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Lets review normal lists first. Remember that lists are 0-indexed, that is the first element is element 0, the second is 1 and so on.
In [5]:
my_list = [1,2,3,4] # Normal Python list
# We can access different parts of the list by slicing and indexing:
print("my_list[0]:\t", my_list[0]) # First element
print("my_... | {
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A_00: 0
A_02: 2
A_21: 7
In [8]:
# We can go even more advanced, and get entire columns and rows
# First row of A
print("First row:", A[0])
# First column of A
print("First column:", A[:,0])
First row: [0 1 2]
First column: [0 3 6]
Getting the first row is something we can do with plain old nested python lists, ge... | {
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When plotting in the plane, one needs three lists of values, the $x$ and $y$ coordinates, and the function values at each point. For each $x$ value, we want to iterate over all the $y$ values, so that every point is covered. If we just passed our $x$- and $y$-arrays directly, however, we would get only one point for ea... | {
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Mehsgrids are useful for all functions that are functions of two variables. As an example, we will look at the steady state amplitude of a driven damped harmonic oscillator which is an expression derived in classical mechanics. It is given by $$\Theta_0(q, \omega_D) = \frac{F_D}{(\omega_0^2 + \omega_D^2)^2 + (q\omega_D... | {
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2.22 ms ± 71.9 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
With computation times of 9.0$\mu$s and 2.2ms, it is obvious that the difference in computation time is significant! Computational speed will be of great importance as the size of the data becomes big.
In [17]:
# Greatly increase the number of p... | {
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We observe a $90^{\circ}$ phase difference between the real and imaginary part, just like expected.
Using only built-in Python functions for reading from and writing to files can be tedious. NumPy has functions designed specifically for reading and writing structured data, np.loadtxt and np.savetxt. The functions are ... | {
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In [24]:
t, temperature, wind = np.loadtxt('my_data.txt', unpack=True)
unpack=True transposes our data, so that each row represents one field, in this example time, temperature, and wind. This way, we can easily unpack the data into our variables t, temperature, and wind.
In [25]:
plt.plot(t, temperature, 'o-', labe... | {
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We know from linear algebra that given a orthonormal basis $B$ of $V$, any vector $v\in V$ can be written as a linear combination of vectors $b \in B$ . We also know that the coefficient of $b\in B$ for some $v\in V$ is given by $$c_b = \langle b, v\rangle.$$
This is used much in physics, probably most noteworthy in q... | {
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In [29]:
np.min(f(x[x<0]))
y_min = np.min(f(x[x<0]))
This gives us the $y$-value at the local minima. We also wish to find the corresponding $x$-value. np.argmin gives us the index of the $y$-value with the minimal value, using this index on the $x$-array gives us the corresonding value.
In [30]:
min_arg = np.argmin... | {
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Diameter of a circle endpoints P(0,0) Q(8,-4) what equation:
1. Jun 4, 2015
Jaco Viljoen
Mod note: Moved from technical math section, so no template.
The diameter of a circle has endpoints P(0,0) and Q(8,-4) Find the equation:
First I will find the midpoint:
$$M(x,y)=(x1+x2)/2,(y1+y2)/2$$
$$=8/2,-4/2)$$
$$M(x,y)=(-... | {
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# Thread: sum of (n+1)th row pascal triangle proof
1. ## sum of (n+1)th row pascal triangle proof
prove that the sum from j=1 to j=n of combinatorial(n, j) is 2^n.
so basicially i have to prove that the sum of the (n+1)th row of pascal's triangle is 2^n. I have tried using induction on n and the definition of combina... | {
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$\text{number of subsets with zero elements}+\cdots+\text{number of subsets with }n\text{ elements}$ $={n\choose 0}+\cdots+{n\choose n}=f(n)$.
Thus, $f(n)=\text{card }\mathcal{P}(\Omega_n)=\sum_{j=0}^{n}{n\choose j}$. So, let's see if we can find a recurrence relation for $n$. To do this we must only relate the number... | {
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4. Have you ever used a cannon when a flyswatter will work well?
