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### The sad state of Perl documentation

by SuperCruncher (Pilgrim)
 on Sep 14, 2000 at 02:08 UTC ( #32376=perlmeditation: print w/ replies, xml ) Need Help??

I have a number of problems with Perl's documentation. The way I see it, the documentation is currently provided through a system of antiquated man pages or automated conversions from these man pages to HTML. Documentation is not provided in formats suitable for printing such as PDF. At one point on my long search around Perl.com I was able to find a PostScript version of the documentation. This was for Perl 5.003 or something old like that. Postscript?! Until starting university, I had not encountered files in Postscript format before (I had heard of it, but not encountered). It seems to be a very common thing on UNIX platforms. So Perl started out on UNIX, maybe it's time to loosen the ties? Granted PS can be converted to PDF using the ps2pdf program, but not everyone has the time or knowledge to get this program.

Perl's documentation has its problems, and even worse is that Perl's competitors (well, I don't consider them competitors as I think Perl is far superior, but..) all have documentation available in many formats. PHP has documentation available in many formats and languages. With Python, it's a similar story. I recognise though that Python might not be an option for the free software purists since according to RMS it is incompatible with the GPL.

I had hoped that new sites like Perl FAQ and Perldoc would improve the situation, but perldoc especially seems to be a simple a re-hashing of the standard docs.

I also have problems with the actual documentation itself in some cases (not the actual format). In this node I complained about how I consider the documentation for exec() in perlfunc to be misleading. More recently, I also had the misfortune of trying to figure out the documentation for chmod(), also in perlfunc. As is common throughout the Perl documentation, the documentation for chmod assumes much knowledge of UNIX. I have used Linux for around 2-3 years now (albeit not on my own PC) and I have never once used the octal numbers as a mode for chmod. The authors of the chmod documentation in perlfunc consider mentioning how to calculate mode values is beyond the scope of the documentation. It's bad enough that Perl doesn't support the symbolic a+r etc. notation, and it's even worse that the documentation for Perl's chmod function doesn't at least include a table with the values of the most commonly used symbolic modes.

Perl does have good documentation provided by O'Reilly in the form the llama and camel books. However, I'm sure most monks will realise that these books (although very useful) are quite costly. At Amazon.co.uk, Learning Perl and Programming Perl currently cost £15.96 and £26.80 respectively.

I'd be willing to try and help out with the Perl documentation, does anyone know who to ask?

And BTW, I know this post would probably have belonged in the "Rant" section of PerlMonks if there was one, but to me it's the one thing of Perl that really needs major improvement

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RE (tilly) 1: The sad state of Perl documentation
by tilly (Archbishop) on Sep 14, 2000 at 02:33 UTC
If you have specific changes, feel free to download the current development version and submit patches. You will need a working diff program, a slow one is available free from PPT. (I recommend using the "-u" option.)

As for the other complaints, I would suggest modifying POD::Parser to spit out other kinds of documentation.

However for the record I started out with less Unix background than you have, and by and large I found the online documentation quite readable and informative. There were sections I had to go back to, nor was it particularly easy, but I am impressed with the quality of the information.

RE: The sad state of Perl documentation
by mdillon (Priest) on Sep 14, 2000 at 03:04 UTC
it's pretty easy to make PDF's from POD using fairly common tools. watch:
$perldoc -u HTTP::Request > HTTP_Request.pod$ pod2latex HTTP_Request.pod
$perl -e 'open TEX, "+< HTTP_Request.tex" or die; @l = <TEX>; seek TEX, 0, 0; truncate TEX, 0; print TEX "\\documentclass[10pt]{article}\\begin{document}",$/,
print TEX map { s/^%\\/\\/; $_ } @l; print TEX "\\end{document}",$/;
close TEX;'
$pdflatex HTTP_Request.tex [download] here are HTTP::Request and strict. PostScript is similarly easy to generate by replacing pdflatex with pslatex. Or even skip all the other steps and just use Pod::Pdf directly! {grin} I think you're largely missing the point. Sure, I can convert the document in whatever format using module Foo::Bar, but I shouldn't have to. What if I want to get a nice clean print-out of part of the perl documentation? The quickest way to do this is to download a PDF, and print the desired pages. The solution should not be that I am forced to write a script to do the converting for me. Also imagine poor Perl newbies trying to figure out how to install module Foo::Bar required for converting into whatever format and then trying to figure out how to actually write the script. if you're really so worried about the "poor newbies", then undertake this task yourself. don't wait for "the leaders" of the "Perl community" to do it for you. you don't even have to write a script. you can just use the one i wrote, or take merlyn's suggestion and use pod2pdf or its underlying module Pod::Pdf. then, find space on a web server somewhere and keep available an up-to-date version of your compiled documentation. i haven't missed any points. you have in assuming that someone else should be creating this documentation that you want. more power to whomever should decide that printable documentation is important enough to themselves and other users of Perl to implement and maintain a resource such as this. RE: The sad state of Perl documentation by extremely (Priest) on Sep 14, 2000 at 03:50 UTC I think you'll find revamping the documentation to be a multi-month full-time job. That is why people like Randal wind up selling books, documentation is HARD! If you think you can do better that the current state of things, MORE POWER TO YA! Me, I've found the docs to be almost exactly what I want. I rarely consult the books these days in favor of the perldoc pod system. (well, and 'less' for the more inscrutable modules. =) Maybe you should consider championing a perl6-docu list for proposed changes in how POD should be used/written in Perl6. If you get something going come back here and shout it from the bell towers! I'll stick my nose in on something like that. --$you = new YOU;
honk() if \$you->love(perl)

RE: The sad state of Perl documentation
by Maclir (Curate) on Sep 14, 2000 at 04:29 UTC
Generally, there are two types of documentation:

1) Stuff that tells you how to do things. This assumes that you know what you want to do in the first place ("Hmmm - is the recurse option on chmod -R or -r?"). We call these "reference manuals" - the Camel book is a fine example of such a beast.

2) Stuff that tells you what things to do. This means you have an understanding of the end result that is needed, but not sure of the way to do it. The "Perl Cookbook" is a fine example of such a manual. These are far more useful for use beginning programmers.

But - much of perl's power lies in the great array of modules that are constantly being developed. The documentation of these varies from wonderful to woeful. But, what is missing is an extension of the 'how to' approach to incorporate these modules. For example, how many of the questions posed here are answered by:

You should use module Foo::Bar::WDNNS_Documentation

Now maybe that is what more attention needs to be focussed on. There are lots of resources on the web - merlyn's Web techniques column is one that immediately comes to mind. At least the experts here can regularly point us to such sources.

But - much of perl's power lies in the great array of modules that are constantly being developed. The documentation of these varies from wonderful to woeful.
True, but I am unhappy with the standard documentation, not the documentation for other modules. There's no way to force authors of modules to write documentation (except for maybe not putting them into CPAN unless they are accompanied by documentation, but I don't think this would be a good thing), and I'm not too worried about that, but I think that the basic core of the language should be well documented.

I will have to completely disagree here.

Perl itself comes with a huge amount of docs. I am sure it's harder to find your ways around it that let's say a CD player, but that's only because using Perl is a little more complicated than using a CD player.

And I am really unhappy with the sad state of the documentation for all VCR's I have ever owned by the way ;--)

I think enough people care about, and work on the core docs that it is quite comprehensive.</p

The problem is really with modules, as most of the time they are written, supported and documented by te same poor guy, usually better at (and more interested in) writing code than docs.

Plus the author is not necessarily the best person to write an introduction to a complex module. Things that seem easy and natural for someone who has written socket handling software for 10 tears might not be quite as natural to Joe Poor Perl Hacker who was writing COBOL up until last month.

So I think one of the best ways for module users to give feedback to the author is probably to write a piece on how they used the module, or just give some feedback on the doc.

I was documenting religiously and in great detail every single method in XML::Twig, until I realized through a random email on a mailing list stating that a potential user had given up after the first 2 screens (out of 22!). I needed to write a tutorial introducing the basics of the module (but more complex than the usual simple synopsis). Darn! But I'm really glad I caught that feedback. I hate coding stuff that will never be used because nobody looking for an easy way to solve a simple problem did not make it to the 44th method in the 3rd class of the module which was exactly what they needed.

So really I whished module docs were better and I think feedback could really help. But I'm happy with the core docs, thank you.

And stop killing trees and use perldoc, man and grep!

RE: The sad state of Perl documentation
by jplindstrom (Monsignor) on Sep 14, 2000 at 15:19 UTC
I remember when first learning Perl that the problem for me was knowing which part of the documentation to read. Reading the FAQ a number of times as I got more comfortable with Perl was very useful, but the most important man page for beginners is IMHO perlfunc.

I still have a special link to that page in my local docs structure simply because most times when I need to refer to the documentation, that's the page I'm after.

RE: The sad state of Perl documentation
by dmmiller2k (Chaplain) on Sep 14, 2000 at 17:49 UTC
> Perl does have good documentation provided by O'Reilly in the form the
> llama and camel books.

Perhaps you've noticed, perhaps not. The Camel book is essentially a printed version of the perlfunc perldoc/manpage. Which means in particular that the exec function is just as obscurely documented for £26.80 as for the free machine readable version.

As to the state of the documentation, you should be aware that the evolution of perl over the past two or three years (since 5.003) has been in great leaps and bounds, with the number of modules in the CPAN more or less keeping pace. The amount of documentation is awe inspiring, if some of the individual examples of it perhaps are not.

Now that 5.6 is out, give the documentation a year or so to catch up.

In the meantime, sites like this one can fill the gap, at least in terms of pointing out which of the hundreds of "things" to read might be helpful apropos of a particular problem.

