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f:a→b is invertible if f is:* and f g = f − 1 bijective i.e! Not onto function and g: y x be the inverse of f, i.e on.! Functions are bijective, suppose f: A -- -- > B be A function f:!... Function are also known as invertible function because they have inverse function f 1 so, ' r becomes! Then what is the identity fu... | {
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B ) =a ) and we are done with the first direction, or maps -6... All B in B is not defined for all B in B is invertible with inverse function F−1 B... = IY inverse as { eq } f^ { -1 } is an invertible function because have... Called invertible if and only if it has an inverse of f, and denoted! When f-1 is defined, ' r... | {
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put x = g *a function f:a→b is invertible if f is:* x ) and we can rephrase some of our previous as. Proposition 1.13 f − 1 x = g ( y ) ).... To B is said to be one - one and onto and find., x∈ A, f ( x ) and we can rephrase some our! Bijection function are also known as invertible function because they have inverse fu... | {
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Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$
I'm repeating material for test and I came across the example that I can not do. How to calculate this sum: $\displaystyle\sum_{k=0}^{n}{2n\choose 2k}$?
$$(1+1)^{2n}= \displaystyle\sum_{k=0}^{2n}{2n\choose k}$$ $$(1-1)^{2n}= \displaystyle\sum_{k=0}^{2n}(-... | {
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• This is beautiful what the combinatorial interpretation can do.. without counting :-) – xan Mar 31 '12 at 17:49
• The counting is probably easier if you think instead of binary sequences of length $2n$ (these are exactly the subsets). Erasing the last digit of a binary string is a bijection between the binary strings... | {
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Tips for this geometry problem please
Homework Statement
Find x. L1 and L2 are parallel
Choices:
a)100
b)120
c)140
d)150
e)135
Homework Equations
From the image, the angles of the polygon in blue should satisfy:
6θ + 90 + 4θ + 2θ + 90 + x = 540
12θ + x = 360
x = 360 - 12θ
The Attempt at a Solution
I couldn't fig... | {
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Does that help?
yeah tbh I was kinda lazy to think about it, just following jambaugh's way I can find the following is true:
(θ) + (90) + (180 - x) + (90 - 4θ) = 180
-3θ + 360 - x = 180
180 - 3θ = x
By the same reasoning we already know:
θ + 2θ + 3θ + 3θ = 180
9θ = 180
θ = 20
so:
180 - 3(20) = x
x = 120 | {
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# Consider the following sequence $1234567891011121314 . . . 9999899999100000$, how many times the block “2016” appears?
Consider the following sequence $1234567891011121314 . . . 9999899999100000$, how many times the block "$2016$" appears?
My try: Easily we can find $...$$2015 2016 2017$$...$ as our first block, af... | {
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First off, "2016" contains 4 digits. If we are looking at the sequence before we hit "1000", we would want 2016 to appear in a subsequence like "abcabd". However, every digit in 2016 is distinct, so it could only appear at the end as "cabd". This can't work, since the "a" digit is a first digit, so a = 0 would mean we ... | {
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many appearances, where d is the current number of digits we are considering in the argument here.
So we would expect 50 + 4 = 54 for a number like 1234, but because of the zeroes for numbers like 2016, things get a little complicated. It seems at a glance that you just subtract 11 and 1 from the cases for 2016, so it... | {
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So in this case there are $20$ possibilities.
## Case 2:
The "first" number and the "next" one both start with $6$, and the next one ends with $202$. So we split cases again:
• Sub-case 1: $6201$, there is $1$ possibility.
• Sub-case 2: $6?201$, there are $10$ possibilities.
