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Angela Stazzone

7 Jun 2001 7 Jun '01

5:38 p.m.

Hi Pat, can you help me to find a formal proof (or an example it is not true) to the following assertion?

EXTREMAL RAYS ARE STABLE WHEN COMBINED WITH ANY VECTOR OF THE LINEALITY SPACE.

I found it in H. Leverge - A note on Chernikova's Algorithm - Publication Interne 635 - February 1992 - page 9.

I guess "stable" means that the combination between an extremal ray with an element of the lin. space is an extremal ray.

I think it is intuitive that it's true, but I can't find a formal proof!

Also, I found (in Leverge page 4) this result:

if the set G = cone{y} + lin.space (C) is a face of C, then y is called an extremal ray of C and G a minimal proper face of C;

where, given a set X cone{X} is the set of all positive combination of all vectors in X.

Can this help to demonstrate the assertion above?

Thanks, Angela.

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P M Hill

7 Jun 7 Jun

5:46 p.m.

Hi,

-- On Thu, 7 Jun 2001, Angela Stazzone wrote: > Hi Pat, > can you help me to find a formal proof (or an example it is not true) to > the following assertion? > > EXTREMAL RAYS ARE STABLE WHEN COMBINED > WITH ANY VECTOR OF THE LINEALITY SPACE. > > I found it in H. Leverge - A note on Chernikova's Algorithm - > Publication Interne 635 - February 1992 - page 9. > Where can I find this? ciao, Pat > I guess "stable" means that the combination between an extremal ray with > an element of the lin. space is an extremal ray. > > I think it is intuitive that it's true, but I can't find a formal proof! > > Also, I found (in Leverge page 4) this result: > > if the set G = cone{y} + lin.space (C) is > a face of C, then y is called an extremal ray > of C and G a minimal proper face of C; > > where, given a set X cone{X} is the set of all positive combination of > all vectors in X. > > Can this help to demonstrate the assertion above? > > Thanks, > Angela. > > _______________________________________________ > PPL-devel mailing list > PPL-devel@cs.unipr.it > http://www.cs.unipr.it/mailman/listinfo/ppl-devel >

P M Hill

8:45 p.m.

Hi Angela,

-- On Thu, 7 Jun 2001, Angela Stazzone wrote: > Hi Pat, > can you help me to find a formal proof (or an example it is not true) to > the following assertion? > > EXTREMAL RAYS ARE STABLE WHEN COMBINED > WITH ANY VECTOR OF THE LINEALITY SPACE. > > I found it in H. Leverge - A note on Chernikova's Algorithm - > Publication Interne 635 - February 1992 - page 9. > > I guess "stable" means that the combination between an extremal ray with > an element of the lin. space is an extremal ray. > > I think it is intuitive that it's true, but I can't find a formal proof! It can't be true. Have a look at the definition of extreme ray. It is extreme if it is NOT the positive combination of two other rays. Note that a line is just two rays in opposite directions. I would be interested to know what Leverge means by "stable". I've been trying to guess, but without any success. ciao, Pat

Roberto Bagnara

11:28 p.m.

P M Hill wrote:

...

On Thu, 7 Jun 2001, Angela Stazzone wrote:

...
I found it in H. Leverge - A note on Chernikova's Algorithm - Publication Interne 635 - February 1992 - page 9.

I would be interested to know what Leverge means by "stable". I've been trying to guess, but without any success.

Pat,

do you have that paper? I guess not: we had to request it to IRISA (Herve Leverge died tragically in 1994).

Angela or anyone else being at the department tomorrow,

if Pat does not have that paper, please make a copy and send it (just ask Paola or Francesca to make a copy, put it into an envelope, lookup Pat's address from her homepage, and then ask Paola or Francesca to send it with the quickest method not requiring a signature by the ministry of research. (I know I am being pedantic, you also know why ;-) While you are at it, you may check if Pat is missing something else that is instead in our carpet and is not available electronically.

Elisa,

would I be you, I would bring the paper to Pat personally, possibly on foot ;-) Ciao,

Roberto

-- Roberto Bagnara Computer Science Group Department of Mathematics, University of Parma, Italy http://www.cs.unipr.it/~bagnara/ mailto:bagnara@cs.unipr.it

P M Hill

8 Jun 8 Jun

6:52 a.m.

-- On Thu, 7 Jun 2001, Roberto Bagnara wrote: > P M Hill wrote: > > On Thu, 7 Jun 2001, Angela Stazzone wrote: > > > I found it in H. Leverge - A note on Chernikova's Algorithm - > > > Publication Interne 635 - February 1992 - page 9. > > > > > I would be interested to know what Leverge means by "stable". > > I've been trying to guess, but without any success. > > Pat, > > do you have that paper? I guess not: we had to request it > to IRISA (Herve Leverge died tragically in 1994). I don't have it and would be grateful for a copy. Thanks & ciao, Pat

Angela Stazzone

8:44 a.m.

P M Hill wrote:

...

--

On Thu, 7 Jun 2001, Roberto Bagnara wrote:

...
P M Hill wrote:

...
On Thu, 7 Jun 2001, Angela Stazzone wrote:

...
I found it in H. Leverge - A note on Chernikova's Algorithm - Publication Interne 635 - February 1992 - page 9.

I would be interested to know what Leverge means by "stable". I've been trying to guess, but without any success.

Pat,

do you have that paper? I guess not: we had to request it to IRISA (Herve Leverge died tragically in 1994).

I don't have it and would be grateful for a copy.

I'm sending it to you!

Ciao, Angela.

...

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Angela Stazzone
P M Hill
Roberto Bagnara