Ag.algebraic geometry – characteristic polynomial of a symmetrical $ 8 times 8 $ matrix with indefinite entries in relation to octonionic multiplication

I look at $ 1, i, j, k, l, m, n, o $ the standard base of the (complexed at will) octon ions ($ mathbb {O} $ for the octon ions).
To let $ a = x_1.1 + ldots + x_8.o $. $ b = x_9.1 + ldots + x_ {16} .o $ and $ c = x_ {17} .1+ ldots + x_ {24} .o $, Where $ x_1, ldots, x_ {24} $ are indefinite about the basic field (pron $ mathbb {C} $).

I denote by $ L_a $ the $ 8 times $ 8 Matrix that represents the left multiplication with $ a $ in the $ mathbb {O} simeq mathbb {C} ^ 8 $ and $ R_a $ the $ 8 times $ 8 Matrix that represents the correct multiplication with $ a $, Similar names for $ b $ and $ c $, I want to calculate the characteristic polynomial of the symmetric matrix:
$$ S = R_a L_b L_c + {} ^ {t} (R_a L_b L_c), $$
Where $ {} ^ {t} X $ is the transpose of $ X $,

I tried Macaulay2 and this calculation seems to go far beyond what my computer (which is supposed to be a fairly powerful portable workstation) offers.

A simple reformulation of the eigenvalue problem on a well-chosen basis (namely let $ mathbb {H} $ be the quaternionic subalgebra of $ b $ and $ c $, Splits $ mathbb {O} $ how $ mathbb {H} bigoplus mathbb {H} .e $, Where $ e $ is orthogonal to $ mathbb {H} $ and take a base adapted to this decomposition) shows that:
$$ (T – mathrm {Re} (( overline {b} c) overline {a})) ^ 4 textrm {divides} det (, $$
Where $ mathrm {Re} (z) $ is the real part of $ z in mathbb {O} $,

I put $ f (T) = dfrac {det (S-Tid)} {(T – mathrm {Re} ((bc) overline {a}) ^ 4} $, A variety of calculations over finite fields and specialization of the $ x_i $ random values ​​suggests that $ f (T) $ is indeed a square, we say $ f (T) = g (T) ^ 2 $, Where $ g $ is a quadratic polynomial in $ T $,

I would like a closed expression of $ g (T) $, May it be a clean formula $ a, b $ and $ c $ or a dirty "in coordinates" polynomial. I would be very happy about any suggestion. I would also be interested in a theoretical argument that shows that $ f (T) $ is indeed a square.

Many thanks!

magento2 – CSV Uploader randomly throws errors for identical entries

I am trying to upload 12000 SKUs (1000 each) using the CSV uploader.

I get some random errors for identical data.

Example: Each cell in the attribute_set_column column has a 9, but I get this error

 Invalid value for Attribute Set column (set doesn't exist?) in row(s): 575, 577

Is there a way to fix such inconsistencies or get less vague error messages?

Thank you very much

views – Calculates the total number of field entries

I have a content type called "Books" and a number field called "Ratings" and an entity reference called "Author Name". I created several nodes with the book titles, the author's name and the corresponding rating.

I would like to display the author name and the total number of ratings of this author with 10, 9.5, 9 etc. in a table view. I tried using Views Conditional to indicate that & # 39; if this field = Rating & # 39; & # 39; equal to 10 & # 39; & # 39; Then output this … & # 39; & # 39; is, and my only options are {{field_rating}}. What I want to spend is the total time the author has received 10. So the table would show: Jon Doe 10 (3), 9.5 (6), 9 (5) and so on.

I have created a table view that contains the fields "Views: Views" for each possible rating (10, 9.5, 9, 8.5, 8, 7.5, 7) together with the fields "Content: Rating" and "Content: author name" is used. I thought Views Aggregation Plus might work, but I don't seem to be able to get it to work.

Maybe I should try to create a custom view ad?

Rename and group Google Sheets like entries

I want to create the "Nuts", "Dairy" and "Nuts and Dairy" answers without duplications. I only want the three answers, but my data currently shows about 20 with additional information that I don't actually need or want. How would I rename everything so that an observation like "nuts, peanuts, almonds" only returns "nuts"?

This has to be done in Google Sheets.