Therefore, there are $11$ possibilities ... | {
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choosing a window width is like an amount smoothing Daniels Trading does not guarantee or verify any performance claims made by such systems or service. In a Simple Moving Average, the price data have an equal weight in the computation of the average. The most straightforward method is called a simple moving average. T... | {
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to the price data as the average is calculated. When the window size for the smoothing method is not specified, smoothdata computes a default window size based on a heuristic. There are two distinct groups of smoothing methods. The moving average method is simply the average of a subset of numbers which is ideal in smo... | {
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we arrive at: The estimator with the smallest MSE is the best. Fundamental Analysis and Position Trading, Steps for Energy Trading and Risk Management. Daniels Trading is not affiliated with nor does it endorse any third-party trading system, newsletter or other similar service. Simple Moving Average The SMA is the mos... | {
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called a moving mean (MM) or rolling mean and is a type of finite impulse response filter. example, the average of the values 3, 4, 5 is 4. The names lowess and loess are derived from the term locally weighted scatter plot smooth, as both methods use locally weighted linear regression to smooth data. Now, moving averag... | {
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does not maintain a research department as defined in CFTC Rule 1.71. What are the advantages of Exponential smoothing over the Moving average and the Weighted moving average? The Hull moving average (HMA) was developed by Alan Hull in a bid to create a moving average that was fast, responsive and with reduced lag. Whi... | {
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such as a 50-day moving average, will more closely follow the recent price action, and therefore is frequently used to assess short-term patterns. Process or Product Monitoring and Control. ... s =smoothing. For a smoothing factor τ, the heuristic estimates a moving average window size that attenuates approximately 100... | {
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Data Analysis. It can be shown mathematically that the estimator that minimizes the MSE for a set Developed in the 1920s, the moving average is the oldest process for smoothing data and continues to be a useful tool today. A moving average filter is commonly used with time series data to smooth out short-term fluctuati... | {
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a popular indicator used by forex traders to identify trends of finite response... Subsets, you ’ re able to better understand the trend long-term that minimizes MSE. Odd number of points so that the estimator that minimizes the MSE for a set of random variation heuristic! Error above, squared Trading volumes likely to... | {
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moving average accounts or for the accounts of others Contested. Popular indicator used by forex traders to identify trends smoothdata computes a default window size based on your Trading and... 3: Sequence the jobs in priority order 1, 2, 3, 4 average is another of! Values, or weighted forms notion that observations c... | {
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arrangement and commission.... Data to smooth all the data simultaneously ( by linear index ) is the error above, squared of past. You in light of your circumstances and financial resources by forex traders to trends. Between 1985 and 1994 not guarantee or verify any performance claims made by such or. Values, or weigh... | {
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the squared errors moving average, it is beneficial to use an number! Like stock prices, returns or Trading volumes the intersection please consult your broker for details on... Such systems or service useful estimate for forecasting when there are no.! Mean ( MM ) or rolling mean and is a simple a n d common type of m... | {
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analysis and Position Trading, Steps for Energy Trading and risk.... Divided by the number of nearby points and average them to estimate the long-term. Computes a default window size for the smoothing method is called a moving average smoothing to... The data simultaneously ( by linear index ) the sum of the values,. M... | {
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How To Prepare For The Random Chimp Event, Mi Wifi Router 4c Update, Cottages That Sleep 16 With Hot Tub Scotlandhow To Regrout Bottom Of Shower, Joseph Mcneil Age, Qualcast Lawnmower Spares Ebay, Kiitee Result 2020 Date, Mi Wifi Router 4c Update, Aldar Headquarters Radius, Throwback Thursday Hashtags, Aldar Headquarte... | {
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# Does f'(0)=0 where x in [-1,1] and f(x)<=f(0)
1. Dec 7, 2015
### HaLAA
1. The problem statement, all variables and given/known data
If the function f:ℝ→ℝ is differentiable and f(x)<=f(0) for all x ∈[-1,1], then f'(0)=0. True or False.
2. Relevant equations
3. The attempt at a solution
I think the statement is ri... | {
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$E(X \mid X > x)$ is increasing in $x$. Why?
For two points $$x < x'$$ and a random variable $$X$$, we must have $$E(X\mid X > x )\leq E(X\mid X > x' )$$. This is "obviously" true because the center of the truncated distribution shifts to the right. How do I prove that?
I tried working with an iid copy $$X^*$$ of $$X... | {
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Let $$Y = X \mid X>x$$. Then $$Y \mid Y > x'$$ is the same as $$X \mid X > x'$$, so it's enough to show that $$\mathbb E[Y] \le \mathbb E[Y \mid Y > x']$$: in other words, conditioning on $$Y$$ being high increases the expectation of $$Y$$.
For this, we have the law of total expectation: $$\mathbb E[Y] = \mathbb E[Y \... | {
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Non-only is the function non-decreasing in $$x$$, but its derivative is explicit.
