As promised in the last post, here is the beginning of the theory behind the images from the previous post:
The three sets correspond respectively to the yellow, blue and the separating areas on the figures in the last post.
Exercises:
1) Prove that [2u]=[u].
2) For any p define -p, so that it is well defined.
3) To which of the three sets -p belongs?
4) Show that
-Mp+ = Mp-,
-Mp- = Mp+ ,
-∞p
= ∞p.
5) Prove that Mp+ and Mp- are non-empty.
P.S.1. Friday January 27, 9:30 AM: replaced yesterday's version of the file, changing a little bit the notation.
P.S.2. Saturday, January 28, 15:44. Added the last section - dealing with the construction work. It is being constructed right now. Pieces added and removed, until the foundation is firm so that it can survive even earthquakes.
Actually it ends with a "Hypothesis". I will try to check now if it is true or not? If it will prove to be correct, it will be a plus for my intuitions. If it will prove to be wrong - it will mean that my intuitions are not yet good, and that there is is an extra work needed in this particular area. And that is important: to state specific, clear and bold hypotheses, and to work hard to prove or disprove them!
P.S.3. Time "flows" backward in anti-space M_p^-. Whatever that means. That's evident to me. It must be so. Otherwise the Universe would be unbalanced - a dreadful thought.
P.S.4. 9:45 AM. If "our" universe is S^1xS^3, then there should be also the next level universe, which is S^1xS3xS^7.
P.S.5. I will post the new version of the pdf file on Monday afternoon. The "Hypothesis" will then become a "Proposition".
P.S.6. Today I had the following conversation on my old Polish blog:
30.01.2023 X: I do not fit in your blog, so let me ask here.
"Logically, the sentence: if 1=2
then 3=33 is true. "Did the three squeeze into you, or am I
missing something ?
30.01.2023 Arkadiusz Jadczyk
don't see the
question....
X:30.01.2023
From the falsehood is
false - "3=33", and the blog is that we get a true
sentence.
30.01.2023 Arkadiusz Jadczyk
The whole sentence "If 1=2 then
3=33 " is logically true.
Whereas without this "if",
the " 3=33" itself is false.
Context is important.
X: 30.01.2023
Mother of one, such a compromise of my favorite logic. I hope this remains a secret. ;)
The intuitive absurdity of the implication triumphed over memory, or maybe it was the stupefaction of discussing the integral from dx.
Thank you for your reply.
30.01.2023 Arkadiusz Jadczyk
Success with dx integrals. Thank you for your question. He who asks does not wander.
Source: https://www.ruspeach.com/en/handbook/proverbs/1878/
© www.ruspeach.com - Russian for foreigners
This causes me to think: such situations are not just restricted to abstract logic. They happen in real life. Suppose you tell your friend: If you do this, then I will do that. Time goes on, and, after a while, your friend goes around complaining about you: "He had promised me that he will do that, and he has failed to keep his promise!" The "If" part, that he has failed to fulfil, is conveniently forgotten. His friends will never learn about the "if" part, and even if some of them do learn, it will be neglected as unimportant. Such is our tendency to look blameless, smart, and innocent in our own eyes. While such a tendency is probably hardwired in our brains - it only shows that our actions do obey the good old Aristotelian logic that we supposedly know so well.
(Warning: no one should take the above as a "hidden message". Nothing is hidden there, and I did not have in mind any particular person. I am writing about what I consider a general useful wisdom.)
P.S.7. Monday 15:50: I am still not happy with the actual version of the pdf. It still needs verification and corrections. I am currently working on it.
P.S.8 Monday 18:50: I am not finished yet, as I did not yet describe how the images has been created. This will come tomorrow.
P.S.9. Tuesday 7:50: Today is this "tomorrow". I decided to leave the creation of the images as an exercise for the Reader. All the theory that is needed is already described in details. More important is now to connect the SO(4,2) description with the SU(2,2) description. In their ingenious paper (see the reference in the pdf file) Wojciech Kopczyński and Lech Stanisław Woronowicz provided such a connection.
W. Kopczyński and L.S. Woronowicz, A geometrical approach to the twistor formalism, Rep. Math. Phys. Vol 2, pp. 35-51 (1971). It is a paper of an indescribable beauty!!! A true Spring Sonata. Therefore in the next posts I will write about my own little variations on this subject. My main addition, my little twist, will be: adding the orientation. In the SU(2,2) description, till now, there is no place for "anti-space". Anti-space, anti-time, anti-matter, anti-gravity - that's what I love! There would be no life, no stability without these "antics". P.S.10. Tuesday 16:15: Change of mind. I still have to define the conformal structure (i.e. metric up to a variable scale factor). And then to study null geodesics (light rays). This can be done without using SU(2,2). Though SU(2,2) will add an extra perspective. But that's for later. Practical question: can light travel between space and anti-space? We need to know!
