Coxeter groups – Characterization of Alcove addresses of weak order reference request

Leave $ Phi $ be the root system of type $ A $. Leave $ mathcal {A} $ Be a niche of the corresponding affine disposition. the address (or Coordinates of shi) of $ mathcal {A} $ It is a function $ k: Phi ^ + rightarrow mathbb {Z} $ satisfactory
$$ k ( epsilon_i – epsilon_j) <( lambda, epsilon_i – epsilon_j) <k ( epsilon_i – epsilon_j) + 1, $$
for all $ lambda in mathcal {A} $. (Here $ ( cdot, cdot) $ It is the standard internal product in $ mathbb {R} ^ n $.)

Leave $ leq_R $ denote the correct weak order in $ widetilde {A} _n $. by $ w in widetilde {A} _n $, leave $ k_w $ denote the address of the niche corresponding to $ w $. The weak order is characterized by
$$ u leq_R v iff | k_u | leq | k_v | text {y} text {sgn} (k_u) = text {sgn} (k_v), $$
where the signs are considered equal whenever $ k_u ( epsilon_i – epsilon_j) = 0 $.

I am looking for a reference for this characterization.

One can prove this in general by applying Theorem 4.5 of Humphreys Reflection groups and cooxter groups.. Also, connecting Theorem 4.1 of Shi & # 39; s On in two presentations of the Weyl-like groups of classical types with Theorem 5.3 of Bj$ ddot { text {o}} $The related permutations of Rner and Brenti type A work. Is there anything more direct in literature?

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Ordinary differential equations: calculation of the variable speed in time and the final speed through second order EDO

I have to work with an example of physics to strengthen my understanding of the second law of Newton's movement through second-order ODE.

I have been given the following configurations: that the parachutist is descending under a parachute at a constant rate, where t is time; and I have to assume displacement and zero velocity at time zero.

$$ frac {d ^ 2y} {dt ^ 2} + frac {k} m ^ {} frac {dy} {dt} = g $$

I have come to the following general solution and (t):

$$ y (t) = A + Be ^ {- frac {kx} {m}} + g frac {m} {k} t $$

If I set the displacement and velocity at time zero, I got:

$$ A = -B = -g frac {m ^ 2} {k ^ 2} $$

From that point, how can I calculate:

  1. Vertical speed variable in time?
  2. Final velocity corresponding to the velocity of the parachutist when descending at a constant speed?

In Trello, is there any way to see all the cards assigned to me within a specific team and then assign them an order?

I am a member of the Trello team, where the meetings correspond to individual projects, each list of a project corresponds to a type of task. (for example, errors, characteristics, etc.), and each card is an individual task. The cards that I need to complete are assigned to me.

These cards do not have an established expiration date, but there is a priority order for them that is not currently represented in Trello and I would like to fix it. I'm not sure what method I can use to see them in one place and then organize them in that order of priority to be aware of what I have to do.

Is there any way I can see all the cards assigned to me in one place, Y change your order to the order in which I need to complete them, regardless of which table or list they come from? Basically, I want a personal list of "all the cards" that I must complete.

I am open to using power-ups or third-party software to achieve this.

I am also willing to talk to my project manager and suggest a completely different method to organize the cards in Trello if what I am looking for is to go too much against the grain. I will say that I like this particular way of organizing things: I like that my conceptual separations are discrete and are visually represented, and the use of different tables / lists in this way makes the labels do not, although the labels are very different . More powerful and flexible. However, not having a single canonical task list is a serious problem.

8 – Allow the Order item type field to be displayed in some Commerce 2 products, but hidden in others?

I have two types of products on my website: Shipping Products and Non-Shipping Products. I have several shipping products established. One of those Shippable Products needs a custom field to collect a serial number; The other shipping products do not need this field.

I have set a text field "Serial number" in the default "Order item type" and added it to the Add to cart form screen. Now it is displayed on all the pages of products that can be sent. How can I show it only in the enviable Product that needs it and avoid showing it in other products that do not need it?

Or is there a different way to do it?

Thank you

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mysql: the order of the three tables is correct but the values ​​are false

I have those tables

                    Table 1
| id | name |
| 1 | ax
| 2 | arc
| 3 | car |
| 4 | dart


table2 table3
| t1_id | number | | t1_id | letter |
| 1 | 5 | | 1 | a |
| 1 | 6 | | 1 | b |
| 1 | 2 | | 1 | c |
| 2 | 2 | | 2 | a |
| 2 | 2 | | 2 | c |
| 2 | 3 | | 2 | r |
| 3 | 8 | | 3 | and |
| 3 | 3 | | 3 | i |
| 3 | 1 | | 3 | a |
| 4 | 8 | | 4 | a |
| 4 | 9 | | 4 | b |
| 4 | 10 | | 4 | c |

Y table1 (id) is linked with table2 (t1_id), table3 (t1_id)

I execute it to order them by the highest account_letter party then by the highest average number match to get this appropriate result http://www.sqlfiddle.com/#!9/69086b/8/0

SELECT
t1.id,
t1.name

SINCE
table1 t1

UNIT INTERNALLY
table2 t2
IN t2.t1_id = t1.id

Join on the left
table3 t3
IN t3.t1_id = t1.id
AND t3.letter IN (& # 39 ;, & # 39; b & # 39 ;, & # 39; c & # 39;)

GROUP BY
t1.id

ORDER BY
COUNT (t3.letter) DESC,
AVG (t2.number) DESC

| id | name |
| 4 | dart
| 1 | ax
| 2 | arc
| 3 | car |

and everything is working well


but when I wanted to check if there is any problem with the query, I decided to verify the letter_count Y avg_number so I used this query

SELECT
t1.id,
t1.name,
COUNT (t3.letter) LIKE letter_count,
AVG (t2.number) AS avg_number

SINCE
table1 t1

UNIT INTERNALLY
table2 t2
IN t2.t1_id = t1.id

JOIN LEFT
table3 t3
IN t3.t1_id = t1.id
AND t3.letter IN (& # 39 ;, & # 39; b & # 39 ;, & # 39; c & # 39;)

GROUP BY
t1.id

ORDER BY
letter_count DESC,
avg_number DESC

what I expected to be the result

| id | name | letter_count | avg_number |
| 4 | Dart 3 | 9 |
| 1 | ax 3 | 4.3333333333 |
| 2 | arch 2 | 2.3333333333 |
| 3 | car | 1 | 4 |

but the result I got was http://www.sqlfiddle.com/#!9/69086b/3/0

| id | name | letter_count | avg_number |
| 4 | dart 9 | 9 |
| 1 | ax 9 | 4.3333333333 |
| 2 | arch 6 | 2.3333333333 |
| 3 | car | 3 | 4 |

I'm supposed to be worried about how the letter_count it multiplies three times in the future or it just happens when I want to select the values ​​and nothing bad will happen if I keep it ORDER BY?


I just want them to be ordered correctly, which is what I got, but checking the values ​​showed me that letter_count it multiplies many times, so I'm confused about its performance or the letter_count Can it be ignored and the performance will not be affected?

Google Sheets: Matrix formula to add letters after duplicate numbers in order

Late,

The following formula adds a letter after a sequential number, A, B, C, D, etc. for the same number until you find a new number, then start again from A.

Do I need help to convert this into a matrix formula, if possible?

= J3 & CHAR (96 + COUNTIF (J $ 3: J3, J3))

I tried it this way, and depending on if I modify J3: J3, do I get A or D at the end of the number?

= {""; arrayformula (J3: J & CHAR (96 + COUNTIF (J $ 3: J, J3: J)))}

The formula as a single line pushes down the next one.

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