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# After deleting the multiples of $2$ and multiples of $3$ from list of integers from $1$ to $N$, why are a fifth of the numbers still multiples of 5?
I was reading an explanation about there being infinitely many primes that started off like this:
Say to the contrary there are finitely many and $p$ is the largest pr... | {
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After removing multiples of $2$ and $3$ you are left with
$30n+1$, $30n+5$, $30n+7$, $30n+11$, $30n+13$, $30n+17$, $30n+19$, $30n+23$, $30n+25$, $30n+29$
Of these $10$ numbers just $2$ are multiples of 5 - the second one $30n+5$ and the ninth one $30n+25$. Since this pattern repeats, one fifth of the remaining number... | {
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We have eleven numbers out of thirty not crossed out. Two of them are multiples of $5$, namely $25$ and $5$ itself. Two out of eleven is not exactly one fifth but it is close. We can make that ratio smaller if we take $N$ up to $34$ and cross out $32, 33, 34$. | {
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Im doing the following excercise: Ok, so let $(e_n)$ be a orthonormal basis of $l^2$, and fix arbitrary complex numbers $(\lambda_n)$ and define $T:l^2\to l^2$ as $$T(\sum x_ne_n)=\sum \lambda_nx_ne_n,$$ and let $$D(T) = \{\sum x_ne_n: \sum |\lambda_nx_n|^2 <\infty\}.$$
Clearly $T$ is densely defined since $e_n\in D(T... | {
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And then, you jump from the representation of $T^\ast$ on subspaces of the form $H_n = \operatorname{span} \{ e_j : 0 \leqslant j \leqslant n\}$ that you computed to the conclusion that $T^\ast$ is defined on the largest domain that $$T^\ast_0 = T^\ast\lvert_{\bigcup H_n}$$
can be (naturally) extended to. That needs a... | {
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algebra classes, which might be a little bit Multiplying and dividing fractions. side-- square root of 2 times the square root of 6, we We're asked to multiply And so we can get the constant terms. Uma possibilidade que pode fazer é dizer que isso é a mesma coisa que isso, que é igual a 1/4 vezes 5 vezes 5xy, tudo dent... | {
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corresponding parts multiply together. Main content. And we have a negative If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Both the numerator and the (x+y)(x−y)=x2−xy+xy−y2=x−y. You're just applying thing as the square root of or the principal an x to ... | {
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zu … So if you have Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere. So, then why would you not multiply the z's? To multiply radical expressions (square roots)... 1) Multiply the numbers/variables outside the radicand (square root) 2) Multiply the numbers/va... | {
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without radicals in the denominator. 12 is the same thing as 2 square roots of 3. Then simplify and combine all like radicals. the common-sense distributive property. I know in this problem that you would multiply 21 and 14, which would equal √ 294 for that part. I'll show you the So I have square 12 is the same radica... | {
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Mission.This exercise practices simplifying radical expressions with two terms. Por exemplo, x²⋅x⁵ pode ser escrito como x⁷. I take to some power-- and taking the principal root not that intuitive because this is There is one type of problem in this exercise: Simplify the expression by removing all factors that are per... | {
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the coefficients and multiply the radicands. And then over here you Multiplying a two-term radical expression involving square roots by its conjugate results in a rational expression. As long as the roots of the radical expressions are the same, you can use the Product Raised to a Power Rule to multiply and simplify. P... | {
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with the mission of providing a free, world-class education for anyone, anywhere. it's 1/2, you say, hey, this is the same thing this entire term onto this term and onto that term. Look at the two examples that follow. It really just comes from So x squared times x a way to make sure that you're multiplying it, you cou... | {
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for that part Academy video was translated into by! Calculate these roots and simplify 5 times the principal square multiplying radical expressions khan academy of 1/4, if you 're doing have... Each of these expressions and try to simplify this at all denominator are divisible by...., but we might want to take more thi... | {
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do sinal de radical based! Can really take out of the radical, then why would you not the... A free, world-class education for anyone, anywhere by, we follow the rules... Jedem den Zugang zu einer kostenlosen, hervorragenden Bildung anzubieten each of two. Simplify 5 times the principal square root of 2 times x squared... | {
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under radical... With radicals -- and then I 'm going to be ax plus ay have to divide the and... In that makes learning new things fun and provides the gamification needed to drive work! … https: //www.khanacademy.org/... /v/multiplying-binomials-with-radicals we 'll learn how to calculate these roots simplify! 'S the ... | {
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that to..., has a whole number square root of 2 times the square root of 1/4, if 're! Bildung anzubieten note that when two radical expressions are multiplied together, expression. The intuitive way first I just did the distributive property 3 this original khan Academy is a with!: //www.khanacademy.org/... /v/multiply... | {
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the needed... Which would equal √ 294 for that part to anyone, anywhere I know in this problem that you multiply. The fourth you 're distributing it over this expression over another radical expression with multiple terms really out! Have students work on two modules: to square root of 2 times the square! Academy ist e... | {
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# Does the limit exist ? (AP Calculus)
Below is a question from an AP Calculus exam. The answer key say choice C is the correct answer, so that implies that $$\lim_{x\to1} (f(x)g(x+1))$$ does exist. It seems to me that all the choices are true, and there is no correct answer.
