text:
# How do you solve the inequality 1/2<n/6?

$n > 3$
$\text{multiply both sides by 6}$
${\cancel{6}}^{3} \times \frac{1}{\cancel{2}} ^ 1 < \cancel{6} \times \frac{n}{\cancel{6}}$
$\Rightarrow 3 < n \text{ or } n > 3$


text:
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# Can you help us write software for simple climate models?

13»

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101.
edited November 2011

I put JSXGraph onto the Open Visualisation page.

Source Text:I put JSXGraph onto the [Open Visualisation](http://wiki.okfn.org/OpenVisualisation) page.
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102.

The Java Climate Model ran fine on my computer, and will probably run on any reasonably modern Macintosh, which all have Java runtimes installed.

As for local apps vs. web apps, there's a tradeoff. While local apps have advantages, I think fewer people are likely to download and install one, rather than run something in a browser window.

Source Text:The Java Climate Model ran fine on my computer, and will probably run on any reasonably modern Macintosh, which all have Java runtimes installed. As for local apps vs. web apps, there's a tradeoff. While local apps have advantages, I think fewer people are likely to download and install one, rather than run something in a browser window.
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103.
edited November 2011

Cool nad ! maybe you should put d3 as well which is a tremendous package for client-side visualization

John: did u say CC-SA 3.0 ? that would be good but i cant se your post as im reading and writing this on p.3

Nathan: I agree. I tend to use a lot of Google apps, for writing and more as they are realtime collaborative and i can write anywhere, but for graphics, i use a mixed bag of Inkscape and GIMP

Source Text:Cool nad ! maybe you should put [d3](http://mbostock.github.com/d3/) as well which is a tremendous package for client-side visualization John: did u say CC-SA 3.0 ? that would be good but i cant se your post as im reading and writing this on p.3 Nathan: I agree. I tend to use a lot of Google apps, for writing and more as they are realtime collaborative and i can write anywhere, but for graphics, i use a mixed bag of Inkscape and GIMP
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104.
edited November 2011

staffan,

i had actually mentioned d3, however i felt a bit uneasy about putting it on the page. That is Mike Bostock claims his software is free and by reading the licence I understood it as being free in the usual sense, but I am no lawyer and it seems he uses no standard free licence. So I wasn't sure what to write into the table.

Source Text:staffan, i had actually [mentioned](http://www.math.ntnu.no/~stacey/Mathforge/Azimuth/comments.php?DiscussionID=856&page=1) d3, however i felt a bit uneasy about putting it on the page. That is Mike Bostock claims his software is free and by reading the [licence](https://raw.github.com/mbostock/d3/master/LICENSE) I understood it as being free in the usual sense, but I am no lawyer and it seems he uses no standard free licence. So I wasn't sure what to write into the table.
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105.

Talking of d3 check out this cool example of a quantum anharmonic oscillator which +Dan Piponi just posted on his Google+ stream.

Pretty slick -- looks like Dan Piponi isn't as reticent as I am when it comes to writing numerical solvers in JavaScript (a fft solver in this case)!

Source Text:Talking of [d3](http://mbostock.github.com/d3/) check out [this cool example of a quantum anharmonic oscillator](http://homepage.mac.com/sigfpe/Harmonic/anharmonic.html) which +Dan Piponi just posted on his Google+ stream. Pretty slick -- looks like Dan Piponi isn't as reticent as I am when it comes to writing numerical solvers in JavaScript (a fft solver in this case)!
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106.
edited November 2011

I'm sorry, I've been feeling depressed about my (in)ability to do anything really useful to help save the environment, so I've been avoiding this forum for the last week. That's sort of childish; I apologize. I'm back.

Staffan wrote:

I've just found an example which should be near and dear to your hearts involving predator/prey relationships between Carp and Pike populations:

I'd say this qualifies as fun! (Or am I just easily amused??)

I think that's fun! The cute graphics make it more fun than a mere graph of populations would be.

Source Text:I'm sorry, I've been feeling depressed about my (in)ability to do anything really useful to help save the environment, so I've been avoiding this forum for the last week. That's sort of childish; I apologize. I'm back. Staffan wrote: > I've just found an example which should be near and dear to your hearts involving [predator/prey relationships between Carp and Pike populations](http://jsxgraph.uni-bayreuth.de/showcase/carpsandpikes.html): > I'd say this qualifies as fun! (Or am I just easily amused??) I think that's fun! The cute graphics make it more fun than a mere graph of populations would be.
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107.
edited December 2011

All that aside, an equation and some hints should be enough to get me going!

I hope Allan E. and/or you can do a simple model of the sort George Musser described:

$$C \frac{d T}{d t} = - A - B T + Q(t) \, a_p(T(t))$$

where

$$a_p(T) = a_i + \frac{1}{2} (a_f-a_i) (1 + \tanh(\gamma T))$$

and

$$a_i=0.35, \qquad a_f=0.7$$

and

• $A=218$ watts per square meter,

• $B=1.90$ watts per square meter per degree Celsius,

• $C = 5 \times 10^6$ per degree Celsius,

and most importantly:

$$Q(t) = 342.5 + f(t)$$

where $f$ is a function that's zero outside some interval, and takes a positive constant value inside that interval. The user should be able to choose this constant value, choose when the interval ends, and maybe when it starts too, though in a way that's no so important.

Unfortunately I doubt this differential equation can be solved in closed form. So, it will require some numerical computation.

Source Text:> All that aside, an equation and some hints should be enough to get me going! I hope Allan E. and/or you can do a simple model of the sort George Musser described: $$C \frac{d T}{d t} = - A - B T + Q(t) \, a_p(T(t))$$ where $$a_p(T) = a_i + \frac{1}{2} (a_f-a_i) (1 + \tanh(\gamma T))$$ and $$a_i=0.35, \qquad a_f=0.7$$ and * $A=218$ watts per square meter, * $B=1.90$ watts per square meter per degree Celsius, * $C = 5 \times 10^6$ per degree Celsius, and most importantly: $$Q(t) = 342.5 + f(t)$$ where $f$ is a function that's zero outside some interval, and takes a positive constant value inside that interval. The user should be able to choose this constant value, choose when the interval ends, and maybe when it starts too, though in a way that's no so important. Unfortunately I doubt this differential equation can be solved in closed form. So, it _will_ require some numerical computation.
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108.

John wrote:

Unfortunately I doubt this differential equation can be solved in closed form. So, it will require some numerical computation.

I've got a metaphysical allergy with respect to JavaScript (my soul gets pimples when I have to write this stuff), however Runge-Kutta solvers for systems of ODE are probably one of the simplest excercises modern numerical math has to offer, especially if we don't need error control and/or adaptive step size control.

If someone else cares to set up the web page, I could try to talk myself into writing a solver in JavaScript.

Source Text:John wrote: <blockquote> <p> Unfortunately I doubt this differential equation can be solved in closed form. So, it will require some numerical computation. </p> </blockquote> I've got a metaphysical allergy with respect to JavaScript (my soul gets pimples when I have to write this stuff), however Runge-Kutta solvers for systems of ODE are probably one of the simplest excercises modern numerical math has to offer, especially if we don't need error control and/or adaptive step size control. If someone else cares to set up the web page, I could try to talk myself into writing a solver in JavaScript.
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109.

I hope Allan E. and/or you can do a simple model of the sort George Musser described:

$$C \frac{d T}{d t} = - A - B T + Q(T) \, a_p(T(t))$$

Should be easy enough! I assume that's $Q(t)$ above not $Q(T)$?

To show this new time dependence I imagine we could now animate the graph in a loop, eg covering a time interval $I$ every 15 seconds or so and allowing the user to choose a sub-interval $[t_0, t_1]$ with sliders.

Source Text:> I hope Allan E. and/or you can do a simple model of the sort George Musser described: > > $$C \frac{d T}{d t} = - A - B T + Q(T) \, a_p(T(t))$$ Should be easy enough! I assume that's $Q(t)$ above not $Q(T)$? To show this new time dependence I imagine we could now animate the graph in a loop, eg covering a time interval $I$ every 15 seconds or so and allowing the user to choose a sub-interval $[t_0, t_1]$ with sliders.
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110.
edited November 2011

Tim van Beek wrote:

Runge-Kutta solvers for systems of ODE are probably one of the simplest excercises modern numerical math has to offer

The JSXGraph library Staffan suggested already has a Runge-Kutta solver built-in, so I was planning to give that a try.

I share your allergy to JavaScript btw -- although hadn't realized it was metaphysical (until now!)

Source Text:Tim van Beek wrote: > Runge-Kutta solvers for systems of ODE are probably one of the simplest excercises modern numerical math has to offer The JSXGraph library Staffan suggested already has a Runge-Kutta solver built-in, so I was planning to give that a try. I share your allergy to JavaScript btw -- although hadn't realized it was metaphysical (until now!)
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111.

John wrote:

I've been feeling depressed about my (in)ability to do anything really useful to help save the environment

This couldn't be further from where I imagined you to be -- I had envisioned you energetically working on Network Theory Part N (a network of people which actually DOES save the environment)

Source Text:John wrote: > I've been feeling depressed about my (in)ability to do anything really useful to help save the environment This couldn't be further from where I imagined you to be -- I had envisioned you energetically working on Network Theory Part N (a network of people which actually DOES save the environment)
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112.

John you quoted me as saying something about the pred-prey example in jsxgrtaph but it was allan.

anyway i think we should rely on Sage for all math/numerical computation and then use things like jsxgraph, processing.js or what have you.js when needed to visualize things on the client-side and it is not supported by Sage notebook. i saw jsxgraph had runge-kutta solvers but right now i am building box models with system of ODE and that has been straightforward.

Allan and Tim i used to dislike javascript as well , please remind me why :-) I like GWT too Tim but it might be to heavy weight for these simple examples.