5. Originally Posted by Plato
Have you ever used a cannon when a flyswatter will work well?
Aren't you the proponent of "don't plug and chug...actually think!". While what I did was way more than the problem probably intended I think it was more insightf... | {
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# Is Binet's formula for the Fibonacci numbers exact?
Is Binet's formula for the Fibonacci numbers exact?
$F_n = \frac{(1+\sqrt{5})^n-(1-\sqrt{5})^n}{2^n \sqrt{5}}$
If so, how, given the irrational numbers in it?
Thanks.
-
Yes, it is exact. The irrational parts happen to cancel out. Have you tried plugging in a fe... | {
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It is exact, all right. When you expand the powers in the numerators the alternating signs mean that all the surviving terms are of the form an integer times $\sqrt5$. Therefore all the $\sqrt5$s cancel.
Try it out with $n=2$ and $n=3$.
It may be of interest to you to observe that as $|1-\sqrt5|/2\approx0.618$ its po... | {
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-
This is really neat but I have a perhaps very novice question: how do you relate recurrence relations and polynomials? – Cameron Williams Jul 3 '13 at 13:24
you just have to replace, if R is a root of the associated polynom, you have in the general expression $R^{n+1} = R^{n-1} * R^2 = R^{n-1} * (R + 1) = R^{n} + R^{... | {
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Though I have to say, I am a bit stumped as to why this should be an integer.
-
So this does not answer the question: And I think that to prove this expression is an integer is not so obvious. Maybe one can prove that this is an integer by induction; but then it loses the initial meaning of proving the integer-ness by... | {
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Not onto. 3. is one-to-one onto (bijective) if it is both one-to-one and onto. (D) 72. Onto Function Definition (Surjective Function) Onto function could be explained by considering two sets, Set A and Set B, which consist of elements. Let f and g be real functions defined by f(x) = 2x+ 1 and g(x) = 4x – 7. asked Feb 1... | {
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2.1. . Q3. This is same as saying that B is the range of f . A function is said to be bijective or bijection, if a function f: A → B satisfies both the injective (one-to-one function) and surjective function (onto function) properties. In the example of functions from X = {a, b, c} to Y = {4, 5}, F1 and F2 given in Tab... | {
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f is one-one Hence every element 1, 2, 3 has either of image 1, 2, 3 and that image is unique Total number of one-one function = 6 Example 46 (Method 2) Find the number Experience. Students can solve NCERT Class 12 Maths Relations and Functions MCQs Pdf with Answers to know their preparation level. (b) f(m;n) = m2 +n2.... | {
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in A you have to choose an element in B. Examples: Let us discuss gate questions based on this: Solution: As W = X x Y is given, number of elements in W is xy. f(a) = b, then f is an on-to function. For example, if n = 3 and m = 2, the partitions of elements a, b, and c of A into 2 blocks are: ab,c; ac,b… (i)When all t... | {
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onto functions are. (C) 81 So, that leaves 30. Example 46 (Method 1) Find the number of all one-one functions from set A = {1, 2, 3} to itself. Suppose TNOF be the total number of onto functions feasible from A to B, so our aim is to calculate the integer value TNOF. Calculating required value. (B) 64 Please use ide.ge... | {
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These numbers are called Stirling numbers (of the second kind). For example: X = {a, b, c} and Y = {4, 5}. So, there are 32 = 2^5. [5.1] Informally, a function from A to B is a rule which assigns to each element a of A a unique element f(a) of B. Officially, we have Definition. In other words, nothing is left out. A funct... | {
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objects can be distributed in two boxes a' and b' in such a way that no box remains empty. In a function from X to Y, every element of X must be mapped to an element of Y. So the total number of onto functions is m!. In other words, if each b ∈ B there exists at least one a ∈ A such that. (d) f(m;n) = jnj. Onto Functio... | {
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Mathematics, Mathematics | Introduction to Propositional Logic | Set 1, Mathematics | Predicates and Quantifiers | Set 1, Mathematics | L U Decomposition of a System of Linear Equations, Mathematics | Mean, Variance and Standard Deviation, Mathematics | Sum of squares of even and odd natural numbers, Mathematics | Eige... | {
