Hierarchical Reserves II

Andre Willers

7 Feb 2013

Synopsis :

Hierarchies create reserves . Optimal Reserves are known . Thus, optimal hierarchies can be calculated .

 

Discussion :

Please disregard the previous version of this post . That version of the author has been re-educated.

 

1.Some nifty footwork is required .

2.Hierarchies :

These are defined as a first approximation as Sierpinsky multi-dimensional triangles . (Tetrahedrons)

I use Tetrahedrons because 3-dimensions are the minimum-necessary sufficient to describe any projections of higher dimensions .

Also because it is what we observe in our Universe . Tetrahedrons are stable .

Most old hierarchical organizations have evolved into tetrahedrons if you look at their organograms . Sets of three interlocking influence centers .

 

 

See :

http://en.wikipedia.org/wiki/Dominance_hierarchy

http://en.wikipedia.org/wiki/Sierpinski_triangle

http://en.wikipedia.org/wiki/Tetrahedron

 

 

3. The Area of the Sierpinsky tetrahedron is defined as

Area=L^2 * (3)^0.5

Area has the surprising property of being independent of the degree of iteration of the fractal

Which is why we use it .

3.1The Model :

The Area A describes number of executive administrators controlling the Volume , described as the administree's .

This has some delicious consequences .

3.1.2 You can administer any number of people with the same number of Civil Servants .

See http://en.wikipedia.org/wiki/Imperial_Civil_Service where it was actually done . Chinese Mandarins also approached these levels .

A far cry from today .

 

3.1.3The model is fractal :

The Administrators have a similar structure . You can then calculate the exact number of entry-level managers (L) to give unlimited managerial capability .

A= 1263 This is the optimal Civil Service , for any number of people .

L= ( 1263/(3)^(1/2) )

= 27

Augustus used a similar system when he set up the Roman Empire , especially iro the military commands .

So did the initial Chinese Emperors . And the Incas .

 

I chose 27 because this is the minimum-sufficient number of dimensions to describe a full Arith System .

See "The inside of Zero" in Appendix Omega (right at the end .)

Notice that the Indian Imperial civil Service limped along with slightly less , which led to their demise .

 

3.1.4 An organization , any organization , with more than 1263 executives will fragment .

 

3.2 That was the easy part . In real life , it did "nearly" happen that way . Usually remembered as Golden Ages .

Governments were small . Individual Freedom was equally small if you were not in Administration .(Usually forgotten)

For the system to work , orders had to be followed . The only free reserves tended to accumulate outside the reach of the Administration .

By definition . When this reaches 2/3 of the systems reserves , a revolution occurs . The system has insufficient reserves to withstand a sustained attack . If management is smart , they merge (like England and Netherlands in 1688 AD)

3.3 The reach of administration is when things get sticky .In a Small-World network , that is when there are more than 6 steps . See Appendix I .

The system tries to extend it's reach by increasing the Sierpinsky sets . But , remember the Hausdorff !

All they do is control more-and-more of less-and-less . This is what the Hausdorff-index means . The system collapses eventually from complexification (as per Tainter)

Or , the off-book economy tootles off to do it's own thing . (Jared Diamond Collapse)

Does the Small-world network apply in the case of social media like Facebook ?

 

The critical point would be : Who is left out ?

If more than 2/3 are left out , it is unstable .

1/3 to 2/3 : some problems .

Less than 1/3 : it would be like literacy . Very hard to hide really large reserves .

 

3.4 The Trick :

Make the edges of the Sierpinsky Tetrahedrons fuzzy to Degree 6 (See Appendix I) . This immediately gives the reach of supervision and the degrees of freedom outside supervision . Note that the top levels watch each other incessantly (many duplications) . Goes with the volume .

This is not at all like the classical Sierpinsky model . Everybody watches everybody else and generates new sierpinsky fractals .

Things go to a hell-basket very quickly . As the 6-limit is exceeded , communication degrades . The left-hand does not know what the right-hand is doing . A classical case was the 9/11 attack on the US , where the different agencies had the whole picture , but could not put it together in time .

3.4.1 I don't know what the Hausdorff dimension would be , but it won't be less than 2

3.4.2 That means there would be an optimal A/V for a certain fractal iteration .

But where ?

3.4.3 " I have ways of tickling your Sierpinsky tetrahedrons that you have not even imagined !" Would James Bond fall for this line ?

3.4.4 In time honoured tradition , look at the bits that the normal bits don't reach .

3.4.5 In the model , look at the black tetrahedrons . Compensate for the 6 degrees reach , calculate the volume and compare to total pyramid volume .

 

4 The Volume V(n) = L^3 /(6*(2)^0.5) *(1/(2)^n)) of sierpinsky tetrahedrals

Where n is the fractal iteration . In other words , our hierarchical level without fuzziness . Now add fuzziness to level 6 .

4.1 V(n)= (L+12)^3 /(6*(2)^0.5) *(1/(2)^n)) an approximation . L is extended on both sides by 6 , giving 12 .

4.2 The Breakpoint :

A/V(n) =L^2/(L+12)^3 * 14.696933 *(2^n)

4.2.1Setting A/V =1/3 = n^2/(n+12)^3 * 14.696933 *(2^n)

Gives n(1)= 3.0371 or -10.4917 see http://www.wolframalpha.com/input/?i=x%5E2%2F%28x%2B12%29%5E3*2%5Ex-0.0222680469%3D0

4.2.2 Setting A/V= 2/3 , gives n(2)= 3.66985 or -10.8782

4.3 What does this mean ?

As usual , the problem lies in the interpretation . The system is not linear .

The 2/3 are not double the 1/3 levels .

N(2)/n(1) = 1.20834 or 1.0368386

I used 2/3 , because the system has not been defined to be biased . It does not know which reserves are which .

The above means that the system is biased by (0.5- 0.20834) =0.29166 or (0.5-0.0368386)=0.46316

4.3.1 :What does this mean ?

Between n= 3.0371 and n= 3.6685 , your civilization flourishes .Else it is boom-bust .

I did not expect the optimal level to be so low . This is Parent-child-Grandchild + fractal level .

Will facebook and other social media change this ?

A calculation left for the reader .(hint –Small-world model )

4.3.2 What bothers me more is that negative solution of -10.xxx

What on earth does this mean ?

Looking at the graph on wolfram , it seems to indicate below n=-10 levels , reserves build up rapidly . In other words , intelligence .

Between n= -10 and n = 3 , we have unicellular organisms . The genes are still there (note the very large number of genes in simple organisms . )

 

All the need is some devolution to get n<10 and some really interesting things will start happening . I won't say Lovecraft if you won't .

At the fractal interfaces of n , you should be able to engineer nutritious jellyfish and their ilk .

But I can intuit that , if there were old intelligences that didn't mind waiting a few billion years , no human containment will work .

But a bargain can be struck . Because (A+~A<Universum ) , a knowledge trade can always be made .

