### 数学代写|数论作业代写number theory代考|The Regular Hexagon

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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数论作业代写number theory代考|Symmetries and Spectra

Let $\mathcal{H}$ denote the interior of the regular hexagon with center at the origin, and sides of unit length. The diagonals $D_{i}, i=1,2,3$, joining opposite vertices, and the medians $M_{j}, j=1,2,3$, joining the mid-points of opposite sides, are lines of mirror symmetry of the hexagon $\mathcal{H}$, see Fig. 11 .

We consider the diagonals $D_{1}$ and $M_{2}$, and the associated mirror symmetries of $\mathcal{H}$. They commute,
$$M_{2} \circ D_{1}=D_{1} \circ M_{2}=R_{\pi}$$
and we can therefore apply the methods of Sect. 2.1.
It follows that $D_{1}^{}$ leaves the subspaces $\mathcal{S}{M{2} . \pm}$ globally invariant, and that $M_{2}^{}$ leaves the subspaces $\mathcal{S}{D{1}}, \pm$ globally invariant. As a consequence, we have the following orthogonal decomposition of $L^{2}(\mathcal{H})$,
$$L^{2}(\mathcal{H})=\mathcal{S}{+,+} \stackrel{\perp}{\oplus} \mathcal{S}{-,-} \stackrel{\perp}{\oplus} \mathcal{S}{+,-} \stackrel{\perp}{\oplus} \mathcal{S}{-,+},$$
where
$$\mathcal{S}{\sigma, \tau}:=\left{\phi \in L^{2}(\mathcal{H}) \mid D{1}^{} \phi=\sigma \phi \text { and } M_{2}^{} \phi=\tau \phi\right},$$
for $\sigma, \tau \in{+,-}$.
Similar decompositions hold for the Sobolev spaces $H^{1}(\mathcal{H})$ and $H_{0}^{1}(\mathcal{H})$, which are used in the variational presentation of the Neumann (resp. Dirichlet) eigenvalue

problem for the hexagon. Since the Laplacian commutes with the isometries $D_{1}$ and $M_{2}$, such decompositions also hold for the eigenspaces of $-\Delta$ in $\mathcal{H}$, with the boundary condition $\mathfrak{b} \in{\mathfrak{o}, \mathfrak{n}}$ on the boundary $\partial \mathcal{H}$.

In the following figures, anti-nodal lines are indicated by dashed lines, and nodal lines by solid lines. Figure 12 displays the nodal and anti-nodal lines common to all functions in $H^{1}(\mathcal{H}) \cap \mathcal{S}{\sigma, \tau}$, where $\sigma, \tau \in{+,-}$ Denote by $R$ the rotation $R{\frac{2 \pi}{3}}$,
$$\left{\begin{array}{l} R=D_{2} \circ D_{1}=M_{2} \circ M_{1}=\ldots \ R^{-1}=D_{1} \circ D_{2}=M_{1} \circ M_{2}=\ldots \end{array}\right.$$
This is an isometry of $\mathcal{H}$, and the action $R^{*}$ of $R$ on functions is an isometry of $L^{2}(\mathcal{H})$ with respect to the $L^{2}$-inner-product.

## 数学代写|数论作业代写number theory代考|Symmetries and Boundary Conditions on Sub-domains

Let $\mathcal{Q}$ (resp. $\mathcal{P}$ ) denote the interior of the quadrilateral (resp. the pentagon) which appears in Fig. 14. Let $\mathcal{R}$ (resp. $\mathcal{T}{h}$ ) denote the interior of the quadrilateral (resp. of the hemiequilateral triangle) which appears in Fig. 15. Then, $\overline{\mathcal{Q}}$ (resp. $\overline{\mathcal{P}}$ ) is a fundamental domain of the action of the mirror symmetry $D{1}$ (resp. $M_{2}$ ), and $\overline{\mathcal{R}}$ is a fundamental domain for the action of the group generated by $D_{1}$ and $M_{2}$.

The boundary decompositions for the domains $\mathcal{P}, \mathcal{Q}, \mathcal{R}$, and for the hemiequilateral triangle $\mathcal{T}_{h}$, are illustrated in Figs. 14 and $15 .$

Consider the eigenvalue problem $(\mathcal{H}, \mathfrak{c})$ for the hexagon, with $\mathfrak{c} \in{\mathfrak{n}, \mathfrak{o}}$. Let $\mathcal{E}(\mu, \mathrm{c})$ be an eigenspace of $-\Delta$ for $(\mathcal{H}, \mathrm{c})$. If $\phi \in \mathcal{E}(\mu, \mathrm{c}) \cap \mathcal{S}{\sigma, \tau}$, then the restriction $\phi \mid \mathcal{R}$, of the function $\phi$ to the domain $\mathcal{R}$, is an eigenfunction of $-\Delta$ in $(\mathcal{R}, \varepsilon(\sigma) \varepsilon(\tau) \mathfrak{c})$, where $$\varepsilon(+)=\mathfrak{n} \text { and } \varepsilon(-)=\mathfrak{o}$$ associated with the same eigenvalue $\mu$. Conversely, let $\psi$ be an eigenfunction of $-\Delta$ in $(\mathcal{R}, \mathfrak{a b c})$, associated with the eigenvalue $\mu$, where $\mathfrak{c}$ is the given boundary condition on $\partial \mathcal{H}$, and $\mathfrak{a}, \mathfrak{b} \in{\mathfrak{d}, \mathfrak{n}}$ are boundary conditions on the sides $M{2}, D_{1}$. Extend $\psi$ to a function $\breve{\psi}$ defined on $\mathcal{H}$, by symmetry (resp. anti-symmetry) with respect to $M_{2}$, if $\mathfrak{a}=\mathfrak{n}$ (resp. if $\mathfrak{a}=\mathfrak{d}$ ), and by symmetry (resp. anti-symmetry) with respect to $D_{1}$, if $\mathfrak{b}=\mathfrak{n}$ (resp. if $\mathfrak{b}=\mathfrak{d}$ ). Then, the function $\tilde{\psi}$ is an eigenfunction of $-\Delta$ for $(\mathcal{H}, \boldsymbol{c})$, associated with the eigenvalue $\mu$, and belongs to $\mathcal{S}_{\sigma, \tau}$ with $\varepsilon(\sigma)=\mathfrak{a}$ and $\varepsilon(\tau)=\mathfrak{b}$.

