Questions on discrete mathematics generally: "the study of mathematical structures that are fundamentally discrete rather than continuous"
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Replace a continuous probability distribution with a discrete one
When trying to fit data to a given normal distribution, one wants to minimize the sum of weighted error, which is commonly given by:
$$\text{Weighted Error}=\sum_x\left(\frac{x-\mu}{\sigma}\right)^2$$
...
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0answers
16 views
Implementing Explicit formulation of 1D wave equation in Matlab
so the theory is straight forward.
we have:
$\frac{d^2U}{(\Delta t)^2}=c^2 \frac{d^2U}{(\Delta x)^2}$
discretizing it gives:
$\frac{U(i+1,j)- 2U(i,j) + U(i-1,j)}{(\Delta t)^2} = c^2 ...
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2answers
24 views
Could graph theory aid in the understanding of comparison sorting algorithms?
I am interested in computing the exact number of comparisons that are needed to sort a list. See this wikipedia article.
Up to $n=15$, we know how many comparisons between elements one must make to ...
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1answer
43 views
Proof of divisibility by 2 and 3 if and only if divisible by 6
I can't find a way of proving that:
For integer a, a is divisible by 2 and divisible by 3 if and only if a is divisible by 6.
I’m not sure where to go from here. Any help would be great!
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1answer
26 views
diameter and radius of a regular graph
I am trying to find the radius and diameter of a regular graph $G$ with $d(v_i) < (n-1)/2$. I know for $d(v) \geq (n-1)/2$, $\rm{diam}(G) \leq 2$ and $\rm{radius}(G)=\rm{diam}(G).$ If we are not ...
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1answer
27 views
Number of circular combinations with no adjacent members.
Suppose I have to place 3 identical letters on a circular table which has 7 slots in such a way that no two letters are in consecutive slots. In how many ways can I do this?
Can this be generalized ...
2
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1answer
46 views
Solving the equation for $x$ in $Z_n$
How do you solve for x in the the $Z_n$ specified? For example, for the equation:
1) $3\odot x\oplus8\equiv1(\rm{mod} 10)$ or
2) $342\odot x\oplus 448\equiv73(\rm{mod}1003)$
How would you solve for ...
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2answers
35 views
Counting problem: Assigning students to dorm rooms
This was a question on a recent test and I was hoping for a conclusive answer and reasoning behind it.
A local university housing office has a problem. It has 11 students to squeeze into 3 dorm ...
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3answers
44 views
Prove that Statements forms are tautologies
Given variable statement forms $A$ and $B$. How to prove that if $(A\land B)$ is a tautology then $A$ and $B$ are tautologies too?.
Mi approach would be a proof by contradiction, something like: If ...
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2answers
34 views
Number of rectangles with odd side lengths on a chess board?
Given an 8x8 chess board, how do we find the total number of rectangles with odd side lengths?
(Both sides have odd length).
In general, what would be an elegant method to deal with problems like ...
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1answer
45 views
How to prove the identity $(n-k)! \sum _{i=0}^{n-k} \frac{(k+i-1)!}{i!} = \frac{n!}{k}$?
I am stuck in proving the following :
$$(n-k)! \sum _{i=0}^{n-k} \frac{(k+i-1)!}{i!} = \frac{n!}{k}$$
NOTE: I don't want any combinatorial proof. I think it is some algebraic manipulation.
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1answer
42 views
Proof by Induction solution not understood
Here is a question and solution but I don't understand what's happening after $m = m+1$.
How does $(3(m+1))!$ equal $(3m)!(3m+1)(3m+2)(3m+3)$? Should it not be $(3m+3)!$?
Same thing with the ...
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vote
1answer
29 views
True/False cardinal question in Discrete Maths
I would like some help on these claims. Thanks in advance!
True or false? If true, give a proof, if not, give a counterexample.
If $A$ is a set of functions $\Bbb N \rightarrow \Bbb N$ and for ...
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2answers
21 views
Solving Linear Congruence Equation: Finding the Nonnegative Integer Representation
I have a question for a part of the following problem: Solve the linear congruence 7x ≡ 6(mod 29)
I understand how to find the linear combination equality using ...
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0answers
19 views
Simplification of a dervived binary tree with n nodes [duplicate]
hi I need help with this problem how do simplify this equation and what are the steps and approaches to this problem
