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(Suggested reading. Cox's book quoted above and Serre's paper: $\Delta=b^2-4ac$)
-
While interesting, I think you are over complicating the problem. For a complete solution, and a way to explicitly write down the number of representations, see math.stackexchange.com/questions/44139/… – Eric Naslund Oct 19 '11 at 16:05... | {
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# coefficient of a polynomial
Show that the coefficient of $$[x^nu^m]$$ in the bivariate generating function $$\dfrac{1}{1-2x+x^2-ux^2}$$ is $${n+1\choose n-2m}.$$
I tried to do this by using the multinomial theorem (an extension of the binomial theorem), which basically states that for terms $$x_1,\cdots, x_r, n\in ... | {
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It might be more helpful to factorize the quadratic expression first (taking it as a variable in $$x$$). This way we can extract coefficient of $$x^n$$ ($$u$$ taken as a constant) and then coefficient of $$u^m$$ (in other words $$[x^n u^m]f(x,u)=[u^m]([x^n]f(x,u))$$. So, by factorization of denominator we arrive at $$\... | {
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# num_permutations: number of permutations for creating subsequences of k elements
A function to calculate the number of permutations for creating subsequences of k elements out of a sequence with n elements.
from mlxtend.math import num_permutations
## Overview
Permutations are selections of items from a collectio... | {
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import itertools
import sys
import time
from mlxtend.math import num_permutations
items = {1, 2, 3, 4, 5, 6, 7, 8}
max_iter = num_permutations(n=len(items), k=3,
with_replacement=False)
for idx, i in enumerate(itertools.permutations(items, r=3)):
# do some computation with itemset i
time.sleep(0.01)
sys.stdout.write(... | {
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Closed form of recurrent arithmetic series summation
Knowing that $$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$ how can I get closed form formula for $$\sum_{i=1}^n \sum_{j=1}^i j$$ or $$\sum_{i=1}^n \sum_{j=1}^i \sum_{k=1}^j k$$ or any x times neasted summation like above
• You won't be able to solve this just by using the ... | {
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We can write the last multiple sum as \begin{align*} \color{blue}{\sum_{i_1=1}^n\sum_{i_2=1}^{i_1}\sum_{i_3=1}^{i_2}i_3} &=\sum_{i_1=1}^n\sum_{i_2=1}^{i_1}\sum_{i_3=1}^{i_2}\sum_{i_4=1}^{i_3} 1\\ &=\sum_{1\leq i_4\leq i_3\leq i_2\leq i_1\leq n}1\tag{1}\\ &\,\,\color{blue}{=\binom{n+3}{4}}\tag{2} \end{align*} In (1) we ... | {
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Now, the difference between this and regular combinations is that in a regular combination every chosen number must be distinct; but if we have an ordered list $$\langle k, l, m\rangle$$ of the (not necessarily distinct) numbers we've chosen between $$1$$ and $$n$$ then we can turn this into an ordered list of not nece... | {
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# Number of possible combinations of x numbers that sum to y
I want to find out the number of possible combinations of $x$ numbers that sum to $y$. For example, I want to calculate all combination of 5 numbers, which their sum equals to 10.
An asymptotic approixmation is also useful. This question seems to be very cl... | {
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• is there a way to do this when the numbers being added are between two values, say 1-26.. for example, how many ways can 7 numbers being between 1 and 26 inclusive add up to 55 Dec 21, 2017 at 2:41
• @0TTT0 Integer solutions are essentially the problem of distributing identical units into distinct boxes, and these ar... | {
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• Just in case. Alessandro's reasoning is good. Another way to get confidence in the answer is to view the positive integer n as n indistinguishable balls, which supposed to be placed into k distinguishable boxes so that some boxes can remain empty. By "well known", I mean, for instance, Riordan, John. An Introduction ... | {
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Probability of no Ace but at least one King
So let me try to explain my reasoning:
Since we know there are no Ace, there are 48 cards left. Now, we know that we have at least one Kings which mean we have (exactly one or two or three or four kings) while choosing the remaining cards from 44 leftover cards.
$$\frac{{4... | {
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• What is the question? Probability of picking king before ace? – астон вілла олоф мэллбэрг Apr 13 '16 at 4:54
• It sounds like he's talking about a hand of five cards, but this should have been specified in the question. – browngreen Apr 13 '16 at 4:55
• In your first answer, you made a typo, the final term on the num... | {
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Since there are four aces in a deck of $52$ cards, the number of cards that are not aces is $52 - 4 = 48$. Therefore, the number of five card hands that contain no aces is $\binom{48}{5}$ since we must select five cards from the $48$ cards that are not aces.
Since there are four aces and four kings in a $52$ card deck... | {
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Find the probability that a five card hand contains no aces and no queens but at least one king.
