You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'With our recently proposed effective Hamiltonian via Monte Carlo, we are able to compute low energy physics of quantum systems. The advantage is that we can obtain not only the spectrum of ground and excited states, but also wave functions. The previous work has shown the success of this method in (1+1)-dimensional quantum mechanical systems. In this work we apply it to higher dimensional systems.'
address:
- |
    Department of Physics, Zhongshan University, Guangzhou 510275, China\
    and National Center for Theoretical Sciences, Hsinchu, Taiwan 30043, China
- 'Department of Physics, Foshan Science and Technology College, Foshan 528000, China'
- 'Department of Physics, Guangdong Institute of Education, Guangzhou 510303, China'
- 'Département de Physique, Université Laval, Québec, Québec G1K 7P4, Canada'
- 'Department of Mathematics, Statistics and Computational Science, Dalhousie University, Halifax, Nova Scotia B3H 3J5, Canada'
author:
- 'Xiang-Qian Luo , C.Q. Huang , J.Q. Jiang , H. Jirari $^{\rm{c}}$, H. Kröger , K. J.M. Moriarty'
title: Quantum Theory with Many Degrees of Freedom from Monte Carlo Hamiltonian
---

Introduction
============

Monte Carlo (MC) method with importance sampling is an excellent technique to calculate high dimensional integrals, and has successfully been applied to Lagrangian lattice gauge theory. Recently, we proposed an algorithm to compute the MC Hamiltonian [@mch]. The advantage, in comparison with the standard Lagrangian MC approach, is that one can obtain not only the spectrum of ground and excited states, but also the wave functions. The method has been tested in quantum mechanics (QM) in 1+1 dimensions, where the exact results were reproduced with high precision [@mch; @lat99_2; @Huang]. In this paper, we apply this algorithm to QM in 2+1 dimensions. The motivation is as follows. (a) A number of algorithms for solving the Schrödinger equation in lower dimensions (e.g. Runge-Kutta) do not work in 2+1 dimensions. Our method works in 2+1 dimensions, it is likely to work also in higher dimensions. (b) The ultimate goal of the algorithm is to study structure function, the $S$ matrix, QCD at finite density as well as systems with many degrees of freedom. QM in 2+1 dimensions is a simple example.

Algorithm
=========

Let’s review briefly the basic ideas [@mch]. According to Feynman’s path integral approach to QM, the (imaginary time) transition amplitude between an initial state at position $x_i$, and time $t_i$, and final state at $x_f$, $t_f$ is related to the Hamiltonian $H$ by $$\begin{aligned}
<x_{f},t_{f} | x_{i},t_{i}> 
= <x_{f} | e^{-H(t_{f}-t_{i})/\hbar} | x_{i}>\end{aligned}$$ $$\begin{aligned}
= \sum_{\nu=1}^{\infty} <x_{f} | E_{\nu} > 
e^{-E_{\nu} T/\hbar} < E_{\nu} | x_{i}>,\end{aligned}$$ where $T=t_f-t_i$. The starting point of our method, as described in detail in [@mch] is to construct an effective Hamiltonian $H_{eff}$ (finite $N \times N$ matrix) by $$\begin{aligned}
<x_{f} | e^{-H(t_{f}-t_{i})/\hbar} | x_{i}> \approx 
<x_{f} | e^{-H_{eff} T/\hbar} | x_{i}>\end{aligned}$$ $$\begin{aligned}
=\sum_{\nu=1}^{N} < x_{f} | E^{eff}_{\nu}> e^{-E^{eff}_{\nu} T/\hbar} 
< E^{eff}_{\nu} | x_{i} > .\end{aligned}$$ $H_{eff}$ can be found by MC simulation using the following procedure:

\(a) Discretize the continuous time.

\(b) Generate configurations $[x]$ obeying the Boltzmann distribution $$\begin{aligned}
P(x) = { \exp(- S[x]/ \hbar)  \over
       \int [dx] \exp(- S[x]/ \hbar)}.\end{aligned}$$ (c) Calculate the transition matrix elements $$\begin{aligned}
M_{fi} = 
<x_{f} | e^{-H_{eff} T/\hbar} | x_{i}>\end{aligned}$$ between $N$ discrete $x_i$ points and $N$ $x_f$ points. Note that the matrix $M$ is symmetric.

\(d) Diagonalize $M$ by a unitary transformation $$\begin{aligned}
M &=& U^{\dagger}DU,\end{aligned}$$ where $D =diag (e^{-E^{eff}_{1}T/\hbar},..., e^{-E^{eff}_{N}T/\hbar})$. Steps (a) and (b) are the same as the standard MC method. Step (c) is the essential ingredient of our method, from which we can construct $H_{eff}$, and obtain the eigenvalues $E^{eff}_{\nu}$ and wavefunction $\vert E^{eff}_{\nu} >$ through step (d). Once the spectrum and wave functions are available, all physical information can be retrieved. Since the theory described by $H$ is now approximated by a theory described by a finite matrix $H_{eff}$, the physics of $H$ and $H_{eff}$ might be quite different at high energy. Therefore we expect that we can only reproduce the low energy physics of the system. This is good enough for our purpose.

Results
=======

We consider the following examples in $D=2$:

Uncoupled harmonic oscillator
-----------------------------

The Euclidean action is given by $$\begin{aligned}
S={\int_{0}^{T}{dt}\left[{m \over 2}(\dot{x}^2+\dot{y}^2)
+{m{\omega}^2 \over 2} (x^2+y^2) \right]}.\end{aligned}$$ The spectrum is degenerate and exactly known: $$\begin{aligned}
E_{n_1,n_2}=\omega \hbar (n_1+n_2 +1),~n_1,n_2 = 0,1,2,\ldots
\label{Uncoupled}\end{aligned}$$

Coupled harmonic oscillator
---------------------------

The Euclidean action is given by $$\begin{aligned}
S={\int_{0}^{T}{dt}\left[{m \over 2}(\dot{x}^2+\dot{y}^2)+{m{\omega}^2 
\over 2}(x^2+y^2)
+ \lambda x y \right]},\end{aligned}$$ $$\begin{aligned}
\rm{for} ~ 0<\lambda < m \omega^2.\end{aligned}$$ The spectrum is also exactly known: $$\begin{aligned}
E_{n_1,n_2}=\omega_1 \hbar (n_1+{1\over 2})
+\omega_2 \hbar (n_2+{1\over 2}),\end{aligned}$$ $$\begin{aligned}
n_1,n_2= 0,1,2,\ldots,\end{aligned}$$ where $\omega_1=\sqrt{\omega^2+{\lambda \over m}}$ and $\omega_2=\sqrt{\omega^2-{\lambda \over m}}$.

