Re #13: My apologies if they should not have been removed. Here is the content. The pages can be put back if requested. The source compiles correctly in the actual nLab, the differences are due to the fact that only the old renderer is used on the nForum; I have lightly edited some syntax to make it display reasonably here.

An *hyperanalytic function* is a function that is locally given by a converging power series with a speed that corresponds to tetration.

Let $V$ and $W$ be complete Hausdorff topological vector spaces, let $W$ be locally convex, let $c$ be an element of $V$, and let $(a_0,a_1,a_2,\ldots)$ be an infinite sequence of homogeneous operators from $V$ to $W$ with each $a_k$ of degree $k$.

Given an element $c$ of $V$, consider the infinite series

$\sum_k a_k(x - c)^k$(a power series). Let $U$ be the interior of the set of $x$ such that this series converges in $W$; we call $U$ the **domain of convergence** of the power series. This series defines a function from $U$ to $W$; we are really interested in the case where $U$ is inhabited, in which case it is a balanced neighbourhood of $c$ in $V$ (which is Proposition 5.3 of Bochnak–Siciak).

Let $D$ be any subset of $V$ and $f$ any continuous function from $D$ to $W$. This function $f$ is **hyperanalytic** if, for every $c \in D$, there is a power series as above with inhabited domain of convergence $U$ such that

for every $x$ in both $D$ and $U$ and just **rare $a_k$ are not equal zero**. (That $f$ is continuous follows automatically in many cases, including of course the finite-dimensional case.)

{#Examples}

It is known that there is a fundamental connection between analyticity of the function and the convergence of its Fourier coefficients. The better the function, the faster its coefficients tend to zero, and vice versa. The power decrease of Fourier coefficients is inherent in functions of the $C^{k}$ class while exponential to analytical functions. Here there is a possibility of existence of the hyperanalytic functions, for which the decrease of the Fourier coefficients corresponds to tetration.

Natural hyperanalytic function occurs when considering reticulum with a step L, in which nodes there are not defined yet objects. The distribution of center’s objects can be described using the reticulum functions (RF). The definition of a one-dimensional RF is based on the following identity:

$\frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-\frac{1}{2}(\frac{x}{\sigma})^{2}}dx=\frac{1}{\sigma\sqrt{2\pi}}\int_{-\frac{L}{2}}^{\frac{L}{2}}\sum_{n=-\infty}^{\infty}e^{-\frac{1}{2}(\frac{x-nL}{\sigma})^{2}}dx=1.$From here *RF* is

It is obvious that the RF can not be laid out in the Fourier series because it does not have antiderivative that can be expressed as elementary functions. By virtue of this RF cannot be decomposed into even and odd functions, while an arbitrary analytic function $f$ can be only presented in the form of sum of odd and even functions in the interval $[a,b]$:

$f\left(x\right)=g\left(x\right)+h\left(x\right),$where

$g\left(x\right)=\frac{f\left(x-a\right)-f\left(b-x\right)}{2},$ $h\left(x\right)=\frac{f\left(x-a\right)+f\left(b-x\right)}{2}.$Due to this the RF can be laid out in an endless row of two primitive hyperanalytic functions by sequential attempts to decompose on even and odd functions. Thus, the RF can be decomposed by the simplest way, but such a series is not one like the orthonormal basis of Fourier series.

As it follows from (1.1) the mean value of RT is 1. However as will be seen from the further, it is expedient to choose the greater value of the decomposition’s constant member. Introduce the following definitions:

$\mathbb{R}\left(0\right)$ is

$\mathbb{R}\left(0\right)=\mathbb{R}_{max}=\frac{1}{\sigma\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-\frac{1}{2}\left(\frac{-n}{\sigma}\right)^{2}},$$\mathbb{R}\left(1/2\right)$ is

$\mathbb{R}\left(1/2\right)=\mathbb{R}_{min}=\frac{1}{\sigma\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-\frac{1}{2}\left(\frac{1/2-n}{\sigma}\right)^{2}}.$Then $A_{0}$ is the mean value of RF:

$A_{0}=\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}.$One can approximate first difference by the following way:

$A_{1}\left(x\right)=\frac{\mathbb{R}_{max}-\mathbb{R}_{min}}{2}cos\left(2\pi x\right).$Let introduce parameter of the fine structure $\alpha$ as function of $\sigma$:

$\alpha\left(\sigma\right)=\frac{1}{2}\frac{\mathbb{R}_{max}-\mathbb{R}_{min}}{\mathbb{R}_{max}+\mathbb{R}_{min}}.$Now $A_{1}\left(x\right)$ can be expressed:

$A_{1}\left(x\right)=\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}\left(2\alpha\left(\sigma\right)cos\left(2\pi x\right)\right).$The choice of the name and symbol of this parameter is due to the fact that

$\alpha\left(0.4992619105929628\right)=\alpha=\frac{e^{2}}{4\pi\epsilon_{0}\hbar c}$is the value known in physics as a fine structure constant.

Even differences are a primitive hyperanalytic function $\overline{\mathbb{V}}(2i\times2\pi x)$, which is quasisymmetric relative to the point $\text{x=0.25}$.

Its symmetrical part approximated in the following way:

$A_{2i}\left(x\right)=c_{2i}\left(cos\left(2i\times2\pi x\right)-1\right)$and

$\sum_{i=1}^{\infty}c_{2i}=\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}-1=2 * \sum_{k=1}^{\infty} \alpha^{4^{k}}.$Using the value

$\mathbb{R}\left(1/4\right)=\mathbb{R}_{1/4}=\frac{1}{\sigma\sqrt{2\pi}}\sum_{n=-\infty}^{\infty}e^{-\frac{1}{2}\left(\frac{1/4-n}{\sigma}\right)^{2}}$define the amplitude for $c_{2}$: $\frac{1}{2}\left(\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}-\mathbb{R}_{1/4}\right)=2\alpha^{4}$. This definition allows to select approximation $A\left(x\right)$ in the form:

$A\left(x\right)=\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}\left(1+2\alpha\left(\sigma\right)cos\left(2\pi x\right)\right)+2\alpha^{4}\left(cos\left(2\times2\pi x\right)-1\right).$Odd differences are a primitive hyperanalytic function $\mathbb{W}\left((2i-1)\times2\pi x\right)$, which is quasiantisymmetric relative to the point $\text{x=0.25}$.

Quasiantisymmetry of $\mathbb{W}\left(2\pi x\right)$ follows from the fact that the integral of $A\left(x\right)$ differs from 1:

$\int_{-1/2}^{1/2}\text{A}\left(x\right)\text{dx}-1=\frac{1}{4}\left(\mathbb{R}_{max}+\mathbb{R}_{min}\right)+\frac{1}{2}\mathbb{R}_{1/4}-1\simeq1.02E-34.$Thus function $\mathbb{W}\left((2i-1)\times2\pi x\right)$ should be decomposed in the even and odd function. Its even part is:

$\mathbb{W}^{\text{qs}}\left((2i-1)\times2\pi x\right)=\frac{\mathbb{W}\left((2i-1)\times2\pi x\right)+\mathbb{W}\left((2i-1)\times2\pi\left(0.5-x\right)\right)}{2}=\overline{\mathbb{V}}(2(i+1)\times2\pi x).$However, as shown above, $\overline{\mathbb{V}}(2i\times2\pi x)$ is not an even function.

The odd part of $\mathbb{W}\left((2i-1)\times2\pi x\right)$ is no longer a hyperanalytic function and is equal to:

$W^{\text{qa}}\left((2i-1)\times2\pi x\right)=\frac{\mathbb{W}\left((2i-1)\times2\pi x\right)-\mathbb{W}\left((2i-1)\times2\pi\left(0.5-x\right)\right)}{2}.$It can be approximated with any degree of accuracy following way:

$A(W^{\text{qa}}\left((2i-1)\times2\pi x\right))=\beta(cos\left(3(2i-1)\times2\pi x\right)-cos\left((2i-1)\times2\pi x\right)),$where $\beta$ is a normalizing multiplier.

