Mathematics | Insight Maker

THE BROKEN LINK BETWEEN SUPPLY AND DEMAND CREATES TURBULENT CHAOTIC DESTRUCTION

The existing global capitalistic growth paradigm is totally flawed

Growth in supply and productivity is a summation of variables as is demand ... when the link between them is broken by catastrophic failure in a component the creation of unpredictable chaotic turbulence puts the controls ito a situation that will never return the system to its initial conditions as it is STIC system (Lorenz)

The chaotic turbulence is the result of the concept of infinite bigness this has been the destructive influence on all empires and now shown up by Feigenbaum numbers and Dunbar numbers for neural netwoirks

See Guy Lakeman Bubble Theory for more details on keeping systems within finite working containers (villages communities)

THE BROKEN LINK BETWEEN SUPPLY AND DEMAND CREATES CHAOTIC TURBULENCE (+controls)

Guy Lakeman

20

The Logistic Map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

Mathematically, the logistic map is written

where:

is a number between zero and one, and represents the ratio of existing population to the maximum possible population at year n, and hence x0 represents the initial ratio of population to max. population (at year 0)r is a positive number, and represents a combined rate for reproduction and starvation.
For approximate Continuous Behavior set 'R Base' to a small number like 0.125To generate a bifurcation diagram, set 'r base' to 2 and 'r ramp' to 1
To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

The Logistic Map

Guy Lakeman

26

The simulation integrates or sums (INTEG) the Nj population, with a change of Delta N in each generation, starting with an initial value of 5.

The equation for DeltaN is a version of

Nj+1 = Nj + mu (1- Nj / Nmax ) Nj

the maximum population is set to be one million, and the growth rate constant mu = 3.

Nj: is the “number of items” in our current generation.

Delta Nj: is the “change in number of items” as we go from the present generation into the next generation. This is just the number of items born minus the number of items who have died.

mu: is the growth or birth rate parameter, similar to that in the exponential growth and decay model. However, as we extend our model it will no longer be the actual growth rate, but rather just a constant that tends to control the actual growth rate without being directly proportional to it.

F(Nj) = mu(1‐Nj/Nmax): is our model for the effective “growth rate”, a rate that decreases as the number of items approaches the maximum allowed by external factors such as food supply, disease or predation. (You can think of mu as the growth or birth rate in the absence of population pressure from other items.) We write this rate as F(Nj), which is a mathematical way of saying F is affected by the number of items, i.e., “F is a function of Nj”. It combines both growth and all the various environmental constraints on growth into a single function. This is a good approach to modeling; start with something that works (exponential growth) and then modify it incrementally, while still incorporating the working model.

Nj+1 = Nj + Delta Nj : This is a mathematical way to say, “The new number of items equals the old number of items plus the change in number of items”.

Nj/Nmax: is what fraction a population has reached of the maximum "carrying capacity" allowed by the external environment. We use this fraction to change the overall growth rate of the population. In the real world, as well as in our model, it is possible for a population to be greater than the maximum population (which is usually an average of many years), at least for a short period of time. This means that we can expect fluctuations in which Nj/Nmax is greater than 1.

This equation is a form of what is known as the logistic map or equation. It is a map because it "maps'' the population in one year into the population of the next year. It is "logistic'' in the military sense of supplying a population with its needs. It a nonlinear equation because it contains a term proportional to Nj^2 and not just Nj. The logistic map equation is also an example of discrete mathematics. It is discrete because the time variable j assumes just integer values, and consequently the variables Nj+1 and Nj do not change continuously into each other, as would a function N(t). In addition to the variables Nj and j, the equation also contains the two parameters mu, the growth rate, and Nmax, the maximum population. You can think of these as "constants'' whose values are determined from external sources and remain fixed as one year of items gets mapped into the next year. However, as part of viewing the computer as a laboratory in which to experiment, and as part of the scientific process, you should vary the parameters in order to explore how the model reacts to changes in them.

POPULATION LOGISTIC MAP (WITH FEEDBACK)

Guy Lakeman

8

Simulation of MTBF with controls

F(t) = 1 - e ^ -λt

Where

• F(t) is the probability of failure

• λ is the failure rate in 1/time unit (1/h, for example)

• t is the observed service life (h, for example)

The inverse curve is the trust time
On the right the increase in failures brings its inverse which is loss of trust and move into suspicion and lack of confidence.
This can be seen in strategic social applications with those who put economy before providing the priorities of the basic living infrastructures for all.