4.3.3 Will that work with a virus ? Has anybody ever tried ? Maybe if you devolve the virus to levels of n=-10 . Make the virus smart, then conclude a treaty with n>3 entities . The mind boggles .

4.3.4 . Can viruses be smarter ? Looking at HIV and the lot , and how humans have to jump handsprings to explain their conduct in mechanistic terms , maybe a bit of devolution is called for to get interpreter for n<10 organisms .

5. How to create a world-beating organization :

5.1 No more than 1263 executives , with no more than 27 entry-level posts .

5.2.The organizational levels must lie between n= 3.0371 or 3.66985 using sierpinsky fractals .

5.3 Leave the negative n levels for the experts or the suicidal . Read Carles Stross or HP Lovecraft .

 

6. An interesting aside:

You can tweak unicellullars to produce high-energy proteins by regressing their evolutionary systems to n<-10 .

Nice jellyfish steak , anyone ?

 

To summate :

Cthulhu would have to be declared a protected species . Otherwise he would end up as sushi .

 

Regards

Andre .

 

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Appendix I

 

Six degrees of separation . Derivation .

Andre Willers

27 Jan 2013

Synopsis :

We derive Reliability Percentages for Degrees of Separation from first principles of Small World Networks . Six Degrees has at most 98.85% reliability .

 

Discussion :

See Para 15 below for a real surprise .

 

1.Small-world Networks :

http://en.wikipedia.org/wiki/Small-world_network

"A small-world network is a type of mathematical graph in which most nodes are not neighbors of one another, but most nodes can be reached from every other by a small number of hops or steps. Also known as degrees of separation ."

 

The major principle is that not all points(nodes) are interconnected . The nodes are clustered , with clusters connected by long connections

2.Examine an existing Small-world network in 2 dimensions . Sketch it on a piece of paper (2 dimensional) . The connections are shown as lines .

3.Define a Start node . The sender of the original message .

4.Then connect the Start Node with every other node , counting the number connections as Hops . This is computationally possible (not Hard) because not all nodes are connected .

5.Now , rewrite the Network Map in terms of Hops , Hops being the radius units from the Start node .

This gives a map of circular bands around the Start node as origin . The bands include all the nodes outside the Start node , with many duplicates .

The duplicates are important . Cluster n-hops are in them .

This is basically a Riemann-orbital system . See Appendix H .

6. The Trick !

We minimize the system by squeezing any wandering test path on the system to return to the Starter Node .

In effect , Riemann's theorem . We know that it returns to the origin if the real exponent = -1/2 .

(The only-if part exp=-1/2 is evident , but the theoreticians and red-pencil brigade have too much fun nit-picking)

7. The upperbound :

We can then calculate the Upperbound and Reliability percentage :

The actual Zeta = Sigma(x(i) / i^(1/2) , i=1 , 2, 3 ….

Where -1<= x(i) <=+1 at Beth(0) random for any i .

For the upper boundary , we set x(i) = 1 .

This gives :

Zeta (upper) = 1+(1/2)^0.5 + (1/3)^0.5 + (1/4)^0.5 + …

The Sum(n) = 1+(1/2)^0.5 + (1/3)^0.5 + (1/4)^0.5 + …+(1/n)^0.5 , the sum of Zeta(upper) to the n'th term . This is increasing .

The Diff(n) = (1/n)^0.5 – (1/(n+1))^0.5 is the difference between successive terms .This is decreasing

The Ratio(n) = Diff(n) / Sum(n) and is decreasing .

The Percentage Reliability PR(n) = (1-Ratio(n)) x 100

8.Table of Upperbound Reliability percentages for minimized Small-world Networks :

N denotes number of degrees (hops) in PR(n) below

PR(1) = 70.71%

PR(2) = 92.40%

PR(3) = 93.97%

PR(4) = 97.02%

PR(5) = 98.47%

PR(6) = 98.89%

PR(7) = 99.11%

 

9.What does this mean ?

See http://en.wikipedia.org/wiki/Six_degrees_of_separation .

It means "you pays your money and you takes your pick "

You get the most bang for your buck at about n=6 .

Small-world networks are very efficient at distributing information .

Remember , this is a very general derivation .

 

10 .What if just one node is left out ?

I suspect that the above will still hold .

It is a topological suspicion (the worst kind) . Even one little puncture causes a major shift in topological classifications .

In the class of "a little bit pregnant"

This has major consequences in the real world .

Plan on that there is always somebody on the "need-to-know" list being left out . This might be you .

 

11. What if many nodes are left out ?

Surprisingly , the system will default to 70% efficiency . Civil Service level . But remember that this is only the upperbound .

But is this then the Lowerbound for PR(2) and better ? I Think so , but some deeper analysis is called for .

No wonder civilization manages to struggle along .

Notice how close it is to 67% of Appendix H Infinite Probes .

 

12.The Family Paradox :

"My spouse does not understand me ."

Of course they don't . You are at PR(1) = 70% reliability as long as you communicate directly (eg language) . Just about bottommed out .

Add some more nodes . Usually this is children , grandparents , friends , hobbies , interests .

But different levels like emails , notes , flowers , presents , etc will work .

Even saving the Dodo .

Join the Dodo Liberation Army .

 

13. This is equally applicable to your work environment .

 

14. A surprising application .

How to turn your personal network into a Small-world network .

(Then you can use the system as described above )

14.1 Get long-range contacts . Pen-friends . Internet friends (but be careful) .

Even one will change the topology of your network .

You don't have to frantically network .

14.2 Virtual Topologies :

An imaginary friend .

https://www.facebook.com/MyImaginaryFriendLLC

These are available .

Notice the interesting addition of Internet at the bottom of Maslow's hierarchy of needs .

A Facebook account for an imaginary friend with strict access control will bounce parental communication with a child from 70% to 90% .

 

15.Marital Problems :

Or even between couples who have communication problems . A virtual friend on Facebook where only the two partners have access will increase communication by a whopping 20% !

What a surprise !

Not exactly what I was expecting .

 

15. Upgrading to higher dimensional networks .

Project the many hops onto a two-dimensional space as above . Then crumple them . See Appendix C .

This collapses into the same Riemannian space as discussed above

 

16. Virtual Topologies :

 

 

Viva La Dodo Liberation Army !

Andre

 

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Appendix H

Infinite Probes 2

Andre Willers

30 April 2008

 

http://andreswhy.blogspot.com "Infinite Probes'

 

From discussing this with various recipients , there seems to be a need for a simpler explanation . I thought I had explained it in the simplest fashion possible . The subject matter is inherently complex .

 

But , here goes .

 

How much must you save ?

If you save too little , a random fluctuation can wipe you out .

If you save too much , you lose opportunity costs . If you are in competition , this loss can be enough to lose the competition (ie you die)

 

Intuitively , you can realize there is an optimum level of saving .

