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Your first approach is correct if you permit $4$ of a kind and a pair as well as $4$ of a kind and two different values for the other two dice. In your second approach, you do indeed need to account for the different arrangements of the dice. There are $\binom{6}{4} = 15$ orders in which the same number could occur on ... | {
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# Limit involved in derivative of exponential function
1. Feb 8, 2009
### symbolipoint
Can a convenient value for a be found without resorting to substituting numerical values for h in this expression?
EDIT: I am trying to indicate, "as h approaches zero".
EDIT: neither of the formattings worked; hopefully someone ... | {
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Setting h= 1/n, h goes to 0 as n goes to infinity and that says that eh is approximately equal to 1+ h so that eh-1 is approximately equal to h and (eh-1)/h goes to 1 as h goes to 0.
Of course, for the general case, use the fact that ah= eh ln(a).
6. Feb 11, 2009
### lurflurf
so we define exp(x) to be a function su... | {
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# What's the probability of a an outcome after N trials, if you stop trying once you're “successful”?
This follows on from this question about being hit by a bus.
In this question, there is a 1/1000 chance of being hit and the question was about the probability of being hit if you cross the road 1000 time.
I wondere... | {
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At that point, what difference does it make whether I attempt another $950$ crossings or
never try to cross that street again?
An obvious difference is that if I continue crossing, I could be hit by a bus again.
I could be hit several times before the $1000$th attempt to cross.
This will make a difference to the expect... | {
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They are explicitly the same situation: the probability of being hit at least once before finishing 1000 crossings. Whether or not you stop searching after the event occurs should make no difference to the probability of the event occurring. There’s no foreknowledge of future trials in the scenario.
On the other hand,... | {
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# fluid concentration, workhour problem
• April 17th 2008, 05:17 AM
agus hendro
fluid concentration, workhour problem
I need help for these problems :
1. 1000 kgs of chemical is stored in a container. The chemical is made up of 99% water and 1% oil. Some water is evaporated from the chemical until the water content i... | {
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I think if I got the clue for the second problem, this new problem will be easy for me. Please help......
• April 20th 2008, 11:19 AM
Soroban
Hello, agus hendro!
I think I've solved it . . .
Quote:
Teams $X$ and $Y$ work separately on two different projects.
On sunny days, team $X$ can complete the work in 12 days,
... | {
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Therefore, there were $\boxed{10\text{ rainy days}}$
• April 22nd 2008, 05:54 AM
agus hendro
still don't understand
Thank you for the answer. But please explain why do you add s/12 and r/24 and how come the result of the addition equal to 1?
(Wondering) | {
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# Uniform Convergence Preserves Continuity
Briefly, the definitions of point-wise convergence (PWC) and uniform convergence (UC) for a sequence of functions $f_n:[a,b]\to\mathbb{R}$ in my mind are recorded as
\begin{align*} &\text{Point Wise Convergent on $[a,b]$} \iff \\ &\forall x\in [a,b]\,\forall\epsilon\gt0\,\ex... | {
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\begin{align*} &\forall\epsilon_1\gt0\,\exists \delta_1=\Delta_1(\epsilon_1,x_0,n)\gt0,\,|x-x_0|<\delta_1 \implies |f_n(x)-f_n(x_0)|<\epsilon_1 \\ \\ &\forall x\in [a,b]\,\forall\epsilon_2\gt0\,\exists N=\mathcal{N}(\epsilon_2)\gt0, n\ge N \implies |f_n(x)-f(x)|<\epsilon_2. \end{align*} Finally, choosing any $\epsilon_... | {
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SOLVED Hp Prime - CAS inconsistent derivatives of sin, cos, tan
05-29-2014, 10:28 AM (This post was last modified: 05-29-2014 01:19 PM by CR Haeger.)
Post: #1
CR Haeger Member Posts: 275 Joined: Dec 2013
SOLVED Hp Prime - CAS inconsistent derivatives of sin, cos, tan
In CAS I notice now that taking first derivative of... | {
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Regards,
Chris
Correct
$$(g\circ f)'=g'\circ f*f'\\ \sin { (a*x)'=cos(a*x)*a }$$
While converting from degree to radian you get the conversion factor
$$a=\frac { \pi }{ 180 }$$
05-29-2014, 01:22 PM (This post was last modified: 05-29-2014 01:26 PM by CR Haeger.)
Post: #4
CR Haeger Member Posts: 275 Joined: Dec 2013
RE... | {
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Start typing.
