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# Water Balloon Fight
#### veronica1999
##### Member
7 people are having a water ballon fight. At the same time, each of the 7 people throws a water balloon at one of the other 6 people, chosen at random. What is the probability that there are 2 people who throw the balloon at each other?
I am having a lot of troubl... | {
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And so on...
#### Opalg
##### MHB Oldtimer
Staff member
This is not an easy one.
I thought so too, until I saw that veronica1999 was actually well on the way to a solution.
7 people are having a water ballon fight. At the same time, each of the 7 people throws a water balloon at one of the other 6 people, chosen at ... | {
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#### chisigma
##### Well-known member
May be it is comfortable to proceed step by step as follows. If n is the number of players, then the possible plays are $n\ (n-1)$. Let's start...
a) n=2. We call the players A and B and the 2 possible plays are...
AB BA
... and the only 'good pair' is AB + BA, so that P=1...
... | {
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You have started in exactly the right way. There are $6^7$ possible outcomes, and there are $21\times6^5$ ways of getting a pair. But, as you realised, that includes some double counting. You can eliminate that by using the inclusion-exclusion method. First, you need to subtract the number of ways of getting two pairs ... | {
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It took me a long time to figure out that I had to divide by 2 and 6 for the 2 pairs and 3pairs.
I hope my understanding is correct.
Once again, thanks!!!!! | {
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# Integrate sinx*cox* using double angle
1. Aug 5, 2013
### Jude075
1. The problem statement, all variables and given/known data
∫ Sinx cos x dx
2. Relevant equations
3. The attempt at a solution
If you integrate it using substitution, you get -cos2(x)/2but if you use double angle formula to rewrite the problem, ... | {
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# Homework Help: Determine the sum of the given series:
1. Jul 22, 2014
### ybhathena
1. The problem statement, all variables and given/known data
Sum starting from n=1 to infinity for the expression, (3/4^(n-2))
What the solutions manual has done is multiply the numerator and the denominator by 4.
12/(4^(n-1))
... | {
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10. Jul 22, 2014
### gopher_p
They multiplied both the numerator and denominator of the term inside the sum by 4, which is the same as multiplying the whole term inside the sum by 4/4=1. Multiplying by 1 doesn't change anything, so there is nothing to "undo". It goes $$\sum\limits_{n=1}^\infty\frac{3}{4^{n-2}}=\sum\l... | {
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# Factor $a^4-3a^2-2ab+1-b^2$
In order to factor $a^4-3a^2-2ab+1-b^2$, I find that $a=1, b=-1$ makes the value of the expression 0. Thus, I assume $b=-a$.
I rewrite the expression on the assumption as: $$a^4-3a^2-2ab+1-b^2$$ $$=a^4-3a^2+2a^2+1-a^2$$ $$=a^4-2a^2+1$$ $$=(a^2-1)^2$$
Then I insert $a+b$, which is anothe... | {
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Here is probably the most simple method to factor it.
Note that $$a^2-b^2=(a-b)(a+b)$$So $$a^4-3a^2-2ab+1-b^2=a^4-2a^2+1-a^2-2ab-b^2=(a^2-1)^2-(a+b)^2$$ So we can factor it into $$(a^2+a+b-1)(a^2-a-b-1)$$ | {
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# Statistics of 7 game playoff series
Background: a friend of mine makes a hobby (as I imagine many do) of trying to predict hockey playoff outcomes. He tries to guess the winning team in each matchup, and the number of games needed to win (for anyone unfamiliar with NHL hockey a series is decided by a best of 7). His... | {
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Possible 5 game series (8):
LWWWW WLLLL
WLWWW LWLLL
WWLWW LLWLL
WWWLW LLLWL
Possible 6 game series (20):
LLWWWW WWLLLL
LWLWWW WLWLLL
LWWLWW WLLWLL
LWWWLW WLLLWL
WLLWWW LWWLLL
WLWLWW LWLWLL
WLWWLW LWLLWL
WWLLWW LLWWLL
WWLWLW LLWLWL
WWWLLW LLLWWL
Possible 7 game series (40):
LLLWWWW WWWLLLL
LLWLWWW WWLWLLL
LLWWLWW WWL... | {
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For a team to win [the series] in game N, they must have won exactly 3 of the first N-1 games. For game seven, there are $\binom{6}{3} = 20$ ways to do that. There are 2 possible outcomes for game seven, and 20 possible combinations of wins for each of the teams that can win, so 40 possible outcomes. For an N-game seri... | {
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Think of the way a game plays out with points going to each player as a path. Each path that leads to a win has a probability given by the product of the probabilities for the number of wins and losses. The following image tries to illustrate an example.