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views – Calculates the total number of field entries

I am trying to find a solution for a Drupal 8 table view I have created that outputs book ratings per author. I want to display a table that shows the author name with the total number of ratings of 10.0, 9.5, 9.0, etc. Not the sum, but the "number" of times that a given author received a 10, 9.5, 9, etc. The ideal output would be the following example:

(field_author_name) Jon Doe received 2 x 10 reviews, 3 x 9.5 reviews, 5 x 9 reviews, etc.

Hopefully I'll explain that correctly. (See attached screenshot for a visual representation) Does anyone know if this is possible with Views Conditional? Or is there a better solution out there?

Many thanks to the Drupal community!
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index – indexing only for the latest entries

Consider the following table:

| raffle                                              |
| id | shuffle |  user_id | ... |           notify_at |
| 1  | 4D6G8Z1 |      542 | ... | 2019-12-01 14:00:00 |
| 2  | 64G264D |        6 | ... | 2019-12-28 14:00:00 |
| 3  | 4IPF93D |       58 | ... | 2020-01-01 14:00:00 |
| 4  | D25LF03 |       58 | ... | 2020-01-14 14:00:00 |
| 5  | G04LDWE |      684 | ... | 2020-03-02 13:00:00 |

In this table, most inquiries are not made to the id Column but on the user_id and created_atThis is a 64-bit timestamp (not a 2038 error):

SELECT * FROM (raffle) WHERE (user_id) = ? and (created_at) = ?

The table grows by the minute, but that's not the problem, it's the records for them notify_at in the current month, most are accessed as the rest. An index of the user_id and notify_at Totals 160MB, of which only 1% are heavily accessed.

Is there any way to optimize an index (or other strategy) to get the records for the current month faster?

oracle – queries all audit entries that are issued with the & # 39; audit & # 39; have been defined

Depending on the method used for the check, you will find this information in the following views:


The content of the audit trail, also depending on the configuration and version, can be found in the following views:


r – Replace column entries starting with a specific word

I am editing a data frame in R in which one of the columns is dates, but there are text entries in the same column that prevent the date from having the format YYYY-MM-DD. Since these texts are not important, my idea was to replace them with NA so that I could enter the date correctly, but I cannot make the replacement.

The texts in this column always start with the word "Legal". So my idea was to code a function to identify these characters and replace all occurrences. I thought of the following code:

UHE$Ato(UHE$Ato == '^Ofício') <- NA

But when I run this line of code, nothing changes in the data frame.
Would anyone know a way to solve this?

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python – City Construction Solution requires a timeout for extensive entries


The country of hacker land has n Cities connected by m
unidirectional streets. The cities are numbered by 1 to n,
The government recently decided to build new cities in the hacker land.

Your job is to simulate q Events. An event can be of two types
described below:

  • 1 x d : A new city n + 1 is built in the hacker land and connected to the city x, If d = 0 the direction of the new road
    is of x to n + 1, If d = 1 The direction of the new street is
    from n + 1 to x,

  • 2 x y : To press Yes If it is possible to move out of the city x to the city y, to press No Otherwise.


0 <n, m <= 50000

0 <q <= 10000

x, y always correspond to an existing city in the hackerland.

The total number of cities in the Hackerland will not exceed 50,000

A full description of the problem can be found here

My efforts

I tried to use one BFS here, however, all test cases fail except for the default due to a timeout. For these test cases n >= 4000. m >= 20000,


n, m = map(int, raw_input().split())
roads = {}
for _ in xrange(m):
    u, v = map(int, raw_input().split())
    if u in roads:
        children = roads(u)
        children = set()
        roads(u) = children

q = input()
for _ in xrange(q):
    c, x, d = map(int, raw_input().split())

    if c == 1:
        n = n + 1
        y = n
        if d == 0:
            if x in roads:
                children = roads(x)
                children = set()
                roads(x) = children
            children = set()
            roads(y) = children
        y = d
        visited = set()
        current = list()
        answer = "No"
        while current:
            parent = current.pop(0)
            if parent == y:
                answer = "Yes"
            if parent in roads:    
                for i in roads(parent):
                    if i not in visited:
        print answer 


Does anyone have a suggestion on how to optimize the solution?