- Dear Discussion groups,

I've been working on population models for a long time:

this approach is new and seems to have important implications and

applications:

http://www.crosswinds.net/~translight/population.html

Respectfully,

Roger L. Bagula

tftn@...

11759Waterhill Road

Lakeside,Ca 92040-2905

tel: 619-5610814

URL: http://home.earthlink.net/~tftn

URL: http://www.geocities.com/ResearchTriangle/Thinktank/7279/

URL: http://www.angelfire.com/ca2/tftn/index.html

URL: http://members.xoom.com/RogerLBagula/index.html

URL: http://sites.netscape.net/rlbtftn/index.html

URL: http://victorian.fortunecity.com/carmelita/435/

URL: http://members.tripod.com/tftnrlb/index.html

Chat Room URL:http://planetall.homestead.com/tftn/index.html

URL: http://fractals.jumpfun.com/

URL: http://members.spree.com/education/tftn9/

URL: http://www.crosswinds.net/~translight/index.html

URL: http://www.freestation.com/ca/tftnroger/index.shtml - In looking in my books on Banach space I find the shift operator:

f(z)-->f(z)*(z-i)/(z+i)

Shift operators of the form:

f(z)-->f(z)*(a*z+b)/(c*z+d)

are also well known and definable.

The function

g(z)=z*(a*z+b)/(c*z+d)

has a form of such a shift operator.

What a shift operator does is define a transform for which

the Banach space is invariant, I think. Bilinear operators have

this kind of invariance property, just as linear operators do. As long as

the operator is well defined, there is no problem with such functions.

Modular forms are based on just such a shift operator mechanism I think.

When instead of the complex plane we reduce it to the real number line,

as with the population function, it is much easier to use and define.

The type of function that can be defined is:

g(t)=K*exp(-k*t)*(a*exp(-k*t)+b)/(c*exp(-k*t)+d)

with matrix

M={{a,b},{c,d}}; detM<>0

(m=M/detM^(1/2) as the unitary matrix)

f(x)=Integrate[g(t),{t,0,x}]

when

I=Integrate[g(t),{t,0,Infinity}]< Infinity

This definition is very similar to how wave functions are defined in quantum

mechanics.

It has been theorized that there are quantum levels of cultures that have

sustainable population levels per area in kinds like:

hunting

gathering

herding

agricultural

industrial

If they form orthogonal sets, I have no idea. But we had no

real way to make models for them before my kind of function.

It is a kind of progress in understanding that I have been hoping for.

ROGER BAGULA wrote:

> Dear David C. Ullrich,

--

> On Banach space : the radial part gives the disk radius

> ( usually taken on {0,1]) and the angular part

> is usually used to give Fourier type Hilbert space of orthogonal

> functions, but other

> kinds of orthogonals are possible in Banach spaces as well. A good example

> is

> a Banach space made from an Hermite vibrational set of orthogonal

> functions: like

> phi(r,n,t)=K*r*H(t,n)*exp(-I*t^2/4) which is normalized to one on the unit

> disk.

> I hope this isn't wrong: I picked up Banach spaces by reading. But I had a

> course in

> quantum mechanics that covered Hilbert space and some operator spaces

>

> I had worked on these very same population

> formulas before I had studied Klein

> groups and couldn't make the connection I did this time!

> The connection to Klein groups

> gives a new way to control the kind of equation that results!

> Instead of using the powers of the von Bertalanffy allometric equation ,

> a matrix is the controlling factor

> and is can be "switched" to give two to n

> elements in the generating group. It is essentially a new technology for

> predicting population dynamics. When I invented it I had no idea it

> could be generalized like that! Go ahead make you worse critical comments,

>

> it's a good new approach and original to me.

>

> "David C. Ullrich" wrote:

>

> > In article <398DC4C3.44E4B99F@...>,

> > tftn@... wrote:

> > > Dear Discussion groups,

> > > I've been working on population models for a long time:

> > > this approach is new and seems to have important implications and

> > > applications:

> > > http://www.crosswinds.net/~translight/population.html

> >

> > Fascinating. Of course I know nothing about traditional

> > population dynamics so I'm not qualified to judge its

> > importance.