The pattern for the coefficients is given by (i+N-1) choose (i) ... | {
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An alternate way to look at would be a binomial distribution: You need x=3 (exactly 3 successes) in n = 6 (trails) , so if the probability of winning a game is .5 (both teams equally likely) , binomial would say: P(x=3) = 6C3 * (.5)^3 * (.5)^3 = .3125 This would mean there is 31.25% chance of going to a 7 game series. ... | {
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## Description
example
pade(f,var) returns the third-order Padé approximant of the expression f at var = 0. For details, see Padé Approximant.
If you do not specify var, then pade uses the default variable determined by symvar(f,1).
example
pade(f,var,a) returns the third-order Padé approximant of expression f at ... | {
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syms x
p11 = subs(p11,x,vpa(1/5))
p11 =
x
p11 =
0.2
Find the approximation error by subtracting p11 from the actual value of tan(1/5).
y = tan(vpa(1/5));
error = y - p11
error =
0.0027100355086724833213582716475345
Increase the accuracy of the Padé approximant by increasing the order using Order. Set Order to [2 2],... | {
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Expansion point, specified as a number, or a symbolic number, variable, function, or expression. The expansion point cannot depend on the expansion variable. You also can specify the expansion point as a Name,Value pair argument. If you specify the expansion point both ways, then the Name,Value pair argument takes prec... | {
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collapse all
By default, pade approximates the function f(x) using the standard form of the Padé approximant of order [mn] around x = x0 which is
$\frac{{a}_{0}+{a}_{1}\left(x-{x}_{0}\right)+...+{a}_{m}{\left(x-{x}_{0}\right)}^{m}}{1+{b}_{1}\left(x-{x}_{0}\right)+...+{b}_{n}{\left(x-{x}_{0}\right)}^{n}}.$
When Order... | {
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# The floor and ceiling functions are equal for integer numbers
The exercise
Proof the following directly: Let $x \in \mathbb{R}$. If $x \in \mathbb{Z}$, then $\lfloor x \rfloor = \lceil x \rceil$.
My problem
I mostly fail completely on the formal part of any kind of proof (here a direct proof). This means I've mos... | {
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• Thanks! The second part ($x = \max A$) seems to be too easy. Should this be enough? Beside that have you any hints about my first question (how to get the formally correct form or how to improve it)? Sep 14 '14 at 17:08
• Your approach is fine. As for $x=\max A$, what I wrote should be enough. Sep 14 '14 at 17:13
Th... | {
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• Thanks! It looks like you're using a proof by contradiction for $x = \max\{k \leq x\}$. Is this okay as a part of a direct proof? Beside that have you any hints about my first question (how to get the formally correct form or how to improve it)? Sep 14 '14 at 17:11
• The proof by contradiction is a bit of an over-kil... | {
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# How to find the reaction force from the joint from a bar supported to a wall where a block is resting?
#### Chemist116
The problem is as follows:
The figure from below shows a system at equilibrium. The bar is homogeneous and uniform and has a weight of $14\,N$ and the block which is labeled $Q$ has a weight equal... | {
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$\sum F_x = 0 \implies T\sin(53) = R_x \implies R_x = 14 \text{ N}$
$\sum F_y = 0 \implies T\cos(53) + R_y - (28-T) - 14 \implies R_y = 14 \text{ N}$
$R = 14\sqrt{2} \text{ N}$
Can you please include a FBD for this thing?. I'm getting tangled with the direction of the vectors acting in the object.
My major confusion... | {
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# We choose $5$ numbers from $1$ to $100$. We order them by value. What is the expected difference between the second and the third?
I came across this peculiar problem.
We choose $$5$$ numbers from $$1$$ to $$100$$ (with repetition). We order them in decreasing order by value. What is the expected difference between... | {
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Doing the random choices in a different order shouldn't change anything, so we still get the sought-for answer. One more reformulation:
We have $$n+1$$ chairs arranged in a circle. Select six chairs at random. Start from a random chair among the selected six chairs and walk clockwise. Count how many chairs appear betw... | {
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For $$n=20, 25, 30$$, this is $$\{\frac{53333}{16000},\frac{65104}{15625},\frac{809999}{162000}\}$$.
@Roman notes that this is $$\frac{n^4-1}{6n^3}$$, so by engineer's induction, the answer is $$\frac{33333333}{2000000}$$, or precisely $$16.6666665$$.
