probability or statistics – Odd order moments and Symmetricity of the Distribution function

The question I had in an examination was to prove or disprove the following:

If all odd-order central moments of a distribution function are zero, the distribution
is symmetric.

I proved the statement but, the instructor cut the marks showing me the following counter-example:
enter image description here

I understand the counter-example is valid but I can’t figure out what’s wrong with my proof given below.
enter image description here
enter image description here
Note that equation 1 mentioned in the proof is the given condition that all order central moments are zero.
Please help..

graphics3d – `GraphPlot3D` does not preserve the order of points/edges. How to identify coordinates of those points then?

Here is my code.

data contains list of edges of a graph. For example the first edge is 5->8 which means the edge has end points 5 and 8. The order of edges is important for me, although the graph would look the same even if the order of edges was different.

gr contains the plot of the graph. You can use FullForm[gr] to see what it consists of.

coordinates contains coordinates of points extracted form the plot.

edges contains edges extracted from the plot.

data = {5 -> 8, 8 -> 11, 11 -> 14, 14 -> 17, 17 -> 20, 20 -> 30, 
   30 -> 32, 32 -> 34, 34 -> 36, 36 -> 38, 38 -> 3, 3 -> 6, 6 -> 9, 
   9 -> 12, 12 -> 17, 17 -> 19, 19 -> 21, 21 -> 23, 23 -> 25, 
   25 -> 35, 35 -> 38, 38 -> 1, 1 -> 4, 4 -> 7, 7 -> 10, 10 -> 20, 
   20 -> 22, 22 -> 24, 24 -> 26, 26 -> 28, 28 -> 33, 33 -> 36, 
   36 -> 39, 39 -> 2, 2 -> 7, 7 -> 9, 9 -> 11, 11 -> 13, 13 -> 15, 
   15 -> 25, 25 -> 28, 28 -> 31, 31 -> 34, 34 -> 37, 37 -> 40, 
   40 -> 10, 10 -> 12, 12 -> 14, 14 -> 16, 16 -> 18, 18 -> 23, 
   23 -> 26, 26 -> 29, 29 -> 32, 32 -> 37, 37 -> 39, 39 -> 1, 1 -> 3, 
   3 -> 5, 5 -> 15, 15 -> 18, 18 -> 21, 21 -> 24, 24 -> 27, 27 -> 30, 
   30 -> 40, 40 -> 2, 2 -> 4, 4 -> 6, 6 -> 8, 8 -> 13, 13 -> 16, 
   16 -> 19, 19 -> 22, 22 -> 27, 27 -> 29, 29 -> 31, 31 -> 33, 
   33 -> 35, 35 -> 5};
gr = GraphPlot3D[data, Method -> "SpringEmbedding"]
coordinates = gr[[1, 1]]
edges = gr[[1, 2, 1, 2, 1]]

Output:

enter image description here

{{2.15099, 1.18307, 3.11637}, {3.04554, 0.934916, 2.56025}, {3.55454, 
  1.21541, 1.90224}, {3.64482, 1.99867, 1.19894}, {3.2115, 2.38972, 
  0.549773}, {2.28274, 2.36187, 0.00260647}, {1.28767, 2.28706, 
  0.00272485}, {0.379458, 2.18177, 0.552232}, {0., 1.64516, 
  1.20573}, {0.226403, 0.985251, 1.912}, {0.946206, 0.603301, 
  2.56797}, {1.83231, 0.329511, 2.56276}, {2.65923, 0.220807, 
  1.90275}, {3.17392, 0.63046, 1.19994}, {3.26151, 1.46478, 
  0.547847}, {3.0135, 3.17375, 1.21467}, {2.42955, 3.54417, 
  1.92239}, {1.82077, 3.30175, 2.57303}, {1.50578, 2.44273, 
  3.11857}, {1.19737, 1.49462, 3.11698}, {1.21934, 0.0929668, 
  1.91297}, {1.99693, 0., 1.20222}, {2.38452, 0.434066, 
  0.545737}, {2.3541, 1.36368, 0.}, {2.18052, 3.26543, 
  0.558385}, {1.64804, 3.64083, 1.21331}, {0.991134, 3.41471, 
  1.91654}, {0.612497, 2.69602, 2.57153}, {0.339418, 1.81279, 
  2.56705}, {0.632563, 0.468973, 1.20897}, {1.46275, 0.38174, 
  0.550977}, {3.31226, 1.81971, 2.56268}, {2.45549, 2.13245, 
  3.11539}, {0.0986677, 2.42307, 1.91563}, {0.434602, 1.25784, 
  0.551964}, {1.36011, 1.29101, 0.00233058}, {3.42145, 2.65132, 
  1.9113}, {2.70536, 3.02741, 2.57249}, {0.471524, 3.01036, 
  1.21356}, {1.25737, 3.2109, 0.556519}}

