Tim,
PS perhaps you should ask George Sheldrick whether he has ever found
himself constrained as to the algorithms he is able to program by the
semantics of Fortran.
I suspect his answer will be the same as mine!
Cheers
-- Ian
On Sat, Oct 16, 2010 at 8:50 AM, Tim Gruene <[log in to unmask]> wrote:
> Dear Ian,
>
> maybe you should switch from Fortran to C++. Then you would not be forced to
> make nature follow the semantics of your programming language but can adjust
> your code to the problem you are tackling.
> The question you post would nicely fit into a first year's course on C++ (and of
> course can all be answered very elegantly).
>
> Cheers, Tim
>
> On Fri, Oct 15, 2010 at 11:55:54PM +0100, Ian Tickle wrote:
>> On Fri, Oct 15, 2010 at 8:11 PM, Douglas Theobald
>> <[log in to unmask]> wrote:
>> > Vectors are not only three-dimensional, nor only Euclidean -- vectors can be
>> > defined for any number of arbitrary dimensions. Your initial comment
>> > referred to complex numbers, for instance, which are 2D vectors (not 1-D).
>> > Obviously scalars are not 3-vectors, they are 1-vectors. And contrary to
>> > your earlier assertion, you can always represent complex numbers as vectors
>> > (in fortran, C, on paper, or whatever), and it is possible to define many
>> > different valid types of multiplication, exponentiation, logarithms, powers,
>> > etc. for vectors (and matrices as well).
>>
>> I didn't say that vectors are only 3D or only Euclidean (note my
>> qualification 'Euclidean *or otherwise*'). I was stating that we're
>> not talking here specifically about 1D (or 2D) vectors; my use of 3D
>> is only an example, since my original example referred to structure
>> factors which are usually defined in 3D reciprocal space. Quite
>> obviously vectors can be generalised to any number of dimensions (as
>> in the example below).
>>
>> Let's take a simple example of an operation that's trivial to express
>> using complex numbers as scalars. Suppose we have 2 vectors of
>> complex structure factors of equal dimension (n) and we want to form
>> the (complex) scalar product. This kind of equation arises in, for
>> example, the theory of direct methods.
>>
>> Let F = (F1, F2, F3, ... Fn)
>> and G = (G1, G2, G3, ... Gn)
>>
>> Then the scalar product F.G = F1*G1 + F2*G2 + F3*G3 + ... Fn*Gn (or
>> SUM [j=1 to n] Fj*Gj),
>> where '+' and '*' here are the normal addition and multiplication
>> operators on scalars (here complex of course).
>> Note that the result of a scalar product of 2 vectors is by definition
>> a scalar and here it's complex!
>>
>> Also note that the RHS can be programmed exactly as written, in fact
>> something like:
>>
>> COMPLEX F, G, S
>> S = 0
>> 1 READ (*,*,END=2) F, G
>> S = S + F*G
>> GOTO 1
>> 2 PRINT *,S
>> END
>>
>> Now, how would you express the scalar product F.G in a way that could
>> be programmed, using vector notation for all the complex numbers, and
>> obviously you can only use operators that are defined for vectors,
>> namely addition, subtraction, scalar multiplication, scalar and
>> exterior product?
>>
>> Then when you've done that, how would you express a ratio of complex
>> numbers (say F1/G1), again using only vector notation?
>>
>> -- Ian
>>
>> >
>> > On Oct 15, 2010, at 12:40 PM, Ian Tickle wrote:
>> >
>> >> Any vector, whether in the 'mathematical' or 'physical' sense as
>> >> defined in Wikipedia, and which is defined on a 3D vector space
>> >> (Euclidean or otherwise - which I hope is what were talking about),
>> >> has by definition 3 elements (real or complex). This clearly excludes
>> >> all scalars (real or complex) which have only 1 whatever the dimension
>> >> of the space. Therefore it's plainly impossible for an entity in 3D
>> >> space to be both a scalar and a vector at the same time. Your
>> >> conclusion that scalars and complex numbers fulfil the axioms of a
>> >> vector space is applicable only in the case of a 1D vector space, and
>> >> therefore is not relevant. My original observation which started this
>> >> thread was intended to be general one, not for a particular special
>> >> case.
>> >>
>> >> -- Ian
>> >>
>> >> On Fri, Oct 15, 2010 at 5:17 PM, Douglas Theobald
>> >> <[log in to unmask]> wrote:
>> >>> On Oct 15, 2010, at 11:37 AM, Ganesh Natrajan wrote:
>> >>>
>> >>>> Douglas,
>> >>>>
>> >>>> The elements of a 'vector space' are not 'vectors' in the physical
>> >>>> sense.
>> >>>
>> >>> And there you make Ed's point -- some people are using the general vector definition, others are using the more restricted Euclidean definition.
>> >>>
>> >>> The elements of a general vector space certainly can be physical, by any normal sense of the term. And note that physical 3D space is not Euclidean, in any case.
>> >>>
>> >>>> The correct Wikipedia page is this one
>> >>>>
>> >>>> http://en.wikipedia.org/wiki/Euclidean_vector
>> >>>>
>> >>>>
>> >>>> Ganesh
>> >>>>
>> >>>>
>> >>>>
>> >>>> On Fri, 15 Oct 2010 11:20:04 -0400, Douglas Theobald
>> >>>> <[log in to unmask]> wrote:
>> >>>>> As usual, the Omniscient Wikipedia does a pretty good job of giving
>> >>>>> the standard mathematical definition of a "vector":
>> >>>>>
>> >>>>> http://en.wikipedia.org/wiki/Vector_space#Definition
>> >>>>>
>> >>>>> If the thing fulfills the axioms, it's a vector. Complex numbers do,
>> >>>>> as well as scalars.
>> >>>>>
>> >>>>> On Oct 15, 2010, at 8:56 AM, David Schuller wrote:
>> >>>>>
>> >>>>>> On 10/14/10 11:22, Ed Pozharski wrote:
>> >>>>>>> Again, definitions are a matter of choice....
>> >>>>>>> There is no "correct" definition of anything.
>> >>>>>>
>> >>>>>> Definitions are a matter of community choice, not personal choice; i.e. a matter of convention. If you come across a short squat animal with split hooves rooting through the mud and choose to define it as a "giraffe," you will find yourself ignored and cut off from the larger community which chooses to define it as a "pig."
>> >>>>>>
>> >>>>>> --
>> >>>>>> =======================================================================
>> >>>>>> All Things Serve the Beam
>> >>>>>> =======================================================================
>> >>>>>> David J. Schuller
>> >>>>>> modern man in a post-modern world
>> >>>>>> MacCHESS, Cornell University
>> >>>>>> [log in to unmask]
>> >>>>>
>> >>>>>
>> >>>>>
>> >>>>>
>> >>>
>> >>>
>> >
>> >
>
> --
> --
> Tim Gruene
> Institut fuer anorganische Chemie
> Tammannstr. 4
> D-37077 Goettingen
>
> phone: +49 (0)551 39 22149
>
> GPG Key ID = A46BEE1A
>
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