This applies to policies and strategic decisions as well as physical equipment.
A) Equipment wears out through friction and preventive maintenance can increase the useful lifetime,
B) Policies/working practices/guidelines have to be updated to reflect changes in the external environment and eventually be replaced when for instance a population rises too large (constitutional changes are required to keep pace with evolution, e.g. the concepts of the ancient Greeks, 3000 years ago, who based their thoughts on a small population cannot be applied in 2013 except where populations can be contained into productive working communities with balanced profit and loss centers to ensure sustainability)

Early Life

If we follow the slope from the leftmost start to where it begins to flatten out this can be considered the first period. The first period is characterized by a decreasing failure rate. It is what occurs during the “early life” of a population of units. The weaker units fail leaving a population that is more rigorous.

Useful Life

The next period is the flat bottom portion of the graph. It is called the “useful life” period. Failures occur more in a random sequence during this time. It is difficult to predict which failure mode will occur, but the rate of failures is predictable. Notice the constant slope.

Wearout

The third period begins at the point where the slope begins to increase and extends to the rightmost end of the graph. This is what happens when units become old and begin to fail at an increasing rate. It is called the “wearout” period.

BATHTUB MEAN TIME BETWEEN FAILURE (MTBF) RISK

Guy Lakeman

7

The Binary Adder:

Andy Long

Spring, 2020 - Year of Covid-19

Having constructed a working example of a finite state machine (FSM), from Gersting's 7th edition (p. 730, Example 29), I decided to create a more useful one -- a binary adder (p. 732). It works!

Subject to these rules:

Your two binary numbers should start off the same length -- pad with zeros if necessary. Call this length L.
Now pad your two binary numbers with three extra 0s at the end; this lets the binary-to-decimal conversion execute.
numbers are entered from ones place (left to right).
In Settings, choose "simulation start" as 1, your "simulation length" as L+2 -- two more than the length of your initial input number vectors. (I wish that the Settings issues could be set without having to explicitly change it each time -- maybe it can, but I don't know how.)

Be attentive to order -- start with 1s place, 2s place, 4s, place, etc., and your output answer will be read in the same order.

To understand why we need three additional inputs of 0s:

For the useless first piece of output -- so n -> n+1
For the possibility of adding two binary numbers and ending up with an additional place we need to force out: 111 + 111 = 0 1 1 1
For the delay in computing the decimal number: it reads the preceding output to compute the decimal value.

Mat385 Finite State Machine (Binary Adder)

Andrew E Long

An infinite number of mathematicians walk into a bar. The first one tells the bartender he wants a beer. The second one says he wants half a beer. The third one says he wants a fourth of a beer. The bartender puts two beers on the bar and says “You guys need to learn your limits.” From Wild About Math

Infinite drinkers

Geoff McDonnell

OVERSHOOT GROWTH GOES INTO TURBULENT CHAOTIC DESTRUCTION

The existing global capitalistic growth paradigm is totally flawed

The chaotic turbulence is the result of the concept of infinite bigness this has been the destructive influence on all empires and now shown up by Feigenbaum numbers and Dunbar numbers for neural netwoirks

See Guy Lakeman Bubble Theory for more details on keeping systems within finite limited size working capacity containers (villages communities)

OVERSHOOT GROWTH INTO TURBULENCE

Guy Lakeman

7

Path of a ball either dropped or thrown up vertically

Vertical path of a ball

Jean Caillé

FORCED GROWTH GROWTH GOES INTO TURBULENT CHAOTIC DESTRUCTION
BEWARE pushing increased growth blows the system!
(governments are trying to push growth on already unstable systems !)

The existing global capitalistic growth paradigm is totally flawed

The chaotic turbulence is the result of the concept and flawed strategy of infinite bigness this has been the destructive influence on all empires and now shown up by Feigenbaum numbers and Dunbar numbers for neural netwoirks

See Guy Lakeman Bubble Theory for more details on keeping systems within finite limited size working capacity containers (villages communities)

FORCED GROWTH INTO TURBULENCE

Guy Lakeman

5

object is projected with an initial velocity u at an angle to the horizontal direction.

We assume that there is no air resistance .Also since the body first goes up and then comes down after reaching the highest point , we will use the Cartesian convention for signs of different physical quantities. The acceleration due to gravity 'g' will be negative as it acts downwards.

h=v_ox*t-g*t^2/2

l=v_oy*t

Launched at an Angle

Tsogbadrakh Banzragch

4

This insight implements integration as an InsightMaker model.

It is important to use Euler's method, with Simulation Length equal to n, in Seconds.

Fun to try a couple of different cases, so I have built four choices into this example. You can choose the function ("Function Choice" of 0, 1, 2, or 3) using the slider.

Andy Long

Spring, 2020

Integral of a function

Andrew E Long

This is an example of a finite state machine (FSM), from Gersting's 7th edition (p. 730, Example 29).

It's the first time I've used vectors (for input); and there are a few interesting twists on models that are different for the Finite State Machine.

So it's been an interesting learning experience for me....

It works!

