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# Number of binary numbers with two consecutive zeros
This is an unsolved question in my textbook (so full answers are ok too). We are asked to count the number of binary numbers of length 15, beginning with '1', that have a total of ten 1s, five 0s, and two consecutive zeros.
My thoughts are, we can begin by orderin... | {
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We solve (a). We first count the ways to have no two consecutive $0$'s. Put the ten $1$'s in a row as in the post. There are $10$ "gaps" immediately to the right of a $1$, including the gap at the right end. We must choose $5$ of these gaps for the $0$'s. This can be done in $\binom{10}{5}$ ways.
Next we count the tot... | {
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$2.$ We can attack interpretation (a) in other ways. One approach is to list the types of ways we could have one or more consecutive $0$'s, count, and add. We could have $5$ consecutive $0$'s; r we could have $4$ and a singleton; or we could have a $3$ and a $2$; or $3$ and two singletons; or two $2$'s and a singleton;... | {
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T:=Filtered(PermutationsList([1,1,1,1,1,1,1,1,1,1,0,0,0,0,0]),L->L[1]=1);;
Size(T);
• 1750: The number of binary numbers with 10 ones and 5 zeroes with two consecutive zeroes somewhere. Computed by the following:
S:=Filtered(T,L->ForAny([1..14],i->L[i]=0 and L[i+1]=0));;
Size(S);
• 1200: The number of binary number... | {
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Q2:=Filtered(T2,L->Size(Filtered([1..14],i->L[i]=0 and L[i+1]=0))=1);;
Size(Q2);
While it's a good idea to understand the material, the computer can provide a handy check of one's argument (and a means of coming up with the argument in the first place; and a means of pinpointing a flaw, if one's argument is invalid).... | {
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# Find the remainder of $3^{333}$ divided by $100$
Find the remainder of $$3^{333}$$ divided by $$100$$
So I can find that $$100=2^2\cdot 5^2$$
Then I want to find $$3^{333}$$ mod $$4$$ and mod $$25$$ and use chinese remainder theorem to find a solution mod $$100$$.
I can find that $$3^{333}\equiv (-1)$$ mod $$4$$
... | {
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So, if $$x=3^{333}$$, $$x=-2+25k \equiv-2+k \equiv-1 \pmod4$$
Hence $$k \equiv1 \pmod 4$$
Therefore $$x=-2+25(1+4k')=23+100k' \equiv23 \pmod{100}$$
Use the totient function. $$2$$ and $$5$$ are distinct primes so $$\phi(100)=\phi(2^25^2)=\phi(2^2)\phi(5^2)=[2(2-1)][5(5-1)]=40.$$
Now since $$\gcd(3,100)=1$$ we have ... | {
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Use Euler.
$$\phi(25) = 20$$ so $$8^{20}\equiv 1 \pmod {25}$$
So we need to figure out $$8^{17}\equiv 8^{-3}$$ which is which will be equivalent to $$k^3$$ where $$k$$ is the multiplicative inverse of $$8$$. $$8*3=24\equiv -1$$ so $$8*(-3)\equiv 1 \pmod{25}$$ and $$8^{17} \equiv (-3)^3 \equiv -27\equiv -2\equiv 23 \p... | {
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1. ## Investing Fundamentals
John recently won a large sum of money which will be disbursed through two alternative payments:
• Option 1: A one off payment of RM100,000 that will be paid in 10 years.
• Option 2: Three uneven payments of RM10,000, RM30,000 and RM40,000 that will be paid in year 1, 5 and 10, respective... | {
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The comparison can then be made between the two present values to confirm that indeed the second investment is worth more than the first
PV = 100000 x (1+8%)^-10
PV = 100000 x 0.46319
PV = 46,319.35
Code:
T CF x PVIF CF x PVIF Present Value
1 10000 x (1+8%)^-1 10000 x 0.92593 9,259.26
5 ... | {
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7. ## Re: Investing Fundamentals
Once we have the NIV - net intermediate value, we can confirm the NFV - net future value as shown in the other reply and the NPV - net present value as shown in the second last reply
To find NPV from NIV, we shall discount it by 5 years at 8%
Code:
NIV factor factor NIV x factor ... | {
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The RM10,000 that he gets "in year 1" can be invested, at 8% interest, for 10 years (I assume compounded annually) and so will be worth $10000(1.08)^{10}= 21589.26$ at the end of the 10 years. The RM30,000 that he gets "in year 5" can be invested at 8% interest for 5 years and so will be worth $30000(1.08)^{5}= 44079.8... | {
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# Does there exist a function such that the preimage of $x ^ { 2 } + y ^ { 2 } \leq 1$ is the closed interval $[-1,1]?$
Does there exist a continuous function $$f : \mathbb { R } \rightarrow \mathbb { R } ^ { 2 }$$ such that the preimage of the closed unit disk $$x ^ { 2 } + y ^ { 2 } \leq 1$$ is the closed interval $... | {
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f:R -> R×R, x -> (x,0) is continuous.