05-31-2014, 01:29 AM
Post: #8
rprosperi Senior Member Posts: 5,066 Joined: Dec 2013
RE: SOLVED Hp Prime - CAS inconsistent derivatives of sin, cos, tan
(05-29-2014 01:53 PM)CR Haeger Wrote: ...Id prefer not to retype all this back in.
(05-29-2014 02:11 PM)Michael de Estrada Wrote: Start typing.
Yup! ... | {
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# transient and steady-state response of first order system
Considering this general 1st order transfer function
$$H(z) = \frac{b_0 + b_1z^{-1}}{1-az^{-1}}$$
How to find (analytically) the transient and steady-state responses?
With steady-state response I mean response to a sinusoid. I'm particularly interested in ... | {
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with
$$A=\frac{1}{1-e^{-j\omega_0}}=H(e^{j\omega_0})\tag{6}$$
The inverse $$\mathcal{Z}$$-transform of $$(5)$$ is
$$\tilde{y}[n]=H(e^{j\omega_0})e^{j\omega_0n}u[n]+H^*(e^{j\omega_0})u[n]\tag{7}$$
The response to the step-modulated sinusoid $$(1)$$ is easily obtained from $$(7)$$ by taking its imaginary part:
\begi... | {
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• ah, so I should have focused on the Fourier transform instead of the z-transform. Is there any chance that you can explain/derive or provided a reference to the added pole expressions for $H_{-1}$ and $H_{1}$? Can one say that the delta function in the $H_{1}$ expression change the magnitude response but not the phas... | {
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# Thread: help with probability!!
1. ## help with probability!!
I need help with the following questions, i attempted to solve them, and i also put my answers here too.
When you are answering them, please show all your steps as i am kinda lost in this field.
1. suppose S = {1,2,3} and P({1,2}) = 1/3 and P({2,3}) = 2... | {
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5. Suppose 55% of students are female and 45% are male.
44% of females have long hair and 15% of males have long hair.
What is the probability that a random student will either be female or have long hair (or both)?
chance of female: 55%
chance of long hair: (55%*44%) + (45%*15%) = 0.2420 + 0.0675 = 0.3095 = 30.95%
cha... | {
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# Basic induction proof that all natural numbers can be written in the form $2a + 3b$
The theorem given is:
If $n$ is a natural number then $n$ can be written in the form $2a + 3b$ for some integers $a$ and $b$.
How would I prove this by induction? I've had a go at proving this but I don't know if my technique is so... | {
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Your proof is perfectly good. You can use whatever integer $b$ you like as the base case, to prove some proposition $P(n)$ is true for all integers $n\ge b$. $0$ and $1$ are both very common base cases. You can also use induction in the other direction (e.g., for negative numbers) to prove that every integer below $b$ ... | {
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1. ## function
The function g is defined by g(x)=(x+1)/(x-2), x is not equal to k and m, find the values of k and m
of course, I know that x is definitely not equal to 2, but how about the other one?
2. ## Re: function
That's it -- x cannot equal 2. Are you sure you wrote the function correctly?
3. ## Re: function... | {
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So if the only restriction on g(x) is that it is a real-valued function, x = 2 is the only number necessarily outside its domain. Of course the definition of g may exclude any other real number.
yeah, that's it. The answer given is k=2, m=5. I can't understand
8. ## Re: function
Originally Posted by Trefoil2727
yeah,... | {
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# Name of random variable that's +1 or -1 with equal probability?
Is there a name for this distribution: $$P(X = 1) = P(X = -1) = 0.5?$$
I'm currently writing $$2X-1$$ where $$X \sim \text{Ber}(0.5)$$.
• You can call it “uniform distribution on $\{-1, 1\}$”. Sep 27, 2021 at 11:44
• You can also refer to it as the si... | {
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# Prove existence of unique fixed point
Let $f(x)$ be a strictly decreasing function on $\mathbb{R}$ with $|f(x)-f(y)|<|x-y|$ whenever $x\neq y$. Set $x_{n+1}=f(x_n)$. Show that the sequence $\{x_n\}$ converges to the root of $x=f(x)$.