Your approach of subtracting the number of hands that contain no aces, no queens, and no kings from the number of hands that contain no aces and no queens is correct. However, there are $52 - 3 \cdot 4 = 40$ cards that con... | {
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# Rank of a matrix
1. Mar 17, 2008
### proxyIP
1. The problem statement, all variables and given/known data
Find all possible values of rank(A) as a varies.
A=[1, 2, a]
[-2, 4a, 2]
[a, -2, 1]
A is 3x3, just merge the 3 row vectors.
3. The attempt at a solution
I have the solution but I create this thread to fin... | {
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3. Mar 18, 2008
### HallsofIvy
Staff Emeritus
The simplest way to determine the rank of a matrix is to "row-reduce". The rank is the number of rows that contain non-zero entries.
In the case you give
$$A= \left(\begin{array}{ccc}1 & 2 & a \\-2 & 4a & 2\\ a & -2 & 1\end{array}\right)$$
Add twice the first row to the s... | {
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# Functions continuous in each variable
Suppose we have a map $f:X \times Y \rightarrow Z$, where $X,Y$, and $Z$ are topological spaces. Are there any conditions on $X$,$Y$, and $Z$ that would allow one to determine that $F$ is continuous if it was known that it was continuous in each variable? It seems like there sho... | {
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Letting $S^1=\{z\in\mathbb{C}\colon\vert z\vert=1\}$ be the unit circle, consider the map $F\colon S^1\times I\to S^1$ given by $$F(e^{2\pi\theta i},s) = e^{2\pi\theta^si}$$ for $0 < \theta\le 1$ and $s\in I$. Then $F$ is continuous in each variable, $F(z,1)=z$ and $F(z,0)=1$. So, if you only required continuity in the... | {
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# Finding formula for nth partial sum
I need to find a formula for the nth partial sum of the series:
$2 + \frac 23 + \frac 29 + \frac {2}{27} + .... + \frac {2}{3^{n-1}} + ...$
then I need to use the formula to find the series' sum if the series converges.
The answer from the back of the book is (the formula):
$$... | {
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First, it may be helpful to factor out $2$ from your series; it will only clutter up your actual work. So really, you are out for a formula of the $n^{th}$ partial sum of $\sum_{n=0}^\infty \frac{1}{3^n}$. As the comment on your post points out, this is just a special instance of a geometric progression $\sum_{n=0}^\in... | {
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# Solving inequality $\frac{2x}{x-2}>1$
I'm trying to solve
$$\frac{2x}{x-2}>1$$
but I can't seem to get the correct answer. I'm doing something wrong but I don't know what; that is why I'm asking. This is what I've got:
$$\frac{2x}{x-2}>1$$
Since we do not know if the denominator is positive or negative, we can't... | {
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Another common approach is to reduce your inequality to something of the form $\displaystyle\frac{N(x)}{D(x)}\lesseqgtr 0$.
You then have to study the sign of the numerator and the sign of the denominator, i.e. solving some non fractional inequalities. Finally, you can study the sign of the ratio of $N(x)$ and $D(x)$ a... | {
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$$2x(x-2) > (x-2)^2 \iff 2x^2 - 4 > x^2 - 4x + 4 \iff (x-2)(x+2) > 0 .$$
We see that $(x-2)(x+2) > 0$ if and only if $x-2$ and $x+2$ have the same signs. Clearly $x = -2$ and $x = 2$ are important values. In the region $(-\infty,-2)$ we have $x\pm 2<0$. In the region $(-2,2)$ we have $x-2 < 0$ and $x+2 > 0$, in the re... | {
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# Math Help - Can any help with this one please
1. ## Can any help with this one please
Find the third degree polynomial whose graph is shown in the
figure.
choices are
f(x) = x^3 - x^2 - 2x + 2
f(x) = 1/4(x)^3 - 1/2(x)^2 - x+2
f(x) = 1/4(x)^3 -1/4(x)^2 + 2x + 2
f(x) = 1/2(x)^3 - 1/2(x)^2 - x + 2
2. The answer ... | {
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$\begin{array}{ccc}(a)\;f(x) & = &x^3 - x^2 - 2x + 2 \\
(b)\;f(x) &= &\frac{1}{4}x^3 - \frac{1}{2}x^2 - x + 2 \\
(c)\;f(x) & = & \frac{1}{4}x^3 -\frac{1}{4}x^2 + 2x + 2 \\
(d)\;f(x) & = & \frac{1}{2}x^3 - \frac{1}{2}x^2 - x + 2\end{array}$
The graph has an x-intercept at -2.
. . The function has a factor of: $(x + ... | {
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My Math Forum (http://mymathforum.com/math-forums.php)
IlanSherer March 23rd, 2018 08:32 PM
Statistics
Hello :)
The average weight of the cakes is 750 grams, and the standard deviation is 25 grams.
Each cake is divided into 6 slices of equal size, and each slice is added exactly 5 grams of cherry.
