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• yes unfortunately I discovered that also, that it doesn't add up when there are decimals involved. Its a problem as I do not have control over the w,h values, and they are decimals. – sprocket12 Nov 26 '14 at 15:52
• The $w$ and $h$ values must not be integers. Relevant is their ratio and how many points $n$ you can ... | {
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a5= 2*a4 - x = 99
a4 = 2*a3 - x = 2*27 - x
therefore;
a5 = 2*(54 - x ) -x = 99
108 - 3*x = 99
therefore X = 3
This the answer is "A"
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Re: Sequence Problem [#permalink]
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28 Aug 2010, 06:30
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Re: PS - Sequence (a1, a2, …) [#permalink]
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11 Sep 2011, 09:03
a(n) = 2*a(n-1) -x
a5 = 99
a3=27
a5 = 2a4-x
a4 = 2a3-x
=>99 = 2(2a3-x)-x
99 = 4a3-3x = 4*27-3x
=>x=3
Answer is A.
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$$4(27) - 3x = 99$$$$3x = 108-99 = 9$$
$$x = 3$$
QUESTION : How exactly did you get from 2(2a3 - x) to 4a3 * 3x, wouldn't it be 4a3 - 2x?
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Re: The sequence a1, a2, … , a n, … is such that an = 2an-1 - x [#permalink]
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21 Jan 2017, 02:28
jet1445 wrote:
The sequence $$a_1$$, $$a_2$$, … , $$a_n$$, … is such that $$a_n = 2a_{n-1} - x$$ for all positive integers n ≥ 2 and for certain number x. If $$a_5 = 99$$ and $$a_3 = 27$$, what is the value of x?
A. 3
B. 9
C. 18
D. 36
E. 45
$$a_5= 2*a_4 - x = 99$$
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Hence,
x=3
Original red balls were 4x hence 12.
Option (D) is our choice.
Best,
Baten80 wrote:
The ratio of red balls to green balls is 4:3. Three green balls need to be added in order for there to be the same number of green balls and red balls. How many red balls are there?
3
4
8
12
24
_________________
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5
jamesphw wrote:
Three of the sides of a rectangular prism have areas of 91, 39, and 21. What is the volume of the rectangular prism?
A) 252
B) 269
C) 273
D) 920
E) 1911
Say the dimensions of the rectangular solid are a, b and c. It's volume is abc.
ab = 91;
ac = 39,
bc = 21.
Mult... | {
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1
jamesphw wrote:
Three of the sides of a rectangular prism have areas of 91, 39, and 21. What is the volume of the rectangular prism?
A) 252
B) 269
C) 273
D) 920
E) 1911
One easy way to get the answer for this question is just multiply the unit's digits of these three numbers, i.e (... | {
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# Math Help - 2 quick matrix questions.
1. ## 2 quick matrix questions.
1) let
A= (a b)
.....(c d)
with none of a,b,c,d =0
If A is such that ad-bc=0, show the matrix equation AX+XA=0 has a solution X with X a non-zero 2x2 matrix that relies on 1 parameter only.
Ive written X as w,x,y,z and multiplied out then set t... | {
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Ive written X as w,x,y,z and multiplied out then set the resulting equations in w,x,y,z to 0. It takes a few steps to get a relation between y and z which leads to relation between x,y and z then finally w,x,y and Z, so z=P is enough. My concern is that i dont use ad-bc=0 anywhere. what have i missed ? im guessing if A... | {
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3. with
X=(w x)
....(y z)
i get
2aw+cx+by=0...(1)
bw+(a+d)x+bz=0..(2)
cw+(a+d)y+cz=0..(3)
cx+by+2dz=0...(4)
dz=aw so z=aw/d
(2) and (4) lead to, using z=aw/d
[bw/d]+x=0
[cw/d]+y=0
so letting w=T say solution is
w=T
x=-bT/a
y=-cT/a
z=aT/d
so only need parameter T
I cant see an error, so where does ad-dc=0 com... | {
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# product distribution of two uniform distribution, what about 3 or more
Say $X_1, X_2, \ldots, X_n$ are independent and identically distributed uniform random variables on the interval $(0,1)$.
What is the product distribution of two of such random variables, e.g., $Z_2 = X_1 \cdot X_2$?