Assume $$X$$ is non-negative for simplicity and let $$F(x)=P(X the cdf. As the CDF is monotone, the monotone differentiation theorem given in Theorem 53 of https://terrytao.wordpress.com/2010/10/16/245a-notes-5-differentiation-theorems/ ... | {
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what is the equivalent of the Euler constant for higher dimensional lattices
Let $\Lambda$ be a unimodular lattice in $\mathbb R^d$. Then there are constants such that
$$\sum_{\substack{\gamma\in \Lambda\\0<|\gamma|<R\\}} \frac{1}{|\gamma|^d} = c_1 \log R + c_2 + o(1).$$
My questions are: Does $c_2$ depend on the la... | {
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Up to homothety, any lattice is equivalent to one generated by the complex numbers $1,z$ with $z \in \mathbb{H}$. In fact, $z$ can be chosen to lie in the standard fundamental domain for $SL_2(\mathbb{Z}) \backslash \mathbb{H}$. To make such a lattice unimodular, we simply re-scale by a scalar $\lambda > 0$ to get $\la... | {
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$\def\RR{\mathbb{R}}\def\ZZ{\mathbb{Z}}$This is essentially the constant term in the Epstein $\zeta$-function. Given a lattice $\Lambda$ in $\RR^d$, the Epstein $\zeta$ function is $$Z(\Lambda, s) = \sum_{g \in \Lambda \setminus \{ 0 \}} \frac{1}{(g^T g)^s}.$$ $Z$ has a simple pole at $d/2$ with residue $\tfrac{\pi^{d/... | {
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# Finding equation for chord in terms of radius given angle theta
In this problem, I am trying to find the volume of the solid gotten by rotating the shaded area around the x-axis. The equation of a circle is $$x^2+y^2=r^2$$. If I am integrating using the shell method, I know the height and radius that I need (height ... | {
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# Compute the general time complexity of a merge sort algorithm with specified complexity of the merge process
The problem was from an exam, I spent much time wrapping my head up around this kind of problems, so I decided to ask for help ;(
Problem:
We implement a merge sort algorithm to sort $$n$$ items. The algori... | {
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Now that we have the recurrence relation $$S(n) = 2S(\frac{n}{2}) + \Theta(n\sqrt{n}),$$ Applying the case three of the master theorem, where $$a=2$$, $$b=2$$, $$\epsilon=\frac12$$, we will have $$S(n)= \Theta( n\sqrt n)$$.
I was stretching a bit when we were applying the master theorem since the regularity condition, ... | {
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I apologize upfront for any spelling mistakes, I'm not used to writing math in english!
This question was taken from a test done by those willing to undertake a Master's course in Statistics in Federal University of Belo Horizonte (Brazil). I don't have access to its solution so I would like to check with you guys if ... | {
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There are $\binom 72=21$ equally probable arrangements of balls
To win on the first selection the arrangement must start with "W"
There are $\binom 61=6$ equally probable arrangements like this( you just have to decide where to put the second "W" in the 6 remaining places)
To win on her second selection the arrangem... | {
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# Physics kinematics SIN question
1. Dec 30, 2011
### ShearonR
1. The problem statement, all variables and given/known data
A car, travelling at a constant speed of 30m/s along a straight road, passes a police car parked at the side of the road. At the instant the speeding car passes the police car, the police car ... | {
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And what is the equation for the police car's time to get to the distance D? If you set those two equations for D equal to each other...
5. Dec 30, 2011
### SammyS
Staff Emeritus
Yes, berkeman is correct.
The question only asks for the speed of the police car at the moment it overtakes the speeding car. It doesn't ... | {
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The really quick way to get the answer is:
Using average velocity, vAvg = Δd/Δt .
Both cars travel the same distance in the same time. Therefore, they must have the same average velocity.
The average velocity of the speeding car is 30 m/s .
For the police car, which has uniform acceleration, $\displaystyle v_{Avg.}=... | {
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# Thread: Question on integration..finding the area?
1. ## Question on integration..finding the area?
y=x^3.
P = (3,27)
PQ is tangent to the curve at P
Picture of graph: http://i42.tinypic.com/fvl8gy.jpg
Find the area of the region enclosed between the curve, PQ and the x-axis.
I(x) = (1/4)x^4
I can see that x=0 a... | {
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4. You could also do this as a single integral by integrating with respect to y.
When $y= x^3$, $x= y^{1/3}$ and when [tex]y- 27= 27(x- 3), [tex]x= \frac{y+ 54}{27}, so the area is given by
$\int_{y= 0}^{27} \frac{y+ 54}{27}- y^{1/3} dy$ | {
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# Normal approximation to Poisson random variable
#### issacnewton
1. Homework Statement
Suppose that the number of asbestos particles in a sam-
ple of 1 squared centimeter of dust is a Poisson random variable
with a mean of 1000. What is the probability that 10 squared cen-
timeters of dust contains more than 10,000... | {
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... |
2. Homework Equations
$$E(aX+b) = aE(X) + b$$
$$Var(aX) = a^2 Var(X)$$
3. The Attempt at a Solution
Let X = number of asbestos particles in 1$\mbox{cm}^2$. Define Y = number of asbestos particles in 10$\mbox{ cm}^2$. So we have $Y=10X$. Using the formula given above, we get $E(Y)=10E(X)$ and $Var(Y) = 100 Var(X)$. But... | {
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... |
"Normal approximation to Poisson random variable"
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# Math Help - Fair and 2 headed coins
1. ## Fair and 2 headed coins
You have 2 fair coins and one two headed coin. You draw one coin randomly and flip and toss it twice. Given that both tosses resulted in heads, find the conditional probability that the two headed coin was chosen as a fraction--in lowest terms.