What is the aver... | {
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IlanSherer March 24th, 2018 02:08 PM
Quote:
Originally Posted by mathman (Post 590709) Variances add, so the standard deviation of one slice should be $\sqrt{104}$ unless the slicing is exact.
"Each cake is divided into 6 slices of equal size", so I think the slicing is exact.
All times are GMT -8. The time now is ... | {
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# Cover $\{1,2,...,100\}$ with minimum number of geometric progressions?
In another question, posted here by jordan, we are asked whether it is possible to cover the numbers $$\{1,2,\ldots,100\}$$ with $$20$$ geometric sequences of real numbers. Naturally, we would like to extend the question:
Problem: What is the mi... | {
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Switching from proving lower bounds to finding upper bounds, I played around with the list of progressions a bit and came up with another small improvement:
The $$16$$ progressions $$(1, 2, 4, 8, 16, 32, 64)\\ (3, 6, 12, 24, 48, 96)\\ (5, 10, 20, 40, 80)\\ (7, 14, 28, 56)\\ (9, 18, 36, 72)\\ (11, 22, 44, 88)\\ (13, 26... | {
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Length 5: 5,10,20,40,80
Length 4: 7,14,28,56$\qquad \ \$ 9,18,36,72$\qquad \ \$ 11,22,44,88
There are 35 numbers which are not in a geometric progression of length three or more:
29,31,37,39,41,43,47,51,53,55,57,58,59,61,62,65,67,69, 71,73,74,77,78,79,82,83,85,86,87,89,91,93,94,95,97
So you need 6 progressions to c... | {
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Why not 9,27,81 included? Not to spend three "squares" for a single triplet.
• Nice work. Maybe a computer program is the way to go. If I have time I'll write it. Still, it would be much more pleasing to see a clever proof! Aug 7, 2015 at 12:34
• It is interesting to point out that these progressions fit the scheme of... | {
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Update 2015-08-08: I just saw that my reasoning is a bit flawed; see my comment below.
• Brilliant. The bound shrinks yet again! Aug 7, 2015 at 13:10
• This could lead to another question: how many distinct geometric progressions (not proper subsequences of longer ones) of lenght $k$ are there in the set $\{1, ..., n\... | {
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# Disproving the claim that the numbers 1+2+4, 1+2+4+8, 1+2+4+8+16… alternate between prime and composite
I am working through an elementary number theory book and I have come across the following problem.
Show the following claims are wrong:
Claim 1: The sequence 1+2+4, 1+2+4+8, 1+2+4+8+16, ... is alternately prime... | {
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For Claim 1: Note that $2^{ab}-1$ is always divisible by $2^a-1$ (since $x-1$ divides $x^b-1$ in $\Bbb{Z}[x]$, now let $x=2^a$); hence $2^8-1, 2^9-1, 2^{10}-1$ are all composite.
• A more concrete proof of 2 is to take $m=7$, so that you have 6 consecutive composite numbers. – Ian Aug 20 '14 at 21:15
• $m!$ will be a ... | {
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(Which is just $(217,215)$, and indeed the pair are divisible by 7 and 5 respectively).
This argument extends to pairs of the form $(kn+1, kn-1)$ for $k>2$ (though the case for $k$ odd is obvious anyway, since both members will be even).
For the fun of it, I constructed the smallest counter-examples to both claims by... | {
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Since I was not sure that this solution $(203, 205)$ was as small as possible, I repeated the process for the pair of factors $(5, 11)$. Reusing the congruence $7\text{ or }23 \mod 30$ for the multiple of $11$, I found the smallest multiples $187$ and $143$.
At this point, my smallest counter-example was $(143, 145)$.... | {
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# What's $P$ and what's $Q$ in this classic proof of the irrationality of $\sqrt 2$?
In this proof extracted from the Wikipedia
A classic proof by contradiction from mathematics is the proof that the square root of 2 is irrational. If it were rational, it could be expressed as a fraction $a/b$ in lowest terms, where ... | {
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What we can do is to think of the assertion to be proven as a simple "atomic" proposition: $\,Q.\,$ Then $\,\lnot Q\,$ is the statement to the effect:
Suppose $\,\sqrt 2\,$ is not irrational. $\;$ Put differently, suppose $\,\sqrt 2\,$ is rational.$\quad(\lnot Q)$
The proof then proceeds, after having supposed $\,\ln... | {
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• Why was this down-voted? – JLA Jul 15 '13 at 3:24
• Thanks! I figured specifying what Q is and then inferring what not(Q) is makes more sense than just saying what not(Q) is. – JLA Jul 15 '13 at 18:44
• And you were correct, but during my time here I have noticed downvoting here is not about correct/incorrect answers... | {
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# How to prove that $\frac{\cos(x)-\cos(2x)}{\sin(x)+\sin(2x)} = \frac{1-\cos(x)}{\sin(x)}$ in a simpler way.
