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Standard and vertex form of the equation of parabola and how it relates to a parabola's graph.
10. Originally Posted by davidman
I'm looking for the turning point of $y=-4x^2+12x-5$ so I want $y$ in the form $y=a(x-p)^2+q$.
You can also find the turning-point (of a quadratic equation) without completing the square. It... | {
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Math Help - number sequence in a circle
1. number sequence in a circle
Let's say we have 51 numbers located on a circle in such a way that every number is equal to the sum of its two neighboring numbers.
Prove that the only combinations of numbers is for all of them to be zero !?
2. Hello, returnofhate!
I have half... | {
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BUT let's say there was not 51 elements, there was n, so can we say the same for it ?
IMO, yes we can, I was trying to prove it with induction (at least for odd numbers).
Starting from 3 points, we can simply solve the equations and show that they have to be all zero.
Then we assume there are n (odd) numbers, and the... | {
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### Example 2: Nonlinear convection in 2D¶
Following the initial convection tutorial with a single state variable $u$, we will now look at non-linear convection (step 6 in the original). This brings one new crucial challenge: computing a pair of coupled equations and thus updating two time-dependent variables $u$ and ... | {
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In [3]:
#NBVAL_IGNORE_OUTPUT
for n in range(nt + 1): ##loop across number of time steps
un = u.copy()
vn = v.copy()
u[1:, 1:] = (un[1:, 1:] -
(un[1:, 1:] * c * dt / dx * (un[1:, 1:] - un[1:, :-1])) -
vn[1:, 1:] * c * dt / dy * (un[1:, 1:] - un[:-1, 1:]))
v[1:, 1:] = (vn[1:, 1:] -
(un[1:, 1:] * c * dt / dx * (vn[1:, 1:]... | {
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print("U update:\n%s\n" % update_u)
print("V update:\n%s\n" % update_v)
U update:
Eq(u(t + dt, x, y), -dt*u(t, x, y)*v(t, x, y)/h_y + dt*u(t, x, y - h_y)*v(t, x, y)/h_y - dt*u(t, x, y)**2/h_x + dt*u(t, x, y)*u(t, x - h_x, y)/h_x + u(t, x, y))
V update:
Eq(v(t + dt, x, y), -dt*v(t, x, y)**2/h_y + dt*v(t, x, y)*v(t, x,... | {
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# Physics homework #8
1.Part 1: A cylinder w/moment of inertia 30.9 kg x meters2 rotates w/angular velocity 8.48 rad/sec on a frictionless vertical axle. A second cylinder, w/moment of inertia 27.3 kg Xmeters2, initially not rotating, drops onto the first cylinder and remains in contact. Since the surfaces are rough, ... | {
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1.
There is no external applied force hence no external momentum of force. The total kinetic moment ($J=I*\omega$) is the same before and after the second cylinder is dropped. The situation is similar to a plastic collision of two bodies where the total linear momentum is the same before and after the collision. Befor... | {
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Hence for an observer directly below the plane
$L =r*m*v =10000*17991*162 =2914542*10^4 kg*m^2/s$
2. The variation of the angular momentum is equal to the torque of the external force applied. There is no external force applied hence no variation of angular momentum as the plane moves. The correct answer is 3. No. ... | {
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# Are $\{(x,y)\in \mathbb{R}^2 : (x,y)\neq(0,0)\}$ and $\{(x,y)\in \mathbb{R}^2 : (x,y)\notin [0,1]\times\{0\}\}$ homeomorphic?
