Triplets in math

[Mr Skeptic]

In math there is a lot of things that go in pairs, addition and subtraction, multiplication and division, real and imaginary, a function and its inverse, quite a few things that go in pairs. But I can't really think of any math structures that go in triplets. It should be possible since the idea of quark triplets should have a mathematical basis, but I can't think of any.

[Xittenn]

> = <

 

Couldn't one technically say that multiplication and division by zero does not exist or that it is in fact an operator onto itself? Similarly so for addition and subtraction ....

[TonyMcC]

I don't know whether you would call this a triplet but it is very common for "natural" formulae encountered in physics and all branches of engineering to be a mixture of whole numbers, squares and/or square roots (never cubes or cube roots). The only exception that I can think of are those concerning 3 phase electricity which I consider "man made" rather than "natural". You will now probably get a list of formulae that don't follow this observation! lol.

Come to think of it, I don't know of any formula that uses all three parameters at once so perhaps any such formula can only be considered a further example of a doublet.

Edited January 27, 2011 by TonyMcC

[Xittenn]

none, real, imaginary

[imatfaal]

numerical term, power of x term, derivative of x term

 

you can also have a certain concept that generates one single and one three fold state from a set of four. I am vaguely remembering this happening with two particles' spin - but the exact circumstances escape me.

 

SU(3) works on gluon/quark colours by reducing from 9 combinations to 8 because one of the superpositions is a seperate singlet state that does not exist - thus there are 8 gluons not the 9 which one might expect. I am sure there is a parallel ie a good example of a 4 output solution being reduced by one to make a triplet solution and a single solution.

Edited January 27, 2011 by imatfaal

[michel123456]

Maybe like in catalysis, 2 substances react with the help of a third one (catalyst).

[the tree]

Positive, negative and imaginary on a Poincaire half plane?

 

Vector triple products?

 

Scalar triple products?

[Mr Skeptic]

Positive, negative and imaginary on a Poincaire half plane?

 

Wouldn't this also have negative imaginary making it a quadruplet, or would there be a good reason to exclude that possibility?

 

Vector triple products?

 

Scalar triple products?

 

That's pretty good. These can all be composed by multiple binary operations though. Would that necessarily be the case?

 

numerical term, power of x term, derivative of x term

 

you can also have a certain concept that generates one single and one three fold state from a set of four. I am vaguely remembering this happening with two particles' spin - but the exact circumstances escape me.

 

SU(3) works on gluon/quark colours by reducing from 9 combinations to 8 because one of the superpositions is a seperate singlet state that does not exist - thus there are 8 gluons not the 9 which one might expect. I am sure there is a parallel ie a good example of a 4 output solution being reduced by one to make a triplet solution and a single solution.

 

Is this what results in the triplet of quarks, or of the three "colors" of quarks? I'm kind of curious how that comes about.

[Xittenn]

Wouldn't this also have negative imaginary making it a quadruplet, or would there be a good reason to exclude that possibility?

 

 

 

That's pretty good. These can all be composed by multiple binary operations though. Would that necessarily be the case?

 

 

 

Is this what results in the triplet of quarks, or of the three "colors" of quarks? I'm kind of curious how that comes about.

 

My reply was avoided and I feel your replies to the other replies are the same as mine in nature. Any and all discussions of set theory include NULL. I didn't say that there are no real solutions to addition of NULL, or imaginary ones for that matter, just that it is neither positive or negative and represents a solution onto itself.

 

I'm curious as to what answer the OP was seeking precisely? Is there a certain elegance required here? I do understand what the OP is trying to achieve in assigning distinct operators to quark triplets but is there some set of logic that you could define in how one would assess the validity of a response?

 

This is an honest question to you and the question you have asked is one I am asking myself routinely ....

[the tree]

Positive, negative and imaginary on a Poincaire half plane?

Wouldn't this also have negative imaginary making it a quadruplet, or would there be a good reason to exclude that possibility?

The Half Plane doesn't include the real axis or anything below it, in the same way that the Poincaré disk doesn't include the edge or anything outside of it.

[Mr Skeptic]

My reply was avoided and I feel your replies to the other replies are the same as mine in nature. Any and all discussions of set theory include NULL. I didn't say that there are no real solutions to addition of NULL, or imaginary ones for that matter, just that it is neither positive or negative and represents a solution onto itself.

