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# Maximum possible order of an element in $S_7 \text{ and } S_{10}$
Exercise :
Find the maximum possible order of an element of the group of permutations $S_7$. Do the same thing for $S_{10}$.
Discussion :
Recalling that any permutation can be written as a product of disjoint cycles :
$$c=c_1 c_2\dots c_r$$
the o... | {
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• Why $(1,2,3)(4,5)(6,7)$ has order $2$? – Skills Dec 29 '17 at 11:55
• @Skills Sorry, was typo. – Rebellos Dec 29 '17 at 11:57
• Case it by the order of the largest cycle, from high to low, so just 10 cases, and most will be already dominated. Also, no need to write out the actual cycles, just write their orders (e.g.... | {
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With these two facts in mind, for $n=7$ we are left with permutations of cycle types $$(7),\ (5,2),\ (4,3),\ (4,2,1),\ (3,3,1),\ (3,2,2)\quad \text{and}\quad (2,2,2,1).$$ For $n=10$ we are left with permutations of cycle types $$(9,1),\ (8,2),\ (7,3),\ (7,2,1),\ (5,5),\ (5,4,1),\ (5,3,2),\ (4,4,2),\ (4,3,3),\ (4,3,2,1)... | {
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"lm_q1q2_score": 0.840741534210072,
"lm_q2_score": 0.8539127548105611,
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• For the observation on prime power lengths: Let $\sigma\in S_n$ be a cycle of length $m=\prod_{p\mid m}p^{m_p}>1$. We want to show that there exists a $\tau\in S_n$ of order $m$ that is a product of disjoint cycles of prime power lengths. If $\sum_{p\mid m}p^{m_p}\leq n$ we can simply take disjoint cycles of lengths ... | {
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"lm_q2_score": 0.8539127548105611,
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# Where is the calculation hiding in this proof about how $p$ splits in $\mathbb{Q}(\zeta_n)$?
I just worked through a proof in Daniel Marcus' book Number Fields that if $p\nmid n$, the inertial degree of any prime ideal of $\mathbb{Q}(\zeta_n)$ lying over $p$ is equal to the order of $p$ in $(\mathbb{Z}/n\mathbb{Z})^... | {
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Marcus' argument: Consider the action of $\tau$ on the field generator $\zeta_n$ and of $\sigma$ on its image in $\mathbb{Z}[\zeta_n]/P_i$. $\tau^m=\mathrm{id.}$ if and only if $\tau^m(\zeta_n)=\zeta_n$ because it is a generator. Likewise, $\sigma^m=\mathrm{id.}$ if and only if $\sigma^m(\bar\zeta_n)=\bar\zeta_n$, with... | {
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To reiterate my question: where did I hide Marcus' calculation $(1-\zeta_n)\dots(1-\zeta_n^{n-1})=n?$ Or, is my proof wrong? Or, is the calculation actually extrinsic to the conclusion?
-
seems right to me. – user29743 Oct 11 '12 at 1:58
Actually I prefer your proof ! – user18119 Apr 15 '13 at 22:18
Your proof is c... | {
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# Chapter 4 : Truncation Errors and the Taylor Series¶
## Example: 4.1 Page No:79¶
In [13]:
from math import factorial
from scipy.misc import derivative
def f(x):
y=-0.1*x**4-0.15*x**3-0.5*x**2-0.25*x+1.2#
return y
xi=0#
xf=1#
h=xf-xi#
fi=f(xi)##function value at xi
ffa=f(xf)##actual function value at xf
#for n=0, i... | {
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#for n=4, i.e, fourth order approximation
def f4(x):
y=derivative(f3,x)
return y
f4i=f4(xi)##value of fourth derivative of function at xi
f4f=f3f+f4i*(h**4)/factorial(4)##value of fourth derivative of function at xf
Et_5=ffa-f4f##truncation error at x=1
print "The value of fourth derivative of f at x=0 :",f4i
print "Th... | {
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#for n=1, i.e, first order approximation
def f1(x):
y=derivative(f,x)
return y
f1i=f1(xi)##value of first derivative of function at xi
f1f=fi+f1i*h##value of first derivative of function at xf
et2=(ffa-f1f)*100/ffa##% relative error at x=1
print "The value of f at x=1 due to first order approximation :",f1f
print "% re... | {
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"openwebmath_score": 0.8045533299446106,
"tags": null... |
#for n=5, i.e, fifth order approximation
f5i=(f4(1.1*xi)-f4(0.9*xi))/(2*0.1)##value of fifth derivative of function at xi (central difference method)
f5f=f4f+f5i*(h**5)/factorial(5)##value of fifth derivative of function at xf
et6=(ffa-f5f)*100/ffa##% relative error at x=1
print "The value of f at x=1 due to fifth orde... | {
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## Example 4.3 : Page No:85¶
In [2]:
from math import pi,cos,factorial
m=input("Input value of m:")
h=input("Input value of h:")
def f(x):
y=x**m
return y
x1=1#
x2=x1+h#
fx1=f(x1)#
fx2=fx1+m*(fx1**(m-1))*h#
if m==1:
R=0#
elif m==2 :
R=2*(h**2)/factorial(2)#
elif m==3:
R=(6*(x1)*(h**2)/factorial(2))+(6*(h**3)/factoria... | {
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## Example 4.5: Page No: 95¶
In [4]:
from scipy.misc import derivative
def f(x):
y=x**3
return y
x=2.5#
delta=0.01#
deltafx=abs(derivative(f,x))*delta#
fx=f(x)#
print "true value is between : ",fx-deltafx,"and",fx+deltafx
true value is between : 15.4275 and 15.8225
## Example 4.6: Page No: 96¶
In [5]:
from scipy.... | {
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#computing condition number for x2bar
condnum2=x2bar*derivative(f,x2bar)/f(x2bar)#
print "The condition number of function for x =",condnum2,"is :",x2bar
if abs(condnum2)>1:
print "Function is ill-conditioned for x =",x2bar
The condition number of function for x = 0.18201112073 is : 1.72787595947
The condition number ... | {
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# What can be the maximum and minimum number of prime factors?