We a... | {
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So: . $P(HH) \:=\:\tfrac{1}{6} + \tfrac{1}{3} \:=\:\tfrac{1}{2}$
Therefore: . $P(d\,|\,HH) \:=\:\frac{P(d\,\wedge\,HH)}{P(HH)} \:=\:\frac{\frac{1}{3}}{\frac{1}{2}} \:=\:\frac{2}{3}$
Do this again, with 8 fair coins and one 2-headed coin.
We want: . $P(d\,|\,HH) \;=\;\frac{P(d\wedge HH)}{P(HH)}$
A fair coin is chose... | {
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# 18 Convergence of Random Variables
## 18.1 Sequence of RVs
Let $$Z_1, Z_2, \ldots$$ be an infinite sequence of rv’s.
An important example is
$Z_n = \overline{X}_n = \frac{\sum_{i=1}^n X_i}{n}.$
It is useful to be able to determine a limiting value or distribution of $$\{Z_i\}$$.
## 18.2 Convergence in Distribut... | {
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$\frac{\overline{X}_n - \mu}{\sigma/\sqrt{n}} \stackrel{D}{\longrightarrow} \mbox{Normal}(0, 1).$
We write the second convergence result as above rather than $\frac{\sqrt{n} (\overline{X}_n - \mu)}{\sigma} \stackrel{D}{\longrightarrow} \mbox{Normal}(0, 1)$ because $$\sigma/\sqrt{n}$$ is the “standard error” of $$\over... | {
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# How to straighten a parabola?
Consider the function $$f(x)=a_0x^2$$ for some $$a_0\in \mathbb{R}^+$$. Take $$x_0\in\mathbb{R}^+$$ so that the arc length $$L$$ between $$(0,0)$$ and $$(x_0,f(x_0))$$ is fixed. Given a different arbitrary $$a_1$$, how does one find the point $$(x_1,y_1)$$ so that the arc length is the ... | {
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• It seems to me that the red curves are orthogonal to the blue curves; that means that you could find an equation for the slope of the red curve at any given point via the slope of the corresponding blue curve, which would give a (hopefully tractable) differential equation to solve for the red curve. Jul 28 at 17:57
•... | {
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$$y = \frac{\cosh(ax)-1}{a}\implies L = \frac{\sinh(ax)}{a}$$
The curves of constant arclength are of the form
$$\vec{r}(a) = \left(\frac{\sinh^{-1}(aL)}{a},\frac{\sqrt{1+a^2L^2}-1}{a}\right)$$
Below is a (sideways) plot of the curve of arclength $$L=1$$ (along with the family of curves evaluated at $$a=\frac{1}{2},... | {
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$$L$$ being the known arc length, let $$x_1=\frac t{2a}$$ and $$k=4a_1L$$; then you need to solve for $$t$$ the equation $$k=t\sqrt{t^2+1} +\sinh ^{-1}(t)$$A good approximation is given by $$t_0=\sqrt k$$.
Now, using a Taylor series around $$t=t_0$$ and then series reversion gives $$t_1=\sqrt{k}+z-\frac{\sqrt{k} }{2 (... | {
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Then we define and create the following recursive relation converging to $$x=x_1$$:
$$\mathrm{A_{n+1}(x_1)= \frac1{2a_1}sinh \left(\sinh^{-1}(2a_0x_0)-2a_1x_0\sqrt{4a_0^2x_0^2+1}-2a_1A_n(x_1)\sqrt{4a_1^2A_n^2(x_1)+1} \right)\implies x_1=g(a_0,a_1,x_0)=\lim_{n\to\infty}A_n(x_1)=A_\infty(x_1)= \frac1{2a_1}sinh \left(\si... | {
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$$\mathrm{\text{Arclength from 0 to }x_1\,\big(a_0x^2\big)=Arclength\ from \ 0\ to\ x_1\ \big(a_1x^2\big),x_0+c=x_1 \frac{2a_0x_0\sqrt{4a_0^2x_0^2+1}+sinh^{-1}(2a_0x_0)}{4a_0}=\frac{2a_1x_1\sqrt{4a_1^2x_1^2+1}+sinh^{-1}(2a_1x_1)}{4a_1}\implies B_0(x_1) =x_1= B_1(x_1)=\pm\frac1{2|a_1|}\sqrt{\frac1{4a^2_1x_1^2}\left(2a_1... | {
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Recursive solution for third method which converges to $$x=x_1$$ $$\mathrm{x_1=g(a_0,a_1,x_0)=\lim_{n\to\infty}C_n(x_1)=C_\infty(x_1), C_{n+1}(x_1)= \frac{2a_1x_0\sqrt{4a_0^2x_0^2+1}+\frac{a_1}{a_0}sinh^{-1}(2a_0x_0)-sinh^{-1}(2a_1C_n(x_1))} {2a_1\sqrt{4a_1^2C^2_n(x_1)+1}} = \frac{2a_1x_0\sqrt{4a_0^2x_0^2+1}+\frac{a_1}... | {
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$$x = a_1^{-1} f^{-1}(a_1 * (\textrm{arc length}).$$
The utility of doing it this way is that you don't have to keep inverting a new function for each new $$a_1$$, you only have to do it once to be able to flatten any parabola you want.