 

You're right, somehow I forgot that zero was neither positive nor negative. So zero goes between positive and negative among the reals, integers, rationals, etc, so all of those might be considered triplets. On the other hand, I know that some people consider there to be two zeros, positive zero and negative zero. It pretty much only matters when taking a limit.

 

I'm curious as to what answer the OP was seeking precisely? Is there a certain elegance required here? I do understand what the OP is trying to achieve in assigning distinct operators to quark triplets but is there some set of logic that you could define in how one would assess the validity of a response?

 

This is an honest question to you and the question you have asked is one I am asking myself routinely ....

 

Well I'm kind of curious for a mathematical analogue to quarks, but also as to whether there is something that is fundamentally a triplet in math (whether or not quarks actually follow that).

 

The Half Plane doesn't include the real axis or anything below it, in the same way that the Poincaré disk doesn't include the edge or anything outside of it.

 

Oops. I thought that was something else. Thanks for the link.

[Xittenn]

 

Well I'm kind of curious for a mathematical analogue to quarks, but also as to whether there is something that is fundamentally a triplet in math (whether or not quarks actually follow that).

 

 

 

And I'm kind of curious as to what you would define as fundamental!

[Mr Skeptic]

And I'm kind of curious as to what you would define as fundamental!

 

Me too!

[Xittenn]

I mean division and multiplication are abbreviations for addition and subtraction right?

 

I started with Null and got the Null Set of which the Power Set is comprised of Null and the Null set which is the single set of two Null elements or Null itself but there is still one, two or more elements that can be deferred to. And so we have the natural numbers right? I think there was more, something about all intersections of two sets who subset another set when intersected can always be evaluated if not to Null. With the set of Natural numbers comes the operator of addition(+) and the integers(-) soon follow ...

 

Could we say that to be fundamental one must superset a subset in some formal manner wherein the subset cannot reach? This is to say we can not formally extend the set of fundamentals by drawing deeper into nothing. I'm sure you or Tree have a much nicer way of saying all of this and this is why I asked the question! It's really hard to define an answer that has no defined requisites to begin with. But I'm sure when a truly elegant statement is made with regard all will be understood.

 

Forgive me if I'm being verbose, I am not a very good conversationalist!

[ajb]

In math there is a lot of things that go in pairs...

 

Thinking a little more generally, the number "two" appears important. Lots if classical objects are of "order two". Let me list some well-known ones: Metrics and inner products, composition of maps, Lie brackets, symplectic forms, Poisson structures, commutators, Laplacians, curvature two forms...

 

Part of my work is in the area of "higher structures", which sits part in mathematics and part in physics. The general idea is that analogues of classical "binary" structures exist and are useful. One nice way in which some of these thing arise in in homotopy theory, where one takes a classical structure and relaxes the definitions/properties to hold only up to homotopy. (Motivation on physics for these comes in "cohomological physics", for instance the BRST formulation of gauge theories.)

 

Great examples of this are the Loo-algebras (L-infinity), which we can think of as Lie algebras in which one relaxes the Jacobi identity up to homotopy, or in more conventional language up to something closed. This is then controlled by terms that themselves are controlled by higher consistent homotopies, and so on.

 

A more basic way to think of them is as Lie algebras with not just one binary bracket, but a whole series of n-order (n>0 or sometimes including zero) brackets that satisfy a higher order generalisation of the Jacobi identity.

 

There are plenty of other nice examples of higher algebraic and geometric objects. One I have worked on myself is higher Poisson and Schouten structures.

 

The idea of higher structures is maybe 20 years old, and is still an emerging field of study.

[TonyMcC]

In electronics there is (at least) one formula that uses 3 different kinds of numbers. That is the formula for resonance. It uses a rational number (2), an irrational number (pi) and a number which may or may not be rational (square root). Since the universe may be oscillating, things our size may be oscillating and the electron may be considered a sort of oscillating wave function this formula may have some special significance.

[michel123456]

I thought the OP question was not only about numbers or entities, but about operation.

 

In math there is a lot of things that go in pairs, addition and subtraction, multiplication and division, real and imaginary, a function and its inverse, quite a few things that go in pairs. But I can't really think of any math structures that go in triplets. It should be possible since the idea of quark triplets should have a mathematical basis, but I can't think of any.

