不同参数下Galton板实验的探讨及仿真

介绍了 Galton板实验的实验现象和物理背景 ,建立了细致的概率模型对实验进行了分析 ,并基于模型对不同参数下的 Galton板实验进行了探讨 ,还利用 MATLAB编制了仿真软件 ,对结论进行了验证 .     

Stochastic nonlinear dynamics of interpersonal and romantic relationships

Current theories from biosocial (e.g., the role of neurotransmitters in behavioral features), ecological (e.g., cultural, political, and institutional conditions), and interpersonal (e.g., attachment) perspectives have grounded interpersonal and romantic relationships in normative social experiences. However, these theories have not been developed to the point of providing a solid theoretical understanding of the dynamics present in interpersonal and romantic relationships, and integrative theories are still lacking. In this paper, mathematical models are used to investigate the dynamics of interpersonal and romantic relationships, via ordinary and stochastic differential equations, in order to provide insight into the behaviors of love. The analysis starts with a deterministic model and progresses to nonlinear stochastic models capturing the stochastic rates and factors (e.g., ecological factors, such as historical, cultural and community conditions) that affect proximal experiences and shape the patterns of relationship. Numerical examples are given to illustrate various dynamics of interpersonal and romantic behaviors with particular emphases placed on sustained oscillations and transitions between locally stable equilibria that are observable in stochastic models (closely related to real interpersonal dynamics), but absent in deterministic models.     

Modeling treatment of cancer using virotherapy with generalized logistic growth of tumor cells

Abstract

Virotherapy is an effective strategy in cancer treatment. It eliminates tumor cells without harming the healthy cells. In this article, a deterministic mathematical model to understand the dynamics of tumor cells in response to virotherapy is formulated and analyzed by incorporating cytotoxic T lymphocytes (CTLs). The basic reproduction number and the immune response reproduction number are computed and different equilibria of the proposed model are found. The local stability of different equilibria is discussed in detail. Further, the proposed model is extended to stochastic model. Numerical simulation is performed for both deterministic and stochastic models. It is observed that when both the reproduction numbers are greater than one, which corresponds to existence of unique nontrivial equilibrium point, dynamics of deterministic and stochastic models are almost same. The deterministic model shows a very complex dynamics when one or both the reproduction numbers are below one. The system exhibits both backward bifurcation and Hopf-bifurcation for suitable sets of parameters and in this situation it is not easy to predict the dynamics of cancer cells and virus particles. The existence of backward bifurcation demonstrates the fact that partial success of virotherapy can be achieved even if the immune response reproduction number is less than one.     

Random self-similar trees and a hierarchical branching process

We study self-similarity in random binary rooted trees. In a well-understood case of Galton–Watson trees, a distribution on a space of trees is said to be self-similar if it is invariant with respect to the operation of pruning, which cuts the tree leaves. This only happens for the critical Galton–Watson tree (a constant process progeny), which also exhibits other special symmetries. We extend the prune-invariance setup to arbitrary binary trees with edge lengths. In this general case the class of self-similar processes becomes much richer and covers a variety of practically important situations. The main result is construction of the hierarchical branching processes that satisfy various self-similarity definitions (including mean self-similarity and self-similarity in edge-lengths) depending on the process parameters. Taking the limit of averaged stochastic dynamics, as the number of trajectories increases, we obtain a deterministic system of differential equations that describes the process evolution. This system is used to establish a phase transition that separates fading and explosive behavior of the average process progeny. We describe a class of critical Tokunaga processes that happen at the phase transition boundary. They enjoy multiple additional symmetries and include the celebrated critical binary Galton–Watson tree with independent exponential edge length as a special case. Finally, we discuss a duality between trees and continuous functions, and introduce a class of extreme-invariant processes, constructed as the Harris paths of a self-similar hierarchical branching process, whose local minima has the same (linearly scaled) distribution as the original process.     

Computation of symbolic dynamics for one-dimensional maps

In this paper we design and implement rigorous algorithms for computing symbolic dynamics for piecewise-monotone-continuous maps of the interval. The algorithms are based on computing forwards and backwards approximations of the boundary, discontinuity and critical points. We explain how to handle the discontinuities in the symbolic dynamics which occur when the computed partition element boundaries are not disjoint. The method is applied to compute the symbolic dynamics and entropy bounds for the return map of the singular limit of a switching system with hysteresis and the forced Van der Pol equation.     

