TITLE: Is the Archimedean principle necessary to prove the density of $\mathbb Q$ in $\mathbb R$?
QUESTION [4 upvotes]: I've noticed that most proofs of the density of $\mathbb Q$ in $\mathbb R$ use the Archimedean principle. For example see @Arturo Magidin's checked answer here. Density of irrationals
I'm puzzled by the fact that in the hyperreals, the standard rationals must be dense in the standard reals; yet the hyperreals are not Archimedean. This would seem to imply that the Archimedean principle is not a necessary assumption.
I can see that in the hyperreals, if $\epsilon$ is infinitesimal then there is no standard rational between $\epsilon$ and $2 \epsilon$; so that the standard rationals are not dense in the hyperreals.
But now I'm confused. Wouldn't the hyper-rationals be dense in the hyperreals? And wouldn't it still be a theorem that the standard rationals are dense in the standard reals?
Clearly the answer must be in some subtlety involving standard/nonstandard reals and the transfer principle. Can anyone shed any light? Is the Archimedean principle necessary to prove that the rationals are dense in the reals? Why aren't the hyperreals a counterexample?

REPLY [2 votes]: The truncation $10^{-n}\lfloor10^n x\rfloor$ of a real $x$ at rank $n$ of its decimal expansion is rational and implies that the rationals are dense in the reals.  This argument works also in the hyperreals and does not use the Archimedean property.


text:
\section{DISCUSSION}
\label{sec:discussion}

Of the methods presented, the McPIC model shows the most promising results for both failure path and time to failure predictions.
This is the only method that considered crack pairs as the unit of interest, highlighting the importance of capturing these interactions.
The method not only had a high number of correct exact failure path predictions, but also shows the ability to capture coalescence outside of the dominant failure pathway and crack branching behavior.
This is the only method that did not purely consider geometric factors, but also included crack intensity factors as variables in the feature vectors for the ML algorithms to train with.
This likely impacted the both the crack propagation and pathway behavior and, subsequently, the time at which failure was predicted.
We expect that the model would see improvements if tertiary (or higher order) crack interactions were considered rather than just pairwise interactions.
The initial preexisting cracks are relatively close to one another, hence higher order interactions may have a notable impact on the growth rate of the crack network.
If this were explicitly accounted for in the algorithm, improvement in the accuracy of predictions would be expected.

The OP method also produced good results, particularly in calculating the failure pathway.
For this metric, this model was only outperformed by the McPIC method in the case of predicting the dominant crack pathway in simulations where the sample was not yet failed.
It also only slightly underpredicted the time at which failure occurs (~0.0042s vs HOSS 0.00446s).
These results essentially show that the accounting for the orientation dependence with varying crack lengths is a reasonable assumption, as long as average times for failure are adequate for the application.
However, this assumption is only applicable for Mode I failure, and would not be acceptable for shear or combined loading conditions.
If interactions with neighboring cracks above and below the horizontal growth path were accounted for in the crack growth rate, time to failure would likely be predicted more accurately.

The EPZ method produced interesting results.
While this method did not outperform others in a significant way, it was able to capture coalescence of cracks outside of the dominant failure pathway.
It was also able to capture reasonable crack curvature and potentially also crack branching behavior.
These behaviors are likely a result of the elliptical PZs that exist at each crack tip and evolve in shape and orientation over time and with the location of the crack tip.
This provides a large amount of variability in the direction in which cracks can grow, which is something that the other methods do not allow for.
However in the EPZ method, it seems that much of the crack propagation is overpredicted in comparison to the HOSS results.
In the EPZ images shown in Figure \ref{fig:six}, many of the cracks that are predicted to propagate also are shown to propagate in the HOSS results, however in the EPZ model they seem to propagate too quickly resulting in incorrect coalescence and sometimes incorrect failure path predictions.
This perhaps reflects a dependence on one or more material factors within the posited crack growth factor (Equation \ref{eq:Cf}) that is too strong.
In addition, accounting for stress state information at the crack tips could also impact the crack growth rate, improving results.

The NFPZ approach is somewhat a hybrid of the OP and EPZ models, and consequently it performs better than the EPZ method and not as well as the OP method for failure path prediction.
This model does assume, based on Mode I failure arguments, that horizontal cracks will grow the fastest similar to the OP method.  This idea is used to determine a zone where failure is likely.
This method also assumes PZs present at crack tips, similar to the EPZ approach.
The PZs are used to determine which cracks are likely to coalesce, and this also informs the likely location of failure.
Unlike the EPZ method, the PZs are not recalculated with time and all crack tips have the same PZ size (which is true as long as $a_1$=$a_2$).
Adding some variability to the PZ calculation (e.g., dependence on orientation, stress state, etc.) could improve results.
In addition, the likely region of failure may be too narrowly defined.
Other methods (specifically the OP and McPIC methods) that did not reduce the system to a local failure zone present better results.

It is worth noting that the SPA approach did produce reasonable results for calculating the failure path based solely on a shortest fracture pathway, without any use of ML algorithms.
Of course, similar to the OP method, the SPA method is based on assumptions only applicable for Mode I failure scenarios.
Future modifications to this algorithm could include weighting the shortest path by geometric as well as physical considerations.
For example, through the consideration of pairwise interactions the probability of two cracks coalescing could be higher when the distance between tips is small.
This can be incorporated into the weights while computing shortest paths which may improve results.

\begin{figure*}
\includegraphics[width=\textwidth]{Figures/eight.eps}
\centering
\caption{The five models presented in this study represented on a spectrum that ranges from purely data-driven to incorporating heuristic physics information. }
\label{fig:eight}
\end{figure*}

Comparison between these methods would be incomplete if it did not provide us with insights into the relative importance of using a combination of physics intuition and training data.
Figure~\ref{fig:eight} shows where the different methods fall on this spectrum of training features that go from purely data-driven to physics informed constraints.
Methods at either end of this spectrum performed poorly in terms of the metrics considered here.
The methods on the left had an insufficient ability to learn from the training data with the SPA approach not utilizing the training data at all and the OP approach using only one feature (crack length) to predict crack growth.
In contrast, the methods on the right included more physics, but the physics was simplified which limited their ability make accurate predictions.
The rigid manner in which physics was incorporated in these models also made it difficult for the learning to overcome the simplifications.
The McPIC algorithm struck a balance between these two approaches by including a rich feature set including physical information (such as the stress intensity factor) while utilizing ML algorithms to learn the behavior of the system based on these features.
This research exercise further confirms our notion that there exists a ``Goldilocks'' region, where the judicious use of physics constraints in combination with training data is more amenable to learning the underlying complex physics of the fracture propagation problem.

config: arxiv


TITLE: Constructing a Fractional Linear Map
QUESTION [1 upvotes]: I am working on a practice prelim question:
"Construct a nonlinear fractional map $\phi(z) = \frac{az+b}{cz+d}$ with $c \ne 0$ such that $\phi(\phi(\phi(z))) = z$.  I feel like I just need to take the composition and just work it out to solve for the constants $a,b,c$ and $d$.  Is this the right idea?  Or is there a short cut?

REPLY [1 votes]: Why don’t you just take $z\mapsto\omega z$, where $\omega$ is a primitive cube root of unity, and conjugate your linear map by a fractional linear that’s so nonlinear that the result is nonlinear? You’re surely looking for a $120^\circ$-rotation about some point in the plane.
EDIT: That was too complicated. Just take a real fractional-linear that maps $0\mapsto1\mapsto\infty\mapsto0$.