Summary
-------

Tabs. 1 and 2 compare the spectrum from $H_{eff}$ with the exact results. Figs. 1-4 show the first two wave functions. They are in very good agreement with the exact ones, at least in the low energy domain. In Ref. [@Jiang], more results for wave functions of higher execited states, and thermodynamical quantities such as the average energy and specific heat, were reported. However, the CPU time is much longer than in 1-dimensional systems. We need to find a better (stochastic) way to select the basis in Hilbert space in step (c) of Sect. 2, before extending the method to quantum field theory. We are curious to see if our method can resolve the long standing problem of QCD at finite density. It is worth mentioning the recent analytical [@Luo] and numerical [@creutz] efforts in Hamiltonian formulation.

X.Q.L. is supported by the National Science Fund for Distinguished Young Scholars, supplemented by National Natural Science Foundation, fund for international cooperation and exchange, Ministry of Education, and Hong Kong Foundation of Zhongshan University Advanced Research Center. We also thank the Lat’99 organizers and NCTS, Taiwan for partial support. H.K. would like to acknowledge support by NSERC Canada.

[99]{}

H. Jirari, H. Kröger, X.Q. Luo, K. Moriarty, [*Phys. Lett.*]{} [**A258**]{} (1999) 6.

H. Kröger, these proceedings.

C.Q. Huang [*et al.*]{}, in preparation.

J.Q. Jiang [*et al.*]{}, in preparation.

X.Q. Luo [*et al.*]{}, in preparation.

M. Creutz, hep-lat/9905024.

   $n_1, n_2$     $E_{n_1n_2}^{MC}$    $E_{n_1n_2}^{Exact}$
  ------------- --------------------- ----------------------
      0, 0       0.9975 $\pm$ 0.0258          1.0000
      0, 1       1.9902 $\pm$ 0.0420          2.0000
      1, 0       2.0022 $\pm$ 0.0468          2.0000
      0, 2       2.9499 $\pm$ 0.0393          3.0000
      2, 0       2.9815 $\pm$ 0.0550          3.0000
      1, 1       3.0014 $\pm$ 0.0546          3.0000
   $n_1+n_2=3$   3.9718 $\pm$ 0.0496          4.0000
   $n_1+n_2=3$   3.9853 $\pm$ 0.0626          4.0000
   $n_1+n_2=3$   4.0547 $\pm$ 0.0611          4.0000
   $n_1+n_2=3$   4.0713 $\pm$ 0.0851          4.0000

  : Spectrum of [*uncoupled*]{} hamonic oscilator.

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(930,558)(0,0) =cmr10 at 10pt (220.0,113.0)

------------------------------------------------------------------------

(198,113)[(0,0)\[r\][-0.6]{}]{} (846.0,113.0)

------------------------------------------------------------------------

(220.0,221.0)

------------------------------------------------------------------------

(198,221)[(0,0)\[r\][-0.4]{}]{} (846.0,221.0)

------------------------------------------------------------------------

(220.0,328.0)

------------------------------------------------------------------------

(198,328)[(0,0)\[r\][-0.2]{}]{} (846.0,328.0)

------------------------------------------------------------------------

(220.0,436.0)

------------------------------------------------------------------------

(198,436)[(0,0)\[r\][0]{}]{} (846.0,436.0)

------------------------------------------------------------------------

(274.0,113.0)

------------------------------------------------------------------------

(274,68)[(0,0)[-5]{}]{} (274.0,470.0)

------------------------------------------------------------------------

(382.0,113.0)

------------------------------------------------------------------------

(382,68)[(0,0)[-3]{}]{} (382.0,470.0)

------------------------------------------------------------------------

(489.0,113.0)

------------------------------------------------------------------------

(489,68)[(0,0)[-1]{}]{} (489.0,470.0)

------------------------------------------------------------------------

(597.0,113.0)

------------------------------------------------------------------------

(597,68)[(0,0)[1]{}]{} (597.0,470.0)

------------------------------------------------------------------------

(705.0,113.0)

------------------------------------------------------------------------

(705,68)[(0,0)[3]{}]{} (705.0,470.0)

------------------------------------------------------------------------

(812.0,113.0)

------------------------------------------------------------------------

(812,68)[(0,0)[5]{}]{} (812.0,470.0)

------------------------------------------------------------------------

(220.0,113.0)

------------------------------------------------------------------------

(866.0,113.0)

------------------------------------------------------------------------

(220.0,490.0)

------------------------------------------------------------------------

(45,301)[(0,0)[$\psi_{01}$]{}]{} (543,23)[(0,0)[$x$]{}]{} (543,535)[(0,0)[$T=\omega=m=\hbar=1, N=10 \times 10$]{}]{} (220.0,113.0)

------------------------------------------------------------------------

(736,325)[(0,0)\[r\][$Exact$]{}]{} (758.0,325.0)

------------------------------------------------------------------------

(220,436) (351,434.67)

------------------------------------------------------------------------

(351.00,435.17)(3.000,-1.000)[2]{}

------------------------------------------------------------------------

(220.0,436.0)

------------------------------------------------------------------------

(370,433.67)

------------------------------------------------------------------------

(370.00,434.17)(3.500,-1.000)[2]{}

------------------------------------------------------------------------

(377,432.67)

------------------------------------------------------------------------

(377.00,433.17)(3.000,-1.000)[2]{}

------------------------------------------------------------------------

(383,431.67)

------------------------------------------------------------------------

(383.00,432.17)(3.500,-1.000)[2]{}

------------------------------------------------------------------------

(390,430.17)

