PatrickT

Dr. Patrick T

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14 years, 91 days

MaplePrimes Activity


These are replies submitted by PatrickT

 Stay with 1D:  very sound advice.  1D input is more intuitive, once you learn the basics. There are big differences between 1D and 2D input, confusing the two can quickly turn into a nightmare. One of the FAQs here is related to "implicit multiplication" whereby a space is interpreted as a multiplication (a genius must have come up with the idea, because it's a nightmare for regular brains)

 Stay with 1D:  very sound advice.  1D input is more intuitive, once you learn the basics. There are big differences between 1D and 2D input, confusing the two can quickly turn into a nightmare. One of the FAQs here is related to "implicit multiplication" whereby a space is interpreted as a multiplication (a genius must have come up with the idea, because it's a nightmare for regular brains)

tip: (not a great tip, just my preference) I prefer to call the packages as I use them, this way 1) you remember which package does what, 2) if you copy-paste commands across your own worksheets, this ensures you also get the correct package loaded/called, 3) sometimes different packages use the same commands, so confusion could arise, you will get a "the blabla command is not part of the blibli package" which is helpful in keeping track of what you do 4) when you post for help on mapleprimes it's a lot clearer what you do where. I prefer to write in full LinearAlgebra:-Multiply, etc..

and in your case, if the with(LinearAlgebra); doesn't, for some reason, work properly, hopefully the "long" call to the command will.

 

 

tip: (not a great tip, just my preference) I prefer to call the packages as I use them, this way 1) you remember which package does what, 2) if you copy-paste commands across your own worksheets, this ensures you also get the correct package loaded/called, 3) sometimes different packages use the same commands, so confusion could arise, you will get a "the blabla command is not part of the blibli package" which is helpful in keeping track of what you do 4) when you post for help on mapleprimes it's a lot clearer what you do where. I prefer to write in full LinearAlgebra:-Multiply, etc..

and in your case, if the with(LinearAlgebra); doesn't, for some reason, work properly, hopefully the "long" call to the command will.

 

 

@Markiyan Hirnyk  Is that a limit cycle (similar to Lorenz) in the middle there or just a mess?

@Markiyan Hirnyk  Is that a limit cycle (similar to Lorenz) in the middle there or just a mess?

Many users here agree with you, but as Alejandro has said the point has come up repeatedly since mapleprimes2 was launched and nothing has been done about it. 

I have posted at least twice about this and have ranted in passing more often than twice. Alejandro has also been vocal, even opening a new account so his counter would be reset to 0, and trying hard not to get votes (620 votes are what Alejandro gets when he tries not to get voted!) Many others have expressed opinions just like yours. 

When mapleprimes2 was launched there were many problems and complaints (as you would expect when a new thing is launched), most were fixed within a short time (I recall asking to be able to make your own post a favourite without incrementing your own counter and that was implemented immediately), and it was then said that a revamped version of mapleprimes would be launched which would rethink post orderings and the like, but then the whole project seems to have gone dead.

A plausible theory is that it was discovered that mapleprimes2 was basically flawed but fixing it would be too much work. Quite possibly the people in charge have been reassigned to other projects or moved on, etc..

@Preben Alsholm   yes, Preben, that may well be all there is to it, but what the center manifold theory alerts us to is the possibility of a situation where the center manifold could be locally stable in the saddle-path sense (e.g. one stable trajectory out of an infinity of unstable ones) even when the Jacobian exhibits a positive eigenvalue. 

If the dynamic system is a representation of a phenomenon observed in one of the physical sciences, chances are that such a stable path is of no practical relevance: The odds of an atom (or whatever) starting with just the exact initial values are essentially zero.

By contrast, in economics, for instance, the critical values (the stationary-state continuum, as economists usually call the indeterminate critical values) could represent a situation of maximum profit, in the context of a situation where rational, forward-looking, perfectly-informed maximizers would do everything in their power to get there, including selecting initial values that put them on a trajectory that gets there (if the choice of these initial values is up to them, the case for flow values but not generally for stock values). Thus in practice, it could, perhaps, be important to know of the existence of a stable trajectory on a center-manifold projection when the Jacobian exhibits a positive eigenvalue.

Well this is my limited understanding of the issues involved. My references are the 1981 book by Jack Carr (Applications of Centre Maniford Theory) and the 1990 (revised 2003) Springer book by Wiggins (Introduction to Applied Nonlinear Dynamical Systems and Chaos). Edit: Please do correct me if my interpretation is flawed.

@Preben Alsholm   yes, Preben, that may well be all there is to it, but what the center manifold theory alerts us to is the possibility of a situation where the center manifold could be locally stable in the saddle-path sense (e.g. one stable trajectory out of an infinity of unstable ones) even when the Jacobian exhibits a positive eigenvalue. 

If the dynamic system is a representation of a phenomenon observed in one of the physical sciences, chances are that such a stable path is of no practical relevance: The odds of an atom (or whatever) starting with just the exact initial values are essentially zero.

By contrast, in economics, for instance, the critical values (the stationary-state continuum, as economists usually call the indeterminate critical values) could represent a situation of maximum profit, in the context of a situation where rational, forward-looking, perfectly-informed maximizers would do everything in their power to get there, including selecting initial values that put them on a trajectory that gets there (if the choice of these initial values is up to them, the case for flow values but not generally for stock values). Thus in practice, it could, perhaps, be important to know of the existence of a stable trajectory on a center-manifold projection when the Jacobian exhibits a positive eigenvalue.

Well this is my limited understanding of the issues involved. My references are the 1981 book by Jack Carr (Applications of Centre Maniford Theory) and the 1990 (revised 2003) Springer book by Wiggins (Introduction to Applied Nonlinear Dynamical Systems and Chaos). Edit: Please do correct me if my interpretation is flawed.

@Markiyan Hirnyk 

you are absolutely correct, the stability analysis of the two systems is in the referenced article, and the worksheet I posted has the animations. 

 

 

@Markiyan Hirnyk 

you are absolutely correct, the stability analysis of the two systems is in the referenced article, and the worksheet I posted has the animations. 

 

 

@Alejandro Jakubi 

 `^` is an imperfect constructor

fascinating examples you've given here Alejandro. 

@Alejandro Jakubi 

 `^` is an imperfect constructor

fascinating examples you've given here Alejandro. 

@Alejandro Jakubi 

If I may IMPERFECTLY summarize the issues discussed in the exchange linked by Alejandro, a short mnemonic is "power dumb, square-root clever."

To selectively and partially quote acer,

 

"There's nothing clever in (b^2)^(1/2), that can take advantage of b>0 say. The cleverness resided in sqrt(), the function. Evaluating (b^2)^(1/2) at b= will not call sqrt(^2) once more. Once sqrt() returns its result, the opportunity for sqrt() to be clever has gone."

@Alejandro Jakubi 

If I may IMPERFECTLY summarize the issues discussed in the exchange linked by Alejandro, a short mnemonic is "power dumb, square-root clever."

To selectively and partially quote acer,

 

"There's nothing clever in (b^2)^(1/2), that can take advantage of b>0 say. The cleverness resided in sqrt(), the function. Evaluating (b^2)^(1/2) at b= will not call sqrt(^2) once more. Once sqrt() returns its result, the opportunity for sqrt() to be clever has gone."

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