Proving NP-completeness of an extension in List Coloring Problem

In the List Coloring Problem (LCP), one is given an undirected graph $ G(V,E)$ , each vertex $ v \in V$ is given a list of permissible colors, and one tries to assign colors to vertices such that each vertex is assigned a color from its own list, with adjacent vertices getting different colors.

In my problem: Given a graph $ G(V,E)$ an integer $ k$ , for tractability, we take over the coloring using natural number from 1 to k. Every vertex is required to be colored by a number. The objective is to minimize the sum of the coloring number of $ v \in V$ in $ G(V,E)$ while satisfying no any two adjacent vertices are colored using same number.

I want to prove the above problem (called P1) is $ \mathcal{NP}$ -complete.

Answer: Reduce the $ f$ -coloring List Coloring Problem to P1.

I now define a decision version problem (called P2): Given an undirected graph $ G(V,E)$ and an integer $ k$ in List Coloring Problem, we ask whether a $ k$ -coloring can be found in $ G(V,E)$ .

In my opinion, to solve P1, we have to solve P2 firstly and then find the minimum sum from the feasible solutions set. Since LCP is a generalization of graph coloring problem(GCP), i.e., the latter is a special case of the former, and it is well known that the $ k$ -coloring problem in GCP is $ \mathcal{NP}$ -complete. Thus, we can say that the $ k$ -coloring problem in LCP is also $ \mathcal{NP}$ -complete, i.e., P2 is $ \mathcal{NP}$ -complete. This concludes that P1 is also $ \mathcal{NP}$ -complete. Is this reduction correct?

I would sincerely appreciate if someone can give any useful suggestions!

Extension of a “statistical limit functional”

$ c=$ The set of all real convergent sequences

$ l_\infty=$ The set of all real bounded sequences

Clearly $ c\subset l_\infty$

$ f:c\to \mathbb R$ is called limit functional defined by $ f(x)=\lim\limits_{n\to\infty} x_n$

There exist infinitely many extensions of $ f$ to $ l_\infty$ . From here main concept of Banach Limit functional comes.

With the help of this Banach limits the notion of almost convergence is defined.

My Question : What happens if we take extensions of the statistical limit functional $ g$ ?

(where $ g:st\to \mathbb R$ is defined by $ g(x)=stat\lim\limits_{n\to\infty} x_n$ )

Here, $ st=$ The set of all bounded statistically convergent sequences.

May we define a new kind of convergence in this approach? Is it not available in literature?

Schengen visa – Duration of stay Extension – Approach

I applied for Schengen visa thru Germany embassy for travel from 10th August to 24th August, staying maximum days in Germany (5). I got my visa for duration of stay as 15 days (same as itinerary). Some plans have changed at my end and I may plan to Visit Spain as well as part of the trip with Spain being the maximum stay duration. Can i apply for another Schengen visa at Spain embassy giving the exact reasons so that my effective duration of stay covers the next 7 days? OR do I have to request the German embassy for duration of stay extension?

Is there a Chrome extension that would fetch the content from the meaningless social media emails?

I’m deeply disgusted by receiving emails that contain no content, they tell me something has happened, visit the platform to see it.

It should be possible to create Chrome Extension that would launch an invisible iframe and fetch the content for me without leaving Gmail.

MAYBE SUCH THING ALREADY EXISTS?

Not calling it an anti-pattern. If you are not paying for the product, you are the product, increasing shareholders profit requires using various techniques to keep people on the website.

Examples

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How to edit .imovielibrary extension as project?

On iMovie 10.1.6 for Macbook Pro High Sierra 10.13.6— I’m trying to edit a project/clip which has been autosaved by iMovie with a .imovielibrary extension.

When I try to open it to edit in iMovie, it shows me the saved movie clip in My Media and allows me to drag to panel below. Doing so seems to have saved none of the transitions, overlaid music, etc necessary to make additional edits. Checking My Movie above in Project Media is the same story.

Question: How can I recover / open the video clip with the transitions, overlays, etc so I can begin where I left off?