------------------------------------------------------------------------

(390.00,431.17)(3.302,-2.000)[2]{}

------------------------------------------------------------------------

(396.00,428.95)(1.355,-0.447)[3]{}

------------------------------------------------------------------------

(396.00,429.17)(4.855,-3.000)[2]{}

------------------------------------------------------------------------

(403.00,425.95)(1.132,-0.447)[3]{}

------------------------------------------------------------------------

(403.00,426.17)(4.132,-3.000)[2]{}

------------------------------------------------------------------------

(409.00,422.94)(0.920,-0.468)[5]{}

------------------------------------------------------------------------

(409.00,423.17)(5.340,-4.000)[2]{}

------------------------------------------------------------------------

(416.00,418.93)(0.599,-0.477)[7]{}

------------------------------------------------------------------------

(416.00,419.17)(4.796,-5.000)[2]{}

------------------------------------------------------------------------

(422.00,413.93)(0.581,-0.482)[9]{}

------------------------------------------------------------------------

(422.00,414.17)(5.824,-6.000)[2]{}

------------------------------------------------------------------------

(429.59,406.37)(0.482,-0.671)[9]{}

------------------------------------------------------------------------

(428.17,407.69)(6.000,-6.685)[2]{}

------------------------------------------------------------------------

(435.59,398.45)(0.485,-0.645)[11]{}

------------------------------------------------------------------------

(434.17,399.73)(7.000,-7.725)[2]{}

------------------------------------------------------------------------

(442.59,388.54)(0.482,-0.943)[9]{}

------------------------------------------------------------------------

(441.17,390.27)(6.000,-9.270)[2]{}

------------------------------------------------------------------------

(448.59,377.50)(0.485,-0.950)[11]{}

------------------------------------------------------------------------

(447.17,379.25)(7.000,-11.251)[2]{}

------------------------------------------------------------------------

(455.59,363.43)(0.482,-1.304)[9]{}

------------------------------------------------------------------------

(454.17,365.72)(6.000,-12.717)[2]{}

------------------------------------------------------------------------

(461.59,348.79)(0.485,-1.179)[11]{}

------------------------------------------------------------------------

(460.17,350.89)(7.000,-13.895)[2]{}

------------------------------------------------------------------------

(468.59,331.88)(0.482,-1.485)[9]{}

------------------------------------------------------------------------

(467.17,334.44)(6.000,-14.440)[2]{}

------------------------------------------------------------------------

(474.59,315.32)(0.485,-1.332)[11]{}

------------------------------------------------------------------------

(473.17,317.66)(7.000,-15.658)[2]{}

------------------------------------------------------------------------

(481.59,297.08)(0.485,-1.408)[11]{}

------------------------------------------------------------------------

(480.17,299.54)(7.000,-16.539)[2]{}

------------------------------------------------------------------------

(488.59,277.33)(0.482,-1.666)[9]{}

------------------------------------------------------------------------

(487.17,280.16)(6.000,-16.163)[2]{}

------------------------------------------------------------------------

(494.59,259.08)(0.485,-1.408)[11]{}

------------------------------------------------------------------------

(493.17,261.54)(7.000,-16.539)[2]{}

------------------------------------------------------------------------

(501.59,239.60)(0.482,-1.575)[9]{}

------------------------------------------------------------------------

(500.17,242.30)(6.000,-15.302)[2]{}

------------------------------------------------------------------------

(507.59,223.03)(0.485,-1.103)[11]{}

------------------------------------------------------------------------

(506.17,225.01)(7.000,-13.013)[2]{}

------------------------------------------------------------------------

(514.59,207.71)(0.482,-1.214)[9]{}

------------------------------------------------------------------------

(513.17,209.86)(6.000,-11.855)[2]{}

------------------------------------------------------------------------

(520.59,194.98)(0.485,-0.798)[11]{}

------------------------------------------------------------------------

(519.17,196.49)(7.000,-9.488)[2]{}

------------------------------------------------------------------------

(527.59,184.65)(0.482,-0.581)[9]{}

------------------------------------------------------------------------

(526.17,185.82)(6.000,-5.824)[2]{}

------------------------------------------------------------------------

(533.00,178.94)(0.920,-0.468)[5]{}

------------------------------------------------------------------------

(533.00,179.17)(5.340,-4.000)[2]{}

------------------------------------------------------------------------

(357.0,435.0)

------------------------------------------------------------------------

(546.00,176.60)(0.920,0.468)[5]{}

------------------------------------------------------------------------

(546.00,175.17)(5.340,4.000)[2]{}

------------------------------------------------------------------------

(553.59,180.00)(0.482,0.581)[9]{}

------------------------------------------------------------------------

(552.17,180.00)(6.000,5.824)[2]{}

------------------------------------------------------------------------

(559.59,187.00)(0.485,0.798)[11]{}

------------------------------------------------------------------------

(558.17,187.00)(7.000,9.488)[2]{}

------------------------------------------------------------------------

(566.59,198.00)(0.482,1.214)[9]{}

------------------------------------------------------------------------

(565.17,198.00)(6.000,11.855)[2]{}

------------------------------------------------------------------------

(572.59,212.00)(0.485,1.103)[11]{}

------------------------------------------------------------------------

(571.17,212.00)(7.000,13.013)[2]{}

------------------------------------------------------------------------

(579.59,227.00)(0.482,1.575)[9]{}

------------------------------------------------------------------------

(578.17,227.00)(6.000,15.302)[2]{}

------------------------------------------------------------------------

(585.59,245.00)(0.485,1.408)[11]{}

------------------------------------------------------------------------

(584.17,245.00)(7.000,16.539)[2]{}

------------------------------------------------------------------------

(592.59,264.00)(0.482,1.666)[9]{}

------------------------------------------------------------------------

(591.17,264.00)(6.000,16.163)[2]{}

------------------------------------------------------------------------

(598.59,283.00)(0.485,1.408)[11]{}

------------------------------------------------------------------------

(597.17,283.00)(7.000,16.539)[2]{}

------------------------------------------------------------------------

(605.59,302.00)(0.485,1.332)[11]{}

------------------------------------------------------------------------

(604.17,302.00)(7.000,15.658)[2]{}

------------------------------------------------------------------------

(612.59,320.00)(0.482,1.485)[9]{}

------------------------------------------------------------------------

(611.17,320.00)(6.000,14.440)[2]{}

------------------------------------------------------------------------

(618.59,337.00)(0.485,1.179)[11]{}

------------------------------------------------------------------------

(617.17,337.00)(7.000,13.895)[2]{}

------------------------------------------------------------------------

(625.59,353.00)(0.482,1.304)[9]{}

------------------------------------------------------------------------

(624.17,353.00)(6.000,12.717)[2]{}

------------------------------------------------------------------------

(631.59,368.00)(0.485,0.950)[11]{}

------------------------------------------------------------------------

(630.17,368.00)(7.000,11.251)[2]{}

------------------------------------------------------------------------

(638.59,381.00)(0.482,0.943)[9]{}

------------------------------------------------------------------------

(637.17,381.00)(6.000,9.270)[2]{}

------------------------------------------------------------------------

(644.59,392.00)(0.485,0.645)[11]{}

------------------------------------------------------------------------

(643.17,392.00)(7.000,7.725)[2]{}

------------------------------------------------------------------------

(651.59,401.00)(0.482,0.671)[9]{}

------------------------------------------------------------------------

(650.17,401.00)(6.000,6.685)[2]{}

------------------------------------------------------------------------

(657.00,409.59)(0.581,0.482)[9]{}

------------------------------------------------------------------------

(657.00,408.17)(5.824,6.000)[2]{}

------------------------------------------------------------------------

(664.00,415.59)(0.599,0.477)[7]{}

------------------------------------------------------------------------

(664.00,414.17)(4.796,5.000)[2]{}

------------------------------------------------------------------------

(670.00,420.60)(0.920,0.468)[5]{}

------------------------------------------------------------------------

(670.00,419.17)(5.340,4.000)[2]{}

------------------------------------------------------------------------

(677.00,424.61)(1.132,0.447)[3]{}

------------------------------------------------------------------------

(677.00,423.17)(4.132,3.000)[2]{}

------------------------------------------------------------------------

(683.00,427.61)(1.355,0.447)[3]{}

------------------------------------------------------------------------

(683.00,426.17)(4.855,3.000)[2]{}

------------------------------------------------------------------------

(690,430.17)

