Ryan, for you and other ULers ....., a linear, first order and separable differential equation with initial conditions.

Enjoy.

Edited by romandial on 12/10/2006 13:56:46 MST.

Roman, I'll have to get back to you on that one. I'm number 7 on the wait list for Diff Eq next semester. We have been doing some DE's in Circuit Analysis, but nothing quite like that. Hopefully I like Diff Eq better than Calc....

Adam

First, is there a right answer to this? And, if we find where the hiker is at the end of 24 days do we get a gift?

Second, is this in the same vein as the edible gear. And, does it make a difference if the edible gear is packed in a vacuum sealed canister or not? i.e. does the vacuum lighten the load? And, what would be the shelf life of said edible gear if dt>ds and we hold it constant for s(0)=0?

You bet there's a correct answer! And with all the education and sophistication among the posters and lurkers at BPL, at least a dozen of you should be able to solve it handily.

As for prizes......?

Ryan,
I have an Edirol R-09, too. I'm a composer and do a lot of electronic manipulation, so I use it for that and recordings for ethnomusicological research. I hadn't thought of taking it backpacking, but now that you mention it.... I'm definitely going to try it! The sound is absolutely great for such a small device, and with a 2GB card, you can fit a lot of audio.

About those recordings... is BPL thinking about starting a podcast or something similar? That would be great, and I'd love to help. I know quite a lot about recording technology, audio editing, music composition, etc.

Well done Roman, this question seems to be a model for a recent unsupported long distance hike.

It haven't solved these equations in perhaps 7 or 8 years, but my initial calculation gave this as a result:
s(t)=14t + t²/2 with s(24)=624.

Toms right:

Here's some further explanation

Its first order separable, linear equation that does exist and have a unique solution.

First we move the dt to the right side, which gives us

ds = 14dt + tdt

Then you take the integral of each term, which gives us

s = 14t + 1/2t^2 + c

The constant of integration was left out by Tom, but since it comes out to be zero izzz all good.

Plug in the initial condition.

0 = 14(0) + 1/2(0)^2 + C

C = 0

s = 14t + 1/2t^2

s = 14(24)+1/2(24)^2
s = 336 + 288

s = 624 miles

Come on Roman is that all you've got?? Send me one that is at least a second order linear that only exists around a regular singlular point....ok??

:D

Yes 624, but what is the connection between 624 and 24 and Roman?

624 was the easy part.

Edited by rogerb on 12/10/2006 15:49:01 MST.

You guys are great. I love this site.

24 days and 624 miles were the end points of my experience on the Arctic 1000.

While I had hoped to eat ~2lbs/day and improve our distance by ~2 miles/day, as I'd found from adventure racing, it actually came out to reduce by 2 lbs/day meant that we could go 1 mile more per day.

Possibly this latter result was because we made 12 hour days the norm rather than 18+ hours as in adventure racing.

And Crazy Pete: very well done...but know I have yet to come up with a good second order DE that we might feel pertains to us....but I will work on it!

Edited by romandial on 12/10/2006 17:10:01 MST.

After Ryan mentioned taking mechanical pencils in another thread, I found some mini mechanical pencils at a Walmart Neighborhood grocery store. They are in a pack of 5 pencils, just under 4 inches and are Foohy brand...for less than 2 bucks.

I got the following after first Romanizing the equation. Starting with the original Romanesque Equation I first got:

DS/DT = XIV + T

Divide and multiply by D/D, then (S/T = XIV + T) BY D/D.

Assuming D is not 0, then D/D = I. Then I times (XIV + T) = ITSELF or UNITY. In other words it is a unified disintegrated differential equations.

Then S/T = XIV + T OR T = (XIV + T)/S, if you divide both sides of the equation by S.

Thus, s(0) = O could not be true unless T = 0. So the only possible answer is that given by other scholars and trekkers who would know what the starting factors, location, and variables were (which appear to be Roman's appetite and food budget for trip in the Arctic). One would, therefore, make the assumption that the range of probable, if not possible answers would include the farthest distances one could go and the least simultaneously -- fulfilling the requirement of diminishing simultaneous returns in closed dietary pack systems, not otherwise providing for the possibility of negative distances or 0 -- requiring that there be a positiev but uncertain resolution to the equation.

If the equation were to be subjected to further testing and field experience, possibly deserving an article here or in the print magazine, then the numerical probability would probably be more certain and one could then hypothesize both an absolute maximum, the mean, median, and even negative limitation points -- such as those sugggested by Pondering in the unpublished treatise "Negative Distance Effects of Hunger: The Simultaneous Distance Quandry of Positive Distance Traveled Total Miles Trekked Diminishing In Returning for More Food Events."