Let D be the closed unit disk.
The preimage of D by f is
f$$^{-1}$$(D) = { x : f(x) in D } = [-1,1].
• Good example, but you really should format it better. – zhw. Dec 20 '18 at 17:25 | {
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# Summation Of Product Of Fibonacci Numbers
Im trying to find out a general term for the following summation of products of fibonacci numbers:--
# $$\sum_{k=4}^{n+1} F_{k}F_{n+5-k}\; , n \geq 3$$
I tried using Binet's equation but I am getting stuck at a certain point. So, I would be very glad if someone could post ... | {
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Using the closed form for the Fibonacci numbers, $$F_n=\frac{\phi^n-(-1/\phi)^n}{\sqrt{5}}\tag{1}$$ and Lucas numbers $$L_n=\phi^n+(-1/\phi)^n\tag{2}$$ we get \begin{align} F_iF_{n-i} &=\frac{\phi^i-(-1/\phi)^i}{\sqrt{5}}\frac{\phi^{n-i}-(-1/\phi)^{n-i}}{\sqrt{5}}\\ &=\frac{\phi^n+(-1/\phi)^n-(-1)^i\left(\phi^{n-2i}+(-... | {
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# Have I used induction correctly in this proof of $x<y \implies x^n<y^n$?
A while ago I posted an attempt at a proof of $x<y \iff x^n<y^n$. It was pointed out that I hadn't actually used induction, and had instead done a direct proof. Below is the link to the question, so please do not mark this question as a duplica... | {
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In my opinion you should work better out where and how you use the inductive claim (I. C.).
$x^{n+1}=x\cdot x^n\stackrel{I.C}{<}x\cdot y^n\stackrel{x<y}{<}y\cdot y^n=y^{n+1}$
• I am not sure I follow your superscript notation, could you possibly explain that in more detail? – Benjamin Aug 8 '18 at 15:54
• Sure: The s... | {
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Difficult integration question
Sudharaka
Well-known member
MHB Math Helper
Flipflop's question from Math Help Forum,
Determine the following:
a)$\int\frac{1}{e^x+2}\,dx$
b)$\int\frac{\sqrt{16x^2-9}}{x}\,dx$
How do I do part (b). I've been stuck at it for hours, even after looking at all the different types of int... | {
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# What is the method to compute $\binom{n}{r}$ in a recursive manner?
How do you solve this?
Find out which recurrence relation involving $\dbinom{n}{r}$ is valid, and thus prove that we can compute $\dbinom{n}{r}$ in a recursive manner.
I appreciate any help. Thank You
-
Is $C(n,r)$ the binomial coefficient? Most ... | {
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-
It might be useful to mention that that recurrence defines Pascal's Triangle. – robjohn May 9 '12 at 18:32
@robjohn Right. I was trying to draw a diagram of the Pascal's triangle. I will update the post soon. – user17762 May 9 '12 at 18:38
@J.M. Thanks. I was not aware of \dbinom. – user17762 May 9 '12 at 18:42
No wo... | {
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Question
# If the sum to infinity of the series $$1+4x+7x^2+10x^3+\cdots$$ is $$\dfrac{35}{16}$$, then $$x=$$
A
15
B
25
C
37
D
17
Solution
## The correct option is A $$\displaystyle \frac{1}{5}$$Let$${ S }_{ \infty }=1+4x+7{ x }^{ 2 }+10{ x }^{ 3 }+...\infty$$ $$...(1)$$Now, multiply by $$x$$ throughout in equ... | {
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# the value of $\lim\limits_{n\rightarrow\infty}n^2\left(\int_0^1\left(1+x^n\right)^\frac{1}{n} \, dx-1\right)$
This is exercise from my lecturer, for IMC preparation. I haven't found any idea.