A proof of this fact is left as an exercise to the reader.
• I wr... | {
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$$D = (100)^5.$$
$$N$$ will be calculated by going through all of the various summations, and attaching a weight to them. In the double summation below, the weight is $$(b-a)$$ and the number of entries represents the number of ways having exactly 2 numbers $$\leq a$$ and 1 number $$\geq b$$.
$$\binom{5}{3} \times \s... | {
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# Can a nowhere continuous function have a connected graph?
After noticing that function $$f: \mathbb R\rightarrow \mathbb R$$ $$f(x) = \left\{\begin{array}{l} \sin\frac{1}{x} & \text{for }x\neq 0 \\ 0 &\text{for }x=0 \end{array}\right.$$ has a graph that is a connected set, despite the function not being continuous a... | {
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Abstract:
Cauchy discovered before 1821 that a function satisfying the equation $$f(x)+f(y)=f(x+y)$$ is either continuous or totally discontinuous. After Hamel showed the existence of a discontinuous function, many mathematicians have concerned themselves with problems arising from the study of such functions. However... | {
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• Very nice examples. They show how easy is to break some continuity with additional dimensions while retaining enough of it to maintain connectedness of the graph. So I've added another case to the question, $f: \mathbb R \rightarrow \mathbb R$, in which such methods won't work. Do you think a function in this case is... | {
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So, then, here is the construction. Fix an enumeration $$(U_\alpha,V_\alpha)_{\alpha<\mathfrak{c}}$$ of all pairs of open subsets of $$\mathbb{R}^2$$. By a transfinite recursion of length $$\mathfrak{c}$$ we define values of a function $$f:\mathbb{R}\to\mathbb{R}$$. At the $$\alpha$$th step, we add a new value of $$f$$... | {
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If this is not possible, then $$U_\alpha$$ and $$V_\alpha$$ must partition $$A\times\mathbb{R}$$ where $$A\subseteq\mathbb{R}$$ is the set of points where we have not yet defined $$f$$. Since $$\mathbb{R}$$ is connected, this means we can partition $$A$$ into sets $$B$$ and $$C$$ (both open in $$A$$) such that $$U_\alp... | {
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At the end of this construction we will have a partial function $$\mathbb{R}\to\mathbb{R}$$ such that by construction, its graph is not separated by any pair of open subsets of $$\mathbb{R}^2$$, and the same is guaranteed to hold for any extension of our function. Extending to a total function, we get a total function ... | {
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• Without the last sentence, you might have ended up with $f(x)=0$ ;) – Hagen von Eitzen Jun 26 at 6:49
• @HagenvonEitzen: There's not actually a need to do anything additional to make the function nowhere continuous. The construction directly implies that that every Jordan curve in the plane will intersect the graph, ... | {
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Great question, and I don't have an answer for you, but I've got some small thoughts:
By summing up weighted and displaced copies of $$f$$, you can get discontinuities at many places. For instance, you could write $$F(x) = \sum_{n \in \Bbb Z} \frac{f(x-n)}{1+n^2}$$ That'll have an $$f$$-like discontinuity at every int... | {
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But I have a feeling that sliding over to the uncountable-set territory is going to be a lot harder.
• This is a good way to get functions which are discontinuous at many points, but are the graph of $F$ and and the graph of $G$ still connected? – Adam Chalumeau Jun 25 at 14:46
• Well...they could only be disconnected... | {
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# How does variable ordering in expressions work when creating functions from an equation?
I'm having a really hard time understanding some aspects of functions, i've tried looking around on Khan academy and haven't quite found something to answer my question, i'm sure i'm overlooking something stupid but wanted to kn... | {
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• Given that $x_1\succ x_2\succ x_3\succ\dots$, comparing a term $ax_1^{\alpha_1}x_2^{\alpha_2}x_3^{\alpha_3}\cdots$ to $bx_1^{\beta_1}x_2^{\beta_2}x_3^{\beta_3}\cdots$ the term to be written first is determined by comparing powers of $x_1$. Whichever has the higher power of $x_1$ should be written first. If equal then... | {
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• If it's just personal preference and my answer is still correct then why would khan academy say it's wrong? May 23, 2016 at 1:29
• @zack6849 Poorly coded homework submission software will search for a specific way or ways of writing an answer. Essentially running a "is inputstring*=*answer" check. Better software wil... | {
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# Probability of drawing 5 cards from a deck of 52 that will have the same suit?
A standard deck of cards has 52 members consisting of 4 suits each with 13 members. Five cards are dealt from the randomly mixed deck. What is the probability that all cards are the same suit?
