Dear Brian,
Many thanks; it's really good of you to be so considerate,
despite all your exam papers!
Replies embedded.
>>>> Take x and y to be integers. If we have a set P(m)
>>>> comprising the first m primes beneath a value z, and a
>>>> general subset J of of P(m), then I am constructing a
>>>> matrix M_1 in which the members of J index the rows.
>
> Do you really mean this, or do you mean that the only rows
> of M_1 whose elements are defined (or non-empty, if you
> think of M_1 as an array in a computer program, for
> instance) are the ones indexed by the members of J?
>
What I define is what I have given in that paragraph above; and anything
undefined I do not want to be part of my matrix M_1.
>>>> I am also constructing an array M_[x,y] comprising black
>>>> cells and white cells. The number of black cells in a
>>>> column is given by t(n,M_[x,y]). Will it be acceptable
>>>> in a mathematical paper to devise a form of notation
>>>> referencing an interval [x,y] in such a way that the
>>>> values of t(n,M_[x,y]) over [x,y] conform precisely to
>>>> the divisibility distribution that is exhibited in an
>>>> interval [x,y] M_1?
>
>>> I can't tell from this description just what you're trying
>>> to do.
>
>> Let a(J) be the product of all members of J.
>
>> I'm trying to construct an array in which the black cells
>> do not necessarily conform to the distribution of
>> occupied matrix components in the matrix M_1.
>
> What do you mean by 'conform to'?
If u divides v, and u is a member of J, then there is a single black cell
on
the u-th row and in the v-th column. Otherwise, there is none such. I am
making one exception, and that is for the 0-th column, in which there are
J
black cells. This is what I mean by conforming to the same distribution as
the occupied matrix components of M_1.
'Occupied matrix
> components' is also problematic, unless you're thinking of
> M_1 as an array (e.g., in a computer program) whose cells
> may or may not have defined contents rather than as a
> (mathematical) matrix.
>
The distribution of occupied matrix components in M_1, in the words laid
out
above: if u is a member of j, and u divides v, then there is a single
occupied matrix component on the u-th row and in the v-th column.
Otherwise,
there is none such.
> My best guess at this point is that you have a partially
> populated array M_1 = [m(i, j)]. Now let
> P = {(i, j) : m(i, j) is defined}; in other words, P
> contains the addresses of the populated cells of M_1. You
> now want to construct an array A = [a(i, j)] of black and
> white cells; presumably it's an array of the same dimensions
> as M_1. If B = {(i, j) : a(i, j) is black}, B is not
> necessarily equal to P. (I've used A for simplicity; I
> gather that this is your M_[x,y].)
>
>> I am considering subintervals [x,y] in the interval that
>> is the product of all members of J.
>
> But the product of the members of J isn't an interval: it's
> simply an integer. Do you mean subintervals of [1, a(J)],
> or perhaps of [0, a(J)]?
Sorry, I meant a subinterval [1,a(J)].
>> But I am also wondering if I can devise a form of
>> notation in which I indicate that the distribution of
>> black cells in my array, for some [x,y], do indeed
>> conform to the distribution found for [x,y] in M_1.
>> Ideally I want to have an array in which all the black
>> cells in columns indexed by integers 0 to 2a(J) conform
>> to the same distribution as do the occupied matrix
>> components M_1.
>
> This is still pretty opaque, because I've no idea how the
> indexing works. But if I've guessed correctly what you mean
> by 'conforms' in this context, you're asking whether there
> is some notational way to indicate (in the notation that I
> used above) that B = P.
>
That looks just right.
>> But I also want there to be other subintervals in the
>> array such that this conformity does not apply. This
>> will mean I don't have to keep switching between a matrix
>> and an array.
>
> This sounds like you're making some distinction between
> 'matrix' and 'array', but I can't tell what it is, unless
> you just mean switching back and forth between M_1 and what
> I called A.
Yes, that's what I mean. In my array, I consider the indexing of rows to
be
redundant; column height is the im****tant concern.
It is also not clear to me what you mean by a
> 'subinterval in the array'
>
I take an array comprising a number of columns the highest integer
indexing
any of which is f, and I am considering intervals that are within [1,f].
>> It's just that I don't know whether that sort of thing is
>> done.
>
> And I can't tell you, because I still haven't any real idea
> of what you want to do. Perhaps I'm being uncommonly dense,
> but I suspect that you don't realize how very much
> background you're taking for granted. E.g., in the other
> thread your answer to my final question is useless, because
> I have no idea what the interval [x', y'] has to do with
> anything that precedes that point.
I'll see what I can do about that. Thanks for bringing it to my attention.
>
> At any rate we're not communicating: I still have only the
> foggiest idea of what you're trying to do, let alone what
> your question about it really is, and it's pretty clear that
> I've not adequately conveyed the nature and extent of my
> confusion(s). Since I've a small mountain of exams to mark
> (and more to prepare), I should probably just bow out, but
> if you want to try again with a more detailed explanation,
> I'll try to make time to have one more go at it.
Thanks. My ultimate issue is one of notation. If you think of the iron
railings at the Brighton seafront, and imagine that sometimes the
post-and-rail pattern takes the form of a distinctive x-shape-between-two
uprights* and, if you walk far enough, they eventaully get haphazard in
their appearance (as - who knows? - perhaps once they were), I'm trying to
devise an array such that I can point to a part of it and say 'in this
case,
we have the distinctive pattern' and at other places, ' in this case, it's
all haphazard'.
Perhaps there are more professional ways of doing it....
* For this pattern, read 'divisibility distribution for members of J'.
All the best.
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