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Feb
25
comment Decoding used in Algorithms
As it stands this question is rather light on context to expect a good answer. If you have the definitions then it should just be a case of working through them. If you don't, go back to the person who set the question and complain.
Feb
12
comment What is graph theory interpretation of this linear programming problem?
@Graphth, wow, thanks for the bounty! I was slightly surprised that the previous one wasn't awarded automatically, but it really wasn't necessary to do this.
Jan
2
comment Petersen graph prolems
I think 4 says that the largest independent set is of size 4.
Dec
15
comment Chinese Remainder Theorem result varies
I have sent a message through the contact page on calc2's site to notify the webmaster.
Dec
15
comment Chinese Remainder Theorem result varies
@HenningMakholm, I can reproduce the value given with calc 2.
Dec
15
comment Chinese Remainder Theorem result varies
@HenningMakholm has hit the nail on the head. I just checked the first one, but it is indeed giving a value larger than $95\cdot 97\cdot 99\cdot 99$ and congruent to $3267154$. It's not clear why that calculator doesn't reduce the result.
Dec
15
comment What is graph theory interpretation of this linear programming problem?
Actually on further thought I think it should be a universal quantifier rather than an existential one.
Dec
15
comment What is graph theory interpretation of this linear programming problem?
He uses $j$ in $y_i^{(j)}$ (or $[g_i \in C_j]$, using an Iverson bracket), and he constrains its value to be between 1 and $s$ inclusive, but he doesn't bind it as a parameter of anything. An alternative to my previous comment is to add an existential quantifier and define $$P_G = \{(x_1, \ldots, x_n) \mid \exists j : \sum_{i = 1}^n [g_i \in C_j] x_i \leq 1 \wedge \forall i : x_i \geq 0 \wedge 1 \leq j \leq s\}$$
Dec
15
comment What is graph theory interpretation of this linear programming problem?
Given the way he's using it, $j$ must be a parameter, so I think he should really define $$P_{G,C} = \{(x_1, \ldots, x_n) \mid \forall i:x_i \geq 0 \wedge \sum_{g_i \in C} x_i \leq 1\}$$ where $C$ is a clique of $G$.
Dec
1
comment Sunflower combinatorics
$l$, not $p$. If there are $(p-1)$ sets there can't be $p$ of them which form a sunflower. (Although $p=1$ is also a counterexample, because if there are $0$ sets you can't choose that single set to be a $1$-petal sunflower).
Dec
1
comment Sunflower combinatorics
What's the constraint on $l$? There must be one, because otherwise $l=1$ is a trivial counterexample.
Nov
27
comment Myhill Nerode - is language regular or not?
Are you asking for help in proving that your $L$ is regular? If so, you're in trouble, because it isn't.
Nov
17
comment What's wrong with an irregular digon?
I think you mean "vertices" rather than "edges".
Nov
4
comment How many coordinates are necessary to determine a sphere?
In how many dimensions is your sphere embedded?
Nov
3
comment Closed Formula Expression for Sum of Combinatorics
You don't need to worry about defining K precisely: $$\binom{a}{b} = 0$$ when $a,b \in \mathbb{N}$ and $b > a$.
Oct
10
comment Prove by induction $\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}$ for $n\ge1$
+1 I don't know why so many people want to expand the $(n+1)^3$ when this approach is far clearer.
Sep
10
comment Find an expression for the $n$-th derivative of $f(x)=e^{x^2}$
More generally, $$ \frac{d^n}{dx^n}e^{x^p} = \left(\sum_{k \in \mathbb{Z}}a_{n,k;p}x^{kp-n} \right) e^{x^p} $$ where the coefficients $a_{n,k;p}$ form the exponential Riordan array $A_p = [1, (1+x)^p - 1]$. This can be proved using some of the techniques from math.stackexchange.com/questions/18284 and is given as an exercise (although not in those terms) in Comtet's Advanced Combinatorics.
Aug
30
comment Calculate $\pi$ to an accuracy of 5 decimal places?
-1 Incomplete. You need to show that the terms smaller than 1E-5 don't sum to more than 1E-5. In fact, actually it's even more complicated than that, because the answer should be correctly rounded. You also potentially have loss of significance due to adding from the largest elements in the series rather than from the smallest.
Aug
11
comment What does asymptotically optimal mean for an algorithm?
@Geek, depends on what you average over. And worst case is what's relevant unless otherwise specified.
Jul
5
comment Quantization of angular momentum: is Dirac's proof wrong?
With step 2 as written (and BTW it would be easier to follow without overloading $j$ - it was the first element of the ket earlier, and now it's the second), isn't the existence of the eigenket $\lvert \beta, j\rangle$ true by definition of $j$ as "the maximum allowable value of $m$ for fixed $\beta$"?