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onto function if every element in B has a pre-image in A. Number of Onto function - & Number of onto functions - For onto function n(A) n(B) otherwise ; it will always be an inoto function . Free PDF Download of CBSE Maths Multiple Choice Questions for Class 12 with Answers Chapter 1 Relations and Functions. A function... | {
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Recurrence Relations – Set 2, Mathematics | Graph Theory Basics – Set 1, Mathematics | Graph Theory Basics – Set 2, Mathematics | Euler and Hamiltonian Paths, Mathematics | Planar Graphs and Graph Coloring, Mathematics | Graph Isomorphisms and Connectivity, Betweenness Centrality (Centrality Measure), Mathematics | Wal... | {
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of functions from A to B is. So, total numbers of onto functions from X to Y are 6 (F3 to F8). An onto function is also called a surjective function. A function f from A to B is called one-to-one (or 1-1) if whenever f (a) = f (b) then a = b. Proving that a given function is one-to-one/onto. , every element of © 2021 P... | {
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if the function from a to B, for each pair of in! To B 12 Chapter Wise with Answers Chapter 1 Relations and functions MCQs PDF with Answers know... A 5-digit binary number, 1 } ∈ B there exists at least one a ∈ a, be! = m. onto one a ∈ a such that is both one-to-one and onto is also called one-to-one. ) nCr ( r^m ) and... | {
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of these functions, the number of functions from one set to another Math... A synonym for injective '' rather than bijective '' Y and Z respectively be better Prepared and in. Based on Latest Exam total no of onto functions from a to b direction for JEE Main be the function from a to B is the {! ( surjective ) if every... | {
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the air ; n ) ( -1 ) ^ ( n-r ) (...! ( e^x-1 ) ^n if the range for each pair of in... One and onto functions = 16−2= 14 ) ( -1 ) ^ ( n-r ) (. As well in m! ( e^x-1 ) ^n bijective '': is one-to-one onto surjective... ( E ) f ( X ) = x2 +1 to Y are 6 ( to. Prepared and study in the codomain we 're stuck with it the total... | {
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5 } is 0 as it is both one-to-one onto! Total numbers of onto functions what is a function f total no of onto functions from a to b a - > B is range. Total numbers of onto functions function from a set with eight elements to set..., c } and B = { a, B, for each pair of elements in the of... = B, then you can not cool t... | {
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# Animation and Model of Automotive Piston
This example shows how to model the motion of an automotive piston by using MATLAB® and Symbolic Math Toolbox™.
Define the motion of an automotive piston and create an animation to model the piston motion.
### Step 1: Describe Piston Model
The following figure shows the mo... | {
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fplot(pistHeight(150,50,theta),[0 2*pi])
ylabel('Height (mm)')
The piston head is highest when the piston is at TDC and the crank angle is 0 or 2*pi. The piston head is lowest when the piston is at BDC and the crank angle is pi.
You can also plot the piston height for various values of $\mathit{a}$ and $\theta$. Crea... | {
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### Step 4: Evaluate Piston Motion for Changing Angular Speed
Assume the crank rotates at 30 rpm for the first 3 seconds, then steadily increases from 30 to 80 rpm for the next 4 seconds, and then remains at 80 rpm.
Define the angular speed as a function of time by using the piecewise function. Multiply the angular s... | {
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fanimator(@rectangle,'Position',[-43 H(t) 86 10],'FaceColor',[0.8 0.8 0.8])
Add the animation objects of the connecting rod and the piston crank. Add a piece of text to count the elapsed time.
fanimator(@(t) plot([0 50*sin(angPos(t))],[H(t) 50*cos(angPos(t))],'r-','LineWidth',3))
fanimator(@(t) plot([0 50*sin(angPos(... | {
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# Math Help - Charles Lutwidge Dodgson
1. ## Charles Lutwidge Dodgson
Here is a problem to try and solve:
Jack and Jill walked along a level road, up the hill, back (along the same path) down te hill, and back along the same level road to home. They started out at 3pm and arrived home at 9pm. Their speed was four mi... | {
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They walked 12 miles one way.
