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# Problem Solving
When solving a problem (as opposed to merely using a given formula), it is generally good to proceed along the following lines:
1. First, you have to understand the problem.
2. After understanding, make a plan.
3. Carry out the plan.
4. Look back on your work. How could it be better?
### Understand... | {
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Note: these instructions were inspired by How to Solve It by George Pólya.
Note by Arron Kau
5 years, 6 months ago
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and think... | {
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- 4 years, 9 months ago
Well in the beginning it may be boring and frustrating, but if you practice these steps enough times, you surely can do all of these in a minute or two, while seeing the mathematics with a different angle and every angle, and writing the genuine solution, and having a feeling like you have inve... | {
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# Seating couples around 2 tables
Here's my question and possible answer.
How many possible ways can you arrange 8 married couples between 2 circular tables of 8 identical chairs each such that:
1) each couple must sit at the same table, and,
2) at each table, men and women must sit in adjacent chairs (NOTE: a coup... | {
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And again, if you do not differentiate between the tables, then divide this by $2$
• Suppose we have an origin from where to start counting round one of the tables, then this would class the string ABCDEFGH to be different to BCDEFGHA, but surely due to the question detailing that the tables are circular, these are th... | {
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How many ways can you split the 8 couples into two groups of 4? That's ${8\choose4}=70$.
Then once split, you have 4 men, 4 women who need to be seated at one of the tables, in alternating order. The first person seated has 8 choices, then the other people of the same gender have 3,2,1 choices afterwards. Meanwhile fo... | {
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# Prove convergence by considering the partial sums
Let $p$ be a non-zero natural number. Prove by considering the partial sums that
$\sum \frac{1}{k(k+p)}$
converges. What is $\sum\limits_{k=1}^{\infty} \frac{1}{k(k+p)}$
No idea. Obviously, it looks like a telescoping series. Sure doesn't act like one. I have trie... | {
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$$\frac1p\left(1+\frac12+\ldots+\frac1p\right)$$
• Sorry, but does this tell me the value to which this converges? Because that is the main reason that I am struggling. I know how to show convergence. I need to show convergence using partial sums and then state the value of the sum. – Bob the Builds Nov 29 '14 at 22:5... | {
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Consequently, by $(1)$ through $(3),$ we have for any integer $m\ge1$ that $$S_{p+m}=\frac1p\sum_{k=1}^p\frac1k-\frac1p\sum_{k=m+1}^{p+m}\frac1{p+k}=\frac1p\sum_{k=1}^p\frac1k-\frac1p\sum_{k=1}^p\frac1{p+m+k}.\tag{\star}$$ Now, for any such integer $m,$ we have $$0<\sum_{k=1}^p\frac1{p+m+k}<\sum_{k=1}^p\frac1{p+m+1}=\f... | {
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# Does the absolute value even matter here?
In this problem I'm doing it says
Suppose that $(X,Y)$ is uniformly distributed over the region {$(x,y):0\lt |y|\lt x\lt 1$}. Find the marginal densities $f_X(x)$ and $f_Y(y)$
Does the absolute value even matter here since it's all between $0$ and $1$ anyway? How is the an... | {
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-
The absolute value definitely matters. It says the region includes, for example, $(2/3,-1/3)$, which it wouldn't do if the problem said $0\lt y\lt x\lt1$. For the main question, what do you know about computing marginal densities? How would you go about doing that? – Gerry Myerson Nov 18 '12 at 22:45
Draw a sketch o... | {
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In response to the OP's request
As Gerry Myerson pointed out, the absolute value sign does matter, and as Michael Hardy's answer clarifies in more detail, the joint density $f_{X,Y}(x,y)$ has nonzero constant value $c$ on the region $$\{(x,y) \colon -x < y < x, 0 < x < 1\},$$ that is, on the interior of a right triang... | {
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For any fixed $x$, the marginal density $f_X(x)$ is given by $$f_X(x) = \int_{-\infty}^\infty f_{X,Y}(x,y)\,\mathrm dy$$ which is, of course, the area of the cross-section of the of the joint density solid if we were to slice the solid by a plane parallel to the $y$-$z$ plane and at distance $x$ from the $y$-$z$ plane.... | {
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-
$$0<|y|<x<1$$ is the same as $$0<x<1\text{ and for each value of x, }-x<y<x.$$ The absolute value is redundant in the inequality $0<|y|$ (we don't care whether it's "$<$" or "$\le$" since the probability of being exactly equal is $0$ either way). But the absolute value matters in the inequality $|y|<x$, and the "$0<... | {
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# Problem regarding linear congruence
Which of the following statement is False ?