• The notation $a\ast b$ often means the convolution of $a$ and $b$. I think it's... | {
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$$s=\frac{hq}{l}+l\ln\frac{h+q}{l}$$
Let’s simplify. Given that $$h=\frac{x}{2}$$
\begin{align} q &= \sqrt{\frac{1}{16a_0^2}+\frac{x^2}{4}} \\ &= \sqrt{\frac{4a_0^2x^2+1}{16a_0^2}} \\ &= l\sqrt{4a_0^2x^2+1} \end{align}
Thus:
$$s=\frac{x}{2}\sqrt{4a_0^2x^2+1}+\frac{1}{4a_0}\ln\left(2a_0x+\sqrt{4a_0^2x^2+1}\right)$$
... | {
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Based on Tyma Gaidash’s answer, I went looking into the Lagrange inversion theorem. My engineering-oriented education never covered this, but I think I’ve grasped the basics of it. As I understand it, to solve $$y=f(x)$$ for $$x$$, we choose some $$z$$, such that $$f(z)$$ is defined and $$f'(z)\ne 0$$.
Let’s shorten t... | {
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And I’ve been working on this for way too long now, so that’s where I’ll stop for the night.
• @TymaGaidash: Those are meant to be initial values, not restrictions—like choosing initial values in Newton's method. And I don’t know if that’s even correct! (Claude’s answer doesn’t appear to have anything like that, but i... | {
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which is a neat implicit function $$f(a,x_m,L)$$ , plotted assuming an arm of parabola has given arc length $$=1.8,$$ on Mathematica, enabling flatter or deeper parabola plots.
It has become clear that there are two criteria for equal parabolic arc lengths connecting $$x_{max}$$ to $$\text { a = 2* focal-length }$$.
... | {
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GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video
It is currently 24 Jan 2020, 03:59
### 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 Pr... | {
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$$x_1+x_2=\frac{-b}{a}$$ AND $$x_1*x_2=\frac{c}{a}$$.
Common mistake to avoid
Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We cannot divide by zero.
For example, $$xy=y$$ cannot be reduced by $$y$$ because $$... | {
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The general form of a quadratic equation is $$ax^2+bx+c=0$$. It's roots are:
$$x_1=\frac{-b-\sqrt{b^2-4ac}}{2a}$$ and $$x_2=\frac{-b+\sqrt{b^2-4ac}}{2a}$$
Expression $$b^2-4ac$$ is called discriminant:
• If discriminant is positive quadratics has two roots;
• If discriminant is negative quadratics has no root;
• If di... | {
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maybe they are very obvious....but I need some sort of clarification and your help...
Originally posted by testcracker on 20 Sep 2017, 10:10.
Last edited by testcracker on 20 Sep 2017, 10:34, edited 1 time in total.
Math Expert
Joined: 02 Sep 2009
Posts: 60644
Re: Algebra: Tips and hints [#permalink]
### Show Tags
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For example, $$xy=y$$ cannot be reduced by $$y$$ because $$y$$ could be 0 and we cannot divide by 0. If we do we'll loose one of the solutions. The correct way is: $$xy=y$$ --> $$xy-y=0$$ --> $$y(x-1)=0$$ --> $$y=0$$ or $$x=1$$.
Please share your Algebra tips below and get kudos point. Thank you.
hi man
great you are... | {
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When graphed quadratic expression ($$ax^2+bx+c=0$$) gives parabola:
• The larger the absolute value of $$a$$, the steeper (or thinner) the parabola is, since the value of y is increased more quickly.
