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Introduction
#1
Hello.

I am a graduate student working toward my PhD in Physics. Aside from physics, I enjoy programming, camping, and parkour (though I don't have much time for the last 2 activities these days).

I admit that I come bearing an idea that will undoubtedly require guidance in the future. That being said, I am also trying to read as many related posts as I can on this forum so that I have a good idea of what progress has already been made in related areas. The problem with that is that I don't understand much of what I read on the forum. Is the language something that I should expect to pick up over time? Is the type of jargon used on the forum common to mathematicians, or only to tetration enthusiasts? Are there any specific references that you can recommend as starting points?

My idea involves a hyperoperator of which addition, multiplication, exponentiation, are cross-sections. While I am still chiseling at it, I am learning a lot as I go. At first, I was perfectly happy to make generalizations from the top down: Formulating conjectures by observing mathematical patterns and symmetries. But, since then, I have realized that conjectures are not adequate: Some of my conjectures, which seemed obvious from observation, were inconsistent with each other. The problem is that a conjecture can be wrong, and there is no particular reason that a conjecture should be true. Lately, I am been trying to take a more axiom-based approach. An axiom is true by definition, and I have realized that some of the conjectures I was coming up with (from mere observations of patterns) were actually already built in to the axioms I was using. I just hadn't figured out how to prove them yet. One of my current problems is evaluating my expressions: As I am learning, proving a relation between 2 hyperoperators does not necessarily mean that you can evaluate either of them. Around every turn, I find more questions than answers, and as my methods become more rigorous, the road becomes more treacherous. As I evolve, I gain new insights, but this means that my goal-posts are constantly moving. Each insight leads me to realize that I am further from my goal than I previously thought.

So far, this forum is the greatest resource I have found. There are mathematical concepts that, for the most part, are of interest to all members of this forum from what I can tell, and yet cannot be found anywhere else on the internet. There are conversations held here that are not held anywhere else. For people interested in certain concepts, there is nowhere else to go, it seems.

I am currently typing up an overview of my idea in LaTeX, so I will start a thread on that in a few weeks or so, depending on how much time I have to work on it. In my write-up, I am trying to be as clear and as thorough as possible, while also being as concise as possible. I would like to say that I am being as thorough as possible, but I would have to amend that I am only being as thorough as possible for a physicist. I hope that the progression of my document is not unbearable. More so, I hope that my current hyperoperator framework is thoughtful so as to be irrefutable (if that is possible). Another thing I would like to know is whether or not there are (among forum members) any accepted properties that a hyperoperator SHOULD have. For example, I read in more than one thread that

.
(and similar claims at this, and this. Why is a beautiful result?)

Where is this coming from? Is this an axiom or is it derivable?

In the meantime, I will try to read through and understand the threads on related topics. I would like to post a comment or question in each of these threads, if only to confirm that I read/understood it. I think that my biggest barrier to making use of this forum right now is understanding the language, as I mentioned before.
Some threads that seem related:
1) http://math.eretrandre.org/tetrationforu...r+operator
This sounds very similar to what I am aiming for, though I can't say I understand it yet.
2) http://math.eretrandre.org/tetrationforu...hp?tid=546
Ideally, the end product of my formulation is an actual plot of values for intermediate operations, such as the one at the above link.
3) http://math.eretrandre.org/tetrationforu...r+operator

Sorry for the long introduction. I look forward to interesting discussions on this forum in the future, and I look forward to better getting to know the community here.

hixidom
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#2
Hello hixidom -

(03/03/2014, 09:13 AM)hixidom Wrote: Hello.

I am a graduate student working toward my PhD in Physics. Aside from physics, I enjoy programming, camping, and parkour (though I don't have much time for the last 2 activities these days).

"Parkour" - is that this climbing through streets and garages et al about what my doughter has told me some years ago? Seems a pretty wild, nice hobby... :-)

(03/03/2014, 09:13 AM)hixidom Wrote: (...)
Sorry for the long introduction. I look forward to interesting discussions on this forum in the future, and I look forward to better getting to know the community here.

hixidom

I like your "long" introduction much. It allows to have an imagination of some person instead of an avatar. So for my part here, stay to be as explanative as shown here.
Unfortunately
1) there are not many threads here which do an explicite conclusion (for instance usable as compressed overview over the basic idea, its main techniques, main results (and main limits/pertainingproblems)) of the plentiness of here presented proposals
2) my own investigative power has reduced much in the few recent years so I'll have difficulties to even improve my own overviews/conclusions (which were always meant to serve as most-possible self-contained treatizes to invite and make the reader curious....)

In shorter words - as you go through the material here and scan the discussion and their (even if only partial) results: why not make notes in that process and help the tetration-forum later with summaries from these? (Just some vague idea... :-) )

All in all - welcome here and thanks for the introduction -

Gottfried
Gottfried Helms, Kassel
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#3
(03/03/2014, 09:13 AM)hixidom Wrote: Hello.

