mm-4189
===
Subject: Re: Infinitesimal Arithmetic
> And just saying they
> do doesn't demonstrate the truth of how that happens.
>True.
> So we can't
> really say whether it happens or not.
>But I have demonstrated how that happens, so I can.
> Hmmm. I saw the assertion that it does but missed the
demonstration.
Right. You don't subscribe to sci.logic, so you haven't seen it.
> As I recall you claimed that for a given middle term such as B in
A
> is C because A is B and B is C the middle B drops out with
which I
> agree. However I don't recollect as you demonstrated how or why
that
> occurs in mechanically exhaustive terms.
>Because I haven't.
> Well I seem to recollect as you claim to have demonstrated why the
> middle term drops out.
Your recollection is in error.
Whatever.
> I don't mean to be captious
You captious boy, you!
> but perhaps you could refresh my memory because this is exactly
where
> the whole ball of wax in formal mechanical logic comes to rest.
>Really? I don't see that.
> Of course you don't.
Of course I don't.
Whatever. Tendentious nonsense of zero probative value.
It certainly is.
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
Bytes: 1746
> but perhaps you could refresh my memory because this is exactly
where
> the whole ball of wax in formal mechanical logic comes to rest.
Really? I don't see that.
Of course you don't.
>Of course I don't.
> Whatever. Tendentious nonsense of zero probative value.
It certainly is.
Then at least we agree on something.
~v~~
===
Subject: Re: Infinitesimal Arithmetic
>The premises upon which the logician's claim to demonstrate
>true conclusions from true premises rests are not claimed,
>in turn, to be demonstrated.
> So where does the truth of their claims come from?
example, it's plainly true that I have a red car
>and I have a blue car is true if and only if I
>have a red car is true and I have a blue car is
>true.
Self evidential nature? Self evidential nature?? Is this a joke?
> In other words plainly the truth of their claims is just so much
> guesswork in terms of what they consider self evidential? Truly
> appalling that a serious thinker
> wouldn't even think twice about
> extraneous appeals and special pleading as a substitute for truth.
What, you don't think that I have a red car and I have a blue
car is true if and only if I have a red car is true and I have
a blue car is true?
Or that if _I_ think that, it's only guesswork on my part?
>It's plainly true, as you have acknowledged, that
>John has a big dog implies John has a dog.
I acknowledged no suchuvva thing.
/quote/
> Does not John has a big dog imply John has a dog?
> Sure.
/endquote/
> What I acknowledged was that
> whatever is implied the one thing no formal logician, mathematiker, or
> empiric could ever demonstrate true was what that implication was
> because, just as you acknowledge above, all operate on implicit
> assumptions of truth and guesswork rather than demonstrations of
> truth.
Now you're saying the truth of that implication is not evident?
Or that its self-evidence is not sufficient grounds for the
recognition and acknowledgement of its truth?
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
Bytes: 5391
The premises upon which the logician's claim to demonstrate
>true conclusions from true premises rests are not claimed,
>in turn, to be demonstrated.
So where does the truth of their claims come from?
>example, it's plainly true that I have a red car
>and I have a blue car is true if and only if I
>have a red car is true and I have a blue car is
>true.
> Self evidential nature? Self evidential nature?? Is this a joke?
> In other words plainly the truth of their claims is just so much
> guesswork in terms of what they consider self evidential? Truly
> appalling that a serious thinker
>
Oh well every dog has his day I expect, even serious thinkers.
> wouldn't even think twice about
> extraneous appeals and special pleading as a substitute for truth.
What, you don't think that I have a red car and I have a blue
>car is true if and only if I have a red car is true and I have
>a blue car is true?
Or that if _I_ think that, it's only guesswork on my part?
Certainly it's only guesswork if you think you possess some mystical
necessary transcendental comprehension of the predicates involved.
What if blue meant not car? Or do you have some self evidential
and self reverential insight to the contrary?
>It's plainly true, as you have acknowledged, that
>John has a big dog implies John has a dog.
> I acknowledged no suchuvva thing.
/quote/
> Does not John has a big dog imply John has a dog?
Sure.
/endquote/
> What I acknowledged was that
> whatever is implied the one thing no formal logician, mathematiker, or
> empiric could ever demonstrate true was what that implication was
> because, just as you acknowledge above, all operate on implicit
> assumptions of truth and guesswork rather than demonstrations of
> truth.
Now you're saying the truth of that implication is not evident?
Of course not. Not without special knowledge of the predicates
involved. In other words unless you share some special insight not
evident in the proposition itself you can hardly claim to have shown
whether it's true. The point being that the propositional predicates
involved are never self evident unless one is of a mystical bent and
is on an intergalactic need to know basis with the universe at large.
>Or that its self-evidence is not sufficient grounds for the
>recognition and acknowledgement of its truth?
Come, come, Herb. We all at one time or another believe ourselves to
be in mystic communion with the necessary and essential meaning of
everything great and small. Doesn't mean it's true however. If it were
you wouldn't have made the rookie mistake of assuming you knew what
predicates like red blue and so on necessarily had to mean to the
exclusion of other meanings in the first place.
That's exactly why we need demonstrations and not just self evidential
self reverential assumptions of truth to manage our meager store of
actual knowledge with some kind of truth for a change. We're hardly
ever in a position to judge every aspect of every predicate and when
some implicit self contradiction crops up we're forced to reconsider
our perspective on the universe at large. Just look what happened
when Galileo insisted on revisiting the farmer's almanac perspective
on the perceptually obvious rotation of the sun around the earth.
Or furthermore consider the significance of not contradiction or
differences. Almost everyone just sees these as nuisance predicates
to be unceremoniously shoveled off to one side in pursuit of holistic
truth. Yet when actual demonstrations of truth are considered they
turn out to be of primary and ultimate significance to any conceivable
demonstration of truth.
~v~~
===
Subject: Re: Infinitesimal Arithmetic
The premises upon which the logician's claim to demonstrate
>true conclusions from true premises rests are not claimed,
>in turn, to be demonstrated.
So where does the truth of their claims come from?
>example, it's plainly true that I have a red car
>and I have a blue car is true if and only if I
>have a red car is true and I have a blue car is
>true.
> Self evidential nature? Self evidential nature?? Is this a
joke?
> In other words plainly the truth of their claims is just so much
> guesswork in terms of what they consider self evidential? Truly
> appalling that a serious thinker
> Oh well every dog has his day I expect, even serious thinkers.
> wouldn't even think twice about
> extraneous appeals and special pleading as a substitute for truth.
What, you don't think that I have a red car and I have a blue
>car is true if and only if I have a red car is true and I have
>a blue car is true?
Or that if _I_ think that, it's only guesswork on my part?
Certainly it's only guesswork if you think you possess some mystical
> necessary transcendental comprehension of the predicates involved.
It's only necessary to know what and means.
> What if blue meant not car?
What if pigs were pigeons?
> Or do you have some self evidential
> and self reverential insight to the contrary?
Yeah. It's called speaking English.
>It's plainly true, as you have acknowledged, that
>John has a big dog implies John has a dog.
> I acknowledged no suchuvva thing.
/quote/
> Does not John has a big dog imply John has a dog?
> Sure.
/endquote/
> What I acknowledged was that
> whatever is implied the one thing no formal logician, mathematiker, or
> empiric could ever demonstrate true was what that implication was
> because, just as you acknowledge above, all operate on implicit
> assumptions of truth and guesswork rather than demonstrations of
> truth.
Now you're saying the truth of that implication is not evident?
Of course not. Not without special knowledge of the predicates
> involved.
No, the particular predicates don't matter. Haven't you
ever heard that an inference is valid by virtue of its form?
> In other words unless you share some special insight not
> evident in the proposition itself you can hardly claim to have shown
> whether it's true. The point being that the propositional predicates
> involved are never self evident unless one is of a mystical bent and
> is on an intergalactic need to know basis with the universe at large.
That's cute. I like that.
>Or that its self-evidence is not sufficient grounds for the
>recognition and acknowledgement of its truth?
Come, come, Herb. We all at one time or another believe ourselves to
> be in mystic communion with the necessary and essential meaning of
> everything great and small. Doesn't mean it's true however. If it were
> you wouldn't have made the rookie mistake of assuming you knew what
> predicates like red blue and so on necessarily had to mean to the
> exclusion of other meanings in the first place.
What does the meaning of red and blue have to do with
John has a big dog implying John has a dog?
> That's exactly why we need demonstrations and not just self evidential
> self reverential assumptions of truth to manage our meager store of
> actual knowledge with some kind of truth for a change. We're hardly
> ever in a position to judge every aspect of every predicate and when
> some implicit self contradiction crops up we're forced to reconsider
> our perspective on the universe at large. Just look what happened
> when Galileo insisted on revisiting the farmer's almanac perspective
> on the perceptually obvious rotation of the sun around the earth.
Galileo gave a logical demonstration of the fact that the earth
revolves about the sun?
> Or furthermore consider the significance of not contradiction or
> differences. Almost everyone just sees these as nuisance predicates
> to be unceremoniously shoveled off to one side in pursuit of holistic
> truth. Yet when actual demonstrations of truth are considered they
> turn out to be of primary and ultimate significance to any conceivable
> demonstration of truth.
If you're saying that contradictions are necessarily false, every-
one knows that.
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
Bytes: 8095
>Kindly take the trouble to _carefully_ re-read the sentence I
have
>written. It is about _all_ sentences compounded by means of
truth-
>functional operators, not just sentences that purport to be
inferences.
> I replied to this in my preceeding post, Herb. I suppose it's
possible
> to have purely anecdotal assemblages of predicates but that's not
what
> I thought you had in mind.
Yes, well, at this point I'm trying to establish the concept of
>a truth-function -- a function that assigns the value true or
>the value false to compound sentences.
But the functions you refer to don't determine truth they only
> manipulate bits.
>They determine the truth of compound sentences from the
>truth-value (truth or falsehood) of the sentences of which
>they are compounded.
> Just as I said.They don't determine truth they only manipulate
bits.
What, compound sentences aren't true or false?
Predicate reversals are both true and false. Elephants are animals
> is true but animals are elephants is false.
What, compound sentences aren't true or false?
Besides, Elephants are animals and Animals are elephants
are obviously two different propositions.
And they're not compound sentences anyway.
>More generally, in an n-valued system a truth-function
>determines which of the n truth-values a compound sentence
>takes from the truth-values of its components.
> Sure. And in an n-valued home run system Babe Ruth hits home runs.
Jeez, Lester, I only included the bit about n-valued system
>because you maintain that sentences take three truth values:
>true, false, and problematic.
So what? That isn't quite the same as saying a n-value truth system
> has an n-bit binary representation.
Well, forget it, then.
> You're missing the point about the
> truth of predicate relations and reversals.
No doubt.
> Just adding extra bits
> doesn't address or resolve the problem of demonstrations of truth in
> non binary terms.
I'll take your word for it.
> [...]
>Material implication is a binary truth function equivalent to
>Either not-A or B.
Well from my perspective, Herb, absent truth there are no
truth
> functions.
>That's true, Lester.
> So show us some of this truth truth functions function.
I already have. You have already agreed that the truth
>of I have a red car and I have a blue car is a function
>of the truth of I have a red car and the truth of I have
>a blue car:
Well you've already agreed while I've agreed that your just saying
> something is so doesn't demonstrate that it is so as attested in the
> present instance by your saying I've agreed to something I haven't
> except in highly restrictive terms involving specialized knowledge.
>Surely you would agree that the truth of I have a red car and
>I have a blue car is a function of the truth of I have a red
>car and the truth of I have a blue car? Specifically, it is
>true if and only if its two component sentences are true, and
>it is false otherwise -- no?
> Yes certainly for this highly restrictive example.
I assume that by Yes certainly you meant That is true, Herb.
No I don't, Herb. I call it a highly restrictive example and you take
> it to be a general agreement which it is not unless you and I both
> have specialized knowledge regarding the meaning of the predicates
> involved which you seem to think we do when in point of fact we don't.
I interpret you saying Yes certainly for this highly restrictive
example as saying Yes certainly for this example.
--
hz
>So there's some truth for you, Lester.
The situation is, of course, not altered in principle
>if we allow a third truth value: Will you maintain
>that I have a red car and I have a blue car is
>problematic when either of I have a red car or
>I have a blue car is problematic? What if one
>of I have a red car and I have a blue car is
>false, and one is problematic?
The use of problematic here, Herb, is not to indicate just that we
> don't know the truth in a given instance but rather we don't know if
> the predicate ordering is correct. That's the whole problem. There is
> nothing in the predicates themselves to indicate what the ordering
> should necessarily be.
What if I have a steel idea and I have a red idea? Applying
> analogous logic pro forma you might be inclined to look for ideas in a
> materials warehouse. And without non pro forma specialized knowledge
> regarding the significance of the various predicates and the relations
> between and among them that's exactly what might happen.
Or are these not the sort of statements that can be
>problematic?
All predicate relations are problematic except for not not and the
> like because we have no way to tell necessarily what the ordering of
> predicates should be just based on the statement of the problem alone.
Finally (for the moment) what will be the case if
>some statement A is problematic: will A or not-A
>be problematic, true, or false?
Predicate relations are always problematic. That was the point I tried
> to make in connection with your statement of sets of predicates. They
> don't just exist in what I would, for want of a better word, call flat
> sets or groups. The relations between and among predicates represent
> more a polydimensional liason between and among predicates which might
> or might not be correct under particular conditions.
Hell truth itself and demonstrations of truth are relatively mundane
> and straight forward. The problem and complexity arise in connection
> with deciphering the significance of predicates in relation to one
> another. So just nailing one predicate to another with some kind of
> conjunction and demanding it exhibit only a single binary truth value
> completely misrepresents the mechanics involved in determining truth.
~v~~
--
===
Subject: Re: Infinitesimal Arithmetic
>Surely you would agree that the truth of I have a red car and
>I have a blue car is a function of the truth of I have a red
>car and the truth of I have a blue car? Specifically, it is
>true if and only if its two component sentences are true, and
>it is false otherwise -- no?
Yes certainly for this highly restrictive example.
>I assume that by Yes certainly you meant That is true, Herb.
> No I don't, Herb. I call it a highly restrictive example and you take
> it to be a general agreement which it is not unless you and I both
> have specialized knowledge regarding the meaning of the predicates
> involved which you seem to think we do when in point of fact we don't.
I interpret you saying Yes certainly for this highly restrictive
>example as saying Yes certainly for this example.
Certainly not for this example because the highly restrictive
qualification denotes specialized knowlege and a common understanding
for the predicates involved which cannot just be assumed.
It appears to me you want easy yes-no, true-false answers to questions
which beg the questions by presuming the specialized knowledge and a
common understanding of terms needed for agreement. In other words
they're complex questions to which you want simple answers. Then you
want to argue the general validity of such techniques extended to all
analogous questions as if the terms involved in those questions were
also intuitively obvious to the casual observer as well.
It doesn't happen that way because you can't claim answers which
already presuppose your own way of looking at logic purely in terms of
yes-no, true-false architecture. In fact if you tried to posit such a
system without the ability to demand simple answers from others you'd
find out in a hurry that all you'd really wind up with would be a
purely personal and arbitrary exposition which relied on your own
prejudices regarding the meaning of logic and various predicates.
~v~~
===
Subject: Re: Infinitesimal Arithmetic
<j6a873lerqbbe865eqhbqeink49dc3a9p6@4ax.com>
<4674D729.DCD1BA5E@gmail.com>
<co1b731est94hvh8n8jioc3buds0nmb816@4ax.com>
<4677946E.DF15C3AC@gmail.com>
<d1si73dob98s0ihddmdkip43gloqh8bevr@4ax.com>
<467B154A.CF567C04@gmail.com>
<fc3o73t9hm73chjp7guuqra2h9a0ofj4es@4ax.com>
<467D8A33.F79272E9@gmail.com>
<vafr73d30dtouhpdqu51s5o33fb9us8s73@4ax.com>
<467F47E9.D88F2EE4@gmail.com>
<vp40831sp6ads5hvk99mtft1s29ob4se5c@4ax.com>
<46823429.3C55A1C4@gmail.com>
<moc583tqmp2oiodsmm5jr2upboacfqf7mc@4ax.com>
Bytes: 4308
>Surely you would agree that the truth of I have a red car and
>I have a blue car is a function of the truth of I have a red
>car and the truth of I have a blue car? Specifically, it is
>true if and only if its two component sentences are true, and
>it is false otherwise -- no?
Yes certainly for this highly restrictive example.
>I assume that by Yes certainly you meant That is true, Herb.
> No I don't, Herb. I call it a highly restrictive example and you take
> it to be a general agreement which it is not unless you and I both
> have specialized knowledge regarding the meaning of the predicates
> involved which you seem to think we do when in point of fact we don't.
I interpret you saying Yes certainly for this highly restrictive
>example as saying Yes certainly for this example.
Certainly not for this example because the highly restrictive
> qualification denotes specialized knowlege and a common understanding
> for the predicates involved which cannot just be assumed.
It appears to me you want easy yes-no, true-false answers to questions
> which beg the questions by presuming the specialized knowledge and a
> common understanding of terms needed for agreement. In other words
> they're complex questions to which you want simple answers. Then you
> want to argue the general validity of such techniques extended to all
> analogous questions as if the terms involved in those questions were
> also intuitively obvious to the casual observer as well.
It doesn't happen that way because you can't claim answers which
> already presuppose your own way of looking at logic purely in terms of
> yes-no, true-false architecture. In fact if you tried to posit such a
> system without the ability to demand simple answers from others you'd
> find out in a hurry that all you'd really wind up with would be a
> purely personal and arbitrary exposition which relied on your own
> prejudices regarding the meaning of logic and various predicates.
~v~~
For every observation made by a freak observer spontaneously
materializing from Hawking radiation or thermal fluctuations, there
are trillions upon trillions of observations made by regular observers
who have evolved on planets like our own and who make veridical
observations of the universe.
===
Subject: Re: Infinitesimal Arithmetic
>Surely you would agree that the truth of I have a red car and
>I have a blue car is a function of the truth of I have a red
>car and the truth of I have a blue car? Specifically, it
is
>true if and only if its two component sentences are true, and
>it is false otherwise -- no?
> Yes certainly for this highly restrictive example.
>I assume that by Yes certainly you meant That is true, Herb.
> No I don't, Herb. I call it a highly restrictive example and you take
> it to be a general agreement which it is not unless you and I both
> have specialized knowledge regarding the meaning of the predicates
> involved which you seem to think we do when in point of fact we
don't.
>I interpret you saying Yes certainly for this highly restrictive
>example as saying Yes certainly for this example.
> Certainly not for this example because the highly restrictive
> qualification denotes specialized knowlege and a common understanding
> for the predicates involved which cannot just be assumed.
> It appears to me you want easy yes-no, true-false answers to questions
> which beg the questions by presuming the specialized knowledge and a
> common understanding of terms needed for agreement. In other words
> they're complex questions to which you want simple answers. Then you
> want to argue the general validity of such techniques extended to all
> analogous questions as if the terms involved in those questions were
> also intuitively obvious to the casual observer as well.
> It doesn't happen that way because you can't claim answers which
> already presuppose your own way of looking at logic purely in terms of
> yes-no, true-false architecture. In fact if you tried to posit such a
> system without the ability to demand simple answers from others you'd
> find out in a hurry that all you'd really wind up with would be a
> purely personal and arbitrary exposition which relied on your own
> prejudices regarding the meaning of logic and various predicates.
> ~v~~
For every observation made by a freak observer spontaneously
>materializing from Hawking radiation or thermal fluctuations, there
>are trillions upon trillions of observations made by regular observers
>who have evolved on planets like our own and who make veridical
>observations of the universe.
Howzzatagain??
~v~~
===
Subject: Re: Infinitesimal Arithmetic
Bytes: 4301
>Surely you would agree that the truth of I have a red car and
>I have a blue car is a function of the truth of I have a red
>car and the truth of I have a blue car? Specifically, it is
>true if and only if its two component sentences are true, and
>it is false otherwise -- no?
Yes certainly for this highly restrictive example.
>I assume that by Yes certainly you meant That is true, Herb.
> No I don't, Herb. I call it a highly restrictive example and you take
> it to be a general agreement which it is not unless you and I both
> have specialized knowledge regarding the meaning of the predicates
> involved which you seem to think we do when in point of fact we don't.
I interpret you saying Yes certainly for this highly restrictive
>example as saying Yes certainly for this example.
Certainly not for this example because the highly restrictive
> qualification denotes specialized knowlege and a common understanding
> for the predicates involved which cannot just be assumed.
It just relies on a common understanding of the meaning and usage
of the sentential connective and.
And Yes certainly for this highly restrictive example certainly
means Yes certainly for this example as anyone who speaks English
understands, despite your denial of the obvious.
> It appears to me you want easy yes-no, true-false answers to questions
> which beg the questions by presuming the specialized knowledge and a
> common understanding of terms needed for agreement.
The sentential connective and means the same thing regardless of
the terms involved in the component sentences.
> In other words
> they're complex questions to which you want simple answers. Then you
> want to argue the general validity of such techniques extended to all
> analogous questions as if the terms involved in those questions were
> also intuitively obvious to the casual observer as well.
Whether the meanings of the terms involved in the component
sentences are known to the casual observer or not, the
meaning of the connective and is not thereby changed.
> It doesn't happen that way because you can't claim answers which
> already presuppose your own way of looking at logic purely in terms of
> yes-no, true-false architecture.
So _you_ mean something different by the sentential connective and?
> In fact if you tried to posit such a
> system without the ability to demand simple answers from others you'd
> find out in a hurry that all you'd really wind up with would be a
> purely personal and arbitrary exposition which relied on your own
> prejudices regarding the meaning of logic and various predicates.
Well, it's nice to know that you see the value of consensus here.
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
> I replied to this in my preceeding post, Herb. I suppose it's
possible
> to have purely anecdotal assemblages of predicates but that's
not what
> I thought you had in mind.
>Yes, well, at this point I'm trying to establish the concept of
>a truth-function -- a function that assigns the value true or
>the value false to compound sentences.
> But the functions you refer to don't determine truth they only
> manipulate bits.
They determine the truth of compound sentences from the
>truth-value (truth or falsehood) of the sentences of which
>they are compounded.
Just as I said.They don't determine truth they only manipulate
bits.
>What, compound sentences aren't true or false?
> Predicate reversals are both true and false. Elephants are animals
> is true but animals are elephants is false.
What, compound sentences aren't true or false?
If you mean necessarily either true or false then the compound
sentence it is an animal and it is an elephant are both true and
false according to how the relation between the predicates animal
and elephant is established. Even assuming it is an elephant
animal is true the converse it is an animal elephant is false.
>Besides, Elephants are animals and Animals are elephants
>are obviously two different propositions.
But animals are elephants represents the combination of predicates
and sentences it is animal and it is elephant with the ordering of
predicates ambiguous.
>And they're not compound sentences anyway.
Sure they are. Every predicate represents a sentence when prefixed by
it is or they are or whatever.
~v~~
===
Subject: Re: Infinitesimal Arithmetic
> I replied to this in my preceeding post, Herb. I suppose it's
possible
> to have purely anecdotal assemblages of predicates but that's
not what
> I thought you had in mind.
>Yes, well, at this point I'm trying to establish the concept of
>a truth-function -- a function that assigns the value true
or
>the value false to compound sentences.
> But the functions you refer to don't determine truth they
only
> manipulate bits.
They determine the truth of compound sentences from the
>truth-value (truth or falsehood) of the sentences of which
>they are compounded.
Just as I said.They don't determine truth they only manipulate
bits.
>What, compound sentences aren't true or false?
> Predicate reversals are both true and false. Elephants are animals
> is true but animals are elephants is false.
What, compound sentences aren't true or false?
If you mean necessarily either true or false then the compound
> sentence it is an animal and it is an elephant are both true and
> false
The compound sentence It is an animal and it is an elephant is
not at once both true and false.
It is just ambiguous, and it is clearly true or clearly false
just as soon as one settles upon what meaning one intends.
This is no more problematical than noting that the word
kid is ambiguous, since it can mean child, baby goat,
or joking.
It is no more problematical than noting that the word
or has an exclusive sense (Latin: aut) and an inclusive
sense (Latin: vel).
It is a simple case of amphiboly: an ambiguous grammatical
structure in a sentence such as I once shot an elephant
in my pajamas (Definition and example from Wikipedia).
> according to how the relation between the predicates animal
> and elephant is established. Even assuming it is an elephant
> animal is true the converse it is an animal elephant is false.
The truth of it is an elephant animal and the truth of it
is an animal elephant is established just as soon as one settles
the ambiguity of their grammatical construction by deciding upon
what meanings are intended.
This in no way has any bearing on the fact that compound sentences
are true or false.
It just is a plain fact that sentences, like words, sometimes
have more than one meaning.
>Besides, Elephants are animals and Animals are elephants
>are obviously two different propositions.
But animals are elephants represents the combination
What the combination? There are several ways to combine
the sentences It is an animal and It is an elephant --
you'll have to be more specific.
> of predicates
> and sentences it is animal and it is elephant with the ordering
of
> predicates ambiguous.