 

Methods exist of calculating this optimum in very specific instances (ie portfolios of shares ,eg Kelly criteria , or tactics in war eg MiniMax ) .

 

The General Case

We need to hold a Reserve in case Something goes wrong . But we do not know what thing goes wrong .

 

Infinite Probes tries to answer the general case . What is really , really surprising is that a answer is possible .

 

The Infinite bit comes from using the mathematical expansion of the Definition of Eulers Constant e = ( 1 + 1/2! + 1/ 3! + … 1/n! + …)

Where n!= n*(n-1)*(n-2)*(n-3)*…*(1)

 

This approaches a constant , widely used in mathematics and physics .

(e = 2.718…) .

 

All we need is a system that can be subdivided indefinitely (to infinity) .

 

First , we divide by 1

Then 2

Then 3

Then 4

 

And so forth till infinity .

 

What is important is not that we do know what these divisions are , only that they are possible . We also do not know which one element goes wrong .

 

The other critical insight is that it is the relation between elements that is important . (Permutations) (The failure of an element in total isolation cannot affect the whole system by definition .)

 

We can count the number of relationships where there is failure of one element .

It is n! , where n is number of divisions where only one failure .

 

Multiple failures are handled by summing :

 

Our Reserve(R ) is divided by n to infinity and summed .

 

TotReserve= R*( 1 + 1/2! + 1/ 3! + … 1/n! + …)

TotReserve= R * e

 

To find the boundaries of our Reserve , we set TotReserve = Cost

 

Then

 

R = 1/e * Cost

R ~ 0.37 * Cost

 

What does this mean?

This method measures the upper boundary of the reserve needed to survive failures in any element of the Cost-Universe . Ie , internal fluctuations .

 

This is the surprising bit . Any society that keeps at least 37% reserves , can only be destroyed by something outside it's envelope . It is internally stable , no matter what .

 

Empires like the Ancient Egyptians , Romans , Chinese are possible , as long as there is no climactic fluctuation , new inventions , diseases ,etc . Rare events . Hence the technological stasis of old civilizations . The two are synonymous .

 

This is true at any scale (except quantal , by definition.) .

 

Individuals too . Humans can be seen as empires of noospheres .

 

The upper boundary does not take any double-counting into account . It is true for any system whatsoever .

 

A truly remarkable precise result from such general axioms .

 

The Lower Boundaries .

This is where it gets interesting .

Remember , we are just counting the number of ways in which permutations of one element can fail . We then sum them to get the effect of the failures of other elements

 

The easiest is the business that just starts and is not selling anything . It fails on n elements on every term . It's floor capital must then be

R=Cost/(1+e)

R=0.27*Cost

 

This is the initial reserve to get off the ground .

This is true in any ecosystem . This is why it is so difficult to start a new business , or why a new species cannot succeed . Or why waves of pandemics are scarcer .

 

For the epidemically minded , this 10% difference is responsible for the demise of the Black Death ( smallpox outcompeted bubonic plaque variants for the CR5 access site.Ironically , the reason why we have only a limited HIV plague is the high competition for this site , probably some flu vectors . As one would expect , the incidence of HIV then becomes inversely proportional to connectivity (ie flights) .

 

A cessation of airplane flights will then lead to a flare-up of diseases like these .

Not exactly what anybody has in mind . )

 

When we find that we really need the spread of infectious vectors to stay healthy , then we know we have really screwed ourselves .

 

These are the two main boundaries .

The literature is full of other limits the series can approach . Keep a clear head on what the physical significance is .

 

Fat

I cannot leave the subject without the thing closest to human hearts : appearance .

Fat and fitness .

Sadly , the present fad for leanness is just that . The period of superfluous food is coming to an end .

Rich individuals can afford to be lean because the reserves are in the monetary wealth Women have to bear children individually , so they cannot store the needed reserves externally . Hence their fat storage is close to the theoretical optimum even in Western societies (33%) . In other societies the percentage is about 37-40% .

 

Human males have been bred (Mk III humans) for muscle and little fat (8% in a superbly fit male) . He does not have reserves to withstand even garrison duty (even little diseases will lay him low .) Note the frequent references to diseases laying whole armies low .

 

Note what is left out : the camp-followers . They survived The women and babyfat children . Every army seeded the invaded area with women and children .

 

The bred soldier has to eat a high-carb food frequently : not meat or fat , his body cannot store it . This is the definition of a wheat-eating legionary .

 

Ho ! Ho! Ho!

The Atkinson diet .

No wonder it does not make sense in evolutionary terms .

Mesomorphic humans have been bred not to transform expensive proteins and fats into bodymass .

 

The soldier-class were kept on a carbohydrate leash , which could only be supplied by farming .

 

 

The Smell of Horses .

 

Horses exude pheromones that promote body-leanness in humans . This has an obvious advantage to horses . Horses are breeding jockeys .

 

The time-span is enough : at least 8 000 years . (400 generations)

 

Because pregnant women cannot ride horses , there was a selection pressure to breed horses who have a pheromone that block female dominance pheromones , especially since females have to weigh more because of fat-reserve considerations .

 

Outside a farming environment , horses will sculpt their riders as much as the riders are sculpting them .

 

Small Mongolian ponies , small Mongolians .

 

This is why alpha-males like horses and horse-dominated societies were able to conquer and keep matriarchies .

 

Note the effect of the pheromones on women riders . Androgeny .

On males it becomes extreme blockage of oestrogen . It seems like a surge of male hormones , but it is just an imbalance . (If too much male hormones , the men just kill their horses )

 

This is why the auto-mobile had such a big sociological effect . No horses , so the men became more effeminate .

 

Want to be Lean and Mean ?

Sniff Horse sweat pheromone .

Perfumiers take note .

 

Dogs

 

The other leg of the human-horse-dog triumvirate .

Dogs accept female pack-leaders and have evolutionary reasons for blocking horse inhibitions of human female pheromones .

 

While the males are away , the females look after and rely on the dogs .

(The reason why Mongols ride from yurt to yurt: they are too scared of the dogs.)

 

With dogs around , the male testosterone activity is ameliorated . This is a well known effect , especially if horses are around .

 

Hence the female love of lap-dogs . They are actually quite ferocious , and exude large amounts of pheromones that soothes the savage male breast .

 

Your attention is drawn to the Pekinese lapdog , which has had a disproportionately large effect on human history .

 

If this sounds convoluted , it is because this is exactly how this type of bio-system operates : by inhibitions of inhibitions of inhibitions ,etc .

 

Andre

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Appendix IV A

Orders of Randomness 2

Andre Willers

15 Aug 2008

 

See http://andreswhy.blogspot.com : "Orders of Randomness"

 

I have been requested to expand a little on orders of Randomness and what it means .

