domain/subdomain
Jim Parish
jparish at SIUE.EDU
Tue Oct 11 17:09:18 UTC 2011
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.)
> 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.
> 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!)
> 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.
Jim Parish
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