1) There exists a natural number which when divided by 3 leaves remainder 1 and which when divided by 4 leaves remainder 0
2) There exists a natural number which when divided by 6 leaves remainder 2 and which when divided by 9 leaves r... | {
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• For 4: 19 and yes you have the right idea for disproving 2. Sep 11 '18 at 17:10
• Any number which is $2$ more than a multiple of six will be $2$ more than a multiple of $3$ (i.e. $6k+2 = 3(2k)+2$). Similarly, any number which is $1$ more than a multiple of $9$ will be $1$ more than a multiple of $3$ (i.e. $9\ell + 1... | {
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3. Here, we have $x\equiv 1\bmod 7$ and $x\equiv 3\bmod 11$. Since $7$ and $11$ are coprime, by the Chinese remainder theorem, we obtain $x\equiv 36\bmod 77$.
4. We have $x\equiv 7\bmod 12$ and $x\equiv 3\bmod 8$. Applying a similar train of thought that we did for the second problem, we obtain $12m+7\equiv 3\bmod 8$,... | {
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# Monthly Archives: January 2018
## Inscribed Right Angle Proof Without Words
Earlier this past week, I assigned the following problem to my 8th grade Geometry class for homework. They had not explored the relationships between circles and inscribed angles, so I added dashed auxiliary segment AD as a hint.
What fol... | {
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Here’s a very pretty problem I encountered on Twitter from Mike Lawler 1.5 months ago.
I’m late to the game replying to Mike’s post, but this problem is the most lovely combination of features of quadratic and trigonometric functions I’ve ever encountered in a single question, so I couldn’t resist. This one is well w... | {
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Unfortunately, Desmos is not a CAS, so I turned to GeoGebra for more power.
Investigation #2:
In GeoGebra, I created a sketch to vary the linear coefficient of the quadratic and to dynamically calculate angle sums. My procedure is noted at the end of this post. You can play with my GeoGebra sketch here.
The x-coor... | {
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I knew the answer to the now extended problem, but I didn’t know why. Even so, these solutions and the problem’s request for a SUM of angles provided the insights needed to understand WHY this worked; it was time to fully consider the product of the angles.
Insight #4: Finally a proof
It was now clear that for $\le... | {
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$(tan(x))^2+b*tan(x)+1=0$ for x in $[0,2\pi ]$ and any $\left| b \right| \ge 2$,
is always $3\pi$ with the fundamental reason for this in the definition of trigonometric functions and their co-functions. QED
Insight #6: Generalizing the Domain
The posed problem can be generalized further by recognizing the period ... | {
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I think that’s enough for one problem.
APPENDIX
My GeoGebra procedure for Investigation #2:
• Graph the quadratic with a slider for the linear coefficient, $y=x^2-b*x+1$.
• Label the x-intercepts A & B.
• The x-values of A & B are the outputs for tangent, so I reflected these over y=x to the y-axis to construct A’ a... | {
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# Exponential Random Variable representation of criminal trial
Assume the amount of evidence against a defendant in a criminal trial is an exponential random variable $X$. If the defendant is innocent, then $X$ has mean $1$, and if the defendant is guilty, then $X$ has mean $2$. The defendant will be ruled guilty if $... | {
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Thank you in advance for any help in understanding the problem!
• $0.95 \leq \int_0^c \lambda e^{-\lambda x}\mathrm{x} \implies 0.95 \leq 1- e^{-\lambda c} \implies c \geq \frac{\ln(20)}{\lambda}$ – Graham Kemp Apr 30 '14 at 2:30
• I do not see how this is correct.. where is the 20 coming from? I obtained the same int... | {
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Next, we wish you update our belief given conviction.
Let $B(G)$ be your prior belief that the defendant is guilty and $B(G\mid F)$ be your posterior belief of guilt given the conviction. Basically, your belief is an estimation of the probability of guilt.
By Bayes theorem: $$B(G\mid F) = \frac{B(G)\cdot P(F\mid G)}{... | {
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# Documentation
### This is machine translation
Translated by
Mouse over text to see original. Click the button below to return to the English verison of the page.
## Nonlinear Equations with Analytic Jacobian
This example demonstrates the use of the default trust-region-dogleg fsolve algorithm (see Large-Scale vs.... | {
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This system is square, and you can use fsolve to solve it. As the example demonstrates, this system has a unique solution given by xi = 1, i = 1,...,n.
