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Jack Stone
science forum beginner

Joined: 08 Jul 2006
Posts: 3

Posted: Mon Jul 10, 2006 3:18 pm    Post subject: Properties of weakly continuous maps in a Hilbert space

Let f(x) and g(x) be weakly continuous maps in a Hilbert space. Is f(g(x)) weakly continuous?

Is it true that every weakly continuous map from a Hilbert space to itself has a fixed point? (As I know it is true for the unit ball ||x||<=1 but I cannot prove or disprove this for the whole space.)

David C. Ullrich
science forum Guru

Joined: 28 Apr 2005
Posts: 2250

Posted: Mon Jul 10, 2006 3:38 pm    Post subject: Re: Properties of weakly continuous maps in a Hilbert space

On Mon, 10 Jul 2006 11:18:33 EDT, Jack Stone <stone1292@mail.com>
wrote:

 Quote: Let f(x) and g(x) be weakly continuous maps in a Hilbert space. Is f(g(x)) weakly continuous?

The composition of two continuous maps is continuous in _any_
topological space.

 Quote: Is it true that every weakly continuous map from a Hilbert space to itself has a fixed point? (As I know it is true for the unit ball ||x||<=1 but I cannot prove or disprove this for the whole space.)

Say v is a non-zero vector in H. Let f(x) = x + v.

************************

David C. Ullrich
James Burns
science forum beginner

Joined: 05 May 2005
Posts: 48

Posted: Tue Jul 11, 2006 12:59 am    Post subject: Re: Properties of weakly continuous maps in a Hilbert space

"David C. Ullrich" wrote:
 Quote: On Mon, 10 Jul 2006 11:18:33 EDT, Jack Stone

That doesn't seem at first to answer the question, which
was about /weakly/ continuous functions. However, the two
proofs are so similar in structure that one might consider
it a good clue.

If I'm right in thinking that /weakly/ continuous functions
f and g will map weakly convergent series to weakly convergent
series and limit point to limit point, then g will take
a weakly convergent series as input, then pass a weakly
convergent series to f, which returns yet another weakly
convergent series. Likewise for limit points. Therefore,
yes, the composition of weakly continuous functions is
weakly continuous.

I didn't know what "weakly continuous" meant, so I attempted
to google it. "Weakly convergent" is pretty easy to find,
"weakly continuous function" less so. (The series x_n is
weakly convergent if the inner product < x_n, y > converges
for all y in the Hilbert space H.).

However, I strongly suspect the definition "weakly continuous
function" is analogous to "continuous function", as I
expressed it above. If I'm wrong, I would appreciate being
told. (Too late if I've made a fool of myself, of course.)

Jim Burns
Ronald Bruck
science forum Guru

Joined: 05 Jun 2005
Posts: 356

Posted: Tue Jul 11, 2006 2:30 am    Post subject: Re: Properties of weakly continuous maps in a Hilbert space

In article <44B2F7EE.62178FE@osu.edu>, Jim Burns <burns.87@osu.edu>
wrote:

 Quote: "David C. Ullrich" wrote: On Mon, 10 Jul 2006 11:18:33 EDT, Jack Stone

You're wrong. That's a weakly SEQUENTIALLY continuous function.

 Quote: then g will take a weakly convergent series as input, then pass a weakly convergent series to f, which returns yet another weakly convergent series. Likewise for limit points. Therefore, yes, the composition of weakly continuous functions is weakly continuous. I didn't know what "weakly continuous" meant, so I attempted to google it.

Weak continuity means, between the weak topologies. The weak topology
for a Hilbert space has as a subbase for the neighborhoods of 0, the
sets of the form {x : (w,x) < 1}. This means you take finite
intersections of sets of this form (to get the base of the neighborhood
system at 0), then translates of these, then arbitrary unions of THOSE,
to get the topology.

David's comment was meant to convey the fact that compositions of
continuous mappings (between compatible topologies) are ALWAYS
continuous.

There are many variations on weak continuity: demicontinuity [from
strong to weak topology], hemicontinuity [from strong to weak on line
segments--or weak to weak, if you prefer (same thing)], etc. etc. But
one thing you need to understand: these variations were introduced
BECAUSE THERE ARE VERY FEW WEAKLY CONTINUOUS FUNCTIONS!!