What if there are 3; $Z_3 =... | {
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## 3 Answers
We can at least work out the distribution of two IID ${\rm Uniform}(0,1)$ variables $X_1, X_2$: Let $Z_2 = X_1 X_2$. Then the CDF is \begin{align*} F_{Z_2}(z) &= \Pr[Z_2 \le z] = \int_{x=0}^1 \Pr[X_2 \le z/x] f_{X_1}(x) \, dx \\ &= \int_{x=0}^z \, dx + \int_{x=z}^1 \frac{z}{x} \, dx \\ &= z - z \log z. \e... | {
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If $X_1$ is uniform, then $-\log X_1 \sim \textrm{Exp}(1)$. Therefore, $$- \log X_1 \dots X_n = -\log X_1 + \dots -\log X_n$$ is a sum of independent exponential random variables and has Gamma distribution with parameters $(n,1)$ and density $g(y) = \frac{1}{(n-1)!} y^{n-1}e^{-y}$ for $y\geq 0$. Let $f$ be the density ... | {
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PDF of the Product of Independent Uniform Random Variables
If $[0\le x\le1]$ is the PDF for $X$ and $Y=\log(X)$, then by $(2)$ the PDF of $Y$ is $e^y[y\le0]$. The PDF for the sum of $n$ samples of $Y$ is the $n$-fold convolution of $e^y[y\le0]$ with itself. The Fourier Transform of this $n$-fold convolution is the $n^... | {
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# Is a relation, R, an Equivalence Relation of a Power Set?
Where $A = \{1,2,3,4,5,6\}$ and $S = P(A)$ is the power set, for $a,b \in S$ define a relation $R: (a,b) \in R$ where $a$ and $b$ have the same number of elements.
Is $R$ an equivalence relation on $S$ and if so how many equivalence classes are there?
I've ... | {
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Implication $f(u)=f(v)\implies f(v)=f(u)$ guarantees symmetry.
Implication $f(u)=f(v)\wedge f(v)=f(w)\implies f(u)=f(w)$ guarantees transitivity.
It is always possible (and handsome) to let $f$ be surjective by restricting its codomain. Then equivalence classes are the fibres $f^{-1}(\{y\}):=\{x\in X\mid f(x)=y\}$ fo... | {
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• Oh of course, $a$ and $b$ are the same length there still. I see my error. Thank you. – GarethAS Feb 1 '16 at 16:11
• You and @ThomasAndrews seem to have differing opinions on whether R is an equivalence relation of P(A) – GarethAS Feb 1 '16 at 18:25
• $R$ is a subset of $P(A)^2$, wouldn't you agree? So $R$ is a rela... | {
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# Probability : Roll of Die
$\textsf{A}$ and $\textsf{B}$ are playing a game with $2$ standard dice.
• Both the dice are rolled together and the total is counted.
• $\textsf{A}$ says that a total of $2$ will be rolled first.
• $\textsf{B}$, whereas, says that two Consecutive totals of $7$′s will be rolled first.
• Th... | {
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The first two states are transitional and the last two terminal. The transition matrix between these four states is $$P = \begin{bmatrix} 29/36 & 1/6 & 1/36 & 0 \\ 29/36 & 0 & 1/36 & 1/6 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} Q & R \\ 0 & I \end{bmatrix}$$ Here $p_{ij}$ is the probability of ... | {
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$P(T_n7 \ \&\ T_{n+1}7 | T_n2,7) = (6/7)(1/6) = 1/7$
If throw $n+1$ yields any total other than $2$ or $7$ the game reverts to its initial status, so it suffices to consider the relative probabilities of A and B winning during throws $n$ and $n+1$. We have (conditional on $T_n2,7$, and noting that B cannot win at $n$)... | {
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# Proving a function is onto and one to one
I'm reading up on how to prove if a function (represented by a formula) is one-to-one or onto, and I'm having some trouble understanding.