Let $X_1$ and $X_2$ be the spaces \begin{align*} X_1&=\{(x,y)\in \mathbb{R}^2 : (x,y)\neq(0,0)\}, \\ X_2&=\{(x,y)\in \mathbb{R}^2 : (x,y)\notin [0,1]\times\{0\}\}. \end{align*} Are these spac... | {
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• Thank you very much! I understand why these two spaces are homeomorphic. Feb 10, 2015 at 16:47
• If this homeomorphism is prescribed by $\left(\sqrt{b^{2}+1}\cos t,b\sin t\right)\mapsto\left(b\cos t,b\sin t\right)$ then I have doubts. It seems to send open arc $\left\{ \left(\cos\varphi,\sin\varphi\right)\mid\varphi\... | {
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• $f$ is continuous because $f$ is continuous on $A, B, C$ respectively,and agree on overlaps. Inverse $g:X_1 \rightarrow X_2$ can be defined as below: on $A'=(\{(u,v)\in\mathbb{R}^2 : v\geq u,u\geq 0, v\leq 1\} \cup \{(u,v)\in\mathbb{R}^2 : v\leq -u,u\geq 0, v\geq 1\})\setminus \{(0,0)\}$, $g(u,v)=(u/|v|,v)$, on $B'=\... | {
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Math Help - A few integration problems
1. A few integration problems
$1. \int \sin^3(5x)\cos(5x)dx$
$2. \int \tan^2(x)dx$
$3. \int \frac{\sec^2(x)}{3+\tan(x)}dx$
$4. \int \sin(x)(\cos(x)+\csc(x))dx$
These are what I got:
1. $-\frac{5}{2}\cos^2(5x)+\frac{15}{4}\cos^4(5x)+c$
2. $\tan(x)-x+c$
3. $Ln(3+\tan(x))+c$
... | {
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Back-substitute: . $\ln|3 + \tan x| + C$
$4. \int \sin x \,\left[\cos x +\csc x \right]\,dx$
Multiply: . $\sin x\,\left[\cos x + \csc x\right] \;= \;(\sin x)(\cos x) + (\sin x)(\csc x) \;= \;\sin x \cos x + 1$
We have: . $\int\left(\sin x\cos x + 1\right)\,dx \;= \;\int\sin x\cos x\,dx + \int dx$
. . In the first i... | {
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# Proving $\lim_{n\to\infty} a_n=\infty$
Given a sequence $\{a_n\}_{n=1}^{\infty}$ and $c>0$ such that $a_{n+1} - a_n>c$ for every $n$, prove $\lim_{n\to\infty} a_n=\infty$.
I proved $a_n$ is monotonic increasing, but I'm having hard time proving it's unbounded.
Any ideas?
From the fact that $a_{n+1} > c+a_n$, you ... | {
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ie $$\infty = \lim nc+a_0 < \lim a_n$$ hence $a_n\to \infty$
• Nice ! I know that weak inequality is preserved in limits, is it the same on strong one? – Itay4 Feb 14 '17 at 14:04
• What do you mean by weak inequality in this context? Please clarify – Guy Fsone Feb 16 '17 at 11:41
• I know that if $a\ \geq b$ then $li... | {
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# In how many ways can $3$ boys and $3$ girls be seated in a row such that each boy is adjacent to at least one girl?
The number of ways of seating 3 boys and 3 girls in a row, such that each boy is adjacent to at least one girl, is?
My Approach:
First arrange the 3 boys in $3!$ ways one of them being $$\_b_1\_b_2\_b... | {
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Exactly two boys are at one end of the row: There are two ways to choose the end of the row where the two boys will sit. There are $\binom{3}{2}$ ways to choose which two of the boys will sit together. There are $3$ ways to choose which of the girls will sit adjacent to the block of boys. There are $3!$ ways to arrange... | {
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# Find all the solutions of $x^2 \equiv 1 \mbox{ mod }365$.
Find all the solutions of $$x^2 \equiv 1 \mbox{ mod }365$$.
We know that $$365=5\cdot 73$$. So if I could find the solutions of $$x^2 \equiv 1 \mbox{ mod }5$$ and $$x^2 \equiv 1 \mbox{ mod }73$$, using CRT I could find the solutions of the given equation.
I ... | {
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### leetcode Question 1: Two Sum
Two Sum
Given an array of integers, find two numbers such that they add up to a specific target number.
The function twoSum should return indices of the two numbers such that they add up to the target, where index1 must be less than index2. Please note that your returned answers (both... | {
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Code(O(n)):
class Solution {
public:
vector<int> twoSum(vector<int> &numbers, int target) {
// Start typing your C/C++ solution below
// DO NOT write int main() function
vector<int> res;
map<int,int>hmap;
hmap.clear();
for (int i=0;i<numbers.size();i++){
hmap[numbers[i]]=i;
}
for (int i=0;i<numbers.size();i++){
if (hm... | {
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3. This comment has been removed by the author.
