Matthew He

Differential equation of Appell polynomials via the factorization method

Let {Pn(x)}n=0∞ be a sequence of polynomials of degree n. We define two sequences of differential operators Φn and Ψn satisfying the following properties: Φn(Pn(x)) = Pn-1(x), Ψn(Pn(x)) = Pn+1(x). By constructing these two operators for Appell polynomials, we determine their differential equations via the factorization method introduced by Infeld and Hull (Rev. Mod. Phys. 23 (1951) 21). The differential equations for both Bernoulli and Euler polynomials are given as special cases of the Appell polynomials.

Eigenfunctions of laguerre-type operators and generalized evolution problems

We consider eigenfunctions of a class of differential operators generalizing the Laguerre derivative. Applications in the framework of generalized evolution problems are also derived.

Mathematics of Bioinformatics : Theory, Practice, and Applications

Preface. About the Authors. 1. Bioinformatics and Mathematics. 1.1 Introduction. 1.2 Genetic Code and Mathematics. 1.3 Mathematical Background. 1.4 Converting Data to Knowledge. 1.5 Big Picture: Informatics. 1.6 Challenges and Perspectives. References. 2. Genetic Codes, Matrices, and Symmetrical Techniques. 2.1 Introduction. 2.2 Matrix Theory and Symmetry Preliminaries. 2.3 Genetic Codes and Matrices. 2.4 Genetic Matrices, Hydrogen Bonds and the Golden Section. 2.5 Symmetrical Patterns, Molecular Genetics and Bioinformatics. 2.6 Challenges and Perspectives. References. 3. Biological Sequences, Sequence Alignment, and Statistics. 3.1 Introduction. 3.2 Mathematical Sequences. 3.3 Sequence Alignment. 3.4 Sequence Analysis and Further Discussions. 3.5 Challenges and Perspectives. References. 4. Structures of DNA and Knot Theory. 4.1 Introduction. 4.2 Knot Theory Preliminaries. 4.3 DNA Knots and Links. 4.4 Challenges and Perspectives. References. 5. Protein Structures, Geometry, and Topology. 5.1 Introduction. 5.2 Computational Geometry and Topology Preliminaries. 5.3 Protein Structures and Prediction. 5.4 Statistical Approach and Discussions. 5.5 Challenges and Perspectives. References. 6. Biological Networks and Graph Theory. 6.1 Introduction. 6.2 Graph Theory Preliminaries and Network Topology. 6.3 Models of Biological Networks. 6.4 Challenges and Perspectives. References. 7. Biological Systems, Fractals, and Systems Biology. 7.1 Introduction. 7.2 Fractal Geometry Preliminaries. 7.3 Fractal Geometry in Biological Systems. 7.4 Systems Biology and Perspectives. 7.5 Challenges and Perspectives. References. 8. Matrix Genetics, Hadamard Matrix, and Algebraic Biology. 8.1 Introduction. 8.2 Genetic Matrices and the Degeneracy of the Genetic Code. 8.3 The Genetic Code and Hadamard Matrices. 8.4 Genetic Matrices and Matrices of Hypercomplex Numbers. 8.5 Some Rules of Evolution of Variants of the Genetic Code. 8.6 Challenges and Perspectives. References. 9. Bioinformatics, Living Systems and Cognitive Informatics. 9.1 Introduction. 9.2 Emerging Pattern, Dissipative Structure, and Evolving Cognition. 9.3 Denotational Mathematics and Cognitive Computing. 9.4 Challenges and Perspectives. References. 10. Evolutionary Trends and Central Dogma of Informatics. 10.1 Introduction. 10.2 Evolutionary Trends of Information Sciences. 10.3 Central Dogma of Informatics. 10.4 Challenges and Perspectives. References. Appendix A. Bioinformatics Notation and Databases. Appendix B. Bioinformatics/Genetics/Timeline. Appendix C. Bioinformatics Glossary. Index.

The Faber polynomials for circular lunes

Abstract We study the Faber polynomials F n ( z ) generated by a circular lune symmetric about both axes with vertices at the points z = ± α (0 α ≤ 2) and exterior angle απ. An explicit expression of F n ( z ) was obtained by computing the coefficients via a Cauchy integral formula. We also illustrate the location of the zeros of Faber polynomial and of its derivative. Our results include a circle and a segment as special cases when α = 1, 2, respectively.

The Faber polynomials for m -fold symmetric domains

Abstract The Faber polynomials for a region of the complex plane are of interest as a basis for polynomial approximations to analytic functions. In this paper we study the Faber polynomials associated with m -fold symmetric domains. Explicit formulae of Faber polynomials both for symmetric and nonsymmetric lemniscates are derived. In addition, we use a new determinant representation which relates the zeros of Faber polynomials to the eigenvalues of a certain matrix to compute the zeros of Faber polynomials for lemniscates.

Irreplaceable Amino Acids and Reduced Alphabets in Short-Term and Directed Protein Evolution

In this paper we extend codon volatility definition to amino acid reduced alphabets to characterize mutations that conserve physical-chemical properties. We also define the average relative changeability of amino acids in terms of single-base codon self-substitution frequencies (identities). These frequencies are taken from an empirical codon substitution matrix [14]. It is shown that this index splits the amino acids into two groups: replaceable and irreplaceable. The same grouping is obtained from the size/complexity index introduced by Dufton [32]. Also, a 71 % agreement is obtained with residues in mutually persistent conserved (MPC) positions [31]. These positions play a key role in fold and functional determination. The residual 29 % can be readily explained. 75 % of residues with highest rank according to MPC positions have the highest probability of causing disease if mutated.

Asymptotics of the zeros of relativistic Hermite polynomials

The relativistic Hermite polynomial (RHP) is a class of orthogonal polynomials associated with varying weights. We study the asymptotics of the zeros of the RHP when both degree n of polynomials and relativistic parameter N approach infinity.

The Relativistic Szego Polynomials

A new relativistic-type polynomiasl system is defined by means of the Relativistic Polynomial (RLP) system , recently introduced by P. Natalini. These polynomials, denoted by , are called Relativistic Generalized Hermite Polynomials (RGHP) because they reduce, in the non-relativistc limit , to the Generalized Hermite Polynomials first considered by G. Szego. Some properties of this new set of orthogonal polynomials are derived.

Asymptotic Distribution of Zeros of Weighted Fibonacci Polynomials

Exploiting the relation between the classical Fibonacci polynomials {F n(z)} and a certain weighted Faber polynomials {B n(z,g)} associated with a domain E in complex plane and a weight function g(z), we define weighted Fibonacci polynomials in complex domain. Applying fundamental properties of weighted Faber polynomials, we extend basic properties of Fibonacci polynomials to complex plane. Using potential theoretic methods, we determine the asymptotic distribution of the zeros of the weighted Fibonacci polynomials.

Variation of gaussian curvature under conformal mapping and its application

We characterize conformal mapping between two surfaces, S and S∗, based on Gaussian curvature before and after motion. An explicit representation of the Gaussian curvature after conformal mapping is presented in terms of Riemann-Christoffel tensor and Ricci tensor and their derivatives. Based on changes in surface curvature, we are able to estimate the stretching of non-rigid motion during conformal mapping via a polynomial approximation.