(132) is not a full cycle and just interchanges the last two elements so

you indeed get (udu). The state of an up quark?

If (12)[uud] = [uud] then (132)[uud] = [udu].

If u is an up quark and d is a down quark then [uud] is a proton.

Dear Dr. Weisz and Dr. Hoxi,

Really it depends on that what kind of definition one uses for permutations. I use the definition of Emil Kiss: Introduction in the abstract algebra (in Hungarian), it means that the product of two cycles with a common element is defined as

(ab)(bc)=(abc)

and a cycle (132) acts on a list [123] as

(132)[123]=[312].

Really, I have found this problem in particle physics u is an up quark and d is a down quark. I think the result cannot depend on the definition of the permutation rules.

Ok. You use another definition. But if I symmetrize the state [uud] according to a Young tableau

u d

u

I should get null. E.g.

(id-(12))(id+(13))[uud]=(id-(12)+(13)-(132))[uud]=[uud]-[uud]+[duu]- ?

If I use [duu] instead of ? I get null. (This tableau is not allowed for the state [uud].) If I use [udu] the result is not null and I am in discrepancy with the particle physics.

... and also with the product rules of the Young tableaux i think...

The solution of this problem is that it is not the same if we apply first the symmetry on the state and then transform its points or reverse. The symmetry (12) does not commute with the transformation (132), and it can be dealed as identity only if the orbits of the points do not change under applying the symmetry.

https://sites.google.com/site/knyshus1/home/ver2

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https://sites.google.com/site/knyshus1/home/ver2

There are symmetries, the first main symmetry of the Decart,

its coordinates, the second symmetry is the symmetry of the

function — here we are Dostoevsky-Lebesgue.Function monoton-this it

настоящея симметрия функции.

I enjoy being part of this discussion but the intricacies of symmetry elude me. There are conventions that I gather are not consistent among those who use them. Is this an example of a "broken symmetry"?

Dear Dr. Knyshev, thank you for your invaluable answer. I did not know that there are also interesting symmetries of functions.

Dear Dr. Hoxie,

I do not know. If we label the points of the list [uud] as 1,2,3 then the symmetry stays after the transformation under (132), because the result is [duu] e.g 3,1,2. So if we change (12) then we get 3,2,1 e.g. [duu] again.

But if we consider the places of these points so the first and the second places are not symmetric in the list yet.

Dear Dr. Breuer,

Thank you for answering me.

1. As I have seen in mathematics there are more definitions for permutations. I have learned my def from my teacher Dr. Kiss. Another way is for example to multilpy cycles as (12)(13)=(123), but then one must be consequent.

We define the action of (132) on the list 1, 2, 3 as

3, 1, 2.

2. In physics one talk about symmetry if a system stays invariant under a transformation, for example the state [uud] is invariant under (12).

3. Regarding on the commutativity...ok, it was a bit mixed. So, if we act with (12) first and then with (132), it means we act with (132)*(12) e.g with (23). If we act first with (132) and then with (12), we act essentially with (13), and we get not the same state at the end.

If we take (12) like the identity on the state [uud] we make a mistake, if we act with a permutation after that. So it is not allowed to compute the product (23)(12) as (23)*id =(132)(12) on the state [uud] because it leads to a wrong result. We must utilize the symmetry of this state only after acting of the permutation (132).

Interesting question.

Really grateful to all researchers for their contribution towards knowledge.

Скажу абсолютную неправду.Когда мы жили на плоскости, то древнее греки придумали полярные координаты.Когда мы подросли, опять же те же самые древние греки опять придумали цилиндрическую систему координат.