Application of Gray codes to the study of the theory of symbolic dynamics of unimodal maps

In this paper we provide a closed mathematical formulation of our previous results in the field of symbolic dynamics of unimodal maps. This being the case, we discuss the classical theory of applied symbolic dynamics for unimodal maps and its reinterpretation using Gray codes. This connection was previously emphasized but no explicit mathematical proof was provided. The work described in this paper not only contributes to the integration of the different interpretations of symbolic dynamics of unimodal maps, it also points out some inaccuracies that exist in previous works.     

Well-posedness of Hamilton–Jacobi equations in population dynamics and applications to large deviations

We prove Freidlin–Wentzell type large deviation principles for various rescaled models in populations dynamics that have immigration and possibly harvesting: birth–death processes, Galton–Watson trees, epidemic SI models, and prey–predator models.The proofs are carried out using a general analytic approach based on the well-posedness of a class of associated Hamilton–Jacobi equations. The notable feature for these Hamilton–Jacobi equations is that the Hamiltonian can be discontinuous at the boundary. We prove a well-posedness result for a large class of Hamilton–Jacobi equations corresponding to one-dimensional models, and give partial results for the multi-dimensional setting.     

Time-series analysis of TCP/RED computer networks,an empirical study

Packet-level observations show that the TCP/RED congestion control systems exhibit complex non-periodic oscillations which vary with the network/RED parameter variations. In this paper, it is investigated whether such complex behaviors are due to nonlinear deterministic chaotic dynamics or do they originate from nonlinear stochastic dynamics. To do this, various methods of linear and nonlinear time series analyses have been applied to the packet-level data gathered from a typical network simulated in ns-2. The results of the analysis for a wide range of variations in averaging weight of RED (as the most important bifurcation factor in TCP/RED networks) show that such behaviors are not due to deterministic chaos in the system, but originate from the stochastic nature of the network.     

A global MINLP approach to symbolic regression

Symbolic regression methods generate expression trees that simultaneously define the functional form of a regression model and the regression parameter values. As a result, the regression problem can search many nonlinear functional forms using only the specification of simple mathematical operators such as addition, subtraction, multiplication, and division, among others. Currently, state-of-the-art symbolic regression methods leverage genetic algorithms and adaptive programming techniques. Genetic algorithms lack optimality certifications and are typically stochastic in nature. In contrast, we propose an optimization formulation for the rigorous deterministic optimization of the symbolic regression problem. We present a mixed-integer nonlinear programming (MINLP) formulation to solve the symbolic regression problem as well as several alternative models to eliminate redundancies and symmetries. We demonstrate this symbolic regression technique using an array of experiments based upon literature instances. We then use a set of 24 MINLPs from symbolic regression to compare the performance of five local and five global MINLP solvers. Finally, we use larger instances to demonstrate that a portfolio of models provides an effective solution mechanism for problems of the size typically addressed in the symbolic regression literature.     

The road coloring problem

A synchronizing word of a deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton to a single state. A coloring of edges of a directed graph is synchronizing if the coloring turns the graph into a deterministic finite automaton possessing a synchronizing word. The road coloring problem is the problem of synchronizing coloring of a directed finite strongly connected graph with constant outdegree of all its vertices if the greatest common divisor of lengths of all its cycles is one. The problem was posed by Adler, Goodwyn and Weiss over 30 years ago and evoked noticeable interest among the specialists in the theory of graphs, deterministic automata and symbolic dynamics. The positive solution of the road coloring problem is presented.     

Measuring complexity in a business cycle model of the Kaldor type

The purpose of this paper is to study the dynamical behavior of a family of two-dimensional nonlinear maps associated to an economic model. Our objective is to measure the complexity of the system using techniques of symbolic dynamics in order to compute the topological entropy. The analysis of the variation of this important topological invariant with the parameters of the system, allows us to distinguish different chaotic scenarios. Finally, we use a another topological invariant to distinguish isentropic dynamics and we exhibit numerical results about maps with the same topological entropy. This work provides an illustration of how our understanding of higher dimensional economic models can be enhanced by the theory of dynamical systems.     

Interpreting supply chain dynamics: A quasi-chaos perspective

Chaotic phenomena, chaos amplification and other interesting nonlinear behaviors have been observed in supply chain systems. Chaos can be defined theoretically if the dynamics under study are produced only by deterministic factors. However, deterministic settings rarely present themselves in reality. In fact, real data are typically unknown. How can the chaos theory and its related methodology be applied in the real world? When the demand is stochastic, the interpretation and distribution of the Lyapunov exponents derived from the effective inventory at different supply chain levels are not similar to those under deterministic demand settings. Are the observed dynamics of the effective inventory random, chaotic, or simply quasi-chaos? In this study, we investigate a situation whereby the chaos analysis is applied to a time series as if its underlying structure, deterministic or stochastic, is unknown. The result shows clear distinction in chaos characterization between the two categories of demand process, deterministic vs. stochastic. It also highlights the complexity of the interplay between stochastic demand processes and nonlinear dynamics. Therefore, caution should be exercised in interpreting system dynamics when applying chaos analysis to a system of unknown underlying structure. By understanding this delicate interplay, decision makers have the better chance to tackle the problem correctly or more effectively at the demand end or the supply end.     