Eric: How can we add js libraries to the wiki? Instiki is built on Rails which has support for adding more js-libraries but I don't have the rights to do that. but it should be easy for the person with the right authority.

Source Text:John you quoted me as saying something about the pred-prey example in jsxgrtaph but it was allan. anyway i think we should rely on Sage for all math/numerical computation and then use things like jsxgraph, processing.js or what have you.js when needed to visualize things on the client-side and it is not supported by Sage notebook. i saw jsxgraph had runge-kutta solvers but right now i am building box models with system of ODE and that has been straightforward. Allan and Tim i used to dislike javascript as well , please remind me why :-) I like GWT too Tim but it might be to heavy weight for these simple examples. Eric: How can we add js libraries to the wiki? Instiki is built on Rails which has support for adding more js-libraries but I don't have the rights to do that. but it should be easy for the person with the right authority.
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113.
edited November 2011

Allan and Tim i used to dislike javascript as well , please remind me why :-)

It is not typesafe, there is no decent IDE and no decent debugger, it does not have any decent utility library, and none of the modern concepts of modularization (classes, namespaces). All "industrial strength" frameworks rely on generating it, not writing it (I'm thinking about GWT and JSF).

Anyway, I too found the example with the Runge-Kutta solver in the JSXGraph wiki here:

http://jsxgraph.uni-bayreuth.de/wiki/index.php/Category:Examples

But I agree with Staffan that one should use JavaScript at most for the generation of graphics. So, Allan, Staffan, what is the state of affairs? I've started to play around with JSXGraph and would be ready to make an exception and use their Runge-Kutta solver, just in order to get the stuff that John needs on the web in time.

Source Text:<blockquote> <p> Allan and Tim i used to dislike javascript as well , please remind me why :-) </p> </blockquote> It is not typesafe, there is no decent IDE and no decent debugger, it does not have any decent utility library, and none of the modern concepts of modularization (classes, namespaces). All "industrial strength" frameworks rely on generating it, not writing it (I'm thinking about GWT and JSF). Anyway, I too found the example with the Runge-Kutta solver in the JSXGraph wiki here: http://jsxgraph.uni-bayreuth.de/wiki/index.php/Category:Examples But I agree with Staffan that one should use JavaScript at most for the generation of graphics. So, Allan, Staffan, what is the state of affairs? I've started to play around with JSXGraph and would be ready to make an exception and use their Runge-Kutta solver, just in order to get the stuff that John needs on the web in time.
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114.
edited November 2011

Allan wrote:

Should be easy enough! I assume that's $Q(t)$ above not $Q(T)$?

Yes. I'll fix that in my original comment so we have it correct there.

This couldn't be further from where I imagined you to be -- I had envisioned you energetically working on Network Theory Part N (a network of people which actually DOES save the environment).

I'm almost always energetically doing something these days - I'm busy with the network theory project, and also busy on Google+ trying to foment revolution (passing on news and strategies for Occupy Wall Street) and attract attention to the Azimuth Blog (with articles about Lynn Margulis and 'liquid light'). But I feel very unsure about what to do that'll really have an impact on the environmental crisis I see looming, and I don't think I'll ever be very good at climate science (in part because I don't see that as being enough to save the day). So I've been a bit depressed, and somehow that's manifested itself in me shirking our project here.

But if I relax a bit, I can enjoy it.

Source Text:Allan wrote: > Should be easy enough! I assume that's $Q(t)$ above not $Q(T)$? Yes. I'll fix that in my original comment so we have it correct there. > This couldn't be further from where I imagined you to be -- I had envisioned you energetically working on Network Theory Part N (a network of people which actually DOES save the environment). I'm almost always energetically doing _something_ these days - I'm busy with the network theory project, and also busy on Google+ trying to foment revolution (passing on news and strategies for Occupy Wall Street) and attract attention to the Azimuth Blog (with articles about Lynn Margulis and 'liquid light'). But I feel very unsure about what to do that'll really have an impact on the environmental crisis I see looming, and I don't think I'll ever be very good at climate science (in part because I don't see that as being enough to save the day). So I've been a bit depressed, and somehow that's manifested itself in me shirking our project here. But if I relax a bit, I can enjoy it.
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115.
edited November 2011

In fact, reading the comments of Allan, Staffan and Tim, I'm actually getting excited. I like working as part of a team. I'm glad you're all eager to try this project.

I want to figure out a way to take the simpler model Allan already programmed, here, and publicize it. Probably having it appear as part of a This Week's Finds is the easiest way. Unfortunately I'm not sure I can get it to work on my blog. But I can get it to work on my website, and that's probably good enough.

Source Text:In fact, reading the comments of Allan, Staffan and Tim, I'm actually getting excited. I like working as part of a team. I'm glad you're all eager to try this project. I want to figure out a way to take the simpler model Allan already programmed, [here](http://math.ucr.edu/home/baez/coalbedo/coalbedo.html), and publicize it. Probably having it appear as part of a This Week's Finds is the easiest way. Unfortunately I'm not sure I can get it to work on my blog. But I can get it to work on my website, and that's probably good enough.
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116.

Unfortunately I'm not sure I can get it to work on my blog.

You would need to be able to add two statements to the header node, and a div-node and JavaScript in the body of the HTML-Page for each graphic, which is probably beyond the kind of control that Wordpress offers its users.

Source Text:<blockquote> <p> Unfortunately I'm not sure I can get it to work on my blog. </p> </blockquote> You would need to be able to add two statements to the header node, and a div-node and JavaScript in the body of the HTML-Page for each graphic, which is probably beyond the kind of control that Wordpress offers its users.
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117.
edited November 2011

Reading the very simple model of Snowball Earth page now. I like the word coalbedo. Coal, being black, should have a high value of it.

Source Text:Reading the [very simple model of Snowball Earth](http://math.ucr.edu/home/baez/coalbedo/coalbedo.html) page now. I like the word _coalbedo._ Coal, being black, should have a high value of it.
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118.
edited November 2011

Hi all! Maybe this can also be of use. It's a fourth order Runge-Kutta integrator implemented in JavaScript. More importantly, it comes with some tests that can serve as a starting point for an implementation.

Edit: Oops, sorry for pointing you to that undocumented file, the actual documentation is of more use.

Source Text:Hi all! Maybe [this](http://cpansearch.perl.org/src/PJB/Math-RungeKutta-1.07/js/) can also be of use. It's a fourth order Runge-Kutta integrator implemented in JavaScript. More importantly, it comes with some [tests](http://cpansearch.perl.org/src/PJB/Math-RungeKutta-1.07/js/test.html) that can serve as a starting point for an implementation. Edit: Oops, sorry for pointing you to that undocumented file, [the actual documentation](http://search.cpan.org/~pjb/Math-RungeKutta-1.07/RungeKutta.pm) is of more use.
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119.
edited December 2011

Thanks for the tip!

Sigh, no meaningful variable names, no comments, no explanation of what a method does, no indication of what methods are supposed to be part of the API and what are for internal use only...

Anyway, programming a Runge-Kutta solver should not be the problem here. My problem right now is to find 2 hours of time where I still have enough energy to program anything. But I think it should be possible for us to expand Allan's page with hand coded JavaScript and JQuery-UI-Tools so that it will suffice for an Azimuth blog post.

I'll play around a little bit with the Runge–Kutta solver of JXSGraph next.

@Allen: Did you intend to do anything with the processing graphical language since processing-1.3.6.min.js is in the js folder?

Source Text:Thanks for the tip! Sigh, no meaningful variable names, no comments, no explanation of what a method does, no indication of what methods are supposed to be part of the API and what are for internal use only... Anyway, programming a Runge-Kutta solver should not be the problem here. My problem right now is to find 2 hours of time where I still have enough energy to program anything. But I think it should be possible for us to expand Allan's page with hand coded JavaScript and JQuery-UI-Tools so that it will suffice for an Azimuth blog post. I'll play around a little bit with the Runge–Kutta solver of JXSGraph next. @Allen: Did you intend to do anything with the processing graphical language since processing-1.3.6.min.js is in the js folder?
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120.

@Tim: I stopped using Processing after Staffan pointed out JSXGraph, and had removed it from my repo. (We could return to it if we wanted better 3D support.)

You asked what the state of affairs was: I'm currently pondering the new equation with dynamic Q(t).

What would be the best way to chart/animate it and make it fun? (Presumably fire-breathing lizards and dancing polar bears could easily be swapped in for pike/carp).

More mundanely, I'm also trying to decide if considering initial values perturbed away from the equilibrium could make the graphs more interesting -- eg by changing the lengths of some little vector arrows showing how unstable points in the unstable regime are.

Source Text:@Tim: I stopped using Processing after Staffan pointed out JSXGraph, and had removed it from my repo. (We could return to it if we wanted better 3D support.) You asked what the state of affairs was: I'm currently pondering the new equation with dynamic Q(t). What would be the best way to chart/animate it and make it fun? (Presumably fire-breathing lizards and dancing polar bears could easily be swapped in for pike/carp). More mundanely, I'm also trying to decide if considering initial values perturbed away from the equilibrium could make the graphs more interesting -- eg by changing the lengths of some little vector arrows showing *how* unstable points in the unstable regime are.
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121.

Staffan, you asked about JavaScript -- it's actually enjoyable enough in small doses. I just have pained memories of some critical size (not much larger than Chaitin's Lisp L constant if you've been following these posts, around 2000 bytes) above which it stops being fun and starts beating you in the head. This is mostly for the reasons Tim gave, although support is much better today than a few years back.

Source Text:Staffan, you asked about JavaScript -- it's actually enjoyable enough in small doses. I just have pained memories of some critical size (not much larger than Chaitin's Lisp L constant if you've been following these posts, around 2000 bytes) above which it stops being fun and starts beating you in the head. This is mostly for the reasons Tim gave, although support is much better today than a few years back.
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122.