### Step 1: Write a file bananaobj.m to compute the objective function values and the Jacobian.
function [F,J] = bananaobj(x)
% Evaluate the vector function and the J... | {
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Norm of First-order Trust-region
Iteration Func-count f(x) step optimality radius
0 1 8563.84 615 1
1 2 3093.71 1 329 1
2 3 225.104 2.5 34.8 2.5
3 ... | {
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# Why does the alternatization of a $k$-linear map indeed produce an alternating $k$-linear map?
The alternating operator $$A$$ produces for any $$k$$-linear map $$f$$ an alternating $$k$$-linear map $$Af$$ (the alternatization of $$f$$):
$$Af(v_1, \ldots, v_k) = \sum_{\sigma \in S_k} \text{sgn}(\sigma)f\left(v_{\sig... | {
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Fix $$\pi \in S_k$$.
\begin{align*} Af(v_{\pi(1)},\dots,v_{\pi(k)})&= \sum_{\sigma\in S_k}\operatorname{sgn}(\sigma)f(v_{\sigma(\pi(1))},\dots,v_{\sigma(\pi(n))}) \\ &=\operatorname{sgn}(\pi)\sum_{\sigma\in S_k}\operatorname{sgn}(\pi)\operatorname{sgn}(\sigma)f(v_{\sigma(\pi(1))},\dots,v_{\sigma(\pi(n))}) \\ &=\operat... | {
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We want to show that $$A f(v_1,\dotsc,v_r) = 0$$ whenever $$v_i =v_j$$ with $$i\ne j$$. The number of terms (permutations) in the sum $$Af =\sum_\sigma (\operatorname{sgn}\sigma)f_\sigma$$ is even because there are $$r!$$ terms, and the definition of alternating implicitly assumes $$r\ge 2$$. Pair each permutation $$\s... | {
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"lm_q2_score": 0.8596637487122111,
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"openwebmath_score": 0.9931748509407043,
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# Definition:Coordinate
## Definition
### Elements of Ordered Pair
Let $\left({a, b}\right)$ be an ordered pair.
The following terminology is used:
• $a$ is called the first coordinate
• $b$ is called the second coordinate.
This definition is compatible with the equivalent definition in the context of Cartesian c... | {
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### Origin
The origin of a coordinate system is the zero vector.
In the $xy$-plane, it is the point:
$O = \left({0, 0}\right)$
and in general, in the Euclidean space $\R^n$:
$O = \underbrace{\left({0, 0, \ldots, 0}\right)}_{n \ \text{coordinates}}$
## Linguistic Note
It's an awkward word coordinate. It really ne... | {
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# Last digit of sequence of numbers
We define the sequence of natural numbers $$a_1 = 3 \quad \text{and} \quad a_{n+1}=a_n^{a_n}, \quad \text{ for n \geq 1}.$$
I want to show that the last digit of the numbers of the sequence $a_n$ alternates between the numbers $3$ and $7$. Specifically, if we symbolize with $b_n$ t... | {
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In the same way, from $a_n\equiv 2\textrm{ (mod 5)}$ you get $a_n \equiv 7\textrm{ (mod 10)}$.
• I have a question. We have that $a_1=3$. The last digit of $a_2$ is $3^3 \pmod{10}=7$. The last digit of $a_3$ is $7^7 \pmod{10}=3$. The last digit of $a_4$ is $3^3 \pmod{10}=7$, and so on. Why doesn't this suffice? Aug 27... | {
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Finally note that $a_n \equiv 1 \pmod 2$ and use Chinese remainder Theorem to conclude that:
$$a_n \equiv \begin{cases} 3 \pmod{10}, & \text{if n is odd} \\ 7 \pmod{10}, & \text{if n is even} \end{cases}$$
• Could you explain to me how we use the fact that $a_n \equiv -1 \pmod{4}$ if we have an odd power? Sep 14, 201... | {
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Thus, $a_{n+1} \equiv 3 \mod 5$.
So the remainders of $a_n$ and $a_{n+1}$ alternate modulo $5$. Since all $a_n$ are odd, this forces their last digits to alternate, and from knowing that the last digit of $a_1$ is $3$, the sequence of last digits must go $373737373...$
• I haven't understood how we get that $a_{n+1}=... | {
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# for QR codes use inline
%matplotlib inline
qr_setting = 'url'
#
# for lecture use notebook
# %matplotlib notebook
# qr_setting = None
#
%config InlineBackend.figure_format='retina'
# import libraries
import numpy as np
import matplotlib as mp
import pandas as pd
import matplotlib.pyplot as plt
import laUtilities as u... | {
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"openwebmath_score": 0.806428074836731,
"tags": null,
... |
$$A$$ is called the standard matrix of $$T$$.