Andy Long

Spring, 2020

Year of Covid-19

Clone of Mat385 Discrete Math FSM -- First Example

ovil

Example of mathematical modeling leading to known functions: sine and cosine

Calculus - Periodic functions

Jean Caillé

THE BROKEN LINK BETWEEN SUPPLY AND DEMAND CREATES TURBULENT CHAOTIC DESTRUCTION

The existing global capitalistic growth paradigm is totally flawed

Growth in supply and productivity is a summation of variables as is demand ... when the link between them is broken by catastrophic failure in a component the creation of unpredictable chaotic turbulence puts the controls ito a situation that will never return the system to its initial conditions as it is STIC system (Lorenz)

The chaotic turbulence is the result of the concept of infinite bigness this has been the destructive influence on all empires and now shown up by Feigenbaum numbers and Dunbar numbers for neural netwoirks

See Guy Lakeman Bubble Theory for more details on keeping systems within finite working containers (villages communities)

Clone of THE BROKEN LINK BETWEEN SUPPLY AND DEMAND CREATES CHAOTIC TURBULENCE (+controls)

Sugiharto

MAT375: Non-linear Exam....

This insight implements Newton's method as an InsightMaker model.

It is important to use Euler's method, with step-size of 1. That's what allows us to get away with this!:)

Fun to try a couple of different cases, so I have built four choices into this example. You can choose the function ("Function Choice" of 0, 1, 2, or 3) using the slider.

Andy Long

Spring, 2020

Clone of Newton's Method

Samuel Kaelin

MAT375: Non-linear Exam....

This insight implements Newton's method as an InsightMaker model.

It is important to use Euler's method, with step-size of 1. That's what allows us to get away with this!:)

Fun to try a couple of different cases, so I have built four choices into this example. You can choose the function ("Function Choice" of 0, 1, 2, or 3) using the slider.

Andy Long

Spring, 2020

Newton's Method

Andrew E Long

27

The Logistic Map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

Mathematically, the logistic map is written

where:

is a number between zero and one, and represents the ratio of existing population to the maximum possible population at year n, and hence x0 represents the initial ratio of population to max. population (at year 0)r is a positive number, and represents a combined rate for reproduction and starvation. To generate a bifurcation diagram, set 'r base' to 2 and 'r ramp' to 1
To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

Clone of Clone of The Logistic Map

Bryan Potter

Problem of the sliding chain

Sliding Chain

Jean Caillé

51

The simulation integrates or sums (INTEG) the Nj population, with a change of Delta N in each generation, starting with an initial value of 5.

The equation for DeltaN is a version of

Nj+1 = Nj + mu (1- Nj / Nmax ) Nj

the maximum population is set to be one million, and the growth rate constant mu = 3.

Nj: is the “number of items” in our current generation.

Delta Nj: is the “change in number of items” as we go from the present generation into the next generation. This is just the number of items born minus the number of items who have died.

mu: is the growth or birth rate parameter, similar to that in the exponential growth and decay model. However, as we extend our model it will no longer be the actual growth rate, but rather just a constant that tends to control the actual growth rate without being directly proportional to it.

F(Nj) = mu(1‐Nj/Nmax): is our model for the effective “growth rate”, a rate that decreases as the number of items approaches the maximum allowed by external factors such as food supply, disease or predation. (You can think of mu as the growth or birth rate in the absence of population pressure from other items.) We write this rate as F(Nj), which is a mathematical way of saying F is affected by the number of items, i.e., “F is a function of Nj”. It combines both growth and all the various environmental constraints on growth into a single function. This is a good approach to modeling; start with something that works (exponential growth) and then modify it incrementally, while still incorporating the working model.

Nj+1 = Nj + Delta Nj : This is a mathematical way to say, “The new number of items equals the old number of items plus the change in number of items”.

Nj/Nmax: is what fraction a population has reached of the maximum "carrying capacity" allowed by the external environment. We use this fraction to change the overall growth rate of the population. In the real world, as well as in our model, it is possible for a population to be greater than the maximum population (which is usually an average of many years), at least for a short period of time. This means that we can expect fluctuations in which Nj/Nmax is greater than 1.

This equation is a form of what is known as the logistic map or equation. It is a map because it "maps'' the population in one year into the population of the next year. It is "logistic'' in the military sense of supplying a population with its needs. It a nonlinear equation because it contains a term proportional to Nj^2 and not just Nj. The logistic map equation is also an example of discrete mathematics. It is discrete because the time variable j assumes just integer values, and consequently the variables Nj+1 and Nj do not change continuously into each other, as would a function N(t). In addition to the variables Nj and j, the equation also contains the two parameters mu, the growth rate, and Nmax, the maximum population. You can think of these as "constants'' whose values are determined from external sources and remain fixed as one year of items gets mapped into the next year. However, as part of viewing the computer as a laboratory in which to experiment, and as part of the scientific process, you should vary the parameters in order to explore how the model reacts to changes in them.

