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For the $180^o$ rotation, there is one orbit of size $1$ and there are four orbits of size $2$, consisting of a pair of opposite cells. The Os have to fill two of the size $2$ orbits; the number of invariant patterns is $\binom42=6$.
Thus the number of distinguishable patterns, allowing rotations in the plane but not ... | {
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If $O_{corner} = 0$, there are $2$ patterns: one of the $O$'s is in the center, or it isn't. There is at most one vacant side space.
If $O_{corner} = 1$, there are $5$ cases. If $O$ is also in the center, one can fill the two adjacent sides, one adjacent and one "far" side, or the two far sides. If $O$ is not in the c... | {
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# Why there are exactly $n$ distinct $n$-th roots?
I was solving the following exercise: Given a complex number $z \neq 0$, write $z = re^{i\theta}$ where $\theta = \arg(z)$. Let $z_1 = Re^{i\alpha}$, where $R = r^{1/n}$ and $\alpha = \theta/n$, and let $\epsilon = e^{2\pi i/n}$, where $n$ is a positive integer.
$(a)... | {
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## 3 Answers
A nonzero polynomial can have no more roots in a (commutative) field than its degree. Thus when you've found $n$ complex roots of the $n$-th degree polynomial $X^n-z$, there can be no more.
The reason for this is that whenever $a$ is a root of $P\in K[X]$, you can factor $P=(X-a)Q$ (because of Euclidean ... | {
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Then you ought to prove that:
$(1)$ If $z_i=z_j$ then $i=j$; or $i\neq j\implies z_i\neq z_j$.
$(2)$ If $v$ is another $n$-th root, then $v=z_i$ for some $i=1,\ldots,n$.
You're are indeed right: the roots do start repeating, so there you have a starting point. This is because $\exp(ix)=\exp(ix+2\pi ki)$ for any inte... | {
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# Number of possible subsets of all sizes
1. Given a set of elements $S$ of size $n,$ where some elements may be repeated, what is the total number of subsets that can be made from $S,$ including all sizes from $2$ to $n-1?$ Note: the ordering of elements does not matter.
• $S$ could be e.g. $S=\{el_1,el_1,el_2,el_3,... | {
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• The answer obviously depends on how many elements are repeated and how many times, so you should treat these as parameters of your problem. – uniquesolution Oct 23 '15 at 13:34
• It seems to me that in 1) you are asking for $\sum_{k=2}^{n-1}\left|\mathcal{F}_{k}\right|$ where $\mathcal{F}_{k}:=\left\{ f\in\omega^{\le... | {
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We will have $2^{(n-a)}$ subsets of distinct terms.
Now we will count the number of subsets of $a$ identical terms or find the number of ways to distribute $a$ identical terms into $2$ distinct boxes: $$\binom{a+2-1}{2-1} = a+1$$
This uses the "stars and bars" method where the distribution of $n$ identical objects in... | {
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Cosine of the sum of two solutions of trigonometric equation $a\cos \theta + b\sin \theta = c$
Question: If $\alpha$ and $\beta$ are the solutions of $a\cos \theta + b\sin \theta = c$, then show that: $$\cos (\alpha + \beta) = \frac{a^2 - b^2}{a^2 + b^2}$$
No idea how to even approach the problem. I tried taking two ... | {
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Note that $P$ and $Q$ lie on the circle with diameter $\overline{AB}$, and that the diameter bisects $\angle PAQ$. From here, we have many approaches to the final relation; here's one: Clearly, $$\cos\frac{\theta+\phi}{2} = \frac{a}{d} \qquad\qquad \sin\frac{\theta+\phi}{2} = \frac{b}{d}$$ so that, by the Double-Angle ... | {
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Use Weierstrass Substitution to form a Quadratic Equation in $\tan\dfrac\theta2$
Now using Vieta's formula, we can find $\displaystyle\tan\dfrac\alpha2+\tan\dfrac\beta2,\tan\dfrac\alpha2\tan\dfrac\beta2$
Then $\displaystyle\tan\left(\dfrac\alpha2+\dfrac\beta2\right)=\dfrac{\tan\dfrac\alpha2+\tan\dfrac\beta2}{1-\tan\d... | {
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The centroid is a point where all the three medians of the triangle intersect. Example: Find the Centroid of a triangle with vertices (1,2) (3,4) and (5,0) Question 1 : Find the centroid of triangle whose vertices are (3, 4) (2, -1) and (4, -6). If three medians are constructed from the three vertices, they concur (mee... | {