EDIT: Preferably a LHS = RHS proof, where you work on one side only then yield the other side.
My way is as follows:
Prove: $$\frac{\cos(x)-\cos(2x)}{\sin(x)+\sin(2x)} = \frac{1-\cos(x)}{\sin(x)}$$
I use th... | {
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$$\color{red}{\sin(-x)}+\sin x=0$$
and we're done by the double implications all through (and assuming the first expression is well defined, of course)
Check all the cancellations are correct and check all the trigonometric identities used above.
Another way: We begin with the left side, again: assuming it is well d... | {
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Here's a trigonographic demonstration:
• Not a proof to conceive in 2-2.5 minutes of the exam, but a wonderful one. Aug 3 '20 at 13:21
• I would also put in evidence the orange triangle in this copy of your image: drive.google.com/file/d/1e_wnuvau4j0Z12s8HNBKWMZjuSJCQBjB/… Aug 3 '20 at 13:29
• @enzotib: Thanks for the... | {
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which is true.
If you are familiar with complex number:
Let $$z$$ be a complex number satisfying $$|z|=1$$. We want to prove that $$z-\frac{1}{z^{2}}=k\left(1-\frac{1}{z}\right)$$ where $$k$$ is real. After some algebraic manipulation, we obtain $$k=z+1+\frac{1}{z}$$ which is always real. | {
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# How do I find the inductive definition of the set defined as $\{2n+3m+1|n,m\in\mathbb N\}$?
$\lbrace 2n+3m+1:n,m\in N\rbrace$ is the set of all positive integers except for $0$ and $2$. I need to know how to write its inductive definition.
This is part of the introduction on learning how to develop recursive functi... | {
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A solution could be found by starting with $1$. Then, if $n\in S$, you require $n+f(n)\in S$ too, where $f(n)$ is 2 for $n=1$ and $1$ otherwise. One way to do this is to make use of the step function $\frac{x}{|x|}$, which is $-1$ for negative input and $+1$ for positive input. Inserting a horizontal shift so that the ... | {
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How's this: 1 and 4 are in $S$; if $n$ is in $S$, then $n+2$ is in $S$.
-
The problem is that I need one definition that works for every number in the set. A small sample of the set is (1,3,4,5,6,7,8,9,10...). With n+2 starting at 1, I would only get the odd numbers and miss out on all the even numbers that are in the... | {
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• $1 \in S$;
• if $k \in S$, then $k+2 \in S$; and
• if $k \in S$, then $k+3 \in S$.
This is not the most efficient recursive description of $S$, but it is the one that most directly matches the definition that you’ve been given. After you prove that $S$ is in fact the set of all positive integers except $2$, you can ... | {
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# If people sharing the same birthday raise their hand, how many hands do you expect to see raised?
The following question is taken from an interview book assuming that no calculator is provided.
Question: There are $$25$$ people at a party. One person asks everybody to annouycne their birthday, and for anyone who ha... | {
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The probability that a given person raises his hand is the probability that someone else at the party has the same birthday as he does: $$1-\left({364\over365}\right)^{24}$$ The expected number of hands raised is $$25-25\left({364\over365}\right)^{24}\approx1.593,$$ by linearity of expectation.
• This is actually how ... | {
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Apparently the flaw in your formula is that while estimating the probability of a group of $$x$$ people with the same birthday, you neglect to account for the fact that in order for the size of the group to be exactly $$x,$$ nobody else among the $$25$$ people can have the same birthday as these $$x$$ people. The proba... | {
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To evaluate that sum (i.e., the correct one) without a calculator, I think the easiest method is to convert it back to the original problem and then observe that that problem is solved by evaluating $$25\left(1 - \left(\frac{364}{365}\right)^{24}\right),$$ so you now have a much simpler calculation. As a first approxim... | {
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# Probability of getting two pair in poker
I was looking at this website http://www.cwu.edu/~glasbys/POKER.HTM and I read the explanation for how to calculate the probability of getting a full house. To me, the logic basically looked like you figure out the number of possible ranks and multiply by the number of ways t... | {
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• I think the part that is tripping me up is why my solution for the two pair solution assumes an order, but the full house solution does not. In other words, why doesn't the full house solution count say, QQQKK differently from KKQQQ?