 Quote: "Weakly convergent" is pretty easy to find, "weakly continuous function" less so. (The series x_n is weakly convergent if the inner product < x_n, y > converges for all y in the Hilbert space H.). However, I strongly suspect the definition "weakly continuous function" is analogous to "continuous function", as I expressed it above. If I'm wrong, I would appreciate being told. (Too late if I've made a fool of myself, of course.) Jim Burns
David C. Ullrich
science forum Guru

Joined: 28 Apr 2005
Posts: 2250

Posted: Tue Jul 11, 2006 10:55 am    Post subject: Re: Properties of weakly continuous maps in a Hilbert space

On Mon, 10 Jul 2006 20:59:26 -0400, Jim Burns <burns.87@osu.edu>
wrote:

 Quote: "David C. Ullrich" wrote: On Mon, 10 Jul 2006 11:18:33 EDT, Jack Stone

How can you possibly even _conjecture_ that this does
not answer the question, since, as you say below, you
don't know what "weakly continuous" means?

 Quote: However, the two proofs are so similar in structure that one might consider it a good clue. If I'm right in thinking that /weakly/ continuous functions f and g will map weakly convergent series to weakly convergent series and limit point to limit point, then g will take a weakly convergent series as input, then pass a weakly convergent series to f, which returns yet another weakly convergent series. Likewise for limit points. Therefore, yes, the composition of weakly continuous functions is weakly continuous. I didn't know what "weakly continuous" meant, so I attempted to google it. "Weakly convergent" is pretty easy to find, "weakly continuous function" less so. (The series x_n is weakly convergent if the inner product < x_n, y > converges for all y in the Hilbert space H.). However, I strongly suspect the definition "weakly continuous function" is analogous to "continuous function", as I expressed it above. If I'm wrong, I would appreciate being told. (Too late if I've made a fool of myself, of course.)

If you don't know what "weakly continuous" means probably
you should determine that first, before attempting to
figure out questions regarding weakly continuous
functions?

Yes, you've made a fool of yourself. Nothing foolish
about not knowing what "weakly continuous" means - if
you'd asked someone would have said, no problem. But
_before_ finding a definition of the phrase, and
without asking for the definition, that's definitely
making a fool of yourself. As is conjecturing that
an answer is incorrect, before learning the
definition of the terms involved.

A Hilbert space has a weak topology. A weakly continuous
function is precisely a function which is continuous with
respect to this topology. And hence what I said _does_

 Quote: Jim Burns

************************

David C. Ullrich
James Burns
science forum beginner

Joined: 05 May 2005
Posts: 48

Posted: Thu Jul 13, 2006 2:42 am    Post subject: Re: Properties of weakly continuous maps in a Hilbert space

Ronald Bruck wrote:
 Quote: In article <44B2F7EE.62178FE@osu.edu>, Jim Burns burns.87@osu.edu> wrote: I didn't know what "weakly continuous" meant, so I attempted to google it. Weak continuity means, between the weak topologies. The weak topology for a Hilbert space has as a subbase for the neighborhoods of 0, the sets of the form {x : (w,x) < 1}. This means you take finite intersections of sets of this form (to get the base of the neighborhood system at 0), then translates of these, then arbitrary unions of THOSE, to get the topology. David's comment was meant to convey the fact that compositions of continuous mappings (between compatible topologies) are ALWAYS continuous. There are many variations on weak continuity: demicontinuity [from strong to weak topology], hemicontinuity [from strong to weak on line segments--or weak to weak, if you prefer (same thing)], etc. etc. But one thing you need to understand: these variations were introduced BECAUSE THERE ARE VERY FEW WEAKLY CONTINUOUS FUNCTIONS!!

Thank you for trying to help me. I have added the varieties
of convergence and continuity to my list of things I want
to loom into mrore closely, when I get around to it.

Jim Burns
James Burns
science forum beginner

Joined: 05 May 2005
Posts: 48

Posted: Thu Jul 13, 2006 2:42 am    Post subject: Re: Properties of weakly continuous maps in a Hilbert space

"David C. Ullrich" wrote:
 Quote: On Mon, 10 Jul 2006 20:59:26 -0400, Jim Burns burns.87@osu.edu> wrote: That doesn't seem at first to answer the question, which was about /weakly/ continuous functions. How can you possibly even _conjecture_ that this does not answer the question, since, as you say below, you don't know what "weakly continuous" means?