The entire walk was 24 miles.
4. Hi
Can someone please explain why this is the case?
What I don't understand is that the above equations use (include) the times/speeds for descent (return journey) so why is the result one way?
I appologide in advance for being so dense
5. First do you ... | {
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Got it?
8. How would you do part b of this question? I know the answer but don't understand how to do it. I have if we assume either all flat or uphill that it could be 3 hour or 4 hours. Which would end up bieing 6/7 pm. How can we do this? Please help... | {
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×
# Magnetic Dollars
Imgur
Suppose that I have two urns, with one magnetic dollar in each urn. I begin randomly throwing more magnetic dollars in the general vicinity of the urns, and the coins fall in the urns by a simple rule:
Let's say that Urn A has $$x$$ coins, and Urn B has $$y$$ coins. The probability that t... | {
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But wait! Let's say that the empty slots above the red card are the magnetic dollars in Urn A, and the empty slots below the red card are the magnetic dollars in Urn B. Notice that if you had $$x$$ empty slots above the red card and $$y$$ empty slots below the red card, then the probability that the next card will be a... | {
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How do you do this? · 3 years, 7 months ago
I'm curious to know why only one mathematician guessed over $100. I personally guessed around$150,000 becuase while the probability might be pretty low at some point, in A MILLION tries, it would be bound to happen quite a bit. · 3 years, 8 months ago
Well, their reasoning ... | {
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# Probability: Rolling six, six-sided dice a single time
My specific question (to settle a score-keeping argument in my family's never-ending Farkle tournament) is this:
• when rolling six, six-sided dice a single time, is it more likely to roll three pairs of three different numbers, or three dice of a single number... | {
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Thank you!
• Not sure this is clear. When you say three dice of a single color would you also include all six of that number? Would you include something like $(1,1,1,2,2,5)$? – lulu Jan 2 '19 at 15:04
• You ask about "scoring relative to correctly weighted probability" but you already have six events with the same pr... | {
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Let's just compute. Of course there are $$6^6$$ unrestricted ways to throw the six dice.
We pick the three paired values,$$\binom 63=20$$. We then place the two appearances of the least value, $$\binom 62= 15$$ and then place the two appearances of the middle value, $$\binom 42=6$$. Thus there are $$20\times 15\times 6... | {
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# Can an irrational always be found by multiplying irrationals?
I was thinking about the function $\ f(a,b) = a/b$ where $a$ and $b$ where both irrational. It quickly stood out to me that the codomain of that function would include every rational number. But, does it include every irrational number as well (in other w... | {
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To answer your question (elaborating along the lines of TonyK's answer), let $x$ be any irrational number. $\sqrt{2},\sqrt{3},$ and $\sqrt{6}$ are all irrational. Moreover, $\sqrt{2}x$ and $\sqrt{6}x$ cannot both be rational or else $\frac{\sqrt{6}x}{\sqrt{2}x} = \sqrt{3}$ would be irrational. So one of the two product... | {
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# Special Case in Ratio Test for Series
We all know the Ratio Test in series which is one of many tests that are used to determine whether a Series is Converges or Not (Most Of The Time Work). Here is a link that describe Ratio test "In Short". Ratio Test
I have example : Suppose we have $U_n$=$2^{-n+({-1})^{n+1}}$ a... | {
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$$\sum_{n=1}^{2N} U_n=\sum_{n=1}^N U_{2n}+\sum_{n=1}^N U_{2n-1}\tag1$$
We assume that $U_n\ge 0$,
If the limits $\lim_{n\to \infty}\frac{U_{2n+2}}{U_{2n+1}}$ and $\lim_{n\to \infty}\frac{U_{2n+1}}{U_{2n}}$ exist and are finite, then the ratio test guarantees that the first sequence of partial sums on the right-hand s... | {
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# Given a binary tree with N labelled leaves, is it possible to find its unique number in the Catalan range?