Well, just as soon as your resolve the ambiguities, so is the truth
or falsehood of the sentences Animals are elephants and Elephants
are animals resolved.
>And they're not compound sentences anyway.
Sure they are. Every predicate represents a sentence when prefixed by
> it is or they are or whatever.
If they are compound, it is not for that reason. It smokes is no
more a compound sentence than Jones smokes is compound.
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
> It is a simple case of amphiboly: an ambiguous grammatical
> structure in a sentence such as I once shot an elephant
> in my pajamas (Definition and example from Wikipedia).
Captain Jeffrey T. Spaulding, _Animal_Crackers_, [1930].
--
Michael Press
===
Subject: Re: Infinitesimal Arithmetic
I on the other hand just say that in some compounding of
differences
> such as (A - B) - (C - B) B drops out because B - B =0
and that
> the whole of logic, syllogistic, mathematical, or empirical
represents
> only the truth of the compounding of differences in various
contexts.
Since the whole of logic, syllogistic, mathematical, or empirical
represents only the truth of the compounding of differences in
various contexts, perhaps you would care to state plainly what a
difference A - B is supposed to be.
For the kids, Lester. Do it for the kids.
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
>
I on the other hand just say that in some compounding of
differences
> such as (A - B) - (C - B) B drops out because B - B =0
and that
> the whole of logic, syllogistic, mathematical, or empirical
represents
> only the truth of the compounding of differences in various
contexts.
>Since the whole of logic, syllogistic, mathematical, or empirical
>represents only the truth of the compounding of differences in
>various contexts, perhaps you would care to state plainly what a
>difference A - B is supposed to be.
The basis of the whole of logic, syllogistic, mathematical, or
empirical. If you find this a trifle circular the best I can suggest
is that a difference is that whose alternatives, in different from
differences not not or the contradiction of contradiction are
self contradictory and together form a mechanically exhaustive
reduction for everything true and false. Otherwise ask the kids.
>For the kids, Lester. Do it for the kids.
You mean for the goats?
~v~~
===
Subject: Re: Infinitesimal Arithmetic
 I on the other hand just say that in some compounding of
differences
> such as (A - B) - (C - B) B drops out because B - B =0
and that
> the whole of logic, syllogistic, mathematical, or empirical
represents
> only the truth of the compounding of differences in various
contexts.
>Since the whole of logic, syllogistic, mathematical, or empirical
>represents only the truth of the compounding of differences in
>various contexts, perhaps you would care to state plainly what a
>difference A - B is supposed to be.
The basis of the whole of logic, syllogistic, mathematical, or
> empirical. If you find this a trifle circular
Just a tad.
> the best I can suggest
> is that a difference is that whose alternatives, in different from
> differences not not or the contradiction of contradiction are
> self contradictory and together form a mechanically exhaustive
> reduction for everything true and false.
So when you say that the difference B - B = 0, you're saying that
the alternatives to B - B (and hence 0) are self-contradictory?
> Otherwise ask the kids.
Sounds like good advice.
>For the kids, Lester. Do it for the kids.
You mean for the goats?
It's nice that we can kid around like this.
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
 I on the other hand just say that in some compounding of
differences
> such as (A - B) - (C - B) B drops out because B - B
=0 and that
> the whole of logic, syllogistic, mathematical, or empirical
represents
> only the truth of the compounding of differences in various
contexts.
>Since the whole of logic, syllogistic, mathematical, or empirical
>represents only the truth of the compounding of differences in
>various contexts, perhaps you would care to state plainly what a
>difference A - B is supposed to be.
> The basis of the whole of logic, syllogistic, mathematical, or
> empirical. If you find this a trifle circular
Just a tad.
> the best I can suggest
> is that a difference is that whose alternatives, in different from
> differences not not or the contradiction of contradiction are
> self contradictory and together form a mechanically exhaustive
> reduction for everything true and false.
So when you say that the difference B - B = 0, you're saying that
>the alternatives to B - B (and hence 0) are self-contradictory?
Yeah I think so. My main problem is that this is a tough question that
gets right to the heart of cognitive mechanics and I don't know
exactly how to interpret those mechanics. Obviously B - B = 0 and in
that sense becomes useless for further interpretive purposes in terms
of anything that B may be. However I'm reluctiant to attach the term
self contradictory to that.
When I say self contradictory I'm basically referring to expressions
such as not not the contradiction of contradiction different from
differences or even - - which deny themselves in mechanical terms.
I don't see that B - B = 0 does that. So I suppose cognitively what we
wind up with is just something which can be mechanized but without
information available in the form of differences.
In the past I've toyed with the notion that if through some mechanism
we inadvertently predicate something of itself we get as a result self
contradiction since there is supposed to be some difference between
subject and predicate. In other words if in the context of cognitive
processing we wind up with for example A is A then the result is
both that A is A and A is not A because in general terms a subject
is its predicate but a predicate is not its subject.
I'm not completely satisfied with this explanation but at the moment
it's the best I can offer.However I'd like to emphasize here that what
I'm after is a fully mechanical explanation for all these effects and
not just miscellaneous terminology having no mechanical significance.
> Otherwise ask the kids.
Sounds like good advice.
>For the kids, Lester. Do it for the kids.
> You mean for the goats?
It's nice that we can kid around like this.
Better than being a horse's ass I expect as long as we focus on truth.
~v~~
===
Subject: Re: Infinitesimal Arithmetic
Bytes: 2608
> At the very least you need to demonstrate
> the truth of how premises lead to conclusions.
>Yes. That's why we study logic.
> Agreed. But I don't see a recognition in formal and mathematical
logic
> of any need to demonstrate ones premises and hence conclusions
true.
Right. It's very freeing.
I don't doubt that at all. Too bad student demonstrations of
theorems
> aren't comparably freed.
>But they are.
> Student theorems are freed from demonstrating the truth of their
> theorems? Curiouser and curioser.
What, you proved the truth of Euclid's parallel postulate when
>you were a student?
Postulates aren't theorems. But I wouldn't mind having a go at it
> today.
I see. As a student, you demonstrated the truth of Euclid's
theorems _without_ demonstrating the truth of his postulates.
That _is_ impressive. How did you do that?
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
> At the very least you need to demonstrate
> the truth of how premises lead to conclusions.
>Yes. That's why we study logic.
> Agreed. But I don't see a recognition in formal and mathematical
logic
> of any need to demonstrate ones premises and hence conclusions
true.
>Right. It's very freeing.
> I don't doubt that at all. Too bad student demonstrations of
theorems
> aren't comparably freed.
But they are.
Student theorems are freed from demonstrating the truth of their
> theorems? Curiouser and curioser.
>What, you proved the truth of Euclid's parallel postulate when
>you were a student?
> Postulates aren't theorems. But I wouldn't mind having a go at it
> today.
I see. As a student, you demonstrated the truth of Euclid's
>theorems _without_ demonstrating the truth of his postulates.
That _is_ impressive. How did you do that?
The same way everyone else did, by assuming the truth of the postulate
and proceeding apace.
~v~~
===
Subject: Re: Infinitesimal Arithmetic
> At the very least you need to demonstrate
> the truth of how premises lead to conclusions.
>Yes. That's why we study logic.
> Agreed. But I don't see a recognition in formal and
mathematical logic
> of any need to demonstrate ones premises and hence conclusions
true.
>Right. It's very freeing.
> I don't doubt that at all. Too bad student demonstrations of
theorems
> aren't comparably freed.
But they are.
Student theorems are freed from demonstrating the truth of their
> theorems? Curiouser and curioser.
>What, you proved the truth of Euclid's parallel postulate when
>you were a student?
> Postulates aren't theorems. But I wouldn't mind having a go at it
> today.
I see. As a student, you demonstrated the truth of Euclid's
>theorems _without_ demonstrating the truth of his postulates.
That _is_ impressive. How did you do that?
The same way everyone else did, by assuming the truth of the postulate
> and proceeding apace.
'Nuff said.
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
Bytes: 2373
3) It does not follow that not demonstrating the truth of
> premises or conclusion precludes demonstating how the
> truth of the conclusion follows from true premises in
> a valid argument.
Except that is a premise whose truth cannot be demonstrated either.
>If so, then so is that.
> Is that true or just a guess?
Are those mutually exclusive alternatives?
What are mutually exclusive alternatives are assumptions of truth as
> opposed to demonstrations of truth.
That's just an assumption.
> You seem to think, Herb, that any conclusion you reach is not also a
> premise subject to further inference.
I can't imagine what I might have said to make you think that, since
I don't, in fact, think that.
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
Bytes: 2912
>3) It does not follow that not demonstrating the truth of
> premises or conclusion precludes demonstating how the
> truth of the conclusion follows from true premises in
> a valid argument.
> Except that is a premise whose truth cannot be demonstrated
either.
If so, then so is that.
Is that true or just a guess?
>Are those mutually exclusive alternatives?
> What are mutually exclusive alternatives are assumptions of truth as
> opposed to demonstrations of truth.
That's just an assumption.
So self evidential and self reverential assumptions of truth are okay
where you're concerned but not so okay where us kids are concerned? Or
does that just mean us kids have got your goat?
> You seem to think, Herb, that any conclusion you reach is not also a
> premise subject to further inference.
I can't imagine what I might have said to make you think that, since
>I don't, in fact, think that.
Except I'm having some difficulty determining what you do think. Then
your demonstrations of truth for conclusions in terms of premises are
not in fact true? Or is that not what you think either? I mean if your
conclusions do represent premises subject to further inference surely
that would mean all premises would also be subject to demonstrations
of truth in their own right?
~v~~
===
Subject: Re: Infinitesimal Arithmetic
>3) It does not follow that not demonstrating the truth of
> premises or conclusion precludes demonstating how the
> truth of the conclusion follows from true premises in
> a valid argument.
> Except that is a premise whose truth cannot be demonstrated
either.
If so, then so is that.
Is that true or just a guess?
>Are those mutually exclusive alternatives?
> What are mutually exclusive alternatives are assumptions of truth as
> opposed to demonstrations of truth.
That's just an assumption.
So self evidential and self reverential assumptions of truth are okay
> where you're concerned but not so okay where us kids are concerned?
Ah! When others make assumptions, that's bad, but Lester need not
hold himself to the same standard.
> Or
> does that just mean us kids have got your goat?
You're not a kid, you old goat. ;-)
> You seem to think, Herb, that any conclusion you reach is not also a
> premise subject to further inference.
I can't imagine what I might have said to make you think that, since
>I don't, in fact, think that.
Except I'm having some difficulty determining what you do think. Then
> your demonstrations of truth for conclusions in terms of premises are
> not in fact true?
If the premises are true, then the conclusion is. I thought
that point was understood.
> Or is that not what you think either? I mean if your
> conclusions do represent premises subject to further inference surely
> that would mean all premises would also be subject to demonstrations
> of truth in their own right?
How did premises subject to further inference become premises
subject to demonstrations of truth?
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
>3) It does not follow that not demonstrating the truth of
> premises or conclusion precludes demonstating how the
> truth of the conclusion follows from true premises in
> a valid argument.
> Except that is a premise whose truth cannot be demonstrated
either.
>If so, then so is that.
> Is that true or just a guess?
Are those mutually exclusive alternatives?
What are mutually exclusive alternatives are assumptions of truth as
> opposed to demonstrations of truth.
>That's just an assumption.
> So self evidential and self reverential assumptions of truth are okay
> where you're concerned but not so okay where us kids are concerned?
Ah! When others make assumptions, that's bad, but Lester need not
>hold himself to the same standard.
You're suggesting demonstrations of truth are the same as assumptions
of truth? You're the one who claims self evidential assumptions as the
basis for your conclusions of truth. I prefer demonstrations of truth.
> Or
> does that just mean us kids have got your goat?
You're not a kid, you old goat. ;-)
Just as long as you don't get my goat.
> You seem to think, Herb, that any conclusion you reach is not also a
> premise subject to further inference.
>I can't imagine what I might have said to make you think that, since
>I don't, in fact, think that.
> Except I'm having some difficulty determining what you do think. Then
> your demonstrations of truth for conclusions in terms of premises are
> not in fact true?
If the premises are true, then the conclusion is. I thought
>that point was understood.
But you offer no way to determine the truth of premises except self
evidential mystical communion. So if you in fact don't consider
conclusions mechanically different from premises and you can't
demonstrate the truth of premises then I really don't see how you can
claim the truth of conclusions. And all I'm trying to do is just get
the picture straight.
> Or is that not what you think either? I mean if your
> conclusions do represent premises subject to further inference surely
> that would mean all premises would also be subject to demonstrations
> of truth in their own right?
How did premises subject to further inference become premises
>subject to demonstrations of truth?
They're one and the same. Premises are both subject to further
inference and subject to demonstrations of truth as well. The only
difference for conclusions is that they already are demonstrated of
various premises.
If you can't demonstrate the truth of premises and you consider
conclusions premises subject to further inference then I don't
understand how you can claim to establish the truth of conclusions by
means of premises whose truth you can't demonstrate. Just saying if
premises are true doesn't prove the truth of conclusions based on
those premises, nor, for that matter does it prove the truth of that
basic contention either.
~v~~
===
Subject: Re: Infinitesimal Arithmetic
>Well, I certainly _do_ agree with the following:
> the truth of conclusions is not
> established regardless of formalisms used to draw conclusions from
> premises which are not themselves demonstrated true.
>except I might quibble over demonstrated true: I'd say instead
>established as true or some such.
> Established true? And what is that when it's at home?
established means the same as what it meant in the sentence
>of yours I was responding to.
The point being that your and my understandings of established
> appear to be entirely different. Mine is mechanically tautological and
> I can't tell what you mean by established as true.My use of the word
> was specifically in regard to formalisms which don't establish truth
> and not in opposition to demonstrated true whereas your use of the
> word appears to be interpreted as an alternative to demonstrations.
> So I hardly see that your meaning can be the same as mine.
Since propositions can be established as true other than by
logical demonstration, my use of the word establish is
wider than and inclusive of the meaning of demonstrate.
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
Well, I certainly _do_ agree with the following:
> the truth of conclusions is not
> established regardless of formalisms used to draw conclusions from
> premises which are not themselves demonstrated true.
except I might quibble over demonstrated true: I'd say instead
>established as true or some such.
Established true? And what is that when it's at home?
>established means the same as what it meant in the sentence
>of yours I was responding to.
> The point being that your and my understandings of established
> appear to be entirely different. Mine is mechanically tautological and
> I can't tell what you mean by established as true.My use of the word
> was specifically in regard to formalisms which don't establish truth
> and not in opposition to demonstrated true whereas your use of the
> word appears to be interpreted as an alternative to demonstrations.
> So I hardly see that your meaning can be the same as mine.
Since propositions can be established as true other than by
>logical demonstration, my use of the word establish is
>wider than and inclusive of the meaning of demonstrate.
No since about it. If propositions could be established true other
than by logical demonstration you might have a point. However you'd
first have to demonstrate that's true which needless to say you don't.
But the point is meretricious because my point entailed establishment
by means of demonstration and my question concerning your comment
entailed what other means of establishment you envisioned apart from
demonstration. So obviously you cannot then revert your meaning to my
own preceding comment since the two cannot be the same if you object
to my use of establishment to mean establishment by demonstration.
So far what you've contended is that establishment would represent
some form of self evidential apparency but what kind and to whom is
less than apparent. On the other hand if you continue to expect me to
intuit your meaning without clarifications and demonstrations I can
only stand by more or less educated guesswork despite the demurrers.
~v~~
===
Subject: Re: Infinitesimal Arithmetic
Well, I certainly _do_ agree with the following:
> the truth of conclusions is not
> established regardless of formalisms used to draw conclusions
from
> premises which are not themselves demonstrated true.
except I might quibble over demonstrated true: I'd say instead
>established as true or some such.
Established true? And what is that when it's at home?
>established means the same as what it meant in the sentence
>of yours I was responding to.
> The point being that your and my understandings of established
> appear to be entirely different. Mine is mechanically tautological and
> I can't tell what you mean by established as true.My use of the
word
> was specifically in regard to formalisms which don't establish
truth
> and not in opposition to demonstrated true whereas your use of the
> word appears to be interpreted as an alternative to
demonstrations.
> So I hardly see that your meaning can be the same as mine.
Since propositions can be established as true other than by
>logical demonstration, my use of the word establish is
>wider than and inclusive of the meaning of demonstrate.
No since about it. If propositions could be established true other
> than by logical demonstration you might have a point.
I might at that.
> However you'd
> first have to demonstrate that's true which needless to say you don't.
I'm afraid you'll have to demonstrate that that's true, which needless
to say, you don't.
> But the point is meretricious because my point entailed establishment
> by means of demonstration and my question concerning your comment
> entailed what other means of establishment you envisioned apart from
> demonstration. So obviously you cannot then revert your meaning to my
> own preceding comment since the two cannot be the same if you object
> to my use of establishment to mean establishment by
demonstration.
Since I did suspect that you meant established as meaning solely
established by demonstration, I will grant that I cannot revert
my meaning to your preceeding use.
I certainly agree with your orginal assertion, whichever sense
of established you meant. I'd change demonstrated true to
established true in my sense, or some such, however.
> So far what you've contended is that establishment would represent
> some form of self evidential apparency
You are prone this sort of mis-representation. What I asserted
was that propositions can be established as true other than by
logical demonstration.
> but what kind and to whom is less than apparent.
Well, _you've_ agreed on when I have a red car and I have a
blue car is true, and you've agreed that John has a big dog
implies that John has a dog. I suspect that if you were frank
you'd agree that a proposition A is true if and only if not-A is
false. We can go a long way on what we have so far.
> On the other hand if you continue to expect me to
> intuit your meaning without clarifications and demonstrations I can
> only stand by more or less educated guesswork despite the demurrers.
It's funny that _you_ would say that to _me_.
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
>Well, I certainly _do_ agree with the following:
> the truth of conclusions is not
> established regardless of formalisms used to draw conclusions
from
> premises which are not themselves demonstrated true.
>except I might quibble over demonstrated true: I'd say
instead
>established as true or some such.
> Established true? And what is that when it's at home?
established means the same as what it meant in the sentence
>of yours I was responding to.
The point being that your and my understandings of established
> appear to be entirely different. Mine is mechanically tautological
and
> I can't tell what you mean by established as true.My use of the
word
> was specifically in regard to formalisms which don't establish
truth
> and not in opposition to demonstrated true whereas your use of
the
> word appears to be interpreted as an alternative to
demonstrations.
> So I hardly see that your meaning can be the same as mine.
>Since propositions can be established as true other than by
>logical demonstration, my use of the word establish is
>wider than and inclusive of the meaning of demonstrate.
> No since about it. If propositions could be established true other
> than by logical demonstration you might have a point.
I might at that.
> However you'd
> first have to demonstrate that's true which needless to say you don't.
I'm afraid you'll have to demonstrate that that's true, which needless
>to say, you don't.
Except it was your hidden assumption not mine and it was concealed by
the use of since.
> But the point is meretricious because my point entailed establishment
> by means of demonstration and my question concerning your comment
> entailed what other means of establishment you envisioned apart from
> demonstration. So obviously you cannot then revert your meaning to my
> own preceding comment since the two cannot be the same if you object
> to my use of establishment to mean establishment by
demonstration.
Since I did suspect that you meant established as meaning solely
>established by demonstration, I will grant that I cannot revert
>my meaning to your preceeding use.
I certainly agree with your orginal assertion, whichever sense
>of established you meant. I'd change demonstrated true to
>established true in my sense, or some such, however.
Except I'm still trying to understand in what sense you mean
established true. You say in your sense but I don't understand what
sense that is that isn't in my sense of established by demonstration.
So far all you've suggested is some self evidential establishment of
truth which you don't explain in any mechanically definitive terms
comparable to demonstration.
> So far what you've contended is that establishment would represent
> some form of self evidential apparency
You are prone this sort of mis-representation. What I asserted
>was that propositions can be established as true other than by
>logical demonstration.
Well then by all means have at it. I'm all ears. Nor do I think I
misrepresented your original remarks on establishment of truth.
> but what kind and to whom is less than apparent.
Well, _you've_ agreed on when I have a red car and I have a
>blue car is true, and you've agreed that John has a big dog
>implies that John has a dog. I suspect that if you were frank
>you'd agree that a proposition A is true if and only if not-A is
>false. We can go a long way on what we have so far.
Well now, Herb, you continue to maintain I've agreed when I haven't
and repeatedly pointed out exactly how and why I do and don't agree.
So I don't really have a lot else to say on this particular subject
and would appreciate it if you didn't continue to repeat this claim
that I agree.
As to your final contention it has some limited truth but only that. A
predicate is universally true if its alternative is universally self
contradictory (which I take to mean false). The difficulty is finding
predicates of appropriate universal applicability.
If you simply use any particular predicate then the proposition is not
true because not A can be true or false without making A true. For
example B is certainly not A and not B certainly doesn't prove A true.
On the other hand if you use a predicate of universal applicability
such as not difference or contradiction then their denial is in
fact self contradictory and false hence the alternative is necessarily
true and true of everything universally.
> On the other hand if you continue to expect me to
> intuit your meaning without clarifications and demonstrations I can
> only stand by more or less educated guesswork despite the demurrers.
It's funny that _you_ would say that to _me_.
Tendentious casuistry or in the modern parlance yadayada whatever.
~v~~
===
Subject: Re: Infinitesimal Arithmetic
Or is your position that, though you agree that John has a big
>dog implies John has a dog, you don't know whether the argument
>John has a big dog, therefore John has a dog is valid or not?
Once again, not being in your particular valid loop I can't comment
> on the validity of what you guess. What I can comment on is that you
> can't comment on the truth of whether John has a big dog or whether
> John has any dog at all because you haven't defined the truth of
> validity in any way that's particularly true.Perhaps when you can I
> shall.
I'll take that as a yes.
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
>Or is your position that, though you agree that John has a big
>dog implies John has a dog, you don't know whether the argument
>John has a big dog, therefore John has a dog is valid or not?
> Once again, not being in your particular valid loop I can't comment
> on the validity of what you guess. What I can comment on is that you
> can't comment on the truth of whether John has a big dog or whether
> John has any dog at all because you haven't defined the truth of
> validity in any way that's particularly true.Perhaps when you can I
> shall.
I'll take that as a yes.
No doubt in your particular validity loop, Herb, you'd take a truth
valuable no as a yes too.
~v~~
===
Subject: Re: Infinitesimal Arithmetic
>This is an attractive interpretation, but I'm not sure that
>it can be sustained. It assumes that operands A, B, C, etc.,
>have truth-values, though it is not clear that they are
>propositions, and so possess truth-values. The bulk of
>his writings suggest that they are predicates, and he
>maintains that what he is formalizing is predicate logic.
And how pray tell is a predicate not a proposition? If we prefix it
> is to any predicate we have a proposition which can be true or false
That's true, but I wasn't sure whether at this stage of the
exposition we had arrived at propositions yet. At the beginning
of the essay there is the proposition P:[not], but at this
point we're dealing with expressions that do not have full
colons or square brackets such as not(not A not B) which for
all I know is just a compound predicate, not a proposition.
I'm not a mind reader, y'know.
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
>This is an attractive interpretation, but I'm not sure that
>it can be sustained. It assumes that operands A, B, C, etc.,
>have truth-values, though it is not clear that they are
>propositions, and so possess truth-values. The bulk of
>his writings suggest that they are predicates, and he
>maintains that what he is formalizing is predicate logic.
> And how pray tell is a predicate not a proposition? If we prefix it
> is to any predicate we have a proposition which can be true or false
That's true, but I wasn't sure whether at this stage of the
>exposition we had arrived at propositions yet. At the beginning
>of the essay there is the proposition P:[not], but at this
>point we're dealing with expressions that do not have full
>colons or square brackets such as not(not A not B) which for
>all I know is just a compound predicate, not a proposition.
So having just noted above and eliciting your agreement that a
predicate is a proposition you now however suggest a compound
predicate would somehow not be a proposition? Curiouser and curiouser.
>I'm not a mind reader, y'know.
And who exactly are you to complain about mind reading when I'm forced
to intuit what you mean by establishment that doesn't entail
demonstration?
~v~~
===
Subject: Re: Infinitesimal Arithmetic
>This is an attractive interpretation, but I'm not sure that
>it can be sustained. It assumes that operands A, B, C, etc.,
>have truth-values, though it is not clear that they are
>propositions, and so possess truth-values. The bulk of
>his writings suggest that they are predicates, and he
>maintains that what he is formalizing is predicate logic.
> And how pray tell is a predicate not a proposition? If we prefix it
> is to any predicate we have a proposition which can be true or false
That's true, but I wasn't sure whether at this stage of the
>exposition we had arrived at propositions yet. At the beginning
>of the essay there is the proposition P:[not], but at this
>point we're dealing with expressions that do not have full
>colons or square brackets such as not(not A not B) which for
>all I know is just a compound predicate, not a proposition.
So having just noted above and eliciting your agreement that a
> predicate is a proposition you now however suggest a compound
> predicate would somehow not be a proposition? Curiouser and curiouser.
Well, Alice, agreeing with If we prefix it is to any predicate
we have a proposition is not agreeing to a predicate is a
proposition.