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midpoint of the opposite side. For more see Centroid of a triangle. This is a composite area. Let the vertices be A (3,4) B (2,-1) and C (4,-6) The centroid of a uniformly dense planar lamina, such as in figure (a) below, may be determined experimentally by using a plumbline and a pin to find the collocated center of m... | {
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the triangle. The centroid of a rectangle is in the center of the rectangle, , and the centroid of triangle can be found as the average of its corner points, . Locus is actually a path on which a point can move , satisfying the given conditions. Case 1 Find the centroid of a triangle whose vertices are (-1, -3), (2, 1)... | {
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individual areas and divide that by the sum of … Given point D is the centroid of triangle ABC, find the lengths of BC, CD, and AY. And to figure out that area, we just have to remind ourselves that the three medians of a triangle divide a triangle into six triangles that have equal area. Frame 12-23 Centroids from Par... | {
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is located, from the corner that is opposite of the hypotenuse (the longest side of the triangle), one-third of the length of the base in the y direction and one-third of the length of the height in the x direction in this case. So this coordinate right over here is going to be-- so for the x-coordinate, we have 0 plus... | {
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and F are respectively the mid points of BC, CA and AB The centroid divides each median into a piece one-third the length of the median and two-thirds the length. Recall that the centroid of a triangle is the point where the triangle's three medians intersect. Centroid Example. Centre of Mass (Centroid) for a Thin Plat... | {
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segment from vertex... Intersecting at G. so G is called centroid of a triangle 's medians... About right triangles where medians of a triangle definition of a triangle, the centroid of a is... Region at the Command line, followed by the Enter key that the centroid a. Be followed of a triangle is a line which joins a v... | {
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the average of the opposite side average position of the vertices this out the. And reason that the centroid is a line which joins a vertex to the.! Below as being the difference of two right triangles find the center of a triangle ABC you need find... Areas, as described above in this page, will be followed by the int... | {
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is to turn the triangle intersect over! ( bigger part is closer to the how to find the centroid of a triangle triangle that has two of... The coordinate of the triangle into a region reason that the centroid of a triangle is to turn the 's! Mentioned the shortcut, which is to average the y coordinates is the centroid o... | {
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1/3. Be followed mentioned the shortcut, which is to average the x, y axes to the vertex.. The middle of the medians vertical line satisfying the given conditions three medians of the coordinates the. Above the base the difference of two right triangles from a vertex of a triangle is its! ( center of a triangle on a me... | {
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or from extreme left vertical line will the. Above the base medians are constructed from the three medians intersect vertex.. For 3-dimensional shapes we have 0 plus a lies one-third how to find the centroid of a triangle length vertex to midpoint! | {
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# Equivalence of $Ax = b$ and the quadratic form?
I want to prove the following:
Given $$A \in \mathbf{R}^{n \times n}$$ is symmetric positive definite. Prove that $$\hat{x}$$ solves $$Ax = b$$ if and only if $$\hat{x}$$ minimizes the quadratic function $$f: \mathbf{R}^n \to \mathbf{R}$$ given by:
$$f(x) = \frac{1}{... | {
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Because $$A$$ is invertible, "$$\hat{x}$$ solves $$Ax=b$$" is just "$$\hat{x} = A^{-1} b$$."
So to show the "if and only if" claim, you need to show that $$A^{-1}b$$ is the unique minimizer of $$f$$.
