Understanding the rules of parsing table construction

I’m studying top-down parsing, especially LL(1) parsing. However, I cannot understand what is the meaning of rules.

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Example grammer is like this

S → ( S ) S | $ \epsilon$

How can I derive like below table?

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I read all explanations of textbook, but I cannot totally understand the process of table construction. Are there anyone who can explain more easily about this process?

Writing down a recursive construction

The standard practice when doing a recursive construction seems to be to list all of the desired properties for the construction first:

We construct $ (x_n)$ , $ (y_n)$ , $ (z_n)$ such that

(1)

(2)

(3)

… (a bunch of properties about the individual items $ x_n,z_n,y_n$ and their relationships to $ x_k, y_k,z_k$ ‘s when $ k<n$ )

and then explicitly write out the construction for $ n=0$ , and then do the “recursive step”.

My question is, if the list of items is sufficiently short and the construction is not too complex, is it necessary to write the construction like this? Would it be acceptable to simply begin the construction, do a few cases at the beginning, $ n=0,1,2$ until you get the feel for things, and then just say “continue in this manner”? And then when proving things about the construction, just say “By construction,…”.

When I say acceptable, I mean acceptable for publication in a good journal.

Are there any mathematical advantages to actually writing the conditions (1),(2),(3),… beforehand?

How to hide a construction site without hindering the workers?

Let’s say you have a building site that could fit in a roughly 100 feet square. And you want to hide it without hindering workers.

Now you could use the mirage arcane spell to hide the building but, a.) the workers would not be hidden, and b.) even the workers would not see the building (which seems counter-productive).

Or, you could use Mordenkainen’s private sanctum, which would hide the build site in a not at all conspicuous 100 feet square fog (insert sarcasm sign here).

So, can you use hallucinatory terrain to hide the fog, by making the original terrain there? If yes, how would that work for the workers? Or could you create a terrain that hides the fog and is even more inconspicuous? Or am I thinking too much into that and there is a completely different approach to this? Is it even possible in 5e?

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Construction of a K(pi_1)- space?

I was advised to post my question here. A colleague suggested a proof of a fact which I have hard time to believe. Since I am not a topologist by training I wonder if this can be true in such a generality.

Consider any finite CW-complex structure on the 𝑑-dimensional sphere $ 𝑆^𝑑$ . Let $ 𝐾$ be $ 𝑆^𝑑$ with all strata of codimension at least 2 removed, i.e. we keep just 𝑑 and (𝑑−1)-dimensional strata of the complex under consideration.

Claim. K is contractible to a graph (the dual graph of the preserved strata). Thus it is a $ K(\pi_1, 1)$ – space where $ \pi_1$ is a free group.

The claim is trivially true for d=2, but looks suspicious for higher d. Also if it is true then it might work for more general CW-complexes as well.

Best regards, Boris Shapiro/Stockholm University/

Coordinate-free B.Feix construction of a hyperkähler metric

In the 2001’s paper ‘Hyperkähler metrics on cotangent bundles’ B.Feix gives a construction of a hyperkähler metric on a neighbourhood of zero section in $ T^*X$ where $ X$ is a real analytic Kähler manifold.

As for me, this construction is awful. It’s not difficult to understand the general steps of it and I’m really inspired by them but if you look at details it’s just awful. Almost everything is written in local coordinates without explaining why everything can be glued globally, the article starts with choosing a local antiholomorphic involution on $ X$ without mentioning it properly, she calls ‘the complexified Kähler form on $ X^\mathbb C$ ‘ a form which is not the analytic continuation of the Kähler form on $ X$ etc.

I’ve spent several days trying to rewrite the paper in a coordinate-free way changing some steps as it’d be more similar to constructions of a Kähler metric on a neighbourhood of zero section in $ T^*X$ where $ X$ is a real analytic Riemannian manifold due to Lempert, Szöke and Guillemin, Stenzel. I’m not completely successful yet, though. So I ask, perhaps this paper is already written? I know about D.Kaledin’s paper but his approach is rather different, I want something which looks like Feix’s construction.

Class construction: Data loader

I’m writing some code which loads a dataset and then performs some transformations on this (relatively large, ~5 GB) dataset. I need different sub-sets of this data for use with analysis code I have wrote in a Jupyter notebook. I’m approaching the problem as follows, but have been learning about different decorators (specifically, class methods) and was wondering if there is a better way of doing the following, particularly could load_data be a class method?

class Data():      def __init__(self, filename='/path_to_data'):         self.df = self._load_data(filename)         self._transform_df()      def _load_data(self, filename):         """doc string to write"""         data = pd.read_csv(filename)         # do other stuff to data         return data     def _transform_df(self):         """doc string to write"""         self.df['Date'] =  pd.to_datetime(self.df['Date'], format="%Y/%m/%d %H:%M:%S")     def get_subset_data(self, p1, p2):        """doc string to write"""        mask = ((self.df['value_1']==p1) &                 (self.df['value_2']==p2)                )        return self.df[mask]      def __repl__():         return 'Data({})'.format(filename)       # more definitions  

Then what I will do is load the data once and generate the different subsets when needed:

global_df = Data() subset_1_df = global_df.get_subset_data(p1=1, p2=2) subset_2_df = global_df.get_subset_data(p1=10, p2=20)  ... 