------------------------------------------------------------------------

(690.00,429.17)(3.302,2.000)[2]{}

------------------------------------------------------------------------

(696,431.67)

------------------------------------------------------------------------

(696.00,431.17)(3.500,1.000)[2]{}

------------------------------------------------------------------------

(703,432.67)

------------------------------------------------------------------------

(703.00,432.17)(3.000,1.000)[2]{}

------------------------------------------------------------------------

(709,433.67)

------------------------------------------------------------------------

(709.00,433.17)(3.500,1.000)[2]{}

------------------------------------------------------------------------

(540.0,176.0)

------------------------------------------------------------------------

(729,434.67)

------------------------------------------------------------------------

(729.00,434.17)(3.000,1.000)[2]{}

------------------------------------------------------------------------

(716.0,435.0)

------------------------------------------------------------------------

(735.0,436.0)

------------------------------------------------------------------------

(736,380)[(0,0)\[r\][$MC$]{}]{} (780,380) (274,436) (328,436) (382,433) (435,401) (489,278) (543,174) (597,278) (651,401) (705,433) (758,436) (812,436) (758.0,380.0)

------------------------------------------------------------------------

(758.0,370.0)

------------------------------------------------------------------------

(824.0,370.0)

------------------------------------------------------------------------

(274,436) (264.0,436.0)

------------------------------------------------------------------------

(264.0,436.0)

------------------------------------------------------------------------

(328,436) (318.0,436.0)

------------------------------------------------------------------------

(318.0,436.0)

------------------------------------------------------------------------

(382,433) (372.0,433.0)

------------------------------------------------------------------------

(372.0,433.0)

------------------------------------------------------------------------

(435.0,401.0) (425.0,401.0)

------------------------------------------------------------------------

(425.0,402.0)

------------------------------------------------------------------------

(489.0,278.0) (479.0,278.0)

------------------------------------------------------------------------

(479.0,279.0)

------------------------------------------------------------------------

(543.0,172.0)

------------------------------------------------------------------------

(533.0,172.0)

------------------------------------------------------------------------

(533.0,176.0)

------------------------------------------------------------------------

(597.0,278.0) (587.0,278.0)

------------------------------------------------------------------------

(587.0,279.0)

------------------------------------------------------------------------

(651.0,401.0) (641.0,401.0)

------------------------------------------------------------------------

(641.0,402.0)

------------------------------------------------------------------------

(705,433) (695.0,433.0)

------------------------------------------------------------------------

(695.0,433.0)

------------------------------------------------------------------------

(758,436) (748.0,436.0)

------------------------------------------------------------------------

(748.0,436.0)

------------------------------------------------------------------------

(812,436) (802.0,436.0)

------------------------------------------------------------------------

(802.0,436.0)

------------------------------------------------------------------------

   $n_1, n_2$    $E_{n_1n_2}^{MC}$    $E_{n_1n_2}^{Exact}$
  ------------ --------------------- ----------------------
      0, 0      0.9640 $\pm$ 0.0245          0.9659
      0, 1      1.6693 $\pm$ 0.0335          1.6730
      1, 0      2.1843 $\pm$ 0.0535          2.1907
      0, 2      2.3650 $\pm$ 0.0363          2.3801
      1, 1      2.8818 $\pm$ 0.0553          2.8978
      0, 3      3.0883 $\pm$ 0.0447          3.0872
      0, 3      3.0883 $\pm$ 0.0447          3.0872
      2, 0      3.4049 $\pm$ 0.0532          3.4154
      1, 2      3.6275 $\pm$ 0.0526          3.6049
      0, 4      3.7414 $\pm$ 0.0327          3.7944

  : Spectrum of [*coupled*]{} hamonic oscilator.

(930,558)(0,0) =cmr10 at 10pt (220.0,128.0)

------------------------------------------------------------------------

(198,128)[(0,0)\[r\][0]{}]{} (846.0,128.0)

------------------------------------------------------------------------

(220.0,200.0)

------------------------------------------------------------------------

(198,200)[(0,0)\[r\][0.1]{}]{} (846.0,200.0)

------------------------------------------------------------------------

(220.0,273.0)

------------------------------------------------------------------------

(198,273)[(0,0)\[r\][0.2]{}]{} (846.0,273.0)

------------------------------------------------------------------------

(220.0,345.0)

------------------------------------------------------------------------

(198,345)[(0,0)\[r\][0.3]{}]{} (846.0,345.0)

------------------------------------------------------------------------

(220.0,418.0)

------------------------------------------------------------------------

(198,418)[(0,0)\[r\][0.4]{}]{} (846.0,418.0)

------------------------------------------------------------------------

(220.0,490.0)

------------------------------------------------------------------------

(198,490)[(0,0)\[r\][0.5]{}]{} (846.0,490.0)

------------------------------------------------------------------------

(274.0,113.0)

------------------------------------------------------------------------

(274,68)[(0,0)[-5]{}]{} (274.0,470.0)

------------------------------------------------------------------------

(328.0,113.0)

------------------------------------------------------------------------

(328,68)[(0,0)[-4]{}]{} (328.0,470.0)

------------------------------------------------------------------------

(382.0,113.0)

------------------------------------------------------------------------

(382,68)[(0,0)[-3]{}]{} (382.0,470.0)

------------------------------------------------------------------------

(435.0,113.0)

------------------------------------------------------------------------

(435,68)[(0,0)[-2]{}]{} (435.0,470.0)

------------------------------------------------------------------------

(489.0,113.0)

------------------------------------------------------------------------

(489,68)[(0,0)[-1]{}]{} (489.0,470.0)

------------------------------------------------------------------------

(543.0,113.0)

------------------------------------------------------------------------

(543,68)[(0,0)[0]{}]{} (543.0,470.0)

------------------------------------------------------------------------

(597.0,113.0)

------------------------------------------------------------------------

(597,68)[(0,0)[1]{}]{} (597.0,470.0)

------------------------------------------------------------------------

(651.0,113.0)

------------------------------------------------------------------------

(651,68)[(0,0)[2]{}]{} (651.0,470.0)

------------------------------------------------------------------------

(705.0,113.0)

------------------------------------------------------------------------

(705,68)[(0,0)[3]{}]{} (705.0,470.0)

------------------------------------------------------------------------

(758.0,113.0)

------------------------------------------------------------------------

(758,68)[(0,0)[4]{}]{} (758.0,470.0)

------------------------------------------------------------------------

(812.0,113.0)