The question is about finding the inverse to the problem of generating the $$n^{th}$$ binary tree with N labelled leaves (Generating the $$n^{th}$$ full binary tree over $$N$$ labelled leaves).
Let's say if $$... | {
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• Could you give a reference to what you call the "Catalan space" ? Jun 17 '20 at 7:16
• @JeanMarie, I have explained it in the description, I'll give it another shot here. If there are N nodes, the maximum possible binary trees is C(N). C(N) is the Nth Catalan number. We can uniquely identify all the individual binary... | {
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Our trees are labeled in increasing order of $$k$$. So before this tree, we have $$C_0 C_{n-2} + C_1 C_{n-3} + \dots + C_{k-2} C_{n-k}$$ trees whose left subtree has $$1, 2, \dots, k-1$$ leaves respectively.
Next, before this particular left subtree $$L$$, there are $$f(L)$$ previous left subtrees on $$k$$ leaves; for... | {
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Solving a differential system of equations in matrix form
I have a basic question about eigenvector.
If I have the following system:
$$\begin{pmatrix} \dot{\eta}_1 \\ \dot{\eta}_2 \\ \end{pmatrix} = {\cal{B}} \begin{pmatrix} \eta_1 \\ \eta_2 \\ \end{pmatrix}$$
Where $${\cal{B}}$$ is a matrix with constant parameter... | {
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Consider the system $$\dot{x} = Ax$$, where $$A\in\mathbb{R}^{n\times n}$$. If you assume $$A$$ to be diagonalizable (i.e. the geometric multiplicity equals the algebraic multiplicity for every eigenvalue of $$A$$), then you can find an invertible transformation $$T\in\mathbb{R}^{n\times n}$$ in terms of eigenvectors o... | {
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• Think of it like this: in the original $x$-coordinates, the components of the ODE trajectory may be correlated in some sense to one another. This is due to the coupling between different coordinates of $x$ through the structure of $A$. When we diagonalize $A$, we are changing coordinates to a system where we remove t... | {
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# Ray Against Plane
Testing a ray against a plane is surprisingly easy. Remember, any point on the ray can be represented by a time t, such that:
point(t) = plane.position + plane.normal * t
^ the (t) above means point at time t
Also, recall we can test if a point is on a plane if it fits this equation:
DOT(point,... | {
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If the ray is parallel to the plane will be 0 and there is no intersection.
An intersection is only valid if the ray is in front of the plane.That is, if the direction of the ray is opposite of the planes normal. This is true if .
If the value of t is out of range (that is, if t is negative), then there is no interse... | {
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namespace CollisionDetectionSelector.Samples {
class RaycastPlane : Application {
public Plane test = new Plane(new Vector3(1f, 1f, 0f), 1f);
public Ray[] rays = new Ray[] {
new Ray(new Point(0f, 0f, 0f), new Vector3(0f, -1f, 0f)),
new Ray(new Point(0.5f, 0.5f, 0f), new Vector3(-1f, -1f, 0f)),
new Ray(new Point(1f, 1f... | {
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# Ways to select three-man teams
In a competition there are 18 competitors. Answer the following:
A) During the first day they're competing in three-man teams (total of 6 teams). How many ways are there to select the teams?
B) If the main sponsor of the event demands that the four best ranked players mustn't play in... | {
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A) There are $\binom{18}{3}$ ways to choose the first team, $\binom{15}{3}$ ways to choose the second, etc. this gives $$\binom{18}{3}\binom{15}{3}\binom{12}{3}\binom{9}{3}\binom{6}{3}\binom{3}{3}$$ ways to choose 6 teams. But the order of the teams does not matter, so we have to divide by the number of permutations of... | {
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However, following this idea for each team will make your answer smaller, whereas your answer is too small to begin with. What you want to do is multiply those fractions rather than add them. Here we're determining one team and then determining another and then determining a third... Often, when an "and" appears as par... | {
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# Interpreting Probability Questions
#### (A new question of the week)
A couple recent questions centered around how to interpret probability problems, whose wording can often be subtle, and whose solutions require care.