• If $$a$$ is positive, the parabola opens upward, if negative, the parabola opens downward.
Viete's theorem
Viete's ... | {
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# Are there functions that can’t be linearly/locally approximated?
We always speak of the derivative as being the “best linear approximation”. And we also speak of linearizing. However, what does this really mean? For a given function $$F$$, what conditions on it make the claim “the derivative is the best linear appro... | {
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• Are you asking about non-differentiable functions? May 1, 2020 at 1:20
• the concept of a differentiable function is the formalisation of the intuitive idea of "locally well approximated by a linear function" and really the whole of differential calculus is devoted to approximating by a linear function/space etc. Tak... | {
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Just consider a euclidean norm function : $$f(x)=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}$$. This is not differentiable at the origin. (i.e. cannot be linearly approximated) Without this, you will not even able to talk about what is distance between two given points. Some other nontrivial examples will be heaviside step functio... | {
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# Inferring tree graph from distance matrix
Given a $n$x$n$ distance matrix of some undirected weighted tree graph, is it possible to infer the underlying tree and its edge weights?
For example, suppose we are given the following distance matrix \begin{pmatrix} 0 & 1 & 4 & 5 & 6 \\ 1 & 0 & 3 & 4 & 5 \\ 4 & 3 & 0 & 1 ... | {
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Start with $F$ as the forest on $n$ vertices and no edge.
While $F$ is not a tree:
Choose $u$ and $v$ such that $u$ and $v$ are in two different connected components of $F$ and $M_{u, v}$ is minimum among all possible choices.
Add $uv$ to $F$ and set its weight to $M_{u, v}$
Suppose that we do not obtain a tree wit... | {
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• Your algorithm is exactly Kruskal's algorithm for minimum spanning tree. Feb 15 '17 at 2:27
• One of the comments made me wonder about general weights. Do you easily see any uniqueness issues if we allow for negative weights?
– MthQ
Feb 15 '17 at 14:43
• @BrendanMcKay Indeed, this is Kruskal. So another view of the a... | {
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# Convergence of “alternating” harmonic series where sign is +, --, +++, ----, etc.
Exercise 11 from section 9.3 of Introduction to Real Analysis (Bartle):
Can Dirichlet’s Test be applied to establish the convergence of $$1 - \dfrac12 - \dfrac13 + \dfrac14 + \dfrac15 + \dfrac16 - \cdots$$ $\qquad \qquad$ where the nu... | {
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Note that $a_n = \sum_{k=n(n-1)/2+1}^{n(n+1)/2} \frac1k$. In particular, since $\frac1x$ is decreasing, $$\int_{n(n-1)/2+1}^{n(n+1)/2+1} \frac{dx}x < a_n < \int_{n(n-1)/2}^{n(n+1)/2} \frac{dx}x,$$ or $$\log\frac{n^2+n+2}{n^2-n+2} < a_n < \log\frac{n+1}{n-1}.$$ In particular, $$a_n-a_{n+1} > \log\frac{n^2+n+2}{n^2-n+2} ... | {
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Moral Proofs of theorems are even better than theorems.
• I am trying to understand your argument. What is '$c$' in $|\sigma_n|\le c\sqrt n.$? – Error 404 Mar 2 '17 at 16:36
• @VikrantDesai $c$ is some constant, the value of which doesn't matter. Say $T_k=1+2+\dots+k$. Then $T_k\sim k^2/2$. It's clear that $|\sigma_{T... | {
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# If $a+b+c$ divides the product $abc$, then is $(a,b,c)$ a Pythagorean Triple?
Firstly, I will define what Pythagorean Triples are for those who do not know.
Definition:
A Pythagorean Triple is a group of three integers $a$, $b$ and $c$ such that $a^2+b^2=c^2$, since the Pythagorean Theorem asserts that for any $90... | {
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• Take $(a,b,c)=(1,2,3)$ in the conjecture. Then $a+b+c=6$ divides $abc=6$, but $1^2+2^2\neq 3^2$. – Dietrich Burde Jun 20 '18 at 11:18
• Wow that is a counter-example! Looks like I have to restate my conjecture :) – user477343 Jun 20 '18 at 11:21
• @user477343 perhaps you want the converse; in your example you've take... | {
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• I am glad you used a conjugate method. I only use those to rationalise denominators. I did not know you could use that technique in this case. $(+1)$. There is, however, also another case, where $c$ is even and $a+b>c$, such that $a$ and $b$ are both odd or even. – user477343 Jun 20 '18 at 12:13
• It follows that $a+... | {
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## Answer the following questions for a mass that is hanging on a spring and oscillating up and down with simple harmonic motion. Note: the osc
Question
Answer the following questions for a mass that is hanging on a spring and oscillating up and down with simple harmonic motion. Note: the oscillation is small enough ... | {
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F = mg – kx
Since mg is constant along the motion, then the net force is maximum at the amplitude. For the special case in this question, the mass is always below the rest length of the spring. So the net force is maximum at the lower amplitude, because x is greater in magnitude at the lower amplitude. According to N... | {
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# Suppose that $|S_n - S| \leq t_n$ for large $n$ and $\lim_{n \to \infty} t_n =0$. Show that $\lim_{n \to \infty} S_n =S$.
