algorithms – Why do we need MinHeap for Meeting Rooms 2 Leetcode problem

I came across this problem online which is a Leetcode premium problem. Most of the people are solving this using Minheap. To me using minHeap for this seems like repetitive way to solve a problem if we can achieve the same thing just by sorting the array based on startTimes & than checking if meeting(i).endTime > meeting(i+1).startTime. Increment if the aforementioned condition is true.

I don’t have access to leetcode premium so I can’t verify if the following solution would suffice.
Am I missing something here?

Problem Statement:
enter image description here

public int minMeetingRooms(List<Interval> intervals) {
        // Write your code here
        
        if(intervals == null || intervals.size() == 0)
            return 0;
            
        Collections.sort(intervals, (meeting1, meeting2) -> meeting1.start-meeting2.start);
        
        int rooms = 1;
        
        for(int i=0; i<intervals.size()-1; i++) {
            if(intervals.get(i).end > intervals.get(i+1).start)
                rooms++;
        }
        
        System.out.println("rooms required: "+ rooms);
        
        return rooms;
    }

real analysis – A series convergence problem about Gamma function

In Apostol Mathematical Analysis Exercise 10.31, the question want us to show that
$$Gamma(x)=sum_{n=0}^infty dfrac{(-1)^n}{n!}dfrac1{n+x}+sum_{n=0}^infty c_nx^n$$
for $x>0$ where $c_n=(1/n!)int_1^infty t^{-1}e^{-t}(log t)^ndt$. This is an easy one. The question after it is asking to show that the complex series
$$sum_{n=0}^infty c_nz^n$$
converges for $zin mathbb C$. I find this very difficult for me to prove, maybe I’m missing something.

My approach is using the ratio test
$$left|dfrac{c_{n+1}z^{n+1}}{c_nz^n}right|=dfrac{c_{n+1}}{c_n}|z|<1$$
so we need to show that
$$0=lim_{ntoinfty}dfrac{c_{n+1}}{c_n}=lim_{ntoinfty}dfrac1{n+1}dfrac{int_1^infty t^{-1}e^{-t}(log t)^{n+1}dt}{int_1^infty t^{-1}e^{-t}(log t)^ndt}$$
so the ratio test will always return $<1$ for every $zinmathbb C$.

Notice the integrands in both integrals, I’ve come up an idea which let $f(t)=t^{-1}e^{-t}(log t)^n, g(t)=log t$, then I use the Intermediate value theorem for integrals
$$int_1^infty f(t)g(t)dt=g(c)int_1^infty f(t)dt$$
for some $c>1$. For this I’m trying first not to consider that this integral is improper, then find out the value of $c$, or some reasonable bound of $c$, but I’m stuck from here. My expectation is that the ratio of integrals is of order $log n$, but logically speaking it is good enough if the ratio is $o(n)$.

Another idea of mine is to show
$$0=lim_{ntoinfty}sqrt(n){c_n}=lim_{ntoinfty}sqrt(n){frac1{n!}int_1^infty t^{-1}e^{-t}(log t)^ndt},$$
but this seems even harder.

Problem with Taylor expansion of a function

I need to make a Taylor expansion of the following expression:

(ρ^2 (-z + z2 + Sqrt((z - z2)^2 + ρ^2)) (-z + z4 + Sqrt((z - z4)^2 + ρ^2)))/
   ((-z + z1 + Sqrt((z - z1)^2 + ρ^2)) (-z + z3 + Sqrt((z - z3)^2 + ρ^2)))

about ρ = 0 for z1 <= z <= z2 && z3 <= z <= z4, where z1 <= z2 <= z3 <= z4.