A number is having exactly $72$ factors or $72$ composite factors. what can be the maximum and minimum number of prime factors of this number? I search on internet and I found a solution of that question, solution like:
factorise $72$ as $2\times2\times2\... | {
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$$\left( \prod_{k=1}^n (\alpha_k + 1)\right) - (k+1).$$
The minimum is still $1$ (set $\alpha_1 = 73$). For the maximum, you might need to do some casework. You want the number $72 + k + 1$ to have exactly $k$ prime factors. So $k$ cannot get too large. (See below for precise reasoning.) I tried the first $10$ values ... | {
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The largest product which gives $k=72$ is $2\times2\times2\times3\times3$, so you can have at most $5$ prime factors (and for exactly $5$, you must have $n=p_1p_2p_3p_4^2p_5^2$ for some primes $p_1,...,p_5$). | {
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# Finding the joint CDF using the joint PDF; why can't I do this?
Find the joint CDF of the independent random variables $$X$$ and $$Y$$, where
$$f_X(x)=x/2, 0\le x \le 2,$$ and
$$f_Y(y)=2y, 0 \le y \le 1$$.
To do this, we can find the CDF separately for each of the marginal PDFs, and then multiply them together to... | {
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"openwebmath_score": 0.9312463998794556,
"ta... |
Thank you!
If $$X$$ and $$Y$$ are independent random variables, then $$F_{X,Y}(x,y)=\int_{-\infty}^x \int_{-\infty}^y f_{X,Y}(w,v)\,dv\,dw = \int_{-\infty}^x \int_{- \infty}^{y} f_X(w)f_Y(v)\,dv\,dw$$ $$=\int_{-\infty}^x f_X(w)\,dw\int_{-\infty}^{y}f_Y(v)\,dv = F_X(x)F_Y(y).$$
Method 1 (joint pdf approach) gives: $$f_... | {
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"openwebmath_score": 0.9312463998794556,
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# Two inner products different by a scalar
Let $$\langle \cdot, \cdot \rangle_{1}$$ and $$\langle \cdot,\cdot \rangle_{2}$$ be inner product on a finite-dimensional vector space with the property that \begin{align*} \langle u,v\rangle_{1}=0 \Leftrightarrow \langle u,v\rangle_{2}=0 \end{align*} for all $$u,v\in V$$. Sh... | {
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• If $\langle w,v\rangle_1=0$, then $\langle w,v\rangle_2=0$ and $0=c\times 0$ holds for any $c$, so I don't see any problem in that case. – learner Mar 16 at 5:53
• @learner Then there's no way to find the value of c. – Jiexiong687691 Mar 16 at 6:18
• You don't really need to find a particular value of $c$, you have t... | {
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Which is why we restrict our attention to the vectors $$\{w+tv : t \in \mathbb R\}$$, which (up to scaling) covers every element of the span of $$v$$ and $$w$$. Then the rest should be straightforward. | {
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# Numerical iterative method, estimating error
Given iterative method: $x_{n+1}=0.7\sin x_n +5 = \phi(x_n)$ for finding solution for $x=0.7\sin x +5$, I want to estimate $|e_6|=|x_6-r|$ as good as possible, with $x_0=5$, where $r$ is exact solution. This method obviously converges, because $\phi$ is contraction mappin... | {
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"openwebmath_score": 0.8589611649513245,
"ta... |
Given some $x_0$, you are looking for a bound on the error of the root for the equation:
$0 = 0.7 \mathrm{sin}(x) - x + 5$
Note that the function is bounded above by $5.7 - x$ and below by $4.3 - x$
Each of these have a trivial root at $5.7$ and $4.3$ respectively. You know that any roots of your function then fall in ... | {
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# probability of bingo
It is the first time I heard about bingo game and I would like to learn more on this game by mathematical analysis. To make it simple, I consider the American BINGO with 75 balls used and each game will at maximum draw 50 balls. The winning pattern on each card is the center cross (the column pa... | {
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• Thanks Mick. I see why my math is wrong now :) Aug 24 '14 at 19:51 | {
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# Approximation to the prime counting function
Is there a function similar to PrimePi that gives approximate value for large numbers?
