Stephen S.-T. Yau

On a necessary and sufficient condition for finite dimensionality of estimation

Ever since the technique of the Kalman–Bucy filter was popularized, there has been an intense interest in finding new classes of finite dimensional recursive filters. In the late seventies, the concept of the estimation algebra of a filtering system was introduced. It has proven to be an invaluable tool in the study of nonlinear filtering problems. In this paper, a simple algebraic necessary and sufficient condition is established for an estimation algebra of a special class of filtering systems to be finite-dimensional. Also presented is a rigorous proof of the Wei–Norman program which allows one to construct finite-dimensional recursive filters from finite dimensional estimation algebras.

A Fourier Characteristic of Coding Sequences: Origins and a Non-Fourier Approximation

The 3-base periodicity, identified as a pronounced peak at the frequency N/3 (N is the length of the DNA sequence) of the Fourier power spectrum of protein coding regions, is used as a marker in gene-finding algorithms to distinguish protein coding regions (exons) and noncoding regions (introns) of genomes. In this paper, we reveal the explanation of this phenomenon which results from a nonuniform distribution of nucleotides in the three coding positions. There is a linear correlation between the nucleotide distributions in the three codon positions and the power spectrum at the frequency N/3. Furthermore, this study indicates the relationship between the length of a DNA sequence and the variance of nucleotide distributions and the average Fourier power spectrum, which is the noise signal in gene-finding methods. The results presented in this paper provide an efficient way to compute the Fourier power spectrum at N/3 and the noise signal in gene-finding methods by calculating the nucleotide distributions in the three codon positions.

Optimal fast tracking observer bandwidth of the linear extended state observer

In current industrial control applications, the proportional + integral + derivative (PID) control is still used as the leading tool, but constructing controller requires precise mathematical model of plant, and tuning the parameters of controllers is not simple to implement. Motivated by the gap between theory and practice in control problems, linear active disturbance rejection control (LADRC) addresses a set of control problems in the absence of precise mathematical models. LADRC has two parameters to be tuned, namely, a closed-loop bandwidth and observer bandwidth. The performance of LADRC depends on the quick convergence of a unique state observer, known as the extended state observer, proposed by Jinqing Han (1994). Only one parameter, observer bandwidth, significantly affects the tracking speed of extended state observer. This paper studies numerically the optimal fast tracking observer bandwidth and the absolute tracking error estimation for a class of non-linear and uncertain motion control problems by finite difference method.

Gorenstein quotient singularities in dimension three

Introduction Classification of finite subgroups of

A Novel Construction of Genome Space with Biological Geometry

SL(3,\mathbb C)

Kohn-Rossi Cohomology and its Application to the Complex Plateau Problem, I

The invariant polynomials and their relations of linear groups of

Association rule and quantitative association rule mining among infrequent items

SL(3,\mathbb C)

Efficient association rule mining among both frequent and infrequent items

Gorenstein quotient singularities in dimension three.

Structure and classification theorems of finite-dimensional exact estimation algebras

A genome space is a moduli space of genomes. In this space, each point corresponds to a genome. The natural distance between two points in the genome space reflects the biological distance between these two genomes. Currently, there is no method to represent genomes by a point in a space without losing biological information. Here, we propose a new graphical representation for DNA sequences. The breakthrough of the subject is that we can construct the moment vectors from DNA sequences using this new graphical method and prove that the correspondence between moment vectors and DNA sequences is one-to-one. Using these moment vectors, we have constructed a novel genome space as a subspace in RN. It allows us to show that the SARS-CoV is most closely related to a coronavirus from the palm civet not from a bird as initially suspected, and the newly discovered human coronavirus HCoV-HKU1 is more closely related to SARS than to any other known member of group 2 coronavirus. Furthermore, we reconstructed the phylogenetic tree for 34 lentiviruses (including human immunodeficiency virus) based on their whole genome sequences. Our genome space will provide a new powerful tool for analyzing the classification of genomes and their phylogenetic relationships.

A protein map and its application.

Let