Thus, the approximation of $\mathbb{R}(x)$ is:

$A\left(x\right)=\frac{\mathbb{R}_{max}+\mathbb{R}_{min}}{2}(1+2\alpha cos\left(2\pi x\right)) +2\sum_{i=1}^{\infty}\alpha^{4^{i}}\left(cos\left(2i\times 2\pi x\right)-1\right)+\frac{2}{\mathbb{W}_{max}}\sum_{i=1}^{\infty}\alpha^{9{i}^2}\left(cos\left(3 \times (2i-1)\times 2\pi x\right)-cos\left((2i-1) \times 2\pi x\right)\right),$where $\mathbb{W}_{max}$ is a normalizing multiplier.

Three-dimensional RF $\mathbb{R}\left(x,y,z\right)$ can be obtained from the definition $\left(1.2\right)$:

$\mathbb{R}\left(x,y,z\right)=\mathbb{R}_{max}^{2}\mathbb{R}\left(x\right).$Thus, the approximation of the three-dimensional RF is also the series of the fine structure constant $\alpha$ along any axis of the reticulum three-dimensional space, and the constant itself is a function of the dimensionless parameter $\sigma$, which is equal to quotient of the “diameter” of some physical object, located in each cell, to the grid step L.

To quantize the time the direct use of the lattice idea is too formal. It is therefore appropriate to use a definition of derivative with respect to time but without moving to the limit. Let $\mathbb{R}\left(t\right)$ is RF on a unit interval $\left[-\text{T/2},\text{T/2}\right]$ and $\tau=\sigma$ и $\text{T}=1$:

$\mathbb{R}\left(t\right)=\frac{1}{\tau\sqrt{2\pi}}\sum_{i=-\infty}^{\infty}\left[\exp\left(-\frac{1}{2}\left(\frac{t+\text{T/4}-i}{\tau}\right)^{2}\right)-\exp\left(-\frac{1}{2}\left(\frac{t-\text{T/4}-i}{\tau}\right)^{2}\right)\right].$By consistently subtracting sinuses, one can show that the approximation of the $\mathbb{R}\left(t\right)$ has the following form:

$A\left(t\right)=\sum_{k=0}^{\infty}\left(-1\right)^{k+1}a_{k}sin\left(2\pi\left(2k+1\right)t\right).$Let use $k+1$ equations with different values of $l$ to determine the coefficient’s values $a_{k}$:

$\sum_{i=0}^{k}\left(-1\right)^{i}a_{i}sin\left(\frac{2i+1}{2l+1}\frac{2\pi}{4}\right)=\mathbb{R}\left(\frac{1}{4\left(2l+1\right)}\right).$Given that $A\left(1/4\right)$ is numerically equal to $2\left(\mathbb{R}_{max}\left(\tau\right)+\mathbb{R}_{min}\left(\tau\right)\right)\alpha\left(\tau\right)$, equation can be written as follows:

$\alpha_{eff}\left(t,\tau\right)=\frac{1}{2\left(\mathbb{R}_{max}\left(\tau\right)+\mathbb{R}_{min}\left(\tau\right)\right)}\sum_{k=0}^{\infty}\left(-1\right)^{k+1}a_{k}sin\left(2\pi\left(2k+1\right)t\right).$$\mathbb{R}\left(t\right)$ is also a hyperanalytic function, as the next approximation takes place:

$\alpha_{eff}\left(t,\tau\right)=\sum_{k=0}^{\infty}\left(-1\right)^{k+1}\alpha^{(2k+1)^{2}}sin\left(2\pi\left(2k+1\right)t\right).$The theory of hyperanalytic function was constructed to some extent by A. Rybnikov (2014) http://www.gaussianfunction.com/.

]]>Zoran, I'm sorry that you feel that way. I understand that you don't like changing a system that you've gotten used to. I do feel that you make life a little complicated for yourself:

I click on latest changes, get a link to forum, log in, respond to Urs's entry which concerns the item I added on, click back, get just the entry related to that entry, not the list of all recent udpates like in latest changes before.