Also, as a secondary numeric anomaly, since T is iself equal to itself, a period of Time spent hiking for 24 days, with s(0) = 0, can only lead to a conclusion that d = d during the particular time as empirically measured and not deduced.

Therefore, the concept or mathematical idea that s or t leads to a change in location (d) is only a theoretical possibility in a range of locations from 0 to 624, based upon actual calculations and observations by trekkers in this time-space reference frame.

The ultimate conclusion: 624 is the best available solution, but is subject to human error and will require both verification by the BPL Staff who should weigh in with a study with thermometers, calibrated walking devices, and photographic evidence. The more likely conclusion, based on this equation is that nobody is getting anywhere fast. Even that is questionable within the ranges of possible, though not probable sets of whole integers resolving the equation for s(t) at the conclusion of 24 days.

BD-- I am not exactly sure what you are doing, and I believe that you have made some simple mathmatical errors. First off, what exactly are you doing multiplying/dividing by D/D?? Are you just saying that the Ds cancel each other out because they are superimposed over a fraction line?? Because those are demarcations of the process of derivation, not variables, and to multiply by a D is incorrect, unless you are somehow intergrating somewhere else in the equation.

Secondly "S/T = XIV + T OR T = (XIV + T)/S" is NOT true, because when solving for T one would get

S = T *(XIV + T)
S/(XIV + T) = T

Thus your statement T = (XIV + T)/S is wrong.

And if you were actually solving for T in that equation you would factor it out, otherwise the entire excercise would be fruitless.

Thirdly, a solution DOES exist thanks to the Existence and Uniqueness Theorem, and a quick look at the graph shows it to be continuous in the examined area. I am thus confused at your statement that 624 is the best available solution, because it IS the solution for the IVP with given input value.

Crazy Pete,

Oh. Oops. Sorry:) (Do you want help off that rock? Great avatar, by the way.)

I thought the processes or process of derivation was a constant, but the variability over the set of possible processes might itself vary, although not in this case -- given that it is apparently a function or process of Roman's appetite, or any other trekkers or the multiple variabilities of either other trekkers or sets of trekkers dietary requirements or choices. Kind of like econometric regression models of consumer behaviors in the case of scarcity models based on multiple variabilities and differential processes. (*See below comment on the Existence and Uniqueness Theorum.)

Thus the overall apparent aberations from mathematical norms and rules. Also, while the use of S may appear to dictate your version of the equation it is also possible that speed will dimish the dt portion of the equation IMO, thus I chose the diminutive form of dividing by S to find the solution over any given range of T.

Sorry bout that. I still firmly believe that best choice of an answer (your solution) is 624, based on the previously described simultaneous returns quandary. But, I am willing to go along with any solution so long as I do not think about this when trying to go to sleep tonite.

* Re the Existence and Uniqueness Theroum from the University of Texas at the Permian Basin: "There are two important points to make now. First the theorem does not tell us anything about how to find a solution. Finally, we have no idea of how small the region may have to be to accommodate the theorem. In other words the interval of existence for the solution may include little more than the initial value." at http://www.utpb.edu/scimath/wkfield/mod3/Exuni.htm

Edited by bdavis on 12/10/2006 21:45:21 MST.

BD-------
"Then S/T = XIV + T OR T = (XIV + T)/S, if you divide both sides of the equation by S.

Thus the overall apparent aberations from mathematical norms and rules. Also, while the use of S may appear to dictate your version of the equation it is also possible that speed will dimish the dt portion of the equation IMO, thus I chose the diminutive form of dividing by S to find the solution over any given range of T. "
-----End BD

S/T = XIV + T
(S/T)/S = (XIV + T)/S

1/T = (XIV+T)/S
Not T = (XIV + T)/S

T = (-14 plusminus (196-4s)^.5)/2

But this is all subiderary to the fact that what the equation states is that the rate position is changing with respect to the change in time (velocity), is equal to 14 plus t. Which means that the velocity is changing, which is the whole point of a variable slope. The velocity increases as T increases---FOR ROMAN. Whether or not the equation applies to others has no effect on the correctness of the postulate. Say for someone sitting all the time, the equation would be DS/DT = K, where K would be a constant velocity, in this case zero.

As to the E and U Theorem, it does not tell us how to solve for a solution. It gives whether one exists.

The graph of the equation proves continuity:




The interval of existance (-infinity, infinity).

Perhaps I've never learned what you are doing with the derivative--but treating the D as a variable automatically removes it from the equation, just as 3/3 = 1. D/D would be one as well, except it means the first derivative of position with respect to its corresponding first order time. So in essence, I am still confused by what you did...

And waaaaaaiiit a minute--- time is the INDEPENDANT variable here. You don't calculate time based on distance, but rather distance based upon time.

Edited by crazypete on 12/10/2006 22:36:46 MST.

Crazy Pete,

How about you? Have you done any UL modeling you can share with us?