I tried to express the conditions as

$Assumptions := z1 <= z <= z2 && z3 <= z <= z4 && z1 <= z2 <= z3 <= z4

and I evaluated:

Series(
  (ρ^2 (-z + z2 + Sqrt((z - z2)^2 + ρ^2)) (-z + z4 + Sqrt((z - z4)^2 + ρ^2))) /
     ((-z + z1 + Sqrt((z - z1)^2 + ρ^2)) (-z + z3 + Sqrt((z - z3)^2 + ρ^2))),
  {ρ,0,1})

However, I did not get the correct expansion.

How should I adjust my code to get the correct expressions?

Is MAX-averageSAT a well-known problem?

Is there any variant of the Boolean SAT or Max-SAT problem that has a flavor of maximizing or minimizing the average of the weights of the satisfied clauses of a WCNF formula? Any literature on an optimization problem of similar flavor would be appreciated. Thanks.

java – i have problem with android.view.View.FindViewById();

Caused by: java.lang.NullPointerException: Attempt to invoke virtual method ‘android.view.View android.view.View.findViewById(int)’ on a null object reference
at com.example.azerhi.My_List_View.onCreate(My_List_View.java:28)

walidList.findViewById(R.id.walid_listee);

reference request – Restricted partition problem into parts with a given set of prime factors

I need a reference for the following question:

Let $mathcal{P}$ be a finite set of $k$ primes and let $f(n)$ be the number of partitions of $n$ into parts whose prime factors are restricted to the set $mathcal{P}$. Then $f(n)<Ce^{ln(n)^{k+1}}$ for some constant $C$.

It feels very classical, but I have not managed to track it down.

plugins – Ultimate Member Basedir/Baseurl modification problem

I’m trying to modify Ultimate Members plugin behaviour so I can upload files outside public website domain folder (usually www folder).

I’ve achieved it by modifying class-uploader.php in plugin’s folder, but that’s not optimal as changes will break at every plugin update.

I tried using um_upload_basedir_filter filter inside my child theme’s functions.php as stated in their docs but it isn’t working, as this successfully creates the intended folder but doesn’t uploads files there as requested, instead it does it inside the default upload/ultimatemember folder.

Any help will be appreciated!

Code example of what I’ve tried:

add_filter( 'um_upload_basedir_filter', 'custom_upload_basedir', 10, 1 );
function custom_upload_basedir( $basedir ) {
  return "NEW BASE PATH HERE";
}

formal languages – Is the problem that determines whenever the word member $in$ L(M) decidable or not?

Given a Turing machine M on alphabet {m,e,b,r} we’re asked to determine if member $in$ L(M).
You must realize that M is not one specific machine and can be any turing Machine with the same alphabet.
My goal is to determine whenever this problem is decidable or not.

My idea was to use mapping reducibility. The goal was to see if we can translate all problems from $A_{TM}$ which is known to be undecidable into our current problem. This would make our current problem undecidable by contagion. However I’m struggling in doing so because I’m not sure if it’s possible. $A_{TM}$ is defined as a Turing machine M that accepts the word w.

Any help to get unstuck would be appreciated.

How to solve this infinite square root problem?

Question:

Find $$2.,sqrt{3+sqrt{3+sqrt{2+sqrt{2+sqrt{2+…}}}}} $$

See picture.


I am calculating the answer as $sqrt{3+sqrt{5}}$, but the solution says it has to be $sqrt{10}+sqrt{2}$.

multivariable calculus – A Surface integral problem using Spherical coordinate system

For the integral $$int!!!intlimits_S {({x^2} + {y^2})} dS,S:{x^2} + {y^2} + {z^2} = 2z$$The
correct answer is $${{8pi } over 3}$$ I used Spherical coordinate system,it turns to $$int_0^{2pi } {dtheta int_0^{{pi over 2}} {({r^2}{{sin }^2}varphi } )({r^2}sin varphi )dvarphi } ,r = 2cos varphi $$Then use $r = 2cos varphi$,it turns to $$32pi int_0^{{pi over 2}} {{{sin }^3}varphi {{cos }^4}varphi dvarphi } = {{64} over {35}}pi $$Doesn’t match the answer,I wonder where am I wrong.