Why not have a tab open at the forum and keep yourself logged in, then it's no more clicks than the old system. And to get the full list, simply click on the words 'latest changes' wherever they appear (apart from the title of this discussion, of course), that takes you to the category view where you get a summary of all that's going on.

Indeed, I'm sure that some smart cookie who knows a little about javascript could write a "smart bookmarklet" that took the title of the page that is being edited and started a new discussion here on that subject all with one click.

However, I realise that your situation is somewhat unique so I don't want to make you feel hassled about this.

Mike, things like that can be tweaked. This forum has themes and stylesheets that can be set up to give almost any presentation that is desired. However, I don't regard this system as The System. A much better one would be a forum-like set-up integrated into Instiki. However, I think that this is a necessary step on the way to that as it gives us a chance to see what we would like from such an integrated system.

Jon, that applies to your comment as well (though I don't understand your comment about not knowing if you've seen what you needed to see, the inference is that that was *better* in the old system). It's all very well pipedreaming, but some things are easy to implement and others are hard. And it's a long road between saying "wouldn't it be great to have section-local editing" to an actual implementation - just look over the threads about redirects to see the sort of things that can go on. For example, you can have "section-local editing" right now: simply make a master page with lots of includes of sub-pages. But that wouldn't work for latest changes because everyone would have to keep editing the master page to add their new link to the subpage that they'd just created to record their latest change. Plus there'd be much more to remember and many more steps to remember.

We are trying this out for a month. There will be no major changes during that month. After that month, we can discuss what the consensus is.

]]>I find the new system a little easier, I think, because I can view the various updates more independently.

This would not work if I didn't have an account. But since I do, and since I accept the cookie to stay logged in (until it disappears after a month or something and I need to log in again), the main page tells me where new posts are, and clicking on ‘New’ takes me right where I want to be.

]]>Well, this is just a test run to see how we like it. We can still switch back.

But I don't at the moment see why you find it harder to follow recent changes here. I find it easier, even without RSS. All the new links are at the top, easily visible. no?

]]>Thanks, Ariadne, that was just the clue! — but I have to say it was a mite further back than I normally pull.

Theseus

]]>Actually, I don't know whether I would like the current setup if I weren't using a feed-reader. For purposes of replies, I think it would be better to have a threaded view of everything that's been written, rather than just a view of the first post in each discussion. What do others think?

Jon, presumably you're referring to this post which was a reply to this one. I don't usually have any trouble finding the replied-to post on the cafe, but on the rare occasions when it is a problem, what I do is switch to thread view and scroll up until the vertical line immediately to the left of the reply first disappears. Possibly what the cafe could really use is a collapsible thread-view.

]]>Yes, the forum-based system of logging latest changes is turning into a regular blog-bog where you can't be sure whether you've seen what you need to see — like just yesterday in the cafe someone said something was "very clever" and I wasted a ½ hour cycling around between the find-on-page and chronological and thread views trying to figure out what it was that was "very clever" and I never did find it — and this will only get multiplicatively worse as the number of headings hydrafy.

I think a better thing to think about is whether we can get section-local editing on the wiki — this would reduce the chances of lockouts and edit conflix globally, not just on the latest changes page.

]]>Okey-dokey, it was just a thought.

I'm done messing around with the category view for latest changes. It displays the *first* comment (i.e. the one that it was all about) if there are *any* unread comments. So you get an overview of what the discussion is about and thus whether or not it's worth going deeper. Also, the Instructions post is *not* visible from the main page (since otherwise it would always be the first post there and would look like the instructions for the *whole* forum). I've also installed the 'mark all as read' plugin. I haven't experimented with this too much, but I think that it marks *everything* as read in the forum, so use At Your Own Risk!

I like the current situation with one discussion per "created X". It is quick and easy to type and quick and easy to follow.

]]>Consider it stuck.

I've also modified the view for the 'latest changes' category in line with what I said above. In the *category* view, and only for 'latest changes', it displays the *first* comment of each discussion. One could consider truncating these, but none seem to be too long. Also, maybe they could be hidden if they are already marked as read - that shouldn't be hard to do.