I hope you will believe me when I tell you I was not trying to
upset you, or irritate you or get you. I'm sorry if I have.

As to how I could _conjecture_ (or, as I would prefer,
available and had failed at finding out what "weakly
continuous" meant (see below). At that point, my choices were
to leave the post for another day (which my history shows
often means "never") or post with incomplete information
and make it clear how incomplete my information was.

Below you write "if you'd asked someone would have said,
no problem." But, part of the reason I posted was to get a
clarification of what "weakly continuous" meant -- I just
didn't use question marks. I have noticed over the years that
it's much easier to get a correction on USEnet than an answer
to a question. Perhaps that's not true of you -- then, good
for you, but I'd still maintain you are not typical.

What would you have done if you were me? Notice that I am not
asking what you would have done in my place, but to imagine
what it's like to be me and be faced with a similar choice.
You are a professional mathematician, I am no kind of academic
at all. You apparently take your appearance on USEnet very
seriously, while I, having very little in the way of
also have, beyond your mental mathematical resources, texts,
references and colleagues practically at your fingertips.
Well, I don't.

Would you just dropped the topic and moved on? Maybe I would
have, too, if I hadn't already invested half an hour or so
in trying to beat a definition of "weakly continuous
function" out of the World Wide Web.

I hope you're not upset that I sorta-kinda-maybe-if-God-
willing-I'm-right accused you of making a mistake. You
do make mistakes, you know, though it's very rare, I'm sure.
This was not one of them, but anyone who knew the correct
definition would know that, including the original poster.
Yes, I thought of that before I posted. If I had had a chance
of misleading him, I would have let it drop. But if, by
chance, I were right, I could have done a bit of good.

Yours,
Jim Burns

 Quote: However, the two proofs are so similar in structure that one might consider it a good clue. If I'm right in thinking that /weakly/ continuous functions f and g will map weakly convergent series to weakly convergent series and limit point to limit point, then g will take a weakly convergent series as input, then pass a weakly convergent series to f, which returns yet another weakly convergent series. Likewise for limit points. Therefore, yes, the composition of weakly continuous functions is weakly continuous. I didn't know what "weakly continuous" meant, so I attempted to google it. "Weakly convergent" is pretty easy to find, "weakly continuous function" less so. (The series x_n is weakly convergent if the inner product < x_n, y > converges for all y in the Hilbert space H.). However, I strongly suspect the definition "weakly continuous function" is analogous to "continuous function", as I expressed it above. If I'm wrong, I would appreciate being told. (Too late if I've made a fool of myself, of course.) If you don't know what "weakly continuous" means probably you should determine that first, before attempting to figure out questions regarding weakly continuous functions? Yes, you've made a fool of yourself. Nothing foolish about not knowing what "weakly continuous" means - if you'd asked someone would have said, no problem. But asking a question about weakly continuous functions _before_ finding a definition of the phrase, and without asking for the definition, that's definitely making a fool of yourself. As is conjecturing that an answer is incorrect, before learning the definition of the terms involved. A Hilbert space has a weak topology. A weakly continuous function is precisely a function which is continuous with respect to this topology. And hence what I said _does_ answer the question. Jim Burns ************************ David C. Ullrich
James Burns
science forum beginner

Joined: 05 May 2005
Posts: 48

Posted: Thu Jul 13, 2006 2:43 am    Post subject: Re: Properties of weakly continuous maps in a Hilbert space

"David C. Ullrich" wrote:
 Quote: On Mon, 10 Jul 2006 20:59:26 -0400, Jim Burns burns.87@osu.edu> wrote: That doesn't seem at first to answer the question, which was about /weakly/ continuous functions. How can you possibly even _conjecture_ that this does not answer the question, since, as you say below, you don't know what "weakly continuous" means?

I hope you will believe me when I tell you I was not trying to
upset you, or irritate you or get you. I'm sorry if I have.

As to how I could _conjecture_ (or, as I would prefer,
available and had failed at finding out what "weakly
continuous" meant (see below). At that point, my choices were
to leave the post for another day (which my history shows
often means "never") or post with incomplete information
and make it clear how incomplete my information was.

