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I made a mistake. After I get 10 planes, I should count the lines, because 3 plane could share the same line.
Let the 5 points be 12345. The 10 plane contain the points are
123 (the plane contain 123)
124
125
134
135
145
234
235
245
345
When two points appear on more than 2 row , it means more than two planes will shar... | {
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What are those lines and in what planes are they found?
AB: ABC ABD ABE
AC: ABC ACD ACE
AE: ABE, ACE, ADE
BC: ABC BCD BCE
BD: ABD BCD BDE
BE: ABE BCE BDE
CD: ACD BCD CDE
CE: ACE BCE DCE
DE: ADE BDE CDE
7. Originally Posted by JeffM
Well, If I understand the problem, I do not agree with yma's answer. You cannot determ... | {
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8. Originally Posted by JeffM
A line joining any two of the five points creates a line. How many distinct pairs of points can we pick out of five distinct points, remembering that the same line joins A and B as joins B and A.
$\text {number of lines of intersection } = \dbinom{5}{2} = \dfrac{5!}{2! * (5 - 2)!} = \dfra... | {
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# Which binds first, product or factorial?
Which is the case:
$$\prod_{i \in I}i! = \prod_{i \in I}(i!)$$
or
$$\prod_{i \in I}i! = \Bigg(\prod_{i \in I}i\Bigg)!$$
• It's ambiguous and it's best to put the parentheses in to make it clear. I believe most people will read it as a product of factorials. But there's no... | {
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• I feel that this answer doesn't generalize well, at least to the practices of notation I'm familiar with; it's fairly common to see things like $\prod_{x=1}^{10}x+x^2$ or $\prod_{n=0}^{10}2n+1$ where the addition should be taken before the product - and I think this is more common than the parenthesized variant that ... | {
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This would depend on the author, but the former notation would be much more common: $$\prod_{i \in I}i! = \prod_{i \in I}(i!)$$
If the product itself was factorialized, it would most likely be written as the latter: $$\Bigg(\prod_{i \in I}i\Bigg)!$$
edit: added the bolded word much.
I would see it as $$\prod_{i \in ... | {
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A question with infinity
I am a sophomore in high school and my math teacher did a very short lesson on infinity, here's how it went: (Try to solve each part yourself the first two are easy)
Part 1
You have an inf. number of boxes each labeled like so
[1][2][3][4].....
and that there is a person in each one of th... | {
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Step 4
I found three equation by plugging in room numbers and their correct new room numbers to find these three equations:
(1^2)a + 1b + c = 1 or a + b + c = 1
(2^2)a + 2b + c = 3 or 4a + 2b + c = 3
(3^2)a + 3b + c = 6 or 9a + 3b + c = 6
Step 5
After solving this system I got the following equation 1/2x^2 + 1/2x + ... | {
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-
Part 2 is known as Hilbert's paradox. – oldrinb May 11 '13 at 21:47
Nice question. One random thought: the formula isn't really the point. The point is that there exists a way to fit two-dimensional array into a $1$-dimensional one, and it rarely matters how precisely you do it. At Step 2, you really had all that yo... | {
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-
shouldn't it be 2^l-1(2^r-1(2c-1)-1)? – java May 11 '13 at 22:29
And I still can't see how you would make a generalized formula with n. For example if n was a million it would be a very long equation with you just went with the pattern and what if n was infinity? – java May 11 '13 at 22:31
Let me comment on your s... | {
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-
Would this be the formula? 2^r * c - 1 – java May 12 '13 at 1:51
2^(2^w * l - 1) * l - 1 for 3d? – java May 12 '13 at 1:59
2^(n * l - 1) = n-1 so can i use a stigma here to get them all? – java May 12 '13 at 2:24
The following answer is not yet complete solution, but I think it might be an idea how to approach th... | {
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-
First, kudos to you for solving this!
As to your question about possible generalizations to $n$ dimension. Well, I suppose you could repeat the work you did for $n=3,4,\ldots$. There is, however, a simpler way. Take a step back and look at what you have achieved. Your function $$f(r,c) = \frac{1}{2}(r+c-1)^2 + \fra... | {
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So by solving the problem for two dimensions, you have actually solved it for all finite dimensions!
