discrete mathematics – Split all subsets of $mathcal{P}(mathbb{N})$ into 2 groups $A dotcup B =mathcal{P}mathbb(N)$ s.t. no 2 neighboring sets are in the same group

So I’m trying to come up with a proof of the above action being possible or not. I feel like I could use the compactness theorem on this one but I’m not quite sure how to split up the finite subsets of $mathbb{n}$ to make the condition true.

I’d really appreciate some help on this :] Thanks so much!

discrete mathematics – How would you deal with equivalence relation and equivalence classes with functions??

Suppose a function f:A→B is given. Define a relation ~ on A as follows:

a_1~a_2 ⟺ f(a_1 )=f(a_2)

Since ~ is an equivalence relation, it induces a partition of A into equivalence classes. Describe these equivalence classes in each of the following cases. (R is the set of real numbers).

(c) A=B=R, f(x)=x^2

(d) A=B=R, f(x)=|x|

(f) A=R×R,B=R, f(x,y)=x+y

my approach:

(a): the equivalence class contains no elements because the square of different numbers is different.

(b) : the equivalence class contains the set of real numbers.

(c): don’t know about c

Is my approach anywhere near to correct? I am not so sure about the answers. please help!

performance – Parallel 3D Discrete Cosine Transformation Implementation in Matlab

I am trying to implement 3D Discrete Cosine Transformation calculation in Matlab with parallel computing parfor. The formula of 3D Discrete Cosine Transformation is as follows.

The experimental implementation

The experimental implementation of 3D Discrete Cosine Transformation function is DCT3D.

function X=DCT3D(x)

N1=size(x,1);
N2=size(x,2);
N3=size(x,3);
X=zeros(N1,N2,N3);

for k1=0:N1-1
    for k2=0:N2-1
        for k3=0:N3-1
            sumResult=0;            
            parfor n1=0:N1-1
                for n2=0:N2-1
                    for n3=0:N3-1
                        sumResult=sumResult+...
                            x(n1+1,n2+1,n3+1)*...
                            cos(pi/(2*N1)*(2*n1+1)*k1)*...
                            cos(pi/(2*N2)*(2*n2+1)*k2)*...
                            cos(pi/(2*N3)*(2*n3+1)*k3);
                    end
                end
            end
            X(k1+1,k2+1,k3+1)=8*sumResult*CalculateK(k1)*CalculateK(k2)*CalculateK(k3)/(N1*N2*N3);            
        end
    end
end

Moreover, the used function CalculateK:

function output = CalculateK(x)
  output = ones(size(x));
  output(x==0) = 1 / sqrt(2);

Test cases

%% Create test cells
testCellsSize = 10;

test = zeros(testCellsSize, testCellsSize, testCellsSize);
for x = 1:size(test, 1)
    for y = 1:size(test, 2)
        for z = 1:size(test, 3)
            test(x, y, z) = x * 100 + y * 10 + z;
        end
    end
end

%% Perform test
result = DCT3D(test);

%% Print output
for z = 1:size(result, 3)
    fprintf('3D DCT result: %d planen' , z);
    for x = 1:size(result, 1)
        for y = 1:size(result, 2)
            fprintf('%ft' , result(x, y, z));
        end
        fprintf('n');
    end
    fprintf('n');
end

The output of the test code above:

3D DCT result: 1 plane
1726.754760 -80.720722  -0.000000   -8.646042   -0.000000   -2.828427   0.000000    -1.143708   -0.000000   -0.320717   
-807.207224 0.000000    0.000000    -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   
-0.000000   -0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   
-86.460422  -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   0.000000    -0.000000   
-0.000000   -0.000000   0.000000    0.000000    -0.000000   0.000000    -0.000000   -0.000000   -0.000000   -0.000000   
-28.284271  -0.000000   0.000000    -0.000000   -0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    
0.000000    0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   0.000000    0.000000    
-11.437076  -0.000000   0.000000    0.000000    -0.000000   0.000000    -0.000000   -0.000000   0.000000    -0.000000   
-0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    
-3.207174   -0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    

3D DCT result: 2 plane
-8.072072   0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   
-0.000000   -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   0.000000    -0.000000   
-0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    
-0.000000   -0.000000   -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    -0.000000   
-0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   
-0.000000   0.000000    -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   -0.000000   -0.000000   
0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   0.000000    -0.000000   -0.000000   0.000000    
-0.000000   0.000000    0.000000    -0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   -0.000000   
0.000000    0.000000    0.000000    0.000000    0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    
0.000000    -0.000000   -0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    -0.000000   0.000000    

