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 <t...@shelx.uni-ac.gwdg.de> 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 >> <dtheob...@brandeis.edu> 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 >> >> <dtheob...@brandeis.edu> 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 >> >>>> <dtheob...@brandeis.edu> 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 >> >>>>>> schul...@cornell.edu >> >>>>> >> >>>>> >> >>>>> >> >>>>> >> >>> >> >>> >> > >> > > > -- > -- > Tim Gruene > Institut fuer anorganische Chemie > Tammannstr. 4 > D-37077 Goettingen > > phone: +49 (0)551 39 22149 > > GPG Key ID = A46BEE1A > > > -----BEGIN PGP SIGNATURE----- > Version: GnuPG v1.4.9 (GNU/Linux) > > iD8DBQFMuVlgUxlJ7aRr7hoRAj9xAKCViaYrKDaXMstIUJFANe/n5hhrkwCgw5iE > ISYjMLOKXbM58ByD/7vGLsA= > =s4ZA > -----END PGP SIGNATURE----- > >