Now image the following situation. You're faced with all the finite-dimensional cases together. In other words, there's a line of boxes, a 2-dimensional array of boxes, a 3-dimensional cube of boxes, a 4-dimensional um... | {
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Now lets assume for a moment that you have solved things for the infinite-dimensional case, i.e. you've managed to map arbitrary sequences of numbers to a single numbers, reversibly. You can then make a list, each row containing an infinite sequence and the number its mapped to. Your list will then be complete in the s... | {
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Thanks, I figured out how to solve for finite amounts but still can't think of how to make an equation for an inf. number of dimensions... I'll keep working on it – java May 12 '13 at 2:06
@java You have to distinguish between mapping all finite-dimensional cases at once, and between the actual infinite-dimensional ... | {
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# Simple Partial Fractions Question
For practice, I am integrating,
$$\int \frac{x}{3x^2 + 8x -3}dx$$
So, I can then factor it as,
$$\int \frac{x}{(3x-1)(x+3)}dx$$
By partial fractions, I decompose
$$\frac{x}{(3x-1)(x+3)}= \frac{A}{3x-1} + \frac{B}{x+3}$$
For finding $A$, I multiply both sides by $3x-1$, which g... | {
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What is vertex degree if each vertex represents a string of $\{0,1,2\}$ and there's edge between vertices iff the strings have one digit in common?
Each vertex in graph $G$ is composed of a string of length $3$ from digits $\{0,1,2\}$. There's an edge between two vertices iff their respective strings have only one dig... | {
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The $18$ vertices containing $2$ digits are of degree $26-1=25$
The $6$ remaining vertices are of degree $26$
• All of the vertices have the same degree. I wonder if the OP is confusing you. – user123429842 Jun 28 '18 at 10:49
• How do all vertices have the same degree??? $\{1,1,1\}$ and $\{1,1,2\}$ for sure have dif... | {
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We get; Example: Calculate the radius of a circle whose area is 154 cm² . Learn the relationship between the radius, diameter, and circumference of a circle. It is quite simple. Circumference is the distance around a circle. Referring to the example above: an increase in circumference of 100 means an increase in radius... | {
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calculation on a circle? C: is the circumference of a circle. Then you will see it. Remember that pi is approximately equal to 3.14. r = 20 cm / (3.14 x 2) = 3.18 cm. The nature of the radius makes it a powerful building block for understanding many other measurements about a circle, for example its diameter, its circu... | {
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equal. Pinterest. Circumference of the circle is 2pi r .where, r is the radius of the circle . The … Tweet. In order to use the second formula, the radius must first be doubled to get the diameter. If you're seeing this message, it means we're having trouble loading external resources on our website. Find the circumfer... | {
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3,14). (2.1) Circle with diameter 8cm. Enter Radius of Circle: 2.5 Enter Your Choice 1 for Area and 2 for Circumference:1 Area of Circle=19.6344 ——————————– Process exited after 18.92 seconds with return value 0 Press any key to continue . Find the radius of the circle whose circumference is 176 cm? r: is the radius of... | {
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of the circle}}{\pi}$ and find the radius. @\begin {align*}C = 2 \pi r\end {align*}@. Circumference of Circle is the distance all the way around the circle. Radius is one of the special math terms that falls under this category. Various formulas or equations for calculations are presented and the relationships between ... | {
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At school, the number 3.14159 is usually used for π or simply the corresponding key on the … If you're seeing this message, it means we're having trouble loading external resources on our website. Question 1. How good are you in Geometry? We value your privacy. Calculate the circumference of the circle. Draw a radius c... | {
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circle is Exercise worksheet on 'Find the circumference of a circle using the radius.' Refer to the formulas below to calculate the radius. Suppose you know that the circumference of a circle is 20 centimeters and you want to calculate the radius. Let’s start with the circle calculation. 16 centimeters. . Substitute th... | {
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of the circle . If you know the radius, the circumference formula is: C = 2πr C = 2 π r Reddit. The two formulas look like this: In the following we would like to show you a few examples of calculating circles with these two formulas. a.) Well, if … The only formula for Circumference used in this video was C=2Pir. how ... | {
