数学代写|计算复杂度理论代写Computational complexity theory代考| Replicator Dynamics

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我们提供的计算复杂度理论Computational complexity theory及其相关学科的代写,服务范围广, 其中包括但不限于:

  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等概率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (Generalized) Linear Models 广义线性模型
  • Statistical Machine Learning 统计机器学习
  • Longitudinal Data Analysis 纵向数据分析
  • Foundations of Data Science 数据科学基础
数学代写|计算复杂度理论代写Computational complexity theory代考| Replicator Dynamics

数学代写|计算复杂度理论代写Computational complexity theory代考|Replicator Dynamics

In the classical cases above mentioned, individuals may belong only to one species, e.g., a predator cannot become a prey. In doing so, a competition process can

lead to the extinction of a species, i.e., to the complete removal of its individuals, without to affect the amount of individuals belonging to the winning species (whose value might depend on other parameters, like the growth factor, etc.). Instead, in EGT, individuals might change their group of belonging, and the total size of the population is conserved over time. Actually, as we will see, in EGT we do not focus on the agents from a physical point of view (e.g., growth mechanisms), but on their strategies that constitute the parameter of the system that varies over time. Here, the benefits (or gains) deriving from interactions can be interpreted as fitness. So, again, what that reproduces/extincts in these dynamics are not the individuals but their strategies. The equivalence payoff-fitness, and the relation between the payoff of an individual and its strategy (and those of its neighbors), allows to introduce an analytical description of the system. For instance, in two-strategy games, i.e., with individuals that can adopt the strategy $C$ and the strategy $D$, we can write
\frac{d C}{d t}=C\left(\pi_{C}-\phi\right) \
\frac{d D}{d t}=D\left(\pi_{D}-\phi\right)
with $\phi=C \pi_{C}+D \pi_{D}$, and $C+D=1$. The solution of this system can lead to different equilibria, as the extinction of a strategy, as well as the coexistence of both. This approach leads to the so-called replicator dynamics, which considers a population with $n$ strategies, and a $n \times n$-matrix, named “payoff matrix,” with elements $a_{i j}$ representing the gain individuals receive according to their actions (i.e., strategies). The general differential equation, i.e., the replicator equation is defined as
\frac{d x_{i}}{d t}=x_{i}\left(\pi_{i}-\phi\right)
with $i$ going from 1 to $n, x_{i}$ representing the density of the $i$-th strategy in the population, $\pi_{i}$ payoff (or fitness) of the $i$-th strategy computed as $\pi_{i}=\sum_{j=1}^{n} a_{i j} x_{j}$, and $\phi$ average payoff equal to $\phi=\sum_{i} \pi_{i} x_{i}$. The Eq. (2.4) shows a deterministic dynamics, and, as reported in several previous studies, the behavior of the system strongly depends on the value of $n$, i.e., the number of strategies. Notably, in the most simple case with $n=2$, we can observe the prevalence of one strategy, their coexistence, and bistable behaviors. While for $n>2$, different behaviors can be observed as limit cycles and chaos.

数学代写|计算复杂度理论代写Computational complexity theory代考|The Ising Model

Phase transitions are critical collective phenomena constituting one of the most important topics of Physics. Despite their underlying complexity, a number of models, for describing their dynamics, has been proposed. Among them, the Ising