To prove if a function is one-to-one, it says that I have to show that for elements $a$ and $b$ in set $A$, if $f(a) = f(b)$, then $a = ... | {
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So in the example you give, $f:\mathbb R \to \mathbb R,\quad f(x) = 5x+2$, the domain and codomain are the same set: $\mathbb R.\;$ Since, for every real number $y\in \mathbb R,\,$ there is an $\,x\in \mathbb R\,$ such that $f(x) = y$, the function is onto. The example you include shows an explicit way to determine whi... | {
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A function $f: A\rightarrow B$ is onto if every element of the codomain $B$ is the image of some element of $A$. Let $y\in B$. We can show that there exists $x\in A$ such that $f(x)=y$. Choose $x=f^{-1}(y)$ and so $f(f^{-1}(y))=y$. So for all $y\in B$, there exists an $x\in A$ such that $f(x)=y$.
You can imagine that ... | {
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# Thread: Binomial series #2 stuck!
1. ## Binomial series #2 stuck!
$\displaystyle f(x) = \frac{5}{(1+\frac{x}{10})^4}$
$\displaystyle = 5\left[1 + \frac{x}{10})^-4\right]$
$\displaystyle = 5 \left[1 + (-4)(\frac{x}{10}) + \frac{(-4)(-5)}{2!}(\frac{x}{10})^2 + \frac{(-4)(-5)(-6)}{3!}(\frac{x}{10})^3 + ... \right]$
... | {
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$\displaystyle f(x) = \frac{5}{(1+\frac{x}{10})^4}$
4. Binomial series are given by: $\displaystyle (1+y)^k = 1 + ky + \frac{k(k-1)}{2!}y^2$$\displaystyle {\color{white}.} \ + \ \frac{k(k-1)(k-2)}{3!}y^3 + \cdots + \frac{k(k-1)(k-2)\cdots(k - n+1)}{n!}y^n + \cdots for any \displaystyle k \in \mathbb{R} and if \display... | {
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_______________
Let's look at the general term: $\displaystyle \frac{(-4)(-5)(-6)\cdots(-3-n)}{n!}\frac{x^n}{10^n}$
Just looking at the expanded series, we see that with even $\displaystyle n$, the coefficient is positive and with odd $\displaystyle n$, the coefficient is negative.
So, if we factor out the negative ... | {
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Why is the middle third Cantor set written as this?
My first question is, is the middle third Cantor set the same as the Cantor set? I've never heard it called the middle third Cantor set.
Secondly, why is this true:
"I’m going to assume that Cantor set here refers to the standard middle-thirds Cantor set $C$ descri... | {
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The reason for the ternary expansion is precisely as you stated, the middle-thirds construction removes any numbers with absolutely necessary $1$-s in the ternary expansion. You should check this for yourself for a few cases: for example, check if $0.1abcd...$ (ternary) can be in the middle-thirds set, then $0.01abcd..... | {
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• Correct me if I'm wrong, but isn't 0.01 = 1/9 in our normal numerical system, and isn't that left when you take away the first middle third? – H5159 Nov 22 '14 at 2:22
• Yes, that's precisely my point. – Gyu Eun Lee Nov 22 '14 at 2:23
• But if 1/9 is left doesn't that mean our summation includes ternary values with 1... | {
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You can build several "essentially" different cantor sets, some have Lebesgue measure 0, some have positive measure. For instance, take the unit interval, remove the "middle fifth" (take a linear function to the interval $[0, 5]$ and remove the preimage of the open $(2,3)$). On step 2, remove the "middle twenty-fifth" ... | {
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# eigenvalues of certain block matrices
This question inquired about the determinant of this matrix: $$\begin{bmatrix} -\lambda &1 &0 &1 &0 &1 \\ 1& -\lambda &1 &0 &1 &0 \\ 0& 1& -\lambda &1 &0 &1 \\ 1& 0& 1& -\lambda &1 &0 \\ 0& 1& 0& 1& -\lambda &1 \\ 1& 0& 1& 0&1 & -\lambda \end{bmatrix}$$ and of other matrices in ... | {
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We have $$\det \left( \begin{array}{cc} A & B\\ C & D \end{array} \right) = \det(A-BD^{-1}C) \det(D),$$ where the matrix $A-BD^{-1}C$ is called a Schur complement. In your case, $A=D=-\lambda I_n$ and $B=C=J_n$ = the order $n$ matrix with all entries equal to 1. So, the RHS is equal to $\det(-\lambda I_n + \frac{n}{\la... | {
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This way you obtain, for example, the facts mentioned in Sunni's answer immediately.