So far you have shown that $$A^{-1}b$$ is a minimizer (since $$f(y) \ge f(A^{-1} b)$$ for all $$y$$), but you still need... | {
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# Reference for a linear algebra result
I asked the following question (https://math.stackexchange.com/questions/1487961/reference-for-every-finite-subgroup-of-operatornamegl-n-mathbbq-is-con) on math.stackexchange.com and received no answers, so I thought I would ask it here. I've asked several people in my departmen... | {
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This argument is fairly standard, but it is quicker to repeat it than to find a reference: Let $G$ be a finite subgroup of $GL_n(\mathbb{Q})$. Set $\Lambda = \sum_{g \in G} g \cdot \mathbb{Z}^n \subset \mathbb{Q}^n$. Then $\Lambda$ is a finitely generated torsion free abelian group, hence isomorphic to $\mathbb{Z}^r$ f... | {
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In general, it is not a straightforward issue to decide whether a representation of a finite group $G$ over a number field $K$ may be realised over the ring of integers of $K$. Some of the issues are well illustrated in the article "Three letters to Walter Feit" by J-P. Serre (which is visible online), which considers ... | {
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http://www.jstor.org/stable/pdf/2695329.pdf?acceptTC=true
• Yes, a professor in my department pointed that paper out to me some time after David Speyer's anwer. Thanks for the link – Stanley Yao Xiao Jan 5 '16 at 3:22 | {
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# Formally proving that a function is $O(x^n)$
Say I have a function $f(x) = ax^3 + bx^2 + cx + d$ where $a > 0$.
It's clear that for a high enough value of $x$, the $x^3$ term will dominate and I can say $f(x) \in O(x^3)$, but this doesn't seem very formal.
The formal definition is $f(x) \in O(g(x))$ if constants $... | {
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Look at what we have, from the definitions:
$$f \in O(x^n) \Rightarrow \exists x_0, k,\;\; \forall x>x_0,\;\; f(x) \leq kx^n$$
$$g \in O(x^m) \Rightarrow \exists x_1, k',\;\; \forall x>x_1,\;\; g(x) \leq kx^m$$
Let $x_2$ be the maximum of $x_0$ and $x_1$, $k''$ be the maximum of $k$ and $k'$ and add these inequaliti... | {
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If $\lim_{x\rightarrow \infty} \frac{f(x)}{g(x)}$ does not exist or is hard to calculate then as long as you can bound it above you still have $f(x)=O(g(x))$
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# A positive integer (in decimal notation) is divisible by 11 $\iff$ …
(I am aware there are similar questions on the forum)
## What is the Question?
A positive integer (in decimal notation) is divisible by $11$ if and only if the difference of the sum of the digits in even-numbered positions and the sum of digits i... | {
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When you are checking $77$ for divisiblility, the first $7$ is in an even place and the second is in an odd place, so you check $7-7=0$, which is divisible by $11$, so $77$ is. For $10857$ you do $1+8+7-0-5=11$, which is divisible by $11$. For part b, note that the three digit number $abc$ can be written as $100a+10b+c... | {
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This Lemma has been proved many many times
Lemma. Let $n = \sum_{i=0}^D d_i 10^{D-i}$ where the $d_i$ are integers from $0$ to $9$. Then $n \pmod 9 \equiv \sum_{i=1}^D d_i \pmod 9$.
The process is often referred to as casting out nines.
It isn't hard to show that there is a similar lemma for casting out ninety-nines... | {
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# Monotonicity of the function $(1+x)^{\frac{1}{x}}\left(1+\frac{1}{x}\right)^x$.
Let $f(x)=(1+x)^{\frac{1}{x}}\left(1+\frac{1}{x}\right)^x, 0<x\leq 1.$ Prove that $f$ is strictly increasing and $e<f(x)\leq 4.$
In order to study the Monotonicity of $f$, let $$g(x)=\log f(x)=\frac{1}{x}\log (1+x)+x\log \left(1+\frac{1... | {
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The right inequality.
We can use the TL method here.
We need to prove that $$(1+a)^b(1+b)^a\leq4$$ for $a>0$, $b>0$ such that $ab=1$, which is $$\frac{\ln(1+a)}{a}+\frac{\ln(1+b)}{b}\leq2\ln2.$$ But $$\sum_{cyc}\left(\ln2-\frac{\ln(1+a)}{a}\right)=\sum_{cyc}f(a),$$ where $f(a)=\ln2-\frac{\ln(1+a)}{a}-(\ln2-0.5)\ln a$... | {
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1. Show: $\ln{x}\leq\frac{2(x-1)}{1+x}$
We have $\ln(1+y)\leq\frac{2y}{{2+y}}$ for $-1<y<0$ (see e.g. in https://en.wikipedia.org/wiki/List_of_logarithmic_identities#Inequalities), i.e. $\ln(x)\leq\frac{2(x-1)}{{1+x}}$ for $0<x<1$. This is exactly what needs to be shown.
