Re: ? linear, semiL, pseudoL, and NL
- From: dwarfppe2004@xxxxxxxxxxxxxx (Ulysse Keller)
- Date: Wed, 15 Jun 2005 10:49:43 GMT
On Mon, 13 Jun 2005 15:55:38 -0700, "Cheng Cosine"
<acosine@xxxxxxxxxxxxxxx> wrote:
> I don't have my DE books at hand, so the below are from my
> memory. Correct me if there is any mistake.
> In defining linear, semilinear, pseudolinear, and nonlinear DEs,
> when there is no nonlinear term, the DE is linear; when the one
> with highest derivative term is nonlinear, the DE is nonlinear;
> when any non-highest derivative term is nonlinear, the DE is
> pseudolinear; when the nonlinear term involves no derivative
> the DE is semilinear.
>
> are there any other Math branch use these terms: linear,
> semilinear, pseudolinear, and nonlinear? If so, how are they
> defined? (...)
>
A) Yes, *linear algebra* uses the term "linear" - which isn't very
astonishing ;-) *and* the definition of a linear mapping (and of
a linear equation) given in linear algebra is fundamental for
the use of the word in all other parts of math. where it is used
(including diff. equations ...) - with only few exceptions, which
BTW are less in number in other languages I know, like French
and German. These are the main exceptions I can think of now:
1) speaking of ordered sets, one sometimes says that an ordering
is linear instead of calling it a total order (if this isn't clear for
you, you will easily find details in any basic text about ordered
sets) - it's because such an order makes the set look kind of
one-dimensional like a (geometric) line; AFAIK this is specifically
English ...
2) in affine geometry (and geometry depending on it like euclidian
geom.) as well as in projective geometry the word "linear" has
a slightly different sense, when used for a mapping:
when a linear space (also called a vector space) is considered as
an affine space in the standard way, one says "linear homogenous"
instead of just "linear" in the linear algebra sense, because "linear"
is slightly more general (linear in the affine sense = linear
homogenous + a constant / translation, where "translation"
- at least in French ... - means a mapping of the kind x->x+a,
a being a constant vector)
in projective geometry, I think that the word "linear" is not so often
used; when it is, it means the result of passing a true (homogen.)
linear mapping of the vector spaces "above" the considered projective
spaces to the quotient(set)s, i.e. to the latter spaces (passing a map
to the quotients means completing a commutative diagram of mappings
in which there are the original map and the canonical maps from the
original sets - between which the orig. map goes - to the quotient
sets; when this is possible, the map is said to be compatible with
the equivalence relations used) -> to understand this, you must know
the essentials about equivalence relations in set theory; and what
I say here depends on the model of a projective space as derived
from a linear space by considering 2 non-zero vectors as equivalent
iff one is the product of the other by a scalar (the 0 is excluded
from the vector space to get the set in which an equivalence relat.
is defined) - the quotient set *is* the projective space.
B) Nonlinear usually just means "not linear" - so this is different
from 'your' definition - above (it is not very standard, I think ...)
C) Then, the term "semi-linear" is also used in connection with linear
algebra especially in the context of the theory of hermitian forms,
but the meaning has nothing to do with 'your' definition (except if
the missing hyphen seems essential to you): semi-linear is like
linear, except that in the formula f(ax) =af(x) which is used in one
'half' of the def. of "linear" the 'a' in the righthand site is
replaced by a*, where a* is the image of a by some (fixed)
automorphism of the base field (if instead of a field one has
a non-commutative [division] ring like the ring of quaternions,
then one will use an anti-automorphism). If fields (and rings)
- and then also their (anti-)automorphisms - as defined in modern
algebra are not known to you, think of the (most important) case
where complex numbers are used as scalars, then a' will mean
the complex conjugate of the number a (usually written as
the letter 'a' with a bar over it). Hermitian forms are like scalar
products of vectors (which are defined as bilinear symmetric in
the case of real scalars [only] - plus usually a special property
known as "definite positive") again with a* instead of a in one
of the formulas defining "bilinear" and 'symmetric' is replaced by
anti-symmetric defined like 'symmetric' but with the same function
a->a* applied to one side of the formula for symmetry (*not* by
skew-symmetric which would be with a->-a). Hermitian forms
have become important since they are inherent to Hilbert spaces
(yes the thing used in quantum physics)
.
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- From: Cheng Cosine
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