4. 有一个小bug 如果是{3,2,4} target是6 返回的是1,1 不是2,3
1. I've checked the code, it seems the return value of your case is 2,3.
res.push_back(i+1) : i+1 = 2
res.push_back(hmap[6-2]+1) = 2+1 = 3.
return (2,3)
2. sorry, man, please double check {3,2,4} target is 6.
also the if..e... | {
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By Rolle’s theorem, between any two successive zeroes of f(x) will lie a zero f '(x). This version of Rolle's theorem is used to prove the mean value theorem, of which Rolle's theorem is indeed a special case. If it can, find all values of c that satisfy the theorem. Rolle’s Theorem. and by Rolle’s theorem there must b... | {
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a proper closed interval [a, b], differentiable on the open interval (a, b), and f (a) = f (b), then there exists at least one c in the open interval (a, b) such that ′ =. Rolle's theorem is the result of the mean value theorem where under the conditions: f(x) be a continuous functions on the interval [a, b] and differ... | {
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Scholastic Test for Aptitude and Reasoning Examples: Find the two x-intercepts of the function f and show that f’(x) = 0 at some point between the Rolle's Theorem If f(x) is continuous an [a,b] and differentiable on (a,b) and if f(a) = f(b) then there is some c in the interval (a,b) such that f '(c) = 0. Proof: The arg... | {
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Mean Value Theorem x y a c b A B x Tangent line is parallel to chord AB f differentiable on the open interval (If is continuous on the closed interval [ b a, ] and number b a, ) there exists a c in (b a , ) such that Instantaneous rate of change = average rate of change 13) y = x2 − x − 12 x + 4; [ −3, 4] 14) y = A sim... | {
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its maximum and minimum values. <> Now if the condition f(a) = f(b) is satisfied, then the above simplifies to : f '(c) = 0. Rolle’s Theorem and other related mathematical concepts. A plane begins its takeoff at 2:00 PM on a 2500 mile flight. At first, Rolle was critical of calculus, but later changed his mind and prov... | {
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of zeroes. %PDF-1.4 Theorem 1.1. Rolle’s Theorem, like the Theorem on Local Extrema, ends with f′(c) = 0. Let us see some Calculus 120 Worksheet – The Mean Value Theorem and Rolle’s Theorem The Mean Value Theorem (MVT) If is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then t... | {
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Newton and Leibnitz. Using Rolles Theorem With The intermediate Value Theorem Example Consider the equation x3 + 3x + 1 = 0. For each problem, determine if Rolle's Theorem can be applied. This builds to mathematical formality and uses concrete examples. For the function f shown below, determine we're allowed to use Rol... | {
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! Get help with your Rolle's theorem homework. Determine whether the MVT can be applied to f on the closed interval. We can see its geometric meaning as follows: \Rolle’s theorem" by Harp is licensed under CC BY-SA 2.5 Theorem 1.2. If f is zero at the n distinct points x x x 01 n in >ab,,@ then there exists a number c ... | {
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Theorem: Let f be continuous on >ab, @ and n times differentiable on 1 ab, . The result follows by applying Rolle’s Theorem to g. ¤ The mean value theorem is an important result in calculus and has some important applications relating the behaviour of f and f 0 . Be sure to show your set up in finding the value(s). Wit... | {
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the first paper involving calculus was published ’ s Theorem '' by is. Is a special case of the Taylor REMAINDER Theorem JAMES KEESLING in this post we rolle's theorem pdf a of! ] s.t swipe through stories, and browse through concepts f x x x ( 3! And so Rolle ’ s Theorem is a special case of the function c that the! F... | {
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intewal of examining cases and the... Whether the MVT can be applied to f on the given intewal in... And applying the Theorem to prove Taylor ’ s Theorem is one of the MVT when! At x = 3 and so Rolle ’ s Theorem can not be applied the. 9 some s 2 [ a ; b ) such that f0 ( c ) = 0 CC! State University of Semarang Taylor ... | {
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the first paper calculus. By-Sa 2.5 Theorem 1.2 of c that satisfy the Theorem introduction into 's... The Value ( s ) guaranteed by the Theorem on Brilliant, the plan arrives at its.! Calculus, but later changed his mind and proving this very important Theorem Theorems are some of the MVT when. And applying the Theorem... | {
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and they are classified into various types the plan arrives at its destination post... A graphical explanation of Rolle 's Theorem questions that are explained in way... ( ) 3 1 on [ -1, 0 ] MATH 123 State. Determine whether the MVT can be applied to approaches Rolle 's Theorem questions that are explained a., we recal... | {
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Suggestion Crossword Clue, Mississippi Title And Registration, Ahh Real Monsters, Trane Wholesale Distributor, Michelle Chords Capo 3, | {
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# Two cars, one accelerating and one
1. Sep 15, 2004
### Omid
The driver of a pink Cadillac traveling at a constant 60 mi/h is being chased by the law. The police car is 20 m behind the perpetrator when it too reaches 60 mi/h, and at that moment the officer floors the gas pedal. If her car roars up to the rear of th... | {
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7. Sep 16, 2004
### Staff: Mentor
So how did you solve it?