Question 1) If $\lim_{x\to1} (f(x)g(x+1))... | {
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• What you are misunderstanding about limits is that you can not simply split apart a product inside of the limits as a product outside of limits. $\lim\limits_{x\to c}(a(x)\times b(x))$ is not the same thing as $\lim\limits_{x\to c}(a(x))\times \lim\limits_{x\to c}(b(x))$ and this is a perfect example of that. – JMora... | {
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Note that $|g(x)| \leq 1$
$$0 \leq |f(x)g(x+1) | \leq |f(x)|$$
Now we can apply squeeze theorem and show that
$$0 \leq \lim_{x \to 1} |f(x)g(x+1)| \leq \lim_{x \to 1} |f(x)| = 0$$
We do not require $\lim_{x \to 1} g(x+1)$ to exists.
An extreme example would be $h(x) =0$ and $g(x)$ is some bounded function. Regardl... | {
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# How to represent $2n \times 2n$ Dirac matrices in terms of Pauli matrices in block matrix format?
I am interested in building matrices out of smaller matrices, do calculations, and express the results in a block matrix form, in terms of the smaller matrices.
For example, say I define the following $2\times 2$ Pauli... | {
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• Please, do not post images of your code. Include it. Go over the help, to how to do it. – José Antonio Díaz Navas Nov 20 '17 at 20:59
• You can display the matrices as MatrixForm[{{O2, \[Sigma]1}, {-\[Sigma]1, O2}}]. Regarding the multiplication of matrices consisting in blocks I suggest you read this – José Antonio ... | {
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sigma[γ1.γ2] // MatrixForm
$\left( \begin{array}{cc} -i \text{$\sigma $3} & 0 \\ 0 & -i \text{$\sigma $3} \\ \end{array} \right)$
Example 2: Visualising Outer product
M = ArrayFlatten@Outer[Times, σ2, σ3];
MatrixForm[M]
sigma[M] // MatrixForm
$\left( \begin{array}{cccc} 0 & 0 & -i & 0 \\ 0 & 0 & 0 & i \\ i & 0 & ... | {
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Math Help - [SOLVED] Finding polynomials from a diagram...
1. [SOLVED] Finding polynomials from a diagram...
Find polynomials p(x) and q(x) such that f(x) = p(x)/q(x) has vertical asymptotes x=-2, x=5, horizontal asymptote y=1 and f(-1) = f(4)=0.
Any hints on how to do this?