Camel 3 has been updated to document the non-existent (yet) 5.6.1. Llama 2 is independant of the underlying Perl version, since I'm just showing the basics parts of Perl, which change far less from release to release. {grin}

Llama 3 will show a few new additions like our, but nearly all the examples and exercises would have been just as home on 5.4 from two or three years ago.

RE: The sad state of Perl documentation
by gumpu (Friar) on Sep 14, 2000 at 17:56 UTC

Postscript?! Until starting university, I had not encountered files in Postscript format before (I had heard of it, but not encountered). It seems to be a very common thing on UNIX platforms.

To put things in perspective. Postscript is not such on odd format. Pdf is basicially a souped-up form of postscript. Furthermore a lot of printer use postscript. A lot of programs can generate postscript, even on Win32 (for instance Word, IE). There is also a free postscript viewer (Ghostscript), this also runs on Win32.

Also, while pdf is nice (seach and hyperlinks are useful) it is also a proprietary format, the readers are free, but you have to pay money for the distiller.

Have Fun

And besides Postscript is a cool programming language! I haven't written any programs in Postscript in a while, though. Using Perl to generating Postscript for drawing complex, detailed graphs with lots of labels (basically maps) was pretty easy (no modules involved, I don't think there were Postscript modules way back then) and let me view and print the graphs lots of places. But the world has change a lot since then...

- tye (but my friends call me "Tye")
(redmist) RE: The sad state of Perl documentation
by redmist (Deacon) on Sep 14, 2000 at 22:43 UTC
"So Perl started out on UNIX, maybe it's time to loosen the ties?"

For the love of all that is good and wholesome, no! Under a *nix, developing Perl (and pretty much anything else) is soooooo much easier and better than under Windows. I am not trying to be a bigot (although I am probably being one anyway), but under *nix you have a real Command Line Interface in which you can use utilities like man, grep, tail, and perldoc to accomplish what you need to do in a more efficient manner. Plus you have multiple terminals (which you could have in Windows, but it would hog the task bar and screen in addition to the fact that it would be more difficult to keep track of the content on each terminal).

It's not really *one* thing or another, but many little things that make *nix a better platform to develop on than Windows.

Perl's association with UNIX is important because some of the functionality and use of utilities in *nix has carried over into perl itself. Grep (and regular expressions) for example. In addition to this, Perl culture has been influenced by *nix culture enough that it is important that we not forget our roots. There is no reason why Perl should become more disassociated from UNIX.

As for the quality of the docs, I think that the majority of the docs are fine as far as quality is concerned. Some are a bit misleading or are not as clear as I would like, but that's the nature of the beast. With as much documentation as Perl has, you cannot expect it to be perfect.

On the format issue, I don't know why you are so attached to the pdf format. If you want to print out some docs, nothing is stopping you. If you want docs with links, then docs are also available in html form. I don't understand why this is such a big issue for you.

Cheers, redmist
redmist.dyndns.org
email::redmist
Actually, red, I gotta disagree with you. I feel, and I know many others do as well, that UNIX is NOT a good development platform. Sure, you have this powerful Command Line Interface - so what? Humans do not work that way. The reason the GUI was considered such a huge advance is because it WAS. Even the ability to open four small term windows, each a complete CLI, and see all of them at once is a huge advantage. Add things like inline spell-checking, good Copy/Paste systems, and an interface that automates as much as possible, and you make for an interface that makes almost ANYTHING more efficient than it is with a CLI, including writing code. CLI is powerful, yes, but it's not perfect. It's a hammer, and a good one, but sometimes you need a scalpel.

As it happens, I think the documentation that comes with the core of Perl is pretty good. On the other hand, I *DO* think Perl should dissociate itself from Unix. Why? Because Perl is a tool. It's a great tool, as perfect for its purpose as any - but its purpose is to make live easier for systems admins and web developers. And despite what the hype would have you believe, the internet is NOT UNIX. It never will be. The internet is ANY operating system. Perl should be, too. Take with it what we like about Unix, certainly. But just because Unix was the cradle of Perl doesn't mean Perl should stay there. You can't live in the cradle forever.

- email Ozymandias
OK, message recieved, zero distortion...I still disagree with you though.

"that UNIX is NOT a good development platform. Sure, you have this powerful Command Line Interface - so what? Humans do not work that way."

I don't think that your point about CLI and humans not working in the same way is valid. Why? Because computers and humans don't work in the same way. But it's ok this way. I have to adapt to the machine, because it won't adapt to me (at least until really, really good AI comes along). Even if the machine *could* adapt to me, I don't think that I would want it that way. I have learned to work with CLI, and have come to love it.

One of the reasons why I think that CLI can be so great is that the rules are clearly defined (not intuitively, but through documentation) and both you and the SW that you are interacting with "knows" that you are two seperate entities, with different ways of working. In this type of environment, the SW or interface will not try to anticipate what you want or how you want to do something. Anticipation of human action usually results in it doing the opposite of what it's original purpose was: to help you and make you more efficient.

Once you have adapted yourself to the CLI (or another non-human-anticipating interface), you can reap the results of such an interface.

"And despite what the hype would have you believe, the internet is NOT UNIX. It never will be. The internet is ANY operating system. Perl should be, too."

I don't want Perl to be tied to UNIX in such a way that would inhibit or diminish it's use on other platforms either. I just think that is important to keep the ties to UNIX that are already there. We should *transcend* the cradle.

I realize that the Internet is not UNIX, but a damn big part of it is...and UNIX is also a big part of other networks and computing as a whole.

redmist
redmist.dyndns.org
email::redmist
PerlMonks isn't the place for OS wars, but I have a few slightly on-topic comments:

Actually, red, I gotta disagree with you. I feel, and I know many others do as well, that UNIX is NOT a good development platform.

I disagree with you.

powerful Command Line Interface

Unix != CLI. The Unix (and really I'm talking about all the Unix spinoffs and clones here) philosophy has to do with operating system structure, and nothing to do with GUI vs CLI.

I run Linux, running X windows, and Keep 10-12 xterm (CLI) windows open. The GUI allows this. This is not different than Windows or Macintosh in any way. However I have more tools (Thanks to GNU) available, I have compilers available (Does Windows ship with a C compiler?), etc. ActiveState has to do a fair amount of work for their Windows version of Perl, whereas the conversion to BSD or Linux or other Unix-like systems is far easier.

Windows and Mac restrict access and reduce flexibility. That is why they are a poor development environment, and why Perl stays "in it's cradle".

The reason the GUI was considered such a huge advance is because it WAS.

Absolutely. And it was a huge advance YEARS before Windows was ever thought of. But it isn't a replacement, it's an extension. Some tasks are very difficult to do in a CLI, while others are very difficult to do in a GUI.

Perl is a tool. Perl is flexible. It is and should be system independant.

Sure, you have this powerful Command Line Interface - so what? Humans do not work that way.

/me is surprised to learn that he is not human.

I hate to reach for the mouse because is not well designed for use by humans. It was apparently a design stolen from the race in "The Mote in God's Eye". I'm a programmer so I have to actually type quite a bit and I really prefer to do it fast so I actually use two hands for it. Everytime I have to reach for the mouse (and it usually amazes people how infrequently I'm forced to) I have to take one hand away from the keyboard. I hate that and it slows me down.

The great advantage to the GUI is that it makes the learning curve more shallow. This is lovely for things that you've never done before or things that you only rarely do. For things that you do all of the time and that you are supposed to be an expert at, a GUI almost always gets in the way and slows things down. On the other hand, I like climbing steep curves so I'm probably a freek as well as being from another planet (certainly many of my cow orkers don't bother to become "experts" at editting, to my eye). On the gripping hand, there is more than one way to do user interfaces (TIMTOWTDUI) and no one way is the best for all situations.

- All in fun, Love Tye
RE: The sad state of Perl documentation
by little (Curate) on Sep 15, 2000 at 16:29 UTC
open your eyes,
Perldoc as PDF 3.0 MB
currently covering Perl Version 5.005_02, so it's in general quite up to ones needs, and if you look further into the document info you will not by chance and not by surprise see that it's been made with pod2ps v1.0.1, which give's you and all of us the ability to convert whole directorys in one handstrike to a new PDF-manual.
So, if you want to help, mail to Andy Dougherty.
RE: The sad state of Perl documentation
by ScottSaddison (Initiate) on Sep 15, 2000 at 23:57 UTC
I am very happy with the documentation that comes with perl. It is an excellent format and reference for perl without the fluff. In fact, last week I began making standard POD templates for the documentation we use in my department.

One of the exceptional features of perldoc that sets it apart from the crowd is simplicity and integration. It's very nice to write in-line formatted documentation and be able to translate that into other formats (i.e. html). Granted, before that, our documentation structure consisted of a box of sand and a stick... but... you get the idea. I have not experienced any of the POD modules yet, but I'm sure they are just as intriguing.

Scott Saddison

in a perfect world, all the servers are UNIX.
RE: The sad state of Perl documentation
by odie (Sexton) on Sep 18, 2000 at 21:28 UTC
And opinions filled the room...
I may as well add a few things:

man / perldoc:
These are not intended (for the most part) to be a tutorial. If you want to learn, you buy a book, like with most other things. Perldoc is however an excellent reference manual. It's not very useful until you actually know perl. On the other hand, when you DO know perl, it is an excellent place to "look something up". The great advantage is that practically everyone installs perldoc along with perl. If you program perl, you always have perldoc. It is however , in my humble opinion, utterly useless when printed. The sections are HUGE and quite impossible to navigate without some form of search/matching. Furthermore, it is nice to know that perldoc is up to date. I do not consider this sad. As soon as you buy a book or print a reference manual, it will become old.