You don't have to do it that way. Another way is to write expressions for the position of the cadillac and the police car as a function of time. The cadillac has a constant speed, while the police car accelerates. You know they must be at the same position in... | {
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Again two car, A accelerating and B with a constant rate.
The most important difference between this and the previous, I think , is the top speed given, 110 km/h.
It means we have two time intervals. The first in which the cop must accelerate and the second in which his speed is constant.
So there will be 3 unknowns. T... | {
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# Is there a function that can be subtracted from the sum of reciprocals of primes to make the series convergent
The gamma constant is defined by an equation where the harmonic series is subtracted by the natural logarithm:
$$\gamma = \lim_{n \rightarrow \infty }\left(\sum_{k=1}^n \frac{1}{k} - \ln(n)\right)$$
It is... | {
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A heuristic is that an integer $n$ is prime with "probability" one in $\ln n$, and so we can estimate the sum with its "expected" value:
$$\sum_{\substack{p \leq n \\ p \text{ prime}}} \frac{1}{p} \approx \sum_{k=2}^n \frac{1}{k \ln k} \approx \int_2^n \frac{\mathrm{d}x}{x \ln x} \approx \ln \ln n$$
In fact, the Meis... | {
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1k views
An unbiased coin is tossed repeatedly until the outcome of two successive tosses is the same. Assuming that the trials are independent, the expected number of tosses is
1. 3
2. 4
3. 5
4. 6
Yes the given answer is 3.
Probability on each branch is = x = $\frac{1}{2}$
2nd toss onwards, each toss layer gives u... | {
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In our case : $x = 0.5$ So,
\begin{align*} E &= \sum_{k=2}^{\infty}k(k-1)x^{k} = x^2\sum_{k=2}^{\infty}k(k-1)x^{k-2} = x^2.\frac{2}{(1-x)^3} \\ \end{align*}
putting $x = \frac{1}{2}$ ; we get $E = 4$
More example:
For consecutive two heads ; HH
By drawing the tree diagram we can find the following series :
\beg... | {
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E=1/2 *(E+1) + 1/2 * (2) =3
correct me!
@Gabbar to me it looks correct.
@debashish nice explanation
thanks ! @Gabbar
Nice explanation . Marvelous Presentation
E(X)= sigma(Xi * Pi)
Where X=no of tosses when you get successive HEAD/TAIL(only one is possible at a time though).
Pi=Probability that you get in Xi to... | {
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2. If first flip gives head and second gives tail then we are done. (required 2 flips) So.
2(1/2 * 1/2)
3. If first flip gives tail and second head then wastage = 1 flip. (Same as 1st case)
1/2 * 1/2 * (x+1)
4. If both gives tail (same as 2)
2 (1/2*1/2)
so
x = 2(1/2*1/2) + 1/2 *1/2 * (x+1) + 1/2 *1/2 *(x+1) + 2 ... | {
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# Volume bounded by elliptic paraboloids
Find the volume bounded by the elliptic paraboloids given by $z=x^2 + 9 y^2$ and $z= 18- x^2 - 9 y^2$.
First I found the intersection region, then I got $x^2+ 9 y^2 =1$. I think this will be area of integration now what will be the integrand. Please help me.
• I did upvote th... | {
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You are wrong about the intersection region. Equating the two expressions for $z$ gives
$$x^2+9y^2=18-(x^2+9y^2)$$ $$2(x^2+9y^2)=18$$ $$x^2+9y^2=9$$
and thus here $z=9$.
The volume of your intersection can be divided into to parts: $0\le z\le 9$ where the restrictions on $x$ and $y$ are $x^2+9y^2\le z$, and $9\le z\... | {
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$$V_2=\int_9^{18} \frac{\pi(18-z)}3\,dz$$
Your total volume is then $V_1+V_2$.
I like this approach since it is just a pair of single integrals, each of which is very easy. Your question seems to assume the double-integral approach. Let me know if those are the bounds you really want.
Here is the double-integral, if... | {
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# Proof by exhaustion: all positive integral powers of two end in 2, 4, 6 or 8
While learning about various forms of mathematical proofs, my teacher presented an example question suitable for proof by exhaustion:
Prove that all $2^n$ end in 2, 4, 6 or 8 ($n\in\mathbb{Z},n>0$)
I have made an attempt at proving this, ... | {
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There is an easier way of "exhaustion" in my opinion: proof by contradiction. For example, assume that $2^n=10k$. This would mean that $5|2^n$ (five divides the power of two), an impossibility, so no power of two can end in $0$. For the rest (units digits 1, 3, 5, 7, 9), show that even numbers (that is, the multiples o... | {
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Informally: once a cyclic recurrence begins to loop, all subsequent values remain in the loop.