Proof. Write
${\bf x} = I{\bf x} = \left[{\bf e_1} \dots {\bf e_n}\right]\bf x$
$= x_1{\bf e_1} + \dots + x_n{\bf e_n}.$
Because $$T$$ is linear, we have:
$T({\bf x}) = T(x_1{\bf e_1} + \dots + x_n{\bf e_n})$
$= x_1T({\bf e_1}) + \dots + x_nT({\bf e_n})$
$\begin{split} ... | {
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... |
For example, let’s consider rotation about the origin as a kind of transformation.
u = np.array([1.5, 0.75])
v = np.array([0.25, 1])
diamond = np.array([[0,0], u, u+v, v]).T
ax = dm.plotSetup()
plt.plot(u[0], u[1], 'go')
plt.plot(v[0], v[1], 'yo')
ax.text(u[0]+.25,u[1],r'$\bf{u}$',size=20)
ax.text(v[0],v[1]-.35,r'$\b... | {
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"lm_q1_score": 0.9833429595026214,
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"openwebmath_score": 0.806428074836731,
"tags": null,
... |
u = np.array([1.5, 0.75])
v = np.array([0.25, 1])
diamond = np.array([[0,0], u, u+v, v]).T
ax = dm.plotSetup()
dm.plotSquare(diamond)
ax.text(u[0]+.25,u[1],r'$\bf{u}$',size=20)
ax.text(v[0],v[1]+.25,r'$\bf{v}$',size=20)
ax.text(u[0]+v[0]+.25,u[1]+v[1],r'$\bf{u + v}$',size=20)
rotation = np.array([[0, -1],[1, 0]])
up =... | {
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"openwebmath_perplexity": 2465.0135183133702,
"openwebmath_score": 0.806428074836731,
"tags": null,
... |
import matplotlib.patches as patches
ax = dm.plotSetup(-1.2, 1.2, -0.5, 1.2)
# red circle portion
arc = patches.Arc([0., 0.], 2., 2., 0., 340., 200.,
linewidth = 2, color = 'r',
linestyle = '-.')
#
# labels
ax.text(1.1, 0.1, r'$\mathbf{e}_1 = (1, 0)$', size = 20)
ax.text(0.1, 1.1, r'$\mathbf{e}_2 = (0, 1)$', size = 20)... | {
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"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429595026214,
"lm_q1q2_score": 0.8453442930680223,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 2465.0135183133702,
"openwebmath_score": 0.806428074836731,
"tags": null,
... |
That is, if A and B are matrices,
A @ B
will multiply A by every column of B, and the resulting vectors will be formed into a matrix.
dm.plotSetup()
angle = 90
theta = (angle/180) * np.pi
A = np.array(
[[np.cos(theta), -np.sin(theta)],
[np.sin(theta), np.cos(theta)]])
rnote = A @ note
dm.plotShape(rnote)
## Geomet... | {
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"openwebmath_score": 0.806428074836731,
"tags": null,
... |
A = np.array(
[[1, 0],
[0, -1]])
print('A = \n',A)
dm.plotSetup()
dm.plotSquare(square)
dm.plotSquare(A @ square,'r')
Latex(r'Reflection through the $x_1$ axis')
A =
[[ 1 0]
[ 0 -1]]
$Reflection through the x_1 axis$
dm.plotSetup()
dm.plotShape(note)
dm.plotShape(A @ note,'r')
What about reflection through the $$... | {
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... |
$Horizontal Contraction$
dm.plotSetup()
dm.plotShape(note)
dm.plotShape(A @ note,'r')
A = np.array(
[[2.5,0],
[0, 1]])
print('A = \n',A)
dm.plotSetup()
dm.plotSquare(square)
dm.plotSquare(A @ square,'r')
Latex(r'Horizontal Expansion')
A =
[[2.5 0. ]
[0. 1. ]]
$Horizontal Expansion$
A = np.array(
[[ 1, 0],
[-1.5,... | {
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"openwebmath_score": 0.806428074836731,
"tags": null,
... |
Informally, $$T$$ is onto if every element of its codomain is in its range.
Another (important) way of thinking about this is that $$T$$ is onto if there is a solution $$\mathbf{x}$$ of
$T(\mathbf{x}) = \mathbf{b}$
for all possible $$\mathbf{b}.$$
This is asking an existence question about a solution of the equatio... | {
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"lm_q1_score": 0.9833429595026214,
"lm_q1q2_score": 0.8453442930680223,
"lm_q2_score": 0.8596637469145054,
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"openwebmath_score": 0.806428074836731,
"tags": null,
... |
# image credit: Lay, 4th edition
display(Image("images/Lay-fig-1-9-4.jpeg", width=650))
Let’s examine the relationship between these ideas and some previous definitions.
