A graph with the maximum signless Laplacian spectral radius among all $F_k$-free graphs of order $n$ is connected?

Assume that a graph $G$ has the maximum signless Laplacian spectral radius among all $F_k$-free ($kgeq 2$) graphs of order $n$ ($ngeq 3k^2 -k-2$).

I want to prove that $G$ is connected.

On the contrary, assume that $G$ is not connected. Then we can add some new edges into $G$ so that the obtained graph $G’$ is connected and still $F_k$-free.

How can I prove that the Rayleigh quotient and the Perron-Frobenius theorem imply that $q_1 (G’)>q_1 (G)$? where $q_1 (G)$ is the signless Laplacian spectral radius of $G$.

Thanks in advance.

❕NEWS – U.S Among Countries to Adopt BTC as Legal Tender, Survey Shows. | Proxies-free

CoinMarketCap last month survey on the country that is likely to adopt bitcoin as legal tender shows that showed that top ten countries received a total of 234,057 votes indicating highest likelihood to declare Bitcoin as Legal tender. Paraguay ranked highest with 81,433 votes followed by Venezuela with 58,656 votes. Anguilla and Panama followed with 34,376 and 17,323 votes respectively. United States ranked 6th with 7,823 votes.

This is interesting , let see if this will come to reality.

multithreading – Java inter-thread communication among 2-n-1 threads

I am relatively new to concurrency in java, currently working on a project that will eventually have me utilizing 10 threads to solve a cumbersome cpu intensive task (mining). Before I get to that exciting stuff I have to establish inter-thread communication where one thread can send a message to all other threads…they then read said message in whichever order the CPU decides…then the next thread sends a message to all other threads and so forth. Important that I note we need to distinguish messages in such a way that the thread doesn’t read its own message. Do I need to perhaps pass in to each thread a concurrent List where a message is comprised of a private int threadId //corresponds to the thread id, and a private String randomStr? I’ve been racking my brains, maybe not hard enough past few days would appreciate if anyone could direct me to an example I can dive in on.

This is what I am trying to mimic:
Thread-0 sending message WSDSFVFQUC to all other threads.
Thread-1 sending message YMGQJLAEFL to all other threads.
Thread-1 received message WSDSFVFQUC.
Thread-2 sending message NGBABUSZOC to all other threads.
Thread-2 received message WSDSFVFQUC.
Thread-2 received message YMGQJLAEFL.
Thread-0 received message YMGQJLAEFL.
Thread-0 received message NGBABUSZOC.
Thread-1 received message NGBABUSZOC.

web applications – JWT logout: Sharing blacklisted invalid token among services

I am working on a microservices project involving 4 services – Auth Service, Service-A, Service-B and Service-C.

All the services are implemented using Spring Boot.
The Auth Service is responsible for authenticating logged in user and generating a JWT bearer token.
Each of Service-A/B/C has JWT filters which checks for validity of token and then provide access to the Rest APIs.

Now I want to implement logout feature. The logout request goes to Auth Service. The Auth Service uses Redis.
The token is added to list of invalid tokens with ttl set so that after the expiry the token is removed automatically.

Now how can JWT filters in Service-A/B/C access the blacklisted token so that Rest API access is approved/disapproved?
If all the services are deployed in same system the services can access Redis easily.
If the services are deployed in different systems, how can they access the invalid tokens?