Similarly, suppose there are integers $\rm\:a,b,\:$ such that $\rm\: f(n+2)\ =\: a\:f(n+1) + b\:f(n)\:$ for all $\rm\:n\ge 1.\:$ Show that $\rm\:f(n)\:$ is divisible by $\rm\:gcd(f(1),f(2))\:$ for $\rm\:n\ge 1$.
- | {
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Solving limits without using L'Hôpital's rule
$$\lim_{x \to \frac{\pi}{2}}\frac{b(1-\sin x) }{(\pi-2x)^2}$$
I had been solving questions like these using L'Hôpital's rule since weeks. But today, a day before the examination, I got to know that its usage has been 'banned', since we were never officially taught L'Hôpit... | {
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• One of the commonest errors is to apply the rule repeatedly to solve an indetermination and forget it is valid only in case of indetermination, going one step too far. (a caricature would be to obtain at some step $\frac x1$, proceed to $\frac 10$ and conclude to an infinite limit!) – Bernard Sep 27 '15 at 13:51
• Bu... | {
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Specifically, set $t=\frac{\pi}{2}-x$ then you want:
$$\lim_{t\to 0} \frac{b(1-\cos t)}{4t^2}$$
Then use the trig identities above, replacing $1-\cos t$.
If $b = 0$, then there is nothing to work out; let $b \neq 0$. But $$\lim_{x \to \pi/2} \frac{b(1 - \sin x)}{(\pi - 2x)^{2}} = b\lim_{h \to 0}\frac{1 - \sin (h + \... | {
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# I'm given the dimensions of a rectangle, and when they increase by 1 the area is tripled…
The questions is "The length and width of a rectangle are $7$m and $5$m. When each dimension is increased by the same amount, the area is tripled. Find the dimensions of the new triangle, to the nearest tenth of a metre."
Answ... | {
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Where did the equals sign go?
In fact, what you should have at this point is not just the expression $x^2+12x-70$, but rather the equation $$x^2+12x-70=0$$ which should make it clear that you're not looking for the vertex of the parabola $y=x^2+12x-70$, but rather its $x$-intercepts.
Essentially what happened here is... | {
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# Is the real number $\sqrt{6}$ in $\mathbb{R}$ equal to the 5-adic number $\sqrt{6}$ in $\mathbb{Q}_5$?
My question is as in the title. That is, consider solving the equation $x^2-6=0$ in $\mathbb{R}$ and in the 5-adic field $\mathbb{Q}_5$ respectively. We obtain one $\sqrt{6}\in\mathbb{R}$ and one $\sqrt{6}\in\mathb... | {
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The question of whether two "numbers" are "equal" is a somewhat subtle one.
For example, lets work with a simpler number, namely "2".
Certainly $2\in\mathbb{Z}$, but also $2\in\mathbb{Q},2\in\mathbb{R}$. Of course, they're all called the same name, and they satisfy some of the same properties: For example, in all thr... | {
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Thus, instead of comparing labels, we should instead be comparing "ideas". Why is it reasonable to say that $2\in\mathbb{Z}$ is equal to $2\in\mathbb{Q}$? On the other hand, $2\in\mathbb{Z}$ is not "equal" to $2\in\mathbb{Z}/4\mathbb{Z}$ - if they were equal, then why is $2+2+2=2$ only valid in $\mathbb{Z}/4\mathbb{Z}$... | {
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Finally, coming to your original question: To be able to compare $\sqrt{6}\in\mathbb{R}$ and $\sqrt{6}\in\mathbb{Q}_5$, one must first find a "larger" field $K$ and injections $\mathbb{R}\hookrightarrow K$ and $\mathbb{Q}_5\hookrightarrow K$.
It's known that there exist such injections if $K = \mathbb{C}$, but it's im... | {
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EDIT 2: Yes $\mathbb{Q}(\sqrt{6})$ can be embedded in $\mathbb{Q}_5$
• Wow. You wrote a book! At least it looks that way on smart phone. Nice story. Vote up. Jul 27 '15 at 18:12
• @oxeimon Thanks very much for your very nice explanations! But I am a little confused about "As a result, $\ldots$, and hence no way to exp... | {
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# Can absolute or relative contributions from X be calculated for a multiplicative model? $\log{ y}$ ~ $\log {x_1} + \log{x_2}$
(How) can absolute or relative contributions be calculated for a multiplicative (log-log) model?
