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Is it possible to deploy a .NET 5 Web API solution inside of a VM hosted on Azure?

Is it possible to deploy a .NET 5 Web API solution inside of a VM hosted on Azure? – Software Engineering Stack Exchange

complex numbers – Is there a method to verify whether you have the correct solution for greatest/least values of arg z?

I often solve questions that are about finding the greatest or least arg z values for points lying in a region and I was just wondering whether there is a way to verify or double-check if you’ve come up with the right solution or not (either during an exam or for practice purposes)?

For example, I had this particular question on one practice paper:

On an Argand diagram, sketch the locus of points representing complex numbers u satisfying
the relation |z-u| = 1, where u is -1+ 4√3i. Determine the greatest value of arg z for points on this locus.

I know the center is (-1, 4√3) and the radius is 1 and I can use the tan formula to get arctan (1/4√3) and then multiply the answer by 2 and add pi/2 to get the final answer that is = 1.86 radians, but is there any way I could verify whether my final answer is correct or incorrect during the exam?

Are there any methods that can help with minimizing errors as much as possible?
I do practice questions a lot but I honestly would like to know if there is any way I could make sure I do not make mistakes during an actual exam?

Any suggestions would be greatly appreciated!

algorithm – Is there any way to optimize this backtracking solution?

I have just solved the problem 1087 (Brace Expansion) on Leetcode using backtracking and was wondering if there is any way to optimize the solution? I have tried to come up with different ways but nothing seems to work. If possible, would it also be able to ask for help in analyzing the runtime of such a solution, as it seems very recursion-heavy? Thanks a lot! Code is provided below:

class Solution {
    public String() expand(String s) {
        if(s.length() == 0 || s == null)
            return new String(0);
        
        List<String> options = new ArrayList();
        int length = s.length();
        
        for(int i = 0; i < s.length(); i++) {
            if(s.charAt(i) == '{') {
                int j = i + 1;
                StringBuilder tempContents = new StringBuilder();
                while(j < length && s.charAt(j) != '}') {
                    if(s.charAt(j) == ',') {
                        j++;
                        continue;
                    }
                    tempContents.append(s.charAt(j));
                    j++;
                }
                options.add(tempContents.toString());
                i = j;
            } else {
                options.add(s.charAt(i) + "");
            }
        } //roughly O(n) runtime as it loops through and parses all the strings
        
        List<String> permutations = new ArrayList();
        calculatePermutations(permutations, options, new StringBuilder(), 0);
        
        String() result = new String(permutations.size());
        for(int i = 0; i < result.length; i++) {
            result(i) = permutations.get(i);
        }
        
        Arrays.sort(result);
        
        return result;
    }
    
    public static void calculatePermutations(List<String> permutations, List<String> options, StringBuilder tempContents, int index) {
        if(tempContents.length() == options.size()) {
            permutations.add(tempContents.toString());
            return;
        }
        
        for(char c : options.get(index).toCharArray()) {
            tempContents.append(c);
            calculatePermutations(permutations, options, tempContents, index + 1);
            tempContents.setLength(tempContents.length() - 1);
        }
    }
}

Where is my mistake in my solution?

If $2^x$ = $3^y$ = 216, find $1/x + 1/y$.

Now, $216 = 2^3 * 3^3$.
So, $2^x * 3^y = 216 * 216 = 2^3 * 3^3 * 2^3 * 3^3 = 2^6 * 3^6$

By fundamental theorem of arithmetic and given:

$2^x * 3^y = 2^6 * 3^6$, I conclude that $x=6$ and $y=6$. Therefore $1/x + 1/y = 1/6 + 1/6 = 1/3$. The correct answer!

However, $2^6 neq 3^6 neq 216$.

What have I done wrong?

ap.analysis of pdes – Green kernel vs fundamental solution

Let $L$ being the Laplacian for a given Lie group $G$. I would like to know what is the difference between the two notions in relation to the operator $L$:

  • The fundamental solution $Gamma(x)$ of $L$;
  • And the green kernel $mathcal G(x)$of $L$.

In which case, we have $Gamma(x)=mathcal G(x)$ or the contrary ?