Below you write "if you'd asked someone would have said,
no problem." But, part of the reason I posted was to get a
clarification of what "weakly continuous" meant -- I just
didn't use question marks. I have noticed over the years that
it's much easier to get a correction on USEnet than an answer
to a question. Perhaps that's not true of you -- then, good
for you, but I'd still maintain you are not typical.

What would you have done if you were me? Notice that I am not
asking what you would have done in my place, but to imagine
what it's like to be me and be faced with a similar choice.
You are a professional mathematician, I am no kind of academic
at all. You apparently take your appearance on USEnet very
seriously, while I, having very little in the way of
also have, beyond your mental mathematical resources, texts,
references and colleagues practically at your fingertips.
Well, I don't.

Would you just dropped the topic and moved on? Maybe I would
have, too, if I hadn't already invested half an hour or so
in trying to beat a definition of "weakly continuous
function" out of the World Wide Web.

I hope you're not upset that I sorta-kinda-maybe-if-God-
willing-I'm-right accused you of making a mistake. You
do make mistakes, you know, though it's very rare, I'm sure.
This was not one of them, but anyone who knew the correct
definition would know that, including the original poster.
Yes, I thought of that before I posted. If I had had a chance
of misleading him, I would have let it drop. But if, by
chance, I were right, I could have done a bit of good.

Yours,
Jim Burns

 Quote: However, the two proofs are so similar in structure that one might consider it a good clue. If I'm right in thinking that /weakly/ continuous functions f and g will map weakly convergent series to weakly convergent series and limit point to limit point, then g will take a weakly convergent series as input, then pass a weakly convergent series to f, which returns yet another weakly convergent series. Likewise for limit points. Therefore, yes, the composition of weakly continuous functions is weakly continuous. I didn't know what "weakly continuous" meant, so I attempted to google it. "Weakly convergent" is pretty easy to find, "weakly continuous function" less so. (The series x_n is weakly convergent if the inner product < x_n, y > converges for all y in the Hilbert space H.). However, I strongly suspect the definition "weakly continuous function" is analogous to "continuous function", as I expressed it above. If I'm wrong, I would appreciate being told. (Too late if I've made a fool of myself, of course.) If you don't know what "weakly continuous" means probably you should determine that first, before attempting to figure out questions regarding weakly continuous functions? Yes, you've made a fool of yourself. Nothing foolish about not knowing what "weakly continuous" means - if you'd asked someone would have said, no problem. But asking a question about weakly continuous functions _before_ finding a definition of the phrase, and without asking for the definition, that's definitely making a fool of yourself. As is conjecturing that an answer is incorrect, before learning the definition of the terms involved. A Hilbert space has a weak topology. A weakly continuous function is precisely a function which is continuous with respect to this topology. And hence what I said _does_ answer the question. Jim Burns ************************ David C. Ullrich
James Burns
science forum beginner

Joined: 05 May 2005
Posts: 48

Posted: Thu Jul 13, 2006 2:52 am    Post subject: Re: Properties of weakly continuous maps in a Hilbert space

"David C. Ullrich" wrote:
 Quote: On Mon, 10 Jul 2006 20:59:26 -0400, Jim Burns burns.87@osu.edu> wrote: That doesn't seem at first to answer the question, which was about /weakly/ continuous functions. How can you possibly even _conjecture_ that this does not answer the question, since, as you say below, you don't know what "weakly continuous" means?

I hope you will believe me when I tell you I was not trying to
upset you, or irritate you or get you. I'm sorry if I have.

As to how I could _conjecture_ (or, as I would prefer,
available and had failed at finding out what "weakly
continuous" meant (see below). At that point, my choices were
to leave the post for another day (which my history shows
often means "never") or post with incomplete information
and make it clear how incomplete my information was.

Below you write "if you'd asked someone would have said,
no problem." But, part of the reason I posted was to get a
clarification of what "weakly continuous" meant -- I just
didn't use question marks. I have noticed over the years that
it's much easier to get a correction on USEnet than an answer
to a question. Perhaps that's not true of you -- then, good
for you, but I'd still maintain you are not typical.

What would you have done if you were me? Notice that I am not
asking what you would have done in my place, but to imagine
what it's like to be me and be faced with a similar choice.
You are a professional mathematician, I am no kind of academic
at all. You apparently take your appearance on USEnet very
seriously, while I, having very little in the way of
also have, beyond your mental mathematical resources, texts,
references and colleagues practically at your fingertips.
Well, I don't.