Note that the condition is weaker than what is required in the contracting mapping... | {
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• I rehashed my answer to remove a lot of unnecessary details and clarify the main point. – 6005 Sep 7 '16 at 10:01
• Thanks! Great answer, and the intuition too. It's called a cobweb plot, as I recall. – Zhang Edison Sep 7 '16 at 10:08
• @ZhangEdison yeah kinda reminded me of the microeco lesson I had. – Vim Sep 9 '16... | {
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# Probability question about panda births and statistical tests
I am self-learning statistics, and I have a question about how to do the following problem:
There are two species of panda bear, A and B. Both are equally common in the wild and live in the same places. A veterinarian has a new genetic test that can iden... | {
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From the problem information we have
\begin{align} \mathrm{prob}(\mathrm{test} = A | \mathrm{species} = A, \mathcal{I}) &= 0.8 \\ \mathrm{prob}(\mathrm{test} = B | \mathrm{species} = B, \mathcal{I}) &= 0.65 \\ \mathrm{prob}(\mathrm{species} = A | \mathcal{I}) &= 0.5 \\ \mathrm{prob}(\mathrm{species} = B | \mathcal{I})... | {
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# What is the shortest distance between skew lines in N dimensions?
I have two skew lines in $\mathbb{R}^N$ ($N > 2$) defined as $\vec{x} = \vec{x}_A + \vec{d}_A t$ and $\vec{x} = \vec{x}_B + \vec{d}_B s$ ($t, s \in \mathbb{R}$). Now, I'd like to calculate the shortest distance between those lines. In 3D, this seems t... | {
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Hint: Even easier, use $d(s,t)^2$.
• Thanks for your answer! I though of minimizing $d(s, t)^2$ too. This would give me two points $P_A$ and $P_B$ that lie on either line, and the vector connecting the points $\vec{v}_{AB}$ would give the direction. However, I still have a problem with the uniqueness of $\vec{v}_{AB}$... | {
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Compared to cross product solution for the 3-dimensional case, you'll note that projecting onto the cross product of $$u$$ and $$v$$ is equivalent to the projection operator $$1 - \hat{u} \hat{u}^T - \hat{v} \hat{v}^T /(\hat{v}^T \hat{v})$$ when $$u$$ and $$v$$ are independent. More generally, if $$P$$ is an orthogonal... | {
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# Let $n$ be a 6-digit number, perfect square and perfect cube. If $n-6$ is not even or a multiple of 3, find $n$
Let $n$ be a 6-digit number, perfect square and perfect cube. If $n-6$ is not even or a multiple of 3, find $n$.
My try
Playing with the first ten perfect squares and cubes I ended with:
The last digit ... | {
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# Trying to prove that $\lim_{n\rightarrow\infty}(\frac{\Gamma '(n+1)}{n!} -\log(n))=0$
In my attempt to prove that $\Gamma'(1)=-\gamma$, I've reduced the problem to proving that $\lim_{n\rightarrow\infty}(\frac{\Gamma '(n+1)}{n!} -\log(n))=0$.
Where $\gamma$ is the Euler-Mascheroni constant, and $\log$ denotes the n... | {
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-
@C.Williamson $\Gamma(n+2) = (n+1) \Gamma(n+1)$, hence $\log\Gamma(n+2) = \log\Gamma(n+1) + \log(n+1)$. – Sasha Aug 16 '12 at 23:30
Yeah, I deleted my comment after seeing my mistake. I like the answer! – C. Williamson Aug 16 '12 at 23:32
Wow, this is a really clever use of the MVT, thanks. – Thoth Aug 16 '12 at 2... | {
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$$-\frac{n}{{n + 1}}{H_{n + 1}} = \int_0^n {{{\left( {1 - \frac{m}{n}} \right)}^n}} \log mdm - \frac{n}{{n + 1}}\log n$$
So we get
$$\frac{n}{{n + 1}}\left( {\log n - {H_{n + 1}}} \right) = \int_0^n {{{\left( {1 - \frac{m}{n}} \right)}^n}} \log mdm$$
Now, by letting $n\to \infty$, we get
\eqalign{ & \mathop {\lim }... | {
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Can a finite set with a non prime number of elements be a field?
I understand that as typically defined (using modular arithmetic) finite fields require a prime number of elements.
But I recall hearing someone say that if you modify the way addition and multiplication is defined on a set with a non-prime number of el... | {
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In either case, $$x^2+x$$ is reduced to $$1$$.