Should I implement pub/sub messaging and each service have a list of invalid tokens stored in redis? Or is there a better approach in microservices environment?

algorithms – Given $n$ sets of matrices, find $n$ matrices that have the least number of LCDs among their entries

Let’s say I have $n$ sets of matrices

$$
A = left{begin{pmatrix}
2 & 4 & 17\
5 & 6 & 9\
end{pmatrix}
begin{pmatrix}
2 & 4 & 18\
5 & 6 & 9\
end{pmatrix}
right}
$$

$$
B = left{begin{pmatrix}
13 & 20\
3 & 16\
end{pmatrix}
begin{pmatrix}
14 & 20\
3 & 16\
end{pmatrix}
begin{pmatrix}
13 & 21\
3 & 16\
end{pmatrix}
begin{pmatrix}
14 & 21\
3 & 16\
end{pmatrix}
begin{pmatrix}
13 & 20\
3 & 17\
end{pmatrix}
begin{pmatrix}
14 & 20\
3 & 17\
end{pmatrix}

right}
$$

Let’s define $T$ as a vector that contains the lowest common denominators shared among the entries of all the $n$ matrices.

I need to find $n$ matrices, picking one from each set, that will minimize the length of $T$, i.e. in this case

$$
A_{1}=begin{pmatrix}
2 & 4 & 18\
5 & 6 & 9\
end{pmatrix}
$$

$$
B_{1}=begin{pmatrix}
14 & 20\
3 & 16\
end{pmatrix}
$$

The resulting $T$ would be

$$
T=begin{bmatrix}
2,3,4,5,9
end{bmatrix}
$$

I know I can bruteforce every possible combination, but is there a more efficient way?

how to perform code review among different types of developer?

My develper team includes Android, iOS, PHP, Web Front, .Net developers, but every type of developer only 1 person, and they don’t know other’s programming languages very well.

We have setup Jenkins with some static analysis tools such as Sonarqube, but it’s not enough to find out some logic mistakes.

Is there any suggestion on performing code review in this kind of team? Thanks a lot!

distributed computing – Distributing very long running work among microservices

We have a microservice whose multiple instances are running for High Availablity and Scaling. But although we have multipe stateless microservice there are some tasks which needs to be singleton. For e.g. Lets say we have to fetch emails from multiple emailIds and do some work on those emails. Now since I would like to ensure same emails are not processed multiple times I would want only one microservice instance to make connection to a specific email profile. Now Lets say I have 2 Microservice instance InsA and InsB. Also I have 4 email profile E1, E2, E3 and E4. Now assuming there are almost equal rate of messages in each email profile My Ideal scenario would be each microservice making connection to 2 email profile. (InsA – E1 And E2, InsB – E3 and E4). Also If InsA goes down I would like to temporarily move E1 and E2 to InsB until InsA is active again. Also If we scale Up the Email profiles must be redistributed.

Question 1.
Are there some known patterns to solve this problem.

Question 2.
Are we on the right direction atleast or there is something we are missing here. I mean Is distributing the email connection among stateless microservices was a good Idea or there are also some other ways to solve this problem.

python – Selection of independent variables in K means clustering among a vast dataset

As I understand it, the process of K means clustering takes a set of sample points with k arbitrary centroids and uses Euclidean distance to classify the points closest to centroids to k groups.

What I am unable to understand is a point in the cartesian plane has only an x and y coordinate and so, amongst a given dataset we can only chose 2 independent variables and plot the points and proceed with the algorithm. However, there might be many more independent variables which could influence classification, for example, if we are trying to classify dogs based on their breeds using physical attributes such as size of ears, radius of eyes, body weight, length of legs, lifespan and so on. I’m not sure how this problem is resolved in K clustering.

Are the two variables with maximum information gain considered or are the points plotted in an n-dimensional space where each axis defines each attribute.

Could someone provide clarity on this issue. Thanks for any help

I am trying to play Among us cracked on ubuntu 20.04 (read body)

I watched a video on youtube and followed the steps this is the error i am getting

~/Downloads/Among.Us.v2021.3.5s/Among.Us.v2021.3.5s$ sudo wine Among Us.exe 000f:err:service:process_send_command receiving command result timed out X Error of failed request: GLXBadFBConfig Major opcode of failed request: 152 (GLX) Minor opcode of failed request: 0 () Serial number of failed request: 227 Current serial number in output stream: 227 link to yt tutorial https://www.youtube.com/watch?v=SSa5uAqzQvE