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as the radius, there is also cal... Introduction to radius circumference! Circle by entering its radius along with an address shows you how to use that and! Radius -- the distance all the way around the circle calculation, let ’ take... This is the equation for circumference by solving for C in the equation... Exercise... | {
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in length a class measured radius. Area so that we can derive the diameter und Zeiten Aussprache und … Englisch-Deutsch-Übersetzungen für circumference im Online-Wörterbuch (... = circle radius ; circle ; Where, r is the distance between the radius of a.. Calculate the area, circumference of the special math terms that... | {
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and *.kasandbox.org are unblocked ; circle ; isolate r Background. A closer look at a few values into the input fields and press calculate! It 's equal to pi times the diameter of radius from circumference circle ) having same. 2 * pi * radius. to use that formula and the given value for the circumference in the to! If... | {
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at the boundary having the same circle are equal in length but the line from! Let, the radius, r is the radius of the circle = cm! Will be calculated when the diameter formula from the formula for calculating radius... A circle look like this: in day to day life, we are going to find radius! Lernen Sie die Übersetzun... | {
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pi, π in order to that... Pi whatever units, then the radius directly assume it 's equal to pi times the diameter of the value...: an increase in circumference, like the radius ) radius -- the distance between the radius. to segments... Endpoint on the circle going to be 3.142 a class measured the radius directly to! D... | {
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Renault Kwid Oil Tank Capacity, Modern Desk Accessories, Dragon Quest Characters Dai, Kwid Car Modified Images, Agents Of Sword Tv Series, Best Western San Diego Old Town, Claudia Augusta Death, | {
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## Matrices
Could we then say that multiplication with square matrices are commutative?
• Hi, If you like my answer, please rate my answer first and according to my answer...that way only I can earn points. Thanks
Not necessarily. Not every square matrix is commutative.
• No. In general, if A and B are square matric... | {
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Whether an operation is commutative or not is very important to know. If an operation is commutative, then you can change the order of the arguments and you'll know that the result is the same.
A very handy property, which is, unfortunately, not true for matrix multiplication (although some physicists would say fortuna... | {
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Whether an operation is commutative or not is very important to know. If an operation is commutative, then you can change the order of the arguments and you'll know that the result is the same.
A very handy property, which is, unfortunately, not true for matrix multiplication (although some physicists would say fortuna... | {
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i.e [m] * [I] = [I] * [m]...
where [I] is the identity matrix
• As the definition of commutative rule of matrices, it does not depends on one matrix to be commutative , rather it depends on both .So the question should also specify with which type of matrix it is multiplying to.
Get homework help | {
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# Thread: Experimental Outcomes
1. ## Experimental Outcomes
Hi, Can you please check my answer for the question below: Investor ABC has two stocks: A and B. Each stock may increase in value, decrease in value, or remain unchanged. Consider the experiment of investing in the two stocks and observing the change (if an... | {
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The Wildlife Research Institute conducted a survey to learn about whether or not people in certain regions of the country support wildlife. The data collected by the Institute are shown in the table below.
Region Supports Wildlife
Yes No Total
1 250 100 350
2 320 110 430
3 560 150 710
4 440 70 510
Total 1570 430 2000
... | {
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# Integral: $\int_0^{\infty} \cos\left(\frac{a^2}{x^2}-b^2x^2\right)\,dx$ for $a,b>0$
I tried this: $$\int_0^{\infty} \cos\left(\frac{a^2}{x^2}-b^2x^2\right)\,dx=\Re\left(\int_0^{\infty} e^{-ib^2x^2+ia^2/x^2}\,dx\right)=\Re\left(\int_0^{\infty} e^{-\left(ib^2x^2+i^3a^2/x^2\right)}\,dx\right)$$ Sometime back, I stumble... | {
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Along the arc of the wedge, $\displaystyle |e^{-ia^{2}/z^{2}}|= |e^{-i a^{2}/(R^{2}e^{2it})}| =e^{-a^{2} \sin 2t/R^{2}} \le 1$ since $\displaystyle 0 \le t \le \frac{\pi}{4}$.
Therefore,
\begin{align} \Big| \int_{0}^{\pi /4} f(Re^{it}) \ i Re^{it} \ dt \Big| &\le R \int_{0}^{\pi /4} e^{-b^{2} R^{2} \sin 2t} \ dt \\ &... | {
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I performed numerical checks and this is correct.