## 数学代写|数论作业代写number theory代考|Numerical Computations

Numerical approximations for the Dirichlet eigenvalues of the regular hexagon have been obtained by several authors, see for example $[5,13,16]$, or the recent paper [17].
The main idea, in order to make the identification of multiple Dirichlet eigenvalues of $\mathcal{H}$ easier, is to take the symmetries of $\mathcal{H}$ (see Sect. 3.2) into account from the start. For this purpose, one computes the eigenvalues of the domains $\mathcal{R}$ and $\mathcal{T}_{h}$, for mixed boundary conditions $\mathfrak{a} \mathfrak{b} \mathfrak{o}$, with $\mathfrak{a}, \mathfrak{b} \in{\mathfrak{d}, \mathfrak{n}}$.

Table 8 displays the first four eigenvalues of $(\mathcal{R}, \mathfrak{a} \mathfrak{b})$, as computed with MATLAB, and contains some useful relations between these eigenvalues.

Remark 3.7 The eigenvalues in Table 8 are partially ordered ‘vertically’. Indeed, for $i \geq 1$, we have the strict inequalities,
$$\left{\begin{array}{l} \mu_{i}(\mathcal{R}, \mathfrak{n n d})<\mu_{i}(\mathcal{R}, \mathfrak{d} \mathfrak{D} \mathfrak{J})<\mu_{i}(\mathcal{R}, \mathfrak{J} \mathfrak{J}) \ \mu_{i}\left(\mathcal{R}, \mathfrak{n n \mathfrak { J } )}<\mu_{i}(\mathcal{R}, \mathfrak{n d J})<\mu_{i}(\mathcal{R}, \mathfrak{d} \mathfrak{J})\right. \end{array}\right.$$
which follow from Proposition 2.2, see [22, Proposition 2.3]. These inequalities are indicated in the table by the (rotated) strict inequality signs. Note that it is general not possible to compare the eigenvalues $\mu_{i}(\mathcal{R}, \mathfrak{d} \mathfrak{n d})$ and $\mu_{i}(\mathcal{R}, \mathfrak{n d} \mathfrak{J})$. This is indicated in the table by the black question marks.

Table 9 displays some eigenvalues of $\left(\mathcal{T}_{h}, \mathfrak{a} \mathfrak{b} \mathfrak{d}\right)$, for $\mathfrak{a}, \mathfrak{b} \in{\mathfrak{d}, \mathfrak{n}}$. The lower bound in the second line follows from Dirichlet monotonicity (see Sect. 3.3.2). In the third line, we have used the fact due to Pólya (see [19]) that the first Dirichlet eigenvalue of a kite-shape is bounded from below by the first Dirichlet eigenvalue of a square with the same area. In the last two lines, the eigenvalues are known explicitly.

## 数学代写|数论作业代写number theory代考|Symmetries and Spectra

\mathcal{S}{\sigma, \tau}:=\left{\phi \in L^{2}(\mathcal{H}) \mid D{1}^{} \phi=\sigma \phi \文本 { 和 } M_{2}^{} \phi=\tau \phi\right}，\mathcal{S}{\sigma, \tau}:=\left{\phi \in L^{2}(\mathcal{H}) \mid D{1}^{} \phi=\sigma \phi \文本 { 和 } M_{2}^{} \phi=\tau \phi\right}，

$$\左{ R=D2∘D1=米2∘米1=… R−1=D1∘D2=米1∘米2=…\正确的。$$

## 数学代写|数论作业代写number theory代考|Symmetries and Boundary Conditions on Sub-domains

e(+)=n 和 e(−)=○与相同的特征值相关联μ. 反之，让ψ是一个特征函数−Δ在(R,一个bC)，与特征值相关联μ， 在哪里C是给定的边界条件∂H， 和一个,b∈d,n是边上的边界条件米2,D1. 延长ψ到一个函数ψ˘定义于H，通过对称性（分别是反对称）关于米2， 如果一个=n（分别如果一个=d），并通过对称（resp.反对称）相对于D1， 如果b=n（分别如果b=d）。然后，函数ψ~是一个特征函数−Δ为了(H,C)，与特征值相关联μ, 并且属于小号σ,τ和e(σ)=一个和e(τ)=b.

## 数学代写|数论作业代写number theory代考|Numerical Computations

$$\left{ μ一世(R,nnd)<μ一世(R,dDĴ)<μ一世(R,ĴĴ) μ一世(R,nnĴ)<μ一世(R,ndĴ)<μ一世(R,dĴ)\正确的。$$

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## MATLAB代写

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