Find the value of
$$\lim_{n\rightarrow\infty}n^2\left(\int_0^1 \left(1+x^n\right)^\frac{1}{n} \, dx-1\right)$$
Thank you
... | {
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-
Nice work! Now it's proven :) – Start wearing purple Jul 3 '13 at 10:29
Very nice! The value of $\pi^2/12 = .822...$ falls within my bounds of .75 to .875 which is comforting. – marty cohen Jul 3 '13 at 13:33
@sos440 here is a shortcut I noticed. By DCT $\lim_{n\to\infty}\int_{0}^{1} n \left(1-y^{1/n}\right) (1 + y... | {
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We now look at $\int_0^v (1+x^n)^{1/n}\, dx$ to see what happens as $v \to 1$.
\begin{align} \int_0^v (1+x^n)^{1/n}\, dx &=\sum_{k=0}^{\infty} \binom{1/n}{k} \int_0^v x^{kn}\, dx\\ &=\sum_{k=0}^{\infty} \binom{1/n}{k} \frac{v^{kn+1}}{kn+1}\\ &=v+\frac{v^{n+1}}{n(n+1)}+\sum_{k=2}^{\infty} \binom{1/n}{k} \frac{v^{kn+1}}... | {
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Mathematica evaluates the integral to $$\int_0^{1}(1+x^n)^{1/n}dx={}_2F_1\left(-\frac{1}{n},\frac1n,1+\frac1n;-1\right).\tag{1}$$ Next, let us write the standard series representation for the hypergeometric function $$_2F_1(a,b,c;t)=\sum_{k=0}^{\infty}\alpha_kt^k,\qquad \alpha_k=\frac{\Gamma(a+k)\Gamma(b+k)\Gamma(c)}{k... | {
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# Is a certain subset of $M_2(\mathbb{R})$ closed?
I am trying to prove that the set M of all matrices in the normed linear space $M_2(\mathbb{R})$ such that both eigen values are real is closed (under metric topology; metric induced by the norm).
Following is my attempt. I recall the property that finite dimensional... | {
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So if you can prove that $\Delta(a,b,c,d)$ is continuous, you've won (do you see why?)
• discriminant turns out to be Tr(M)^2 - 4*det(M) which is continuous as it is a linear combination of continuous functions (Tr^2 is composition of two continuous functions). Thus my original set is the inverse image of [0,\infty) u... | {
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# Probability of winning games at tournament
I know it's a simple problem but apparently I am doing something wrong: The probability of winning every single game at a tournament is 0.4. There is only win and lose - no draw. Find the probability of winning exactly 2 games by playing at most 6 games.
Since winning and ... | {
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Probability of playing 3 games and winning 2 is $$2\cdot(0.4)^2(0.6)^1$$
Probability of playing 4 games and winning 2 is $$3\cdot(0.4)^2(0.6)^2$$
Probability of playing 5 games and winning 2 is $$4\cdot(0.4)^2(0.6)^3$$
Probability of playing 6 games and winning 2 is $$5\cdot(0.4)^2(0.6)^4$$
We add all these and we ... | {
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# Why is $\nabla_x x^T A x = 2 A x$?
Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ and $f(x) = x^T A x$. I will denote $\nabla_x$ or $\nabla$ as the gradient to some vector-valued variable and $\nabla^2$ or $H$ as the Hessian.
The lecturer postulated that $\nabla f(x) = 2 A x$, and that $\nabla^2 f(x) = 2A$.
It's not... | {
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• So symmetry of $A$ is an absolutely necessary condition for $\nabla_x f = 2Ax$? Oct 23, 2016 at 12:43
• @Jasper Yes. See the edit. Oct 23, 2016 at 12:48
• > Therefore, the derivative is given by 𝑓′𝑥=⟨𝐴𝑥,⋅⟩+⟨𝑥,𝐴⋅⟩. can someone explain how this was found? Dec 23, 2021 at 0:28 | {
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# Conceptual reason for why the volume of an ocahedron is four times the volume of a tetrahedron
The image below shows that a regular octahedron can be scaled by a factor of $$2$$ (resulting in a $$2^3$$ factor in volume) and decomposed as six octahedra and eight tetrahedra.
If $$V_o$$ and $$V_t$$ respectively repres... | {
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Is there a conceptual reason why this relationship is "nice"?
• When you say "an octahedron" and "a tetrahedron" what is the implied normalization? Do you require that they both have edges of length $1$? (Anyway, I think your image is already quite a nice explanation!) – Qiaochu Yuan Sep 4 '20 at 4:36
• @QiaochuYuan, ... | {
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Consider a cube with a tetrahedron inside it sharing four of its vertices. The cube dissects into this tetrahedron, and four identical triangular pyramids. Look at this picture of the cube standing on one vertex:
A body diagonal of the cube, vertical in this picture, is split into three equal parts by the heights of t... | {
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For example:
$
6x^2 +11x -35
$
I've been doing this by trial and error but it takes too long. I tried googling "factoring quadratics" but when it comes to examples like this, they just use the quadratic equation to find the values of x assuming the whole equation was equal to zero in the first place. I don't think thi... | {
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Next, try to come up with 2 factors that multiply to get -210 and sum to +11.