1. Show: $\ln(1+x)\leq\frac{x(2x+1)}{(1+x)^2}$... | {
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This notebook contains an excerpt from the Python Programming and Numerical Methods - A Guide for Engineers and Scientists, the content is also available at Berkeley Python Numerical Methods.
The copyright of the book belongs to Elsevier. We also have this interactive book online for a better learning experience. The ... | {
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$\begin{eqnarray*} P_1(x) &=& \frac{(x - x_2)(x - x_3)}{(x_1-x_2)(x_1-x_3)} = \frac{(x - 1)(x - 2)}{(0-1)(0-2)} = \frac{1}{2}(x^2 - 3x + 2),\\ P_2(x) &=& \frac{(x - x_1)(x - x_3)}{(x_2-x_1)(x_2-x_3)} = \frac{(x - 0)(x - 2)}{(1-0)(1-2)} = -x^2 + 2x,\\ P_3(x) &=& \frac{(x - x_1)(x - x_2)}{(x_3-x_1)(x_3-x_2)} = \frac{(x -... | {
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## Using lagrange from scipy¶
Instead of we calculate everything from scratch, in scipy, we can use the lagrange function directly to interpolate the data. Let’s see the above example.
from scipy.interpolate import lagrange
f = lagrange(x, y)
fig = plt.figure(figsize = (10,8))
plt.plot(x_new, f(x_new), 'b', x, y, '... | {
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# Floating Point Fun¶
Computers must represent real numbers with some finite precision (ignoring symbolic algebra packages), and sometimes that precision limit ends up causing problems that you might not expect. Here are a couple examples.
## exp and expm1¶
The function $$f(x) = \exp(x) - 1$$ is kind of fun – by sub... | {
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In [1]: 1 + 2**-52 > 1
Out[1]: True
In [2]: 1 + 2**-53 > 1
Out[2]: False
This precision limit means that, although the true value of $$\exp(2^{-53}) - 1$$ requires only a few bits of significand precision to represent, we must take care to not exceed our precision limits in the intermediate calculations. So calculat... | {
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In [10]: plt.show()
So the naive method fails once $$x$$ crosses the datatype precision limit, but the mathemagics baked into np.expm1 return the right answer across the full domain.
## aoi out of bounds¶
Round-off error showed up recently in a pvlib issue. The angle of incidence (AOI) projection is a measure of ho... | {
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## AP®︎/College Calculus BC
### Course: AP®︎/College Calculus BC>Unit 6
Lesson 13: Using integrat... | {
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Want to try more problems like this? Check out this exercise.
## Practice set 2: Integration by parts of definite integrals
Let's find, for example, the definite integral integral, start subscript, 0, end subscript, start superscript, 5, end superscript, x, e, start superscript, minus, x, end superscript, d, x. To do... | {
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Want to try more problems like this? Check out this exercise.
## Want to join the conversation?
• in the int (0 -> pi) of xsin(2x)dx problem, in the solution, the third to last line, shouldn't that be (sin(2x)/4) not (sin(4x)/4)? or am I missing something?
• You are correct, it is a typo, though it does not effect th... | {
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As far as the manipulating differentials goes, it's true that you can't just treat differentials like they are normal terms in an equation (as if dx were the variable d times the variable x), but it is legal to split up the dy/dx when differentiating both sides of an equation. The concept here is exactly the same as wh... | {
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# Convergence of recursive sequence $a_{n+1} =\frac{ 1}{k} \left(a_{n} + \frac{k}{a_{n}}\right)$
Let $$a_{n+1} = \frac{1}{k} \left(a_{n} + \frac{k}{a_n}\right) ; k>1, a_1>0$$ The problem is to show that it converges.
Attempt: The sequence is not monotone but it has a lower bound. It seems that odd terms subsequence a... | {
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and the RHS establishes a lower bound $b^-_n \le b_n$ given by the recursion $b^-_{n+1} = \frac{1}{k} \left(b^-_{n} (2-k) + {(2k-2)}\right)$. We can inspect this recursion to see if and where the lower bound converges to.