Besides, how am I to know that your formalism does not allow
or compel predicates shorn of the it is prefix at some stage?
>I'm not a mind reader, y'know.
While we're on the subject, if I say It is fast and it is a car
should I or should I not assume that the pronoun it refers
to the same thing at each of its occurrances? Or does that
depend on the predicate? Or what?
> And who exactly are you to complain about mind reading when I'm forced
> to intuit what you mean by establishment that doesn't entail
> demonstration?
That must have been brutal. My apologies.
How about clueing me in on the grammatical differences between P:[not]
and not(not A not B)?
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
>This is an attractive interpretation, but I'm not sure that
>it can be sustained. It assumes that operands A, B, C, etc.,
>have truth-values, though it is not clear that they are
>propositions, and so possess truth-values. The bulk of
>his writings suggest that they are predicates, and he
>maintains that what he is formalizing is predicate logic.
And how pray tell is a predicate not a proposition? If we prefix it
> is to any predicate we have a proposition which can be true or
false
>That's true, but I wasn't sure whether at this stage of the
>exposition we had arrived at propositions yet. At the beginning
>of the essay there is the proposition P:[not], but at this
>point we're dealing with expressions that do not have full
>colons or square brackets such as not(not A not B) which for
>all I know is just a compound predicate, not a proposition.
> So having just noted above and eliciting your agreement that a
> predicate is a proposition you now however suggest a compound
> predicate would somehow not be a proposition? Curiouser and curiouser.
Well, Alice, agreeing with If we prefix it is to any predicate
>we have a proposition is not agreeing to a predicate is a
>proposition.
A distinction without a difference.
>Besides, how am I to know that your formalism does not allow
>or compel predicates shorn of the it is prefix at some stage?
Obviously it does. It's all predicates and predicate combinations. No
special mystery there that I can see.
>I'm not a mind reader, y'know.
While we're on the subject, if I say It is fast and it is a car
>should I or should I not assume that the pronoun it refers
>to the same thing at each of its occurrances? Or does that
>depend on the predicate? Or what?
The subject it always depends on the predicates because the
predicates are the only thing that give it meaning. And if you
compound predicate clauses conjunctively there is no way to assume or
know it refers to one and the same thing. That's why I don't conjoin
predicate clauses.
In order to assure you're using one and the same it you really have
to compound predicates in one proposition which specify the ordering
of predicates without compounding independent predicate clauses. For
example you could say it is a fast car. Otherwise there is no way to
conjoin subject and predicates uniquely in any unambiguous way.
> And who exactly are you to complain about mind reading when I'm forced
> to intuit what you mean by establishment that doesn't entail
> demonstration?
That must have been brutal. My apologies.
How about clueing me in on the grammatical differences between P:[not]
>and not(not A not B)?
There isn't any intentional mystery here, Herb. Parens, brackets, and
braces just specify various groupings. I've used them all in different
contexts and I don't consider any one kind essential to any particular
context. It's just a matter of convention. All they do is isolate some
material from other material. I've also used quote marks as well as
apostrophes as occasions demand.
~v~~
===
Subject: Re: Infinitesimal Arithmetic
>Do you mean that Q:[not not] is the negation of P:[not]?
Yes. I also mean that P:[not] is the negation of Q:[not not].
Yes, that would follow.
I had a thought today that maybe the : in P:[not] was
some sort of symbol to denote naming -- that P:[not]
could be read something like Let the proposition [not]
be called P, or something like that.
Can you clarify this point?
[...]
>How about (not (not A B) B)?
What is not A B?
Well, it's your symbolism, so if you don't understand it,
I guess it's an ill-formed expression. In the absence
of any stated syntax rules, I'm just guessing at what's
correct.
It was intended to denote the disjunction of not-A and B.
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
>Do you mean that Q:[not not] is the negation of P:[not]?
> Yes. I also mean that P:[not] is the negation of Q:[not not].
Yes, that would follow.
I had a thought today that maybe the : in P:[not] was
>some sort of symbol to denote naming -- that P:[not]
>could be read something like Let the proposition [not]
>be called P, or something like that.
Can you clarify this point?
I see nothing wrong with this interpretation. P Q et cetera are
only intended as names referring to the content of the expressions.
>[...]
>How about (not (not A B) B)?
> What is not A B?
Well, it's your symbolism, so if you don't understand it,
>I guess it's an ill-formed expression. In the absence
>of any stated syntax rules, I'm just guessing at what's
>correct.
Well my only claims and syntax referred to P, Q, and mechanical
reductions for boolean conjunctions AND and OR in terms of
not(A) and not(B) so I don't quite understand where secondary
occurrences of the B are supposed to come from especially if you
consider them to designate the same variable in both instances. In
other words I guess I just don't understand what the expression is
supposed to represent.
>It was intended to denote the disjunction of not-A and B.
Except I don't understand here either what you mean by disjunction.
I don't say your question is meaningless in more advanced contexts,
Herb. I just can't quite make it out. If I read your meaning correctly
you're trying to formulate not(A) and B which I suppose would be
not(not(not(A)) not(B))but I'm frankly leery of being held too close
to form here not so much because I'm skeptical of the form but mainly
because I'm not really sure how to derive predicates like B except by
not(not(B)) and I am skeptical of then using B in other instances.
~v~~
===
Subject: Re: Infinitesimal Arithmetic
>Do you mean that Q:[not not] is the negation of P:[not]?
> Yes. I also mean that P:[not] is the negation of Q:[not not].
Yes, that would follow.
I had a thought today that maybe the : in P:[not] was
>some sort of symbol to denote naming -- that P:[not]
>could be read something like Let the proposition [not]
>be called P, or something like that.
Can you clarify this point?
I see nothing wrong with this interpretation. P Q et cetera are
> only intended as names referring to the content of the expressions.
OK, good.
So the expression [not] stands for a proposition.
Do square brackets indicate that an expression inside them is a
proposition? Do square brackets mean anything beyond their normal
function, in English, of just being a variant notation for parentheses?
Is an empty pair of square brackets [] a meaningful expression?
Can you clarify the function that square brackets serve in
your symbolism?
>[...]
>How about (not (not A B) B)?
> What is not A B?
Well, it's your symbolism, so if you don't understand it,
>I guess it's an ill-formed expression. In the absence
>of any stated syntax rules, I'm just guessing at what's
>correct.
Well my only claims and syntax referred to P, Q, and mechanical
> reductions for boolean conjunctions AND and OR in terms of
> not(A) and not(B) so I don't quite understand where secondary
> occurrences of the B are supposed to come from especially if you
> consider them to designate the same variable in both instances. In
> other words I guess I just don't understand what the expression is
> supposed to represent.
It was intended to denote the disjunction of not-A and B.
Except I don't understand here either what you mean by disjunction.
> I don't say your question is meaningless in more advanced contexts,
> Herb. I just can't quite make it out.
In conventional terminology, or is disjunction, not is negation,
and is conjunction.
> If I read your meaning correctly
> you're trying to formulate not(A) and B
No, I'm trying to formulate not(A) OR B. (I'm just capitalizing
for emphasis.)
> which I suppose would be
> not(not(not(A)) not(B))
Right, this symbolizes not(A) and B.
What would you suppose not(A) OR B would be in your symbolism?
> but I'm frankly leery of being held too close
> to form here not so much because I'm skeptical of the form but mainly
> because I'm not really sure how to derive predicates like B except by
> not(not(B)) and I am skeptical of then using B in other instances.
'K. One step at a time:
1) I said How about (not (not A B) B)?
2) You said What is not A B?
3) So I want to talk about the expression not A B, that is,
(not A B), or better, (not(A) B), in isolation from the
larger expression (not (not(A) B) B) in which it is embedded
as a part.
4) Assuming that (not(A) B) can itself be a meaningful expression
in your symbolism, I would say that it is intended to mean
whatever (not(A) not(not(B))) means when you eliminate the
even pairing of not's in front of the B.
So if (not(A) not(not(B))) is meaningful in your symbolism,
so hopefully is (not(A) B).
We note that there are no secondary occurrances of either
A or of B in these last two expressions.
6) So the question for you at this point would be,
is (not(A) not(not(B))) a meaningful expression
in your symbolism, and if so, is (not(A) B)?
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
<7sb4731e64f4s15nao5u3rpl5tqa3dj5q8@4ax.com>
<46737A95.E9014DD5@gmail.com>
<j6a873lerqbbe865eqhbqeink49dc3a9p6@4ax.com>
<4674D729.DCD1BA5E@gmail.com>
<7d3b731mbpo40394n45uhc8p9fp30it7p3@4ax.com>
<467A2D52.D2B2D581@gmail.com>
<teel73ts8r8el368pqifnl2726eku8vamb@4ax.com>
<467B58E6.CA13397F@gmail.com>
<c57o73d8t90orhnbgp9totupuqeq37ll6h@4ax.com>
<467D8B52.3A8C0ED4@gmail.com>
<cbgr73dbicdka8k9bk6jmqeku7p6vvr9d9@4ax.com>
<467F57D2.BB803272@gmail.com>
<653083pmkuobt7h8ap00rmo3a6n70nfjj0@4ax.com>
Bytes: 3653
> truth-functionalists run around pretending there
> is only a binary truth relation between predicates
Please rephrase this clause. I'm pretty sure it
isn't true. I'm pretty sure that truth-functionalists
hold that there are consistent sets of rules of
inference for some truth valued functions that take
truth valued arguments. Also, predicate logicians
hold that certain predicates are truth valued functions.
> when in point of
> fact there is also a different truth relation between predicates which
> are reversed.
I don't see your point. Not all truth valued functions
take unordered sets of arguments. The IsA function takes
an ordered pair of arguments neither of wich is a truth
valued function.
> For example if elephants are animals there is a true-
> false relation associated with the statement but also a false relation
> associated with predicates taken in reverse order as in animals are
> elephants.
What are you taking as predicates in your example?
> And nothing in the predicates themselves shows what the
> correct ordering of terms must necessarily be. So we are invariably
> left to consider issues of predicate ordering as another alternative.
> In other words you can't just take two predicates and assume true-
> false binary truth values for the combined predicate proposition.
I take the predicate to be the overloaded function
IsA where certain quantity restictions apply, that
is IsA(John, Man) and IsA(elephants, animals) are
acceptable use but IsA(John, animals) is not.
Sorry for deleting the rest but I couldn't get the meaning of it.
You seem to be taking Animals as a predicate. It doesn't
seem to be a truth valued predicate.
===
Subject: Re: Infinitesimal Arithmetic
> truth-functionalists run around pretending there
> is only a binary truth relation between predicates
Please rephrase this clause. I'm pretty sure it
>isn't true. I'm pretty sure that truth-functionalists
>hold that there are consistent sets of rules of
>inference
Of course there are. Truth functionalists just don't seem to know what
the rules of inference are or how to demonstrate their truth.
In my preceding post I mentioned that in given predicate combinations
there are differences between subject and predicate. For example in a
combination elephants are animals there is some difference between
elephants on the one hand and animal on the other even if the
combination is true. And I cited as evidence of this the fact that if
there were no difference we could use one word for both predicates.
Now the difficulty with this is that if differences between predicates
in fact represent a necessary and essential component of predicate
logic generally we have no choice but to recognize that differences
cannot just be manipulated ad hoc independent of predicate order.
In other words summation is a commutative process but differences are
not. It makes no difference which way we take sums but it makes a huge
difference which way we use differences. Let's suppose we have various
differences such as A - B and B - C et cetera. If we order terms
appropriately as in [(A - B) - (C - B)] we can determine a difference
A - C' effectively by factoring out the common middle term B since
it occupies the same place in both differences. But if we don't pay
attention to ordering the result is a meaningless hash of differences.
That's the problem with ordinary approaches to rules for formal
inferential logic. They suggest it makes no difference which way
predicate differences are considered or taken and that all that
matters is some fixed n-value logic values, true, false, or whatever
which make it possible to deal with truth values in commutative
terms of binary cardinal arithmetic instead of ordinal arithmetic
required for the manipulation and calculation of differences.
> for some truth valued functions that take
>truth valued arguments. Also, predicate logicians
>hold that certain predicates are truth valued functions.
Well good luck with that. The problem seems to be that predicate
logicians and truth functionalists just call things truth value
functions without a clue as to what constitutes truth in those
truth valued functions.
> when in point of
> fact there is also a different truth relation between predicates which
> are reversed.
I don't see your point. Not all truth valued functions
>take unordered sets of arguments.
The problem is that in real truth situations you don't necessarily
have any ability to specify ordering.
> The IsA function takes
>an ordered pair of arguments neither of wich is a truth
>valued function.
I really can't tell what this is supposed to mean.
> For example if elephants are animals there is a true-
> false relation associated with the statement but also a false relation
> associated with predicates taken in reverse order as in animals are
> elephants.
What are you taking as predicates in your example?
Anything you say, talk or think with or about are predicates.
> And nothing in the predicates themselves shows what the
> correct ordering of terms must necessarily be. So we are invariably
> left to consider issues of predicate ordering as another alternative.
> In other words you can't just take two predicates and assume true-
> false binary truth values for the combined predicate proposition.
I take the predicate to be the overloaded function
>IsA where certain quantity restictions apply, that
>is IsA(John, Man) and IsA(elephants, animals) are
>acceptable use but IsA(John, animals) is not.
Tell me something, Gary, do you really talk this way when you're
trying to ascertain and demonstrate the truth of what you say? I mean
IsA(true, Gary) really an acceptable discursive technique for
probative purposes?
>Sorry for deleting the rest but I couldn't get the meaning of it.
>You seem to be taking Animals as a predicate. It doesn't
>seem to be a truth valued predicate.
Animal is certainly a predicate but I have no idea whether it is a
truth valued predicate because I have no idea what a truth valued
predicate is and no one seems to be able to explain the concept in
demonstrably probative terms.
~v~~
===
Subject: Re: Infinitesimal Arithmetic
The problem seems to be that predicate
> logicians and truth functionalists just call things truth value
> functions without a clue as to what constitutes truth in those
> truth valued functions.
So enlighten us, Lester. What constitutes truth in those truth
valued functions?
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
> The problem seems to be that predicate
> logicians and truth functionalists just call things truth value
> functions without a clue as to what constitutes truth in those
> truth valued functions.
So enlighten us, Lester. What constitutes truth in those truth
>valued functions?
Well there's really no way for me to say, Herb, since I don't do
truth valuable functions. I rather expect you might be able to
obtain a more truthful answer from those who do. Or not.
~v~~
===
Subject: Re: Infinitesimal Arithmetic
> The problem seems to be that predicate
> logicians and truth functionalists just call things truth value
> functions without a clue as to what constitutes truth in those
> truth valued functions.
So enlighten us, Lester. What constitutes truth in those truth
>valued functions?
Well there's really no way for me to say, Herb, since I don't do
> truth valuable functions.
In that case, how do you know that predicate logicians and truth
functionalists are clueless as to what constitutes truth in
those horrible truth valued functions (like and or not) which
you don't do?
> I rather expect you might be able to
> obtain a more truthful answer from those who do. Or not.
Or?? not?? What could you possibly intend here, I wonder?
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
The problem seems to be that predicate
> logicians and truth functionalists just call things truth value
> functions without a clue as to what constitutes truth in those
> truth valued functions.
>So enlighten us, Lester. What constitutes truth in those truth
>valued functions?
> Well there's really no way for me to say, Herb, since I don't do
> truth valuable functions.
In that case, how do you know that predicate logicians and truth
>functionalists are clueless as to what constitutes truth in
>those horrible truth valued functions (like and or not) which
>you don't do?
Oh well, Herb, some is just hyperbolic irony as a rhetorical device.
But there are two specific reasons. One, predicate logicians and truth
functionalists don't offer demonstrations of truth for their claims.
(Actually apart from moi no one does so I suppose that must include
predicate logicians and truth functionalists.) And, two, I and boolean
logicians also use conjunctions such as and and or without to the
best of my knowledge referring to them as truth valued functions. So I
really don't understand what you're getting at with that terminology
that is anything other than ordinary conjunctive logic. (Bear in mind
also that I've shown the mechanical origin and significance of boolean
conjunctions in terms of not alone and I consider that explanation
true but I haven't run into any truth valued function except not.)
In fact lately I've begun to wonder if what you really mean by truth
valued functions is not so much ordinary conjunctive logic such as
and and or but more on the order of something like either . . .
or and the excluded middle for predicate pairs. At the least that's
the best I can make of what you would appear to be getting at.
> I rather expect you might be able to
> obtain a more truthful answer from those who do. Or not.
Or?? not?? What could you possibly intend here, I wonder?
Oh just more truthful logic I expect.
~v~~
===
Subject: Re: Infinitesimal Arithmetic
> The problem seems to be that predicate
> logicians and truth functionalists just call things truth value
> functions without a clue as to what constitutes truth in those
> truth valued functions.
> So enlighten us, Lester. What constitutes truth in those truth
> valued functions?
> --
> hz
If you have not caught on yet, Lester will never actually give
you a straight answer about or a demonstration of truth, despite
all his diatriabes about such demonstrations.
Stephen
===
Subject: Re: Infinitesimal Arithmetic
Bytes: 2973
The problem seems to be that predicate
> logicians and truth functionalists just call things truth value
> functions without a clue as to what constitutes truth in those
> truth valued functions.
> So enlighten us, Lester. What constitutes truth in those truth
> valued functions?
> --
> hz
If you have not caught on yet, Lester will never actually give
>you a straight answer about or a demonstration of truth, despite
>all his diatriabes about such demonstrations.
And when indeed have I not given you a straight answer about or a
demonstration of truth, Stephen? Was it my demonstration of truth in
universal terms through tautological reduction to self contradictory
alternatives which bothers you? Or mayhap when I pointed out the
absence of any real number line? Or the use of infinitesimals in
modern mathematics? Or the analytical origin of Planck's constant,
terms of quantum effects and Euler's analysis of angular mechanics? Or
the correct explanation for Michelson-Morley and various SR effects?
Or the exhaustive reduction of boolean conjunctions to truth in
mechanical terms? I mean really, Stephen, you can't possibly be such a
pissant you can't even bother to criticize intelligently what you
insist on reading? Or maybe you can.
~v~~
===
Subject: Re: Infinitesimal Arithmetic
> The problem seems to be that predicate
> logicians and truth functionalists just call things truth value
> functions without a clue as to what constitutes truth in those
> truth valued functions.
So enlighten us, Lester. What constitutes truth in those truth
> valued functions?
--
> hz
If you have not caught on yet, Lester will never actually give
> you a straight answer about or a demonstration of truth, despite
> all his diatriabes about such demonstrations.
Stephen
But Lester is all about truth. He bears it as his standard.
Surely he will not dissimulate here.
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
>
The problem seems to be that predicate
> logicians and truth functionalists just call things truth value
> functions without a clue as to what constitutes truth in those
> truth valued functions.
> So enlighten us, Lester. What constitutes truth in those truth
> valued functions?
> --
> hz
> If you have not caught on yet, Lester will never actually give
> you a straight answer about or a demonstration of truth, despite
> all his diatriabes about such demonstrations.
> Stephen
But Lester is all about truth. He bears it as his standard.
Surely he will not dissimulate here.
Dissembling is not my long suit, Herb, just as truth is not the long
suit of truth functionalists.
~v~~
===
Subject: Re: Infinitesimal Arithmetic
The problem seems to be that predicate
> logicians and truth functionalists just call things truth value
> functions without a clue as to what constitutes truth in
those
> truth valued functions.
> So enlighten us, Lester. What constitutes truth in those truth
> valued functions?
> --
> hz
> If you have not caught on yet, Lester will never actually give
> you a straight answer about or a demonstration of truth, despite
> all his diatriabes about such demonstrations.
> Stephen
But Lester is all about truth. He bears it as his standard.
Surely he will not dissimulate here.
Dissembling is not my long suit, Herb,
Long suit or short pants, I'm sure there'll be no dissimulating.
> just as truth is not the long suit of truth functionalists.
Yes, but they're playing with a full deck.
--
hz
--
===
Subject: Re: Infinitesimal Arithmetic
> The problem seems to be that predicate
> logicians and truth functionalists just call things truth value
> functions without a clue as to what constitutes truth in
those
> truth valued functions.
So enlighten us, Lester. What constitutes truth in those truth
> valued functions?
--
> hz
If you have not caught on yet, Lester will never actually give
> you a straight answer about or a demonstration of truth, despite
> all his diatriabes about such demonstrations.
Stephen
>But Lester is all about truth. He bears it as his standard.
>Surely he will not dissimulate here.
> Dissembling is not my long suit, Herb,
Long suit or short pants, I'm sure there'll be no dissimulating.
The only lies I tell are true lies.
> just as truth is not the long suit of truth functionalists.
Yes, but they're playing with a full deck.
Yeah, sure, Herb, a deck full of jokers.
~v~~
===
Subject: Re: Infinitesimal Arithmetic
<4674D729.DCD1BA5E@gmail.com>
<7d3b731mbpo40394n45uhc8p9fp30it7p3@4ax.com>
<467A2D52.D2B2D581@gmail.com>
<teel73ts8r8el368pqifnl2726eku8vamb@4ax.com>
<467B58E6.CA13397F@gmail.com>
<c57o73d8t90orhnbgp9totupuqeq37ll6h@4ax.com>
<467D8B52.3A8C0ED4@gmail.com>
<cbgr73dbicdka8k9bk6jmqeku7p6vvr9d9@4ax.com>
<467F57D2.BB803272@gmail.com>
<653083pmkuobt7h8ap00rmo3a6n70nfjj0@4ax.com>
<56g283l76rredn8d5uflqev1fkm4hs8agf@4ax.com>
Bytes: 7939
> truth-functionalists run around pretending there
> is only a binary truth relation between predicates
Please rephrase this clause. I'm pretty sure it
>isn't true. I'm pretty sure that truth-functionalists
>hold that there are consistent sets of rules of
>inference
Of course there are. Truth functionalists just don't seem to know what
> the rules of inference are or how to demonstrate their truth.
In my preceding post I mentioned that in given predicate combinations
> there are differences between subject and predicate. For example in a
> combination elephants are animals there is some difference between
> elephants on the one hand and animal on the other even if the
> combination is true. And I cited as evidence of this the fact that if
> there were no difference we could use one word for both predicates.
Now the difficulty with this is that if differences between predicates
> in fact represent a necessary and essential component of predicate
> logic generally we have no choice but to recognize that differences
> cannot just be manipulated ad hoc independent of predicate order.
In other words summation is a commutative process but differences are
> not. It makes no difference which way we take sums but it makes a huge
> difference which way we use differences. Let's suppose we have various
> differences such as A - B and B - C et cetera. If we order terms
> appropriately as in [(A - B) - (C - B)] we can determine a difference
> A - C' effectively by factoring out the common middle term B since
> it occupies the same place in both differences. But if we don't pay
> attention to ordering the result is a meaningless hash of differences.
That's the problem with ordinary approaches to rules for formal
> inferential logic. They suggest it makes no difference which way
> predicate differences are considered or taken and that all that
> matters is some fixed n-value logic values, true, false, or whatever
> which make it possible to deal with truth values in commutative
> terms of binary cardinal arithmetic instead of ordinal arithmetic
> required for the manipulation and calculation of differences.
for some truth valued functions that take
>truth valued arguments. Also, predicate logicians
>hold that certain predicates are truth valued functions.
Well good luck with that. The problem seems to be that predicate
> logicians and truth functionalists just call things truth value
> functions without a clue as to what constitutes truth in those
> truth valued functions.
> when in point of
> fact there is also a different truth relation between predicates which
> are reversed.
I don't see your point. Not all truth valued functions
>take unordered sets of arguments.
The problem is that in real truth situations you don't necessarily
> have any ability to specify ordering.
The IsA function takes
>an ordered pair of arguments neither of wich is a truth
>valued function.
I really can't tell what this is supposed to mean.
> For example if elephants are animals there is a true-
> false relation associated with the statement but also a false relation
> associated with predicates taken in reverse order as in animals are
> elephants.
What are you taking as predicates in your example?
Anything you say, talk or think with or about are predicates.
> And nothing in the predicates themselves shows what the
> correct ordering of terms must necessarily be. So we are invariably
> left to consider issues of predicate ordering as another alternative.
> In other words you can't just take two predicates and assume true-
> false binary truth values for the combined predicate proposition.
I take the predicate to be the overloaded function
>IsA where certain quantity restictions apply, that
>is IsA(John, Man) and IsA(elephants, animals) are
>acceptable use but IsA(John, animals) is not.
Tell me something, Gary, do you really talk this way when you're
> trying to ascertain and demonstrate the truth of what you say? I mean
> IsA(true, Gary) really an acceptable discursive technique for
> probative purposes?
Sorry for deleting the rest but I couldn't get the meaning of it.
>You seem to be taking Animals as a predicate. It doesn't
>seem to be a truth valued predicate.
Animal is certainly a predicate but I have no idea whether it is a
> truth valued predicate because I have no idea what a truth valued
> predicate is and no one seems to be able to explain the concept in
> demonstrably probative terms.
Animal isn't a predicate in any language of which I'm aware.