{{1,2},{1,12},{1,20},{1,33},{2,3},{2,13},{2,32},{3,4},{3,14},{3,32},{4,5},{4,15},{4,37},{5,6},{5,15},{5,16},{6,7},{6,24},{6,25},{7,8},{7,36},{7,40},{8,9},{8,35},{8,39},{9,10},{9,34},{9,35},{10,11},{10,29},{10,30},{11,12},{11,20},{11,21},{12,13},{12,21},{13,14},{13,22},{14,15},{14,23},{15,24},{16,17},{16,25},{16,37},{17,18},{17,26},{17,38},{18,19},{18,27},{18,38},{19,20},{19,28},{19,33},{20,29},{21,22},{21,30},{22,23},{22,31},{23,24},{23,31},{24,36},{25,26},{25,40},{26,27},{26,40},{27,28},{27,39},{28,29},{28,34},{29,34},{30,31},{30,35},{31,36},{32,33},{32,37},{33,38},{34,39},{35,36},{37,38},{39,40}}

Now you can see that order of the edges that was in my original data is different in output edges.
But I need to know what coordinates belong to what point. For example what are coordinates of end points of my edge 5->8?

I consider it to be a bug, that the order of edges was nor preserved, otherwise I would be able to identify coordinate of each point/edge.
Is there some way to do it automatically? I can do it manually but that is very time consuming.

Magento 2 Order Email Template subTotal tfoot colspan Edit

Magento 2 Order Email Template subTotal tfoot.

Where Can I edit the Order Email Template subTotal tfoot ??? I need to edit colspan 2 to 3. please check the image

dnd 5e – What opportunities exist in D&D 5e in order to turn into beasts, while maintaining personality and mental characteristics?

Options that keep mental scores

Wild Shape allows you to retain your mental ability scores, as you mentioned.

Additionally, shapechange, a 9th level druid or wizard spell, works (though it only affects you). To affect others, you could use the 8th level druid spell animal shapes.

Options that don’t work

Polymorph explicitly calls out that the mental ability scores are changed to match the new form, as does true polymorph.

Customers admin page add grid column as order link in orders grid

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2.3 how create guest order on backoffice

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  2. Why when orders are created on admin panel magento don’t send a new order email? I know email sendig is working fine because in storefront it sends the email

Thanks 😀

custom order please ask first then order for $1

custom order please ask first then order

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magento2 – Magento 2 – Create order using REST API – Third Party Payment Gateway such as Paypal or other 3rd party payment gateway

I need to use the Magento REST API to create an order from an API. In my case, I am trying to implement the payment directly using API to Magento. What I need to do is to create an order by setting the payment method as a Paypal or can say RazorPay or any other 3rd Party Payment method and to perform a guest checkout. Which API(s) should I use to achieve this?

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REST API design: communicating order of inputs is important to the user (in POST request)

Let’s say I have a REST endpoint where the order of inputs for a given field is important:

POST Request: api.myrestendpoint.com/evolution-timeline/
inputs: {'this_list_should_be_ordered' : ( 'reptiles', 'birds', 'mammals') }

I’d like suggestions on how one can specify the contract so its clear to the client that the order of inputs is important in order to get sensible results. (i.e. passing in the input as (‘mammals’, ‘reptiles’, ‘birds’) wont give a sensible output).

Notes: Couple of obvious solutions are make the field name so it suggests ordering is important or having the swagger/any other documentation tool state the ordering is important…

here’s obviously some validation that will go into figuring whether the ordering is important on the server side, but in interest of making the api faster, i’d like to make it obvious to the user that the ordering does not make sense/and do some lightweight validation of inputs…