Please note that human endeavours at this date use only randomness of the order of flipping a coin ( Beth(0) )

 

Aleph is the first letter of the Hebrew Alphabet . It was used by Cantor to denote

Classes of Infinity (ie Aleph(0) for Rational numbers , Aleph(1) for Irrational Numbers , etc

 

Beth is the second letter of the Hebrew Alfabet . It means "House"

 

I will first repeat the derivation of Orders of Randomness from http://andreswhy.blogspot.com : "Orders of Randomness" because it is so important .

 

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Start Quote:

First , simple Randomness .

Flip of a coin .

Heads or Tails . 0 or 1

Flip an unbiased coin an infinite number of times ,write it down below each other and do it again .

All possible 0 and 1's

 

An example : Beth(0)

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Flips(1) 0,1,1,1,1,… etc

Flips(2) 0,1,1,1,0,… etc

.

Flips(infinity) 0,0,0,0,0,0,…etc

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This describes all possible states in a delineated binary universe .

"delineated binary" means a two sided coin which cannot land on it's side .

 

Now draw a diagonal line from the top left of Flips(1) to Flips(infinity) .

At every intersection of this diagonal line with a horizontal line , change the value .

The Diagonal Line of (0,1)'s is then not in the collection of all possible random

Horizontal coin-Flips(x) .

 

This means the Diagonal Line is of a stronger order of randomness .

This is also the standard proof of an Irrational Number .

 

This is the standard proof of aleph numbers .

Irrational numbers ,etc

Since any number can be written in binary (0,1) , we can infer that the order of randomness is the same as aleph numbers .

 

This means we can use number theory in Randomness systems .

Very important .

 

Google Cantor (or Kantor)

 

Define coin-flip Randomness as Beth(0) , analogous to Aleph(0)

Then we have at least Beth(1) , randomness an order stronger than flipping a coin .

Then we can theorize Beth(Omega) <->Aleph(Omega) .

 

End Quote

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Cardinal Numbers .

 

The cardinal number is the index x of Aleph(x) .

Cantor proved that

Aleph(n+1) = 2 ^ Aleph( n )

 

Where n is the cardinal number of the infinity .

 

Tying them together :

He also proved that

P(A) = 2^ n

Where A is any set , P(A) is the PowerSet of A and n is the cardinal number of set A

Thus , Cardinal Number of P(A) =(n+1)

 

The PowerSet of A = the Set of all subsets of A .

This sounds fancy , but it is simply all the different ways you can combine the elements of set A . All the ways you can chop up A .

You can see it easily in a finite binomial expansion (1+1)^n = P(A) = 2^n

 

See http://andreswhy.blogspot.com : "Infinite Probes"

There we also chop and dice , using infinite series .

 

Can you see how it all ties together ?

 

Why 2 ?

 

This derives from the Delineation Axiom . Remember , we can only talk about something if it is distinct and identifiable from something else . This gives a minimum of 2 states : part or non-part .

 

That is why the Zeta-function below is described on a 2-dimensional plane , or pesky problems like Primes always boil down to 2 dimensions of some sort .

 

This is why the irrational numbers play such an important part in physics .

Z=a+ib describes a 2-dimensional plane useful for delineated systems without feedback systems

 

Its in the axiom of Delineation , dummy .

 

But we know that Russell proved that A+~A<Universum .

The difference can be described as the Beth sequences . Since they are derivatives of summation-sequences(see below) , they define arrows usually seen as the time-arrows .

 

These need not to be described a-la-dunne's serial time , as different Beth levels address the problem adequately without multiplying hypotheses .

 

 

 

Self-referencing systems and Beth sequences .

 

A Proper Self-referencing system is of one cardinal Beth number higher than the system it derives from .

Self-referencing systems (feedback systems) can always be described as sequences of Beth systems . Ie as Beth(x) <-> Beth(y) . The formal proof is a bit long for inclusion here .

 

The easiest way to see it is in Bayesian systems . If Beth(x) systems are included , Bayesian systems become orders of magnitude more effective .

 

Life , civilization and markets are such . See below .

 

Conservation Laws :

By definition , these can always be written in a form of

SomeExpression = 0

 

Random (Beth(0) Walk in Euclidean 2-dimensions

 

This is a powerful unifying principle derived from the Delineation Axiom .

 

In Random Walk the Distance from the Center is = d * (n)^0.5 . This is a property of Euclidean systems .

(Where d = step , n=number of random beth(0) steps)

 

Immediately we can say that the only hope of the Walker returning to the center after an infinity of Beth(0) steps is if d ~ 1/(n)^0.5 . This is the Riemann Hypothesis .

 

Now , see a Universum of 2-dimensional descriptors z=a+ib

 

Sum all of them . Add together all the possible things that can be thus described .

 

This can be done as follows :

From z=a+ib Raise both sides to the e

e^(z) = e^(a) . e^i(b)

Raise both sides to the ln(j) power where j is real integers.

j^(z) = j^(a) . e^(b/ln(j))

 

Now , sum them :

Zeta=Sum of j^(z) for j=1 to infinity

 

Now we extract all possible statements that embody some Conservation Law . Beth(1)

 

This means that Zeta is zero for the set of extracted statements if and only if (b/ln(j)) is of the order of Beth(0) and a=(-1/2)

 

Tensors .

The above is a definition of a tensor for a discontinous function .

 

Riemann's Zeta function.

This can describe any delineated system .

If Zeta = 0 , conservation laws apply .

 

Zeta = Sigma(1/j )^z for j=1,2,3,…,infinity and z=a+ib , where z is complex and i =(-1)^0.5

The z bit is in two dimensions as discussed above .

 

This function has a deep underlying meaning for infinite systems .

If you unpack the Right-Hand side on a x-yi plane you get a graph that looks like a random walk .

 

If every point is visited that a random walk would visit over infinity (ie all) , without clumping , then Zeta can only be non-trivially zero if a=(-1/2) .

 

Why (x – yi) plane ? See "Why 2 " above . The system is fractal . Two dimensions are necessary in any delineated system .

 

Remember , randomwalk distance from origin = step*sqrt(number of steps) .

So if the steps = 1/ ( sqrt(number of steps) ) , then the Origin might be reached if and only if a= -1/2

This is easily proven .

 

If a= - 1/2 , then b can be any real function . This would include Beth(0) and Beth(1) , but not higher orders of beth .

 

If a= -1/2 and b is an unreal number , then a cannot be equal to -1/2 anymore . Conservation cannot hold at any level .

 

Consequences:

Conservation Laws can only hold for Beth(0) and Beth(1) systems .

 

This is forced by the two dimensions of delineation .

 

Mathematically , this means that Beth(2+) systems of feedbacks can only be described in terms of attractors or/and fractal systems (ie not in isolation)

 

Physically , conservation of energy and momentum need not hold for Beth(2+) systems .

 

This has an interesting corollary in decryption (unpacking) . A Beth(2) mind unpacking Beth(0) or Beth(1) encryption is functionally equivalent to Non-Conservation of Energy

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Appendix B

For Blast

Blast from the Past

Andre Willers

13 Jan 2012

 

Synopsis:

High intensities of gamma- and X-ray radiation on Earth from the center of our galaxy can be expected during 2013 AD , due to an infalling gas cloud into the central Milky Way black hole .