• Thank you. Very well explained. Oct 15, 2019 at 21:29
In fact for any power $$p^n$$ of a prime $$p$$ you can find a finite field, usually denoted by $$\mathbb F_{p^n}$$ or $$GF(p^n)$$. You can construct these by finding an integer polynomial $$p \in \mathbb Z[X]$$ of d... | {
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Conversely, for any natural number $$k$$ and any prime, there exists a field, denoted $$\mathbf F_{p^k}$$, with $$p^k$$ elements, and this field is unique up to a field isomorphism. Furthermore, the field $$\mathbf F_{p^k}$$ is (isomorphic to) a subfield of the field $$\mathbf F_{p^l}$$ if and only if $$k$$ divides $$l... | {
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# For any finite abelian group $G$, there is an integer $m$ with $G$ isomorphic to a subgroup of $U(\mathbb{Z}_{m})$.
I want to prove if the following assertion from Rotmans Advanced Algebra page 205 is true:
For any finite abelian group $$G$$, there is some integer $$m$$ with $$G$$ isomorphic to a subgroup of $$U(\m... | {
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Now, take such a finite abelian group $$G$$. We have $$G = \prod_{i=1}^n \mathbb Z/n_i \mathbb Z$$, some integers $$n_i$$. Let $$p_i$$ be a prime such that $$p_i \equiv 1 \text{ mod } n_i$$. This exists by the second fact listed above. In fact, as there are infinitely many such primes we can take these $$p_i$$ to be di... | {
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• And I'm wondering why Rotman would say otherwise. Apr 25 '19 at 1:18
• You didn't explicitly answer the question "yes" or "no" (it is conventional to do so at the start of the answer). Apr 25 '19 at 3:01
• @BillDubuque I've added this, thanks for catching that. Apr 25 '19 at 4:08
• @QuangHoang In the copy of Rotman I... | {
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## Sequences, series
Forum for the GRE subject test in mathematics.
CoCoA
Posts: 42
Joined: Wed Sep 03, 2008 5:39 pm
### Sequences, series
What is
lim_{n->\infty}{(n!)^(1/n)}?
Nameless
Posts: 128
Joined: Sun Aug 31, 2008 4:42 pm
Use the Sterling formula : lim (n!/[sqrt(2pi * n)e^(-n)n^(n)])=1 when n goes to infinit... | {
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CoCoA
Posts: 42
Joined: Wed Sep 03, 2008 5:39 pm
in I is m a constant or a typo?
typo
corrected
sorry!
Posts: 24
Joined: Sun Oct 05, 2008 1:41 am
How about one from the 05 practice test...
Find the set of real numbers for which the series converges:
Sum[1 to inf] [n!*x^(2n)]/[n^n(1+x^(2n))]
CoCoA
Posts: 42
Joined:... | {
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# Four married couples attend a party. Each person shakes hands with every other person, except their own spouse, exactly once. How many handshakes?
Four married couples attend a party. Each person shakes hands with every other person, except their own spouse, exactly once. How many handshakes?
My book gave the answe... | {
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$$8$$ people. Each experiences handshakes with $$6$$ people. There are $$6\times 8=48$$ experiences of handshakes. Each handshake is experienced by two people so there $$48$$ experiences means $$48\div 2=24$$ handshakes.
Suppose the spouses were allowed to shake each other's hands. That would give you $$\binom{8}{2} =... | {
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$$k$$ couples entails $$2k$$ people. If we imagine each couple going in sequential order, couple 1 will each have to shake $$2k-2$$ couple's hands for each individual, or $$4k-4$$ handshakes for couple 1 total. Since there is 1 fewer couple every time a new couple shakes hands, there will be $$4k-4i$$ handshakes by the... | {
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There are $$4\text{ couples}=8\text{ people}$$. The statement, " Each person shakes hands with every other person, except their own spouse, exactly once," means that $$8$$ people shook hands with $$6$$ other people. That yields $$8\times6=48$$. It would be $$7$$ others but a spouse and the handshaker are both excluded ... | {
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# Complex Integration - Poles on the Imaginary axis
1. Apr 27, 2012
### knowlewj01
1. The problem statement, all variables and given/known data
evaluate the integral:
$I_1 =\int_0^\infty \frac{dx}{x^2 + 1}$
by integrating around a semicircle in the upper half of the complex plane.
2. Relevant equations
3. The at... | {
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we have:
$F(z) = \frac{1}{Z^2+1}=\frac{1}{(z+i)(z-i)}$
$\oint_C F(z)dz =\int_0^R F(z)dz + \int_\omega F(z)dz + \left[\int_{iR}^{i+\delta}F(z)dz+\int_{i-\delta}^0 F(z)dz\right] + \int_\lambda F(z)dz =0$
as F(z) is analytic everywhere inside the contour we demand that the integral = 0 by Green's theorem.
we know from... | {
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# Why must a radical be isolated before squaring both sides?