You may have to fiddle around a little to find them, but this one came to me right away. It's +21 and -10.
Replace the middle coefficient (-11) with these two numbers:
$6x^2+21x-10x-35$
Now, group the first two terms and the last 2 terms ... | {
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# Eigenvectors for an eigenvalue of $0$
Without calculation, find one eigenvalue and two linearly independent eigenvectors of $A = \begin{bmatrix} 2 && 2 && 2 \\ 2 && 2 && 2 \\ 2 && 2 && 2 \end{bmatrix}$.
This matrix is non-invertible because its columns are linearly dependent. So the number $0$ is an eigenvalue of $... | {
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Why is it that any nonzero vector in ($\Bbb R^3$) whose entries sum to $0$ will work?
• "why is it that any nonzero vector whose entries sum to $0$ will work?" Try multiplying such a vector to it and see what happens. Since all entries in $A$ are equal, you could simply factor out a common $2$. "how do you obtain..." ... | {
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On the other hand, supposing that $x_1+x_2+x_3=k\neq 0$, you would have $Ax = a\begin{bmatrix}k\\k\\k\end{bmatrix}\neq \overrightarrow{0}=0x$ and so $x$ would not be an eigenvector for the eigenvalue of zero.
In general, any nonzero multiple of the $Ones_{n\times n}$ matrix, say $a\cdot Ones_{n\times n}$ (matrix where... | {
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• Why does it matter that the entries must sum to $0$ when $\lambda x = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}$ due to lambda being $0$? – user1766555 Oct 19 '15 at 1:47
• @user1766555 Suppose that they didn't sum to zero, but instead summed to $k$. Then going through the matrix multiplication, you see that $Ax = \b... | {
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# Divisibilty vs. division, for negative divisors
I'm learning some basic number theory from Strayer's 'Elementary Number Theory.' I've arrived at what seems to be a very basic problem, albeit complete with a nasty twist:
Let $$a,b,c \in \mathbb Z$$ with $$c \neq 0$$. Prove that $$a|b$$ if and only if $$ac|bc$$.
The... | {
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• I know something called the Euclidean algorithm. I am not sure what Division Algorithm means, or what it accomplishes... Dec 15 '18 at 1:32
• @Will It's a common name for Euclidean Division (with Remainder) (algorithm) Dec 15 '18 at 1:35
• @Bill fair enough. What I was thinking about was that the algorithm I had in m... | {
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• Great insight Bill. I always love an algebraic approach. Dec 15 '18 at 18:49
The division algorithm isn't the definition of divisibility. It's simply a statement that unique divisors and remainder pairs exist.
The definition is that $$a|b$$ if there exists an integer $$m$$ so that $$am=b$$. Sign has nothing to do w... | {
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# Is Euler-Lagrange equation necessary and sufficient for minimization in a variational problem?
To the best of my knowledge, the function that minimizes of the integral posed in calculus of variations must also satisfy the Euler-Lagrange equations. In other words, the Euler equations are a necessary condition for fin... | {
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• So, why do I see people solving these minimization problems as if it is sufficient (like the example I cited) ? Or is there a gap in my understanding about why they are allowed to solve it that way? – makansij Jan 3 '18 at 0:53
• sure. here's an example taken from gilbert strang's textbook: ocw.mit.edu/courses/mathem... | {
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GMAT Question of the Day - Daily to your Mailbox; hard ones only
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A. 45
B. 48
C. 49
D. 50
E. 52
Originally posted by Fabino26 on 08 Feb 2014, 05:55.
Last edited by Fabino26 on 09 Feb 2014, 00:08, edited 1 time in total.
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08 Feb 2014, 12:30
2
1
Fabino26 wrote:
The perimeter of square S is 40. Square T is inscribed in square AS. What is the least possible value of the area of square T ?
A. 45
B. 48
C. 49
D. 50
E. 52
Consider this picture:
The perimeter of square S is 40 implies each side of S is 10, which also means that di... | {
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18 Jun 2014, 05:35
1
Can anybody please explain me that why are we not considering the option of a 7 * 7 square inscribed into the square S as dis would give us the area less than 50.