Postscript:
This is not only very common on UNIX platforms. It is also quite common on other systems such as Windows. Printers really like PostScript.
So, following some form of logical deduction, one might say that PostScript is a good format to store text in, if it is intended for printing. This causes no problems, as it can also be viewed with a variety of programs.
) However, following some form of illogical deduction, Windows does things a bit differently. It uses PS. A lot. There are however few programs that can display a simple PS file. Instead they have choosen to use every other format imaginable to store files, ready for printing. They use rtf, doc, pdf and and a variety off odd (and always commercial) formats. This strikes me as very odd. Having dozens of non-free formats that do the exact same thing is not a bright move. Even more so as the first thing that happens when you print it, is a conversion to PS, so the printer will understand it.

When I learned Perl there were no books about Perl. I learned by reading the documentation that came with Perl. I was quite happy with it.

Soon after that the first book about Perl came out and we all bought copies. I read it, enjoyed it, and thought it was a good book. I did learn a few things from reading the book but I think most of those I would have also learned from rereading the standard documentation.

Since then I've bought several books on Perl. Several of them are very good books. I've never finished reading any of them and I don't ever turn to them when I need to look something up.

When I need to look something up I read *.pod files, the source code to modules, the source code to Perl, try something (usually in the Perl debugger), or search the internet or some part of it.

So I think there are quite a few fine books that you could buy to help you learn Perl. But I don't think there is any need to buy a book.

There are certainly some parts of the standard Perl documentation that I believe could stand some major improvements. For example, I find the explanations on pack() and unpack() confusing and worse. But it isn't hard to submit patches to the documentation so I don't bitch much since it just means I haven't bothered to fix the problem. (:

- tye (but my friends call me "Tye")
RE: The sad state of Perl documentation
by footpad (Monsignor) on Sep 20, 2000 at 00:53 UTC

I wouldn't say perl is poorly documented, but that it, like many languages, relies heavily on some basic proficiency with the underlying OS and its rules.

What many Unix/Linux advocates seem to forget is that Windows has a different mindset, one that needs to be unlearned or discarded when dealing with *nix os's.

For example, I'm still pretty new to perl and had a project dropped in my lap. Write, in way too little time, a form-based emailer. Simple stuff for you folks, I'm sure, but since the lionshare of my development experience is with Windows languages, it wasn't something I felt entirely comfortable doing.

So I fired up several search engines, blundered around until I found some relevant material and managed to scrape something together. I did manage to meet the deadline, but I would never consider posting the code because it's flat out awful.

In finishing the project, I did run into several hurdles that would have been easier to solve had they been documented more clearly.

For example, you all know (I'm sure) that here documents are, by default, interpolated. Granted, it's not the best way to include a stylesheet, but I was against a time crunch. However, the stylesheet had external file references...

As another example, I didn't find clear examples of printing scalars with leading zeros or sending email attachments from mtulti-part forms until I heard about this site today. (If I'd only found the site a few months back...)

Each of these three behaviors took ~two days to puzzle out. I would have been far happier if I'd seen, very clearly in "Learning Perl," a warning about the interpolation issue. I did manage to figure it out using "Learning Perl" and "Programming with Perl," but it took a bit of page flipping.

You've all paid similar dues, I'm sure.

I have three main points:

1. If you write documentation, web articles, pod, or even answers to newsgroup/discussion messages, please take a moment to step back and review the technical overhead your information requires. I consider myself a literate, intelligent person, but half the man pages are too obscure for me. Yes, I can eventually puzzle them out, but I really don't have the time to invest. (I've got five projects due by the end of next month.) If you obfuscate behind a dizzying array of details, I get lost and annoy the heck out of my sysadmin..

At the very least, provide some cross-references for those of us who aren't as experienced as you are. (Remember when "man printf(3)" caused you errors?)

2. If you don't like the way one author has put it and you can't find one you like, then this is an opportunity for you to step up. For example, I am in the process of writing short web articles for a site I'm bringing online. Nothing that would appeal to you folks, I know, but it's stuff I ran into while learning the language, the rules, and the OS. You can do the same. There are several places where you can get free server space; use it.

3. If you provide links, web articles, or other online content, please review those from time to time. Some folks haven't touched their pages since '96. 'Nuff said.</P

Compared to other languages, perl is reasonably well documented. that's not to say there aren't areas where improvements could be made, but the docs are out there.

(P.S. In general, _please_ design your pages so they can be easily printed. Far too many sites clip content when printed.)
Re: The sad state of Perl documentation
by wazoox (Prior) on Jul 20, 2006 at 10:32 UTC
The complete Perl 5.8.8 documentation is available online at http://perldoc.perl.org/, and on the main page you may download both the HTML (with a very useful included javascript search engine) and the PDF version. ( http://perldoc.perl.org/perldoc.tar.gz). I don't think other languages are doing any better in this regard.

update: BTW if you want to learn Perl, there are very good docs available here : http://www.techbooksforfree.com/perlpython.shtml
"Extreme Perl" and "Learning Perl the hard way" are especially worth reading.

update: I don't disagree anymore :)

wazoox, please keep in mind that you're responding to a post which is almost 6 years old. That's a lot of water under the bridge.

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I'm afraid I forgot to look at the date, shame on me :)

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# Mechatronics:Japan’s Newest Threat by V. Daniel Hunt

By V. Daniel Hunt

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Extra info for Mechatronics:Japan’s Newest Threat

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An automated factory can break even at 25 percent capacity, while a conventional factory breaks even at 60 to 65 percent of capacity. The challenge presented to the developer of factory automation systems will be to gather the emerging technologies of smart robots, machine vision, process planning, computer-aided design and manufacturing functions, and to interconnect and control their performance so that the entire sequence is optimized. Even though there are manyacceptable components for building an automated factory, it is rare when the components can be found to have common interfaces.

Product concept development in mechatronics begins long before the factory operation, with the concept defined and refined by thoughtful iteration and interaction with the product users. In this phase the team will develop the product concept, and optimize the product design. Computer-aided engineering in conjunction with the computeraided design and drafting function will define, refine, and optimize product design by utilizing computer-aided engineering design and analysis tools built into the product design computer base.

Vertically integrated manufacturing is what is most commonly meant by mechatronics. Many experts would consider horizontal or "shop floor" integration to be only partially computer-integrated or mechatronic manufacturing. Figure 2-12 is a conceptual framework for mechatronics which illustrates the role of some of the CAD/CAM technologies at various levels of factory control. A vertically integrated factory usually implies maximum use and coor- Graph iCS and CAD Language CAD / CAM Computer Facility Matena ls Requirement Plann ing System NC Robot CNC Figure 2-12.

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# Fitting a model#

We can fit a model to individual participant data in a free-recall dataset by maximizing the probability of the data according to the model. This involves using a search algorithm to adjust the model parameters until the probability, or likelihood (see Evaluating a model), of the data is maximized.

First, load some sample data to fit:

In [1]: from cymr import fit, cmr

In [2]: data = fit.sample_data('Morton2013_mixed').query('subject <= 3')

## Search Definition#

Next, we need to define our search parameters. There are two types of parameters used specifically for searches:

fixed

Parameters that have a fixed value. These parameters are not searched.

free

Parameters that may vary to fit a dataset. For a search, must specify a range to be searched over.

We’ll also use two other types of parameters that set properties of the model based on a given parameter set:

dependent

Parameters that are derived from other parameters. These parameters are specified using an expression that generates them from other parameters.

weights

Parameters that define weighting of different patterns in the model.

We can organize these things by creating a Parameters object. To run a simple and fast search, we’ll fix almost all parameters and just fit one, $$\beta_\mathrm{enc}$$. For a real project, you may want to free other parameters also to fit individual differences in the primacy effect, temporal clustering, etc.

In [3]: par = cmr.CMRParameters()

In [4]: par.set_fixed(T=0.1, Lfc=0.15, Lcf=0.15, P1=0.2, P2=2,
...:               B_start=0.3, B_rec=0.9, X1=0.001, X2=0.25)
...:

In [5]: par.set_free(B_enc=(0, 1))

In [6]: par.set_dependent(Dfc='1 - Lfc', Dcf='1 - Lcf')

To simulate free recall using the context maintenance and retrieval (CMR) model, we must first define pre-experimental weights for the network. For this example, we’ll define localist patterns, which are distinct for each presented item. They can be represented by an identity matrix with one entry for each item. See Evaluating a model for details.

In [7]: n_items = 768

In [8]: study = data.query("trial_type == 'study'")

In [9]: items = study.groupby('item_index')['item'].first().to_numpy()

In [10]: patterns = {'items': items, 'vector': {'loc': np.eye(n_items)}}

In [13]: par.set_weights('fc', weights)

In [14]: par.set_weights('cf', weights)

We can print the parameter definition to get an overview of the settings.

In [15]: print(par)
fixed:
T: 0.1
Lfc: 0.15
Lcf: 0.15
P1: 0.2
P2: 2
B_start: 0.3
B_rec: 0.9
X1: 0.001
X2: 0.25

free:
B_enc: (0, 1)

dependent:
Dfc: 1 - Lfc
Dcf: 1 - Lcf

dynamic:

sublayers:

weights:

sublayer_param:

The to_json() method of CMRParameters can be used to save out parameter definitions to a file. The output file uses JSON format, which is both human- and machine-readable and can be loaded later to restore search settings:

In [16]: par.to_json('parameters.json')

## Using search output#

To use the output from the search for evaluating the model on new data or running simulations, we must first convert the results DataFrame into a dictionary.