I am a graduate student working toward my PhD in Physics. Aside from physics, I enjoy programming, camping, and parkour (though I don't have much time for the last 2 activities these days).

I admit that I come bearing an idea that will undoubtedly require guidance in the future. That being said, I am also trying to read as many related posts as I can on this forum so that I have a good idea of what progress has already been made in related areas. The problem with that is that I don't understand much of what I read on the forum. Is the language something that I should expect to pick up over time? Is the type of jargon used on the forum common to mathematicians, or only to tetration enthusiasts? Are there any specific references that you can recommend as starting points?

In my opinion (I may be wrong) the main topic of this forum, the hyper-4 aka Tetration, has almost achieved his standard notation and terminology with the work of this forum even if maybe not the same holds for the hyperoperations families (sequences):

For what I know the main terminology about came from Goodstein and from the work on this forum and from works of K.A.Rubtsov and G.F.Romerio (Not sure on the order and the connections, I'm not good at history Wink ).

I did many researches and even if there is not an universal terminology and notation, because the study of hyperoperations is a not very common topic and only few authors work(ed) on it, thanks to this forum the terminology is getting a little bit more homogeneous (at lest for the standard hyperoperation family with the [n] notation).
Actually all these concepts can be easily translated in the language of "common mathematicians" since we are talking about differend kind of recursions, indexed families of binary functions and their extensions.

The things i think you should read are:

"NATURAL" HYPEROPERATIONS
-HYPEROPERATIONS http://en.wikipedia.org/wiki/Hyperoperation
-Ackermann Funtion http://en.wikipedia.org/wiki/Ackermann_function
-G.F.Romerio's Hyperoperations terminology http://math.eretrandre.org/tetrationforu...hp?aid=208
-Geisler's site http://tetration.org/Ackermann/index.html
-Trappmann,Robbins Tetration FAQ http://math.eretrandre.org/tetrationforu...hp?aid=189
-Rubtsov,Romerio's New arithmetical Operations http://www.rotarysaluzzo.it/filePDF/Iper...%20(1).pdf
-Trappmann's arborescent numbers: Higher Arithmetic operations http://opus.kobv.de/ubp/volltexte/2007/1...n_diss.pdf (here there is a detailed formalization of the different bracketing in hyperoperations)

these are the basis imho.

Quote:My idea involves a hyperoperator of which addition, multiplication, exponentiation, are cross-sections. While I am still chiseling at it, I am learning a lot as I go. At first, I was perfectly happy to make generalizations from the top down: Formulating conjectures by observing mathematical patterns and symmetries. But, since then, I have realized that conjectures are not adequate: Some of my conjectures, which seemed obvious from observation, were inconsistent with each other. The problem is that a conjecture can be wrong, and there is no particular reason that a conjecture should be true. Lately, I am been trying to take a more axiom-based approach. An axiom is true by definition, and I have realized that some of the conjectures I was coming up with (from mere observations of patterns) were actually already built in to the axioms I was using. I just hadn't figured out how to prove them yet. One of my current problems is evaluating my expressions: As I am learning, proving a relation between 2 hyperoperators does not necessarily mean that you can evaluate either of them. Around every turn, I find more questions than answers, and as my methods become more rigorous, the road becomes more treacherous. As I evolve, I gain new insights, but this means that my goal-posts are constantly moving. Each insight leads me to realize that I am further from my goal than I previously thought.
I'm glad to hear that because it happens to me too xD. Everytime seems i'm going to get the "unification" theory of hyperoperations I undestand that I've forgot something and "something" usually is another field of mathematic that I didn't know. The positive side that i'm closer to that unification now than some years ago.

Mathematics is a very big... Big and wonderfull landscape. with Big i mean beyond every known magnitude and cardinal number.

Quote:So far, this forum is the greatest resource I have found. There are mathematical concepts that, for the most part, are of interest to all members of this forum from what I can tell, and yet cannot be found anywhere else on the internet. There are conversations held here that are not held anywhere else. For people interested in certain concepts, there is nowhere else to go, it seems.

agree!

Quote: I hope that the progression of my document is not unbearable. More so, I hope that my current hyperoperator framework is thoughtful so as to be irrefutable (if that is possible). Another thing I would like to know is whether or not there are (among forum members) any accepted properties that a hyperoperator SHOULD have. For example, I read in more than one thread that
At the moment I'm working on a general theory of hyperoperations families and I came at some conclusions for the property that this class of families should have but I'll open a topic here when I'll be able to explain and discuss all my ideas about it.


Quote:In the meantime, I will try to read through and understand the threads on related topics. I would like to post a comment or question in each of these threads, if only to confirm that I read/understood it. I think that my biggest barrier to making use of this forum right now is understanding the language, as I mentioned before.

Was difficult for me too at the beginning, but if you train alot with the function notation, indexed family of functions, function composition and iteration will be easier for you to translat the notations... i hope.
MathStackExchange account:MphLee
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#4
Thanks for all of those wonderful resources MphLee and for the moral support. I am in awe of what you all have accomplished here.
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