 

Discussion :

The discovery was made by Stefan Gillesson of the Max Planck Institute for Extraterrestrial Physics

in Garching , Germany .

References :

1. "Nature" , DOI 10.1038/nature10652
   
2. "New Scientist" 17 Dec 2011 p16 "Cloud suicide could transform black hole"
   

 

A gas cloud of an estimated three earth masses is expected to impact the supermassive , rotating black hole at the center of our galaxy ("Sagittarius A*") in 2013 AD . Note that radiation from this event can be expected shortly afterward in local time .

 

A large part of this mass will be converted to electro-magnetic energy .

 

Now , three earth masses is not a lot of energy in the galactic scheme of things , but the way it is distributed does .

 

Briefly , the energy release is characterized by very short wavelenghts (gamma or x-ray) and lobe-shaped distributions of the pulse-wavefronts in the galactic ecliptic .

 

There is a combination of relativistic and quantum effects in the last few cm's before the event horizon .

 

1.There is a slingshot effect , because the gas cloud must be lumpy , even if only at atomic scale . The gas cloud vanishes in a few Planck -time units , emitting very short-wave radiation due to differential tidal friction .

 

2.But the black hole is large and rotating , which forms time-bands of slower time , which gives enough time for feedback . Depending on the size and rotation speed of the black hole , this will lead to 2n lobes of wavefronts on the rotation equator , where n=1,2,3,...

 

3.Hawking Burps .

This process in para 2 above "foams" the event horizon , leading to a drastically increased Hawking radiation . (The "foam" would be something like Sierpinsky chaotic triangles)

The density and time-band concentrations of energy separates particle-anti-particle pairs to give Hawking radiation .

 

This burp of energy sweeps matter from the neighborhood of the black hole and reduces the mass of the black hole , preventing black holes from swallowing everything . Elegant .

 

4.Turning a Hawking Burp into a White Hole .

There is a narrow window where the Burp becomes self-sustaining . A large black hole then evaporates in an awesome release of energy . Even the tiniest random fluctuation (eg matter-antimatter) will get magnified and perpetuated .

Not the place to take your mother for a picnic .

 

5.Foaming Space-time .

Well , space-time is already foamed , but only to a simple level . (Quantum foam)

Call it Foam(0)

We can churn this to higher orders of foam (Foam(1) , Foam(2) , Foam(3) ,...,Foam(b) )

where b can be derived from Kantor's Aleph classes combined with chaos theory .

 

Taken to the logical extreme , this leaves only Foam ordered into Branes .

 

The smile on the Cheshire cat just after he brushed his teeth .

 

So what does this mean to us ?

A full frontal impact from a lobe will give a definitive answer to Fermi's Paradox : "Where are they ?" Why , extinct .

 

More likely is an immersion in the higher radiation levels alongside the lobes for a long time .

Can be survived , given enough warning .

Apply to the Max Planck Institute for Extraterrestrial Physics for more .

 

Friday the thirteenth can be quite a gas .

 

Andre

 

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Appendix C

Crumpling Paper and Space-Time

Andre Willers

23 Feb 2012

 

"The moving finger writes , and having writ , crumples it in random ruins."

With apologies to Omar Khayyam .

 

Synopsis:

Crumpled paper gives a good approximation of spacetime as a membrane with clumpy masses .

"Empty" spaces not occupied by the membrane gives an impression of dark matter .

We derive an expression to give this ratio using Infinite Descent and Beth(0) Random Walk .

 

Discussion :

1.The Crumpled paper :

Consider a paper disk of radius r and thickness d .

It's volume is then Vp=pi * r^2 * d

Draw a line from the center to the edge , in steps of length d , over the edge , then back to the center Let nu=r/d , a measure of the thickness of the paper . Note that it is a pure number .

 

The number of steps in the line is then n0=(2r/d)+1

But the number of steps to the edge of the original paper disk is n1=r/d=(n0-1)/2

r=d*(n0-1)/2

n0=2*nu+1

 

Vp=(pi*d^3*(n0-1)^2 )/4

 

Crumple it up in a way that is as random as flipping a coin (ie Beth(0) )

 

The Trick : The line we have drawn up above breaks up into random vectors by rotating through a third dimension = crumpling into a ball .

 

We thus have a continuous line of random steps of known number of steps .

In 3 dimensions , the mean square distance from the center then is known

R = d * (n0)^0.5 …. See true for all dimensions as long as all are of Beth(0) order of randomness.

 

 

Volume of crumpled ball Vb=4/3*pi*R^3

The Ratio Vb/Vp = mu then gives the ratio of crumpled ball space to volume of paper mass .

 

Mu={4/3*pi*d^3 *n0^(3/2)} / (pi*d^3*(n0-1)^2 )/4

Notice the d^3 term and pi cancels out . This has profound physical implications .

This simplifies to

 

Mu=4*4/3*(n0^3/2/(n0-1)^2)

Expressed as thickness of paper , nu , which is a pure number independent of metric chosen .

mu=4*4/3(2*nu+1)^3/2 / (2*nu)^2

mu=4/3*(2nu+1)^3/2 / nu^2

This gives a quartic equation in nu , which can be solved exactly algebraically .

(mu)^2*(nu)*4 – 2^7/3^2 *(nu)^3 – 2^6/3^2
(nu)^2 – 2^5/3^2 * (nu)^1 - 2^4/3^2 =0

 

 

Test it on A4 paper:

A4 paper has thickness d~0,1 mm and r~150 mm

nu=150/0,1

nu=1500

mu=4/3*(3001^3/2)/(1500^2)

mu=0.097421589

mu= 1- 0.90257841

This means that the crumpled A4 paper ball encloses about 90% empty space .

This agrees with experimental results . See NewScientist.

 

Note that the force applied does not matter . As long as the paper is untorn , mu will be the same .

 

How many times can it be folded ?

Solving the above (see below) gives mu=1 for about nu=14.7 to 14.8 .

This means there are no empty spaces left to fold into .

 

This can get complicated , so I will keep it simple .

Take a piece of paper and fold it . You then have a new piece of paper .The test-circle of same r will have double the thickness .

Ie , nu will double .

Between 7 and 8 folds , nu will hit the ceiling of mu=1 , regardless of the starting value of nu .

This is the maximum number of paper folds , as confirmed from other sources .

 

Physical interpretations :

Take an m-dimensional space . Randomness of order Beth(0) applies equally to all . The underlying equalizer . Collapse it to three dimensions and let the third one approach single Planck lengths .

Then we can use the above paper approximation . Notice how d cancels out except for an addition of 1 in final ratio .

 

What does it mean ?

See the physical universe as a brane (ie sheet of paper) in a multiverse . Crumpling it means it has mass and singularities . Both are aspects of the same thing .