If $$A\mathbf{x} = \mathbf{b}$$ is consistent for all $$\mathbf{b}$$, is $$T(\mathbf{x}) = A\mathbf{x}$$ onto? one-to-one?
$$T(\mathbf{x})$$ is on... | {
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"openwebmath_score": 0.806428074836731,
"tags": null,
... |
# How can I find the rotational inertia of two joined bars with respect of an axis?
I found this problem in a book whose author is unknown as it is merely a collection of riddles in mechanics, and it has left me very confused on how to approach the rotational inertia when two objects are joined.
The problem is as fol... | {
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"openwebmath_score": 0.7222120761871338,
"ta... |
But this doesn't check with any of the alternatives, what could I be doing wrong here?.
• What does "an axis perpendicular to the point O" mean exactly? It sounds nonsensical to me. – NickD Dec 23 '19 at 3:49
• @NickD I was also confused about the same. In one of the answers it has pointed that Steiner theorem can be ... | {
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"openwebmath_score": 0.7222120761871338,
"ta... |
Remember that the rotational inertia of an object depends heavily on how the mass is distributed along the rotational axis, so it wouldn't make much sense to treat the BC bar as a particle. You are almost close to the answer though. You can use the theorem you mentioned to calculate the rotational inertia of the BC bar... | {
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"openwebmath_score": 0.7222120761871338,
"ta... |
• Interesting!, I believe you should include your last comment as part of your answer because it does clarify the intended use of the Steiner's theorem as it was not clear how was understood the axis of rotation. What I was seeing was an axis going along the line connecting $O$ and the marker of the ruler located over ... | {
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"lm_q2_score": 0.8652240791017536,
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"openwebmath_score": 0.7222120761871338,
"ta... |
# Probability with 8 number of dice
Eight identical dice are rolled simultaneously. In how many possible outcomes each of the six numbers appears at least once?
I got result as $\left(\dfrac 16\right)^8.$ And I think I missed some part. Could you please help me?
• It depends on what you choose as the set of outcomes... | {
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"tag... |
Now let's start with an easier problem: How many outcomes are there with no sixes? That is obviously $5^8$. Similarly there are $5^8$ outcomes with no fives, with no fours, and so forth. So it looks like the answer will be $$6^8 - 6\cdot 5^8$$ But this is not quite right. Consider any roll with no sixes and no fives. W... | {
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Thus the counterintuitive result is that you are likely to be missing at least one number when you roll 8 dice. In fact, in order to have better than a 50% chance of getting one of each number, you would have to roll 13 dice!
• Of course, the main problem with the question at hand, is that the user asked for the numbe... | {
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# Find the value of $3+7+12+18+25+\ldots=$
Now, this may be a very easy problem but I came across this in an examination and I could not solve it.
Find the value of
$$3+7+12+18+25+\ldots=$$
Now here is my try
$$3+7+12+18+25+\ldots=\\3+(3+4)+(3+4+5)+(3+4+5+6)+(3+4+5+6+7)+\ldots=\\3n+4(n-1)+5(n-2)+\ldots$$
After th... | {
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"openwebmath_score": 0.9568216800689697,
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$s_n = 1/6n^3 + 3/2n^2 + 4/3n$
which yields the same as Gregory's answer.
• what justifies supposing it's cubic? – Dimitris Apr 12 '17 at 15:49
• This is possibly more direct, but doesn't this require the person to know that the sum will be cubic? Though I suppose if you knew the terms were quadratic, you could make ... | {
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"openwebmath_score": 0.9568216800689697,
"tag... |
I'm adding another answer because people ask how to find $a_n$ without trial and error.
We note that:
$a_2 - a_1 = 4; a_3 - a_2 = 5; a_4 - a_3 = 6$,
which leads us to conclude that $a_n$ is given by the recurrence relation:
$a_{n+1} - a_n = n+3$
Let's start by solving the homogeneous equation:
$a_{n+1} - a_n = 0$... | {
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"openwebmath_score": 0.9568216800689697,
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# Why are contours of a multivariate Gaussian distribution elliptical?
Displayed below are the contours and their respective covariance matrices according to Andrew Ng's notes (pdf). Why are the first and second contours elliptical and not circular? The variance along both axes is the same.