> >

> > Towards the end you say

> >

> > "When the function:

> > 18) h(t)=a*exp(-b*t)=r*exp(i*theta)=z

> > we have a disk function that can be expressed as a Banach space type

> > of production. "

> >

> > What's a "Banach space type of production"? See, I _do_ know

> > what a Banach space is, and I don't see how anything you

> > write has anything to do with Banach spaces. If you could

> > explain this that would be good. (Or you could rant about

> > how I'm NOBODY instead - that would be good too.)

> >

> > At the very end you say

> >

> > "it opens up tools developed in non-Euclidean

> > geometry to the use of population studies"

> >

> > This fascinates me as well, this time because I _don't_

> > know anything about non-Euclidean geometry. What's an

> > example of a tool developed in non-Euclidean geometry

> > that this work of yours makes applicable to population

> > dynamics?

> >

> > > Respectfully,

> > > Roger L. Bagula

> > > tftn@...

> > > 11759Waterhill Road

> > > Lakeside,Ca 92040-2905

> > > tel: 619-5610814

> > > URL: http://home.earthlink.net/~tftn

> > > URL: http://www.geocities.com/ResearchTriangle/Thinktank/7279/

> > > URL: http://www.angelfire.com/ca2/tftn/index.html

> > > URL: http://members.xoom.com/RogerLBagula/index.html

> > > URL: http://sites.netscape.net/rlbtftn/index.html

> > > URL: http://victorian.fortunecity.com/carmelita/435/

> > > URL: http://members.tripod.com/tftnrlb/index.html

> > > Chat Room URL:http://planetall.homestead.com/tftn/index.html

> > > URL: http://fractals.jumpfun.com/

> > > URL: http://members.spree.com/education/tftn9/

> > > URL: http://www.crosswinds.net/~translight/index.html

> > > URL: http://www.freestation.com/ca/tftnroger/index.shtml

> > >

> > >

> >

> > --

> > Oh, dejanews lets you add a sig - that's useful...

> >

> > Sent via Deja.com http://www.deja.com/

> > Before you buy.

>

> --

> Respectfully,

> Roger L. Bagula

> tftn@...

> 11759Waterhill Road

> Lakeside,Ca 92040-2905

> tel: 619-5610814

> URL: http://home.earthlink.net/~tftn

> URL: http://www.geocities.com/ResearchTriangle/Thinktank/7279/

> URL: http://www.angelfire.com/ca2/tftn/index.html

> URL: http://members.xoom.com/RogerLBagula/index.html

> URL: http://sites.netscape.net/rlbtftn/index.html

> URL: http://victorian.fortunecity.com/carmelita/435/

> URL: http://members.tripod.com/tftnrlb/index.html

> Chat Room URL:http://planetall.homestead.com/tftn/index.html

> URL: http://fractals.jumpfun.com/

> URL: http://members.spree.com/education/tftn9/

> URL: http://www.crosswinds.net/~translight/index.html

> URL: http://www.freestation.com/ca/tftnroger/index.shtml

Respectfully,

Roger L. Bagula

tftn@...

11759Waterhill Road

Lakeside,Ca 92040-2905

tel: 619-5610814

URL: http://home.earthlink.net/~tftn

URL: http://www.geocities.com/ResearchTriangle/Thinktank/7279/

URL: http://www.angelfire.com/ca2/tftn/index.html

URL: http://members.xoom.com/RogerLBagula/index.html

URL: http://sites.netscape.net/rlbtftn/index.html

URL: http://victorian.fortunecity.com/carmelita/435/

URL: http://members.tripod.com/tftnrlb/index.html

Chat Room URL:http://planetall.homestead.com/tftn/index.html

URL: http://fractals.jumpfun.com/

URL: http://members.spree.com/education/tftn9/

URL: http://www.crosswinds.net/~translight/index.html

URL: http://www.freestation.com/ca/tftnroger/index.shtml - Dear Discussion group,