Looking at the content of most of them, I suspect that having a separate discussion for each page is a little over the top when most of them are just "created X". Perhaps if all that is wanted is to flag new pages, one could just write a "New Pages" discussion and list all the new ones that you've added in the past hour/day/whatever (since titles can be non-unique, and the person's name and the date are automatically added, just "New Pages" would be enough).

These are just thoughts from the first day. I'm sure I'll change my mind many times over the month!

]]>If you care to make the Instructions thread sticky, Andrew, here it is.

]]>By my first point, I meant that as I tend to work on the fringes of the n-lab (content-wise), most of what goes by on latest changes doesn't concern me. But it's still good to glance through it, occasionally I learn something and even more occasionally I can contribute somewhere that I didn't think that I could. At the moment, I have to click on every discussion to see what's going on. If there was a view that simply showed the first comment in each discussion then that would fix it.

I don't think that there's any way to synchronise a feed reader with the forum. It would require information going in both ways, whereas at the moment it goes in just one. On the other hand, a "mark everything as read" might help you since if it did annoy you to have things out of synch, you could read all the ones you wanted to in your feed reader and then just click the "mark everything as read" over here. But if you primarily use your reader, it probably doesn't matter at all.

]]>Since the first post in a discussion is likely to be fairly short, I'd like to be able to see all the unread ones in one go.

I don't know what you mean by that, can you elaborate?

I'd like to be able to quickly mark a load of discussions as read.

The "read or not read" flag on the forum is totally out of whack for me since I read most things just via the rss feed, which doesn't mark them as read by me. Whereas my feed reader does keep track of what I've read there. There's probably no hope of synchronizing those, so this one doesn't matter much to me.

]]>Yes, I felt like going ahead. I also thought that some role model might be helpful. :-)

I am already enjoying the new system: it's quicker to use. I don't have to retype my name, don't have to worry about formatting, I just go new-entry, didthisandthat, done. Nice.

]]>So I see that Urs has started using this a day early! Now that I see it in use, I can think of two minor improvements:

- Since the first post in a discussion is likely to be fairly short, I'd like to be able to see all the unread ones in one go.
- I'd like to be able to quickly mark a load of discussions as read.

There's a plugin for the second, but I think the first might take a little hackery, which is why I'll not bother if only I think it's worth it.

]]>Probably, in which case it's your job, Mike! I think it'll be useful to have a specific feedback discussion in the latest changes category (mainly to keep all such feedback in one place and avoid cluttering up the latest changes category), but there's no reason why someone wanted to comment on the implementation should have to wade through all the discussion that lead to it if they don't want to.

]]>Actually, maybe now this discussion should be moved back to a different, more meta-, category?

]]>OK, I made another announcement, this time at the top of latest changes. I will move most of that to a new ‘Instructions’ post (which Andrew will make sticky, I guess) in about 23 hours, so edit it now if you find it unclear. And then I will archive the 2009 September changes, giving us a new latest changes page that basically just tells people to come to the Forum.

]]>Urs, Toby volunteered to do it already with the sensible suggestion that it happen at the usual monthly rollover. Of course, if you are at a loose end I'm sure he'd be happy for you to do some of it!

]]>By the way, I'll try to get around doing it. But if anyone else feels to do what it takes to "set up the new system", don't feel obliged to wait for me...

]]>I've cleaned out the 'latest changes' category. Regular readers will be pleased to know that I moved the discussions that were there to a new category (Trash) rather than delete them. This discussion stays, obviously. The others were all people trying out the guest posting.

Once someone writes a post with instructions then I'll make it "sticky" which means it will always be at the top of the page.

]]>But given how much we like to talk, perhaps something formal like that is unnecessary.

That's partly what I meant. The other part is that, for me, the whole n-lab has the feel of a "test run".

]]>Zoran's last comment said ‘It is simpler that you all ignore my complaints.’. Let's give it a test.

Since the changes are currently being broken up monthly, it might be convenient to say that our test is for October. Put a standout box at the top of latest changes announcing this fact (I know, another announcement, but the old one's been up so long that people are probably ignoring it now), and I'll follow Andrew's steps Wednesday afternoon (as I see it), when the date changes UTC.

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