In [22]: subj_param = best.T.to_dict()

As an example of using the best-fitting parameters, we can use them to confirm the likelihood values from the search.

The group_param input to likelihood() sets parameters that are the same for all participants, while the subj_param sets subject-specific parameters. Here, we’ll just set everything through the subj_param input.

In [23]: group_param = {}

In [24]: model.likelihood(
....:     data,
....:     group_param,
....:     subj_param=subj_param,
....:     param_def=par,
....:     patterns=patterns,
....: )
....:
Out[24]:
logl    n
subject
1        -954.976072  373
2       -1109.066717  426
3        -983.635155  379

In Generating simulated data, we’ll use a set of parameter values to generate simulated data for analysis.


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# To Bet Or Not To Bet

### Solution

As a matter of fact, we always end up with the same number of cards. Thus you always lose $\$1.\$

### Acknowledgment

This is problem 317 from Five Hundred Mathematical Challenges by E. J. Barbeau et all (MAA, 1995).


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# .argonne-logo-style0{fill:#231F20;} .argonne-logo-style1{fill:#FFFFFF;} .argonne-logo-style2{fill:#007836;} .argonne-logo-style3{fill:#0082CA;} .argonne-logo-style4{fill:#0B1F8F;} .argonne-logo-style5{fill:none;} .argonne-logo-style6{fill:#A12B2F;} .argonne-logo-style7{fill:#CD202C;} .argonne-logo-style8{fill:#77B300;} Argonne National Laboratory

Feature Story | Argonne National Laboratory

# JCESR accomplishments halfway through its five-year charter

Today’s lithium-ion batteries hold more than twice the energy of those released in 1991, and they continue to improve. But even when brought to their full theoretical energy storage potential, lithium-ion batteries fall short of some of our energy storage needs. Next-generation batteries are necessary to meet these needs.

For this reason, the Joint Center for Energy Storage Research (JCESR), a U.S. Department of Energy (DOE) Innovation Hub led by Argonne National Laboratory, looks exclusively beyond lithium-ion. With JCESR at its halfway point, 2 1/2 years into its five-year charter, this is a good time to step back and look at the big picture: how far we have come, what we have learned and where we are going.

#### Looking Back

We launched JCESR with a bold vision: to create game-changing next-generation battery technologies that will transform the transportation sector and the electric grid the way lithium-ion batteries transformed personal electronics. This bold vision addresses pressing national needs to reduce carbon emissions, increase energy efficiency, lower our dependence on foreign oil, accelerate deployment of renewable solar and wind electricity on the grid, and modernize the grid with new operating concepts that strengthen its flexibility, reliability, and resilience.

Next-generation energy storage could meet these needs for transportation and the grid with a single dramatic innovation: batteries that deliver five times the energy density at one-fifth the cost.  Such batteries would allow inexpensive electric cars to drive five times farther on a single charge, rivaling the 400-mile range of conventional gasoline cars, and they would make storing and releasing electricity on the grid just as cheap as generating it with natural gas turbines.

Next-generation storage has the potential to replace traditional, century-old fossil fuel technologies with newer, more sustainable, and cleaner alternatives. The market implications are equally impressive: transportation and the grid account for nearly 70 percent of U.S. energy use, compared to 2 percent for personal electronics powered by today’s lithium-ion batteries. In energy terms, next-generation batteries could have at least 10 times the market reach of today’s batteries. In dollars and cents, channeling half the transportation and grid energy through storage would create a market 10 times larger than personal electronics powered by lithium-ion batteries, which currently stands at $15-$20 billion.

#### Where We Stand

For the past 2 1/2 years, we have been pursuing three energy storage concepts: multivalent intercalation,” which involves replacing singly charged lithium ions with doubly or triply charged working ions; chemical transformation,” that improves how we store energy in chemical bonds; and redox flow,” whereby energy is stored in liquid electrodes.

The first concept would create high-capacity, high-voltage batteries using multivalent intercalation.  Replacing lithium ions, which carry a single positive charge (Li+), with ions having two or three positive charges could increase the battery energy storage capacity by a factor of two or three. We conducted a computer-driven screening of 1,800 candidate multivalent host structures, the largest such systematic survey ever undertaken. This effort identified the most promising ions as calcium and magnesium, both of which have a two-plus charge (Ca2+ and Mg2+).

Moving from the computer to the laboratory, we then demonstrated that the Mg2+ ion could be inserted into and released from manganese oxide spinel cathodes. This discovery is the first substantive experimental evidence that the multivalent battery could one day reach its theoretical promise. Just as important, we have gained a fundamental understanding of the molecular-level interactions that govern the properties of multivalent electrolytes.

For the second concept, careful design studies using techno-economic modeling showed that storing energy in lithium-oxygen chemical bonds is less feasible than previously thought because of the cost and energy density penalty of dealing with a high-purity gaseous oxygen electrode.  We thus decided to shift emphasis from lithium-oxygen to lithium-sulfur, focusing on developing a viable lithium metal electrode and preventing the polysulfide intermediates that form in the cathode from migrating to the anode and discharging the battery.

The classic challenge for lithium metal anodes is the growth of dendrites, which reach across the electrolyte and short out the cathode, severely shortening the life of the battery. JCESR researchers have reported on a new electrolyte that suppresses dendrite growth over thousands of charge-discharge cycles. In addition, techno-economic modeling indicated that high sulfur concentration compared with the volume of electrolyte in the lithium-sulfur cell (a lean” electrolyte) is a promising route to achieve JCESR performance and cost targets.

For the third concept, we have been investigating the replacement of solid electrodes with energy-dense liquids that charge and discharge as they flow through the battery. These redox flow batteries” store large amounts of energy inexpensively and are well suited to the grid. A critical component is the separator between the positive and negative liquid electrodes. We developed new concepts for flow batteries — storing energy in liquid suspensions containing  polymer or colloid active materials up to a micron in size, which are easily separated by a standard off-the-shelf membrane.

To measure progress, we have created a set of metrics that will track the ongoing productivity of JCESR over the course of the remainder of the project. The report card to date looks like this:

• Published more than one hundred journal articles. In the last year, the ISI Web of Science has designated 13 JCESR publications as Highly Cited Papers.”
• Filed 28 invention disclosures as part of the patent application process
• Calculated key properties of more than 16,000 candidate organic molecules for use in electrolytes and as active materials in flow batteries, the analysis of which is being used to guide future research directions
• Completed seven techno-economic modeling studies, the projections from which have guided the research directions on the three storage concepts
• Designed five research prototypes suitable for testing
• Held seven regional outreach events

Halfway through our first five years, we see the need and opportunity to focus more strategically and intently on achieving our three legacies:

1. A library of fundamental science of the materials and phenomena of energy storage at the atomic and molecular level;
2. Two prototypes, one for transportation and one for the grid that, when scaled to manufacturing, have the potential to deliver five times the energy density at one-fifth the cost; and
3. A new paradigm for battery R&D that integrates discovery science, battery design, research prototypes, and manufacturing collaboration in a single highly interactive organization.

We have already launched our new paradigm, which integrates talent from 20 partner and funded collaborator organizations, comprising more than 180 researchers including students, postdocs, early career researchers, senior scientists, and engineers. This team provides a critical mass large enough to embrace the full spectrum of diverse disciplines and specialties needed to pursue beyond-lithium-ion battery R&D.

JCESR’s new paradigm takes mission-driven research implemented through strategic planning and nimble change as its guiding principles, based on frequent and regular in-person communication at all levels. We have continuously refined our new operational paradigm since its introduction 2 1/2 years ago, as we identify structural and functional challenges and opportunities.

In the coming 2 1/2 years, we will narrow our focus through convergent” research, focusing on the most promising proof-of-principle prototypes that can be transitioned to manufacturing. JCESR’s increased focus on strategy may require shifting resources to implement our strategic directions.

As part of our strategy, however, we will also continue to emphasize exploratory or divergent” research on alternative approaches for prototypes, knowing from experience in battery R&D that many excellent ideas fail for collateral reasons, often driven by the diversity of competing side chemical reactions that inevitably occur at electrochemical interfaces. JCESR intentionally maintains a strategic portfolio of divergent and convergent research so that alternate approaches are readily available to move into convergent positions when our most promising directions encounter unexpected challenges.

#### Outlook

At this the halfway point, we have an impressive record of accomplishment, and we are finalizing the strategic research directions likely to bear fruit for each of the three energy storage concepts and for achieving our first two legacies. In the next two years, these exciting research directions for science and prototypes will take shape and mature.  Stay tuned.

Argonne National Laboratory seeks solutions to pressing national problems in science and technology. The nation’s first national laboratory, Argonne conducts leading-edge basic and applied scientific research in virtually every scientific discipline. Argonne researchers work closely with researchers from hundreds of companies, universities, and federal, state and municipal agencies to help them solve their specific problems, advance America’s scientific leadership and prepare the nation for a better future. With employees from more than 60 nations, Argonne is managed by UChicago Argonne, LLC for the U.S. Department of Energy’s Office of Science.

The U.S. Department of Energy’s Office of Science is the single largest supporter of basic research in the physical sciences in the United States and is working to address some of the most pressing challenges of our time. For more information, visit the Office of Science website.

The Joint Center for Energy Storage Research (JCESR), a DOE Energy Innovation Hub, is a major partnership that integrates researchers from many disciplines to overcome critical scientific and technical barriers and create new breakthrough energy storage technology. Led by the U.S. Department of Energy’s Argonne National Laboratory, partners include national leaders in science and engineering from academia, the private sector, and national laboratories. Their combined expertise spans the full range of the technology-development pipeline from basic research to prototype development to product engineering to market delivery.