I came on to give the answers to the functional problem for the

generalized functions for this population model:

g(t)=a*k*Exp(-k*t)*(a*exp(-k*t)+b)/(c*exp(-k*t)+d)

f(x)=Integrate[g(t),{t,0,x}]

f(x)=a^2/c-a^2*Exp(-k*x)/c-a*(detM)*log(c+d)/c^2+a*detM*log(d+c*exp(-k*t))/c^2

I=Integrate[g(t),{t,0,Infinity}]

I=(a/c^2)*(a*c+detM*(log(d)-log(c+d)))

As long as detM=a*d-b*c<>0 and I<>0 ,

the function can be normalized as a probability.

ROGER BAGULA wrote:

> In looking in my books on Banach space I find the shift operator:

--

> f(z)-->f(z)*(z-i)/(z+i)

> Shift operators of the form:

> f(z)-->f(z)*(a*z+b)/(c*z+d)

> are also well known and definable.

> The function

> g(z)=z*(a*z+b)/(c*z+d)

> has a form of such a shift operator.

>

> What a shift operator does is define a transform for which

> the Banach space is invariant, I think. Bilinear operators have

> this kind of invariance property, just as linear operators do. As long as

> the operator is well defined, there is no problem with such functions.

>

> Modular forms are based on just such a shift operator mechanism I think.

>

> When instead of the complex plane we reduce it to the real number line,

> as with the population function, it is much easier to use and define.

> The type of function that can be defined is:

> g(t)=K*exp(-k*t)*(a*exp(-k*t)+b)/(c*exp(-k*t)+d)

> with matrix

> M={{a,b},{c,d}}; detM<>0

> (m=M/detM^(1/2) as the unitary matrix)

> f(x)=Integrate[g(t),{t,0,x}]

> when

> I=Integrate[g(t),{t,0,Infinity}]< Infinity

> This definition is very similar to how wave functions are defined in quantum

> mechanics.

> It has been theorized that there are quantum levels of cultures that have

> sustainable population levels per area in kinds like:

> hunting

> gathering

> herding

> agricultural

> industrial

> If they form orthogonal sets, I have no idea. But we had no

> real way to make models for them before my kind of function.

> It is a kind of progress in understanding that I have been hoping for.

> ROGER BAGULA wrote:

>

> > Dear David C. Ullrich,

> > On Banach space : the radial part gives the disk radius

> > ( usually taken on {0,1]) and the angular part

> > is usually used to give Fourier type Hilbert space of orthogonal

> > functions, but other

> > kinds of orthogonals are possible in Banach spaces as well. A good example

> > is

> > a Banach space made from an Hermite vibrational set of orthogonal

> > functions: like

> > phi(r,n,t)=K*r*H(t,n)*exp(-I*t^2/4) which is normalized to one on the unit

> > disk.

> > I hope this isn't wrong: I picked up Banach spaces by reading. But I had a

> > course in

> > quantum mechanics that covered Hilbert space and some operator spaces

> >

> > I had worked on these very same population

> > formulas before I had studied Klein

> > groups and couldn't make the connection I did this time!

> > The connection to Klein groups

> > gives a new way to control the kind of equation that results!

> > Instead of using the powers of the von Bertalanffy allometric equation ,

> > a matrix is the controlling factor

> > and is can be "switched" to give two to n

> > elements in the generating group. It is essentially a new technology for

> > predicting population dynamics. When I invented it I had no idea it

> > could be generalized like that! Go ahead make you worse critical comments,

> >

> > it's a good new approach and original to me.

