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I asked a question yesterday regarding numbers defined on graph structures that I call graph numbers. I posted the algorithm I was using to define graph numbers, which are simply a natural extension of linear numbers we use all the time. Nolion's response was close but not quite right. The equations are actually:

$$d_{m+n}(v) = \biggl (d_{n}(v) + d_{m}(v) + \sum_{(u,v)\in{E}}{d_{{m+n},{carry over}}(u)} \biggr ) \mod {2}$$

$$d_{{m+n},carryover}(v) = \biggl \lfloor \frac {d_{n}(v) + d_{m}(v) + \sum_{(u,v)\in{E}}{d_{{m+n},{carry over}}(u)} } {2} \biggr \rfloor$$

After reading nolion's response, I was able to come up with these equations. Notice, they are added in $Z$, which is why I need the $\mod 2$ to bring them back to $Z/2Z$. However, the digits of m and n are chosen from $Z/2Z$ (just as in normal binary addition of more than 2 numbers). But the order of evaluation of $d_{m+n}$ matters (which, again, I am not sure how to describe mathematically) starting from the "root" nodes and evaluating as in a Breadth First Search. Notice, that if the Directed Acyclic Graph is a linear connection of nodes, this becomes normal arithmetic in base 2. My question is, has there been any work done on such numbers?

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    I just need to know if I can take this to my professor without looking like an idiot. Any help would be greatly appreciated. I am a Computer Science grad student, and if no work has been done in this area, I am considering doing my thesis on this.2011-02-19
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    Thanks for clarifying the question somewhat. In yesterday's question you also wrote that you defined multiplication on these numbers -- how is that defined?2011-02-19
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    That gets *really* complicated. For the subset of numbers that are isomorphic to the regular integers over addition, you can convert them to integers, multiply, and convert them back to these graph numbers. For those numbers that are not part of that subset, I have developed a process for specific types of graphs (square, cubic, etc..., and polyhedral, triangular, square, pentagonal, etc... and similar types of graphs), but again, I don't know how to describe it mathematically, because it requires "shifting" the graph.2011-02-19
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    Basically, any graph with a "fractal" structure, has a nice algorithm associated with multiplication. Multiplication can essentially be derived from repeated addition, in a way similar to that of normal linear numbers.2011-02-19
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    @Fred: If all you are worried about is looking bad for your professor, then what you need to do is try to define these things *clearly*. Even if it's been done before, you won't look like an idiot for putting forth an idea that is carefully thought out (not your fault if somebody else got there first). Whether or not you can do your thesis on this depends on just what these things are or what they are good for, especially if this is computer science. Theoretical structures may be well and good, but surely CS will require some *use* out of them. That's up to your advisor.2011-02-19
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    @Magidin: I am not really sure how one can ask if a number is useful or not. If it has never been examined before, and consistent, surely it *must* be useful, right? As for clearly defined ideas, I cannot clearly define the idea if I don't know of any mathematical notation to describe them (or if it does not exist). Addition is fairly simple. Multiplication and powers are not so simple. And on top of my mathematical handicap (because, again, I am not a mathematician; I do this in my spare time), I am a very busy student.2011-02-19
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    @Fred: Just because something has not been examined before but is consistent does not mean that it *must* be useful; of course, it does not mean that it *cannot* be useful. It just means it hasn't been done before, and as the films in the Jackass! series will show, that by itself does not necessarily mean it *must* be done. That said: bringing an honest query born of honest curiosity, leavened by hard work (which seems very much to be the case here) will **not** make you look like an idiot to your professor (and if (s)he says it does, then switch professors; (s)he's not worth your time).2011-02-19
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    @Magidin: I, respectfully, disagree with conclusions regarding novel mathematical concepts. Something that is truly novel, is necessarily useful by definition. (Your Jackass! analogy, while colorful, does not apply.) As for bringing the query to my professor, I agree with your conclusions on that. I probably should have done that 3 years ago, but I figured that mathematicians had surely already considered these basic concepts. It seems I was (possibly?) wrong. I would, however, like to make a suggestion that there be an outlet for people (like me) with ideas to share.2011-02-19
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    @Fred: I neither gave nor few any *conclusions*; in fact, what I said is that you **cannot** make conclusions as to the usefulness of something based *only* on the fact that it is new. Your (undisclosed) definition of "useful" is at odds with the way the vast majority of the world uses that word, and I'll wager it is at odds with the way 100% of Computer Science people use the word. And since you say you are in Computer Science, and are thinking about this for a dissertation, then it is the CS-notion of "useful" that is at issue. (cont...)2011-02-19
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    @Fred (cont..) If this were a Math thesis, the "originality" is the key issue (either original work, or work that casts an old idea in an original way that sheds new light into it, is the usual way to phrase it). In that situation, certainly "new" is a key concern. But even then, you would not call it "useful", you would call it "new". Plenty a mathematical thesis/paper/book lies collecting dust in university libraries, stuffed to the gills with truly novel concepts that have proven, up to now, utterly useless (except in providing their writers with a degree and a job, I guess).2011-02-19
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    @Magidin: I highly doubt that 100% of Computer Science people use the word in the same way. But the sense in which I am using the word "useful", would be for the exact same reasons that numbers are considered "useful". There are many thing that you can do with numbers that have no practical applications, but certainly, numbers are still useful.2011-02-19
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    @Fred: I'll wager that *none* of the many ways in which CS people use the word "useful" is as a synonym for "nobody has done it before", which is how you are proposing to use it. And numbers are considered useful not **because** of the many thing you can do that have no practical applications, but **in spite** of those, and because of all the ways that they **do** have practical applications. Just because the sky is blue doesn't mean that everything which is above your head is blue.2011-02-19
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    @Fred: But this is neither here nor there. Organize your ideas, and present them to your professor. (S)he'll be the one to tell you if they can make a thesis or not, and be in a better position to direct you to someone who can tell you if it is new or not (since you'll be far more able to explain yourself interactively in person than here). My only relevant advice is two-fold: (i) Don't worry about it having been done before or not when you present it to him; that's not going to be the issue; and (ii) don't take it personally if it *has*, or if he tells you the idea seems useless to him.2011-02-19
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    @Magidin: I guess we'll just have to agree to disagree on the "usefulness" issue. It's more of a philosophical point anyways.2011-02-19