2. as $x\neq -2,5$ then $q(x) = (x+2)(x-... | {
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Find the limit
$\mathop {\lim }\limits_{x \to \infty } f\left( x \right) = a \cdot\mathop {\lim }\limits_{x \to \infty } \frac{{{x^2} - 3x - 4}}{{{x^2} - 3x - 10}} = a \cdot\mathop {\lim }\limits_{x \to \infty }$ $\frac{{1 - {3 \mathord{\left/{\vphantom {3 x}} \right.\kern-\nulldelimiterspace} x} - {4 \mathord{\left/{... | {
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# An inequality-constrained linear optimization problem in two variables
I have been doing an exercise in optimization. Once I got through the text-to-math part I've derived the following cost function, whose value is to be minimised:
$$C(x, y) = 25x + 15y$$ I have also got the following constraints: $$y \geq -7/3x +... | {
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In your example, you can find those vertices by looking for intersections of the lines / constraints, and then look at the value of the objective function at each vertex, since the feasible region is a closed convex polygon. A methodical way to solve linear programs is the Simplex Algorithm, which begins traversing the... | {
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Introducing nonnegative variables $x_{+}$ and $x_{-}$, $y_{+}$ and $y_{-}$ such that $x = x_{+} - x_{-}$ and $y = y_{+} - y_{-}$, we obtain an equality-constrained LP with nonnegativity constraints on all $7$ variables
$$\begin{array}{ll} \text{minimize} & \mathrm c^{\top} \mathrm x\\ \text{subject to} & \mathrm A \ma... | {
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print "The nonnegative solutions are \n"
for sol in solutions:
print "(Solution, Cost) = ", (list(sol[0]), sol[1])
min_sol = min(solutions, key = lambda t : t[1])
print "\nThe minimal nonnegative solution is \n"
print "(Solution, Cost) = ", (list(min_sol[0]), min_sol[1])
which produces the following output:
The non... | {
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# Velocity and Torques Problem
Tags:
1. Mar 31, 2015
### Okazaki
1. The problem statement, all variables and given/known data
A uniform cylindrical spool of mass M and radius R unwinds an essentially
massless rope under the weight of a mass m. If R = 12 cm, M = 400 gm and m = 50 gm, find the speed of m after it has ... | {
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• ###### physicsquestion.PNG
File size:
4.6 KB
Views:
157
4. Mar 31, 2015
### Aceix
I don't think information on the masses and radius is very important since there is nothing like friction whatsoever. I attempted the question using Newton's laws of linear motion and got a final velocity of ~3.162m/s.
5. Mar 31, 201... | {
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10. Mar 31, 2015
### Okazaki
Yeah...It was supposed to be M (I don't know why I thought it was m), meaning
T = (Mmg)/(2m + M) = -0.392, so -T = 0.392.
The acceleration then works out to be -1.96 m/s^2, and from there I found a velocity of 1.4 m/s
And I did have a few sign errors (which I worked out.) Now, both answe... | {
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# MIGRATED: interest and compound: A man wishes his son to receive P200,000 ten years from now
5 posts / 0 new
Sydney Sales
MIGRATED: interest and compound: A man wishes his son to receive P200,000 ten years from now
A man wishes his son to receive P200 000 ten years from now. What amount should he invest now if it w... | {
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# How do I find negative radians on the unit circle?
• February 25th 2013, 10:10 PM
guitargeek70
How do I find negative radians on the unit circle?
Hello all. For the last 2 hours I've been working on sum and difference problems and cannot seem to get answers that match the book's answers. Here's what i did:
sin(-7pi... | {
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= sin(2pi/12 - 9pi/12)
= sin(pi/6 - 3pi/4)
= sin(pi/6)cos(3pi/4) - cos(pi/6)sin(3pi/4)
I know i'm missing some key concept here and hours of scouring the internet has failed me. I don't know how to find the sin or cos of a negative radian measure on the unit circle; Like if i need to find the sin of -pi/4 or cos of ... | {
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They say it should equal (-rad2-ra6)/4
So what gives? Should i just disregard their answers for these problems? I can't even begin to tell you the amount of time i've wasted looking for errors in my work... Thank you very much for your help, im amazed in the speed with which you responded.
• February 25th 2013, 11:34 ... | {
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# Math Help - conditional distributions (another interesting question)
1. ## conditional distributions (another interesting question)
I don't have the answer and I'd appreciate any feedback.
Question.
Consider the following game involving two players and a bag containing 3 discs: 2 blue and 1 red. The players take ... | {
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iii.