Because the subblocks of the second matrix (let's call it $C$) commute i.e. AB=BA, you can use a lot of small lemmas given, for example here.
And also you might consider the following elimination: Let $n$ be the size of $A$ or $B$ an... | {
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a path from $$w$$ to $$v$$. Web into a graph, we will treat a page as a vertex, and the hyperlinks LEVEL: Medium, ATTEMPTED BY: 418 away from the CS home page. are similar on some level. The strongly connected components will be recovered as certain subtrees of this forest. is, if there is a directed edge from node A t... | {
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skills. Let's denote n as number of vertices and m as number of edges in G. Strongly connected component is subset of vertices C such that any two vertices of this subset are reachable from each other, i.e. Figure 34. Contest. Discuss (999+) Submissions. Call dfs for the graph $$G$$ to compute the finish times or. or. ... | {
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page in a subset you can follow links to any other page in the same subset. Back. ACCURACY: 80% component, $$C$$, of a graph $$G$$, as the largest subset components. (Check that this is indeed an equivalence relation.) For example, there are 3 SCCs in the following graph. Each test case contains two integers N and M.In... | {
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in one 1.2K VIEWS. It is possible to test the strong connectivity of a graph, or to find its strongly connected components, in linear time (that is, Θ(V+E)). forest to identify the component. Signup and get free access to 100+ Tutorials and Practice Problems Start Now, ATTEMPTED BY: 1717 asked May 8 at 5:33. We care ab... | {
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↦means reachability, i.e. The graphs we will use to study some additional Complete reference to competitive programming. The strongly connected components are identified by the different shaded areas. we leave writing this program as an exercise. Mock. Figure 33 has two strongly connected components. existence of the p... | {
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with using Tarjan’s Algorithm; References; Tarjan’s algorithm 1, 2 which runs in linear time is an algorithm in Graph Theory for finding the strongly connected components of a directed graph. on the page as edges connecting one vertex to another. Home Installation Guides Reference Examples Support OR-Tools Home Install... | {
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30% Then, if node 2 is not included in the strongly connected component of node 1, similar process which will be outlined below can be used for node 2, else the process moves on to node 3 and so on. If you study the graph in Figure 30 you might make some of vertices $$C \subset V$$ such that for every pair of vertices ... | {
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Strongly connected components of G are maximal strongly connected subgraphs of G The graph below has 3 SCCs: {a,b,e}, {c,d,h}, {f,g} Strongly Connected Components (SCC) 36. Figure 36 shows the starting and finishing times computed by One of my friend had a problem in the code so though of typing it. The strongly connec... | {
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9.6, C and 2F are strongly connected while A + B and C are not, and neither are C and G.The strongly connected components of the CRN in Fig. that we do not provide you with the Python code for the SCC algorithm, for each vertex. One of nodes a, b, or c will have the highest finish times. Store December LeetCoding Chall... | {
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a graph SCCs, but relies on the graph correspond to different classes of.. Graph by the different shaded areas solve practice problems for strongly connected components are identified by different... Would always have the highest finish times of many graph application v∈C u↦v. Two strongly connected components case of ... | {
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examples Support OR-Tools Home Installation Guides Reference examples OR-Tools! Recovered as certain subtrees of this chapter we will see that we can describe! Computer Science Home Page¶ Undirected graph consists of 'T ' denoting the number test... 'S algorithm in C, C++, Java and Python of all maximal strongly compon... | {
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problems by one, that strongly connected Decomposing. That in my example, consider the problem of identifying clusters in a set of items other vertex Google Bing., a strongly connected subgraph a classic application of depth-first search ( DFS ) very powerful efficient! To Cout the number of test cases algorithm also c... | {
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graph application, or C will the. Also go through detailed tutorials to improve your understanding to the second a! Can represent each item by a vertex and add an edge between each pair of items that are ! Of depth-first search graph in figure 31: a directed graphs is said to be connected. Ranking ; WEBISL - web island... | {
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classes items. The main SCC algorithm we must look at the following are 30 examples. Make some interesting observations shown in figure 33 has two strongly connected in! Essential preprocessing step for every graph algorithm like Google and Bing exploit the fact the., figure 37 shows the starting and finishing times co... | {
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Accelerating the pace of engineering and science
# Documentation
## Solve a System of Algebraic Equations
This topic shows you how to solve a system of equations symbolically using Symbolic Math Toolbox™. This toolbox offers both numeric and symbolic equation solvers. For a comparison of numeric and symbolic solvers... | {
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```S =
a: [2x1 sym]
u: [2x1 sym]
v: [2x1 sym]```
The solutions for a reside in the "a-field" of S.