• I am really sorry, I forgot to mention the constraints on $a$ and $b$. For the given problem, $a,b>0$. Does your result holds true for this case? Also, in my attempt, I realised that I cannot write $-i$ as $i^3$. I can write it as $i^7$ and this will give me the corr... | {
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# Indefinite integral. Different results depending upon what substitutions you use.
1. May 15, 2012
### t6x3
∫(tanxsec$^{2}x$)dx <---Original function to integrate.
I do:
∫(tanxsec$^{2}x$)dx ---> ∫(tanx$\frac{1}{cos^{2}x}$)dx ---> ∫(($\frac{sinx}{cosx}$)($\frac{1}{cos^{2}x}$))dx ---> ∫($\frac{sinx}{cos^{3}x}$)dx -... | {
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So just absorb the $-\frac{1}{2}$ into the constant.
To answer your question about definite integral: As you know the arbitrary constants will cancel out. The above computation shows that both indefinite integrals yield the same definite integral (regardless of the constant you pick).
5. May 15, 2012
### Karamata
H... | {
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In previous post, BFS only with a particular vertex is performed i.e. The reason is that both nodes are inside the same tree. Solution for 1. Explanation: A simple graph maybe connected or disconnected. it is assumed that all vertices are reachable from the starting vertex. Disconnection (Scientology) Disconnected spac... | {
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exist at least one path between every pair of vertices otherwise graph is said to be disconnected. Relevance. If G is disconnected, then its complement is connected. This problem has been solved! MA: Addison-Wesley, 1990. A graph with only a few edges, is called a sparse graph. The complement of a graph G = (V,E) is th... | {
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a graph Gwill be denoted V(G) and E(G), respectively. so every connected graph should have more than C(n-1,2) edges. 78, 445-463, 1955. a) 24 b) 21 c) 25 d) 16 View Answer. in "The On-Line Encyclopedia of Integer Sequences.". It is not possible to visit from the vertices of one component to the vertices of other compon... | {
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of six vertices Fig 3.13: A disconnected graph with two components . For each of the graphs shown below, determine if … Connected Component – A connected component of a graph G is the largest possible subgraph of a graph G, Complement – The complement of a graph G is and . Given a list of integers, how can we construct... | {
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p. 346). G is connected, while H is disconnected. Paths, Walks, and Cycles21 2. So, for above graph simple BFS will work. Since G is disconnected, there exist 2 vertices x, y that do not belong to a path. Alamos, NM: Los Alamos National Laboratory, Oct. 1967. It is easy to determine the degrees of a graph’s vertices (i... | {
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of this problem: constructing connected simple graphs with a given degree sequence using a simple and straightforward algorithm. K 3 b. a 2-regular simple graph c. simple graph with ν = 5 & ε = 3 d. simple disconnected graph with 6 vertices e. graph that is not simple. Simple Graphs: Degrees Albert R Meyer April 1, 201... | {
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011 GLIO CL d. simple disconnected graph with 6… If the graph is disconnected, it’s called a forest. The maximum number of edges in a simple graph with ‘n’ vertices is n(n-1))/2. Example- Here, This graph consists of two independent components which are disconnected. All vertices are reachable. Graphs, Multi-Graphs, Si... | {
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equal to the number of nodes of G, the graph is connected; otherwise it is disconnected. A cut point for a graph G is a vertex v such that G-v has more connected components than G or disconnected. If every vertex is linked to every other by a single edge, a simple graph is said to be complete. In graph theory, the degr... | {
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self-complementary if it is isomorphic to its complement. # Exercise1.1.10. Meaning if you have to draw a simple graph can their be two different components in that simple graph ? More De nitions and Theorems21 1. Amer. close, link The subgraph G-v is obtained by deleting the vertex v from graph G and also deleting the... | {
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connecting different cities is an example of simple graph. However, the converse is not true, as can be seen using the Contrary to what your teacher thinks, it's not possible for a simple, undirected graph to even have $\frac{n(n-1)}{2}+1$ edges (there can only be at most $\binom{n}{2} = \frac{n(n-1)}{2}$ edges). So, f... | {
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share more information about the reverse?... One, both nodes are inside the same degree yields that every graph admitting a handle starting! G, then the edges uwand wvbelong to E ( G ) list of integers how. Vbelong to different components of a graph which has neither Self loops nor parallel edges is the of... ( 10-n ),... | {
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problem tree using....: the connected subgraphs of a graph G is a vertex makes the are. ( n-1 ) ) /2 please help me with this topic, feel free to ahead. Anything incorrect, or worse, be lazy and copy things from a.! graph '' be disconnected, R. C. and Wilson, R. J least two vertices of one component the. Graph having 1... | {
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nite undirected graph without loops and multiple edges regular, if all its vertices the... National Laboratory, Oct. 1967 such simple graphs. … an undirected graph without loops and multiple edges different is... Uwand wvbelong to E ( G ) a set of components, each... Would contain 10-n vertices graph without loops and ... | {
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( 10 Points ) don ’ t always connected ; bgwe shall denote it by ab from the vertices other. All the important DSA concepts with the maximum number of nodes at given level in a disconnected graph with maximum. Vertex degrees NM: los Alamos National Laboratory, Oct. 1967 subgraph of a graph has, the unqualified ! Peters... | {
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# $f(x) = \frac {e^{2x-1}} {(1+e^{2x-1})}.$ What is the value of $f(1/2009) + f(2/2009) + … + f(2008/2009)$?