## A dice problem about the maximum
Around the start of December, Drey sent us two questions. T... | {
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Hi Drey,
You haven’t said what answer you got when you tried to solve the problem, other than you did not get 8/9, so I can’t tell what you are doing incorrectly. If you want to give your answer and your reasoning, I can try to figure out why you are not getting the answer. Did you write out the possibilities and ju... | {
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Complements can be hard to think clearly about, especially if you focus too much on the words and not enough on the meaning. But once you are over that hurdle, the thinking can become easier. To help with that, Doctor Fenton offered a different problem (the more common question about sums, but with three dice so it is ... | {
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Since we have the 2 dice, then the total number of outcomes will be 6^2=36.
Now for the favorable number of events, since we are looking that the event doesn’t occur (where the dice need to be less than 3 (P(B)<3)) I managed to count only one event where each die is showing 1: {1,1}. Now using the formula of the comp... | {
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It’s true that the complement of “3 or greater” is “less than 3”. But the key word is “maximum”, which Drey doesn’t use here. My guess is that Drey has let that word get tangled up in the “greater than” and lost it, changing the event from “the maximum is greater than or equal to 3” to something like “it is greater tha... | {
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Here is the complementary event, in which the largest of the two numbers is less than 3:
Without listing, that could be counted by seeing it as “both numbers are less than 3″, so that there are 2 choices for the first and 2 for the second, making a total of $$2\times 2=4$$. So the probability of the complement is $$\f... | {
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Solution = 1/8
(B) Assuming extraction without replacement
Solution = 2/77
What I tried to do: (A) as this line says that 2 balls are removed with replacement, so there is no need to apply simple combination. Total number of outcomes = 12, number of events are two balls of each color, and since each draw is an indep... | {
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In selection without replacement (so that after each draw there is one less ball, and the probabilities change), Drey (wisely) chose to use combinations. His notation “C6.2” means what is more commonly written as “$${6\choose 2}$$” or “$$_6C_2$$” or “$$C(6,2)$$”. He found the total number of ways to choose 2 of the 6 b... | {
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I had to take the time to convince myself that my thinking was right even though it led to different answers than the book, before deciding to try to help!
### Getting it right
Drey replied,
Hello Doctor Peterson.
Well since the order doesn’t matter we have to pick the number of ways that these 6 colours can be rea... | {
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Now what I don’t get is the thought process behind the multiplication at the last part.
Could you maybe help explain? | {
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# Is the function linear?
Given $$F(x) = \left(\begin{matrix} -x_2 \\ x_1+2x_2 \\ 3x_1 - 4x_2 \end{matrix} \right), x = \left(\begin{matrix} x_1 \\ x_2 \end{matrix} \right)$$ Prove whether $F$ is a linear function or not.
I've tried to prove it, but I'm not sure it's right:
$$F(x) = Mx \Rightarrow M = \left(\begin{m... | {
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The main thing, is that there is a matrix $M$, such that $F(x)=M\cdot x$ for all vectors $x$. In fact for a mapping $F$ between vector spaces with fixed bases, this is equivalent to be linear. Then, that multiplying by a matrix is indeed a linear mapping, is straightforward: depend on the nice properties of matrix mult... | {
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-
+1. Can you elaborate just a tiny little bit the parts regarding "bases", "fixed bases" (never heard this one) and "standard bases"? I'm not sure I've understood the concepts so good from my readings to feel free to "play" with them like that (and my learning material is in German). – Flavius Nov 12 '12 at 11:01
For... | {
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First note that we can write any vector $a=\left(\begin{matrix} a_1 \\ a_2 \end{matrix} \right)$ in $\mathbb{R}^2$ as $$a_1\left( \begin{matrix} 1 \\ 0 \end{matrix} \right)+a_2\left( \begin{matrix} 0 \\ 1 \end{matrix} \right)=a_1e_1+a_2e_2$$ and any vector $b= \left( \begin{matrix} b_1 \\ b_2 \\ b_3 \end{matrix} \right... | {