EDIT: How I went about it before posting thi... | {
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• How many ways are there of drawing five cards from the same suit? Note that there are four suits, so the number of ways of drawing five cards from the same suit is four times, say, the number of ways of drawing five clubs. And how many ways are there of drawing five cards in general? – joeb Apr 5 '17 at 1:05
• When y... | {
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We know that the first card determines the suit, the second must be one of the $12$ out of $51$ remaining cards with that suit, the third must be one of the $11$ out of $50$ remaining cards with the same suit, etc. The reason the the numerator and denominator keep decrementing is because each time we choose a card, it ... | {
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Now multiply with 4 as we have 4 suits.
$$4 \times \frac{\binom {13}5}{\binom{52}{5}}$$
• I went ahead and tried your solution (answer: 33/16660), and for some reason it's still saying it's wrong. I do believe this solution (along with others) are correct, but maybe the TA's setup the question wrong online... Will up... | {
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# Name of the union of a set with its holes
Given an arbitrary connected and compact set $S$ with holes in it, is there a name for the simply connected set formed by the union of $S$ and its holes?
For example, let $S = \{x\in \mathbb{R}^n\ |\ 0 < a \leq||x||^2_2 \leq b\}$, its hole is $H = \{x\in \mathbb{R}^n\ |\ ||... | {
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• What exactly is meant by a hole ? – Shailesh May 14 '16 at 2:49
• I tried to edit my question to address yours. This is how I would describe the process of making a set with holes: take a simply connected set, remove subsets in its interior so that the resulting set is still connected. This subsets are now the so cal... | {
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I assume you mean a compact set in $\mathbb{R}^n$ (or some metric space) so that "bounded" makes sense. Also, that in your clarification you probably meant to say something like the following (but see the NOTE at the end)
"take a COMPACT simply connected set $T$ and remove ( open) subsets of the interior to get a comp... | {
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• Yes, the set I described in that comment should be compact. Besides, your definition is very nice. I guess that I could also define a hole $H_i$ of $S$ as a bounded connected subset of the complement of $S$. Then $H(S)$ is the union of all such sets. Would this be equivalent to the definition I originally posted? – j... | {
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# $A \times B$ is an open set in $\Bbb R^2 \implies A$ and $B$ are both open in $\Bbb R$; $A,B \neq \emptyset$
I am studying Analysis on my own. Reading The Elements of Real Analysis by Bartle. Came across the above problem and I came up with the following solution but am very unsure about it. Would be very grateful i... | {
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• I think a lot of your doubt comes from the fact your grasp of both the cartesian product and the product topology is still shaky.A big flaw in your arguement is you simply assumed A and B are not the empty set-there's nothing in your arguement that shows that. By definition, if either A or B is the empty set, then A ... | {
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• As for the hint, for every z,x in A, your definition is clearly an open ball in A. But you made z arbitrary in A. You defined Bx, b={(z1,z2) | ∣∣∣∣(z1,z2)−(x,b)∣∣∣∣<rx, b} is contained in A×B. We know for each ordered pair in A X B, the first coordinate lies in A by definition of the Cartesian product. So now let z=z... | {
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I think the proof is, logically, just fine.
Stylistically, I would make a few changes, none of them major or all that significant:
First, it seems weird to use $x$ and $b$ as your chosen elements of $A$ and $B$. I'd use $a$ and $b$.
Second, the notation is subscript heavy. For example, I'd probably just write $r$ in... | {
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# Do there exist pro-$p$ groups with finite quotients of non $p$ power order?
We define a pro-$p$ group to be a projective (i.e. inverse) limit of $p$-groups.
My question is exactly as stated in the title:
If a subgroup $H$ of a pro-$p$ group $G$ has finite quotient, must $|G/H|$ be some power of $p$?
If we restric... | {
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The problem: indices are only defined for closed subgroups.
If $|A|=p^nm$ with $(p,m)=1$ and select $a\in A$ such that $|a|\big| m$, which is possible by Cauchy's theorem. Name the projection map $\pi:G\to G/N\cong A$ (first isomorphism theorem) is surjective, we may select a lift $\stackrel{\sim}{a}\in G$, and by def... | {
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• Doesn't the universal property go the other way? – Alexander Jan 30 '15 at 4:31
• But doesn't this only work for normal subgroups? – Alexander Jan 30 '15 at 5:14
• @Alexander You're assuming that $A$ is the image of $G$ under a homomorphism, and kernels of homomorphisms are always normal... And even so, the open norm... | {
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# Analyzing the graphs of Greatest Integer Functions
Gold Member
## Homework Statement
Consider $u\left(x\right)=2\left[\frac{-x}{4}\right]$
(a) Find the length of the individual line segments of the function,
(b) Find the positive vertical separation between line segments.