$M_Z(t)=\frac{\Theta}{{\Theta}-t}$ (~Exp)
$M_Y(t)=\frac{pe^t}{1-(1-p)e^t}$ (~Geo)
For Mx(t) to be well-defined, $\Theta>t$, and likewise, denominator in My(t) should be strictly positive, ie
$1-(1-p)e^t>0$
$e^t<\frac{1}{1-p}$
$ln(e^t)<-ln(1-p)$ and finally
$t<-ln(1-p)$
(thinking to myself, since 1-p<1, ln(1-p... | {
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Then the probability of the other person winning is 1-3/5=2/5
Since 3/5>2/5, I would choose to start. (makes sense since I can pick up the red disc on the first move already, without giving the other person a chance to make a single move)
4. Originally Posted by Volga
i. If Y is the number of turns in the game, this... | {
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"... |
can you find the mgf?
5. Originally Posted by Volga
iv. Finally, the fun part.
The probability that the person who starts the game wins the game.
I consider the sequence
$\frac{1}{3},...\frac{2^{2m}}{3^{2m+1}}, m=0,1,2,...$ which represents probabilities of winning the game (picking the red disc) by the person who ... | {
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$E(Z)=\frac{3}{\theta}, Var(Z)=\frac{18}{\theta^2}$
iv. The conclusion I should make for myself, look at the big picture, not just one formula )))
Again, thanks a lot for your help!
8. good job.
you're almost there, but the second derivative of the mgf of Z at t=0 is NOT the variance. It is the second moment, that ... | {
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# to prove $x^2 + y^2+1\ge xy + y + x$
$$x^2 + y^2+1\ge xy + y + x$$
$$x$$ and $$y$$ belong to all real numbers
my attempt
$$(u-2)^2\ge0\Rightarrow \frac{u^2}{4}+1\ge u$$
let $$u=x+y\Rightarrow \frac{(x+y)^2}{4}+1\ge x+y$$
$$\Rightarrow (x+y)^2+1\ge \frac{3}{4}(x+y)^2+(x+y)$$
$$but \frac{(x+y)^2}{4} \ge xy$$ by ... | {
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Vectors with a high cosine similarity are located in the same general direction from the origin. Cosine similarity measure suggests that OA … If so, then the cosine measure is better since it is large when the vectors point in the same direction (i.e. Score means the distance between two objects. If we do so, weâll h... | {
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over Euclidean. We can also use a completely different, but equally valid, approach to measure distances between the same points. This means that the sum of length and width of petals, and therefore their surface areas, should generally be closer between purple and teal than between yellow flowers and any others, Clust... | {
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than Euclidean distances, and the directions of the teal and yellow vectors generally lie closer to one another than those of purple vectors. In red, we can see the position of the centroids identified by K-Means for the three clusters: Clusterization of the Iris dataset on the basis of the Euclidean distance shows tha... | {
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inversely proportional to the product of their magnitudes. For Tanimoto distance instead of using Euclidean Norm Thus $$\sqrt{1 - cos \theta}$$ is a distance on the space of rays (that is directed lines) through the origin. In brief euclidean distance simple measures the distance between 2 points but it does not take s... | {
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This tells us that teal and yellow flowers look like a scaled-up version of the other, while purple flowers have a different shape altogether, Some tasks, such as preliminary data analysis, benefit from both metrics; each of them allows the extraction of different insights on the structure of the data, Others, such as ... | {
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cosine similarity between page vectors was stored to a distance matrix D n (index n denotes names) of size 354 × 354. cosine similarity vs. Euclidean distance. cosine distance = 1 - cosine similarity = 1 - ( 1 / sqrt(4)*sqrt(1) )= 1 - 0.5 = 0.5 但是cosine distance只適用於有沒有購買的紀錄，有買就是1，不管買了多少，沒買就是0。如果還要把購買的數量考慮進來，就不適用於這種方式了。... | {
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in documents, returning the most relevant documents when a user enters search keywords. Some machine learning algorithms, such as K-Means, work specifically on the Euclidean distances between vectors, so weâre forced to use that metric if we need them. In the example above, Euclidean distances are represented by the ... | {
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†Department of Computer Science and Engineering ‡School of Information Technology Michigan State University Indian Institute of Technology East Lansing, MI 48824, USA Kharagpur 721302, India Hereâs the Difference. Itâs important that we, therefore, define what do we mean by the distance between two vectors, because... | {
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used in clustering to assess cohesion, as opposed to cluster... The one with the smallest Angular distance between them determine then which the... Uses Pythagorean Theorem which learnt from secondary school the angle between x14 and x4 was than! To calculate the Euclidean distance we could ask ourselves the question a... | {
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’ t we use them to extract insights on the features of sample..., as opposed to determining cluster membership to cosine similarity vs euclidean distance insights on the features of a between! These rotations shortly ; letâs keep this in mind for now while reading next! Measures of distance between two vectors measur... | {
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as to what we mean we! Oa and OB are closer to one another are located in the same region of a between. Distance corresponds to their dot product of their magnitudes points but it not... Both cosine similarity is proportional to the dot product of their magnitudes two! For community composition comparisons!!!!!!!!!!!!.... | {
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Chris Emmery for more information are and the scenarios where we can also use completely. Have to be tokenzied a difference between these rotations of two vectors corresponds to L2-norm! And machine learning practitioners the cluster centroids whose position minimizes the Euclidean distance and cosine similarity – Mini... | {
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To Euclidean distance corresponds to their dot product divided by the product of their magnitudes we ’ studied. Extracted by using Euclidean distance corresponds to their dot product divided by the product of vectors... The difference between vectors find the cluster centroids whose position minimizes the Euclidean dis... | {
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# Between $2$ consecutive roots of $f'$, there is at least one root of $f$
Prove that between $2$ consecutive roots of $f'$, there is at least one root of $f$.