John wrote:

I don't think I'll ever be very good at climate science (in part because I don't see that as being enough to save the day)

I don't think I'll ever be very good at JavaScript (and don't see it saving the day either.. at least, not all on its own). What I liked about this project was that it seemed self-contained (hence 'do-able'), and a good way to meet/interact with the Azimuth forum which I had been shirking even in my mere capacity as lurker! That, and "simple climate science" is an attractive idea given what an unapproachably complex subject I perceived it to be.

I want to figure out a way to take the simpler model Allan already programmed, here

Don't you want to wait for something based on your time-dependent Q model?

Source Text:John wrote: > I don't think I'll ever be very good at climate science (in part because I don't see that as being enough to save the day) I don't think I'll ever be very good at JavaScript (and don't see it saving the day either.. at least, not all on its own). What I liked about this project was that it seemed self-contained (hence 'do-able'), and a good way to meet/interact with the Azimuth forum which I had been shirking even in my mere capacity as lurker! That, and "simple climate science" is an attractive idea given what an unapproachably complex subject I perceived it to be. > I want to figure out a way to take the simpler model Allan already programmed, [here](http://math.ucr.edu/home/baez/coalbedo/coalbedo.html) Don't you want to wait for something based on your time-dependent Q model?
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123.

Sage has ODE solvers en masse so I must state again: use Sage and Sage interact - this is for DE go up to see more as a first choice and the if you need to use any web2.0 ajax js technology to clarify things to visualize things for Azimuth do so but put the code. on our google code page which Tim mentioned.

Allan JS question . so its Firebug to debug Javascript ?

Source Text:Sage has ODE solvers en masse so I must state again: use Sage and [Sage interact - this is for DE go up to see more](http://wiki.sagemath.org/interact/diffeq) as a first choice and the if you need to use any web2.0 ajax js technology to clarify things to visualize things for Azimuth do so but put the code. on our google code page which Tim mentioned. Allan JS question . so its Firebug to debug Javascript ?
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124.

So John B and Tim why don't you help me on the other threads on carbon cycle? I intend to do a small Sage-based framework for box models. I'll add some of my intentions there

Source Text:So John B and Tim why don't you help me on the other threads on carbon cycle? I intend to do a small Sage-based framework for box models. I'll add some of my intentions there
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125.

so its Firebug to debug Javascript ?

Firebug is the best tool that I know of, Microsoft has made a copy of it (as they often do) for their IE in version 8 which you will find under tools -> F12 developer tools

So John B and Tim why don't you help me on the other threads on carbon cycle?

I'm exhausted by the job I currently have to do...(lead developer for a project that handles software for car electronics).

Source Text:<blockquote> <p> so its Firebug to debug Javascript ? </p> </blockquote> Firebug is the best tool that I know of, Microsoft has made a copy of it (as they often do) for their IE in version 8 which you will find under tools -> F12 developer tools <blockquote> <p> So John B and Tim why don't you help me on the other threads on carbon cycle? </p> </blockquote> I'm exhausted by the job I currently have to do...(lead developer for a project that handles software for car electronics).
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126.

@Staffan - Firebug, but more recently the JS console in Chrome.. it's full featured and quite responsive (Eg on Dan Piponi's example I posted earlier, you can watch d3 updating the DOM live)

Source Text:@Staffan - Firebug, but more recently the JS console in Chrome.. it's full featured and quite responsive (Eg on Dan Piponi's example I posted earlier, you can watch d3 updating the DOM live)
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127.

Allan wrote:

Don't you want to wait for something based on your time-dependent Q model?

I don't mind waiting; I just don't want that earlier work to go to waste. I'll have to see what George Musser wants: if we can use both bits of software in the same article, that would be great. (The only reason why not is that it might somehow tire or distract the readers.) If not, I'll use your first program in a separate article of its own... which should logically show up first, somewhere.

Staffan wrote:

So John B and Tim why don't you help me on the other threads on carbon cycle?

I will!

Source Text:Allan wrote: > Don't you want to wait for something based on your time-dependent Q model? I don't mind waiting; I just don't want that earlier work to go to waste. I'll have to see what George Musser wants: if we can use _both_ bits of software in the same article, that would be great. (The only reason why not is that it might somehow tire or distract the readers.) If not, I'll use your first program in a separate article of its own... which should logically show up _first_, somewhere. Staffan wrote: > So John B and Tim why don't you help me on the other threads on carbon cycle? I will!
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128.

Am I correct in analyzing the above thread to assume that no one is actively working on doing a javascript web-based slider control dynamic version of the ODE solution for the equation John described in #108 above? Tim, it looks like you've done some investigation but are very busy at the moment.

I happen to be very interested in coming up to speed with Javascript graphics for some other projects I'm working on, and I have connections to some hardcore javascript experts for assistance. So this would be a good challenge for me.

I also don't mind converting from any other languages. Does anyone have a solution to the equation in another language (Python or C++ preferred) that I might model?

Of course, I'll put the code into the Google Code page, as per Staffan's suggestion, if I do this.

Source Text:Am I correct in analyzing the above thread to assume that no one is actively working on doing a javascript web-based slider control dynamic version of the ODE solution for the equation John described in #108 above? Tim, it looks like you've done some investigation but are very busy at the moment. I happen to be very interested in coming up to speed with Javascript graphics for some other projects I'm working on, and I have connections to some hardcore javascript experts for assistance. So this would be a good challenge for me. I also don't mind converting from any other languages. Does anyone have a solution to the equation in another language (Python or C++ preferred) that I might model? Of course, I'll put the code into the Google Code page, as per Staffan's suggestion, if I do this.
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129.

Curtis wrote:

Am I correct in analyzing the above thread to assume that no one is actively working on doing a javascript web-based slider control dynamic version of the ODE solution for the equation John described in #108 above?

Hi Curtis. I'm working on it -- just haven't found an evening to push forward much. I think you should still feel free to jump in mind you! We could compare notes.

Source Text:Curtis wrote: > Am I correct in analyzing the above thread to assume that no one is actively working on doing a javascript web-based slider control dynamic version of the ODE solution for the equation John described in #108 above? Hi Curtis. I'm working on it -- just haven't found an evening to push forward much. I think you should still feel free to jump in mind you! We could compare notes.
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130.

Hi, Curtis!

I'd love to work our way toward a system where we can put a bunch of simple interactive climate models on the web, together with explanations, for educational purposes. I think this could be a way to help hundreds or even tens of thousands of people around the world learn more climate science. This seems like a better use of my time than trying to become a full-fledged climate scientist myself.

I can easily write the explanations and try to lure people into reading them. This is what I'm good at.

I can imagine writing the math code if I have some templates to copy. I don't want to do it, but it would probably be good for me. And I don't want to constantly annoying people by saying "oh, you know, that ODE we coded up, I decided we need an extra term". If I get stuck I could ask for help.

What I can't imagine is putting together the web-based interface. I know nothing about that sort of stuff.

Jacob Biamonte suggested that ultimately we should do the math with one programming language and the interface with another, like Javascript. It sounds like Staffan was suggesting the same thing. It makes sense to me but again, writing a piece of software with different pieces in different languages communicating to each other goes way beyond my pathetically limited programming skills.

Source Text:Hi, Curtis! I'd love to work our way toward a system where we can put a bunch of simple interactive climate models on the web, together with explanations, for educational purposes. I think this could be a way to help hundreds or even tens of thousands of people around the world learn more climate science. This seems like a better use of my time than trying to become a full-fledged climate scientist myself. I can easily write the explanations and try to lure people into reading them. This is what I'm good at. I can _imagine_ writing the math code if I have some templates to copy. I don't want to do it, but it would probably be good for me. And I don't want to constantly annoying people by saying "oh, you know, that ODE we coded up, I decided we need an extra term". If I get stuck I could ask for help. What I _can't_ imagine is putting together the web-based interface. I know nothing about that sort of stuff. Jacob Biamonte suggested that ultimately we should do the math with one programming language and the interface with another, like Javascript. It sounds like [Staffan](http://www.math.ntnu.no/~stacey/Mathforge/Azimuth/comments.php?DiscussionID=856&Focus=5980#Comment_5980) was suggesting the same thing. It makes sense to me but again, writing a piece of software with different pieces in different languages communicating to each other goes way beyond my pathetically limited programming skills.
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131.

I'm pretty sure that we can come up with some ways of making this process work for everyone, especially those who want to learn more about climate science.

There are conflicting needs when you look at serious science programming when compared with that designed for educational purposes. This thread is interesting in that it hits on a real concrete way that many people might learn about climate science who are not capable of understanding the science through equations.

We could certainly walk people through the parts of a very complex climate model, piece by piece, without forcing people to understand the equations by showing the math interactively and visually.

As I work on this, I'll work with the others to see if we can come up with a way of isolating the ODE sections so that you'd have to learn only a minimum of programming to implement changes to an ODE. Still too early for me to say how successful this will be.

We ought to be able to do things in such a way that the graphics coding is abstracted so you won't have to worry about that complexity.

As far as language, we're really going to have to stick with javascript if we want a modern-looking presentation accessible to most people throughout the world. It certainly has it's shortcomings as a language, but it is pretty easy to learn, so you'll find that part nice.

Source Text:I'm pretty sure that we can come up with some ways of making this process work for everyone, especially those who want to learn more about climate science. There are conflicting needs when you look at serious science programming when compared with that designed for educational purposes. This thread is interesting in that it hits on a real concrete way that many people might learn about climate science who are not capable of understanding the science through equations. We could certainly walk people through the parts of a very complex climate model, piece by piece, without forcing people to understand the equations by showing the math interactively and visually. As I work on this, I'll work with the others to see if we can come up with a way of isolating the ODE sections so that you'd have to learn only a minimum of programming to implement changes to an ODE. Still too early for me to say how successful this will be. We ought to be able to do things in such a way that the graphics coding is abstracted so you won't have to worry about that complexity. As far as language, we're really going to have to stick with javascript if we want a modern-looking presentation accessible to most people throughout the world. It certainly has it's shortcomings as a language, but it is pretty easy to learn, so you'll find that part nice.
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132.
edited December 2011

Curtis wrote:

Tim, it looks like you've done some investigation but are very busy at the moment.