The Construction of a Basis of Holomorphic Differential 1-forms for a given Planar Curve

Working over the complex numbers, consider a function $ F\left(x,y\right)$ and a curve $ C$ defined by $ F\left(x,y\right)=0$ .

I know that to construct the Jacobian variety associated to $ C$ , one integrates a basis of global holomorphic differential forms over the contours of the curve’s homology group. I’m looking for information that is oriented toward actually computing things for given concrete examples; everything I’ve seen so far, however, has been uselessly abstract or non-specific. Note: I’m new to this—I’m an analyst who knows next to nothing about algebra and even less about differential geometry or topology.

In my quest for a sensible answer, I turned to a H.F. Baker’s wonderful (though densely written) text from the start of the 20th century. Just reading through the first few pages makes it abundantly clear that there is a general procedure for constructing a basis of holomorphic differential forms for a given curve. Ted Shifrin’s comment on this math-stack-exchange problem only makes me more certain than ever that the answers I seek are out there, somewhere.

Broadly speaking, my goals are as follows. In all of these, my aim is to be able to use the answers to these questions to compute various specific examples, either by hand, or with the assistance of a computer algebra system. So, I’m looking for formulae, explanations and/or step-by-step procedures/algorithms, and/or pertinent reference/reading material.

(1) In the case where $ F$ is a polynomial, what is/are the procedure(s) for determining a basis of holomorphic differential 1-forms over $ F$ ? If the procedure varies depending on certain properties of $ F$ (say, if $ F$ is an affine curve, or a projective curve, or of a certain form, or some detail like that), what are those variations?

(2) In the case where $ F$ is a polynomial of $ x$ -degree $ d_{x}$ , $ y$ -degree $ d_{y}$ , and $ C$ is a curve of genus $ g$ , I know that the basis of holomorphic differential 1-forms for $ C$ will be of dimension $ g$ . In the case, say, where $ C$ is an elliptic curve, with:

$ $ F\left(x,y\right)=4x^{3}-g_{2}x-g_{3}-y^{2}$ $

the classical Jacobi Inversion Problem arises from considering a function $ \wp\left(z\right)$ which parameterizes $ C$ , in the sense that $ F\left(\wp\left(z\right),\wp^{\prime}\left(z\right)\right)$ is identically zero. Using the equation: $ $ F\left(\wp\left(z\right),\wp^{\prime}\left(z\right)\right)=0$ $ we can write: $ $ \wp^{-1}\left(z\right)=\int_{z_{0}}^{z}\frac{ds}{4s^{3}-g_{2}s-g_{3}}$ $ and know that the multivaluedness of the integral then reflects the structure of the Jacobian variety associated to $ C$ .

That being said, in the case where $ C$ is of genus $ g\geq2$ , and where we can write $ F\left(x,y\right)=0$ as: $ $ y=\textrm{algebraic function of }x$ $ nothing stops us from performing the exact same computation as for the case with an elliptic curve. Of course, this computation must be wrong; my question is: where and how does it go wrong? How would the parameterizing function thus obtained relate to the “true” parameterizing function—the multivariable Abelian function associated to $ C$ ? Moreover, how—if at all—can this computation be modified to produce the correct parameterizing function (the Abelian function)?

(3) My hope is that by understanding both (1) and (2), I’ll be in a position to see what happens when these classical techniques are applied to non-algebraic plane curves defined but with $ F$ now being an analytic function (incorporating exponentials, and other transcendental functions, in addition to polynomials). Of particular interest to me are the transcendental curves associated to exponential diophantine equations such as: $ $ a^{x}-b^{y}=c$ $ $ $ y^{n}=b^{x}-a$ $

That being said, I wonder: has this already been done? If so, links and references would be much appreciated.

Even if it has, though, I would still like to know the answers to my previous questions, even if it’s merely for my personal edification alone.

Thanks in advance!

Are “moustache” distortions in rectilinear ultrawides the result of a fisheye+pincushion construction?

If I wanted to try and build an ultrawide, I guess I would try building a fisheye first, then try tearing the frame back into shape with a group that introduces pincushion distortion.

The kind of “wavy” distortions found in some ultrawides seem very similar to what you would get as remaining distortion if you actually did that….

Is this how some of these lenses actually work, or is this a coincidence?