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]


Q:

Chebyshev net in 3D

I would like to know the reasons why the existance of Chebyshev net in 3D-case is problematic.
This question boils down to the PDE described below.
(I do not know much about PDEs, so feel free to say something trivial.)
Set
$$\mathbb W^2 =\{\,(x,y,z)\in \mathbb R^3\mid x+y+z=0\,\}.$$
Given a smooth map $f: \mathbb W^2\to \mathbb R^3$ which is $C^\infty$-close
to the identity map,
I need to extend $f$ to a neighborhood of $\mathbb W^2$ so that it satisfies the following three equations:
$$\left|\frac{\partial f}{\partial x}\right|^2=\left|\frac{\partial f}{\partial y}\right|^2=\left|\frac{\partial f}{\partial z}\right|^2=1.$$
Comments.

The 2D-case has a solution, this is due to Chebyshev and it is more than 100 years old.
In general, 3D Chebyshev net is a solution of 3 equasions as above where $f$ maps a domain in $\mathbb R^3$ in a 3-dimensional Riemannian manifold and $|{*}|$ is defined by its metric. I guess this case is just as hard as the special case described above.
Assuming that $x+y+z>0$ and $f(x,y,z)$ is defined,
it is easy to see that $f(x,y,z)$ depends only on the values
$f(x',y',z')$ for $(x',y',z')$ in the triangle $\Delta\subset\mathbb W^2$ defined by the inequalities $$x'\le x,\  y'\le y,\ z'\le z.$$

A:

The question asked is a special case of the following more general one: Let
$$
\mathbb{W} = \{x^1 + \cdots + x^n = 0\} \subset \mathbb{R}^n.
$$
Given an $n$-dimensional Riemannian manifold $M$ and a smooth map $f: \mathbb{W} \rightarrow M$, can $f$ be extended to a map $f: \mathbb{R}^n \rightarrow M$ such that
$$
|\partial_1f|^2= \cdots = |\partial_nf|^2 = 1\ ?
$$
Here, note that $\partial_if(x) \in T_{f(x)}M$, and the norm is taken with respect to the Riemannian metric on $M$.
The first remark, which I'm sure Anton and Robert already know, is that if $f$ happens to be an embedding (which is the case in the original question), then we can view it as a choice of all $n$ co-ordinate functions along a hypersurface in $M$ and the question is whether these functions can be extended to co-ordinate functions where the diagonal elements of the metric tensor written with respect to these co-ordinates are identically equal to $1$.
To determine what kind of PDE this system is, we can linearize it. Let $\dot{f}$ denote an infinitesimal variation of $f$. The linearized system is given by
$$
2\partial_if\cdot\partial_i\dot{f} = \dot{h}_i.
$$
If $f$ is an immersion, then $\partial_1f(x), \ldots, \partial_nf(x)$ are always a basis of $\mathbb{R}^n$. Therefore, solving the linearized system for $\dot{f}$ is equivalent to solving for the functions
$$
u_1 = \partial_1 f\cdot \dot{f}, \ldots, u_n = \partial_nf\cdot \dot{f}.
$$
``Differentiating by parts'', the linearized system can be rewritten as
$$
2\partial_i u_i - g^{jk}u_j\cdot\nabla_i\partial_kf = \dot{h}_i.
$$
Note that the top order terms fully uncouple into ODE's in each of the co-ordinate directions. The coupling that occurs in the zero-th order terms prevents this system from being purely a system of ODE's. However, it is easy to check that this system is indeed a symmetric hyperbolic system, where initial smooth data posed on $\mathbb{W}$ can be extended to a unique smooth solution on all of $\mathbb{R}^n$. You can find more about this in a paper I wrote with DeTurck (Duke Math. J. Vol.51, No. 2
(1984), 243-260).
Using appropriate regularity estimates and the inverse function theorem with the appropriate Banach norm, this implies existence and uniqueness of a solution to the original nonlinear system on a tubular neighbhorhood of $\mathbb{W}$. I do not know whether there is a global solution; this requires a more careful analysis of this particular system.
ADDED: Actually, since the system is fully nonlinear, you cannot use the Banach space implicit function theorem to solve the system as given. You can, however, do one of two thing: Either "prolong" (which means roughly differentiate the system and add the partial derivatives of $f$ as unknown functions) the system into a quasilinear system or just use the Nash-Moser iteration argument. Either way, you still get what I describe above.
ADDED (in response to Robert's comment): Robert is right that if you specify $f$ along $\mathbb{W}$, the extension of $f$ is not uniquely determined, due to the full nonlinearity. Robert indicates that there are exactly 2 choices when $n = 3$. You have to do the linear algebra carefully along $\mathbb{W}$ to see this and count the number of possibilities in arbitrary dimension. I haven't done this yet.
ADDED: If along $\mathbb{W}$, you set $v = \frac{1}{n}(\partial_1 f+\cdots \partial_n f)$ and $u_i = \partial_if - v$, then the $u_1, \dots, u_n$ are tangential derivatives of $f$ along $\mathbb{W}$ and therefore given by the initial data. These vectors along $\mathbb{W}$ satisfy the equations
$$
u_i\cdot v = \frac{1}{2}[1 - |u_i|^2 - \frac{1}{n}(1 - \sum_i |u_i|^2)]
$$
$$
v\cdot v = \frac{1}{n}(1 - \sum_i |u_i|^2)
$$
Clearly, a necessary condition for a solution is that
$$
|u_1|^2 + \cdots + |u_n|^2 \le 1.
$$
If strict inequality holds on $W$, I believe that there are always exactly two solutions on $W$. These translate into a corresponding inequality that the tangential derivatives of $f$ along $\mathbb{W}$ must satisfy. And if strict inequality holds, then there are exactly two ways to extend $f$ onto a tubular neighborhood of $\mathbb{W}$ such that $f$ defines a Chebyshev net.

A:

Well, there's no problem if the map $f$ is real-analytic and close enough to the identity.  Then it essentially reduces to the Cauchy-Kovalevskaya Theorem. For the smooth theory, you'll need something a bit more subtle, but it may be OK anyway.  The characteristic variety (again, assuming that the initial conditions are sufficiently close to the identity) factors into three linear factors, which is something that Dennis DeTurck and Deane Yang know quite a bit about.  It's possible that this problem is what they call `symmetric hyperbolic', in which case, the initial value problem will be OK.  I'd check with one of them.
Oh:  I should have explained that, because the characteristic variety is not smooth (being the union of three lines in general position in the projectivized cotangent space of each point of the domain, the initial value problem is not symmetric hyperbolic in the strict sense, so the simple hyperbolic existence theory for smooth initial data cannot be applied.  That doesn't mean (as mentioned above) that a refinement of the symmetric hyperbolic theory won't work.  The difference between this and the 2D case is that, in that case, the characteristic variety is 2 (real distinct, multiplicity one) points in projectivized cotangent space of each domain point, so, of course, it is smooth, and the standard hyperbolic existence theory applies.
By the way, you don't have uniqueness.  When the obvious inequalities on the initial data are satisfied (so that the hyperbolic theory can be applied), there will be two (and only two) distinct solutions for each choice of initial data $f$ that you prescribe along $\mathbb{W}$.