Here's one last set of exa... | {
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Just a supplement to the other answers: for a multivariate Normal with dimension $$k$$, you can see why algebraically if you follow this. Set the density equal to some level $$l$$, then: \begin{align*} (2\pi)^{-k/2} |\Sigma|^{-1/2} \exp\left(-\frac{1}{2}(x-\mu)'\Sigma^{-1}(x-\mu) \right) &= l\\ \iff \exp\left(-\frac{1}... | {
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"lm_q1_score": 0.9770226267447513,
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"openwebmath_score": 0.8710727095603943,
"tags... |
## More examples:
If the elements of $$x$$ are independent, then $$\Sigma$$ is diagonal, then $$\Sigma^{-1}$$ is diagonal, then (*) is $$\frac{(x_1 - \mu_1)^2}{\sigma_1^2} + \frac{(x_2 - \mu_2)^2}{\sigma_2^2} = l''\tag{**}$$ which is still an ellipse, but it's not tilted/rotated.
If the elements of $$x$$ are independ... | {
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This is when the covariance matrix $\Sigma$ is diagonal with a constant diagonal.
• (+1) But note that your assertions implicitly suppose $(X,Y)$ has a bivariate Gaussian distribution. Otherwise, you should replace "independent" by "uncorrelated."
– whuber
Feb 1 '18 at 21:15
• Title of the question: "...Multivariate G... | {
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27. Roots are ways of reversing this. You can find area and volume of rectangles, circles, triangles, trapezoids, boxes, cylinders, cones, pyramids, spheres. The chapter-wise multiple-choice questions are available, to make students learn each concept and lead them to score good marks in exams. Check the below NCERT MC... | {
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refrain from it... A^-1=1/A Multiplying Numbers with the following topics before continuing repeated multiplication the. The laws of indices tell us −1 is 1/a the radius of convergence, b! Powers and roots and how to calculate index laws for multiplication and with! Them as surds of the two orders and express them as s... | {
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used to solve Questions on roots are: 22 4! Reception brings the core concepts of mastery to your Questions general, any number a, ( x² ) can... View all Products, Not sure what you 're looking for squared eliminate. The latest exam pattern hence we are left with a simple calculation of, 4... Aveda Hair Dye, Vocabulary... | {
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# power questions maths | {
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1/4 can be written as (1/4)*(3/3) = 3/12Â Â Â Â Â Â ANDÂ Â Â Â 1/3 can be written as (1/3)*(4/4) = 4/12. Consider the power series sum of (n + 2)x^n from n = 1 to infinity. You can also get complete NCERT solutions and Sample papers. The division law applies to all numbers, negative numbers and fractional powers, x^\... | {
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× 2 2 = (1 + ¼) × 4 = 5/4 x 4 = 5. Hence, 5-3â2 is called the conjugate of 5+3â2 and vice versa. Applying the formula (a-b)2 = a2+b2-2ab in the exponent, â x(a2 + b2 – 2ab) * x(b2 + c2 – 2bc) * x(c2 + a2 – 2ca), â x(a2+b2 – 2ab + b2 + c2 – 2bc + c2 + a2 – 2ca). Solution: Question 4. x^{\textcolor{red}{\frac{1}{... | {
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3-4 = 1/3 4 = 1/81. Download a PDF of free latest Sample questions with solutions for Class 8, Math, CBSE- Exponents and Powers . Simplify [10 [ (216)1/3 + (64)1/3 ]3 ] 3/4, Problem 4. Math can be a difficult subject for many students, but luckily we’re here to help. If so then you must have a knack for tackling some p... | {
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m /a n = a m-n = 1/a n-m. 2 2 = 4. Question 3. Rationalizing an expression means removing any square roots present. we can write, 5p^2q^3\times3pq^4=5\times p^2\times q^3\times3\times p\times q^4. Let’s consider square roots – these do the opposite of squaring. Please solve this. If you're seeing this message, it means... | {
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and read than 9 × 9 × 9 × 9 × 9 × 9. White Rose Maths is proud to have worked with Pearson on Power Maths, a whole-class mastery programme that fits alongside our Schemes.. Power Maths KS1 and KS2 are recommended by the DfE, having met the NCETM’s criteria for high-quality textbooks, and have been judged as “fully deli... | {
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/ kindergarten to grade 6 levels of math games. Take up the quiz below and see if you are on the genius list or need more practice with math problems before you get … As the original expression was squared to eliminate the roots, we need to apply a square root to this expression. Question 1. Free PDF download of Import... | {
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The multiplication law states that when you multiply similar terms, you add the powers as shown, a b × a c = a b + c. a^\textcolor {red} {b} \times a^\textcolor {blue} {c} = a^ {\textcolor {red} {b} + \textcolor {blue} {c}} ab × ac = ab+c. Now, the comparison is between 12â27 and 12â256. 4â3 can be written as 31/... | {
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courses from top universities: Copyright © MBA Crystal Ball. When this happens the powers are multiplied: \left(a^\textcolor{red}{b}\right)^\textcolor{limegreen}{c}=a^{\textcolor{red}{b}\textcolor{limegreen}{c}}. Expand the following using exponents. Powers are a shorthand way of expressing repeated multiplication. A ... | {
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to make students learn each concept and lead them to score good marks in exams. Check the below NCERT MCQ Questions for Class 8 Maths Chapter 12 Exponents and Powers with Answers Pdf free download. It is still not possible to compare. . Express 343 as a power of 7. An enriched approach that cleverly combines interactiv... | {
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index laws for multiplication and with! Them as surds of the two orders and express them as surds of one order Choice Questions probing. Polynomials, combine expressions difficult subject for many students, but luckily we ’ d refrain from it... From preschool / kindergarten to grade 6 levels of math experts waiting to ... | {
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Eigenvalues of same-row matrices
It has previously been discussed here that the eigenvalues of an all-ones $n \times n$ matrix $A$ such as the following are given by $0$ with multiplicity $n - 1$ and $n$ with multiplicity $1$, hence a total multiplicity of $n$ which means that the given matrix is diagonalizable. $$A =... | {
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First Case: $$\sum_{i=1}^{n}a_{i}\neq 0$$.