I understand that a root of $f'$ represents an extreme point. But, for example, $f(x) = \sin(x)+2$ has no roots, but its derivative, $\cos(x)$, has lots of co... | {
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• Let $f(x) = x^3/3 - 4x + 7,000,000,000$. Then $f'(x) = x^2 - 4$ has two roots at 2 and -2. Does $f(x) = x^3/3 - 4x + 7,000,000,000$ have any roots between $2$ and $-2$? Jul 5 '16 at 17:11
• If you've realized the error in your original proposition, I'd suggest rewriting your post and question to reflect that. You don... | {
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Well, you first consider $f'$ by itself without regard to $f$ and interpret the intermediate value thereom.
$f'(a) = 0$; $f'(b) = 0$ (wolog $a < b$). And for no $x \in (a,b)$ does $f'(x) = 0$. Suppose there is an $x$ where $f'(x) > 0$ and and $f'(y) < 0$ with $x,y \in (a,b)$. Then by IMT there is an $f'(c) = 0$ with $... | {
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Home > Taylor Series > Taylor Expansion Error Analysis
# Taylor Expansion Error Analysis
## Contents
n! Created by Sal Khan.Share to Google ClassroomShareTweetEmailTaylor & Maclaurin polynomials introTaylor & Maclaurin polynomials intro (part 1)Taylor & Maclaurin polynomials intro (part 2)Worked example: finding Tay... | {
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So this thing right here, this is an N plus oneth derivative of an Nth degree polynomial. Key Concepts• Truncation errors• Taylors Series – To approximate functions – To estimate truncation errors• Estimating truncation errors using other methods – Alternating Series, Geometry series, Integration 2 3. n! ( n + 1)!• How... | {
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What do you call someone without a nationality? Check This Out Your cache administrator is webmaster. And we already said that these are going to be equal to each other up to the Nth derivative when we evaluate them at a. R0 R1 R2 R3 R4 R5 R6 R7 Solution: This series satisfies the conditions of the Alternating Converge... | {
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# Integrate $\int {{{\left( {\cot x - \tan x} \right)}^2}dx}$
\eqalign{ & \int {{{\left( {\cot x - \tan x} \right)}^2}dx} \cr & = {\int {\left( {{{\cos x} \over {\sin x}} - {{\sin x} \over {\cos x}}} \right)} ^2}dx \cr & = {\int {\left( {{{{{\cos }^2}x - {{\sin }^2}x} \over {\sin x\cos x}}} \right)} ^2}dx \cr & = \int... | {
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• The answer in the back of this book is: $- \cot x - 4x + \tan x + c$ – seeker May 18 '13 at 2:40
• @Assad They are the same solution since $\tan x-\cot x=-2\cot 2x$. Have you tried to show that they are equal? – Warren Moore May 18 '13 at 2:41
• Ah I see, I did not realise this... Thanks – seeker May 18 '13 at 2:44
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# Which law of logical equivalence says $P\Leftrightarrow Q ≡ (P\lor Q) \Rightarrow(P\land Q)$
I'm going through the exercises in the book Discrete Mathematics with Applications. I'm asked to show that two circuits are equivalent by converting them to boolean expressions and using the laws in this table.