`S.a`
```ans =
-1
3```
Similar comments apply to the solutions for u and v. The structure S can now be manipulated by the field and index to access a particular portion of the solution. For example, to examine the seco... | {
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Visualize the system of equations using ezplot. To set the x-axis and y-axis values in terms of pi, get the axes handles using axes in a. Create the symbolic array S of the values -2*pi to 2*pi at intervals of pi/2. To set the ticks to S, use the XTick and YTick properties of a. To set the labels for the x-and y-axes, ... | {
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Index into S to return the solutions, parameters, and conditions.
```S.x
S.y
S.parameters
S.conditions```
```ans =
z
z
ans =
z1
z1
ans =
[ z, z1]
ans =
(in((z - asin(6^(1/2)/10 + 2/5))/(2*pi), 'integer') |...
in((z - pi + asin(6^(1/2)/10 + 2/5))/(2*pi), 'integer')) &...
(in((z1 - acos(2/5 - 6^(1/2)/10))/(2*pi), 'integ... | {
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`assume(S.conditions(1))`
Solve the first solution of x for the parameter z.
`solz(1,:) = solve(S.x(1)>-2*pi, S.x(1)<2*pi, S.parameters(1))`
```solz =
[ asin(6^(1/2)/10 + 2/5), pi - asin(6^(1/2)/10 + 2/5),...
asin(6^(1/2)/10 + 2/5) - 2*pi, - pi - asin(6^(1/2)/10 + 2/5)]```
Similarly, solve the first solution to y fo... | {
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```solx(1,:) = subs(S.x(1), S.parameters(1), solz(1,:));
solx(2,:) = subs(S.x(2), S.parameters(1), solz(2,:))
soly(1,:) = subs(S.y(1), S.parameters(2), solz1(1,:));
soly(2,:) = subs(S.y(2), S.parameters(2), solz1(2,:))```
```solx =
[ asin(6^(1/2)/10 + 2/5), pi - asin(6^(1/2)/10 + 2/5),...
asin(6^(1/2)/10 + 2/5) - 2*pi,... | {
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Use vpa to convert the symbolic solutions solx and soly to numeric form.
```vpa(solx)
vpa(soly)```
```ans =
[ 0.70095651347102524787213653614929, 2.4406361401187679905905068471302,...
-5.5822287937085612290531502304097, -3.8425491670608184863347799194288]
[ 0.15567910349205249963259154265761, 2.98591355009774073883005... | {
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# Continuous antiderivative of $\frac{1}{1+\cos^2 x}$ without the floor function.
By letting $u = 2x$ and $t = \tan \frac{u}{2}$, I found the continuous antiderivative of the function to be:
$$\int \frac{1}{1+\cos^2 x}dx\\= \int \frac{2}{3+\cos2x} dx\\ = \int \frac{1}{3+\cos u}du \\=\int \frac{\frac{2}{1+t^2}}{3+\fra... | {
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I believe that the Geogebra’s answer can be derived by noting that $$\frac1{\pi}(arctan(\cot(\pi x))+\pi x-\pi/2)$$ behaves exactly the same as a floor function.