P.S.11. Wednesday 9:11: By "chance", searching the net for "graph of a unitary operator" I have found this amazing video demonstrating how linear algebra can enhance and lighten our lives:
It also shows that in fact there exists a left-handed anti-universe.
1) Prove that [2u]=[u]
ReplyDeleteAd 1)
[2u] = [w] \Leftrightarrow 2u = \mu w \forall \mu > 0.
Hence,
u = (\mu/2)w \Leftrightarrow [u] = [w].
So,
[2u] = [w] = [u].
Similarly, we can show that [nu] = [u], n=1,2,3,...
Further proofs later.
Additional observation: Before (9) you have u \in pi^1(p). You should replace this with u \in pi^-1(p).
ReplyDeleteOne more comment:
ReplyDeleteMaybe make the document available in tex as well.
2) For any p define -p, so that it is well defined.
ReplyDeleteAd 2)
I see it like this:
V_{-p}^{+} = {w \in V: w.(-u) >0} = {-w \in V: (-w).u >0} = -V_p^+.
At the same time:
V_{-p}^{+} = {w \in V: w.(-u) >0} = {w \in V: (w).u <0} = V_p^-.
Moreover,
V_{-p}^{-} = {w \in V: w.(-u) <0} = {-w \in V: (-w).u <0} = -V_p^-
and
V_p^+ = {w \in V, w.u >0}.
Is this correct?
3) To which of the three sets -p belongs?
ReplyDeleteAd 3) It depends on which set p belongs to. What kind of answer do you expect here?
4) Show that
-Mp+ = Mp-,
-Mp- = Mp+ ,
-∞p = ∞p.
Ad 4)
-M_p^+ = -pi(V_p^+) = pi(-V_p^+) = pi(V_p^-) = M_p^-
-M_p^- = -pi (V_p^-) = pi(-V_p^-) = pi(V_p^+) = M_p^+
- \infty_p = -pi(V_p^0) = pi(-V_p^0) = pi(V_p^0) = \infty_p
The last one is very easy, because V_p^0 = -V_p^0. Why? Because:
V_p^0 = {w \in V: w.u = 0} = {-w \in V: w.u = 0} = -V_p^0.
Ad (A task that has disappeared).
ReplyDeleteSets M_p^+, M_p^- are open and \infty_p is closed in M.
Theorem (Topologia ogólna, part 1., Roman Duda) – I will rewrite it in our markings and in English:
Let V be a topological space and pi: V -> M be a suriection of this space to the set M.
The set A \subset M is open (closed) in topology introduced to the set M by mapping pi iff its preimage pi^-1(A) is open (closed) in V.
Now just look at the definitions and you can see that these two must be open and this one closed.
Ad 5)
For these sets to be empty then the equivalence classes would have to be empty…
One more thing: Remove formula (16). There is nothing there. It is empty.
ReplyDelete"here would be no life, no stability without these "antics" "
ReplyDeleteif we go into dualism - let's not forget about anti-mathematics :)))
https://youtu.be/pATX-lV0VFk
Creating a video recording software requires a lot of mathematics Creating a computer chip needed to play the recording requires even more mathematics. Therefore "close but no cigar".
DeleteIf it were not for the life in the musicians, there would be no work and all this technology alone means nothing and can create nothing. You criticized X (Robaka?) for not understanding the meaning of "modus ponendo ponens" and I guess you yourself also fell into these traps of "if" :))))
Delete"If it were not for the life in the musicians, there would be no work"
DeleteI don't see how these musicians help creating work or technology. What they create is an entertainment. Sometimes useful sometimes helpful, but let us not exaggerate ...
"and I guess you yourself also fell into these traps of "if" "
DeleteYou see, you consider "life" as something "given", that does not require understanding or explanation. I am, on the other hand, asking: life is a result of "something". What is this "something".
What is this "P" in "If P" that makes "thus Q=life" possible.
"I don't see how these musicians help creating work or technology."
DeleteMaybe because they don't create technology. If you can't receive their message - it's probably because you are looking for it - in the wrong place - in the area of mathematics :)))
"You see, you consider "life" as something "given", that does not require"
DeleteUntil I consider - it is my life that I take for granted.