Когда мы поняли, что живем на круглой земле, именно тогда придумали сферические координаты.И тут оказалось, что живем мы на круглой земельке, а что вокруг нас творится.На самом деле мы живем вот тут a+bx+cy+dz+exy+fxz+gyz+hxyz.Это не сфера, на которой мы живем, это нечто другое.Поверхностей второго порядка-только три-эллипс, гипербола, порабола.А этих недоработанных-аж не менее 28 вариантов вселенной.тьфу

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I will say the absolute untrue. When we lived on a plane,

ancient Greeks came up with the polar coordinates. When we

grew up, again the same ancient Greeks came up with a

coordinate system again. When we realized that we live on a round earth, it was then

that we came up with spherical coordinates. And it turned

out that we live on a round earth and what’s going on around

us. In fact, we live here

a + bx + cy + dz + exy + fxz + gyz + hxyz. This is not the

sphere in which we live, it is something else. Second-order

surfaces are only three-ellipse, hyperbole, parabola.

And these unfinished ones are at least 28 options of

the universe.

Thank you for the invaluable answers.

Exercise 2.

Let we prove that the two products are equal to each other:

(12)(23)(31)(34) and (34)(24).

Ева, вопросы корректно надо ставить.А так любой дурак задаст один вопрос-и 1001 мудрец не ответит.Вы из Венгрии?Если это так, то я хотел бы вступить в общество почитателей вашего Яноша Больяи, математика вашего знаменитого.Если этого нет, то надо это сделать.Я буду одним из первых его почитателей.

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Eva, the questions must be correctly posed. And so any fool

will ask one question, and 1001 wise men will not answer.

Are you from Hungary? If so, then I would like to join the

society of admirers of your Janos Bolyai, the mathematician

of your famous one. I need to do this. I will be one of his

first admirers.

Dear Dr. Knyshev,

You have a good idea, mathematicians will be happy to meet you. The Russian mathematician Lobacsevszkij was as so famous as Bolyai at that area.

Although I think my question is very important, and i am not fool.

Dr. Breuer is also right but I think he uses another definition. He permutes the places (koordinates) under the elements of the list. It is like in the relativity theory: we can move on a train or the world under us can also move into the opposite direction, we cannot notice any differences between the both cases.

Во-первых, я никакой ни дохтур.Дорогая Ева, вы не ответили

----------

У вас есть сообщество почитателей Яноша Больяи?

-----

Если этого нет, то я буду вторым, после вас.

----------

First, I am not a doctor. Dear Eve, you did not answer ---------- Do you have a community of admirers of Janos Bolyai? ----- If not, then I will be the second, after you.

Dear Mr. Knyshev,

... and I am not a mathematician. So I will be the second after you.

Дорогая Ева, у нас таких дохуя, хатя , спасибо з внимание.

--------

У вас есть , в Вегнирии ,

---------------

скажу сразу -я есть такой же как и Ваш ЯНОШ БОЛЬЯИ

все просто Я ВМЕСТЕ СНИМ. очень просто.

Beste Eva, ons het sulke Doha, Khatya, dankie vir u aandag. -------- Jy het in Vegnië, --------------- Ek sal dadelik sê - ek het dieselfde as jou YANOSH BOLYAI alles is eenvoudig, ek kom saam. baie eenvoudig.

Dear Dr. Breuer,

I understand your logic. If one solves the exercise 2. above (It is from the book of E. Kiss), one can do it in two different ways according to the two different definitions of permutations. I think that is the deviance in our treatment.

Вася Кныш, а мне каицца ,што лечить надо именно ВАС.Лечить , штоб вы не забыли вашего НАСТОЯЩЕГО вашего всенароднолюбимого тнащего ЯНОША БОЛЯИ

----------------------

Vasya Knysh, and I think it is necessary for me to treat it.

YOU. Treat. Shtob, you didn’t forget your PRESENT your

nationally-loved, youngster YANOSHA BOLYA

My Dear Talking Partners,

Thank you for the ingenious discussion.

Best wishes.

Производная-это скорость.А вторая?Это -градиент некоторой функции.

-------

А третья?Что это такое?

----------

The derivative is speed. And the second? This is the gradient

of some function. ------- And the third? What is it?

Svetlana A. Budochkina аждваво, спасибо Сереге А

головой думал, про недостачу , недостал, письки про вихри кык сожаленню у меня нет смиешанных производных, вихрей, недостеьский

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head thought, about the shortage, not enough, the pussy

about the whirlwinds kkk regret I have no mixed derivatives,

whirlwinds, the unfortunate