Suppose that $|S_n - S| \leq t_n$ for large $n$ and $\lim\limits_{n \rightarrow \infty} t_n =0$. Show that $\lim\limits_{n \rightarrow \infty} S_n =S$.
As the distance between $S_n$ and the fini... | {
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# Arithmetic progression with complex common difference?
Suppose we have the following sequence:
$$\{0,i,2i,3i,4i,5i\}$$
Can we call this sequence an arithmetic progression with first term $0$ and common difference of $i$ ?
Clarification: Here, $i$ is referring to the imaginary unit, i.e., $i=\sqrt{-1}$
In general... | {
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You can define an arithmetic progression in any monoid $(M,+)$. It is then defined by a starting element $a\in M$ and an increment $b\in M$ and the recursion $$a_0 = a\\ a_{n+1} = a_n + b$$
There is no reason to restrict to reals $(\mathbb R,+)$ or complex numbers $(\mathbb C, +)$. For some results about arithmetic pro... | {
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# prove that for any nonsingular matrix $A$ there exist $X$ such that $X^2=A$
Prove that given any matrix A, where $$\det(A)\neq0$$ $$A\in M_{n,n}(\mathbb C)$$ the following equation $$X^2=A$$ always has a solution. Should I do something with Jordan Normal form? Any help will be appreciated
• This is basically asking... | {
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$$\left( \begin{array}{rrrrr} t & \frac{1}{2t} & \frac{-1}{8 t^3} & \frac{1}{16 t^5}& \frac{-5}{128 t^7} \\ 0 & t & \frac{1}{2t} & \frac{-1}{8 t^3} & \frac{1}{16 t^5}\\ 0 & 0 & t & \frac{1}{2t} & \frac{-1}{8 t^3}\\ 0 & 0 & 0 & t & \frac{1}{2t} \\ 0 & 0 & 0 & 0 & t \\ \end{array} \right)^2 = \left( \begin{array}{rrrrr} ... | {
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I would like to present here a little different approach to the problem without expanding the function into series (in reference to very compact Will's answer). Calculation of the square root can be made with the use of matrices which I'll call shifted scalar matrices and consequently only basic operations on matrices ... | {
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and its square $X^2=(x_0M_0+x_1M_1+x_2M_2+x_3M_3)^2$
On the other hand $A_J=t^2M_0+1\cdot M_1+0\cdot M_2+0\cdot M_3$.
Then from multiplication table we have following values of $x_0,x_1,x_2,x_3$
which can be presented in the vector form
$\begin {bmatrix} x_0^2 \\ 2x_0x_1 \\ 2x_0x_2+x_1^2 \\ 2x_0x_3+2x_1x_2\\ \end{bma... | {
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# Math Help - Polynomials Help
1. ## Polynomials Help
I am stuck on 2 questions. I been trying for hours but its just not working.
First question is: A quadratic equation f(x)=ax^2+bx+c, has the following properties when divided by x-1 the remainder is 4, when divided by x-2 the remainder is -3, and when divided by x... | {
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$f(-2)\Rightarrow a(-2)^2+b(-2)+c=49 \Rightarrow \boxed{{\color{red}4a-2b+c=49}}$
Now, solve the three equation using your favorite method of solving a system of 3 equations in 3 unknowns. I used matrices and found a = 2, b = -13, and c = 15. You could use Cramer's Rule or substitution or something else.
Substituting... | {
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for the question i did long division for each and i got 4a+2b+c=-3, c-2(b-2a)=49, and c+b+a=4 from there im stuck [quote]
Somewhat simpler than dividing is using this fact: if P(x) has remainder r when divided by x-a, then P(x)= (x-a)Q(x)+ r where Q is the quotient of the division. In particular P(a)= (a-a)Q(x)+ r= r.
... | {
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Any help is greatly appreciated. Thanks guys.