Clone of POPULATION LOGISTIC MAP (WITH FEEDBACK)

Roman Knaus

The Binary Adder:

Andy Long

Spring, 2020 - Year of Covid-19

Having constructed a working example of a finite state machine (FSM), from Gersting's 7th edition (p. 730, Example 29), I decided to create a more useful one -- a binary adder (p. 732). It works!

Subject to these rules:

Your two binary numbers should start off the same length -- pad with zeros if necessary. Call this length L.
Now pad your two binary numbers with three extra 0s at the end; this lets the binary-to-decimal conversion execute.
numbers are entered from ones place (left to right).
In Settings, choose "simulation start" as 1, your "simulation length" as L+2 -- two more than the length of your initial input number vectors. (I wish that the Settings issues could be set without having to explicitly change it each time -- maybe it can, but I don't know how.)

Be attentive to order -- start with 1s place, 2s place, 4s, place, etc., and your output answer will be read in the same order.

To understand why we need three additional inputs of 0s:

For the useless first piece of output -- so n -> n+1
For the possibility of adding two binary numbers and ending up with an additional place we need to force out: 111 + 111 = 0 1 1 1
For the delay in computing the decimal number: it reads the preceding output to compute the decimal value.

Clone of Mat385 Finite State Machine (Binary Adder)

Santiago Nicolás

One of my favorite topics in Discrete Math is Finite State Machines (FSM). We study deterministic machines without feedback, which can be used to

* recognize certain types of strings

* perform certain tasks (e.g. binary addition).

This particular example of an FSM recognizes input that include multiples of 3 0s (including none). If the nth input represents a complete multiple of 3 0s, then the output will be 1; otherwise 0.

Andy Long

Spring, 2020

Year of Covid-19

Mat385 Discrete Math FSM -- Recognition

Andrew E Long

The Logistic Map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.

Mathematically, the logistic map is written

where:

is a number between zero and one, and represents the ratio of existing population to the maximum possible population at year n, and hence x0 represents the initial ratio of population to max. population (at year 0)r is a positive number, and represents a combined rate for reproduction and starvation. To generate a bifurcation diagram, set 'r base' to 2 and 'r ramp' to 1
To demonstrate sensitivity to initial conditions, try two runs with 'r base' set to 3 and 'Initial X' of 0.5 and 0.501, then look at first ~20 time steps

Clone of Clone of The Logistic Map

Aaron Bouchard

Simulation of MTBF with controls

F(t) = 1 - e ^ -λt

Where

• F(t) is the probability of failure

• λ is the failure rate in 1/time unit (1/h, for example)

• t is the observed service life (h, for example)

The inverse curve is the trust time
On the right the increase in failures brings its inverse which is loss of trust and move into suspicion and lack of confidence.
This can be seen in strategic social applications with those who put economy before providing the priorities of the basic living infrastructures for all.

This applies to policies and strategic decisions as well as physical equipment.
A) Equipment wears out through friction and preventive maintenance can increase the useful lifetime,
B) Policies/working practices/guidelines have to be updated to reflect changes in the external environment and eventually be replaced when for instance a population rises too large (constitutional changes are required to keep pace with evolution, e.g. the concepts of the ancient Greeks, 3000 years ago, who based their thoughts on a small population cannot be applied in 2013 except where populations can be contained into productive working communities with balanced profit and loss centers to ensure sustainability)

Early Life

If we follow the slope from the leftmost start to where it begins to flatten out this can be considered the first period. The first period is characterized by a decreasing failure rate. It is what occurs during the “early life” of a population of units. The weaker units fail leaving a population that is more rigorous.

Useful Life

The next period is the flat bottom portion of the graph. It is called the “useful life” period. Failures occur more in a random sequence during this time. It is difficult to predict which failure mode will occur, but the rate of failures is predictable. Notice the constant slope.

Wearout

The third period begins at the point where the slope begins to increase and extends to the rightmost end of the graph. This is what happens when units become old and begin to fail at an increasing rate. It is called the “wearout” period.

Clone of Clone of Clone of Clone of BATHTUB MEAN TIME BETWEEN FAILURE (MTBF) RISK

Alena Peskova

OVERSHOOT GROWTH GOES INTO TURBULENT CHAOTIC DESTRUCTION

The existing global capitalistic growth paradigm is totally flawed

The chaotic turbulence is the result of the concept of infinite bigness this has been the destructive influence on all empires and now shown up by Feigenbaum numbers and Dunbar numbers for neural netwoirks

See Guy Lakeman Bubble Theory for more details on keeping systems within finite limited size working capacity containers (villages communities)

Clone of OVERSHOOT GROWTH INTO TURBULENCE

George J F Mackintosh

Mathematics Models