------------------------------------------------------------------------

(812,68)[(0,0)[5]{}]{} (812.0,470.0)

------------------------------------------------------------------------

(220.0,113.0)

------------------------------------------------------------------------

(866.0,113.0)

------------------------------------------------------------------------

(220.0,490.0)

------------------------------------------------------------------------

(45,301)[(0,0)[$\psi_{00}$]{}]{} (543,23)[(0,0)[$x$]{}]{} (488,535)[(0,0)[$T=\omega=m=\hbar=1,\lambda=1/2, N=10 \times 10$]{}]{} (220.0,113.0)

------------------------------------------------------------------------

(736,425)[(0,0)\[r\][$Exact$]{}]{} (758.0,425.0)

------------------------------------------------------------------------

(220,128) (370,127.67)

------------------------------------------------------------------------

(370.00,127.17)(3.500,1.000)[2]{}

------------------------------------------------------------------------

(220.0,128.0)

------------------------------------------------------------------------

(383,128.67)

------------------------------------------------------------------------

(383.00,128.17)(3.500,1.000)[2]{}

------------------------------------------------------------------------

(390,129.67)

------------------------------------------------------------------------

(390.00,129.17)(3.000,1.000)[2]{}

------------------------------------------------------------------------

(396,130.67)

------------------------------------------------------------------------

(396.00,130.17)(3.500,1.000)[2]{}

------------------------------------------------------------------------

(403,132.17)

------------------------------------------------------------------------

(403.00,131.17)(3.302,2.000)[2]{}

------------------------------------------------------------------------

(409,134.17)

------------------------------------------------------------------------

(409.00,133.17)(3.887,2.000)[2]{}

------------------------------------------------------------------------

(416.00,136.60)(0.774,0.468)[5]{}

------------------------------------------------------------------------

(416.00,135.17)(4.547,4.000)[2]{}

------------------------------------------------------------------------

(422.00,140.60)(0.920,0.468)[5]{}

------------------------------------------------------------------------

(422.00,139.17)(5.340,4.000)[2]{}

------------------------------------------------------------------------

(429.00,144.59)(0.599,0.477)[7]{}

------------------------------------------------------------------------

(429.00,143.17)(4.796,5.000)[2]{}

------------------------------------------------------------------------

(435.00,149.59)(0.581,0.482)[9]{}

------------------------------------------------------------------------

(435.00,148.17)(5.824,6.000)[2]{}

------------------------------------------------------------------------

(442.59,155.00)(0.482,0.581)[9]{}

------------------------------------------------------------------------

(441.17,155.00)(6.000,5.824)[2]{}

------------------------------------------------------------------------

(448.59,162.00)(0.485,0.645)[11]{}

------------------------------------------------------------------------

(447.17,162.00)(7.000,7.725)[2]{}

------------------------------------------------------------------------

(455.59,171.00)(0.482,0.943)[9]{}

------------------------------------------------------------------------

(454.17,171.00)(6.000,9.270)[2]{}

------------------------------------------------------------------------

(461.59,182.00)(0.485,0.874)[11]{}

------------------------------------------------------------------------

(460.17,182.00)(7.000,10.369)[2]{}

------------------------------------------------------------------------

(468.59,194.00)(0.482,1.214)[9]{}

------------------------------------------------------------------------

(467.17,194.00)(6.000,11.855)[2]{}

------------------------------------------------------------------------

(474.59,208.00)(0.485,1.103)[11]{}

------------------------------------------------------------------------

(473.17,208.00)(7.000,13.013)[2]{}

------------------------------------------------------------------------

(481.59,223.00)(0.485,1.179)[11]{}

------------------------------------------------------------------------

(480.17,223.00)(7.000,13.895)[2]{}

------------------------------------------------------------------------

(488.59,239.00)(0.482,1.575)[9]{}

------------------------------------------------------------------------

(487.17,239.00)(6.000,15.302)[2]{}

------------------------------------------------------------------------

(494.59,257.00)(0.485,1.332)[11]{}

------------------------------------------------------------------------

(493.17,257.00)(7.000,15.658)[2]{}

------------------------------------------------------------------------

(501.59,275.00)(0.482,1.575)[9]{}

------------------------------------------------------------------------

(500.17,275.00)(6.000,15.302)[2]{}

------------------------------------------------------------------------

(507.59,293.00)(0.485,1.332)[11]{}

------------------------------------------------------------------------

(506.17,293.00)(7.000,15.658)[2]{}

------------------------------------------------------------------------

(514.59,311.00)(0.482,1.395)[9]{}

------------------------------------------------------------------------

(513.17,311.00)(6.000,13.579)[2]{}

------------------------------------------------------------------------

(520.59,327.00)(0.485,1.179)[11]{}

------------------------------------------------------------------------

(519.17,327.00)(7.000,13.895)[2]{}

------------------------------------------------------------------------

(527.59,343.00)(0.482,1.123)[9]{}

------------------------------------------------------------------------

(526.17,343.00)(6.000,10.994)[2]{}

------------------------------------------------------------------------

(533.59,356.00)(0.485,0.798)[11]{}

------------------------------------------------------------------------

(532.17,356.00)(7.000,9.488)[2]{}

------------------------------------------------------------------------

(540.59,367.00)(0.482,0.581)[9]{}

------------------------------------------------------------------------

(539.17,367.00)(6.000,5.824)[2]{}

------------------------------------------------------------------------

(546.00,374.59)(0.710,0.477)[7]{}

------------------------------------------------------------------------

(546.00,373.17)(5.630,5.000)[2]{}

------------------------------------------------------------------------

(377.0,129.0)

------------------------------------------------------------------------

(559,377.17)