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# zbMATH — the first resource for mathematics

## Journal of Natural Science of Heilongjiang University

 Short Title: J. Nat. Sci. Heilongjiang Univ. Parallel Title: Heilongjiang Daxue Ziran Kexue Xuebao Publisher: Editorial Department of Journal of Natural Science of Heilongjiang University, Harbin ISSN: 1001-7011 Online: http://www.airiti.com/CEPS/ec_en/ecJnlIntro.aspx?Jnliid=1136&newIssueiid=59916&jnlcattype=1&jnlptype=3&jnltype=472 Comments: Vol.12, No.4 (1995) - Vol.19, No.4 (2002) were indexed with the original title: Heilongjiangdaxue Ziran Kexue Xuebao
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# Cost Per Order (CPO)

## Contents

Cost per order (CPO) refers to the costs incurred during a purchase, an order placement, or when a lead is given in e-commerce. In online marketing, the CPO is used to calculate all costs incurred in the course of an order or when a lead is generated. These include any advertisement costs, subscription charges, and obligatory shipping costs. It is a measure used to determine the effectiveness of marketing measures and is often used in affiliate marketing and internet advertising. The cost per order is also referred to as cost per sale or cost per lead.

### Calculating the CPO

The CPO is calculated as follows: The total costs of an action are divided by the number of the reactions.

CPO = Action costs / No. of reactions

When determining the net income, the online campaign budget is important for the use of the cost per order. The CPO is, however, not used to determine the net income in a business sense. Rather, it is used to determine the effectiveness of marketing campaigns with respect to the budget used. [1] Here, the focus is in the number of reactions that resulted from a specific advertising campaign.

## Use of CPO

The CPO is most commonly used in online advertisement. If a text or banner advertisement is displayed, many users get to see it, but not every impression draws a click (see cost per click (CPC)) or a purchase. This is why the CPO and other models from the cost per area field are used. With the CPO, it becomes possible to measure the actual success of the advertisement.

The value for the cost per order is more than a measure in affiliate marketing. It is a billing model that implies a campaign, a tracking, and the payment of commissions within this model. As a rule, the CPO is connected to a fee that is paid by the seller of a product for the placement of advertisements on the websites of his/her affiliates and partners. The CPO often involves a fixed amount or percentage of the sales that is paid out to the partners.

## Importance for online marketing

The CPO is an accounting method that is based on actual sales. This has a significant advantage for product sellers: They are able to scale their advertisement measures in a better way and only pay for the actual sales made. Although a certain percentage is paid to the affiliates, it is the affiliates that provide their inventory for the presentation of advertisement to potential customers. The CPO is also oriented to the achieved margin: Companies that sell expensive products can generally afford higher CPO values, whereas small sales margins are rather compatible with low CPO values. Basically, it is the goal of the company to keep the CPO value as low as possible. The CPO model is superior to other concepts, which do not take the company’s actual value of the advertisement measures into account. This is because the traffic, page impressions, and click-through rates do not necessarily draw sales, even when these are important from a search engine-marketing point of view. The CPO therefore reduces the risk of spending too much on advertisement at a low utility value and is particularly crucial for small and mid-size companies that have a smaller budget but wish to advertise effectively.

## References

1. What is Cost Per Order (CPO)? brickmarketing.com. Accessed on 07/15/2014


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PDF e-Pub

## Section: New Results

### A fully labelled proof system for intuitionistic modal logics

Participants : Lutz Straßburger, Marianela Morales.

In this paper we present a labelled sequent system for intuitionistic modal logics such that there is not only one, but two relation symbols appearing in sequents: one for the accessibility relation associated with the Kripke semantics for normal modal logics and one for the preorder relation associated with the Kripke semantics for intuitionistic logic. This puts our system in close correspondence with the standard birelational Kripke semantics for intuitionistic modal logics. As a consequence it can encompass a wider range of intuitionistic modal logics than existing labelled systems. We also show an internal cut elimination proof for our system [30].


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# How to find acceleration with two direction forces and a mass?

0 votes
136 views
I want to find acceleration of a rope with two forces going opposite directions (5.0N right) (30.0N left) with a rope mass of 2.5 kg.  May you please tell me how you have solved the problem also?
asked Mar 31, 2014

## 1 Answer

0 votes

Assume the gravitational force won't be bothered as the rope could be supported vertically by the smooth ground and or a table.

To find acceleration, you would need to always find the net force. In this case, the vertical forces don't need to be considered as we assumed that they will all cancel out. Then the net force will be calculated based on the horizontal forces. Define left is your positive direction, then your net force would be $$F_net=30N-5N=25N$$

Then the acceleration would be $$a=\frac{F_{net}}{m}$$

answered Apr 10, 2014 by (21,750 points)


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# imbalanced dataset in text classififaction

I have a data set collected from Facebook consists of 10 class, each class have 2500 posts, but when count number of unique words in each class, they has different count as shown in the figure

Is this an imbalanced problem due to word count , or balanced according number of posts. and what is the best solution if it imbalanced?

update My python code:

data = pd.read_csv('E:\cluster data\One_File_nonnormalizenew2norm.txt', sep="*")

data.columns = ["text", "class1"]
data.dropna(inplace=True)
data['class1'] = data.class1.astype('category').cat.codes
text = data['text']

y = (data['class1'])
sentences_train, sentences_test, y_train, y_test = train_test_split(text, y, test_size=0.25, random_state=1000)
from sklearn.feature_extraction.text import CountVectorizer
num_class = len(np.unique(data.class1.values))

vectorizer = CountVectorizer()
vectorizer.fit(sentences_train)

X_train = vectorizer.transform(sentences_train)
X_test  = vectorizer.transform(sentences_test)

model = Sequential()
max_words=5000
model.summary()
model.compile(loss='sparse_categorical_crossentropy',
optimizer='rmsprop',
metrics=['accuracy'])

model.fit(X_train, y_train,batch_size=150,epochs=10,verbose=2,validation_data=(X_test,y_test),shuffle=True)
predicted = model.predict(X_test)
predicted = np.argmax(predicted, axis=1)
accuracy_score(y_test, predicted)
predicted = model.predict(X_test)
predicted = np.argmax(predicted, axis=1)
accuracy_score(y_test, predicted)

0.9592031872509961

• Could you please post your approach/code here? – Sunil Feb 6 '19 at 14:37
• which code??I ask a general question based on number of samples?? – mtesta010 Feb 6 '19 at 15:18

Thank you for your message Ahmed. There are things to point out:

1. Is this an imbalanced problem? Which problem? THIS is not a problem. This is data.
2. What analysis is going to be done? In some cases you need posts and in some you need these keywords.
3. What method is going to be done for that analysis? Some methods get keywords as input and some get posts.

But about the numbers themselves; Not necessarily. The smallest class has 20% of the largest population and moreover, the scale is pretty high (20000 samples). So it is not necessarily an imbalanced class distribution. Again, see what you want to do with this data. That determines the answer much more accurate.

Hope it helped. If you write about the task you want to do I can post the solution here.

Cheers,

## UPDATE

Well, then the problem is pretty straight-forward. These unique words are probably not much meaningful here. I certainly recommend that you try BoW models first (TF-IDF and classic BoW) for modeling your corpus. Then tune the hyperparameters of models and using a simple Multinomial Naive Bayes you will get an acceptable result.

Data is not counted that imbalanced. I had a problem in which some classes had 3000-4000 samples and some only 20! That is certainly called imbalanced but here you still have enough data to represent your minority class and also you will use Precision-Recall for evaluation instead of Accuracy so you will be fine. I strongly recommend you to have a look at this for Python implementation and also seeing some imbalanced data in practice.

The DL thing is answered in the comment.

• thank you Kasra for your answer, 1-the problem i mean that this data-set will be used in text classification system, 2 , 3-I will apply text preprocessing technique and some ML algorithms (NB,svm ,..) and Deep learning (Keras). – mtesta010 Feb 7 '19 at 10:44
• when I use Deep Learning text classification there is no improvement occurred comparing with traditional ML ??!! – mtesta010 Feb 7 '19 at 10:51
• First the last: 1) Doing Deep Learning is tricky (be sure your setting is right) 2) Not necessarily it is supposed to improve. Depends on your data, the problem and the architecture you use. – Kasra Manshaei Feb 7 '19 at 13:42
• OK , I will try it thank you very much – mtesta010 Feb 7 '19 at 14:10

I don't now wether I got your question right. But if you count all words within a class, for example, the word "the" is counted everytime it appears. However, if you count the unique words the word "the" is counted once. This is why your counts differ from your plot. Each class can have a different number of unique words.

• count of unique words after remove stop words,the count differ because posts lengths are different – mtesta010 Feb 6 '19 at 15:55

Here's a simple way to deal with the word-count imbalance.

You could first convert the representation of the tokens using word embedding. There are two publicly available models that are very popular: Word2Vec and GloVe. Word embeddings can be very useful because they capture latent semantic and lexical information that is not usually available in vanilla BOW models.

Next, using the word embeddings, take a mean over the set for each example in you data set. That will balance the classes in some sense because you are essentially reducing the problem to an average embedding representation for each example. Assuming that the number of examples per class are balance then your balance problem is solved. Also consider doing stop-word filtering to remove useless terms.

You can then use that representation as input to you classification model of choice - SVM, Random Forest, Logistic Regression etc... but that is it's own problem to solve.