In the following equation:
$$\sqrt{2x + 1} + 1 = x$$
You are supposed to isolate the radical:
$$\sqrt{2x + 1} = x - 1$$
And then proceed by squaring both sides.
If you start by solving the equation this way, you will eventually complete the square and g... | {
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Well, it doesn't lead to the wrong path: if you square the equation right away you get that
$$(2x+1)+2\sqrt{2x+1}+1=x^2$$
And because $\sqrt{2x+1}=x-1$, you get the equation
$$(2x+1)+2(x-1)+1=x^2$$
This is $x^2-4x=0$ which has solutions $x=0,4$, but the solution $x=0$ doesn't do it because $\sqrt{1}$ is taken to be... | {
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""$(3x +2)/3 = 11/3$
The answer is we can do what we darned well like:
$x + \frac 23 = \frac {11}3$
$x = \frac {11}3 - \frac 23$.
Nothing wrong with that... but nothing right with it either.
We isolate terms, for whatever operation, for the purpose of isolating them so we can work directly with them.
That's all.
... | {
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The main problem with your approach is that squaring will still leave a radical, whereas isolating the radical won't. So it's mostly about simplicity, rather than doing some compulsory transformation.
However, squaring is not sufficient. You can only do $$(\sqrt{2x+1}+1)^2=x^2$$ under the assumption that $x\ge0$, or y... | {
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There is no need to isolate the radical. However, you usually want to isolate the radical in order to simplify computations. From $$\sqrt{2x+1}+1=x=\frac{\left(\sqrt{2x+1}\right)^2-1}{2}\,,$$ we have $$\left(\sqrt{2x+1}\right)^2-2\,\sqrt{2x+1}-3=0\,.$$ Thus, $$\left(\sqrt{2x+1}-3\right)\,\left(\sqrt{2x+1}+1\right)=0\,.... | {
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# Rate of change with $f(x)=4x^2-7$ on $[1,b]$
As part of a textbook exercise I am to find the rate of change of $$f(x)=4x^2-7$$ on inputs $$[1,b]$$.
The solution provided is $$4(b+1)$$ and I am unable to arrive at this solution.
Tried:
$$f(x_2)=4b^2-7$$
$$f(x_1)=4(1^2)-7=4-7=-3$$
If the rate of change is $$\frac... | {
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# Number of ways in which balls are distributed
In how many ways can we distribute 5 different balls into 4 different boxes, given that order does not matter inside the boxes and empty boxes are not allowed?
My attempt
First I chose $$4$$ balls out of $$5$$ and arranged them for the $$4$$ boxes: $$\binom 54 \times 4... | {
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Using inclusion-exclusion principle,
$$4^5 - ^4C_1 3^5 + ^4C_2 2^5 - ^4C_3 1^5$$
Subtract cases when all 5 objects go in 3 boxes (exclusion) then include when objects go in 2 boxes, because they are subtracted more than the number of times such case comes. Then again exclude when all objects go in one box.
Two metho... | {
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# plot3
3-D point or line plot
## Syntax
``plot3(X,Y,Z)``
``plot3(X,Y,Z,LineSpec)``
``plot3(X1,Y1,Z1,...,Xn,Yn,Zn)``
``plot3(X1,Y1,Z1,LineSpec1,...,Xn,Yn,Zn,LineSpecn)``
``plot3(___,Name,Value)``
``plot3(ax,___)``
``p = plot3(___)``
## Description
example
````plot3(X,Y,Z)` plots coordinates in 3-D space. To plot ... | {
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```t = 0:pi/50:10*pi; st = sin(t); ct = cos(t); plot3(st,ct,t)```
Create two sets of x-, y-, and z-coordinates.
```t = 0:pi/500:pi; xt1 = sin(t).*cos(10*t); yt1 = sin(t).*sin(10*t); zt1 = cos(t); xt2 = sin(t).*cos(12*t); yt2 = sin(t).*sin(12*t); zt2 = cos(t);```
Call the `plot3` function, and specify consecutive `XY... | {
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`plot3(xt1,yt1,t,xt2,yt2,t,'--')`
Create vectors `t`, `xt`, and `yt`, and plot the data in those vectors. Return the chart line in the output variable `p`.