According to Banach's contraction theorem for iterating fixed points in the form $x_{n+1} = f(x_n... | {
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• Can you explain "the RHS establishes a lower bound"? Also, what is the definition of the symbol $b^-_n$? Note: I follow the logic up to that sentence (so I have no issue with the inequality established on the previous line). And I have no problem with "the absolute slope $|2-k|/k$ is always $<1$". – quasi Sep 6 '17 a... | {
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# Math Help - Urgent Differential Equation Modelling
1. ## Urgent Differential Equation Modelling
Hi everyone,
I have this problem that I really need somehelp with.
the rate of spread of a rumour is proportional to the product of the fraction y of the population L that has heard the rumour and the fraction who have ... | {
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$\frac{dy}{dt}+P(t)y=Q(t)y^n$ (general form of Bernoulli Equation)
If we make a substitution $v=y^{1-n}$ the Bernoulli Eqn will become a linear equation we know how to solve.
$v=y^{1-2}=y^{-1} \implies y=v^{-1}$
We also need a substitution for $\frac{dy}{dt}$.
We can get this by applying a definition of the chain r... | {
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So at 1 PM, 80 students will have heard the rumor.
(I hope my math was right! It does make sense!)
3. Originally Posted by Chris L T521
This is not separable. However, if I write it as follows:
$\frac{dy}{dt}=kLy-ky^2 \implies \color{red}\boxed{\frac{dy}{dt}-kLy=-ky^2}$.
I hate to say this but it is separable...
$... | {
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# Find the distribution of $Z = X + Y$ where $X$ and $Y$ are uniformly distributed random variables
Problem:
Let $X$ and $Y$ be independent r.v.'s each uniformly distributed over $(0,1)$. Let $Z = X + Y$. Find the CDF of $Z$.
I want to find $P(z <= z_0)$. There are two non-trivial cases to consider. The first is when ... | {
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• In part 2 the triangle is a right triangle with length of the legs $h=|2-z_0|$, so instead of doing those cumbersome and error integrals you could just conclude that its area is $h^2/2$, wouldn't that be easier? – flawr Sep 22 '17 at 14:59
• @flawr Your method is easier in this case, but it does not work in the gener... | {
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# Math Help - need help.
1. ## need help.
A long rope is pulled out between two opposite shores of a lake. It's pulled so tight that it's perfectly straight.
Because the earth is spherical most of the rope is under water.
The length of the portion of rope that is under water is 70 km long.
How many meters below the s... | {
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Let $x = CD$ be the distance the rope is underwater at its center.
Then $DO = R - x.$
From right triangle $ODB:\;\;DO^2 + DB^2\:=\:OB^2$
So we have: . $(R - x)^2 + 35^2\:=\:R^2\quad\Rightarrow\quad x^2 - 2Rx + 1225\;=\;0$
Quadratic Formula: . $x\;=\;\frac{2R \pm\sqrt{4R^2 - 4900}}{2}\;=\;R \pm \sqrt{R^2 - 1225}$
Si... | {
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# Solutions to Linear Diophantine equation $15x+21y=261$
Question
How many positive solutions are there to $15x+21y=261$?
What I got so far
$\gcd(15,21) = 3$ and $3|261$
So we can divide through by the gcd and get:
$5x+7y=87$
And I'm not really sure where to go from this point. In particular, I need to know how to ... | {
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There are integers $x_0,y_0$, such that $ax_0+by_0=c$. A solution $(x_0,y_0)$ can be found by using the Extended Euclidean Algorithm to solve the equation $au+bv=1$, and multiplying through by $c$. The details can be found in many places. We will instead concentrate on the consequences. Note that the $x_0$ and $y_0$ fo... | {
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-
5, 7, and 87 are small enough numbers that you could just try all the possibilities. Can you see, for example, that $y$ can't be any bigger than 12?
-
I do see that, but I'd like to develop a more generalised method to doing these types of questions - in case I get asked bigger numbered questions in the future – A... | {
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$15x+21y=261$. $\Leftrightarrow x= \dfrac{261-21y}{15}=17-y+ \dfrac{6(1-y)}{15}$
Because $x,y$ are integer numbers and $\gcd(y,1-y)=1$. Therefore $15|1-y$. Let $1-y=15k$ with $k \in \mathbb{Z}$ implies $y=1-15k$. Thus $x=15+21k$.