– Curt
Jan 14, 2014 at 6:05
• Because the formula given above for the full house doe... | {
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$$P = \frac{{13\choose2}{4\choose2}{4\choose2}{11\choose1}{4\choose1}}{52\choose5}$$
The difference between this solution and that for the full house is that there is more "symmetry" for the two pair: both pairs are groups of two. With the full house, one is a group of three, and the other is a group of two. Aces over... | {
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I find permutation more intuitive to follow for this kind of problems. For people like me:
We have five slots to fill: - - - - - . The first slot can take all 52 cards. The second slot can take only three cards so that they can make a pair. Similarly, the third and fourth slots can take 48 and 3 cards, respectively. T... | {
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• I came up with this method on my own while trying to do a problem for Combinatorics, you're the only person I found online talking about this method so thanks for that! But I don't understand the part where you divide by 2!*2!*2!, I feel like it has to do with the implicit separation of the cards into different cases... | {
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• Card 1: 52 cards
• Card 2: 3, since it must match the suite of Card 1
• Card 3: 48, since we can't match the suite of Card 1
• Card 4: 3, since we can't match the suite of Card 3
• Card 5: 44, since we can't match the suite of the other cards
This will give us all orderings of the form XXYYZ. We then notice two issu... | {
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# What are different ways to compute $\int_{0}^{+\infty}\frac{\cos x}{a^2+x^2}dx$?
I am interested to compute the following integral
$$I=\int_{0}^{+\infty}\frac{\cos x}{a^2+x^2}dx$$
where $a\in\mathbb{R}^+$. Let me explain my first idea. As the integrand is an even function of $x$ then
$$2I=\int_{-\infty}^{+\infty}... | {
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• a beautiful way: math.stackexchange.com/a/2480371/515527 – Zacky Apr 12 '18 at 16:28
• This is the most beautifully explained question I've seen on the math.stackexchange. $(+1)$ if I did not reach my daily voting limit... – Mr Pie Apr 12 '18 at 16:28
• – Zacky Apr 12 '18 at 16:33
• @zacky: Thanks for the link. How d... | {
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The first integral vanishes, the second after taking the limit $L \to \infty$, gives us $-2 \pi t$. In the same way we find the other integral with $(y-t)^2$.
So the two 'problematic' integrals give us:
$$-2 \pi \int_{-\infty}^\infty \frac{\cos at ~dt}{4+t^2}$$
Grouping the other terms we get:
$$\frac{3t}{(y+t)^2+1... | {
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Another way is to prove $$\int_\mathbb{R}\dfrac{a}{\pi}\dfrac{e^{ikx}}{a^2+x^2}=\exp -a|k|$$for $a>0$, by noting we're just trying to compute the characteristic function of a Cauchy distribution. The inversion theorem implies we need only check this characteristic function gives the right pdf. To prove $$\int_\mathbb{R... | {
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# Is my solution to the system of equations correct?
If I'm told that $T(\vec x)=A\vec x=\vec b$ and $A=\left[\begin{matrix}1&-3&2\\ 3&-8&8\\ 0&1&2\\ 1&0&8\\\end{matrix}\right]$ and that $\vec b=\left[\begin{matrix}1\\6\\3\\10\end{matrix}\right]$. I need to find some vector $\vec x$ that whose image under $T$ is $\vec... | {
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The solutions are then given by $\begin{bmatrix}10\\3\\0\end{bmatrix}+ x_3 \begin{bmatrix}-8\\-2\\0\end{bmatrix}$, $x_3 \in \mathbb{R}$.
- | {
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# Tell if $(\mathbb Z_6, \odot)$ is a semigroup and if the identity element belongs to it
Let the operation $\odot$ be defined in $\mathbb Z_6$ as follows:
$$a \odot b = a +4b+2$$
check if $(\mathbb Z_6, \odot)$ is a semigroup and if the identity element belongs to it.
This is the way I have solved this exercise:
... | {
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Since there can be only one identity element, there cannot be left identity. But we can check this anyway.
Existence of left identity $e$ would mean that for each $a$ we have $e\odot a=a$, i.e. $$e+4a+2=a\\e=4+3a.$$ The expression $4+3a$ has various values for various $a$'s (namely the values $1$ and $4$), so there is... | {
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# Two equivalent definitions Riemann integral
Definition 7.1 in Apostol's book gives the following definition for the Riemann integral. Let $P=\{x_0,x_1,\ldots,x_n\}$ be a partition of $[a,b]$ and $t_k$ any point in $[x_{k-1},x_k]$. Denote $S(P,f)=\sum_{k=1}^n f(t_k)(x_k-x_{k-1})$.
Definition 1: We say that $f$ is Ri... | {
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There exist points $\alpha_j, \beta_j \in I_j$ such that
$$m_j \leqslant f(\alpha_j) < m_j + \frac{\epsilon}{4(b-a)}, \\ M_j - \frac{\epsilon}{4(b-a)} < f(\beta_j) \leqslant M_j.$$
Multiplying by $(x_j - x_{j-1})$, summing over $j$ and using (*) we get
$$A - \epsilon/4<S(P_N,f, \{\alpha_j\})< L(P_N,f) + \epsilon/4, ... | {
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• Why is the number $N$ the same for all tags $\{t_j\}$? A priori, the rapidness of convergence depends on the tag. – user39756 Feb 5 '18 at 19:01
• @user39756: That's the way I'm reading the statement and that is what I'm proving. You won't find this definition 2 in any book that I've seen so I'm not sure how convolut... | {
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# Limit of infinite sequence from partial sum
I think there was a rule in Calculus that mentions this, but I am not sure.