## Homework Equations
The output of Grea... | {
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If that is the case then the function value is piecewise constant and steps down at each multiple of 4, ie at ....,-8, -4, 0, 4, 8, ....
So each line segment is horizontal and extends for the period that x takes to increase by 4. So its length is 4.
How much does it step down by each time? Well $\lfloor\frac{-x}4\rflo... | {
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# Is there an “and” symbol?
I was wondering if there was a symbol for "and"? For example, I want to say something like
$\therefore a = b \, \text{and} \, c = d$
Then can I replace "and" with a symbol?
Thanks!
• Though your mathematical argument may be much more legible if you write "Therefore, $a=b$ and $c=d$." (U... | {
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In your case, you can express $a = b\;\; \text{and}\;\; c = d$ as: $$a = b\, \land\, c = d$$
• For formatting in latex/mathjax: $P \land Q$ is given by P \land Q or by P \wedge Q. – amWhy Aug 17 '13 at 15:05
• +1 How does this symbol wedge spell in speaking, Amy? Can I spell it as "$P$ and $Q$"? – mrs Aug 18 '13 at 5:... | {
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# Taylor Series for $e^x$ where $x = 1$, estimating the Error
I'm trying to calculate $e$ to a certain number of digits. The Maclaurin Series expansion of $\displaystyle e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$. When $x = 1$ we can approximate the value of $e$ by evaluating $\displaystyle\sum_{n=0}^m\frac1{n!}$. How ca... | {
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$$g(x) = x f(x) = \sum_{n=0}^{+\infty} \frac{x^{2n+2}}{(2n+1)!}$$
Now, define
$$h(x) = g'(x) = \sum_{n=0}^{+\infty} \frac{(2n+2) x^{2n+1}}{(2n+1)!}$$
So, we can apply the ideas of estimating the error of a Taylor series towards the calculation of $h(1)$, where
$$h(x) = \sinh x + x \cosh x = \frac{1}{2} \left( e^x -... | {
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But note that $$\sum_{n=m+1}^\infty\frac{1}{n!}<\sum_{n=1}^\infty\frac{1}{(m+1)^n}=\frac{1}{m}.$$ What is an acceptable(if the idea is just to get some bound, not necessarily a good bound) error bound.
-
Improving @Integral's bound:
\begin{align} \sum_{n=m+1}^\infty \frac{1}{n!} &= \frac{1}{(m+1)!} \left( \frac11 + ... | {
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# Symmetric matrices and eigenvalues
If the eigenvalues of a symmetric matrix $A$ are greater than 0, show that $v^{\top}Av > 0$ for every $v \ne 0$
I am trying to prove this as follows:
If $v$ is an eigenvector of $A$, then $Av = \lambda v$ and we can rewrite the expression as follows: \begin{align*} v^\top A v = v... | {
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Let $U$ be the orthogonal matrix that diagonalizes $A$, such that $UU^T=U^TU=I$ and $(U^TAU)_{ij}=\lambda_i\delta_{ij}$, where $\lambda_i$ is the $i$'th eigenvalue of $A$ and $\delta_{ij}$ is the Kronecker Delta.
Using tensor notation, with summation implicit over repeated indices, we have
\begin{align} (v^TAv)_{in}&... | {
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# Picard group under base change for algebraically closed fields
Let $k \subset K$ be an extension of algebraically closed fields of characteristic $0$ (e.g. $\overline{\mathbb{Q}} \subset \mathbb{C}$).
Let X be a smooth variety over $k$. Then is the natural map $$\mathrm{Pic}(X)\,/\,n \to \mathrm{Pic}(X_K)\,/\,n$$ a... | {
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Proposition: Let $X/k$ be a qcqs scheme with $k$ separably closed. Let $K/k$ be a separable extension with $K$ separably closed. Let $n \in \mathbf{N}$ be a positive integer that is prime to the characteristic of $k$. Then the natural map $$\frac{\operatorname{Pic}(X)}{n\operatorname{Pic}(X)} \to \frac{\operatorname{Pi... | {
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To prove this, write $K$ as a direct limit of smooth $k$-algebras. Since cohomology commutes with direct limits for qcqs schemes, if a (cohomological) Brauer class $\alpha$ in $H^2(X,\mathbf{G}_m)$ dies in $H^2(X_K,\mathbf{G}_m)$ it must die in $H^2(X_R, \mathbf{G}_m)$ for some smooth $k$-algebra $R$. By smoothness and... | {
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# Does a positive semidefinite matrix always have a non-negative trace?