Encarta defines predicate this way:
1. grammar--part of sentence excluding subject:
a word or combination of words, including the verb,
objects, or phrases governed by the verb that make
up one of the two main parts of a sentence
A predicate must have a verb. It can be more than the verb.
There is also a definition as used in logic but I like this
one from the Wiktionary better:
(logic) A statement that may be true or false depending on
the values of its variables.
I didn't define the IsA function but I thought it clear
IsA(true, Gary) isn't a legal construct.
not WellFormed(IsA(true, Gary))
Do I think this way? Quite a bit when more than surface
thinking is needed.
I will communicate with others with what I take to be
sufficient percision. I will increase formal discourse
with increase misunderstanding, either intentional or
not. I'm pretty informal so getting to common terms
takes some time and is pretty ad hoc unless we already
know a common formal language.
===
Subject: Re: Infinitesimal Arithmetic
Bytes: 8491
> truth-functionalists run around pretending there
> is only a binary truth relation between predicates
>Please rephrase this clause. I'm pretty sure it
>isn't true. I'm pretty sure that truth-functionalists
>hold that there are consistent sets of rules of
>inference
> Of course there are. Truth functionalists just don't seem to know what
> the rules of inference are or how to demonstrate their truth.
> In my preceding post I mentioned that in given predicate combinations
> there are differences between subject and predicate. For example in a
> combination elephants are animals there is some difference between
> elephants on the one hand and animal on the other even if the
> combination is true. And I cited as evidence of this the fact that if
> there were no difference we could use one word for both predicates.
> Now the difficulty with this is that if differences between predicates
> in fact represent a necessary and essential component of predicate
> logic generally we have no choice but to recognize that differences
> cannot just be manipulated ad hoc independent of predicate order.
> In other words summation is a commutative process but differences are
> not. It makes no difference which way we take sums but it makes a huge
> difference which way we use differences. Let's suppose we have various
> differences such as A - B and B - C et cetera. If we order terms
> appropriately as in [(A - B) - (C - B)] we can determine a difference
> A - C' effectively by factoring out the common middle term B since
> it occupies the same place in both differences. But if we don't pay
> attention to ordering the result is a meaningless hash of differences.
> That's the problem with ordinary approaches to rules for formal
> inferential logic. They suggest it makes no difference which way
> predicate differences are considered or taken and that all that
> matters is some fixed n-value logic values, true, false, or whatever
> which make it possible to deal with truth values in commutative
> terms of binary cardinal arithmetic instead of ordinal arithmetic
> required for the manipulation and calculation of differences.
> for some truth valued functions that take
>truth valued arguments. Also, predicate logicians
>hold that certain predicates are truth valued functions.
> Well good luck with that. The problem seems to be that predicate
> logicians and truth functionalists just call things truth value
> functions without a clue as to what constitutes truth in those
> truth valued functions.
> when in point of
> fact there is also a different truth relation between predicates
which
> are reversed.
>I don't see your point. Not all truth valued functions
>take unordered sets of arguments.
> The problem is that in real truth situations you don't necessarily
> have any ability to specify ordering.
> The IsA function takes
>an ordered pair of arguments neither of wich is a truth
>valued function.
> I really can't tell what this is supposed to mean.
> For example if elephants are animals there is a true-
> false relation associated with the statement but also a false
relation
> associated with predicates taken in reverse order as in animals are
> elephants.
>What are you taking as predicates in your example?
> Anything you say, talk or think with or about are predicates.
> And nothing in the predicates themselves shows what the
> correct ordering of terms must necessarily be. So we are invariably
> left to consider issues of predicate ordering as another alternative.
> In other words you can't just take two predicates and assume true-
> false binary truth values for the combined predicate proposition.
>I take the predicate to be the overloaded function
>IsA where certain quantity restictions apply, that
>is IsA(John, Man) and IsA(elephants, animals) are
>acceptable use but IsA(John, animals) is not.
> Tell me something, Gary, do you really talk this way when you're
> trying to ascertain and demonstrate the truth of what you say? I mean
> IsA(true, Gary) really an acceptable discursive technique for
> probative purposes?
>Sorry for deleting the rest but I couldn't get the meaning of it.
>You seem to be taking Animals as a predicate. It doesn't
>seem to be a truth valued predicate.
> Animal is certainly a predicate but I have no idea whether it is a
> truth valued predicate because I have no idea what a truth valued
> predicate is and no one seems to be able to explain the concept in
> demonstrably probative terms.
Animal isn't a predicate in any language of which I'm aware.
You've got to be kidding. So if I say An elephant is an animal
animal is not a predicate? Then what is animal that is not a
predicate?
>Encarta defines predicate this way:
> 1. grammar--part of sentence excluding subject:
> a word or combination of words, including the verb,
> objects, or phrases governed by the verb that make
> up one of the two main parts of a sentence
A predicate must have a verb. It can be more than the verb.
>There is also a definition as used in logic but I like this
>one from the Wiktionary better:
(logic) A statement that may be true or false depending on
> the values of its variables.
So why exactly did you ask me to define predicate? Do dictionaries
define predicates of predication and a demonstrable theory of truth as
well?
>I didn't define the IsA function but I thought it clear
>IsA(true, Gary) isn't a legal construct.
>not WellFormed(IsA(true, Gary))
Not a legal construct? Not well formed? Well we seem to have all kinds
of rules to accommodate your private definitions and special
pleadings, don't we?
>Do I think this way? Quite a bit when more than surface
>thinking is needed.
How about when only surface (whatever that means) thinking is needed
to demonstrate truth? I strongly suspect what you mean by surface
thinking is thinking whose truth you can't demonstrate so you move
on to obscure subsurface thinking whose truth you can't demonstrate
either but at least it will prove too abstruse to allow others to
penetrate it murkiness.
>I will communicate with others with what I take to be
>sufficient percision. I will increase formal discourse
>with increase misunderstanding, either intentional or
>not. I'm pretty informal so getting to common terms
>takes some time and is pretty ad hoc unless we already
>know a common formal language.
How about if we already have a common informal language? Don't you
expect your utterances in generic language to be demonstrably true or
false?
~v~~
===
Subject: Re: x^2-y^2=p, and p*x-y=x^2+y^2=p_1
>if p = prime numbers
>x^2-y^2=p
>when
>p*x-y=x^2+y^2
>is p_1 ? (p_1=prime)
Of course,
if x^2+y^2=p
then
(p-x)/y=x^2-y^2.
Example
10^2+9^2=181
(181-10)/9=19=10^2-9^2
Vincenzo Librandi
===
Subject: Re: x^2-y^2=p, and p*x-y=x^2+y^2=p_1
x^2 + y^2 = (k+1)^2 + k^2 = 2k^2 + 2k + 1
On the other hand,
p*x - y = (2k+1)(k+1) - k = 2k^2 + 3k + 1 - k = >2k^2+2k+1, so this
equality always holds.
Yes,it's right.
>What is p_1 supposed to be?
p_1=(p^2+1)/2
p_1=sqrt(8n+9)
(for some values of n=0,2,14,44,104,...,)
Vincenzo Librandi
===
Subject: Re: x^2-y^2=p, and p*x-y=x^2+y^2=p_1
On 25 Jun, 21:41, Vincenzo Librandi <vincenzo.librandw...@alice.it
> if p = prime numbers
> x^2-y^2=p
when
p*x-y=x^2+y^2
is p_1 ? (p_1=prime)
Jan Kristian Haugland remember:
>http://www.research.att.com/~njas/sequences/A048161
Yes,
more than a list,
I would want one mathematical theory.
Vincenzo Librandi
===
Subject: Re: x^2-y^2=p, and p*x-y=x^2+y^2=p_1
<16495498.1182839740634.JavaMail.jakarta@nitrogen.mathforum.org>
Bytes: 1781
On 26 Jun, 08:35, Vincenzo Librandi <vincenzo.librandw...@alice.it Jan
Kristian Haugland remember (sic):
>http://www.research.att.com/~njas/sequences/A048161
Yes,
> more than a list,
> I would want one mathematical theory.
I am sure you would, but number theory hasn't evolved that far yet. At
best we can give a heuristic estimate of the density of primes with
the desired property (but I won't bother to calculate it).
---
J K Haugland
http://home.no.net/zamunda
===
Subject: Injective Functions
let f:[0,1]->R be a continuous function, with f(0)<f(1).
Can you find a subset X of [0,1] such that:
(I) the restriction of f to X is injective and
f(X)=f([0,1])
(II)for every x in X, if I={y in X| y<x}, S={y in X|y>x},
f(supI)=f(x)=f(infS) ?
If such a set exist, can we find a strictly increasing surjective map
g:[0,1]->X? Would fg:[0,1]->R a continous
function?
Hmm ... I have no idea for now: every help is welcome!
thank you very very much for your attention.
Maury
===
Subject: Re: Injective Functions
> let f:[0,1]->R be a continuous function, with f(0)<f(1).
> Can you find a subset X of [0,1] such that:
(I) the restriction of f to X is injective and
> f(X)=f([0,1])
(II)for every x in X, if I={y in X| y<x}, S={y in X|y>x},
> f(supI)=f(x)=f(infS) ?
This is almost certainly nonsense,
but first we can assume that the maximum and minimum of f(x)
are attained at the end-points,
by restricting to a sub-interval [a,b] of [0,1].
Suppose f(0) = min f(x), f(1) = max f(x).
Now for each t in R = range(f), let x(t) be the smallest x
for which f(x) = t.
Then the set {x(t): t in R} would have the required property.
Maybe ...
--
Timothy Murphy
e-mail (<80k only): tim /at/ birdsnest.maths.tcd.ie
tel: +353-86-2336090, +353-1-2842366
s-mail: School of Mathematics, Trinity College, Dublin 2, Ireland
===
Subject: Re: Injective Functions
let f:[0,1]->R be a continuous function, with
> f(0)<f(1).
> Can you find a subset X of [0,1] such that:
(I) the restriction of f to X is injective and
> f(X)=f([0,1])
(II)for every x in X, if I={y in X| y<x}, S={y in
> X|y>x},
> f(supI)=f(x)=f(infS) ?
This is almost certainly nonsense,
> but first we can assume that the maximum and minimum
> of f(x)
> are attained at the end-points,
> by restricting to a sub-interval [a,b] of [0,1].
> Suppose f(0) = min f(x), f(1) = max f(x).
Now for each t in R = range(f), let x(t) be the
> smallest x
> for which f(x) = t.
> Then the set {x(t): t in R} would have the required
> property.
> Maybe ...
--
> Timothy Murphy
your help!
I apologize for my odd and confused attempt to traduce
my ideas in a mathematical form ...
Anyhow, my problem suggests an interesting (I think)
question: if X is an uncountable subset of [0,1], is
there a strictly increasing surjective map g:[0,1]->X?
Maury
Anyhow,
===
Subject: Re: Injective Functions
<29452806.1182960550937.JavaMail.jakarta@nitrogen.mathforum.org>
Bytes: 2680
> let f:[0,1]->R be a continuous function, with
> f(0)<f(1).
> Can you find a subset X of [0,1] such that:
> (I) the restriction of f to X is injective and
> f(X)=f([0,1])
> (II)for every x in X, if I={y in X| y<x}, S={y in
> X|y>x},
> f(supI)=f(x)=f(infS) ?
This is almost certainly nonsense,
> but first we can assume that the maximum and minimum
> of f(x)
> are attained at the end-points,
> by restricting to a sub-interval [a,b] of [0,1].
> Suppose f(0) = min f(x), f(1) = max f(x).
Now for each t in R = range(f), let x(t) be the
> smallest x
> for which f(x) = t.
> Then the set {x(t): t in R} would have the required
> property.
> Maybe ...
--
> Timothy Murphy
your help!
I apologize for my odd and confused attempt to traduce
> my ideas in a mathematical form ...
> Anyhow, my problem suggests an interesting (I think)
> question: if X is an uncountable subset of [0,1], is
> there a strictly increasing surjective map g:[0,1]->X?
Maury
Anyhow,
Let X = ]0,1]
If g:[0,1]->X is strictly increasing, it cannot be surjective
as no x in [0,1] could map to anything below f(0).
hagman
===
Subject: Re: Injective Functions
Bytes: 1988
> let f:[0,1]->R be a continuous function, with f(0)<f(1).
> Can you find a subset X of [0,1] such that:
(I) the restriction of f to X is injective and
> f(X)=f([0,1])
(II)for every x in X, if I={y in X| y<x}, S={y in X|y>x},
> f(supI)=f(x)=f(infS) ?
If such a set exist, can we find a strictly increasing surjective map
g:[0,1]->X? Would fg:[0,1]->R a continous
> function?
Hmm ... I have no idea for now: every help is welcome!
> thank you very very much for your attention.
> Maury
Is there some practical reason you'd want to do such a thing, or is it
just woolgtahering? 6 billion people need to be fed!
===
Subject: Re: Coefficients and Roots of a set of Polynomials
Raymond Manzoni a .8ecrit :
> The pari/gp functions I used (probably corresponding to your routines):
/*your initial polynomials :*/
f(n)={local(H=matrix(n-1, n-1), i, j); for(i=1, n-1, for(j=1, i,
H[i,j]=(n*i^3-3*n*(n+1)*i^2/2+n*(3*n+1)*i/2+(n^4-n^2)/2)/6-
(i^2-(2*n+1)*i+n*(n+1))*(j-1)*j/4; H[j, i]=H[i, j]; ); );charpoly(H)}
/*presented as a square matrix once divided by -(-n)^(n-i-2) */
m(nn)=ma=matrix(nn,nn);for(n=1,nn,pol=f(n);for(i=0,poldegree(pol),
ma[n,i+1]=-polcoeff(pol,i)/(-n)^(n-i-2)));ma
Hi Gerry,
Let's come back to the point where m was rather simple :
> m(12)
[1 0 0 0 0 0 0 0 0 0 0 0 ]
[1 2 0 0 0 0 0 0 0 0 0 0 ]
[1 8 3 0 0 0 0 0 0 0 0 0 ]
[1 14 34 4 0 0 0 0 0 0 0 0 ]
[1 20 101 104 5 0 0 0 0 0 0 0 ]
[1 26 204 *504* 259 6 0 0 0 0 0 0 ]
[1 32 343 1420 1966 560 7 0 0 0 0 0 ]
[1 38 518 3068 7610 6412 1092 8 0 0 0 0 ]
[1 44 729 5664 20971 33564 18238 1968 9 0 0 0 ]
[1 50 976 9424 47125 117026 127100 46552 3333 10 0 0 ]
[1 56 1259 14564 92444 318592 556045 425476 108823 5368 11 0 ]
[1 62 1578 21300 164596 734616 1824824 2316080 1286115 236522 8294 12]
Note that the (x,y) term under the diagonal may be obtained from
the terms of the previous line starting with the term over (x,y) (the
hidden terms are considered 0).
Example : 504 = 1*104+ 6*101- 17*20+ 134*1
So that instead of an infinite tower of triangles we have this
simple(?) sequence :
1, 6, -17, 134, -1333, 14890, -178394, 2240268,...
I'll let you play with that!
Raymond
/* Scripts to get more terms */
nn=12;
mm= m(nn);
g(x,y)= if(x>0 && y>0, mm[x,y], 0);
/* starting with matrix(nn,nn,x,y,g(x-1,y)+6*g(x-1,y-1)-g(x,y))
* and inserting new terms before the final -g(x,y) in this case
* -29101197*g(x-1,y-8) and so on... */
matrix(nn,nn,x,y,g(x-1,y)+6*g(x-1,y-1)-17*g(x-1,y-2)+134*g(x-1,y-3)
-1333*g(x-1,y-4)+14890*g(x-1,y-5)-178394*g(x-1,y-6)+2240268*g(x-1,y-7)-g(x,y

))
(P.S.: you may have a look at my recent answers at sci.math)
===
Subject: Re: Coefficients and Roots of a set of Polynomials
Gerry a .8ecrit :
For the degree n=8 the SumB(b(j,7))=0 for

j=(24,120,144,240,264,720,744,840,864,960,984,1440,1464,1560,1584,1680,1704,
2
160,2184,2280,2304,2400,2424,5040,

5064,5160,5184,5280,5304,5760,5784,5880,5904,6000,6024,6480,6504,6600,6624,6
7
20,6744,7200,7224,7320,7344,7440,7464,

10080,10104,10200,10224,10320,10344,10800,10824,10920,10944,11040,11064,1152
0
,11544,11640,11664,11760,11784,

12240,12264,12360,12384,12480,12504,15120,15144,15240,15264,15360,15384,1584
0
,15864,15960,15984,16080,16104,

16560,16584,16680,16704,16800,16824,17280,17304,17400,17424,17520,17544,2016
0
,20184,20280,20304,20400,20424,

20880,20904,21000,21024,21120,21144,21600,21624,21720,21744,21840,21864,2232
0
,22344,22440,22464,22560,22584,40320)
>
Hi Gerry,
To find a pattern you may write this as sum of factorials :
4! 5! 6!
24 1
120 1
144 1 1
2
1 2
1
1 1
1 1
1 1 1
2 1
1 2 1
2
1 2
1 2
1 1 2
2 2
1 2 2
3
1 3
1 3
1 1 3
2 3
1 2 3
1
and so on...
The set 'seems' to be the set of numbers obtained by taking a finite
sum of factorials for n>= 4 with n! taken from 0 to (n-3) times
(including or excluding 0 as a result of the sum)
I'll let you see if this pattern works ad infinitum...
Hoping you liked that!
Raymond
===
Subject: Poisson Process Problem
A Poisson process has rate lambda = 1. Let N be the number of integers k
such that k is one of 0, 1, 2,..., 99 and there is at least one observation
in [k,k+2]. (So for example if the process has hits at 0.7 and 25.8 and no
other hits in [0,101] then N=3).
Find E[N] and E[N^2].
I'm stuck on this. I've tried using Markov chains and treating N as a sum
of Bernoulli RV's, but since the intervals are not independent, I'm having 
a
tough time.
===
Subject: Re: Poisson Process Problem
Bytes: 2334
> A Poisson process has rate lambda = 1. Let N be the number of integers k
such that k is one of 0, 1, 2,..., 99 and there is at least one observation
in [k,k+2]. (So for example if the process has hits at 0.7 and 25.8 and no
other hits in [0,101] then N=3).
Find E[N] and E[N^2].
I'm stuck on this. I've tried using Markov chains and treating N as a sum
of Bernoulli RV's, but since the intervals are not independent, I'm having 
a
tough time.
If T is the time of a Poisson event, let K(T) be the number of hits
associated with that T. If 0 < T < 1, only k=0 will work, so K(T) = 1.
If 1 < T < 2, both k=0 and k=1 will work, so K(T) = 2. Similarly, K(T)
= 2 if 2 < T < 3 or 3 < T < 4 or ... or 100 < T < 101. The case of
integer T is a bit different: for T = 1, both k=0 and k=1 work, so
K(T) = 2; and for T = 2,3,4,..., K(T) = 3. So, every non-integer
arrival in (0,1) contributes 1, while every non-integer arrival in
(1,101) contributes 2. I will let you worry about the contribution of
the integer arrival times to the probability distribution of N.
R.G. Vickson
===
Subject: Re: Poisson Process Problem
That's not true. If, for instance, there are arrivals at 1.5 and 1.7, you
do not add anything for the second arrival.
===
Subject: Re: Poisson Process Problem
<20827891.1182969304932.JavaMail.jakarta@nitrogen.mathforum.org That's not
true. If, for instance, there are arrivals at 1.5 and 1.7, you
do not add anything for the second arrival.
There is one more question to add to the one I asked in message #7. If
we have arrivals at, say 24.3 and 25.3, does this contribute a count
of 3 or 4 towards N? 24.3 is in [23,25] and [24,26], so k = 23 and 24.
25.3 is in [24,26] and [25,27], so k = 24 and 25. Do we count the k =
24 twice, or only once? If it counts twice, the problem is easy, but
if it counts only once the problem is quite hard.
R.G. Vickson
===
Subject: Re: Poisson Process Problem
It counts only once. That makes the increments dependent, which makes the
problem tricky.
===
Subject: Re: Poisson Process Problem
<20827891.1182969304932.JavaMail.jakarta@nitrogen.mathforum.org>
Bytes: 1839
> That's not true. If, for instance, there are arrivals at 1.5 and 1.7, you
do not add anything for the second arrival.
OK, so any arrivals at all in (0,1) give k = 0 and hence contribute 1
toward N. Any arrivals at all in (1,2) have k = 0 and k = 1, so
contribute 2 towards N, etc. (At least, I assume so, following the
counting you gave in your first example.) The number of arrivals in
(1,2) is unimportant; it matters only whether or not it is >=1. Do I
have it now? (If so, it looks like a fairly easy problem.)
R.G. Vickson
===
Subject: Re: Poisson Process Problem
<20827891.1182969304932.JavaMail.jakarta@nitrogen.mathforum.org>
Bytes: 1547
> That's not true. If, for instance, there are arrivals at 1.5 and 1.7, you
do not add anything for the second arrival.
OK, you should be able to figure out the answer easily enough. It will
be even simpler than what I indicated.
R.G. Vickson
===
Subject: Re: Poisson Process Problem
Bytes: 1856
> A Poisson process has rate lambda = 1. Let N be the number of integers k
such that k is one of 0, 1, 2,..., 99 and there is at least one observation
in [k,k+2]. (So for example if the process has hits at 0.7 and 25.8 and no
other hits in [0,101] then N=3).
Find E[N] and E[N^2].
If the Xi are iid exponential(1), then N = largest n such that X1 + X2
+ ... + Xn <= 101. This looks a lot like a renewal theory problem.
R.G. Vickson
I'm stuck on this. I've tried using Markov chains and treating N as a sum
of Bernoulli RV's, but since the intervals are not independent, I'm having 
a
tough time.
===
Subject: Re: Poisson Process Problem
Bytes: 2489
A Poisson process has rate lambda = 1. Let N be the number of integers
k such that k is one of 0, 1, 2,..., 99 and there is at least one 
observation
in [k,k+2]. (So for example if the process has hits at 0.7 and 25.8 and no
other hits in [0,101] then N=3).
Find E[N] and E[N^2].
If the Xi are iid exponential(1), then N = largest n such that X1 + X2
> + ... + Xn <= 101. This looks a lot like a renewal theory problem.
R.G. Vickson
Sorry: that is not quite right. Let T be the time of a Poisson event,
and K(T) the number of hits associated with that event. We can neglect
the event that T = integer. If 0 < T < 1, only k = 0 works, so K(T) =
1. If 1 < T < 2, both k=0 and k=1 both work, so K(T) = 2. If 2 < T <
3, k = 1 and 2 both work, so K(T) = 2. Similarly, K(T) = 2 if 3 < T <
101. So, every arrival in [0,1) congtributes 1, while every arrival in
[1,101) contributes 2. I think you can take it from here.
R.G. Vickson
I'm stuck on this. I've tried using Markov chains and treating N as a
sum of Bernoulli RV's, but since the intervals are not independent, I'm
having a tough time.
===
Subject: Re: Poisson Process Problem
<3073655.1182946707760.JavaMail.jakarta@nitrogen.mathforum.org>,
> A Poisson process has rate lambda = 1. Let N be the number of integers k
> such that k is one of 0, 1, 2,..., 99 and there is at least one
observation
> in [k,k+2]. (So for example if the process has hits at 0.7 and 25.8 and
no
> other hits in [0,101] then N=3).
Find E[N] and E[N^2].
I'm stuck on this. I've tried using Markov chains and treating N as a sum
of
> Bernoulli RV's, but since the intervals are not independent, I'm having a
> tough time.
Use indicator random variables.
--
A.
===
Subject: Re: Poisson Process Problem
> <3073655.1182946707760.JavaMail.jakarta@nitrogen.mathforum.org>,
A Poisson process has rate lambda = 1. Let N be the number of integers
k
> such that k is one of 0, 1, 2,..., 99 and there is at least one
observation
> in [k,k+2]. (So for example if the process has hits at 0.7 and 25.8 and
no
> other hits in [0,101] then N=3).
Find E[N] and E[N^2].
I'm stuck on this. I've tried using Markov chains and treating N as a
sum
> of
> Bernoulli RV's, but since the intervals are not independent, I'm having
a
> tough time.
Use indicator random variables.
More specifically, let Ik be the indicator of the event that the Poisson
process has at least one mark in the time interval [k,k+2].
Then E[Ik] = P[Ik = 1] = 1- P[Ik = 0] =1 - e^{-2}.
Consequently, as N = sum Ik, k=0,1,2,...,99, we must have
E[N] = sum E[Ik] = 100(1-e^{-2}).
To compute E[N^2] you need to compute
E[Ij*Ik] = P[Ij = Ik = 1].
The value of this expectation depends upon whether |j-k|
is equal to 0, to 1, or is greater than 1. Once found, it can be used
to compute E[N^2], which is equal to the double sum
sum_j sum_k E[Ij*Ik].
--
A.
===
Subject: Re: Poisson Process Problem
===
Subject: Re: Poisson Process Problem
Could you be more specific? I attempted to use indicator RV's (same as sum
of bernoulli's, correct?) and I couldn't figure it out.