------------------------------------------------------------------------

(559.00,378.17)(3.887,-2.000)[2]{}

------------------------------------------------------------------------

(566.59,374.65)(0.482,-0.581)[9]{}

------------------------------------------------------------------------

(565.17,375.82)(6.000,-5.824)[2]{}

------------------------------------------------------------------------

(572.59,367.45)(0.485,-0.645)[11]{}

------------------------------------------------------------------------

(571.17,368.73)(7.000,-7.725)[2]{}

------------------------------------------------------------------------

(579.59,357.26)(0.482,-1.033)[9]{}

------------------------------------------------------------------------

(578.17,359.13)(6.000,-10.132)[2]{}

------------------------------------------------------------------------

(585.59,345.03)(0.485,-1.103)[11]{}

------------------------------------------------------------------------

(584.17,347.01)(7.000,-13.013)[2]{}

------------------------------------------------------------------------

(592.59,329.16)(0.482,-1.395)[9]{}

------------------------------------------------------------------------

(591.17,331.58)(6.000,-13.579)[2]{}

------------------------------------------------------------------------

(598.59,313.32)(0.485,-1.332)[11]{}

------------------------------------------------------------------------

(597.17,315.66)(7.000,-15.658)[2]{}

------------------------------------------------------------------------

(605.59,295.32)(0.485,-1.332)[11]{}

------------------------------------------------------------------------

(604.17,297.66)(7.000,-15.658)[2]{}

------------------------------------------------------------------------

(612.59,276.60)(0.482,-1.575)[9]{}

------------------------------------------------------------------------

(611.17,279.30)(6.000,-15.302)[2]{}

------------------------------------------------------------------------

(618.59,259.32)(0.485,-1.332)[11]{}

------------------------------------------------------------------------

(617.17,261.66)(7.000,-15.658)[2]{}

------------------------------------------------------------------------

(625.59,241.16)(0.482,-1.395)[9]{}

------------------------------------------------------------------------

(624.17,243.58)(6.000,-13.579)[2]{}

------------------------------------------------------------------------

(631.59,225.79)(0.485,-1.179)[11]{}

------------------------------------------------------------------------

(630.17,227.89)(7.000,-13.895)[2]{}

------------------------------------------------------------------------

(638.59,209.43)(0.482,-1.304)[9]{}

------------------------------------------------------------------------

(637.17,211.72)(6.000,-12.717)[2]{}

------------------------------------------------------------------------

(644.59,195.74)(0.485,-0.874)[11]{}

------------------------------------------------------------------------

(643.17,197.37)(7.000,-10.369)[2]{}

------------------------------------------------------------------------

(651.59,183.26)(0.482,-1.033)[9]{}

------------------------------------------------------------------------

(650.17,185.13)(6.000,-10.132)[2]{}

------------------------------------------------------------------------

(657.59,172.45)(0.485,-0.645)[11]{}

------------------------------------------------------------------------

(656.17,173.73)(7.000,-7.725)[2]{}

------------------------------------------------------------------------

(664.59,163.37)(0.482,-0.671)[9]{}

------------------------------------------------------------------------

(663.17,164.69)(6.000,-6.685)[2]{}

------------------------------------------------------------------------

(670.00,156.93)(0.492,-0.485)[11]{}

------------------------------------------------------------------------

(670.00,157.17)(5.962,-7.000)[2]{}

------------------------------------------------------------------------

(677.00,149.93)(0.491,-0.482)[9]{}

------------------------------------------------------------------------

(677.00,150.17)(4.962,-6.000)[2]{}

------------------------------------------------------------------------

(683.00,143.94)(0.920,-0.468)[5]{}

------------------------------------------------------------------------

(683.00,144.17)(5.340,-4.000)[2]{}

------------------------------------------------------------------------

(690.00,139.95)(1.132,-0.447)[3]{}

------------------------------------------------------------------------

(690.00,140.17)(4.132,-3.000)[2]{}

------------------------------------------------------------------------

(696.00,136.95)(1.355,-0.447)[3]{}

------------------------------------------------------------------------

(696.00,137.17)(4.855,-3.000)[2]{}

------------------------------------------------------------------------

(703,133.17)

------------------------------------------------------------------------

(703.00,134.17)(3.302,-2.000)[2]{}

------------------------------------------------------------------------

(709,131.17)

------------------------------------------------------------------------

(709.00,132.17)(3.887,-2.000)[2]{}

------------------------------------------------------------------------

(716,129.67)

------------------------------------------------------------------------

(716.00,130.17)(3.000,-1.000)[2]{}

------------------------------------------------------------------------

(722,128.67)

------------------------------------------------------------------------

(722.00,129.17)(3.500,-1.000)[2]{}

------------------------------------------------------------------------

(553.0,379.0)

------------------------------------------------------------------------

(735,127.67)

------------------------------------------------------------------------

(735.00,128.17)(3.500,-1.000)[2]{}

------------------------------------------------------------------------

(729.0,129.0)

------------------------------------------------------------------------

(742.0,128.0)

------------------------------------------------------------------------

(736,380)[(0,0)\[r\][$MC$]{}]{} (780,380) (274,128) (328,128) (382,129) (435,149) (489,246) (543,378) (597,330) (651,190) (705,135) (758,128) (812,128) (758.0,380.0)

------------------------------------------------------------------------

(758.0,370.0)

------------------------------------------------------------------------

(824.0,370.0)

------------------------------------------------------------------------

(274,128) (264.0,128.0)

------------------------------------------------------------------------

(264.0,128.0)

------------------------------------------------------------------------

(328,128) (318.0,128.0)

------------------------------------------------------------------------

(318.0,128.0)

------------------------------------------------------------------------

(382,129) (372.0,129.0)

------------------------------------------------------------------------

(372.0,129.0)

------------------------------------------------------------------------

(435.0,148.0) (425.0,148.0)

------------------------------------------------------------------------

(425.0,149.0)

------------------------------------------------------------------------

(489.0,244.0)

------------------------------------------------------------------------

(479.0,244.0)

------------------------------------------------------------------------

(479.0,248.0)

------------------------------------------------------------------------

(543,378) (533.0,378.0)

------------------------------------------------------------------------

(533.0,378.0)

------------------------------------------------------------------------

(597,330) (587.0,330.0)

------------------------------------------------------------------------

(587.0,330.0)

------------------------------------------------------------------------

(651.0,189.0) (641.0,189.0)

------------------------------------------------------------------------

(641.0,190.0)

------------------------------------------------------------------------

(705,135) (695.0,135.0)

------------------------------------------------------------------------

(695.0,135.0)

------------------------------------------------------------------------

(758,128) (748.0,128.0)

------------------------------------------------------------------------

(748.0,128.0)

------------------------------------------------------------------------

(812,128) (802.0,128.0)

------------------------------------------------------------------------

(802.0,128.0)

------------------------------------------------------------------------

(930,558)(0,0) =cmr10 at 10pt (220.0,167.0)

------------------------------------------------------------------------

(198,167)[(0,0)\[r\][0]{}]{} (846.0,167.0)

------------------------------------------------------------------------

(220.0,275.0)

------------------------------------------------------------------------

(198,275)[(0,0)\[r\][0.2]{}]{} (846.0,275.0)

------------------------------------------------------------------------

(220.0,382.0)

------------------------------------------------------------------------

(198,382)[(0,0)\[r\][0.4]{}]{} (846.0,382.0)

------------------------------------------------------------------------

(220.0,490.0)

------------------------------------------------------------------------

(198,490)[(0,0)\[r\][0.6]{}]{} (846.0,490.0)

------------------------------------------------------------------------

(274.0,113.0)

------------------------------------------------------------------------

(274,68)[(0,0)[-5]{}]{} (274.0,470.0)

------------------------------------------------------------------------

(382.0,113.0)

------------------------------------------------------------------------

(382,68)[(0,0)[-3]{}]{} (382.0,470.0)