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I had a nice, long answer written out that has been subsumed by all the comments above. Oh well.

The only part that isn't redundant is that for multiplication, you want to look at whether the set of endomorphisms of your DAG acts vertex transitively. Ideally, it would act simply vertex-transitively, because otherwise, you would have to include choices of endomorphisms explicitly in the structure. Anyway, that's the mathematical terminology for your "fractal structure".

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    I think I understand you. I have "shown" that for certain graphs, multiplication is transitive and vertex transitive on these graph numbers. It is also commutative, and distributive over addition.2011-02-19
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    Now, I am going to take (what some would consider) an enormous leap. I would say that these numbers (or, the lack thereof) are the reason why Fermat's Last Theorem and Fermat's Polygonal Number theorem are so difficult to solve. You can describe "cubic" or high powers with these numbers, but in a different form. You can also describe polygonal numbers. And Fermat was specifically investigating these types of numbers and claimed he had a proof for his (also considered Cauchy's) Polygonal Number Theorem. Now, I don't put much weight on this line of reasoning, but it does make some sense.2011-02-19
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    @Aubrey da Cunha: Can you point me toward the work that has been done for addition? Thanks.2011-02-20
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    I'm not sure exactly what work you mean. As far as I know, nobody has defined _exactly_ this structure before. Under appropriate assumptions about the underlying graph, you shouldn't have any trouble showing that these things form a [ring](http://en.wikipedia.org/wiki/Ring_%28mathematics%29), but math is awash with examples of rings. Arturo's comment about the usefulness of these rings boils down to an explanation of why anyone should care about _these_ rings. Maybe they have interesting computational properties?2011-02-20
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    @Fred I should add that I agree with Arturo's assessment: _do_ show this to your professor and don't sweat the response too much.2011-02-20
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    @Aubrey da Cunha: Thanks. I will definitely show this to my professor.2011-02-20