===
Bytes: 1914
I can confirm that we had been working with Fintan UK however we are
absolutely in no way associated with the company in any way
whatsoever
anymore. Nor will we ever be doing business with them or any other
incarnation of the company again. Keep up the good work but please
leave our name out of this as we do not wish to be associated in any
way with Mr Ingram or his business activities
===
Bytes: 3125
> I can confirm that we had been working with Fintan UK however we are
> absolutely in no way associated with the company in any way
> whatsoever
> anymore. Nor will we ever be doing business with them or any other
> incarnation of the company again. Keep up the good work but please
> leave our name out of this as we do not wish to be associated in any
> way with Mr Ingram or his business activities
ADHERES TO THE BLUEPRINT OF BUSINESS FAILURE. FINTAN'S ABOUT TO GO
DOWN THE DRAIN ANYWAY, DUE TO THE MORON'S INCOMPETENT FINANCIAL
MANAGEMENT, AND UTTER CLUELESSNESS ABOUT HOW TO RUN A BUSINESS. LEE
COMES TO BUSINESSES. FINTAN UK IS THE LATEST INCARNATION IN HIS
LIFETIME OF FAILURE, AND IS ABOUT TO GO DOWN THE TOILET AS A RESULT OF
WORK OF FICTION (THE DA VINICI CODE), WHICH HE BLATANTLY AND
CAREER IN THE SECURITY SECTOR - NOTHING SOPHISTICATED MIND YOU - HE
WAS A TYPE OF GLORIFIED BOUNCER, THE ONES THAT HAVE AN IQ OF A SINGLE
DIGIT. WHAT MADE THIS MORON THINK HE COULD START A BUSINESS I HAVE NO
OF THE GOLDFISH ON THE FINTAN HOMEPAGE :-)
FINTAN - THE CELTIC GOD OF CHANGE APPARENTLY.
===
Bytes: 1916
I can confirm that we had been working with Fintan UK however we are
absolutely in no way associated with the company in any way
whatsoever
anymore. Nor will we ever be doing business with them or any other
incarnation of the company again. Keep up the good work but please
leave our name out of this as we do not wish to be associated in any
way with Mr Ingram or his business activities
===
Bytes: 1917
I can confirm that we had been working with Fintan UK however we are
absolutely in no way associated with the company in any way
whatsoever
anymore. Nor will we ever be doing business with them or any other
incarnation of the company again. Keep up the good work but please
leave our name out of this as we do not wish to be associated in any
way with Mr Ingram or his business activities.
===
I can confirm that we had been working with Fintan UK however we are
absolutely in no way associated with the company in any way whatsoever
anymore. Nor will we ever be doing business with them or any other
incarnation of the company again. Keep up the good work.
===
Subject: Re: Matrix derivatives
> Suppose x is a column vector, and v' denotes its transpose, for a
quadratic scalar function f
> f = x' * A * x
> I want to calculate df/dx. Assuming that A is symmetric, we can get
> df/dx = 2Ax
> However, if A is a function of x, i.e A = g(x),
> how can I computer df/dx?
The result that I get is
df/dx
= Ax + A^T x +
( sum_{i=1}^n ( d/dx[a_1i a_2i ... a_ni] )x_i ) x
where A is an nxn matrix, A^T is A transpose, and x is an n-element
column vector in R^n.
Of course if matrix A is independent of x, then this reduces to
df/dx = Ax + A^T x
And if A is symmetric, then this further reduces to
df/dx = 2Ax
, as you said.
If you are interested in the derivation of the above result, you can
download this 431kByte pdf document:
http://banyan.cm.nctu.edu.tw/~dgreenhoe/msd/matcalc.pdf
(or http://banyan.cm.nctu.edu.tw/%7Edgreenhoe/msd/matcalc.pdf )
I wish you all the best in your matrix calculus work =)
Dan Greenhoe
===
Subject: to JSH and all : tommy's prime factoring tricks
i promised to post on factoring yesterday, and i am a man of my word.
since JSH wasted a part of his life here on this forum (and the ones 
reading
it)and spend (wasted) a lot of time on factoring , he might actually learn
something today about factoring.
btw he already admitted his algoritm is bad or slow for big integers , so 
in
a way he admitted it not being good.
certain algoritms are too complicated to explain to JSH and even others 
here
, so i will keep it relatively simple.
there is no superduper factoring algoritm and im not going to deny the
working of other algoritms , nor claim to have a superduper factoring
algoritm.
but there certainly are some intresting properties for certain cases , and
the cases that can be factored into them.
working with modulus can help factoring.
knowing the amount of divisors can help factoring too , but usually that is
not the case.
between any two squared integers lies at least one prime.
( not proven but equivalent to a RH proof )
as for coupling divisors that is simply a diophantine equation :
a^2 + k a = b
sow : a(a+k)=b
where a and (a+k) are the coupling relations and a and
(a+k) are the divisors of b.
and other couplings are of course similar although hidden in sum form 
rather
than product.
a demonstration of mod is trivial :
n^2 + n or equivalently n ( n + 1)
is always devidable bye 2 ( because of mod 2 or parity )
also trivial is factoring certain numbers in polynomial form
so a number (n^4-1) always has a factor.
all this is basic and even known bye most adult crankpots and amateurs. (
apart from gene ray from timecube.com :p )
also digit recognizing of polynomial forms from an imput number can
therefore be usefull.
e.g. ( 10^10 - 1 )/9 is easily recognizable from its digits and more
advanced methods/examples exist.
less well known , but still basic.
what if a numbers equals a polyomial , which has none of the basic
properties ??
and so all of its zero's are unreal or nonintegers ?
anything we can know about their factors ??
yes !!
so here ARE some prime factoring tricks ( for polynomial correspondance or
factorable in certain polynomials) ,
primes are denoted p_n.
..and there more where that came from...
TOMMY1729 PRIME FACTORING TRICKS
TRICK ONE : a bit probalistic though.
let a = (p_1)^2 -1
then a is very likely to be highly composite !
( trick can work in reverse too ! )
and a = p_2 (or +1/-1) + p_3
(where p_2 can be a negative prime too although there always exists a
positive one.)
with abs (p_2) <<< p_3 and smaller abs (p_2) having higher probability than
bigger ones(probability ordered bye size of abs)
TRICK TWO : perfectly deterministic !
let n be an integer
factor : 5 nnnn - 10 nnn + 20 nn - 15 n + 11.
all of its primefactors end in 1.
in other words all prime factors of this polynomial number are of the form
10m+1.
before you ask : dont all number ending on 1 give primefactors ending on 1
??
no 21 = 7 * 3
and 221 = 17 * 13
these tricks are also nice for testing the consistancy of ( unproven )
factoring algoritms or at least their javacodes.
the one and only
tommy1729
===
Subject: Re: to JSH and all : tommy's prime factoring tricks
Bytes: 2132
On Wed, 27 Jun 2007 08:40:08 EDT, tommy1729 <tommy1729@gmail.comlet n be an
integer
factor : 5 nnnn - 10 nnn + 20 nn - 15 n + 11.
all of its primefactors end in 1.
in other words all prime factors of this polynomial number are of the form
10m+1.
That's interesting.
I don't see immediately how to prove it.
But I'll make a conjecture ...
Let f be a univariate polynomial with integer coefficients and let S
be the set of primes which divide f(n) for some integer n. If there
exists an integer m such that all elements of S are congruent mod m,
then they are all congruent to 1 mod m.
I think the above conjecture should be easy to prove.
A harder question is this ...
For which integers m>1 does there exist such an f?
Obviously m=2,5,10 work.
What about m=3? Does there exist a univariate polynomial f with
integer coefficients such that, for every integer n, all prime factors
of f(n) are congruent to 1 mod 3?
quasi
===
Subject: Re: to JSH and all : tommy's prime factoring tricks
>On Wed, 27 Jun 2007 08:40:08 EDT, tommy1729 <tommy1729@gmail.com
>let n be an integer
>factor : 5 nnnn - 10 nnn + 20 nn - 15 n + 11.
>all of its primefactors end in 1.
>in other words all prime factors of this polynomial number are of the 
form
10m+1.
That's interesting.
I don't see immediately how to prove it.
But I'll make a conjecture ...
Let f be a univariate polynomial with integer coefficients and let S
>be the set of primes which divide f(n) for some integer n. If there
>exists an integer m such that all elements of S are congruent mod m,
>then they are all congruent to 1 mod m.
I think the above conjecture should be easy to prove.
or maybe easier to disprove: suppose f(n) = 1537, then S = { 29, 53 }
and both 29 and 53 are congruent to 5 mod 24.
>A harder question is this ...
For which integers m>1 does there exist such an f?
Obviously m=2,5,10 work.
Suppose f(n) = 119, then S = { 7, 17 } and both 7 and 17 are congruent
to 2 mod 5.
>What about m=3? Does there exist a univariate polynomial f with
>integer coefficients such that, for every integer n, all prime factors
>of f(n) are congruent to 1 mod 3?
f(n) = 1; there are no prime factors, so the statement is vacuously
true.
f(n) = 7; the only prime factor of f(n) is 1 mod 3.
f(n) = 91; the prime factors of f(n) are 7 and 13, both are 1 mod 3.
I've not looked at the case of polynomials of degree greater than 0.
Did you have something in mind?
Rob Johnson <rob@trash.whim.org>
take out the trash before replying
===
Subject: Re: to JSH and all : tommy's prime factoring tricks
Bytes: 3143
>On Wed, 27 Jun 2007 08:40:08 EDT, tommy1729 <tommy1729@gmail.comlet n be 
an
integer
factor : 5 nnnn - 10 nnn + 20 nn - 15 n + 11.
all of its primefactors end in 1.
in other words all prime factors of this polynomial number are of the
form 10m+1.
>That's interesting.
>I don't see immediately how to prove it.
>But I'll make a conjecture ...
>Let f be a univariate polynomial with integer coefficients and let S
>be the set of primes which divide f(n) for some integer n. If there
>exists an integer m such that all elements of S are congruent mod m,
>then they are all congruent to 1 mod m.
>I think the above conjecture should be easy to prove.
or maybe easier to disprove: suppose f(n) = 1537, then S = { 29, 53 }
>and both 29 and 53 are congruent to 5 mod 24.
>A harder question is this ...
>For which integers m>1 does there exist such an f?
>Obviously m=2,5,10 work.
Suppose f(n) = 119, then S = { 7, 17 } and both 7 and 17 are congruent
>to 2 mod 5.
>What about m=3? Does there exist a univariate polynomial f with
>integer coefficients such that, for every integer n, all prime factors
>of f(n) are congruent to 1 mod 3?
f(n) = 1; there are no prime factors, so the statement is vacuously
>true.
f(n) = 7; the only prime factor of f(n) is 1 mod 3.
f(n) = 91; the prime factors of f(n) are 7 and 13, both are 1 mod 3.
I've not looked at the case of polynomials of degree greater than 0.
>Did you have something in mind?
Yes, in my conjecture, as well as in my followup questions, I intended
(but forgot to specify) that the polynomial f is required to be
_nonconstant_.
quasi
===
Subject: all prime factors of f(n) are congruent to 1 mod 3
Bytes: 1624
What about m=3? Does there exist a univariate polynomial f with
>integer coefficients such that, for every integer n, all prime factors
>of f(n) are congruent to 1 mod 3?
Say p = 2 mod 3 is prime, consider the quadratic character of -3 mod p.
If p = 1 mod 4 then (-3|p) = (3|p) = (p|3) = (2|3) = -1,
where (a|p) is the Legendre symbol.
If p = 3 mod 4 then (-3|p) = - (3|p) = (p|3) = -1.
So n^2 + 3 can't be divisible by any such prime p.
--
Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email)
===
Subject: Re: all prime factors of f(n) are congruent to 1 mod 3
Bytes: 2180
>What about m=3? Does there exist a univariate polynomial f with
>integer coefficients such that, for every integer n, all prime factors
>of f(n) are congruent to 1 mod 3?
Say p = 2 mod 3 is prime, consider the quadratic character of -3 mod p.
If p = 1 mod 4 then (-3|p) = (3|p) = (p|3) = (2|3) = -1,
>where (a|p) is the Legendre symbol.
If p = 3 mod 4 then (-3|p) = - (3|p) = (p|3) = -1.
So n^2 + 3 can't be divisible by any such prime p.
Right.
But the prime factors are not all congruent to 1 mod 3 (since 3 itself
can be a prime factor) so it almost works, but not quite.
quasi
===
Subject: Re: all prime factors of f(n) are congruent to 1 mod 3
Bytes: 2445
>What about m=3? Does there exist a univariate polynomial f with
>integer coefficients such that, for every integer n, all prime factors
>of f(n) are congruent to 1 mod 3?
>Say p = 2 mod 3 is prime, consider the quadratic character of -3 mod p.
>If p = 1 mod 4 then (-3|p) = (3|p) = (p|3) = (2|3) = -1,
>where (a|p) is the Legendre symbol.
>If p = 3 mod 4 then (-3|p) = - (3|p) = (p|3) = -1.
>So n^2 + 3 can't be divisible by any such prime p.
Right.
But the prime factors are not all congruent to 1 mod 3 (since 3 itself
>can be a prime factor) so it almost works, but not quite.
Also, n^2+3 has the prime factor 2 when n is odd. The problem requires
that _all_ prime factors be congruent to 1 mod 3.
quasi
===
Subject: Re: all prime factors of f(n) are congruent to 1 mod 3
Bytes: 2654
>What about m=3? Does there exist a univariate polynomial f with
>integer coefficients such that, for every integer n, all prime
factors
>of f(n) are congruent to 1 mod 3?
Say p = 2 mod 3 is prime, consider the quadratic character of -3 mod p.
If p = 1 mod 4 then (-3|p) = (3|p) = (p|3) = (2|3) = -1,
>where (a|p) is the Legendre symbol.
If p = 3 mod 4 then (-3|p) = - (3|p) = (p|3) = -1.
So n^2 + 3 can't be divisible by any such prime p.
>Right.
>But the prime factors are not all congruent to 1 mod 3 (since 3 itself
>can be a prime factor) so it almost works, but not quite.
Also, n^2+3 has the prime factor 2 when n is odd. The problem requires
>that _all_ prime factors be congruent to 1 mod 3.
>
Replace n by 6n+2. That gives us 36n^2 + 24n + 7 divisible only by
primes congruent to 1 mod 3.
RS
===
Subject: Re: all prime factors of f(n) are congruent to 1 mod 3
Bytes: 3010
On Thu, 28 Jun 2007 09:42:56 -0700, Robert Sheskey
>What about m=3? Does there exist a univariate polynomial f with
>integer coefficients such that, for every integer n, all prime
factors
>of f(n) are congruent to 1 mod 3?
>Say p = 2 mod 3 is prime, consider the quadratic character of -3 mod p.
>If p = 1 mod 4 then (-3|p) = (3|p) = (p|3) = (2|3) = -1,
>where (a|p) is the Legendre symbol.
>If p = 3 mod 4 then (-3|p) = - (3|p) = (p|3) = -1.
>So n^2 + 3 can't be divisible by any such prime p.
Right.
But the prime factors are not all congruent to 1 mod 3 (since 3 itself
>can be a prime factor) so it almost works, but not quite.
>Also, n^2+3 has the prime factor 2 when n is odd. The problem requires
>that _all_ prime factors be congruent to 1 mod 3.
Replace n by 6n+2. That gives us 36n^2 + 24n + 7 divisible only by
>primes congruent to 1 mod 3.
Yes, that fixes it perfectly.
In fact, that shows how to do it in general, I think.
I'm on my way out -- more later.
quasi
===
Subject: Re: to JSH and all : tommy's prime factoring tricks
> i promised to post on factoring yesterday, and i am a
> man of my word.
since JSH wasted a part of his life here on this
> forum (and the ones reading it)and spend (wasted) a
> lot of time on factoring , he might actually learn
> something today about factoring.
btw he already admitted his algoritm is bad or slow
> for big integers , so in a way he admitted it not
> being good.
certain algoritms are too complicated to explain to
> JSH and even others here , so i will keep it
> relatively simple.
there is no superduper factoring algoritm and im not
> going to deny the working of other algoritms , nor
> claim to have a superduper factoring algoritm.
but there certainly are some intresting properties
> for certain cases , and the cases that can be
> factored into them.
working with modulus can help factoring.
> knowing the amount of divisors can help factoring too
> , but usually that is not the case.
> between any two squared integers lies at least one
> prime.
> ( not proven but equivalent to a RH proof )
> as for coupling divisors that is simply a diophantine
> equation :
a^2 + k a = b
sow : a(a+k)=b
> where a and (a+k) are the coupling relations and a
> and
> (a+k) are the divisors of b.
and other couplings are of course similar although
> hidden in sum form rather than product.
a demonstration of mod is trivial :
n^2 + n or equivalently n ( n + 1)
is always devidable bye 2 ( because of mod 2 or
> parity )
also trivial is factoring certain numbers in
> polynomial form
so a number (n^4-1) always has a factor.
all this is basic and even known bye most adult
> crankpots and amateurs. ( apart from gene ray from
> timecube.com :p )
also digit recognizing of polynomial forms from an
> imput number can therefore be usefull.
e.g. ( 10^10 - 1 )/9 is easily recognizable from its
> digits and more advanced methods/examples exist.
less well known , but still basic.
what if a numbers equals a polyomial , which has none
> of the basic properties ??
> and so all of its zero's are unreal or nonintegers ?
> anything we can know about their factors ??
yes !!
so here ARE some prime factoring tricks ( for
> polynomial correspondance or factorable in certain
> polynomials) ,
> primes are denoted p_n.
...and there more where that came from...
> TOMMY1729 PRIME FACTORING TRICKS
TRICK ONE : a bit probalistic though.
let a = (p_1)^2 -1
then a is very likely to be highly composite !
( trick can work in reverse too ! )
and a = p_2 (or +1/-1) + p_3
(where p_2 can be a negative prime too although there
> always exists a positive one.)
> with abs (p_2) <<< p_3 and smaller abs (p_2) having
> higher probability than bigger ones(probability
> ordered bye size of abs)
> TRICK TWO : perfectly deterministic !
let n be an integer
factor : 5 nnnn - 10 nnn + 20 nn - 15 n + 11.
all of its primefactors end in 1.
in other words all prime factors of this polynomial
> number are of the form 10m+1.
before you ask : dont all number ending on 1 give
> primefactors ending on 1 ??
no 21 = 7 * 3
> and 221 = 17 * 13
> these tricks are also nice for testing the
> consistancy of ( unproven ) factoring algoritms or at
> least their javacodes.
the one and only
> tommy1729
damn , you seem to know more about factoring than my teacher :-D
in wich year will i learn that kind of math ??
what is RH ?
its kind a silly , but i feel good now.
i finally understood something not superevident.
i think i want to become a mathematician :-)
well if i dont become a singing girl.
and if im smart enough :p:p:p:p :/
hihi lol
what kind of music does a mathematician listen too ?
is it classical ?
people told me mathematcians listen to classical music , is this true ?
or did the *.8e(*.8e(*.8e assholes fooled me again.
kiss
amy
===
Subject: Re: to JSH and all : tommy's prime factoring tricks
<31443027.1182972275601.JavaMail.jakarta@nitrogen.mathforum.org>
Bytes: 2496
what kind of music does a mathematician listen too ?
> is it classical ?
people told me mathematcians listen to classical music , is this true ?
> or did the *?(*?(*? assholes fooled me again.
hello evil one
girls interested in math
have to listen to different music than the guys
you'll find out why eventually i'm sure
so if you really truly are interested in math
i'd say you should be listening to some
anarcho
japanoise
no wave
glitch
idm
when it gets that time maybe taking in a little
bonni prince billy
nick cave
( sneak in some album leaf - but don't let it out )
and the eventual decay to ani and tori
stay as far away from math rock as you possibly can
its not really very good or very bad
but it is about as lame as possible
and more lame than one could possibly expect
then mix in some musique concrete and flamenco
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
currently playing: architect, the analysis of noise trading
===
Subject: Re: to JSH and all : tommy's prime factoring tricks
<31443027.1182972275601.JavaMail.jakarta@nitrogen.mathforum.org girls
interested in math
> have to listen to different music than the guys
you'll find out why eventually i'm sure
so if you really truly are interested in math
> i'd say you should be listening to some
[...]
> idm
Hell yeah. Autechre are the best band ever. (I'm not a girl btw).
===
Subject: Re: to JSH and all : tommy's prime factoring tricks
<31443027.1182972275601.JavaMail.jakarta@nitrogen.mathforum.org>
Bytes: 1664
> what kind of music does a mathematician listen too ?
> is it classical ?
people told me mathematcians listen to classical music , is this true ?
> or did the *?(*?(*? assholes fooled me again.
kiss
> amy
I'm no mathematician (yet!), but I like indie rock and indie pop. Of
Montreal, Belle & Sebastian, Deerhoof, etc.
===
Subject: Re: to JSH and all : tommy's prime factoring tricks
<31443027.1182972275601.JavaMail.jakarta@nitrogen.mathforum.org>
Bytes: 1543
> what kind of music does a mathematician listen too ?
Mathematicians all listen to the new Kindzadza album:
http://saikosounds.com/english/display_release.asp?id=6543
(OK, so they don't. But they should!)
===
Subject: Re: to JSH and all : tommy's prime factoring tricks
Bytes: 1497
On Wed, 27 Jun 2007 15:24:05 EDT, amy666 <tommy1729@hotmail.comwhat kind of
music does a mathematician listen too ?
>is it classical ?
For me, it's blues -- deep, down-home, rural blues.
I also like certain types of traditional international folk music.
quasi
===
Subject: Re: to JSH and all : tommy's prime factoring tricks
>TRICK TWO : perfectly deterministic !
>let n be an integer
>factor : 5 nnnn - 10 nnn + 20 nn - 15 n + 11.
>all of its primefactors end in 1.
>in other words all prime factors of this polynomial >number are of the
form 10m+1.
n=(1,2,3,4,5,7,9,...,)
your polynomial:
p=(11,61,281,911,2311,9461,27011,...,)
all=1 mod.10
Vincenzo Librandi
===
Subject: Re: to JSH and all : tommy's prime factoring tricks
>TRICK TWO : perfectly deterministic !
>let n be an integer
>factor : 5 nnnn - 10 nnn + 20 nn - 15 n + 11.
>all of its primefactors end in 1.
>in other words all prime factors of this polynomial
>number are of the form 10m+1.
> n=(1,2,3,4,5,7,9,...,)
your polynomial:
p=(11,61,281,911,2311,9461,27011,...,)
all=1 mod.10
Vincenzo Librandi
dammed !!!
i EXPLICITELY EXPLAINED THAT THE ENDING ON 1 DOES NOT NECCESARILY MEAN
PRIMEFACTORS ENDING ON ONE ...
eg : 21 = 7*3
221 = 17*13
REREAD THE ORIGINAL AGAIN !!!!
I ANTICIPATED THIS MISUNDERSTANDING YET IT STILL OCCURS.
RETARD !!
sorry , i exaggerated.
your not a retard.
but you do frustrate me with this misunderstanding !!!!!
i explicitly told 21 = 7 * 3 as an example !!
are you posting this on purpose ???
well uhm
lol this seems strange after what i just said :p
greets
tommy1729
===
Subject: Re: to JSH and all : tommy's prime factoring tricks
is it bedtime in all the other countries maybe ...
===
Subject: Re: to JSH and all : tommy's prime factoring tricks
no replies yet hmmm
JSH why dont you reply ??
you pretended to be intrested in factoring ??
guess your not willing to learn anything hmm.
then i fear its a complete waste of your time here.
===
Subject: Poll: would you rather...
Bytes: 1343
Live on a deserted island the rest of your life...
Not have sex or masturbate the rest of your life...
or
Not be able to talk for the next 10 years?
Just an interesting hypothetical I'm putting out there :)
For recording purposes, please vote here:
http://www.thepollspace.com/polls.php?pollid=1689
===
Subject: IBRAHIM SHAFIULLAH
Cc: mishafiullah@gmail.com
Muhammad Ibrahim SHAFIULLAH
http://www.youtube.com/mishafiullah
===
Subject: solutions manual
Bytes: 1178
Hello everyone,
I would need the solutions manual to Probability and Statistics,
DeGroot, Schervish (3rd
edition)
Does anybody know how to get one.
I appreciate your help
===
Subject: Basic Concepts of Mathematics
Bytes: 1163
I want to build sound knowledge in the fundaments of physics. I'm from
===
Subject: Re: Basic Concepts of Mathematics
Bytes: 1949
I'm a student from India pursuing masters in human biology.
Since childhood, I've been fascinated with research in science. For
last four years I almost devoted myself for research in theoretical
biology & bioinformatics. During this process I had to ignore my
studies. As a result my overall concept in mathematics or physics is
poor although I've good knowledge in my research areas. Recently I
realized that I need to have sound knowledge in fundamental sciences
in order to do good research and my mentors would expect the same from
me. Also I failed to do anything substantial research in my area.
Earlier I thought that I'm in the right path but now I realized my
mistake. Please guide me so that I can follow your paths in research.
===
Subject: Re: Basic Concepts of Mathematics
> I want to build sound knowledge in the fundaments of
> physics. I'm from
> anticipation.
