for some . By Nakayama’s Lemma, there is some

with $\latex aI^n=0$. Thus there is a with nonzero

but and so has a nonzero annihilator. ]]>

My challenge (which is maybe a bit silly) is: is there a completely elementary proof of the fact that any proper ideal in a finite commutative unital ring has non-zero annihilator? It looks like something that should be completely obvious …

]]>of finite local rings is a special case of the theorem that

any Artinian ring is a finite direct product of Artinian local rings.

This is a a standard theorem: it is Theorem 8.7 in Atiyah and

Macdonald’s Introduction to Commutative Algebra. ]]>

We have to come to terms with the fact that a small but disproportionately powerful sector of society gets what it desires — and what it desires knows no bounds — by painting and propagating a false picture of reality and getting the rest of society to act on that picture.

]]>When you ask if the connecting lines are “there”, are you talking about lines drawn with a tool of some sort, or idealized pure math lines that are 1 point wide? If the former then there are some purely mechanical difficulties in completing your cover of the positive half of the real line. If the latter then in what way are they there? I think we would need to set up some kind existence definitions in order to be sure we are talking about the same thing…

]]>“The process [basis construction] will terminate if n^2(L) < 2^j(L) -1 since at level L there are only n^2(L) linearly independent matrices available (and not all non-singular)."

The vector space of bit strings of length J forms a Vector Space spanned by a basis of independent unit vectors {2^0,2^1,2^2,…2^(J-1)} and thus we can guarantee the existence of J independent square binary matrices of dimensions JxJ whose eigenvectors are these unit vectors. These can be found by un-diagonalizing the unit matrix as I mentioned in a previous post. So the claim is completely false.

The reason the construction stops is that there are not enough bits in 256 x 256 to create a 2^127 -1 bit string; ie: 256×256 bits =2^16 bits << 2^127 – 1 bits. The key understanding is that an independent basis of size k requires strings of at least k bits. So I can now shout Hallelujah! The particular Combinatorial Hierarchy in question does stop naturally which is somewhat useful to the particular theory of Noyes et al. So Noyes, Bastin and Amson's papers are correct where they rely on the basis construction stopping at the fourth level but not for the reason they give in that particular paper and others too numerous to mention.

We can see that we could continue the construction if we used a square matrix of at least 2^127-1 bits to construct the fifth level for example but this represents a different Combinatorial Hierarchy in which the number of bits is chosen to exactly cover the maximal DCss (strict subset) at that level. Thus the sequence of bit lengths would be 2,3,7,127, 2^127 – 1, 2^(2^127 – 1) – 1,… rather than 2,4,16,256, stop.

There seems to be no a priori reason to prefer one sequence to another and indeed we see the same crucial number 137=3+7+127=(2^2-1)+(2^3-1)+(2^7-1) embedded quite naturally in the alternative sequence which does not terminate. Since the probability calculations are not affected by the particular bit length chosen to represent the vectors (provided they are long enough to represent the basis vectors) but only the cardinal size of the basis set, these two hierarchies give the same probability calculations if we ignore the smaller terms from the second; but as the second does not stop I can only conclude that there are an unlimited number of structural coupling constants and thus forces but that they are so weak they have as yet remained undetected [the next would be 2^(2^127 – 1)]. These of course would provide an explanation and calculation for Dark Matter/Energy if the theory is a correct model for quantum physics and provide corrections to the coupling constants already calculated by Rhodes, Noyes, Bastin, Amson et al. The failure of the Combinatorial Hierarchy to stop may not be crucial at all but instead, explanatory!

]]>“Does mathematics combine the mystical and the rational, as no other subject can? Maybe.

“Or maybe not. I had a student in Oxford who read Mathematics and Philosophy because he had read Robert M. Pirsig’s Zen and the Art of Motorcycle Maintenance and decided that he would find the answers to the ultimate questions in the region between mathematics and philosophy. What he found, of course, was a very substantial dose of logic!”

I hope that you do not mind this question…

Draw a horizontal line from zero towards infinity. Then draw lines from numbers below one to their reciprocals above one. Obviously, these lines connecting reciprocals cannot be distinguished from either the original positive number line nor the connecting lines. But those connecting lines are nonetheless there, yes? If so, without losing the reciprocal connections, to see those connections pivot a last portion of the original line vertically up off of the horizontal such that a first portion is going from zero to one is still horizontal, while a last portion is now vertically going from one towards infinity. If you do so, is there be any connecting line vertically up from zero or vertically down from infinity? I do not think so, and I am hoping you confirm that. I am also hoping you can tell me what area of mathematics might pertain to my observations that there is no connecting line whose diagonal becomes vertical no matter how close to infinity someone draws a connection from a point closer and closer to zero and to its reciprocal; plus this second observation: the distance of the top of a connector at infinity to non-connector line vertically above zero is dependent on the place of the pivot.

Warmest regards

P.S. Per the above unconventional metaphor for the positive real numbers, here might be where something religious for some mathematicians interested in religion might connect with something mathematical: if you approach 1 horizontally through the proper fractions from 0 at 1, you reach 1 as wholeness. Whereas, if you go vertically down from infinity through decreasing multiplicities, you reach 1 as oneness. Thus, wholeness and oneness are both the same and different in some kind of mathematical way, yes? As is 1 as unity, the union of wholeness and oneness. Sir, while I doubt any of this is of any great mathematical significance, I think it a way of using mathematics to highlight terms found in religion which are rarely seen as also mathematical. But I may be all wet, which is why I am reaching out to you for you unfettered response.

]]>I contribute to the free software Sage, where we build plenty of data. I implemented a *lot* of known constructions of designs, in the hope that every design that is known to exist (given a set of parameters) will eventually be obtainable by a simple command. Similarly, we try these days to build all strongly regular graphs indexed in Andries Brouwer’s website in Sage. Nothing better than having an example on your computer to make sure that this or that object exists.

But Sage, like Magma or GAP, is no database though it can produce data. Installing any of them (and learning how it works) is not entirely trivial, and in my experience it does not replace something as simple as raw data. And we *must* find a way to index the mathematical databases.

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