3D DCT result: 3 plane
-0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    0.000000    
0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    
-0.000000   -0.000000   0.000000    -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   0.000000    
0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   0.000000    
0.000000    0.000000    0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    -0.000000   -0.000000   
-0.000000   0.000000    0.000000    -0.000000   0.000000    0.000000    -0.000000   -0.000000   -0.000000   -0.000000   
-0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    
0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    0.000000    0.000000    
-0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    0.000000    
0.000000    0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    0.000000    

3D DCT result: 4 plane
-0.864604   -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    -0.000000   
-0.000000   -0.000000   -0.000000   0.000000    0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   
-0.000000   -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   0.000000    
-0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    -0.000000   -0.000000   -0.000000   
-0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   0.000000    0.000000    -0.000000   0.000000    
-0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   
-0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    
0.000000    -0.000000   0.000000    0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    
-0.000000   -0.000000   -0.000000   -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    0.000000    
0.000000    -0.000000   0.000000    -0.000000   -0.000000   -0.000000   -0.000000   0.000000    0.000000    -0.000000   

3D DCT result: 5 plane
-0.000000   0.000000    0.000000    0.000000    0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    
0.000000    -0.000000   -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    0.000000    
-0.000000   0.000000    -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   0.000000    0.000000    
0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   0.000000    
0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   
0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    
0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    0.000000    -0.000000   
-0.000000   -0.000000   0.000000    0.000000    -0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    
0.000000    0.000000    -0.000000   0.000000    0.000000    -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   
-0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   0.000000    -0.000000   

3D DCT result: 6 plane
-0.282843   -0.000000   -0.000000   -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   
-0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    
-0.000000   0.000000    0.000000    -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   
-0.000000   -0.000000   0.000000    -0.000000   -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    
-0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    
-0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    
0.000000    -0.000000   -0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    
0.000000    -0.000000   -0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   
0.000000    -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    
0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   

3D DCT result: 7 plane
0.000000    -0.000000   0.000000    -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   0.000000    -0.000000   
-0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    0.000000    -0.000000   0.000000    0.000000    
0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    0.000000    -0.000000   -0.000000   
-0.000000   -0.000000   -0.000000   0.000000    0.000000    -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   
0.000000    0.000000    -0.000000   0.000000    0.000000    0.000000    0.000000    0.000000    -0.000000   -0.000000   
-0.000000   0.000000    0.000000    -0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   -0.000000   
-0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   0.000000    -0.000000   -0.000000   
0.000000    -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    
0.000000    -0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    -0.000000   0.000000    -0.000000   
-0.000000   -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    0.000000    

3D DCT result: 8 plane
-0.114371   -0.000000   -0.000000   0.000000    -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   
-0.000000   -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    0.000000    0.000000    -0.000000   
-0.000000   -0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    
-0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    
0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    -0.000000   -0.000000   -0.000000   -0.000000   
-0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    
-0.000000   0.000000    0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    
-0.000000   0.000000    -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   -0.000000   
-0.000000   0.000000    0.000000    -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   0.000000    
0.000000    -0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    0.000000    

3D DCT result: 9 plane
-0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   
0.000000    0.000000    0.000000    0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    0.000000    
0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   -0.000000   
0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    -0.000000   
-0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   0.000000    -0.000000   
0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    
-0.000000   0.000000    0.000000    -0.000000   -0.000000   -0.000000   0.000000    -0.000000   0.000000    -0.000000   
0.000000    0.000000    0.000000    0.000000    0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   
-0.000000   -0.000000   0.000000    0.000000    -0.000000   0.000000    -0.000000   0.000000    -0.000000   -0.000000   
0.000000    -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    

3D DCT result: 10 plane
-0.032072   -0.000000   0.000000    -0.000000   -0.000000   0.000000    0.000000    0.000000    -0.000000   0.000000    
-0.000000   -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    -0.000000   -0.000000   0.000000    
0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    0.000000    0.000000    0.000000    
-0.000000   0.000000    0.000000    -0.000000   -0.000000   0.000000    0.000000    -0.000000   0.000000    0.000000    
-0.000000   -0.000000   -0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   -0.000000   0.000000    
-0.000000   -0.000000   0.000000    -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   -0.000000   
0.000000    0.000000    -0.000000   -0.000000   0.000000    -0.000000   0.000000    0.000000    -0.000000   -0.000000   
-0.000000   0.000000    -0.000000   -0.000000   0.000000    -0.000000   0.000000    -0.000000   -0.000000   0.000000    
0.000000    -0.000000   0.000000    -0.000000   0.000000    0.000000    0.000000    -0.000000   -0.000000   0.000000    
-0.000000   0.000000    -0.000000   0.000000    0.000000    0.000000    0.000000    0.000000    0.000000    0.000000    