>
and dont waste to many time on the twin paradox or the uncertainty
principle.
modern physics is an unfinished science...
im waiting for the LHC in 2008 ...
and the probe B experiment.
there are still many paradoxes in physics...
e.g. certain probalistic decays where even the probability is not as
expected.
( tau decays )
well at least thats my opinion ...
===
Subject: Re: Basic Concepts of Mathematics
> I want to build sound knowledge in the fundaments of physics. I'm from
>
Perhaps make the request in a physics newsgropup?
===
Subject: Point of Condensation
Bytes: 1172
Please, help me with this.
Prove that 0 is the ONLY point of condensation for {1/n, n=1,2...}.
===
Subject: Re: Point of Condensation
Bytes: 1550
> Please, help me with this.
Prove that 0 is the ONLY point of condensation for {1/n, n=1,2...}.
Best way to start is to provide the definition. There are so many books
and so many variations ... accumulation point, adherent point, limit
point ... what definition of condensation point are you using? Once you
write out the definition clearly, the problem will solve itself.
===
Subject: Re: Point of Condensation
Bytes: 2266
Please, help me with this.
Prove that 0 is the ONLY point of condensation for {1/n, n=1,2...}.
Best way to start is to provide the definition. There are so many books
> and so many variations ... accumulation point, adherent point, limit
> point ... what definition of condensation point are you using? Once you
> write out the definition clearly, the problem will solve itself.
The Definition
Let x0 is an element of E. We'll say that it is a point of
condensation for E, iff there's at least one point in any of x0's
neighbourhoods, i.e. for any epsilon>0, we can find x from E{x0},
such that |x-x0|<epsilon.
It's easy to prove that 0 is a point of condensation for E={1/n,
n=1,2...}.
The problem is I can't prove, that it's the only condensation point.
===
Subject: Re: Point of Condensation
Bytes: 2605
On 27 Jun., 19:01, Narek Saribekyan <narek.saribek...@gmail.com
> Please, help me with this.
> Prove that 0 is the ONLY point of condensation for {1/n, n=1,2...}.
Best way to start is to provide the definition. There are so many books
> and so many variations ... accumulation point, adherent point, limit
> point ... what definition of condensation point are you using? Once you
> write out the definition clearly, the problem will solve itself.
The Definition
Let x0 is an element of E. We'll say that it is a point of
> condensation for E, iff there's at least one point in any of x0's
> neighbourhoods, i.e. for any epsilon>0, we can find x from E{x0},
> such that |x-x0|<epsilon.
It's easy to prove that 0 is a point of condensation for E={1/n,
> n=1,2...}.
> The problem is I can't prove, that it's the only condensation point.
Alternatively, how about showing the following
LEMMA: If U is open neighbourhood of x0 and has only finitely many
points in common with E, then x0 is not a point of condensation.
Then for x0>0 consider U = ]x/2,infinity[
===
Subject: Re: Point of Condensation
Bytes: 2754
On 27 Jun., 19:01, Narek Saribekyan <narek.saribek...@gmail.com
> Please, help me with this.
> Prove that 0 is the ONLY point of condensation for {1/n, n=1,2...}.
Best way to start is to provide the definition. There are so many books
> and so many variations ... accumulation point, adherent point, limit
> point ... what definition of condensation point are you using? Once you
> write out the definition clearly, the problem will solve itself.
The Definition
Let x0 is an element of E. We'll say that it is a point of
> condensation for E, iff there's at least one point in any of x0's
> neighbourhoods, i.e. for any epsilon>0, we can find x from E{x0},
> such that |x-x0|<epsilon.
It's easy to prove that 0 is a point of condensation for E={1/n,
> n=1,2...}.
Proving this *cannot* be easy since 0 fails to be an element of E
although your definition requires it.
> The problem is I can't prove, that it's the only condensation point.
Now consider any point x on the real line that is not 0.
Contemplate the following cases:
(i) x < 0
(ii) x > 1
(iii) x = 1/n for some n (or put the other way round: 1/x in N)
(iv) 0<x<1 and 1/x is not in N
See if you can handle some of these cases.
hagman
===
Subject: Re: Point of Condensation
Bytes: 3491
> On 27 Jun., 19:01, Narek Saribekyan <narek.saribek...@gmail.com
> Please, help me with this.
> Prove that 0 is the ONLY point of condensation for {1/n, n=1,2...}.
> Best way to start is to provide the definition. There are so many
books
> and so many variations ... accumulation point, adherent point, limit
> point ... what definition of condensation point are you using? Once
you
> write out the definition clearly, the problem will solve itself.
The Definition
Let x0 is an element of E. We'll say that it is a point of
> condensation for E, iff there's at least one point in any of x0's
> neighbourhoods, i.e. for any epsilon>0, we can find x from E{x0},
> such that |x-x0|<epsilon.
It's easy to prove that 0 is a point of condensation for E={1/n,
> n=1,2...}.
Proving this *cannot* be easy since 0 fails to be an element of E
> although your definition requires it.
The problem is I can't prove, that it's the only condensation point.
Now consider any point x on the real line that is not 0.
> Contemplate the following cases:
> (i) x < 0
> (ii) x > 1
> (iii) x = 1/n for some n (or put the other way round: 1/x in N)
> (iv) 0<x<1 and 1/x is not in N
> See if you can handle some of these cases.
hagman- Hide quoted text -
- Show quoted text -
I can handle (i)and (ii) easily. Let's try to do something with (iii).
Suppose, 1/n is a condensation point, so for any epsilon>0 we can find
0<x<1, s.t. 1/x in N, i.e. x = 1/m, that
|1/m-1/n|<epsilon
|m-n|/(m*n)<epsilon
We have that |m-n|>1, so epsilon>1/(m*n). So I'm getting that it's
really possible 1/n to be a condesation point, because we can take m
as big as we want for any fixed n.
Where's my mistake?
===
Subject: Re: Point of Condensation
Bytes: 3847
On 27 Jun., 20:12, Narek Saribekyan <narek.saribek...@gmail.com
> On 27 Jun., 19:01, Narek Saribekyan <narek.saribek...@gmail.com
> Please, help me with this.
> Prove that 0 is the ONLY point of condensation for {1/n,
n=1,2...}.
> Best way to start is to provide the definition. There are so many
books
> and so many variations ... accumulation point, adherent point,
limit
> point ... what definition of condensation point are you using? Once
you
> write out the definition clearly, the problem will solve itself.
> The Definition
> Let x0 is an element of E. We'll say that it is a point of
> condensation for E, iff there's at least one point in any of x0's
> neighbourhoods, i.e. for any epsilon>0, we can find x from E{x0},
> such that |x-x0|<epsilon.
> It's easy to prove that 0 is a point of condensation for E={1/n,
> n=1,2...}.
Proving this *cannot* be easy since 0 fails to be an element of E
> although your definition requires it.
> The problem is I can't prove, that it's the only condensation point.
Now consider any point x on the real line that is not 0.
> Contemplate the following cases:
> (i) x < 0
> (ii) x > 1
> (iii) x = 1/n for some n (or put the other way round: 1/x in N)
> (iv) 0<x<1 and 1/x is not in N
> See if you can handle some of these cases.
hagman- Hide quoted text -
- Show quoted text -
I can handle (i)and (ii) easily. Let's try to do something with (iii).
> Suppose, 1/n is a condensation point, so for any epsilon>0 we can find
> 0<x<1, s.t. 1/x in N, i.e. x = 1/m, that
|1/m-1/n|<epsilon
|m-n|/(m*n)<epsilon
We have that |m-n|>1,
Strictly, |m-n| >= 1, but nevertheless ...
> so epsilon>1/(m*n).
... remains correct, though inefficient.
Note: if m =/= n, we have either m>=n+1 or m<=n-1,
hence either 1/m <= 1/(n+1) or 1/m >= 1/(n-1).
> So I'm getting that it's
> really possible 1/n to be a condesation point, because we can take m
> as big as we want for any fixed n.
Where's my mistake?
===
Subject: Re: Point of Condensation
Bytes: 2269
Please, help me with this.
Prove that 0 is the ONLY point of condensation for {1/n, n=1,2...}.
Best way to start is to provide the definition. There are so many books
> and so many variations ... accumulation point, adherent point, limit
> point ... what definition of condensation point are you using? Once you
> write out the definition clearly, the problem will solve itself.
The Definition
Let x0 is an element of E. We'll say that it is a point of
condensation for E, iff there's at least one point in any of x0's
neighbourhoods, i.e. for any epsilon>0, we can find x from E{x0},
such that |x-x0|<epsilon.
It's easy to prove that 0 is a point of consideration for E={1/n,
n=1,2...}.
The problem is I can't prove, that it's the only consideration point.
===
Subject: Re: Point of Condensation
Please, help me with this.
Prove that 0 is the ONLY point of condensation for {1/n, n=1,2...}.
Best way to start is to provide the definition. There are so many books
> and so many variations ... accumulation point, adherent point, limit
> point ... what definition of condensation point are you using? Once you
> write out the definition clearly, the problem will solve itself.
The Definition
Let x0 is an element of E. We'll say that it is a point of
consideration for E, iff there's at least one point in any of x0's
neighbourhoods, i.e. for any epsilon>0, we can find x from E{x0},
such that |x-x0|<epsilon.
It's easy to prove that 0 is a point of consideration for E={1/n,
n=1,2...}.
The problem is I can't prove, that it's the only point of condensation.
===
Subject: Re: Point of Condensation
Bytes: 2271
Please, help me with this.
Prove that 0 is the ONLY point of condensation for {1/n, n=1,2...}.
Best way to start is to provide the definition. There are so many books
> and so many variations ... accumulation point, adherent point, limit
> point ... what definition of condensation point are you using? Once you
> write out the definition clearly, the problem will solve itself.
The Definition
Let x0 is an element of E. We'll say that it is a point of
consideration for E, iff there's at least one point in any of x0's
neighbourhoods, i.e. for any epsilon>0, we can find x from E{x0},
such that |x-x0|<epsilon.
It's easy to prove that 0 is a point of consideration for E={1/n,
n=1,2...}.
The problem is I can't prove, that it's the only consideration point.
===
Subject: Re: Point of Condensation
Bytes: 2254
Please, help me with this.
Prove that 0 is the ONLY point of condensation for {1/n, n=1,2...}.
Best way to start is to provide the definition. There are so many books
> and so many variations ... accumulation point, adherent point, limit
> point ... what definition of condensation point are you using? Once you
> write out the definition clearly, the problem will solve itself.
The Definition
Let x0 is an element of E. We'll say that it is a point of
consideration for E, iff there's at least one point in any of x0's
neighbourhoods, i.e. for any epsilon>0, we can find x from E{x0},
such that |x-x0|<epsilon.
It's easy to prove that 0 is a point of consideration for E={1/n,
n=1,2...}.
The problem is I can't prove, that it's the only consideration point.
===
Subject: A closed form method for the closest approximation of x! for the
first 15 factorials.
Bytes: 2858
Going with this enhanced approximation formula --
x! ~ x^x*e^((1/(12*x))- x)*sqrt(2pix)
Note: (2pix) is just (2*pi*x).
x
--
1! = x^x*e^((1/(12*1))- x)*sqrt(2pix) = 1!=
1.0022744491822266585
2! = x^x*e^((1/(12*2))- x)*sqrt(2pix) =
2.00065204769096582997
3! = x^x*e^((1/(12*3))- x)*sqrt(2pix) =
6.00059914246899352327
4! = x^x*e^((1/(12*4))- x)*sqrt(2pix) =
24.00102389134971443835
5! = x^x*e^((1/(12*5))- x)*sqrt(2pix) =
120.00263708619696414538
6! = x^x*e^((1/(12*6))- x)*sqrt(2pix) =
720.00918730600425143784
7! = x^x*e^((1/(12*7))- x)*sqrt(2pix) =
5040.04058203608770218505
8! = x^x*e^((1/(12*8))- x)*sqrt(2pix) =
40320.21778522545760932557
9! = x^x*e^((1/(12*9))- x)*sqrt(2pix) =
362881.37788575317066188753 (Floor) just +1.
10! = x^x*e^((1/(12*10))- x)*sqrt(2pix) =
3628810.05142693352994116531... just +10
11! = x^x*e^((1/(12*11))- x)*sqrt(2pix) =
39916883.11036904547429087637... just +83.
12! = x^x*e^((1/(12*12))- x)*sqrt(2pix) =
479002368.48071891461010082264... just +768.
13! = x^x*e^((1/(12*13))- x)*sqrt(2pix) =
6227028659.88917915456467479342...just +7859
14! = x^x*e^((1/(12*14))- x)*sqrt(2pix) =
87178379323.31547159867855728656... just +88123
15! = x^x*e^((1/(12*15))- x)*sqrt(2pix) =
1307675442913.47043986439248446528... Just +1074913
etc.
What is interesting here is in the part of the formula
(12*x) then (12*x) + 1 results are <x! or
(12*x) - 1 results are >x! for any (x).
My question is, why does 12 have such a profound influence
on this formula that approximate factorials?
I know there are better formulas for larger factorials but
still I found this one interesting because of the integer
(12) involved in all x! approximations.
Dan
===
Subject: Re: A closed form method for the closest approximation of x! for
the first 15 factorials.
Bytes: 3655
> Going with this enhanced approximation formula --
x! ~ x^x*e^((1/(12*x))- x)*sqrt(2pix)
Note: (2pix) is just (2*pi*x).
If we let G(y) = gamma function of y, then G(y+1) = y! for integer y
(and this can be used as a definition of y! for non-integer y). It is
known that
G(x) = x^(x- 1/2) e^(-x) sqrt(2 pi) F, where
e^[1/(12 x + 6/7)] < F < e^[1/(12 x)] (L. Gordon, Amer. Math. Monthly
(1994), Vol. 101, pp. 858--865). The approximation you use replaces 1/
[12(x+1)] by the slightly larger 1/[12 x] and then uses its
exponential as an estimate of F.
R.G. Vickson
x
> --
> 1! = x^x*e^((1/(12*1))- x)*sqrt(2pix) = 1!=
1.0022744491822266585
2! = x^x*e^((1/(12*2))- x)*sqrt(2pix) =
2.00065204769096582997
3! = x^x*e^((1/(12*3))- x)*sqrt(2pix) =
6.00059914246899352327
4! = x^x*e^((1/(12*4))- x)*sqrt(2pix) =
24.00102389134971443835
5! = x^x*e^((1/(12*5))- x)*sqrt(2pix) =
120.00263708619696414538
6! = x^x*e^((1/(12*6))- x)*sqrt(2pix) =
720.00918730600425143784
7! = x^x*e^((1/(12*7))- x)*sqrt(2pix) =
5040.04058203608770218505
8! = x^x*e^((1/(12*8))- x)*sqrt(2pix) =
40320.21778522545760932557
9! = x^x*e^((1/(12*9))- x)*sqrt(2pix) =
362881.37788575317066188753 (Floor) just +1.
10! = x^x*e^((1/(12*10))- x)*sqrt(2pix) =
3628810.05142693352994116531... just +10
11! = x^x*e^((1/(12*11))- x)*sqrt(2pix) =
39916883.11036904547429087637... just +83.
12! = x^x*e^((1/(12*12))- x)*sqrt(2pix) =
479002368.48071891461010082264... just +768.
13! = x^x*e^((1/(12*13))- x)*sqrt(2pix) =
6227028659.88917915456467479342...just +7859
14! = x^x*e^((1/(12*14))- x)*sqrt(2pix) =
87178379323.31547159867855728656... just +88123
15! = x^x*e^((1/(12*15))- x)*sqrt(2pix) =
1307675442913.47043986439248446528... Just +1074913
etc.
What is interesting here is in the part of the formula
> (12*x) then (12*x) + 1 results are <x! or
> (12*x) - 1 results are >x! for any (x).
My question is, why does 12 have such a profound influence
> on this formula that approximate factorials?
I know there are better formulas for larger factorials but
> still I found this one interesting because of the integer
> (12) involved in all x! approximations.
Dan
===
Subject: Re: A closed form method for the closest approximation of x! for
the first 15 factorials.
Bytes: 3989
Going with this enhanced approximation formula --
x! ~ x^x*e^((1/(12*x))- x)*sqrt(2pix)
Note: (2pix) is just (2*pi*x).
If we let G(y) = gamma function of y, then G(y+1) = y! for integer y
> (and this can be used as a definition of y! for non-integer y). It is
> known that
> G(x) = x^(x- 1/2) e^(-x) sqrt(2 pi) F, where
> e^[1/(12 x + 6/7)] < F < e^[1/(12 x)] (L. Gordon, Amer. Math. Monthly
> (1994), Vol. 101, pp. 858--865). The approximation you use replaces 1/
> [12(x+1)] by the slightly larger 1/[12 x] and then uses its
> exponential as an estimate of F.
R.G. Vickson
x
> --
> 1! = x^x*e^((1/(12*1))- x)*sqrt(2pix) = 1!=
1.0022744491822266585
2! = x^x*e^((1/(12*2))- x)*sqrt(2pix) =
2.00065204769096582997
3! = x^x*e^((1/(12*3))- x)*sqrt(2pix) =
6.00059914246899352327
4! = x^x*e^((1/(12*4))- x)*sqrt(2pix) =
24.00102389134971443835
5! = x^x*e^((1/(12*5))- x)*sqrt(2pix) =
120.00263708619696414538
6! = x^x*e^((1/(12*6))- x)*sqrt(2pix) =
720.00918730600425143784
7! = x^x*e^((1/(12*7))- x)*sqrt(2pix) =
5040.04058203608770218505
8! = x^x*e^((1/(12*8))- x)*sqrt(2pix) =
40320.21778522545760932557
9! = x^x*e^((1/(12*9))- x)*sqrt(2pix) =
362881.37788575317066188753 (Floor) just +1.
10! = x^x*e^((1/(12*10))- x)*sqrt(2pix) =
3628810.05142693352994116531... just +10
11! = x^x*e^((1/(12*11))- x)*sqrt(2pix) =
39916883.11036904547429087637... just +83.
12! = x^x*e^((1/(12*12))- x)*sqrt(2pix) =
479002368.48071891461010082264... just +768.
13! = x^x*e^((1/(12*13))- x)*sqrt(2pix) =
6227028659.88917915456467479342...just +7859
14! = x^x*e^((1/(12*14))- x)*sqrt(2pix) =
87178379323.31547159867855728656... just +88123
15! = x^x*e^((1/(12*15))- x)*sqrt(2pix) =
1307675442913.47043986439248446528... Just +1074913
etc.
What is interesting here is in the part of the formula
> (12*x) then (12*x) + 1 results are <x! or
> (12*x) - 1 results are >x! for any (x).
My question is, why does 12 have such a profound influence
> on this formula that approximate factorials?
I know there are better formulas for larger factorials but
> still I found this one interesting because of the integer
> (12) involved in all x! approximations.
Dan- Hide quoted text -
- Show quoted text -
Dan
===
Subject: Re: Transformation of complex arguments in polynomials?
Bytes: 1335
J.9frgen Will <jwill@onlinehome.de> schrieb im Newsbeitrag
> Real zeros should remain real. They could become the same or other reals.
> They should persist on the real axis.
O excuse me. It is a mistake. Real zeros should change to imaginary zeros.
They shoud be mapped onto the imaginary axis.
===
Subject: Re: Transformation of complex arguments in polynomials?
<9625722.1182696994241.JavaMail.jakarta@nitrogen.mathforum.org>
<f5p6dd$4sd$1@online.de>
<f5u2vr$ufd$1@online.de>
Bytes: 1942
> They should persist on the real axis.
O excuse me. It is a mistake. Real zeros should change to imaginary
zeros.
> They shoud be mapped onto the imaginary axis.
Given a real polynomial P(X), consider the polynomial Q(X)=P(iX)*P(-
iX).
It has real coefficients, whenever a+ib is a root of P, we have
that b+ia is a root of Q and in the generic case, Q is of minimal
degree with these conditions.
===
Subject: Re: Transformation of complex arguments in polynomials?
O excuse me. It is a mistake. Real zeros should
> change to imaginary zeros.
> They shoud be mapped onto the imaginary axis.
Given a real polynomial P(X), consider the polynomial
> Q(X)=P(iX)*P(-
> iX).
> It has real coefficients, whenever a+ib is a root of
> P, we have
> that b+ia is a root of Q and in the generic case, Q
> is of minimal
> degree with these conditions.
>
lets see :
p(x) = x**3-x**2+x-2
q(x) = (-i x**3 +x**2 +ix -2)(-i x**3 +x**2 -ix -2)
= -x^6 -2 i x^5 +x^4 + 4i x^3 -3x^2 + 4
I SEE 2 IMAGINAIRY COEFFICIENTS so no deal for the real coefficients man !!
also the degree of Q is twice that of P !!
and sometimes Q can be reduced to a smaller minimum polynomial (which
actually has nothing to do with Q ! )
or Q needs to be extended to remove the imaginary and give a bigger minimum
polynomial (also not really related to Q but just the roots of P ! )
tommy1729
===
Subject: Re: Transformation of complex arguments in polynomials?
<9716823.1182975942175.JavaMail.jakarta@nitrogen.mathforum.org>
Bytes: 3462
> O excuse me. It is a mistake. Real zeros should
> change to imaginary zeros.
> They shoud be mapped onto the imaginary axis.
Given a real polynomial P(X), consider the polynomial
> Q(X)=P(iX)*P(-
> iX).
> It has real coefficients, whenever a+ib is a root of
> P, we have
> that b+ia is a root of Q and in the generic case, Q
> is of minimal
> degree with these conditions.
lets see :
> p(x) = x**3-x**2+x-2
> q(x) = (-i x**3 +x**2 +ix -2)(-i x**3 +x**2 -ix -2)
Start again from
q(x) = (-i x**3 + x**2 + ix -2)(i x**3 + x**2 -ix +2)
= -x^6 -2 i x^5 +x^4 + 4i x^3 -3x^2 + 4
I SEE 2 IMAGINAIRY COEFFICIENTS so no deal for the real coefficients man
!!
Note that P(conj(z)) = conj(P(z)) by assumtion.
Therfore
Q(conj(z)) = P(i conj(z)) P(-i conj(z))
= P(conj(-iz)) P(conj(iz))
= P(-iz) P(iz)
= Q(z)
as well.
also the degree of Q is twice that of P !!
Necessarily (at least in the generic case).
Requirement was: If a,b are real numbers such that P(a+ib)=0
then Q(b+ia)=0.
If for example P(3+2i)=0, the condition is fulfilled for (a,b)=(3,2).
But as P is real, we also have P(3-2i)=0, so the condition is
fulfilled
as well for (a,b)=(3,-2).
Therefore we must have Q(2+3i)=Q(-2+3i)=0.
Again, Q should be real, thus Q(2-3i)=Q(-2-3i)=0 as well.
> and sometimes Q can be reduced to a smaller minimum polynomial (which
actually has nothing to do with Q ! )
Note that I used the term minimal polynomial not in the sense that
Q divides all real polynomials with one of the given roots.
Q has minimal degree among all real polynomials such that
Q(b+ia)=0 whenever P(a+ib)=0 at least in the generic case, i.e.
if P(z)=0 implies P(iz) != 0 and P'(z) != 0.
Even when it is not minimal, the failure is predictable
as the multiplicities behave accordingly.
If you wish, you may calculate gcd(Q,Q') to find multiple roots
that one can cast out.
or Q needs to be extended to remove the imaginary and give a bigger
minimum polynomial (also not really related to Q but just the roots of P ! 
)
tommy1729
===
Subject: Re: Transformation of complex arguments in polynomials?
Bytes: 1801
tommy1729 <tommy1729@gmail.com> schrieb im Newsbeitrag
My last answer was for us at all.
> since if half the zero's have imaginairy part the ones without the
conjugate
> generate a polynomial of half degree from the original.
> since its conjugates are missing whether or not a polynomial exists for
it
> depends on the minimum polynomials of the zero's.
I already know: Erasing the zeros of P(z) only in one half plane by
radicals
is not possible in general in accordance to Abel. I want to transform all
zeros of P(z) into other zeros of another polynomial therefore.
Are the conjugates missing?
> of course sometimes you can use multiplicity of roots.
Could you please declare that? I'm not a mathematician. I'm a natural
scientist.
===
Subject: Re: Transformation of complex arguments in polynomials?
> tommy1729 <tommy1729@gmail.com> schrieb im
> Newsbeitrag
> athforum.org...
> My last answer was for us at all.
since if half the zero's have imaginairy part the
> ones without the
> conjugate
> generate a polynomial of half degree from the
> original.
> since its conjugates are missing whether or not a
> polynomial exists for it
> depends on the minimum polynomials of the zero's.
> I already know: Erasing the zeros of P(z) only in one
> half plane by radicals
> is not possible in general in accordance to Abel. I
> want to transform all
> zeros of P(z) into other zeros of another polynomial
> therefore.
> Are the conjugates missing?
of course sometimes you can use multiplicity of
> roots.
> Could you please declare that? I'm not a
> mathematician. I'm a natural
> scientist.