An estimate of the number of singularities can be made from edges and points in crumpled paper .

 

Can we crumple the paper to a ball that is just paper ?

That is a particle .

The answer is "Yes" .

 

Such crumpling means that mu=1 (no empty space in any dimension )

This gives an quartic equation in nu that solves to four values , other dimensions than three denoted by i=(-1)^0.5

 

See http://www.1728.org/quartic.htm for a calculator

nu1= 14.722181 (this makes the physical particle universe possible . Mass .

Nu2= - 0.004167 + i*0.49558 (Rotation :Spin :charge and magnetism)

nu3 = - 0.004167 - i*0.49558 (Rotation :Spin :charge and magnetism) notice the minus sign .

Nu4= - 0.49164542 (quantum effects as the particles dither. Inertia?)

 

What does a negative nu mean ?

nu=r/d . A negative nu means one of r or d must be negative .

1.If r is negative , it can be interpreted as curled up dimensions , inside the "outside" dimensions as defined by i . See http:andreswhy.blogspot.com " The inside of zero" Aug 2009

2.If d is negative , it can be interpreted as quantum effects . A particle does not "occupy" all the space . Likes hopscotch .

3.But notice the the two are interrelated .The notorious observer effect . Where we place the minus sign between r or d .

 

There should be relationships between nu2 , nu3 and nu4 . Various rotations between macro- and micro dimensions .

This means the contraption is not symmetrical But we already know that ,

 

Physical constants :

Things like charge , mass , etc should be derivable from these basics . Hint:use lots of crumpled paper .

 

There is hope . The fact that it is quartic equation , which is always solvable , means that the Universe can be understood . Complicated and perverse , but as long as you stick to Beth(0) randomness , it can be understood . For higher orders of randomness , good luck .

 

Dark Matter :

I nearly forgot . Using Planck units , we can define the ratio of thickness of the brane as

nu=c*PlanckTime/(1*Planck Time)

nu=c = 3*10^8

This gives a

Mu=4/3*(2c+1)^3/2 / c^2

Simplifying (c is very large) . This gives the approximation

mu=4/3* 2^1.5 / c^0.5

mu=2.1773242 * 10^ (-4)

mu = 1-0.999783357

This means that 99.9783357 % of the universe can be interpreted as being "Dark Matter".

Ie with attractive and repulsive qualities . Basically empty space .

May you have joy of that .

 

An interesting aside :Creative artists .

How many pieces of paper does an artist need to crumple up and throw away before he finds something acceptable ?

Something acceptable would translate to mu=1 . Thus , we can say 7-8 truly random foldings should give a result .

The same holds for cryptanalysis or any attempt to find an unknown .

Algorithm :

Try 8 times , crumple , then put it aside and try again later .

There is a quantum connection , strange as it might seem .

 

And what about a nice little Crumpling App for smartphones ?

But the randomness should be from truly random tables , not pseudo-random generators .

 

Randomly yours.

Andre

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Appendix Omega

 

The Inside of Zero .

Andre Willers

7 Aug 2009

 

Synopsis :

A system of 13 Diophantine equations with 26 unknowns are the necessary sufficient to describe Arith I systems relative to Arith II system , with a Degree of Complexity = 10 .

These are used to describe a mathematical vacuum , with some physical consequences .

 

Discussion :

See Appendix A , B , Recursive Theory .

See previous posts , where Arith I and Arith II systems were discussed in detail .

 

The problem lies in discussing Non-Aristotelian systems using Aristotelian concepts of delineation (ie True , not-True ) .

 

Infinity .

The alert reader would have noticed that most of the problems come from processes continued indefinitely , which is taken as infinity . But is it ? Kantor already proved that varieties of infinity exists . It immediately follows that the software-computer we call mathematics and logic needs some revision .

The works of Russell , Godel , Matiyasevich et al pointed out some further contradictions in the Aristotelian model .

Can a theorem be true only for Aleph0 but not for Aleph1 ?

This is analogous to the problems with parallel lines continuing "infinitely" , that led to non-Euclidean geometries .

 

Recursive Genesis .

The standard axioms of arithmetic needs only a tweak on one axiom to generate the necessary revisions .

Generate new numbers by adding 1 to any number a .

 

Arith II

The Standard Set (call it Arith II) states that a+1<>a , where a is a previous number . The number line does not loop back on itself .

 

Arith I

The number line can loop back on itself . A circular number-line is formed . In a certain sense , we are discussing the topology of circular number loops in a Arith II space and their relationships .

 

The metric has not been defined . The question then becomes :

How many Arith I systems (= ArithI(m) ) plus one ArithII system (we only need one ArithII) are necessary sufficient to describe this particular Universum ?

 

Rotational Translations (spin) .

This is actually moving from one dimension to another , regardless of the frame of reference . Every ArithI system then actually needs a spin indicator : ie , which way it is curving in an (n-1) dimensional space .

I draw your attention to the curious fact that the angle in 2-dim is 2pi , while in 3 dim it is 4pi . More of this anon .

 

The Degree :

The maximum exponent in an equation if you change all the variables into one variable . This is important because it indicates the number of dimensions we have to use to describe the equation . Do not confuse it with the number of variables .

 

Minimum Necessary Sufficient .

The Ball-Breaker . The description defines reality .

 

This has been called many things :

Principle of Least effort , time , distance ,

Entropy .

Occam's Razor .

Collapse of the wave-function .

Economy of effort , etc .

 

The trade-off :

Matiyasevich et al has shown that there is a relationship between the Degree and the Number of variables necessary to describe an item in an Universum using a related number of equations .

 

Boundaries :

The following relationships has been proved :

Degree = 4 , variables 58

Degree = 10 , variables = 26 , equations =13

Degree = 10^45 , variables = 10

 

Is there a minimum number number of degrees ?

I doubt very much whether a Degree lower than 4 will be found . See Physical significance below .

See previous posts .

 

Physical Significance .

"Everything that can be , will be . But not all at once ." AW

The Degree can be described as the number of dimensions . You will notice the correlation with string theory .

Sadly , a Theory of Everything is impossible . But we can creep up on it .

 

Delicious !

Degree = 10 , variables = 26 , equations =13 , Spin =2

The numbers 26 =2x13 , and spin =2 should be knocking at the jaded doors of your mind .

 

Cards .

A pack of 13x4 = 52 cards forms a very good analogue of the Mathematical Process of a Universum .

You can work out for yourself why humans have a good use for a very good analogue of the universe .

And the Jokers ?

Remember , the Joker can take on any value . A good decription of a trans-luminal , low-probability event .

The most popular string theory uses 10 dimensions .

 

And the rest of the Tarot pack ?

Remember , we are talking about necessary sufficient without straining human capabilities too much .

 

Prime Numbers :

A prime number is simply an ArithI system (in ArithII measurements) that cannot be chopped up .

A mathematical atom , relative to ArithII . The number we need is related to the number of variables .