A simple question:
If $A$ is a positive semidefinite matrix ($A\succeq 0)$, does it imply that $\mbox{Tr}(A)\geq 0$, where the $\mbox{Tr}(\cdot)$ denotes the trace.
If not, any counter-example? Thanks.
• Suppose a matrix has a negative diagonal... | {
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# Derivatives of $\frac{\csc x}{e^{-x}}$ and $\ln\left(\frac{3x^2}{\sqrt{3+x^2}}\right)$
I have tried to mainly ask thoughtful conceptual questions here, but now I am reduced to asking for help on a specific problem that I have been wrestling with for over an hour.
Disclaimer: I am not a lazy student trying to get fr... | {
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Answer key says choice (2): $$\frac{x^2+6}{x^3+3x}$$
-
Please, try to make the title of your questions more informative. Note that our maths renderer MathJax also works in titles. E.g., Why does $a\le b$ imply $a+c\le b+c$? is much more useful for other users than A question about inequality. For more information on c... | {
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Then using the standard derivative rules for logs we get: $g'(x)=\frac{2}{x}-\frac{x}{x^2+3}=\frac{x^2+6}{x^3+3x}$, as required.
-
Thank you for the replies.
For #1, it looks like the question meant to say $csc(2x)$ Looks like I solved the original question correctly, and I redid the revised problem to match the giv... | {
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## Monday, January 11, 2010
### SICP Exercise 1.16: Fast Exponentiation
From SICP section 1.2.4 Exponentiation
Exercise 1.16 asks us to design an exponentiation procedure that evolves an iterative process using successive squaring and uses a logarithmic number of steps. A hint tells us to use the observation that (b... | {
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Here are a few sample runs of the procedure:
> (fast-expt 2 2)4> (fast-expt 2 3)8> (fast-expt 2 10)1024> (fast-expt 3 3)27> (fast-expt 5 5)3125
You can try running the procedure with large values of n to verify that it really is fast.
Related:
For links to all of the SICP lecture notes and exercises that I've done s... | {
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# Construct a function on a bounded interval on $\Bbb{R}$ which is continuous everywhere but differentiable only at irrationals.
Initially, I constructed a function with the help of famous Thomae's function, but later I found it to be wrong. Firstly, I did-
Let $f:[0,1]\to\Bbb{R}$ be the Thomae's function defied by $... | {
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Case 1: $x \notin D$. Let $n_0 \in \mathbb{N}$. For $y$ in some neighborhood of $x$, $[x,y]$ does not contain $a_1,...,a_{n_0}$. Then $\frac{f(y)-f(x)}{y-x} = \sum\limits_{a_n > \max(x,y)} \frac{1}{2^n} + \sum\limits_{a_n < \min(x,y)} \frac{-1}{2^n} + \sum\limits_{a_n \in [x,y]} \frac{|y-a_n|-|x-a_n|}{2^n}$. And $\big|... | {
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• +1 I haven't checked the details, but I'm pretty sure this works, and I'm surprised I'd forgotten its history. Schwarz gave an example of such a function in 1873, using roughly the same method (which is basically an application of Hankel's 1870 method of "condensation of singularities"). My next 3 comments quote from... | {
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Yes, such a function exists. In fact, Zahorski [1] (see also p. 103 here) proved that if $E$ can be written as $E = G \cup Z,$ where $G$ is a $\mathcal{G}_{\delta}$ set and $Z$ is a $\mathcal{G}_{\delta \sigma}$ Lebesgue measure zero set, then there exists a continuous function $f: {\mathbb R} \rightarrow {\mathbb R}$ ... | {
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Continuous functions are differentiable on a measurable set?
Set of zeroes of the derivative of a pathological function
Characterization of sets of differentiability
Points of differentiability of $f(x) = \sum\limits_{n : q_n < x} c_n$
Monotone Function, Derivative Limit Bounded, Differentiable – 2 | {
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# If $0.9999\ldots=1$, then why is $\lim_{n\to\infty}\frac{\tan(89.[n\,\text{"$9$"s}]^\circ)}{\tan(89.[(n-1)\;\text{"$9$"s}]^\circ)}$ not equal to $10$?
