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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|黎曼几何代写Riemannian geometry代考|Application to Stochastic Calculus

Brownian motion is the term applied to random motion of particles in a medium, originally observed by Robert Brown in 1827 . The motion is probabilistic, continuous in time, but not differentiable (with probability one). Such motion, applied to a large number of particles to make the result more obvious, gives a diffusion. The Itô and Stratonovich calculi are pathwise probabilistic constructions used to describe diffusion by Brownian motion or more general stochastic effects on manifolds. They are both ‘first-order calculi’ in some sense. Here we shall not present any probabilistic construction, but rather an algebraic construction of a calculus in our sense of a DGA, which works to all orders and whose first order part has the same form as the stochastic differentials in Itô and Stratonovich theory.

Stochastic calculus deals with the evolution in time of functions on a manifold under the twin effects of diffusion and drift. Our approach will be to model a manifold $M$ through its exterior algebra $\Omega(M)$ and to model diffusion and drift via a homotopy deformation. In fact our construction is purely algebraic and works for any $\operatorname{DGA}(\Omega, \mathrm{d}, \wedge)$ on an algebra $A=\Omega^{0}$, which need not be commutative. First we add a time variable by taking the tensor product of $\Omega$ and the classical differential calculus on the line $\mathbb{R}$, where we take $t$ to be the coordinate on the line. This gives
$$\Omega_{\mathbb{R}}^{n}=\left(\Omega^{n} \otimes C^{\infty}(\mathbb{R})\right) \bigoplus\left(\Omega^{n-1} \otimes C^{\infty}(\mathbb{R})\right) \wedge \mathrm{d} t .$$

## 数学代写|黎曼几何代写Riemannian geometry代考|Basic Examples of Hopf Algebras

We start with a handful of classical examples where one hardly needs Hopf algebras but where Hopf algebras unify disparate classical concepts.

Example $2.9$ Let $G$ be a group. Its group algebra $\mathbb{k} G$ (consisting of finite linear combinations of elements of $G$ ) becomes an algebra with the group product of $G$ extended linearly. Here $1=e$ (the algebra identity is equal to the group identity). This forms a Hopf algebra with
$$\Delta x=x \otimes x, \quad \epsilon x=1, \quad S x=x^{-1}$$
for all $x \in G$. An element of a coalgebra with such a diagonal coproduct is said to be grouplike. Over $\mathbb{C}$, we have a Hopf $$-algebra with x^{}=x^{-1}. The reader should be able to see that a \mathbb{k} G-module algebra is an algebra on which G acts by algebra automorphisms, while a k G-comodule algebra is a G-graded algebra A in the sense that$$
A=\bigoplus_{g \in G} A_{g}, \quad A_{h} \cdot A_{g} \subseteq A_{h g} .
$$If G is finité thên k(G), the algebrà of functions on G with pointwise multiplication, is also a Hopf algebra, with$$
$$for all x, y \in G, f \in \mathbb{k}(G), where we identify \mathbb{k}(G) \otimes \mathbb{k}(G)=\mathbb{k}(G \times G). Over \mathbb{C}, we have a Hopf$$-algebra with $f^{}(x)=\overline{f(x)}$. A left $\mathbb{k}(G)$-module algebra is a $G$-graded algebra $A$ where $f \triangleright a=f(g) a$ when $a \in A_{g}$. A left $\mathbb{k}(G)$-comodule algebra means an algebra on which $G$ right acts by algebra automorphisms, the two being related by
$$\Delta_{L}(a)=\sum_{g \in G} \delta_{g} \otimes a \triangleleft g .$$

# 黎曼几何代考

## 数学代写|黎曼几何代写Riemannian geometry代考|Application to Stochastic Calculus

$$\Omega_{\mathbb{R}}^{n}=\left(\Omega^{n} \otimes C^{\infty}(\mathbb{R})\right) \bigoplus\left(\Omega^{n-1} \otimes C^{\infty}(\mathbb{R})\right) \wedge \mathrm{d} t .$$

## 数学代写|黎曼几何代写Riemannian geometry代考|Basic Examples of Hopf Algebras

$$\Delta x=x \otimes x, \quad \epsilon x=1, \quad S x=x^{-1}$$

-algebrawith $\$ x=x^{-1} \$$.Thereadershouldbeabletoseethata\⿺𠃊G – modulealgebraisanalgebraonwhich \$$ Gactsbyalgebraautomorphisms, whilea $\$$A= \backslash bigoplus_{g \in G} A_{g}, \quad A_{ {h} \backslash c d o t A_{-}{g} \backslash subseteq A_{ {h g} 。 (Delta f)(x, y)=f(x y), \quad \epsilon f=f(e), \backslash quad (S f)(x)=f \backslash l e f t\left(x^{\wedge}{-1} \backslash r i g h t\right)$$ \text { forall\$x, } y \in G, f \in \mathbb{k}(G) \$\text {, whereweidentify } \$ \mathbb{k}(G) \otimes \mathbb{k}(G)=\mathbb{k}(G \times G) \$\text {. Over } \$ \mathbb{C} \text { \$, wehaveaHopf } $$-代数与 f(x)=\overline{f(x)}. 一个左 \mathbb{k s}(G)-模代数是 G 分级代数 A 在哪里 f \triangleright a=f(g) a 什么时候 a \in A_{g}.个左 \mathbb{k s}(G)-comodule algebra 指代数 G 代数自同构的正确行 为，两者通过$$ \Delta_{L}(a)=\sum_{g \in G} \delta_{g} \otimes a \triangleleft g .$\$

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