There are tradeoffs to every type of feature engineering that you do. Be aware of that and do your due diligence to evaluate what type of systematic effects (if any) this type of preprocessing will have on your result.


text:
MathSciNet bibliographic data MR1215439 33D45 (17B37 33D80) Koornwinder, Tom H. Askey-Wilson polynomials as zonal spherical functions on the ${\rm SU}(2)$${\rm SU}(2)$ quantum group. SIAM J. Math. Anal. 24 (1993), no. 3, 795–813. Article

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.


text:
## Symmetric monopoles.(English)Zbl 0846.53016

Monopoles considered in this paper are pairs $$(A, \varphi)$$, consisting of an $$su(2)$$-valued connection 1-form $$A$$ on $$\mathbb{R}^3$$ and an $$su (2)$$-valued Higgs field $$\varphi$$, which satisfy the Bogomolny equations $$^*F_A = \nabla_A \varphi$$ with suitable additional boundary conditions at infinity. The moduli space of framed monopoles of topological charge $$k$$ modulo gauge equivalence is a hyper-Kähler manifold $${\mathcal M}_k$$ of dimension $$4k$$ [M. Atiyah and N. J. Hitchin, The geometry and dynamics of magnetic monopoles, Princeton Univ. Press (1988; Zbl 0671.53001)].
This interesting paper is concerned with classes of $$k$$-monopole solutions which are invariant under various symmetry groups. As it is hard to study monopoles directly in terms of the fields $$(A, \varphi)$$, the authors carry out their analysis mostly in terms of other types of mathematical objects associated in a unique way to monopoles: the spectral curve arising from the twistorial approach [N. J. Hitchin, Commun. Math. Phys. 89, 145-190 (1983; Zbl 0517.58014)], the rational curve defined by S. K. Donaldson [ibid. 96, 387-407 (1984; Zbl 0603.58042)], and Nahm’s equations, resulting by a clever adaptation of ADHM construction of instantons. Monopoles invariant under inversion in a fixed plane, monopoles having cyclic symmetry, and monopoles invariant under symmetry groups of regular solids are carefully analyzed.
Although a $$k$$-monopole can be physically described as the superposition of $$k$$ unit charge magnetic monopoles, it has a well-defined centre and total phase. In connection with these notions, strongly centered $$k$$-monopoles are defined in Section 5, and it is proved that their moduli space is a geodesic submanifold of the whole moduli space $${\mathcal M}_k$$.
Major results are the proof of the existence of a tetrahedrally symmetric 3-monopole and of an octahedrally symmetric 4-monopole, together with the explicit description of the corresponding spectral curves. On the other hand, in spite of some evidence to the contrary, it is proved the nonexistence of an icosahedrally symmetric monopole of charge 6.
In the last section of this rich paper, the authors describe the rational maps associated to strongly centered cyclic symmetric $$k$$-monopoles and discover a novel type of nonplanar $$k$$-monopole scattering process.

### MSC:

 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) 32L05 Holomorphic bundles and generalizations 81T13 Yang-Mills and other gauge theories in quantum field theory

### Citations:

Zbl 0671.53001; Zbl 0517.58014; Zbl 0603.58042
Full Text:


text:
Consider the continuous piece-wise linear function $f:\mathbb{R}\rightarrow\mathbb{R}$, with slope $f'(x)=-1$ or $f'(x)=+1$ according whether $\lfloor x/\sqrt 2\rfloor=0\ \mathrm{ mod} 3$, or not (like this). So $f(x)=x/3+O(1)$ as $|x|\to\infty$ and it is therefore surjective; precisely, any of its fiber has three points, and the arithmetic mean of some two of these is an odd multiple of $\sqrt{2}$. This means that the equation $f(x)=c$ has always an irrational solution, whatever is $c\in\mathbb{R}$.
Consider the continuous piece-wise linear function $f:\mathbb{R}\rightarrow\mathbb{R}$, with slope $f'(x)=-1$ or $f'(x)=+1$ according whether $\lfloor x/\sqrt 2\rfloor=0\ \mathrm{ mod} 3$, or not. So $f(x)=x/3+O(1)$ as $|x|\to\infty$ and it is therefore surjective; precisely, any of its fiber has three points, and the arithmetic mean of some two of these is an odd multiple of $\sqrt{2}$. This means that the equation $f(x)=c$ has always an irrational solution, whatever is $c\in\mathbb{R}$.


text:
# Tag Info

5

EDIT: Per comment below by YALMIP deveeloper Johan Lofberg, the bug in YALMIP was a typo, which has now been corrected and available at https://github.com/yalmip/YALMIP/archive/develop.zip . Using this newest develop version of YALMIP, Mosek should now be able to solve the problem. Edited to reflect YALMIP developer's acknowledgement that there is a bug in ...

3

The following is from Section 2.2 of Semidefinite Programming for Combinatorial Optimization by Christoph Helmberg. Let's define \begin{align*} p^* &= \inf\{\langle C,X\rangle\,:\,X\text{ primal feasible}\},\\ d^* &= \sup\{\langle b,y\rangle\,:\,y\text{ dual feasible}\}. \end{align*} In particular, $p^*=\infty$ if the primal is infeasible, \$d^*=-\...

1

I now believe the answer is yes. I.e., at least one optimal solution will lie at an extreme point. See Hermann Schichl's answer here. For a minimization problem, you need a convex objective function on a convex domain. For a maximization problem, the objective function must instead be concave. Since SDPs have a linear objective function and a convex feasible ...

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text:
# Tag Info

5

First of all, may I point out two big misperceptions that you may have: Implied Volatility (IV) is the input to any vanilla option pricing model (not just Black Scholes (BS) that impacts the pricing the most. You can verify this by flipping through the different risk exposures (greeks and higher order sensitivities) and study mean volatilities in such risk ...

4

There are several reasons, maybe the most important and also quite intuitive one: Implied volatility more or less assumes that the stock price is driven by Brownian motion and thus moves in a continuous fashion. What we observe is that stocks can jump (usually downwards, sometimes upwards) which needs to be modelled using something like a jump process ...

4

I suggest you avoid using the VIX for implied vols. Why? One has to consider that the VIX is not simply solely dependant on the dynamics on the S&P 500 anymore because the VIX can be traded via options, etc. Thus many more parameters affect the trajectory of the VIX. The VIX has to equal the ATM option vol because this is where arbitrage assumption ...

4

There is a known expansion of implied volatility in moments (I'll find the reference) $$\textrm{IV} = \textrm{vol} * (1 + \frac{\textrm{skew}}{6} * \textrm{LMM} + \frac{\textrm{kurt}}{24}*(\textrm{LMM}^2-1))$$ where log-moneyness is \textrm{LMM} = ...

4

Actually, closing options prices can be downloaded from the exchange, so the data necessary to get the skew is available. If for some reason you don't want to use those closing prices, it is possible to obtain a vol skew from VIX and SKEW. You would need to fit the parameters of a stochastic volatility model (such as Heston's) to the VOL and SKEW data. ...

3

I think you would find the following paper very useful. It compares different pricing models applied to VIX options. You can use it as starting point to apply to VSTOXX options and see where it gets you. The Performance of VIX Option Pricing Models: EmpiricalEvidence Beyond Simulation The following models were tested: Whaley (1993) Grunbichler and ...

2

There is another approach to compute Implied Volatility, namely the Model Free Implied Volatility (MFIV). According to this link: "Unlike the traditional concept of implied volatility, where the implied volatility is estimated numerically from an option pricing model, the model free implied volatility (MFIV) is not dependent on any option pricing ...

1

The volatility smile is the result of market forces knowing form experience that out of the money option pay out more often that what would be expected by a normal (Gaussian) distribution. For years Quants speculated why the market drove the out of the money options higher that the price of the Black-Scholes model. The best theory speculates that the ...

1

it's difficult to say that they are not popular. Some people definitely use them for live pricing. I'd say the real question is "why are they not popular in the academic literature"? One answer would simply be that most the questions that arise in their use are ones of fiddliness which do not make good papers.

1

The focus on volatility comes about because all price changes "look like" volatility, no matter their source. Improvements in volatility treatment are therefore conflated with improvements in the model, and typically when people consider altered models, they first look to how well the alterations do in providing prices that explain skew for the classical ...

1

This article discusses the problem on the German electricity market. They arrive at the following conclusion:"When the B&S model is used to calculate implied volatilities one often obtain different numbers for different values of K and T. In particular, a "smile" or "smirk" shape is often observed in the plot of implied volatility versus strike price. ...

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text:
#### Archived

This topic is now archived and is closed to further replies.

# Float or Double

This topic is 5517 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

What are the pro''s and con''s of using floats or doubles in 3D gfx? Which should i use?

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The obvious:

double
- Higher precision

float
- Faster (not always)
- Less memory

For graphics, precision doesn''t matter all that much, which is why floats are often used.

Cédric

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The FPU should handle floats and doubles at the same speed. It''s the extra size (read: memory bandwidth used) that causes the difference. When using SIMD instructions, twice as many floats can be processed as doubles.

I agree with Cedric... you most likely won''t need the added precision. Try it out. If you get precision problems, you can always switch to doubles. I''d recommend a simple:

  typedef float real_t;

Use real_t everywhere. If you find the need to use extra precision, all you need to do is change the typedef.

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How does double multiplication not take more cycles than float multiplication if they are same speed?

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AFAIK, the FPU converts from 32bit (or 64bit) to a standard precision (80bit on Intel processors, 64bit on some, though not all, AMD processors) and converts back upon storing. The multiplication is done entirely using those registers, taking 3 clock ticks. Funnily, depending on the state of the floating point control word, division does take a different number of ticks for each precision (19 ticks for 24bit precision, 33 ticks for 53bit precision, 39 ticks for 64bit precision)

These timings are for Pentium/Pentium MMX. The situation is analogous (if a little more complicated) for more recent processors.