If $$0.9999\ldots=1$$, then why is this limit not equal to $$10$$? $$L = \lim_{n \to \infty} \frac{\tan(89.\overbrace{9999...}^{\text{n times}} \space ^\circ)}{\tan(8... | {
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$$\lim_{x\to a}f(x)$$ you don't care about $$f(a)$$ (which could be defined or undefined), but only about $$f(x)$$ for $$x\ne a$$.
We cannot simply assume that both of the limits $$\lim_{n \rightarrow \infty}f(x)$$ and $$\lim_{n \rightarrow \infty}g(x)$$ exist, so the formula $$\lim_{n \rightarrow \infty} \dfrac{f(x)}{... | {
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# Prove that $6|2n^3+3n^2+n$
My attempt at it: $\displaystyle 2n^3+3n^2+n= n(n+1)(2n+1) = 6\sum_nn^2$ This however reduces to proving the summation result by induction, which I am trying to avoid as it provides little insight.
-
if you want to use induction then use the fact that for your polynomial $p(n)=n(n+1)(2n+1... | {
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-
Yet another way to look at it is as follows:
First, as several others have already noted, $n(n+1)$ is divisible by $2$, so we just need to check for divisibility by $3$. Now, $n \equiv 0,1, \text{ or } 2 (\text{mod } 3)$. In the case of $n \equiv 0 (\text{mod } 3)$, the problem is trivial. In the case of $n \equiv ... | {
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Here is a general collection of results that can be applied in cases like this.
Define the combinatorial polynomial of degree $k$: $C_k(n)=\binom{n}{k}$. Let $L_k$ be the set of integral linear combinations of combinatorial polynomials of degree at most $k$. That is, $$f\in L_k \Leftrightarrow f=\sum\limits_{j=0}^ka_k... | {
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Proof: Suppose $P$ is a polynomial of degree $k$ and $P:\{m,m+1,m+2,\dots,m+k\}\mapsto\mathbb{Z}$. The Claim above assures that there is a $Q\in L_k$ so that $Q(j)=P(m+j)$ for $j=0,1,2,\dots,k$. Since a polynomial of degree $k$ is determined by its values at $k+1$ points, we must have that $P(j)=Q(j-m):\mathbb{Z}\mapst... | {
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If $n$ is even then 2 divides $n$ and $n+1$ will be odd so $n+1$ can be $3k+2$ or $3k$ where $k$ is some integer.
If $3k+1 = n+1$ as it would make $n$ itself a multiple of 6 as our assumption that $n$ is even
So if $n+1$ is $3k$ it can be divided by 3 so no problem. But if $n+1 = 3k+2$ then $2n+1$ will be $2(3k+1) +1... | {
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# Sum to infinity of a Series
Problem: Find the sum of the series $$x+2x^2+3x^3+4x^4+... = \sum\limits_{n=1}^\infty nx^n$$
My solution is as follows: Let P be the above series. $$P=x(1+2x+3x^2+4x^3+...) =x(1+x+x^2+x^3+x^4+...x+2x^2+3x^3+4x^4) =x\left(\frac{1}{1-x}+P\right)$$
Then by some trivial algebraic manipulati... | {
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Let $$P_N=\sum\limits_{n=1}^Nnx^n$$ Note that $$P=\lim\limits_{N\to\infty}P_N$$
We have $$P_N=x\cdot \sum\limits_{n=1}^N nx^{n-1} = x \cdot \left(\underbrace{1+x+...+x^{N-1}}+\underbrace{x+2x^2+...+(N-1)x^{N-1}}\right) =$$ $$= x \cdot \left( \sum\limits_{n=0}^{N-1} x^n + \sum\limits_{n=1}^{N-1} nx^n\right) = x \cdot \... | {
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# Two forces, two angles; Net force = ?
1. Feb 3, 2009
### MaximaMan
Greetings everyone! Thanks in advance for the help!!!
1. The problem statement, all variables and given/known data
Two parts to the problem. For both parts it asks for the magnitude of the net force and the direction of angle from the X axis.
Pa... | {
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3. Feb 3, 2009
### v0id19
I agree with your magnitude in Part A, but like Hannisch said, I don't think your angle is right. Remember that $$arctan(\frac{opposite}{adjacent})=\theta$$. Also, it is imperative that you define the reference point for your angle! Is it 24o above the horizontal, below the horizontal, or is... | {
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# Prove $\bigcap (A_n \cup B_n) \supset (\bigcap A_n)\cup(\bigcap B_n)$
Prove that:
$$\bigcap (A_n \cup B_n) \supset \left(\bigcap A_n\right) \cup \left(\bigcap B_n\right)$$
Also find an example when there is no equality. Ok, so
$$\bigcap A_n := \{a \mid \forall Y \in A: a \in Y\}$$
$$\bigcap B_n := \{b \mid \fora... | {
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# How do I write a trig function that includes inverses in terms of another variable?