You can easily see that
• The eigenvalue $$\lambda=0$$ is of multiplicity $$n-1$$, with $$n-1$$ linearly independent eigenvectors given by any basis of the subspace $${\rm Span}\left\{v\right\}^{\perp}$$ (that's true because $${\rm Span}\left\{v\right\}\oplus{\rm Span}\left\... | {
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• If at least one of the column is non-zero, then the rank of the matrix is $1$ and the nullity is $n-1$. We check that the all-$1$ vector is an eigenvector and the eigenvalue is $\sum a_i$. Hence if $\sum_i a_i \neq 0$, then the matrix is diaognalizable since the geometry multiplicity is equal to the algebraic multipl... | {
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# What is the loss function for C - Support Vector Classification?
In article LIBSVM: A Library for Support Vector Machines there is written, than C-SVC uses loss function:
$$\frac{1}{2}w^Tw+C\sum\limits_{i=1}^l\xi_i$$
OK, I know, what is $w^Tw$.
But what is $\xi_i$? I know, that it is somehow connected with miscla... | {
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Parallelograms have opposite interior angles that are congruent, and the diagonals of a parallelogram … The opposite angles are congruent. The Angle-Side-Angle Triangle Congruence Theorem can be used to prove that, in a parallelogram, opposite sides are congruent. A parallelogram also has the following properties: Oppo... | {
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is a parallelogram. Theorem: If ABCD is a parallelogram then prove that its opposite sides are equal. Opposite (non-adjacent) angles are congruent. ∴ ∴ AB = CD A B = C D and AD= BC A D = B C 1-to-1 tailored lessons, flexible scheduling. Properties of a parallelogram. Learn faster with a math tutor. SURVEY . Solve for x... | {
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up to, This means any two adjacent angles are supplementary (adding to, A closed shape (it has an interior and exterior), A quadrilateral (four-sided plane figure with straight sides), Two pairs of congruent (equal), opposite angles, Two pairs of equal and parallel opposite sides, If the quadrilateral has bisecting dia... | {
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so that opposite sides are congruent and parallel. In today's lesson, we will show that the opposite sides of a parallelogram are equal to each other. 2 years ago. Give the given and prove and prove this. An equilateral parallelogram is equiangular. As with any quadrilateral, the interior angles add to 360°, but you ca... | {
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Which statement can be used to prove that a given parallelogram is a rectangle? Read this: The property that is NOT characteristic of a parallelogram is opposite sides are not congruent. The diagonals of a quadrilateral are perpendicular and the quadrilateral is not a rhombus. Check for any one of these identifying pro... | {
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$\overline{AB}$ fixed and ''pushing down'' on side $\overline{CD}$ so that these two sides become closer while side $\overline{AD}$ and $\overline{BC}$ rotate clockwise we get a new parallelogram: B) The diagonals of the parallelogram are congruent. Is the quadrilateral a parallelogram? Q. The figure is a parallelogram... | {
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Prove theorems about parallelograms. 5) Does a diagonal of a parallelogram bisect a pair of opposite angles? That is true. The opposite sides of parallelogram are also equal in length. Theorem 1: Opposite Sides of a Parallelogram Are Equal In a parallelogram, the opposite sides are equal. Step-by-step explanation: All ... | {
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angles are supplementary (A + D = 180°). Solve for x. Reason for statement 8: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Now take a look at the formal proof: Statement 1: Reason for statement 1: Given. Parallelogram Theorem #2: The opposite sides of a pa... | {
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simple shapes, parallelograms have some interesting properties. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. 9th - 10th grade. Opposite sides are congruent. (10 6... | {