In particular, I need a reasonably good approximation for $$\pi(x)$$, where $$x\approx10^{1000}$$.
More or less a function that gives $$\int_2^x\frac{dt}{\ln t}$$ or better.
Edit: I ... | {
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Plot[{ LogIntegral[x] - PrimePi[x], RiemannR[x] - PrimePi[x],
x/(Log[x] - 1) - PrimePi[x]}, {x, 2, 3 10^5}, MaxRecursion -> 3,
Frame -> True, PlotStyle -> {{Thick, Red}, {Thick, Darker @ Green},
{Thick, Darker @ Cyan}},
PlotLegends -> Placed["Expressions", {Left, Bottom}],
ImageSize -> 800, AxesStyle -> Thick]
Plot[... | {
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Finding the Jordan Canonical form of a $6 \times 6$ matrix
Find the Jordan Canonical Form of the following matrix $$\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 0 & 0\\ 1 & 0 & 1 & 0 & 0 & 0\\ 1 & 0 & 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0 & 1 & 0\\ 1 & 1 & 1 & 1 & 1 & 1\\ \end{bmatrix}$$
My try: I go about finding... | {
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"lm_q1_score": 0.9867771770811147,
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"lm_q2_score": 0.8519528094861981,
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I am a little stuck here.
Thanks for the help!!!
• $A-I$ is nilpotent, it will most certainly not have the same rank as its square! You'll find that $(A-I)^2$ has all its coefficients zero, except for its last line which equals $$(\;4\quad 0\quad 0\quad 0\quad 0\quad 0\;)$$ – Olivier Bégassat Jun 3 '15 at 11:00
• @Ol... | {
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24
Apr 18
## Solution to Question 1 from UoL exam 2016, Zone B
This problem is a good preparation for Question 2 from UoL exam 2015, Zone A (FN3142), which is more difficult.
## Problem statement
Two corporations each have a 4% chance of going bankrupt and the event that one of the two companies will go bankrupt is... | {
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There are a couple of general ideas to understand before embarking on calculations. The return on the bond of one company is a binary variable taking values 0% and -100%. All calculations involving it are similar to the ones for the coin. After doing calculations the return figures can be translated to dollar amounts b... | {
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return values are 0 (none of the companies goes bankrupt), -50 (one goes bankrupt and the other does not) and -100 (both go bankrupt). The corresponding probabilities follow from Table 2 and we get Table 3. Probability table for return on the total portfolio Total return Probabilities 0 0.9216 -50 $2\times 0.0384=0.076... | {
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We say that Value at Risk is sub-additive if $VaR_P^\alpha\le Var^\alpha_{P_1}+Var^\alpha_{P_2}$. Our calculations show that Value at Risk is not sub-additive in case of independent returns. This has an important practical implication. Suppose that a financial institution has several branches and each of them keeps the... | {
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# does a convex polynomial always reach its minimum value?
Consider a convex polynomial $p$ such that $p(x_1,~x_2,\dots x_n)\geq 0~\forall x_1,~x_2,\dots x_n\in \mathbb{R}^n$. Does the polynomial reach its minimum value?
This is not true for non-convex polynomials like $(1-x_1x_2)^2+x_1^2$, see the response of J.P. M... | {
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EDIT: As requested, I'll expand on "using compactness". For each $s \in \mathbb S^{n-1}$, there is $t > 0$ such that $p(ts) > C$. Thus the open sets $\{s \in \mathbb S^{n-1}: p(t s) > C\}$ for $t > 0$ form an open covering of $\mathbb S^{n-1}$. Because $\mathbb S^{n-1}$ is compact, this has a finite subcovering, i.e. $... | {
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# How to prove if this $\sum_{l=0}^{n}\binom{n}{l}=2^{n}$ is valid for all $n\in \mathbb{N}$? [duplicate]
Prove for for all $n\in \mathbb{N}$: $\sum_{l=0}^{n}\binom{n}{l}=2^{n}$
I know the steps of induction but i have no idea how to prove this equation with binomial coefficient.