------------------------------------------------------------------------

(489.0,113.0)

------------------------------------------------------------------------

(489,68)[(0,0)[-1]{}]{} (489.0,470.0)

------------------------------------------------------------------------

(597.0,113.0)

------------------------------------------------------------------------

(597,68)[(0,0)[1]{}]{} (597.0,470.0)

------------------------------------------------------------------------

(705.0,113.0)

------------------------------------------------------------------------

(705,68)[(0,0)[3]{}]{} (705.0,470.0)

------------------------------------------------------------------------

(812.0,113.0)

------------------------------------------------------------------------

(812,68)[(0,0)[5]{}]{} (812.0,470.0)

------------------------------------------------------------------------

(220.0,113.0)

------------------------------------------------------------------------

(866.0,113.0)

------------------------------------------------------------------------

(220.0,490.0)

------------------------------------------------------------------------

(45,301)[(0,0)[$\psi_{01}$]{}]{} (543,23)[(0,0)[$x$]{}]{} (488,535)[(0,0)[$T=\omega=m=\hbar=1,\lambda=1/2, N=10 \times 10$]{}]{} (220.0,113.0)

------------------------------------------------------------------------

(736,425)[(0,0)\[r\][$Exact$]{}]{} (758.0,425.0)

------------------------------------------------------------------------

(220,167) (351,165.67)

------------------------------------------------------------------------

(351.00,166.17)(3.000,-1.000)[2]{}

------------------------------------------------------------------------

(220.0,167.0)

------------------------------------------------------------------------

(370,164.67)

------------------------------------------------------------------------

(370.00,165.17)(3.500,-1.000)[2]{}

------------------------------------------------------------------------

(357.0,166.0)

------------------------------------------------------------------------

(383,163.67)

------------------------------------------------------------------------

(383.00,164.17)(3.500,-1.000)[2]{}

------------------------------------------------------------------------

(390,162.67)

------------------------------------------------------------------------

(390.00,163.17)(3.000,-1.000)[2]{}

------------------------------------------------------------------------

(396,161.67)

------------------------------------------------------------------------

(396.00,162.17)(3.500,-1.000)[2]{}

------------------------------------------------------------------------

(403,160.67)

------------------------------------------------------------------------

(403.00,161.17)(3.000,-1.000)[2]{}

------------------------------------------------------------------------

(409,159.17)

------------------------------------------------------------------------

(409.00,160.17)(3.887,-2.000)[2]{}

------------------------------------------------------------------------

(416,157.67)

------------------------------------------------------------------------

(416.00,158.17)(3.000,-1.000)[2]{}

------------------------------------------------------------------------

(422,156.17)

------------------------------------------------------------------------

(422.00,157.17)(3.887,-2.000)[2]{}

------------------------------------------------------------------------

(429,154.17)

------------------------------------------------------------------------

(429.00,155.17)(3.302,-2.000)[2]{}

------------------------------------------------------------------------

(435,152.17)

------------------------------------------------------------------------

(435.00,153.17)(3.887,-2.000)[2]{}

------------------------------------------------------------------------

(442,150.17)

------------------------------------------------------------------------

(442.00,151.17)(3.302,-2.000)[2]{}

------------------------------------------------------------------------

(448,148.67)

------------------------------------------------------------------------

(448.00,149.17)(3.500,-1.000)[2]{}

------------------------------------------------------------------------

(377.0,165.0)

------------------------------------------------------------------------

(461,149.17)

------------------------------------------------------------------------

(461.00,148.17)(3.887,2.000)[2]{}

------------------------------------------------------------------------

(468,151.17)

------------------------------------------------------------------------

(468.00,150.17)(3.302,2.000)[2]{}

------------------------------------------------------------------------

(474.00,153.59)(0.710,0.477)[7]{}

------------------------------------------------------------------------

(474.00,152.17)(5.630,5.000)[2]{}

------------------------------------------------------------------------

(481.00,158.59)(0.492,0.485)[11]{}

------------------------------------------------------------------------

(481.00,157.17)(5.962,7.000)[2]{}

------------------------------------------------------------------------

(488.59,165.00)(0.482,0.762)[9]{}

------------------------------------------------------------------------

(487.17,165.00)(6.000,7.547)[2]{}

------------------------------------------------------------------------

(494.59,174.00)(0.485,0.874)[11]{}

------------------------------------------------------------------------

(493.17,174.00)(7.000,10.369)[2]{}

------------------------------------------------------------------------

(501.59,186.00)(0.482,1.304)[9]{}

------------------------------------------------------------------------

(500.17,186.00)(6.000,12.717)[2]{}

------------------------------------------------------------------------

(507.59,201.00)(0.485,1.332)[11]{}

------------------------------------------------------------------------

(506.17,201.00)(7.000,15.658)[2]{}

------------------------------------------------------------------------

(514.59,219.00)(0.482,1.756)[9]{}

------------------------------------------------------------------------

(513.17,219.00)(6.000,17.025)[2]{}

------------------------------------------------------------------------

(520.59,239.00)(0.485,1.637)[11]{}

------------------------------------------------------------------------

(519.17,239.00)(7.000,19.183)[2]{}

------------------------------------------------------------------------

(527.59,261.00)(0.482,2.027)[9]{}

------------------------------------------------------------------------

(526.17,261.00)(6.000,19.610)[2]{}

------------------------------------------------------------------------

(533.59,284.00)(0.485,1.713)[11]{}

------------------------------------------------------------------------

(532.17,284.00)(7.000,20.065)[2]{}

------------------------------------------------------------------------

(540.59,307.00)(0.482,2.027)[9]{}

------------------------------------------------------------------------

(539.17,307.00)(6.000,19.610)[2]{}

------------------------------------------------------------------------

(546.59,330.00)(0.485,1.637)[11]{}

------------------------------------------------------------------------

(545.17,330.00)(7.000,19.183)[2]{}

------------------------------------------------------------------------

(553.59,352.00)(0.482,1.756)[9]{}

------------------------------------------------------------------------

(552.17,352.00)(6.000,17.025)[2]{}

------------------------------------------------------------------------

(559.59,372.00)(0.485,1.179)[11]{}

------------------------------------------------------------------------

(558.17,372.00)(7.000,13.895)[2]{}

------------------------------------------------------------------------

(566.59,388.00)(0.482,1.123)[9]{}

------------------------------------------------------------------------

(565.17,388.00)(6.000,10.994)[2]{}

------------------------------------------------------------------------

(572.59,401.00)(0.485,0.645)[11]{}

------------------------------------------------------------------------

(571.17,401.00)(7.000,7.725)[2]{}

------------------------------------------------------------------------

(579.00,410.60)(0.774,0.468)[5]{}

------------------------------------------------------------------------

(579.00,409.17)(4.547,4.000)[2]{}

------------------------------------------------------------------------

(585,412.67)