model covers a special relevance, being at the same time both simple, powerful, and, in addition, useful for investigating also the behavior of nonphysical systems (e.g., see the area named Sociophysics). The Ising model considers a lattice of dimension $D$ composed of $N$ cells (see Fig. 2.1), each one provided with a spin $\sigma=\pm 1$. Since a cell may, in general, represent different kinds of objects (e.g., atoms, neurons, etc.), this simple model can be adopted in a wide variety of contexts. Accordingly, the variable $\sigma$ takes a meaning whose value depends on the related scenario, e.g., a magnetic moment, an opinion, an agent state, and so on. Here, it is worth to emphasize that the restricted range of $\sigma$, i.e., $\pm 1$ in the majority of cases, is reflected in the descriptive power of the model. However, a number of problems can be successfully faced by using this very simple modelization. Then, in the defined lattice, a pair of cells (e.g., $(i, j)$ ) forms a bond $J$, which represents their interaction. The whole set of bonds can be denoted as $B$, and for each element of the set, we have an energy of value $-J \sigma_{i} \sigma_{j}$. In doing so, the interaction energy is equal to $-J$ for $\sigma_{i}=\sigma_{j}$ and to $J$ in the opposite case. If $J$ is positive, the case $\sigma_{i}=\sigma_{j}$ has an energy smaller than the case $\sigma_{i}=-\sigma_{j}$, so the former is more stable. Positive interactions (i.e., $J>0$ ) are defined as “ferromagnetic,” while negative interactions as “antiferromagnetic.” In addition, some sites of the lattice can have an own energy of value $-h \sigma_{i}$ (here $h$ may represent an external field). So, the Hamiltonian of the Ising model reads
H=-J \sum_{(i j) \in B} \sigma_{i} \sigma_{j}-h \sum_{i=1}^{N} \sigma_{i}
Once defined the Hamiltonian function, it is possible to compute the expected value of a physical quantity, of the system under consideration, by using the

Gibbs-Boltzmann distribution. For instance, it is interesting to compute the average spin configuration $\Sigma$ at a given temperature $T$. To this end, we compute the distribution
P(\Sigma)=\frac{e^{-\beta H(\Sigma)}}{Z}
with $Z$ representing the partition function, and $\beta=\frac{1}{k_{b} T}$, i.e., inverse of the product between the Boltzmann constant $k_{b}$ and the system temperature $T$. As for the partition function $Z$, its role is normalizing the distribution (i.e., Eq. (2.6)) and, in general, takes the following form
Z=\sum_{i} e^{-\beta H\left(\Sigma_{i}\right)}
with the summation extended to all possible spin configurations. It is important to emphasize that, unfortunately, the explicit definition of $Z$ is not always trivial. Like previously mentioned, when spins $\sigma$ have values different from $\pm 1$, other models can be considered (e.g., the $X Y$ model). Among the quantities that can be measured in the Ising model, the parameter called magnetization is often particularly useful, allowing to have a high level overview on the system. In particular, the magnetization is defined as
m=\frac{1}{N}\left\langle\sum_{i=1}^{N} \sigma_{i}\right\rangle
In the thermodynamic limit, i.e., for $N \rightarrow \infty$, this parameter (i.e., $m$ ) measures the order of a system. Notably, the magnetization vanishes when the amount of positive spins is equal to that of negative ones, i.e., full disorder, and it is maximized when all spins are aligned in the same direction. It is then interesting to evaluate how the temperature affects the state of order of a system. Notably, at low temperatures (i.e., for $\beta \gg 1$ ), Eq. (2.6) suggests that low-energy configurations have a probability higher than high-energy configurations. Moreover, in absence of external fields (i.e., for $h=0$ ), low-energy states of the Ising model have all spins pointing in the same direction, so that the magnetization $m$ has a (absolute) value close to 1 . Now, increasing the temperature $T$, spin configurations with various energies emerge with equal probabilities. Accordingly, the macroscopic state of the Ising model becomes disordered, and its magnetization goes to zero. Therefore, it is possible to identify a relation between $m$ and $T$ and, most importantly, a critical temperature $T_{c}$. The latter entails that for $TT_{c}$ the magnetization reduces until its value goes to zero. The phenomenon here briefly described is known as “order-disorder phase transition,” and it has a deep relevance both in Physics and in the related applications to complex systems. Eventually.