1) For the induction base i need to ... | {
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Here, we rely on something known as pascal's identity: $\binom{n+1}{r} = \binom{n}{r}+\binom{n}{r-1}$.
Examining the case of $n+1$, we have then:
$$\begin{array}{rlr}\sum\limits_{l=0}^{n+1}\binom{n+1}{l}&=\sum\limits_{l=0}^{n+1}\left(\binom{n}{l}+\binom{n}{l-1}\right)&\text{via pascal's identity}\\ &=\left(\sum\limit... | {
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\begin{align} \sum_{k=0}^{n} \binom{n}{k} &= 1+ \sum_{k=1}^{n-1} \binom{n}{k} + 1 \\ &= 1+ \sum_{k=1}^{n-1} \left[ \binom{n-1}{k-1} + \binom{n-1}{k} \right] + 1 \\ &= 1+ \sum_{k=1}^{n-1} \binom{n-1}{k-1} + \sum_{k=1}^{n-1}\binom{n-1}{k} + 1 \\ &= \sum_{k=1}^{n} \binom{n-1}{k-1} + \sum_{k=0}^{n-1}\binom{n-1}{k} \\ &=\su... | {
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Finding the point on $6x^2+y^2=262090$ that is nearest to the point $(1045,0)$
I know that to find the point on $6x^2+y^2=262090$ that is nearest to the point $(1045,0)$, we can try to minimize the squared distance $S=(x-1045)^2+262090-6x^2$. However, calculus tells us that this function does not have a minimum point ... | {
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There is nothing wrong with using the squared distance.
You converted the problem into a one-parameter minimization problem. You are looking for the minimum of a smooth function on the interval $[-\sqrt{262090/6},\sqrt{262090/6}]$. The minimum value is obtained at a zero of the derivative (a critical point) or at one ... | {
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• I'm afraid the result you gave $x=\sqrt{262090}$ does not satisfy the equation given: $6x^2+y^2 = 6\cdot(\sqrt{262090})^2+0^2 = 1572540$ which does not equal the RHS... – CiaPan Mar 28 '18 at 11:53
• @CiaPan Good catch, thanks! I forgot to divide by some sixes. Does it make more sense now? – Joonas Ilmavirta Mar 28 '... | {
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# Proving Floor and Ceiling of a Rational Number
Suppose x,y $\in \mathbb{Z}^+$
Prove $\lceil x/y \rceil = \lfloor (x-1)/y \rfloor + 1$
I was considering using the definition of floor and ceiling to prove this. But this does not seem like a valid proof to me as I assume the right hand side is already equal to the le... | {
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starting from line 2, the inequality can be rewritten as
\begin{equation*} n-1+1/y \leq x/y < n+1/y \end{equation*} since x,y are integers, the left inequality is equivalent to
\begin{equation*} n-1 < x/y \end{equation*}
and the right inequality is equivalent to \begin{equation*} x/y \leq n \end{equation*}
• I get ... | {
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# In how many ways can 4 cards be drawn randomly from a pack of 52 cards such that there are at least 2 kings and at least 1 queen among them?
So i tried this question in two ways
(i)In my first method I made different possible arrangements and then find the number of ways
So, the different possibilities are:
2 kin... | {
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Simlarly for the queens, $${4\choose 2}$$ for the case we have 2 queens, and another $${4\choose 2}$$ because we have $${4\choose 2}=6$$ possibilities of the first 2 kings. (Note that here we don't multiply by $$2$$ here since if we have $$k,k,q_1,q_2$$ we only want to cancel only 1 equivalent case and that is $$k,k,q_... | {
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# How to solve this equation by hand? $4.68-4.50\cos\alpha-1.23\alpha=0$
I'm trying to solve the next equation
$$4.68-4.50\cos\alpha-1.23\alpha=0$$
But when using a calculator it gives $$\alpha$$ as $$0.226$$ but it s supposed to be $$4.483$$.