> >

> > "David C. Ullrich" wrote:

> >

> > > In article <398DC4C3.44E4B99F@...>,

> > > tftn@... wrote:

> > > > Dear Discussion groups,

> > > > I've been working on population models for a long time:

> > > > this approach is new and seems to have important implications and

> > > > applications:

> > > > http://www.crosswinds.net/~translight/population.html

> > >

> > > Fascinating. Of course I know nothing about traditional

> > > population dynamics so I'm not qualified to judge its

> > > importance.

> > >

> > > Towards the end you say

> > >

> > > "When the function:

> > > 18) h(t)=a*exp(-b*t)=r*exp(i*theta)=z

> > > we have a disk function that can be expressed as a Banach space type

> > > of production. "

> > >

> > > What's a "Banach space type of production"? See, I _do_ know

> > > what a Banach space is, and I don't see how anything you

> > > write has anything to do with Banach spaces. If you could

> > > explain this that would be good. (Or you could rant about

> > > how I'm NOBODY instead - that would be good too.)

> > >

> > > At the very end you say

> > >

> > > "it opens up tools developed in non-Euclidean

> > > geometry to the use of population studies"

> > >

> > > This fascinates me as well, this time because I _don't_

> > > know anything about non-Euclidean geometry. What's an

> > > example of a tool developed in non-Euclidean geometry

> > > that this work of yours makes applicable to population

> > > dynamics?

> > >

> > > > Respectfully,

> > > > Roger L. Bagula

> > > > tftn@...

> > > > 11759Waterhill Road

> > > > Lakeside,Ca 92040-2905

> > > > tel: 619-5610814

> > > > URL: http://home.earthlink.net/~tftn

> > > > URL: http://www.geocities.com/ResearchTriangle/Thinktank/7279/

> > > > URL: http://www.angelfire.com/ca2/tftn/index.html

> > > > URL: http://members.xoom.com/RogerLBagula/index.html

> > > > URL: http://sites.netscape.net/rlbtftn/index.html

> > > > URL: http://victorian.fortunecity.com/carmelita/435/

> > > > URL: http://members.tripod.com/tftnrlb/index.html

> > > > Chat Room URL:http://planetall.homestead.com/tftn/index.html

> > > > URL: http://fractals.jumpfun.com/

> > > > URL: http://members.spree.com/education/tftn9/

> > > > URL: http://www.crosswinds.net/~translight/index.html

> > > > URL: http://www.freestation.com/ca/tftnroger/index.shtml

> > > >

> > > >

> > >

> > > --

> > > Oh, dejanews lets you add a sig - that's useful...

> > >

> > > Sent via Deja.com http://www.deja.com/

> > > Before you buy.

> >

> > --

> > Respectfully,

> > Roger L. Bagula

> > tftn@...

> > 11759Waterhill Road

> > Lakeside,Ca 92040-2905

> > tel: 619-5610814

> > URL: http://home.earthlink.net/~tftn

> > URL: http://www.geocities.com/ResearchTriangle/Thinktank/7279/

> > URL: http://www.angelfire.com/ca2/tftn/index.html

> > URL: http://members.xoom.com/RogerLBagula/index.html

> > URL: http://sites.netscape.net/rlbtftn/index.html

> > URL: http://victorian.fortunecity.com/carmelita/435/

> > URL: http://members.tripod.com/tftnrlb/index.html

> > Chat Room URL:http://planetall.homestead.com/tftn/index.html

> > URL: http://fractals.jumpfun.com/

> > URL: http://members.spree.com/education/tftn9/

> > URL: http://www.crosswinds.net/~translight/index.html

> > URL: http://www.freestation.com/ca/tftnroger/index.shtml

>

> --

> Respectfully,

> Roger L. Bagula

> tftn@...

