linear algebra – Are vectors equal when their product is equal?

When we have such a simple outer products on two sides of the equation as below:

$aa^T$ = $bb^T$

where a and b are vectors,does it necessarily means a=b?

If yes, consider the equation:
$1/n aa^T$ = $m^2bb^T$
where, m and n are constant values. Now what does $a$ equal to?

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magento2 – cuustom product tab not displayed in Magento 2 enterprise edition

I have created a custom tab in product detail page used the following code in layout file

Vendor/Module/view/frontend/layout/catalog_product_view.xml

<referenceBlock name="product.info.details">
        <block class="VendorModuleBlockProductViewAskQuestion" ifconfig="productqa_tab/productqa_setting/productqa_active" name="product.question" template="product/view/ask_question.phtml" group="detailed_info">
            <container name="form.additional.info" label="Form Additional Info">
                <block class="MagentoCaptchaBlockCaptcha" name="captcha" after="-" >
                    <action method="setFormId">
                        <argument name="formId" xsi:type="string">product_qa</argument>
                    </action>
                    <action method="setImgWidth">
                        <argument name="width" xsi:type="string">230</argument>
                    </action>
                    <action method="setImgHeight">
                        <argument name="width" xsi:type="string">50</argument>
                    </action>
                </block>
            </container>

            <arguments>
                <argument translate="true" name="title" xsi:type="string">Question And Answer</argument>
                <argument name="sort_order" xsi:type="string">20</argument>
            </arguments>               
    </referenceBlock>

The above code displaying tab in product detail page for existing products.

if any new product created, none of the tabs are displayed.

This issue happening only in enterprise edition(2.4.1). Working perfectly in community edition.

Can anyone please suggest me how to resolve this issue. Thanks in Advance!!

mac mini – What results in the price difference for seemingly the same product?

I am considering buy the latest Apple Mac Mini M1 with 512GB storage from Apple or from Amazon. Apple lists it at $899 while Amazon lists it at $859. What is the hardware/software different that results in Amazon being $40 cheaper than from Apple directly?

system.xml – How to create a dropdown option as a configuration section to update the options from backend and show it from product details page?

I want to add a select box to the product details page.
Which has several options but this is not a product attribute. It should configure as a global variable.
Admin should able to add new options too from the backend.

I tried the system.xml way and I couldn’t update the new option value for the select box I created.

Can someone help me to do this, please?

linear algebra – Inner product space – simple proof

This seems like a simple proof but I haven’t been able to do it.

Problem: Suppose $V$ is an inner-product space with $u, v in V$, with $v neq 0$. Provided that $u_{perp} = u – u_{par}$ and that $u_{par} = frac{<u, v>}{||v||^2}v$, prove that $<u_{perp}, u_{par}> = 0$.

My attempt:

Note that $<u_{perp}, u_{par}>$ = $<u – u_{par}, u_{par}>$ = $<u, u_{par}> – <u_{par}, u_{par}>$ = $<u, u_{par}>$

In a similar fashion, $<u_{perp}, u_{par}>$ = $<u_{perp}, u – u_{perp}>$ = $<u_{perp}, u> – <u_{perp}, u_{perp}>$ = $<u_{perp}, u>$

Therefore: $<u_{perp}, u_{par}>$ = $<u, u_{par}>$ = $<u_{perp}, u>$

$$implies <u, u_{par}> = <u_{perp}, u>$$

$$implies <u, u_{par}> – <u_{perp}, u> = 0$$

$$implies <u – u_{perp}, u_{par} – u> = 0$$

$$implies <u_{parr}, -u_{perp}> = 0$$

$$implies -<u_{parr}, u_{perp}> = 0$$

$$implies <u_{parr}, u_{perp}> = 0$$

$$implies -<u_{perp}, u_{parr}> = 0$$

$$implies <u_{perp}, u_{parr}> = 0$$

QED.

I can’t help but think that I’m using wrong reasoning or my proof has some flaw. In fact, I’m pretty sure there is a flaw when expanding the inner products. I tried to use the fact that $||u_{perp} + u_{par}||^2 = ||u_{perp}||^2 + ||u_{par}||^2$ implies orthogonality, but the RHS here is $||u||^2$ whereas the LHS after expanding is $2<u_{par}, u_{par}> – 2<u, u_{par}> + ||u||^2$, which I’m unable to simplify further.

Any assistance or even hints are much appreciated.

magento2 – How to display total qty product in list.pthml

Under product name I need display total product stock qty.

I use code:

   <?php
                            $_productNameStripped = $block->stripTags($_product->getQty(), null, true);
                            ?>
                                    <?php /* @escapeNotVerified */
                                    echo $_helper->productAttribute($_product, $_product->getQty(), 'quantity_and_stock_status'); ?>
                                
                            

But return 0 for all.