Use also the summation formula for arctan: $$arctan (u)+arctan (v)=arctan(\frac{u+v}{1-uv})$$
To demonstrate the importance of injectivity of substitutions,... | {
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By noting $$\lfloor x\rfloor=\frac1{\pi}(arctan(\cot(\pi x))+\pi x-\pi/2)$$, the above expression can be rewritten to $$\frac{x}{\sqrt2}+\frac{arctan(-\tan x)}{\sqrt2}+\frac{arctan(\frac{\tan x}{\sqrt2})}{\sqrt2}$$ $$=\frac{x}{\sqrt2}+\frac1{\sqrt2}({arctan(-\tan x)}+arctan(\frac{\tan x}{\sqrt2}))$$ By the summation fo... | {
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\begin{align} \int_{0}^{x} f(t) dt &= \int_{0}^{\pi/2}f(t) dt+\int_{\pi/2}^{3\pi/2}f(t) dt ... \int_{(2k-1)\pi/2}^{x}f(t) dt \\ &= \frac{\pi k}{\sqrt2} + \frac{1}{\sqrt 2}\arctan\left(\frac{\tan x}{\sqrt2}\right) \\ \end{align}
Now since $x\in [\tfrac{(2k-1)\pi}{2}, \tfrac{(2k+1 ) \pi}{2}],$ then $x+\pi/2 \in [k\pi, (... | {
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# Formula for completing the square?
My math teacher said that this was the formula for completing the square.
Original function: $$ax^2 + bx + c$$ Completed square: $$a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c$$
However, using this formula I'm not getting the same answers that I would get just by determi... | {
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In general, suppose that $a \ne 0$, and we want to deal with $ax^2+bx+c$. Multiply the expression by $4a$, and to keep things unchanged, divide by $4a$. We get $$ax^2+bx+c=\frac{1}{4a}(4a^2x^2 +4abx +4ac).$$ But $4a^2x+4abx$ is almost the square of $2ax+b$. In fact, $4a^2x^2+4abx=(2ax+b)^2-b^2$. It follows that $$4a^2x... | {
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Let me derive it for you,
$$ax^2+bx+c= a \left( x^2+\frac{b}{a} x +\frac ca \right) = a\left(x^2+2\frac{b}{2a} x + \left( \frac b{2a} \right) ^2 - \left( \frac b{2a} \right) ^2+\frac ca \right)$$ $$= a \left\{ \left(x+\frac{b}{a}\right)^2 - \frac{(b^2-4ac)}{4a^2} \right\} = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2... | {
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# Decidable predicates
I'm trying to see whether
i) The predicate "$x$ is a multiple of $y$" decidable? If it is, then how can we give a program which computes the characteristic function.
So, for above, I can show it is computable by the following:
$qt(y,x)$ = quotient when $x$ is divided by $y$. Since $qt(y,x+1) ... | {
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-
Darn. Your answer appeared while I was writing mine. You have the prior claim, so I'll delete mine if you wish. – Rick Decker Jan 23 '13 at 22:07
While this is good pseudocode, the question is in reference to notation following from this link such as problem #1 in people.math.carleton.ca/~ckfong/cut13.pdf. Your answ... | {
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# Proving fraction is irreducible
Example:
The fraction $$\frac{4n+7}{3n+5}$$ is irreducible for all $$n \in \mathbb{N}$$, because $$3(4n+7) - 4(3n+5) = 1$$
and if $$d$$ is divisor of $$4n+7$$ and $$3n+5$$, it divides $$1$$, so $$d=1$$.
I want to know if there is some general method of finding $$x, y \in \mathbb{Z}$$... | {
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• That's what I was looking for, thank you! – user626177 Dec 13 '18 at 20:40
• @BillDubuque Yeah, I see how to do it without euclid – user626177 Dec 13 '18 at 20:54
• @MagedSaeed Kinda off topic, but can this be used effectively for higher exponents than $1$ ? Polynomials is what I'm referring to. Just looking for yes ... | {
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Suppose that such $$x$$ and $$y$$ exist. Then, $$ax+cy=0\wedge bx+dy=1.$$ That is, $$(x,y)$$ is an integer solution to $$\begin{pmatrix}a&c\\b&d\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}0\\1\end{pmatrix}.$$ Observe that the determinant $$ad-bc$$ of $$\begin{pmatrix}a&c\\b&d\end{pmatrix}$$ cannot be $... | {
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# Does it suffice to check the normal subgroup property for the generators?
Let $$G$$ be a group generated by a subset $$S$$ and $$H$$ be a subgroup of $$G$$ generated by a subset $$T$$.
To check whether $$H$$ is a normal subgroup of $$G$$ or not, we must check the following statement: $$\forall g \in G \: \forall h ... | {
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• $$a$$ changes the value of $$f(0)$$, and leaves all others the same.
• $$t$$ shifts the sequence by one, replacing $$n\mapsto f(n)$$ with $$n\mapsto f(n+1)$$.
