Group valued flows: A possible example

@IMSc, Chennai

Am here as an attendee for the NBHM nurture programme. Today, we had a talk by one of the participants, ADG on flows in graphs (ref. Diestel’s Graph Theory). In his talk, he defined the concept of group valued flows. Let there be a function , where is the edge set of the multigraph and is an abelian group. Hence, assigns to each edge of the edgeset , an element in . This concept is an extension of the concept of integral flow in graphs, with in the latter. I won’t go much into the theory itself, but I have some different objectives here.

After the lecture, one of the attendees complained that this notion of group valued flow is too abstract. So, a possible non-trivial example happened to occur to me in which the abelian group doesn’t equal . Consider, a hypothetical mode of communication (I don’t know if one such already exists), in which information happen to travel in the form of nuclei spin in the medium carrying it. So, the source or the transmitter orients the nuclei of the medium those are just adjacent to it, in a particular direction and the fraction of nuclei in that cross section which are oriented in that particular direction determine the magnitude of the spin vector. These nuclei then transfer this spin to their neighbours and reorient themselves randomly in all the directions such that their net spin vector again becomes zero. In this fashion, the information about the nuclei spin can flow through the medium. This flow can hence be thought of as a vector valued flow in which flow in each edge of the graph is given by a vector . Hence, the group happens to be the vector space having the usual addition operation of the vector space as its abelian operation.

P.S. 1) I’ve written this post primarily so that I can refer back to this idea whenever I might need it at some future point of time.

2) This is the first post belonging to the category math on my blog .

3) To write equations and other stuff in in your WordPress blog posts, type $latex yourcodehere $. Though I feel that WordPress doesn’t adjust the sentences involving LaTeX that good, but I have no other option .

Inline figures in a LaTeX document

I am trying to do some stuff in LaTeX, in which I wanted to insert figures into my amsart documentclass … Naturally, I could not find that in any standard book and went bout googling for it … still there was not a site which explained it in full although Piet van Oostrum’s documentation about picins was quite helpful. I am writing out here one possible way of doing the same which worked for me, there are other ways too like using wrapfig package. The .sty file can downloaded from here, and put it where your LaTeX compiler looks for packages, then in your .tex file do the following things:

documentclass{article}%I have seen it working wd amsart also

 . . .

usepackage{... , picins, ...}%where ... indicates other packages

then write the following command wherever you need you figure to be there

parpic(width,height)(x-offset,y-offset)[Options][Position]{Picture}

All parameters except the Picture are optional. The picture can be
positioned left or right, boxed with a rectangle, oval, shadowbox,
dashed box, a caption can be given which includes it in the List of
figures.

If the width and height are not given [in which case also no offsets
can be given] or if they are given as 0pt, the actual size of the
Picture is used.

Options can be (default is l):
l – put picture on left side of the paragraph
r – put picture on right side of the paragraph

f – frame the picture
d – put a dashed frame aroud the picture
o – put an \oval around the picture
s – put a shadowbox around the picture
x – put a 3D box around the picture
Max one of each group can be given

Position:
l – put the picture left in the box
r – put the picture right in the box
t – put the picture in the top of the box
b – put the picture in the bottom of the box
Default is centering. It only makes a difference if the width and
height given are bigger than the actual picture. If offsets are
given Position is not used.

Picture can be any LaTeX construct.

I have just now “tested” that the \piccaption option for writing caption to such an inline figure doesn’t work properly and isn’t properly typesetted. The other options mentioned in Piet’s documentation viz. \piccaptioninside, \piccaptionoutside, \piccaptionside and \piccaptiontopside doesn’t work at all on my computer. Also, even without captions, this seems to be an only somewhat satisfying solution and does throw lots of problems like improper alignment of text … text over text, text over figure, problems with size adjustment of figure and so on.

Please note that I have borrowed most of the material from Piet’s documentation itself and have written here so that it might appear on search results more easily. Please comment freely as if this information was of some use to you and if you would like to tell us of some new improvements in a non-technical language, of course