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Joined: 02 Sep 2009
Posts: 53066
Re: The perimeter of square S is 40. Square T is inscribed in square S. [#perma... | {
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01 Jul 2015, 18:04
If $$x^{2}$$ is area of square, then find x, one side of the square. If square is inscribed, then diagonal is the length of larger square and therefore the diagonal is $$10$$. To determine the side, the formula also includes the area of the square, $$x^{2}$$. So, if $$2x^{2} = 100$$ then $... | {
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# Why is zero factorial equal to 1?
1. Nov 13, 2005
### quasi426
Why is zero factorial equal to 1?
2. Nov 13, 2005
### bomba923
As they say :
___________________________
!
*Edit: Sorry about that! Here's why:
Three reasons, from my perspective:
1) 0! is defined to be equal to one.
2) (This will sound weird. Oh ... | {
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*there's more than one equivalent way to define factorial of course
7. Nov 14, 2005
### quasi426
Thanks guys, I was going with the explanation that "0! is just equal to 1 because it is." But I thought about the definition of n! in the book of n(n-1)(n-2).....and I couldn't see exactly how it was so that 0! = 0. So I... | {
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13. Nov 22, 2005
### leon1127
The special case 0! is defined to have value 0!==1, consistent with the combinatorial interpretation of there being exactly one way to arrange zero objects (i.e., there is a single permutation of zero elements, namely the empty set emptyset).
from the MathWorld
14. Nov 23, 2005
### ma... | {
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# How to do exponential and logarithmic curve fitting in Python? I found only polynomial fitting
I have a set of data and I want to compare which line describes it best (polynomials of different orders, exponential or logarithmic).
I use Python and Numpy and for polynomial fitting there is a function polyfit(). But I... | {
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Note that Excel, LibreOffice and most scientific calculators typically use the unweighted (biased) formula for the exponential regression / trend lines. If you want your results to be compatible with these platforms, do not include the weights even if it provides better results.
Now, if you can use scipy, you could us... | {
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• @Tomas: Right. Changing the base of log just multiplies a constant to log x or log y, which doesn't affect r^2. – kennytm Aug 8 '10 at 11:20
• This will give greater weight to values at small y. Hence it is better to weight contributions to the chi-squared values by y_i – Rupert Nash Aug 8 '10 at 16:54
• This solutio... | {
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• Nice. Is there a way to check how good a fit we got? R-squared value? Are there different optimization algorithm parameters that you can try to get a better (or faster) solution? – user391339 May 20 '16 at 3:32
• For goodness of fit, you can throw the fitted optimized parameters into the scipy optimize function chisq... | {
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"""
Print the coefficients and plot the funcion.
"""
plt.plot(x, func(x, *popt), label="Fitted Curve") #same as line above \/
#plt.plot(x, popt[0]*x**3 + popt[1]*x**2 + popt[2]*x + popt[3], label="Fitted Curve")
plt.legend(loc='upper left')
plt.show()
the result is: a = 0.849195983017 , b = -1.18101681765, c = 2.24... | {
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This results in the following graph:
• Is there a saturation value the fit approximates? If so, how can on access it? – Ben Jul 19 at 9:08 | {
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# Let $S=\{x~|~x\in G$ and $x^2 \in H\}$. Show that $S$ is a subgroup of $G$ for $H<G$, $G$ abelian.
The question states:
Let $$G$$ be an Abelian group with subgroup $$H < G$$. Let $$S=\{x~|~x\in G$$ and $$x^2 \in H\}$$. Show that $$S$$ is a subgroup of $$G$$.