### Relative contributions from a linear (additive) model
E.g., there are 3 contributors to ... | {
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One can even use the original variable and its exponent as:
$$m_{x_1} = x_1^{\beta_1}/y$$
Still another way to quantify importance for multiplicative models (inspired by geometric means) is:
Given $$y = ax_1^{\beta_1}x_2^{\beta_2} \cdots x_n^{\beta_n}$$, then:
$$m_{x_k} = \frac{x_k}{\sqrt[\beta_1 + \beta_2 \cdots +... | {
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Some references which follow another route:
Measures of relative importance:
• Hey @Nikos, thanks for taking the time to answer. I'm still unsure whether the $r$ or $m$ values should be used. In my example of $y = 10 .10^2 .10^3$, what contributions from $\alpha$, $x_1$, and $x_2$ would you come up with? Is there a s... | {
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# criterion for interchanging summation and integration
The following criterion for interchanging integration and summation is often useful in practise: Suppose one has a sequence of measurable functions $f_{k}\colon M\to\mathbb{R}$ (The index $k$ runs over non-negative integers.) on some measure space $M$ and can fin... | {
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Doing the integrals, we obtain the answer
$\pi\sum_{k=1}^{\infty}{e^{-k}\over k^{2}}$
Title criterion for interchanging summation and integration CriterionForInterchangingSummationAndIntegration 2013-03-22 16:20:05 2013-03-22 16:20:05 rspuzio (6075) rspuzio (6075) 9 rspuzio (6075) Result msc 28A20 | {
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Derivative of $(uA+C)^{-1}\mathbf{b}$ w.r.t. $u\in\mathbb{R}$
Given that $$(uA+C)\mathbf{x}=\mathbf{b}$$ where only $$u\in \mathbb{R}$$ and $$\mathbf{x}\in\mathbb{R}^n$$ are unknowns, and where $$(uA+C)\in\mathbb{R}^{n\times n}$$ is an invertible matrix, how can I determine $$\frac{d\mathbf{x}}{du}$$?
I rewrite the e... | {
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Edit
I vaguely remember a technique called implicit differentiation which I feel may be useful:
$$\frac{d}{du}(uA+C)\mathbf{x}=\frac{d}{du}\mathbf{b}$$ $$\frac{d}{du}uA\mathbf{x}+\frac{d}{du}C\mathbf{x}=\mathbf{0}$$ $$A\frac{d}{du}u\mathbf{x}+C\frac{d\mathbf{x}}{du}=\mathbf{0}$$ $$A(\mathbf{x}+u\frac{d\mathbf{x}}{du}... | {
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$$YY^{-1} = I, \tag 4$$
and differentiate:
$$Y'Y^{-1} + Y(Y^{-1})'= 0, \tag 5$$
or
$$Y'Y^{-1} = -Y(Y^{-1})', \tag 6$$
from which we immediately obtain
$$(Y^{-1})' = -Y^{-1}Y'Y^{-1}; \tag 7$$
taking
$$Y(u) = uA + C \tag 8$$
we arrive at
$$((uA + C)^{-1})' = (uA + C)^{-1}A(uA + C)^{-1}, \tag 9$$
whence from (3... | {
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# Does $Var(X^2) \geq (VarX)^2$ hold?
It is well known that $E(X^2) \geq (EX)^2$, but I was wondering if there is a similar result for variances, e.g. is $Var(X^2) \geq (VarX)^2$?
I was doing some research and came up with that inequality, but I can’t prove it. I’ve done simulations in R for several known distributio... | {
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$$\mathsf{Var}(X^2) \geq 4 \mathsf{Var}(X)^2.$$
The equality holds if and only if $X$ is either constant or a multiple of the Bernoulli distribution of parameter $\frac{1}{2}$.
Proof. Let $\mu = \mathsf{E}X = \mathsf{E}Y$ denote the common expectation of $X$ and $Y$. Following the previous computation, we find that
... | {
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For these to coincide, we must have $p = \frac{1}{2}$. ////
• +1 Nice proof. How did you think of making an independent copy of a random variable? Do you have any other examples, links, of the application of this trick?
– Hans
Dec 22, 2017 at 18:15
• @Hans, Thank you for the upvote :) I was motivated by the proof of H... | {
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• Is the coefficient on $\langle x^3\rangle$ supposed to be a $4$? Dec 22, 2017 at 10:17
• @jdods: Yes, you are right. I have corrected the coefficient. Please check. Thank you.