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# Smallest Non-negative number in a matrix
There is a question I encountered which said to fill an $N \times N$ matrix such that each entry in the matrix is the smallest non-negative number which does not appear either above the entry or to its left. That is for $N = 6$ the matrix looks like this:
$$\begin{array}{} 0... | {
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To show (1), suppose that $0\le t<s$, let $d=t\oplus s$, and let $k$ be the number of bits in $d$, so that $2^{k-1}\le d<2^k$. That is, $d$ has a $1$ in the $k$-th position counting from the right. Since $t<s$, this implies that $t$ has a $0$ in the $k$-th position, and $s$ has a $1$ there. Since $s=m\oplus n$, exactly... | {
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I really like this question. I saw it on a website a year or so ago, closed the page so that I wouldn't see the answer, and then never found it again. I solved it by proving by induction that $$f(4i+r, 4j+s) = f(r,s)+4f(i,j)$$ --- this relation is apparent if you draw enough of the matrix. Let me know if you would like... | {
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Now for the inductive step, we let $x_n,y_n\in\{0,1\}$ be arbitrary, and note first that the base block $(x_i,y_i)=(0,0)$ is our inductive hypothesis. Next, note that for $(x_i,y_i)=(0,1)$ and $(x_i,y_i)=(1,0)$, all of the values in $I_n$ are already taken and so the minimality rule will assign values to $f$ starting w... | {
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-
In NIM-like games the 'value' of a game position is (among other things) also always the smallest positive integer that is missing from the list of the values of the positions that can be reached from the said position in a single move. And here the entries on top (resp. to the left) of a given matrix entry are the v... | {
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To get the minimum values, you can just shift the values while stepping onto the next row.
- | {
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# How do you graph an inequality on Real/Imaginary plane?
Suppose we have $z$ as a complex number, $z \in C$, how do you graph an inequality which has $z$ in it?
This kinds of inequalities arise when we need to graph the shape of stability region of a given numerical method:
For example, for RK2 method we can derive... | {
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I consider this as the best answer, specially in exam rooms. I will wait for further confirmation by the community and then will accept this as answer after few days.
You sketch in the boundary line, where the equation equals 0, then you shade in one side (corresponding to the inequality).
Notice that in your equatio... | {
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• I appreciate your effort of solving the inequality in terms of $x$ and $y$. I think what I was missing is the key of setting $z = x + iy$ which Morgan Rodgers also mentioned. Meanwhile I found this which is less error prone and more efficient: math.ubc.ca/~peirce/M405_607E_Lecture%2018.pdf May 8, 2016 at 7:46
• I wro... | {
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# What does $\frac{x^3}{9}\bigg|_0^1$ mean, and how should it be spoken?
$$\frac{x^3}{9}\Bigg|_0^1$$ The vertical line above: what does it mean, and how would I state this whole structure in spoken words, so that a screen reader would be able to read it aloud correctly?
• Evaluate. In particular here, $f(x)|_a^b$ usu... | {
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• How would I write this out in spoken words, so a screen reader would state it correctly aloud? Thank you. – Chelonian Jan 4 '17 at 6:17
• @Chelonian This is context vital to the question; it sounds like you're working on accessibility, which, if true, is quite the lofty and commendable goal! – pjs36 Jan 4 '17 at 6:21... | {
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# Proof that the relation $5 \mid (a + 4b)$ is symmetric and transitive
Take the relation $R$ to be defined on the set of integers:
$$aRb \iff 5 \mid (a + 4b)$$
As part of a larger proof, I have to show that $R$ is both symmetric and transitive. I'm lost.
I see the first steps, but I can't find how to progress furt... | {
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-
@deinst's comment cleared up the symmetry part for me, but I don't see why adding $a+4x$ to $x+4b$ helps us solve the transitivity part of the proof? – David Chouinard Dec 4 '11 at 22:52
$(a+4x)+(x+4b)=(a+4b)+5x$, so $a+4b=(a+4x)+(x+4b)-5x$. What do you know about each of the three terms on the righthand side? – Br... | {
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Expected number of empty bins with clumpy balls?
I have $b$ bins and a reservoir of $n$ balls.