[edited by - Kippesoep on November 6, 2002 11:06:52 AM]

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text:
NETCDF-CF_PROFILE

Overview

This importType is available in DELFT-FEWS versions after 28-10-2009 (FEWS version 2009.02)

It is only available in timeSeriesImportRun and not for example in the importNetcdfActivity of the GeneralAdapter

Imports profile time series data from NetCDF files which comply to the CF standard. More information about the cf standards can be found at: http://cfconventions.org/cf-conventions/v1.6.0/cf-conventions.html

See also the following two other types of NetCDF-CF imports that are available:

In DELFT-FEWS versions 2011.02 and later this import type can also be used to import data using OPeNDAP, see Import data using OPeNDAP.

Configuring the import

An example of the NETCDF-CF_PROFILE import will be given here.

From 2021.01 it is possible to import NETCDF attributes as timeSeriesProperties into FEWS. This can be configured in the properties (see example below). Using the example config, the value of the NETCDF attribute ‘systemid’ will be imported as a timeSeriesProperty ‘rivieren2’.

Since 2021.02 it is also possible to export timeSeriesProperties as NETCDF attributes. You can find a config example here:  https://publicwiki.deltares.nl/display/FEWSDOC/NETCDF-CF_GRID+Export

ImportNetcdf_Profile 1.00 default.xml
```<?xml version="1.0" encoding="UTF-8"?>
<timeSeriesImportRun xmlns="http://www.wldelft.nl/fews" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.wldelft.nl/fews http://fews.wldelft.nl/schemas/version1.0/timeSeriesImportRun.xsd">
<import>
<general>
<importType>NETCDF-CF_PROFILE</importType>
<folder>$IMPORT_FOLDER$/NETCDF</folder>
<failedFolder>$IMPORT_FAILED_FOLDER$</failedFolder>
<backupFolder>$IMPORT_BACKUP_FOLDER$</backupFolder>
<idMapId>IdImportNetCDF</idMapId>
</general>
<properties>
<string key="netcdfattribute:systemid" value='timeseriesproperty:rivieren2'/>
</properties>
<timeSeriesSet>
<moduleInstanceId>ImportNetcdf_Profile</moduleInstanceId>
<valueType>longitudinalprofile</valueType>
<parameterId>H.fx</parameterId>
<locationSetId>SobekProfiles_WL</locationSetId>
<timeSeriesType>external historical</timeSeriesType>
<timeStep unit="day" multiplier="1"/>
</timeSeriesSet>
</import>
</timeSeriesImportRun>
```

An example of the IdMapping used for the NETCDF-CF_PROFILE import will be given below.
Note that in the IdMapping of the parameters, the external name must match the variable names as used by the netcdf file exactly (case sensitive). The locations that are mapped refer to branch id's which are defined in the Branches.xml.

IdImportNetCDF 1.00 default.xml
```<?xml version="1.0" encoding="UTF-8"?>
<idMap xmlns="http://www.wldelft.nl/fews" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.wldelft.nl/fews http://fews.wldelft.nl/schemas/version1.0/idMap.xsd" version="1.1">

<parameter internal="H.fx" external="waterlevel"/>

<location internal="Maastakken_NDB(Haringvliet)" external="Maastakken_NDB(Haringvliet)"/>
<location internal="Rijntakken_NDB_NWW" external="Rijntakken_NDB_NWW"/>
<location internal="Rijntakken_AmsterdamRijnkanaal" external="Rijntakken_AmsterdamRijnkanaal"/>
<location internal="Rijntakken2_NDB2(NieuweWaterweg)" external="Rijntakken2_NDB2(NieuweWaterweg)"/>
<location internal="Rijntakken_IJssel" external="Rijntakken_IJssel"/>
<location internal="IJssel_IJsselmeer" external="IJssel_IJsselmeer"/>
<location internal="Markermeer_VeluweRandmeren" external="Markermeer_VeluweRandmeren"/>

</idMap>
```

An example of the branches file is shown below.

Branches 1.00 default.xml
```<?xml version="1.0" encoding="UTF-8"?>
<branches xmlns="http://www.wldelft.nl/fews" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.wldelft.nl/fews http://fews.wldelft.nl/schemas/version1.0/branches.xsd" version="1.1">
<geoDatum>Rijks Driehoekstelsel</geoDatum>
<branch id="Maastakken_NDB(Haringvliet)">
<branchName>Maastakken_NDB(Haringvliet)</branchName>
<startChainage>1030</startChainage>
<endChainage>321624</endChainage>
<pt chainage="1030" label="R_MS_001_1" x="176029.1129" y="308594.236" z="40.32" z_rb="51.34"/>
<pt chainage="2061" label="R_MS_001_2" x="176631.808" y="309427.7428" z="41.79" z_rb="50.92"/>
...
<pt chainage="321624" label="N_NDB_92" x="57935.1" y="436953" z="-7.82" z_rb="2.79"/>
</branch>
...
<branch id="Markermeer_VeluweRandmeren">
...
</branch>
</branches>
```

The locationSetId used by the ImportNetcdf_Profile.xml must contain the branches defined in the above IdMapping.

LocationSets 1.00 default.xml
```<?xml version="1.0" encoding="UTF-8"?>
<locationSets xmlns="http://www.wldelft.nl/fews" version="1.1" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.wldelft.nl/fews http://fews.wldelft.nl/schemas/version1.0/locationSets.xsd">
<locationSet id="SobekProfiles_WL" name="Sobek Profiles WL">
<locationId>Maastakken_NDB(Haringvliet)</locationId>
<locationId>Rijntakken_NDB_NWW</locationId>
<locationId>Rijntakken_AmsterdamRijnkanaal</locationId>
<locationId>Rijntakken2_NDB2(NieuweWaterweg)</locationId>
<locationId>Rijntakken_IJssel</locationId>
<locationId>IJssel_IJsselmeer</locationId>
<locationId>Markermeer_VeluweRandmeren</locationId>
</locationSet>
</locationSets>
```
• No labels


text: ABSTRACT: This paper investigates a Description Logic, namely $\mathcal{SHI}_+$, which extends $\mathcal{SHI}$ by adding  transitive closure of roles. The resulting logic $\mathcal{SHI}_+$ allows transitive closure of roles to occur not only in concept inclusion axioms but also in role inclusion axioms. We show that $\mathcal{SHI}_+$ is decidable by devising a sound and complete algorithm for deciding satisfiability of concepts in $\mathcal{SHI}_+$ with respect to a set of concept and role inclusion axioms.


text:
×

## Limits of Sequences and Series

Infinitely many mathematicians walk into a bar. The first says "I'll have a beer". The next ones say "I'll have half of the previous guy". The bartender pours out 2 beers and says "Know your limits".

# Level 2

Tetration is defined as

$\Large {^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_{n \ a\text{'s}}.$

Find the value of

$\lim_{n\rightarrow\infty}{^{n}}\left(\sqrt {2}\right).$

Consider the sequence $$a_1,a_2,\ldots$$ where $$a_n = \cos(π\sqrt{n^{2}+n})$$. Find $\large \lim_{n\rightarrow\infty} a_n.$

Consider the sequence $$\left\{a_n\right\}$$ where $$a_n=\left(\frac{2x-1}{5}\right)^n.$$ What is the sum of all integers $$x$$ for which this sequence converges?

Let $$P_{n}$$ be the product of the numbers in the $$n$$th row of Pascal's Triangle.
$\lim_{n\to\infty}\frac{P_{n-1} P_{n+1}}{P_{n}^2} =$

The sequence $$\{a_n\}$$ follows the recursion $$a^2_{n+1}=2a_n+3$$ with $$a_1=7.$$

Determine $$\displaystyle \lim_{n \to \infty} a_n$$.

×

Problem Loading...

Note Loading...

Set Loading...


text:
Workshop on Symplectic Dynamics

June 27-July 01, 2022
Instituto Superior Técnico
Lisboa, Portugal

# Titles and Abstracts

Luca Asselle

Title: The dynamics of strong magnetic fields on surfaces: periodic orbits, trapping regions, and rigidity of Zoll systems

Abstract: How does a magnetic field influence the motion of a charged particle on a surface? Are there periodic orbits or trapping regions for the particle? How difficult is it to construct a magnetic field for which all orbits are periodic? In this talk we will see that, if the magnetic field is strong, a normal form going back to the Russian school allows us to use the Poincaré-Birkhoff theorem and KAM theory to tackle these questions. This is joint work with Gabriele Benedetti.

Gabriele Benedetti

Title: A bi-invariant Lorentz-Finsler structure in contact geometry

Abstract: Contact systolic geometry studies the relationship between the minimal period of closed orbits of (autonomous) Reeb flows and the volume of a contact manifold. Extending this study to non-autonomous Reeb flows gives rise to a bi-invariant Lorentz-Finsler structure on the group of contactomorphisms. Although this structure is too flexible to produce non-trivial measurements in general, we will see that it leads to interesting local rigidity phenomena for certain classes of contactomorphisms. This is joint work with Alberto Abbondandolo and Leonid Polterovich.

Frédéric Bourgeois

Title: Generating families and augmentations for Legendrian submanifolds

Abstract: Generating families provide a convenient way to describe wide classes of Legendrian submanifolds in 1-jet spaces, but they can also be used to define a homological invariant via Morse theory. I will  describe a degeneration process that can be used to better understand the differential of the corresponding complex. It turns out that this approach can also lead to a relation between generating families for a Legendrian submanifold and augmentations for its Chekanov-Eliashberg differential graded algebra.