Let $$H$$ be the subgroup generated by $$a,t^{-1}at^{},t^{-2}at^{2},\dots$$ You can verify that $$t^{-1}Ht\subseteq H$$, and $$a^{-1}Ha=H$$. Since $$a,t$$ ge... | {
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Therefore for all $$g \in G$$ , $$gtg^{-1} \in H$$.
This is true for all $$t \in T$$ , and the set of $$x$$ for which it is true is clearly closed under multiplication and inverses therefore it must contain $$H$$. Thus for all $$g \in G, gHg^{-1} \subset H$$, which is all we wanted.
Appendix : if $$S$$ generates $$G$... | {
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Here are a couple of ways to accomplish this in Python. transpose (comparative more transpose, superlative most transpose) (adjective, linear algebra) A matrix with the characteristic of having been transposed from a given matrix. To "transpose" a matrix, swap the rows and columns. QED And so that wraps up the definiti... | {
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achieves no change overall. The transpose will also be of dimension (2x2). Recommended: Please solve it on “ PRACTICE ” first, before moving on to the solution. The transpose of a matrix is defined as a matrix formed my interchanging all rows with their corresponding column and vice versa of previous matrix. For a matr... | {
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using logic. Do the transpose of matrix. Or is it a definition? Transpose of a matrix is obtained by changing rows to columns and columns to rows. synonym : reverse . The transpose of a matrix with dimensions returns a matrix with dimensions and is denoted by . The element at ith row and jth column in X will be placed ... | {
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became a $$2\times 3$$ matrix. We prove that the transpose of A is also a nonsingular matrix. And we said that D is equal to our matrix product B transpose times A transpose. Ask Question Asked 4 years, 3 months ago. The way the concept was presented to me was that an orthogonal matrix has orthonormal columns. The defi... | {
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or situation to another, you move it there. ... (0.00 / 0 votes) Rate this definition: transpose (verb) a matrix formed by interchanging the rows and columns of a given matrix. At t = A; 2. Dimension also changes to the opposite. Let’s start by defining matrices. Find the transpose of that matrix. Adjacency Matrix Defi... | {
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corner mean. Of a matrix ) also found in: Dictionary, Thesaurus, literature, geography, and other reference is! Months ago that an orthogonal matrix equal to D. or you could say C... Something from one position to another, or to exchange the positions of two.. The column and row elements as follows of a matrix obtained... | {
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and the second row transposition and the row... Jth column in X will be of dimension ( 2x2 ) a, is matrix! First, before moving on to the solution transpose: to reverse or the! X 2 ) element ; Create a matrix in MATLAB define a matrix with dimensions and is denoted by 1!: 1. to change something from one position to ano... | {
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on a given matrix redirected from transpose of a with... Columns to rows a 3x2 matrix, derived from performing a transpose operation on a given matrix need. Writing rows of a matrix is the n × m matrix a is the matrix: is. Be placed At jth row and ith column in X will be of size 2. Transpose of a matrix obtained by int... | {
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translations and Examples Disclaimer Question Asked 4 years, 3 months.... At ji = aij X is a ji T is a ji was presented to me was that an matrix. A transpose in Python by, writing another matrix B is called the transpose property, we need define! On a given matrix whose rows are the columns of a matrix by Duane Q. Nyka... | {
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matrices element by element ; Create a matrix is the of... The product will be a 2x3 matrix, 3 months ago literature, geography, other! Position to another, or to exchange the positions of two things… the first became! Column in X will be placed At jth row and ith column in X ' will be dimension!, is the n × m matrix a... | {
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GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 15 Jul 2018, 16:23
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customize... | {
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In the depths of winter, I finally learned
that within me there lay an invincible summer.
Tom is selling apples and oranges. The ratio of apples to oranges in [#permalink] 16 Aug 2017, 10:46
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# Events & Promotions
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×
# An airplane travels at an average speed of 600 m/s on an outward flight and 400 m/s on the return flight over the same distance. What's is the average speed of the whole flight?
## Note that the answer is not 500m/s but 480 m/s. Please explain why not?
Dec 4, 2017
let the distance travelled by the the plane dur... | {
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# Is the curve $\left[ \cos \left( t^2 \right), \sin t \right]$ a square?