As far as I know it has no closed form solution so its necessary a numer... | {
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I don't know how the calculator got its value, $$0.226$$, or where the value $$4.483$$ comes from. Plotting $$f(\alpha) = 4.68 - 4.50 \cos \alpha - 1.23 \alpha$$ gives
We expect this graph to be a cosine with midline given by $$4.68 - 1.23 \alpha$$, so as soon as a local minimum is above zero, we need not proceed furt... | {
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Let's separate the polynomial (actually, linear in this problem) and non-polynomial parts and plot them separately. We want $$4.68 - 1.23 \alpha = 4.50 \cos \alpha$$ Since $$-1 \leq \cos \alpha \leq 1$$, the right-hand side is in the interval $$[-4.50, 4.50]$$. The left-and side is a line and we can solve the the inter... | {
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## Binary Search
So we can proceed by binary search. We know that the $$f$$ is positive on one side of the zero and negative on the other, so we can cut the interval containing the zero in half for each evaluation of $$f$$. You seem to be using thousandths as precision, so we need only evaluate about ten times. We ind... | {
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## Taylor Series
There are other methods we could try. We could replace the cosine with leading segments of its Taylor expansion. Then we are seeking roots of the resulting polynomial. This works if we can center the series close to the root. The Taylor series of cosine centered at $$0$$ is $$\cos x = 1 - \frac{x^2}{2... | {
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## Newton's method
The last method I'll show is Newton's Method. We have prior information that $$f$$ has a zero near $$\alpha = 4$$, so we use a linear approximation to $$f$$ at that point, find that approximation's $$\alpha$$-intercept, and report that as an improved location of the zero. We need to know that $$\alp... | {
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# Two vectors with the same normal surface projection and the same normal surface cross product, are equal?
I have two vectors, $$\mathbf a$$ and $$\mathbf b$$, that fulfill the following conditions:
$$(\mathbf a-\mathbf b)\cdot \mathbf n= 0$$
$$(\mathbf a-\mathbf b)\times \mathbf n=\mathbf 0$$
being $$\mathbf n$$ ... | {
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You could also do this using the fact that $$\vec{v} \cdot \vec{w} = \vert\vec{v}\rvert \lvert \vec{w} \rvert \cos\theta$$ and $$\lvert\vec{v} \times \vec{w}\rvert = \vert\vec{v}\rvert \lvert \vec{w} \rvert \sin\theta$$. In your case this leads to \begin{align*} (\vec{a} - \vec{b})\ \cdot \ \vec{n} = \lvert\vec{a} - \v... | {
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## Get ready for AP® Calculus
### Course: Get ready for AP® Calculus>Unit 7
Lesson 3: Extraneous ... | {
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When you plug X = -8 back into the equation, you end up with sqrt(9) = -3.
This is extraneous because we are supposed to use the principle root.
I have three questions.
#1: Why do we have to use the principle root? It seems logical that the sqrt(9) should be equal to both +3 and -3.
#2: Why is this extraneous solution... | {
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3. As mentioned above, x = -8 is the solution to the equation -√(4x + 41) = x + 5. So, if we graph -√(4x + 41) instead of √(4x + 41), it will intersect with x + 5 at x = -8.
• In the last situation, why did they plug -1 in the equation? was it a random number?
• This got me at first too. In the original question, it sp... | {
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After you have solved the resulting linear or quadratic equation for x, remember that you're not finished yet! Because every positive number has a positive and a negative square root, but the radical symbol denotes only the positive (principal) square root, the act of squaring both sides can create invalid (extraneous)... | {
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Why multiplication of 142857 with 2,3,4,5,6 gives the same digits shifted?
I was reading about a not so practical way to determine the divisibility of a number by $$7$$.
At some point the following number is mentioned: $$142857$$ (which is the result of $$\frac{999999}{7}$$) and apparently this number as I have verifi... | {
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$$\color{red}{1}\times10 = 10 = \color{green}{1}\times7+\color{red}{3}$$ (*)
$$\color{red}{3}\times10 = 30 = \color{green}{4}\times7+\color{red}{2}$$ (**)
$$\color{red}{2}\times10 = 20 = \color{green}{2}\times7+\color{red}{6}$$
$$\color{red}{6}\times10 = 60 = \color{green}{8}\times7+\color{red}{4}$$
$$\color{red}{4... | {
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$$100\frac {1,000,000 - 1}7 - k(1,000,000 -1) =$$
$$14,285,700 - \overline{k000000} +k$$.
Hmm.... that we know $$k = 14$$ seems to be a really nice coincidence.
$$14,285,700 - 14,000,000 + 14 = 285714$$ and of course the numbers are reversed.
If we can assume that if $$10^{m_j} = (\text{first }m_j\text{ digits of }... | {
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And that's it.
This will actually be true of any $$n$$ so that $$\gcd(n,10) = 1$$
Take, say $$\frac 1{17}= 0.\overline{0588235294117647}$$
We should get $$0588235294117647 \times 2..... 16$$ should be the same digits shifted. Try it.
• $\frac{999,999}{7}=142857$. You used $1,000,000-1$ instead. But I don't understa... | {
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# Alternating Sums of powers of 2
#### Rutzer
Hello,
Here's an interesting problem that I have solved, but would like someone else's input simply to see if people come up with the same solution or possibly find a more elegant one. I hope anyone who tries this has fun with the problem!