```t = linspace(-10,10,1000); xt = exp(-t./10).*sin(5*t); yt = exp(-t./10).*cos(5*t); p = plot3(xt,yt,t);```
Change the line width to `3`.
`p.LineWidth = 3;`
S... | {
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## Input Arguments
collapse all
x-coordinates, specified as a scalar, vector, or matrix. The size and shape of `X` depends on the shape of your data and the type of plot you want to create. This table describes the most common situations.
Type of PlotHow to Specify Coordinates
Single point
Specify `X`, `Y`, and `Z`... | {
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"openwebmath_perplexity": 2288.381022143978,
"openwebmath_score": 0.9287667870521545,
"tags": ... |
Multiple sets of points
(using matrices)
Specify at least one of `X`, `Y`, or `Z` as a matrix, and the others as vectors. Each of `X`, `Y`, and `Z` must have at least one dimension that is same size. For best results, specify all vectors of the same shape and all matrices of the same shape. For example:
`plot3([1 2 3... | {
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Line style, marker, and color, specified as a character vector or string containing symbols. The symbols can appear in any order. You do not need to specify all three characteristics (line style, marker, and color). For example, if you omit the line style and specify the marker, then the plot shows only the marker and ... | {
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For a custom color, specify an RGB triplet or a hexadecimal color code.
• An RGB triplet is a three-element row vector whose elements specify the intensities of the red, green, and blue components of the color. The intensities must be in the range `[0,1]`; for example, ```[0.4 0.6 0.7]```.
• A hexadecimal color code ... | {
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Marker size, specified as a positive value in points, where 1 point = 1/72 of an inch.
Marker outline color, specified as `'auto'`, an RGB triplet, a hexadecimal color code, a color name, or a short name. The default value of `'auto'` uses the same color as the `Color` property.
For a custom color, specify an RGB tri... | {
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`[0.3010 0.7450 0.9330]``'#4DBEEE'`
`[0.6350 0.0780 0.1840]``'#A2142F'`
Marker fill color, specified as `'auto'`, an RGB triplet, a hexadecimal color code, a color name, or a short name. The `'auto'` option uses the same color as the `Color` property of the parent axes. If you specify `'auto'` and the axes plot box i... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9787126513110865,
"lm_q1q2_score": 0.8450122899970595,
"lm_q2_score": 0.8633916082162403,
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"tags": ... |
`[0.3010 0.7450 0.9330]``'#4DBEEE'`
`[0.6350 0.0780 0.1840]``'#A2142F'`
## Tips
• Use `NaN` or `Inf` to create breaks in the lines. For example, this code plots a line with a break between `z=2` and `z=4`.
` plot3([1 2 3 4 5],[1 2 3 4 5],[1 2 NaN 4 5])`
• `plot3` uses colors and line styles based on the `ColorOrde... | {
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# Magnetic field on current loop, involves rotational energy
1. ### cdingdong
3
1. The problem statement, all variables and given/known data
A rectangular loop of sides a = 0.3 cm and b = 0.8 cm pivots without friction about a fixed axis (z-axis) that coincides with its left end (see figure). The net current in the l... | {
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did i do something wrong? what i did makes sense in my mind. is their answer wrong? why did they not not take a square root at the end? is kinetic energy not = 1/2J$$\omega^{2}$$, but instead 1/2J$$\omega$$ without the square?
### Staff: Mentor
What's the final potential energy, when θ = 0?
3. ### cdingdong
3
wow, ... | {
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# Minimum variance for sum of three random variables
#### skijunkie
##### New Member
Hi all,
I have been working on the following problem:
Given you have VarX = 1, VarY = 4, and VarZ = 25, what is the minimum possible variance for the random variable W = X + Y + Z, or min Var(X+Y+Z)?
My first thought is to complet... | {
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$100 + 2\sigma_{XY}\sigma_{XZ}\sigma_{YZ} - 25\sigma_{XY}^2 - 4\sigma_{XZ}^2 - \sigma_{YZ}^2 \geq 0$
So any covariances satisfy the above two inequalities will be valid. The remaining optimization can be done by KKT multiplier, see
http://en.wikipedia.org/wiki/Karush–Kuhn–Tucker_conditions
P.S. One additional thing ... | {
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# What is $\vert 0 \rangle \otimes \vert + \rangle$?