If I need to find $$\lim_{n \to \infty} a_n$$ and I am only given the nth partial sum: $$S_n =\sum_{k=1}^{n} a_k = f(n)$$
To find $$\lim_{n \to \infty} a_n$$ I just have to find $$\lim_{n \to \in... | {
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$$\lim_{n\to \infty}S_n=\sum_{k=1}^{\infty} a_k=L \implies S_n-S_{n-1} =a_n \to 0$$
• So the limit of my sequence is always 0!? If partial sums exist? – glockm15 Sep 28 '18 at 14:25
• @StackUser With reference to the OP we have that $a_n=f(n)-f(n-1)$ therefore $$\lim_{n \to \infty} a_n=\lim_{n \to \infty} f(n)-f(n-1)$... | {
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# choosing poker hand with a specific card
How many ways can you choose at least one A from a deck of card in a poker hand?
I just wanted to double check my answer, would it be
C(52,5)- C(48,5)
Help is much appreciated,
• Yes, your calculation gives the number of hands with at least one Ace. – André Nicolas Apr 23... | {
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# Divergent subsequence of an unbounded sequence
Let $(a_n)$ be a sequence of real numbers that is unbounded above. Show that $\exists$ a subsequence $(a_{n_k})_{k \ge 1}$ such that $\lim_{k \rightarrow \infty} a_{n_k} = + \infty$.
Working so far:
Since $(a_n)$ is not bounded above, this means that there exists no $... | {
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EDIT:
So to summarize, the above construction produces a subsequence $(a_{n_k})_{k \ge 1}$ such that for all $n_k < n_{k+1}$, we have $a_{n_k} < a_{n_{k+1}}$ and $a_{n_k} > k$. To show that this subsequence diverges, just pick $k$ large enough so that $k \ge H$ and then set $N = n_k$.
construction part is correct and... | {
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# 8.3. Inverse iteration¶
Power iteration finds only the dominant eigenvalue. We next show that it can be adapted to find any eigenvalue, provided you start with a reasonably good estimate of it. Some simple linear algebra is all that is needed.
Theorem 8.3.1
Let $$\mathbf{A}$$ be an $$n\times n$$ matrix with eigenv... | {
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(8.3.1)$|\lambda_n-s| \ge \cdots \ge |\lambda_2-s| > |\lambda_1-s|.$
Then it follows that
$|\lambda_1-s|^{-1} > |\lambda_{2}-s|^{-1} \ge \cdots \ge |\lambda_n-s|^{-1},$
and power iteration on the matrix $$(\mathbf{A}-s\mathbf{I})^{-1}$$ converges to $$(\lambda_1-s)^{-1}$$, which is easily solved for $$\lambda_1$$ it... | {
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Function 8.3.3 : inviter
Shifted inverse iteration for the closest eigenvalue
1"""
2 inviter(A,s,numiter)
3
4Perform numiter inverse iterations with the matrix A and shift
5s, starting from a random vector. Returns a vector of
6eigenvalue estimates and the final eigenvector approximation.
7"""
8function inviter(... | {
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The observed linear convergence rate is found from the data.
@show observed_rate = err[22]/err[21];
observed_rate = err[22] / err[21] = 0.33326532173735623
We reorder the eigenvalues to enforce (8.3.1).
The sortperm function returns the index permutation needed to sort the given vector, rather than the sorted vect... | {
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s = β
x = y/y[1]
y = (A-s*I)\x
β = x[1]/y[1] + s
0.5612761406172997
Still not much apparent progress. However, in just a few more iterations the results are dramatically better.
for k in 1:4
s = β
x = y/y[1]
y = (A-s*I)\x
@show β = x[1]/y[1] + s
end
β = x[1] / y[1] + s = 0.5964312884753865
β = x[1] / y[1] + s = 0.... | {
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(a) $$s=1\quad$$ (b) $$s=2\quad$$ (c) $$s=1.6$$
3. ✍ Why is it a bad idea to use unshifted inverse iteration with the matrix $$\displaystyle \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$$? Does the shift $$s=-1$$ improve matters?
4. ✍ When the shift $$s$$ is very close to an eigenvalue of $$\mathbf{A}$$, the matrix ... | {
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surface(reshape(v,n,n))
6. ⌨ This problem explores the use of dynamic shifting to accelerate the inverse iteration.
(a) Modify Function 8.3.3 to change the value of the shift $$s$$ to be the most recently computed value in the vector $$\beta$$. Note that the matrix B must also change with each iteration, and the LU f... | {
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# Six-card hand with $2$ cards of each suit - did I calculate this correctly?