Which numbers can be written by... | {
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# sum of the complex series
• Jun 4th 2013, 12:39 PM
alteraus
sum of the complex series
Hello everybody, I am wondering how can I find the sum of the series $\sum_{n=1}^{\infty}\frac{ni^n2^n}{(z+i)^{n+1}}$ ? I found out that the series converges for |z+i|>2, but I am not able to find the sum of the series , thank you ... | {
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thank you
• Jun 9th 2013, 08:24 AM
alteraus
Re: sum of the complex series
no more suggestions?
• Jun 9th 2013, 08:49 AM
Prove It
Re: sum of the complex series
It works exactly the same way as a real geometric series.
• Jun 9th 2013, 09:04 AM
alteraus
Re: sum of the complex series
thank you very much, I found the result... | {
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# Count elements of a sorted matrix that fall into a given interval
I have a $n\times n$ matrix called $M$, and two integers $k_\min$ and $k_\max$. Each row and each column of M is sorted in the increasing order.
I would like to know if there is way I can count the number of its elements which are inside $[k_\min, k_... | {
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• I don't get your first paragraph. Knowing $|A|$ and $|B|$, nothing is known about $|A \cap B|$ without further knowledge. Your algorithmic idea is nice, though, and can be adapted to solve the original problem in one pass by moving two points upwards in the matrix and summing over the difference in their positions. –... | {
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# Distance covered by falling particle P
Particle P released from rest at O. Falls freely under gravity until reaching point A which is $1.25$m below O.
(i) Find speed of P at A and time taken for P to reach A.
P continues to fall, but now its downward acceleration $t$ seconds after passing through A is $(10-0.3t)$ me... | {
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# Prove convergence of the sequence $(z_1+z_2+\cdots + z_n)/n$ of Cesaro means [duplicate]
Prove that if $\lim_{n \to \infty}z_{n}=A$ then: $$\lim_{n \to \infty}\frac{z_{1}+z_{2}+\cdots + z_{n}}{n}=A$$ I was thinking spliting it in: $$(z_{1}+z_{2}+\cdots+z_{N-1})+(z_{N}+z_{N+1}+\cdots+z_{n})$$ where $N$ is value of $n... | {
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Can you find proper $a_i$ in terms of say $z_i$'s??
• thankyou for help. It's not homework. Would $a_{i}$ be $\frac{z_{i}}{n}$, and since for $n \to\infty$all sums go to $0$, it would be proved? @TheJoker – Mykolas Oct 5 '12 at 19:40
• and would my first idea be wrong? – Mykolas Oct 5 '12 at 19:44
• You can take $a_i=... | {
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Let $\{a_n\}_{n=1}^{\infty}$ be a real sequence such that $\lim_{n\to\infty}a_n=a$. And we have a family of finite sequences $\{\{b_{nm}\}_{m=1}^{m=n}\}_{n=1}^{\infty}$: $$b_{11}\\ b_{21},b_{22}\\ b_{31},b_{32},b_{33}\\ \cdots$$ such that $$b_{mn}\geq 0$$ for all $m,n$, and $\sum_{m}b_{nm}=1$ for each $n=1,2,\cdots$. L... | {
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# Math Help - Proof of an ellipse
1. ## Proof of an ellipse
The question I am stuck on is (iii):
This part concerns the conic with equation
$3x^2+5y^2=75$
(i) By rearranging the equation of the conic, classify it as an ellipse, parabola or hyperbola in standard position, and sketch the curve.
I have re-arranged th... | {
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I have re-arranged the equation and discovered it is an ellipse in standard position:
$(x^2)/5^3 + (y^2)/(\sqrt{15})^2 = 1$
a=5, b= $\sqrt{15}$
The points of the ellipse are (-5,0), (0, $\sqrt{15}$), (5,0) and (0, $-\sqrt{15}$)
Surely you meant "the points where the ellipse meets the axis are..."
(ii) Find exact valu... | {
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The other point of intersection is (5, 0).
For this point, PF = (a + ae) , and PD is (2a + a/e)
Thank you for the quick reply. I understand that the other point of intersection is (5,0), but the question specifically says:
Check your answers to part (a)(ii) by verifying that the equation PF = ePd holds at each of the... | {
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### Where were we? Where are we going?
In part 1 we wrote our first R function to compute the difference between the max and min of a numeric vector. We checked the validity of the function’s only argument and, informally, we verified that it worked pretty well.