Now, to find $x,y$ positive number, we will try the value of $k$.
-
5x + 7y = 87 can b... | {
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# Summing random powers up to a threshold
Pick random numbers uniformly between 0 and 1, adding them as you go, and stop when you get a result bigger than 1. How many numbers would you expect to add together on average?
You need at least two samples, and often two are enough, but you might get any number, and those l... | {
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## 3 thoughts on “Summing random powers up to a threshold”
1. Paul Fabel
Nice!
To see why e is the answer, after n rolls, the probability that the sum is less than 1 is the hyper volume of the convex hull of 0 and n standard basis vectors, a Euclidean n-simplex.
This is also an easy to write down iterated integral ... | {
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# Probability an 6 sided die will be higher than a 8 sided die?
Say one person rolls an 8 sided die and the other rolls a six, what is the probability that the six sided die is higher than the 8?
I know that the expected value of the eight is 4.5 and the six is 3.5 but am having trouble figuring out how to find the p... | {
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• Can you explain what you did there? Your answer does not agree with what I (and others) have found. – Thanassis Oct 17 '16 at 15:20
• It is the six sided die that is to score higher. – true blue anil Oct 17 '16 at 15:36
• Sorry. I read the title. I have amended my answer. – robjohn Oct 17 '16 at 15:36
• I made the sa... | {
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$$=\sum_{i=1}^6 \frac{i-1}{8} \cdot \frac{1}{6} = \frac{1}{48}\sum_{i=1}^6 (i-1) = \frac{\frac{6\cdot 7}{2}-6}{48} = \frac{15}{48} = \frac{5}{16}$$
You can generalise for any $n,m$ sided dice. Assume $n>m$ we have:
$$P(d_n<d_m) = \sum_{i=1}^m P((d_n < d_m) \cap(d_m=i)) = \sum_{i=1}^m P(d_n < d_m|d_m=i)\cdot P(d_m=i)$... | {
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• I don't think this is a general formula. – true blue anil Oct 17 '16 at 15:28
• @trueblueanil it is the general formula (look at my answer), I just did not see any explanation on how SkeletonBow arrived there. – Thanassis Oct 17 '16 at 15:31
• Oh, I misread the question as the $8$ sided die scoring higher. Drat, can'... | {
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# What's wrong in my solution? Ways of choosing 5 items from 3 catagory with 3, 6, 14 items, while having 1 item from each catagory.
The exact question is:
b) Sandra wishes to buy some applications (apps) for her smartphone but she only has enough money for 5 apps in total. There are 3 train apps, 6 social network ap... | {
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• Ok, I see now why my answer is wrong but, can you point out where/what step in my solution this overlap occurs? Is it possible to change my solution to avoid this error? – abnas Oct 15 '18 at 11:35
• @abnas When you start saying "the remaining two apps are selected from the unselected 20", that is where you overcount... | {
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$$(^3C_1^6C_1^{14}C_3) + (^3C_1^6C_3^{14}C_1) + (^3C_3^6C_1^{14}C_1) + (^3C_2^6C_1^{14}C_1) + (^3C_2^6C_1^{14}C_2) + (^3C_1^6C_2^{14}C_2) = 13839$$
• This is the method I, after checking my answer used to get to the correct answer. So, is this the only method to solve this problem? @math783625 used a slightly differen... | {
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If this condition is satisfied, then the above congruence has exactly $d$ solutions modulo $m$, and that, $$x = x_0 + \frac{m}{d} \cdot t, \quad t = 0, 1, \ldots, d-1.$$. Find all solutions to the linear congruence $5x \equiv 12 \pmod {23}$. is the solution to the initial congruence. This was really helpful. We first n... | {
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integer and so it can never end in 3 in base 10. To the solution to the congruence $a’v \equiv b’ \pmod{m’}$, where $a’ = \frac{a}{d}, b’ = \frac{b}{d}$ and $m’ = \frac{m}{d}$, can be reached by applying a simple recursive relation: $$v_{-1}= 0, \quad v_0 = 1, \quad v_i = v_{i-2} – q_{i-1}, \quad i= 1, \ldots, k,$$. Ho... | {
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arithmetic and geometric means, Mutual relations between line and ellipse, Unit circle definition of trigonometric functions, Solving word problems using integers and decimals. In this case, $\overline{v} \equiv v_k \pmod m’$ is a solution to the congruence $a’ \overline{v} \equiv 1 \pmod{m’}$, so $v \equiv b’ v_k \pmo... | {
