### 计算机代写|量子计算代写Quantum computing代考|Ternary S/D Trees

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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

## 计算机代写|量子计算代写Quantum computing代考|their Inclusive Forms

The following Sect. defines the ternary Shannon and ternary Davio decision trees over GF(3). As analogous to the binary case, we can have expansions that are mixed of Shannon (S) for certain variables and Davio $\left(D_{0}, D_{1}\right.$, and $\left.D_{2}\right)$ for the other variables. This will lead, analogously to the binary case, to the Kronecker TDT. Moreover,the mixed expansions can be extended to include Pseudo Kronecker TDT. (Full discussion of these TDTs that correspond to various expansions, as well as their hierarchy will be included in Sect. 3.5). The basic $S, D_{0}, D_{1}$, and $D_{2}$ ternary expansions (i.e., flattened forms) over GF(3) can be represented in Ternary DTs (TDTs) and the corresponding varieties of Ternary DDs (TDDs) (according to the corresponding reduction rules that are used). For one variable (one level), Fig. $3.3$ represents the expansion nodes for $S, D_{0}, D_{1}$, and $D_{2}$, respectively.

## 计算机代写|量子计算代写Quantum computing代考|Ternary S/D trees and Inclusive Forms

In correspondence to the binary S/D trees, we can produce the Ternary S/D Trees. To define the Ternary S/D Trees we will define the Generalized Davio expansion over GF(3) as shown in Fig. 3.4:

Our notation here is that $(x)$ corresponds to the three possible shifts of the variable $x$ as follows:
$$x \in\left{x, x^{\prime}, x^{\prime \prime}\right} \text { over } G F(3) .$$

Definition 3.1. The ternary tree with ternary Shannon and ternary Generalized Davio expansion nodes, that generates other ternary trees, is called the Ternary Shannon/Davio (S/D) tree.

Utilizing the definition of ternary Shannon (Fig. 3.3a) and ternary generalized Davio (Fig. 3.4), we obtain the ternary Shannon/Davio trees (ternary S/D trees) for two variables as shown in Fig. 3.5. From the ternary S/D DTs shown in Fig. 3.5, if we take any S/D tree and multiply the second-level cofactors (which are in the TDT leaves) each by the corresponding path in that TDT, and sum all the resulting cubes (terms) over GF(3), we obtain the flattened form of the function $\mathrm{f}$, as a certain GFSOP expression. For each TDT in Fig. 3.5, there are as many forms obtained for the function $f$ as the number of possible permutations of the polarities of the variables in the second-level branches of each TDT.

## 计算机代写|量子计算代写Quantum computing代考|Enumeration of Ternary Inclusive Forms

Each of the S/D trees shown in Fig. $3.5$ is a generator of a set of flattened forms (TIFs). Each one of these TIFs is merely a Kronecker-based transform as can be obtained from Eqs. (2.23) through (2.26). The numbers of these TIFs generated by the corresponding S/D trees are shown on the top of each S/D tree for two variables in Fig. 3.5.
Example 3.2.
3.2a. For the S/D trees in Fig. 3.5a, and by utilizing the notation from Eq. (3.2), we obtain for Figs. 3.6a and 3.7a, the ternary trees in Figs. $3.6 \mathrm{~b}-3.6 \mathrm{~d}$ and Figs. $3.7 \mathrm{~b}-3.7 \mathrm{~d}$, respectively.

3.2b. Let us produce some of the ternary trees for the S/D tree in Fig. 3.5b. Utilizing the notation from Eq. (3.2), we obtain, for the S/D tree in Fig. 3.8a, the ternary trees in Figs. 3.8b, 3.8c, and $3.8 \mathrm{~d}$, respectively.
The generalized IFs (GIFs) can be defined as the union of both IFs.
Definition 3.3. The family of forms, which is created as a union of sets of TIFs for all variable orders, is called Ternary Generalized Inclusive Forms (TGIFs).

## 计算机代写|量子计算代写Quantum computing代考|Ternary S/D trees and Inclusive Forms

x \in\left{x, x^{\prime}, x^{\prime \prime}\right} \text { over } GF(3) 。


## 计算机代写|量子计算代写Quantum computing代考|Enumeration of Ternary Inclusive Forms

3.2a。对于图 3.5a 中的 S/D 树，并利用公式中的符号。（3.2），我们得到图。3.6a和3.7a，图3中的三叉树。3.6 b−3.6 d和无花果。3.7 b−3.7 d， 分别。

3.2b。让我们为图 3.5b 中的 S/D 树生成一些三叉树。利用方程式中的符号。(3.2)，对于图 3.8a 中的 S/D 树，我们得到图 3.8a 中的三叉树。3.8b、3.8c 和3.8 d， 分别。

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## MATLAB代写

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