Thank you in advance.

ap.analysis of pdes – Solution of parabolic partial differential equation using singular perturbation method

Consider the following parabolic partial differential equation (PDE)

begin{align}
label{eq:42}
left(cospsi frac{partial}{partial r} + frac{gamma}{r} sinpsi frac{partial}{partial psi} + epsilon frac{partial^{2}}{partial psi^{2}} right)u(r, psi) = -1,
end{align}

where $u(r, psi): (0,1)times(0,2pi) to mathbb{R}^+$ and $epsilon,gamma in mathbb{R}^+$ are constant parameters.
The boundary conditions are Dirichlet $u(r, psi)|_{r=1} = 0$ and periodic $u(r, psi+2pi) = u(r, psi)$.

When $epsilonto 0$, the above second order PDE reduces to first order PDE, the boundary conditions generally cannot be fulfilled anymore, entailing singular perturbation method to handle the equation.

How to solve the above the equation with $epsilonto 0$ using singular perturbation method?

I mainly refer to the book written by J. Kevorkian and J.D. Cole titled ‘‘Multiple Scale and Singular Perturbation Methods’’. Singular perturbation methods for elliptic and parabolic PDEs are introduced there. However, I still have no clue of how to handle the above parabolic PDE.
In case it helps, the (projected) characteristic curve of the reduced PDE when $epsilon=0$ is

begin{align}
label{eq:62}
r = |sinpsi|^{frac{1}{gamma}} C,
end{align}

where $C$ is constant, which can be visualized in the following plot ($gamma=1$).

enter image description here

My background is theoretical physics. Please let me know if there is something mathematically inaccurate in the above problem formulation. Any suggestion or recommendation of references would be greatly appreciated. Thanks.

web part – Data from the email, the content is not showing in the solution, it’s showing blank

I am developing a help desk software in SharePoint. I am using the out of box web parts as well as some customized one. Anyway, I have set up a flow that automatically creates a ticket for emails coming in at support. The flow is running OK and ticket is creating, but the problem is data from the email, the content is not showing in the solution, its showing blank. I don’t understand why this is happening. Has anyone else faced this kind of issue? Please do share your thought on how I can fix this.

nt.number theory – Prove that the equation $2^a – 2^b – 1=3^c$ has no integral solution with $a,bgeq 3$

I was looking for a natural power of 3 that could be written like

Binary format:

11..(N times)..11011..(M times)..11

Example: 1111110111111111111111 (…isn’t a power of 3)

Or could also be written like

3^x = 2^a – 2^b – 1

(x is arbitrary, “a” and “b” are natural numbers, a = N-M-1, b = M, and the single zero in binary format is a must)

But couldn’t find any, so I thought there might be some proof that there’s no such numbers (altho that would contradict intuition) or maybe it can be proven that there might be such numbers?

differential equations – how we can extract the value of the solution of a PDE in a point x? (NDSolve)

Please I need your help, I calculate the solution of heat equation using methode of line
This is my code:

n = 10    
grid = 1/n  Range[0, n];
d1 = NDSolve`FiniteDifferenceDerivative[Derivative[1], grid];
d2 = NDSolve`FiniteDifferenceDerivative[Derivative[2], grid];
M1 = d1["DifferentiationMatrix"];
M2 = d2["DifferentiationMatrix"];
y00[t_] := Sqrt[2] Sin[Pi t];
T= 0.02;
tab = Table[u[i]
tab1 = Table[u[i], {i, Length[grid]} ];
ux = M1.tab;
uxx = M2.tab;
solu1   = D[u[1]
solu2 = D[u[n + 1]
solu3 = Table[D[u[i]
solu4 = Table[u[i][0] == y00[grid[[i]]], {i, Length[grid]}];
sol1 = NDSolve[{solu1, solu2, solu3, solu4}, tab1, {t, 0, T}, 
   Method -> {"MethodOfLines", 
     "SpatialDiscretization" -> {"TensorProductGrid", 
       "MaxPoints" -> 25}}];

Now that I have calculated the solution “sol1”. I need to approximate the value of this solution in $t = T$ in the space $ grid$ something like

h = Table [sol1 [grid[[i]],T],{i,Length[grid]}]

but this is not working for me, can anyone help me with that please??

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