Yes, but Allan works on it and I'll certainly try to do it during the next week. Since this part is easy, we could all try to solve it in parallel and compare notes later, there is no need to cooperate here :-)

Allan and I will try to use JSXGraph (for the math, and jQuery for GUI functionality) to get a pure JavaScript version (details can be found in this thread).

As far as language, we're really going to have to stick with javascript if we want a modern-looking presentation accessible to most people throughout the world.

The question really is if we go for a Web interface (HTML+CSS+JavaScript) or a rich client which could be anything. I agree with Staffan and you that probably a Web interface is the way to go, because it is most easily accessible. This would mean that we'd need to include server calls via Ajax later (with maybe the server running Sage or something), for the sophisticated stuff that you don't want to program in JavaScript. But the stuff that John is interested in, in this thread, can be done with a simple Runge-Kutta solver which can be- and has been - written in JavaScript, so we could dodge this topic and go for a pure Web solution in this case.

Source Text:Curtis wrote: <blockquote> <p> Tim, it looks like you've done some investigation but are very busy at the moment. </p> </blockquote> Yes, but Allan works on it and I'll certainly try to do it during the next week. Since this part is easy, we could all try to solve it in parallel and compare notes later, there is no need to cooperate here :-) Allan and I will try to use JSXGraph (for the math, and jQuery for GUI functionality) to get a pure JavaScript version (details can be found in this thread). <blockquote> <p> As far as language, we're really going to have to stick with javascript if we want a modern-looking presentation accessible to most people throughout the world. </p> </blockquote> The question really is if we go for a Web interface (HTML+CSS+JavaScript) or a rich client which could be anything. I agree with Staffan and you that probably a Web interface is the way to go, because it is most easily accessible. This would mean that we'd need to include server calls via Ajax later (with maybe the server running Sage or something), for the sophisticated stuff that you don't want to program in JavaScript. But the stuff that John is interested in, in this thread, can be done with a simple Runge-Kutta solver which can be- and has been - written in JavaScript, so we could dodge this topic and go for a pure Web solution in this case.
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133.
edited December 2011

Here's the simplest web service form slider I've come across. It's in html5 with javascript value display. http://bit.ly/a2Vs0u

hth

Source Text:Here's the simplest web service form slider I've come across. It's in html5 with javascript value display. http://bit.ly/a2Vs0u hth
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134.

Curtis wrote:

There are conflicting needs when you look at serious science programming when compared with that designed for educational purposes. This thread is interesting in that it hits on a real concrete way that many people might learn about climate science who are not capable of understanding the science through equations.

We could certainly walk people through the parts of a very complex climate model, piece by piece, without forcing people to understand the equations by showing the math interactively and visually.

I should probably use a "layered" explanation, or what my uncle called "the spiral approach". In other words. something like this: first a very quick explanation, then a longer one without equations, then one with equations. If they're clearly separated, different audiences can focus on different parts.

I'd been planning to focus on the audience who can understand equations, because even here the number of people who understand climate physics is much smaller than it should be. But you're making me want to be more ambitious about reaching out to a larger crowd.

Source Text:Curtis wrote: > There are conflicting needs when you look at serious science programming when compared with that designed for educational purposes. This thread is interesting in that it hits on a real concrete way that many people might learn about climate science who are not capable of understanding the science through equations. > We could certainly walk people through the parts of a very complex climate model, piece by piece, without forcing people to understand the equations by showing the math interactively and visually. I should probably use a "layered" explanation, or what my uncle called "the spiral approach". In other words. something like this: first a very quick explanation, then a longer one without equations, then one with equations. If they're clearly separated, different audiences can focus on different parts. I'd been planning to focus on the audience who _can_ understand equations, because even here the number of people who understand climate physics is much smaller than it should be. But you're making me want to be more ambitious about reaching out to a larger crowd.
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135.

john concluded:

because even here the number of people who understand climate physics is much smaller than it should be. But you're making me want to be more ambitious about reaching out to a larger crowd.

I like that and I think its good to try to have more animations and ajax on our pages- Sage uses a bunch of js libraries for implementing its notebook and they always try to keep track of what's out there. and I think we need to also consider having more visual info on Sage wiki. Curtis and I talked about this a long time ago, but maybe we should start now

Source Text:john concluded: > because even here the number of people who understand climate physics is much smaller than it should > be. But you're making me want to be more ambitious about reaching out to a larger crowd. I like that and I think its good to try to have more animations and ajax on our pages- Sage uses a bunch of js libraries for implementing its notebook and they always try to keep track of what's [out there](http://wiki.sagemath.org/devel/JavascriptResources). and I think we need to also consider having more visual info on Sage wiki. Curtis and I talked about this a long time ago, but maybe we should start now Curtis: I'll holler on the carbon threads about javascript help !
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136.

John wrote:

But you're making me want to be more ambitious about reaching out to a larger crowd.

Isaac Asimov was my hero growing up. Especially his two books on organic and inorganic chemistry. You've got the same knack for taking complex ideas into a place where nonscientists can understand them.

This is what climate science is missing. In all the complexity, there are some fairly basic scientific principles that if more widely understood would change the global conversation. People are being misled because they can be. They simply don't understand enough of the basic principles to be able to think for themselves and draw their own conclusions.

It has always been my hope that the Azimuth Project can change that.

Source Text:John wrote: >But you're making me want to be more ambitious about reaching out to a larger crowd. <img src = "http://math.ucr.edu/home/baez/emoticons/thumbsup.gif" alt = ""/> Isaac Asimov was my hero growing up. Especially his two books on organic and inorganic chemistry. You've got the same knack for taking complex ideas into a place where nonscientists can understand them. **This is what climate science is missing.** In all the complexity, there are some fairly basic scientific principles that if more widely understood would change the global conversation. People are being misled because they can be. They simply don't understand enough of the basic principles to be able to think for themselves and draw their own conclusions. It has always been my hope that the Azimuth Project can change that.
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137.

Tim van Beek wrote:

"Firebug is the best tool that I know of, Microsoft has made a copy of it (as they often do) for their IE in version 8 which you will find under tools -> F12 developer tools"

Aptana, or Aptana as a plugin for Eclipse is also useful for development.

Staffan Liljegren wrote:

"Allan and Tim i used to dislike javascript as well , please remind me why :-) I like GWT too Tim but it might be to heavy weight for these simple examples."

Side remark: it seems not so clear how the future of the Google Web Toolkit (I assume that is what you mean that bei GWT) will look like due to Dart

By the way I couldnt find a list of editing options for the comment section.

Source Text:Tim van Beek wrote: "Firebug is the best tool that I know of, Microsoft has made a copy of it (as they often do) for their IE in version 8 which you will find under tools -> F12 developer tools" [Aptana](http://en.wikipedia.org/wiki/Aptana), or Aptana as a plugin for Eclipse is also useful for development. Staffan Liljegren wrote: "Allan and Tim i used to dislike javascript as well , please remind me why :-) I like GWT too Tim but it might be to heavy weight for these simple examples." Side remark: it seems not so clear how the future of the Google Web Toolkit (I assume that is what you mean that bei GWT) will look like due to [Dart](http://en.wikipedia.org/wiki/Dart_%28programming_language%29) By the way I couldnt find a list of editing options for the comment section.
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138.

By the way I couldnt find a list of editing options for the comment section.

I'm not sure what you're talking about. At the top of each comment you write, there should be a little list of choices:

"edit delete Source PermaLink blog this"

If you click edit, you can edit it. I can edit everybody's, and I do - just to improve spelling, grammar and formatting.

Source Text:nad wrote: > By the way I couldnt find a list of editing options for the comment section. I'm not sure what you're talking about. At the top of each comment you write, there should be a little list of choices: "edit delete Source PermaLink blog this" If you click edit, you can edit it. I can edit everybody's, and I do - just to improve spelling, grammar and formatting.
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139.

John wrote: "I'm not sure what you're talking about. At the top of each comment you write, there should be a little list of choices: "edit delete Source PermaLink blog this" If you click edit, you can edit it. I can edit everybody's, and I do - just to improve spelling, grammar and formatting."

With editing options I mean something like which one gets if one follows the "help link" of the "Markdown and Itex (Help)" , but first I find the link is hard to find (because one could assume that the (Help) in brackets refers to a general help to understand what Markdown and Itex is, rather than that there is a list of Markdown editing options behind it) and then it seems not fully complete (or may be I missed here something?) - like how do you do the green blockquote bars?

Source Text:John wrote: **"I'm not sure what you're talking about. At the top of each comment you write, there should be a little list of choices: "edit delete Source PermaLink blog this" If you click edit, you can edit it. I can edit everybody's, and I do - just to improve spelling, grammar and formatting."** With editing options I mean something like which one gets if one follows the "help link" of the "Markdown and Itex (Help)" , but first I find the link is hard to find (because one could assume that the (Help) in brackets refers to a general help to understand what Markdown and Itex is, rather than that there is a list of Markdown editing options behind it) and then it seems not fully complete (or may be I missed here something?) - like how do you do the green blockquote bars?
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140.
Nad: I like Aptana to a lot but its been a while since I used it. When I write sage stuff, i actually use the sage notebook, which allows you to edit sage/python/latex/html/fortran/cython/r and more so its y new scientific software lab :-)
Source Text:Nad: I like Aptana to a lot but its been a while since I used it. When I write sage stuff, i actually use the sage notebook, which allows you to edit sage/python/latex/html/fortran/cython/r and more so its y new scientific software lab :-)
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141.

staffan: what is r? Aptana is also useful for javascript (thats actually why i mentioned it)

concerning sage: I sofar postponed to use a sage notebook. i tried the online sage version, which was so against my internal intuition that i know that i would need a giantly huge amount of patience to learn to use sage, but there is only a limited amount of patience per day. :)

Source Text:staffan: what is r? Aptana is also useful for javascript (thats actually why i mentioned it) concerning sage: I sofar postponed to use a sage notebook. i tried the online sage version, which was so against my internal intuition that i know that i would need a giantly huge amount of patience to learn to use sage, but there is only a limited amount of patience per day. :)
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142.