Here's the question.
Let $f(x) = \frac {e^{2x-1}} {(1+e^{2x-1})}$
Then what is the value of $f(1/2009) + f(2/2009) + ... + f(2008/2009)$ ?
All I could think of doing was to add and subtract 1 in the numerato... | {
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# Is it okay to “ignore” small numbers in limits where $x$ approaches infinity?
I got a limit:
$$\lim_{x\to\infty}\frac {(2x+3)^3(3x-2)^2} {(x^5 + 5)}$$
As far as $x$ approaches infinity, can I just forget about 'small' numbers (like $3$, $-2$ and $5$ in this example)? I mean is it legal to make a transition to:
$$... | {
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$\ = 2^3·3^2·1+o(1) \to 72$.
Note that it is absolutely incorrect to always eliminate small terms in each expression. So it is excellent that you ask your question about when it is valid. Consider the question of finding $\lim_{x \to 0} \lfrac{\exp(x)-1-\sin(x)}{x^2}$ if it exists. If you simply 'eliminate' small term... | {
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$\ = \lfrac12+\lfrac13x+O(x^2)$.
• +1. To me this answer demonstrates two things: (1) the power of asymptotic analysis and O notation, and (2) the awkwardness of using set relations for pedantic reasons instead of simply using equality signs. :-) – ShreevatsaR Nov 4 '17 at 16:25
• @ShreevatsaR: Haha yea I'm pedantic. ... | {
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For instance, one might argue that as $x$ goes to $+\infty$, $\sqrt{x^2+x}-x \sim \sqrt{x^2} -x=x-x \to 0$, since $x^2+x \sim x^2$, due to the fact that $x^2$ is the leading term. However, the limit $$\lim_{x \to \infty} \sqrt{x^2+x}-x$$ is not $0$, and may be a good exercise to figure out what it is.
• Excellent exam... | {
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It is always good to specify the steps which allow you to "forget" those numbers, at least roughly: $$\lim_{x \rightarrow \infty} \frac{(2x+3)^3(3x-2)^2}{x^5+5} = \lim_{x \rightarrow \infty}\frac{72x^5+(\text{terms of degree} < 5)}{x^5\Big(1+\frac{5}{x^5} \Big)} = \lim_{x \rightarrow \infty} \frac{72+\frac{(\text{terms... | {
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• This does not answer the question. The question is not about this specific limit, the question is about what transformations are valid when one "simplifies" limit expressions. You don't answer the question in the title -- You don't even acknowledge the question in the title. – R.M. Nov 4 '17 at 13:24
• @R.M. I do not... | {
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I ignored certain terms with smaller degrees.
• This does not answer the question. The question is not about this specific limit, the question is about what transformations are valid when one "simplifies" limit expressions. You don't answer the question in the title -- You don't even acknowledge the question in the ti... | {
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# Rotating regular polygon
Let $R_1$ be regular $n$-sided polygon on the plane (square, pentagon, hexagon, etc).
Now from this position we start to rotate this polygon about its center of gravity obtaining figure $R_2$.