>
simple, suppose you have a certain zero twice as root , that gives you an
additional degree 2 with just 1 distinct zero.
or let the zeros have simple relationships , than you can easily find your
polynome.
however you are of course intrested in the transformation of polyno. where
the roots dont have rootform and no multiplicity ( repeat of the same zero 
)
tommy1729
===
Subject: Re: Transformation of complex arguments in polynomials?
Bytes: 1843
> Hallo,
a real polynomial P(z) with real coefficients and complex arguments z is
> given. The zeros of P(z) are z0[k] = a[k]+/-b[k]*i (k: index). Is there a
> transformation of P(z) which transforms the polynomial P(z) into a
> polynomial P1(z1) with the zeros z10[k] = b[k]+/-a[k]*i, without solving
> P(z)=0?
For example, this would have to map
X^2 + a X + b (with a^2<4b)
to
X^2 + sqrt(4b-a^2) X + b
One might be able top deduce similar formulae
for e.g. 4th degree, but I'm not even sure what you think
is supposed to happen if P as a real root, e.g. X^2-X-1
===
Subject: Re: Transformation of complex arguments in polynomials?
Bytes: 1461
> For example, this would have to map ...
I'm interested in the general case of arbitrary degrees.
> but I'm not even sure what you think
> is supposed to happen if P as a real root, e.g. X^2-X-1
Real zeros should remain real. They could become the same or other reals.
They should persist on the real axis.
===
Subject: Re: Transformation of complex arguments in polynomials?
<f5rhr2$tk7$2@online.de
> I'm interested in the general case of arbitrary degrees.
I just wanted to handle the basic case of irreducible polynomials
once and for all.
Any real polynomial with no real roots an be written as a product
of quadratics.
This makes it reasonable to postulate that your transformation be
multiplicative, i.e. T(P*Q) = T(P)*T(Q) should hold for
polynomials P,Q.
but I'm not even sure what you think
> is supposed to happen if P as a real root, e.g. X^2-X-1
Real zeros should remain real. They could become the same or other reals.
> They should persist on the real axis.
In that case
P = X^2 + aX + b
should be transformed to
T(P) = X^2 + sqrt(4b-a^2) X + b if a^2<4b
but
T(P) = some polynomial with two real zeroes if a^2 >= 4b.
It would be most natural to postulate T(X-c) = X-c, therefor
T( X^2 + aX + b ) = X^2+aX+b if a^2>=b.
Then T(T(P)) = P if 0<a^2<4b, but T(T(P)) = X^2 + 2sqrt(b) X + b if
a=0.
This make T look somewhat weird to me.
Moreover, what I said above ignores the sign of the sqrt, i.e.
one could just as well have
T(P) = X^2 - sqrt(4b-a^2) X + b
for P=X^2+aX+b with a^2<4b
By mixing variants for higher degrees, you might thus have
T(X^4+4) = X^4+4 or (X^2-2X+2)^2 or (X^2+2X+2)^2.
===
Subject: Re: Transformation of complex arguments in polynomials?
> On 26 Jun., 19:20, J?rgen Will <j...@onlinehome.de hagman
<goo...@von-eitzen.de> schrieb im
> oglegroups.com...
> On 22 Jun., 21:02, J?rgen Will
> have to map ...
I'm interested in the general case of arbitrary
> degrees.
I just wanted to handle the basic case of irreducible
> polynomials
> once and for all.
> Any real polynomial with no real roots an be written
> as a product
> of quadratics.
> This makes it reasonable to postulate that your
> transformation be
> multiplicative, i.e. T(P*Q) = T(P)*T(Q) should hold
> for
> polynomials P,Q.
 but I'm not even sure what you think
> is supposed to happen if P as a real root, e.g.
> X^2-X-1
Real zeros should remain real. They could become
> the same or other reals.
> They should persist on the real axis.
In that case
> P = X^2 + aX + b
> should be transformed to
> T(P) = X^2 + sqrt(4b-a^2) X + b if a^2<4b
> but
> T(P) = some polynomial with two real zeroes if a^2
> 2 >= 4b.
> It would be most natural to postulate T(X-c) = X-c,
> therefor
> T( X^2 + aX + b ) = X^2+aX+b if a^2>=b.
Then T(T(P)) = P if 0<a^2<4b, but T(T(P)) = X^2 +
> 2sqrt(b) X + b if
> a=0.
> This make T look somewhat weird to me.
Moreover, what I said above ignores the sign of the
> sqrt, i.e.
> one could just as well have
> T(P) = X^2 - sqrt(4b-a^2) X + b
> for P=X^2+aX+b with a^2<4b
By mixing variants for higher degrees, you might thus
> have
> T(X^4+4) = X^4+4 or (X^2-2X+2)^2 or (X^2+2X+2)^2.
thats an interisting point of view hagman.
but doesnt factoring in degree 2 ( to do the transform) , require solving
the polynomial ?
yes.
but then it is transformed again and the question becomes
CAN the transform be simplified in terms of solving the original polynomial
??
my guess is no.
but im open for ideas.
also note that it is mentioned before that there are 2 transforms , which
you have forgotten i believe.
its an intresting idea , but i think wrong ...
tommy1729
===
Subject: Re: @ timothy golden
> On Jun 4, 6:16 pm, tommy1729 <tommy1...@gmail.com > Timothy , have you 
ever
considered infinite
> dimensional polysigned numbers ?
i guess not ...
btw i meant uncountable infinite.
Since the set of signs is discrete, you could only
> have a countably
> infinite number of signs.
Darren
>
well , yes and no , i was also considering transcendental roots of one ,
therefore the entire circle of radius 1.
and hence uncountable.
however considering countable infinity is probably sufficient.
since the difference between points of infinite countable and infinite
uncountable is infinite ** small.
** countable ??
never mind thats not so important now, but couldnt help asking
intuitively...
===
Subject: Selling off math books
Bytes: 1278
My mom died last year and finally we sell off her books. She collected
a lot and was kind a packrat. I dont have a list but she really
collected everything Send me an email and ask elakenad@yahoo.com
ps sorry if this not appropiate here, but i couldnt figure ebay
===
Subject: Re: Selling off math books
Bytes: 2269
> My mom died last year and finally we sell off
> her books. She collected a lot and was kind a
> packrat. I dont have a list but she really
> collected everything Send me an email and
> ask elake...@yahoo.com
ps sorry if this not appropiate here, but
> i couldnt figure ebay
You really need to make a list. I doubt anyone
is going to randomly ask about books without knowing
if there's even a ghost of a chance that you have it.
For example, I would be interested if she had a copy
of Fine Topology Methods in Real Analysis and Potential
Theory by Jaroslav Lukes, Jan Maly, and Ludek Zajicek,
but I seriously doubt she's that much of a packrat to
have all (or most) of the Springer Verlag Lecture Notes
in Mathematics volumes. I'm also particularly interested in
getting a copy of Eduard Cech's 1936 book Bodov? Mnoziny
(or at least a copy of the appendix written by V. Jarnik
on Baire category results for the Dini derivate behavior
of most continuous functions), but I'm not going to
hold my breath.
Dave L. Renfro
===
Subject: Re: Selling off math books
My mom died last year and finally we sell off
> her books. She collected a lot and was kind a
> packrat. I dont have a list but she really
> collected everything Send me an email and
> ask elake...@yahoo.com
ps sorry if this not appropiate here, but
> i couldnt figure ebay
You really need to make a list. I doubt anyone
> is going to randomly ask about books without knowing
> if there's even a ghost of a chance that you have it.
> For example, I would be interested if she had a copy
> of Fine Topology Methods in Real Analysis and Potential
> Theory by Jaroslav Lukes, Jan Maly, and Ludek Zajicek,
You can now download the pdf from Springer's website, if
your library has access.
--
A.
===
Subject: Re: Selling off math books
<artfldodgr-D1601A.15075427062007@individual.net>
Bytes: 2573
> For example, I would be interested if she had a copy
> of Fine Topology Methods in Real Analysis and Potential
> Theory by Jaroslav Lukes, Jan Maly, and Ludek Zajicek,
> You can now download the pdf from Springer's website,
> if your library has access.
Well, I'm not affiliated with a university (anymore),
but there is a university in town that I visit often. Since
all the Springer Lecture Notes series are on the library
shelves, I've never had the occasion to see if .pdf copies
of them are available. Nonetheless, it would be nice to have
a bound print copy of my own (compact size, can write notes
in it, etc.). I've had a photocopy of it since 1991 or 1992
(done on a very good copy machine, with pages maximally
enlarged for increased readability), but it fills three
loose notebooks and thus it's a bit awkward to work with
when all I want to do is quickly double-check something
in it. (Because of space considerations, it's not even in
my workroom at home, but instead it's in some overflow
bookshelves in my apartment's living room.) Still, a .pdf
file can useful, because you can do electronic searches
in it. Thus, it might be worth my while to download the
.pdf file and e-mail it to myself sometime.
Dave L. Renfro
===
Subject: thinking about metric spaces
Bytes: 1045
===
Subject: Re: thinking about metric spaces
Bytes: 1334
Oops! This was meant to go in the other topic. I hit the wrong button.
Sorry guys.
===
Subject: Sci.
Bytes: 2358
Hi all,
I would like to form a set theory that can surve as the basis for
logically consistent scientific theories. I would like this theory to
have non logical primitive predicates like 'x is a man' ,
'x is a proton' etc.... , I will call this theory 'Sci'.
Here is a simple trial to do that:
Sci is the set of all sentences entailed by Lsci ( see below )
from the following axioms:
Language is Lsci which is: first order logic with identity and the
following non logical primitive predicates P1,P2,P3,....,Pn
and the primitive constant M.
Axioms:
1) Foundation: Ax( Ey yex -> Ey( yex & ~Ec(cey & cex))).
2) Schema of Ur-elements: for every i
Ax( Pi(x) -> xeM)
is an axiom.
3) Emptyness: Ax( xeM -> Ay(~yex))
4) Extensionality: AxAy((~xeM&~yeM)->(Az(zex<->zey) -> x=y)).
5) Pairing: ArAsExAy(yex<->(y=r v y=s)).
6) Union: Ar Ex ( ~xeM & Ay(yex<->Ez(zer&yez)) ).
7) Power: Ar Ex ( ~xeM & Ay(yex<->Az(zey->zer)) ).
8) Separation: if F is a formula in which x is not free, then all
closures of
Ar Ex ( ~xeM & Ay(yex<->(yer & F(y))))
are axioms.
9) Finity: Ax( x is finite ).
x is finite <-> EREconvR( R is well ordering on x & convR is well
ordering on x ).
10) Replacement: if F is a formula in which b is not free, then all
closures of
AxE!y(F(x,y)) -> ArEb(~beM & Ay(yeb<->Exer(F(x,y))))
are axioms.
/
Zuhair
===
Subject: Re: Sci.
> Hi all,
I would like to form a set theory that can surve as
> the basis for
> logically consistent scientific theories. I would
> like this theory to
> have non logical primitive predicates like 'x is a
> man' ,
> 'x is a proton' etc.... , I will call this theory
> 'Sci'.
Here is a simple trial to do that:
Sci is the set of all sentences entailed by Lsci (
> see below )
> from the following axioms:
Language is Lsci which is: first order logic with
> identity and the
> following non logical primitive predicates
> P1,P2,P3,....,Pn
> and the primitive constant M.
Axioms:
1) Foundation: Ax( Ey yex -> Ey( yex & ~Ec(cey &
> cex))).
2) Schema of Ur-elements: for every i
Ax( Pi(x) -> xeM)
is an axiom.
3) Emptyness: Ax( xeM -> Ay(~yex))
4) Extensionality: AxAy((~xeM&~yeM)->(Az(zex<->zey)
> -> x=y)).
5) Pairing: ArAsExAy(yex<->(y=r v y=s)).
6) Union: Ar Ex ( ~xeM & Ay(yex<->Ez(zer&yez)) ).
7) Power: Ar Ex ( ~xeM & Ay(yex<->Az(zey->zer)) ).
8) Separation: if F is a formula in which x is not
> free, then all
> closures of
Ar Ex ( ~xeM & Ay(yex<->(yer & F(y))))
are axioms.
9) Finity: Ax( x is finite ).
x is finite <-> EREconvR( R is well ordering on x &
> convR is well
> ordering on x ).
10) Replacement: if F is a formula in which b is not
> free, then all
> closures of
AxE!y(F(x,y)) -> ArEb(~beM & Ay(yeb<->Exer(F(x,y))))
are axioms.
/
Zuhair
>
hi ,
what is a set theory ?
amy
===
Subject: Irregular Pyramid
We were getting into a little debate at work about how to find the
height of an irregular pyramid just by knowing A) the lengths of all
sides, and B) when looking down from the top of the pyramid the edges
are orthognal. Is there a way to find the height of this shape or do I
need more info?
Matt
===
Subject: Re: Irregular Pyramid
Bytes: 1739
We were getting into a little debate at work about how to find the
> height of an irregular pyramid just by knowing A) the lengths of all
> sides, and B) when looking down from the top of the pyramid the edges
> are orthognal. Is there a way to find the height of this shape or do I
> need more info?
Matt
just the lengths (of every side) should do, that will completely
define the 4 triangles and one quadrilateral.
if you are pricing a new pyramid however, pay someone to do a proper
survey!
===
Subject: Re: Irregular Pyramid
Bytes: 2140
We were getting into a little debate at work about how to find the
> height of an irregular pyramid just by knowing A) the lengths of all
> sides, and B) when looking down from the top of the pyramid the edges
> are orthognal. Is there a way to find the height of this shape or do I
> need more info?
Matt
>
Yes, this would appear to be enough information to find the
height of the pyramid. Let the apex of the pyramid be at
the origon O of XYZ axes in a 3D coordinate system.
Let the points of known length (a,b,c) be in a positive direction
along each axis. Call the points A,B,C,
Point A is along X axis at (a, 0, 0)
Point B is along Y axis at (0, b, 0)
Point C is along Z axis at (0, 0, c)
the equation of the plane ABC which forms the
base of the pyramid OABC. Google technique..
Equation of Plane given 3 points
Once we have the equation of base plane then
we can compute the shortest distance from a
point (O) to the plane. This will be the height
of the pyramid. Again, Google technique...
Shortest Distance Point to Plane
hth Mick.
===
Subject: Re: Irregular Pyramid
Bytes: 1556
We were getting into a little debate at work about how to find the
>height of an irregular pyramid just by knowing A) the lengths of all
>sides, and B) when looking down from the top of the pyramid the edges
>are orthognal. Is there a way to find the height of this shape or do I
>need more info?
Can you explain more clearly what (B) means?
quasi
===
Subject: amy's holes (no porn !! :p)
this is about my holes eh no !!! :p
i mean holes in math.
recently i told : math is just algebra and geometrie.
steve told me there is also topolicology or something ?
its basicly about holes in figures.
i understand a multivariable polynomial can describe a figure. ( hurrah for
me :-) )
but how do you simply compute the number of holes in it ?
holes f(x;y;...) = ???
especially 3d , 4d and 5d interest me.
is this topothingy used for computers ?
kiss
amy
===
Subject: Re: amy's holes (no porn !! :p)
Bytes: 1269
Question: tommy1729, why are you using a pseudonym?
I saw amy666's email address.
===
Subject: What is the effect of adding same constant to numerator and
denominator
Bytes: 1161
What is the effect of adding same constant to numerator and
denominator?
alg
===
Subject: Re: What is the effect of adding same constant to numerator and
denominator
Bytes: 1442
> What is the effect of adding same constant to numerator and denominator?
If x is added, (a + x)/(b+x) - a/b = x (b -a) /( b (b+x)), so it may
increase or decrease in value.
===
Subject: Re: What is the effect of adding same constant to numerator and
denominator
> What is the effect of adding same constant to numerator and
> denominator?
> alg
a/b becomes (a+c)/(b+c).
You could write that as a/b + c (b-a)/(b (b+c))
--
Robert Israel israel@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: What is the effect of adding same constant to numerator and
denominator
> What is the effect of adding same constant to numerator and
> denominator?
> alg
>
The larger the constant, the closer to 1 the result.
Bob Kolker
===
Subject: Re: What is the effect of adding same constant to numerator and
denominator
days. My association with the Department is that of an alumnus.
>What is the effect of adding same constant to numerator and
>denominator?
It changes the rational, except when you add 0 to both, or when the
original rational was 1.
--
It's not denial. I'm just very selective about
what I accept as reality.
--- Calvin (Calvin and Hobbes by Bill Watterson)
Arturo Magidin
magidin-at-member-ams-org
===
Subject: Re: What is the effect of adding same constant to numerator and
denominator
<f5uf41$1el5$1@agate.berkeley.eduWhat is the effect of adding same constant 
to
numerator and
>denominator?
It changes the rational, except when you add 0 to both, or when the
> original rational was 1.
>
Also (if the numerator, denominator and the added constant are
positive), it brings the resulting fraction closer to 1.
===
Subject: Re: What is the effect of adding same constant to numerator and
denominator
Also (if the numerator, denominator and the added constant are
> positive), it brings the resulting fraction closer to 1.
>
...assuming the numerator, denominator, and the constant you added are
all positive...
--
G. A. Edgar
http://www.math.ohio-state.edu/~edgar/
===
Subject: logarithm question
Bytes: 1393
Hi
Please can I get some help with the following question:
Express log(2sqrt(10))-1/3log(0.8)-log(10/3) in the form c + log(d)
where c and d are rational numbers and the logarithms are to base 10.
Looking at the back of the book, the answer is log(3)-1/6 , but I
would really like to know how to get there.
===
Subject: Re: logarithm question
Bytes: 2457
> Hi
Please can I get some help with the following question:
Express log(2sqrt(10))-1/3log(0.8)-log(10/3) in the form c + log(d)
> where c and d are rational numbers and the logarithms are to base 10.
Looking at the back of the book, the answer is log(3)-1/6 , but I
> would really like to know how to get there.
Consider the following.
How is log(A B) related to log(A) and log(B)?
How is x log(A) related to log(A^x) ?
Here A^x is A to the power of x.
How is log(A) related to log(1/A)?
Read your text to discover these relationships.
Then use them to simplify the sum you have been
given.
Heh heh: Reminds me of an old joke. This was told
to me back in the days when highschool math classes
still taught sliderule use. This Park Ranger was
walking through the woods, and he noticed that this
particular kind of snake was not producing any young
snakes, and they were getting very rare. But he knew
exactly what to do. He selected some trees, and cut
them down, leaving the fallen wood on the ground.
The snakes then used them as hiding places for their
mating activities. And then, there were plenty of
young snakes the next season.
You see, logs allow adders to multiply.
Socks
===
Subject: Re: logarithm question
days. My association with the Department is that of an alumnus.
>Hi
Please can I get some help with the following question:
Express log(2sqrt(10))-1/3log(0.8)-log(10/3) in the form c + log(d)
You are imprecise in your writing. The second term could be
1/(3*log(0.8)),
or it could be (1/3)*log(0.8).
the latter.
>where c and d are rational numbers and the logarithms are to base 10.
The answer is very much non-unique. There are many expressions that
would satisfy the requirement.
>Looking at the back of the book, the answer is log(3)-1/6 , but I
>would really like to know how to get there.
Use the properties of the logarithm.
a*log(b) = log(b^a)
so you can change the second term to something of the form log(x).
Also, log(x) - log(y) = log(x/y). So once you have
log(x) - log(y) - log(z) you could change it to a single
logarithm.
Alternatively, to get the answer in the back, you can do the
following: remembering that log(10)=1, you can replace the first
logarithm with a sum of logarithms first (using the identity log(ab) =
log(a) + log(b)), and then replace the second with a single rational
using log(b^a) = a*log(b) and log(10)=1.
Since 0.8 = 8/10, you can do something similar there. Then remember
that 8 = 2^3 and do it again. Finally, use log(x/y) = log(x)-log(y) to
simplify log(10/3). Then add it all up and simplify.
--
It's not denial. I'm just very selective about
what I accept as reality.
--- Calvin (Calvin and Hobbes by Bill Watterson)
Arturo Magidin
magidin-at-member-ams-org
===
Subject: Re: logarithm question
<f5ufpf$1ert$1@agate.berkeley.edu>
Bytes: 1473
Hi
the question was:
express log(2*sqrt(10)-1/3*log(0.8)-log(10/3) in log c + d form
following your advise:
log2 + 1/2 - 1/3*log(2^3) + 1/3 - 1 + log3
=log2 - log2 + log3 + 1/2 - 2/3
=log3 + 3/6 - 4/6
=log3 - 1/6
===
Subject: Re: logarithm question
>Hi
>Please can I get some help with the following question:
>Express log(2sqrt(10))-1/3log(0.8)-log(10/3) in the form c + log(d)
You are imprecise in your writing. The second term could be
1/(3*log(0.8)),
or it could be (1/3)*log(0.8).
the latter.
>
be more precisely (1/3)*log(0.8). Multiplication and division are
usually taken serially, and at the same level of precedence.
If the expression had been written 1/3 log(0.8), I doubt one would
think that the log(0.8) were in the denominator, and I don't think
the extra space changes the interpretation.
Rob Johnson <rob@trash.whim.org>
take out the trash before replying
===
Subject: Re: Stable forms for iterates
these guys bought that topic since they are alone here :p
kiss
amy
===
Subject: Re: Stable forms for iterates
<22851509.1182796993564.JavaMail.jakarta@nitrogen.mathforum.org>
Bytes: 3531
> On 23 juin, 20:52, alainvergh...@yahoo.fr
> <alainvergh...@yahoo.fr > We may easily show the stability of parity for
> iterated functions.
> f being even f^[n] is even ;f being odd , f^[n]
> is odd.
> Extension to a four term decomposition is possible:
> f(x) = E(x) + O(x) ,E(x) =e1(x) +e2(x), O(x) =o1(x)
> +o2(x)
> and e1(x) = {f(x) + f(-x) +f(Ix) +f(-Ix) }/4
> e2(x) = {f(x) + f(-x) - f(Ix) - f(-Ix) }/4
> i1(x) = {f(x) - f(-x) - I*f(Ix)+
> I*f(-Ix)}/4 (1)
> i2(x) = {f(x) - f(-x)+ I*f(Ix) -
> I*f(-Ix)}/4
> Forms E(x), e1(x), O(x), o1(x) are stable under
> integer iteration.
> Example:
> -------------
> g(x) =x +x^5/5! + x^9/9! +....
> function g is 'ultra odd' form i1(x) .
> This function may be built by formula (1) from
> any function f(x) = (1-h(x))*exp(x)+h(x)*sin(x) .
> MY SEARCH
> Are there any other stable forms under iteration
> known?
> your ideas and comments please,
> Alain
I do want your ideas and suggestions on this thema,
> Alain
then plz clarify your idea...
> what do you mean bye ultra odd
> what iteration ??
> parity like in fourrier series ??
> stable ?? what kind of stability ?
> wasnt my answer what you wanted ??
plz clarify ...- Masquer le texte des messages pr?c?dents -
- Afficher le texte des messages pr?c?dents -
***
Bonjour,
I am interested since a long time by iteration
of functions.Very often iteration leads to cumbersome,
messy expressions.
Here I've started from a simple observation for
polynomials(and analytical functions) when the
function ,say f is even so is f o f o f ...
f odd
We may consider an extension to four terms
Even part e(x) = e1(x) +e2(x), o(x) = o1(x) + o2(x) .
g(x) 'ultraodd' corresponds to
g(x) = o1(x) = {f(x)- f(-x)-I*f(Ix)+ I*f(-Ix)}/4
Namely if g is a polynomial you've got only x^(4*n+1) terms.
In fact forms e(x) ,e1(x),o(x) and o1(x) are - in conditions
to be defined - kept while iterating ,
Alain
===
Subject: Re: Stable forms for iterates
> On 25 juin, 20:42, tommy1729 <tommy1...@gmail.com > On 23 juin, 20:52,
alainvergh...@yahoo.fr
> <alainvergh...@yahoo.fr > We may easily show the stability of parity for
> iterated functions.
> f being even f^[n] is even ;f being odd ,
> f^[n]
> is odd.
> Extension to a four term decomposition is
> possible:
> f(x) = E(x) + O(x) ,E(x) =e1(x) +e2(x), O(x)
> =o1(x)
> +o2(x)
> and e1(x) = {f(x) + f(-x) +f(Ix) +f(-Ix) }/4
> e2(x) = {f(x) + f(-x) - f(Ix) - f(-Ix)
> }/4
> i1(x) = {f(x) - f(-x) - I*f(Ix)+
> I*f(-Ix)}/4 (1)
> i2(x) = {f(x) - f(-x)+ I*f(Ix) -
> I*f(-Ix)}/4
> Forms E(x), e1(x), O(x), o1(x) are stable under
> integer iteration.
> Example:
> -------------
> g(x) =x +x^5/5! + x^9/9! +....
> function g is 'ultra odd' form i1(x) .
> This function may be built by formula (1) from
> any function f(x) = (1-h(x))*exp(x)+h(x)*sin(x)
> .
> MY SEARCH
> Are there any other stable forms under
> iteration
> known?
> your ideas and comments please,
> Alain
> I do want your ideas and suggestions on this
> thema,
> Alain
then plz clarify your idea...