My proof is different than what is in the book, I procee... | {
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• You used the fact that $G$ is abelian in going from $(ab^{-1})^2\in H$ to $a^2(b^2)^{-1}\in H$. You are using $(xy)^2 = x^2y^2$, which is not generally true in nonabelian groups. Yes; your argument seems correct. – Arturo Magidin Apr 23 at 4:00
• @ArturoMagidin True, as you can see I'm still having trouble with detai... | {
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# Rational and Irrational Values of Powers
Here’s a cute little existence argument that I was exposed to as an undergraduate and have never forgotten. It shows that there must be irrational positive reals $\alpha$ and $\beta$ for which $\alpha^\beta$ is rational. Furthermore, it’s done by showing that one of two very ... | {
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Just for fun, here are elementary examples of all possible choices of rational and irrational choices in taking powers:
$\alpha^\beta$ rational irrational
${rational\ }^{rational}$ $2^3 = 8$ $2^{1/2} = \sqrt{2}$
${irrational\ }^{rational}$ ${\sqrt{2}}^2 = 2$ ${\sqrt{2}}^3 = 2 \sqrt{2}$
${rational\ }^{irrational}$ $2^{... | {
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$\sqrt{3}$ is irrational. Assume it’s not. Then there are integers $m,n$ for which $\sqrt{3} = \frac{m}{n}$ i.e. for which $m^2 = 3 n^2$ (and, of course, $n \ne 0$). We show that both $m,n$ have to be divisible by $3$, and that’s impossible for all $m,n$ with $n \ne 0$ satisfying $\sqrt{3} = \frac{m}{n}$ because by rem... | {
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$\alpha + \beta$ rational irrational
${rational\ }+{\ rational}$ $2+3=5$ cannot happen
${irrational\ }+{\ rational}$ cannot happen $({\sqrt{2}}-1) + 1 = \sqrt{2}$
${rational\ }+{\ irrational}$ cannot happen $1+({\sqrt{2}}-1) = \sqrt{2}$
${irrational\ }+{\ irrational}$ $(3-\sqrt{2})+ \sqrt{2} = 3$ $\sqrt{2} + \sqrt{2} =... | {
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# Eigenvalues of block matricies
If the eigenvalues of a matrix $A$ are $\lambda_1,\lambda_2,\dots,\lambda_n$, what are the eigenvalues of the matrix?
$\begin{bmatrix}0 &A\\A&0\end{bmatrix}$
From some numerical examples I have found that the eigenvalues are just $\lambda_1$,$\lambda_2,\dots,\lambda_n$ and $-\lambda_... | {
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Let $$B=\begin{bmatrix} 0 & A \\ A & 0 \\ \end{bmatrix}.$$ We observe that $$\begin{bmatrix} 0 & A \\ A & 0 \\ \end{bmatrix}\begin{bmatrix} \mathbf{u} \\ \mathbf{v} \\ \end{bmatrix}=\lambda\begin{bmatrix} \mathbf{u} \\ \mathbf{v} \\ \end{bmatrix}$$ implies that $A\mathbf{v}=\lambda\mathbf{u}$ and $A\mathbf{u}=\lambda\m... | {
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# $T_2$ topologies that are "as disjoint as possible"
Let $X$ be an infinite set. Are there Hausdorff topologies $\tau_1, \tau_2$ on $X$ such that $\tau_1\cap\tau_2 = \{\emptyset\} \cup \{U\subseteq X: X\setminus U\text{ is finite}\}$?
(That is, the intersection is as small as it can get.)
And what about the special... | {
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Proof: We need the following lemma:
Lemma: Let $Y=(Y,\nu)$ be an infinite topological space such that $$|\overline{E}^\nu|\ge 2^{\omega}$$ for all $E\in [Y]^\omega$. Then there is an injective map $f:\mathbb R\to Y$ such that $$\forall D\in [\mathbb R]^\omega \text{ if }\overline D\ne \mathbb R \text{ then } \exists x... | {
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To show that $\tau$ is as required assume that $\emptyset\ne U\in \varepsilon$ such that $F=\mathbb R\setminus U$ is infinite. Let $D$ be a countable $\varepsilon$-dense subset of $F$. Then there is $x_D\in \mathbb R\setminus \overline D$ such that
$f(x_D)\in\overline{f[D]}^{\nu}$. Since $f$ is a homeomorphism, $x_D\in... | {
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Thus we proved the proposition.
• Very nice! And this argument seems to apply to lots of other familiar spaces as well (any perfect Polish space, for example, or, generalizing from $\omega^*$ to $U(\kappa)$, any space of the form $[0,1]^\kappa$ or $2^\kappa$, if $\kappa$ is regular). Mar 23, 2016 at 17:28
• Wonderful,... | {
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They answer Dominic's first question in the negative:
Proposition 3.2: If $X$ is countably infinite, then no two Hausdorff topologies on $X$ intersect in only the co-finite sets.
They are also able to strengthen this result under various extra hypotheses:
Theorem 3.3: It is consistent (it follows from $\neg$CH plus ... | {
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• Beautiful, thanks Will, also the comments concerning $\mathbb{R}$! Mar 22, 2016 at 19:47
If $\tau_1$ and $\tau_2$ meet your requirement, then they are called $T_1$-independent. The following paper may contain useful information: Shakhmatov, D.; Tkachenko, M.; Wilson, R. G. Transversal and $T_1$-independent topologie... | {
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# Selecting council members for a committee - elementary combinatorics.