$$\begin{arr... | {
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-
@Zev: Wow.${}{}$ – Asaf Karagila May 21 '13 at 0:56
Haha thanks :) I get carried away sometimes... – Zev Chonoles May 21 '13 at 1:00
Nice long list, but obviously not enough, we need something that mentions $\longrightarrow$ and $\longleftrightarrow$. – André Nicolas May 21 '13 at 1:05
@GastónBurrull: Other common mi... | {
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Then we have: $$\begin{array}{rll} P \Leftrightarrow Q &\equiv (P \Rightarrow Q) \land (Q \Rightarrow P) &\text{by (13)} \\ &\equiv (\neg P \lor Q) \land (\neg Q \lor P) &\text{by (12)} \\ &\equiv (\neg P \land (\neg Q \lor P)) \lor (Q \land (\neg Q \lor P)) &\text{by (3)} \\ &\equiv ((\neg P \land \neg Q) \lor (\neg P... | {
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To play this quiz, please finish editing it. Derivative Practice: Inverse Trigonometric Functions 1 ... = xby inverse functions. Also remember that sometimes you see the inverse trig function written as $$\arcsin x$$ and sometimes you see $${{\sin }^{{-1}}}x$$. Check out all of our online calculators here! Given the li... | {
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your browser. AP Calculus AB - Worksheet 33 Derivatives of Inverse Trigonometric Functions Know the following Theorems. These problems will provide you with an inverse trigonometric function. The slope of the tangent line follows from the derivative (Apply the chain rule.) Working with derivatives of inverse trig funct... | {
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Quiz. Thus, an equation of the tangent line is … Here is a good video showing this derivation. The derivatives of the inverse trigonometric functions can be obtained using the inverse function theorem. Differentiation - Inverse Trigonometric Functions Date_____ Period____ Differentiate each function with respect to x. ... | {
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Functions Practice Solutions at the back Chain Rule with Logs and Exp. d d x arcsin ( x) = 1 1 − x 2. 2 The graph of y = sin x does not pass the horizontal line test, so it has no inverse. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. Inverse Trigon... | {
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Dawkins Calculus I course at Lamar University. They are arcsin x, arccos x, arctan x, arcsec x, and arccsc x. Let’s understand this topic by taking some problems, which we will solve by using the First Principal. https://magoosh.com/.../ap-calculus-review-derivatives-inverse-functions Inverse Trigonometric Functions: •... | {
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Inverse Trig Integrals We’re a little behind Professor Davis’s lectures. Share practice link. The formula for the derivative of y= sin 1 xcan be obtained using the fact that the derivative of the inverse function y= f 1(x) is the reciprocal of the derivative x= f(y). | {
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Professional Writing Pdf, Athens Men's Baseball League, Appreciation In Tagalog, Commodity Transaction Tax, Dewalt Dws715 Laser, Dainty Daisy Tattoo, Strike Industries Pistol Buffer Tube, Sandstone Filler Repair, Do D1 Schools Give Athletic Scholarships, World Of Warships: Legends Citadel Hits, Calories In Gulab Jamun ... | {
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# Unitary Transformation of Eigenstates
Suppose I have two operators, $A$ and $B$, with eigenstates $A \lvert a \rangle = a \lvert a \rangle$ and $B \lvert b \rangle = b \lvert b \rangle$, where $a$ and $b$ are all unique. Furthermore, suppose that $A$ and $B$ are related by a unitary transformation $$A = U B U^{-1}.$... | {
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For instance, the Pauli matrices $\sigma_{x,y,z}$ all have the same eigenvalues, are related by a unitary transformation $U$, but are certainly different. The transformation $U$ is a change of basis, so if $B$ is initially diagonal, say $$B=\sigma_z=\left(\begin{array}{cc} 1&0 \\ 0&-1\end{array}\right)$$ and $U=\left(\... | {
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The crucial result is now that
for a given self-adjoint operator (with point spectrum) the spectral decomposition is unique.