数学代写|计算复杂度理论代写Computational complexity theory代考|Mean Field

In principle, the Gibbs-Boltzmann distribution defined in Eq. (2.6) indicates that it is possible to compute the expected value of any physical quantity. However, due to the huge amount of sums over $2^{N}$ terms in the partition function (i.e., Eq. (2.7)), sometimes this task is actually almost impossible (in a limited amount of time). Thus, in such cases, the utilization of opportune methods of approximation becomes mandatory, like, for instance, the mean-field approach now described. In few words, the mean-field approach neglects fluctuations of variables around their mean values. Notably, we assume $m=\frac{\sum_{i}\left(\sigma_{i}\right\rangle}{N}=\sum_{i}\left\langle\sigma_{i}\right\rangle$, and the deviation $\delta \sigma_{i}=\sigma_{i}-m$, in addition the second-order term with respect to the fluctuation $\delta \sigma_{i}$ is assumed to be small enough to be neglected. Accordingly, the Hamiltonian can be rewritten as
H &=-J \sum_{(i j) \in B}\left(m+\delta \sigma_{i}\right)\left(m+\delta \sigma_{j}\right)-h \sum_{i=1}^{N} \sigma_{i} \
& \sim-J m^{2} N_{B}-J m \sum_{(i j) \in B}\left(\delta \sigma_{i}+\delta \sigma_{j}\right)-h \sum_{i=1}^{N} \sigma_{i}
with $N_{B}$ number of elements in the set $B$. Here $\delta \sigma_{i}$ and $\delta \sigma_{j}$, which refer to the extrema of each bond, are summed up $z$ times, with $z$ number of bonds starting from a site. In doing so, the Hamiltonian can be finally reduced to the following form:
H=N_{B} J m^{2}-(J m z+h) \sum_{i=1}^{N} \sigma_{i}
Becoming much more easy for analytical calculations.

数学代写|计算复杂度理论代写Computational complexity theory代考| Replicator Dynamics


数学代写|计算复杂度理论代写Computational complexity theory代考|Replicator Dynamics


导致一个物种的灭绝,即完全消除其个体,而不影响属于获胜物种的个体数量(其值可能取决于其他参数,如生长因子等)。相反,在 EGT 中,个人可能会改变他们的归属群体,并且人口的总规模会随着时间的推移而保持不变。实际上,正如我们将看到的,在 EGT 中,我们并不关注从物理角度(例如,增长机制)的代理,而是关注它们构成随时间变化的系统参数的策略。在这里,来自交互的好处(或收益)可以解释为适应度。因此,再一次,在这些动态中复制/灭绝的不是个人,而是他们的策略。等价收益-适应度,以及个人的收益与其策略(及其邻居的策略)之间的关系,允许引入对系统的分析描述。例如,在双策略博弈中,即与可以采用该策略的个人C和策略D, 我们可以写成

dCd吨=C(圆周率C−φ) dDd吨=D(圆周率D−φ)\正确的。

\frac{d x_{i}}{dt}=x_{i}\left(\pi_{i}-\phi\right)
与一世从 1 到n,X一世表示密度一世-人口战略,圆周率一世的收益(或适应度)一世-th 策略计算为圆周率一世=∑j=1n一个一世jXj, 和φ平均收益等于φ=∑一世圆周率一世X一世. 方程。(2.4)显示了确定性动力学,并且正如之前的几项研究中所报道的,系统的行为在很大程度上取决于n,即策略的数量。值得注意的是,在最简单的情况下n=2,我们可以观察到一种策略的流行,它们的共存和双稳态行为。而对于n>2,不同的行为可以被观察为极限环和混沌。

数学代写|计算复杂度理论代写Computational complexity theory代考|The Ising Model


模型涵盖了特殊的相关性,同时既简单又强大,此外,对于研究非物理系统的行为也很有用(例如,参见名为社会物理学的领域)。Ising 模型考虑了维数格D由…组成的ñ细胞(见图 2.1),每个细胞都有一个旋转σ=±1. 由于一个单元通常可以代表不同种类的对象(例如原子、神经元等),所以这个简单的模型可以在广泛的环境中采用。因此,变量σ取值取决于相关场景的含义,例如磁矩、意见、代理状态等。在这里,值得强调的是,限制范围σ, IE,±1在大多数情况下,反映在模型的描述能力上。但是,使用这种非常简单的建模可以成功解决许多问题。然后,在定义的格子中,一对单元格(例如,(一世,j)) 形成债券Ĵ,代表他们的互动。整个债券集可以表示为乙,并且对于集合的每个元素,我们都有一个价值能量−Ĵσ一世σj. 这样做时,相互作用能等于−Ĵ为了σ一世=σj并Ĵ在相反的情况下。如果Ĵ为正,此案σ一世=σj能量小于外壳σ一世=−σj,所以前者更稳定。积极的互动(即,Ĵ>0) 被定义为“铁磁性”,而负相互作用被定义为“反铁磁性”。此外,格子的一些位点可以有自己的价值能量−Hσ一世(这里H可能代表一个外部场)。因此,伊辛模型的哈密顿量为