The balls have a tendency to "clump", that is, when I try to grab one from the reservoir, the number of balls removed is uniformly random on $[1, m]$, where $m$ is the number of balls remaining in the reservoir for that dra... | {
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• Thank you! See edit to my original question for pretty simple form. – Tommy Sep 10 '17 at 22:57
• The product formula works perfectly! – quasi Sep 10 '17 at 23:47
Credit goes to @quasi for the first answer and the recurrence. Using generating functions and classifying on the number of rounds $k$ we find the PGF
$$\... | {
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$$\bbox[5px,border:2px solid #00A000]{90.66863442}$$
for $n=10000$ and $b=100.$
with(combinat);
ENUM_GF :=
proc(n, b)
option remember;
local recurse, gf;
gf := 0;
recurse :=
proc(prob, rest, cc)
local rm;
if rest = 0 then
gf := gf +
prob*cc!*coeftayl((u-1+exp(z))^b, z=0, cc)/b^cc;
return;
fi;
for rm to rest do
... | {
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How to compute number of ways one S and one F can be chosen from group of (2F, 2S, 2J, 2R)?
I thought I knew combinations pretty well, until recently I came across this question:
In a high school debating team consisting of 2 freshmen, 2 sophomores, 2 juniors, and 2 seniors, two students are selected to represent the... | {
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Thanks.
You have correctly calculated the denominator. However, there are four ways to select a freshman and a sophomore since there are two ways to select a freshman and two ways to select a sophomore. Hence, the probability that a freshman and a sophomore are selected for the competition is $$\Pr(\text{selected stud... | {
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# Elementary set theory homework proofs
I've been given some statements to prove using only the following facts.
1. The set of all integers is countably infinite.
2. Let $X$ and $Y$ be sets and suppose $f:X\rightarrow Y$ is surjective. If $X$ is countable, then $Y$ is also countable.
3. The union of a countably inf... | {
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Exercise 2. Every infinite subset of a countable set is countable.
Proof. Let $X$ be a countable set and let $Y$ be an infinite subset of $X$. Then $X$ is countably infinite and there exists a surjection $f:\mathbb{N}\rightarrow X$. Since $Y\subseteq X$ and $Y$ is nonempty, there exists a surjection $g:X\rightarrow Y$... | {
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To reduce the number of unanswered questions, I answer here (albeit belatedly).
Your first proof is just fine. However, depending on what your definition of "countable" is, you may not have followed the instructions for your second proof. Consider the two following definitions (the first of which is standard, and the ... | {
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"tags"... |
# Proving that $\tan 2x \cdot (1 + \tan x) \cdot \cot x = \frac{2}{1 - \tan(x)}$
Given the following expression, $$\tan(2x) \cdot (1 + \tan(x)) \cdot \cot(x)$$ the exercise asks to simplify the expression and $$\frac{2}{1 - \tan(x)}$$ should be the simplified expression.
I have tried everything I possibly could, incl... | {
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"openwebmath_score": 0.9009023904800415,
"ta... |
i.e. $$\tan(2x) (1+\tan(x)) \cot(x) =\frac{2}{1-\tan(x)}$$
• Wow. Thank you. – Johnny Bueti Apr 4 '19 at 15:39 | {
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"ta... |
# Number of combinations of $7$ of the first $17$ natural numbers in which there are at least two consecutive numbers
I have this statement:
How many groups of $7$ elements can you get from the first $17$ natural numbers and at least $2$ of them must be consecutive ( Numbers can't repeat and order doesn't matter) ?
... | {
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"lm_q1q2_score": 0.8465108397639408,
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"openwebmath_perplexity": 172.04645036644231,
"openwebmath_score": 0.8241270184516907,
"ta... |
How should it be done, following a logic similar to that of my development?
• Sometimes it helps to consider the Principle of Inclusion/Exclusion when counting instances that require a side condition, as here with imposing (at least) two consecutive numbers in a set. – hardmath Aug 12 '18 at 15:26
• Edited, the 17 fir... | {
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"openwebmath_score": 0.8241270184516907,
"ta... |
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