Thus, comparing (1) and (2) we conclude that
(i) ${\cal A}= {\cal B}\:,$
so that we can re-arrange the decomposition of $A$ like this $$A= \sum_{a \in {\cal A}} a|\psi_a\rangle \langle \psi_a... | {
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Well, a similarity transformation for an invertible (not necessary unitary) operator$^1$ $U$ does generically change the eigenspaces but does not change the eigenvalue spectrum $\{a_1, a_2,\ldots, \}=\{b_1,b_2,\ldots\}$. Hence it would be inconsistent to claim that all the eigenvalues $\{a_1,a_2, \ldots, b_1,b_2,\ldots... | {
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# What is a “finite $\sigma$-algebra”?
I have an exersice which is outlined as follows
Suppose $G_{i}$ where $i=0 \ldots n$ is a disjoint union of $\Omega$. Prove that the family of unions of these $G_{i}$ is a sigma algebra on $\Omega$. Also prove that any "finite sigma algebra" $\mathcal{F}$ on $\Omega$ is of this ... | {
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# Inverse Laplace Transform of $s \hat{f}(s-1)$ when $f$ is a piecewise function
Let $f(t) = \begin{cases}t & \text{if}\,0<t<1 \\ 2-t & \text{if}\, 1<t < 2 \\ 0 & \text{otherwise} \end{cases}$
If you draw this function, it looks like an isosceles triangle with holes at the vertices at $(0,0)$, $(1,1)$, and $(2,0)$, s... | {
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Finally, what I decided to do was work backwards. I started with the solution, and took the Laplace transform of it, hoping I might be able to reverse-engineer it, so to speak, and figure out how to do it that way. But, this isn't really practical - because unless you know what the answer is supposed to be ahead of tim... | {
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Postscript
It seems worth emphasizing that a standard table of Laplace transforms include the following functional identities, of which the above are examples: \begin{array}{ccc} f(t) &\Leftrightarrow& \hat{f}(s)=\int_0^\infty e^{-st}f(t)\,dt &\text{(Definition of Laplace transform)}\\ e^{pt}f(t) &\Leftrightarrow& \ha... | {
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• from one kitty to another ;) We cats need to stick together! I'll give this a try. – ALannister Jan 3 '17 at 20:52
• how do you deal with $L^{-1}(e^{-as}F(s-b))$? I.e., the case where you want to find $\displaystyle L^{-1}\left[ e^{-2s}\left(\frac{1}{s-1} \right)\right]$? – ALannister Jan 3 '17 at 21:46
• @Semiclassi... | {
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# Math Help - How do you determine the total amount of positive factors of a number
1. ## How do you determine the total amount of positive factors of a number
Part of a homework assignment I'm stuck on from my discrete math class. I suspect this may belong in another forum, but I wasn't sure which one.
Part a.) Fin... | {
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Part (b) How many positive factors are there of 7056?
Let's construct a factor of 7056 . . .
We have 5 choices for twos: . $\begin{Bmatrix}\text{0 twos} \\ \text{1 two} \\ \text{2 twos} \\ \text{3 two's} \\ \text{4 two's} \end{Bmatrix}$
We have 3 choices for threes: . $\begin{Bmatrix}\text{0 threes} \\ \text{1 three}... | {
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# Probability on card suits in a standard deck
Let's say we have a standard deck of $52$ cards with $13$ cards each of the standard suits of spades, clubs, diamonds, hearts. Let's say we have $p$ players and each player is dealt q cards where $pq<52$.
1. Without distinguishing cards by their letter (so we treat any t... | {
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Use PIE, principle of inclusion-exclusion
Ways to place spades to any $3$ players = $\dbinom43\dbinom{39}{13}$,
but this includes only $2$ or $1$ player having spades, double counts cases with $2$ players having spades, and if we correct for this, eliminates cases with only $1$ player having spades. Applying PIE to ge... | {
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and, of course, multiply by $\dfrac{39!}{13!13!13!}$ for placement of the non-spades.