------------------------------------------------------------------------

(585.00,413.17)(3.500,-1.000)[2]{}

------------------------------------------------------------------------

(592.00,411.93)(0.599,-0.477)[7]{}

------------------------------------------------------------------------

(592.00,412.17)(4.796,-5.000)[2]{}

------------------------------------------------------------------------

(598.59,405.45)(0.485,-0.645)[11]{}

------------------------------------------------------------------------

(597.17,406.73)(7.000,-7.725)[2]{}

------------------------------------------------------------------------

(605.59,395.74)(0.485,-0.874)[11]{}

------------------------------------------------------------------------

(604.17,397.37)(7.000,-10.369)[2]{}

------------------------------------------------------------------------

(612.59,382.16)(0.482,-1.395)[9]{}

------------------------------------------------------------------------

(611.17,384.58)(6.000,-13.579)[2]{}

------------------------------------------------------------------------

(618.59,366.55)(0.485,-1.255)[11]{}

------------------------------------------------------------------------

(617.17,368.78)(7.000,-14.776)[2]{}

------------------------------------------------------------------------

(625.59,348.33)(0.482,-1.666)[9]{}

------------------------------------------------------------------------

(624.17,351.16)(6.000,-16.163)[2]{}

------------------------------------------------------------------------

(631.59,330.08)(0.485,-1.408)[11]{}

------------------------------------------------------------------------

(630.17,332.54)(7.000,-16.539)[2]{}

------------------------------------------------------------------------

(638.59,310.05)(0.482,-1.756)[9]{}

------------------------------------------------------------------------

(637.17,313.03)(6.000,-17.025)[2]{}

------------------------------------------------------------------------

(644.59,291.32)(0.485,-1.332)[11]{}

------------------------------------------------------------------------

(643.17,293.66)(7.000,-15.658)[2]{}

------------------------------------------------------------------------

(651.59,272.60)(0.482,-1.575)[9]{}

------------------------------------------------------------------------

(650.17,275.30)(6.000,-15.302)[2]{}

------------------------------------------------------------------------

(657.59,256.03)(0.485,-1.103)[11]{}

------------------------------------------------------------------------

(656.17,258.01)(7.000,-13.013)[2]{}

------------------------------------------------------------------------

(664.59,240.43)(0.482,-1.304)[9]{}

------------------------------------------------------------------------

(663.17,242.72)(6.000,-12.717)[2]{}

------------------------------------------------------------------------

(670.59,226.74)(0.485,-0.874)[11]{}

------------------------------------------------------------------------

(669.17,228.37)(7.000,-10.369)[2]{}

------------------------------------------------------------------------

(677.59,214.54)(0.482,-0.943)[9]{}

------------------------------------------------------------------------

(676.17,216.27)(6.000,-9.270)[2]{}

------------------------------------------------------------------------

(683.59,204.45)(0.485,-0.645)[11]{}

------------------------------------------------------------------------

(682.17,205.73)(7.000,-7.725)[2]{}

------------------------------------------------------------------------

(690.59,195.65)(0.482,-0.581)[9]{}

------------------------------------------------------------------------

(689.17,196.82)(6.000,-5.824)[2]{}

------------------------------------------------------------------------

(696.00,189.93)(0.581,-0.482)[9]{}

------------------------------------------------------------------------

(696.00,190.17)(5.824,-6.000)[2]{}

------------------------------------------------------------------------

(703.00,183.94)(0.774,-0.468)[5]{}

------------------------------------------------------------------------

(703.00,184.17)(4.547,-4.000)[2]{}

------------------------------------------------------------------------

(709.00,179.94)(0.920,-0.468)[5]{}

------------------------------------------------------------------------

(709.00,180.17)(5.340,-4.000)[2]{}

------------------------------------------------------------------------

(716.00,175.95)(1.132,-0.447)[3]{}

------------------------------------------------------------------------

(716.00,176.17)(4.132,-3.000)[2]{}

------------------------------------------------------------------------

(722,172.17)

------------------------------------------------------------------------

(722.00,173.17)(3.887,-2.000)[2]{}

------------------------------------------------------------------------

(729,170.67)

------------------------------------------------------------------------

(729.00,171.17)(3.000,-1.000)[2]{}

------------------------------------------------------------------------

(735,169.17)

------------------------------------------------------------------------

(735.00,170.17)(3.887,-2.000)[2]{}

------------------------------------------------------------------------

(455.0,149.0)

------------------------------------------------------------------------

(749,167.67)

------------------------------------------------------------------------

(749.00,168.17)(3.000,-1.000)[2]{}

------------------------------------------------------------------------

(742.0,169.0)

------------------------------------------------------------------------

(762,166.67)

------------------------------------------------------------------------

(762.00,167.17)(3.000,-1.000)[2]{}

------------------------------------------------------------------------

(755.0,168.0)

------------------------------------------------------------------------

(768.0,167.0)

------------------------------------------------------------------------

(736,380)[(0,0)\[r\][$MC$]{}]{} (780,380) (274,167) (328,167) (382,165) (435,153) (489,163) (543,316) (597,412) (651,282) (705,185) (758,168) (812,167) (758.0,380.0)

------------------------------------------------------------------------

(758.0,370.0)

------------------------------------------------------------------------

(824.0,370.0)

------------------------------------------------------------------------

(274,167) (264.0,167.0)

------------------------------------------------------------------------

(264.0,167.0)

------------------------------------------------------------------------

(328,167) (318.0,167.0)

------------------------------------------------------------------------

(318.0,167.0)

------------------------------------------------------------------------

(382,165) (372.0,165.0)

------------------------------------------------------------------------

(372.0,165.0)

------------------------------------------------------------------------

(435.0,152.0) (425.0,152.0)

------------------------------------------------------------------------

(425.0,153.0)

------------------------------------------------------------------------

(489,163) (479.0,163.0)

------------------------------------------------------------------------

(479.0,163.0)

------------------------------------------------------------------------

(543.0,315.0) (533.0,315.0)

------------------------------------------------------------------------

(533.0,316.0)

------------------------------------------------------------------------

(597.0,412.0) (587.0,412.0)

------------------------------------------------------------------------

(587.0,413.0)

------------------------------------------------------------------------

(651.0,280.0) (641.0,280.0)

------------------------------------------------------------------------

(641.0,283.0)

------------------------------------------------------------------------

(705.0,184.0) (695.0,184.0)

------------------------------------------------------------------------

(695.0,186.0)

------------------------------------------------------------------------

(758,168) (748.0,168.0)

------------------------------------------------------------------------

(748.0,168.0)

------------------------------------------------------------------------

(812,167) (802.0,167.0)

------------------------------------------------------------------------

(802.0,167.0)

------------------------------------------------------------------------