Suppose you have a deck of $$36$$ cards - $$3$$ different suits, $$12$$ cards per suit. If you draw a $$6$$-card hand, what is the chance of a hand with $$2$$ cards of each suit ($$2-2-2$$)?
I would do $$\frac{\left(\binom{12}2\right)^3}{\b... | {
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Choose the suit from which three cards will be drawn, choose three of the twelve cards of that suit, choose from which of the two remaining suits two cards will be drawn, choose two of the twelve cards of that suit. The remaining card must be drawn from the remaining suit. Choose one of the twelve cards of that suit. $... | {
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# Evaluate ⌊x⌋ + ⌊y⌋ + ⌊z⌋
#### anemone
##### MHB POTW Director
Staff member
If $x, y, z$ are real numbers such that $x+y+z=6$, $xy+yz+xz=9$, find the sum of all possible values of the expression $\lfloor x\rfloor+\lfloor y\rfloor+\lfloor z\rfloor$.
#### Klaas van Aarsen
##### MHB Seeker
Staff member
Re: Evaluate ⌊... | {
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"lm_q1_score": 0.9852713831229043,
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"openwebmath_score": 0.9394071698188782,
"ta... |
Edit. I like Serena beat me by just seconds!
#### Klaas van Aarsen
##### MHB Seeker
Staff member
Re: Evaluate ⌊x⌋+⌊y⌋+⌊z⌋
Edit. I like Serena beat me by just seconds!
We can only see that we posted in the same minute.
Let $x$ be the time I like Serena posted in minutes, and let $y$ be the time Opalg posted.
Then we... | {
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# Difference between revisions of "2019 AMC 8 Problems/Problem 3"
## Problem 3
Which of the following is the correct order of the fractions $\frac{15}{11},\frac{19}{15},$ and $\frac{17}{13},$ from least to greatest?
$\textbf{(A) }\frac{15}{11}< \frac{17}{13}< \frac{19}{15} \qquad\textbf{(B) }\frac{15}{11}< \frac{19}... | {
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# 2. Poisson Processes¶
## 2.1. Overview¶
Counting processes count the number of “arrivals” occurring by a given time (e.g., the number of visitors to a website, the number of customers arriving at a restaurant, etc.)
Counting processes become Poisson processes when the time interval between arrivals is IID and expo... | {
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As a function of $$t$$, the process $$N_t$$ is called a counting process.
The jump times $$(J_k)$$ are sometimes called arrival times and the intervals $$J_k - J_{k-1}$$ are called wait times or holding times.
### 2.2.2. Exponential Holding Times¶
A Poisson process is a counting process with independent exponential ... | {
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fig, ax = plt.subplots()
ax.plot(np.insert(Js, 0, 0)[:-1], Ys, 'o')
ax.hlines(Ys, np.insert(Js, 0, 0)[:-1], Js, label='$N_t$')
ax.vlines(Js[:-1], Ys[:-1], Ys[1:], alpha=0.25)
ax.set(xticks=[],
yticks=range(Ys.max()+1),
xlabel='time')
ax.grid(lw=0.2)
ax.legend(loc='lower right')
plt.show()
## 2.3. Stationary Indepe... | {
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Let $$\hat N_t$$ count the number of visits by time $$t$$, as shown in the next figure.
($$V_i = 1$$ is indicated by a vertical line at $$t_i = i h$$.)
fig, ax = plt.subplots()
np.random.seed(1)
T = 10
p = 0.25
B = np.random.uniform(size=T) < p
N = np.cumsum(B)
m = N[-1] # max of N
t_grid = np.arange(T)
t_ticks = [... | {
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In particular, increments are stationary (the distribution depends on $$t$$ but not $$s$$).
The approximation also illustrates independence of increments, since, in the approximation, increments depend on separate subsets of $$(V_i)$$.
## 2.4. Uniqueness¶
What other counting processes have stationary independent inc... | {
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"lm_q1q2_score": 0.8451226167552972,
"lm_q2_score": 0.8596637559030338,
"openwebmath_perplexity": 719.5748317107717,
"openwebmath_score": 0.9237521886825562,
"tags": ... |
$M_{t'} - M_t = N_{s+t'} - N_{s + t} \sim \text{Poisson}((t' - t) \lambda)$
Hence $$(M_t)$$ has stationary increments and, using the relation $$M_{t'} - M_t = N_{s+t'} - N_{s + t}$$ again, the increments are independent as well.