It is like zero

 

The Inside of Zero .

Degree = 10 , variables = 26 , equations =13 , Spin =2

If we plug in 26 prime numbers into the Diophantine polynomial generational equation in AppendixA below (and there are an infinity to choose from) , we get 26 ArithI systems , which have a mathematical vacuum inside them . No numbers .

A very interesting place . Note that the resultant is also a prime atom . It is recursive . Only the spin remains free .

Like the inside of a singularity .

 

Physically , this will have some very interesting effects .

There are no quantum fluctuations inside zero . The metric does not exist , even at Planck level .

 

Super-conductivity :

Purely an effect of the number of atoms crowded together .

It does not matter which atoms . They just have to be in certain configurations . Hence the present confusion in the field .

 

Disintegration of matter

(cold-fusion or cold-fission) .

But observational systems really like conservation laws . Energy release can then be only through particle or EM means .

If the geometries are chosen correctly , we can constrain the output mainly to electron/proton or electron/EM .

Direct electrical energy from matter . Very good power generation in our Universum .

 

Quantum Epigenetics .

The patterns on the surface of zero are constrained by trans-luminal effects inside zero . The outside patterns dictate the quantum-fluctations , as well as trans-luminal and super-luminal effects from all over .

 

The spin of Zero will thus drag creation of quantum fluctuations around it . This will affect things not only on a small scale , but on a large scale as well . The Drags do not balance out .(cf Relativistic rotation drag)

This can actually easily be calculated in the standard way by wave functions and General Relativity .

 

Rotating around a point

Note that there is a difference between spin and a particle rotating about center .

This can be constrained by using the fact that angular radians in 2 dimensions is 2pi and in 3 dimensions is 4pi .

Physically , in our descriptions , it means the particle does not really know whether it is orbiting in 2 dimensions or is spread over a surface in 3 dimensions (cf h/2pi)) , but we can constrain the geometries (and do in our quantum devices !)

 

God's sense of humour .

Degree = 10 , variables = 26 , equations =13 , Spin =2

Each degree (ie dimension) can take on +1, 0, -1 spins . Thus 10^3 number of states .

(We do not worry about minimum necessary sufficient spins , only state what is .)

This gives a polynomial of 27 integers of degree 10 with a value of 3 spins . See Appendix A below .

 

The Fine structure constant of our universe is

1/alpha = h/2pi * c / e^2

=137.035 999 070 (98)

where h is Planck's Constant , c is lightspeed in vacuo (see above) and e is electron charge , all in dimensionless electrostatic units . The value is dimensionless (ie the same for any definition of units)

It shows the relationship between h (Plancks constant , which includes the definition of mass) , spin (the pi , but there has to be compensation for dimensional drifting between dim2 and dim3 as discussed above) , observational speed (c ) and electric charge (e) .

It means that spinning mass and charge are intimately related to the number of dimensions it has to rotate through .

So , it is no surprise to find that

Beta = (1/10 + 1/27) * 10^3

=1000*(0.1 + 0.037037037…)

= 137 . 037 037 …

The difference in the sixth decimal can be attributed to drag effects and dimensional compensations , which have not been taken into account .

 

Biological Epigenetics .

The same type of argument can be applied to biological cells and denizens of multicellular organism . While they might not rotate , they definitely do partially rotate to-and-fro .

 

Three magnetic fields at right angles to each other or twistor-EM waves will have definite biological effects .

Do not try this at home .

 

Does nothing matter ?

The Zero knows .

 

Andre .

 

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Appendix A

 

From http://mathworld.wolfram/com/PrimeDiophantineEquations.html

From http://en.wikipedia.org/wiki/Formula_for_Primes

Formula based on a system of Diophantine equations

A system of 14 Diophantine equations in 26 variables can be used to obtain a Diophantine representation of the set of all primes. Jones et al. (1976) proved that a given number k + 2 is prime if and only if the following system of 14 Diophantine equations has a solution in the natural numbers:

α₀ = wz + h + j − q = 0

α₁ = (gk + 2g + k + 1)(h + j) + h − z = 0

α₂ = 16(k + 1)³(k + 2)(n + 1)² + 1 − f² = 0

α₃ = 2n + p + q + z − e = 0

α₄ = e³(e + 2)(a + 1)² + 1 − o² = 0

α₅ = (a² − 1)y² + 1 − x² = 0

α₆ = 16r²y⁴(a² − 1) + 1 − u² = 0

α₇ = n + l + v − y = 0

α₈ = (a² − 1)l² + 1 − m² = 0

α₉ = ai + k + 1 − l − i = 0

α₁₀ = ((a + u²(u² − a))² − 1)(n + 4dy)² + 1 − (x + cu)² = 0

α₁₁ = p + l(a − n − 1) + b(2an + 2a − n² − 2n − 2) − m = 0

α₁₂ = q + y(a − p − 1) + s(2ap + 2a − p² − 2p − 2) − x = 0

α₁₃ = z + pl(a − p) + t(2ap − p² − 1) − pm = 0

The 14 equations α₀, …, α₁₃ can be used to produce a prime-generating polynomial inequality in 26 variables:

ie: PrimeNumber = (k+2) ( 1- a0^2 - … a13^2) )

This is equal to the polynomial

(k + 2)(1 −

[wz + h + j − q]² −

[(gk + 2g + k + 1)(h + j) + h − z]² −

[16(k + 1)³(k + 2)(n + 1)² + 1 − f²]² −

[2n + p + q + z − e]² −

[e³(e + 2)(a + 1)² + 1 − o²]² −

[(a² − 1)y² + 1 − x²]² −

[16r²y⁴(a² − 1) + 1 − u²]² −

[n + l + v − y]² −

[(a² − 1)l² + 1 − m²]² −

[ai + k + 1 − l − i]² −

[((a + u²(u² − a))² − 1)(n + 4dy)² + 1 − (x + cu)²]² −

[p + l(a − n − 1) + b(2an + 2a − n² − 2n − 2) − m]² −

[q + y(a − p − 1) + s(2ap + 2a − p² − 2p − 2) − x]² −

[z + pl(a − p) + t(2ap − p² − 1) − pm]²)

> 0

is a polynomial inequality in 26 variables, and the set of prime numbers is identical to the set of positive values taken on by this polynomial inequality as the variables a, b, …, z range over the nonnegative integers.

In other words , we have a single Diophantine polynomial equation with 27 variables based on 14 sub-equations .

Eliminating one variable (n) as discussed above , leaves us with 26 variables based on 13 equations , but the Exponential Order (Degree) is unchanged .