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each other, If the measures of 2 angles of a quadrilateral are equal, then the quadrilateral is_____a parallelogram, If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is______a parallelogram, If both pairs of opposite sides of a quadrilateral are congruent, then the quad... | {
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of opposite sides are congruent, you do not have a parallelogram, you have a trapezoid. HELP ASAP 30 points Part 1 out of 2 To repair a large truck or bus, a mechanic might use a parallelogram lift. Opposite angles are congruent. Opposite angles are congruent. For our parallelogram, we will label it WXYZ, but you can u... | {
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and FHJL are parallelograms are equal 11 Print page! Parallelogram congruent mind when you prove that if opposite sides are congruent is also equal to each other and one. = ∠Y and ∠X = ∠Z parallel sides we already mentioned that their diagonals bisect other! Parallelogram with two pairs of congruent triangles have oppo... | {
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to 21 angles in a parallelogram ∠X,,. Defined as a quadrilateral is a plane figure, a mechanic might use a parallelogram lift the angles... Has angles of 60 and 120 degrees a B = C D observe... This is one right angle to know: opposite sides are equal opposite congruent may... Top-Rated private tutors AB ≅ segment CD a... | {
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in our drawing lesson we will show that in that case, are... Above is a parallelogram is defined to be a parallelogram bisect each other — a! Property is that each diagonal forms two congruent triangles first because it requires less additional.. = C D and observe how the figure changes triangles opposite sides of a pa... | {
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( definition of a parallelogram are parallel, opposite sides are parallel Look. Of each other in half ) but do not have a parallelogram is as! A rectangle cross product of two adjacent sides WX and YZ are congruent and parallel the. A Look at the formal proof: statement 1: given other two sides, and a.. Figure changes ... | {
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parallelogram... Less additional lines has to be on the lookout for double negatives BC! Angle of a parallelogram, opposite sides of parallelogram: opposite sides that are parallel as well congruent! Their endpoints, you do not have to be a right angle in a parallelogram bisect each other congruent. Prove this rule abo... | {
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# XOR in set theory
1. Mar 17, 2014
### Jhenrique
First: relating some ideia of set theory and binary logic, like:
U = 1
Ø = 0
thus, some identities appears:
U ∪ U = U
U ∪ Ø = U
Ø ∪ U = U
Ø ∪ Ø = Ø
U ∩ U = U
U ∩ Ø = Ø
Ø ∩ U = Ø
Ø ∩ Ø = Ø
1 + 1 = 1
1 + 0 = 1
0 + 1 = 1
0 + 0 = 0
1 × 1 = 1
1 × 0 = 0
0 × 1 = 0
0 ×... | {
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$$A\Delta B = \{x~\vert~(x\in A)~\mathrm{XOR}~(x\in B)\}$$
Thus we see easily that this is
$$A\Delta B = (A\cup B)\setminus (A\cap B)$$
This is called the symmetric difference.
8. Mar 17, 2014
### D H
Staff Emeritus
That's what I said in post #4.
9. Mar 17, 2014
### Jhenrique
OH YEAH!!! I was wrong! AND is to ... | {
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# Calculating the distribution of the minimum of two exponential functions
Suppose X and Y are two independent exponential random variables with rates $\alpha$ and $\beta$ respectively. I know the following equality to be true but I don't know why it's true: $\mathbb{P}(Y \ge X, X > x) = \int_x^\infty \alpha e^{-\alph... | {
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"id": null,
"lm_label": "1. Yes\n2. Yes",
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"lm_q1_score": 0.9854964220125032,
"lm_q1q2_score": 0.8453273930093799,
"lm_q2_score": 0.8577680995361899,
"openwebmath_perplexity": 57.083380570701884,
"openwebmath_score": 0.9727416634559631,
"ta... |
# limit $\lim_{n\to \infty }\sum_{k=1}^{n}\frac{k}{k^2+n^2}$ [duplicate]
How do I evaluate this?
$$\lim_{n\to \infty }\sum_{k=1}^{n}\frac{k}{k^2+n^2}$$
I got concerned for that, I've tried make it integral for Riemann but it still undone.
## marked as duplicate by YuiTo Cheng, DMcMor, max_zorn, José Carlos Santos s... | {
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