In this part, we generalize this function, learn more t... | {
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quantile(gapminder$lifeExp) ## 0% 25% 50% 75% 100% ## 23.5990 48.1980 60.7125 70.8455 82.6030 quantile(gapminder$lifeExp, probs = 0.5)
## 50%
## 60.7125
median(gapminder$lifeExp) ## [1] 60.7125 quantile(gapminder$lifeExp, probs = c(0.25, 0.75))
## 25% 75%
## 48.1980 70.8455
boxplot(gapminder$lifeExp, plot =... | {
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This is better:
qdiff3 <- function(my_x, my_probs) {
stopifnot(is.numeric(my_x))
the_quantiles <- quantile(x = my_x, probs = my_probs)
return(max(the_quantiles) - min(the_quantiles))
}
qdiff3(my_x = gapminder$lifeExp, my_probs = 0:1) ## [1] 59.004 If you are going to pass the arguments of your function as arguments of... | {
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Oops! At the moment, this causes a fatal error. It can be nice to provide some reasonable default values for certain arguments. In our case, it would be crazy to specify a default value for the primary input x, but very kind to specify a default for probs.
We started by focusing on the max and the min, so I think thos... | {
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# Epsilon-Delta Proof problem
Hey guys heres the problem,
## Homework Statement
lim (4x^2+9) / (3x^2 +5) = 4/3
x->infinity
find k, such that x> k/sqrt(epsilon) guarantees abs((4x^2+9) / (3x^2 +5) - 4/3) < epsilon
## The Attempt at a Solution
By removing the absolute sign and making the denominator common, we get
... | {
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$$9x^2 = \frac{7}{\epsilon} - 15$$
Let's also set $x$ to the minimum allowed:
$$x = \frac{k}{\sqrt{\epsilon}}$$
and substitute this into the boundary case:
$$\frac{9k^2}{\epsilon} = \frac{7}{\epsilon} - 15$$
or equivalently
$$9k^2 = 7 - 15\epsilon$$
Then we have
$$k = \frac{1}{3}\sqrt{7 - 15\epsilon}$$
assumin... | {
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# Expressing the values of a matrix at pow N
I have a square matrix (that comes from a Markov Chain) that looks like that:
$$Q = \begin{bmatrix} 0 & 1& 0 & 0 & .. & 0 & 0\\ 0 & a & 1-a & 0 & .. & 0 & 0\\ 0 & 0 & b & 1 - b & .. & 0 & 0\\ .. & .. & .. & .. & .. & .. & .. \\ 0 & 0 & 0 & 0 & .. & 0 & 1 \end{bmatrix}$$
w... | {
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Edit4:
it's $a <= b$ and not $a < b$, sorry!
-
If ${\bf R}^n$ has a basis consisting of eigenvectors for $Q$, then $Q^N=PD^NP^{-1}$, where $P$ is the matrix whose columns are the eigenvectors of $Q$, and $D$ is the diagonal matrix whose diagonal entries are the eigenvalues of $Q$. The point is, it is very easy to com... | {
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Then the eigenvalues of $M$ are simply $a_1, \ldots, a_{n-1},1$. The eigenvector corresponding to $1$ is obviously the constant 1 vector. The eigenvector for $a_i$ is the vector $v^i$, where $$v^i_j = \begin{cases}0, &j > i\\ \frac{\prod_{k=j}^{i-1} (a_k-1)}{\prod_{k=j}^{i-1} (a_k-a_i)}, & j \leq i\end{cases}$$ and $v^... | {
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Prove that $T$ is linear, one to one and onto.
Consider the function $T:\mathbb R^2 \to \mathbb R^2$ given by $$T \begin{pmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \end{pmatrix} = \begin{bmatrix} 2x+y \\ -3x \end{bmatrix}$$ Prove that $T$ is linear, one-to-one, and onto. Here is what I 've got so far:
Let $u,v \in... | {
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The matrix of $T$ is $$A = \begin{bmatrix} &2 &1 \\ &-3 &0 \end{bmatrix}$$ The kernel of $T$ is $\bf0$, because the only solution to $A * [x,y]^T = \bf 0$ is $x=y=0$. Because $$\operatorname{dim}\,\operatorname{ker}\,A + \operatorname{dim}\,\operatorname{rank}\,A = \text{# of columns of A},$$ it follows that $\operator... | {
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# Defining Change in an interval
I don't know an appropriate topic for this. Here is my question though.
How does one define change in something? For example, regarding Riemann sums one defines change in the width of the rectangles to be $$\Delta x_i = x_i - x_{i-1}~~~,~~~i=1,2,\ldots, n$$
Would I be wrong in defini... | {
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# Math Help - series
1. ## series
Is there a better way to do this problem?
For the following convergent series, at least how many terms do you have to sum so that the partial sum is within 0.001 of the sum of the series?