A general theorem of Matiyasevich says that if a set is defined by a system of Diophantine equations, it can also be defined by a system of Diophantine equations in only 9 variables. Hence, there is a prime-generating polynomial as above with only 10 variables. However, its degree is large (in the order of 10⁴⁵). On the other hand, there also exists such a set of equations of degree only 4, but in 58 variables (Jones 1982). Jones et al 1976 , Riesel 1994 p40

Appendix B

Diophantine set

From Wikipedia, the free encyclopedia

Jump to: navigation, search

In mathematics, a Diophantine set of j -tuples of integers is a set
S for which there is some polynomial with integer coefficients

f(n₁, ..., n_j, x₁, ..., x_k)

such that a tuple

(n₁, ..., n_j)

of integers is in S if and only if there exist some (non-negative) ^[1] integers

x₁, ..., x_k with

f(n₁, ..., n_j, x₁, ..., x_k) = 0.

Such a polynomial equation over the integers is called a Diophantine equation. In other words, a Diophantine set is a set of the form

where f is a polynomial function with integer coefficients. ^[2]

Matiyasevich's theorem, published in 1970, states that a set of integers is Diophantine if and only if it is recursively enumerable. A set S is recursively enumerable precisely if there is an algorithm that, when given an integer, eventually halts if that input is a member of S and otherwise runs forever. This means that the concept of general Diophantine set, apparently belonging to number theory, can be taken rather in logical or recursion-theoretic terms. This is far from obvious, however, and represented the culmination of some decades of work.

Matiyasevich's theorem effectively settled Hilbert's tenth problem. It implies that Hilbert's tenth problem is unsolvable. This problem is the challenge to find a general algorithm which can decide whether a given system of Diophantine equations has a solution among the integers. David Hilbert posed the problem in his celebrated list, from his 1900 address to the International Congress of Mathematicians.

Contents [hide] 1 Examples 2 Matiyasevich's theorem 2.1 Proof technique 3 Application to Hilbert's Tenth problem 3.1 Logical structure 3.2 Refinements 4 Further applications 5 Footnotes 6 References 7 External links

[edit] Examples

The well known Pell equation

X^2 – d(y +1)^2 = +- 1

is an example of a Diophantine equation with a parameter. As has long been known, the equation has a solution in the unknowns x,y precisely when the parameter d is 0 or not a perfect square. In the present context, one says that this equation provides a Diophantine definition of the set

{0,2,3,5,6,7,8,10,...}

consisting of 0 and the natural numbers that are not perfect squares. Other examples of Diophantine definitions are as follows:

• The equation a = (2x + 3)y defines the set of numbers that are not powers of 2.
  
• The equation a = (x + 2)(y + 2) defines the set of numbers that are not prime numbers.
  
• The equation a + x = b defines the set of pairs (a,b) such that (a<=b)
  

[edit] Matiyasevich's theorem

Matiyasevich's theorem says:

Every recursively enumerable set is Diophantine.

A set S of integers is recursively enumerable if there is an algorithm that behaves as follows: When given as input an integer n, if n is a member of S, then the algorithm eventually halts; otherwise it runs forever. That is equivalent to saying there is an algorithm that runs forever and lists the members of S. A set S is Diophantine precisely if there is some polynomial with integer coefficients f(n, x₁, ..., x_k) such that an integer n is in S if and only if there exist some integers x₁, ..., x_k such that f(n, x₁, ..., x_k) = 0.

It is not hard to see that every Diophantine set is recursively enumerable: consider a Diophantine equation f(n, x₁, ..., x_k) = 0. Now we make an algorithm which simply tries all possible values for n, x₁, ..., x_k, in the increasing order of the sum of their absolute values, and prints n every time f(n, x₁, ..., x_k) = 0. This algorithm will obviously run forever and will list exactly the n for which f(n, x₁, ..., x_k) = 0 has a solution in x₁, ..., x_k.

[edit] Proof technique

Yuri Matiyasevich utilized an ingenious trick involving Fibonacci numbers in order to show that solutions to Diophantine equations may grow exponentially. Earlier work by Julia Robinson, Martin Davis and Hilary Putnam had shown that this suffices to show that every recursively enumerable set is Diophantine.

[edit] Application to Hilbert's Tenth problem

Hilbert's tenth problem asks for a general algorithm deciding the solvability of Diophantine equations. The conjunction of Matiyasevich's theorem with a result discovered in the 1930s implies that a solution to Hilbert's tenth problem is impossible. The result discovered in the 1930s by several logicians can be stated by saying that some recursively enumerable sets are non-recursive. In this context, a set S of integers is called "recursive" if there is an algorithm that, when given as input an integer n, returns as output a correct yes-or-no answer to the question of whether n is a member of S. It follows that there are Diophantine equations which cannot be solved by any algorithm.

[edit] Logical structure

Here an argument taking exactly the form of an Aristotelian syllogism is of interest:

(Major premise): Some recursively enumerable sets are non-recursive.

(Minor premise): All recursively enumerable sets are Diophantine.

(Conclusion): Therefore some Diophantine sets are non-recursive.

The conclusion entails that Hilbert's 10th problem cannot be solved. The most difficult part of the argument is the proof of the minor premise, i.e. Matiyasevich's theorem, which itself is much stronger than the unsolvability of the Tenth Problem.

[edit] Refinements

Later work has shown that the question of solvability of a Diophantine equation is undecidable even if the equation only has 9 natural number variables (Matiyasevich, 1977) or 11 integer variables (Zhi Wei Sun, 1992).

[edit] Further applications

Matiyasevich's theorem has since been used to prove that many problems from calculus and differential equations are unsolvable.

One can also derive the following stronger form of Gödel's first incompleteness theorem from Matiyasevich's result:

Corresponding to any given consistent axiomatization of number theory,^[3] one can explicitly construct a Diophantine equation which has no solutions, but such that this fact cannot be proved within the given axiomatization.

[edit] Footnotes

1. ^ The two definitions are equivalent. This can be proved using Lagrange's four-square theorem.
   
2. ^ Note that one can also use a simultaneous system of Diophantine equations to define a Diophantine set, because the system
   

f1 =0 , …,fk =0

is equivalent to the single equation

f1^2 + f2^2 + … + fk^ = 0

1. ^ More precisely, given a -formula representing the set of Gödel numbers of sentences which recursively axiomatize a consistent
   theory extending Robinson arithmetic.
   

[edit] References

• Yuri Matiyasevich. "Enumerable sets are Diophantine." Doklady Akademii Nauk SSSR, 191, pp. 279-282, 1970. English translation in Soviet Mathematics. Doklady, vol. 11, no. 2, 1970.
  
• M. Davis. "Hilbert's Tenth Problem is Unsolvable." American Mathematical Monthly 80, pp. 233-269, 1973.
  
• Yuri Matiyasevich. Hilbert's 10th Problem Foreword by Martin Davis and Hilary Putnam, The MIT Press. ISBN 0-262-13295-8
  
• Zhi-Wei Sun, Reduction of unknowns in Diophantine representations, Sci. China Ser. A, 35:3 (1992), pp. 257–269.
  

[edit] External links

• Matiyasevich theorem on Scholarpedia.
  

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Categories: Diophantine equations | Hilbert's problems

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