A simple question that I cannot seem to figure-out why I cannot achieve the correct result. When I evaluate $$\vert 0 \rangle \otimes \vert + \rangle,$$ I end up with $$\begin{bmatrix}1\\0\end{bmatrix} \otimes \begin{bmatrix}\tfrac{1}{\sqrt{2}}\\\tfrac{1}{\sqrt{2}}\... | {
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Why is my solution doesn't align with algassert?
• Just note, you used a tag entanglement. There is nothing about entanglement by definition because your state is described by tensor product. This means that both states are separable and not entangled. Therefore, I removed the tag. Apr 10 '20 at 21:52
• So given $\ver... | {
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• Thank you for this comment! I would have never thought the convention they are using is being used practically. Would it be possible to change their convention to the one I am using? I might be looking for some "clever gate" where I apply it first before starting my circuit and it will achieve my desire? Apr 11 '20 a... | {
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# Confusion regarding $\log(x)$ and $\ln(x)$
I was solving an integral and I encountered in some question
$$\displaystyle \int_{2}^{4}\frac{1}{x} \, \mathrm dx$$
I know its integration is $\log(x)$. But my answer comes correct when I use $\ln(x)$ instead. What is this confusion? How do I know which one to use? Thank... | {
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$\log(x)$ has different meanings depending on context. It can often mean:
• any logarithm if the base is not important (e.g. in general proofs about logarithms)
• the natural logarithm $\log_e(x) = \ln(x)$ (usual convention in mathematics)
• the decadic logarithm $\log_{10}(x)$ (usual convention in chemistry, biology ... | {
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The problem then is what you make of an expression of $\log x$? That's ambiguous since the base is not specified. However in some cases it would be assumed that the default base is $e$ (but in some cases it maybe different).
Bottom line is if you don't want confusion you should normally use $\ln$ instead, or maybe $\l... | {
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# How should students say in words the notation for a limit?
$$\lim_{x\rightarrow a} f(x)=L$$
Which way should students best get in the habit of?
1. The limit of $$f(x)$$, as $$x$$ approaches $$a$$, equals $$L$$
2. The limit of $$f(x)$$ equals $$L$$, as $$x$$ approaches $$a$$
3. The limit, as $$x$$ approaches $$a$$,... | {
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• Thank you for the first reference---it is fantastic. I had never considered that someone might have codified spoken mathematics, and am quite chuffed to know that it exists. – Xander Henderson Sep 5 '19 at 16:03
• I am not sure what is the purpose of the three pages dedicated to Roman letters, as they offer no pronun... | {
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Indeed, you could argue that $$\displaystyle\lim_{x \to a}$$ is an operation you perform on the function $$f(x).$$ That is, you can change what $$x$$ the limit is being found at and also what function you are doing it to.
Secondly the “$$\lim_{x \to a} f(x)$$” is a thing all by itself, and so it makes sense to say it ... | {
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As $$x$$ approaches $$a$$, $$f(x)$$ approaches $$L$$.
First, we emphasize what is happening to the independent variable, then we explain the consequence. I think that this phrasing is concise and easy to understand. It is clean and efficient. This is essentially (3), but I think that the sub-clause "The limit..." is u... | {
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f(x) lähestyy L:ää, kun x lähestyy a:ta
and sometimes I instead say it more colloquially as
f(x):n raja on L, kun x on a.
The inverted
Kun x lähestyy a:ta, f(x) lähenee L:ää
is also fine and in use.
These would correspond to 2) and 4) in English. I would avoid linguistic complexity, such as a side clause embedded... | {
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• This seems more like a comment than an answer to the question, mostly because you only say things like "I prefer..." when the main question is about which one a student should use. – Brendan W. Sullivan Sep 4 '19 at 15:41
• Title has different wording. At the end of the day, it's not a super important question and di... | {
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GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 14 Oct 2019, 01:18
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customize... | {
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Genralizing the formula:
$$\frac{x}{y}$$, $$\frac{x+a}{y+b}$$, $$\frac{x+2a}{y+2b}$$,$$\frac{x+3a}{y+3b}$$...$$\frac{x+na}{y+nb}$$
Then $$\frac{x+na}{y+nb}$$ is greatest among all given fractions.
1. Both numerator and denominator increase in constant values.(Numerator by a, denominator by b)
2. ($$a>=b$$)
what if $... | {
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### Show Tags
05 Nov 2014, 15:40
Thank you. This was enlightening.
