Numerical encoding plays an important role in DNA sequence analysis via computational methods, in which numerical values are associated with corresponding symbolic characters. After numerical representation, digital signal processing methods can be exploited to analyze DNA sequences. To reflect the biological properties of the original sequence, it is vital that the representation is one-to-one. Chaos Game Representation (CGR) is an iterative mapping technique that assigns each nucleotide in a DNA sequence to a respective position on the plane that allows the depiction of the DNA sequence in the form of image. Using CGR, a biological sequence can be transformed one-to-one to a numerical sequence that preserves the main features of the original sequence. In this research, we propose to encode DNA sequences by considering 2D CGR coordinates as complex numbers, and apply digital signal processing methods to analyze their evolutionary relationship. Computational experiments indicate that this approach gives comparable results to the state-of-the-art multiple sequence alignment method, Clustal Omega, and is significantly faster. The MATLAB code for our method can be accessed from: www.mathworks.com/matlabcentral/fileexchange/57152
The entanglement quantification and classification of multipartite quantum states is an important research area in quantum information. In this paper, in terms of the reduced density matrices corresponding to all possible partitions of the entire system, a bounded entanglement measure is constructed for arbitrary-dimensional multipartite quantum states. In particular, for three-qubit quantum systems, we prove that our entanglement measure satisfies the relation of monogamy. Furthermore, we present a necessary condition for characterizing maximally entangled states using our entanglement measure.
We give an overview of our philosophy of pictures in mathematics. We emphasize a bidirectional process between picture lan- guage and mathematical concepts: abstraction and simulation. This motivates a program to understand different subjects, using virtual and real mathematical concepts simulated by pictures.
We show that if X is a uniformly perfect complete metric space satisfying
the finite doubling property, then there exists a fully supported measure with lower regularity
dimension as close to the lower dimension of X as we wish. Furthermore, we show that, under
the condensation open set condition, the lower dimension of an inhomogeneous self-similar set EC
coincides with the lower dimension of the condensation set C, while the Assouad dimension of
EC is the maximum of the Assouad dimensions of the corresponding self-similar set E and the
condensation set C. If the Assouad dimension of C is strictly smaller than the Assouad dimension
of E, then the upper regularity dimension of any measure supported on EC is strictly larger than
the Assouad dimension of EC. Surprisingly, the corresponding statement for the lower regularity
Jianlian CuiDepartment of Mathematics, Tsinghua University,Chi-KwongLiDepartment of Mathematics, College of William and MaryNung-SingSzeDepartment of Applied Mathematics, The Hong Kong Polytechnic University
It is known that every complex square matrix with nonnega-tive determinant is the product of positive semi-definite matrices. There are characterizations of matrices that require two or five positive semi-definite matrices in the product. However, the characterizations of matrices that require three or four positive semi-definite matrices in the product are lacking. In this paper, we give a complete characterization of these two types of matrices. With these results, we give an algorithm to determine whether a square matrix can be expressed as the product of kpositive semi-definite matrices but not fewer, for k=1, 2, 3, 4, 5.