Any solution?

product identification – Help me identify these 20-sided dice with assorted numbers from 4 to 72

I went and Googled for the numbers, and the first (and currently only) result was US patent 7815191B2 titled “Equals: the game of strategy for the basic facts”. The abstract reads:

“An open rectangular prism with rotating cubes on dowel rods, two 12-sided dice, and three 20-sided dice invented with an accompanying method of use to function as a game to assist students in remembering the basic math facts including addition, subtraction, multiplication, and division.”

and further on, in the “detailed description of the invention”, the dice are described as follows (emphasis mine):

“5. The dice: The 12-sided dice are a different color from the 20-sided dice. The numbers are clear so that there is a way to understand the difference between the numbers on the dice. Dice 1 and 2 are dodecahedrons. Dice 3, 4, and 5 are icosahedrons.

  • A. Dice 1 printed numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 3, 5, & 7
  • B. Dice2 printed numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 4, 6, & 8
  • C. Dice3 printed numbers: 4, 5, 7, 10, 12, 15, 16, 18, 20, 21, 24, 28, 35, 36, 42, 49, 54, 56, 64, & 72
  • D. Dice 4 printed numbers: 1, 2, 3, 6, 8, 9, 12, 14, 18, 24, 25, 27, 30, 32, 36, 40, 45, 48, 63 & 81
  • E. Dice 5 printed numbers: 4, 6, 8, 9, 12, 16, 21, 25, 27, 28, 32, 35, 36, 42, 48, 49, 54, 56, 64, & 72″

So I guess that’s where they’re from.


Ps. While there seem to be quite a few math games named “Equals” (such as this one), searching for the full title of the patent works better and turns up a bunch of sites that sell (or at least used to sell) the game in question.

Alas, it seems like the game’s original web site (playequals.com) no longer works, but the Wayback Machine does have an archived copy.

There also (thanks, Someone_Evil) appears to be a new site located at playequals.jimdofree.com which includes some YouTube videos (#1, #2) demonstrating the gameplay — although, alas, apparently only the simplest game mode, using only the two 12-sided dice. The site even has a combined PDF product flyer for all of its products, which contains the best picture of the actual game that I’ve been able to locate so far, including all of the dice:

Picture of "Equals: the game of strategy for the basic facts"


Pps. The patent describes four different game modes: addition, subtraction, multiplication and division. Of these, only the multiplication mode actually uses the 20-sided dice:

“C. Multiplication:

To set up the board: Have each player choose a side. Each side has 2 sets of numbers with a symbol in the middle. Choose the multiplication symbol in the middle of the sets of numbers so that it faces up. Choose the numbers 1-9 on both sides of the multiplication symbol so that they face up.

To play: Roll dice 3, 4, and 5 at the same time. (These dice have 20 sides). Select one of the numbers rolled to be the product of two numbers on opposite sides of the multiplication symbol. Choose a number on each side of the multiplication symbol that when multiplied will equal one of the numbers rolled. (…)”

Given this, we can have some insight into how the numbers on the 20-sided dice were chosen: they’re all products of two numbers between 1 and 9 inclusive, with the factors distributed somewhat uniformly across the interval:

begin{array}{|c|c|c|}
hline
textbf{Die 3} &
textbf{Die 4} &
textbf{Die 5} \
hline
begin{aligned}
4 &= 1 times 4 = 2 times 2 \
5 &= 1 times 5 \
7 &= 1 times 7 \
10 &= 2 times 5 \
12 &= 2 times 6 = 3 times 4 \
15 &= 3 times 5 \
16 &= 2 times 8 = 4 times 4 \
18 &= 2 times 9 = 3 times 6 \
20 &= 4 times 5 \
21 &= 3 times 7 \
24 &= 3 times 8 = 4 times 6 \
28 &= 4 times 7 \
35 &= 5 times 7 \
36 &= 4 times 9 = 6 times 6 \
42 &= 6 times 7 \
49 &= 7 times 7 \
54 &= 6 times 9 \
56 &= 7 times 8 \
64 &= 8 times 8 \
72 &= 8 times 9 \
end{aligned} &
begin{aligned}
1 &= 1 times 1 \
2 &= 1 times 2 \
3 &= 1 times 3 \
6 &= 1 times 6 = 2 times 3 \
8 &= 1 times 8 = 2 times 4 \
9 &= 1 times 9 = 3 times 3 \
12 &= 2 times 6 = 3 times 4 \
14 &= 2 times 7 \
18 &= 2 times 9 = 3 times 6 \
24 &= 3 times 8 = 4 times 6 \
25 &= 5 times 5 \
27 &= 3 times 9 \
30 &= 5 times 6 \
32 &= 4 times 8 \
36 &= 4 times 9 = 6 times 6 \
40 &= 5 times 8 \
45 &= 5 times 9 \
48 &= 6 times 8 \
63 &= 7 times 9 \
81 &= 9 times 9 \
end{aligned} &
begin{aligned}
4 &= 1 times 4 = 2 times 2 \
6 &= 1 times 6 = 2 times 3 \
8 &= 1 times 8 = 2 times 4 \
9 &= 1 times 9 = 3 times 3 \
12 &= 2 times 6 = 3 times 4 \
16 &= 2 times 8 = 4 times 4 \
21 &= 3 times 7 \
25 &= 5 times 5 \
27 &= 3 times 9 \
28 &= 4 times 7 \
32 &= 4 times 8 \
35 &= 5 times 7 \
36 &= 4 times 9 = 6 times 6 \
42 &= 6 times 7 \
48 &= 6 times 8 \
49 &= 7 times 7 \
54 &= 6 times 9 \
56 &= 7 times 8 \
64 &= 8 times 8 \
72 &= 8 times 9 \
end{aligned} \
hline
end{array}

(FWIW, the only numbers that occur on all three dice are 12 = 2 × 6 = 3 × 4 and 36 = 4 × 9 = 6 × 6.)