Fluid dynamics models and their applications in transportation and pricing

Fluid dynamics models provide a powerful deterministic technique to approximate stochasticity in a variety of application areas. In this paper, we study two classes of fluid models, investigate their relationship as well as some of their applications. This analysis allows us to provide analytical models of travel times as they arise in dynamically evolving environments, such as transportation networks as well as supply chains. In particular, using the laws of hydrodynamic theory, we first propose and examine a general second-order fluid model. We consider a first-order approximation of this model and show how it is helpful in analyzing the dynamic traffic equilibrium problem. Furthermore, we present an alternate class of fluid models that are traditionally used in the context of dynamic traffic assignment. By interpreting travel times as price/inventory–sojourn-time relationships, we are also able to connect this approach with a tractable fluid model in the context of dynamic pricing and inventory management.     

Quincunx fundamental refinable functions and quincunx biorthogonal wavelets

We analyze the approximation and smoothness properties of quincunx fundamental refinable functions. In particular, we provide a general way for the construction of quincunx interpolatory refinement masks associated with the quincunx lattice in . Their corresponding quincunx fundamental refinable functions attain the optimal approximation order and smoothness order. In addition, these examples are minimally supported with symmetry. For two special families of such quincunx interpolatory masks, we prove that their symbols are nonnegative. Finally, a general way of constructing quincunx biorthogonal wavelets is presented. Several examples of quincunx interpolatory masks and quincunx biorthogonal wavelets are explicitly computed.

     

Evaluación probabilista de la capacidad,fragilidad y daño sísmico de edificios de hormigón armado

Currently, many structures existing in seismic areas are highly vulnerable because they have been built without the use of seismic design codes or by using outdated codes. Often, methods for assessing the vulnerability of the structures do not take into account that their seismic behavior is dynamic and highly nonlinear and, moreover, that the structural characteristics and action have large uncertainties. This article aims to assess the vulnerability of structures taking into account that the mechanical properties of materials and the seismic action are random variables, by using advanced techniques based on the Monte Carlo method and on the nonlinear stochastic dynamics. The results obtained with these techniques are compared with those corresponding to a standard vulnerability assessment, based on deterministic models, in order to highlight the differences between both approaches. The main conclusion of this work is the need to address the vulnerability assessment problem from a probabilistic perspective which, combined with advanced nonlinear static and dynamic structural analysis techniques, provides a powerful tool giving information impossible to be captured by means of deterministic models. Finally, detailed results obtained for a building with waffle slabs, which is a structural typology widely used in Spain, are included and discussed.     

Stochastic volatility Gaussian Heath-Jarrow-Morton models

This paper extends the class of deterministic volatility Heath-Jarrow-Morton models to a Markov chain stochastic volatility framework allowing for jump discontinuities and a variety of deformations of the term structure of forward rate volatilities. Analytical solutions for the dynamics of the volatility term structure are obtained. Semimartingale decompositions of the interest rates under a spot and forward martingale measures are identified. Stochastic volatility versions of the continuous time Ho-Lee and Hull-White extended Vasicek models are obtained. Introducing a regime shift in volatility that is an exponential function of time to maturity leads to a Vasicek dynamics with regime switching coefficients of the short rate.     

Dynamics of Kolmogorov systems of competitive type under the telegraph noise

This paper studies the dynamics of Kolmogorov systems of competitive type under the telegraph noise. The telegraph noise switches at random two Kolmogorov competition-type deterministic models. The aim of this work is to describe the omega-limit set of the system and investigates properties of stationary density.     

传递子集的性质及其在符号动力系统中的应用

讨论了传递子集的一些性质,并且应用这些性质研究符号动力系统的弱混合子集和传递子集之间的关系,给出了符号动力系统的传递子集是弱混合子集的一个充分条件.     

Mathematical modeling of complex mechanical systems

Mathematical modeling of mechanical systems based on multibody system models is a well tested approach. Generating the equations of motion for complex multibody systems with a large number of degrees of freedom is difficult with paper and pencil. For this reason methods for automatic equation generation have been developed. Most methods result in numerical equations of motion without explicit information about the parameters. In this paper a method is described resulting in symbolic equations of motion. The method allows also the determination of the constraint forces which are important for design purposes. The inverse problem of dynamics is also easily solved.