Intern
Joined: 14 Feb 2019
Posts: 2
Re: Fractions : Faster calculation [#permalink]
### Show Tags
01 Mar 2019, 00:37
great. It is very healpful
Re: Fractions : Faster calculation [#permalink] 01 Mar 2019, 00:37
Display posts from previous: Sort by | {
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It does this by representing the function in infinite sums of cosines and sines. The convergence of the Fourier series of g is uneventful, and after a few steps it is hard to see a difference between the partial sums, as well as between the partial sums and g. But from the Sequence of Terms Divergence Criterion for Inf... | {
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Analysis by NPTEL. Inverse Fourier Transform 10. It is permissible to have a finite number of finite discontinuities in one period. x(t) = x(t + p). For this example, this average is non-zero. Fourier Series 97 Absolutely Convergent Fourier Series Theorem. So by Bessel's inequality we have that the series $\displaystyl... | {
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notes are friendly enough for A-Level - please see preview. We also construct orthonormal bases for the Hilbert. Materials include course notes, lecture video clips, practice problems with solutions, a problem solving video, and problem sets with solutions. If f(x) is any function define d for−π < x≤π, then there is a u... | {
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Transforms Mark Rodwell, Doluca Family Chair, ECE Department University of California, Santa Barbara [email protected] We highly recommend you to follow your syllabus and then read these resources if you are under R15 regulation and for R13 Regulation we have provided important questions as per their. It stresses throu... | {
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is based on certain criteria. (a) The function and its Fourier series 0 0. Also a simple sin function did not work. The first part of this course of lectures introduces Fourier series, concentrating on their. Convolutions and correlations and applications; probability distributions, sampling theory, filters, and analysi... | {
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function and integrating the result. Ferroptosis is a form of regulated cell death with clinical translational potential, but the efficacy of ferroptosis-inducing agents is susceptible to many endogenous factors when administered alone, for which some cooperating mechanisms are urgently required. It is through this ave... | {
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the Fourier series representation of several continuous-time periodic wave-forms. These series had already been studied by Euler, d'Alembert, Bernoulli and others be-fore him. 5 Adding sine waves. Signals and systems: Part II. Notice that t he first equation is exactly the same as we got when considering the Fourier Cos... | {
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a sum of constituent harmonics. INTRODUCTION TO FOURIER TRANSFORMS FOR PHYSICISTS 5 and the inverse transform : (15) ψ(~k) = 1 (2π)32 Z ∞ −∞ ψ(~x)e−i(~k·~x)d3x We note that every time we go up in dimension, we tag on an extra scaling factor of 1 2π 1 2. Mathematica for Fourier Series and Transforms Fourier Series Perio... | {
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necessarily belong to the linear span, as the span of a family of vectors is de ned as nite linear combinations of vectors from the family. An important consequence of orthonormality is that if s= P n k= n c ke. Fourier series is used to decompose signals into basis elements (complex exponentials) while fourier transfo... | {
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approach. 2) which has frequency components at. As I was going through Arthur Mattuck’s excellent differential equations course at MIT’s Open Courseware , the Fourier series clicked for me, so I thought I’d distill this out. 1: The cubic polynomial f(x)=−1 3 x 3 + 1 2 x 2 − 3 16 x+1on the interval [0,1], together with ... | {
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modes as trial functions and solving the flow and the salt transport equations simultaneously in the spectral space. Conventions and first concepts The purpose of these notes is to introduce the Fourier series of a. Besides the textbook, other introductions to Fourier series (deeper but still elementary) are Chapter 8 ... | {
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Author: Recami E. In fact, one way of. as will be seen below. In these notes we de ne the Discrete Fourier Transform, and give a method for computing it fast: the Fast Fourier Transform. Determine Power. As a result, the summation in the Discrete Fourier Series (DFS) should contain only N terms: xe. 6 (C,1)-Summability... | {
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Jean Baptiste Joseph Fourier (21 March 1768 - 16 May 1830) Fourier series. Without even performing thecalculation (simplyinspectequation2. Jean Baptiste Joseph Fourier (21 March 1768 - 16 May 1830) Fourier series. Note that because the modulus was taken after averaging Fourier coefficients, our derivation of amplitude ... | {
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in harmonic analysis. • finance - e. Media in category "Fourier analysis" The following 111 files are in this category, out of 111 total. Note, for instance, that if we set χ = 7r/2 in (1) and χ = π in (4), we obtain the respective results. However, periodic complex signals can also be represented by Fourier series. FOU... | {
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