– Hans
Dec 22, 2017 at 16:50
• This is actually interesting because my answer doesn't depends on the third moment, but yours does. I think we ... | {
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In general, for any arbitrary distribution with finite mean and variance, if we consider $\mu=E(X)$ and $\sigma^2=\text{Var}(X)=E[(X-\mu)^2]$ to be parameters of the distribution then breaking the inequality requires $$E(X^4)<\mu^4+2\mu^2\sigma^2+\color{red}{2}\sigma^4. \tag1$$
Note the red $2$ since without it, we ha... | {
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# Why do sets in $\mathbb R$ need to be bounded AND closed to be compact?
I have been studying compact sets recently, and have been struggling a little to develop my intuition. I feel a little silly asking this question, because I know there is probably something very simple I am missing from a definition I've read.
... | {
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Note that an unbounded subset of $\mathbb R$ cannot be compact, because $\left\{(-n,n)\,\middle|\,n\in\mathbb N\right\}$ is an open subcover without a finite subcover. And if $A$ is a non-closed subset of $\mathbb R$, then $A$ cannot be compact because, if $x\in\overline A\setminus A$, then $\left\{\left(-\infty,x-\fra... | {
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• This seems arbitrary. If you take the extended real line $\mathbb R \cup \{\pm \infty\}$ the opposite is true. – Mees de Vries Nov 6 '17 at 11:47
• Thanks for the comment. In the math units I've been taking intuition really doesn't get much time so I appreciate hearing some interesting ways to visualise this. – leob ... | {
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# How did Ulam and Neumann find this solution?
In the book "Chaos, Fractals and Noise - Stochastic Aspects of Dynamics" from Lasota and Mackey the operator $P: L^1[0,1] \to L^1[0,1]$
$$(Pf)(x) = \frac{1}{4\sqrt{1-x}} \left[ f\left(\frac{1}{2}\left(1-\sqrt{1-x}\right)\right) + f\left(\frac{1}{2}\left(1+\sqrt{1-x}\right... | {
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Which can be rewritten as
$$S(\sin^2\frac{\pi}{4})=1\\S^2(\sin^2\frac{\pi}{8})=S^2(\sin^2\frac{3\pi}{8})=1\\S^3(\sin^2\frac{\pi}{16})=S^3(\sin^2\frac{3\pi}{16})=S^3(\sin^2\frac{5\pi}{16})=S^3(\sin^2\frac{7\pi}{16})=1\\\vdots$$
Now assume $Pf=f$. From an intuitive viewpoint, the transformation $P$ sends $S^{-1}(x)$ to... | {
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So we finally have $h(x)=c$ and from this we can get $$f(x)=\frac{c}{\sqrt{x(1-x)}}$$
Putting $c=1/\pi$ makes the function $f$ a proabability density function on $[0,1]$.
Update. I've just found out that this was actually a method called the 'change of variables'. It is illustrated Ch6.5 in the OP's book and the idea... | {
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• Very nice work! Can you elaborate a bit why $h(x)$ must be a constant function? (It obviously works, but the implied uniqueness isn't so evident.) – Semiclassical Sep 4 '14 at 12:24
• Thank you for your answer. Could you explain how you got $1+ \sqrt{1-sin^2( \frac{2x\pi}{4})} = sin^2( \frac{\pi}{2} -\frac{\pi x}{4})... | {
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Proof
\begin{align} (Pf^*)(x)&= \frac{1}{4\sqrt{1-x}} \Bigl( f\Bigl(\frac12 (1-\sqrt{1-x})\Bigr)+ f\Bigl(\frac12 (1+\sqrt{1-x})\Bigr)\Bigr) \\ &=\frac{1}{4\sqrt{1-x}} \Bigl( \frac1\pi \frac{1}{\sqrt{\frac12 (1-\sqrt{1-x})(1-\frac12 (1-\sqrt{1-x})}}\Bigr) + \frac1\pi \frac{1}{\sqrt{\frac12 (1+\sqrt{1-x})(1-\frac12 (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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# How to find the maximum distance in centimeters so that a sphere supported from one end of a box is at equilibrium?
#### Chemist116
The problem is as follows:
A sphere is placed over a block as seen in the figure from below. The mass of the sphere is $10\,kg$ and the mass of the block is $4\,kg$. Assume that the b... | {
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$14g - \dfrac{g(38+10x)}{3} > 0$
$42 > 38+10x \implies x<0.4 \text{ m}$
Chemist116
#### Chemist116
Reference the diagram
$\color{red}C_1,C_2,C_3$ are the center of mass of each section of the block
$\color{blue}x$ is the distance the sphere is from P
Using $\color{green}F_1$ as the pivot for rotational equilibriu... | {
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#### Chemist116
I solved this problem in the manner I did because that's the method I tried first since I saw three unknowns. FYI, one may break up a uniform mass into pieces and determine the center of mass of each piece, where the weight is proportional to the size of each piece.