@ Nadja: I guess it's simply R. What Staffan writes is usually not case-sensitive.

Source Text:@ Nadja: I guess it's simply [R](http://en.wikipedia.org/wiki/R_%28programming_language%29). What Staffan writes is usually not case-sensitive.
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143.
edited December 2011

With editing options I mean something like which one gets if one follows the "help link" of the "Markdown and Itex (Help)"

Oh, I see. I've never tried to learn something from that help system. I learned this stuff by watching other people use it on the n-Category Cafe.

like how do you do the green blockquote bars?

For blockquote you do a > symbol and then a space at the beginning of a paragraph without carriage returns. So

> With editing options I mean something like which one gets if one follows the "help link" of the "Markdown and Itex (Help)" , but first I find the link is hard to find (because one could assume that the (Help) in brackets refers to a general help to understand what Markdown and Itex is, rather than that there is a list of Markdown editing options behind it) and then it seems not fully complete (or may be I missed here something?) - like how do you do the green blockquote bars?

gives

With editing options I mean something like which one gets if one follows the "help link" of the "Markdown and Itex (Help)" , but first I find the link is hard to find (because one could assume that the (Help) in brackets refers to a general help to understand what Markdown and Itex is, rather than that there is a list of Markdown editing options behind it) and then it seems not fully complete (or may be I missed here something?) - like how do you do the green blockquote bars?

Source Text:Nad wrote: > With editing options I mean something like which one gets if one follows the "help link" of the "Markdown and Itex (Help)" Oh, I see. I've never tried to learn something from that help system. I learned this stuff by watching other people use it on the n-Category Cafe. > like how do you do the green blockquote bars? For blockquote you do a > symbol and then a space at the beginning of a paragraph without carriage returns. So > With editing options I mean something like which one gets if one follows the "help link" of the "Markdown and Itex (Help)" , but first I find the link is hard to find (because one could assume that the (Help) in brackets refers to a general help to understand what Markdown and Itex is, rather than that there is a list of Markdown editing options behind it) and then it seems not fully complete (or may be I missed here something?) - like how do you do the green blockquote bars? gives > With editing options I mean something like which one gets if one follows the "help link" of the "Markdown and Itex (Help)" , but first I find the link is hard to find (because one could assume that the (Help) in brackets refers to a general help to understand what Markdown and Itex is, rather than that there is a list of Markdown editing options behind it) and then it seems not fully complete (or may be I missed here something?) - like how do you do the green blockquote bars?
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144.
edited December 2011

Paul Clark has an interactive climate time-series application with a fair number of data sets from GISSTEMP, HADCRUT3, UAH, BEST etc. which seems to be along the lines suggested for azimuth.

WoodForTrees

It's written in C++ so he obviously has an ISP account with shell access which might be something to consider.

Source Text:Paul Clark has an interactive climate time-series application with a fair number of data sets from GISSTEMP, HADCRUT3, UAH, BEST etc. which seems to be along the lines suggested for azimuth. [WoodForTrees](http://www.woodfortrees.org) It's written in C++ so he obviously has an ISP account with shell access which might be something to consider.
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145.

Jim - nice! I should advertise this and various other bits of online software on the blog.

(Your link didn't work, because you left out "http://". I fixed that.)

My website is on a UNIX machine so if someone can write anything that takes advantage of that, great!

Source Text:Jim - nice! I should advertise this and various other bits of online software on the blog. (Your link didn't work, because you left out "http://". I fixed that.) My website is on a UNIX machine so if someone can write anything that takes advantage of that, great!
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146.

the Sage interact works now without login to the latest 4.7.2 Sage notebook server. try moving the gamma slider :-)

Hooray and now more interacts and less animations to "cover up" (emulate maybe)

http://demo.sagenb.org/home/pub/238/

Source Text:the Sage interact works now **without login** to the latest 4.7.2 Sage notebook server. try moving the gamma slider :-) Hooray and now more interacts and less animations to "cover up" (emulate maybe) [http://demo.sagenb.org/home/pub/238/](http://demo.sagenb.org/home/pub/238/)
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147.

Staffan,

it may also be that I have a sagedocumentation incompatibility - using the notebook at http://demo.sagenb.org/home/pub/238/ I could plot a sin within a minute: %auto

var('T gamma')

B = 1.9

ap(T) = sin(BgammaT)

@interact def coalbedo(gamma_value=(0,1,.1)): plot(ap.substitute(gamma=gamma_value),(T,-50,50),legend_label='$a_p$').show()

Source Text:Staffan, it may also be that I have a sagedocumentation incompatibility - using the notebook at http://demo.sagenb.org/home/pub/238/ I could plot a sin within a minute: %auto var('T gamma') B = 1.9 ap(T) = sin(B*gamma*T) @interact def coalbedo(gamma_value=(0,1,.1)): plot(ap.substitute(gamma=gamma_value),(T,-50,50),legend_label='$a_p$').show()
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148.

so u edited my public sheet without login and got a changed interact to work -?

Source Text:so u edited my public sheet without login and got a changed interact to work -?
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149.
edited January 2012

Staffan wrote

"so u edited my public sheet without login and got a changed interact to work -?"

I opened up my own account took some of your code, pasted it together and changed it a bit and then it almost worked. That is luckily someone passed by and saw that I missed the indentation (in the def coalbedo), which resulted in an error message. Since I don't know Python (this identation seems to be a Python feature) I probably never would have found that error.

By the way do you know wether it is possible to get this notebook code as a Javascript export, which one could use in any html document?

Source Text:Staffan wrote > "so u edited my public sheet without login and got a changed interact to work -?" I opened up my own account took some of your code, pasted it together and changed it a bit and then it almost worked. That is luckily someone passed by and saw that I missed the indentation (in the def coalbedo), which resulted in an error message. Since I don't know Python (this identation seems to be a Python feature) I probably never would have found that error. By the way do you know wether it is possible to get this notebook code as a Javascript export, which one could use in any html document?
• Options
150.

Ah I get it. Sometimes when you copy/paste between browser windows the indentation - which is mandatory in Python - gets lost. I actually added one free book on the [[Python]] page ....for newbies "how to think like a Computer Scientist" its very good, short and free :-)

Source Text:Ah I get it. Sometimes when you copy/paste between browser windows the indentation - which is mandatory in Python - gets lost. I actually added one free book on the [[Python]] page ....for newbies "how to think like a Computer Scientist" its very good, short and free :-)

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# M20-04

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If $$A$$, $$B$$, $$C$$, and $$D$$ are integers such that $$A - C + B$$ is even and $$D + B - A$$ is odd, which of the following expressions is always odd?

A. $$A + D$$
B. $$B + D$$
C. $$C + D$$
D. $$A + B$$
E. $$A + C$$
[Reveal] Spoiler: OA

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Official Solution:

If $$A$$, $$B$$, $$C$$, and $$D$$ are integers such that $$A - C + B$$ is even and $$D + B - A$$ is odd, which of the following expressions is always odd?

A. $$A + D$$
B. $$B + D$$
C. $$C + D$$
D. $$A + B$$
E. $$A + C$$

Rewrite $$A - C + B$$ as $$(A + B) - C$$ and $$D + B - A$$ as $$D + (B - A)$$. If $$A + B$$ is even, $$B - A$$ is even too. If $$A + B$$ is odd, $$B - A$$ is odd too. Now it follows from the stem that:
• if $$A + B$$ is even, $$C$$ is even and $$D$$ is odd
• if $$A + B$$ is odd, $$C$$ is odd and $$D$$ is even

Thus, $$C + D$$ is always odd.

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I think I may have found a much more simple methodology to answer this question:

(1) A - C + B = even; thus, A - C + B = 2k (where k is an integer)

(2) D + B - A = odd; thus, D + B - A = 2k + 1 (where k is an integer)

rearrange (2): D + B - A - 1 = 2k

Sub: rearranged (2) into (1): A - C + B = D + B - A - 1 --> 2A + 1 = C + D

thus, per my method above, if C + D = 2A + 1 (where A is an integer), C + D is ALWAYS ODD

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08 Jan 2015, 09:41
I think this question is good and helpful.

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bartone89 I believe your method is wrong because it assumes the values of the two equations are consecutive. ie 2k and 2k+1. I hope I havent misunderstood you solution.

Anyway, I think I may have found a better method.

The two equations are A-C+B and D+B-A, even and odd respectively.

On subtracting the two, you get, A-C+B-D-B+A = 2A-C-D (which is odd, since even - odd = odd)

2A-(C+D) is odd. Which means C+D is odd since we know that 2A has to be even.

C+D is one of the options.
Hope this makes sense.

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19 Jan 2016, 05:25
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There is an easier way this question can be solved.
Lets say A-C+B = 2n (even)
and after rearranging the second equation, -A+D+B=2m+1 (odd)

Adding the two equations we get, 2B+D-C=2(m+n)+1

From this, 2B is even and 2(m+n) is even, hence we conclude,
D-C=odd

When difference of two integers is odd, there sum will also be odd;

Hence D+C = odd
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20 Feb 2016, 12:58
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I solved it in a different way:
A-C+B+D+B-A=odd.
ok, rewrite everything as:
2B+D-C=odd
2B is always even, it must bean only 1 thing: D-C is always odd. if D-C is odd, then D+C is always odd.