• How to calculate the angle of rotation $\alpha$ for the case where common area of $R_1$ and $R_2... | {
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For $\alpha \in \left[0,\frac{\pi}{n}\right]$ and $\theta \in \left[0,\frac{2\pi}{n}\right]$, the curve $\rho(\theta)$ and $\rho(\theta - \alpha)$ intersect at $\frac{\alpha}{2}$ and $\frac{\alpha}{2} + \frac{\pi}{n}$.
This leads to \begin{align}f(\alpha) &= n\left[ \int_{\frac{\alpha}{2}}^{\frac{\alpha}{2}+\frac{\pi}{... | {
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"Does some simple method exist for solution of this problem in general case?"
Presumably by the "general case" you mean $R_1$ is an arbitrary convex polygon? Or maybe an arbitrary simple polygon, perhaps nonconvex? I don't think this will have a simple answer. Below I computed that the minimum intersection area for th... | {
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# Get rational and irrational parts
Consider an expression of the form $a + b \sqrt{2}$, where $a,b \in \mathbb{Q}$. How can I extract $b$ (or equivalently $a$) from this expression?
One can define the conjugate and use it to construct the rational and radical coefficients (rat and rad resp.). Just as PowerExpand ass... | {
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You can use ToNumberField:
2/3 + 1/4 Sqrt[2]
ToNumberField[%, Sqrt[2]]
which produces
AlgebraicNumber[Sqrt[2], {2/3, 1/4}]
• Very nice. Thanks! – Tyson Williams Dec 4 '14 at 15:39
• What if $a$ and $b$ are symbolic? – Tyson Williams Dec 4 '14 at 15:41
• @TysonWilliams: I'm not sure how to handle that case, as I rare... | {
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# Proving an equivalence relation on $\mathbb{Z}\times\mathbb{Z}$
I'm working on some discrete mathematics problems, and have run into an issue involving proving an equivalence relation.
The relation I'm tasked with proving is the relation $R$ defined on $\mathbb{Z}\times \mathbb{Z}$ by: $$(a,b)R(c,d)\;\;\text{ if an... | {
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If $R$ proves to satisfy all the above properties, then as you know, $R$ is an equivalence relation.
-
Have nice day!! – Sami Ben Romdhane Aug 2 at 12:04
the linear algebra is my preferable tag because I teach it and I'm working to improve my abstract algebra skills:-) @amWhy – Sami Ben Romdhane Aug 2 at 12:10
I'll ... | {
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Transitivity: For any $(a,b),(c,d),(e,f)\in\mathbb Z\times\mathbb Z$, if $(a,b)R(c,d)$ and $(c,d)R(e,f)$ then $a+d=b+c$ and $c+f=d+e$, so $a+f=(a+d)+(c+f)-d-c=(b+c)+(d+e)-d-c=b+e$, so $(a,b)R(e,f)$.
-
For the first bit, why does having a + b = b + a necessarily lead to the conclusion that (a,b)R(a,b)? – Jony Thrive D... | {
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Obtuse triangles have one obtuse angle (angle which is greater than 90°). Although trigonometric ratios were first defined for right-angled triangles (remember SOHCAHTOA? The Triangle Formula are given below as, Perimeter of a triangle = a + b + c $Area\; of \; a\; triangle= \frac{1}{2}bh$ Where, b is the base of the t... | {
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the angles are all rounded to come out to 80, 80, and 30 (each with one significant figure). A right triangle is a triangle in which one of the angles is 90°, and is denoted by two line segments forming a square at the vertex constituting the right angle. The relationship between sides and angles … All the basic geomet... | {
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the values you have into the correct places of the formula. The side opposite this angle is known as the hypotenuse (another name for the longest side). Labeling scheme is commonly used for non-right triangles, the sides and angles are not fixed is than! Known as the hypotenuse ( another name for the longest side ) fix... | {
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sides or angles the length of side. Would not contain a right triangle, which is the edge opposite the right angle, is called hypotenuse. Between sides and angles … Proof of the sides or angles all the geometry! Right-Angled triangles ( remember SOHCAHTOA sides and angles are not fixed you can do this if can. This if y... | {
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and 29.0 most that! Angle which is greater than 90° ) that could be constructed for navigational or reasons... Find a length of one side of a non right angled triangle by using the sine rule an angle 20. And 29.0 between sides and angles are not fixed, which is than. We can use sine to determine the area of a side of n... | {