Would you just dropped the topic and moved on? Maybe I would
have, too, if I hadn't already invested half an hour or so
in trying to beat a definition of "weakly continuous
function" out of the World Wide Web.

I hope you're not upset that I sorta-kinda-maybe-if-God-
willing-I'm-right accused you of making a mistake. You
do make mistakes, you know, though it's very rare, I'm sure.
This was not one of them, but anyone who knew the correct
definition would know that, including the original poster.
Yes, I thought of that before I posted. If I had had a chance
of misleading him, I would have let it drop. But if, by
chance, I were right, I could have done a bit of good.

Yours,
Jim Burns

 Quote: However, the two proofs are so similar in structure that one might consider it a good clue. If I'm right in thinking that /weakly/ continuous functions f and g will map weakly convergent series to weakly convergent series and limit point to limit point, then g will take a weakly convergent series as input, then pass a weakly convergent series to f, which returns yet another weakly convergent series. Likewise for limit points. Therefore, yes, the composition of weakly continuous functions is weakly continuous. I didn't know what "weakly continuous" meant, so I attempted to google it. "Weakly convergent" is pretty easy to find, "weakly continuous function" less so. (The series x_n is weakly convergent if the inner product < x_n, y > converges for all y in the Hilbert space H.). However, I strongly suspect the definition "weakly continuous function" is analogous to "continuous function", as I expressed it above. If I'm wrong, I would appreciate being told. (Too late if I've made a fool of myself, of course.) If you don't know what "weakly continuous" means probably you should determine that first, before attempting to figure out questions regarding weakly continuous functions? Yes, you've made a fool of yourself. Nothing foolish about not knowing what "weakly continuous" means - if you'd asked someone would have said, no problem. But asking a question about weakly continuous functions _before_ finding a definition of the phrase, and without asking for the definition, that's definitely making a fool of yourself. As is conjecturing that an answer is incorrect, before learning the definition of the terms involved. A Hilbert space has a weak topology. A weakly continuous function is precisely a function which is continuous with respect to this topology. And hence what I said _does_ answer the question. Jim Burns ************************ David C. Ullrich
David C. Ullrich
science forum Guru

Joined: 28 Apr 2005
Posts: 2250

Posted: Thu Jul 13, 2006 10:20 am    Post subject: Re: Properties of weakly continuous maps in a Hilbert space

On Wed, 12 Jul 2006 22:42:53 -0400, Jim Burns <burns.87@osu.edu>
wrote:

 Quote: "David C. Ullrich" wrote: On Mon, 10 Jul 2006 20:59:26 -0400, Jim Burns burns.87@osu.edu> wrote: That doesn't seem at first to answer the question, which was about /weakly/ continuous functions. How can you possibly even _conjecture_ that this does not answer the question, since, as you say below, you don't know what "weakly continuous" means? I hope you will believe me when I tell you I was not trying to upset you, or irritate you or get you. I'm sorry if I have. As to how I could _conjecture_ (or, as I would prefer, "guess"), I had already used what resources were readily available and had failed at finding out what "weakly continuous" meant (see below). At that point, my choices were to leave the post for another day (which my history shows often means "never") or post with incomplete information and make it clear how incomplete my information was.

Those were not your only choices. You could have
simply asked what the phrase meant!

 Quote: Below you write "if you'd asked someone would have said, no problem." But, part of the reason I posted was to get a clarification of what "weakly continuous" meant -- I just didn't use question marks. I have noticed over the years that it's much easier to get a correction on USEnet than an answer to a question. Perhaps that's not true of you -- then, good for you, but I'd still maintain you are not typical.

That's not true of sci.math. When you ask a question
involving the phrase "weafly continuous" people are
going to assume you know what the phrase means.
If you ask what it means then people will tell you.

 Quote: What would you have done if you were me? Notice that I am not asking what you would have done in my place, but to imagine what it's like to be me and be faced with a similar choice.

When you put the question that way it's a little silly.
If I were you then by definition everything I do would
be the same as what you did.

What you _should_ have done is simply ask what the term
before explaining why someone else's statements about
it were wrong...)