> 11759Waterhill Road

> Lakeside,Ca 92040-2905

> tel: 619-5610814

> URL: http://home.earthlink.net/~tftn

> URL: http://www.geocities.com/ResearchTriangle/Thinktank/7279/

> URL: http://www.angelfire.com/ca2/tftn/index.html

> URL: http://members.xoom.com/RogerLBagula/index.html

> URL: http://sites.netscape.net/rlbtftn/index.html

> URL: http://victorian.fortunecity.com/carmelita/435/

> URL: http://members.tripod.com/tftnrlb/index.html

> Chat Room URL:http://planetall.homestead.com/tftn/index.html

> URL: http://fractals.jumpfun.com/

> URL: http://members.spree.com/education/tftn9/

> URL: http://www.crosswinds.net/~translight/index.html

> URL: http://www.freestation.com/ca/tftnroger/index.shtml

Respectfully,

Roger L. Bagula

tftn@...

11759Waterhill Road

Lakeside,Ca 92040-2905

tel: 619-5610814

URL: http://home.earthlink.net/~tftn

URL: http://www.geocities.com/ResearchTriangle/Thinktank/7279/

URL: http://www.angelfire.com/ca2/tftn/index.html

URL: http://members.xoom.com/RogerLBagula/index.html

URL: http://sites.netscape.net/rlbtftn/index.html

URL: http://victorian.fortunecity.com/carmelita/435/

URL: http://members.tripod.com/tftnrlb/index.html

Chat Room URL:http://planetall.homestead.com/tftn/index.html

URL: http://fractals.jumpfun.com/

URL: http://members.spree.com/education/tftn9/

URL: http://www.crosswinds.net/~translight/index.html

URL: http://www.freestation.com/ca/tftnroger/index.shtml - Dear Discussion groups,

The Adamson measure for Poincare disks is:

http://tftnrlb.tripod.com/gwa_measure.html

f[z_]=(a*z+b)/(c*z+d)

g=Integrate[f[z],{z,0,1}]

g=

a (b c - a d) Log[d] (b c - a d) Log[c + d]

- - ------------------ + ----------------------

c 2 2

c c

This result seems closely related to the integral:

g(t)=a*k*Exp(-k*t)*(a*exp(-k*t)+b)/(c*exp(-k*t)+d)

I=Integrate[g(t),{t,0,Infinity}]

I=(a/c^2)*(a*c+detM*(log(d)-log(c+d)))

That similarity would lead me to conclude that

there is a relationship between the geometry and the population model

that results. Fixed points in the g(t) function like ( x=exp(-k*t) )

x=x*(a*x+b)/(c*x+d)

give

x=(b-d)/(a-c)----> t= -log((b-d)/(a-c))/k

Thus we have substituted the bilinear behavior of a Poincare disk , for

the differential of powers that the von Bertalanffy Allometrics

gives. It is yet to be seen which is a better approach, but just as the

Poincare

approach is a geometry of velocities, so this new approach

works on changes in population trends.

I had thought of how the Mongols who were herders invaded

the Chinese who were agricultural. It would seem that on kind

of population law can have an influence on another in actual human

history. When a culture based on agriculture is ruled by a culture

based on herding, some sort of mix of the two population rules

has to result?

ROGER BAGULA wrote:

> Dear Discussion groups,

--

> I've been working on population models for a long time:

> this approach is new and seems to have important implications and

> applications:

> http://www.crosswinds.net/~translight/population.html

> Respectfully,

> Roger L. Bagula

> tftn@...

> 11759Waterhill Road

> Lakeside,Ca 92040-2905

> tel: 619-5610814

> URL: http://home.earthlink.net/~tftn

> URL: http://www.geocities.com/ResearchTriangle/Thinktank/7279/

> URL: http://www.angelfire.com/ca2/tftn/index.html

> URL: http://members.xoom.com/RogerLBagula/index.html

> URL: http://sites.netscape.net/rlbtftn/index.html

> URL: http://victorian.fortunecity.com/carmelita/435/

> URL: http://members.tripod.com/tftnrlb/index.html

> Chat Room URL:http://planetall.homestead.com/tftn/index.html

> URL: http://fractals.jumpfun.com/

> URL: http://members.spree.com/education/tftn9/

> URL: http://www.crosswinds.net/~translight/index.html

> URL: http://www.freestation.com/ca/tftnroger/index.shtml

Respectfully,

Roger L. Bagula

tftn@...