Michael Brandenbursky

Title: $C^0$-gap between entropy-zero Hamiltonians and autonomous diffeomorphisms of surfaces

Abstract: Let $\Sigma$ be a surface equipped with an area form. There is a long standing open question by Katok, which, in particular, asks whether every entropy-zero Hamiltonian diffeomorphism of a surface lies in the $C^0$-closure of the set of integrable diffeomorphisms. A slightly weaker version of this question asks: "Does every entropy zero Hamiltonian diffeomorphism of a surface lie in the $C^0$-closure of the set of autonomous diffeomorphisms?" In this talk I will answer in negative the later question. In particular, I will show that on a surface $\Sigma$ the set of autonomous Hamiltonian diffeomorphisms is not $C^0$-dense in the set of entropy-zero Hamiltonians. Explicitly constructed examples of such Hamiltonians cannot be approximated by autonomous diffeomorphisms. (Joint with M. Khanevsky).

Urs Frauenfelder

Title: Lagrange multipliers and adiabatic limits

Abstract: This is joint work with Joa Weber. As one learns in calculus there is a one to one correspondence between critical points of the restriction of a functional to a constraint and the Lagrange multiplier functional. The gradient flow lines of the two functional however are different but can be related by an adiabatic limit procedure. In the talk I will present a compactness theorem for this adiabatic limit problem. I also plan to explain an infinite dimensional motivation behind this result concerning Rabinowitz action functional.

Viktor Ginzburg

Title: On the volume of Lagrangian submanifolds

Abstract: We will discuss the continuity property of the surface area of Lagrangian submanifolds, or to be more precise its lower semi-continuity with respect to the gamma-norm, and connections with integral geometry, Floer theory, and the h-principle for isometric embeddings (the Nash theorem). This is a work in progress with Erman Cineli and Basak Gurel.

Basak Gurel

Title: Topological entropy, barcodes and Floer theory

Abstract: Topological entropy is one of the fundamental invariants of a dynamical system, measuring the orbit complexity. In this talk, we discuss a connection between the topological entropy of compactly supported Hamiltonian diffeomorphisms and Floer theory. We introduce a new invariant associated with the Floer complexes of the iterates of such a diffeomorphism, which we call barcode entropy. We show that barcode entropy is closely related to topological entropy and that these invariants are equal in dimension two. The talk is based on joint work with Erman Cineli and Viktor Ginzburg.

Umberto Hryniewicz

Title: Existence and applications of global Surfaces of section

Abstract: A global surface of section (GSS) reduces the study of a non-singular flow in 3D to that of a surface diffeomorphism. In this talk I will present some existence statements within the class of Reeb flows. Firstly, I will focus on the question of finding a GSS spanned by a given collection of periodic orbits, with no genericity assumptions, and state results obtained in collaboration with Salomão and Wysocki. These statements have applications in Celestial Mechanics. Secondly, I will present recent existence statements for rational GSS’s (Birkhoff sections) under genericity assumptions, and explain how to use them to prove that $C^\infty$ generically a Reeb flow on a closed 3-manifold has positive topological entropy. This is the fruit of joint work with Colin, Dehornoy and Rechtman, and is based on broken book decompositions.

Michael Hutchings

Title: Quantitative closing lemmas

Abstract: We consider closing lemmas which assert roughly speaking that a local $C^\infty$ perturbation (of a symplectomorphism or a contact form) leads to the local creation of a periodic orbit. Quantitative closing lemmas assert roughly speaking that an orbit of period $L$ appears within time $O(L^{-1})$. Spectral gaps in Floer theories, or related invariants, can be used to prove such quantitative closing lemmas. In particular we can use periodic Floer homology to prove quantitative closing lemmas for certain area-preserving surface diffeomorphisms, and embedded contact homology to prove quantitative closing lemmas for contact three-manifolds. Based on joint work with Oliver Edtmair.

Jungsoo Kang

Title: Rabinowitz Floer homology for negative line bundles

Abstract: In my previous work with Peter Albers, we proved that the Rabinowitz Floer homology(RFH) for a circle subbundle of a negative line bundle vanishes if the radius of the circle is small. Later Sara Venkatesh showed that, in this setting, the RFH is not always invariant under the change of radius by computing this for some examples. In this talk, I will explain how one can study the RFH for negative line bundles using the filtration introduced by Urs Frauenfelder (also by Viktor Ginzburg and Jeongmin Shon in the context of positive symplectic homology) and revisit the above computations. This is joint work with Peter Albers.

Marco Mazzucchelli

Title: Existence of global surfaces of section for Reeb flows of closed contact 3-manifolds

Abstract: In this talk, based on joint work with Gonzalo Contreras, I will outline a proof of the existence of global surfaces of section for all Reeb flows of closed contact 3-manifolds satisfying the Kupka-Smale condition: non-degeneracy of the closed orbits, and transversality of the stable and unstable manifolds of the hyperbolic closed orbits. This result, in particular, settles the existence of global surfaces of section for the Reeb vector field of a $C^\infty$ generic contact form on any closed 3-manifold, and even for the geodesic vector field of a $C^\infty$ generic Riemannian metric on any closed surface. As an application, I will provide a new characterization of Anosov Reeb flows of closed contact 3-manifolds, which implies the $C^2$ structural stability conjecture for Riemannian geodesic flows of closed surfaces.

Matthias Meiwes

Title: Braid stability and entropy robustness

Abstract: Topological entropy captures the orbit complexity of a dynamical system with the help of a single non-negative number. Detecting robustness of this number under perturbation is a way to understand stability features of a chaotic system. In my talk, I will address the problem of robustness of entropy for Hamiltonian diffeomorphisms in terms of Hofer's metric. Our main focus lies on dimension 2, where there is a strong connection between topological entropy and the existence of specific braid types of periodic orbits. I will discuss a result that states that any braid of non-degenerate one-periodic orbits with pairwise homotopic strands persists under generic Hofer-small perturbations. One application is a local entropy robustness result for surfaces. My talk is based on joint work with Marcelo Alves.

Alexandru Oancea

Title: Secondary continuation map in Floer theory

Abstract: In situations where Floer continuation maps are not isomorphisms, secondary continuation maps obtained by interpolation carry nontrivial topological and dynamical information. I will explain two such examples, with applications to string topology. The talk will be based on joint work with Kai Cieliebak and Nancy Hingston.

Alexander Ritter

Title: Symplectic cohomology of conical symplectic resolutions

Abstract: In this joint work with Filip Zivanovic, we construct symplectic cohomology for a class of symplectic manifolds that admit ${\mathbb C}^*$-actions and which project equivariantly and properly to a convex symplectic manifold. The motivation for studying these is a large class of examples known as Conical Symplectic Resolutions, which includes quiver varieties, resolutions of Slodowy varieties, and hypertoric varieties. These spaces are highly non-exact at infinity, so along the way we develop foundational results to be able to apply Floer theory. Motivated by joint work with Mark McLean on the Cohomological McKay Correspondence, which we also briefly review in the talk, our goal is to describe the ordinary cohomology of the resolution in terms of a Morse-Bott spectral sequence for positive symplectic cohomology. These spectral sequences turn out to be quite computable in many examples. We also obtain a filtration on ordinary cohomology by cup-product ideals, and interestingly the filtration can be dependent on the choice of circle action.

Stefan Suhr

Title: Recent Progress in the Theory of Lyapunov Functions

Abstract: Lyapunov functions are important distinguishers in the theory of dynamical systems as they dissect the underlying space into an “simple” part, i.e. where the dynamics act "gradient like”, and the “interesting” part, i.e. where recurrence appears. The talk will explain a method to construct smooth and complete Lyapunov functions for vector fields (or multi functions). The method is inspired by the closely related problem of time functions in general relativity. If time permits I will give an outlook on generalizations to Lyapunov $1$-forms.

Otto van Koert

Title: A generalization of the Poincare-Birkhoff fixed point theorem

Abstract: In joint work with Agustin Moreno, we propose a generalization of the Poincare-Birkhoff fixed point theorem. We start with a construction of global hypersurfaces of section in the spatial three-body problem and some related problems, describe some return maps and suggest some generalizations of the Poincare-Birkhoff fixed point theorem. We use symplectic homology in the proof of our theorem.


text:
Parallel between Fourier series and orthogonal projections

My professor made an analogy between Fourier series and orthogonal projections, and I was hoping someone could explain that somewhat more. Basically, as I understand it:

$$a_n = \frac{1}{L} \int_L^L f(x) \cos\left(\frac{ n\pi x}{L}\right) \ dx \longleftrightarrow c_1 = \frac{v \cdot b_i}{b_i \cdot b_i},$$

where $$\dfrac{1}{L}$$ can be thought of as normalizing the projection, and the integrand is the inner product (equivalent to the dot product on the right side).

Am I understanding this correctly? And can someone clarify this for me?

• "Paralell..." Not a pun?
– Pedro
May 1, 2014 at 23:35

He is referring to the inner product induced by the space of square-integrable functions $L^2 [-L, L]$.
$$a_n = \frac{ \left <f(x), \cos \left ( \frac{n \pi x}{L} \right )\right >}{ \left < \cos \left ( \frac{n \pi x}{L} \right ),\cos \left ( \frac{m \pi x}{L} \right )\right >} = \frac{ \int_{-L}^{L} f(x) \cos \left ( \frac{n \pi x}{L} \right ) dx }{ \int_{-L}^{L} \cos \left ( \frac{n \pi x}{L} \right ) \cos \left ( \frac{m \pi x}{L} \right ) dx }.$$
where $\left < \cos \left ( \frac{n \pi x}{L} \right ),\cos \left ( \frac{m \pi x}{L} \right )\right > = \left\{\begin{matrix} 1& \text{if} \;n = m \neq 0\\ 0& \text{if} \;n \neq m \end{matrix}\right.$