吉布斯-玻尔兹曼分布。例如,计算平均自旋配置很有趣Σ在给定温度下吨. 为此,我们计算分布

和从表示配分函数,和b=1ķb吨,即玻尔兹曼常数之间的乘积的倒数ķb和系统温度吨. 至于分区函数从,它的作用是规范化分布(即,方程(2.6)),并且通常采用以下形式


在热力学极限,即对于ñ→∞,这个参数(即,米) 测量系统的阶数。值得注意的是,当正自旋的数量等于负自旋的数量时,磁化消失,即完全无序,并且当所有自旋在同一方向上对齐时磁化最大。然后有趣的是评估温度如何影响系统的有序状态。值得注意的是,在低温下(即,对于b≫1),等式。(2.6) 表明低能配置的概率高于高能配置。此外,在没有外部场的情况下(即,对于H=0),伊辛模型的低能态的所有自旋都指向同一方向,因此磁化米具有接近 1 的(绝对)值。现在,提高温度吨,具有各种能量的自旋配置以相等的概率出现。因此,伊辛模型的宏观状态变得无序,其磁化强度变为零。因此,可以确定两者之间的关系米和吨最重要的是,临界温度吨C. 后者意味着吨吨C磁化强度减小,直到其值变为零。这里简要描述的现象被称为“有序-无序相变”,它在物理学和复杂系统的相关应用中都具有深刻的相关性。最终。

数学代写|计算复杂度理论代写Computational complexity theory代考|Mean Field

原则上,方程式中定义的吉布斯-玻尔兹曼分布。(2.6) 表明可以计算任何物理量的期望值。但由于金额庞大2ñ在配分函数中的术语(即方程(2.7)),有时这个任务实际上几乎是不可能的(在有限的时间内)。因此,在这种情况下,必须使用合适的近似方法,例如现在描述的平均场方法。简而言之,平均场方法忽略了变量围绕其平均值的波动。值得注意的是,我们假设米=∑一世(σ一世⟩ñ=∑一世⟨σ一世⟩, 和偏差dσ一世=σ一世−米, 另外关于波动的二阶项dσ一世假设小到可以忽略不计。因此,哈密顿量可以重写为

H=−Ĵ∑(一世j)∈乙(米+dσ一世)(米+dσj)−H∑一世=1ñσ一世 ∼−Ĵ米2ñ乙−Ĵ米∑(一世j)∈乙(dσ一世+dσj)−H∑一世=1ñσ一世
和ñ乙集合中的元素数乙. 这里dσ一世和dσj, 指的是每个债券的极值, 被总结和次,与和从一个站点开始的债券数量。这样做,哈密顿量最终可以简化为以下形式:


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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



有限元是一种通用的数值方法,用于解决两个或三个空间变量的偏微分方程(即一些边界值问题)。为了解决一个问题,有限元将一个大系统细分为更小、更简单的部分,称为有限元。这是通过在空间维度上的特定空间离散化来实现的,它是通过构建对象的网格来实现的:用于求解的数值域,它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统,以模拟整个问题。然后,有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。





随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中,其中问题和解决方案以熟悉的数学符号表示。典型用途包括:数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发,包括图形用户界面构建MATLAB 是一个交互式系统,其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题,尤其是那些具有矩阵和向量公式的问题,而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问,这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展,得到了许多用户的投入。在大学环境中,它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域,MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要,工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数(M 文件)的综合集合,可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。