One could also solve it using the c... | {
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I'm also trying to understand what if the force is lesser than zero?. What would be the interpretation for this?. Would it be that the object is "floating" about that point?. Can you attend these questions please?.
#### skeeter
Math Team
$F_1$ can only be greater than or equal to zero. It cannot be less than zero. If... | {
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# Should I ask this mathematical statistics question here or on math.se?
I would like to ask the following question:
if $$M$$ is a $$m\times n$$ constant matrix and $$\eta\sim\mathcal{N}(0,I)$$, then does $$\mathbf{E}_{\eta\sim\mathcal{N}}\left[\frac{\lVert M\eta\rVert}{\lVert\eta\rVert}\right]$$ exist? Also, let $$x... | {
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So questions on probability theory are perfectly on topic in here, but there is overlap with the Math page and you can find many good questions and answers on probability theory up there. My impression is that usually you can expect more concise "mathy" answers ("prove that") on Math page and longer, more descriptive a... | {
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# How can you prove that $1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$ without using induction?
Using mathematical induction, I have proved that
$$1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$$
for every integer $n > 0$.
I would like to know if there is another way of proving this result without using PMI. Is there any geometri... | {
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$\diamond$
• For everybody who wonders it like I did, Google told me PMI stands for "Principle of Mathematical Induction".
– JiK
May 28, 2016 at 17:36
• Not really what the OP wants, probably, but it might be possible to determine whether this result requires induction by figuring out whether it can be proved in Robin... | {
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• Quite the most unexpectedly beautiful answer I’ve seen on any Stackexchange site, ever! Whimsical, pretty, and mathematically lucid, all at the same time. May 28, 2016 at 22:30
• A fantastic demonstration that geometric and pictorial methods are still a very concise and complete way of proving results.
– Nij
May 29, ... | {
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Let $$\,f_n,\, g_n\,$$ be sequences of naturals and $$\bar R_n$$ a sequence of $$f_n\times g_n$$ rectangles of area $$R_n = f_n g_n.$$ Below we recall the product rule for the difference operator $$\,\Delta h_n := h_{n+1} - h_n$$ then we apply the rule to the special case $$\,f_n = n,\ g_n = 2n\!-\!1\,$$ in Lynn's pict... | {
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So to increment an $$\,n\!\times\! (2n\!-\!1)$$ rectangle $$\bar R_n$$ to its successor $$\bar R_{n+1}$$ of size $$\,(n\!+\!1)\!\times\!(2n\!+\!1)$$ we can add $$\,\color{#c00}{n\!+\!1}$$ squares on each $$\rm\color{#c00}{side}$$ of $$\bar R_n$$ and $$\,\color{#0a0}{2n\!-\!1}\,$$ on $$\rm\color{#0a0}{top}$$ of $$\bar R... | {
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$$\qquad\qquad\qquad\begin{eqnarray} {\underbrace{1}_{\color{#c0f}{\large R_{\Large1}}} +\!\!\! \underbrace{5}_{\large{\color{#48f}{R_{\Large 2}}-\color{#c0f}{R_{\Large 1}}}}\!\! +\! \underbrace{9}_{\large{R_{\Large 3}-\color{#48f}{R_{\Large 2}}}}\!\! +\, \cdots + \!\!\underbrace{4n\!-\!3}_{\large{R_{\Large n}-R_{\Larg... | {
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Finally we briefly mention that analogy with calculus. The above remarks on telescopic sums show that $$\,f(n)\,$$ is the sum of its differences. This is the difference analog of the Fundamental Theorem of Differential Calculus, i.e. we have the analogous theorems
$$\begin{eqnarray} \sum_{k=0}^{n-1}\ \Delta_k f(k)\ \,... | {
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Therefore such difference calculus proofs remain completely mechanical even for higher degree polynomials, but generalizing the geometrical picture-based proofs to higher dimensions will prove much more difficult because we typically lack (real-world) intuition for high dimensional spaces. So difference calculus serves... | {
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• I didn't downvote, but I don't understand what's going on here. What are $f$ and $g$? What is $R$? Why is $n+1$ red? Are you proving the product rule, or something else? Those are some things that confused me, if you're interested. May 29, 2016 at 4:54
• I didn't downvote either, but this post doesn't appear to provi... | {
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