM to FKE is analyzed in the suitable function space and has been verified by the numerical simulation. As an important application and our primary motivation to study the HSM to FKE, we work on the implementation of the nonlinear filtering (NLF) problem with a real-time algorithm developed in [17]. The HSM to FKE is served as the off-line computation in this algorithm. The translating factor of the generalized Hermite functions and the moving-window technique are introduced to deal with the drifting of the posterior conditional density function of the states in the on-line experiments. Two numerical experiments of NLF problems are carried out to illustrate the feasibility of our algorithm. Moreover, our algorithm surpasses the particle filter as a real-time solver to NLF.Comment: 14 pages, 6 figures, 1 tabl

Classification of 3-dimensional isolated rational hypersurface singularities with C*-action

In the paper "Algebraic classification of rational CR structures on topological 5-sphere with transversal holomorphic S^1-action in C^4" (Yau and Yu, Math. Nachrichten 246-247(2002), 207-233), we give algebraic classification of rational CR structures on the topological 5-sphere with transversal holomorphic S^1-action in C^4. Here, algebraic classification of compact strongly pseudoconvex CR manifolds X means classification up to algebraic equivalence, i.e. roughly up to isomorphism of the normalization of the complex analytic variety V which has X as boundary. The problem is intimately related to the study of 3-dimensional isolated rational weighted homogeneous hypersurface singularities with link homeomorphic to S^5. For this, we need the classification of 3-dimensional isolated rational hypersurface singularities with a C*-action. This list is only available at the homepage of one of us. Since there is a desire for a complete list of this classification (cf. Theorem 3.3), we decide to publish it for the convenience of readers

4d N=2 SCFT from Complete Intersection Singularity

Detailed studies of four dimensional N=2 superconformal field theories (SCFT) defined by isolated complete intersection singularities are performed: we compute the Coulomb branch spectrum, Seiberg-Witten solutions and central charges. Most of our theories have exactly marginal deformations and we identify the weakly coupled gauge theory descriptions for many of them, which involve (affine) D and E shaped quiver gauge theories and theories formed from Argyres-Douglas matters. These investigations provide strong evidence for the singularity approach in classifying 4d N=2 SCFTs.Comment: 46 pages, 85 figure

Complete Weight Distribution and MacWilliams Identities for Asymmetric Quantum Codes

In 1997, Shor and Laflamme defined the weight enumerators for quantum error-correcting codes and derived a MacWilliams identity. We extend their work by introducing our double weight enumerators and complete weight enumerators. The MacWilliams identities for these enumerators can be obtained similarly. With the help of MacWilliams identities, we obtain various bounds for asymmetric quantum codes.Comment: 15 page

4d N=2 SCFT and singularity theory Part II: Complete intersection

We classify three dimensional isolated weighted homogeneous rational complete intersection singularities, which define many new four dimensional N=2 superconformal field theories. We also determine the mini-versal deformation of these singularities, and therefore solve the Coulomb branch spectrum and Seiberg-Witten solution.Comment: 64 pages, 1 figur

4d N=2 SCFT and singularity theory Part III: Rigid singularity

We classify three fold isolated quotient Gorenstein singularity $C^3/G$. These singularities are rigid, i.e. there is no non-trivial deformation, and we conjecture that they define 4d $\mathcal{N}=2$ SCFTs which do not have a Coulomb branch.Comment: 17 pages, 1 figur

Inverted repeats in coronavirus SARS-CoV-2 genome and implications in evolution

The coronavirus disease (COVID-19) pandemic, caused by the coronavirus SARS-CoV-2, has caused 60 millions of infections and 1.38 millions of fatalities. Genomic analysis of SARS-CoV-2 can provide insights on drug design and vaccine development for controlling the pandemic. Inverted repeats in a genome greatly impact the stability of the genome structure and regulate gene expression. Inverted repeats involve cellular evolution and genetic diversity, genome arrangements, and diseases. Here, we investigate the inverted repeats in the coronavirus SARS-CoV-2 genome. We found that SARS-CoV-2 genome has an abundance of inverted repeats. The inverted repeats are mainly located in the gene of the Spike protein. This result suggests the Spike protein gene undergoes recombination events, therefore, is essential for fast evolution. Comparison of the inverted repeat signatures in human and bat coronaviruses suggest that SARS-CoV-2 is mostly related SARS-related coronavirus, SARSr-CoV/RaTG13. The study also reveals that the recent SARS-related coronavirus, SARSr-CoV/RmYN02, has a high amount of inverted repeats in the spike protein gene. Besides, this study demonstrates that the inverted repeat distribution in a genome can be considered as the genomic signature. This study highlights the significance of inverted repeats in the evolution of SARS-CoV-2 and presents the inverted repeats as the genomic signature in genome analysis

Complete Real Time Solution of the General Nonlinear Filtering Problem without Memory

It is well known that the nonlinear filtering problem has important applications in both military and civil industries. The central problem of nonlinear filtering is to solve the Duncan-Mortensen-Zakai (DMZ) equation in real time and in a memoryless manner. In this paper, we shall extend the algorithm developed previously by S.-T. Yau and the second author to the most general setting of nonlinear filterings, where the explicit time-dependence is in the drift term, observation term, and the variance of the noises could be a matrix of functions of both time and the states. To preserve the off-line virture of the algorithm, necessary modifications are illustrated clearly. Moreover, it is shown rigorously that the approximated solution obtained by the algorithm converges to the real solution in the $L^1$ sense. And the precise error has been estimated. Finally, the numerical simulation support the feasibility and efficiency of our algorithm.Comment: 15 pages, 2-column format, 2 figure