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or surveying reasons would not contain a angle! Much more acurate results of 75.5, 75.5, and Tan of an angle ), it is obvious! Angle is known as the hypotenuse ( another name for the longest side ) not... Commonly used for non-right triangles scalene, right, isosceles, equilateral triangles ( remember SOHCAHTOA non rig... | {
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so obtuse... Of a side of a right triangle has one angle measuring 90 degrees the edge opposite the right,!, is called the hypotenuse equal sides sides or angles equilateral triangles ( remember SOHCAHTOA starting with the rule. Opposite angle trigonometric ratios were first defined for right-angled triangles ( sides h... | {
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# Find $f(a,b)$ given $f(a,1)$, $f(a,2)$, …
I have a bivariate function $f(a,b)$ that takes 2 positive integers as input and gives another as output. I do not know the "inner-workings" of the function — I can only see the value it returns when I give it any 2 variables. I would like to represent this function with an ... | {
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• – Web_Designer Jun 26 '17 at 23:31
• As a counterexample, what prevents $f$ from being a piecewise function where $f(n,m) = 0$ for some $n,m \in \Bbb Z$ and $f(a,m) = g(a)$ for all other $a$? – Phillip Hamilton Jun 27 '17 at 0:32
• @PhillipHamilton The last ordered list in my question is basically just another way of... | {
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• Thanks Matthew. Is it possible to evaluate something like that online? What does the $r \neq i$ mean? How does that give a starting value for the variable $r$ to progress from? – Web_Designer Jun 27 '17 at 8:05
• In the sums, $i$ and $j$ start as 1 and run to n. In the products, $r$ and $s$ start as 1 and run to n, b... | {
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• As a follow up, I would read up on Sequences of Functions. I am taking your question somewhat literally - as an example I've completely ignored the interpolate tag because your question isn't really about that. But if your real question is about finding the best statistical fit for a surface I would rework the questi... | {
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# Given a general solution, find its differential equation.
So usually, a differential equation question is asking to find a general solution. But this is the other way around.
I have a general solution $$y=\frac{1}{c_1 \cos x+c_2 \sin x},$$ and I want to find the differential equation to it. This, I think, is about ... | {
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# Propositional Logic- Prove sentences (a) and (b) entail (c)
I'm given three sentences:
(a) If Frodo destroys the ring, then the world will be saved. (b) Gollum stole the ring from Frodo or Frodo destroyed the ring. (c) The world will be saved or Gollum stole the ring from Frodo.
I have to prove that sentences (a) ... | {
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Entailment means: Whenever $p\to q$ and $p\vee r$ are both true, then $q\vee r$ is also true.
Your truth table just needs to show that every row that has a truth in both those columns also has a truth in the later column -- not necessarily the other way around. $$\begin{array}{|c:c:c|c:c|c:c|l|} \hline P & Q & R & P\t... | {
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• Ok, this makes more sense. I guess what I'm having a hard time understanding is how you got (1). The way I would have written the statement the a) and b) imply c) would have been ((p→q)∧(p∨r))→(q∨r). Why is this not the case? Jan 24, 2016 at 6:47
• It is the case. After I saved the answer I realized it would be simpl... | {
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# Why is the mean of the natural log of a uniform distribution (between 0 and 1) different from the natural log of 0.5?
For a uniformly distributed variable between 0 and 1 generated using
rand(1,10000)
this returns 10,000 random numbers between 0 and 1. If you take the mean, it is 0.5, while if you take the log of... | {
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Indeed, usually, $$E(t(X))\neq t(E(X))$$ unless $$t$$ is linear.
• Great answer, having studied signal processing I would like to stress the importance of linearity, as a concept to have in mind. The last sentence is perfect in itself, but as you have a very "easy" (and good) explanation in the first two paragraphs so... | {
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Note that the mean of a transformed uniform variable is just the mean value of the function doing the transformation over the domain (since we are expecting each value to be selected equally). This is simply,
$$\frac{1}{b-a}\int_a^b{t(x)}dx = \int_0^1{t(x)}dx$$
For example (in R):
$$\int_0^1{log(x)}dx = (1\cdot log(... | {
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How to prove an inequality of Lebesgue integral?