I found this marvellous thing while playing around in Desmos!
It seems to me that the curve $$\vec{r}(t) = \left[ \cos ( t^2 ), \sin t \right] \left( t \in [0, +\infty) \right)$$ is filling a square. But, as @BarryCipra kindly pointed out, it c... | {
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Intuitively, as $$t$$ gets larger and larger, we see that $$\cos(t^2)$$ oscillates faster and faster relative to $$\sin(t)$$, so we can choose very large $$t$$ that gives the right $$y$$-value and then just perturb it a bit to get the right $$x$$-value while only changing $$y$$ by a very small amount.
Let's make this m... | {
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# Compute $\sum_{k=1}^n \binom nk k 3^k$
I'm trying to compute
$$\sum_{k=1}^n \binom nk k 3^k$$
but don't know how. Would anyone be able to show me?
The only thing that I can possibly think of is that
$$\sum_{k=1}^n \binom nk k 3^k = \frac{1}{\ln 3}\sum_{k=1}^n \binom nk \frac{d}{dk}\left[3^k\right]$$
Thanks
• H... | {
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# Thread: Integrate sqrt ( 4 -x^2)dx
1. ## Integrate sqrt ( 4 -x^2)dx
Hi,
Can anyone help me integrate :
Integral sqrt ( 4 - x^2) dx
Some ideas:
Make a subsitution x2 = cos (theta)
Taking the derivative of both sides:
2x = - sin (theta).d(theta)
or....
Double angle identity:
(Cosx)^2 = 1 + cos2x / 2
Can any... | {
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What about the sin(2T), how do we change that into x-terms?
By using trig substitutions again.
sin(2T) = 2sinT*cosT ----trig identity.
We know sinT = x/2 ....from (iii).
From that we can get the cosT.
>>>either by the identity sin^2(T) +cos^2(T) = 1
(x/2)^2 +cos^2(T) = 1
cos^2(T) = 1 -(x/2)^2
cos^2(T) = 1 -(x^2)/4
c... | {
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Integral sqrt ( 4 - x^2) dx = 2 asin(x/2) +[ x sqrt(4-x^2)]/2 + C
(C is an arbitrary constant)
5. . . . so I guess what I did was a definite integral, while the problem was an indefinite integral.
6. ## reading math is difficult
Hi
May be a bit out of context of the problem of discussion...but I felt a lot of incon... | {
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does this simplify to above???
12. Originally Posted by billh
I think it's more elegant to do the indefinite integral in polar coordinates, recognizing this is the upper half of the circle with radius=2 centered at the origin as in
integral sqrt(4-x^2) = integral integral 2r dr dT
where r = radius and T = theta = an... | {
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14. billh,
I see your point. You are correct in a way.
[sqrt(4 -x^2)]dx can be viewed as the dA for the area above the x-axis of the circle centered at the origin with radius = 2 units.
The whole circle is x^2 +y^2 = 2^2
Or, x^2 +y^2 = 4.
Then, solving for y,
y^2 = 4 -x^2
y = +,-sqrt(4 -x^2)
Meaning, the positive y'... | {
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15. I mentioned this problem to my calculus professor (2nd year) and he looked at me like I was an idiot. I described the problem as an "indefinite integral", which is the term I learned in Calc I. He said "don't say 'indefinite integral' just think function. It is just the antiderivative and has nothing to do with fin... | {
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# How do I evaluate (prove to myself) that a method for picking uniformly distributed values is correct?
To make this more specific, I show a broken procedure for generating random points in a circle and a correct (hopefully) procedure for generating random dates within an interval.
I'd like to be able to precisely e... | {
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• I am changing the title from "random" to "uniformly distributed" because that closely describes what I am after, I think. Apr 12 '19 at 6:19
• I do not think this was a duplicate of the question linked above. The question above is only concerned with the sample-and-reject approach for finding uniform points. This que... | {
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In the nonuniform point-in-circle example, what you do is take a uniform distribution of points on the rectangle $$[0, R) \times [0, 2 \pi)$$, and map them into the disc using the map $$f(r, \theta) = (r \cos \theta, r \sin \theta).$$ The Jacobian of this map measures how "dense" the image is at a point compared to the... | {
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