I'm trying to figure out how to do the following question, but I got stuck. I just don't see how they are counting these people.
In a student council consisting of 16 persons there are mathematics- and computer science students, freshmen as well ... | {
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Case 2: Two groups each have two representatives and each of the others each has one.
Choose which two of the four groups have two representatives. Choose two of the four members from each of these two groups. Choose one of the four members of each of the remaining two groups. There are $$\binom{4}{2}\binom{4}{2}\bino... | {
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$$\begin{array} \text{reserved spots} & \text{additional members}\\ \hline A_1, B_1, C_1, D_1 & A_2, B_2\\ A_1, B_2, C_1, D_1 & A_2, B_1\\ A_2, B_1, C_1, D_1 & A_1, B_2\\ A_2, B_2, C_1, D_1 & A_1, B_1 \end{array}$$
In how many ways can the committee be selected if every group has at most two representatives on the com... | {
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# Find the 3D region containing the origin bounded by given planes
I'm writing a code to generate the Wigner-Seitz cell of the reciprocal lattice for a given set of lattice translation vectors. For example, consider the Body Centered Cubic (BCC) lattice whose basis translation vectors are given by
a1 = {-1, 1, 1}/2;
... | {
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Solve[#, z] & /@ planes
Some of the equations will have to be solved for $$x$$ or $$y$$ before being turned into inequalities. I think I could find a brute force solution but I'm hoping there's something more elegant.
Ultimately I'd like to obtain the inequalities that define the region so that I can visualize it wi... | {
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a1 = {-1, 1, 1}/2;
a2 = {1, -1, 1}/2;
a3 = {1, 1, -1}/2;
B = {b1, b2, b3} = 2 π*Inverse[Transpose[{a1, a2, a3}]];
pts = Flatten[Table[{b1, b2, b3}.{n1, n2, n3}, {n1, -1, 1}, {n2, -1, 1}, {n3, -1, 1}], 2];
G = NearestNeighborGraph[pts, VertexCoordinates -> pts];
neighbors = Rest[VertexOutComponent[G, {0, 0, 0}, 1]];
rhs... | {
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sols = Flatten[
If[sz = Solve[#, z]; sz != {}, sz,
If[sx = Solve[#, x]; sx != {}, sx, Solve[#, y]]] & /@ planes];
Now construct the inequalities for each side of the plane
ineq = Flatten[{#[[1]] <= #[[2]], #[[1]] > #[[2]]} & /@ sols];
And select the ones which contain the origin
region = Select[ineq, (# /. {x -> ... | {
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# Why does cancelling change this equation?
The equation
$$2t^2 + t^3 = t^4$$
is satisfied by $t = 0$
But if you cancel a $t^2$ on both sides, making it $$2 + t = t^2$$ $t = 0$ is no longer a solution.
What gives? I thought nothing really changed, so the same solutions should apply.
Thanks
• About why the unthin... | {
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Note that the same thing happens in reverse. Multiplying both sides of an equation by zero can result in craziness. We can agree that $1 \neq 2$, but this does not mean $0 \cdot 1 \neq 0 \cdot 2$ because both sides of this are zero. It can be harder to see when one is doing this when using a more complicated expression... | {
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If you leave out the $t^2$ factor you lose the $0,0$ roots.
• That's a different way of looking at it. Interesting – Cruncher Nov 6 '15 at 16:33
• Yes, this is the way to do it. Put it in standard form first (move the whole polynomial to the left of the equal sign, with the highest-ordered term on the left, then the n... | {
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## protected by Community♦Dec 24 '15 at 8:05
Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
Would you like to answer one of these unanswered que... | {
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# Direction of velocity vector in 3D space
According to a well-known textbook (Halliday & Resnick), the direction of a velocity vector, $\vec v$, at any instant is the direction of the tangent to a particle's path at that instant, as is illustrated below in 2D.
According to the same textbook, the same holds for 3D. H... | {
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Now, the vector $$\dfrac{\mathrm d\mathbf{x}}{\mathrm dt}=\Biggl(\dfrac{\mathrm dx_{1}}{\mathrm dt},\dfrac{\mathrm dx_{2}}{\mathrm dt},\dfrac{\mathrm dx_{3}}{\mathrm dt}\Biggr)=\left(-5\sin t,5\cos t,2\right) \tag{02}$$ is tangent to the curve at the point $\:\mathbf{x}\left(t\right)\:$ and well-defined without any ind... | {
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Finally we construct the unit vector $$\mathbf{b}=\mathbf{t}\boldsymbol{\times}\mathbf{n}=\sqrt{\frac{1}{29}} \begin{bmatrix} \mathbf{e}_{1} & \mathbf{e}_{2} & \mathbf{e}_{3}\vphantom{\dfrac{\dfrac{}{}}{}}\\ -5\sin t & 5\cos t & 2\vphantom{\dfrac{\dfrac{}{}}{}}\\ -\cos t & -\sin t & 0 \vphantom{\dfrac{\dfrac{}{}}{}} \e... | {
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${\bf N} \propto \frac{d{\bf T}(t)}{dt}$
${\bf B} = {\bf T \times N}$.