It's been awhile since I've used trig and I feel stupid asking this question lol but here goes:
Given: $z = \tan(\arcsin(x))$
Question: How do I write something like that in terms of $x$?
Thanks! And sorry for my dumb question.
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If you want something like $\csc(\arctan x)$, do something similar. We could let the hypotenuse be $1$. But it may be easier to let the leg adjacent to $\angle A$ be $1$, and the leg opposite $\angle A$ be $x$. Then the hypotenuse is $\sqrt{1+x^2}$, and now you can read off the cosecant from the picture.
-
Ok thank yo... | {
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## Symbolic Summation
Symbolic Math Toolbox™ provides two functions for calculating sums:
• sum finds the sum of elements of symbolic vectors and matrices. Unlike the MATLAB® sum, the symbolic sum function does not work on multidimensional arrays. For details, follow the MATLAB sum page.
• symsum finds the sum of a ... | {
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"lm_q1q2_score": 0.8441800509474202,
"lm_q2_score": 0.8558511524823263,
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"openwebmath_score": 0.7671182751655579,
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Show that the outputs are equal by using isAlways. The isAlways function returns logical 1 (true), meaning that the outputs are equal.
isAlways(S_sum == S_symsum)
ans =
logical
1
For further computations, clear the assumptions.
assume(x, 'clear') | {
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# simple chance problem
• December 17th 2006, 12:33 PM
emptyglasses
simple chance problem
I need help with figuring out what method to use to figure out this problem..
Here is the problem.
A fair coin was flipped 10 times; compute the chance that the number of heads is less than 4.
• December 17th 2006, 01:02 PM
Sor... | {
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"lm_q1_score": 0.986363166722906,
"lm_q1q2_score": 0.8441800493793435,
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"openwebmath_score": 0.8105195164680481,
"tags... |
My approah is to find,
$\int_0^{40} \frac{100!}{40!\cdot \Gamma (41-x)}.5^{100} dx$
Where, $s>0$
$\Gamma (s)=\int_0^{\infty} e^{-t}t^{s-1}dt$ (infamous Gamma function).
I know it looks ugly but programs can do this easily. Taking integral by Simpon's rule. Thus, all you need to do is tell your computer to find this in... | {
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Originally Posted by CaptainBlack
Are you sure that is what you use? What about:
$\int_0^{40} \frac{100!}{\Gamma(x+1)\cdot \Gamma (101-x)}(0.5)^{100} dx$?
RonL
It is a nice excecise to compare the normal approximation, the above
approximation and the exact value. These are 0.0179, 0.0227 and 0.0176
respectivly.
Tho... | {
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# Irrationality of $\sqrt{15}$
Could someone verify the correctness of this proof for the irrationality of $\sqrt{15}$?
Assume $\sqrt{15}\in\mathbf{Q}$, then $\sqrt{15}=\frac{p}{q}$ with $p,q\in\mathbf{Z}$ ($q\ne0$ and $\gcd(p,q)=1$). $\implies 15q^2=p^2 \implies 15\mid p^2 \implies 3\mid p^2 \implies 3\mid p$ (Eucli... | {
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"id": null,
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"openwebmath_perplexity": 147.06837061294925,
"openwebmath_score": 0.9311505556106567,
"ta... |
# A question in Fourier analysis
I recently came across this problem:
Let $T=\mathbb R /2 \pi \mathbb Z$ the circle, with its proabability Haar measure $\mu$. Any integrable function $f : T \rightarrow \mathbb C$ has Fourier coefficients $c_n(f)= \int_T f(t) e^{-int} \frac{dt}{2\pi}$, and we define $$||f||_{A(T)} = \... | {
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"lm_q1_score": 0.9863631619124992,
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"openwebmath_score": 0.9676953554153442,
"tag... |
Also, has this problem, or something related, been studied ? Does $\alpha(E)$ have a name ?
-
Out of curiosity: is it important for your purposes that $f\leq 0$ on $E$, rather than $f=0$ on $E$? – Yemon Choi Jun 22 '13 at 3:44
@Yemon: Yes, kind of. I mean, I am mostly interested in upper bound on $\alpha(E)$, so any ... | {
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"openwebmath_score": 0.9676953554153442,
"tag... |
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