We develop a model for the evolution of wealth in a non-conservative economic environment, extending a theory developed earlier by the authors. The model considers a system of rational agents interacting in a game theoretical framework. This evolution drives the dynamic of the agents in both wealth and economic configuration variables. The cost function is chosen to represent a risk averse strategy of each agent. That is, the agent is more likely to interact with the market, the more predictable the market, and therefore the smaller its individual risk. This yields a kinetic equation for an effective single particle agent density with a Nash equilibrium serving as the local thermodynamic equilibrium. We consider a regime of scale separation where the large scale dynamics is given by a hydrodynamic closure with this local equilibrium. A class of generalized collision invariants (GCIs) is developed to overcome the difficulty of the non-conservative property in the hydrodynamic closure derivation of the large scale dynamics for the evolution of wealth distribution. The result is a system of gas dynamics-type equations for the density and average wealth of the agents on large scales. We recover the inverse Gamma distribution, which has been previously considered in the literature, as a local equilibrium for particular choices of the cost function.

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.

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.

Denote by Mnthe set of n ×ncomplex matrices. Let f:Mn→[0, ∞)be a continuous map such that f(μUAU∗) =f(A)for any complex unit μ, A ∈Mnand unitary U∈Mn, f(X) =0if and only if X=0and the induced map t →f(tX)is monotonically increasing on [0, ∞)for any rank onenilpotent X∈Mn. Characterization is given for surjective maps φon Mnsatisfying f(AB−BA) =f(φ(A)φ(B) −φ(B)φ(A)). The general theorem isthen used to deduce results on special cases when the function is the pseudo spectrum and the pseudo spectral radius.