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Then I thought I should click on All Events then drag them onto the library clip but it won’t let me drag or otherwise add the events:

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Originally I thought I may have lost them, so I checked in the cached folder and it’s the same deal here (no editable “project” files):

~/Users/me/Library/Containers/com.apple.iMovieApp/Data/Library/Caches/iMovie Backups.localized

Extension of Standard English Peg Solitaire

An entire analysis of standard English Peg Solitaire has been given. See

Berlekamp, E. R.; Conway, J. H.; Guy, R. K. (2001) [1981], Winning Ways for your Mathematical Plays (paperback) (2nd ed.), A K Peters/CRC Press, ISBN 978-1568811307.

Consider the standard English version of the game. The center peg is adjacent to the center-most peg of the left-most column of a $ 3 \times 3$ grid of pegs on the right, the center-most peg of the right-most column of a $ 3 \times 3$ grid of pegs on the left, and similar adjacency in the up and down directions from the center peg. I’m curious whether a variation of the game has been considered with, say, $ 2n+1 \times 2n+1$ grids in the four directions from the center peg? Many questions could be asked that were thoroughly answered for the standard game where $ n=1$ , like can jumps be made so that there is only one peg remaining? If so, how many ways are there to win both with and without symmetry? Obviously there are $ 4(2n+1)^2 -4 +1=16n^2+16n+1$ pegs in this variation.

Thanks for your time!

Move Customer Extension Attribute to new tab

I’ve successfully created several customer extension attributes that display on the customer form and save and load data.
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When I try to move these fields to a new custom tab on the customer admin page, they break. The source models still load, but the data isn’t populated or saved. I can see when I inspect that the form name changes from customer['my_attribute'] to my_tab[my_attribute].
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I suspect I need to change my UpgradeData.php script to use a custom form, but I’m having trouble finding resources on how to go about this.

Relevant snippet from UpgradeData.php:

        $  exemptionTypeCode = 'my_exemption_type';         $  eavSetup->addAttribute(             CustomerMetadataInterface::ENTITY_TYPE_CUSTOMER,             $  exemptionTypeCode,             [                 'group' => 'General',                 'type' => 'varchar',                 'label' => 'Exemption Type',                 'input' => 'select',                  'required' => false,                 'visible' => true,                 'user_defined' => true,                 'position' => 501,                 'system' => 0,                 'sort_order' => 50,                 'default' => 'non_exempt',                  'source' => 'Custom\Module\Model\Attribute\Source\CustomerExemptionType',                 'backend' => 'Custom\Module\Model\Attribute\Backend\CustomerExemptionType',                 'frontend' => 'Custom\Module\Model\Attribute\Frontend\CustomerExemptionType',                 'global' => 'Magento\Eav\Model\Entity\Attribute\ScopedAttributeInterface::SCOPE_GLOBAL',                  'is_used_in_grid' => false,                 'is_visible_in_grid' => false,                 'is_filterable_in_grid' => false,                 'is_html_allowed_on_front' => true,                 'visible_on_front' => true             ]         );         $  eavSetup->addAttributeToSet(             CustomerMetadataInterface::ENTITY_TYPE_CUSTOMER,             CustomerMetadataInterface::ATTRIBUTE_SET_ID_CUSTOMER,             null,             $  exemptionTypeCode);         $  exemptionType = $  this->eavConfig->getAttribute(             CustomerMetadataInterface::ENTITY_TYPE_CUSTOMER,             $  exemptionTypeCode);         $  exemptionType->setData('used_in_forms', ['adminhtml_customer']);         $  exemptionType->getResource()->save($  exemptionType); 

Snippet from customer_form.xml, changing fieldset name from ‘customer’ to ‘my_tab’ breaks the extension attributes.

<form xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:module:Magento_Ui:etc/ui_configuration.xsd"> <fieldset name="customer">     <field name="my_exemption_type" formElement="select">         <argument name="data" xsi:type="array">             <item name="config" xsi:type="array">                 <item name="label" xsi:type="string" translate="true">Exemption Type</item>                 <item name="source" xsi:type="string">customer</item>             </item>         </argument>         <settings>             <visible>true</visible>         </settings>     </field> </fieldset> 

Any help with either moving attributes to different tabs, creating a new admin form, or a different approach I haven’t considered are appreciated!