11759Waterhill Road

Lakeside,Ca 92040-2905

tel: 619-5610814

URL: http://home.earthlink.net/~tftn

URL: http://www.geocities.com/ResearchTriangle/Thinktank/7279/

URL: http://www.angelfire.com/ca2/tftn/index.html

URL: http://members.xoom.com/RogerLBagula/index.html

URL: http://sites.netscape.net/rlbtftn/index.html

URL: http://victorian.fortunecity.com/carmelita/435/

URL: http://members.tripod.com/tftnrlb/index.html

Chat Room URL:http://planetall.homestead.com/tftn/index.html

URL: http://fractals.jumpfun.com/

URL: http://members.spree.com/education/tftn9/

URL: http://www.crosswinds.net/~translight/index.html

URL: http://www.freestation.com/ca/tftnroger/index.shtml - Dear Discussion groups,

The peak function is:

g(t)=a*k*Exp(-k*t)*(a*exp(-k*t)+b)/(c*exp(-k*t)+d)

where

g(t)=df(t)/dt

If we use z=exp(-k*t)

then g(z)=df(z)/dt=K*z*(a*z+b)/(c*z+d)

From this we can make a Newton's algorithm for the derivative as:

h(z)=z+K*z*(a*z+b)/(c*z+d)

This has the form of a generalized Julia in iterative terms in complex

dynamics. It is not the most satisfactory way to approach

the nonlinear aspects of the model, but it is one that is practical.

In " The Beauty of Fractals" Verhulst dynamics is substituted for

Logistic dynamics

to introduce the onset of Chaos ( page 23 ff). It uses a one dimensional

rate

against population iteration plot to do this. In the case of this new

model

the rate is controlled by the matrix in the bilinear factor. A complex

dynamics model

is seems a better approach to visualization of this kind of equation.

ROGER BAGULA wrote:

> Dear Discussion groups,

--

> I've been working on population models for a long time:

> this approach is new and seems to have important implications and

> applications:

> http://www.crosswinds.net/~translight/population.html

> Respectfully,

> Roger L. Bagula

> tftn@...

> 11759Waterhill Road

> Lakeside,Ca 92040-2905

> tel: 619-5610814

> URL: http://home.earthlink.net/~tftn

> URL: http://www.geocities.com/ResearchTriangle/Thinktank/7279/

> URL: http://www.angelfire.com/ca2/tftn/index.html

> URL: http://members.xoom.com/RogerLBagula/index.html

> URL: http://sites.netscape.net/rlbtftn/index.html

> URL: http://victorian.fortunecity.com/carmelita/435/

> URL: http://members.tripod.com/tftnrlb/index.html

> Chat Room URL:http://planetall.homestead.com/tftn/index.html

> URL: http://fractals.jumpfun.com/

> URL: http://members.spree.com/education/tftn9/

> URL: http://www.crosswinds.net/~translight/index.html

> URL: http://www.freestation.com/ca/tftnroger/index.shtml

Respectfully,

Roger L. Bagula

tftn@...

11759Waterhill Road

Lakeside,Ca 92040-2905

tel: 619-5610814

URL: http://home.earthlink.net/~tftn

URL: http://www.geocities.com/ResearchTriangle/Thinktank/7279/

URL: http://www.angelfire.com/ca2/tftn/index.html

URL: http://members.xoom.com/RogerLBagula/index.html

URL: http://sites.netscape.net/rlbtftn/index.html

URL: http://victorian.fortunecity.com/carmelita/435/

URL: http://members.tripod.com/tftnrlb/index.html

Chat Room URL:http://planetall.homestead.com/tftn/index.html

URL: http://fractals.jumpfun.com/

URL: http://members.spree.com/education/tftn9/

URL: http://www.crosswinds.net/~translight/index.html

URL: http://www.freestation.com/ca/tftnroger/index.shtml