Crazy Pete,

Quite right.

So if T is independent, then the issue becomes is the 14 correct if the equation is to work for the generalized equation for a trek of many people or individuals or people sitting at home.

One question, how do you get to:

T = (-14 plusminus (196-4s)^.5)/2

That may be my problem, even though this started as a kind of fun spoof thing for me, I take seriously what you are saying because my food could depend on it or people's safety.

I think the issue is, mathematically, you cannot include enough processes, differentials, or variables in different levels of mathematical process to make a formula that will work in this example. Although the usefulness of them is that they will prevent disasters by predicting a range of values which will work.

Again, while I was spoofing at first, thus the reference to the made up unpublished treatise, you guys are right on ... I just don't think the "theorum" or "formula" or "equation" will work in realtity.

Thus, like Godels proof, even within a mathematical system let alone when it is applied to realtiy, the system cannot generate its own proof, or in the case of the general theorum of existence of unique solutions, this may be a kind of tautology. Input in = input out, or confirmation that given the intitial hypothesis there will only be one answer, regardless of the realities the formula is trying to measure. Does that make sense to you?

This is starting to interest me so much I can't stop thinking about it, but no harm meant on my side. I wrote a spoofy answer and got back what is expected at this site, a really great answer.

Update: T may not be independent, because of the length of time travelled each day, or the length of time it takes to travel x distance, or some other function of time, even though it is a constant as we measure it. Thus, c becomes important in your equation, because it may not be zeroed out because s(0)=0. So the question that really starts to loom, and I was afraid of this given my mind and interest in this, is: if c, in your equation cannot be established without empirical data, can it be cancelled out ... and can s(0)=0 for more than one set of facts????

Edited by bdavis on 12/10/2006 23:47:22 MST.

Roman-- Not really theoretical stuff like this would be, but simply stuff like force on tarp panels in various wind speeds.

BD - I used the quadratic formula to solve for T.

The C is the constant of integration; briefly, that means it is similar to a y-intercept in that it moves a line up and down. We graph velocity vs time, where the slope is acceleration to determine total distance traveled by area under the curve. By the initial given condition, that S(0)= 0, C = 0. All this initial condition means is that the 0 distance has been traveled at time 0. If you start at say 5 miles already having been traveled, then S(0) = 5 and C would be a value other than 0 due to the face there is more area under the curve.

The great thing about the integral is that it takes into account that the distance traveled each day can be different, as hmmm let me make a graph.



OK, in the graph you can see that the velocity is constantly changing, just like you said. As such, a different distance is traveled everyday, but this can still be accounted for. The hard part is getting the equation for this line, which Roman provided. Everything else is (somewhat)simple mathmatics.

That said, I would not be able to use this for any acutally applications on reality due to the fact I do not have a formula for DS/DT.

Edited by crazypete on 12/11/2006 00:12:27 MST.

Crazy Pete,

Cool. Trying to put math around this UL stuff is cool, way cool.

I never thought of doing it. But I only have myself for a sample/null set v. sample. Thus you guys are way ahead of me on that one, and much more.

I think the problem is your null set or model will require, mathwise, some 15 people to make the N (or in your equation ultimately "c" valid). Then the "C" or "c" and the 14 will work in combo.

Then you can develope a functional equation that is correlated to reality for specific real groups or individuals. (Given the caveat that it won't work on any given day -- the curve margins.)

My first thought is what needs to be done to make it work, to establish a curve, like Roman has started or suggested, you need to create a model of each function: DS and DT. (Thus your comment that DS/DT does not have a formula or reality at present -- to me that means empirical data + thought.)

I believe it can be done and that is what this site is for, we just have to control for or establish what the ranges are for individual people or sets of people, trekkers, or whatevers that the DS/DT will apply to.

Then they will have to hold to a particular projection or model of behaviour to make the math fit the reality???

Edit: That is why I zeroed out the d function. If it is the same graph/curve for an individual or small group, then it is the same over T.

Further update: The velocity problem is exemplified by the graph you just providedl Given the variabiltiy, draw a line from the bottom and top points the range of values gets to be too great to continue it with any meaning. What I noticed is the line goes downwards. ????

Edited by bdavis on 12/11/2006 00:46:20 MST.

DS and DT is simply a ratio of two different rates. All we need is a representation of how one's velocity changes over time and then one can figure everything else out.

I believe that finding this curve is impossible though.

The C and the 14 are unrelated. The 14 was just a part of the velocity equation Roman stipulated. The equation for others may involve a t cubed or rooted or a t cubed plus t rooted. The C is just a marker acknowleding that there are infinite solutions to a differential equation, and depends completly upon the initial value, and won't change for different people.

What changes is the velocity equation, and as such, their constant of integration depends only upon whatever inital value they plug in.