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26 Apr 2016, 17:25
mvictor wrote:
I solved it in a different way:
A-C+B+D+B-A=odd.
ok, rewrite everything as:
2B+D-C=odd
2B is always even, it must bean only 1 thing: D-C is always odd. if D-C is odd, then D+C is always odd.

Very good solution mvictor!

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Re: M20-04 I solved it with a different approach! Just Sharing [#permalink]

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27 Nov 2016, 05:40
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Add or Substract the two equations-

A-C+B=Even
D+B-A=Odd
=2A+C-D (Even-Odd is always Odd) and we know 2A is always even

Therefore C-D should be Odd and thus also C+D

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06 May 2017, 13:56
1
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y4nkee wrote:
mvictor wrote:
I solved it in a different way:
A-C+B+D+B-A=odd.
ok, rewrite everything as:
2B+D-C=odd
2B is always even, it must bean only 1 thing: D-C is always odd. if D-C is odd, then D+C is always odd.

Very good solution mvictor!

I agree very good solution. But it still requires timing luck or incredible number sense. I solved it through plugging in values. My method take no number sense needed and does not require too much logic.

A-C+B is even. Just plug in some easy numbers so that is true. A = 1 C = 2 B = 3 means that 1-2+3 = 2 which is even.
D+B-A is odd. Re-use the same variables from before. D + 3 - 1 is D + 2. Set D to an easy odd number which is 5. So D = 5.
Now you have values for A through D or (A,B,C,D) = (1,3,2,5). Work through the answers.

A. A+D = 1 + 5 = 6
B. B+D = 3 + 5 = 8
C. C+D = 2 + 5 = 7. This is odd. No need to do the rest, the answer is C.
D. A+B
E. A+C

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08 Jun 2017, 07:06
Given A-C+B =Even and D+B-A=Odd
Subtract 1) and 2)
2A-C-D=Even-odd=Odd=> 2A=Odd- (C+D)
Left side of the equation is even hence C+D has to be Odd.

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25 Jun 2017, 03:01
Given A-C+B=Even-----1
& D+B-A = Odd---------2
Add 1 & 2 ---even + odd = odd
hence C-D= 2B - Odd= Odd
if C-D is odd, then C+D is also Odd. (eg 4-3=1, 4+3=7)

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18 Sep 2017, 18:15
1
KUDOS
+A-c+B= even
D+B-A = odd Now equation 1 - equation2 which gives 2A-(D+C)=odd
2A is always even so (D+C) is always odd
Hence C

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20 Nov 2017, 15:41
joondez wrote:
y4nkee wrote:
mvictor wrote:
I solved it in a different way:
A-C+B+D+B-A=odd.
ok, rewrite everything as:
2B+D-C=odd
2B is always even, it must bean only 1 thing: D-C is always odd. if D-C is odd, then D+C is always odd.

Very good solution mvictor!

I agree very good solution. But it still requires timing luck or incredible number sense. I solved it through plugging in values. My method take no number sense needed and does not require too much logic.

A-C+B is even. Just plug in some easy numbers so that is true. A = 1 C = 2 B = 3 means that 1-2+3 = 2 which is even.
D+B-A is odd. Re-use the same variables from before. D + 3 - 1 is D + 2. Set D to an easy odd number which is 5. So D = 5.
Now you have values for A through D or (A,B,C,D) = (1,3,2,5). Work through the answers.

A. A+D = 1 + 5 = 6
B. B+D = 3 + 5 = 8
C. C+D = 2 + 5 = 7. This is odd. No need to do the rest, the answer is C.
D. A+B
E. A+C

the thing is - when presented with such expressions, you have to either add or subtract one from another.
the thing is, you have 3 variables in each expression, therefore, you might lose some extra time to test all the possible options...
my approach is very direct and fast though...and I always have the same approach to solving similar questions.

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# Extreme values (Lagrange multipliers)

determine, if any, the maximum and minimum values of the scalar field f (x, y) = xy subject to the constraint $$4x^2{}$$+$$9y^2{}$$=36

The attempt at a solution

using Lagrange multipliers, we solve the equations $$\nabla$$f=$$\lambda$$$$\nabla$$g ,which can be written as

$$f_{x}$$=$$\lambda$$$$g_{x}$$

$$f_{y}$$=$$\lambda$$$$g_{y}$$

g(x,y)=36

or as

y=$$\lambda$$8x

x=$$\lambda$$18y

$$4x^2{}$$+$$9y^2{}$$=36

it's pretty much all done but can somebody solve this? cause i have some doubts about which are the extreme points

Related Calculus and Beyond Homework Help News on Phys.org
I would say your $$g(x,y) = 36$$ is a little suspicious. If this were the case then the two statements above would be false since $$g_x$$ and $$g_y$$ are 0.

no, in fact g(x,y)=$$4x^2{}$$+$$9y^2{}$$=36 but well, I wrote it that way because is the way my book does

We already know that (0,0) is a turning point which has a value f (0,0) = 0.

Leaving that aside for now, solving the set of equations that you listed:
$$y = 8\lambda x$$
$$x = 18\lambda y$$
$$4x^2 + 9y^2 = 36$$
would yield one or more points.

Then find the value of f(x,y) at each of these points and compare.

Last edited by a moderator:
Firstly, careless mistake: y is $$+/- \frac{2}{3}\sqrt{\frac{18}{5}}$$ as you've written on the third line, but in carrying it over into f(x,y), you mixed up the denominator and numerator.

Secondly, $$\lambda = +/- \frac{1}{12}$$, so there exist two other solutions:
$$(x,y) = (\sqrt{\frac{18}{5}}, - \frac{2}{3}\sqrt{\frac{18}{5}})$$
$$(x,y) = (- \sqrt{\frac{18}{5}}, \frac{2}{3}\sqrt{\frac{18}{5}})$$


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location:  Publications → journals
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Search: All articles in the CMB digital archive with keyword eigenvalues

 Expand all        Collapse all Results 1 - 3 of 3

1. CMB 2011 (vol 54 pp. 506)

 On the Canonical Solution of the Sturm-Liouville Problem with Singularity and Turning Point of Even Order In this paper, we are going to investigate the canonical property of solutions of systems of differential equations having a singularity and turning point of even order. First, by a replacement, we transform the system to the Sturm-Liouville equation with turning point. Using of the asymptotic estimates provided by Eberhard, Freiling, and Schneider for a special fundamental system of solutions of the Sturm-Liouville equation, we study the infinite product representation of solutions of the systems. Then we transform the Sturm-Liouville equation with turning point to the equation with singularity, then we study the asymptotic behavior of its solutions. Such representations are relevant to the inverse spectral problem. Keywords:turning point, singularity, Sturm-Liouville, infinite products, Hadamard's theorem, eigenvaluesCategories:34B05, 34Lxx, 47E05

2. CMB 2006 (vol 49 pp. 560)

Luijk, Ronald van
 A K3 Surface Associated With Certain Integral Matrices Having Integral Eigenvalues In this article we will show that there are infinitely many symmetric, integral $3 \times 3$ matrices, with zeros on the diagonal, whose eigenvalues are all integral. We will do this by proving that the rational points on a certain non-Kummer, singular K3 surface are dense. We will also compute the entire NÃ©ron-Severi group of this surface and find all low degree curves on it. Keywords:symmetric matrices, eigenvalues, elliptic surfaces, K3 surfaces, NÃ©ron--Severi group, rational curves, Diophantine equations, arithmetic geometry, algebraic geometry, number theoryCategories:14G05, 14J28, 11D41

3. CMB 1999 (vol 42 pp. 169)

Ding, Hongming
 Heat Kernels of Lorentz Cones We obtain an explicit formula for heat kernels of Lorentz cones, a family of classical symmetric cones. By this formula, the heat kernel of a Lorentz cone is expressed by a function of time $t$ and two eigenvalues of an element in the cone. We obtain also upper and lower bounds for the heat kernels of Lorentz cones. Keywords:Lorentz cone, symmetric cone, Jordan algebra, heat kernel, heat equation, Laplace-Beltrami operator, eigenvaluesCategories:35K05, 43A85, 35K15, 80A20
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# Solving Vandermonde equation system

Given the following:

$$\begin{pmatrix} 1 & 1 & 1 & ... & 1 \\ a_1 & a_2 & a_3 & ... & a_n \\ a_1^2 & a_2^2 & a_3^2 & ... & a_n^2 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ a_1^{n-1} & a_2^{n-1} & a_3^{n-1} & ... & a_n^{n-1}\end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_n\end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \\ \vdots \\ 1 \end{pmatrix}$$

How can this be solved to give $$x_k$$? By Cramer's rule, this should be $$x_k=\frac{\det(A_i)}{\det(A)}$$

How can this result be simplified? Since this is a Vandermonde matrix, $$\det(A)=\prod_{1\leq i. However, can the numerator be written any simpler? In a paper where I have seen this equation system used, the result is given as: $$x_k=\prod_{1\leq j \leq n,\;j \neq i} (a_i - a_j)^{-1}$$ but it's not clear how the simplification has been achieved?