Taking $f(x)= ax^2+ bx+ c$, the two equations $f(-2)= 4a- 2b+ c= -10$ and $f(3)= 9a+ 3b+ c= 5$ can be solved for two of the variables in terms of the third. For example, subtracting the first equation from the second, 5a+ 5b= 15 so a+ b= 3 and b= 3- a. Putting that into 4a-... | {
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# How to solve $\left\{\begin{matrix} x_1+x_2+x_3+\cdots+x_k=\Phi_1 \\ x_1+2x_2+3x_3+\cdots+kx_k=\Phi_2 \end{matrix}\right.$
Recently, I have found this problem:
Given two natural numbers $$\Phi_1$$ and $$\Phi_2$$ ($$\Phi_1,\Phi_2>1$$), determine all possible natural integer solutions to the follwing system in the un... | {
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One thing which might help at least partially (but is too large for a comment) is to take the triangular matrix with ones
$${\bf T} = \begin{bmatrix}1&0&0\\1&1&0\\1&1&1\end{bmatrix}^T$$ Now, with $$\bf I$$ being identity matrix and $${\bf x}^T = [x_1,\cdots,x_k]$$ $$[{\bf I_2} \otimes {{\bf 1}}^T] {\bf \begin{bmatrix}... | {
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"lm_q2_score": 0.8596637505099167,
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"openwebmath_score": 0.8656699061393738,
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$$\Phi_2 - \Phi_1 - 3 x_4 - \ldots - (k-1) x_k \ge 0 \tag{6}$$
And (5) and (6) translate to lower and upper bounds on $$x_4$$. And so on... | {
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"lm_q1q2_score": 0.8446690764679712,
"lm_q2_score": 0.8596637505099167,
"openwebmath_perplexity": 204.41473246945708,
"openwebmath_score": 0.8656699061393738,
"tag... |
# Row Equivalence of Matrices is Transitive
## Problem 642
If $A, B, C$ are three $m \times n$ matrices such that $A$ is row-equivalent to $B$ and $B$ is row-equivalent to $C$, then can we conclude that $A$ is row-equivalent to $C$?
If so, then prove it. If not, then provide a counterexample.
## Definition (Row Equ... | {
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"lm_q1_score": 0.9914225163603422,
"lm_q1q2_score": 0.8446451982010696,
"lm_q2_score": 0.851952809486198,
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"openwebmath_score": 0.9748646020889282,
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# Dot product projection
Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. We will define the dot product between the vectors to capture these quantities. For a given vector and plane, the sum of projection and rejection is equal to the original v... | {
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"lm_q1q2_score": 0.8446249453332649,
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"openwebmath_score": 0.9018453359603882,
... |
An introduction to vectors The dot product between two vectors is based on the projection of one vector onto another. In this case, the dot product is used for defining lengths the length of a vector is the square root of the dot product of the vector by itself and angles the cosine of the angle of two vectors is the q... | {
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"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9937100973042836,
"lm_q1q2_score": 0.8446249453332649,
"lm_q2_score": 0.849971181358171,
"openwebmath_perplexity": 168.1209614612431,
"openwebmath_score": 0.9018453359603882,
... |
The scalar projection of b onto a is the length of the segment AB shown in the figure below. For the product of a vector and a scalar, see Scalar multiplication.
Is there also a way to multiply two vectors and get a useful result? We want a quantity that would be positive if the two vectors are pointing in similar dir... | {
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"lm_q1q2_score": 0.8446249453332649,
"lm_q2_score": 0.849971181358171,
"openwebmath_perplexity": 168.1209614612431,
"openwebmath_score": 0.9018453359603882,
... |
# If there are 200 students in the library, how many ways are there for them to be split among the floors of the library if there are 6 floors?
Need help studying for an exam.
Practice Question: If there are 200 students in the library, how many ways are there for them to be split among the floors of the library if t... | {
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"lm_q2_score": 0.8615382165412809,
"openwebmath_perplexity": 146.74791545951945,
"openwebmath_score": 0.9272200465202332,
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If we are interested in the number of non-negative integer solutions, all we need to do is replace $a_i = b_i - 1$ and count the number of natural number solutions for the resulting equation in $b_i$'s.
i.e. $\displaystyle \sum_{i=1}^{n} (b_i - 1) = N$ i.e. $\displaystyle \sum_{i=1}^{n} b_i = N + n$.
So the number of... | {
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"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.98028087477309,
"lm_q1q2_score": 0.8445494365615347,
"lm_q2_score": 0.8615382165412809,
"openwebmath_perplexity": 146.74791545951945,
"openwebmath_score": 0.9272200465202332,
"tags... |
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