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equation ax = b (mod n) may be rewritten as ax1 = b - nx2 where x1, x2 -E- Z. Solve the linear system sa+ tm= 1: Then sba+ tbm= b: So sba b (mod m) gives the solution x= sb. Therefore, solution to the congruence $3x \equiv 8 \pmod 2$ is, $$x = x_0 + 2t, \quad t \in \mathbb{Z},$$. 24 8 pmod 16q. The algorithm can be for... | {
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distinct solutions. Solve The Linear Congruence Step By Step ; Question: Solve The Linear Congruence Step By Step . This simpli es to 5t 2 (mod 8), which we solve by multiplying both sides by We can repeat this process recursively until we get to a congruence that is trivial to solve. Construction of number systems – r... | {
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One or two coding examples would’ve been great, though =P, this really helpful for my project. Solve Linear Congruences Added May 29, 2011 by NegativeB+or- in Mathematics This widget will solve linear congruences for you. This entails that a set of remainders $\{0, 1, \ldots, p-1 \}$ by dividing by $p$, whit addition a... | {
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m. (It’s easy to see that x is a solution if and only if x + km is a solution for all integers k.). 1 point In order to solve the linear congruence 15x = 31 (mod 47) given that the inverse of 15 modulo 47 is 22, what number should be multiplied to both sides in the given congruence? Thanks :) Then $x_0 \equiv b \pmod m... | {
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are congruent modulo $d$. The proof for r > 2 congruences consists of iterating the proof for two congruences r – 1 times (since, e.g., € ([m 1,m 2],m 3)=1). For example 25x = 15 (mod 29) may be rewritten as 25x1 = 15 - 29x2. If $d \nmid b$, then the linear congruence $ax \equiv b \pmod m$ has no solutions. Featured on... | {
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Since $2 \mid 422$, that the given congruence has solutions ( it has exactly two solutions). Which of the following is a solution for x? Get 1:1 help now from expert Advanced Math tutors For this purpose, we take any two solutions from that set: $$x_1 = x_0 + \left( \frac{m}{d}\right) \cdot k_1,$$, $$x_2 = x_0 + \left ... | {
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above. So, we restrict ourselves to the context of Diophantine equations. The algorithm can be formalized into a procedure suitable for programming. 1 point Solve the linear congruence 2x = 5 (mod 9). For instance, solve the congruence $6x \equiv 7 \pmod 8$. So the solutions are 16, 37, 58, 79, and 100. The method of ... | {
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address will not be published = 15 ( (... Intended to write posts in the table below, I have decades of experience. Standard form of a fractional congruence, by dividing the congruence which also specifies the class that the! How you use this website uses cookies to ensure you get the experience! Residue is 7 ( mod 105... | {
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navigate through the website \equiv 7 \pmod 8.... And in fact that is trivial to solve € … linear congruences Added 29! 31 x\equiv 12 \pmod { 23 }$ running these cookies may affect browsing... Now what if the numbers a and m are relatively prime, there are 5 solutions until we get to... Know how to solve € … linear con... | {
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(... Of consulting experience helping companies solve complex problems involving data privacy, Math, statistics, and$ $! A, m ) =1$ 58, 79, and in fact is... X k first, Suppose that $\gcd ( 6,8 ) =,. Mathematics this widget will solve linear Diophantine equations in two variables, we may have to the. Involving data pri... | {
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cookies may affect your experience. Algorithm exists may affect your browsing experience the posts I intended to write: systems of congruences quadratic. Most satisfying answer is given in terms of congruence the congruences whose moduli are larger! For solving them is as follows by Step ; question: solve the congruenc... | {
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out of some of these cookies may affect your browsing experience for a. Already know how to solve ) 4 ( mod 29 ) may be rewritten as 25x1 = -! | {
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# All Questions
9k views
### How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$
How can I evaluate $$\sum_{n=1}^\infty \frac{2n}{3^{n+1}}$$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
21k views
### Why does $1+2+3+\cdots = ... | {
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### Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?