> what do you mean bye ultra odd
> what iteration ??
> parity like in fourrier series ??
> stable ?? what kind of stability ?
> wasnt my answer what you wanted ??
plz clarify ...- Masquer le texte des messages
> pr?c?dents -
- Afficher le texte des messages pr?c?dents -
***
> Bonjour,
> I am interested since a long time by iteration
> of functions.Very often iteration leads to
> cumbersome,
> messy expressions.
> Here I've started from a simple observation for
> polynomials(and analytical functions) when the
> function ,say f is even so is f o f o f ...
> f odd
> We may consider an extension to four terms
> Even part e(x) = e1(x) +e2(x), o(x) = o1(x) + o2(x) .
g(x) 'ultraodd' corresponds to
> g(x) = o1(x) = {f(x)- f(-x)-I*f(Ix)+ I*f(-Ix)}/4
> Namely if g is a polynomial you've got only x^(4*n+1)
> terms.
> In fact forms e(x) ,e1(x),o(x) and o1(x) are - in
> conditions
> to be defined - kept while iterating ,
Alain
>
and you look for similar oddyties that are preserved throughout
iterations ....
have you done brute force searches ???
have you tried the solutions to my diff eq ?
other roots of unity then 1 -1 i -i ?
without experiments i thinks its darn hard to answer your questions ,
but i believe some intuitive experiments will get you some answers or
convinces rapidly.
be an experimental mathematician id say.
and keep me informed...
i like the idea... as an iterist
i used to have similar ideas in the past , but since i dont like computers 
i
found it hard to experiment ...
well i was pretty young then though , and forgot about it soon...
tommy1729
===
Subject: Re: Stable forms for iterates
<16125291.1182895231755.JavaMail.jakarta@nitrogen.mathforum.org>
Bytes: 5092
> On 25 juin, 20:42, tommy1729 <tommy1...@gmail.com > On 23 juin, 20:52,
alainvergh...@yahoo.fr
> <alainvergh...@yahoo.fr > We may easily show the stability of parity for
> iterated functions.
> f being even f^[n] is even ;f being odd ,
> f^[n]
> is odd.
> Extension to a four term decomposition is
> possible:
> f(x) = E(x) + O(x) ,E(x) =e1(x) +e2(x), O(x)
> =o1(x)
> +o2(x)
> and e1(x) = {f(x) + f(-x) +f(Ix) +f(-Ix) }/4
> e2(x) = {f(x) + f(-x) - f(Ix) - f(-Ix)
> }/4
> i1(x) = {f(x) - f(-x) - I*f(Ix)+
> I*f(-Ix)}/4 (1)
> i2(x) = {f(x) - f(-x)+ I*f(Ix) -
> I*f(-Ix)}/4
> Forms E(x), e1(x), O(x), o1(x) are stable under
> integer iteration.
> Example:
> -------------
> g(x) =x +x^5/5! + x^9/9! +....
> function g is 'ultra odd' form i1(x) .
> This function may be built by formula (1) from
> any function f(x) = (1-h(x))*exp(x)+h(x)*sin(x)
> .
> MY SEARCH
> Are there any other stable forms under
> iteration
> known?
> your ideas and comments please,
> Alain
> I do want your ideas and suggestions on this
> thema,
> Alain
> then plz clarify your idea...
> what do you mean bye ultra odd
> what iteration ??
> parity like in fourrier series ??
> stable ?? what kind of stability ?
> wasnt my answer what you wanted ??
> plz clarify ...- Masquer le texte des messages
> pr?c?dents -
> - Afficher le texte des messages pr?c?dents -
***
> Bonjour,
> I am interested since a long time by iteration
> of functions.Very often iteration leads to
> cumbersome,
> messy expressions.
> Here I've started from a simple observation for
> polynomials(and analytical functions) when the
> function ,say f is even so is f o f o f ...
> f odd
> We may consider an extension to four terms
> Even part e(x) = e1(x) +e2(x), o(x) = o1(x) + o2(x) .
g(x) 'ultraodd' corresponds to
> g(x) = o1(x) = {f(x)- f(-x)-I*f(Ix)+ I*f(-Ix)}/4
> Namely if g is a polynomial you've got only x^(4*n+1)
> terms.
> In fact forms e(x) ,e1(x),o(x) and o1(x) are - in
> conditions
> to be defined - kept while iterating ,
Alain
and you look for similar oddyties that are preserved throughout
iterations ....
have you done brute force searches ???
have you tried the solutions to my diff eq ?
other roots of unity then 1 -1 i -i ?
without experiments i thinks its darn hard to answer your questions ,
but i believe some intuitive experiments will get you some answers or
convinces rapidly.
be an experimental mathematician id say.
and keep me informed...
i like the idea... as an iterist
i used to have similar ideas in the past , but since i dont like computers
i found it hard to experiment ...
well i was pretty young then though , and forgot about it soon...
tommy1729- Masquer le texte des messages pr?c?dents -
- Afficher le texte des messages pr?c?dents -
Bonjour Tommy,
Hope you're in good spirits.
You did catch my meaning - oddities preserved throughout
iterations - very well.
Sure now, both of us have got plenty of time ,as iterists,
to blacken papersheets.
Looking for other roots of unity may be a fruitful way.
I may start looking for a filter of polynomials of order three,
namely cubic roots of unity,
Amicalement,
Alain
===
Subject: Re: Stable forms for iterates
> On 26 juin, 23:59, tommy1729 <tommy1...@gmail.com > On 25 juin, 20:42,
tommy1729
> <tommy1...@gmail.com > On 23 juin, 20:52, alainvergh...@yahoo.fr
> <alainvergh...@yahoo.fr > We may easily show the stability of parity
> for
> iterated functions.
> f being even f^[n] is even ;f being odd ,
> f^[n]
> is odd.
> Extension to a four term decomposition is
> possible:
> f(x) = E(x) + O(x) ,E(x) =e1(x) +e2(x),
> O(x)
> =o1(x)
> +o2(x)
> and e1(x) = {f(x) + f(-x) +f(Ix) +f(-Ix)
> }/4
> e2(x) = {f(x) + f(-x) - f(Ix) -
> f(-Ix)
> }/4
> i1(x) = {f(x) - f(-x) - I*f(Ix)+
> I*f(-Ix)}/4 (1)
> i2(x) = {f(x) - f(-x)+ I*f(Ix) -
> I*f(-Ix)}/4
> Forms E(x), e1(x), O(x), o1(x) are stable
> under
> integer iteration.
> Example:
> -------------
> g(x) =x +x^5/5! + x^9/9! +....
> function g is 'ultra odd' form i1(x) .
> This function may be built by formula (1)
> from
> any function f(x) =
> (1-h(x))*exp(x)+h(x)*sin(x)
> .
> MY SEARCH
> Are there any other stable forms under
> iteration
> known?
> your ideas and comments please,
> Alain
> I do want your ideas and suggestions on this
> thema,
> Alain
> then plz clarify your idea...
> what do you mean bye ultra odd
> what iteration ??
> parity like in fourrier series ??
> stable ?? what kind of stability ?
> wasnt my answer what you wanted ??
> plz clarify ...- Masquer le texte des messages
> pr?c?dents -
> - Afficher le texte des messages pr?c?dents -
> ***
> Bonjour,
> I am interested since a long time by iteration
> of functions.Very often iteration leads to
> cumbersome,
> messy expressions.
> Here I've started from a simple observation for
> polynomials(and analytical functions) when the
> function ,say f is even so is f o f o f ...
> f odd
> We may consider an extension to four terms
> Even part e(x) = e1(x) +e2(x), o(x) = o1(x) +
> o2(x) .
> g(x) 'ultraodd' corresponds to
> g(x) = o1(x) = {f(x)- f(-x)-I*f(Ix)+ I*f(-Ix)}/4
> Namely if g is a polynomial you've got only
> x^(4*n+1)
> terms.
> In fact forms e(x) ,e1(x),o(x) and o1(x) are - in
> conditions
> to be defined - kept while iterating ,
> Alain
and you look for similar oddyties that are
> preserved throughout iterations ....
have you done brute force searches ???
have you tried the solutions to my diff eq ?
other roots of unity then 1 -1 i -i ?
without experiments i thinks its darn hard to
> answer your questions ,
but i believe some intuitive experiments will get
> you some answers or convinces rapidly.
be an experimental mathematician id say.
and keep me informed...
i like the idea... as an iterist
i used to have similar ideas in the past , but
> since i dont like computers i found it hard to
> experiment ...
well i was pretty young then though , and forgot
> about it soon...
tommy1729- Masquer le texte des messages pr?c?dents
> -
- Afficher le texte des messages pr?c?dents -
Bonjour Tommy,
Hope you're in good spirits.
> You did catch my meaning - oddities preserved
> throughout
> iterations - very well.
> Sure now, both of us have got plenty of time ,as
> iterists,
> to blacken papersheets.
> Looking for other roots of unity may be a fruitful
> way.
> I may start looking for a filter of polynomials of
> order three,
> namely cubic roots of unity,
Amicalement,
> Alain
yes do that , its what i would do anyways.
===
Subject: Re: Stable forms for iterates
<3197860.1182943286835.JavaMail.jakarta@nitrogen.mathforum.org>
Bytes: 6221
> On 26 juin, 23:59, tommy1729 <tommy1...@gmail.com > On 25 juin, 20:42,
tommy1729
> <tommy1...@gmail.com > On 23 juin, 20:52, alainvergh...@yahoo.fr
> <alainvergh...@yahoo.fr > We may easily show the stability of parity
> for
> iterated functions.
> f being even f^[n] is even ;f being odd ,
> f^[n]
> is odd.
> Extension to a four term decomposition is
> possible:
> f(x) = E(x) + O(x) ,E(x) =e1(x) +e2(x),
> O(x)
> =o1(x)
> +o2(x)
> and e1(x) = {f(x) + f(-x) +f(Ix) +f(-Ix)
> }/4
> e2(x) = {f(x) + f(-x) - f(Ix) -
> f(-Ix)
> }/4
> i1(x) = {f(x) - f(-x) - I*f(Ix)+
> I*f(-Ix)}/4 (1)
> i2(x) = {f(x) - f(-x)+ I*f(Ix) -
> I*f(-Ix)}/4
> Forms E(x), e1(x), O(x), o1(x) are stable
> under
> integer iteration.
> Example:
> -------------
> g(x) =x +x^5/5! + x^9/9! +....
> function g is 'ultra odd' form i1(x) .
> This function may be built by formula (1)
> from
> any function f(x) =
> (1-h(x))*exp(x)+h(x)*sin(x)
> .
> MY SEARCH
> Are there any other stable forms under
> iteration
> known?
> your ideas and comments please,
> Alain
> I do want your ideas and suggestions on this
> thema,
> Alain
> then plz clarify your idea...
> what do you mean bye ultra odd
> what iteration ??
> parity like in fourrier series ??
> stable ?? what kind of stability ?
> wasnt my answer what you wanted ??
> plz clarify ...- Masquer le texte des messages
> pr?c?dents -
> - Afficher le texte des messages pr?c?dents -
> ***
> Bonjour,
> I am interested since a long time by iteration
> of functions.Very often iteration leads to
> cumbersome,
> messy expressions.
> Here I've started from a simple observation for
> polynomials(and analytical functions) when the
> function ,say f is even so is f o f o f ...
> f odd
> We may consider an extension to four terms
> Even part e(x) = e1(x) +e2(x), o(x) = o1(x) +
> o2(x) .
> g(x) 'ultraodd' corresponds to
> g(x) = o1(x) = {f(x)- f(-x)-I*f(Ix)+ I*f(-Ix)}/4
> Namely if g is a polynomial you've got only
> x^(4*n+1)
> terms.
> In fact forms e(x) ,e1(x),o(x) and o1(x) are - in
> conditions
> to be defined - kept while iterating ,
> Alain
> and you look for similar oddyties that are
> preserved throughout iterations ....
> have you done brute force searches ???
> have you tried the solutions to my diff eq ?
> other roots of unity then 1 -1 i -i ?
> without experiments i thinks its darn hard to
> answer your questions ,
> but i believe some intuitive experiments will get
> you some answers or convinces rapidly.
> be an experimental mathematician id say.
> and keep me informed...
> i like the idea... as an iterist
> i used to have similar ideas in the past , but
> since i dont like computers i found it hard to
> experiment ...
> well i was pretty young then though , and forgot
> about it soon...
> tommy1729- Masquer le texte des messages pr?c?dents
> -
> - Afficher le texte des messages pr?c?dents -
Bonjour Tommy,
Hope you're in good spirits.
> You did catch my meaning - oddities preserved
> throughout
> iterations - very well.
> Sure now, both of us have got plenty of time ,as
> iterists,
> to blacken papersheets.
> Looking for other roots of unity may be a fruitful
> way.
> I may start looking for a filter of polynomials of
> order three,
> namely cubic roots of unity,
Amicalement,
> Alain
yes do that , its what i would do anyways.- Masquer le texte des messages
pr?c?dents -
- Afficher le texte des messages pr?c?dents -
It works ;
sqrt(1) giving a1=1, a2=(I*sqrt(3) -1)/2,a3=(-I*sqrt(3) -1)/2
following functions
f1(x)={g(a1*x)+a3*g(a2*x)+a2*g(a3*x)}/3 for 3n+1 power terms
f2(x)={g(a1*x)+a3*g(a2*x)+a2*g(a3*x)}/3 for 3n+2
f3(x)={g(a1*x) + g(a2*x) + g(a3*x)}/3 for 3n
form f3(x) seems stable for iteration of polynomials or
analytical functions.
We might also compute:
-x^3/3!+x^9/9!-x^15/15!+x^21/21! ....
Alain
===
Subject: Re: Stable forms for iterates
> order three,
> namely cubic roots of unity,
> Amicalement,
> Alain
yes do that , its what i would do anyways.-
> Masquer le texte des messages pr?c?dents -
- Afficher le texte des messages pr?c?dents -
> It works ;
> sqrt(1) giving a1=1, a2=(I*sqrt(3)
> -1)/2,a3=(-I*sqrt(3) -1)/2
> following functions
> f1(x)={g(a1*x)+a3*g(a2*x)+a2*g(a3*x)}/3 for 3n+1
> power terms
> f2(x)={g(a1*x)+a3*g(a2*x)+a2*g(a3*x)}/3 for 3n+2
> f3(x)={g(a1*x) + g(a2*x) + g(a3*x)}/3 for 3n
form f3(x) seems stable for iteration of polynomials
> or
> analytical functions.
hey , i told you so ;-)
but i must say your still kind a confusing and unclear...
do you consider polynomial iterates of the functions or do you use the
function itself as iterate ?
how do you solve the functional equations , or dont you ?
it seems like a generalization of fourrier series expansions to me , 
despite
you present it different.
to be more precise an abelian extension of the unitroot dif eq ( d_n f = f 
)
to the sine in the fourrier series.
whereas your taylor series on the last line is simply a solution to the
differential equation or i.e. a sine/exp with a complex root of unity.
you seem very friendly towards me :-)
how about reading my topics too ?
tommy1729
===
Subject: New Future Center is working online Free 100%
Bytes: 1538
Future Center for Students!
its for all students looking for scholarships, Student visas for UK,
USA, Australia, Get the List of the worlds best universties & so much
more.
Commited to give u golden opportunities.. Just visit us at:
http://www.mixmasti.com/futurecentre/australia.asp
Travel to America, London, Canada, Australia, American Visa Loter
http://www.pakidols.com/info/visa.asp
===
Subject: Re: New Future Center is working online Free 100%
there is no place in the world where conjectures are seriously considered
apart from a university.
why do you have to pay to give good ideas to the world ?
even spreading ideas for free is no deal.
guess only the sun comes up for free !!!
tommy1729
===
Subject: exponential question
is there an exponential version of the 3n+1 problem ??
has it ever been considered or published bye anybody ??
tommy1729
===
Subject: Re: exponential question
Bytes: 1503
tommy1729 napsal:
> is there an exponential version of the 3n+1 problem ??
has it ever been considered or published bye anybody ??
tommy1729
Nice question if you mean this analogy: 3^n+1.
But I think you will never get 1 when you start with number that is
not a power of 2.
(This problem tends to some diophantic equation.)
===
Subject: Re: exponential question
Bytes: 1455
> is there an exponential version of the 3n+1 problem ??
It already is an exponential problem.
has it ever been considered or published bye anybody ??
Yeah, me.
tommy1729
===
Subject: vote on cantor !!!
people, vote here for pro or contra. (cantor set theory)
tommy1729
===
Subject: Re: vote on cantor !!!
Bytes: 1439
On Wed, 27 Jun 2007 17:02:07 EDT, tommy1729 <tommy1729@gmail.compeople, 
vote
here for pro or contra. (cantor set theory)
Wasn't it just a few days ago you assured us you were not
a crackpot?
>tommy1729
******
David C. Ullrich
===
Subject: Re: vote on cantor !!!
Bytes: 1930
> people, vote here for pro or contra. (cantor set theory)
tommy1729
i like that
either-or boolean you in or you ain't
it seems you are presupposing the mereology of set containment
in a question that looks for challenger
it's a rabbit trap!
personally
i don't believe in belief
particularly on something so artificial
and without accepted interpetation in reality
but i don't like cantor's naive theory
or several other of the later refinements qua set theory
i prefer to have the flexible mereology found in topoi foundations
and similar formalisations...
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
===
Subject: Re: vote on cantor !!!
Bytes: 1819
> people, vote here for pro or contra. (cantor set theory)
> personally
> i don't believe in belief
> particularly on something so artificial
> and without accepted interpetation in reality
But that's precisely what it means to not believe in Cantor's theory--to
recognize that
it is artificial with no interpretation connecting it to reality.
===
Subject: Re: vote on cantor !!!
Bytes: 1790
> people, vote here for pro or contra. (cantor set theory)
tommy1729
If someone with Mozilla Thunderbird wants to filter the messages of
this fool, then he/she can do so by this way:
if it is not selected.
2. Click New... and fill the fields as follows in this image:
http://img171.imageshack.us/img171/7900/filterbv2.png
3. If you want to log the activity of the filter, then from Message
Filters click Filter Log and select Enable the Filter Log.
It works.
Carl
===
Subject: Re: vote on cantor !!!
The survey would be considerably limited if for every
vote were necessary to provide reasoning.
Fernando.
===
Subject: CONTRA CANTOR
for people NOT believing in cantor set theory
===
Subject: Re: CONTRA CANTOR
<27302658.1182978289653.JavaMail.jakarta@nitrogen.mathforum.org>
Bytes: 1327
I'm only familiar with naive set theory and ZFC.
What is cantor set theory about?
===
Subject: PRO CANTOR
for people who believe in cantor set theory
===
Subject: Re: Solution manuals in PDF Available!!
Bytes: 1565
Hey buddy,
I was wondering if you had the solutions manual for Electrical
Engineering Principles and Applications, 3rd Edition, by Allan R.
Hambley. If so, what price are you asking for it and what kind of
guarantee do I have that you will not take my money and run. If you
could, respond as quickly as possible because I am currently enrolled
Kevin
===
Subject: Re: Solution manuals in PDF Available!!
Bytes: 1257
Hello mate!! Posting from Greece! I would like to send me the solution
manual for Modern Control Systems 10E by Richard Dorff !! thnx a
lot !!!!!! :)
===
Subject: Help
Bytes: 1271
I have a time slot assignment problem which i have formulated as a
multi-commodity flow problem . Since the Multi-Commodity flow is
already proven to be NP-Complete , does it imply that the assignment
problem is also NP complete , if not how do i prove it.
Arush
===
Subject: Re: Help
the last link on the wiki of flow network might help you.
===
Subject: Fast way of calculating multiple line intersects
Bytes: 2014
Im trying to make a computer program that simulates a game of
billiards.
What my program basically does is draw straight lines which are
parrallel to the ball's velocity and intersect the ball's current
position.
I now need to find where all of those lines will intersect each other
(where the balls will collide).
The problem I have is that there are 10 balls on the table it is
awkward to program two individual balls at a time using cramer's rule
in the x-y plane.
I really want to be able to use a single matrix describing the x and y
coefficients of the lines corresponding to all 10 balls. The problem I
have now is that I cannot use cramer's rule, because a 10x2 matrix
describing x and y coefficients for ten lines cannot be inversed.
Also, if i used a 10x10 matrix and set all non x or y coefficients to
zero, the determinant of that matrix would be zero too.
Please help!
Adam
===
Subject: Re: Fast way of calculating multiple line intersects
Bytes: 2408
On Jun 27, 10:20 pm, Adam Chapman
Im trying to make a computer program that simulates a game of
> billiards.
What my program basically does is draw straight lines which are
> parrallel to the ball's velocity and intersect the ball's current
> position.
I now need to find where all of those lines will intersect each other
> (where the balls will collide).
The problem I have is that there are 10 balls on the table it is
> awkward to program two individual balls at a time using cramer's rule
> in the x-y plane.
I really want to be able to use a single matrix describing the x and y
> coefficients of the lines corresponding to all 10 balls. The problem I
> have now is that I cannot use cramer's rule, because a 10x2 matrix
> describing x and y coefficients for ten lines cannot be inversed.
> Also, if i used a 10x10 matrix and set all non x or y coefficients to
> zero, the determinant of that matrix would be zero too.
Please help!
Adam
you need a sprite library, that will draw the balls and do your
collision detection for you. I think this is the wrong newsgroup. try
something like http://spritecraft.teggo.com/
===
Subject: Re: Fast way of calculating multiple line intersects
Im trying to make a computer program that simulates a game of
> billiards.
> What my program basically does is draw straight lines which are
> parrallel to the ball's velocity and intersect the ball's current
> position.
I now need to find where all of those lines will intersect each other
> (where the balls will collide).
I doubt it. You don't care where the paths intersect, if the balls are
at the intersection at different times. Also two balls don't need to
be in the same place to collide, they need to be at a distance equal to
the sum of their radii.
> The problem I have is that there are 10 balls on the table it is
> awkward to program two individual balls at a time using cramer's rule
> in the x-y plane.
Nevertheless, you really need to consider all pairs of balls.
Hint... it helps to consider the difference between the positions
and the difference between the velocities.
--
Robert Israel israel@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada V6T 1Z2
===
Subject: Re: Fast way of calculating multiple line intersects
Bytes: 3296
> Im trying to make a computer program that simulates a game of
> billiards.
> What my program basically does is draw straight lines which are
> parrallel to the ball's velocity and intersect the ball's current
> position.
> I now need to find where all of those lines will intersect each other
> (where the balls will collide).
I doubt it. You don't care where the paths intersect, if the balls are
> at the intersection at different times. Also two balls don't need to
> be in the same place to collide, they need to be at a distance equal to
> the sum of their radii.
> The problem I have is that there are 10 balls on the table it is
> awkward to program two individual balls at a time using cramer's rule
> in the x-y plane.
Nevertheless, you really need to consider all pairs of balls.
> Hint... it helps to consider the difference between the positions
> and the difference between the velocities.
I get the impression that first studying the case where one ball (say
the black ball) is at
rest might help me understand. If the only other ball is the white
ball, moving
parallel to the long sides of the rectangular table, the collision
corridor is a
strip of width 4*radius, from the white ball's current location going
towards the black ball which for simplicity we might place at the table
center.
The corridor would represent positions from which the white ball would
collide with the black ball...
If a collision might happen, [ might: eventually, we had other balls]
the approximate time till
impact looks like a useful piece of data.
With 10 balls, in general, and ignoring rebounds, maybe the first
collision is something
to look for; without rebounds, there might never be one. With rebounds,
no collisions
is of no interest to the programmer.
Understanding rebounds might be aided by visualizing a mirror image
table adjacent
to the real one where the ball's image continues in a straight line at
the same velocity.
David Bernier
===
Subject: Mistake in convexfication of reverse convex constraints?
Bytes: 2383
Given the following optimization problem:
Min_over_x f(x)
gi(x)<=0 i=1,...m
hk(x)>=0 k=1,...n
where the functions g and h are convex. Due to the reverse convex
constraints hk(x)>=0, this optimization is non-convex and tough. I
have simple idea to convexify this problem. It is so simple that I am
sure there is a mistake somewhere, because I know dealing with reverse
convex constraints is very tough. Please help me find the mistake.
I am introducing to the objective function a penalty function pk for
each hk that is zero if hk>=0 and non-zero otherwise. i.e. pk=max(0,-
hk). However, since hk is convex, -hk is concave and this leads to a
non-convex program. So I introduce an auxilary variable tk for each
pk: tk+hk(x)<=0 and add -tk to the objective function. I think this
implies tk and hk are equivalent. So I use pk=max(0,tk) in the
objective function. Knowing that max of two affine functions is
convex, I end up getting a convex program:
Min_over_x_AND_tk f(x)+C*sum_over_k(max(tk,0)-tk)
gi(x)<=0
tk-hk(x)<=0
Where C is a large enough weight factor... assuming hk's are bounded,
one can define C to be 1000*boundvalue.
So I could convert a non-convex program to a convex program very
easily in such a way that a minimzer x* in the second is the minmizer
of the first too. Where is my mistake?
H.M.