I don't know what your book is telling you a tangent is. But what is actually meant, and what will tell you how to draw the velocity vector, is this. Consider a short part of the path the object takes near the point you want the tangent. If you ha... | {
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Find a plane containing smallest part of path curve at that instant. The velocity vector must lie in the same plane.
Note: For exactly straight paths it'll be limiting case. Therefore make those impractical straight paths slightly curved ;), then do the above procedure. | {
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# What happens when I include a squared variable in my regression?
I start with my OLS regression: $$y = \beta _0 + \beta_1x_1+\beta_2 D + \varepsilon$$ where D is a dummy variable, the estimates become different from zero with a low p-value. I then preform a Ramsey RESET test and find that i have some misspesificatio... | {
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Edit: Improving question.
• possible duplicate of Why ANOVA/Regression results change when controlling for another variable – Macro Mar 18 '13 at 13:15
• Probable reason: $x_{1}^2$ and $D$ seem to explain the same variablility in $y$ – steadyfish Mar 18 '13 at 13:17
• One thing that might help is to center $x$ before ... | {
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Taking the derivate w.r.t. $x_1$ yields
$$\frac{\delta y}{\delta x_1} = \beta_1 + 2\beta_2 x_1$$
Solving this equation gives you the turning point of the relationship. As user1493368 explained, this is indeed reflecting an inverse U-shape if $\beta_1<0$ and vice versa. Take the following example:
$$\hat{y} = 1.3 + 0... | {
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A good example of including square of variable comes from labor economics. If you assume y as wage (or log of wage) and x as an age, then including x^2 means that you are testing the quadratic relationship between an age and wage earning. Wage increases with the age as people become more experienced but at the higher a... | {
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# Triple Integral with bounds in first octant
I am really confused on how to get my integrating function because I don't know, even after graphing, how the tetrahedron intersects the x-y-z axis.
I am supposed to find the triple integral for the volume of the tetrahedron cut from the first octant by the plane $6x + 3y... | {
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"openwebmath_score": 0.8837581276893616,
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You have a plane $P:~6x+3y+2z=6$ which cut the axes in the first octant as you see through the below plot. $P$, clearly, intersects $z=0$ in a line: $$z=0\to 6x+3y+2\times 0=6\to 6x+3y=4$$ I don't want to tell you that all triple integrals will start from this point that we did above, but we do for many certain triples... | {
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"openwebmath_score": 0.8837581276893616,
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# How to prove two sequence have a common limit.
We have \begin{align} U_{0} &= 1 &&\text{and} & V_{0} &= 2 \\ U_{n+1} &= \frac{U_{n}+V_{n}}{2} &&\text{and} & V_{n+1} &= \sqrt{U_{n+1}V_n} \end{align} How to prove two sequence have a common limit?
I found $(U_n)$ is increasing and $(V_n)$ is decreasing but I don't kno... | {
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"ta... |
We can start with triangles where $$\,n=3\,$$ and find $$\,a(3) = 3\sqrt{3}\,$$ and $$\,b(3) = a(3)/2.\,$$ We then keep doubling the number of sides indefinitely. The connection between the two recursions is that $$\, U_n = 3\sqrt{3}/a(3\,2^n)\,$$ and $$\, V_n = 3\sqrt{3}/b(3\,2^n)\,$$ since the recursions and initial ... | {
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"ta... |
$$U_n+V_n\in[2,4]\implies U_{n+1}\in[1,2]$$
and
$$U_{n+1}V_n\in[1,4]\implies V_n\in[1,2].$$
Hence both sequences are bounded and both converge.
And if the limits exist, they are equal. Indeed,
$$U=\frac{U+V}2,V=\sqrt{UV}$$ imply $U=V$.
• Do you not need some monotony argument? Maybe I overlooked something, but, w... | {
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"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 174.23483268051766,
"openwebmath_score": 0.9864970445632935,
"ta... |
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