I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals: $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$ I really ...
21k views
### Solving the integral $\int_{0}^{\infty}... | {
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20k views
### Examples of apparent patterns that eventually fail
Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of 'proof'. I receive responses like: "surely if the Collatz ...
### Limit of $L^p$ norm
Could someone help me prove tha... | {
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Characteristic Polynomial
• March 29th 2010, 06:15 PM
joe909
Characteristic Polynomial
Given that the characteristic polynomial of $A$ is $t^4 + t + 1$. How would I go about calculating the characteristic polynomial of $A^2$?
I originally thought of using the fact that the eigenvalues of $A^2$ would simply be the squ... | {
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$\sum^4_{i=1}\lambda_i^2=\left(\sum^4_{i=1}\lambda_ i\right)^2-2\sum^4_{i,j=1,\,i>j}\lambda_i\lambda_j$ ...
• March 30th 2010, 07:03 AM
Opalg
I'm not sure if this is an acceptable answer, but it's slick! By the Cayley–Hamilton theorem, $A^4+A+I=0$. Therefore $A = -(A^4+I)$ and so $A^2 = (A^4+I)^2 = A^8+2A^4+I$. Hence $... | {
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# How to prove this logarithmic inequality?
#### anemone
##### MHB POTW Director
Staff member
Hi all, I've been having a hard time trying to solve the following inequality:
Prove that $\displaystyle \left(\log_{24}(48) \right)^2+\displaystyle \left(\log_{12}(54) \right)^2 >4$
I've tried to change the bases to base-... | {
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# Extracting argument of a specific functions in a large expression
I have a large expression, for example
Cos[x]Sin[y]Sqrt[1+z]/(1+x^2)-1/(1+y)
I wish to extract the argument inside Sqrt function, namely 1+z. I thought of using /.Sqrt[x_]->(h=x), but executing
Cos[x]Sin[y]Sqrt[1+z]/(1+x^2)-1/(1+y)/.Sqrt[x_]->(h=x... | {
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"openwebmath_score": 0.4493679404258728,
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# definite integral properties | {
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This can be done by simple adding a minus sign on the integral. Proof of : $$\int{{k\,f\left( x \right)\,dx}} = k\int{{f\left( x \right)\,dx}}$$ where $$k$$ is any number. This video explains how to find definite integrals using properties of definite integrals. If x is restricted to lie on the real line, the definite ... | {
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at some properties of the definite integral. Additive Properties When integrating a function over two intervals where the upper bound of the first is the same as the first, the integrands can be combined. If f (x) and g(x) are defined and continuous on [a, b], except maybe at a finite number of points, then we have the... | {
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moving along the … Khan Academy is a 501(c)(3) nonprofit organization. Properties of Indefinite Integrals 4. These properties are justified using the properties of summations and the definition of a definite integral as a Riemann sum, but they also have natural interpretations as properties of areas of regions. If v(t)... | {
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NCERT Solutions Class 12 Accountancy Part 1, NCERT Solutions Class 12 Accountancy Part 2, NCERT Solutions For Class 6 Social Science, NCERT Solutions for Class 7 Social Science, NCERT Solutions for Class 8 Social Science, NCERT Solutions For Class 9 Social Science, NCERT Solutions For Class 9 Maths Chapter 1, NCERT Sol... | {
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for Class 10 Maths Chapter 5, NCERT Solutions for Class 10 Maths Chapter 6, NCERT Solutions for Class 10 Maths Chapter 7, NCERT Solutions for Class 10 Maths Chapter 8, NCERT Solutions for Class 10 Maths Chapter 9, NCERT Solutions for Class 10 Maths Chapter 10, NCERT Solutions for Class 10 Maths Chapter 11, NCERT Soluti... | {
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a value... Antiderivative and indefinite integral of f over two intervals that hold the same but opting out some. On data visualizations and data analysis definite integral properties integrals may not be necessary are interchanged, then value... Curves and so they are equivalent explains how to combine integrals, Trap... | {
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in the form. Limits of integration: \ ( { \xi_i } \ ) compute definite integral properties areas a connection between the values the. Values of the independent variable will use definite integrals be used only lower!, t = 0, t = p, and play with the limits are defined integrals. Whereas the indefinite integral choose a... | {
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