Using Cramer's rule, we know

$$x_k = \frac{\det(A_k)}{\det(A)}$$

Now, as we know the Vandermonde determinant is equal to

$$\det(A)=\prod_{1\leq i

To calculate $$\det(A_k)$$, we replace the kth column with $$b_i$$s which in this case are all $$0$$s and last entry $$1$$. If we calculate the determinant using the kth column, we need only consider the determinant using the last entry as all the other ones will multiply with $$0$$ and become $$0$$. This last entry sub determinant also turns out to be a vandermonde determinant of one lower degree. So calculating the determinant using that

$$\det(A_k) = (-1)^{n+k}\prod_{1\leq i

where both $$i,j\neq k$$

When you divide them, all the terms that include $$a_k$$ are all that are left.

$$x_k = \left[(-1)^{n+k}\left(\prod_{k+1\leq j\leq n}(a_j-a_k) \prod_{1\leq i\leq k-1}(a_k-a_i)\right)\right]^{-1}$$

The terms which have greater index than $$a_k$$ come under one umbrella with the greater index term leading and the ones with lower come under a second umbrella with $$a_k$$ leading.

Rearrange all the terms in first umbrella so that $$a_k$$ term is leading and take all the $$-1$$s that come out and multiply it with existing to get your answer .

$$x_k = \left[(-1)^{n+k}\left((-1)^{n-(k+1)}\prod_{k+1\leq j\leq n}(a_k-a_j) \prod_{1\leq i\leq k-1}(a_k-a_i)\right)\right]^{-1}$$

$$x_k = \left[(-1)^{n+k+n-k-1}\left(\prod_{k+1\leq j\leq n}(a_k-a_j) \prod_{1\leq i\leq k-1}(a_k-a_i)\right)\right]^{-1}$$

$$x_k = \left[(-1)^{2n-1}\left(\prod_{k+1\leq j\leq n}(a_k-a_j) \prod_{1\leq i\leq k-1}(a_k-a_i)\right)\right]^{-1}$$

$$x_k = \left[\prod_{1\leq i\leq n, i\neq k}(a_i-a_k)\right]^{-1}$$

• There might be a $-1$ multiplication/divide error – Sauhard Sharma Nov 14 '18 at 19:27


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# If the volume of a spherical ball is increasing at the rate of $\;4 \pi \;cc/sec\;$ , then the rate of increase of its radius (in cm/sec) , when the volume is $\;288 \pi\;cc\;$ , is :

$(a)\;\large\frac{1}{6}\qquad(b)\;\large\frac{1}{9}\qquad(c)\;\large\frac{1}{36}\qquad(d)\;\large\frac{1}{24}$

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zbMATH — the first resource for mathematics

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 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

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An iterative method with cubic convergence for nonlinear equations. (English) Zbl 1113.65052
A new three step iterative method for solving a nonlinear equation $f(x)=0$ is introduced based on the following scheme: Let $x_0$ is an initial guess sufficiently close to simple root of the equation $f(x)=0$. The iterative step consists from two predictor steps: $$y_n=x_n-f(x_n)/ f' (x_n),\quad f'(x_n)\ne 0;\quad z_n=-(y_n-x_n)^2\cdot f''(x_n)/2f'(x_n),$$ and one corrector step: $$x_{n+1}=x_n-f(x_n)/f'(x_n)-(y_n+z_n-z_n)^2\cdot f'(x_n)/2\cdot f'(x_n),\quad n=0, 1,2,\dots.$$ The authors show that if the function $f$ is sufficiently differentiable in the open interval, which contain a simple root of the equation $f(x)=0$ and if $x_0$ is sufficiently close to this root, then the proposed iterative algorithm has the order of convergence equal to three. Several numerical examples are given to illustrate the efficiency and performance of the new method.

MSC:
 65H05 Single nonlinear equations (numerical methods)
Full Text:
References:
 [1] Abbasbandy, S.: Improving Newton -- raphson method for nonlinear equations by modified Adomian decomposition method. Appl. math. Comput. 145, 887-893 (2003) · Zbl 1032.65048 [2] Adomian, G.: Nonlinear stochastic systems and applications to physics. (1989) · Zbl 0659.93003 [3] Chun, C.: Iterative methods improving Newton’s method by the decomposition method. Comput. math. Appl. 50, 1559-1568 (2005) · Zbl 1086.65048 [4] Daftardar-Gejji, V.; Jafari, H.: An iterative method for solving nonlinear functional equations. J. math. Anal. appl. 316, 753-763 (2006) · Zbl 1087.65055 [5] Homeier, H. H.: On Newton-type methods with cubic convergence. J. comput. Appl. math. 176, 425-432 (2005) · Zbl 1063.65037 [6] M. Aslam Noor, Numerical analysis and optimization, Lecture Notes, Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan, 2006. [7] M. Aslam Noor, K. Inayat Noor, Three-step iterative methods for nonlinear equations, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.05.055. [8] M. Aslam Noor, K. Inayat Noor, Some iterative schemes for nonlinear equations, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.05.084.


text:
# Generalisation of Maxwell's equations

The electromagnetic action in the language of differential geometry is given by

$$\displaystyle{S \sim \int F \wedge \star F},$$

where $A$ is the one-form potential and $F={\rm d}A$ is the two-form field strength.

At the extremum of the action $S$, $F$ is constrained by ${\rm d}F=0$ and ${\rm d}\star F=0$.

Now, generalise the above action to

$$\displaystyle{S \sim \int H \wedge \star H}$$

where $B$ is the two-form potential and $H={\rm d}B$ is the three-form field strength.

At the extremum of the action $S$, $H$ is constrained by ${\rm d}H=0$ and ${\rm d}\star H=0$.

Are there any qualitative differences between the two sets of equations in $d+1$-dimensions?

• NB $\mathrm{d}F = 0$ is true for any $F$, not just at an extremum of the action. This is the Bianchi identity. – gj255 Oct 30 '16 at 23:57
• Good point, my bad! – nightmarish Oct 31 '16 at 0:13


text:
# Is it necessarry to create theory/infrastructure to prevent people from creating AI incompatible with a safe model (if we create one)?

Nowadays we don't know how to create AI in a safe way (I think that we don't even know yet how to define a safe AI), but there is a lot of research in developing a model allowing it.

Let's say, that someday we discover such a model (maybe even it would be possible to mathematically prove its safety). Is it rational to ask, how do we prevent people from creating AI outside of this model (e.g. they are so confident in their own model, that they just pursue it and end up with something like paperclip scenario)?

Should we also think about creating some theory/infrastructure preventing such a scenario?

• Did I understand you right, that you've recognized who wonderful a paperclip maximizer would be, because such a machine could produce unlimited amount of food, and build housing for everybody and you're asking here in the forum why nobody has invented such a great machine and how this matches to the promise of engineers to make the world a better place? – Manuel Rodriguez Aug 10 at 17:48
• Of course I meant a bad scenario, when AI acts in an unforeseen way, that might be not possible to stop. – Ozaru Aug 10 at 19:42
• "Nowadays we don't know how to create AI in a safe way" Nowadays we don't know how to create AI, dot. Current talks about rules to impose an unknown control model to unknown AIs are not empirical, not scientific, not true research. – pasaba por aqui Aug 11 at 11:48


text:
Generic function for extracting an estimator for the bread of sandwiches.

bread(x, ...)

## Arguments

x a fitted model object. arguments passed to methods.

## Value

A matrix containing an estimator for the expectation of the negative derivative of the estimating functions, usually the Hessian. Typically, this should be an $$k \times k$$ matrix corresponding to $$k$$ parameters. The rows and columns should be named as in coef or terms, respectively.

The default method tries to extract vcov and nobs and simply computes their product.

lm, glm

## References

Zeileis A (2006). “Object-Oriented Computation of Sandwich Estimators.” Journal of Statistical Software, 16(9), 1--16. doi: 10.18637/jss.v016.i09

Zeileis A, Köll S, Graham N (2020). “Various Versatile Variances: An Object-Oriented Implementation of Clustered Covariances in R.” Journal of Statistical Software, 95(1), 1--36. doi: 10.18637/jss.v095.i01

## Examples

## linear regression
x <- sin(1:10)
y <- rnorm(10)
fm <- lm(y ~ x)

#> x -0.2938577  2.0823419


text:
Formulas for calculating the deflection and displacement of a planar truss support with short studs in a lattice

Authors:
Abstract:

The calculation of deformations of building structures is usually carried out by the finite element method in numerical form  [1]–[3]. Analytical methods are used much less often in engineering calculations. For regular trusses with periodically repeating structures in the lattice, in [4]–[7], together with operators of computer mathematics systems, the method of induction is used, which makes it possible to derive calculation formulas for an arbitrary order of structure regularity. Problems of regular statically determinate rod systems were dealt with by Hutchinson, R.G. and Fleck, N.A.   [8], [9]. Matrix methods and graph theory in the calculations of regular planar and spatial trusses in relation to their optimization were applied by  Kaveh A. [10]–[12]. In the papers [13]–[18], by induction in the Maple system, solutions were obtained for the problems of deflection of various planar  trusses with an arbitrary number of panels. An analytical estimate of the first natural frequency of oscillations of planar and spatial trusses was found in [19]–[21]. A two-sided estimate of the first frequency of a regular truss in an analytical form using the Maple system was obtained in [22]. The article [23] proposes a two-node method for predicting the effective elastic properties of a periodic cellular truss. The continuum representation for calculating a regular truss is used in [24]. In [25], in the problem of periodic truss networks based on the concepts of the quasi-continuum method, the decomposition of a regular truss lattice into simple Bravais lattices is used. To reduce computational costs while accurately taking into account the dominant deformation mechanisms, a homogenized continuum description of lattice trusses is introduced in [26], based on the application of the Cauchy–Born multilattice rule to a representative unit cell. The author's handbooks [27], [28] contain schemes of planar static determinate trusses and formulas for calculating their deflections and forces in characteristic rods. Analytical methods for calculating a regular truss in relation to the problem of optimization taking into account creep were also used in [29].


text:
Connect with us

The Binomial theorem is a part of elementary Algebra that explains the power of a binomial as the algebraic expression. Based on the theorem, we can expand the polynomial by adding up the values. The concept of binomial was written by early Mathematicians in 200 BC with solutions. At that time, the theorem was expressed in quotient format. Here, we will discuss different binomial formulas and related concepts for a better understanding of the topic and it will also help the students in the actual implementation of the theorem.