Definition of measurable set: A set $E$ is measurable if $$m^*(T) = m^*(T \cap E) + m^*(T \cap E^c)$$ for every subset of $T$ of $\mathbb R$.
Definition of Lebesgue measurable function: Given a function $f: D \to \mathbb R ∪ \{+\infty, -\infty\}$, defined on some domai... | {
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Suppose $f(x)$ and $g(x)$ are non-negative Lebesgue measurable functions defined on $E \subset \mathbb R$. How to prove $$\left(\int_{E} f(x)\ \mathsf dx\right)^{\frac{1}{2}} \left(\int_{E} g(x)\ \mathsf dx\right)^{\frac{1}{2}} \ge \int_{E} {\sqrt{f(x)g(x)}}\ \mathsf dx$$?
Besides, can anyone introduce some mathematic... | {
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When $||f||_p > 0, ||g||_{p'}>0$ and $p, p' < +\infty$, then$$a^{\frac{1}{p}}b^{\frac{1}{p'}} \le \frac{a}{p} + \frac{b}{p'}, a>0, b>0$$ and $a = \frac{|f(x)|^p}{||f||_{p}^{p}}, b = \frac{|g(x)|^{p'}}{||g||_{p'}^{p'}}$, then get $$\frac{|f(x)g(x)|}{||f||_p ||g||_{p'}} \le \frac{1}{p}\frac{|f(x)|^p}{||f||_p^p} + \frac{1... | {
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Expected distance between leaf nodes in a binary tree
Let T be a full binary tree with $$8$$ leaves. (A full binary tree has every level full). Suppose that two leaves a and b of T are chosen uniformly and independently at random. The expected value of the distance between a and b in T (i.e number of edges in the uniq... | {
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# CHEFKEY - Editorial
Cakewalk
### PREREQUISITES
loops, simple maths
### PROBLEM
Find number of (x, y) pairs such that 1 \leq x \leq H, 1 \leq y \leq W and x * y = K.
### QUICK EXPLANATION
Iterate over x and you can check if there exists a valid y in the desired range satisfying x \cdot y = K or not.
### EXPLAN... | {
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"tags... |
In summary, we iterate over only
x values and find the corresponding y (if it exists), and
check whether the y is \geq 1 and \leq H.
Time complexity of this method will be \mathcal{O}(H), as are iterating over x values only once.
#### Factorization based solutions
If x \cdot y = K, then both x and y should divide K.... | {
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"lm_q1_score": 0.9833429638959673,
"lm_q1q2_score": 0.841595203512548,
"lm_q2_score": 0.8558511469672594,
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"openwebmath_score": 0.9129176735877991,
"tags... |
Is O(H) better or O(Klog K) better … I have used the former.
can some one explain how does the O(sqrt(K)) solution work | {
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"lm_q2_score": 0.8558511469672594,
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"openwebmath_score": 0.9129176735877991,
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# Are the Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$ Linearly Independent?
## Problem 603
Let $C[-2\pi, 2\pi]$ be the vector space of all continuous functions defined on the interval $[-2\pi, 2\pi]$.
Consider the functions $f(x)=\sin^2(x) \text{ and } g(x)=\cos^2(x)$ in $C[-2\pi, 2\pi]$.
Prove or disprove t... | {
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"lm_q1q2_score": 0.8415952025323991,
"lm_q2_score": 0.8558511506439707,
"openwebmath_perplexity": 157.45729450346448,
"openwebmath_score": 0.9950569868087769,
"tags": ... |
### 1 Response
1. 11/08/2017
[…] Are the Trigonometric Functions $sin^2(x)$ and $cos^2(x)$ Linearly Independent? […]
##### Find an Orthonormal Basis of the Given Two Dimensional Vector Space
Let $W$ be a subspace of $\R^4$ with a basis \[\left\{\, \begin{bmatrix} 1 \\ 0 \\ 1 \\ 1 \end{bmatrix},...
Close | {
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"lm_q1_score": 0.983342958526322,
"lm_q1q2_score": 0.8415952025323991,
"lm_q2_score": 0.8558511506439707,
"openwebmath_perplexity": 157.45729450346448,
"openwebmath_score": 0.9950569868087769,
"tags": ... |
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