domain/subdomain
Victor Steinbok
aardvark66 at GMAIL.COM
Tue Oct 11 17:46:38 UTC 2011
On 10/11/2011 1:09 PM, Jim Parish wrote:
> There are a few misunderstandings here, I think.
>> 4. c. Math. 'In the theory of Functions, the portion of the z-plane
>> within a circle which just does not include a singular point is called
>> the domain of its centre' (H. T. Gerrans).
>> 1893 A. R. Forsyth Theory Functions Complex Variable 55 If the
>> whole of the domain of b be not included in that of a.
> Victor Steinbok wrote:
>
>> The Gerrans definition in 4.c. is essentially meaningless, when taken
>> out of context. What Gerrans described is commonly identified as
>> "neighborhood" of the point (and need not be literally a circle, nor
>> need the point be "the center").
> The key point in the definition is "which just does not include a
> singular point"; in other words,
> the boundary of the disk contains such a point. If an analytic function
> f is defined at a point z0,
> there is a power series, centered at z0, which converges to f anywhere
> inside a disk, and there
> is a singular point on the boundary of the disk. (For example, the
> function log z can be represented
> by a power series Sum[a_n (z-1)^n], which converges as long as |z-1|< 1
> - within a disk centered at
> 1, with radius 1. The singular point 0 is on the boundary of this disk.
> This is the type of situation
> described in the definition.)
I see that I misread the singularity issue here. But I don't recognize
this use of "domain".
>> d. Math. An algebraic system with two binary operations defined by
>> postulates stronger than those for a ring but weaker than those for a
>> field; esp. (more fully * integral domain*), a commutative ring in
>> which the cancellation law holds for multiplication of non-zero
>> elements and (with most writers) which has a unit element for
>> multiplication.
> Victor wrote:
> > 4.d. is partially gibberish--trying to make complicated math
> comprehensible in a non-mathematical
> > dictionary--why not just pull a definition from a math encyclopedia
> and modify it to avoid plagiarism?
> > A domain is a kind of ring with specific additional properties (not
> "stronger than a ring"). And what's
> > with "esp. more fully 'integral domain'"? This is one KIND of domain
> in algebra, not "especially" referred
> > to as "domain". What about the Euclidean domain? Principal ideal
> domain? Are these "nothing special"
> > next to "integral domain"? [Wiki is better--see below.]
>
> The word "domain" is often used to mean "integral domain"; the other
> types of domain Victor mentions
> are special cases of (integral) domains. The definition does not claim
> that the *system* is stronger than a ring,
> but that its *postulates* are stronger than those for a ring - i.e.,
> that every domain is a ring, but not every
> ring is a domain.
My point was--and is--that the phrasing is convoluted and not entirely
accurate. It can be stated much simpler without unnecessary
generalizations. Furthermore, not every type of named domain is a kind
of integral domain. But even if it were, if a math text repeatedly
refers to Euclidean domains as just "domains", that contradicts the
notion that "integral domains" are to be singled out as the type that is
"especially" referred to as "domain". This is a disjoint rather than
nested relationship in terminology, irrespectively of any nested
hierarchy of the actual domains.
> > e. Math. The set of values that an independent variable of a function
> can take; the graphical representation of this
> > set; the set comprising all the first elements of the ordered pairs
> constituting some given set.
>
> Victor writes:
> > On the other hand, 4.e. is too narrowly defined (independent variable
> values--yes; ordered "pairs"--no!).
>
> I see nothing wrong with this, given the common representation of a
> function from A to B as a subset of
> the Cartesian product AxB; the domain is, indeed, the set of first
> elements of the ordered pairs constituting
> this subset. (There are philosophical objections to this representation,
> but it can't be denied that it's quite
> common!)
It depends on what one means by an "ordered pair". OED's closest is
ordered, adj.
> 2. d. Math. Designating a pair, triple, or higher multiple of elements
> a, b, (c,.‥) such that (a, b,…) = (u, v,…) if and only if a = u, b =
> v, etc.
This includes two quotations with "ordered pairs"--no separate entry.
Similarly, there is another quotation from a topology book (under
"rigorously"). But the definition here only works, in general, if either
part of the "ordered pair" is allowed to be a vector or a
set--otherwise, multiple inputs are not allowed. Indeed, there are
philosophical objections to such a definition. And not every definition
one finds in quite a few textbooks is accurate.
> > g. Math. An open connected set of at least one point.
>
> This one I might quibble with; "domain" is often restricted to nonempty
> open *simply* connected sets - roughly
> speaking, open sets without any holes in them.
Oddly enough, this one is the same in every dictionary that lists this
meaning. But I agree that there are multiple possibilities here--again,
depending on the source.
Overall, the definitions, other than 4.g., are arcane and convoluted
and, thus, could use improvement.
VS-)
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