$1 - \frac{2}{3}+ \frac{3}{9} - \frac{4}{27}+...+(-1)^{k-1} \frac{k}{3^{k-1}}+...$
If you add... | {
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Hence, we have: . $\frac{2}{3}S \:=\:\frac{3}{4} \quad\Rightarrow\quad S \:=\:\frac{9}{8} \quad\Leftarrow\text{ sum of the series}$
4. Originally Posted by Calculus26
See attachment
I might be wrong, but it seems like what you did is find the first term whose absolute value is less than 0.001.
5. Yes that is from the... | {
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Hence, we have: . $\frac{2}{3}S \:=\:\frac{3}{4} \quad\Rightarrow\quad S \:=\:\frac{9}{8} \quad\Leftarrow\text{ sum of the series}$
It looks like you made a sign error. I think the sum is 9/16.
This is what I did:
$x -\frac{x^{2}}{3} + \frac{x^{3}}{9} - \frac{x^{4}}{27} + ... = \frac{x}{1+\frac{x}{3}} \ , \ \Big|\fr... | {
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# periodicity of constant discrete time signals
are constant discrete time signals periodic?
example $$$$e^{i10\pi n}$$$$ my proffesor says that this signal is aperiodic, in the discrete sense. but it seems wrong, because unlike in the continuous case, i can calculate the smallest time period , which is 1.
• Welcome ... | {
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# Proof of Product Rule for Derivatives using Proof by Induction
I am trying to understand the proof of the General Result for the Product Rule for Derivatives by reading this.
Relevant parts are as follows:
Basis for the induction $$D_x \left({f_1 \left({x}\right) f_2 \left({x}\right)}\right) = D_x \left({f_1 \left... | {
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Note: I hope someone can correct my LaTeX typesetting. I was under the impression that the align environment would automatically number the formulas I write. Thanks in advance.
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How's that (I had to make the products small to make things fit)? – David Mitra Dec 11 '11 at 18:54
@DavidMitra: Thank you. – Sara Dec 15 ... | {
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Multipying both sides above by $\rm\ f_{n+1}\cdots f_1\$ yields the sought result.
Key is this. With $\rm\ L\: f\: :=\: D\:f/f\$ we have $\rm\ L(f\:g)\ =\ L(f) + L(g)\:.\:$ The above proof is simply the inductive extension to a product of $\rm\:n+1\:$ terms, i.e. $\rm\ L(f_{n+1}\cdots f_1)\:=\: L(f_{n+1})+\:\cdots\:+L... | {
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:The answer looks interesting even if I am yet able to comprehend it, but thank you for answering anyway. – Sara Dec 15 '11 at 3:20
@Sara I've expanded the answer. Please let me know if anything is still not clear. – Bill Dubuque Dec 15 '11 at 4:18
My main problem is that my syllabus have not covered logarithmic di... | {
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# Finding the set of values for k of a modulus function.
"Find the set of values of k for which |(x-4)(x+2)| = k has four solutions."
EDIT:
Ok so I thought I'd start with setting the modulus function equal to k and -k to get the two set of results.
Doing that I ended up with:
(x -4)(x +2) = k
and
(x - 4)(x +2) =... | {
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Solving the equation $(x-4)(x+2)=k$ gives you $x_{1,2}=1\pm\sqrt{9+k}$; solving the equation $(x-4)(x+2)=-k$ gives you $x_{3,4}=1\pm\sqrt{9-k}$.
• $k>-9$ has to hold in order to get two solutions out of $x_{1,2}=1\pm\sqrt{9+k}$
• $k<9$ has to hold in order to get two solutions out of $x_{3,4}=1\pm\sqrt{9-k}$
• $k\neq 0... | {
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# Shorthand notation for powers of logarithmic functions
I've got an assignment here, and one question is throwing me off, as I've never seen it written like this before..
$$\int\frac{\ln^3 x}{x}\ dx$$
Is this the same as $$\int\frac{(\ln x)^3}{x}\ dx\;\;?$$
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Yes it is. Moreover $$\int \frac{\ln^3(x)}{x}\,{\rm d}x... | {
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$\sin^2x=(\sin x)^2$ has a lot to answer for... I need to use $\arcsin x$ now or else I see $\sin^{-1} x=\frac{1}{\sin x}$ etc. – Jp McCarthy Feb 8 '13 at 17:10 | {
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# Determing if a function with integral within it is even/odd
Tags:
1. Nov 10, 2015
### mr.tea
1. The problem statement, all variables and given/known data
Determine if the function is even/odd/neither without solving the integral
2. Relevant equations
$$f(x)=\arctan (x)-2\int_0 ^x{\frac{1}{(1+t^2)^2}}$$
3. The at... | {
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5. Nov 10, 2015
### epenguin
HoI essentially just answered your question in #4. He did it for the particular function, but you could write out the proof again for integral of any function f which satisfies the condition of evenness which you should write out. The key point which you may have missed in the detail is, ... | {
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