 Quote: You are a professional mathematician, I am no kind of academic at all. You apparently take your appearance on USEnet very seriously, while I, having very little in the way of reputation to start with, have less to lose. You probably also have, beyond your mental mathematical resources, texts, references and colleagues practically at your fingertips. Well, I don't.

What's your point? Yes, I'm a professional mathematician.
Happens all the time around here that I see people talking
about things I know nothing about - if I can't find a
definition quickly on the web I ask, and someone usually
explains. sci.math is a _huge_ resource, and your access
to it is the same as mine.

 Quote: Would you just dropped the topic and moved on? Maybe I would have, too, if I hadn't already invested half an hour or so in trying to beat a definition of "weakly continuous function" out of the World Wide Web.

Again, I don't see your point. Lemme see:

"What does it mean to say that a map from a Hilbert
space to itself is `weakly continuous'?"

That didn't take a half hour for me to type.

 Quote: I hope you're not upset that I sorta-kinda-maybe-if-God- willing-I'm-right accused you of making a mistake. You do make mistakes, you know, though it's very rare, I'm sure. This was not one of them, but anyone who knew the correct definition would know that, including the original poster. Yes, I thought of that before I posted. If I had had a chance of misleading him, I would have let it drop. But if, by chance, I were right, I could have done a bit of good.

Just trying to help - that's noble.

But you got it wrong. Here's the deal: sci.math can
be a very useful thing. A lot of people learn about
a lot of things here; I know that _I've_ learned a
lot of interesting stuff here, for example. sci.math
is a useful thing because there are a lot of people
who know what they're talking about, giving away
free information.

But it's also true that there are a lot of people
who have no idea what they're talking about,
posting a lot of nonsense. sci.math would be a lot
more useful if that were not so. When you volunteer
a "correction" even though you _know_ that you
don't know what you're talking about that counts
as being part of the _problem_, not part of the
solution.

Again: yes, anyone can make a mistake. I certainly
do, more often than some posters here. When someone
says something false that's too bad, because it can
cause confusion. Usually errors get corrected, and
we just hope that anyone who sees the error will
also see the correction.

So anyone can make a mistake, there's no way to
ensure that everything posted here is correct.
Fine. _But_ when you say something on a topic
even though you _know_ that you don't know what
you're talking about, that's _not_ in the
anyone-can-make-a-mistake category! Simply not
making assertions about things you know you
know nothing about is a very simple rule that
will prevent you from making _that_ sort of
mistake.

************************

David C. Ullrich
Dave Seaman
science forum Guru

Joined: 24 Mar 2005
Posts: 527

Posted: Thu Jul 13, 2006 11:57 am    Post subject: Re: Properties of weakly continuous maps in a Hilbert space

On Wed, 12 Jul 2006 22:42:53 -0400, Jim Burns wrote:
 Quote: I didn't know what "weakly continuous" meant, so I attempted to google it. "Weakly convergent" is pretty easy to find, "weakly continuous function" less so. (The series x_n is weakly convergent if the inner product < x_n, y > converges for all y in the Hilbert space H.). However, I strongly suspect the definition "weakly continuous function" is analogous to "continuous function", as I expressed it above. If I'm wrong, I would appreciate being told. (Too late if I've made a fool of myself, of course.)

I can understand your confusion. What you didn't know (and what may be
hard to find out just by googling) is that the phrase "weakly continuous"
does not mean a special kind of continuity is involved; it's perfectly
ordinary continuity, but with respect to a particular topology (the weak
topology).

You see, continuity *always* refers to some topology or other, because
the concept is meaningless without it. It's just that in very many
situations, there is only one *standard* topology, and therefore it's
assumed the standard topology is the one intended, in the absence of any
statement to the contrary.

Not so in Hilbert space, where there are several different topologies in
common use. Some of these are called the "strong", "weak", and "norm"
topologies. So, the phrase "weakly continuous" means "continuous in the
weak topology".

I forget now what was the exact context of your question, but I think it
may have been whether a composition of weakly continuous functions is
weakly continuous. The point is, a composition of continuous functions
is always continuous, no matter what the underlying topology may be. Of
course, a function that is continuous in one topology may be
discontinuous in some other topology, but that doesn't matter here. As
long as you use the same topology throughout, any composition of
functions that are continous in the given topology will always be
continuous in that same topology.

--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>

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