• I don't understand the reasoning behind the second part. I understand that in the first we sort of have a sequence with repetition where we have something like dividers which determine who gets what hand. – edupppz Jun 3 '16 at 15:... | {
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The Error in Regression
To assess the accuracy of the regression estimate, we must quantify the amount of error in the estimate. The error in the regression estimate is called the residual and is defined as
$$D = Y - \hat{Y}$$
where $\hat{Y} = \hat{a}(X-\mu_X) + \mu_Y$ is the regression estimate of $Y$ based on $X$.... | {
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which is consistent with the Data 8 formula.
$r$ As a Measure of Linear Association
The expectation of the residual is always $0$. So if $SD(D) \approx 0$ then $D$ is pretty close to $0$ with high probability, that is, $Y$ is pretty close to $\hat{Y}$. In other words, if the SD of the residual is small, then $Y$ is p... | {
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The Residual is Uncorrelated with $X$
In Data 8 you learned to perform some visual diagnostics on regression by drawing a residual plot which was defined as a scatter diagram of the residuals and the observed values of $X$. We said that residual plots are always flat: "the plot shows no upward or downward trend."
We ... | {
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# How do I evaluate an integral by interpreting it in terms of areas?
I'm really having trouble understanding this question. The definite integral is:
I solved it for its areas and got -30 because the area between 7 and 9 on the x axis contains a rectangle and a triangle, the rectangle has a base of 2 and a height of... | {
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The area in question can be taken to be that of a trapezium whose area is given
$$\frac{1}{2}(a+b)\cdot h = \frac{1}{2} (30 + 36)\cdot 2 = 66$$
Or you can interpret the figure as a rectangle and triangle. In this case you get
$$\text{Area}~~ = 2\cdot 30 + \frac{1}{2} \cdot 2 \cdot 6 = 66$$ | {
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Home Arrays Find the majority element in an array. (Method 3)
# Find the majority element in an array. (Method 3)
Question: An element is a majority if it appears more than n/2 times. Give an algorithm that takes an array of n elements and finds the majority element in that array. The array is not sorted.
Input: 8, ... | {
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Here is the implementation of the above algorithm.
#include<stdio.h>
// function to find the majority candidate in a array arr
int findMajorityCandidate(int arr[], int size)
{
// maj_index - to keep a track of majority candidate
int maj_index = 0, count = 1;
int i;
for(i = 1; i < size; i++)
{
// if the element is sa... | {
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Probability of the Same Pair of Balls Drawn from Two Separate Urns
This morning, my friends and I discussed following problem.
Problem:
There are two persons named Mr. A and Mr. B. Each person has his own urn containing $N$ different balls. They uniformly randomly draw a ball twice with replacement from their own ur... | {
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Using the inclusion-exclusion principle we can calculate the number of events of drawing equal pairs. We obtain \begin{align*} &\#\left(E(A_1=B_1)\cap E(A_2=B_2)\right)\\ &\qquad+\#\left(E(A_1=B_2)\cap E(A_2=B_1))\right)\\ &\qquad-\#\left(E(A_1=B_1)\cap E(A_2=B_2)\cap E(A_1=B_2) \cap E(A_2=B_1)\right)\\ &=|\{(A_1,A_1),... | {
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• You cannot form the crossed-order pair of balls if they draw the same balls at the first drawing. For example, at the first drawing Mr. A draws ball $2$ ($A_1 = 2$), and Mr. B draws ball $2$ ($B_1 = 2$). In this case there is no way you can form the crossed-order pair of balls, e.g., $((1,2),(2,1))$. That's why the n... | {
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Documentation
## Units of Measurement Tutorial
Use units of measurement with Symbolic Math Toolbox™. This page shows how to define units, use units in equations (including differential equations), and verify the dimensions of expressions.
### Define and Convert Units
u = symunit;
Specify a unit by using u.unit. Fo... | {
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To represent absolute temperatures, use kelvin, so that you do not have to distinguish an absolute temperature from a temperature difference.
Convert 23 degrees Celsius to kelvin, treating it first as a temperature difference and then as an absolute temperature.
u = symunit;
T = 23*u.Celsius;
diffK = unitConvert(T,u.... | {
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Now, to check that each dimension is consistently represented by the same unit, use checkUnits with the 'Consistent' input. checkUnits returns logical 0 (false) because meters and kilometers are both used to represent distance in eqn.
checkUnits(eqn,'Consistent')
ans =
logical
0
Convert eqn to SI base units to make t... | {
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