We can even calculate the exact probability of each number from 1 to 9 being a possible choice for a multiplicand on the first roll using a quick AnyDice script, which produces the following output:

Results: 1: 52%, 2: 61%, 3: 63%, 4: 68%, 5: 46%, 6: 68%, 7: 59%, 8: 63%, 9: 61%

It turns out the 1 and 5 are the least likely factors to work, which kind of makes sense for a game intended to teach multiplication, since multiplying by those numbers is particularly easy in base 10.

I doubt that any particularly deep statistical analysis went into the game design, though. Most likely the inventor just took a single-digit multiplication table and distributed the products more or less randomly across three 20-sided dice, doubling or tripling specific ones that they considered most pedagogically relevant.

How can product managers ensure software quality?

I am a software engineer and I care deeply about my software quality, while I assume my product manager only cares about the final product. I can’t imagine my product manager would care how I get my work done, only that it is done. How can a product manager ensure the work I do as an engineer is quality work?

A few thoughts:

Schedule Testing
For this it seems like something obvious to just ask that all my code is tested. But while I may test my code it doesn’t account for the quality of the code. I may write terrible code and test it, and we’d only pay the penalty in the future when we add new features.

Small Scope – Catch Time Delays Early
One other idea is to have small chunks of work with estimated times, and if the estimate is too much then it may be a sign of code quality and we can plan to address that early. However, as an engineer I usually already know if my code is good or I’m rushing through it. While this seems to make sense it’s also seems like a difficult idea to sell to the business team.

Hire Better Engineers
This is the only idea I know of. Start with good engineers and make them leads. I’m putting this idea here as a baseline. But not everyone is the best. In my experience any requests from a lead seems like time taken out of the final product, then it’s convincing the product that we need more time to have the same the exact same output except with more reliable code base.

Unless the business team fully understands and budgets the need for quality code, it seems like every project will bloat until it’s too late. What else can be done to have code quality be part of the business?

product identification – Help me identify these 20-sided dice with random numbers from 4 to 72

I went and Googled for the numbers, and the first (and currently only) result was US patent 7815191B2 titled “Equals: the game of strategy for the basic facts”. The abstract reads:

“An open rectangular prism with rotating cubes on dowel rods, two 12-sided dice, and three 20-sided dice invented with an accompanying method of use to function as a game to assist students in remembering the basic math facts including addition, subtraction, multiplication, and division.”

and further on, in the “detailed description of the invention”, the dice are described as follows (emphasis mine):

“5. The dice: The 12-sided dice are a different color from the 20-sided dice. The numbers are clear so that there is a way to understand the difference between the numbers on the dice. Dice 1 and 2 are dodecahedrons. Dice 3, 4, and 5 are icosahedrons.

  • A. Dice 1 printed numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 3, 5, & 7
  • B. Dice2 printed numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 4, 6, & 8
  • C. Dice3 printed numbers: 4, 5, 7, 10, 12, 15, 16, 18, 20, 21, 24, 28, 35, 36, 42, 49, 54, 56, 64, & 72
  • D. Dice 4 printed numbers: 1, 2, 3, 6, 8, 9, 12, 14, 18, 24, 25, 27, 30, 32, 36, 40, 45, 48, 63 & 81
  • E. Dice 5 printed numbers: 4, 6, 8, 9, 12, 16, 21, 25, 27, 28, 32, 35, 36, 42, 48, 49, 54, 56, 64, & 72″

So I guess that’s where they’re from.


Ps. While there seem to be quite a few math games named “Equals” (such as this one), searching for the full title of the patent works better and turns up a bunch of sites that sell (or at least used to sell) the game in question.

Alas, it seems like the game’s original web site (playequals.com) no longer works, but the Wayback Machine does have an archived copy.

There also (thanks, Someone_Evil) appears to be a new site located at playequals.jimdofree.com which includes some YouTube videos (#1, #2) demonstrating the gameplay — although, alas, apparently only the simplest game mode, using only the two 12-sided dice. The site even has a combined PDF product flyer for all of its products, which contains the best picture of the actual game that I’ve been able to locate so far, including all of the dice:

Picture of "Equals: the game of strategy for the basic facts"

Solve this problem of Kronecker Product and Prove the given statement true or not

let A,B,C,D are four complex matrices.let
P=Kronecker Profuct (A,B)
Q=Kronecker Profuct (C,D)

Is it true that P.Q= Kronecker Product ((A.C),(B.D))