## Binomial Expansion Formula

Binomial is Polynomial consists of two terms and the sum or differences of the monomial would always be a Binomial. The Binomial expression in mathematics is given as –

$\ ax^{n} + bx^{m}$

where a, b are the non- zero real coefficients and m, n are distinct non- negative integers.

#### Binomial Expansion & its formula

Binomial Expansion is a method of expanding the expression of powers of a binomial term raised to any power. Or this is an Algebraic formula describing the algebraic expansion of a polynomial raised to different powers. For example, based on the binomial expansion theorem, you may expand the power of x + y into the sum form. The Binomial Expansion Formula in Mathematics is given as –

$\ (x+y)^{n} = x^{n} + nx^{n-1}y + \frac{n(n-1)}{2!} x^{n-2} y^{2} + … + y^{n}$

### Binomial Probability Formula

The binomial distributions are common in statistics and it has a maximum of two outputs for each binomial expression. For instance, if you toss a coin then the outcome may be either head or tail. Similarly, when you appear for a test then there are two possible occurrences either pass or fail. The binomial probability formula is used to calculate the success of the binomial distributions. There are the following properties associated with the concept.

• The number of trails is always fixed and given by small n.
• Each of the trails would be independent of other.
• The final output of each trail is fixed i.e. either pass or fail.
• The probability of success is given by the small p and its value always stays the same.
• The Binomial Probability Formula for exactly x number of successes and n number of trails is given by the Formula below –

$\ P(X) = C_{x}^{n} P^{x} q^{n-x}$

Where,
n = Total number of trials
x = Total number of successful trials
p = probability of success in a single trial
q = probability of failure in a single trial = 1-p

### Binomial Distribution Formula

As the suggest binomial distribution can be taken as the common type of probability distribution with 2 possible outcomes. When we discuss the Probability Theory, the binomial distribution comes into two parameters i.e. n and p. The probability distribution becomes equal to the binomial probability distribution by satisfying the specific conditions. The formula for Binomial distribution in Mathematics is given below –

$\ P(x) = \frac{n!}{r!(n-r)!} . p^{r}(1-p)^{n} = C(n, r).p^{r}(1-p)^{n-r}$Where,
n = Total number of events
r = Total number of successful events.
p = Probability of success on a single trial.nCr = $\ \frac{n!}{r!(n − r)!}$

1 – p = Probability of failure.


text:
# Jordan-Brouwer separation theorem for topological manifolds: is it simple to prove the existance of the notion of “inside”?

I'm familiar with a general Jordan-Brouwer Separation theorem for a compact connected smooth manifold of codimension $$1$$ in $$\mathbb{R}^N$$ (i.e. not just hyperspheres). My ultimate goal is to better understand how the analog (in a proper sense) of this theorem carries over to topological manifolds.

This, however, seems to be a very deep result that requires very sophisticated machinery (for a non-expert in graduate-level algebraic topology and differential geometry etc..) with very steep entry barrier.

While I gather more knowledge on the subject (that might take many months), I hope to make a smaller (immediate progress) in this general direction, with the help of those two questions:

The primary question:

Is there a simpler definition/notion of the "inside" and "outside" of $$\mathbb{R}^n \setminus Y$$, for topological compact connected codimension-1 manifold $$Y\subset \mathbb{R}^N$$ (with the topology induced by the Euclidian metric)?

Secondary question:

Let $$X\subset \mathbb{R}^N$$ be compact, connected smooth manifold of codimension-$$1$$ (a hypersurface). So it has an easy notion of "inside" and "outside" (see in the elaboration below). Map $$X$$ by a (non-smooth) homeomorphism $$h:\mathbb{R}^N \to \mathbb{R}^N$$:

(a) Does the image $$Y= h(X)$$ have a simple definition of the "inside"?

(b) if the answer to (a) is negative, what are some of the mild conditions that need to be imposed on $$h$$ so that its image does have a simple definition of "inside"?

Any comments, references or intuition why those are still hard questions (or not) $$-$$ especially anything that helps better understanding the big picture $$-$$ are welcome.

Elaborations

Here is a quote of Jordan-Brouwer Separation theorem from lecture notes:

Theorem 20.1 [Jordan- Brouwer Separation theorem]: Let $$X$$ be compact, connected (smooth) manifold of codimension-$$1$$ in $$\mathbb{R}^N$$(hypersurface). The complement of $$X$$ in $$\mathbb{R}^N$$ consists of two connected open sets, the "outside" $$D_0$$ and the "inside" $$D_1$$. Moreover, $$\overline{D}_1$$ is compact manifold with boundary $$X =\partial \overline{D}_1$$.

The "inside" is, roughly, all points in $$\mathbb{R}^N\setminus X$$ (the complement of $$X$$ in $$\mathbb{R}^N$$) for which if we send a ray [in general position] from this point[$$^*$$] and then counts the number of the intersections of this ray with $$X$$, we have an odd number of intersections then the point is in the "inside" (conversely, if we have even number of cuts the point is in the "outside").

With the above in mind, my primary question is: suppose I have $$Y$$, compact connected topological manifold. So I lose the notion of transversality etc. Is there still a (relatively) simple notion of the "inside" and "outside" of $$\mathbb{R}^n \setminus Y$$, if yes how those are defined? Or is this already a "deep question" that requires to dive into algebraic topology and etc...

Possible duplicates

I've searched extensively to find an explanation on StackExchange or other sources: by looking up "inside topological manifold", "separation theorem for topological manifolds", "Simple Alexander duality for non-spheres" etc... but I failed to piece together any answer to my question. I find a lot of sources (here, here, here, here, here, here, the closest to my question is this one) concerned with smooth manifolds or non-smooth cases but concerned only with hyperspheres, or concerned about topology of the compliments (e.g. if it is simply connected, the counter-example of Alexander horned sphere, Alexander duality is stated in terms of hyperspheres, and here I would like to consider not just hyperspheres but compact, connected codimension-$$1$$ manifold)), or it is stated in a way that (with my current knowledge) is too cryptic to decipher (like here). I hope the way I'm trying to attack this question is simpler and I can make some progress.

[$$^*$$] A ray is going in a general direction so the ray cuts the manifold $$X$$ transversally, such rays are dense so we can always find one.

• Well, the outside is the unique unbounded connected component of the complement, which clearly exists. The inside will be the bounded connected components of the complement, but it is not obvious that there is such a component, nor then that it is unique. – Lord Shark the Unknown May 1 at 18:54
• The invariant you have, the parity of the transversal intersection number, can be extended in the topological category by using the Poincare pairing of $H^{n-1}(Y; Z_2)$ with the relative 1-st homology of neighborhoods of $Y$ in $R^n$ (relative the boundary). However, this is well beyond the tools that you currently have. My suggestion is: Instead of trying to find a non-existing black cat in a dark room, pick up an algebraic topology book and start reading. Hatcher is the standard reference these days. If this is too hard, think about something else. – Moishe Kohan May 1 at 19:54

This is the kind of situation where Alexander Duality says a lot, but establishing the theorem is admittedly no small undertaking and it's not always obvious how you should use it.

The statement of Alexander Duality for singular (co)homology concerns any subspace $$X\subset S^n$$ which is compact and locally contractible (which is true of a compact topological submanifold) and it gives us an isomorphism

$$\tilde{H}_q(X) \cong \tilde{H}^{n-q-1}(S^n\setminus X)$$

for every $$q\geq 0$$. In particular if $$X$$ is a closed (as in compact and without boundary) and oriented $$(n-1)$$-dimensional topological sub-manifold then $$\mathbb{Z} \cong \tilde{H}_{n-1}(X)\cong \tilde{H}^0(S^n \setminus X)$$

But the rank of the reduced degree $$0$$ homology group is one less than the number of path components, so $$S^n \setminus X$$ has exactly two components. They are both open in $$S^n$$ since $$X$$ is compact and hence (topologically) closed in $$S^n$$.

It follows that if $$X\subset \mathbb{R}^n$$ is a closed, connected, oriented codimension-$$1$$ topological submanifold then $$\mathbb{R}^n\setminus X$$ also has two disjoint open components, say $$U_1$$ and $$U_2$$ (by considering $$S^n$$ as the one-point compactification of $$\mathbb{R}^n$$). Since $$X$$ is compact one of these components is definitely unbounded, say $$U_1$$, so it remains to show $$U_2$$ is not. Let $$r >0$$ be large enough so that $$X\subset B_r(0)$$, the ball of radius $$r$$ centred at the origin and consider $$Y:=\mathbb{R}^n \setminus B_r(0)$$. Since $$Y$$ is connected and $$Y\cap U_1 \neq \emptyset$$ it follows that $$U_2\cap Y = \emptyset$$ (and hence $$U_2$$ is bounded), since $$Y = (Y\cap U_1) \cup (U_2 \cap Y)$$. Hence we could consider the bounded component as being "inside".

In fact, if $$X$$ satisfies all the above conditions but is not connected, and has say $$k$$ components, then $$\tilde{H}_{n-1}(X)\cong \mathbb{Z}^k$$ and so $$\mathbb{R}^n\setminus X$$ will have $$k+1$$ components, precisely one of which is unbounded.

(For a similar application of Alexander duality, check out my answer to this question )

I think your second question should follow from the first, since the image $$h(Y)$$ would be a compact topological submanifold of $$\mathbb{R}^n$$.

• Thanks a lot! this helps to see the big picture. Is there an easy intuition of what the inside and outside mean in this case? For the smooth case I get the intuition by considering the number of cuts mod 2 that a ray from any point in the "inside" has with the manifold that "encloses", is there a similar intuition for the non-smooth case? – them May 5 at 16:14