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.
Protein universe is a complex system with critical problem of protein evolution to be analyzed. Early studies have used geometric distances and polygenetic-trees to solve this problem. However, the traditional methods are bivariate, whose taxonomy classification relies on bivariate branching. This is not sufficient to describe the complex nature of protein universe. Therefore, we propose a novel approach on multivariate protein classification. The new method bases on the theory of information and network, can be used to analyze multivariate relationships of proteins. The new method is alignment-free and have wide-applications to both sequences and 3D structures. We demonstrate the new method on six protein examples, results show that the new method is efficient and can potentially be used for future protein classifications.
We propose a fast local level set method for the inverse problem of gravimetry. The theoretical foundation for our approach is based on the following uniqueness result: if an open set D is star-shaped or x<sub>3</sub>-convex with respect to its center of gravity, then its exterior potential uniquely determines the open set D. To achieve this purpose constructively, the first challenge is how to parametrize this open set D as its boundary may have a variety of possible shapes. To describe those different shapes we propose to use a level-set function to parametrize the unknown boundary of this open set. The second challenge is how to deal with the issue of partial data as gravimetric measurements are only made on a part of a given reference domain . To overcome this difficulty we propose a linear numerical continuation approach based on the single layer representation to find potentials on the boundary of some artificial domain
Recent understandings of molecular evolution, together with the fossil records, have established that there are both linear and nonlinear processes in the creation of novel species, which is strikingly similar to the generation of prime numbers and human creativity. Each creation of a more complex species is like a prime number, unpredictable, discontinuous, and yet can be modeled by a smooth curve in relation to time. The mystery behind the complexity increases in nature and human civilizations might well turn out to be similar to that behind the appearances of prime numbers. Here we show that an algorithm for the creative process of humans can create prime numbers in a lawful and yet unpredictable fashion. The essence of primes is the duality of uniqueness and uniformity together with the creation algorithm. The algorithm consists of the non-linear process of uniformity selection to create the unique and the linear process of uniqueness selection to form the uniformity. The iterations of this algorithm can create an infinite number of primes. The algorithm appears to have been hardwired in the human brain as shown by recent experimental studies. This new understanding can deduce some of the best-known properties of primes and may explain the nearly constant and yet seemingly random creation of novelty in relation to time.
We propose the backward phase flow method to implement the FourierBrosIagolnitzer (FBI)-transform-based Eulerian Gaussian beam method for solving the Schrdinger equation in the semi-classical regime. The idea of Eulerian Gaussian beams has been first proposed in . In this paper we aim at two crucial computational issues of the Eulerian Gaussian beam method: how to carry out long-time beam propagation and how to compute beam ingredients rapidly in phase space. By virtue of the FBI transform, we address the first issue by introducing the reinitialization strategy into the Eulerian Gaussian beam framework. Essentially we reinitialize beam propagation by applying the FBI transform to wavefields at intermediate time steps when the beams become too wide. To address the second issue, inspired by the original phase flow method, we propose the backward phase flow method which allows us to