 

An usual operation like addition goes like this:

 

A+B=C

 

the reverse is

 

C-B=A

 

What would be the form of a triplet operation?

 

Something like

 

A@B#C=D ?

 

Where @ and # are the operational factors that both must be used in order to make a result. (like in catalysis) (or like in chess, the horse moves 2 steps ahead and one step aside)

 

???

Edited January 28, 2011 by michel123456

[the tree]

For a ternary (yes, real terms do exist, people) operation, the typical notation would typically be something along the lines of f(x,y,z).

[michel123456]

For a ternary (yes, real terms do exist, people) operation, the typical notation would typically be something along the lines of f(x,y,z).

 

Thank you. I learned something today.

 

See arity:

 

"Arities greater than 2 are seldom encountered in mathematics, except in specialized areas, and arities greater than 3 are seldom encountered in theoretical computer science. In computer programming there is often a syntactical distinction between operators and functions; syntactical operators usually have arity 0, 1 or 2." from the wikipedia article.

 

-------------------------------

After some deeper thoughts:

 

An operation on the X axis is binary, an operation on the XY axis is ternary, an operation on XYZ axis is quaternary. Is that correct?

Edited January 29, 2011 by michel123456

[ajb]

Thank you. I learned something today.

 

See arity:

 

"Arities greater than 2 are seldom encountered in mathematics, except in specialized areas, and arities greater than 3 are seldom encountered in theoretical computer science. In computer programming there is often a syntactical distinction between operators and functions; syntactical operators usually have arity 0, 1 or 2." from the wikipedia article.

 

Arbitrary up to infinity arity things exits. For example, the Loo-algebras I have mentioned, or the other strong homotopy algebras Coo and Aoo, which are the homotopy relatives of commutative and associative algebras. They are encountered in physics, for example string field theory.

[the tree]

Though don't two algebras of two different arities essentially have the same properties assuming they both have an arity that is somewhere between two and infinity? So whilst they exist, the ones that have been studied are more 0,1,2,3,many,infinity than 0,1,2,3,4....?

[ajb]

Though don't two algebras of two different arities essentially have the same properties assuming they both have an arity that is somewhere between two and infinity? So whilst they exist, the ones that have been studied are more 0,1,2,3,many,infinity than 0,1,2,3,4....?

 

If I understand your question correctly then the answer is yes.

 

You can think of infinity algebras as a "series of n-arity operations" with some rules. Usually you can do everything very formally and consider thing up to infinity.

 

For example, we could could consider an Lie 3-algebra, as a one bracket, two bracket and a three bracket together with some rules generalising the Jacobi identity. Or we would consider any Lie n-algebra with n finite.

 

Doing it all quite formally and considering infinite arity seems to work OK and covers these examples. I have asked Jim Stasheff about this and he said he knows no example where considering infinite fails. You may worry about "convergence" properties of an "infinite number of things going on", if you are I always say just consider n finite! However, all this is usually very formal and we don't worry.

[khaled]

In mathematical logic, we have a tertiary operator: ( C ? A : B )

 

where C is a condition, A and B are statements

 

the operator works like this: if C is TRUE do A, if C is FALSE do B

 

this operator is used in programming languages: C\C++

 

example:

int A = 10 * 2 ;

int B = 40 / 2 ;

char R [8] = ( A == B ? "TRUE" : "FALSE" );

printf(" ( %d == %d ) = %s ", A, B, R );

 

output:

( 20 == 20 ) = TRUE

Edited January 30, 2011 by khaled

[Shadow]

Probably a stupid question, but have you looked at the "number three" article on wiki?

 

Also khaled, just at a glance you're missing the "address-of" operators that should precede the variable names in printf. Not that it makes any difference.

[khaled]

Probably a stupid question, but have you looked at the "number three" article on wiki?

 

Also khaled, just at a glance you're missing the "address-of" operators that should precede the variable names in printf. Not that it makes any difference.

 

I haven't looked any article about "number three" ...

 

in C printf you do not need to give "address-of" to variables, that is needed only when using scanf

 

int A = 0 ;

scanf ( "%d", &A ) ;

printf ( " A = %d ", A ) ;