If there is any possible improvement, please let me know.

discrete mathematics – Explicit quotient set

Let $X$ and $Y$ be finite sets. A funtion $f: X rightarrow Y$ defines the equivalence relation $sim_f $ on X, given by $ i sim j$ iff $f(i)=f(j)$

  1. Explicitly describe the quotient set $Bbb Z_{14}/sim_f $ in the case $f: Bbb Z_{14} rightarrow Bbb Z_{14}$ given by $f(x)=4x$
  2. Fixing $f: Xrightarrow Y$ How many functions $g: Xrightarrow Y$ are such that the equivalence relation $sim_f$ and $sim_g$ are the same?

Can someone help me please giving me a hint on how to proceed? I really don’t understand the question, specially the domain and codomain, how can I build a function with the quotient set, isn’t it infinite? But it says X is finite.

I’m in the first year of college, this is for the discrete math course, we’re using Susanna Epp’s book, it’s pretty basic, but this question was given as a supplementary exercise.

discrete mathematics – Estimating the Number of balls by maximizing the probability.

We have a bucket that has N balls we don’t know what N is. We randomly select n balls and mark them, if we put them back and again select n balls and see k of them are marked; How can we choose $hat{N}$ which is our estimate of the number of balls in the bucket so to maximize the probability of said event?

I tried to maximize $frac{binom{n}{k} binom{hat{N}}{n – k}}{binom{hat{N}}{n}}$ but couldn’t get anywhere.

parameters – Different FourierParameters for different dimensions of discrete Fourier transform

This question asks for a way to implement different FourierParameters for different dimensions of a discrete Fourier transform. The question received no answers, but a workaround was mentioned in the comments for the particular case considered by the OP.

I am faced with essentially the same problem, but in my case I would like to use non-integer values for FourierParameters, for which the aforementioned workaround doesn’t help. Given that several years have passed since the linked question was posed, I wonder if there might now be a better solution to the problem of assigning different FourierParameters to different dimensions of a Fourier transform?

Discrete Math(COUNTING NUMBERS)

#counting

Passwords of length 4 or 5 made up of letters where repeats are not allowed. count the possible numbers?

discrete mathematics – Distinguishable balls and urns

Suppose you have $n_1$ distinguishable red balls and $n_2$ indistinguishable yellow balls. A red ball can be distinguished from a yellow
ball.

How many ways are there to distribute these $n_1 + n_2$ balls into $m$ distinguishable urns so that there is no urn with only yellow balls?

Prove your answer.

discrete mathematics – Does this proof make sense? m | n if and only if m | n^2

Question and proof below. I’m a bit unsure if this proof make sense and want a second opinion. I could find a counter example I guess but I was trying to show that it is the case sometimes but not is not implied for all cases.

Either prove or find a counter-example to the statement $m :|: n$ if and only if $m :|: n^2$. (Here $n$ and $m$ are integers.)

Proof(Direct)
$m :|: n Leftrightarrow n=km$
$m :|: n^2 Leftrightarrow n^2=km$
$n^2 = m^2 k^2 = m (m k^2)$
For any Prime $P$ : $P :|: ab Rightarrow (P :|: a$ or $P :|: b)$
If $m$ is Prime $m :|: n^2 Rightarrow m :|: n$
$therefore :m :|: n notLeftrightarrow m :|: n^2$ $blacksquare$

dg.differential geometry – Discrete spectrum of Dirac operator

It is said that if we take the spacetime manifold to be a sphere $S^d$ of large volume so
that the spectrum of Dirac operator $$igamma^mu D_{mu}$$ is discrete.

For example at least for $d=4$, this has been discussed in p.2 of Witten paper from this Physics Letters B, Volume 117, Issue 5, 18 November 1982, Pages 324-328 Physics Letters B, 117(5), 324–328

My question is simple that

  1. why the spectrum of Dirac operator $igamma^mu D_{mu}$ is discrete? in any $d$? Can the spectrum of Dirac operator be continuum?

In what setup can it be discrete?

In what setup can it be continuum?

  1. under what conditions do we have zero eigenvalues for Dirac operator $igamma^mu D_{mu}$?

p.s. I know this is related to the Atiyah Singer index theorem. But I am trying to ask for an explicit user-friendly down-to-earth answer. I am not asking for a reference only. (The paper I quoted is a reference itself.)