We conclude that $$(N_{s+t} - N_s)_{t \geq 0}$$ is indeed a Poisson process independent o... | {
"domain": "quantecon.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9830850852465429,
"lm_q1q2_score": 0.8451226167552972,
"lm_q2_score": 0.8596637559030338,
"openwebmath_perplexity": 719.5748317107717,
"openwebmath_score": 0.9237521886825562,
"tags": ... |
fig, ax = plt.subplots()
ax.plot(vals, [poisson(v, T * λ) for v in vals],
marker='o', label='poisson')
ax.plot(vals, [np.mean(sample==v) for v in vals],
marker='o', label='empirical')
ax.legend(fontsize=12)
plt.show()
Here is one solution. It shows that the approximation is good when $$n$$ is large and $$\theta$$ i... | {
"domain": "quantecon.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9830850852465429,
"lm_q1q2_score": 0.8451226167552972,
"lm_q2_score": 0.8596637559030338,
"openwebmath_perplexity": 719.5748317107717,
"openwebmath_score": 0.9237521886825562,
"tags": ... |
# How can I calculate the limit without using the L'Hopital's rule
I need to prove this limit without using the L'Hopital's rule:
$$\lim_{x\to 0} \frac{(1+a\,x)^{1/4} - (1+b\,x)^{1/4}}{x} = \frac{a-b}{4}$$
How can I do it in Mathematica?
-
I have decided not to close this question or migrate, because it has a sensi... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9830850892111574,
"lm_q1q2_score": 0.8451226166289374,
"lm_q2_score": 0.8596637523076225,
"openwebmath_perplexity": 1602.559644676707,
"openwebmath_score": 0.5287978053092957,
"tag... |
(Sqrt[1 + a x] - Sqrt[1 + b x]) (Sqrt[1 + a x] + Sqrt[1 + b x]) // Expand
a x - b x
We have multiplied the interesting term twice by 2 and now we take the denominator (i.e. x) of the initial exppession. Thus we can see from the last output that $$\lim_{x\rightarrow 0 }\frac{(1 + a x)^{\frac{1}{4}} - (1 + b x)^{\frac... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9830850892111574,
"lm_q1q2_score": 0.8451226166289374,
"lm_q2_score": 0.8596637523076225,
"openwebmath_perplexity": 1602.559644676707,
"openwebmath_score": 0.5287978053092957,
"tag... |
## Another elementary proof
Proof á la Euclid. [Prompted by @Artes's remark about ancient Greeks.]
Let $BX$ be given with $OA=OB=1$ and $AX = u>0$. Let $OP$ have been drawn perpendicular to $BX$ with $OC=1$. Let $BX$ have been bisected at $Q$. With center $Q$ and distance $QX$, let circle $XPB$ have been described. L... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9830850892111574,
"lm_q1q2_score": 0.8451226166289374,
"lm_q2_score": 0.8596637523076225,
"openwebmath_perplexity": 1602.559644676707,
"openwebmath_score": 0.5287978053092957,
"tag... |
The limit in the lemma is the derivative of $x^{1/4}$ at $x = 1$.
## Code dump for figure
labels[u_] := MapThread[Text, Transpose[{
{"O", {0, 0}, {0, 1.5}},
{"A", {1, 0}, {0, 1.5}},
{"B", {-1, 0}, {-1.8, 1.5}},
{"C", {0, 1}, {1.5, 0}},
{"X", {1 + u, 0}, {-1.8, 1.5}},
{"P", {0, Sqrt[1 + u]}, {1.5, -1}},
{"Q", {u/2, 0}... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9830850892111574,
"lm_q1q2_score": 0.8451226166289374,
"lm_q2_score": 0.8596637523076225,
"openwebmath_perplexity": 1602.559644676707,
"openwebmath_score": 0.5287978053092957,
"tag... |
# Expected value of the absolute value of the difference between two independent uniform random variables? [closed]
I am trying to calculate the expected value of the absolute value of the difference between two independent uniform random variables.
Let $X_1\sim\operatorname{Uniform}(0, 2)$ and $X_2\sim\operatorname{... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9830850897067342,
"lm_q1q2_score": 0.8451226135203715,
"lm_q2_score": 0.8596637487122112,
"openwebmath_perplexity": 266.50579165653255,
"openwebmath_score": 1.0000100135803223,
"ta... |
# Probability of Rolling Exactly 4 of a kind on 6 Dice
Probability of 3 of a kind with 7 dice
I've looked at the link above - along with several others - and dusted off my combinatorics notebook to try to figure out the answer to the problem: If you roll six 6-sided dice, what is the probability of rolling exactly fo... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9830850867332735,
"lm_q1q2_score": 0.8451226127314928,
"lm_q2_score": 0.8596637505099168,
"openwebmath_perplexity": 86.36491287129135,
"openwebmath_score": 0.8042610883712769,
"tag... |
• So, I may have gotten ahead of myself and realized that I hadn't accounted for the arrangements of the dice in the second method. For fun, I drew out the 15 different ways that the dice could be arranged so that I could have a visual to help me see what's going on. Now I'm wondering, is this a situation where order i... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9830850867332735,
"lm_q1q2_score": 0.8451226127314928,
"lm_q2_score": 0.8596637505099168,
"openwebmath_perplexity": 86.36491287129135,
"openwebmath_score": 0.8042610883712769,
"tag... |
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